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ports in g/Etudes et rapports d'hydrologie 16<br />
of<br />
resources projects<br />
with inadequate data<br />
Proceedings of <strong>the</strong> Madrid Symposium<br />
,June 1973<br />
Elaboration des projets<br />
d'utilisation des ressources en eau<br />
c dans données suffisantes<br />
Volume I<br />
Unesco - MIMO - 1AHS<br />
Unesco - OMM - AISH<br />
_-<br />
Actes du colloque de Madrid<br />
Juin I973
Studies and reports in hydrology/Études et rapports d’hydrologie 16
TITLES IN THIS SERIES / DANS CETTE COLLECTION<br />
1. The use of analog and digital computers in hydrology: Proceedings of <strong>the</strong> Tucson Symposium.<br />
June 1966 1 L'utilisation des calculatrices analogiques et des ordinateurs en hydrologie: Actes du<br />
colloque de Tucson, juin 1966. Vol. I & 2. Co-edition <strong>IAHS</strong>-Unesco / Coédition AISH-Unesco.<br />
2.<br />
Water in <strong>the</strong> unsaturated zone: Proceedings of <strong>the</strong> Wageningen Symposium, June 1967 1 L'eau dans<br />
la zone non saturée: Actes du symposium de Wageningen, juin 1967. Edited by 1 edité par P. E.<br />
Rijtema & H. Wassink. Vol. 1 & 2. Co-edition <strong>IAHS</strong>-Unesco 1 Coédition AISH-Unesco.<br />
3. Floods and <strong>the</strong>ir computation: Proceedings of <strong>the</strong> Leningrad Symposium, August 1967 / Les crues<br />
et <strong>le</strong>ur évaluation: Actes du colloque de Leningrad, août 1967. Vol. 1 & 2. Co-edition <strong>IAHS</strong>-Unesco-<br />
WMO 1 Coédition AISH-Unesco-OMM.<br />
4. Representative and experimental basins: An international guide for research and practice. Edited<br />
by C. Toebes and V. Ouryvaev. Ptrblished by Unesco.<br />
4. Les bassins représentatifs et expérimentaux: Guide international des pratiques en matikre de recherche.<br />
Publié sous la direction de C. Toebes et V. Ouryvaey. Publié par l'Unesco.<br />
5. 'Discharge of se<strong>le</strong>cted rivers of <strong>the</strong> world 1 Débit de certain cours d'eau du monde. Published by<br />
Unesco 1 Publié par l'Unesco.<br />
Vol. I : General and régime characteristics of stations se<strong>le</strong>cted / Caractéristiques généra<strong>le</strong>s et<br />
caractéristiques du régime des stations choisies.<br />
Vol. II: Monthly and annual discharges recorded at various se<strong>le</strong>cted stations (from start of obser.<br />
vations up to 1964) / Débits mensuels et annuels enregistrés en diverses stations sé<strong>le</strong>ctionnées<br />
(de l'origine des observations à l'année 1964).<br />
'Vol. III: Mean monthly and extreme discharges (1965-1969) I Débits mensuels moyens et débits<br />
extrêmes (1965-1969).<br />
6. List of International Hydrological Decade Stations of <strong>the</strong> world 1 Liste des stations de la Décennie<br />
h'ydrologique internationa<strong>le</strong> existant dans <strong>le</strong> monde. Published by Unesco 1 Publié par l'Unesco.<br />
7. Ground-water studies: An international guide for practice. Edited by R. Brown, J. Ineson, V. Konoplyantsev<br />
and V. Kova<strong>le</strong>vski. (Will also appear in French, Russian and Spanish 1 Paraitra<br />
éga<strong>le</strong>ment en espagnol, en français et en russe.)<br />
8. Land subsidence: Proceedings of <strong>the</strong> To'kyo Symposium, September 1969 1 Affaisement du sol:<br />
Actes du colloque de Tokyo, septembre 1969. 'Vol. 1 & 2. Co-edition <strong>IAHS</strong>-Unesco / Coédition<br />
AISH-Unesco.<br />
9. Hydrology of deltas: Proceedings of <strong>the</strong> Bucharest Symposium, May 1969 1 Hydrologie des deltas:<br />
Actes du colloque de Bucarest, mai 1969. Vol. 1 & 2. Co-edifion <strong>IAHS</strong>-Unesco I Coédition AISH-<br />
Unesco.<br />
10. Status and trends of research in hydrology 1 Bilan et tendances de la recherche en hydrologie.<br />
Published by Unesco 1 Publié par l'Unesco.<br />
11. World water balance: Proceedings of <strong>the</strong> Reading Symposium, July 1970 1 Bilan hydrique mondial:<br />
Actes du colloque de Reading, juil<strong>le</strong>t 1970. Vol. 1-3. Co-edition <strong>IAHS</strong>-Unesco-WMO / Coédition<br />
AISH-Unesco-OMM.<br />
12. Results OF research on representative and experimental basins: Proceedings of <strong>the</strong> Wellington<br />
Symposium, December 1970 1 Résultats de recherches sur <strong>le</strong>s bassins représentatifs et expérimen-<br />
taux: Actes du cowoque de Wellington, décembre 1970. 'Vol. 1 & 2. Coedition <strong>IAHS</strong>-Unesco 1<br />
Coédition AISH-Unesco.<br />
13. Hydrometry: Proceedings of <strong>the</strong> Kob<strong>le</strong>nz Symposium, September 1970 1 Hydrométrie: Actes du<br />
colloque de Cob<strong>le</strong>nce, septembre 1970. Co-edition <strong>IAHS</strong>-Unesco-WMO 1 Coédition AISH-Unesco-<br />
OMM.<br />
14. Hydrologic information systems. Co-edition Unesco-WMO.<br />
15. Ma<strong>the</strong>matical models in hydrology: Proceedings of <strong>the</strong> Warsaw Symposium, July 1971 1 Les mo-<br />
dè<strong>le</strong>s mathématiques en hydrologie: Actes du colloque de Varsovie, juil<strong>le</strong>t 1971. Vol. 1-3. Co-<br />
edition <strong>IAHS</strong>-Unesco-WMO / Coédition AISH-Unesco-OMM.<br />
16. Design of water resources projects with inadequate data: Proceedings of <strong>the</strong> Madrid symposium,<br />
June 1973 1 filaboration des projets d'utilisation des ressources en eau sans données suffisantes:<br />
Actes du colloque de Madrid, juin 1973. Vol. 1-3. Co-edition Unesco-WMO-<strong>IAHS</strong> / Coédition Unesco-<br />
OMM-AISH.
qesign of<br />
water resources projects<br />
with inadequate data :<br />
Proceeúings of <strong>the</strong> Madrid Symposium<br />
.lune 1973<br />
Elaboration des projets<br />
d’utilisation des ressources en eau<br />
sans données suffisantes<br />
A contribution to <strong>the</strong> Iniernaiional Hydrological Decade<br />
Une contribution a la Décennie hydrologique internationa<strong>le</strong><br />
Con resurnenes en espano1<br />
Volume I<br />
Actes du colloque de Moúrid<br />
.luin 1973<br />
Unesco - WMO - LAHS 1974<br />
Unesco - OMM - AISH
Published jointly by<br />
<strong>the</strong> United Nations Educational, Scientific<br />
and Cultural Organization,<br />
7, Place de Fontenoy, 75700 Paris,<br />
World Meteorological Organization,<br />
41 av. Giuseppe-Motta, Geneva, and<br />
<strong>the</strong> International Association of Hydrological Sciences (President: J.-A. Rodier),<br />
19, rue Eugène-Carrière, 75018 Paris<br />
Publié conjointement par<br />
l’Organisation des Nations Unies pour<br />
l’éducation, la science et la culture,<br />
7, place de Fontenoy, 75700 Paris,<br />
l’organisation météorologique mondia<strong>le</strong>,<br />
41, av. Giuseppe-Motta, Genève, et<br />
l’Association internationa<strong>le</strong> des sciences hydrologiques (président: J.-A. Rodier).<br />
19, rue Eugène-Carrière, 75018 Paris<br />
Impreso por el Centro de Estudios Hidrográficos, Madrid<br />
. PLJ[,dv: ’<br />
-.. __<br />
The se<strong>le</strong>ction and presentation of material and <strong>the</strong> opinions expressed in this publication<br />
are <strong>the</strong> responsibility of <strong>the</strong> authors concerned and do not necessarily ref<strong>le</strong>ct <strong>the</strong><br />
views of <strong>the</strong> publishers.<br />
The designations employed and <strong>the</strong> presentation of <strong>the</strong> material do not imply <strong>the</strong><br />
expression of any opinion whatsoever on <strong>the</strong> part of <strong>the</strong> publishers concerning <strong>the</strong> <strong>le</strong>gal<br />
status of any country or territory, or of its authorities, or concerning <strong>the</strong> frontiers<br />
of any country or territory.<br />
Le choix et la présentation du contenu de cet ouvrage et <strong>le</strong>s opinions qui s’y<br />
expriment n’engagent que ia responsabilité des auteurs et ne correspondent pas<br />
nécessairement aux vues des éditeurs.<br />
Les dénominations employées et la présentation des divers éléments n’impliquent<br />
de la part des éditeurs aucune prise de position à l’égard du statut juridique de l’un<br />
quelconque des pays et territoires en cause, de son régime politique ou du tracé<br />
de ses frontières.<br />
ISBN 92-3401137-1<br />
0 UnescuWMO-<strong>IAHS</strong>-1974<br />
Printed in Spain
PkEFACE<br />
The International Hydrological Decade (IHD) 1965-74 was launched by<br />
<strong>the</strong> General Conference of Unesco at its thirteenth session to promote<br />
international co-operation in research and studies and <strong>the</strong> training of spe-<br />
cialists and technicians in scientific hydrology. Its purpose is to enab<strong>le</strong><br />
all countries to make a ful<strong>le</strong>r assessment of <strong>the</strong>ir water resources and a<br />
more rational use of <strong>the</strong>m as man’s demands for water constantly increase<br />
in face of developments in population, industry and agriculture. In 1974<br />
National Committees for <strong>the</strong> Decade had been formed in 108 of Unesco’s<br />
131 Member States to carry out national activities within <strong>the</strong> programme<br />
of <strong>the</strong> Decade. The imp<strong>le</strong>mentation of <strong>the</strong> programme is supervised by a<br />
Co-ordinating Council, composed of 30 Member States se<strong>le</strong>cted by <strong>the</strong> Ge-<br />
neral Conference of Unesco, which studies proposals for. developments<br />
of <strong>the</strong> programme, recommends projects of interest to all or a large<br />
number of countries, assists in <strong>the</strong> development of national and regional<br />
projects and co-ordinates international co-operation.<br />
Promotion of collaboration in developing hydrological research techni-<br />
ques, diffusing hydrological data and planning hydrological installations<br />
is a major feature of <strong>the</strong> programme of <strong>the</strong> IHD which encompasses all<br />
aspects of hydrological studies and research. Hydrological investigations<br />
are encouraged at <strong>the</strong> national, regional and international <strong>le</strong>vel to streng-<br />
<strong>the</strong>n and to improve <strong>the</strong> use of natural resources from a local and a global<br />
perspective. The programme provides a means for countries well advanced<br />
in hydrological research to exchange scientific views and for developing<br />
countries to benefit from this exchange of information in elaborating re-<br />
search projects and in imp<strong>le</strong>menting recent developments in <strong>the</strong> planning<br />
of hydrological installations.<br />
As part of Unesco’s contribution to <strong>the</strong> achievement of <strong>the</strong> objectives<br />
of <strong>the</strong> IHD <strong>the</strong> General Conference authorized <strong>the</strong> Director-General to<br />
col<strong>le</strong>ct, exchange and disseminate information concerning research on<br />
scientific hydrology and to facilitate contacts between research workers<br />
in this field. To this end Unesco initiated two series of publications: Studies<br />
and Reports in Hydrology and Technical Papers in Hydrology.<br />
The Studies and Reports in Hydrology series, in which <strong>the</strong> present<br />
volume is published, is aimed at recording data col<strong>le</strong>cted and <strong>the</strong> main<br />
results of hydrwlogical studies undertaken within <strong>the</strong> framework of <strong>the</strong><br />
Decade, as well as providing information on research techniques. Also<br />
included in <strong>the</strong> series are proceedings of symposia. Thus, <strong>the</strong> series com-<br />
prises <strong>the</strong> compilation of data, discussions of hydrological research techni-<br />
ques and findings, and guidance material for future scientific investigations.<br />
It is hoped that <strong>the</strong> volumes wil furnish material of both practical and<br />
<strong>the</strong>oretical interest to hydrologists and governments participating in <strong>the</strong><br />
IHD and respond to <strong>the</strong> needs of technicians and scientists concerned<br />
with prob<strong>le</strong>ms of water in all countries.<br />
A number of <strong>the</strong>se volumes have been published jointly with <strong>the</strong> In-<br />
ternational Association of Hydrological Sciences and <strong>the</strong> World Meteoro-<br />
logical Organization which have co-operated with Unesco in <strong>the</strong> imp<strong>le</strong>-<br />
mentation of several important projects of <strong>the</strong> IHD.
PRBFACE<br />
La Conférence généra<strong>le</strong> de l’Unesco, à sa treizième session, a décidé<br />
de lancer, pour la période s’étendant de 1965 à 1974, la Décennie hydrologique<br />
internationa<strong>le</strong> (DHI), entreprise<br />
e<br />
mondia<strong>le</strong> visant a faire progresser la connaissance<br />
en matiere ci’ vdrologie scientifique par un développement de<br />
la coopération inyrnati na<strong>le</strong> et par la formation de spécialistes et de<br />
techniciens. Au moment oìi l’expansion démographique et <strong>le</strong> développement<br />
industriel et agrico<strong>le</strong> provoquent un accroissement constant des besoins<br />
en eau, la DHI permet à tous <strong>le</strong>s pays de mieux évaluer <strong>le</strong>urs ressources<br />
hydrauliques et de <strong>le</strong>s exploiter plus rationnel<strong>le</strong>ment.<br />
I1 existe actuel<strong>le</strong>ment dans 108 des 131 Etats membres de l’Unesco un<br />
comité national qui, pour tout ce qui a tratit au programme de la Décennie,<br />
impulse <strong>le</strong>s activités nationa<strong>le</strong>s et assure la participation de son pays<br />
aux entreprises régiona<strong>le</strong>s et internationa<strong>le</strong>s. L’exécution du programme<br />
de la DHI se fait sous la direction d’un Conseil de coordination composé<br />
de 30 Etats membres désignés par la Conférence généra<strong>le</strong> de l’Unesco; ce<br />
conseil étudie <strong>le</strong>s propositions concernant <strong>le</strong> programme, recommande<br />
l’adoption de projets intéressant l’ensemb<strong>le</strong> des pays ou un grand nombre<br />
d’entre eux, aide à la mise sur pied de projets nationaux et régionaux, et<br />
coordonne la coopération à l’échelon international.<br />
Le programme de la DHI, qui porte sur tous <strong>le</strong>s aspects des études et<br />
des recherches hydrologiques, vise essentiel<strong>le</strong>ment à développer la collaboration<br />
dans la mise au point des techniques de recherches, dans la<br />
diffusion des données hydrologiques, dans l’organisation des installations<br />
hydrologiques. I1 encourage <strong>le</strong>s enquêtes nationa<strong>le</strong>s, régiona<strong>le</strong>s et internationa<strong>le</strong>s<br />
tendant à accroître et à améliorer l‘utilisation des resources naturel<strong>le</strong>s,<br />
dans une perspective loca<strong>le</strong> et généra<strong>le</strong>. Il permet aux pays avancés<br />
en matière de recherches hydrologiques d’échanger des informations; aux<br />
pays en voie de développement, il offre la possibilité de profiter de ces<br />
échanges pour élaborer <strong>le</strong>urs projets de recherches et pour planifier <strong>le</strong>urs<br />
installations hydrologiques en tirant parti des acquisitions <strong>le</strong>s plus récentes<br />
de l’hydrologie scientifique.<br />
Pour permettre a l’Unesco de contribuer au succès de la DHI, la Conférence<br />
généra<strong>le</strong> a autorisé <strong>le</strong> Directeur généra<strong>le</strong> à rassemb<strong>le</strong>r, à échanger<br />
et à diffuser des informations sur <strong>le</strong>s recherches d’hydrologie scientifique<br />
et à faciliter <strong>le</strong>s contacts entre <strong>le</strong>s chercheurs dans ce domaine. A cette<br />
fin, l’Unesco fait paraître deux nouvel<strong>le</strong>s col<strong>le</strong>ctions de publications: «Etudes<br />
et rapports d’hydrologie» et «Notes techniques d’hydrologie,.<br />
La col<strong>le</strong>ction «Etudes et rapports d’hydrologie,, dans laquel<strong>le</strong> est publié<br />
<strong>le</strong> présent ouvrage, a pour objet de présenter <strong>le</strong>s données recueillies et <strong>le</strong>s<br />
principaux résultats des études effectuées dans <strong>le</strong> cadre de la Décennie<br />
et de fournir des informations sur <strong>le</strong>s techniques de recherche. On y trouve<br />
aussi <strong>le</strong>s Actes de colloques réunis sur ce sujet. Cette col<strong>le</strong>ction publie<br />
donc des données, des techniques et des résultats de recherches ainsi<br />
qu’une documentation pour <strong>le</strong>s travaux scientifiques futurs.<br />
On espère que ces volumes apporteront aux hydrologues et aux gouvernements<br />
qui participent à ,la DHI des matériaux d’un intérêt tant pra-
tique que théorique, et qu’el<strong>le</strong> répondra aux besoins des techniciens et<br />
des hommes de science de tous pays qui s’occupent des problèmes de l’eau.<br />
Certains de ces ouvrages sont publibs en coopération avec l’Association<br />
internationa<strong>le</strong> des sciences hydrologiques ou l’organisation météorologique<br />
mondia<strong>le</strong> dans <strong>le</strong> cadre de projets réalisés conjointement par ces orga-<br />
nisations et l’Unesco.
INTRODUCTION<br />
The Symposium on <strong>the</strong> Development of Water Resources Projects with<br />
Inadequate Data was held in Madrid from 4 to 8 June 1973 for <strong>the</strong> purpose<br />
of focusing on <strong>the</strong> methodology for hydrologic studies for water resources<br />
projects with inadequate data and on current practices for <strong>the</strong> assessment<br />
of design parameters.<br />
The Symposium was opened at <strong>the</strong> Palacio de Exposiciones on <strong>the</strong><br />
morning of 4 June by Miniester of Public Workes of Spain Addresses were<br />
<strong>the</strong>n given by Dr. Dumitrescu on behalf of <strong>the</strong> Director General of Unesco,<br />
Professor Nevmec on behalf of <strong>the</strong> Secretary-General of WMO, Dr. Rodier<br />
as President of <strong>IAHS</strong> and by Dr. Briones, on behalf of <strong>the</strong> Spanish Na-<br />
tional Committee for <strong>the</strong> IHD.<br />
The Symposium was attended by 480 participants from 77 countries.<br />
The technical programme, detaim<strong>le</strong>d in <strong>the</strong> Tab<strong>le</strong> of Contents, included<br />
consideration of 3 major areas:<br />
1. Methodology for hydrological studies with inadequate data,<br />
2. Current practices in different countries,<br />
3. Relation between project economics and hydrological data.<br />
Each area was fur<strong>the</strong>r sub-divided into topics for each of which <strong>the</strong><br />
individually contributed papers were abstracted into a general report, orally<br />
presented by an invited expert, and followed by discussion.<br />
Since <strong>the</strong> individual papers were not presented at <strong>the</strong> Symposium orally<br />
by <strong>the</strong> authors, <strong>the</strong>ry are reproduced here in <strong>the</strong> orden in which<br />
<strong>the</strong>y were reported in each general report under each topic.
üesip d water reswrcee projects with inadequate dati: Pmeeedin.p d <strong>the</strong> Madrid 8ympoilUm.<br />
June i973 / Blabontion de# projeu d‘utilhition des ressourcci en eau rans domdes nuttlrantei:<br />
Acui du mlloque de Madrid, juin 1973.<br />
Volume I Contents Tab<strong>le</strong> des matieres<br />
Foreword/Avant-propos<br />
TOPIC 1.1 . TRANSFER OF INFORMATION FROM OBSERVED POINTS TO<br />
POINTS OF INTEREST, ESPECIALLY FOR THE ASSESSMENT<br />
OF THE CHARACTERISTICS OF DISCHARGES.<br />
POINT 1.1 - EXTRAPOLATION DES INFORMATIONS RECUEILLIES AUX<br />
POINTS OBSERVES A DES POINTS PRESENTANT UN INTE-<br />
RET PARTICULIER, NOTAMMENT POUR L’EVALUATION<br />
DES DEBITS CARACTERISTIQUES.<br />
SOKOLOV, A.A. (U.S.S.R.) GENERAL REPORT<br />
ALBINET, M., CASTANY, G., DELAROZIERE-BOUILLIN, O., JONAT, R.,<br />
MARGAT, J. (FRANCE)<br />
Evaluation et répartition des ressources en e au d’une grande région par<br />
<strong>le</strong>s paramétres hydroclimatiques et hydrog6ologiques ...............<br />
BALEK, J. (CZECHOSLOVAKIA)<br />
Use of representative and experimental catchments for <strong>the</strong> assement of<br />
hydrological data of African tropical basins .......................<br />
CORMARY, Y - J.M. MASSON. (FRANCE)<br />
Diverses méthodes convergentes pour l’utihtion de l’information a<br />
I’écheUedgiona<strong>le</strong> .......................................<br />
DUBREUIL, PIERRE. (FRANCE)<br />
Le transfert d’infomtion hydrologique a des bassins versants non obser-<br />
vés ....................................................<br />
GARCIA-AGREDA, R., RASULO, G., VIPARELLI, R. (ITALY)<br />
Pluviometric zones and <strong>the</strong> criteria to define <strong>the</strong>ir boundaries for regions<br />
withscarcedata ............................................<br />
OBERLIN, G.R., GALEA, G.C., TONI, J.T. (FRANCE)<br />
Estimation des étiages de bassins non equipés ....................<br />
TIERCELIN, J.R. (FRANCE)<br />
ParamBtres régionaux relatifs aux ressources en eau. Utilisation. PdciPion<br />
d’estimation ............................................... 125<br />
VAN HYLCKAMA, T.E.A. (U.S.A.)<br />
Estimating evapotranspiration by homoclimates ................ 74<br />
1<br />
15<br />
27<br />
47<br />
61<br />
89<br />
103
VOSKRESENSKI, K.P. (U.S.S.R.)<br />
Prinoiph for <strong>the</strong> computation of tho nuin Ohinctdutb of river wrtm<br />
reaowcea in <strong>the</strong> abuncl of oburvrtiona on <strong>the</strong> bu& of goographid<br />
interpoiation of runoff puameten ..............................<br />
VUGLLNSKI, V.S., SEMENOV, V.A. (U.S.S.R.)<br />
Evaiurtion of water IWOIUWW of mounîdn am11 in cani of rhnce or<br />
inadequacy of datr on runoff ..................................<br />
TOPIC 1.2 - THE IMPROVEMENT OF OVERALL HYDR0UX;IC INFOR-<br />
MATION BY SHORT-TERM ADDITIONAL AND PARTICULAR<br />
OBSERVATIONS AND MEASUREMENTS. INCLUDING THE<br />
PLANNING OF THE ADDITIONAL MEASUREMENT CAM-<br />
PAIGN USING HYDROLOGIC DATA SENSITIVITY ANALYSIS<br />
BASED ON PROJECT ECONOMICS.<br />
POINT 1.2 - AMELIORATION DE L'ENSEMBLE DE L'INFORMATION<br />
HYDROLOGIQUE AU MOYEN DE COURTES CAMPAGNES DE<br />
MESURES COMPLEMENTAIRES ET D'OBSERVATIONS PARTI-<br />
CULIERES, COMPRENANT LA MISE EN OEUVRE DE CAM-<br />
PAGNES DE MESURES ADDITIONNELLES UTILISANT UNE<br />
ANALYSE DE SENSIBILITE DES DONNEES BASEE SUR<br />
L'ECONOMIE DES PROJETS.<br />
RODDA, JOHN (U.K.) GENERAL REPORT<br />
BEARD, LEO R. (U.S.A.)<br />
Hydrological data fiii-in and network design ..................<br />
DELHOMME, J.P., DELFINER, P. (FRANCE)<br />
Application du Krigeage a l'optimisation d'une campagne pluviométrique<br />
enzonearide ..............................................<br />
HALASI-KUN, GEORGE, J. (U.S.A.)<br />
Improvement of runoff records in smal<strong>le</strong>r watersheds based on permeabi-<br />
lity of <strong>the</strong> geological subsurface ...............................<br />
KOVACS, GEORGE. MOLNAR, GEORGE. (HUNGARY)<br />
Determination of snow water equiva<strong>le</strong>nt and snowmelt water by<br />
thickness of snow cover data .................................<br />
MEIJERINK, A.M.J. (NETHERLANDS)<br />
Evaluation of local water resources in semiarid hard rock region by using<br />
photo.hydrological indices ....................................<br />
PANT,P.S.,GUPTA, M.G. (INDIA)<br />
Application of satellite cloud pictures in snow hydrology of <strong>the</strong> Himalayas<br />
and in <strong>the</strong> estimation of rainfall over India during southwest<br />
monsoonseason ............................................<br />
137<br />
145<br />
153<br />
161<br />
17<br />
191<br />
205<br />
217<br />
233
I<br />
TOPIC I.3A - THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-<br />
SIGNED FOR DATA-SCARCE AREAS. STATISTICAL ME-<br />
THODS AND DATA OPERATION.<br />
POINT I.3A - UTILISATION DES TECHNIQUES DE SIMULATION SPE-<br />
CIALEMENT EWIBOREE POUR DES REGIONS OU LES<br />
DONNEES SONT RARES. METHODES STATISTIQUES ET<br />
TRAITEMENT DES DONNEES.<br />
JAMES, IVAN CHARLES. (U.S.A.) GENERAL REPORT<br />
CORMARY, Y . GUILBOT, A. (FRANCE)<br />
Etude des relations pluie-débit sur trois bassins versants d’investigation . .<br />
CHARANIA, S.H. (KENYA)<br />
Extension of runoff records for small catchments in semi-arid regions ...<br />
DAVYDOVA,A.I., KALININ, G.P. (U.S.S.R.)<br />
Simulation of hydrological samp<strong>le</strong>s by natural water flow characteristics<br />
HAMLIN, M.J., KOTTEGODA, N.T. (U.K.)<br />
The preparation of a data set for hydrologic system analysis ..........<br />
LENTON, ROBERTO L., SCHAAKE JR., JOHN C., RODRIGUEZ-ITURBE, IG-<br />
NACIO. (U.S.A.)<br />
Potential application of Bayesian techniques for parameter estimation<br />
withlimiteddata ...........................................<br />
McMAHON, T.A., MEIN, R.G. (AUSTRALIA)<br />
Storage-yield estimates with inadequate streamflow data .............<br />
MARTIN JADRAQUE, VALENTIN. (SPAIN)<br />
Estimation of Gumbel law parameters in small samp<strong>le</strong>s ..............<br />
MOSS, M.E.. DAWDY, D.R. (U.S.A.)<br />
Stochastic simulation for basins with short or no records of streamflow<br />
O’CONNELL, P.E., WALLIS, J.R. (U.S.A.)<br />
Choice of generating mechanism in syn<strong>the</strong>tic hydrology with inadequate<br />
data .....................................................<br />
PORRAS, PEDRO., FLORES, ALFREDO. (VENEZUELA)<br />
Stochastic application in ungauged basins for planning purposes .......<br />
ROCHE, MARCEL. (FRANCE)<br />
Homogdnbisation et interpolation des donndes pour un modè<strong>le</strong> de simula-<br />
tion .....................................................<br />
SHARMA, H.D., BHATTACHARYA, A.P., JINDAL, S.R. (INDIA)<br />
The use of simulation techniques for sequential generation of short-sized<br />
rainfall data and its application in <strong>the</strong> estimation of design flood ......<br />
241<br />
265<br />
281<br />
293<br />
305<br />
321<br />
335<br />
349<br />
365<br />
311<br />
355<br />
407<br />
419
VISSER, J.H. (LEBANON)<br />
The we of rtochutlc mod& in hydroJgricultunl dwdopmrnt projoct<br />
Libbanon ................................................<br />
WALLIS, J.R., MATALAS, N.C. (U.S.A.)<br />
Rehtivr importuice of decidon vuirbiem in fiood frequency uulydi ...<br />
WEISS, G. (U.K.)<br />
Shot nohe models for ayn<strong>the</strong>tic generation of multimite M y munflow<br />
data .....................................................<br />
WOOD, ERIC F. (U.S.A.)<br />
Flood control ddgn with Limited data . A comparinon of <strong>the</strong> chsical<br />
andBayesianapproaches .....................................<br />
TOPIC 1.3B . THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-<br />
SIGNED FOR DATA-SCARCE AREAS. THE USE OF MATHE-<br />
MATICAL MODELS.<br />
POINT I.3B - UTILISATION DES TECHNIQUES DE SIMULATION SPE-<br />
CIALEMENT ELABOFSE POUR DES REGIONS OU LES<br />
DONNEES SONT RARES. UTILISATION DES MODELES MA-<br />
THEMATIQUES.<br />
NASH, J.E. (IRELAND) GENERAL REPORT<br />
BERNIER, J. (FRANCE)<br />
Données inadéquates et mode<strong>le</strong>s mathématiques de la pollution en riviere<br />
COOK, SAMUEL P., MBURU, SAMUEL G. (KENYA)<br />
Regional groundwater recharge estimates via meteorological data ......<br />
DELLEUR, J.W., LEE, M.T. (U.S.A.)<br />
A rainfall-runoff model based on <strong>the</strong> watershed stream network .......<br />
HANN, C.T. (U.S.A.)<br />
Monthly streamflow estimation from limited data ..................<br />
KOREN, V.I., KUTCHMENT, L.S. (U.S.S.R.)<br />
Obtaining deficient information by solving inverse prob<strong>le</strong>ms for ma<strong>the</strong>-<br />
maticalrunoffmodels .......................................<br />
ROFAIL, NABIL. (EGYPT)<br />
The ma<strong>the</strong>matical model of water balance for data-scarce areas ........<br />
VILARO, FRANCISCO., CUSTODIO, EMILIO. (SPAIN)<br />
Data acquisition and methodology for a simulation model of <strong>the</strong> Llobre-<br />
gat Delta (Barcelona, Spain) ...................................<br />
435<br />
449<br />
457<br />
469<br />
485<br />
513<br />
525<br />
53 1<br />
545<br />
551<br />
569<br />
581
Contents<br />
Tab<strong>le</strong> des matieres<br />
Volume I<br />
ForewordIAvant-propos ............................<br />
TOPIC 1.1 - TRANSFER OF INFORMATION FROM OBSERVED POINTS TO<br />
POINTS OF INTEREST, ESPECIALLY FOR THE ASSESSMENT<br />
OF THE CHARACTERISTICS OF DISCHARGES.<br />
POINT I. 1 - EXTRAPOLATION DES INFORMATIONS RECUEILLIES AUX<br />
POINTS OBSERVES A DES POINTS PRESENTANT UN INTE-<br />
RET PARTICULIER, NOTAMMENT POUR L’EVALUATION<br />
DES DEBITS CARACTERISTIQUES.<br />
SOKOLOV, A.A. (U.S.S.R.) GENERAL REPORT<br />
ALBINET, M., CASTANY, G., DELAROZIERE-BOUILLIN, O., JONAT, R.,<br />
MARGAT, J. (FRANCE)<br />
Evaluation et répartition des ressources en eaux d’une grande région par<br />
<strong>le</strong>s paramètres hydroclimatiques et hydrogéologiques ...............<br />
BALEK, J. (CZECHOSLOVAKIA)<br />
Use of representative and experimental catchments for <strong>the</strong> assessment of<br />
hydrological data of African tropical basins .......................<br />
CORMARY, Y - J.M. MASSON. (FRANCE)<br />
Diverses méthodes convergentes pour l’utilisation de l’information à<br />
l’échel<strong>le</strong> régiona<strong>le</strong> ...........................................<br />
DUBREUIL, PIERRE. (FRANCE)<br />
Le transfert d’information hydrologique à des bassins versants non obcer-<br />
vés ......................................................<br />
GARCIA-AGREDA, R., RASULO, G., VIPARELLI, R. (ITALY)<br />
Pluviometric zones and <strong>the</strong> criteria to define <strong>the</strong>ir boundaries for regions<br />
with scarce data ............................................<br />
OBERLIN, G.R., GALEA, G.C., TONI, J.T. (FRANCE)<br />
Estimation des étiages de bassins non equipés .....................
II<br />
TIERCELIN, J. R. (FRANCE)<br />
Parametres régionaux relatifs aux ressources en eau. Utilisation. Précision<br />
d’estimation ...............................................<br />
VAN HYLCKAMA, T.E.A. (U.S.A.)<br />
Estimating evapotranspiration by homoclimates ...................<br />
VOSKRESENSKI, K.P. (U.S.S.R.)<br />
Princip<strong>le</strong>s for <strong>the</strong> computation of <strong>the</strong> main characteristics of river water<br />
resources in <strong>the</strong> absence of observations on <strong>the</strong> basis of geographical<br />
interpolation of runoff parameters ..............................<br />
VUGLINSKI, V.S.,SEMENOV, V.A. (U.S.S.R.)<br />
Evaluation of water resources of mountain areas in casi of absence or<br />
inadequacyofdataonrunoff ..................................<br />
TOPIC 1.2 - THE IMPROVEMENT OF OVERALL HYDROLOGIC INFOR-<br />
MATION BY SHORT-TERM ADDITIONAL AND PARTICULAR<br />
OBSERVATIONS AND MEASUREMENTS. INCLUDING THE<br />
PLANNING OF THE ADDITIONAL MEASUREMENT CAM-<br />
PAIGN USING HYDROLOGIC DATA SENSITIVITY ANALYSIS<br />
BASED ON PROJECT ECONOMICS.<br />
POINT 1.2 - AMELIORATION DE L’ENSEMBLE DE L’INFORMATION<br />
HYDROLOGIQUE AU MOYEN DE COURTES CAMPAGNES DE<br />
MESURES COMPLEMENTAIRES ET D’OBSERVATIONS PARTI-<br />
CULIERES, COMPRENANT LA MISE EN OEUVRE DE CAM-<br />
PAGNES DE MESURES ADDITIONNELLES UTILISANT UNE<br />
ANALYSE DE SENSIBILITE DES DONNEES BASEE SUR<br />
L’ECONOMIE DES PROJETS.<br />
RODDA, JOHN (U.K.) GENERAL REPORT<br />
BEARD, LEO R. (U.S.A.)<br />
Hydrological data fiil-in and network design ......................<br />
DELHOMME, J.P., DELFINER, P. (FRANCE)<br />
Application du Krigeage à l’optimisation d’une campagne pluviométrique<br />
enzonearide ..............................................<br />
HALASI-KUN,<br />
GEORGE, J. (U.S.A.)<br />
Improvement of runoff records in smal<strong>le</strong>r watersheds based on permeabi-<br />
lity of <strong>the</strong> geological subsurface ................................
KOVACS, GEORGE. MOLNAR, GEORGE. (HUNGARY)<br />
Determination of snow water equiva<strong>le</strong>nt and snowmelt water by<br />
thickness of snow cover data ..................................<br />
MEIJERINK, A.M.J. (NETHERLANDS)<br />
Evaluation of local water resources in semiarid hard rock region by using<br />
photo-hydrological indices ....................................<br />
PANT, P.S., GUPTA, M.G. (INDIA)<br />
Application of satellite cloud pictures in snow hydrology of <strong>the</strong> Himalayas<br />
and in <strong>the</strong> estimation of rainfall over India during southwest<br />
monsoonseason ............................................<br />
TOPIC I.3A - THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-<br />
SIGNED FOR DATA-SCARCE AREAS. STATISTICAL ME-<br />
THODS AND DATA OPERATION.<br />
POINT I.3A - UTILISATION DES TECHNIQUES DE SIMULATION SPE-<br />
CIALEMENT ELABOREE POUR DES REGIONS OU LES<br />
DONNEES SONT RARES. METHODES STATISTIQUES ET<br />
TRAITEMENT DES DONNEES.<br />
JAMES, IVAN CHARLES. (U.S.A.) GENERAL REPORT<br />
CORMARY, Y - GUILBOT, A. (FRANCE)<br />
Etude des relations pluie-débit sur trois bassins versants d'investigation . .<br />
CHARANIA, S.H. (KENYA)<br />
Extension of runoff records for small catchments in semi-arid regions ...<br />
DAVYDOVA,A.I.,KALININ,G.P. (U.S.S.R.)<br />
Simulation of hydrological samp<strong>le</strong>s by natural water flow characteristics<br />
HAMLIN, M.J., KOTTEGODA, N.T. (U.K.)<br />
The preparation of a data set for hydrologic system analysis ..........<br />
LENTON, ROBERTO L., SCHAAKE JR., JOHN C., RODRIGUEZ-ITURBE, IG-<br />
NACIO. (U.S.A.)<br />
Potential application of Bayesian techniques for parameter estimation<br />
withlimiteddata ...........................................<br />
McMAHON, T.A.,<br />
MEIN, R.G. (AUSTRALIA)<br />
Storage-yield estimates with inadequate streamflow data .............<br />
III
IV<br />
MARTIN JADRAQUE, VALENTIN. (SPAIN)<br />
Estimation of Gumbel law parameters in small samp<strong>le</strong>s ..............<br />
MOSS, M.E., DAWDY, D.R. (U.S.A.)<br />
Stochastic simulation for basins with short or no records of streamflow<br />
O’CONNELL, P.E., WALLIS, J.R. (U.S.A.)<br />
Choice of generating mechanism in syn<strong>the</strong>tic hydrology with inadequate<br />
data .....................................................<br />
PORRAS, PEDRO., FLORES, ALFREDO. (VENEZUELA)<br />
Stochastic application in ungauged basins for planning purposes .......<br />
ROCHE, MARCEL. (FRANCE)<br />
Homogénéisation et interpolation des données pour un modè<strong>le</strong> de simula-<br />
tion .....................................................<br />
SHARMA, H.D., BHATTACHARYA, A.P., JINDAL, S.R. (INDIA)<br />
The use of simulation techniques for sequential generation of short-sized<br />
rainfall data and its application in <strong>the</strong> estimation of design flood ......<br />
VISSER, J.H. (LEBANON)<br />
The use of stochastic models in a hydro-agricultural development project<br />
inLebanon ................................................<br />
WALLIS, J.R., MATALAS,N.C. (U.S.A.)<br />
Relative importance of decision variab<strong>le</strong>s in flood frequency analysis<br />
WEISS, G. (U.K.)<br />
Shot noise models for syn<strong>the</strong>tic generation of multisite daily streamflow<br />
data .....................................................<br />
WOOD, ERIC F. (U.S.A.)<br />
Flood control design with limited data - A comparison of <strong>the</strong> classical<br />
andBayesianapproaches .....................................<br />
TOPIC I.3B - THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-<br />
SIGNED FOR DATA-SCARCE AREAS. THE USE OF MATHE-<br />
MATICAL MODELS.<br />
POINT I.3B - UTILISATION DES TECHNIQUES DE SIMULATION SPE-<br />
CIALEMENT ELABOREE POUR DES REGIONS OU LES<br />
DONNEES SONT RARES. UTILISATION DES MODELES MA-<br />
THEMATIQUES.
NASH, J.E. (IRELAND) GENERAL REPORT<br />
BERNIER, J. (FRANCE)<br />
Données inadéquates et modè<strong>le</strong>s mathématiques de la pollution en riviere<br />
COOK, SAMUEL P., MBURU, SAMUEL G. (KENYA)<br />
Regional groundwater recharge estimates via meteorological data ......<br />
DELLEUR, J.W., LEE, M.T. (U.S.A.)<br />
A rainfall-runoff model based on <strong>the</strong> watershed stream network .......<br />
HANN, C.T. (U.S.A.)<br />
Monthly streamflow estimation from limited data ..................<br />
KOREN, V.I., KUTCHMENT, L.S. (U.S.S.R.)<br />
Obtaining deficient information by solving inverse prob<strong>le</strong>ms for ma<strong>the</strong>-<br />
matical runoff models .......................................<br />
ROFAIL, NABIL. (EGYPT)<br />
The ma<strong>the</strong>matical model of water balance for data-scarce areas ........<br />
VILARO, FRANCISCO., CUSTODIO, EMILIO. (SPAIN)<br />
Data acquisition and methodology for a simulation model of <strong>the</strong> Llobre-<br />
gat Delta (Barcelona, Spain) ...................................<br />
V
Foreword<br />
Whi<strong>le</strong> <strong>the</strong> need for hydrological and meteorological data of many types<br />
for <strong>the</strong> design of water resources projects is obvious, it is often found,<br />
especially in many developing countries, that such data are ei<strong>the</strong>r lacking<br />
or inadequate.<br />
Recognizing <strong>the</strong> existence of this prob<strong>le</strong>m, <strong>the</strong> Co-ordinating Counci*l of<br />
<strong>the</strong> IHD appointed a group of experts (third session, Paris, June 1967) to<br />
study <strong>the</strong> prob<strong>le</strong>m of design of water resources projects with inadequate<br />
data.<br />
Similarly, <strong>the</strong> Commission for Hydrology of WMlO (third session, Geneva,<br />
September 1968) established a Working Group on Hydrological Design<br />
Data for Water Resources Projects to prepare guidance material on this<br />
subject for <strong>the</strong> WMO Guide to Hydrological Practices and to maintain<br />
liaison with <strong>the</strong> IHD group of experts appointed by <strong>the</strong> Co-ordinating<br />
Council.<br />
As a means of taking stock of <strong>the</strong> work carried out by <strong>the</strong> hydrological<br />
community in coping with project design with scarce data, Unesco and<br />
WMO jointly convened a symposium on this subject. The Symposium was<br />
organized with <strong>the</strong> co-operation of <strong>the</strong> <strong>IAHS</strong> and <strong>the</strong> Spanish National<br />
Committee for <strong>the</strong> IHD and was held in Madrid from 4 to 8 June 1973 at<br />
<strong>the</strong> invitation of <strong>the</strong> Government of Spain.<br />
The Madrid Symposium concentrated on <strong>the</strong> methodology of hydro-<br />
logical studies for water resources projects with inadequate data and on<br />
current practices for <strong>the</strong> assessment of design parameters.<br />
The Minister of Public Works of Spain opened <strong>the</strong> Symposium at <strong>the</strong><br />
Palacio de Exposiciones on <strong>the</strong> morning of 4 June. Addresses were given<br />
by Dr. Dumitrescu on behalf of <strong>the</strong> Director-General of Une,sco, Professor<br />
Nemec on benalf of <strong>the</strong> Secretary-General of WMO, Dr. Rodier as President<br />
of <strong>IAHS</strong> and by Dr. Briones, on behalf of <strong>the</strong> Spanish National Committee<br />
for <strong>the</strong> IHD.<br />
The Symposium was atteneded by 480 participants from 77 countries.<br />
The technical programme, detai<strong>le</strong>d in <strong>the</strong> Tab<strong>le</strong> of Contents, included<br />
consideration of 3 major areas:<br />
1. Methodology for hydrological studies with inadequate data;<br />
2. Current practices in different countries;<br />
3. Relation between project economics and hydrological data.<br />
Each area was fur<strong>the</strong>r sub-divided into topics for each of which <strong>the</strong><br />
individually contributed papers were abstracted into a general report,<br />
orally presented by an invited expert, and followed by discussion.
This volume of proceedings was compi<strong>le</strong>d by <strong>the</strong> Spanish National Com-<br />
mittee for <strong>the</strong> IHD; it includes all <strong>the</strong> general reports and individual<br />
papers presented at <strong>the</strong> Symposium, as well as <strong>the</strong> discussions. It is issued<br />
as a joint Unesco/WMO/<strong>IAHS</strong> publication in <strong>the</strong> spirit in which <strong>the</strong> three<br />
Organizations have collaborated during <strong>the</strong> IHD.<br />
Since <strong>the</strong> individual authors did not present <strong>the</strong>ir papers orally at <strong>the</strong><br />
Symposium, <strong>the</strong> papers are reproduced here in <strong>the</strong> order in which <strong>the</strong>y<br />
are discussed in <strong>the</strong> general report for each topic.<br />
Unesco, WMO and <strong>IAHS</strong> wish to record <strong>the</strong>ir thanks to <strong>the</strong> Spanish<br />
National Committee for <strong>the</strong> IHD for <strong>the</strong> many contributions of its members<br />
towards <strong>the</strong> organization of <strong>the</strong> Symposium, and for <strong>the</strong> Committee's as-<br />
sistance in <strong>the</strong> publication of <strong>the</strong>se proceedings.
AVANT-PROPOS<br />
I1 est évident que, pour élaborer des projets d’utilisation des ressources<br />
en eau il est nécessaire de disposer de données hydrologiques et météoro-<br />
logiques de types très divers; or il apparaît que ces données sont souvent<br />
inexistantes ou insuffisantes, notamment dans beaucoup de pays en voie<br />
de développement.<br />
Conscient de ce problème, <strong>le</strong> Conseil de coordination de la DHI a créé,<br />
lors de sa troisième session (Paris, juin 1967) un groupe d’experts chargé<br />
d’étudier <strong>le</strong>s moyens d’elaborer des projets d’utilisation des ressources<br />
en eau sans disposer de données suffisantes.<br />
De son côté, la Commission d’hydrologie de l’OMM a constitué à sa<br />
troisième session (Genève, septembre 1968) un groupe de travail sur <strong>le</strong>s<br />
données hydrologiques nécessaires à l’élaboration des projets d’aménage-<br />
ment des ressources hydrauliques; ce groupe de travail a été chargé de<br />
formu<strong>le</strong>r des recommandations destinées à figurer dans <strong>le</strong> Guide OMM des<br />
pratiques hydrologiques, et d’assurer la liaison avec <strong>le</strong> groupe d’experts<br />
de la DHI créé par <strong>le</strong> Conseil de coordination.<br />
Afin de faire <strong>le</strong> point des travaux accomplis par la communité hydro-<br />
logique en ce qui concerne l’élaboration de projets pour <strong>le</strong>squels on ne<br />
dispose pas de données suffisantes, l’Unesco et l’OMM ont décidé de réunir<br />
conjointement un colloque consacré à cette question. Ce colloque, organisé<br />
avec la collaboration de 1’AISH et du Comité national espagnol pour la<br />
DHI, s’est tenu à Madrid en juin 1973, à l’invitation du gouvernement es-<br />
pagnol.<br />
Le colloque de Madrid a traité en particulier de la méthodologie des<br />
études hydrologiques sans données suffisantes et des pratiques courantes<br />
utilisées pour l’évaluation des paramètres de calcul.<br />
Le colloque a été ouvert par <strong>le</strong> ministre espagnol des travaux publics,<br />
<strong>le</strong> matin du 4 juin, dans <strong>le</strong> cadre du Palais des expositions. Des allocutions<br />
furent prononcées par M. Dumitriscu, au nom du Directeur général de<br />
l’Unesco, par <strong>le</strong> professeur Nemec, au nom du Secrétaire général de l’OMM,<br />
par M. Rodier, président de I’AISH, et par M. Briones, au nom du Comité<br />
national espagnol pour la DHI.<br />
480 participants, venant de 77 pays, participèrent au colloque.<br />
Le programme technique, dont <strong>le</strong> contenu détaillé figure dans la tab<strong>le</strong><br />
des matières, portait sur trois domaines principaux:<br />
1. Méthodologie des études hydrologiques sans données suffisantes;<br />
2. Les pratiques courantes utilisées dans différents pays;<br />
3. Relation entre <strong>le</strong>s données économiques du projet et <strong>le</strong>s données<br />
hydrologiques.
Chacun de ces domaines était subdivisé en thèmes, et sur chaque thème<br />
un rapport général synthétisant <strong>le</strong>s communications individuell<strong>le</strong>s était pré-<br />
senté par un expert, puis suivi d’une discussion.<br />
Les Actes du colloque, établis par <strong>le</strong> Comité national espagnol pour<br />
la DHI, comprennent l’ensemb<strong>le</strong> des communications individuel<strong>le</strong>s et des<br />
rapports généraux, ainsi que <strong>le</strong> compte rendu des débats auxquels ils ont<br />
donné lieu. Ils constituent une publication conjointe de l’Unesco, de l’OMM<br />
et de I’AISH, reflétant l’esprit dans <strong>le</strong>quel <strong>le</strong>s trois organisations ont col-<br />
laboré pendant la DHI.<br />
Comme <strong>le</strong>s communications individuel<strong>le</strong>s n’ont pas été présentées ora-<br />
<strong>le</strong>ment par <strong>le</strong>urs auteurs, el<strong>le</strong>s sont reproduites dans l’ordre où el<strong>le</strong>s sont<br />
apparues dans <strong>le</strong> rapport <strong>le</strong>s concernant.<br />
Unesco, l’OMM et 1’AISH tiennent à remercier <strong>le</strong> Comité national es-<br />
pagnol pour la DHI du concours qu’il a apporté à l’organisation du colloque<br />
et à la publication de ses Actes.
TRANSFER OF INFORMATION FROM OBSERVATION POINTS TO OTHER<br />
POINTS AND DISSEMINATION OF HYDROLOGICAL INFORMATION TO<br />
UNEXPLORED BAS INS<br />
GENERAL REPORT<br />
Prof. A.A. Sokolov<br />
The netv~ork of !yyi;lr;rolugioaì 3bssrvations efist:Ag at present represents<br />
a discrete field of points ivhioh ref<strong>le</strong>ots only approximately <strong>the</strong> *:onstant<br />
variations of hydrologioai elmeats in space and time.<br />
As quite truly notetl Fierre Dubreuil in his report ("<strong>the</strong> prob<strong>le</strong>mi<br />
af traafer of data from observations on a point to one or emo<strong>the</strong>r area<br />
and <strong>the</strong>ir dissenclnation on territories and subjeots where no observations<br />
wero onrried out, has always been. and is n q one of <strong>the</strong> central prob<strong>le</strong>ms<br />
of hydrologf.<br />
This prob<strong>le</strong>m is erspecdally important for developing countries where<br />
<strong>the</strong> network of hydrologioal stations is still hadequate and <strong>the</strong> existing<br />
series of observationa too brief(ehort).<br />
But wen in developed countries v:ith a well organieed and sufficiently<br />
dense network of statione and<br />
is, and always will be/ a great number of water bodies (or regiodor! whose<br />
regime not enough light is thrown by obsemstion data, as a nwnbre of midd<strong>le</strong>-<br />
sized, and espeoially small water bodies, considerhg <strong>the</strong>ir great qurntiQ,<br />
pnill always be examined only se<strong>le</strong>ctively.<br />
posts, having operated Por a long time, <strong>the</strong>re<br />
In <strong>the</strong> Soviet Unior, for eYaq<strong>le</strong>,accordhg to data Prom detai<strong>le</strong>d<br />
inventarication(2) are numbered about 150 O00 rivera with a <strong>le</strong>ngth Of<br />
more than 10 h(and if ne inolude <strong>the</strong> shortest rivers with a l enw balm<br />
10 h, <strong>the</strong>ir total number will aminit to 2 960 O00 ) and about 40 300 lske~<br />
with an area erneeding i square h(?:iùif lakes with an area of <strong>le</strong>ss thsa<br />
1 sq.km are boluded, <strong>the</strong>ir total number amounts to 2 850 o00 ).The<br />
permanently operathg referenoe network of hpirologioal stations inoludes
2<br />
ebout 6200,and <strong>the</strong> meteomlogioal<br />
-<br />
network more<br />
-<br />
than 10.000 observation points.<br />
The task of hydrology as a soienoe oonsiate h establishing, on<br />
<strong>the</strong> basis of a se<strong>le</strong>ctive stuty of water subjects, natural laws of <strong>the</strong><br />
hyilfolcgioal regime and <strong>the</strong> distribution in spaoe of its ohaxeoteristios,<br />
nllowing with a suffioient reliability and preoiwneea neoessaay for praotioe,<br />
to spread hydrologioal data to subjeots or regions with a soarcity or<br />
<strong>the</strong> absenoe of hydrologioal data.<br />
Here we would like to refer agaiii to <strong>the</strong> already mentioned report tnf<br />
Fierre Dubreuil whioh stresses that <strong>the</strong> most important in <strong>the</strong> prob<strong>le</strong>m of<br />
transfer of hydrological data to unexplored basins is <strong>the</strong> analysis anä atuày<br />
of <strong>the</strong> laws of <strong>the</strong> influenoe of natural and anthropogenous faotors on water<br />
regime aad water balanoe, in <strong>the</strong> establishment of qualitative and quan.t;it&e<br />
relations "Hydrology - environment". The author of <strong>the</strong> report notes that <strong>the</strong><br />
applioation in <strong>the</strong> computation of <strong>the</strong> flcod flow and of o<strong>the</strong>r e<strong>le</strong>ments of<br />
~drologioal reg- Qf numerous empirio formulae, determined for certain<br />
natural oonditions in o<strong>the</strong>r regions with different conditions, often results<br />
in gose errors and misoalculations.<br />
The transfer of hydrological data to unexplored basina(regiœna) is<br />
direotly or indirectly relnted to <strong>the</strong> methodology of mapping <strong>the</strong> oharaoterietioe<br />
of <strong>the</strong> 4drological regime applied in hydrology, since <strong>the</strong> praotioalwcyra and<br />
man8 of such tranefer are generally baeed on <strong>the</strong> mapping of oharaoteristioe<br />
of <strong>the</strong> hydrological regime and ita parr reters.<br />
"mo basic methods of dissemination(transfer) of hydrolngioal daea<br />
on mexplored basins(regions) are used with <strong>the</strong> aid of mapa:<br />
1) Drwiilg of maps of isolinesi 2) Division of a territory into regions<br />
based on <strong>the</strong> uniformity of hydrologioal cha-e.oteriat$oa of <strong>the</strong> regime and<br />
its parameters.<br />
The prinoip<strong>le</strong> of <strong>the</strong> method of isolines is <strong>the</strong> assumption of <strong>the</strong><br />
presenoe in ths nature of a smooth, colistant chenve of <strong>the</strong> oharaoteristiorr<br />
of <strong>the</strong> hydrological regime in space,froin one point to ano<strong>the</strong>r. The division<br />
into regions, on <strong>the</strong> oontrary, proceeds from <strong>the</strong> assuqtion "IIomogeneity"
of larger or smal<strong>le</strong>r territories end O€ 8 sudden, spcqmodio ohange of <strong>the</strong><br />
characteristios of <strong>the</strong> regime between one region and ano<strong>the</strong>r..<br />
In <strong>the</strong> publications on hyckology <strong>the</strong>se h o methods of geographic<br />
generalization are often opposed to one ano<strong>the</strong>r. The appearanoe OC.& critioal<br />
thaf<br />
attitude ooncerning <strong>the</strong> method of isolines proceeds from <strong>the</strong> fact w th <strong>the</strong><br />
development of <strong>the</strong> study of smll basins more and more faotors(data) are in<br />
contradiction with <strong>the</strong> mo<strong>the</strong>sis on which this method is baaed. The smal<strong>le</strong>r<br />
<strong>the</strong> river basin, more tho oharacteristics of its hydrologioal regime may differ<br />
from <strong>the</strong> meaning of <strong>the</strong> isolizes wì-ich suppose <strong>the</strong>ir smooth change throughout<br />
<strong>the</strong> territoryo<br />
--<br />
To this o m be given a greaf number 3f exnrnp<strong>le</strong>ao In <strong>the</strong> USSR, or <strong>the</strong><br />
territory ssturted on <strong>the</strong> <strong>le</strong>ft bank of <strong>the</strong> Vo<strong>le</strong>a, for instanoc, two small<br />
basins louated side by side (1OC-200 8q.h.) have a mean many-years spring<br />
flow of 27 and 97 mn, whi<strong>le</strong> on <strong>the</strong> map of isolines of <strong>the</strong> mean depth of <strong>the</strong><br />
spring flow, at this place is shown an isoline of 6ûnnn.<br />
"aturally, <strong>the</strong> question arises: w ht indioate isolineat what is <strong>the</strong>ir<br />
sigriificanoe and <strong>the</strong>ir meming if <strong>the</strong> runoff of actual basins deviates so much<br />
from <strong>the</strong>m?<br />
In our piiblications (3,u are examined <strong>the</strong> reasons of <strong>the</strong> oontradictory<br />
opinions on <strong>the</strong> effioienoy of <strong>the</strong> utilization of <strong>the</strong> method of isolines and<br />
of <strong>the</strong> method OP division into regions. They are oauaed by a misunderetandhg<br />
and an oppesition of <strong>the</strong> zonaliw(moth variations) and <strong>the</strong> eronali-&(sudden,<br />
looal deviafio<strong>le</strong>) in nature.<br />
- In our opinion sonai and asonal 1-8, as well as <strong>the</strong> method8 of mapping<br />
based on <strong>the</strong>m methods of isolines and of division by regione, do not<br />
but mutually oomp<strong>le</strong>te eaoh o<strong>the</strong>r. The first (isolines) shows <strong>the</strong> general,<br />
zonal law# 00 distribution of <strong>the</strong> characteristios of hydrologioal regime through<br />
<strong>the</strong> territory of closed basins( ooinoidenoe or a small difference of $be aurfaoe<br />
and <strong>the</strong> eubsurfaoe nater divide), dieplqred in <strong>the</strong> murse of &ir averaging<br />
for large areas, for whioh <strong>the</strong> influence of azonal(looa1) factors of <strong>the</strong><br />
environment o m be disregardedzThe seaond. permits to reveel <strong>the</strong> kiternaï,<br />
disorste by its essenoe, structure Of' <strong>the</strong>se avoraged oharaoteristioa,<br />
3
4<br />
conditioned by <strong>the</strong> influence of local fwtors - geological struotures, slopes,<br />
vegetation, spi1 and grounds, which constitubo <strong>the</strong> surface of tho basin, ad.<br />
o<strong>the</strong>rs a<br />
ûno of <strong>the</strong> rundamental proTLsions of <strong>the</strong> <strong>the</strong>orj of hydrological mqping<br />
and of <strong>the</strong> applioation of t he method of extrapolation of data on unexplored<br />
basizs by means of maps of io3linesS cor.8ists in tha fact that <strong>the</strong> data uaed<br />
in this chse, concern basins complying with <strong>the</strong> condition:<br />
A ~ A ( A<br />
ma^ ( 1)<br />
i<br />
-<br />
whsre A - mean value of optimal areas OZ <strong>the</strong> catchment in which is te<strong>le</strong>rated<br />
<strong>the</strong> interpolatcion o: hydrological characteristics by mans of isolineai<br />
- A 6, A max respecti/vely <strong>the</strong> lower and <strong>the</strong> upper limit of <strong>the</strong> catchment<br />
area, whose date are unsuitab<strong>le</strong> for drawing m pa of isolines.<br />
Only relation o those basins wiiich comply with <strong>the</strong> condition(I),<strong>the</strong><br />
goographical Iritqrpolation is parrrlssib<strong>le</strong> and, consequently, <strong>the</strong> hypoaesis O?<br />
a smoth and omstant rwiation of tho oharnoteristics of <strong>the</strong> hydrological reLi-ie<br />
on tho terr%toFj is correcto<br />
f<br />
In <strong>the</strong> oatoi-wnt weas rwging from O to A nLio are found o<strong>the</strong>r laws. They<br />
ar+ l.aC<strong>le</strong>cted in larger or mallsr def<strong>le</strong>ctions of <strong>the</strong> characteristioe of <strong>the</strong><br />
ruioff of small rivers fro,^ zonal(ssa <strong>the</strong> abova exainp<strong>le</strong> i>f 27,37 arid 6ûmi)inevitably<br />
everrrged and as if liberated from <strong>the</strong> influonoe of <strong>the</strong> local factorsíspecificities<br />
oí' <strong>the</strong> environment, according ti, P. hbreuil). Chinlg to <strong>the</strong> fa& that in small<br />
basins individual peculiaruios of <strong>the</strong> conditions of <strong>the</strong> runoff of snow melt<br />
and rainfall plow are sham most sharply(for examp<strong>le</strong>, <strong>the</strong> g mmd o," one basin ia<br />
made of ~and,oi' ano<strong>the</strong>r - of olay, or one basin is open, ano<strong>the</strong>r has a forest cover,<br />
etco) <strong>the</strong> data on <strong>the</strong> runaff of <strong>the</strong>se basins generally are not suitab<strong>le</strong> for a<br />
geographioal awmarizing with mags of isolines. With <strong>the</strong> decreaae of <strong>the</strong> size of<br />
<strong>the</strong> ce.%chment increases <strong>the</strong> probabili% of <strong>the</strong> def<strong>le</strong>otions, as well as <strong>the</strong>ir<br />
importanoeo<br />
The above oan be illustrated by a scheme of def<strong>le</strong>otions o? <strong>the</strong> mean annul<br />
runoff(for m my years) of karat rivers from Its zonal significapoe in relation to<br />
<strong>the</strong> area of <strong>the</strong> oatchment(fig.1) .These "fork-shaped" shhemes of def<strong>le</strong>ctions can<br />
be disclosed also in study%ng <strong>the</strong> iliflwnce of othar Pactors(for examp<strong>le</strong>, <strong>the</strong><br />
dekreb of afforestation) on <strong>the</strong> runoef a d <strong>the</strong>ir relatione to <strong>the</strong> dimension of
of <strong>the</strong> o.ztohment.<br />
Th oqiplltation of <strong>the</strong>se dsf<strong>le</strong>otiona is owried out by mane of gemtic<br />
P<br />
rdlatiora of <strong>the</strong> charaoteristioa of <strong>the</strong> hy.itrologioal regime pli* <strong>the</strong> faotors<br />
LieteriliQ tham by <strong>the</strong> lntroduotion of oorrection factors in th~ zonal oharaoteris-<br />
Lics sf <strong>the</strong> hydrologioal regime, obtained for <strong>the</strong> uriexplcred'bssina tlrougk- <strong>the</strong><br />
mq~S of isolines.<br />
Y!ie princip<strong>le</strong>s of oomputation of <strong>the</strong> main oharcoterietios of water<br />
resources in <strong>the</strong> absence or scarcity of )Ij-1romtrioal drrta, based on <strong>the</strong> above eox:al<br />
ari e.zo-ml geographioal laws of $he rtmDff, ar9 examhed with mre tietails in <strong>the</strong><br />
rop0t.t of Rof K.P.Voekresendqr(5)<br />
In <strong>the</strong> light of <strong>the</strong> atom, tht> oonolusion drawn in <strong>the</strong> report of I.Bdek(e;<br />
euòmitted to th3 present Symposim, beco*:es o<strong>le</strong>ar end convincing, nariiely, that<br />
<strong>the</strong> defiliition of reference hy4rologiaal charaoteristios for mexplored sub jecte<br />
(regions)only on <strong>the</strong> basia of data from r-jpreaentativa and experimental basins,<br />
cwnot be ~eoomaded, i.e. it ie not possib<strong>le</strong> to transfer direotly data Prom<br />
7bswvations on 8-1 catoinnents tr, unexplored large basine.<br />
The author of <strong>the</strong> report, analysing <strong>the</strong> data on experimental and represat-<br />
ativu basins of fropioal Mrioa, where under <strong>the</strong> THP programe were created m re<br />
than 100 represantativ<br />
+<br />
d experimental basins, brawe <strong>the</strong> ooaoluiona that a joint<br />
StUdy(8nalya~s)dfdata from experimental and representative basine and of those Prom<br />
<strong>the</strong> standard network is neoesomy. The differenoe between <strong>the</strong> charaoterietios of <strong>the</strong><br />
runoff, obtained on small experimontai catchumts and aidlar characteristioa o? <strong>the</strong><br />
bash of a standard netmork, should be carefully analysed.<br />
Considering <strong>the</strong> influace of fc-eats on <strong>the</strong> runoff on <strong>the</strong> basis of dooumente<br />
from experimental investigationa ia Xqa, ;r.Balak drws <strong>the</strong> conoluaion that a bamboo<br />
or a high mountain forest reduoes <strong>the</strong> surface runoff a that <strong>the</strong> replaobanent of<br />
forests by aEricultural farm inoreases <strong>the</strong> voliono of <strong>the</strong> runoff. Transevaporation<br />
from forest vegetation mas three times higher than that from aeac;onl;r subnierged<br />
fields.<br />
With <strong>the</strong> increase OP tho 'egrje c? boggiilp;, accorahg to data from obser<br />
vations in <strong>the</strong> basin of <strong>the</strong> river Ilafou, <strong>the</strong> annual runoff deoreases. in this oonna-<br />
ctiori, 7.Ba<strong>le</strong>k underlines that mra attertio- should bo drawn 'to hy~oloy;y of<br />
tropical mmps, as <strong>the</strong>se wrsnps play an i:pcrtoil-t ro<strong>le</strong> in ti-9 foranation of <strong>the</strong><br />
river rmoff<br />
5
6<br />
At ths same time he notes that <strong>the</strong> methods of oomptation of <strong>the</strong> nuirf8<br />
usad at present in <strong>the</strong> temperate climate should Se revised taking into aooount<br />
<strong>the</strong> specific oonditione of tropioal oatohrnents.<br />
In <strong>the</strong> report of VbS. Vuglinsw and V.A. se?mmv(fl are atudied <strong>the</strong><br />
specificities of formation and <strong>the</strong> methods oî .h.anefer of data from observa-<br />
tions to unexplored basins si' <strong>the</strong> runoff in mountain are-, including <strong>the</strong><br />
conmody utilized method of detedning <strong>the</strong> standard of <strong>the</strong> annual runoff,<br />
based on <strong>the</strong> establishment of regional relations of <strong>the</strong> npdulua of <strong>the</strong><br />
annual runoff to tho height of <strong>the</strong> uatchment, aooording to data from explored<br />
bas ias<br />
The authors note that,p<strong>the</strong> computation of <strong>the</strong> runoff of small oatch-<br />
-.axts th utilization of <strong>the</strong> relatlon of <strong>the</strong> nodulus of ruroff to tho height<br />
of tho catohment obtains not always satisfaatory resdtte, whio-h can be<br />
explained by <strong>the</strong> kifluence of looal faotors in <strong>the</strong> mouutains~ In this relation<br />
oatchments with <strong>the</strong> same altitude 0811 differ ccmsir<strong>le</strong>rably by <strong>the</strong> conditions of<br />
<strong>the</strong>ir formation, as well as by <strong>the</strong> volun.3 of <strong>the</strong> ennual runoff.<br />
In suoh cases <strong>the</strong> auL,hors recomnend to determine <strong>the</strong> standards of i ki<br />
annual runoff of mountain catcbments witi sufficient or exoessive misture,<br />
by meau of' a joint solution of tho equation of <strong>the</strong> sate- and heat-balanoe.<br />
Ti<strong>le</strong> runoff is oaloulated bj <strong>the</strong> differenoe between preoipitation and m po-<br />
transpiration. The definition of <strong>the</strong> Etasdard annual preoipitation is oarried<br />
out with <strong>the</strong> applioation of graphs of <strong>the</strong> relation of preoipitationa with <strong>the</strong><br />
altitude, taking into amount <strong>the</strong> crogaphio speoifioities of <strong>the</strong> mear<br />
TT oniputation of <strong>the</strong> etandexde of aruiual evaporation is made by a<br />
i.mre preoi e equation of & ïbhdyko# nihioh takes into aooouut <strong>the</strong> turbu<strong>le</strong>nt<br />
heat exohange th3 baaio paramatera of mhioh are : radiation bCbhaOe, precipi-<br />
tation and turbu<strong>le</strong>nt heat-exohange.<br />
The above sohem of computation is used for catolnnents looated in <strong>the</strong><br />
lower and midd<strong>le</strong> mmtein bolts. For <strong>the</strong> higher iiiomrtaine this method oan<br />
be used also, but ki this o w <strong>the</strong> number OP terma of <strong>the</strong> water bcilanoe<br />
equation i8 Fnoreased(it i ta oaïouïata <strong>the</strong> volume of glacier6<br />
ablation, Qf anow pa& melt and a dietinot oeïoulation of evaporation from
various underlying surfaces of <strong>the</strong> high muutains~o<br />
In this report are also studied <strong>the</strong> methoCs of oaloulation of <strong>the</strong><br />
coeft'ioient of variability of <strong>the</strong> annual ruuof'f - Cv, ueed in momtait~<br />
ter ri to i- ies<br />
In <strong>the</strong> report of <strong>the</strong> group of authors: ML Albit<strong>le</strong>t, G. Castany, Mpe<br />
Delaroziere-Boulllin, R. Jonac et G. &-gat(B) is exposed <strong>the</strong> method of appraisal<br />
of <strong>the</strong> &mers1 water resoiiroos(equstsd by <strong>the</strong> authors to <strong>the</strong> mean m ual r-uiorf)<br />
anci <strong>the</strong> renmab<strong>le</strong> groundwater resources with inadequate data. used by th93 authors<br />
for <strong>the</strong> territory of Franca and Venezuela.<br />
The authors recomiiend to determine <strong>the</strong> general water resourcss(Ptreamf1m)<br />
'51~ means of maps of isolines of preci;>itatkon &Td evapotranspiration, by <strong>the</strong><br />
?ifference precipitation minus<br />
t<br />
evepotransporation calculated by ths method<br />
Thornthwalte or Turc. Th3 value f <strong>the</strong>se differancea are dotarmined by conventional<br />
scpares. ìVhm <strong>the</strong>re is a grmi. Wference of <strong>the</strong> factual evaporation,obtained<br />
>y <strong>the</strong> differeroe precipitatior? minus runoff in a looked discharge seotion line<br />
mid evaporation, calculated by ti- method of Thornthwaite or Turc, a oorreotion<br />
coeri'icient is introduced C, by means of irhich <strong>the</strong> map OP <strong>the</strong> rateu evaporatào.on<br />
13 cor reoted.<br />
yhe authore determina <strong>the</strong> na%iiral resources of groundwaters far <strong>the</strong><br />
smis squares of <strong>the</strong> map as <strong>the</strong> volume of tha goneral runoff by moans of<br />
"Geological coefficients" de%i,eci as a portion of <strong>the</strong> unc<strong>le</strong>rgound flair in <strong>the</strong><br />
generai river ~ f f i<br />
'fl.is approach to th?: c<strong>le</strong>.:iriitiov of <strong>the</strong> volurm of renewed resourods<br />
of groundwaters is also utili.eed in <strong>the</strong> ':SSt? in <strong>the</strong> publioations of B.I.Kudolin<br />
and o.v.Fopov(9)<br />
The method o? computation of <strong>the</strong> total runoff by meam ,>P <strong>the</strong> difference<br />
3recipi :.ation minue evapotranapii.ution arid ths 3ef inition, ori tYAs basig oí' <strong>the</strong><br />
so-. ..allad "Clhtio rUnoff",recomnan
8<br />
+<br />
method onl) in regions xh he differenoe-preoipitation rtlh~us wapotranepiration<br />
is stffioiently important.<br />
In <strong>the</strong> practioal application of <strong>the</strong> method, <strong>the</strong> definition of diïferentiated<br />
rr?min@of <strong>the</strong> "geologiaal coefficient" for eaoh square of <strong>the</strong> map Ocin giv$ise<br />
to difficultiee, especially when <strong>the</strong> territory has been inauffioiently stuclied.<br />
In <strong>the</strong> reporta e€ van E ylch are examined <strong>the</strong> methods of oompukation<br />
of evapotranspiration in regione with identioal olhtic oonditione (11)<br />
The author indioates that <strong>the</strong> existing simplified nude<strong>le</strong> of oalouiation<br />
of evapotranspiration through a limited number OP parameters, for at€UUp<strong>le</strong>, by <strong>the</strong><br />
temperature of <strong>the</strong> aiqmrathwaite ,me@ aid Cridd<strong>le</strong>,and o<strong>the</strong>rr) <strong>le</strong>ad to errore<br />
in <strong>the</strong> evaluation of <strong>the</strong> monthly and annual volumea of evaporation m-hg %O<br />
3~% or mare. Contradictions in t h resulta of oalouiationa mado i>r existing<br />
formulae are shown 3x1 fig.1. The author reoommenda, wh& oaloulating evaporation,<br />
a more detai<strong>le</strong>d method with <strong>the</strong> utilieation of euch parametera M radiation<br />
balance, precipitations, air misture8 wìcity of wind.<br />
To find an isauo to this sAtuation, namely, that <strong>the</strong> mentioned initial data<br />
are not everywhere availab<strong>le</strong>, tho author proposes to utilize <strong>the</strong> idea of homolhate.<br />
The main point of his proposal oo#neists in <strong>the</strong> choice of a well-bonm region<br />
mhere <strong>the</strong> clinlatir; conditions approaoh to <strong>the</strong> mx-. <strong>the</strong> conditions of' <strong>the</strong> area<br />
in whioh <strong>the</strong> oamputation of evaporation m&be oarried out, nnd where <strong>the</strong> hitiril<br />
data are misskiq. According to <strong>the</strong> aseertion of <strong>the</strong> author, it is possib<strong>le</strong> by thi&<br />
homolinatio method to oaloulate <strong>the</strong> monthly and m.nual volrpnes of ewlpotrauepira-<br />
tion, differiiig not more than by i@ from those which were measwed.<br />
The author describes <strong>the</strong> desip sohhome adopted by him for <strong>the</strong> homoolimatio<br />
maluationa of evaporation which is based on tho equation developed ky Penmaw<br />
(i9fflDiq56) and <strong>le</strong>er OD improved by Montet((1963)wd Van Baveiï(1966).<br />
It should be noted that this equatior does not take inb aooount <strong>the</strong><br />
temperature stratifioation ar-d th e mietur6 cq-tmt of' +,he soil. Therefore it<br />
iB applioab<strong>le</strong> only for computation of potential svapotranspiration from soctions<br />
of <strong>the</strong> land with an opthal nmisture(irrigation, oapillary subteraraean feeding,<br />
herbagea oloeed up in <strong>the</strong> stage o f optimel development). Unfortunately <strong>the</strong> report<br />
doeo not metnion this.
An important particularity used by <strong>the</strong> author cf <strong>the</strong> equation oonsista in<br />
<strong>the</strong> possibility to oaloulate instentaneOue(urgent) values of <strong>the</strong> velocity of<br />
?vaToration.me author of <strong>the</strong> (van 'PlC%)of <strong>the</strong> opini.on that <strong>the</strong> Us8 OP<br />
<strong>the</strong> seasonal, montka and even aeeWy mean values OP <strong>the</strong> inftfal meteorologiosl<br />
factors for <strong>the</strong> evaluation of <strong>the</strong> evaporability(%) gives false resultem We have<br />
tr agree ruith thie.<br />
var. iìylcluraa<br />
In <strong>the</strong> reference equation Fropoaed by is taken into aooount <strong>the</strong><br />
resistance of <strong>the</strong> out<strong>le</strong>ta(aooord3ng to bntwt). The reoalaulation of tho ptential<br />
evqoration by <strong>the</strong> equation, in nhioh is taùa into aooount <strong>the</strong> rssisfauoe of<br />
oui<strong>le</strong>te, has prmed that this equation obtains <strong>the</strong> best reaults(8ee lower part<br />
cif figme).<br />
This method shoulü be wed oniy for <strong>the</strong> computation of ahorMenn(hourly)<br />
data. l'o illuetrate his opinio8fn Hylc-s fig.1 in which it o m be seen that<br />
fi oalculated by <strong>the</strong> mean hourly initial data cire nearer to fhoee measured,<br />
than <strong>the</strong> data from oalculation through mean daily values of meteorologiod e<strong>le</strong>ments.<br />
TO conolude, t!ie author riotee that on <strong>the</strong> basis of <strong>the</strong> availab<strong>le</strong> climatio<br />
olassification(maps) it is possib<strong>le</strong> to use <strong>the</strong> homolimati0 method and obtain<br />
reliab<strong>le</strong> evaluations of <strong>the</strong> potential Jvapotrsnspirntion for insuffioiontly axplored<br />
regions<br />
The defeot of <strong>the</strong> proposed method of trm-sfer o? data from one region<br />
to ano<strong>the</strong>r oonaista in a huffioient preciseness of <strong>the</strong> definition of <strong>the</strong> hoim-<br />
climate end <strong>the</strong> absence of reliab<strong>le</strong> homolimatic maps.<br />
We have to atop shortly on two o<strong>the</strong>r reports, although different by <strong>the</strong>ir<br />
oontent, but having m q<br />
oomn features. We have iri mind <strong>the</strong> reports of G.R.<br />
Tiercelin (12) and of <strong>the</strong> maup of airthora R. OaroirnAgreda, G. Raastd.0 and<br />
Viparelli(l3). Their oomnon feature is <strong>the</strong> statistical aepe& of <strong>the</strong> prob<strong>le</strong>in of<br />
trauef er of hydrologioal data to unexplored basins(regi0ne)<br />
h3 notes ir his report G.R.Tieroelii: ,disregarding minor d*ih, <strong>the</strong><br />
methods of ddinition of parameters of <strong>the</strong> runoff oould be divided eseentitrlly<br />
hito *O grOU2jS 8<br />
1) The establishment of a regional dependame of <strong>the</strong> value of <strong>the</strong> defined<br />
9
10<br />
paraueter(man, oodfioient of variation, coeffioient of oorreìation between<br />
adjaoenf temm of a series, eto.) from basio phyaiographio oharaoteristics<br />
(precipitation, evaporation, dimension of <strong>the</strong> area of <strong>the</strong> Oabhmonf, height<br />
above sea ïeveï, forests, etc.).<br />
2) A joint analysis of runoff data by a group of bydrologioal identical<br />
catohments(with a similar condition of formation of <strong>the</strong> runoff).<br />
In <strong>the</strong> first oaae, <strong>the</strong> value of <strong>the</strong> interested parameter for an unexplored<br />
stream is oaloulated by <strong>the</strong> dependence obtained through <strong>the</strong> data of <strong>the</strong> neighbus<br />
h g streams, and in <strong>the</strong> seoonä oase this value is oonsidered a8 equal to <strong>the</strong><br />
arithmetioal mean value f'rorn <strong>the</strong> se<strong>le</strong>oted values of parameters of rfvere studied<br />
jointly.<br />
In <strong>the</strong> work of G.P.Tieroelin is used BP assooiated analysis of data<br />
aooording to som previously se<strong>le</strong>oted and hydrologioahy identiod rivers, with<br />
<strong>the</strong> same periods of observation and having slightly differat se<strong>le</strong>otive vduss<br />
of statistioal parame'ters.In this m ~ ~ is e r determined <strong>the</strong> regimal signjd'ioanoe<br />
of parameters of <strong>the</strong> monthly runoff. Data from 12 stations with 49 years of<br />
observatioiis(Prom 1920 to 1968) are wed and are divided iPt0 ho group80<br />
ñegional aues of ertain parameterskoeffioienti of variatiow&ficiat of<br />
htraouo<strong>le</strong>ar correlation, obtained by means of averaging for "idmtioal"<br />
regions are reoonunended by <strong>the</strong> autbr tQ be trenaferred to irnqlored sl;reama<br />
of a given region.<br />
very importtant in this report is <strong>the</strong> <strong>the</strong>oretical part devoted fo <strong>the</strong><br />
definition of <strong>the</strong> mean-square-error of <strong>the</strong> kraasfer of <strong>the</strong> regiod value of<br />
<strong>the</strong> parameter to a oomp<strong>le</strong>tely ur-explored or Fnsuffioiently studied wateroourse.<br />
The importanoe of <strong>the</strong> mean square error depends on <strong>the</strong> qumtlty and<br />
of<br />
hydroïogioaï data for <strong>the</strong> region(o0oasionaï deviation), as neil as from <strong>the</strong><br />
representativeness of <strong>the</strong> studied river for a given region(deviat5on oaused by<br />
geographioal faotors). The Formulation of this prob<strong>le</strong>m tio a large extent reminds<br />
<strong>the</strong> works of S.N.Kritcky, ?d.F.b&el and $.G.Blokhinov(m in whioh it is also<br />
proposed to oonsider fho comp<strong>le</strong>te dispersion of parrunstere of Joint<br />
f<br />
series a8<br />
<strong>the</strong> result of a oonoerted aotion of <strong>the</strong> abové) oauses. The praotioa pplication
of <strong>the</strong> proposed formulae requires a great oare, as <strong>the</strong>ir utilization implies<br />
<strong>the</strong> sigiif;oance of unknown nctuE1 values of dispersions of paraneters and <strong>the</strong><br />
correlation between <strong>the</strong> selocted paraneters. In <strong>the</strong> presenoe of short series and<br />
1;lieir smdl nimiber <strong>the</strong> substitiition of actual values by se<strong>le</strong>oted values o m in<br />
a nunibre of oases oomiderably distort <strong>the</strong> value of <strong>the</strong> mean-square-error of e.<br />
i.ogio2irsS inportanoe.<br />
In <strong>the</strong> work of G.R.Tiercelil? , th0 choice of a group of catchmante<br />
(ciivisiori by regiom) was cwïied out on <strong>the</strong> basi8 of orly a "Visual'1<br />
comparison of <strong>the</strong> se<strong>le</strong>ctive values of <strong>the</strong> parameters oaloulated for separate<br />
rivers.<br />
In our opinion, a preliminary analysis of <strong>the</strong> conditions of formtion<br />
of tho riinoff on catchments outlined for ct joint stuciy with a consequent applic-<br />
ation for <strong>the</strong> final se<strong>le</strong>ction of statistioal oritoria of similarity, would<br />
be m re aoourate. This more rigid apprcjach to <strong>the</strong> se<strong>le</strong>dion of sMlar regions<br />
is applied in <strong>the</strong> work of R. Garcia-Agrede., G.Rassulo and R.Viparelli(l3),<br />
in which <strong>the</strong> authors propose to se1eot"plwiorietric zones'by <strong>the</strong> oonstruction<br />
of "peridssib<strong>le</strong>" 9% confidenoe intervals of paraiiiaters of distribution, determined<br />
according to data from observations in separate points. This more rigid approaoh<br />
will enab<strong>le</strong> w,in a numbor OP cases, +A avoid tho inclusion by mistake in OW<br />
g-oup catohinontJ with heterogonoous ooI?i?itions of formation of <strong>the</strong> runoff<br />
The arithntetioal mean should hardly be taken always as D regional value. It would<br />
be m re advisab<strong>le</strong> to weigh <strong>the</strong> se<strong>le</strong>ctive values of parametars obtained for<br />
separate rivers. For examp<strong>le</strong>, <strong>the</strong> weight ooeffioiente should be taken in a direat<br />
ratio with <strong>the</strong> areas of attraotion and <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> utilized series.<br />
In apite of <strong>the</strong> great preoiseness of regional parameters noted in <strong>the</strong><br />
work o," Mr. Tieroelin , whioh in <strong>the</strong> opinion of <strong>the</strong> author can be muoh higher<br />
than for parameters obtained in a short series, <strong>the</strong> proposed method oannot,of<br />
coupse, replace a oareful analysis of initid data - whioh was already stressed<br />
in <strong>the</strong> report of P. Dubreuil.<br />
In this oorineotion, we would like to refer to <strong>the</strong> detai<strong>le</strong>d oritioiem<br />
of <strong>the</strong> method of hodostations(interc;anneotion of series) submitted in <strong>the</strong> report<br />
of A.I.ChebotareP. and B.I.Serpik on <strong>the</strong> Leningrad Symposium on Floods and <strong>the</strong>ir<br />
Coniputation(l5), with whose opinion, ooncerning <strong>the</strong> sffioiency of this method,<br />
i quite agreeo<br />
11
12<br />
Besides, <strong>the</strong> author himelf repeatedly atreeeee th0 neoeseity of a oareful<br />
approaoh to <strong>the</strong> interconneotion of eerie8 of obeervationa on rims of <strong>the</strong><br />
so-cal<strong>le</strong>d identioal regione.<br />
To conoluäe, it diould be noted fhat bestigatiom on <strong>the</strong> a;plication<br />
of <strong>the</strong> ma<strong>the</strong>matioal apparatus for <strong>the</strong> Lins of epeoe interpolation of hydrologioal<br />
oharacterietios of <strong>the</strong> multip<strong>le</strong> liuear oorrelation are oarried out at preaeurt(16,17)<br />
At <strong>the</strong> sau~ time one<strong>the</strong> basis of a multip<strong>le</strong> linear regreasion, <strong>the</strong> oonetruc-<br />
tion of a field of isolines of hyärologioal characteristior, imp<strong>le</strong>ppsnted by a<br />
conputor with an ewaluation of <strong>the</strong> preoiseneas of interpolation in q givan point,<br />
ie eventually projeoted.
I. P.Dubreui1. Transfer of hydrologioal information to uuexploreâ river basins<br />
(presented ta <strong>the</strong> Symposiimi )<br />
2. A.P.Dodt~, ReG. Dubrovina, A.I. i8aevat"Rivers and Lakes of <strong>the</strong> USSR"<br />
(reference data) Gidrometeoiedat, 1971.<br />
3e A.A.Sokolov "Zonal m d a~oneï factors of <strong>the</strong> runofP".Coll.nf public. on %drology<br />
No 2, GidrO~teOisdat, 1961.<br />
4. A.A. Sohlov.The <strong>the</strong>ory of hydrologioal mapping. Bull&ln VW, N0.1~1968.<br />
5. K.P.Voükrûs~~e Princip<strong>le</strong>s for <strong>the</strong> computation of <strong>the</strong> basi4 oharaoteristioa<br />
of water resouroes of rivers with inadequate observationa on <strong>the</strong> baais of <strong>the</strong><br />
geographical interpolation of th^ paraniators of <strong>the</strong>' runoff (presented to <strong>the</strong><br />
Splposf tall$<br />
6, 3.Ba<strong>le</strong>k. Utilieation of representativo and experimental catobments for <strong>the</strong><br />
weluation of hydrological datri Sron Aifricm. tropical bas- (presented to <strong>the</strong><br />
Syqo s id<br />
7. V.S.Vuglinsky and V.A.Semonov. fialualiion of water rmources of mountain<br />
territories in <strong>the</strong> absence or soarcity of data of <strong>the</strong> runoff(presonted to <strong>the</strong><br />
symposium)<br />
8. M. Albinef, GICastauy, Mr8r belaroeiercBouillin, R. Jonirc,. J. Margat.<br />
Evaluation and distribution of water resources of large regions on <strong>the</strong> baais of<br />
hydroclimatio and hydrologic oharaoteristioa (presented to <strong>the</strong> ~psimn)<br />
9. G.I.Kudelin, OiVmPOpOV. Influeuce of olimate on <strong>the</strong> natural 1-8 of formation<br />
of <strong>the</strong> groumator fian. Reports OP <strong>the</strong> soviet geeïogistrr to <strong>the</strong> 24th session<br />
of th9 International Congresri on ûeology.%ydrogeology and Engitmerlng bolo&,<br />
"Nauka", baoon, 1Wm<br />
10. M.I.LlvovEtoh. E<strong>le</strong>mnte of water regime of <strong>the</strong> rivers OP %e Earth. hblio.<br />
of KtU cenisa) sa) Board OP <strong>the</strong> Qdrsmiteomlogioal Servioe) Ser.IVlvol.18<br />
Sverdlovsk - Leahgrßddr<br />
13
14<br />
Hylckama<br />
11. T.E.A. V a Computation of evapotrmspiration by region8 with<br />
idontical climatic condWions(presented to <strong>the</strong> Symposium)<br />
12. 1LR.Tiercelb Regional parainstors concarning water resouroes. üsee.heciseneas<br />
of evaïuation(presented to <strong>the</strong> Symposium)<br />
13. R. Garoia-Agreda, G. Rasuulc, R. Viparelli. Pluviornetrio zones and oriteria<br />
for evaluation of <strong>the</strong>ir limits Por region8 w5th insufficient data from observations<br />
(presuntod to <strong>the</strong> Symposium)<br />
14. S.?l.Kritzky, M.F.Menke1. T.bthod of a joint aaolyeis of observation8 of <strong>the</strong><br />
runoff of identical basina. Public. of <strong>the</strong> CCI (State Institute of Hydrology)<br />
vol.180, kMmmeteoizdat, lr;7G.<br />
15. A.I.ChsLotmev and B.;.SerpZc. Of <strong>the</strong> passibility of using <strong>the</strong><br />
intorconnected seriea of hyàrologioal ohaxaoteristios for <strong>the</strong> oomputaticn oi t h<br />
runoff:. Internationo1 Sy-fnposiun on Flscds a d <strong>the</strong>ir ComputationbGidrometeoiedat,<br />
1969<br />
16. A.V. lbjdestvendcy. The exporimco of bringing <strong>the</strong> river runoff to a long-term<br />
period by <strong>the</strong> method of multip<strong>le</strong> linear correlation. Coll. of public. on ~drolog<br />
No .10.Gidromsteoi~dat,1970.<br />
17. A.G. bbanova, A.V. HojdestvensQ. Space-correlation funotiona of <strong>the</strong> river'<br />
raoff of <strong>the</strong> rivers of <strong>the</strong> Dn<strong>le</strong>pr b ash coll. of puklio. on ~dr010gy~1Jo.11<br />
Gi drom tuo izdat , 1973
EVALUATION ET REPARTITION DES RESSOURCES EN EAUX D'UNE GRANDE<br />
ABSTRACT<br />
REGION PAR LES PARAMETRES HYDROCLIMATIQUES ET HYDROGEOLOGIQUES<br />
Par: M. Albinet, G. Castany, Mme O. Delaroziere-bouillin,<br />
R. Jonac et J. Margat.<br />
The evaluation and repartition of total and groundwater resources<br />
or a large unit, country, region or groundwater basin, may be rapidly<br />
made with restricted data, by simp<strong>le</strong> calculation, still obtaining a<br />
satisfactory accuracy.<br />
The total water resources, asimilated to <strong>the</strong> average annual total<br />
runoff rate of <strong>the</strong> water courses may be evaluated by <strong>the</strong> specific<br />
runoff. This is calcu<strong>le</strong>d, ei<strong>the</strong>r directly with hydrometric data, or<br />
in <strong>the</strong> absence of gauging by extrapolation based on hydrogeological<br />
characteristics collated with <strong>the</strong> values by <strong>the</strong> climatological exprez<br />
sions (L.TURC, THORTHWAITE).<br />
The groundwater renouvelab<strong>le</strong> resources are egal to <strong>the</strong> average<br />
annual groundwater flow rate those evaluation tests on <strong>the</strong> division,<br />
with <strong>the</strong> help of an index, of <strong>the</strong> specific runoff. These indes are<br />
worhed out with <strong>the</strong> help of geological characteristics an hydrogeo-<br />
logical characteristics punctually obtained b,y field tests.<br />
Thus with resticted hydrogeological and hydrometric data and<br />
sufficient data concerning <strong>the</strong> precipitations, tempertures and geology,<br />
it is possib<strong>le</strong> to obtain a satisfactory know<strong>le</strong>dge of water resources<br />
which exploitation and planification. Practical results have been<br />
obtained in France and Venezuela.<br />
RESUME<br />
L'évaluation et la répartition des ressources en eaux, globa<strong>le</strong>s<br />
et soutterraines, d'une grande unité, pays, région ou bassin hidro-<br />
géologique, peuvent être effectuées rapidement avec des données res-<br />
treintes, par des calcu<strong>le</strong> simp<strong>le</strong>s, tout en obtenant une précision<br />
satisfaisante.<br />
Les ressources en eaux globa<strong>le</strong>s, assimilées au debit d'écou<strong>le</strong>ment<br />
global annuel moyen des cours d'eau, peuvent être évaluées par <strong>le</strong> mo-<br />
du<strong>le</strong> spécifique d'ecou<strong>le</strong>ment total (i/s.km2). Celui-ci est calculé,<br />
soit directement 2 partir des <strong>le</strong>s données hydrométriques, soit, en<br />
l'absence de jaugeages, par extrapolation basée sur <strong>le</strong>s paramètres<br />
hydrogéologiques et confrontée avec <strong>le</strong>s va<strong>le</strong>urs calculées par <strong>le</strong>s ex-<br />
pressions climatologiques (L.TURC, THORTHWAITE).<br />
Les ressources en eaux souterraines renouvelab<strong>le</strong>s son éga<strong>le</strong>s au<br />
débit de l'écou<strong>le</strong>ment souterrain annuel moyen, dont l'évaluation repose<br />
sur <strong>le</strong> fractionnement, a l'aide d'index, du modu<strong>le</strong> spécifique<br />
d'écou<strong>le</strong>ment total. Ces index sont étab<strong>le</strong>s l'aide des paramètres<br />
geologiques et des caractéristiques hydrogéologiques obtenues ponctuel<strong>le</strong>ment<br />
par des essais sur <strong>le</strong> terrain.<br />
Ainsi avec des données hydrométriques et hydrogéologiques res-<br />
treintes et des données suffisantes sur <strong>le</strong>s précipitations, <strong>le</strong>s tem-<br />
pératures et la géologie, il est possib<strong>le</strong> d'obtenir une estimation<br />
satisfaisante des ressources potentiel<strong>le</strong>s moyennes pour la mise en<br />
va<strong>le</strong>ur et la planification, Une realisation pratique a été obtenue<br />
en France et au Venezuela.
16<br />
1 . INTRODUCTION<br />
1.1. Rappel des notions sur l'écou<strong>le</strong>ment de l'eau dans <strong>le</strong> sol et <strong>le</strong><br />
sous-sol. Répartition de l'eau des précirdtatlons.<br />
Le débit de l'écou<strong>le</strong>ment total QT, mesur8 à la station de<br />
jaugeage d'un cours d'eau, exutoire d'un bassin versant, est la somme<br />
de l'écou<strong>le</strong>ment de surface QR dans <strong>le</strong> réseau hydrographique et de 1'<br />
écou<strong>le</strong>ment souterrain QW, transité par <strong>le</strong>s aquifères du bassin drainé$<br />
L'écou<strong>le</strong>ment de surface, QR, direct, rapide (quelques heures<br />
quelque6 Bows) correspond à la crue de l'hydrogramme d'écou<strong>le</strong>ment.<br />
L'écou<strong>le</strong>ment souterrain, QW, <strong>le</strong>nt,différ$, de parcours com-<br />
p<strong>le</strong>xe dans <strong>le</strong>s aquifères et de longue durée (quelques années à des<br />
centaines, voire des milliers, de millénaires) est à l'origine du<br />
débit des cours d'eau pérennes en absence de précipitations (étiage).<br />
D'oui l'importance de la mesure des débits d'étiage représentant <strong>le</strong><br />
déMt minimal moyen de l'écou<strong>le</strong>ment souterrain.<br />
Le débit de l'écou<strong>le</strong>ment total est alimenté par <strong>le</strong>s préci-<br />
pi ta ti ons e f f i c ac e s, PE, dl f f ér en c e s entre 1 I évapo transpira ti on<br />
réel<strong>le</strong>, ETR et <strong>le</strong>s précipitations tota<strong>le</strong>s, PT ( PE = PT - ER). Eh 1'<br />
absence de variation<br />
des réserves (longue période d'observation) <strong>le</strong><br />
déficit d'écou<strong>le</strong>ment moyen interannuel E"T est égal à PT - QT.<br />
Les débits de llécou<strong>le</strong>ment total et de 88s deux composants,<br />
l'écou<strong>le</strong>ment de surface et l'écou<strong>le</strong>ment souterrain, sont régis par six<br />
groupes de facteurs conàîtionnelst<br />
-<br />
caractéristiques dee précipitations: intensi téídurée, nature ;<br />
caractéristiques géologiques du sol: lithologie des terrains,<br />
perméabilité vertica<strong>le</strong>, structures;<br />
- c arac t éri stiqu e 6 mo rp bolo giqu e s : rnorphom 6 t 15 e, pen tes , reli e f ;<br />
- cmactéristiques hydrogéologiques: humidité de la zone non<br />
eaturée, profondeur de la surface piézométrique, paramètres hydrauli-<br />
ques des roches réservoirs et de l'écou<strong>le</strong>ment et de6 structures hydro-<br />
géologiques;
- caractéristiques de la couverture végéta<strong>le</strong>.<br />
Ces facteurs, interférant, peuvent @tre ramenés B trois grands<br />
ensemb<strong>le</strong>s: hydroclimatologie-hydrométrie, géomorphologie, géologie.<br />
Les caractéristiques géomorphologiques et géologiques du bas-<br />
sin jouent un r8<strong>le</strong> primordial dans <strong>le</strong> fractionnement de l'eau des préci-<br />
pitations, d'o.ii la possibilité d'établir des index, utilisab<strong>le</strong>s pour 1'<br />
évaluation du débit de l'écou<strong>le</strong>ment total et de l'écou<strong>le</strong>ment souterraint<br />
De m8me il est possib<strong>le</strong> d'établir des index climatiques.<br />
Les réservoirs aquiferes ont un r8ie régulateur du débit de 1'<br />
écou<strong>le</strong>ment souterrain par la faib<strong>le</strong> vitesse d'écou<strong>le</strong>ment déterminée par<br />
la transmissivité et par la mise en réserve temporaire d'eaux souterrai-<br />
nes, fonction de la diffusivité ( transmissivité/coefficient d'emmagasi-<br />
nement) et des conditions aux limites. Les réserves en eaux souterraines<br />
sont donc a considérer pour l'évaluation des ressources en eau.<br />
1.2. Débit de l'écou<strong>le</strong>ment moyen interannuel et ressources en eaux<br />
renouvelabl es.<br />
L'écou<strong>le</strong>ment moyen interannuel, QT, est assimilé aux ressour-<br />
ces en eaux renouvelab<strong>le</strong>s, potentiel<strong>le</strong>s, moyennes globa<strong>le</strong>s. I1 est<br />
déterminé sur une période de 5 à 10 ans:<br />
- directement par traitement statistique des données hydrométriques;<br />
- -<br />
indirectement 6. l'aide d'expressions climatiques mensuel<strong>le</strong>s ( TURC<br />
et THORNTHWAITE) résolues manuel<strong>le</strong>ment ou sur ordinateur.<br />
L'écou<strong>le</strong>ment moyen interannuel, QT, erprimé en laine d'eau<br />
2<br />
moyenne, ou modu<strong>le</strong> spécifique d'écou<strong>le</strong>ment total (l/s.km ) permet <strong>le</strong>s<br />
interpolations et extrapolations et l'estimation des ressource8 poten-<br />
tiel<strong>le</strong>s moyennes des bassins non jaugés.<br />
L 'estimation des ressources potentfel<strong>le</strong>s moyennes globa<strong>le</strong>s par<br />
cette méthode est très acceptab<strong>le</strong> pour <strong>le</strong>s besoins de la planification,<br />
comparée aux &mi.uations basées uniquement sur des mesures hydrométri-<br />
ques relatives à de longues périodes.<br />
17
18<br />
1.3. - Débit et distribution spatia<strong>le</strong> de l'écou<strong>le</strong>ment souterrain mq- interannuel<br />
Le débit de l'écou<strong>le</strong>ment souterrain moyen<br />
assidlé au débit moyen interannuel des aquifères dans <strong>le</strong> cours d'eau,<br />
peut être évalué par l'analyse de l'écou<strong>le</strong>ment moyen interannue1,QT.<br />
Une méthode de fractionnement, à l'aiae d'index et étalonnage par des<br />
analyses d'hydrogrammes de bassins représentatifs assez homog&nes, a<br />
áté appliquée. Ces index expriment:<br />
index = écou<strong>le</strong>ment souterrain - - c$w en pour cent<br />
écou<strong>le</strong>ment total QT<br />
2. PRINCIPES DE LA METHODE<br />
Une importance particulière est apportée, dans un souci<br />
de planification et d'aménagement du territoire, à la connaissance,<br />
donc A la cartographie, de la distribution spatia<strong>le</strong> des ressources en<br />
eau, globa<strong>le</strong>s et souterraines. Les données hydrologiques disponib<strong>le</strong>s<br />
sont, dans la plupart des régions, insuffisantes pour permettre<br />
une cartographie. Par ail<strong>le</strong>urs dans bien des cas il serait inte-<br />
ressant de pouvoir estimer <strong>le</strong>s modu<strong>le</strong>s spécifiques d'écou<strong>le</strong>ment<br />
de bassins non jaugés.<br />
C'est dans ces perpectives qu'une méthode simplifiée<br />
d'évaluation des écou<strong>le</strong>ments moyens, total et souterrain, par bassin<br />
versant a été mise au point en vue d'une cartographie à petite échel<strong>le</strong><br />
applicab<strong>le</strong> B l'ensemb<strong>le</strong> d'une région ou d'un pays.<br />
Son principe, 6es modalités d'application et <strong>le</strong>s résul-<br />
tats obtenus sont présentés sur un exemp<strong>le</strong> concret.<br />
La méthode d'évaluation et de cartographie de 1'8cou<strong>le</strong>umt<br />
a été établie de facon à pouvoir Itre traitée automatiquement. <strong>le</strong>, ou<br />
<strong>le</strong>s, bassin6 étudiés étant discrétisés en mail<strong>le</strong>s régulières.<br />
2.1. Données de bases nécessaires à l'application de la méthode.<br />
Ce sont:<br />
- surface du bassin versant
- débit moyen interannuel, QT, de la période p, mesuré à l'exutoire<br />
du bassin versant;<br />
- carte en courbes isohyetes des précipitations moyennes interannuel<strong>le</strong>s<br />
PT, de la période p, sur l'ensemb<strong>le</strong> du bassin versant;<br />
- carte de zonalité de l'évapotranspiration réel<strong>le</strong> moyenne inter-<br />
annuel<strong>le</strong> ETR, de la période p, sur l'ensemb<strong>le</strong> du bassin versant. E$<br />
théorie n'importe quel<strong>le</strong> méthode de calcul d'un indice d' évapotrans-<br />
piration réel<strong>le</strong> a partir des données climatologiques mesurées ponctuel-<br />
<strong>le</strong>ment peut 8tre utilisé. En général, <strong>le</strong>s va<strong>le</strong>urs de llévapotranspi-<br />
ration réBl<strong>le</strong> moyenne interannuel<strong>le</strong> <strong>le</strong>s plus significatives sont ob-<br />
tenues par calcul sur l'pas de temps11 mensuel, soit à partir de la<br />
hauteur des précipitations et de la température par la méthode de<br />
THORNTHWAITE, soit à partir de la hauteur des précipitations, de la<br />
température et de l'insolation par la méthode de TURC mensuel<strong>le</strong>. Dans<br />
ces deux cas, <strong>le</strong>s calculs doivent Btre effectués mensuel<strong>le</strong>ment pour<br />
chacune des années réel<strong>le</strong>s successives de la période choisie. La moyen-<br />
ne interannuel<strong>le</strong> doit Btre évaluée exclusivement à partir des va<strong>le</strong>urs<br />
annuel<strong>le</strong>s de 1' évapotranspiration réel<strong>le</strong> obtenues. Ces opérations<br />
peuvent Etre réalisées automatiquement à l'aideud'un programme de cal-<br />
cul établi au Bureau de recherches géologiques et minières (B.R.G.M.).<br />
2.3. Calcul de l'écou<strong>le</strong>ment total.<br />
Les données de base étant acquises, <strong>le</strong>s cdculs suivants<br />
sont<br />
-<br />
effectués successivement:<br />
calcul de la lame d'eau prkcipitée moyenne interannuel<strong>le</strong> (en mm),<br />
m, sur l'ensemb<strong>le</strong> du bassin versant, par moyenne des lames d'eau précipitées<br />
-<br />
sur chaque mail<strong>le</strong>;<br />
calcul du déficit d'écou<strong>le</strong>ment moyen interannuel (en mm), ETT,<br />
sur 1 'ensemb<strong>le</strong> du bassin versant, par moyenne des déficits d'écou<strong>le</strong>ment<br />
relatifs à chaque mail<strong>le</strong> lorsque l'on a admis une hétérogén6ité de<br />
ETR dans <strong>le</strong> bassin (sinon ETT = PT - QT);<br />
19
20<br />
- comparaison<br />
de la différence, PT - QT (données mesurées) avec<br />
hTT. (données calculées) et calcul d'un coefficient de correction c;<br />
C=(PT-QT)/ETT<br />
Puis calage des "bilans" unitaires de chaque mail<strong>le</strong> ( ETT =<br />
PT - QT) sur <strong>le</strong> débit d'écou<strong>le</strong>ment total du bassin par application du<br />
coefficient de correction, C, à 1'EITR de chaque mail<strong>le</strong>.<br />
- calcul de l'écou<strong>le</strong>ment total unitaire, mail<strong>le</strong> par mail<strong>le</strong> (en mm),<br />
par différence entre la lame d'eau précipitée et la hauteur d'&rapo-<br />
transpiration réel<strong>le</strong> corrigée.<br />
2.3. Evaluation de la distribution spatia<strong>le</strong> de l'écou<strong>le</strong>ment souterrain<br />
La distribution par mail<strong>le</strong> de ltécou<strong>le</strong>ment total pour <strong>le</strong><br />
bassin étudie étant connue1.trois procédures sont appliquées en fOnC-<br />
tion des données disponib<strong>le</strong>s pour l'évaluation et la distribution spa-<br />
tia<strong>le</strong><br />
-<br />
de l'écou<strong>le</strong>ment souterrain.<br />
premier<br />
-<br />
cas:iaxistence d'une carte des index géologques (page 514<br />
l'écou<strong>le</strong>ment souterrain, QW, de chaque mail<strong>le</strong> est obtenu di-<br />
rectement par application des index à la va<strong>le</strong>ur de l'écou<strong>le</strong>-<br />
-<br />
ment total de la mail<strong>le</strong>;<br />
<strong>le</strong> calcul de l'écou<strong>le</strong>ment souterrain total du bassin versant<br />
est effectué par sommation des écou<strong>le</strong>ments souterrains de<br />
chaque mail<strong>le</strong>m C'est cette procédure qui a été utilisée pour<br />
l'étude de la P'ranche-Comté (France) objet du cas concret.<br />
"<br />
- deuxième cas: existence d'une estimation de l'écou<strong>le</strong>ment souterrain<br />
total à l'exutoire du bassin versant (va<strong>le</strong>ur obtenue par analyse des<br />
hydrogrammes, selon une convention appropriée) et d'un bassin litholo-<br />
giquement assez homogène. Cette méthode, concevab<strong>le</strong> en théorie est<br />
rarement applicab<strong>le</strong> en pratique, car <strong>le</strong>s bassins assez grands qu'il<br />
faut considérer ne sont généra<strong>le</strong>ment pas homogènes.<br />
- troisiéme cas: existence d'une carte des index et de la va<strong>le</strong>ur<br />
estimée de,l'écou<strong>le</strong>ment souterrain total à l'exutoire du bassin versant
- une première va<strong>le</strong>ur de l'écou<strong>le</strong>ment souterrain de chaque<br />
mail<strong>le</strong> est obtenue par application des index à la va<strong>le</strong>ur de<br />
l'écou<strong>le</strong>ment total de la mail<strong>le</strong>;<br />
- calcul de l'écou<strong>le</strong>ment souterrain total du bassin versant<br />
par sommation des écou<strong>le</strong>ments souterrains de chaque mail<strong>le</strong>;<br />
- comparaison de l'écou<strong>le</strong>ment souterrain total avec l'écou<strong>le</strong>-<br />
ment total, QT, et calcul d'un coefficient de correction C1:<br />
-<br />
QT QW<br />
- application de ce coefficient de correction C1 aux débits<br />
souterrains de chaque mail<strong>le</strong>.<br />
I1 est important de souligner que <strong>le</strong> débit souterran<br />
calculé pour chaque mail<strong>le</strong> a la signification de l'alimentation spé-<br />
cifique moyenne probab<strong>le</strong> des nappes souterraines dans la mail<strong>le</strong>, par<br />
infiltration de l'eau des précipitations, indépendamment de tout<br />
apport pouvant provenir d'une autre mail<strong>le</strong>.<br />
2.4. Simplifications admises.<br />
La méthode d'éualuation des écou<strong>le</strong>ments, total et souter-<br />
rain, peut fournir des résultats significatifs al el<strong>le</strong> est appliquée<br />
H des bassins versants de dimensions assez grandes, à partir des<br />
données hydroclimatologiques moyennes interannuel<strong>le</strong>s, établies<br />
sur une période suffisamment longue pour que <strong>le</strong> rb<strong>le</strong> des réserves,<br />
superficiel<strong>le</strong>s ou souterraines, puisse $tre négligé.<br />
De plus cette méthode s'adresse aux bassins versants pour<br />
<strong>le</strong>squels il est possib<strong>le</strong> d'admettre que <strong>le</strong> débit des nappes souter-<br />
raines est drainé essentiel<strong>le</strong>ment par <strong>le</strong>s cours d'eau du bassin. Eh<br />
domaine karstique,par exemp<strong>le</strong> il sera nécessaire de grouper <strong>le</strong>s<br />
bassins versants de tel<strong>le</strong> sorte que <strong>le</strong>s transferts d'eau aux limites<br />
des groupements établis soient négligeab<strong>le</strong>s.<br />
terrain,<br />
3. APPLICATION DE LA METHODE A UN CAS CONCRET - POSSIBILITES<br />
D 1 AUTOMATI SATION - PROGRAMME FI,&.<br />
Le calcul et la cartographie des écou<strong>le</strong>ments, total et sou-<br />
ont été réalisés pour <strong>le</strong>s bassins versants du Doubs, de la<br />
21
22<br />
Haute baône et de ltxin, sur une période de référence moyenne de 5<br />
ans. A cet effet, une carte des précipitations moyennes et une carte<br />
de 1' évapotranspiration réel<strong>le</strong> moyenne interannuel<strong>le</strong> (méthode de TURC<br />
mensuel<strong>le</strong>) ont été réalisées.<br />
Trois ensemb<strong>le</strong>s de bassins versants ont été utilisés, en<br />
fonction des re<strong>le</strong>vés hydrométriques disponib<strong>le</strong>s, pour llapplication<br />
du programme I.L$C. Dans <strong>le</strong>ur définition, tous <strong>le</strong>s bassins versants<br />
é1éi;<strong>le</strong>ntdres présentant entre eux des échanges souterrains ont été<br />
groupés de tel<strong>le</strong> sorte que pour chaque ensemb<strong>le</strong> <strong>le</strong>s limites topogra-<br />
phiques et hydrogéologiques soient concordantes. Les groupements<br />
suivants ont donc été traités:<br />
- bassins versants de la riaute-Saône et de l'Ognon limités aux<br />
stations de jaugeage de day-sur-&8ne (sur la aaôiie) et de Pesmes<br />
(sur l'Ognon). Superficie tota<strong>le</strong> : 5 782 km2, débit total mogcpn<br />
(période<br />
-<br />
i964-iY68): 3 093, 7. lo6 m3/an;<br />
bassins versants du Doubs et de la Loue limités aux stations<br />
de jaugeage de Rochefort (mir <strong>le</strong> Doubs) et de Champagne (sur la Loue)t<br />
Superficie tota<strong>le</strong>: 6 350 km'; débit total moyen (période 1964-1968):<br />
5 086,3 .i0 6 m 3 /an;<br />
- bassin versant de l'Ain limité à la station de jaugeage de<br />
r*<br />
Chaaey. Superficie tota<strong>le</strong>: 3 630 km2, débit total moyen (periode<br />
1963-1967): 3 944.i06 J/an.<br />
3.1. MaillaRe des bassins<br />
La planche 1 prhsente <strong>le</strong> maillage adopté pour <strong>le</strong> Calcul<br />
des écou<strong>le</strong>ments. Les mail<strong>le</strong>s, généra<strong>le</strong>ment carrées (sauf aux limites) I<br />
ont une surface de 25 km<br />
2<br />
pbur <strong>le</strong>s bassins du Doubs et de la Haute-<br />
Saône et de 9 km<br />
2<br />
pour celui de l'Ain. Soit au total pour chaque<br />
bassin versant hydrographique: Haute-Sadne et Ognon, 232 mail<strong>le</strong>s;<br />
Doubs et Loue, 254 niail<strong>le</strong>s; Ain, 405 mail<strong>le</strong>s.<br />
3.2. Cartographie de la distribution probab<strong>le</strong> de l'écou<strong>le</strong>ment total<br />
et de 1 'écou<strong>le</strong>ment souterrain.
Les va<strong>le</strong>urs par mail<strong>le</strong> de l'écou<strong>le</strong>ment total sont directement<br />
fournies par l'application du programme de calcul FLØC.<br />
La planche 1 présente la carte de la distribution probab<strong>le</strong> de 1'<br />
écou<strong>le</strong>ment total pour <strong>le</strong> territoire étudié. El<strong>le</strong> donne <strong>le</strong>s va<strong>le</strong>urs en<br />
mm par mail<strong>le</strong> de l'écou<strong>le</strong>ment total moyen interannuel (1964-1968) et<br />
<strong>le</strong>s courbes d' (gal écou<strong>le</strong>ment total''.<br />
3.3. CartograDhie de la distribution probab<strong>le</strong> de l'écou<strong>le</strong>ment souter-<br />
rain.<br />
7<br />
L'application des index aux va<strong>le</strong>urs calculées de l'écou<strong>le</strong>ment<br />
total permet de dresser une carte de la va<strong>le</strong>ur moyenne du débit des<br />
nappes d'eau souterraine de la région étudiée.<br />
Pour l'ensemb<strong>le</strong> du territoire étudié,l'aptitude du sol et du<br />
sous-sol à permettre l'infiltration a été analysée sur la base des<br />
données de la carte lithologique établie spécia<strong>le</strong>ment (fig.2) cl partir<br />
de l'examen des hydrogrammes de quelques cours d'eau. Mais <strong>le</strong>s stationr<br />
de jaugeage dont <strong>le</strong>s données on été utilisées sont généra<strong>le</strong>ment rap-<br />
portées A des bassins versants étendus, climatologiquement et litholo-<br />
giquement hétérogènes. L'analyse de <strong>le</strong>urs hydrogrammes n'a donc pu,<br />
dans la plupart des cas, permettre de définir.<br />
tisfaisante l'importance de l'aptitude du sous-sol A permettre l'in-<br />
filtration. Compte-tenu de ces réserves (et de la possibilitk d'affi-<br />
ner cette analyse lorsque l'on disposera des re<strong>le</strong>vés de nouvel<strong>le</strong>s sta-<br />
tions<br />
-<br />
de jaugeage) deux types de domaines ont été distingués:<br />
<strong>le</strong>s domaines od l'infiltration est nbgligeab<strong>le</strong>, <strong>le</strong> sous-sol pou-<br />
23<br />
avec une précision sa-<br />
vant Btre conddéré comme imperméab<strong>le</strong> à l'échel<strong>le</strong> de cette étude) Pour<br />
ces-dbmalnes, A l'échel<strong>le</strong> du l/ 200 O00 la quad totalité de l'écou<strong>le</strong>-<br />
ment<br />
-<br />
est de l'écou<strong>le</strong>ment de surface. L'écou<strong>le</strong>ment souterrain est nui;<br />
<strong>le</strong>s domaines à réservoirs aqui feres pour <strong>le</strong>sque<strong>le</strong> l'infiltration<br />
est possib<strong>le</strong>. L écou<strong>le</strong>ment souterrain représente alors une proportion<br />
plus ou moins é<strong>le</strong>vée de l'écou<strong>le</strong>ment total. I1 est possib<strong>le</strong> de dis-<br />
tinguer:
24<br />
- <strong>le</strong>s domaines 03. l'écou<strong>le</strong>ment souterrain représente une proportion<br />
moyenne de l'écou<strong>le</strong>ment total (9 %) avec <strong>le</strong>s grès du Permien et du<br />
Trias inférieur, <strong>le</strong>s formations marno-calcaires du Crétacé et <strong>le</strong>s dép8tr<br />
giaci air es et fluvi o- glaci air es ;<br />
- <strong>le</strong>s domaines OC l'ecou<strong>le</strong>ment souterrain représente une forte propor-<br />
tion de l'écou<strong>le</strong>ment total (80 %> avec <strong>le</strong> Crétacé & dominante calcaire<br />
du bassin de l'Ain et <strong>le</strong>s formations alluvia<strong>le</strong>s;<br />
- <strong>le</strong>s domaines OU l'écou<strong>le</strong>ment souterrain représente la totalité de<br />
1 'écou<strong>le</strong>ment: formations calcaires du Nuschelkalk et du Jurassique<br />
supérieur et moyen.<br />
Conclusions - Gomparaisons entre <strong>le</strong>s écou<strong>le</strong>ments<br />
mesurés et estimés- Validité de la méthode.<br />
Différents tests ont été effectués sur plusieurs bassins<br />
afin, connaissant <strong>le</strong>s débits mesurés, d'estimer quel<strong>le</strong> validité ont <strong>le</strong>s<br />
débits calculés. A titre d'exemp<strong>le</strong>, on peut 'citer <strong>le</strong> bassin versant du<br />
2<br />
Dessoubre (568 km inclus dans <strong>le</strong> bassin du Doubs) pour <strong>le</strong>quel ont été<br />
mesurés 14,2 m3/s (1964-1968) et calculée 16,16 m 3 /s.
SRAE - FRANCHE-COMTE<br />
Carte da k dihbutin rribbk da I'iroulmrnt trtilnomlnterannud<br />
(1964-19681
USE OF REPRESENTATIVE AND EXPERIMENTAL CATCHMENTS FOR THE<br />
LSCESSMENT OF HYDROLOGICAL DATA OF AFRICAN TROPICAL BASINS:?<br />
ABSTRACT<br />
J. Ba<strong>le</strong>k<br />
Institute of Hydrodynamics,-Academy of Science,<br />
Prague, Czechoslovakia<br />
Extensive hydrological records in tropical Africa are availab<strong>le</strong><br />
mostly for large basins. Since <strong>the</strong> beginning of IMD observation on<br />
small representative and experimental areas has been started and<br />
valuab<strong>le</strong> short records are already availab<strong>le</strong>. A great number of <strong>the</strong><br />
tropical basins still remain unobserved. Although <strong>the</strong> demand for<br />
hydrological data needed for engineering and agricultural develop-<br />
ment is increasing, in many cases it can be found difficult to<br />
provide reliab<strong>le</strong> estimates of <strong>the</strong> hydrological characteristics.<br />
Considering <strong>the</strong> high fluctuation of rainfall patterns and high<br />
non-uniformity of <strong>the</strong> topographical and vegetational cover on small<br />
tropical catchments it cannot be recommended to establish <strong>the</strong> cal-<br />
culation of <strong>the</strong> data for <strong>the</strong> ungauged areas only on <strong>the</strong> records<br />
from representative/experimental catchments. All <strong>the</strong> data from <strong>the</strong><br />
catchments should be compared and analysed jointly with <strong>the</strong> records<br />
of standard network in an attempt to obtain regional characteristics<br />
typical for certain topographical vegetational and rainfall patterns.<br />
Examp<strong>le</strong>s of <strong>the</strong> calculation of <strong>the</strong> data in <strong>the</strong> Central Africa region<br />
are presented in <strong>the</strong> paper.<br />
RESUMEN<br />
Utilización de las cuencas representativas y experimenta<strong>le</strong>s pa-<br />
ra la evaluación de los datos hidrológicos en las cuencas inobserva<br />
das del Africa tropical.<br />
Los datos más fidedignos y más antiguos existen para las cuen--<br />
cas grandes del Africa tropical. Durante el DHI empezaron las obsef<br />
vaciones de las cuencas experimenta<strong>le</strong>s. Hasta el presente muchas -<br />
cuencas tropica<strong>le</strong>s son inobservadas y los cálculos de las caracte--<br />
rísticas hidrológicas para diversos proyectos técnicos y agrícolas<br />
son difíci<strong>le</strong>s. En las pequeñas cuencas tropica<strong>le</strong>s existen signifi--<br />
cantes variaciones en la distribución de las lluvias, topografia y<br />
vegetación y no es posib<strong>le</strong> calcular las características hidrólogi--<br />
cas solamente por la aplicación de los datos de la cuenca represen-<br />
tativa/experimental más próxima. Hay que utilizar todos los datos -<br />
de las cuencas representativas y experimenta<strong>le</strong>s y de la normal red<br />
regional que existen en la región, para determinar las caratteristi<br />
cas hidrológicas de la misma región, representativas para predomi--<br />
nantes tipos de la topografía, vegetación y distribución de las llu<br />
vias. Algunos ejemplos sobre la determinación de los datos para lac<br />
cuencas del Africa Central se presentan en el artículo.<br />
+: The research was sponsored by <strong>the</strong> National Council for Scientific<br />
Research of Zambia. Some unpublished data were obtained by <strong>the</strong><br />
courtesy of W.M.O.
28<br />
INTRODUCTION<br />
During <strong>the</strong> International Hydrological Decade observations<br />
of several representative and experimental catchments were<br />
started in various parts of <strong>the</strong> African tropics, Data<br />
obtained from <strong>the</strong>se catchments toge<strong>the</strong>r with <strong>the</strong> data from<br />
<strong>the</strong> catchments established as parts of various special<br />
proj ects represent very valuab<strong>le</strong> material for engineering<br />
and agricultural projects in Africa. As a main prob<strong>le</strong>m can<br />
be considered how to make best use of <strong>the</strong> data when <strong>the</strong>y<br />
are applied outside <strong>the</strong> catchment boundaries. As can be<br />
seen from <strong>the</strong> compilation of UNESCO (l), long records for<br />
<strong>the</strong> African tropics are availab<strong>le</strong> in most cases for very<br />
large basins. Obviously such data is of very limited use<br />
because <strong>the</strong> number of big hydrotechnical schemes is ra<strong>the</strong>r<br />
small. More frequently <strong>the</strong> data are required for small<br />
basins as a basis for numerous rural development projects.<br />
For <strong>the</strong> interpolation between <strong>the</strong> data from very large<br />
basins and very small catchments <strong>the</strong>re is no standard<br />
method availab<strong>le</strong>. As listed by Toebes and Ouryvaev (2) <strong>the</strong><br />
main purpose of <strong>the</strong> representative catchments is fundamental<br />
research, studies of natural changes, hydrological prediction,<br />
extension of records and in <strong>the</strong> case of experimental<br />
catchments additional effects of cultural changes. Extension<br />
of <strong>the</strong> records is one of <strong>the</strong> most important tasks in <strong>the</strong><br />
tropics because increasing <strong>the</strong> network density can be very<br />
difficult, owing to such circumstances as <strong>the</strong> river<br />
accessability, staff prob<strong>le</strong>ms, finances, etc. Thus <strong>the</strong> idea<br />
of concentrating <strong>the</strong> effort into small areas well instrumented<br />
and observed according to <strong>the</strong> requested standards, appears<br />
to be very useful, particularly regarding satisfactory results<br />
as obtained in temperate regions. As an examp<strong>le</strong> can be given<br />
Volynka catchment located near to <strong>the</strong> Czechoslovakian,<br />
Austrian and West German borders,
River<br />
Volynka<br />
Sputka<br />
Peklovka<br />
I I I<br />
Drainage<br />
area<br />
km2.<br />
Rainfall Runoff<br />
mm. mm.<br />
I I I<br />
1<br />
385 709 246<br />
105 7 54 304<br />
80 625 143<br />
I I I<br />
331<br />
463<br />
443<br />
25.9<br />
8.8<br />
16. 3<br />
29<br />
I<br />
--q--E-<br />
Evapotrans<br />
Yield<br />
pirat ion<br />
i/ s/ km2.<br />
4.53<br />
mm. - !<br />
The catchments is situated in <strong>the</strong> Sumava mountains. In <strong>the</strong><br />
mountains are also <strong>the</strong> headwaters of three rivers (Fig. 1).<br />
Supposing that no direct observation would be availab<strong>le</strong>, an<br />
estimate can be done according to <strong>the</strong> relationship obtained<br />
from <strong>the</strong> representative catchment (Fig. 2):<br />
River<br />
Drainage<br />
km’.<br />
Otava<br />
Blanice<br />
T.Vltava 347 957 550<br />
However, all three rivers have been observed for a long<br />
period, and actual data calculated:<br />
Drainage<br />
Blanice<br />
T. Vltava 34 7 957 514<br />
1<br />
q-T<br />
- I<br />
Evapotrans<br />
Yield<br />
p irat ion<br />
1 / s / km2.<br />
mm .<br />
17.4<br />
Ev apo tr ans<br />
mm .<br />
Obviously, in a region where very litt<strong>le</strong> is known on <strong>the</strong><br />
hydrological regime of <strong>the</strong> rivers, results as obtained<br />
indirectly can be considered as satisfactory.
30<br />
Because factors such as snow melting, çoil freezing and<br />
thawing etc. complicate hydrological regimes of temperate<br />
catchments, one would expect that in tropical catchments<br />
even better results can be achieved. However, owing to a<br />
high variability of evapotranspiration, such an expectation<br />
is far from being correct. There are several factors<br />
contributing to <strong>the</strong> increased evapotranspiration variability:<br />
Precipitation<br />
Above ra<strong>the</strong>r monotonous topography of tropical Africa slow<br />
changes in annual rainfall totals can be expected. The<br />
raingauge network is not dense enough to provide a more<br />
comp<strong>le</strong>te picture, however, availab<strong>le</strong> records support previous<br />
presumption. From <strong>the</strong> records of a very dense network<br />
established in small areas, follows that <strong>the</strong> distribution<br />
of hourly, daily, monthly and even annual rainfall totals<br />
is highly non-uniform. In Fig. 3 <strong>the</strong> distribution of <strong>the</strong><br />
annual rainfall in four Zambian catchments, each of <strong>the</strong>m<br />
<strong>le</strong>ss than 2 km , has been plotted. The rainfall distribution<br />
was measured by <strong>the</strong> network of about 60 gauges. The<br />
variability has been observed for 5 years (3) and it is has<br />
been proved that <strong>the</strong>re is no relationship between <strong>the</strong><br />
topographical and rainfall pattern. Jackson (4), studying<br />
<strong>the</strong> interception of Tanzanian forest, proved similar high<br />
variability within a small area. This of course makes it<br />
difficult to apply some <strong>the</strong>ories, such as, for examp<strong>le</strong> unit<br />
hydrograph, because <strong>the</strong> centres of <strong>the</strong> storms are ra<strong>the</strong>r<br />
randomly distributed above <strong>the</strong> catchments. Thus, identical<br />
runoff volumes produce different types of hydrographs and<br />
identical rainfall totals produce a great variety of runoff<br />
volumes.
Swamps<br />
Origin, size and location of swamps in tropical basins are<br />
o<strong>the</strong>r factors highly influencing tropical hydrological<br />
regimes. The total area of African swamps is about 340.000<br />
km2. They have not yet been classified, according to origin<br />
vegetation, geomorphology, and thus <strong>the</strong> know<strong>le</strong>dge of <strong>the</strong>ir<br />
hydrological ro<strong>le</strong> is also very limited. Several catchments<br />
containing swamps have been under intensive observation in<br />
<strong>the</strong> tropics. In Uganda <strong>the</strong> evapotranspiration from swampy<br />
vegetation consisting mainly of <strong>the</strong> papyrus, has been<br />
studied, because it highly influences <strong>the</strong> water balance of<br />
<strong>the</strong> Upper Ni<strong>le</strong> basin. In Zambia heaäwater swamps, so cal<strong>le</strong>d<br />
dambos, forming a significant part of Central African water<br />
resources, are under intensive study. By a comparison of<br />
<strong>the</strong> results already availab<strong>le</strong> it can be concluded that <strong>the</strong><br />
influence of <strong>the</strong> swamps varies according to <strong>the</strong>ir storage<br />
capacity, location within <strong>the</strong> basin and vegetation. Seasonal<br />
distribution of rainfall above <strong>the</strong> swamps plays an important<br />
ro<strong>le</strong> as well. In Fig. 4 rainfall-runoff relationships as<br />
depending on <strong>the</strong> percentage of <strong>the</strong> swamps within <strong>the</strong> Kafue<br />
basin have been plotted. River Kafue has a low gradient-much<br />
below 0,001. From <strong>the</strong> graph it can be seen how <strong>the</strong> increased<br />
size of swamps reduces <strong>the</strong> annual runoff. Such a type of<br />
relationship is valid for <strong>the</strong> swamps with unlimited capacity<br />
and located in midd<strong>le</strong> and lower courses of <strong>the</strong> river. On <strong>the</strong><br />
o<strong>the</strong>r side <strong>the</strong> headwater swamps behave differently (5). Owing<br />
to <strong>the</strong> limited storage capacity and high gradient of <strong>the</strong><br />
swamps <strong>the</strong> carryover from year to year is negligib<strong>le</strong> and in<br />
most cases <strong>the</strong> swamps are emptied before <strong>the</strong> next rainy<br />
season starts. The swamps are sorrounded by dense woodland<br />
where no surface runoff can occur and <strong>the</strong> only surface runoff<br />
produced from <strong>the</strong> catchment is from <strong>the</strong> over-storaged<br />
31
32<br />
groundwater aquifer in <strong>the</strong> swampy areas. As compared with<br />
<strong>the</strong> previous type of swamps, <strong>the</strong> time of increased evapotrans<br />
piration is ra<strong>the</strong>r limited in <strong>the</strong> headwaters. However,<br />
representative/experimental catchments are frequently located<br />
in <strong>the</strong> headwaters and thus any extension of <strong>the</strong> results<br />
toward <strong>the</strong> lower reaches is very difficult. In Fig. 5 three<br />
curves characterizing <strong>the</strong> behaviour of <strong>the</strong> headwater<br />
catchments with swamps of various slopes are plotted. The<br />
lowest line represents flat areas covered by Brachystegia<br />
woodland in <strong>the</strong> vicinity of <strong>the</strong> swamps. These areas do not<br />
re<strong>le</strong>ase any runoff at all. The midd<strong>le</strong> curve characterizes<br />
<strong>the</strong> runoff from <strong>the</strong> catchment slope of 3%, containing 6% of<br />
swamps or catchments slope of 6% containing 5% of swamps.<br />
The upper curve represents an area slope of 10% with 20% of<br />
swamps.<br />
The very first attempts to measure <strong>the</strong> evapotranspiration<br />
from swamps were made by Hurst (6) who concluded that <strong>the</strong><br />
evapotranspiration from <strong>the</strong> Ni<strong>le</strong> papyrus can exceed <strong>the</strong><br />
evaporation from free water surface. By some hydrologists<br />
this has been considered as improbab<strong>le</strong>, however recent<br />
measurements support Hurst's conclusion.<br />
Vegetation<br />
As can be seen from <strong>the</strong> map in Fig. 6, changes of <strong>the</strong><br />
vegetational cover generally follow <strong>the</strong> changes of <strong>the</strong><br />
climate. Thus it might be expected that an intensive<br />
observation of catchments established in each of <strong>the</strong><br />
climatical/vegetational belts can provide a full picture of<br />
<strong>the</strong> ro<strong>le</strong> of <strong>the</strong> tropical vegetation. However, a more detail<br />
ed map of any of <strong>the</strong> regions indicates a great variety of<br />
vegetational types. It may not be difficult to find a<br />
catchment with uniform cover dominant in <strong>the</strong> region; <strong>the</strong><br />
question is whe<strong>the</strong>r such a catchment can supply more<br />
representative data than <strong>the</strong> catchment with non-uniform
cover. For examp<strong>le</strong>, in <strong>the</strong> catchments of tropical mountains<br />
<strong>the</strong> vegetation varies accordingly with <strong>the</strong> temperature and<br />
<strong>the</strong>re a catchment covered by all characteristical mountaineous<br />
types is certainly more representative than a catchment<br />
uniformly covered by one type only.<br />
The influence of <strong>the</strong> African vegetation on ‘che hydrological<br />
cyc<strong>le</strong> has been studied for a long time, In 1949 Wicht (7) drew<br />
up a set of conclusions on <strong>the</strong> ro<strong>le</strong> of vegetation, founding<br />
that <strong>the</strong> forest will use more water than grass, <strong>the</strong> consumption<br />
of water by forest depends essentially on <strong>the</strong> amount of water<br />
availab<strong>le</strong> in <strong>the</strong> soil and that <strong>the</strong> removal of vegetation causes<br />
an increased discharge. In Kenya actual evapotranspiration/<br />
potential evaporation ratio Et/Eo from various plantations was<br />
measured (8) and in <strong>the</strong> experimental areas was found that<br />
ei<strong>the</strong>r bamboo or tall montane forest is an ideal protection<br />
against overland flow, whi<strong>le</strong> <strong>the</strong> replacement of trees by<br />
plantation increased <strong>the</strong> runoff.<br />
The evapotranspiration from <strong>the</strong> grassland and woodland has<br />
been measured in Zambian catchments. It has been found that<br />
<strong>the</strong> trees consume approximately three times more water than<br />
seasonally flooded grassland. The short grass roots have only<br />
a limited chance to consume soil water, whi<strong>le</strong> <strong>the</strong> woodland<br />
trees will tap <strong>the</strong> water from <strong>the</strong> groundwater tab<strong>le</strong> during<br />
<strong>the</strong> dry periods. These results were confirmed by soil moisture<br />
measurements and root density analysis (9). It has been proved<br />
also that <strong>the</strong> Et/€, rate fluctuates year by year and month by<br />
month depending on <strong>the</strong> meteorological situation, distribution,<br />
intensity and amount of rainfall and groundwater storage<br />
availab<strong>le</strong> during <strong>the</strong> dry season (3). The following tab<strong>le</strong><br />
indicates <strong>the</strong> fluctuation of evapotranspiration as obtained<br />
for <strong>the</strong> swamp grasses and Brachystegia woodland in 1969/70:<br />
33
Month<br />
October<br />
November<br />
December<br />
January<br />
February<br />
March<br />
April<br />
May<br />
July<br />
August<br />
S ep t em b er<br />
Y ear<br />
Rainfall<br />
mm<br />
43.18<br />
80.01<br />
414.27<br />
311.92<br />
237.49<br />
29.97<br />
55.12<br />
. O0<br />
1.02<br />
.o0<br />
11.18<br />
1184.1 5<br />
Evapotranspiration<br />
I<br />
Woodland<br />
mm<br />
Et/Eo Grass Et/Eo<br />
65.35 .4 20.07 .1<br />
98.55 .6 43.82 .3<br />
128.54 .9 95.28 .7<br />
185.71 1.2 97.43 .6<br />
197.91 1.5 71.93 .5<br />
237.05 1.4 77.69 .5<br />
152.66 1.1 39.37 .3<br />
111.54 .9 16.81 .i<br />
74.01 .7 6.02 .1<br />
63.77 .5 5.40<br />
N<br />
60.04 .4 5.28<br />
N<br />
1457.00 .8 407.64 .3<br />
The year 1969/70 was chosen as an examp<strong>le</strong> because it followed<br />
after a very wet year and <strong>the</strong> evapotranspiration from <strong>the</strong><br />
woodland exceeded <strong>the</strong> precipitation, owing to <strong>the</strong> groundwater<br />
storage accumulated during <strong>the</strong> wet year. The grass in swamps<br />
evapotranspirated approximately <strong>the</strong> same amount of water as<br />
during previous years. The ratio Et/Eo indicates when <strong>the</strong><br />
actual evapotranspiration exceeded potential evaporation.<br />
The values in <strong>the</strong> tab<strong>le</strong> have been determined as limits for<br />
<strong>the</strong> locations fully covered by <strong>the</strong> woodland or by <strong>the</strong> grass.<br />
Supposing <strong>the</strong> data were applied outside such an intensively<br />
observed area, actual vegetational composition has to be<br />
taken into account, because owing to it actual evapotranspiration<br />
can be found anywhere between <strong>the</strong> two extrema1<br />
values. The results as obtained in Zambia are representative<br />
for <strong>the</strong> vegetation of tropical wet and dry highlands. To<br />
obtain a more comp<strong>le</strong>te picture, similar experiments should
e performed at <strong>le</strong>ast with two high montane vegetational types,<br />
four types of medium altitude forest, two types of swamp forest,<br />
two types of forest savanna mosaic, four types of wooded<br />
savanna, two types of thicket, two types of swamp vegetation<br />
and various types of tropical cultivated areas.<br />
Topography<br />
From flat areas covered by <strong>the</strong> Brachystegia woodland nei<strong>the</strong>in<br />
surface nor sub-surface runoff has been observed. An occurrence<br />
of flow was observed only from <strong>the</strong> parts of <strong>the</strong> catchments<br />
having some pronounced gradient. Mixed vegetation found <strong>the</strong>re<br />
suggest <strong>the</strong> idea that <strong>the</strong> vegetational cover is influenced by<br />
<strong>the</strong> gradient as well. Actual influence of <strong>the</strong> catchment slope<br />
can be analysed by a comparison of rainfall-runoff relationships<br />
developed for neighbouring catchments of different gradients.<br />
In Fig. 7 <strong>the</strong>re is a family of graphs developed for <strong>the</strong><br />
equatorial highland region. Very likely, for dry-wet tropical<br />
highlands <strong>the</strong> runoff values will be higheri for <strong>the</strong> same amount<br />
of rainfall, owing to <strong>the</strong> increased rainfall rates of separate<br />
rainfalls.<br />
Attention should be paid also to <strong>the</strong> size of <strong>the</strong> representative/<br />
experimental areas. According to Toebes and Ouryvaev (2) <strong>the</strong><br />
recommended size lies between 1 and 250 km2 and rarely exceeds<br />
1000 km2. Frequently <strong>the</strong> areas <strong>le</strong>ss than 100 km2, so cal<strong>le</strong>d<br />
small catchments, are recommended for experimental catchments,<br />
this being based on <strong>the</strong> presumption that a certain uniformity<br />
can be guaranteed. As follows from <strong>the</strong> previous discussion,<br />
<strong>the</strong> significant factors in <strong>the</strong> tropical hydrological cyc<strong>le</strong> are<br />
highly variab<strong>le</strong> even in small areas and no catchment is small<br />
enough from <strong>the</strong> point of view of <strong>the</strong> uniformity. On <strong>the</strong> o<strong>the</strong>r<br />
hand <strong>the</strong> extension of data from very small areas is not an easy<br />
task. It can be perhaps concluded that in tropical regions where<br />
only observational network of low density is availab<strong>le</strong>, a<br />
catchment of any size can be considered as representative<br />
35
36<br />
providing that a higher accuracy of basic hydrometeorological<br />
data can be obtained from <strong>the</strong>re than from <strong>the</strong> standard<br />
network Several catchments established within <strong>the</strong> main area<br />
can increase <strong>the</strong> amount of information remarkably. A similar<br />
increase can be achieved by <strong>the</strong> observation of several<br />
neighbouring catchments. Sometimes occurrence of two or more<br />
factors in some extremal forms can produce ra<strong>the</strong>r surprising<br />
results. For instance, at one catchment in Kagera basin near<br />
<strong>the</strong> Tanzanian-Ugandan borders, steep mountains are drainaged<br />
into an extensive swamp. As a result, <strong>the</strong> runoff coefficient<br />
reaches almost 30%, which is surprisingly high value for <strong>the</strong><br />
tropics. Data from such a catchment cannot be applied directly<br />
to <strong>the</strong> neighbouring basins, however, since a more dense network<br />
has been established <strong>the</strong>re and <strong>the</strong> effects resulting from <strong>the</strong><br />
combination of two extremal factors can be measured and analysed,<br />
<strong>the</strong> catchment can serve as a representative area as well.<br />
CONCLU SION S<br />
Mis<strong>le</strong>ading results can be obtained from direct application of<br />
<strong>the</strong> data obtained from <strong>the</strong> experimental catchments in <strong>the</strong><br />
tropics. Therefore, whenever possib<strong>le</strong>, data from experimental<br />
and representative catchments should be compared and combined<br />
with <strong>the</strong> data as obtained from <strong>the</strong> standard network. Parti-<br />
cularly basic data, such as annual rainfall, runoff and yield<br />
should be compared before any fur<strong>the</strong>r analysis is carried out.<br />
Any difference between <strong>the</strong> data as obtained from <strong>the</strong> catchments<br />
and from <strong>the</strong> standard network should be fully explained and <strong>the</strong><br />
data developed for any cross section within a basin should fit<br />
with <strong>the</strong> data €or <strong>the</strong> headwater catchments and for <strong>the</strong> lowest<br />
observed point as well. Only equal periods of observation<br />
should be used for <strong>the</strong> comparison, although in some cases it<br />
mean neg<strong>le</strong>cting <strong>the</strong> long term records. The long term records<br />
however, are to be used later on, toge<strong>the</strong>r with long term<br />
precipitation records, for <strong>the</strong> extension of data within a<br />
reg ion.
In tab<strong>le</strong> 1 an examp<strong>le</strong> of <strong>the</strong> basic data for <strong>the</strong> Kafue river<br />
basin is given based on <strong>the</strong> observation of <strong>the</strong> experimental<br />
catchments and standard network as well. Conclusions of <strong>the</strong><br />
research on <strong>the</strong> swamp behaviour served as an additional<br />
source of information. Map indicating rivers and swamps is<br />
in Fig. 7 (Headwater swamps are too small and cannot be<br />
traced in <strong>the</strong> map, however it has been estimated that <strong>the</strong>y<br />
cover at <strong>le</strong>ast 10% of <strong>the</strong> basin headwaters). Once <strong>the</strong> data<br />
€or hydrologically significant cross sections such as<br />
confluences, swamp inflows and outflows, observed cross<br />
sections etc. have been estimated, <strong>the</strong> calculation of <strong>the</strong><br />
data €or any point within <strong>the</strong> main basin is easy and more<br />
reasonab<strong>le</strong>.<br />
According to <strong>the</strong> UNESCO survey and o<strong>the</strong>r sources, more than<br />
one hundred representative/experimental catchments have<br />
been established in various parts of <strong>the</strong> African tropics.<br />
More information can be obtained from <strong>the</strong>m providing that<br />
<strong>the</strong> materials is col<strong>le</strong>cted and analyzed jointly.<br />
More attention, should be paid to <strong>the</strong> hydrology of tropical<br />
swamps, because <strong>the</strong>y play an important ro<strong>le</strong> in tropical<br />
hydrology.<br />
The results availab<strong>le</strong> from severa$ experimental catchments<br />
indicate that various hydrological methods currently used<br />
in temperate regisns need to be reviewed, regarding special<br />
conditions existing in tropical catchments.<br />
37
i.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
38<br />
--<br />
REFERENCES<br />
--_____-__________-_--_----__- 1971. Discharge of se<strong>le</strong>cted<br />
rivers of <strong>the</strong> world. UNESCO, Paris.<br />
Toebes, C., Ouryvaev, V., 1970. Representative and<br />
experimental basins. UNESCO, Paris.<br />
Ba<strong>le</strong>k, J., Perry, J., 1972. Luano catchments, first phace-<br />
final report. National Council for Scientific Research,<br />
TR 28 Zambia.<br />
Ba<strong>le</strong>k, J., Perry, J., 1973. Hydrology of seasonally inundated<br />
African headwater swamps, Journal of Hydrology. In print.<br />
Jackson, I.J., 1971. Prob<strong>le</strong>ms of throughfall and interception<br />
assessment under tropical forest. Journal of Hydrology 12.<br />
Hurst, H.E. Le Nil. Paris.<br />
Wicht, C.L., 1949. Forestry and water supplies in South<br />
Africa. Dept. Agric. S. Afr. Bul. 33, p. 58.<br />
Pereira, H.C. 1962. Hydrological effects of changes in land<br />
use in some East African catchment areas. East Afr. Agric.<br />
Forestry Journal 27.<br />
Maxwell, D., 1972. Root range investigations. National<br />
Council for Scientific Research, TR 26 Zambia.
Y<br />
ri<br />
Lc<br />
O<br />
39
djONnä 1WlNNVjO WU<br />
O m<br />
ul<br />
O<br />
I- O<br />
O
RNNLJQL RFIINFQLL 1966- 67, LUQNO - CQTCHMENTS .-<br />
v Gauging Statlms<br />
Levei at lûûûmm<br />
Fig.3
O<br />
P 8 8 8 a
Fig ,6
ABSTRACT<br />
REGIONAL VALUATION OF IIYUROLOGlCAL INFORMATION<br />
Y. Cormary - J.M. Masson<br />
This information, inadequate by its very nature, is basically obtained as<br />
temporal series of climatic, hydrometric and water <strong>le</strong>vel data, of different<br />
durations.<br />
A diagnosis about data coherency and <strong>the</strong> subsequent possibility of regional<br />
interpolation, might result of different methods.<br />
1.<br />
2.<br />
3.<br />
RESUME<br />
The mere report on a map of <strong>the</strong> parameters (of probability for instance)<br />
related to classical hydrologic variab<strong>le</strong>s. For <strong>the</strong> 12 months of monthly<br />
variab<strong>le</strong>s this yields however too many values. A solution might be <strong>the</strong><br />
fitting to <strong>the</strong> 12 values of a parameter, of a FOURIER of which only <strong>the</strong><br />
2 to 4 first coefficients may be conveniently retained for cartography.<br />
Report on a map of <strong>the</strong> basic stochastic processes parameters. Probability<br />
distribution of many hydrologic var,iab<strong>le</strong>s are derived from those proces-<br />
sed and thus depend on <strong>the</strong> interpolated parameter values. Fur<strong>the</strong>rmore<br />
<strong>the</strong>se processes make a better use of <strong>the</strong> existing information and a gaghg<br />
point. Analysis of coincidences between different recorded series may also<br />
improve <strong>the</strong> estimation parameters on <strong>the</strong> short ones.<br />
Principal componentes analysis (mainly on climatic data) which through an<br />
interpolation of covariance matrix, outline some regional tendancies and<br />
a quantitative interpolation at any point.<br />
4. Analysis of variance of regressions between flows and rainfalls for instance<br />
on n different watersheds, in order to obtain <strong>the</strong> effect or morphological,<br />
geological, vegetation, soil factors and <strong>the</strong>refore transform<br />
a basically qualitative information into a quantitative one.<br />
Each of <strong>the</strong>se methods has been actually employed on <strong>the</strong> rivei- Allier (France).<br />
L'information, insuffisante par nature, est représentée essentiel<strong>le</strong>ment par<br />
des series chronoligiques de données climatiques, hydrométriques et piézométriques<br />
plus ou moins longues et nombreuses. '<br />
Un diagnostic d'interpolation sur la cohésion des mesures et <strong>le</strong>s possibilités<br />
d'interpolation géographique peut s'effectuer de plusieurs manières différentes:<br />
1. Cartographie des paramètres des lois de distribution de variab<strong>le</strong>s hydrologiques<br />
classiques. Pour des variab<strong>le</strong>s mensuel<strong>le</strong>s, ceci aboutit à prendre<br />
en compte un nombre trop important de paramètres. Une solution consiste à<br />
ajuster des séries de FOURIER pour représenter l'évolution des paramètres<br />
au cours de l'année. I1 suffit alors de cartbgraphier 2 à 4 coefficients,<br />
<strong>le</strong>s autres étant des constantes caractéristiques de la région.<br />
2. Cartographie des paramètres des processus stochastiques de base, qui permettent<br />
entre autre de retrouver <strong>le</strong>s lois de distribution des variab<strong>le</strong>s<br />
hydrologiques, mais décrivent mieux qu'el<strong>le</strong>s la totalité du phénomène<br />
(pluies ou crues) et utilisent mieux la totalité de l'information disponib<strong>le</strong>.<br />
3. L'analyse de la concomitance des phénomènes entre séries de meme nature ou<br />
non permet de mieux estimer <strong>le</strong>s paramètres de ces processus (mode<strong>le</strong> de renouvel<strong>le</strong>ment<br />
doub<strong>le</strong>).<br />
3. Analyse des composantes principa<strong>le</strong>s (sur <strong>le</strong>s données climatiques essentiel<strong>le</strong>ment)<br />
qui permet de dégager <strong>le</strong>s quelques tendances régiona<strong>le</strong>s predominantes<br />
et permet une interpolation en tout point.<br />
4. Analyse de variance des coefficients de n régressions ajustées entre variab<strong>le</strong>s<br />
hydrologiques (pluies et débits par exemp<strong>le</strong>) sur n bassins versant dif<br />
férents pour mettre en évidence l'influence des caractéristiques physiqueset<br />
morphologiques (géologie, végétation, pente, ... ) et prendre en compte de<br />
manière quantitative une information de type naturaliste essentiel<strong>le</strong>ment<br />
qualitative.<br />
Chaque méthode a été utilisée au cours d'une étude du bassin versant de la rivi$re<br />
Allier.<br />
-<br />
Yves CORMARY - Ingénieur Agronome - Laboratoire National d'Hydraulique - Professeur<br />
Associé à l'Université des Sciences et Techniques du Languedoc - MONTPELLIER (Fran-<br />
ce).<br />
Jean-Marie MASSON - Inggnieur Agrico<strong>le</strong> - Maître Assistant à l'Université des Scien-<br />
ces et Techniques du Languedoc - MONTPELLIER (France).
48<br />
I - INTRODUCTION<br />
En hydrologie, l'information est constitu6.e esscntiel<strong>le</strong>ment<br />
par des mesures de paramètres hydrométriques, climatiques et piézo-<br />
métriques. Ces mesures sont des variab<strong>le</strong>s liges à l'espace (l'endroit<br />
uù ia mesure a été effectuée) et au temps (l'époque où eiie a et6<br />
effectuée).<br />
Sur un emplacement donné, la mesure d'un paramètre est<br />
faite de manière continue ou à interval<strong>le</strong>s de tmps rh~.,iillc.r. 1.a<br />
suite des mesures au même emplacement constitue line séric rhronologique.<br />
Régiona<strong>le</strong>ment, au cours des années, <strong>le</strong>s emplacemerts des<br />
points de mesure changent et on se trouve fina<strong>le</strong>ment en présence<br />
de séries chronologiques plus ou moins longues et plus WI moins<br />
nombreuses.<br />
Cependant, quand un problhe relatif à l'eau se ;rose -et<br />
<strong>le</strong> nombre des problèmes qui se posent ne cesse d'?iipeii'-er .avec<br />
<strong>le</strong> développement économique de nos régions-. I1 n'mistc p:.iti,uc-<br />
ment jamais à 1' emplacement souhaité <strong>le</strong>s infnrmationn nécessaires<br />
pour résoudre <strong>le</strong> problème, ou tout au moins ces infarnations sont<br />
insuffisantes. Un des moyens de pallier ce manque de donz6es en<br />
quantité suffisante et au bon endroit, consiste à rnobi1is.c.r toute<br />
l'information disponib<strong>le</strong> sur la région environnante.<br />
En France, l'importance de cette valorisation dc l'infor-<br />
mation régiona<strong>le</strong> n'avait pas échappé au conlté "Actior. Concertée<br />
Eau" qui proposait comme sujet d'htude en lu65 d'abord- <strong>le</strong>s<br />
prob<strong>le</strong>mes de ressources de maniere scientifique et ri:.iwalc et<br />
de procéder à une synthèse sur un bassin d'assez vaste fiinen!,ion.<br />
Le souhait du comite se concrétisa par un cf'iltïat d'Etude<br />
passé entre la Dflégation GCnéra<strong>le</strong> .? 12 Xcrkerchc Cr i.'u;i:ifique et<br />
Technique (D.G.R. C.T. ) et <strong>le</strong> tahoratoire F:.:.7ticriai d"1ydrsuliqrie<br />
(E.D.F.). Le bassin choisi fut celui de 1'ALLT.IR (14 O00 km2j et<br />
parmi <strong>le</strong>s sujets étudiés, on re<strong>le</strong>vait : "ia mise sur pied d'iriie<br />
étude des lois de distribution à l'6chel<strong>le</strong> régiona<strong>le</strong>, permettant<br />
de créer de nouveaux nodo<strong>le</strong>s dc repr6sentatinn, d'extrnpo<strong>le</strong>r<br />
l'inforination et de déterminer une hiérarchie de l'iii:>:-&t dc<br />
chaque station".
L'étude effectuée par <strong>le</strong> Groupe Hydrologie du L.N.H. et<br />
la Faculté des Sciences de Montpellier a fait l'objet d'une publication<br />
de syn<strong>the</strong>se sous forme d'un atlas cartonné et illustré intitulé :<br />
"Méthodes d'.études régiona<strong>le</strong>s des reesources en eau".<br />
Nous développons ici quelques pages de cette publication,<br />
pages consacrées des méthodes d'analyses régiona<strong>le</strong>3 de l'information.<br />
-<br />
11.- INTERPOLATION REGIOXALE DE L'INFORXATION ET ANALYSE DE LA COHEREK'CE<br />
SPAT1 ALE.<br />
i 1 - 1 - pf~se_elcom~qe_be_lljnooimsq~oo_Ear c arto9laE-~-_régiona<strong>le</strong> -<br />
A/- Au moyen de processus s tzchastiques simp<strong>le</strong>s.<br />
Une série chronologique sur une station, par exemp<strong>le</strong><br />
la série des observatiors journalières des précipita,tions ou<br />
des débits, présente une structure bien particulière qu'on<br />
peut représenter au moyen d'un processus stochastique.<br />
Le type de processus <strong>le</strong> mieux adapté à beaucoup de<br />
phénomènes est un processus de r~nouvel<strong>le</strong>ment.<br />
4<br />
hrru t eu r<br />
des pluies y1<br />
T1<br />
Tt<br />
y3<br />
49<br />
temps<br />
Pour tenir c'ompte des variations saisonnières, on dé-<br />
coupe l'année en périodes OU <strong>le</strong> processus est à peu près sta-<br />
tionnaire : <strong>le</strong>s Ti successifs, durées entre <strong>le</strong>s événements,<br />
doivent &tre indépendants et suivre la même loi de probabi-<br />
lité de densité f(T). Les Yi successifs doivent être éga<strong>le</strong>-<br />
ment indépendants et suivre la même loi de probabilité de<br />
densité g(y).
50<br />
I1 s'agit donc de trouver ces lois de probabilité des variab!es<br />
fondamenta<strong>le</strong>s T et Y et d'estimer <strong>le</strong>urs paramètres, ce qui nous ramène<br />
aux méthodes classiques de la statistique avec cependant ces différenres<br />
- On mobilise toute l'information. Ainsi pour la pluie, on<br />
tient compte aussi bien des jours secs que des jours pluvieux<br />
- Ces variab<strong>le</strong>s fondamenta<strong>le</strong>s sont observées en grand riombre,<br />
ce qui rend <strong>le</strong>ur analyse statistique beaucoup plus valab<strong>le</strong> que l'ana-<br />
lyse de variab<strong>le</strong>s déduit:., fonction souvent compliquée des variab<strong>le</strong>s<br />
fondamenta<strong>le</strong>s (<strong>le</strong>s pluies ou <strong>le</strong>s débits maximums par exemp<strong>le</strong>).<br />
. Un modè<strong>le</strong> a été ainsi construit pour représenter la sixcessiun<br />
des pluies journalières à une station.<br />
I1 suppose que la hauteur d'une pluie journalière est la SOIIIIIIP<br />
d'averses fictives se produisant à des instants aléatoires et rpportant<br />
des quantités de pluie aléatoires. Si <strong>le</strong> nombre des averses par jour<br />
pluvieux suit une loi de POISSON et que <strong>le</strong>s hauteurs des averses<br />
fictives suivent une loi exponentiel<strong>le</strong>, <strong>le</strong>s hauteurs de pluie journa-<br />
lière doivent suivre une loi des fuites (loi I G ama zéro), ce qui<br />
est bien vérifié siir <strong>le</strong>s stations de l'ALLIER. On peut alors estimer<br />
2 paramètres du processus.<br />
p = hauteur moyenne des averses élémentaires fictives<br />
= nombre moyen d'averses élénientaires fictives par jour de<br />
pluie.<br />
c<br />
Ceci, à partir de la moyenne P 24 et de la variance b224<br />
des pluies de 24 heures non nul<strong>le</strong>s.<br />
p=--- *24<br />
L<br />
P24<br />
P=<br />
-<br />
2 P 242<br />
/I 224<br />
Les deux autres paramètres du modè<strong>le</strong> sont :<br />
TI<br />
T2<br />
durée moyenne des episodes secs<br />
durée moyenne des épisodes pluvieux.<br />
qui permettent de connaître en terme de probabilité ilétat sec ou<br />
pluvieux d'un jour connaissant l'état du jour précédent ; on a en effet<br />
G ér if i é <strong>le</strong> caractère markovien de la matrice de transition des états.<br />
On a :
La cartographie des différents paramètres sur l'ensemb<strong>le</strong><br />
des stations du bassin de l'Allier s'crganise bien et permet l'in-<br />
terpolation des parametres du modè<strong>le</strong> pour n'importe quel point du<br />
bassin, ainsi que <strong>le</strong> montrent <strong>le</strong>s graphiques ci-joints concernant<br />
<strong>le</strong> mois d'octobre.<br />
D'une manière généraìe.on remarque :<br />
- l'influence atlantique (Ti court, T2 long,/U fort, faib<strong>le</strong>).<br />
- l'influence méditerranéenne IT, long, T2 court,,U faib<strong>le</strong>,/=, fort).<br />
Le processus peut s'appliquer aussi aux crues et aux t'empé-<br />
ratures moyennant la fixation d'un seuil.<br />
E/- .Au moyen de processus stochastiques associés.<br />
Exposé simplifié du modè<strong>le</strong> associant crues et pluies (;enou-<br />
vel<strong>le</strong>ment doub<strong>le</strong>).<br />
Chaque processus est défini B partir de la. distribution des durées<br />
ectre événements H1 et de cel<strong>le</strong>s des grandeurs des événements<br />
(z, 1 -<br />
Par convention, il y a un événement "crue" ou "pluie" quand<br />
la variab<strong>le</strong> dépasse un seuil choisi pour que, en particulier, <strong>le</strong>s<br />
caractéristiques de grandeur (volume total, va<strong>le</strong>ur de pointe) enregistrées<br />
pendant que la variab<strong>le</strong> est au-dessus du seuil, et <strong>le</strong>s caractéristiques<br />
de distributibn dans <strong>le</strong> temps (durée séparant 2 év6nements<br />
homologues) ne soient pas autocorrélées niais que <strong>le</strong>s 2 6x76nements<br />
pluies-débits soient aussi fréquemment associés que possib<strong>le</strong>.<br />
- pour certains événements (Xi et X2) il y a concomitance entre 1<br />
et II (Pi cas). Sur <strong>le</strong>s grandeurs (zl et Z2) est évaluée la corré<strong>le</strong>tion<br />
pluies crues<br />
I<br />
x2<br />
S<br />
4<br />
x2 s<br />
4<br />
51
52<br />
- pour d'autres événements (X 3, P3 cas et X4, P4 cas) ii n'y a<br />
pas concomitance. Ceci s'explique entre autre chose par <strong>le</strong> fait<br />
que nous sommes obligés de définir un seuil de dépassement cons-<br />
tant sur chacune des deux variab<strong>le</strong>s pluies et débits quel<strong>le</strong> que<br />
soit la saison ou la saturation du sol.<br />
- Marche h suivre :<br />
Sur une courte période T1 on évalue à la f ois <strong>le</strong>s paramètres<br />
du rencuvel<strong>le</strong>ment simp<strong>le</strong> simultané sur chacun des deux processus, on<br />
estime égz<strong>le</strong>ment la corrélation existant entre <strong>le</strong>s grsideurs hydrométriques<br />
-.t <strong>le</strong>s grandeurs pluviométriques (Zl, Z2) soit & . Sur la<br />
série longue (Ti + T,) on estime <strong>le</strong>s paramètres du proce;ci;s pluies<br />
L<br />
seul.<br />
Soit ûT1 un paramètre donné du processus crues, moyenne du<br />
nombre annuel de crucs par pérjrJde, des débits maximums, ou des voiumes<br />
de crue,évalué sur une série courte de T1 et accompagné de<br />
sa variance d'érhantillocnage, soit Var (8 ). La prise en consid&-<br />
T1<br />
ration du procescus pluies (sur T, + Tz) nous permet alors, d'évaluer<br />
de nouvel<strong>le</strong>s estimations des paramètres du processus crues ;<br />
estimations dites améliorées, qui sont accompagndes de nouvel<strong>le</strong>s Trariances<br />
d'échantillonnage plus réduites appliquhcs da.is la théorie<br />
du renouvel<strong>le</strong>ment.<br />
D'autre part, ccs améliorations seront d'autant plus effica-<br />
ceti que <strong>le</strong>s proportions :<br />
a =<br />
seront plus fortes.<br />
+ +<br />
P1<br />
et b =<br />
P1<br />
p3 pl p4 pl<br />
Le résultat final rend compte de la superpositioii du proces-<br />
sus X, (Z,) et X (Z ) dans <strong>le</strong>s lois déduites sur <strong>le</strong>s crues<br />
3 3<br />
C/- Autres .néthodes d'analyse de la cohérence temporel<strong>le</strong> des séries chro-<br />
nologiques.<br />
Dans certains cas, il suffit de mettre seu<strong>le</strong>ment en évidence<br />
la simultanéité des événements : l'association statistique des épi-<br />
sodes pluvieux h deux stations ou cel<strong>le</strong> des crues et épisodes plu-<br />
vieux sur un bassin. Dans d'autres, <strong>le</strong>s deux séries de données peu-<br />
vent être considérées comme respectivement <strong>le</strong>s entrées et <strong>le</strong>s SOT-<br />
ties d'un système de transformation dont on cherche à identifier à<br />
la fois la structure et <strong>le</strong>s paramètres (transformation des pluies<br />
en débits sur un bassin restreint, des débits amont en débits aval<br />
sur.un bief, de la pluie en variations d'un écou<strong>le</strong>ment issu de nap-
pes). Suivant la nature des hypothèses (ou des connaissances) <strong>le</strong><br />
linéarité, l'invariance dans <strong>le</strong> temps ou suivant <strong>le</strong> niveau des<br />
entrées du système, la solution est faci<strong>le</strong>, diffici<strong>le</strong> ou impossi-<br />
b<strong>le</strong>.<br />
Si <strong>le</strong> système de transformation n'a pas ou ne prétend pas<br />
avoir de signification physique il suffit d? détzrminer la fonc-<br />
tion (dite boîte noire) de transformation la plus efficace. Les<br />
transformations de Laplace, de Fourier, etc. répondent à ce pro-<br />
blème de même que <strong>le</strong>s études dc type 'diener sur <strong>le</strong>s fonctions<br />
aléatoires (en utilisant <strong>le</strong>s estimations des fonctions de corré-<br />
lation et d'nutocorrélation sur plusieurs coup<strong>le</strong>s, entráes-softies.<br />
L'analyse spectra<strong>le</strong> a aussi conme intérêt d'expliciter mieux que<br />
<strong>le</strong>s procédés classiques la structure des corrélations et l'effica-<br />
cité d'un échantillonnage de mesures un pas de temps déterminé.<br />
Le calcul automatique permet essentieliement <strong>le</strong>s raìciils ma-<br />
triciels, <strong>le</strong> tirage au hasard (simulation) et la répétition infinie<br />
des tatonnements fastidieux. Les applications en sont :a généralisa-<br />
tion des calciils d'amélioration des stations cuuztes en fonction de<br />
plusieurs stations longues, i'utilisation des techniques de l'analy-<br />
se factoriel<strong>le</strong> pour étudier <strong>le</strong>s liaisons entre variab<strong>le</strong>s ou entre<br />
groupe de variab<strong>le</strong>s. Ce qui met an évidence <strong>le</strong>s redondances entre<br />
variab<strong>le</strong>s ou entre mesures et permet de retenir <strong>le</strong>s plus significa-<br />
tives (analyse discriminante ou canonique.<br />
Ces méthodes élémentaires en dehors des services apprécia-<br />
b<strong>le</strong>s qu'el<strong>le</strong>s rendent par el<strong>le</strong>s-mêmes font parties intégrantes de<br />
méthodes plus élaborées que l'ordinateur permet de maîtriser et sur-<br />
tout d'appliquer à un ensemb<strong>le</strong> important de données hydrornétéorolo-<br />
giques (plusieurs postes, plusieurs variab<strong>le</strong>s).<br />
11.2- F.lode<strong>le</strong> rigional statistique -___-_- _--_ des pluies _-___-_-_____-_-<br />
mensuel<strong>le</strong>s.<br />
I1 s'agit d'expliciter la cohérence spatia<strong>le</strong> des précipita-<br />
tions mensuel<strong>le</strong>s, afin, par exemp<strong>le</strong> d'optimiser <strong>le</strong> réseau de mesures.<br />
Sur l'Allier 30 stat'ions avaient fonctionné pendant 40 ans simultané-<br />
ment. Une tel<strong>le</strong> information peut faci<strong>le</strong>ment se condenser sous la forme<br />
d'un tab<strong>le</strong>au carré 30 x 30 dont chaque élément m i j représente soit<br />
la variance, soit la covariance, entre la station i et la station j.<br />
Une méthode dite "des composantes principa<strong>le</strong>s'' permet par<br />
un opérateur linéaire matriciel A d'élhments a i de passer des 30 va-<br />
riab<strong>le</strong>s aléatoires initia<strong>le</strong>s X (pluviométrie des 40 années successives)<br />
à 30 variab<strong>le</strong>s aléatoires Y indépendantes, dont la matrice de corréla-<br />
tion est ,cette fois-ci diagona<strong>le</strong>. C'est-à-dire ne comportant que des zé-<br />
ros pour <strong>le</strong>s covariances. De plus, cette matrice rassemb<strong>le</strong> dans <strong>le</strong>s tous<br />
premiers texnies dc la diagoiinlc toutr la variation coiiteniie dans 1 'in-<br />
formation ini tiaie. En revcnant dans 1 'espace initial chaque variab<strong>le</strong><br />
53
54<br />
aléatoire X de départ peut s'écrire sous la forme d'une combinaison<br />
linéaire des 3 ou 4 composantes <strong>le</strong>s plus importantes, dites principa-<br />
<strong>le</strong>s.<br />
Les coefficients de ces combinaisons linéaires indiquent<br />
la part prise par chaque station L'la constitution de chacun de ces 3<br />
ou 4 facteurs essentiels et indépendants. D'autre part chaque station<br />
est pour chaque mois caractérisée par un petit nombre de paramètres<br />
que l'on peut cartographier de manière cohérente.<br />
x =<br />
de réa-<br />
lisa-<br />
tioris<br />
Notons m..<br />
Xl<br />
m.<br />
= variance (Xi)<br />
= covariance (X.,X.)<br />
N = 30<br />
= 40<br />
ij 1 J<br />
X 1,'<br />
On peut rassemb<strong>le</strong>r ces mii, m. en une i.iatrice de covarian-<br />
1j<br />
ce c'est-à-dire un tab<strong>le</strong>au carré<br />
mil' m12' m13 ............................. m<br />
1N<br />
m2,, mZ2 ....................................<br />
............................. m.. ............<br />
Ji<br />
..............................................<br />
%i .......................................... m<br />
N,N<br />
Cette matrice symétrique (m.. = m..) comporte des éléments<br />
a<br />
trop nombreux pour en dégager <strong>le</strong>s partiidlarii!es, et on va tenter par<br />
une transformation de <strong>le</strong>s réduire.<br />
Puisque <strong>le</strong>s dernières composantes principa<strong>le</strong>s Y sont assi-<br />
milab<strong>le</strong>s B des constantes, on peut écrire grâce aux propriétés de la<br />
matrice d'éléments a. (matrice unitaire dont l'inverse éga<strong>le</strong> la trans-<br />
pos6e) : ij<br />
3<br />
k<br />
xi = aij<br />
yj + c
où C est en particulier une constante ùe centrage tenant compte en moyen-<br />
ne des composantes négligées, K étant <strong>le</strong> nombre de composantes principa-<br />
<strong>le</strong>s retenues.<br />
Les Y. n'étant pas corréìés, (3) signifie que ia pluie men-<br />
suel<strong>le</strong> Xi a la station i est une combinaison linéaire i.e, une somme<br />
pondbrée d'effets Y. non corrélés, ce qui suggère que ces effets sont<br />
ceux des régimes climatiques ind6pendants et dominants sur <strong>le</strong> bassin.<br />
Pour caractériser fl, on reporte sur-la carte du bassin aux<br />
N stations longues, <strong>le</strong>s va<strong>le</strong>urs des coefficirnts a. ... On cons-<br />
11<br />
fate que ces va<strong>le</strong>urs s'organisent, présentent une direction systématique<br />
de variation, perturbée, ce qui Pst normal, par <strong>le</strong> relief du bassin. On<br />
procdde. de meme pour Y P4,.puisque dans 1,'Allier ces 4 composantes<br />
occupent 80 à 90 $2iey?i variance tota<strong>le</strong>.<br />
Les cartes obtenues suggèrent qu'on assimi<strong>le</strong> Y1, quiprend k<br />
lu: seul 70 % de la variation, aux influences climatiques dominantes ve-<br />
nant du Nord-Ouest. Y,, avec 10 & 15 $ de la variation et un gradient des<br />
courbes orients Sud-XÕrd, est assimilab<strong>le</strong> aii climat méditerranGen-<br />
Enfin Y et Y reprbsenteraient <strong>le</strong>s influences continenta<strong>le</strong>s<br />
3 4<br />
assez importantes dans <strong>le</strong>s vb!lées de l'Allier.<br />
Ces transformations permettent :<br />
1.- de reconstituer <strong>le</strong>s pluies aux points sans mesures. Les va<strong>le</strong>urs des<br />
Coefficients a; . comme nous 1 'avons vu peuvent s'interpo<strong>le</strong>r Yynopti-<br />
quement. I1 estJalors possib<strong>le</strong> de procéàer d'abord pour chaque année<br />
et à l'aide de la pluviométrie des stations longues au calcul de la<br />
réalisation des quatre ou cinq composantes principa<strong>le</strong>s pour chaque<br />
mois. Ensuite, <strong>le</strong>s coefficients 8. lus sur <strong>le</strong>s K cartes permettent<br />
14 .<br />
inversement de calcu<strong>le</strong>r la pluviom trie en un point quelconque à par-<br />
tir des K (quatre ou cinq) composantes précedenment calculées.<br />
La théorie permet d'expliciter <strong>le</strong>s erreurs résiduel<strong>le</strong>s dues<br />
au fait qu'on se limite aux quatre ou cinq composantes qui expliquent<br />
80 à 90 % de la variance tota<strong>le</strong>.<br />
2.- d'estimer <strong>le</strong>s corrélations entre <strong>le</strong>s pluies mensuel<strong>le</strong>s de deux points<br />
quelconques du bnssin. On calcu<strong>le</strong> d'abord <strong>le</strong>s variances et covariances<br />
pour ces deux points à partir des coefficients aij et de la variance<br />
des composantes principa<strong>le</strong>s (va<strong>le</strong>urs propres) correspondantes- Ce cal-<br />
cul permet celui de corrélation et débouche sur des indications objec-<br />
tives concernant la gestion du reseau (puisqu'on pcut déterminer a la<br />
fois l'information ajoutée par chaque poste & la connaissance de la<br />
lame d'eau et l'étendue ,de la "zone d'influence'' de ce poste).<br />
3.- Le calcul de la loi de probabilité de la lame d'eau sur une surface<br />
quelconque (bassin versant) puisque l'on connaît la pluviométrie dr<br />
toutes <strong>le</strong>s iascs 6I6iiiPntaires quc l'on pcut dbcouper dans <strong>le</strong> bassin<br />
versant eii mPme teinpc que <strong>le</strong>ur corrélation.<br />
55
56<br />
4.- calciil de la loi de probabilité d'une sécheresse simultanée à plu-<br />
sieurs stat
L'autocorrélation des ddbits (et des pluies) s'apprécie<br />
sur l'ensemb<strong>le</strong> des stations et conduit à ui,e fonction annuel<strong>le</strong> lissée.<br />
La simulation se fait par un processus de Markov d'ordre 1<br />
compte tenu de la matrice des intercorrélations, <strong>le</strong>s erreurs étant ti-<br />
idpc dans des lois norma<strong>le</strong>s centrées. Les moyennes, écarts types pour<br />
chaque station et chaque mois se déduisent de la cartographie des quel-<br />
ques paramètres du lissage précédent.<br />
Lorsqu'il s'agit de simu<strong>le</strong>r en tenant compte des séries<br />
historiques de pluies, il faut pallier la faib<strong>le</strong> autocorrélation des<br />
pluies : un tirage des erreurs ayant une forte corrélation avec <strong>le</strong>s<br />
débits générés au mois précédent est substitué au tirage au hasard.<br />
III.- MODELES KEGIOSAL'X AShLYTIQL"o8.<br />
m.1- A l'échel<strong>le</strong> - - annuel<strong>le</strong>.<br />
-___<br />
Sur <strong>le</strong> bassin versant de l'Allier et à condition de consi.dé-<br />
rer la meme période de temps, <strong>le</strong>s débits annuels D sont linéairement<br />
liés aux précipitations annuel<strong>le</strong>s P (23 bussins étudids). Les cocffi-<br />
cients de r6gression et <strong>le</strong>s ordonnies 9 l'origice ont des val:%urs<br />
qui dépendent des caractéristiques physiqucc des bassins. Les carac-<br />
téristiques qui ont <strong>le</strong> plus d'influence sont : la pente moyenne et<br />
<strong>le</strong> pourcentage de terrains incultes.<br />
Ces variab<strong>le</strong>s ont un? signification discutab<strong>le</strong> dans la me-<br />
sure OU el<strong>le</strong>s en intkgrent beaucoup d'autres (géologie, altitude,<br />
etc.).<br />
Une analyse de variance - effectuhr en fonctioii de 2 ou 3<br />
modalités des deux caractéristiques prépondéran-tes - nous a per-<br />
mis de choisir statistiquement <strong>le</strong>s coePfi.cients de régression et<br />
l'ordonnt5e à l'origine a retenir suivant <strong>le</strong>s modalitós qui slave-<br />
rent reprisenter des cas différents.<br />
- Résultats de l'analyse de variaLice sur IPS relations<br />
pluie-débit 1'6ciiel<strong>le</strong> aniiuellr.<br />
30<br />
7 30<br />
I<br />
58<br />
Ces liaisons peuvent être utilisges pour allonger des séries<br />
ou mieux pour obtenir <strong>le</strong>s paramètres des lois de distribution des<br />
débits annuels. Sur un bassin supposé ne comporter aucune mesure,<br />
on retrouve ìn. moyenne à 1 6 près mais on sous-estime ia vnriance<br />
de 40 5.<br />
m.2- A l'échel<strong>le</strong> de la crue.<br />
~<br />
Une crue peut être assimilée à un volume modulé dans <strong>le</strong> temps.<br />
- Le rrridement de la pluie conditionne <strong>le</strong> volume R. Sur l'Allier, <strong>le</strong><br />
meil<strong>le</strong>ur type de liaison trouvé entre R et la pluie tota<strong>le</strong> P est<br />
U = a.Fb. avec Q débit avant ia crue ; b, c et a sont des<br />
coefficirnts dont la'va<strong>le</strong>ur varie d'un bassin à l'autre et peut<br />
etre reliée aux caractéristiques physiques et morphologiques.<br />
Les va<strong>le</strong>urs de a et b dépendent surtout de carartéristiques<br />
morphologiques et physiques : s'. de forêts, de labours, géologie,<br />
surface... C'ne analyse de variance faite eri fonction de plusieurs<br />
modalités de 2 de ces caractéristiques, permet de déterminer <strong>le</strong>s<br />
coefficients iì prendre en considération.<br />
- La modulation dans <strong>le</strong> temps peut être étudiée moyennant une hypo-<br />
thèse de linéaiitl psr la théorie ¿!e l'hydrogramme unitaire.<br />
Sur l'Allier, <strong>le</strong>s hydrogrammes unitaires trouvés sont commo-<br />
dément représentés par l'équation d'une courbe (Pearson III) qui est<br />
définie grâce à deux paramètres o( et K qui sont estimés à partir<br />
des moments H, et M2.<br />
Ces va<strong>le</strong>urs, différentes d'un bassin à l'autre peuvent être<br />
reliées par analyse de variance à des caractéristiques morphologi-<br />
ques tel<strong>le</strong>s que : la longueur et la pente du "rectang<strong>le</strong> équiva<strong>le</strong>nt",<br />
<strong>le</strong> pourcentage de la surface du bassin occupée par <strong>le</strong>s gneiss, l'hyp-<br />
Sométrie et la surface du bassin versant.<br />
Sur un bassin non jaugé, à partir des caractéristiques physi-<br />
ques il est donc théoriquement possib<strong>le</strong>, pour une pluie donnde, a<strong>le</strong>s-<br />
timer <strong>le</strong> rendement et la modulation de la crue correspondmte.<br />
Les mêmes modalités d'approche permettent de relier <strong>le</strong>s pa-<br />
ramètres des corrélations des débits minimums annuels de 30 j (Q30<br />
en l/s/km2) avec un facteur climatique (calculé à partir de P et<br />
1IE:T.P.) à la géologie et Èi la surface des bassin.
IV.- CO?;CLUSION.<br />
Ces dernières méthodes en cours de développe'ment, rejoi-<br />
gnent la théorie du contrô<strong>le</strong> qui est aussi une des bases de l'opti-<br />
misation économique. Ceci contribue à créer un outil et un langage<br />
commun aux économistes et RUX hydrologues. En meme temps, se fait<br />
jour, chez ces derniers en particulier, <strong>le</strong> souci d'expliciter "la<br />
va<strong>le</strong>ur ajoutée" non seu<strong>le</strong>ment de <strong>le</strong>urs méthodes (ou modè<strong>le</strong>s) mais<br />
aussi de <strong>le</strong>urs mesures et même de l'organisation de cel<strong>le</strong>s-ci (ré-<br />
seaux). Cette va<strong>le</strong>ur ne peut s'expliciter qu'à travers une intégra-<br />
tion au plan des décisions économiques, intégration qui met en jeu<br />
d'autres variab<strong>le</strong>s beaucoup plus mal connues que la variab<strong>le</strong> hydro-<br />
logique.<br />
L'exemp<strong>le</strong> actuel <strong>le</strong> plus préoccupant qui peut concréti-<br />
ser ce problème de l'élaboration de l'information pour SR mobilisa-<br />
tion en vue d'un objectif prxcis, est bien entendu celui de la pol-<br />
liition. Dans ce domaine il est clair que l'emploi d'un modè<strong>le</strong> quel<br />
qu'il soit suppose dès <strong>le</strong> départ une méthode d'acquisition des don-<br />
nées conçiies.en fonction du modè<strong>le</strong>. I1 ne peut être seu<strong>le</strong>ment ques-<br />
tion d'utiliser l'information statistique issue d'un paramètre iso-<br />
lé à la significat.ion très fluctuante dans <strong>le</strong> temps et suivant la<br />
va<strong>le</strong>ur d'zutres paramètres et dérivan$ au cours des années sous l'in-<br />
fluence des progrès de l'industrie. Une exploration préalab<strong>le</strong>, par<br />
simulation sur <strong>le</strong> modè<strong>le</strong>' envisagg devrait permettre de définir cet-<br />
te stratégie d'acquisition des données. Celui-ci permet d'explorer<br />
<strong>le</strong>s conséquences en particulier biologiques et écbnomiques des va-<br />
riations de tel ou tel facteur dont l'homne a la maîtrise (soutien<br />
des étiages ou modifications de la charge polluante).<br />
C'est-à-dire l'évolution rapide vers l'intégration et<br />
l'interprétation, dans <strong>le</strong> domaine de l'eau ,des diverses discipli-<br />
nes axées sur l'étude spécifique soit des ressources superficiel<strong>le</strong>s<br />
ou souterraines, soit de la pollution, soit des besoins oil des pro-<br />
blèmes économiques. Ces domaines sont encore assez séparés et il en<br />
résulte un effort d'adaptation permanent.<br />
t *<br />
t<br />
Cette note evoque divers points développés dans une publication de<br />
synthèse du Laboratoire National d'Hydraulique et du Laboratoire<br />
d'Hydrologie de Montpellier, éditée sous l'égide de la Délégation<br />
Généra<strong>le</strong> à la Recherche Scientifique et Technique, ouvrage de 133 pages<br />
intitulé "Méthodes d'gtude régiona<strong>le</strong> des ressources ell eau. Application<br />
au bassin dè l'Allier'', dont <strong>le</strong>s auteurs principaux sont MM. CORMARY,<br />
BERNIER, MASSON, LOBERT, DAUTY, SAUCEROTTE, etc... Cette publication<br />
synthétise un bon nombre d'6tudes méthodologiques dont <strong>le</strong> but est la<br />
valorisation régiona<strong>le</strong> de l'information.<br />
59
ABSTRACT<br />
LE TRANSFERT D'INFORMATION HYDROLOGIQUE<br />
A DES BASSINS VERSANTS NON OBSERVES<br />
Par<br />
Pierre DUBREUI L*<br />
The lack of sufficient hydrological datas is generally more<br />
important in <strong>the</strong> basins of small area and located in poorly<br />
developed countries. To estimate <strong>the</strong> water resources in such<br />
basins, we have to do a transfer of information from "similar<br />
basins" for which we have enough datas. This transfer may be by<br />
analogy when <strong>the</strong> regional density of hydrological information is<br />
too slight; that's made up by a qualitative analysis of <strong>the</strong> geo-<br />
morphological factors, which are similar or not, and of <strong>the</strong>ir<br />
influence on water resources, between <strong>the</strong> project basin and <strong>the</strong><br />
similar ones. When <strong>the</strong> regional density of hydrological datas is<br />
higher -old hydrometric network and/or numerous representatives<br />
basins- <strong>the</strong> transfer will be easier, using stochastic relations<br />
between dependent hydrological variab<strong>le</strong>s and explicative variab<strong>le</strong>s<br />
fo <strong>the</strong> physical environment; practically, in this case, we can<br />
establish and utilize regional graphs and norms. Some practical<br />
examp<strong>le</strong>s show <strong>the</strong> possibilities and limits of <strong>the</strong> two methods of<br />
transver.<br />
--<br />
RE S UME<br />
L'abscence de données hydrologiques suffisantes est d'autant<br />
plus aiguë que <strong>le</strong>s bassins versants sont de faib<strong>le</strong> superficie et<br />
situés dans des contrées peu développées. L'estimation des res-<br />
sources en eau sur de tels bassins exige un transfert d'informa-<br />
tion depuis des bassins de comparaison oh l'on possède des don-<br />
nées. Ce transfert peut être analogique lorsque la densité réeio<br />
na<strong>le</strong> d'information hydrologique est faib<strong>le</strong>; il consiste en une<br />
analyse qualitative des éléments géomorpholofiques comparab<strong>le</strong>s<br />
ou dissemblab<strong>le</strong>s et de <strong>le</strong>urs effets sur <strong>le</strong>s ressources en eau<br />
entre bassin du projet et bassins de comparaison. Lorsque la den<br />
sité régiona<strong>le</strong> d'information hydrologique est é<strong>le</strong>vée -réseau hy-<br />
drométrique ancien et/ou nombreux bassins représentatifs- <strong>le</strong><br />
transfert fait appel aux liaisons stochastiques entre variab<strong>le</strong>s<br />
hydrologiques dépendantes et variab<strong>le</strong>s du milieu physique expli-<br />
catives et se matérialise par des normes ou abaques régionaux.<br />
Des exemp<strong>le</strong>s précis et utilisés des deux méthodes de transfert<br />
illustrent <strong>le</strong>urs possibiiitês et <strong>le</strong>urs limites respectives.<br />
* Chef du Département de la Recherche Appliquée-au Service Hydro<br />
logique de 1'O.R.S.T.O.M. - France.<br />
-
62<br />
Les pays dans <strong>le</strong>squels il y a encore de nos jours absence d'infom-<br />
tion hydrologique sont en quarstité de plus en plus réduite.<br />
L'estinmtion des ressources en eau ne peut y etre faite, a priori,<br />
que d'une manière grossière en procédant par analogie avec d'autres pays dotés<br />
eux d'information hydrologique ; ce transfert analogique d'information est<br />
évidement beaucoup moins stir que celui auquel on peut procéder dans un pays<br />
ou une région non dénué d'infomation hydrologique, la méthodologie restant la.<br />
m& come on <strong>le</strong> verra plus loin.<br />
Mis 3. part ces exceptions, la p1upal-t des pays disposent d'informations<br />
hydrologiques fournies soit pr <strong>le</strong>s réseaux hydrométriques, soit par <strong>le</strong>s<br />
bassins représentatifs. Ces infornations hydrologiques,quel<strong>le</strong> que soit la<br />
densité des dispositifs de mesures,ne concernent qu'un certain nombre de bassins<br />
versants. I1 y a toujours des bassins versants non observés, même dans <strong>le</strong>s pays<br />
dotés d'excel<strong>le</strong>nts réseaux de mesures. Or, <strong>le</strong>s besoins de connaissance de la<br />
ressource en eau se posent aussi bien pour <strong>le</strong>s bassins des réseaux que pour <strong>le</strong>s<br />
bassins non observés. En effet, <strong>le</strong>s réseaux de mesures ont été généra<strong>le</strong>nient miis<br />
en place peu à peu au cours de l'histoire, l'implantation des stations s'effectuant<br />
en considération des besoins médiats. Dans certains cas, une planification<br />
préalab<strong>le</strong> de l'implantation des stations du réseau a pu ¿?tre réalisée en<br />
tenant compte de certains objectifs à moyen terme de l'utilisation des eaux.<br />
Malgré tout, la croissance des besoips en eau est tel<strong>le</strong>,au cours des années<br />
présentes de la seconde moitié du erne sièc<strong>le</strong>,que <strong>le</strong>s ressources en eau sont<br />
recherchées là OU, il y a vingt ou trente ans, il paraissait ne pas y avoir de<br />
problème et oh, par conséquent, aucune station de mesures ne fut implantée.<br />
On peut donc dire aujourd'hui que l'hydrologue doit partager son<br />
temps entre l'analyse des informations col<strong>le</strong>ctées sur <strong>le</strong>s bassins observés et<br />
l'estimation des mems informations sur <strong>le</strong>s bassins non observés.<br />
Si <strong>le</strong> problème de cette estimation se pose sur un grand cours d'eau<br />
drainant un bassin de superficie importante, il est à peu p&s certain que l'on<br />
trouve en amont et en aval du lieu d'estimation - c'est-à-dire du site d'un<br />
projet d'aménagemnt hydraulique - une station d'observation. Dans ces condi-<br />
tions, <strong>le</strong> transfert d'analogie est faci<strong>le</strong> puisque <strong>le</strong>s caractéristiques hydrolo-<br />
giques du lieu d'estination sont comprises entre cel<strong>le</strong>s des stations d'observa-<br />
tions dont el<strong>le</strong>s diffèrent d'ail<strong>le</strong>urs assez peu.<br />
La majorité des problèmes d'estjmation se posentpour des bassins non<br />
observés c'est-à-dire pour des bassins versants de superficie faib<strong>le</strong> à modérée<br />
sur <strong>le</strong>squel<strong>le</strong>s n'existent aucune station de mesure. La résolution de ces<br />
prob<strong>le</strong>mes exige <strong>le</strong> recours à l'information disponib<strong>le</strong> dans des bassins voisins<br />
de la &me région climatique. Le transfert d'information repose sur <strong>le</strong> postulat<br />
selon <strong>le</strong>quel deux bassins auront des caractéristiques hydrologiques identiques<br />
si <strong>le</strong>ur milieu physico-climatique - <strong>le</strong>ur environnemnt - est <strong>le</strong> &m.
Le problème consiste donc à analyser ce milieu physico-cbtique, en dégager<br />
<strong>le</strong>s paramètres susceptib<strong>le</strong>s d'influencer <strong>le</strong>s caractères hydrologiques afin de<br />
mettre en évidence <strong>le</strong> r8<strong>le</strong> de ce milieu sur <strong>le</strong>sdits caractères.<br />
Si l'on dispose, dans une région climatique homogène, d'une informa-<br />
tion hydrologique abondante et de bonne qualité, ia méthode de transfert<br />
consiste à utiliser un ensemb<strong>le</strong> de liaisons numériques ou graphiques établies<br />
entre variab<strong>le</strong>s hydrologiques et paramètres de l'environnement.<br />
Si l'information hydrologique régiona<strong>le</strong> est insuffisante ou si<br />
l'ensemb<strong>le</strong> précédent de liaisons l'hydmlogie-milieult n'a pas été élaboré, la<br />
méthode de transfert est purement analogique et qualitative puisqu'el<strong>le</strong> ne<br />
peut estkr <strong>le</strong>s caractères hydrologiques du lieu d'estimation que par analogie<br />
avec ceux du ou des bassins observés ayant l'environnement <strong>le</strong> plus comparab<strong>le</strong><br />
avec celui du bassin non observé.<br />
On examine successivement ces deux méthodes de transfert de l'informa-<br />
tion hydrologique en s'appuyant sur des exemp<strong>le</strong>s concrets.<br />
1. Relations entre variab<strong>le</strong>s hydrologiques et paramètres de l'environnement<br />
Sur un plan général, <strong>le</strong> problème consiste en l'établissement de<br />
relations entre des variab<strong>le</strong>s hydrologiques V1, V2.. . définies, a priori, et<br />
certains paramètres Pl, P2". P du milieu physico-climatique, de la forme<br />
n<br />
V1 = f (Pl, P2...P<br />
k ) de tel<strong>le</strong> sorte que l'écart résiduel soit minimal.<br />
Le problème n'est pas nouveau. Déjà au début du neme sièc<strong>le</strong>, l'hydrologie<br />
considérée aujourd'hui com classique avait abordé <strong>le</strong> prob<strong>le</strong>m en<br />
élaborant diverses formu<strong>le</strong>s d'écou<strong>le</strong>ment ou explicatives de variab<strong>le</strong>s hydrologiques.<br />
La littérature consacrée à ces formu<strong>le</strong>s est abondante ; on en trouve<br />
un bon catalogue dans l'ouvrage de G. REMENIERAS rl] y<br />
011 peut citer :<br />
a) <strong>le</strong>s formu<strong>le</strong>s donnant <strong>le</strong> déficit d'6cou<strong>le</strong>ment annuel moyen en<br />
fonction des précipitations et de La température annuel<strong>le</strong>s moyennes comme cel<strong>le</strong>,;<br />
Cie COUTACa\IE et TURC ou cel<strong>le</strong> de THOFCNTHWAITE prenant en considération <strong>le</strong> bilan<br />
mensuel entm pluie et évapotranspiration.<br />
b) <strong>le</strong>s formu<strong>le</strong>s donnant h s caractéristiques de l'hydrogram unitaire<br />
de cyue - temps de réponse en fonction de la longueur du bassin, débit de pointa<br />
en fonction de la surface, de la durée de la pluie et de l'état du bassin ...<br />
etc . - come cel<strong>le</strong>s de SEDER établies dans la région des Appalaches aim<br />
U.S.A.<br />
63
64<br />
c) <strong>le</strong>s formu<strong>le</strong>s donnant <strong>le</strong> débit ma-1 d'une cme de fréquence<br />
choisie, soit établie de manière rationnel<strong>le</strong> c om cel<strong>le</strong> de CAQUOT<br />
(Q = KI? Cn A', K fonction de la fréquence, C coefficient de ruissel<strong>le</strong>mnt,<br />
I pente et A surface du bassin), soit établies expérimenta<strong>le</strong>ment come cel<strong>le</strong>s<br />
des italiens GIEWEUI, KENTURA... etc ... qui reliaient débit et surface<br />
de bassin, temps de concentration de l'écou<strong>le</strong>nient et surface et pente du bassin.<br />
L'utilisation abusive de ces formu<strong>le</strong>s a conduit à de nombreux déboires.<br />
I1 faut, en effet, considérer qu'el<strong>le</strong>s ont été établies à partir de données<br />
expérimenta<strong>le</strong>s en quantité limitée et en provenance d'une certaine région et<br />
qu'il était illogique de <strong>le</strong>s appliquer à des bassins situés dans des régions<br />
d'environnemnt différent. En outre, <strong>le</strong>s paramètres explicatifs pris en compte<br />
étaient peu nombreux et pour certaines uniquemnt du domaine climatique ; pzr<br />
conséquent, <strong>le</strong>ur application ne pouvait donner que des résulta-ts d'autant plus<br />
erronés que <strong>le</strong>s particularités de milieu étaient importantes.<br />
On admt aujourd'hui que <strong>le</strong> domine d'utilisation de ces form<strong>le</strong>s<br />
doit @tre limité à la région de laquel<strong>le</strong> proviennent <strong>le</strong>s données expérimenta<strong>le</strong>s<br />
ayant contribué à <strong>le</strong>ur élaboration OU à des régions d'environnement comparab<strong>le</strong>s.<br />
I1 est, en effet, évident que si la liaison proposée est de la forme<br />
V = f (Pl, P2.0ePk)J c'est que <strong>le</strong>s paramètres du milieu P à P ne sont pas<br />
1 kS1 n<br />
influents sur V mais c'est aussi à l'inverse que &.-dite liaison n'est utilisa-<br />
1<br />
b<strong>le</strong> que dans une région OU <strong>le</strong>s va<strong>le</strong>urs de P à<br />
k+l<br />
P<br />
n<br />
ne sont pas différentes de<br />
cel<strong>le</strong>s de la région d'élaboration de laAite liaison.<br />
Au cours de la seconde moitié du erne sièc<strong>le</strong>, <strong>le</strong>s mesures hydrométriques<br />
ont été intensémnt développées tandis que, parallè<strong>le</strong>mnt, <strong>le</strong>s utilisateurs<br />
des eau mnifestaient des exigences croissantes quant à la connaissance de la<br />
ressource disponib<strong>le</strong> - précision accrue, diversification des variab<strong>le</strong>s -o<br />
Les relations régiona<strong>le</strong>s entre variab<strong>le</strong>s hydrologiques et paramètres<br />
du milieu doivent de nos jours etre établies à partir de toute l'infomation<br />
disponib<strong>le</strong> critiquée et s'appuyer sur une analyse poussée du milieu.<br />
Rassemb<strong>le</strong>r, analyser et critiquer 1' information hydrologique régiona<strong>le</strong><br />
disponib<strong>le</strong> est aujourd'hui une opération longue et délicate. A l'O.R.S.T.O.M.,<br />
la mise au point d'une monographie de grand bassin hydrographique demnde 4 à 5<br />
ans (SENEGAL, NIGER, CHARI.. .) , la synthèse de quelques 200 bassins représenta-<br />
tifs demnde encore plus de temps. A partir du moment oh lqon exige une bonne<br />
précision des relations hydrologie-milieu, l'analyse critique de consistance<br />
des données est indispensab<strong>le</strong> quel<strong>le</strong> qu'en soit la durée ou la comp<strong>le</strong>xité. C'est<br />
peu pourquoi ces relations régiona<strong>le</strong>s, tant attendues par <strong>le</strong>s planificateurs<br />
et <strong>le</strong>s utilisateurs de la ressource en eau, ne voient <strong>le</strong> jour que très <strong>le</strong>ntemnt,<br />
beaucoup plus <strong>le</strong>ntement que <strong>le</strong>s formu<strong>le</strong>s précédemment évoquées.
Ceci est d'autant plus regrettab<strong>le</strong> qu'en l'absence de tel<strong>le</strong>s relations, l'utili-<br />
sateur est amené, pour chaque projet, à. solliciter l'avis de l'hydrologue qui<br />
se trouve contraint d'opérer au coup par coup par simp<strong>le</strong> transfert analogique<br />
dont la précision des résultats est moindre. I1 paraft urgent qu'un effort<br />
prioritaire soit décidé en vue de l'établissenient rapide de ces relations<br />
régiona<strong>le</strong>s dans tous <strong>le</strong>s pays possédant déjà une information suffisante.<br />
L'0.R.S.T.O.M. a concentr6 une partie de ses activités sur cet objec-<br />
tif au cours des dix dernières années.<br />
Dès 1965, C. AWRAY et J. RODIEX [2] établissaient un ensemb<strong>le</strong> de<br />
graphiques permettant l'esthtion des crues décenna<strong>le</strong>s à l'issue de bassins<br />
versants de 2 à 200 km2 en Afrique occidenta<strong>le</strong> intertropica<strong>le</strong>, à partir de<br />
l'information col<strong>le</strong>ctée sur quelque 60 bassins représentatifs exploités de 1<br />
à 5 ans.<br />
Le tab<strong>le</strong>au suivant décrit sommairemnt <strong>le</strong> contenu de ces graphiques.<br />
: Variab<strong>le</strong> expliquée : Fonction : Paramètre explicatif :<br />
.--------------------:------------------:---------------------.<br />
: decema<strong>le</strong> : précipitation<br />
: Coefficient de missel-: décroissante : logarithme de la<br />
: <strong>le</strong>nient : surface<br />
Hauteur de l'averse croissante : Hauteur annuel<strong>le</strong> de :<br />
: Temps de montée, temps croissante 11<br />
de base et coefficient :<br />
: de forme de l'hydro- :<br />
' gram.<br />
Le miLieu physique était pris en compte par l'intermédiaire de<br />
groupes climatiques (subdésertique à végétation steppique, tropical à végétation<br />
de savane plus ou mohs arbode, équatorial à végétation forestière) à<br />
l'intérieur de chacun desque1 étaient constitués des sous-groupes homogènes<br />
de relief et perméabilité, ces deux paramètres étant d6finis par rangement en<br />
classes arbitraires d'aptitude croissante RI à R6, Pl à P5. Ainsi rien que<br />
pour <strong>le</strong> coefficient de ruissel<strong>le</strong>ment décennal y avait-il près de 25 relations<br />
graphiques pour <strong>le</strong>s seuls groupes de climats subdésertique et tropical.<br />
65
66<br />
Cette synthèse est actuel<strong>le</strong>nient en cours de révision et d'extension<br />
à partir des infomtions col<strong>le</strong>ctées sur plus de 200 bassins représentatifs,<br />
en essayant d'expliciter numériquement <strong>le</strong>s liaisons graphiques et en intmdui-<br />
sant tous <strong>le</strong>s pardtres du milieu par <strong>le</strong> biais de régressions multip<strong>le</strong>s ou de<br />
composantes principa<strong>le</strong>s. Le problème du choix des variab<strong>le</strong>s et de l'interdépen-<br />
dance des paramètres du milieu a nécessité des études préalab<strong>le</strong>s [3] .<br />
On peut éga<strong>le</strong>ment mentionnéfdeux autres exemp<strong>le</strong>s de synthèses régiona<strong>le</strong>s<br />
élaborées à partir d'informations issues cette fois des réseaux hydrométriques,<br />
après mise en forme de cefis-ci dans des monographies de bassins. Ces<br />
synthèses, ayant conduit B des noms hydrologiques pour aménagements hydrauliques<br />
régionaux, ont été réalisées en collaboration avec J. HERBAUD et G. GIRARD<br />
[4,5] , l'une au CESLRA état du nord-est du BRESIL, l'autre en ALSACE (France).<br />
El<strong>le</strong>s concernaient pour l'une tous <strong>le</strong>s bassins de 100 à 10.000 h2, pour l'autre<br />
tous ceux de 15 à 3.000 km2.<br />
Une quantification aussi accentuée que possib<strong>le</strong> a été effectuée pour<br />
la prise en compte des paramètres du milieu, ce qui a permis d'établir des<br />
abaques à plusieurs paramètres sans que l'on ait systématiquement numériser <strong>le</strong>s<br />
liaisons.<br />
Le tab<strong>le</strong>au suivant donne une vision globa<strong>le</strong> des liaisons établies,<br />
<strong>le</strong> paramètre explicatif principal figurant toujours avant <strong>le</strong>s paramètres<br />
secondaires en corrigeant l'effet e<br />
Les domines d'application de ces deux ensemb<strong>le</strong>s de liaisons régiona-<br />
<strong>le</strong>s sont évidement très différents. Celui du Jaguaribe concerne un climat<br />
tropical austral semi-aride, à 600-1000 mn de pluie et terrains cristallins ou<br />
gréseux sous savane arbustive plus ou moins défrichée. Celui d'Alsace correspond<br />
au climat tempéré semi-continental B hiver net, avec 800 & 2500 m de pluie<br />
(effet modéré de la neige de 1000 à 1800 m d'altitude) sur terrains cristallins<br />
ou gréseux sous cultures ou foflts à conifères dominants.<br />
On constate cependant certaines similitudes dans <strong>le</strong>s paramètres<br />
explicatifs principaux (surface drainée, hauteur annuel<strong>le</strong> de pluie) des principa<strong>le</strong>s<br />
variab<strong>le</strong>s (écou<strong>le</strong>ment annuel et crue décenna<strong>le</strong>) ce qui rejoint et confirm<br />
globa<strong>le</strong>ment l'orientation prise par <strong>le</strong>s awburs de formu<strong>le</strong>s. Mais <strong>le</strong>s influences<br />
secondaires du milieu sont assez spécifiques : r8<strong>le</strong> de la pente et de la for&<br />
en Alsace, de la nature géologique du sous-sol au Brésil. Enfin, <strong>le</strong>s coefficients<br />
ùes équations de liaison sont éga<strong>le</strong>ment spécifiques d'un domaine d'application.<br />
Alors que <strong>le</strong>s formu<strong>le</strong>s appliquées sans discernement peuvent conduire<br />
L des estimations erronées de 100 et 200 $, l'utilisation des ensemb<strong>le</strong>s de<br />
liaisons régiona<strong>le</strong>s %ydrologie-enviromeIi<strong>le</strong>nttr assure une précision de 20 à<br />
50 % dans <strong>le</strong>s résultats.
: Variab<strong>le</strong> expliquée : Paramètres explicatifs : Forme de la liaison :<br />
: A - ALSACE<br />
1. Ecou<strong>le</strong>ment moyen<br />
annuel<br />
1.1. Hauteur annuel<strong>le</strong> de:<br />
précipitations<br />
1.2. Taux de forets :<br />
2. Ecart-type de . ' 2.1. Surface du bassin 1<br />
11 écou<strong>le</strong>ment S<br />
annue 1<br />
4<br />
3. Débit spécifique : 3.1. Surface S<br />
-1 de crue : 3.2. Hauteur annuel<strong>le</strong> de:<br />
décenna<strong>le</strong> Q precipitation<br />
3.3. Taux de for&<br />
4. Rapport des 4.1. Surface<br />
pointes de crue i<br />
centenna<strong>le</strong> et<br />
décenna<strong>le</strong><br />
5. Part de l'écou<strong>le</strong>- : 5.1. Hauteur annuel<strong>le</strong> de:<br />
ment d'été dans : précipitation<br />
1' écou<strong>le</strong>ment<br />
annue 1<br />
: 5.2. Taux de for€%<br />
67<br />
linéaire croissante :<br />
linéaire croissante<br />
se400 h2 : linéaire i<br />
décroissante :<br />
S>~OC b2 : linéaire<br />
constante :<br />
4,33<br />
Q = 1950. S<br />
linéaire croissante<br />
linéaire décroissante :<br />
en dessous d'un certain:<br />
seuil d'indice de pente:<br />
croissante<br />
linéaire croissante<br />
(liaison diff &ente sur:<br />
terrains cristallins et:<br />
sédimentaires)<br />
linéaire croissante si :<br />
s>75 km2<br />
: B - JAGUARIBE<br />
1. Ecou<strong>le</strong>ment moyen I 1.1. Surface du bassin I L : A S-Oj1'<br />
annue 1 S<br />
i 1.2. Hauteur annuel<strong>le</strong> dei A = k P" avec n>l<br />
pr6cipitation P :<br />
1.3. Taux de terrains Linéaire décroissante<br />
sédimentaires<br />
(gres><br />
: 1.4. Degré de défriche- : linéaire croissante :<br />
ment<br />
i
68<br />
2. Variabilité de<br />
l'écou<strong>le</strong>ment<br />
(rapport K entre<br />
une fréquence<br />
donnée et la<br />
moyenne )<br />
3. Débit maxjml<br />
spécifique de<br />
crue décenna<strong>le</strong> Q<br />
4. Rapport de pointe<br />
entre crue<br />
annuel<strong>le</strong> et<br />
décenna<strong>le</strong><br />
: 2.1. Surface<br />
: 2.2. Ecou<strong>le</strong>ment moyen<br />
: 3.1. Surface S<br />
:Q=BS<br />
: 3.2. Hauteur annuel<strong>le</strong> de: B croSt linéairement<br />
précipitation P : avecP<br />
: 3.3. Taux de terrains : linéaire décroissante<br />
sédimentaires<br />
(gres><br />
: 3.4. Fornie du chevelu : effet croissant si<br />
: radial, décroissant si<br />
: I<strong>le</strong>n adte"<br />
: 4.1. Surface<br />
: croissante<br />
croissante<br />
4,484<br />
: croissante<br />
La généralisation de synthèses régiona<strong>le</strong>s de ce type pmttra non<br />
seu<strong>le</strong>mnt de mieux répondre à toutes <strong>le</strong>s demandes des utilisateurs de l'eau<br />
mais éga<strong>le</strong>ment d'améliorer <strong>le</strong>s ensemb<strong>le</strong>s de liaison eux-mêmes en précisant <strong>le</strong>s<br />
limites de <strong>le</strong>ur champ d'application et de comprendre <strong>le</strong>s causes qui font que<br />
crest tel paramètre plut8t que tel autre qui ici ou là explique mieux <strong>le</strong>s<br />
caractéristiques hydrologique s.<br />
2. Transferi, analogique de l'information<br />
Lorsqu'un bassin versant non observé est situé dans une région dans<br />
laquel<strong>le</strong> une synthèse de l'information hydrologique disponib<strong>le</strong> a conduit à un<br />
ensemb<strong>le</strong> de liaisons du type de cel<strong>le</strong>s qui viennent d'@tre décrites, ou s'il<br />
est situé dans une région d'environnement comparab<strong>le</strong>, l'estimation des princi-<br />
pa<strong>le</strong>s caractéristiques hydrologiques de ce bassin est chose aisée. I1 suffit<br />
d'en calcu<strong>le</strong>r <strong>le</strong>s paramètres du milieu utilisés dans <strong>le</strong>s liaisons hydrologie-<br />
environnenient et d'appliquer cel<strong>le</strong> s-ci.
La plus grande prudence s'impose si l'on n'est pas sûr de l'hornogénéit6 de<br />
l'environnenient du bassin avec celui de la région étudiée et si <strong>le</strong>s paramètres<br />
du bassin ont des va<strong>le</strong>urs extérieures au champ couvert par ceux-ci dans ia-<br />
dite région : toute extraplation hors du strict domaine d'application est<br />
risquée et ne peut etre effectuée qu'après une reconnaissance géomorphologique<br />
du bassin et de la région de référence.<br />
Beaucoup plus fréquemment <strong>le</strong> bassin non observé est situé dans une<br />
r6gion pour laquel<strong>le</strong> on ne possède pas de synthèse de Itinfomation hydrologique,<br />
laldite synthèse nécessitant de longs et délicats travaux d'analyse critique.<br />
Ainsi en France, en dehors de l'Alsace, aucune région n'a fait jusqu'ici l'objet<br />
d'une tel<strong>le</strong> synthèse systématique. Certes, l'analyse de l'information hydrolo-<br />
gique n'est pas restée au point zéro et beaucoup d%ydrologues régionaux sont<br />
A I& intuitivement d'esthr des caractères hydrologiques de bassins non<br />
observés. Ce transfert analogique n'a l'inconvénient que de devoir etre refait<br />
?I chaque demande et d'&re dépendant de la qualité ou de l'intuition de<br />
l'hydrologue,donc d'@tre imprécis et inconsistant.<br />
Malgré ces défauts, il reste la seu<strong>le</strong> méthode d'estimation en<br />
l'absence de liaisons régiona<strong>le</strong>s établies.<br />
Le processus opérationnel est <strong>le</strong> suivant :<br />
a) reconnaTtre <strong>le</strong> bassin concerné et analyser son environnenent<br />
physico-climatique,<br />
b) rechercher dans la région des bassins observés ayant des environnements<br />
aussi comparab<strong>le</strong>s que possib<strong>le</strong> avec celui du bassin concerné,<br />
c) analyser <strong>le</strong>s variab<strong>le</strong>s hydrologiques des bassins de comparaison<br />
ainsi sé<strong>le</strong>ctionnés,<br />
d) procéder au transfert analogique des variab<strong>le</strong>s hydrologiques des<br />
bassins de comparaison au bassin concerné.<br />
Ce transfert est la seu<strong>le</strong> opération origina<strong>le</strong> de ce processus. En<br />
réalité, il s'appuye implicitenient sur l'hypo<strong>the</strong>se que <strong>le</strong>s va<strong>le</strong>urs des variab<strong>le</strong>s<br />
hydrologiques vont évoluer des bassins de comparaison au bassin concerné CO~E<br />
el<strong>le</strong>s évoluent dans <strong>le</strong>s régions connues c'est-à-dire en fonction des parmètres<br />
du milieu. I1 s'agit donc jntuitivewnt de déce<strong>le</strong>r <strong>le</strong>s paradtres explicatifs<br />
principaux de l'hydrologie régiona<strong>le</strong>,puis d'estimer <strong>le</strong> sens et l'intensité de<br />
<strong>le</strong>ur action pour transférer <strong>le</strong>s variab<strong>le</strong>s hydrologiques.<br />
69
70<br />
Si l'on peut réaliser cela sans trop de difficulté pour <strong>le</strong>s paramètres classi-<br />
ques tels que la hauteur annuel<strong>le</strong> de précipitation et la surface, il n'en est<br />
pas de mhiie des autres facteurs (pente, perméabilité des terrains, cou~rt<br />
végétal ... ) au sujet desquels on peut simp<strong>le</strong>ment dire que <strong>le</strong>ur effet sera<br />
croissant ou décroissant sans pouvoir préciser de combien. On limite <strong>le</strong>s risques<br />
d'erreur en choisissant, si possib<strong>le</strong>, des bassins de comparaison dont <strong>le</strong>s fac-<br />
teurs principaux - surface, pluie annuel<strong>le</strong> - sont proches de ceux du bassin<br />
concerné, sachant que l'effet des facteurs secondaires est de l'ordre de<br />
grandeur de l'imprécision de l'estimation de la variab<strong>le</strong> hydrologique d'après<br />
<strong>le</strong>s facteurs principaux.<br />
Nous avons été anen6 à plusieurs reprises à réaliser des transferts<br />
analogiques de cette sorte pour des prob<strong>le</strong>ms d'hydraulique agrico<strong>le</strong> en France<br />
l'issue de très petits bassins versants ; par exemp<strong>le</strong> :<br />
- barrage réservoir à l'issue d'un bassin de 300 km<br />
2<br />
pour un Syndicat<br />
intercommunal d'adduction d'eau (région du Centre Ouest de la France)<br />
- barrage en terre pour plan d'eau touristique l'issue d'un bassin<br />
de moins de 20 h2 (versant atlantique des Pyrénées).<br />
N'importe qui aurait pu utiliser à l'occasion une formu<strong>le</strong> classique<br />
d'écou<strong>le</strong>ment j <strong>le</strong> risque d'erreur aurait certainement &é énorme. Le transfert<br />
analogique évite l'erreur grossière bien qu'il ne perniette pas d'atteindre la<br />
précision d'emploi des Liaisons régiona<strong>le</strong>s hydrologie-milieu quand el<strong>le</strong>s existent,mais<br />
à la condition qu'il soit effectué par un hydrologue doté d'un sens<br />
critique aigu, connaissant l'hydrologie régiona<strong>le</strong> et capab<strong>le</strong> de détecter <strong>le</strong>s<br />
effets secondaires de l'environnement (géomorphologie, nature des sols . etc ... ).<br />
3. Conclusion<br />
Les abaques régionaux et <strong>le</strong> transfert analogique dtinformation permettent<br />
d'estimer bs principa<strong>le</strong>s variab<strong>le</strong>s hydrologiques d'un bassin non observé<br />
avec une précision qui peut satisfaire <strong>le</strong> planificateur ou l'utilisateur de<br />
l'eau qui procède<br />
un aménagement simp<strong>le</strong> et modeste. Si l'aménagement est<br />
comp<strong>le</strong>xe - réservoir à but multip<strong>le</strong>, régularisation interannuel<strong>le</strong> - son coût<br />
s'accroft et la precision requise de lthydrologue éga<strong>le</strong>ment. Les méthodes exps6e:<br />
ici deviennent alors caduques au-delà du stade de l'avant-projet OU des études<br />
préliminaires. I1 est alors indispensab<strong>le</strong> de doter <strong>le</strong> site d'aménagemnt d'une<br />
station hydrométrique pour affiner <strong>le</strong>s estimations. Cela est pwsque toujours<br />
possib<strong>le</strong> car, entre <strong>le</strong>s études préliminaires et <strong>le</strong> projet définitif, st6cou<strong>le</strong>nt<br />
bien souvent plusieurs années dont l'hydrologue pourra tirer profit s'il a été<br />
avisé en temps uti<strong>le</strong> du problème et de la précision souhaitée.
Ref érence s bibliographiques<br />
1. RAS G. - i960 -<br />
de l'Ingénieur1' Coll. du Lab.<br />
Nat. d'Hydraulique, Eyrol<strong>le</strong>s édit. Paris<br />
2. RODER J.A., AWRAY C. - 1965 - "Premiers essais d'étude généra<strong>le</strong><br />
du ruissel<strong>le</strong>ment sur <strong>le</strong>s bassins expérimntaux et représentatifs<br />
d'Afrique tropica<strong>le</strong>" A.I.S.H. Symposium de Budapest - Public. no 66,<br />
vol. 1, pp. 12-38<br />
3.<br />
4.<br />
5.<br />
DUBRF;UIL P. - 1970 - '%e rô<strong>le</strong> des paramètres caractéristiques du<br />
milieu physique dans la synthèse et l'extrapolation des données<br />
hydrologique s recueillies sur bassins représentatif A o I. S. H a<br />
Colloque de Wellington, N. Zél., Public. no 96, vol. I, pp 583-590<br />
DUBREUU, P., GIRARD G., HERBAUD J - i968 - 'Nonographie hydrolo-<br />
gique du bassin du Jaguaribe" Coll. 'Némoires de l'ORSTOM1' no 28,<br />
21 x 27, 385 P.<br />
DUBREUIL P., HERBAUD J. - 1970 - Yontribution à la connaissance<br />
quantitative des modifications du régime hydrologique sous l'effet<br />
du taux de boisement à l'aide de deu exemp<strong>le</strong>s : <strong>le</strong> bass+ alsacien<br />
du Rhin, et <strong>le</strong> bassin du Jaguaribe (Brésil)" - S.H.F. XIeme<br />
journées de l'Hydraulique - Paris - Tome I, question III, rapport<br />
8.<br />
71
ABSTRACT<br />
ESTIMATING EVAPOTRANSPIRATION BY HOMOCLIMATES<br />
T.E.A. van Hylckama"<br />
Data for planning of water resources projects in arid or<br />
semi-arid climates are generally inadequate. It is here that<br />
<strong>the</strong> evapotranspiratia term plays an important ro<strong>le</strong> in <strong>the</strong><br />
hydrologic cyc<strong>le</strong>. Estimating this term by various empirical<br />
formulae using only measured or estimared air tgmperatures<br />
and <strong>le</strong>ngth of growing season often <strong>le</strong>ads to erroneous results.<br />
It is better to use parameters, such as net radiation, humidity,<br />
wind speeds and rainfall characteristics, obtained from regions<br />
with climates similar to that of <strong>the</strong> region under study. Such<br />
homoclimatic regions have soils and vegetation of a comparab<strong>le</strong><br />
nature because both are largely a result of <strong>the</strong> climate itself.<br />
Examp<strong>le</strong>s of <strong>the</strong> transfer of parameters to determine evapotrans-<br />
piration by <strong>the</strong> use of homoclimates show that such monthly and<br />
yearly values are, at most, 10 percent larger or smal<strong>le</strong>r than<br />
<strong>the</strong> measured ones, a significant improvement over empirically<br />
determined values which often are more than 30 percent off.<br />
RESUME<br />
Dans <strong>le</strong>s pays arides ou sub-arides, <strong>le</strong>s données nécessaires<br />
2 l'établissement des projets hydrauliques sont presque toujours<br />
insuffisantes. C'est dans ces pays éga<strong>le</strong>ment que <strong>le</strong> terme éva-<br />
potranspiration tient un rô<strong>le</strong> important dans <strong>le</strong> cyc<strong>le</strong> hydrolo-<br />
gique. Son estimation à l'aide de differentes formu<strong>le</strong>s empiri-<br />
ques, qui ne tiennent compte que de la température de l'air et<br />
de la durée de la saison cultura<strong>le</strong>, conduit souvent à des re-<br />
sultats erronés. I1 est préférab<strong>le</strong> d'utiliser <strong>le</strong>s va<strong>le</strong>urs de<br />
parametres plus efficaces, comme <strong>le</strong> rayonnement net, l'humidi-<br />
té, la vitesse du vent et <strong>le</strong> régime pluviométrique, obtenues<br />
dans des régions ayant des climats analogues à celui de la ré-<br />
gion étudiée. On peut penser que des régions de climats voisins<br />
ont des caractéristiques de sols et de végétation voisines, car<br />
ces deux éléments sont en grande partie <strong>le</strong> résultat du climat<br />
lui-même. L'auteur présente des exemp<strong>le</strong>s de transfert de don-<br />
nées pour <strong>le</strong> calcul de l'évapotranspiration, basé sur ces consi<br />
derations. Les va<strong>le</strong>urs mensuel<strong>le</strong>s et annuel<strong>le</strong>s obtenues diffé-<br />
rent de moins de 10% des va<strong>le</strong>urs mesurées, alors que l'emploi<br />
de formu<strong>le</strong>s empiriques simplistes fournit des résultats que<br />
différent souvent de plus de 30%.<br />
>* Research Hydrologist, U.S.Geologica1 Survey<br />
Texas Tech University, Lubbock, Texas.
74<br />
NOMENCLATURE<br />
BV<br />
E<br />
Ea’ Eo<br />
H<br />
L<br />
*a<br />
da<br />
k<br />
P<br />
r<br />
r<br />
a<br />
r<br />
U<br />
a<br />
Z<br />
a<br />
z<br />
A<br />
Y<br />
N Y<br />
turbu<strong>le</strong>nt transfer coefficient (g cm-2 min-l mb-l) or<br />
(10 kg m-2 min-’ 0.1 kPa‘l).<br />
rate of evaporation (g cm-2 min’l, mm hr-1, or cm day-l).<br />
actual (or measured) rates and computed potential rates.<br />
net radiation (cal cm-2 min-l or 41867.4 j m-2 min-1).<br />
latent heat of vaporization (about 585 cal g-1 or 2.46 x 106 j kg-l).<br />
temperature of <strong>the</strong> air at height za (meters) (OC).<br />
saturation pressure deficit of air (mb or kPa + 10) = <strong>the</strong> difference<br />
between saturation and actual vapor pressure.<br />
Von Kármán coefficient taken as 0.41.<br />
ambient pressure, assumed constant for <strong>the</strong> sites discussed at 983 mb<br />
or 98.3 kPa.<br />
correlation coefficient.<br />
external resistance (sec cm-1).<br />
stomatal or canopy resistance (sec cm-1).<br />
windspeed at e<strong>le</strong>vation z (cm min’l).<br />
a<br />
e<strong>le</strong>vation above surface (m or cm).<br />
roughness parameter (cm).<br />
first derivative of saturation vapor pressure versus T (mb OC1).<br />
psychrometric constant (mb<br />
a dimension<strong>le</strong>ss number dependent upon T<br />
for p = 983 mb Aly = -0.32 + exp 0.045<br />
and p;
I INTRODUCTION<br />
Evapotranspiration and hence <strong>the</strong> potential evapotranspiration term plays an<br />
important ro<strong>le</strong> in <strong>the</strong> hydrologic cyc<strong>le</strong>, becoming more important as <strong>the</strong> climate<br />
gets drier. Harrold 111 estimates that 75% of all <strong>the</strong> precipitation falling on<br />
<strong>the</strong> conterminous United States goes to evapotranspiration, but in arid lands this<br />
percentage can approach 100. Hence <strong>the</strong> estimate of potential evapotranspiration<br />
(E,) "is an essential requirement in <strong>the</strong> assessment of total availab<strong>le</strong> water,<br />
regional water balance and irrigation demand" 12 1.<br />
Although <strong>the</strong> parameters governing potential evapotranspiration are well<br />
known, in areas with inadequate hydrological data, <strong>the</strong> model required to estimate<br />
Eo quantitatively becomes difficult to construct. Often ra<strong>the</strong>r serious simpli-<br />
fications have been assumed at <strong>the</strong> cost of accuracy in <strong>the</strong> prediction of <strong>the</strong><br />
information needed in water resources projects 131. Hounam 141 presents a few<br />
examp<strong>le</strong>s of "approximations and over-sirnpliïications with regards to procedures<br />
or data. For examp<strong>le</strong> vapor pressure of <strong>the</strong> bulk air is sometimes substituted<br />
for surface vapor pressure with considerab<strong>le</strong> loss of reliability, net radiation<br />
may be estimated from sunshine or even cloudiness and air temperature, whilst<br />
<strong>the</strong> advective term, which can be quite significant in areal evaporation, is<br />
neg<strong>le</strong>cted in most methods."<br />
The desire to have a quantitative estimate of Eo regard<strong>le</strong>ss of <strong>the</strong> paucity<br />
of parameters has resulted in a p<strong>le</strong>thora of empirical equations (= models) which<br />
are valid only (if at all) for areas or regions where <strong>the</strong> empirical correlation<br />
between Eo and one or more parameters was established.<br />
Blaney-Cridd<strong>le</strong> 15 I , who derived a formula for irrigated areas and Thornthwaite,<br />
whose equatiori is based on data from humid climates 16 l.<br />
75<br />
Two examp<strong>le</strong>s may suffice:<br />
Yet such equations are often used for estimating potential evapotranspira-<br />
tion when <strong>the</strong> information needed is inadequate. One has an idea of mean monthly<br />
temperatures and rainfall, ei<strong>the</strong>r on <strong>the</strong> area under study itself or in <strong>the</strong><br />
neighborhood, and determines Eo by <strong>the</strong> use of one or <strong>the</strong> o<strong>the</strong>r of <strong>the</strong>se empirical<br />
equations.<br />
A fairly convincing examp<strong>le</strong> that such methods <strong>le</strong>ad to unsatisfactory results<br />
is presented in figure 1. This graph, based on data presented and discussed by<br />
Cruff and Thompson 171, illustrates <strong>the</strong> very poor agreement between six different<br />
models used to estimate potential evapotranspiration. The reason for <strong>the</strong> discrepancies<br />
is to be found in <strong>the</strong> characteristics of <strong>the</strong> evapotranspiration<br />
phenomenon. Plants, and to a certain extent soils, respond to such inputs as<br />
radiation, vapor pressure and winds in a rapid and nonlinear fashion. Taking<br />
seasonal, monthly, or even weekly averages of such parameters and use those to<br />
estimate Eo <strong>le</strong>ads necessarily to erroneous results. This is especially true<br />
when <strong>the</strong> advective term, combining wind speed and vapor pressure,becomes dominant.
76<br />
It seems that <strong>the</strong>re is a better approach especially if homoclimatic maps,<br />
such as <strong>the</strong> ones being discussed below, are availab<strong>le</strong>. One searches for an<br />
area with a climate as similar as possib<strong>le</strong> to <strong>the</strong> one of <strong>the</strong> area under study,<br />
but which has, in addition to <strong>the</strong> climatic characteristics, also data availab<strong>le</strong><br />
that allow a computation of potential evapotranspiration with a satisfactory<br />
degree of accuracy.<br />
In <strong>the</strong> following we shall first discuss climate classification and review<br />
<strong>the</strong> literature on homoclimates, <strong>the</strong>n show that <strong>the</strong> so-cal<strong>le</strong>d combination method<br />
enab<strong>le</strong>-s one to obtain very satisfactory estimations of potential evapotranspira-<br />
tion arid finally present an examp<strong>le</strong> of <strong>the</strong> proposed method.<br />
I I HOMOCL INATE S<br />
The earlier climatologists, such as Köppen Island Lang 191, classified<br />
climati,:; mostly by certain relationships between mean annual temperatures and<br />
rain€,'l. Later mean monthly values were taken into account and climates were<br />
Classified by <strong>the</strong> march of temperature and mean monthly rainfall throughout <strong>the</strong><br />
year 1101. Still later aridity indices were used 1111 and in 1948 Thornthwaite<br />
introduced <strong>the</strong> concept of potential evapotranspiration. Climates now are often<br />
characterized by diagrams, combining graphs of temperature and precipitation, or<br />
evapotranspiration and precipitation [ 12 I . Stations having similar climatic<br />
diagrams are cal<strong>le</strong>d homoclimes 1131. By extention <strong>the</strong> term has come to mean a<br />
"region climatically similar to ano<strong>the</strong>r specified region" I14 I .<br />
Meigs 1151 was probably <strong>the</strong> first and maybe <strong>the</strong> only one to use <strong>the</strong> word<br />
homoclimates in this sense. He used <strong>the</strong> 1948 Thornthwaite system and his maps<br />
of <strong>the</strong> homoclimates of arid lads are ra<strong>the</strong>r crude and on a sca<strong>le</strong> (about 1 to 3Q<br />
x LO6) too small to be of much practical value.<br />
In Arid Zone Research XXI UNESCO 1161 presents a much more detai<strong>le</strong>d set of<br />
maps which are cal<strong>le</strong>d bioclimatic maps because "<strong>the</strong> purpose---is to exhibit for<br />
a particular region a syn<strong>the</strong>sis of <strong>the</strong> climatic factors of special importance<br />
to living creatures".<br />
subject in itself'' and mention that 26 meteorological e<strong>le</strong>ments can affect <strong>the</strong><br />
climate, <strong>the</strong> environment and <strong>the</strong>refore, a particular animal or plant species 1171.<br />
They fully realize that, at <strong>the</strong> present time, insufficient information on all<br />
<strong>the</strong>se items is availab<strong>le</strong>, but continue: "fortunately however <strong>the</strong>re is one fact<br />
which is firmly established: namely that of all <strong>the</strong> e<strong>le</strong>ments in <strong>the</strong> environment<br />
those of most importance for living entities, plants in particular, are warmth<br />
and water".<br />
The authors say that "climate is an extremely comp<strong>le</strong>x<br />
Unlike most o<strong>the</strong>r climate geographers, however, <strong>the</strong>y were not<br />
content to use <strong>the</strong> ombro<strong>the</strong>rmic diagrams alone, (ombros = rain), but used a<br />
xero<strong>the</strong>rmic index which includes <strong>the</strong> effects of rainy days, days with mist an6<br />
dew, and allows for atmospheric humidity. This is an attractive compromise<br />
between <strong>the</strong> 26 e<strong>le</strong>ments and <strong>the</strong> use of only monthly averages of temperature and<br />
precipitation. A day, for instance, with 5 centimeters of rainfall in one hour
has an effect vastly different from a day<br />
12-hour period. A day with an average of<br />
same as one with <strong>the</strong> same temperature but<br />
humidity of 80 percent.<br />
with a 5 centimeter drizz<strong>le</strong> during a<br />
15OC under a dry c<strong>le</strong>ar sky is not <strong>the</strong><br />
under clouds and with relative<br />
Using <strong>the</strong> diagrams and indices mentioned above, UNESCO I i6 I distinguishes<br />
33 different climates, ranging from true desert (for instance in Libia) to glacier<br />
climates such as in <strong>the</strong> high mountains of Austria. There are four maps of a<br />
sca<strong>le</strong> of 1 to 10~000,000 of <strong>the</strong> dry regions in South Africa, South America, <strong>the</strong><br />
southwest of North America and <strong>the</strong> sou<strong>the</strong>rn parts of Australia. Two o<strong>the</strong>rs on<br />
a sca<strong>le</strong> of 1 toS,b00,000 cover an area from <strong>the</strong> AtlaIitic (long ?OoW) to points<br />
west of Karachi (Pakistan) (long 72OE) and from nor<strong>the</strong>rn Italy (lat 25ON) to<br />
well into <strong>the</strong> Sahara and also covering <strong>the</strong> Arabian Peninsula (lat 14ON).<br />
Ano<strong>the</strong>r source that, at times, could be used to find homoclimatic regions<br />
are <strong>the</strong> 15 volumes "World Survey of Climatology" 1181. However, due to <strong>the</strong> preferences<br />
of 11 sub-editors and numerous authors, <strong>the</strong> classifications are not<br />
consistent, ranging (for examp<strong>le</strong> in Volume 8) from <strong>the</strong> 1918 Köppen system to<br />
classifications according to dynamic concepts from <strong>the</strong> viewpoint of air-mass<br />
mixing and transformation I19 I .<br />
Terjung's maps of isanomalies 1201 might eventually help to improve homoclimatic<br />
maps. For <strong>the</strong> present time <strong>the</strong> author states: 'I--- <strong>the</strong> ra<strong>the</strong>r crude<br />
maps presented here and <strong>the</strong>ir cursory examination should not be considered<br />
definitive or qualitatively accurate".<br />
It is perhaps unfortunate that in none of <strong>the</strong>se sources useful data on<br />
ioeasured potential evapotranspiration could be found, and an examp<strong>le</strong> of <strong>the</strong><br />
applicability of <strong>the</strong> method of using homoclimates to estimate potential evapo-<br />
transpiration had to be taken from <strong>the</strong> semiarid southwestern parts of <strong>the</strong> United<br />
States.<br />
III THE COMBINATION CONCEPT AND SHE CANOPY RESISTANCE<br />
As mentioned earlier, longtime averages of meteorological parameters are<br />
inadequate to estimate potential evapotranspiration. The dynamic characteristics<br />
and <strong>the</strong> sensitivity of Eo to environmental parameters are most c<strong>le</strong>arly demonstrated<br />
by <strong>the</strong> correlation method, aïso known as <strong>the</strong> eddy flux, eddy transfer or covar-<br />
iance method 14, 221.<br />
In this method measurements have to be taken with a<br />
frequency of a few seconds or <strong>le</strong>ss. Ano<strong>the</strong>r method, not as sensitive to be sure,<br />
but quite suitab<strong>le</strong> for our purpose it seems, is a method that combines <strong>the</strong> energy<br />
budget with a mass-transfer term 123, 24, 251. A complication arises when <strong>the</strong><br />
plants, even under conditions of potential evapotranspiration, react to <strong>the</strong><br />
environment and seem to control transpiration by means of opening or closing <strong>the</strong><br />
stomata 126, 27, 281.
78<br />
It must, first of all, be shown that potential evapotranspiration can be<br />
estimated uite accurately by <strong>the</strong> use of <strong>the</strong> combination formul developed by<br />
Penman 1237 and improved by Monteith 129, 301 and van Bavel 125 . The latter,<br />
following <strong>the</strong> method first used by Penman 1311, derived <strong>the</strong> fol owing expression<br />
for <strong>the</strong> instantaneous evaporation rate:<br />
E = '/L<br />
B in this equation is defined as:<br />
V<br />
(A/y) H + L Bv da<br />
A/Y + 1<br />
cai cm-2 min-'<br />
Because expression (2) is based upon standard wind-profi<strong>le</strong> <strong>the</strong>ory, van Bavel<br />
warns that it applies strictly to adiabatic conditions only. But he points out<br />
that <strong>the</strong> combination model (1) has reduced <strong>the</strong> criticality of (2).<br />
This model predicts potential evapotranspiration from wet bare soil and from<br />
alfalfa covered soil with great accuracy over hourly periods, as was convincingly<br />
shown by van Bavel I25 1 . However, when used to compute evapotranspiration from a<br />
stand of saltcedar (Tamarix pentandra) , I32 1 <strong>the</strong>re were fai-rly large discrepancies<br />
when computed and measured values were compared.<br />
discrepancies was immediately evident. It is a Weil-known fact that over tall<br />
vegetation, <strong>the</strong> roughness <strong>le</strong>ngth (2,) varies with <strong>the</strong> wind speed, 133, 341 and a<br />
zero displacement <strong>le</strong>ngth must be incorporated in equation (2). Alternatively,<br />
a modified roughness <strong>le</strong>ngths can be used, and this was done in <strong>the</strong> present<br />
computations 135 I .<br />
One of <strong>the</strong> reasons for <strong>the</strong><br />
With this modification <strong>the</strong> discrepancies are smal<strong>le</strong>r but <strong>the</strong><br />
results are still not very satisfactory. A typical examp<strong>le</strong> is given at <strong>the</strong> top<br />
of figure 2.<br />
Inspection of <strong>the</strong> data fur<strong>the</strong>r showed that <strong>the</strong> largest deviations between<br />
computed and measured evapotranspiration occured under conditions of high wind<br />
speeds. This indicated that <strong>the</strong>re could possibly be a stomatal or o<strong>the</strong>r type of<br />
resistance inside <strong>the</strong> plants, but most likely a closing of <strong>the</strong> stomata under<br />
conditions of high evaporativity 1241. It is possib<strong>le</strong> to measure diffusion<br />
resistance directly on most broad<strong>le</strong>af plants by <strong>the</strong> use of one or o<strong>the</strong>r type of<br />
porometer 136, 371. These instruments cannot be used on <strong>the</strong> small sca<strong>le</strong>-like<br />
<strong>le</strong>aves of saltcedar which are <strong>le</strong>ss than 2 millimeters long and 1 millimeter wide.<br />
Xonteith (301, however, has shown how external and stomatal resistant-es can he<br />
estimated from microclimatological data. The combined energy budget and mass<br />
transfer equation <strong>the</strong>n becomes:
in which <strong>the</strong> external resistance is:<br />
and <strong>the</strong> stomatal resistance:<br />
r a = (log, (z/zO)l2 / U k2<br />
r = (A/y + 1) (Eo/Ea - 1) x ra (5)<br />
A recomputation of potential evapotranspiration with equation (3) shows that a<br />
muili closer agreement between measured and computed evapotranspiration can be<br />
obtained.<br />
Notice that equation (5) contains <strong>the</strong> potential as well as <strong>the</strong> measured<br />
evapotranspiration, but once rs has been computed, it was found that it very<br />
highly correlated with wind speeds, and also (but <strong>le</strong>ss) with vapor pressure<br />
deficits. Using <strong>the</strong> resistances obtained from <strong>the</strong> equation (5) for one set of<br />
data, potential evapotranspiration could <strong>the</strong>n be computed for o<strong>the</strong>r sets of data<br />
and such values are plotted at <strong>the</strong> bottom of figure 2.<br />
IV AN EXAMPLE<br />
In order to demonstrate how <strong>the</strong> application of homoclimatic data may help to<br />
estimate Eo, a comparison will be made between evapotrenspiration rates measured<br />
in evapotranspirometers near Buckeye, Arizona (lat 33ON, long 113OW) with those<br />
computed with data availab<strong>le</strong> from a homoclimatic area about 50 kilometers to <strong>the</strong><br />
east near Tempe, Arizona. Thus we pretend that <strong>the</strong> needed parameters at Buckeye<br />
were not availab<strong>le</strong> and we use those from a homoclimatic region.<br />
Hourly data for 3 days were availab<strong>le</strong> from technical reports issued by <strong>the</strong><br />
U. S. Water Conservation Laboratory 138, 391. Figure 3 shows a typical examp<strong>le</strong><br />
of hourly values measured at Buckeye, compared with those computed from <strong>the</strong> Tempe<br />
data. Figure 4 presents a comparison between two sets of computed data. Measured<br />
hourly data for 9 April were not availab<strong>le</strong> but, as figure 2 shows, <strong>the</strong> computed<br />
values are quite valid. Note, incidentally, that on this day in early spring<br />
<strong>the</strong>re were a few hours with dew (negative E's). Not only are <strong>the</strong> correlation<br />
coefficients very high (0.94 for figure 3 and 0.92 for figure 4) but <strong>the</strong> regression<br />
equations indicate nearly 1:l relationships. For data of figure 3 we have:<br />
E, = 0,19+0,86 Eo, and for figure 4: Ea = 0,02+0,94 Eo. The t values for both<br />
equations are well above <strong>the</strong> 1% confidence limits: respectively 6.3 and 4.0.<br />
Student's t value for <strong>the</strong> 1% limit and 22 degrees of freedom is 2.8, 1401.<br />
That <strong>the</strong> combination method is valid strictly for short-time data has already<br />
been mentioned. The fact is c<strong>le</strong>arly shown by <strong>the</strong> data in tab<strong>le</strong> 1. In <strong>the</strong> <strong>le</strong>ft<br />
79
80<br />
two columns <strong>the</strong> sum of 24 hourly values of evapotranspiration rates is given as<br />
mi.llimiters per day. In <strong>the</strong> right two columns <strong>the</strong> rates are given as computed<br />
from mean daily averages of <strong>the</strong> parameters in equation (3). As can be seen <strong>the</strong><br />
sum of <strong>the</strong> hourly values are not only very close to ano<strong>the</strong>r but also compare<br />
favorably with <strong>the</strong> measured values.<br />
The data availab<strong>le</strong> did not allow to extend <strong>the</strong> computations to months or<br />
years. However, <strong>the</strong> agreement on hourly and daily bases makes it very likely<br />
that, on monthly and yearly bases, even better agreement can be obtained.<br />
V CONCLUSIONS<br />
Serra 1411, remarks "climatology and hydrology are two very different<br />
disciplines: if <strong>the</strong> first is ãescriptive and 'static', <strong>the</strong> o<strong>the</strong>r studies---<strong>the</strong><br />
'dynamics' of water and working methods used for <strong>the</strong> first will have litt<strong>le</strong><br />
chance to fít <strong>the</strong> second.---The climatologist works with an 'average year'. To<br />
establish his---classification indices he will use <strong>the</strong> average temperature of<br />
each of <strong>the</strong> twelve months of <strong>the</strong> year---. The hydrologist by contrast must<br />
follow from day to day---<strong>the</strong> living reality of a phenomenon." The phenomenon<br />
Serra refers to is,of course,<strong>the</strong> evapotranspiration.<br />
If however, <strong>the</strong> climatic classification is detai<strong>le</strong>d enough and one has in<br />
one part of such an area sufficient information, quantitatively as well as quali-<br />
tatively, on <strong>the</strong> parameters that drive <strong>the</strong> evapotranspiration, <strong>the</strong>n it is reason-<br />
ab<strong>le</strong> to assume that in o<strong>the</strong>r parts of this homoclimate <strong>the</strong> same data are applica-<br />
b<strong>le</strong>, at <strong>le</strong>ast within acceptab<strong>le</strong> limits. It might of course be necessary (and it<br />
is nearly always possib<strong>le</strong>) to correct for latitude, e<strong>le</strong>vation and exposure.<br />
What we are dealing with seems to be an integration between climate and<br />
meteorology, something that Kisiel 1421 had in mind when he wrote: "The future<br />
of hydrology rests on our ability and willingness to undertake <strong>the</strong> last integrative<br />
effort on a continuing and adaptab<strong>le</strong> basis. This effort is particularly<br />
urgent if one accepts <strong>the</strong> <strong>the</strong>sis that each watershed or basin is a law unto<br />
itself. Transferability of laboratory know<strong>le</strong>dge to <strong>the</strong> field and of know<strong>le</strong>dge<br />
from one watershed to ano<strong>the</strong>r or from one climate to ano<strong>the</strong>r rests inexplicably<br />
on our ability to provide a ma<strong>the</strong>matical foundation to <strong>the</strong> cyc<strong>le</strong> of model building<br />
and its parts.''<br />
The present paper shows that, in princip<strong>le</strong>, <strong>the</strong> use of homoclimates is<br />
possib<strong>le</strong> and reliab<strong>le</strong> for effectively estimating evapotranspiration rates with<br />
<strong>the</strong> model presented above. The difficulty lies in <strong>the</strong> fact that <strong>the</strong>re are so<br />
few homoclimatic maps and those few do not always use <strong>the</strong> best method of classi-<br />
fying <strong>the</strong> climates. There obviously is a great need for more and more reliab<strong>le</strong><br />
homoclimatic maps. These maps should show <strong>the</strong> locations of stations where com-<br />
p<strong>le</strong>te sets of microclimatological data can be obtained for estimating <strong>the</strong> poten-<br />
tial evapotranspiration with a desirab<strong>le</strong> degree of accuracy.
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Office, Washington.<br />
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nach ihren Beziehungen zur Pflanzenwelt, Geog. Zeitschr., 6, pp. 593-611 and<br />
657-679.<br />
Lang, R., (1915). Versuch einer exacten Klassifikation der Buden in klimatologischer<br />
and geologischer Hinsicht. Internat. Mitt. Bodenkunde, 5, pp. 312-<br />
346.<br />
Kuppen, W., (1918). Klassifikation der Klimate nach Temperatur, Niederschlag<br />
und Jahreslauf, Petermann's Geog. Mitt., 64, pp. 193-203 and 243-248.<br />
de Martonne, E., (1926).<br />
Paris, 9, pp. 3-5.<br />
L'indice d'aridité. ßull. Rssoc. Géog. frangais,<br />
Thornthwaite, C. W., (1948). An approach toward a rational classification Qf<br />
climate, Geog. Review., 38, pp. 55-94.<br />
Buschhe, R. E., (ed.), (1959). Glossary of meteorology, Am. Meteorol. Soc.,<br />
p. 105.<br />
Gove, P. B., (ed,), (1968). Webster's third international dictionary, G. & C.<br />
Merriam Comp., Springfield, Mass., p. 1084.<br />
Meigs, P., (1951). World distribution of arid and semi-arid homoclimates,<br />
UNESCOfNSfAZf37, Paris.<br />
UNESCO, (1963).<br />
Bioclimatic map of <strong>the</strong> mediterranean zone, Arid Zone Research<br />
XXI, UNESCO, Paris and FAO, Rome.<br />
C:i<strong>le</strong>ad, M., & Rosenan, N., (1958). Climatological observational requirements<br />
in arid zones, in UNESCO Climatology, Arid Zone Research X, Paris, pp. 181-188.<br />
81
82<br />
18.<br />
19.<br />
20.<br />
22 s<br />
23.<br />
24.<br />
25.<br />
26.<br />
27.<br />
28.<br />
29.<br />
30.<br />
31.<br />
32.<br />
33.<br />
34.<br />
3s.<br />
36.<br />
37.<br />
38.<br />
39.<br />
Landsberg, H. E., (ed. in chief), (1969). World survey of climatology,<br />
Elsevier Publishing Company, Amsterdam.<br />
Nagao, T., (1961). Dynamical classification of climate based on <strong>the</strong> airmass<br />
mixing and transformation, Geog. Rev. Japan, 34, pp. 307-320.<br />
Terjung, W. ï., (1968). Some maps of isanomalies in energy balance climatology,<br />
Archives Meteorol. Geophys. Bioclimatology By 16, pp. 279-315.<br />
Gangopadhyaya, PI., Harbeck Jr., G. E. Nordenson, T. J., Omar, M. H., and<br />
Uryvaev, V. A., (1966). Measurement and estimation of evaporation and<br />
evapotranspiration, Techn. Note No. 83, World Meteorol. Organization, Geneva,<br />
Switzerland.<br />
Penman, H. L., (1956). Evaporation: ai? introductory survey, Me<strong>the</strong>rland<br />
Jour. of Agr. Sci., 4, pp. 9-29.<br />
Budyko, M. I., (1956). Teplovoi balans aemnoi poverkhnosti, Translated by<br />
Nina A. Stepanova, 1958: The heat balance of <strong>the</strong> earth's surface, U. S.<br />
Dept. of Commerce.<br />
van Bavel, C. H. M., (1966). Potential evapoiation: <strong>the</strong> combination concept<br />
and its experimental verification, Water Resources Research, 2, pp. 455-467.<br />
van Bavel, C. H. M., Newman, J. E., & Hilgeman, R. H., (1967). Climate and<br />
estimated water use by an orange orchard, Agr. Meteorol., 4, pp. 27-37.<br />
Turner, N. C., (1969). Stomatal resistance to transpiration in three contrasting<br />
canopies, Crop Science, 9, pp. 303-307.<br />
Parlange, J-Y, & Waggoner, i?. E., (1970). Stomatal dimensions and resistance<br />
to diffusion, Plant Physiology, 46, pp. 337-342.<br />
Monteith, J. L., (1963). Gas exchange in plant communities, in Evans, L. T.<br />
(ed.), Environmental control of plant growth, Academic Press, New York,<br />
pp. 95-112.<br />
Monteith, J. L., (1965). Evaporation and environment no. 19: The state and<br />
movement of water and living organisms, Cambridge, Symposia of <strong>the</strong> Society<br />
for Experimental Biology, pp. 205-234.<br />
Penman, H. L., (1948). Natural evaporation from open water, bare soil and<br />
grass, Proc. Royal Soc. (London) A 193, pp. 120-145.<br />
van Hylckama, T. E. A., (1970b). Water use by saltcedar, Water Resources<br />
Research, 6, pp. 728-735.<br />
Baumgartner, A., (1956). Untersuchungen Uber den WBrme- und Wascerhaushalt<br />
eines jungen Waldes. Ber. Deutscher. Wetterdienstes, 5(28), pp. 1-53.<br />
Tajchman, S., (1967). Energie- und Wasserhaushalt verschiedener Pflanzenbest-<br />
Bnde bei Munchen. Univ. Muchen, Meteorol. Inst., Wiss. Mitt., 12, pp. 1-94.<br />
van Hylckama, T. E. A., (1970a). Winds over saltcedar, Agric. Meteorol.,<br />
7, pp. 217-233.<br />
Byrne, G. F., Rose, C. W., & Slatyer, R. O., (1970).<br />
porometer, Agr. Meteorol., 7, pp. 39-44.<br />
An aspirated diffusion<br />
Sti<strong>le</strong>s, W., (1970). A diffusive resistance porometer for field use, Jour.<br />
Applied Ecology, 7, pp. 617-622.<br />
Conaway, J., & van Bavel, C. H. M., (1966). Remote measurement of surface<br />
temperature and its application to energy balance and evaporation studies<br />
of bare soil surfaces. Tech. Rep. U. S. Army E<strong>le</strong>ctronics Conmiand 2-67P-1.<br />
van Bavel, C. H. M., (1967). Surface energy balance of bare soil as<br />
influenced by wetting and drying. Tech. Rept. U. S. Army E<strong>le</strong>ctronics Conmiand<br />
2-67P-2.
40. Fisher, R. A., & Yates, F., (1943). Statistical tab<strong>le</strong>s for biological,<br />
agricultural and medical research. Oliver and Boyd Ltd., London, tab<strong>le</strong> ïïI.<br />
41. Serra, P. L., (1954). Le contro<strong>le</strong> hydrologique d'un bassin versant, in Soc.<br />
Hydrotechnique de France, Pluie, Evaporation, Filtration et Ecou<strong>le</strong>ment,<br />
Compte Rendu des Troisièmes Jourdes de l'Hydraulique, pp. 29-35.<br />
42. Kisiel, C. C., (1969). Ma<strong>the</strong>matical methodology in hydrology, in Chow, V. T.<br />
(Dir.), The progress of hydrology, Proc. of First Internat. Seminar for Hydrol.<br />
Professors, Urbana, Illinois, pp. 362-399.<br />
83
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Fig. 4<br />
Comparison of 24 hourly values of evapotranspiration<br />
computed with equation (5) using Buckeye data (Y> and<br />
Tempe data CX), (9 April '66).<br />
87
88<br />
TABLE 1. Water use by saltcedar in millimeters per day<br />
Sum of 24 Computed from<br />
hourly values Measured mean daily values<br />
Dates Tempe data Buckeye data Buckeye* Tempe data Buckeye data<br />
-<br />
1966<br />
9 April 11.0 11.5 10.4 14.9 14.4<br />
28 April 14.6 15.6 15.4 18.7 18.2<br />
3 May 13.9 12.2 13.5 17.0 15.8<br />
*The lysimeters at Tempe were allowed to dry out so no potential evapotranspira-<br />
tion data were availab<strong>le</strong>.
PLUVIOMETRIC ZONES AND THE CRITERIA TO DEFINE THEIR<br />
ABSTRACT<br />
BOUNDARIES FOR REGIONS WITH SCARCE DATA<br />
by<br />
García-Agreda R., Rasulo G., Viparelli R.<br />
A zone is defined "pluviometric zone" if <strong>the</strong> parameters of<br />
<strong>the</strong> rainfall distribution function assume <strong>the</strong> same value in all<br />
of its points, or vary with continuity from one point to ano<strong>the</strong>r<br />
according to <strong>the</strong>ir location.<br />
Consequently, if it is necessary to estimate <strong>the</strong> rainfall<br />
distribution at a point, only <strong>the</strong> information derived from<br />
pluviometers of <strong>the</strong> same pluviometric zone is useful.<br />
By refering particularly to regions in which <strong>the</strong>re is a<br />
scarcity of data, <strong>the</strong> authors point out that, in order to define<br />
<strong>the</strong> boundaries of <strong>the</strong> pluviometric zone pertaining to a given<br />
point, it is necessary preliminarly to formulate a working<br />
hypo<strong>the</strong>sis based on climatic maps in which also <strong>the</strong> geomorphology,<br />
soils and vegetation are considered.<br />
RESUME<br />
Une zone est défine "zone pluviométrique" si <strong>le</strong>s paramètres<br />
de la loi de probabilité des pluies ont la même va<strong>le</strong>ur dans<br />
toute la région ou ils varient d'une façon continue d'un point<br />
à l'autre.<br />
Par conséquence, s'il faut estimer la répartition statisti-<br />
que des pluies en un point, on peut utiliser seu<strong>le</strong>ment <strong>le</strong>s in-<br />
formations tirées des pluviomètres disposés dans la même zone<br />
pluviométrique.<br />
En particulier, en se rapportant aux régions pour <strong>le</strong>squel-<br />
<strong>le</strong>s on a peu de données, <strong>le</strong>s auteurs soulignent que, dans <strong>le</strong><br />
but de définir <strong>le</strong>s lignes de contour de la zone pluviométrique<br />
qui comprend 1.e point considéré, il fau-t d'abord formu<strong>le</strong>r une<br />
hypothèse de travail qui se base sur des cartes climatologiques<br />
dans <strong>le</strong>squel<strong>le</strong>s on considère aussi la géomorphologie, <strong>le</strong>s sols<br />
et la végétation.
90<br />
Symbols<br />
1: Let us indicate by:<br />
- h : <strong>the</strong> annual rainfall depth at any point;<br />
- y : <strong>the</strong> log of h;<br />
- O{h} and {y} : <strong>the</strong> distribution functions of h and y;<br />
- MChl; 0th) and y{h} = m:<br />
aih}<br />
respectively <strong>the</strong> mean, <strong>the</strong><br />
standard deviation and <strong>the</strong> coefficient of variation of <strong>the</strong><br />
probability distribution of h;<br />
- M {y}, oiy} and 02{y): respectively <strong>the</strong> mean, <strong>the</strong> stan-<br />
dard deviation and <strong>the</strong> variance of <strong>the</strong> probability distribution<br />
of y.<br />
Let us also indicate by:<br />
- hi with 1 c i < n: <strong>the</strong> n values of h registered during<br />
<strong>the</strong> observation period;<br />
- yi with 1 < i c n: <strong>the</strong> n values taken by y = log h;<br />
- h, s{h} and gCh}: respectively <strong>the</strong> estimates of MIh},<br />
o{h} and yth};<br />
- 7, sty} and s’{y): respectively <strong>the</strong> estimates of MCy},<br />
aiy} and 02{y};<br />
- 71 and 72, Sf{y} and s;{y): respectively <strong>the</strong> confidence<br />
limits of and s‘{y] with a tol<strong>le</strong>rance <strong>le</strong>vel of 95%;<br />
Assume h is distributed with a good approximation according<br />
to <strong>the</strong> log-normal law 113 [2].<br />
Consequently, y is distributed according to <strong>the</strong> normal law<br />
<strong>the</strong> parameters M{y} and o{y} which characterize its distribution<br />
are connected to M{h} and y{h} by <strong>the</strong> equation:<br />
and<br />
equations:<br />
By estimating <strong>the</strong> parameters 7 and s2{h} by means of
and<br />
n<br />
I-<br />
t Yi<br />
n<br />
n<br />
n - 1<br />
<strong>the</strong> confidence limits of 7 and s2{hl could be expressed by means<br />
of ecuations:<br />
in which t0,025 and ~0,025, to,g75 and ~0,975 are respectively<br />
<strong>the</strong> percenti<strong>le</strong>s of t and x corresponding to <strong>the</strong> probability 0,025<br />
and 0,975.<br />
In t roduc ti on<br />
(3)<br />
2: From direct measurements taken at each sing<strong>le</strong> point A,<br />
B ... of a region, it is possib<strong>le</strong> to deduce only estimates of<br />
<strong>the</strong> values that M{h} and y{h) assume at <strong>the</strong> said points.<br />
Takin into account <strong>the</strong> fact <strong>the</strong> said estimates could<br />
deviate from <strong>the</strong> real value due to sampling errors and that in<br />
technical prob<strong>le</strong>ms <strong>the</strong> average rainfall depth distribution on<br />
given surface must be known, it is necessary:<br />
a) to improve <strong>the</strong> said estimates by decreasing <strong>the</strong> uncer-<br />
tainty with which <strong>the</strong>y were determined;<br />
b) to estimate M{h) and y{h} and consequently <strong>the</strong> annual<br />
rainfall depth that occurs with a given probability, even at<br />
points where no pluviometers had been instal<strong>le</strong>d.<br />
The two prob<strong>le</strong>ms become greater in regions where only a<br />
few measuring stations are availab<strong>le</strong> and for most of <strong>the</strong>m with<br />
a few years of observation.<br />
91
92<br />
Hydrological Similitude Criteria and Pluviometric Zones<br />
3: The rainfall depth registered, at a generic point A, for<br />
a given event occurs due to <strong>the</strong> evolution of meteorological<br />
conditions that have <strong>the</strong>ir repercussions also on <strong>the</strong> rainfall<br />
depths that occur in <strong>the</strong> same event in a more or <strong>le</strong>ss extended<br />
zone around A. As it is known, for different environmental<br />
conditions, such as those connected with <strong>the</strong> morphology of <strong>the</strong><br />
zone, <strong>the</strong> rainfall depth that occurs during <strong>the</strong> same event in<br />
different points, could be highly different; however, in passing<br />
from one event to ano<strong>the</strong>r, at <strong>le</strong>ast normally, <strong>the</strong> said environ-<br />
mental conditions excercise a differential action that acts<br />
always in <strong>the</strong> same direction.<br />
Finally, <strong>the</strong> rainfall depths h registered at a point A, are<br />
affected both by meteorological factors common to <strong>the</strong> entire zone<br />
and acting with a variab<strong>le</strong> intensity from one rainfall event to<br />
ano<strong>the</strong>r; and by <strong>the</strong> environmental factors that are invariab<strong>le</strong>s in<br />
time, but, normally, variab<strong>le</strong> from one point to ano<strong>the</strong>r,. The<br />
deviations that are observed among <strong>the</strong> values that h assumes in A,<br />
year after year, depend upon <strong>the</strong> variability in time of <strong>the</strong><br />
meteorological factors; whi<strong>le</strong> <strong>the</strong> deviations that are noticed<br />
among <strong>the</strong> values that h assumes, with <strong>the</strong> same probability,<br />
respectively in A and in each of <strong>the</strong> o<strong>the</strong>r points of <strong>the</strong> zone<br />
around A, depend upon <strong>the</strong> variability of environmental conditions.<br />
Consequently, if in a zone characterized by common meteoro-<br />
logical factors k pluviometers are instal<strong>le</strong>d, in agreement with<br />
what has been said by o<strong>the</strong>r authors [l), it is safe to suppose<br />
that in passing from one pluviometer to ano<strong>the</strong>r, <strong>the</strong> variation<br />
coefficient y{h} remains constant.<br />
Therefore, ify{h} is constant, it derives, from equation<br />
(11, that even 02{y} remains constant.<br />
At this point, we will say that a greater number of pluviom<br />
eters belong to <strong>the</strong> same pluviometric zone if <strong>the</strong> variance assumes<br />
a common value u “Cy}.<br />
As it is known, <strong>the</strong> definition of a pluviometric zone and<br />
its connected hypo<strong>the</strong>sis are to be considered in a statistical<br />
way.
Precisely, it cannot be excluded that at each sing<strong>le</strong> point<br />
<strong>the</strong> variance 02iy} could differ from <strong>the</strong> value assumed as<br />
<strong>the</strong> value to characterize <strong>the</strong> zone; however, due to <strong>the</strong> fact that<br />
for each sing<strong>le</strong> point only an estirnate s2{y] of 02{y) could be<br />
had it is evident that:<br />
1) <strong>the</strong> deviation s2{y} - ~''{y), that is observed for sing<strong>le</strong><br />
point between s2{yl and o''{yI, could be caused partly, s*{yI -<br />
- 02{y), by a sampling error (a non-significant part of <strong>the</strong><br />
deviation between s2{y} and o"{y) and partly, 02{y] - ot2{y), by<br />
<strong>the</strong> real difference between 02{y} and or2{y) (<strong>the</strong> significant<br />
part 1 ;<br />
2) that, however, <strong>the</strong> deviation significant part is always<br />
modest and such that s'{y] - a"{y} s2{y} - 02{yl t o'íy} -<br />
- C I ~ ~ would I ~ ] range around values that s'{y) - 02{y} would<br />
assume.<br />
4: To determine <strong>the</strong> pluviometric zo'nes that lie a given<br />
region, <strong>the</strong> methodology to follow could be divided in <strong>the</strong>ree<br />
phases.<br />
An attemp to formulate a working hypo<strong>the</strong>sis delimiting <strong>the</strong><br />
sing<strong>le</strong> zones is made during <strong>the</strong> first phase.<br />
By deducing <strong>the</strong> best estimate of s'2{y} of <strong>the</strong> value that<br />
<strong>the</strong> variance o'2{y) assumes in all <strong>the</strong> points of <strong>the</strong> zone during<br />
<strong>the</strong> second phase, <strong>the</strong> working hypo<strong>the</strong>sis is formulated.<br />
In doing this, <strong>the</strong> different significance that <strong>the</strong> series<br />
of data, obtained in each pluviometer of <strong>the</strong> zone, have, must be<br />
taken into account depending on <strong>the</strong> number of a data that appears<br />
in each one of <strong>the</strong>m.<br />
Particularly, if k pluviometers lie in <strong>the</strong> zone, having<br />
indicated by s:{y}, with r being variab<strong>le</strong> from 1 to k, <strong>the</strong><br />
variance estimated for each sing<strong>le</strong> pluviometer from <strong>the</strong> nr data<br />
registered in it, <strong>the</strong> best estimate of s''{y} could be obtained<br />
by means of equation:<br />
93
94<br />
In <strong>the</strong> third phase, finally, by assuming that ot2{y}=<br />
= s'2(y) we proceed on to <strong>the</strong> proof of <strong>the</strong> working hypo<strong>the</strong>sis<br />
thus formulated, checking by means of equation (6) that <strong>the</strong><br />
sing<strong>le</strong> estimates differ from <strong>the</strong> sing<strong>le</strong> value with differences<br />
that could be attributed so<strong>le</strong>ly to sampling errors.<br />
Naturally, in this process we have supposed that data<br />
col<strong>le</strong>cted in each pluviometer of <strong>the</strong> zone are not correlated<br />
among <strong>the</strong>mselves [3].<br />
5: As an examp<strong>le</strong>, <strong>le</strong>t us refer to Morocco<br />
In fig. 1, <strong>the</strong> assumed working hypo<strong>the</strong>sis<br />
of <strong>the</strong> region in pluviometric zones is reported<br />
of <strong>the</strong> division<br />
In fig. 2, shows for some zones a statistical control test<br />
of <strong>the</strong> validity of <strong>the</strong> working hypo<strong>the</strong>sis.<br />
As it can be observed, <strong>the</strong> proof has been carried out by<br />
reporting on a diagram, whose ordinates represent <strong>the</strong> values of<br />
s2{y} and whose abscissas represent <strong>the</strong> number n of observation<br />
years :<br />
a) <strong>the</strong> estimate st2{y} that characterizes <strong>the</strong> zone;<br />
b) <strong>the</strong> range of confidence delimited by <strong>the</strong> two curves<br />
s: (n) and s', (n) corresponding to <strong>the</strong> said value of s'2{y} or,<br />
briefly, <strong>the</strong> confidence band of s2{y);<br />
of <strong>the</strong> zone.<br />
c) <strong>the</strong> point (n, s2{y}) corresponding to each pluviometer<br />
As it can be observed from <strong>the</strong> diagrams, as a proof of <strong>the</strong><br />
assumed working hypo<strong>the</strong>sis, <strong>the</strong> points lie within <strong>the</strong> confidence<br />
bands.
Adaptability of <strong>the</strong> Climatic Charts for <strong>the</strong> delimitations of<br />
Pluviometric Zones.<br />
6: In fig. 3 are reported, with different simbols, <strong>the</strong><br />
division of Morocco in pluviometric zones, as indicated<br />
previously, and <strong>the</strong> division in climatic zones as it deducted<br />
from <strong>the</strong> Meigs Chart [4].<br />
As it can be observed, if <strong>the</strong> arid zones corresponding to<br />
<strong>the</strong> Massif of Atlas, labe<strong>le</strong>d by <strong>the</strong> indez .(1), and <strong>the</strong> semiarid<br />
zone between Anti Atlas and Hamada du Dra, labe<strong>le</strong>d by <strong>the</strong> index<br />
(21, are excluded, a noticeab<strong>le</strong> correspondence exist between <strong>the</strong><br />
pluviometric zones and <strong>the</strong> climatic zones reported by Meigs.<br />
On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> disagreements mentioned previously<br />
could be easily explained if we consider that climatic charts<br />
are deduced by taking also into account geomorphology, <strong>the</strong> soil<br />
and <strong>the</strong> vegetation.<br />
In fact, for <strong>the</strong> zone (11, <strong>the</strong> lack of vegetation that has<br />
induced Meigs to define it arid, could be attributed not to fewer<br />
precipitations but to <strong>the</strong> presence of a calcareous massif that<br />
prevents <strong>the</strong> formation of a vegetative soil.<br />
On <strong>the</strong> o<strong>the</strong>r hand, for <strong>the</strong> zone (2) constituted by a large<br />
depression delimited by <strong>the</strong> Chain of <strong>the</strong> Massif of Atlas on one<br />
side and by Hamada du Dra on <strong>the</strong> o<strong>the</strong>r side, <strong>the</strong> presence of<br />
vegetation that has induced Meigs to define it as smiarid, could<br />
be attributed not to waters caused by rain that falls directly<br />
on <strong>the</strong> zone, but to waters that rush <strong>the</strong>re from nearby zones.<br />
Pluviometric zones of Bolivia and Saudi Arabia<br />
I: As it has been said in <strong>the</strong> previous paragraph 6, to<br />
formulate working hypo<strong>the</strong>sis regarding <strong>the</strong> delimitations of<br />
pluviometric zones for regions where only few pluviometers are<br />
functioning and, moreover , in most cases, functioning so<strong>le</strong>ly for<br />
a short observation period, it could be useful to use climatic<br />
charts.<br />
95
96<br />
Thus, to define <strong>the</strong> pluviometry of Bolivia and sone zones<br />
of Saudi Arabia, we assumed <strong>the</strong> working hypo<strong>the</strong>sis that <strong>the</strong><br />
pluviometric zones coincide with <strong>the</strong> climatic zones (figs. 4<br />
and 5).<br />
Particularly, for Bolivia we used <strong>the</strong> chart published by<br />
UNESCO for <strong>the</strong> arid and semiarid zones [4], and <strong>the</strong> Trewartha<br />
and Robinson chart for <strong>the</strong> humid and sub-humid zones [5]. For<br />
Saudi Arabia only <strong>the</strong> chart published by UNESCO was considered<br />
*<br />
phase :<br />
As is shown in <strong>the</strong> above mentioned figures, in <strong>the</strong> first<br />
Only pluviometers functioning for a long period of obser-<br />
vation have been considered;<br />
<strong>the</strong> estimates s2{y} have been deduced from <strong>the</strong> data<br />
col<strong>le</strong>cted in each zone;<br />
<strong>the</strong> estimates have been divided in groups and <strong>the</strong> location<br />
of each pluviometer has been labe<strong>le</strong>d with a different symbol<br />
according to where s2{y} lies.<br />
From <strong>the</strong> figures we observe:<br />
1) in passing from one to <strong>the</strong> o<strong>the</strong>r climatic zone, <strong>the</strong><br />
values of s2{y} lie in different groups;<br />
2) if in a zone more pluviometers are functioning, <strong>the</strong><br />
corresponding values of s'{y] lie, ei<strong>the</strong>r in <strong>the</strong> same or conti-<br />
guous groups.<br />
Consequently, by taking into account <strong>the</strong> definition given<br />
of <strong>the</strong> zones, it is safe to assume <strong>the</strong> hypo<strong>the</strong>sis that <strong>the</strong><br />
climatic zones coincide with <strong>the</strong> pluviometric zones even for <strong>the</strong><br />
said regions.<br />
Consequently, in <strong>the</strong> second phase of elaborations, still<br />
taking into account <strong>the</strong> data relative to pluviometers functioning<br />
for a long period of observation, <strong>the</strong> working hypo<strong>the</strong>sis for each
sing<strong>le</strong> zone has been formulated, by assuming as estimate st2{yl<br />
of <strong>the</strong> variation ot2{y} that characterizes <strong>the</strong> zone, <strong>the</strong> value<br />
deduced by means of equation (7).<br />
Finally, in<strong>the</strong> third phase, by using also <strong>the</strong> data fur-<br />
nished by <strong>the</strong> pluviometers functioning for a shorter observation<br />
period, to verify <strong>the</strong> working hypo<strong>the</strong>sis, it was checked that<br />
<strong>the</strong> deviations between, st2{y} and <strong>the</strong> value s2{y} deduced for<br />
each pluviometer could be attributed so,<strong>le</strong>ly to sampling errors.<br />
In both cases, from <strong>the</strong> few data availab<strong>le</strong>, <strong>the</strong> hypo<strong>the</strong>sis<br />
that <strong>the</strong> pluviometric zones coincide with <strong>the</strong> climatic zones is<br />
sufficiently ascerIained.<br />
Pluviometric Sub-zones<br />
8: As it has been stated by o<strong>the</strong>r authors [l] whenever <strong>the</strong><br />
estimates of M{h} deduced for each pluviometer from <strong>the</strong> data<br />
registered during <strong>the</strong> observation period, it has been possib<strong>le</strong><br />
to distinguish, in each zone, one or more sub-zones.<br />
In each of <strong>the</strong> said sub-zones, when passing from one point<br />
to ano<strong>the</strong>r, <strong>the</strong> values of <strong>the</strong> estimates fi ei<strong>the</strong>r show that:<br />
a) <strong>the</strong>y scatter around a sing<strong>le</strong> value M{h}, or that<br />
b) <strong>the</strong>y scatter around values of M(h3 that vary in function<br />
of ei<strong>the</strong>r one of <strong>the</strong> parameters which represent <strong>the</strong> morphology<br />
of <strong>the</strong> sub-zone (particularly, in <strong>the</strong> cases considered, <strong>the</strong> land<br />
e<strong>le</strong>vation (2)).<br />
In <strong>the</strong> first case, each sing<strong>le</strong> pluviometric sub-zone has<br />
been characterized by indicating <strong>the</strong> value fit taken by arithmetic<br />
average of fi corresponding to <strong>the</strong> sing<strong>le</strong> pluviometers. In <strong>the</strong><br />
second case, each individual sub-zone has been characterized by<br />
specifying <strong>the</strong> variation law of M{h} as function of z and by<br />
indicating <strong>the</strong> values 6' and that according to <strong>the</strong><br />
(1)<br />
(2)<br />
mentioned variation law correspond to <strong>the</strong> highest and lowest<br />
e<strong>le</strong>vation of <strong>the</strong> sub-zone pluviometers.<br />
97
98<br />
paper:<br />
In <strong>the</strong> fig. 6 are reported on a diagram on logarithmic<br />
a) <strong>the</strong> points (6, g'{h}) which represent <strong>the</strong> pluviometric<br />
sub-zones, for <strong>the</strong> first case;<br />
b) <strong>the</strong> intervals delimited by <strong>the</strong> points (hl (1) Y g'(h1)<br />
and (E1(*), g'{h]) for <strong>the</strong> second case.<br />
As it hast been found by o<strong>the</strong>r authors (61, when passing<br />
from one region to ano<strong>the</strong>r, and for each region from one pluvio~<br />
etric zone to ano<strong>the</strong>r, <strong>the</strong> variability increases as <strong>the</strong> average<br />
annual rainfall decreases.<br />
Instead, as it has been said previously, in each sing<strong>le</strong><br />
pluviometric zone <strong>the</strong> variability expressed by means of ylh} or<br />
u2{y} is comp<strong>le</strong>tely independent from an eventual variability of<br />
<strong>the</strong> average rainfall.
RE FE RCN C ES<br />
[l] VIPARELLI C. : "Idrologia applicata all'ingegneria".<br />
Parte II Fondazione Politecnica del<br />
Mezzogiorno d'Italia, Napoli (1965).<br />
[2] . MARKOVIC R.D. : "Probability Functions of best fit to<br />
Distributions of annual Precipitation<br />
and Runoff".<br />
Hydrology papers, Colorado State Univer-<br />
sity Fort Collins, Colorado (Aug. 1965).<br />
[3] PENTA A., ROSSI F.: "Objective Criteria to declare a<br />
Series of Data sufficient for technical<br />
Purposesff.<br />
Simposio sobre proyectos de recursos hi-<br />
dráulicos con datos insuficientes. Ma-<br />
drid (1973).<br />
[4] CHOW W.T.: "Hand<strong>book</strong> of Applied Hydrology".<br />
Mc Gram-Hill Book Company. Pg. 24-3 e seg.<br />
(1964).<br />
[5] TREWARTHA G.T.; ROBINSON A.H. , HAMMOND E.H.: "E<strong>le</strong>ments of<br />
Ge o graph y It .<br />
Mc Gram-Hill. Book Company (1967).<br />
[6] HAZEN, ALLEN,: "Variation in annual rainfall".<br />
Eng. News, vol. 75 n. 1 (1916).<br />
The pluviometric data used in this paper have been taken from:<br />
- Institut Scientifique Chérifien du Maroc.<br />
- Servicio Nacional de Meteorología e Hidrología de Bolivia.<br />
- Empresa Nacional de E<strong>le</strong>ctricidad de Bolivia.<br />
- Ministery of Agriculture and Water of Saudi Arabian Kingdon.<br />
99
I<br />
l<br />
i<br />
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l<br />
PLWIO?,ETRIC ZONES AND TIE CRITWIA TO DEYIN2 TRTIR EOiZIDARJ2S ?CI!?<br />
ZEGIONS ::'ITH SCARCE DATA<br />
by<br />
Garcia-Agreda R., Rasulo G., Viparel?.i 9.<br />
Fig. 2<br />
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9<br />
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Garcia-Agreda R., Rasulo G., Viparelli R.<br />
PLLT'IOXIX'RIC ZOPJ'XS AND TFIE CXITWIA TO DEFINE THEIR BOUNDARIES FOR<br />
EEGIONS WITH SCARCE DAIA<br />
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Fig. 4<br />
FLLWIO?3TTiIC ZONES hi TIE3 CRITERIA TO DEFINE THEIR BOUNDARIES FOR<br />
REGIONS WITH SCARCE DATA<br />
Garcia-Agreda R., Resulo G., Viparelli R.<br />
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PLLVIOEETSIC zoms urn THE CRITERIA TO DEFINE THEIR BOUNDARIES BOR<br />
REGIONS WITH SCARCE DATA<br />
O
ABSTRACT<br />
ESTIMATION DES ETIAGES DE BASSINS NON EQUIPES<br />
par G.R. OBERLINB, G.C. GALEA* et J.T. TONI**<br />
The first data col<strong>le</strong>cted on <strong>the</strong> different creeks of <strong>the</strong> Or-<br />
geval representative watershed (104 km2), showed a great disparity<br />
between low water specific discharges. This disparity subsisted<br />
after adjustement of man influences (such as pumping and throws).<br />
The differences contrasted with <strong>the</strong> visib<strong>le</strong> simplicity and homo-<br />
geneity of <strong>the</strong> watershed surface and with <strong>the</strong> supposed favourab<strong>le</strong><br />
outline of <strong>the</strong> ground water catchment. That is to say, even with<br />
<strong>the</strong>se propitious conditions in appraising <strong>the</strong> hydrological charac-<br />
teristics (good network and prob<strong>le</strong>m seemingly easy), <strong>the</strong> represen-<br />
tativity, i.e. <strong>the</strong> extrapolation of <strong>the</strong> results to similar<br />
neeghbouring creeks, was found at fault. To resolve this point<br />
without additive equipment, gauging rounds were undertaken during<br />
low water seasons on a great number of creeks. The comparison of<br />
<strong>the</strong>se groups of instantaneous discharges, measured on <strong>the</strong> same day<br />
at different places, ser off <strong>the</strong> behaviour of each watershed. In<br />
some cases, <strong>the</strong> analyses of <strong>the</strong>se observed differences allowed <strong>the</strong><br />
elaboration of general ru<strong>le</strong>s and <strong>le</strong>d to pratica1 conclusiones, of-<br />
ten quantitative. As some sections of <strong>the</strong> measured creeks belonged<br />
to permanent network, and some o<strong>the</strong>r conditions having been satis-<br />
fied (especially: a great enough number of measurements during<br />
each low water season), <strong>the</strong> unknown characteristics of <strong>the</strong> unex-<br />
plored creeks have been evaluated from <strong>the</strong>se of <strong>the</strong> permanent<br />
stations.<br />
RESUME<br />
Les premieres mesures effectuées sur <strong>le</strong>s divers sous-bassins<br />
constituant <strong>le</strong> bassin représentatif de l'0rgeval (104 km'), ont<br />
fait entrevoir de trss importantes différences de débits spécifi-<br />
ques d'étiage. Cel<strong>le</strong>s-ci subsistaient après correction des in-<br />
fluences humaines (pompages et rejets). Ces différences contras-<br />
taient avec la simplicité et l'homogénéité apparentes du bassin<br />
en matière de caractéristiques physiques et avec 1 'aspect favora-<br />
b<strong>le</strong> de son hydrogéologie. Autrement dit, même dans ces conditions<br />
optima<strong>le</strong>s d'estimation de caractéristiques hydrologiques (bon<br />
équipement de mesure et prob<strong>le</strong>me a priori simp<strong>le</strong>), la représenta-<br />
tivité, c'est-à-dire l'extrapolation de résultats aux bassins<br />
voisins et semblab<strong>le</strong>s, était mise en échec. Pour résoudre <strong>le</strong> pro-<br />
b<strong>le</strong>me sans équipements nouveaux, des campagnes de mesures volan-<br />
tes d'étiage ont été réalisées sur un certain nombre de cours<br />
d'eau. La comparaison de ces ensemb<strong>le</strong>s de débits instantanés, me-<br />
surés simultanément en divers lieux, a précisé <strong>le</strong>s différences de<br />
comportement des bassins. Dans certains cas, l'analyse de ces dif-<br />
férences a pu suivre des règ<strong>le</strong>s généra<strong>le</strong>s et conduire à des con-<br />
clusions dont certaines étaient quantifiab<strong>le</strong>s. Comme plusieurs<br />
stations permanentes (équipées) étaient incluses dans ces cam-<br />
pagnes, et que certaines conditionsavaient été satisfaites (en<br />
particulier: un .?ombre suffisant de jaugeages au cours d'une<br />
même saison d'étiage), <strong>le</strong>s caractéristiques inconnues des bassins<br />
non équipés ont alors pu être estimées.<br />
$; C.T.G.R.E.F., Parc de Tourvoie, F - 92160 ANTONY.<br />
*J. ,. ,. Direction Départementa<strong>le</strong> de l'Agriculture. PARAKOU (Dahomey)
104<br />
INTRODUCTION<br />
Dans l'étude des basses eaux, en hydrologie, on se heurte toujours<br />
à une première difficulté concernant la qualité des données de débits de basses<br />
eaux. Cette difficulté résiste remarquab<strong>le</strong>ment bien aux dliorations apportées<br />
au fonctionnement des stations hydrométriques, même bien équipées. Sur <strong>le</strong>s<br />
bassins versants d' investigation de l'orgeval, par exemp<strong>le</strong> (surfaces variant<br />
de 7 ?i 104 km2), malgré d'efficaces et importants travaux (11, 18 procédure<br />
habituel<strong>le</strong> de mesure des hauteurs d'eau, puis de traduction nhauteur - débit"<br />
n'est pas toujours performante en très basses eaux.<br />
Une solution très généra<strong>le</strong>ment utilisée consiste alors à effectuer un<br />
grand nombre de mesures instantanées de débit (Jaugeages) et h Interpo<strong>le</strong>r ent-e<br />
ces mesures toutes <strong>le</strong>s fois OU cela est possib<strong>le</strong>, c'est-à-dire lorsque la dé-<br />
crue n'est pas influencée par une crue, si minime solt-el<strong>le</strong>. Dans cette procédure,<br />
l'équipement de la station hydrométrique (échel<strong>le</strong>; iimnigraphe. etc.. . )<br />
n'est guère utilisé, sinon de façon qualitative fi]. Par généralisation de la<br />
méthode on peut envisager de réaliser ces campagnes de jaugeages sur des cours<br />
d'eau réel<strong>le</strong>ment non équipés et en espérer des résultats autres que ponctuels.<br />
1. RESEAU DE MESURES EPISODIQUES D'ETIAGES<br />
Les premières mesures conventionneiìes effectuées en lgtj2 SUT <strong>le</strong>s 4<br />
stations équipées du bassin de l'orgeval (Minj stère de l'Agriculture, France)<br />
s'étaient évidemment heurtées awc difficultés hydrométriques citées plus<br />
haut. De plus, el<strong>le</strong>s avaient décelé de très grandes différences dans <strong>le</strong>s dé-<br />
bits spécifiques [g, I1 était diffici<strong>le</strong> de savoir si ces différences avaient<br />
une origine hydroaéologique (hétérogénéité dans la répartition des réservoirs<br />
souterrains) ou étaient simp<strong>le</strong>ment dues aux aléas des Jaugeages, voire des<br />
influences humalnes (pompages, re Jets, retenues, etc.. . ). A priori, l'excep-<br />
tionnel<strong>le</strong> homogénéité du bassin en matière de géologie et de formationsde svr-<br />
face (51,Fi~. la) &ait en contradiction avec la premièrc hypothèse. Néan-<br />
moins, <strong>le</strong>s mesures de type extensives mentionnées dens l'introduction ont<br />
été commencées dès 1963.<br />
La liste des stations concernées par ces mesures est donnée dans <strong>le</strong><br />
tab<strong>le</strong>au 1 et la Fig. lb présente <strong>le</strong>ur répartition sur la surface du bassin.
2. 8EMERS RESULTATS ISSUS DE SINPLES CORRELATlONS INTER-STATIONS<br />
2.1. Préliminaire : Dans tout ce qui suit, nous appel<strong>le</strong>rons débits d'étiages<br />
tout débit provenant du drainage d'un réservoir souterrain (même proche du<br />
sol), à l'exclusion dé tout ruissel<strong>le</strong>ment (de surface, direct, retardé,<br />
105<br />
etc...), et de tout écou<strong>le</strong>ment "hypodermique". Nous faisons l'hypothèse que<br />
nous sommes dans ces conditions lorsque la dernière crue est distante de<br />
plusieurs jours (crues estiva<strong>le</strong>s tr&s modestes), ou de Plusieurs semaines<br />
(crues moyennes et fortes), gour des bassins de 10 à 100 Km?: Ne connais-<br />
sant pas la repré;.ntativité dans <strong>le</strong> temps de ces jaugeages instantanés,<br />
nous corrélons deux h deu <strong>le</strong>c jaugeages de même date. Pour simplifier, nous<br />
n'avons pasétudié toutes <strong>le</strong>s combinaisons 2 à 2 réalisab<strong>le</strong>s avec <strong>le</strong> groupe<br />
des 9 ou 10 stations. Enfin, nous n'avons pas corrélé <strong>le</strong>s débits spécifiques<br />
mais <strong>le</strong>s débits absolus. Les grandes différences de comportement des divers<br />
bassins montrent en effet que la notion de surface du bassin superficiel<br />
nia sans doute pas grand chose h voir avec <strong>le</strong>s dimensions des réservoirs<br />
souterrains générateurs des débits d'étiage.<br />
2.2. Rksultats : En général, sur tous <strong>le</strong>s graphiques construits (environ 20), on<br />
observe une dispersior. assez forte (Fig. 22). El<strong>le</strong> est même très forte sur<br />
toute corrélation concernant Mélarchez. El<strong>le</strong> n'est acceptab<strong>le</strong> qu'à l'inté-<br />
rieur du groupe des 3 stations aval du ru des Avenel<strong>le</strong>s (Gouge, Avenel<strong>le</strong>s,<br />
Theil) et de la station du Croupet.<br />
Même en faisant abstraction de la dispersion propre aux erreurs<br />
de mesure (<strong>le</strong>s jaugeages d'étiage sont délicats et peu précis), l'ensemb<strong>le</strong><br />
des points reste très dispersé.<br />
ia première conclusion à en tirer est ia suivante : <strong>le</strong>s conditions<br />
d'alimentation des différents réservoirs souterrains (à l'origine des débits<br />
d'étiage) ne sont pas homogènes sur l'ensemb<strong>le</strong> du bassin, malgrdla tail<strong>le</strong><br />
-<br />
réduite (100 Km2) de ceiui-ci.<br />
En regardant de plus près on constate que, pour une année donnée,<br />
la dispersion est moins grande, et parfois même faib<strong>le</strong>. D'ou la seconde<br />
conclusion : pour une période d'étiage continue donnée (un été), <strong>le</strong>s c oa-<br />
tions d'alimentation (l'hiver prbcédent) se réve<strong>le</strong>nt stab<strong>le</strong>s dans <strong>le</strong> temps.<br />
Pour la plupart des bassins, cette dernière conclusion doit cepen-<br />
dant être nuancée quand on prend en compte <strong>le</strong>s basses eaux tardives (zutorne).<br />
Ces dernières sont d6jà influencées par <strong>le</strong>s premières pluies d'hiver (<strong>le</strong>s
106<br />
débits remontent, ou bien <strong>le</strong>ur baisse diminue ou s'annu<strong>le</strong>), et <strong>le</strong>s corré-<br />
lations montrent des comportements très différenciés selon <strong>le</strong>s bassins :<br />
<strong>le</strong>s points correspondant à des dates tardives (Octobre &. Décembre) sont<br />
souvent rassemblés d'un Seul côté du nuage de points, pour une année donnée.<br />
Nous dirons que <strong>le</strong>s bassins qui "profitent'' rapidement des premières pluies<br />
d'hiver ont une alimentation plus superficiel<strong>le</strong> que <strong>le</strong>s autres. D'où la troi-<br />
sième conclusion : <strong>le</strong>s bassins amont ont une alimentation ñettement plus super.<br />
ficieìïe que <strong>le</strong>s autres (résultat classique) ; ce caractère supérficieì est<br />
surtout marqué au-dessous de i5 km2 (exutoire situé au-dessus des argi<strong>le</strong>s<br />
vertes) ; surface éga<strong>le</strong>, <strong>le</strong> ru du Rognon est plus "superficiel" que <strong>le</strong> ru<br />
des Avenel<strong>le</strong>s : <strong>le</strong> ru de Bourgogne semb<strong>le</strong> être intermédiaire entre <strong>le</strong>s deux,<br />
mais la probabilité d'une assez forte rétention de surface (forêt) rend cette<br />
conclusion aléatoire.<br />
2.3. Aspect méthodologique : Des observations précédentes nous déduisons un graphiqi<br />
caractéristique (Fig. 23) d'une corrélation entre <strong>le</strong>s étiages instantanés de<br />
deux bassins voisins, mais à système hydrogéologique (d'alimentation d'étiage)<br />
différencié. Il y a une forte dispersion globa<strong>le</strong>, mais l'évolution est cohé-<br />
rente à l'intérieur d'une année donnée (i ou j ou k).<br />
3. CORRELRTION INTER-STATION PAFI WUBLES CUMUL5<br />
3.1. Résultats qualitatifs : Etant donné la forte dispersion des corrélations to-<br />
ta<strong>le</strong>s 2à 2, il était diffici<strong>le</strong> d'en tirer des conclusions sur <strong>le</strong>s abondances<br />
relatives des bassins corrélés. Les courbes de doub<strong>le</strong>s cumuls ont donc été<br />
tracées. El<strong>le</strong>s confirment d'abord <strong>le</strong>s conclusions précédentes : hétérog6néité<br />
non négligeab<strong>le</strong> d'une année à l'autre, bonne homogénéité à l'intérieur d'une<br />
année avec courbure, caractéristique de 1' influence des premières pluies Aiver<br />
Néanmoins, une bonne tendance se dessine sur la plupart des graphiques et per-<br />
met de tirer des conclusions sur l'abondance relative des étiages. D'ou la<br />
quatrième conclusion, en raisonnant en débit spécifique (k surface éga<strong>le</strong>) :<br />
<strong>le</strong> ru des Avenel<strong>le</strong>s est nettement plus abondant que <strong>le</strong> ru du Rognon ; <strong>le</strong> ru de<br />
Bourgogne est <strong>le</strong>gèrement plus abondant que <strong>le</strong> ru du Rognon : dans <strong>le</strong> bassin du<br />
Rognon, <strong>le</strong> ru du Petit Courcy (ferme P<strong>le</strong>ssier) est nettement plus abondant w e<br />
<strong>le</strong> haut Rcgnon ; dins <strong>le</strong> bassin des \venel<strong>le</strong>s, c'est <strong>le</strong> ru de 1'Etang qui<br />
apporte l'essentiel des étiages par rapport un bassin de Mélarchez insignifi;
107<br />
On constate une quasi identité entre <strong>le</strong>s bassins d'étiage faib<strong>le</strong><br />
et ceux qui profitent rapidement des premières pluies d'hiver (rus "super-<br />
ficiels"). I1 en est be même entre ceux b. étiage pur (coeur de l'été) abon-<br />
- dant et ceux dont <strong>le</strong>s eaux restent basses tardivement.<br />
A noter, enfin, que <strong>le</strong> caractère de "superficialité" attribué aux<br />
bassins de Mélarchez et de Pierre Levée et déduit d'une-réponse rapide aux<br />
premières pluies d'hiver, se décè<strong>le</strong> éga<strong>le</strong>ment pour la partie, aval du bassin.<br />
Ce sont <strong>le</strong>s pentes plus fortes dominant l'extrêmité aval des thalwegs (en<br />
particulier au droit de la station des Avenel<strong>le</strong>s et du Champ de Tir) qui<br />
sont probab<strong>le</strong>ment à l'origine de cette (faib<strong>le</strong>) croissance relative des<br />
étiages de fin d'été.<br />
3.2. Aspect méthoddogique : La courbe type d'une corrélation par doub<strong>le</strong> cumul<br />
entre deux bassins voisins a été représentée sur la Fig. 32. On a bien<br />
entendu fait l'hypothèse de l'existence de différences classiques entre<br />
i- i-<br />
<strong>le</strong>s nappes alimentant <strong>le</strong>s étiages (- abondantes, - superficiel<strong>le</strong>s, etc...).<br />
Les caractéristiques d'abondance ont +té mises entre parenthèses<br />
sur la Fig. 32 car la liaison "abondance-profondeur (des nappes)" observée<br />
sur l'ûrgeval, n'est pas généra<strong>le</strong>, même si el<strong>le</strong> est fréquente.<br />
4. ESSAI DE COMPAXAISON QUAhTITATIVE ENTRE BASSINS<br />
Dans <strong>le</strong>s tab<strong>le</strong>aux qui suivent nous avons essayé de consigner, sous<br />
forme condensée et parfois numérique, <strong>le</strong>s conclusions présentées auparavant.<br />
Chaque tab<strong>le</strong>au se rapporte à une des 4 stations principa<strong>le</strong>s du bassin. Les<br />
diverses caractéristiques d'étiage déterminées sur ces 4 bassins principaux<br />
fi] fi] fi] (1/ pourront ainsi être approximativement transformées pour<br />
s'adapter à tel ou tel bassin non observé de manière continue ($9 8 & 9).<br />
Le rapport d'abondance moyenne K noté en colonne (4) est une estimation de la<br />
pente moyenne de la courbe des doub<strong>le</strong>s cumuls (9 3), éventuel<strong>le</strong>ment affranchie<br />
des anomalies de cette courbe. Pour souligner <strong>le</strong>s différences de comporte-<br />
ment des bassins, ce rapport est calculé avec <strong>le</strong>s débits spécifiques.<br />
Ce coefficient K est donc (aux rapports des surfaces près) <strong>le</strong> rap-<br />
port entre <strong>le</strong>s moyennes des mesures d'étiages épisodiques effectuées en 2<br />
stations. ia signification statistique de ces moyennes est a priori tout à<br />
fait particulière ; on verra au § 32 qu'el<strong>le</strong> peut être rattachée à une carac-<br />
téristique généra<strong>le</strong>,
108<br />
4.1. Etiages instantanés comparés à cem de la station équipée de Mélarchez :<br />
--------_-e============;<br />
- ----- ~ -___<br />
Car adere<br />
Cours d'eau Station<br />
(1)<br />
Avenel<strong>le</strong>s<br />
(Fosse Rognon) Mélarchez<br />
.---_I-----_--_..----------<br />
Etang. ........ Croupet<br />
Petit Couroy.. Bibartault<br />
Rognon.. ...... Pierre k v<br />
ßourgogne.. ... Ch. de Tir<br />
Rognon. .......<br />
Avenel<strong>le</strong>s<br />
Fosse Rognon 1.<br />
(2 1<br />
Bibartault<br />
Gouge<br />
Rognon ........ Ch. de.Tir<br />
Avenel<strong>le</strong>s..... Avenel<strong>le</strong>s<br />
Orgeval.. ..... Theil<br />
Etang. ........ Croupet<br />
Rognon.. ...... Ch. de Tir 43,4<br />
Avenel<strong>le</strong>s..... Avenel<strong>le</strong>s 45,7<br />
Orgeval ...... Theil 104<br />
0,3 à 0,4 í (<br />
0,6 (<br />
superficiel<br />
des nappes<br />
(5 1<br />
-<br />
-------..--_-<br />
assez<br />
prof ondes<br />
II II<br />
superfiddks<br />
assez prof.<br />
It 11<br />
% par rapport au bassin de référenc5 Tab<strong>le</strong>au 41<br />
superficiel<br />
un peu +<br />
superficiel<br />
pius<br />
:orréiation<br />
;res mauvaise<br />
4.2. Etiages instantanés comparés à ceux de la station équipée de la Gouge :<br />
Nous n'avons étudié que <strong>le</strong>s bassins de surface pas trop petite par rapport<br />
à cel<strong>le</strong> de la Gouge, pour ne pas introduire de trop grosses différences.<br />
Cours d'eau I Station I I 'Kf=<br />
1<br />
._______r____<br />
Caractere<br />
superficiel Remarques<br />
des nappes"<br />
(5) (6)<br />
Avenel<strong>le</strong>s<br />
f par rapport au bassin de référence. Tab<strong>le</strong>au 42<br />
1<br />
)corrélation<br />
)ves bonne<br />
1
_______-___ _--_-_-----<br />
_-_____-I__ --I---<br />
-----<br />
Cours d'eau Station<br />
(1) (2 1<br />
Avenel<strong>le</strong>s.. Avenel<strong>le</strong>s<br />
_______-__--..-----------<br />
Rognon ..... Ch. de Tir<br />
Orgeval ... Theil<br />
-____________ ---<br />
Grgeval ..., Theil<br />
_________________-------<br />
. Ch. de Tir<br />
Rognon . ___________________----<br />
__ - _______ -_- --<br />
109<br />
4.3. Etiages comparés à ceux des stations équipées des Avenel<strong>le</strong>s et du Theil :<br />
5. INFLUENCE DES UTILISATIONS HUMAINES DE L'EAU<br />
K"<br />
superficiel<br />
des nappes*<br />
Remarques<br />
(4) (5) (6)<br />
plus<br />
Os4 superficiel<br />
0,75 semblab<strong>le</strong><br />
Ces observations portent sur 8 années et, pour chaque été, sur des pé-<br />
riodes de plusieurs mois, il nous semb<strong>le</strong> que, sauf exception (cf. ci-dessous),<br />
<strong>le</strong>s influences humaines sur <strong>le</strong>s étiages (pompages, retenues d'eau, etc ...)<br />
ne peuvent pas avoir faussé <strong>le</strong>s conclusions précédentes. De par <strong>le</strong>ur irrégu-<br />
larité (<strong>le</strong>s équipements se modifient, <strong>le</strong>s lieux de pompage se déplacent, <strong>le</strong>s<br />
volumes pré<strong>le</strong>vés ou rejetés sont très irréguliers dans <strong>le</strong> temps) el<strong>le</strong>s sont<br />
partiel<strong>le</strong>ment & l'origine de la forte dispersion mentionnée au début du 5 2,<br />
mais nous avons veillé à ne tirer des conclusions que sur <strong>le</strong>s tendances,<br />
affranchies des irrégularités loca<strong>le</strong>s et instantanées. I1 faut noter jci que<br />
cette étude avait déjà été envisagée en 1966 avec <strong>le</strong>s mesures réalisées<br />
cette date. Etant donné la dispersion observke, <strong>le</strong>s résultats avaient<br />
tel<strong>le</strong>ment décourageants que ces cûmpacnes de jaugeages épisodiques ont failli<br />
être abandonnées. I1 n'a donc pas fallu moins de 8 ans pour arriver à u'a.f'-<br />
franchjr de ces variabilités loca<strong>le</strong>s et percevoir <strong>le</strong>s tendances.<br />
Des estirriations rapides sur <strong>le</strong>s rejets possib<strong>le</strong>s de la commune de DOUE<br />
(alimentée depuis ].'extérieur du bassin) dans <strong>le</strong> ru de l'Etang, ou sur <strong>le</strong>s<br />
pompages de COUIOhPlIEK3 dans <strong>le</strong> PU du Rognon (en amont du Champ de Tir), ont<br />
abouti & des influences mrixima<strong>le</strong>s de quelques 5, sauf pour <strong>le</strong> ru clii Ro:--i?n<br />
au Champ de Tir dans <strong>le</strong> bassin duquel 11 1/s sont captés en quasi-permanence<br />
par la vil<strong>le</strong> de COULOMMIERS. Ces influences ont été ndgligées dans la<br />
été
11 o<br />
6.<br />
présente étude qui fournit simp<strong>le</strong>ment des ordres de grandeur pour <strong>le</strong>s<br />
comparaisons entre bassins dans <strong>le</strong>ur stade actuel de fonctionnement.<br />
Une autre influence peut-être non négligeab<strong>le</strong> concerne <strong>le</strong> captage<br />
d'une partie des eaux de Pierre Levée, en aval immédiat de la station,<br />
par un puits qui ne devrait d'ail<strong>le</strong>urs servir que pour écrêter <strong>le</strong>s hautes<br />
eaux, c'est-à-dire à partir d'un certain débit, largement supérieur a m<br />
basses eaux concernées dans cette étude. Ceci pourrait cxpliquer la fai-<br />
b<strong>le</strong>sse du ru du Rognon à Bibartault (§ 31). I1 n'est pas posiib<strong>le</strong> actuel-<br />
<strong>le</strong>ment d'estimer <strong>le</strong>s d6bit.s ainsi dérivés et donc de corriger <strong>le</strong>s mesures<br />
faites à Pierre Levée.<br />
Pour <strong>le</strong> ru du Rognon au Champ de Tir, compte tenu de ce que nous ne<br />
savons pas où iraient <strong>le</strong>s 11. l/s captés s'ils étaient libres de s'écou<strong>le</strong>r<br />
naturel<strong>le</strong>ment, nous préférons travail<strong>le</strong>r sur <strong>le</strong>s débits observés.<br />
VARIATIONS DES ETIAGES D'AMONT EN AVAL<br />
Du fait de l'organisation propre d'un réseau hydrographique, <strong>le</strong>quel<br />
est constitué de divers tronçons réunis en des confluents, <strong>le</strong>s surfaces<br />
contrôlées par un ru d'amont en aval subissent des discontinuités (conflu-<br />
ent) qui rendent délicates <strong>le</strong>s études de variation des caractéristiques<br />
hydrologiques d'amont en aval. Ceci est particulièrement vrai lorsque, à<br />
un confluent, se réunissent deux tronçons de caractéristiques très diffé-<br />
rentes. En toute rigueur, il faudrait faire apparaitre ces discontinuités<br />
dans <strong>le</strong>s résultats.<br />
En première approximation, nous négligeons ces nuances et <strong>le</strong>s Fig. 6a<br />
et Gb qui suivent présentent un ordre de grandeur de la variation des débits<br />
d'étiage d'amont en aval sur <strong>le</strong>s deux principaux rus du bassin : Avenel<strong>le</strong>s<br />
et Rognon. Ces courbes ne donnent qu'une indication de la fourchette obser-<br />
vée sur <strong>le</strong>s 8 ans d'observation. La courbe centra<strong>le</strong> (point M) n'est pas une<br />
courbe de vraies moyennes statistiques, <strong>le</strong>squel<strong>le</strong>s ne peuvent être calcu-<br />
lées étant donné la distribution anarchique des dates de jauseaye sur ces<br />
e ans. D'autre part, pour une date donnée, l'ensemb<strong>le</strong> des débits observés<br />
d'amont en aval ne. forme pas nécessairement une courbe "parallè<strong>le</strong>" aux<br />
limites ou B la courbe centra<strong>le</strong> esquissés.<br />
Le caractère plus redressé des courbes du ru du i?ognon s'explique par<br />
<strong>le</strong> fait que, en allant de l'amont vers l'aval, il rencontre des rus pr-ogres-<br />
sivement plus abondants ($ 4 1). Ln situation est ir.versée porir <strong>le</strong><br />
-
u des Avenel<strong>le</strong>s à partir du confluent "ru de 1' Etang - ru de<br />
Fosse Rognon".<br />
111<br />
Les figures 6a et 6b peuvent servir à estimer des moyennes fictives de<br />
débits d'étiages mesur6,s en des stations non observées, ce qui permet d'es-<br />
timer <strong>le</strong>urs rapports K ($ 4).<br />
7. ASPECTS HYDROGEOIDCIQüES<br />
L'abondance des débits du ru des Avenel<strong>le</strong>s à la Gouge était expliquée,<br />
Jucqu'à présent, par <strong>le</strong> fait que, situé au-dessous d'un aff<strong>le</strong>drement d'argi<strong>le</strong>s<br />
vertes, ce cours d'eau récupérait l'essentiel des infiltrations stoppées par<br />
cet horizon imperméab<strong>le</strong> des argi<strong>le</strong>s. Or, nous voyons que <strong>le</strong>s débits sont<br />
déjà importants un peu au-dessus de cet aff<strong>le</strong>urement, sur <strong>le</strong> ru de 1'Etang<br />
à Croupet par exemp<strong>le</strong>.<br />
L'examen de la carte géologique (Fig. la) montrait d'autre part une<br />
forte présence de sab<strong>le</strong>s de Fontaineb<strong>le</strong>au dans la partie aval du ru del'Etan@:<br />
(Ouest et Nord de la butte de DOUE).<br />
De même, ce sab<strong>le</strong> (par ail<strong>le</strong>urs présent çà et là dans tous <strong>le</strong>s limons de<br />
Brie) serait éga<strong>le</strong>ment plus abondant à l'amont du ru du Petit Couroy. Or<br />
dernier est, après <strong>le</strong> ru de l'Etang, <strong>le</strong> second "chateau d'eau'' pour <strong>le</strong>s<br />
étiages du bassin. Ta station de Bibartault est, par ail<strong>le</strong>urs, sitde net-<br />
tement au-dessus de l'aff <strong>le</strong>urement des argi<strong>le</strong>s vertes.<br />
ia liaison entre sab<strong>le</strong> et étiages paraissait à envisager et nous avan-<br />
cions l'hypoth&se que <strong>le</strong>s étiages du bassin de l'0rgeval étaient moins <strong>le</strong><br />
résultat d'un drainage des limons localisé immédiatement au-dessus des argi<strong>le</strong>s<br />
vertes, que <strong>le</strong> résultat du drainage des nappes éparses qui peuvent être loca-<br />
lisdes dans toute l'épaisseur des limons, mais qui sont simp<strong>le</strong>ment plus abon-<br />
dantes dans <strong>le</strong>s zones oÙ <strong>le</strong> sab<strong>le</strong> est plus fréquent.<br />
Quant aux étiages assez abondants du ru de Bourgogne, il pourrait s'agir<br />
d'une influence bénéfique de la forêt sur <strong>le</strong> volume des étiages, résultat qui<br />
commence a etre admis un peu partout (pour la zone tempérée), malgré <strong>le</strong>s nom-<br />
breuses controverses toujours en cours sur ce suJet.<br />
Depuis peu, une campagne geophysique de sondages é<strong>le</strong>ctriques a montré<br />
qu'il n'y avait guère de <strong>le</strong>ntil<strong>le</strong>s de sab<strong>le</strong>, mais des <strong>le</strong>ntil<strong>le</strong>s de calcaire<br />
et meulière de 3rie. La signification géologique change, mais <strong>le</strong> rgsultat est<br />
quasiment <strong>le</strong> même pour 1'hyclrol.ogue. Une campagne de mesures épisodiques ue<br />
niveaux de puits a d'ail<strong>le</strong>urs confirms ces hypo<strong>the</strong>ses.<br />
ce
11 2<br />
Dans tout ceci il faut noter l'extraordinaire différence de comporte-<br />
ment en étiage de bassins qui, en l'absence de ces mesures de débits<br />
partiel<strong>le</strong>s et épisodiques, étaient communément considérés comme remarquab<strong>le</strong>-<br />
ment (voire exceptiotuîeììement ) homogènes. Ceci est un avertissement sérieux<br />
qui doit rendre extrêmement circonspect dans toute interpolation ou extrapo-<br />
lation de résultats à une échel<strong>le</strong> régiona<strong>le</strong>.<br />
8. CARACTEXiISTIQUES D'ETIAGFS DES BASSINS NON EQUIPES<br />
I1 reste à présent à utiliser <strong>le</strong>s résultats ci-dessus pour obtenir<br />
quelques caractéristiques d'étiages sur <strong>le</strong>s bassins secondaires non contrô-<br />
lés en permanence. Ceci suppose d'avoir auparavant élaboré <strong>le</strong>s caractéris-<br />
tiques correspondantes des bassins de référence, équipés.<br />
8.1. Résultats observés sur <strong>le</strong>s bassins équipés :<br />
8.1.1. MéLhgdgs-:<br />
L'étude de la forme des courbes .de tarissement est décevante, ce<br />
qui n'est guère étonnant quand on considère la petite tail<strong>le</strong> des bassins<br />
et l'hétkrogénéité des aquifères d'alimentation : ensemb<strong>le</strong> de nappes<br />
plus ou moins superficiel<strong>le</strong>s et loca<strong>le</strong>s irrégulièrement distribuées dans<br />
l'espace. I1 n'est donc pas possib<strong>le</strong> d'estimer <strong>le</strong>s volumes emmagasinés<br />
avec une précision acceptab<strong>le</strong>.<br />
L'étude des étiages à l'échel<strong>le</strong> de temps du mois civil n'est pas non<br />
PlUS très intéressante, étant donné l'inexistance d'une véritab<strong>le</strong> saison<br />
sèche piuviométrique : ï'ûrgevaï est soumis à un climat où <strong>le</strong>s pluies<br />
d'été sont aussi nombreuses et importantes que cel<strong>le</strong>s d'hiver. Certes,<br />
la fonction de rendement (coefficient d'écou<strong>le</strong>ment) est très basse en<br />
été mais, s'agissant de petits bassins, l'influence de ces petites crues<br />
d'été est fondamenta<strong>le</strong> sur <strong>le</strong>s débits. I1 en résulte que <strong>le</strong>s périodes<br />
d'étiage peu supérieures à 5 ou 10 jours sont relativement rares ; et<br />
cel<strong>le</strong>s de 3 jours consécutifs que l'on peut rencontrer sont toujours 2<br />
cheval sur 2 mois civils.<br />
Les seu<strong>le</strong>s caractéristiques intéressantes sont cel<strong>le</strong>s qui s'expriment<br />
en fonction des débits journaliers. Les plus connus snnt <strong>le</strong>s débits classés,<br />
notés Dc et <strong>le</strong>s minimums de débits moyens sur N jours notés Vcn<br />
n<br />
(N = 355 - n). Sur l'Orgeva1, nous avons. pris l'habitude fi] d'y ajouter<br />
un troisibme type, not8 QCn et appelé "débit caractéristique de période
continue". BI étiages la définition de ces QCn est la suivante :<br />
113<br />
QC d'une année est, <strong>le</strong> minimum des débits Journaliers maximums des<br />
n<br />
périodes de N jours consécutifs (N = 365 - n). Cette définition, un peu<br />
comp<strong>le</strong>xe, recouvre'en fait une caractéristique de type "seuil", faci<strong>le</strong><br />
à déterminer sur un graphique (Fig. 811).<br />
8.1.2. Eésultgtz :<br />
Les distributións des 10 va<strong>le</strong>urs annuel<strong>le</strong>s détehnées, pour chacune<br />
des 4 stations de références, sur la période 19ó2-1g0, e& très irrégu-<br />
lière : non seu<strong>le</strong>ment il n'est pas raisonnab<strong>le</strong> d'y ajuster des lois,<br />
mais l'extrapolation de la simp<strong>le</strong> distribution expérimenta<strong>le</strong> F(Q) n'est<br />
même pas envisageab<strong>le</strong>. Dans ces conditions, <strong>le</strong>s seuls résultats synthé-<br />
tiques que l'on puisse avancer sont <strong>le</strong>s va<strong>le</strong>urs moyennes et extrêmes<br />
observées, en précisant qu'ils sont relatifs k 10 années d'observation,<br />
<strong>le</strong> tab<strong>le</strong>au 812 ci-dessous récapitu<strong>le</strong> <strong>le</strong>s résultats et la Fig. 812 pré-<br />
sente <strong>le</strong>s va<strong>le</strong>urs moyennes.<br />
I1 faut noter que <strong>le</strong>s débits caractéristiques QC et VC pour<br />
n n<br />
n = 335 correspondent k des données "mensuel<strong>le</strong>s", mais pour un mois<br />
"mobi<strong>le</strong>", affranchi des limites civi<strong>le</strong>s de début et fin de mois.<br />
O, 493<br />
O, 576<br />
O, 652<br />
1,oe<br />
Note : pour QCn,et VCn, durée de 1.a Période = (365 - n) jours.<br />
<strong>le</strong>
QCn en 1/s<br />
8.2. Estimation des caractéristiques des bassins non équipés :<br />
En examinarit <strong>le</strong>s r6sultats rgcumks sur la Fig.fi12et en <strong>le</strong>s confrontant<br />
aux termes de comparaisons (K essentiel<strong>le</strong>ment) présentés au $ 4, on cons-<br />
tate que :<br />
- la décroissance de chacune des.3 courbes quand n croît se fait à peu<br />
près selon la même pente pour <strong>le</strong>s 4 bassins équipés ;<br />
- <strong>le</strong>s courbes de va<strong>le</strong>urs moyennes sur 10 ans VC (n) et Dc (n) sont très<br />
n n<br />
proches ; cel<strong>le</strong> de QCn(n) est nettement distincte ;<br />
- la connaissance des 3 points : QC335 , VC335 et Dc permet de dél-imi-<br />
. 365'<br />
ter un triang<strong>le</strong> représentant quasiment tous <strong>le</strong>s résultats du tab<strong>le</strong>au 812;<br />
- <strong>le</strong>s rapports d'abondance spocifique K définis au $ 4 correspondent à peu<br />
près aux rapports K1 des QC (spécifiques) ; ceci est compréhen-<br />
335<br />
sib<strong>le</strong> car de toutes <strong>le</strong>s ca,ractéristiques d'étiages déterminées au $ 81,<br />
ce sont <strong>le</strong>s QC qui sont <strong>le</strong> plus éloignés des minimums instantanés<br />
335<br />
et <strong>le</strong>s moins éloignés donc de cette moyenne d'étiages mesurés qui a<br />
servì au calcul du rapport K. ;<br />
- Les rapports K et K entre <strong>le</strong>s deux o.utres caractéristiques définissant<br />
2 3<br />
<strong>le</strong> "triang<strong>le</strong>" cité précédernent sont différents de K et Kl, mais <strong>le</strong>urs<br />
sont approximativement proportionnels et selon un coefficient indépendant<br />
du bassin à l'intérieur d'un même type de bassins (superficiel ou profond)<br />
ces rapports sont cependant variab<strong>le</strong>s avec <strong>le</strong> bassin de référence uti-<br />
lisé ; ils sont présentés dans <strong>le</strong> tab<strong>le</strong>au 82.<br />
Le tab<strong>le</strong>au 82 peut être complété en utilisant <strong>le</strong>s propriétés notées<br />
ci-dessus et 1es.connaissances qualitatives du bassin (0 $ 2 et 3) : <strong>le</strong>s<br />
rapports K1 à 5 estimés y sont notés entre 2arenthèses. A l'aide de ces<br />
rapports et des résultats des bassins de référence, on a estimé, pour <strong>le</strong>s<br />
6 bassins non équipés, <strong>le</strong>s tro'is débits caractéristiques (QC 335' vc335 et
DC délimitant <strong>le</strong>s triang<strong>le</strong>ybbservés sur la Fig. 812. Certains bassins<br />
365<br />
(Croupet et Rognon au Champ de Tir) non équipés avaient été comparés ?i deux<br />
ou trois bassins de référence et <strong>le</strong>s estimations concordent de maniere satis-<br />
faisante, sauf pour l'estimation du Champ de Tir (Rognon) & partir du Theil<br />
qui est faib<strong>le</strong>,<br />
I1 faut noter que seul un souci d'économie à limité <strong>le</strong>s comparaisons<br />
entre bassins équipés et non équipés ; il y avait en fait des données suffi.-<br />
santes pour comparer chacun des 6 bassins non équipés 5 chacun d:s 4 bassins<br />
équipks de référence.<br />
115<br />
La Fig. 82 récapitu<strong>le</strong> <strong>le</strong>s estimations. Compte tenu du caractère aléatoire<br />
des estimations de K et Y on a éga<strong>le</strong>ment cherché à respecter approximative-<br />
ment la forme des "triang<strong>le</strong>s" qui étaient à peu près égaux sur <strong>le</strong>s données<br />
observées (Fig. 812).<br />
MELARCHEZ<br />
2 5'<br />
Tab<strong>le</strong>au 82 ,<br />
Rapports entre débits caractéristiques d'étiages et<br />
moyennes des étiages instantanés<br />
(<strong>le</strong>s rapports estimés sont entre pareri<strong>the</strong>ses)<br />
' ea11<br />
295 a 3<br />
(1)<br />
3<br />
O,? à O,¡<br />
4 à 5<br />
.<br />
2<br />
4 à 5<br />
fia4<br />
lCUARCHEZ (Y = M) 7 Km2 débits spécifiques<br />
G O U G E (Y = G) 24,7 Km2<br />
I I
11 6<br />
9. DISCUSSION<br />
L'absence de mesures continues sur <strong>le</strong>s bassins non équipés ne permet pas<br />
de tester la précision des estimations faites au $ 82. Néanmoins, la méthode<br />
de comparaison (doub<strong>le</strong>s-cumuls) ayant été appliquée aux 4 bassins observés de<br />
manière continue, on a là un moyen de tester partiel<strong>le</strong>ment la méthode : <strong>le</strong>s<br />
estimations sont bonnes, voire excel<strong>le</strong>ntes, mais il était nécessaire de dis-<br />
poser d'au moins 2 ou 3 bassins pour connaître <strong>le</strong>s relations entre <strong>le</strong>s<br />
Ki (i = 1 à 3) et K.<br />
Compte tenu de la méthode employée, et de 1'irrégul.arité des distribdions<br />
$$ 81.2), il n'est pas bon de l'appliquer aux va<strong>le</strong>urs extrêmes observées<br />
l'on ne pourra suffisamment bien mesurer ni la fréquence des rgsultats (repérés.<br />
ou estimés), ni <strong>le</strong>s interval<strong>le</strong>s de confiance correspondants.<br />
Mises à part <strong>le</strong>s courbes de doub<strong>le</strong>-cumuls qui nous semb<strong>le</strong>nt etre une<br />
&ape nécessaire et fondamenta<strong>le</strong> (el<strong>le</strong>s permettent de c'affranchir des nom-<br />
breuses irrdgularités loca<strong>le</strong>s propres aux étiages et de percevoir la tendance),<br />
la suite de l'analyse pr6scntée ne prétend à aucune originalité et il serait<br />
possib<strong>le</strong> d'utiliser <strong>le</strong>s données autrement, par exemp<strong>le</strong> en étudiant <strong>le</strong>s liai-<br />
sons entre <strong>le</strong>s jaugeages instantanés et <strong>le</strong>s débits mensuels.<br />
Si ces cam-a.gnes de jnugeag'en épisodiques n'avaient -as 4té réa.lis4sJ 1-3<br />
dobits d'étiage auraient été estimés directement & parth- des 4 bassins n'user-<br />
vés, en appliquant la règ<strong>le</strong> habituel<strong>le</strong> d'égalité de débit sphcifique, la<br />
car
117<br />
connaissance géologique conduisant à diviser <strong>le</strong> bassin en deux groupes : type<br />
"MELARCHEZ" pour ceux dont l'exutoire est situé au-dessus du niveau impermé-<br />
ab<strong>le</strong> des "argi<strong>le</strong>s vertes!, type "GOUGE-AVENELJXS-THEIL" pour ceux dont l'exu-<br />
toire est situ6 au-dessous. A titre d'exemp<strong>le</strong>, <strong>le</strong> tab<strong>le</strong>au 9 présente <strong>le</strong>s deu<br />
types d'estimations pour <strong>le</strong> QC 335 *<br />
Bass ins<br />
MELARCHEZ<br />
GOUGE<br />
AVENELLES<br />
THEIL<br />
CROUPET<br />
PIERRE LEWx<br />
BIBARTAULT (P. Courcy)<br />
" (Rognon )<br />
CHAPii &TIR (Bourgogne)<br />
Tab<strong>le</strong>au 9<br />
Comparaison des estimations possib<strong>le</strong>s du QC<br />
335 __---_--_____I<br />
II II II<br />
(Rognon 1<br />
--_-_----_____________<br />
__---_-___<br />
observés<br />
On voit sur <strong>le</strong> tab<strong>le</strong>au 9 qu'en l'absence de ces jaugenges isolés <strong>le</strong>s<br />
estimations de débit d'étiage auraient été complètement fausses pour <strong>le</strong>s bas-<br />
sins du CROUPET et de BIEARTAULT (Petit Couroy), et très médiocres pour <strong>le</strong>s<br />
2 bassins du CHAMP de TIR, l'écart pour <strong>le</strong> ru du Rognon au CHAMP de TLH<br />
n'étant que très partiel<strong>le</strong>ment réduit par une évmtuel<strong>le</strong> correction des<br />
débits (au grand maximum + O,25 l/s.W), suite aux captages de COUI13Kt4TERS.<br />
CONCLUSION<br />
Des camnsgnes de ,jaiiFeages 6pisodiqiies en basses eaux pcrrnettent d'c qf iv-r<br />
certaines caractéristiques d'étiages. de cours d'eau non observés en perniônence,<br />
SOUS réserve Ce satisfaire à un certain nombre de conditions. I1 est d'abord<br />
n6cessaire d'effectuer de nombreuses mecurcs (quasi simultanées en tous <strong>le</strong>s<br />
,
11 8<br />
points étudiés) et pendant un assez grand nombre d'années (cyc<strong>le</strong> saisonnier),<br />
de maniere à s'affranchir des incertitudes propres aux mesures d'étiages et<br />
des hétérogénéit6s d'alimentation des. réservoirs souterrains. Ensuite il faut<br />
gén6ra<strong>le</strong>ment se 1imit.er h l'estimation de caractéristiques moyennes, <strong>le</strong>s irré-<br />
gularités citées ne permettant guère<br />
Enfin <strong>le</strong> jaugeage, lors de ces campagnes, de stations observées par ail<strong>le</strong>urs<br />
en permanence (équipées) est indispensab<strong>le</strong> pour faire dépasser aux résultats<br />
<strong>le</strong> stade sommaire d'une moyenne de mesures instantanée<br />
que d'observer de$ moyennes à terme.<br />
(de sighification<br />
statistique inconnue) et permettre l'estimation de caractéristiques classiques.<br />
REMERCIEMENTS : Nous remercions ici M. HIAVEC Robert, Chef de la Division Hydro-<br />
loffie du CTOREF et. M. DUEFEUIL P., Inspecteur de Recherche & l'ORSTOM, qui ont<br />
6th <strong>le</strong>s instigateurs de ces campapes de mesures épisodiques. Nol.re reconnals-<br />
sance va aussi à MM. TESSIER, TOL?N%, ROSIQüE (.Ta et
BASSIN DE L'ORGEVAL<br />
Lôgende<br />
@ Statione Hydrometrlques<br />
1 Y6lnrchez<br />
2 Gouge<br />
3 Avenel<strong>le</strong>i<br />
4 Thell<br />
5 Croupe<<br />
6 P<strong>le</strong>rrelav&<br />
e .<br />
10 .<br />
FIg 2 Réseau des étiagee<br />
119
W<br />
o<br />
120
\<br />
I-<br />
L :-<br />
121
122<br />
dibitr Q<br />
I I
123
ABSTRACT<br />
PARAMETRES REGIONAUX RELATIFS AUX RESSOURCES<br />
EN EAU. UTILISATION. PRECISION D'ESTIMATION<br />
par J.R. TIERCELIN<br />
Di vis i on H y drologie<br />
Centre Technique du Génie Rural, des Eaux et des Forêts<br />
(C.T.G.R.E.F.)<br />
Ministère de 1'Agricultur.e<br />
et du Développement Rural. France<br />
Experience has shown that some parameters relative to monthly<br />
and annual discharges are often similar between <strong>the</strong> various gauging<br />
stations of a network. Calling "regi'onal value of a parameter" <strong>the</strong><br />
arithmetical mean of <strong>the</strong> values of this parameter in <strong>the</strong> various<br />
stations, one supposes that this regional value can be used even in<br />
places where no measurement are availab<strong>le</strong>. Theory and pratica1<br />
aplication show that some results obtained in this way reach a very<br />
interesting accuracy for peop<strong>le</strong> in charge of water management and<br />
designers of water resources projects.<br />
RESUMEN<br />
Las observaciones muestran que ciertos parámetros sobre los<br />
flujos mensua<strong>le</strong>s y anua<strong>le</strong>s varían poco entre las diferentes estacio<br />
nes hidrométricas de una red. Llamando por definición "valor regio-<br />
nal de un parámetro" a la media aritmética de los valores tomados<br />
por este parámetro en las diferentes estaciones de la red, se hace<br />
la hipótesis de que este valor regional conviene, si se utiliza ba-<br />
jo ciertas condiciones, a sitios sobre los que no existen observa-<br />
ciones. La teoria y la aplicación a un caso concreto muestran que<br />
ciertas estimaciones obtenidas de esta manera son, debido a su pre-<br />
cisión, muy interesantes para los responsab<strong>le</strong>s de la reordenación<br />
del agua y para los proyectistas encargados de idear los equipos hi<br />
drdulicos.
126<br />
L'utilisation des données d'un réseau en vue d'effectuer des synthkses<br />
régiona<strong>le</strong>s pour divers paranktres hycirologiques Ect une méthode pratiquée<br />
depuis longterps tn ce qui concerne <strong>le</strong>s crues, meis égz<strong>le</strong>nent utilisab<strong>le</strong><br />
dans l'estimation des apports en eau [i]. En principe <strong>le</strong>s va<strong>le</strong>urs grises<br />
p.? <strong>le</strong>s lararrètris étudias vprient evec <strong>le</strong>s. conditions physiques et c3.L-<br />
matiquos des aiffé~znts bassins, ce qui conduit à recourir k clas corrxia-<br />
t ions mlt ip<strong>le</strong>s .<br />
L'étude rnenoe dans <strong>le</strong> Sud-Ouest de 12 France montre que certzins ?a?-mktres<br />
ont un? v-lour qui varie très ?eu B'm bôssin versant 5 l'autre,<br />
r21
.2.3. Résultats des estimations :<br />
127<br />
Dans <strong>le</strong>s tab<strong>le</strong>aux qui suivent nous présentons <strong>le</strong>s va<strong>le</strong>urs obtenues<br />
p3ur certains pararetres régionzux, ainsi que l'estimation de l'erreiir<br />
que l'on com3et en appliqurnt une va<strong>le</strong>ur régiona<strong>le</strong> d'un paramètre à un<br />
point de la r5gion concernée. Pour donner une représentation concrète<br />
de chaque va<strong>le</strong>ur de procision, cel<strong>le</strong>-ci est exprimée par la longueur<br />
d'une série d'cbservations qui fournirait la même variance d'erreur<br />
pour <strong>le</strong> même paramètre.<br />
En ce qui concerne d'abord <strong>le</strong>s moyennes des logarithmes des débits<br />
mensuels, la prScision sbtenue est tra? faib<strong>le</strong> pour que <strong>le</strong> résultat ait<br />
de l'intérêt, et ceci en raison de la tro- forte variabilité spatia<strong>le</strong><br />
de la pluviodtrie (il pourrait en être autrement dans une raon moins<br />
accidentée).<br />
Pour ce qui est des vdriances drs logarithmes des dobits mensuels,<br />
<strong>le</strong>s résultats sont complétk par la va<strong>le</strong>ur du coefficient de variation<br />
des débits naturels, lié ?i Is varirnce v des logarithmes par l'expression<br />
:<br />
x=dex?(v) - 1 (cf. zar ex. 121).<br />
~n outre, ï'expression L = ex? (t fi- v/2) fournit <strong>le</strong><br />
rapport d'un débit de fréquence quelconque au modu<strong>le</strong>, en appelant t<br />
la va<strong>le</strong>ur de la variab<strong>le</strong> norma<strong>le</strong> centrée réduite pour cette fréqu, once.<br />
Par ail<strong>le</strong>urs, dans l'exemp<strong>le</strong> traité, <strong>le</strong>s résultats relatifs aux 12<br />
stations 6tudiées se regroupent nettement en deux ensemb<strong>le</strong>s correspondont<br />
respectivement à deux sous-régions : Massif-Central (stations 1, 2, 3, h,<br />
7, 6 du schéma d'ensemb<strong>le</strong>), et Pyrénées (stztions 5, 6, 9, 10, 11, 12).
128<br />
2.4. Conclusion sur <strong>le</strong>s résultats obtenus :<br />
L'utilisation de paramètres régionaux s'avère très fructueuse<br />
pour certains paramètres. Ainsi, dans la région étudiée, et SOUS <strong>le</strong>s<br />
conditions qui seront examinées ci-après, en une station même dépour-<br />
L<br />
cients de variation des dkbits mensuels et des coefficients de corr6-<br />
lation sériels, est la même que si on avait disposé d'une trentaine<br />
d'années d'observations.<br />
En partant de séries courtes, <strong>le</strong> résultat obtenu est encore plus<br />
intéressant en va<strong>le</strong>ur relative. Ainsi avec 10 ans d'observations<br />
(1959-1968) , <strong>le</strong>s variances d'erreurs correspondent à une dizaine<br />
d'années équiva<strong>le</strong>ntes. Pour la variance dans <strong>le</strong> Massif Central, <strong>le</strong><br />
nombre d'années équiva<strong>le</strong>ntes est même égal h 13. Ce résultat surpre-<br />
nant est une illustration concrete de la notion de stations-années.
III - CONDITIONS D'APPLICATION<br />
3.1. Utilisation d'un paramètre régional en une station :<br />
129<br />
Un paramètre régional est estimé à partir d'un réseau de<br />
stations dominées par des bassins versants présentant des carac-<br />
téristiques physiques plus oit moins variées.<br />
Pour avoir <strong>le</strong> droit d'utiliser des paramètres régionaux en<br />
un point de la région étudiée, il'faut qudes caractéristiques<br />
du bassin versant concerné entrent ?i peu près dans la gamme des<br />
caractéristiques physiques des bassins versants dominant<br />
stations du réseau, sinon <strong>le</strong>s variances d'erreurs calculées<br />
n'ont aucune signification.<br />
3.2. Combinaison avec d'autres méthodes d'estimation hydrologique :<br />
On peut faire grief à l'utilisation de va<strong>le</strong>urs régiona<strong>le</strong>s de<br />
ne donner des résultats,intéressants que pour certains paramètres<br />
et de ne pas s'appliquer en particulier à l'estimation de modu<strong>le</strong>s<br />
ou de moyennes de logarithmes des débits (du moins dans l'exemp<strong>le</strong><br />
d'application traité). En fait, il faut observer que ces derniers<br />
paradtres peuvent etre estimés.par diverses autres méthodes, meme<br />
en.des points oh il y a peu ou pas d'observations.<br />
Dans ces conditions , l'utilisation de va<strong>le</strong>urs régiona<strong>le</strong>s<br />
apparaft comme un complément des méthodes existantes pour l'esti-<br />
mation des ressources en eau, en vue de connaftre de façon précise<br />
<strong>le</strong>s paramètres de dispersion et de corrélation sériel<strong>le</strong>, qui sont<br />
en général estimés avec une précision médiocre lorsqu'il y a pe'<br />
ou pas d'observations.<br />
3.3. Application d'autres régions :<br />
La dthode est théoriquement utilisab<strong>le</strong> à partir de n'importe<br />
quel réseau de stations observée$ simultanément. Néanmoins, pour<br />
qFe la précision soit intéressante, il faut ,utiliser des groupes<br />
de stations suffisamment homogènes, ce .qui peut conduire à diviser<br />
la région comme cela a été fait dans l'exemp<strong>le</strong> d'application précédent.<br />
Moyennant cette précaution, il est vraisemblab<strong>le</strong> que la<br />
dthode est applicab<strong>le</strong> ?i n'importe quel<strong>le</strong> région du globe, tant<br />
pour <strong>le</strong>s débits que pour <strong>le</strong>s pluies.<br />
--------&o--------<br />
NOUS tenons h remercier, à. l'occasion de cette public ation :<br />
- Mme OBERLIN, du CTCREF, qui a effectué une grande partie du travail de<br />
programmation sur ordinateur,<br />
n M. BERNIW, d'E<strong>le</strong>ctricité de France, dont <strong>le</strong>s conseils ont permis de mener<br />
h bonne fin <strong>le</strong>s calculs de variance d'erreur,<br />
- M. HLAVEK, Chef de la Division Hydrologie du CTGFtEF, dont <strong>le</strong>s observations<br />
ont conduit B améliorer la rédaction de la note,<br />
- M. de BEAUREGARD, d'E<strong>le</strong>ctricit6 de France, MM. EUICLE et BmIERE des 'Cir-<br />
conscriptions E<strong>le</strong>ctriques Sud-Ouest et Centre-Ouest, qui nous ont fourni<br />
<strong>le</strong>s données nécessaires B l'étude.<br />
<strong>le</strong>s
130<br />
I - PRECISION D'ESTIMATION<br />
A N N E X E<br />
1.1. Position du problème :<br />
Considérons n stations étudiées simultan6ment durant m années.<br />
Nous étudions pour un mois pkticulier p de l'année <strong>le</strong>s débits mensuels<br />
sous la forme d'une variab<strong>le</strong> qui doit 6tse comparab<strong>le</strong> entre <strong>le</strong>s différentes<br />
stations : en pratique il s'agira soit du débit moyen mensuel<br />
spécifique, soit du logarithme de cette grandeur. Soit A cette vari-<br />
ab<strong>le</strong> pour la station de rang J ; &e donne lieu m réalisations<br />
1. i<br />
m<br />
paJ , paJ, ... paJ, à partir desquel<strong>le</strong>s nous déduisons l'estimation<br />
d d'un paramètre (par exemp<strong>le</strong> moyenne ou variance de A ). Les obser-<br />
PJ P J<br />
vations et <strong>le</strong>s variab<strong>le</strong>s aléatoires entrant en Jeu sont figurées dans<br />
<strong>le</strong> tab<strong>le</strong>au ci-dessous.<br />
j=q<br />
Nous posons par définition c o m paramètre régional la va<strong>le</strong>ur<br />
n<br />
pdj/tl, moyenne des va<strong>le</strong>urs relatives aux différentes<br />
.P J<br />
... et a sont des estimations<br />
stations. es va<strong>le</strong>urs iì &, ... P<br />
des va<strong>le</strong>urs théoriques u 1s ... ... et pu.<br />
1 i m<br />
pa1 ..... pai ..... Pal<br />
:1 :i 'm<br />
paJ ..... paJ ..... paJ<br />
:<br />
:1 :I 'm<br />
pan ..... pan ..... Pan<br />
P J<br />
Va<strong>le</strong>ur<br />
théorique<br />
Le problème qui se pose est <strong>le</strong> suivant. En un emplacement J'<br />
de la région concernée, différent des emplacements qui ont servi<br />
h l'estimation de la va<strong>le</strong>ur régiona<strong>le</strong>, on décide d'appliquer <strong>le</strong><br />
paramètre régional 0. En fait, théoriquement, ce qui nous intéresse<br />
est 1a.vraie vaïeur'inconnue u I prise par <strong>le</strong> parmètre à l'empia-<br />
cement J', et <strong>le</strong> problème consJste donc à estimer la variance de
l'erreur - ) commise en attribuant à un bassin quelconque J' la<br />
pUJ '<br />
va<strong>le</strong>ur régiona<strong>le</strong> du paramètre.<br />
L'erreur ( d - ) résulte el<strong>le</strong> même de la composition de deux<br />
erreurs indépenbteg<br />
131<br />
- l'erreur d'adéquation ( u - u), de nature purement physique, provenant<br />
P J' P<br />
du fait que la va<strong>le</strong>ur régiona<strong>le</strong> standard n'est pas parfaitement adaptée<br />
au bassin j' ;<br />
- l'erreur d'échanti'llonnage ( u - a), de nature purement statistique.<br />
P P<br />
1.2. Résultats généraux :<br />
Pour avoir une estimation de l'erreur d'adéquation, nous posons <strong>le</strong><br />
postulat suivant : <strong>le</strong> bassin J' présente vis à vis du standard régional<br />
une différence du &me ordre de grandeur que <strong>le</strong>s bassins 1.. . J.. . .n<br />
en regard de ce standard (Ce p6lnt délicat e t fondamental pour l'application<br />
de la méthode sera discuté au $ ci-après). Dans ces conditions,<br />
nous poso s que l'erreur d'adéquation, exprimc5e par l'écart quadratique<br />
(u - uJi)', est donnée par l'expression :<br />
(1) = (pu - puJ)2/n<br />
J=<br />
Cet écart quadratique moyen, ajouté B la variance de 1'échantIllOnnage<br />
de d; donne l'expression théorique de la variance d'erreur tota<strong>le</strong> :<br />
(2)<br />
A u = u + var ( Q)<br />
P P<br />
Le problème est maintenant de rattacher cette expression aux observations.<br />
Pour cela nous allons calcu<strong>le</strong>r l'expression de l'espérance mathématique<br />
0 )2 en fonction des va<strong>le</strong>urs théoriques u et puJ.<br />
E ( ~ Q - ~ J P<br />
Décomposons <strong>le</strong>s espérances de carrés et de produits en faisant res-<br />
sortir <strong>le</strong>s variances et covariances :<br />
E ( p *) ~ = u2 + var ( Q)<br />
P P<br />
2<br />
E(b )= u2+var(h)<br />
P J P J P J<br />
E ( 0. Q ) = p~.p~J + cov ( Q,<br />
P P J P P<br />
û<br />
J<br />
)<br />
Comme 0 est la moyenne arithmétique des va<strong>le</strong>urs<br />
P<br />
la dernière ligne s'écrit : n<br />
19... pQkS s.<br />
Q<br />
P n'
132<br />
Eh reportant ces expressions dans (3), il vient ,:<br />
ce qui permet d'exprimer ( u- u )* en fonction des observations :<br />
P P J<br />
et en effectuant pour toutes <strong>le</strong>sva<strong>le</strong>urs de J la sommation (1) :<br />
L'erreur tota<strong>le</strong> donnée par (2) s'exprime donc par la formu<strong>le</strong> :<br />
et en remplaçant <strong>le</strong>s expressions théoriques par <strong>le</strong>urs estimations, nous obtenons<br />
en définitive :<br />
Par ail<strong>le</strong>urs, la poursuite des calculs exigera <strong>le</strong> recours à la matrice des co-<br />
variances liant <strong>le</strong>s variab<strong>le</strong>s aléatoires A et relatives à deux stations<br />
quelconques J, k. L'estimation sans biad de PAkcette covariance est fournie<br />
Nous ne pouvons d'ail<strong>le</strong>urs écrire cette expression qu'en admettant en<br />
principe que <strong>le</strong> débit du mois p de l'année i à une station est quasiment indé-<br />
pendant du débit du mois p de. l'année i - 1 ?i la &me station, de faconia dis-<br />
poser pour chaque variab<strong>le</strong> aléatoire A d'une série de réalisations a<br />
indépendantes entre el<strong>le</strong>s. PJ P J<br />
II - VARIANCE D'ECHANTILWNNACE DE LA MOYEN'NE<br />
2.1. Expression de la variance d'échantillonnage :<br />
1<br />
Les<br />
i<br />
paramètres sont donnés ici par l'expression : d -(paj + ...<br />
+ am)/m. PJ P J<br />
paJ '*<br />
P J<br />
m
(6)<br />
ia covariance de a.et û est donnée par ì'expression :<br />
PJ p k<br />
&<br />
m<br />
&<br />
2<br />
i' i")<br />
m CBV =<br />
côv 'PaJ' pak<br />
133<br />
Parmi <strong>le</strong>s termes de la sommation, on peut distinguer ceux pour <strong>le</strong>s-<br />
quels i' = i", et qui correspondent ?i des variab<strong>le</strong>s relatives à la même<br />
année, et ceux pour <strong>le</strong>squels i' # i", correspondant à des années diffé-<br />
rentes. Pour ces derniers, lorsque j = k, nous avons posé l'approximation<br />
d'indépendance (ci. supra) ; nous poserons a fortiori la dm hypothèse<br />
pour J # k.<br />
k<br />
11 vient donc simp<strong>le</strong>ment dans ces conditions : c8v (ptìJ,pûk)=pCJ/m<br />
et en particulier : & ( CI ) = c /m.<br />
P J P J<br />
ce qui d!aprbs (4) permet d'écrire l'expression de la variance d'erreur :<br />
n n<br />
A> = 1) (pO-paJ)2/n -2 pcj/m<br />
j =I J=l<br />
De même N( 0) o servations independantes. hypothétiques donnent pow<br />
variance d'erre& &2. Les variances étant d es <strong>le</strong> rapport des effectifs<br />
d'observations :<br />
nous obtenons en définitive l'expression cherchée :<br />
j,q P P J
(8)<br />
134<br />
111 - YARIANCE D'EcHANTILLONNAGE DE LA VAAIANCE<br />
3.1. Expression de la variance. d'6chantillonnage :<br />
Considérons <strong>le</strong>s définitions de variabies et de paramètres, ainsi<br />
que <strong>le</strong>s résultats obtenus au 5 1 pour l'ensemb<strong>le</strong> des parametres. Pour<br />
qu'il n'y ait pas d'atnbigulté avec <strong>le</strong>s résultats du 5 2 relatifs aux<br />
moyennes , nous remplaçons ici toutes <strong>le</strong>s <strong>le</strong>ttres "u'' par <strong>le</strong>s <strong>le</strong>ttres<br />
Il 11<br />
v.<br />
Le parametre empirique dont nous étudions la variance est l'esti-<br />
mation sans biais de la variance théorique. A partir des observations,<br />
cette estimation sans biais s'écrit :<br />
D'après [3], nous avons pour covariance de 0 et 0 en supposant<br />
norma<strong>le</strong>s <strong>le</strong>s variab<strong>le</strong>s aléatoires A etpAk P. J ~ k '<br />
P J<br />
côv ( 0 o ) = 2 ( Ck)2/m<br />
P J ' P ~ P J<br />
et en particulier<br />
J2<br />
(9) v&? (pvJ) = (pcJ) /m<br />
(10)<br />
d'ou, d'après la form<strong>le</strong> (4) dans laquelLe on remplace <strong>le</strong>s <strong>le</strong>ttres "u"<br />
par des <strong>le</strong>ttres "v" ; la variance d'erreur cherchée :<br />
Si on juge plus commode de se référer aux écarts-types qu'aux variances,<br />
on déduira l'erreur A\ sur l'écart-type à partir de<br />
P<br />
l'erreur &v sur ia variance en posant l'approximation :<br />
P<br />
#2&$/ps , ce qui donne :<br />
i=/<br />
14 # A-v/4pol P<br />
3.2. Nombre d'années équiva<strong>le</strong>ntes :<br />
Le raisonnement est analogue à celui du 0 2.2.
Une série de m années Inddpendantes donne par variance d'erreur en<br />
moyenne (loi norma<strong>le</strong>) :<br />
Compte tenu en outre de ia relation : ,&:V/ q v<br />
= m/N (4)<br />
b nombre d'années fictives cherché est donné en définitive par la<br />
relation :<br />
IV - VARIANCE D'ERREUR DU COEFFICIENT DE COW1ELATION SERIELLE<br />
135<br />
Théoriquement il serait concevab<strong>le</strong> d'appliquer ici des raisonnements<br />
analogues à ceux qui ont été faits pour <strong>le</strong>s moyennes et <strong>le</strong>s variances.<br />
En pratique '<strong>le</strong> calcul paraft Inextricab<strong>le</strong> et surtout nécessite l'obtention<br />
préalab<strong>le</strong> des corrélations croisées entre <strong>le</strong>s observations de chaque<br />
mois p à chaque station avec <strong>le</strong>s observations du mois p-1 à toutes <strong>le</strong>s<br />
autres stations. k coût du calcul serait en définitive hors de proportion<br />
avec son intérêt.<br />
4.1. -ne supérieure des variances d'erreur :<br />
Dans la recherche de la variance d'erreur affectant un paramètre régional<br />
quelconque 0, on peut obtenir une borne supérieure de cette variance,<br />
utilisab<strong>le</strong> pour n'importe quel parametre.<br />
2<br />
En effet, d'une part, l'expression 2 9 - pû,) /n est un maJorat de<br />
J -1<br />
(pu - puJ)<br />
2<br />
l'erreur d'inadéquation U =<br />
/n, puisqu'el<strong>le</strong> résulte de la<br />
P<br />
ComtJinaiSon entre cette erreur d'inadéquation et <strong>le</strong>s erreurs d'échantillonnage<br />
sur pQ, et pûl...pûJ...pûn.<br />
D'autre part, nous obtiendrons une borne supérieure-de la variance<br />
d'échantillonnage de par <strong>le</strong> raisonnement suivant.. Nous avons :<br />
P<br />
n A<br />
var ( û) = var<br />
P [ +. ... + pûJ .... + pûn)/n]<br />
=var (a + .*.<br />
P l<br />
+ .*<br />
2<br />
+ ,fln,/n
136<br />
~ e s stations constituant <strong>le</strong> rbseau sont relativement homogènes ;<br />
ainsi on peut supposer que var . , varpûj,. . var Q . sont du &me<br />
pn<br />
ordre de grandeur K. Dans ces conditions, <strong>le</strong>s expressions tel<strong>le</strong>s que<br />
covar ( ) pour J k auront pour major.ant K. I1 vient dans ces<br />
P J ' P ~<br />
conditions : var < K<br />
Une estimation de l'ordre de grandeur K sera fournie par moyenne des<br />
termes var 0 d'ou :<br />
P J<br />
var (,fi) < i var û / n<br />
j -GI PJ<br />
En définitive, nous obtiendrons l'expression généra<strong>le</strong> suivante,<br />
constituant une borne supérieure de la variance d'erreur d'un parametre<br />
j -1 PJ<br />
1 n --<br />
Références bibliographiques :<br />
[l] BENSON M.A. & MATALAS N.C. (1967) - Syn<strong>the</strong>tic hydrology based on regional<br />
statistical parameters. Water resources research. Vol 3 n" 11,<br />
.<br />
[2] AITCHISON J. & BROWN .J.A.C. - The lognormal distribution. CAMBREGE<br />
University Ress.<br />
[3] ANDERSON - An introduction to multivariate statistical analysis. WIIM.
PRINCIPLES FOR THE COMPUTATION OF THE MAIN CHARACTERISTICS OF<br />
RIVER WATER RESOURCES AT THE ABSENCE OF OBSERVATIONS ON THE<br />
BASIS OF GEOGRAPHICAL INTERPOLATION OF RUNOFF PARAMETERS<br />
ABS T RACT<br />
K .P. Voskresenski<br />
State Hydrological Institute<br />
Leningrad, USSR<br />
At <strong>the</strong> absence of hydrological data river runoff parameters<br />
may be determined by means of geographical interpolation of<br />
<strong>the</strong>ir values computed by observations on o<strong>the</strong>r rivers of <strong>the</strong><br />
given area. Thus it is possib<strong>le</strong> to obtain principal characteris-<br />
tics of runoff determining <strong>the</strong> rate of possib<strong>le</strong> development of<br />
rmiver water resources, category and dimensions of <strong>the</strong> projectei<br />
hydraulic structures, On <strong>the</strong> basis of <strong>the</strong> mentioned princip<strong>le</strong>s<br />
methods for river runoff computation have been developed in <strong>the</strong><br />
USSR for <strong>the</strong> <strong>who<strong>le</strong></strong> territory of <strong>the</strong> country.<br />
RESUME<br />
En l'absence de données hydrologiques directes, <strong>le</strong>s paramè-<br />
tres de l'écou<strong>le</strong>ment peuvent être déterminés par interpolation<br />
géographique des va<strong>le</strong>urs observées sur d'autres rivières de la<br />
même région. On peut obtenir ainsi <strong>le</strong>s principaux paramètres de<br />
l'écou<strong>le</strong>ment qui déterminent <strong>le</strong>s possibilités offertes par l'uti<br />
lisation des ressources en eaux de surface et permetten de fixer<br />
<strong>le</strong>s caractéristiques hydrauliques des aménagements. Sur la base<br />
de ces principes, on met au point en URSS des méthodes de calcul<br />
de l'ecou<strong>le</strong>ment de surface pour l'ensemb<strong>le</strong> du pays.
138<br />
The prob<strong>le</strong>m of river runoff computation in connexion with<br />
water resources development and engineering projects with inade-<br />
quabe observational data is very important for many countries<br />
of <strong>the</strong> world.<br />
It is known, that hydrological observabions are made on<br />
a relatively small number of stations and never cover all <strong>the</strong><br />
rivers intended for water resources development. It is often<br />
difficult to predict what rivers will be used for water manage-<br />
ment in future <strong>the</strong>refore <strong>the</strong>y are ungauged from <strong>the</strong> hydrological<br />
point of view. Thus, in case of any particular prob<strong>le</strong>m on hydrau-<br />
lic engineering difficulties arise because of <strong>the</strong> absence of<br />
long-term hydrological observations on a particular river.<br />
In case of absence or inadequacy of observations basic<br />
characteristics of river runoff may be determined only by in-<br />
direct methods based on <strong>the</strong> use of information on water regime<br />
of o<strong>the</strong>r rivers in <strong>the</strong> given region or onlargerterritory.<br />
In <strong>the</strong> Soviet Union <strong>the</strong>re have been developed and introduced<br />
into practice methods for <strong>the</strong> computation of river runoff para-<br />
meters essential for water management projects in region in-<br />
sufficiently gauged from <strong>the</strong> bydrological point of view.<br />
The developed methods provided computation of river runoff<br />
parameters in any region of <strong>the</strong> USSR with different climatic<br />
conditions from sub-tropics to <strong>the</strong> arctic zone.<br />
In case of absence or inadequacy of hydrological obse.mations<br />
basic water resources characteristics may be determined by<br />
means of geographical interpolation (in some cases - by extra-<br />
polation) of river runoff parameters computed by a small number<br />
of basic points with long-term observation series usually estab-<br />
lished on main rivers of <strong>the</strong> country.<br />
This method is physically based on distinct variations of<br />
climakic features of river runoff, i.e. water balance e<strong>le</strong>ments<br />
according to geographic zones.<br />
Latitudinal climatic zonation is <strong>the</strong> basic law of geographic<br />
environment variations. In general it i8 explained by cosmic<br />
reasons determining <strong>the</strong> amount of solar radiation in different<br />
areas of <strong>the</strong> world; <strong>the</strong> latitudinal zonation to a great extent<br />
also depends on <strong>the</strong> total aiimospheric circula-Lion determining<br />
water cyc<strong>le</strong> on continents and islands, on <strong>the</strong> location of<br />
continents and on <strong>the</strong> direction of sea currents.<br />
In accordance with <strong>the</strong> location of climatic zones in plains<br />
and altitudinal climatic belts in mountains it is possib<strong>le</strong> to<br />
observe latitudinal and altitudinal variations of water balance<br />
e<strong>le</strong>ments, i.e. precipitation, evaporation and runoff. This offers<br />
a basis for <strong>the</strong> plotting of maps of runoff or it5 main parameters<br />
used for <strong>the</strong> determination of river water resources characteris-<br />
tics in case of data absence. Runoff parameters of ungauged rivers<br />
may be also determined by means of direct interpolation of <strong>the</strong>ir<br />
values between <strong>the</strong> values obtained for basic points with long-<br />
term observation series.
139<br />
Hydrological parameters interpolation is made in accordance<br />
with areal change of climatic factors of runoff with <strong>the</strong> account<br />
of non-climatic effect of <strong>the</strong> environment, i.e. topography,<br />
geology, soils and vegetation and permanent morphometric basin<br />
characteristics, i.e. drainage area, slope, etc.<br />
Mean runoff within any region aepending mainly on climatic<br />
features of <strong>the</strong> region may greatly differ from its actual value<br />
within <strong>the</strong> limits of individual river basins. In some cases nonclimatic<br />
factors become predominant and <strong>the</strong> ro<strong>le</strong> of climatic<br />
factors becomes subordinate <strong>the</strong>ref ore mean runoff may exceed<br />
<strong>the</strong> climatic norm or be <strong>le</strong>ss than this nom. Local factors<br />
effect is best revea<strong>le</strong>d on small rivers. With .<strong>the</strong> increase of<br />
river basin <strong>the</strong> effect of local factors is averaged and in case<br />
of its optimal value, runoff depends only on non-climatic<br />
e<strong>le</strong>ments. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> increase of basin area above<br />
some definite limit causes great difference in runoff value<br />
in different basin parts and discrepancy between its averaged<br />
value and <strong>the</strong> climatic norm. This is explained by <strong>the</strong> fact that<br />
large river basins are usually located within several geographic<br />
zones<br />
Thus interpolation of runoff over territory is possib<strong>le</strong><br />
only for rivers with basin areas within <strong>the</strong> limits of<br />
Am7A y A,<br />
me K (1)<br />
where: An+~is mean optimal basin area when runoff interpolation<br />
is possib<strong>le</strong>; Am and AK indicate its upper and lower limits<br />
respectively.<br />
Optimal drainage area is different in various geographic<br />
regions. It depends on a combination of natural conditions determining<br />
river runoff. The optimal drainage area for any region<br />
is established experimentally.<br />
Difference in runoff of individual rivers for any area determined<br />
by <strong>the</strong> map depends not only on <strong>the</strong> basin size but on <strong>the</strong><br />
peculiarities of methods for runoff maps plotting. It substantially<br />
differs from <strong>the</strong> methods of o<strong>the</strong>r water balance e<strong>le</strong>ments<br />
mapping. Unlike maps of precipitation anci evaporation when data<br />
are related to <strong>the</strong> observation points whi<strong>le</strong> plotting <strong>the</strong> maps,<br />
maps of runoff are prepared by its values related to <strong>the</strong> basin<br />
centre since water discharge measured at <strong>the</strong> discharge site is<br />
<strong>the</strong> averaged value of runoff from <strong>the</strong> basin upstream this site.<br />
Therefore a discrepancy is possib<strong>le</strong> in runoff values determined<br />
by <strong>the</strong> map in <strong>the</strong> basin centre and in its periphery areas. The<br />
difference in runoff values will tend to decrease simultaneously<br />
with <strong>the</strong> decrease of basin area.<br />
Thus, within definite limits of basin areas gradation runoff<br />
depends on <strong>the</strong> size of this area. The value of a critical area<br />
in any region above which runoff is subject to no changes,<br />
may be determined by <strong>the</strong> graph of relations between runoff and
140<br />
basin area. It is evident that runoff values are plotted on<br />
<strong>the</strong> graph in relative units, i.e. as depth of runoff from <strong>the</strong><br />
<strong>who<strong>le</strong></strong> basin (in mm) or as specific discharge (in l/sec per<br />
1 sq.km).<br />
Geographical interpolation method may be used to de termine<br />
basic runoff parameters showing <strong>the</strong> rate of possib<strong>le</strong> development<br />
of water resowces of <strong>the</strong> river, as well as <strong>the</strong> types and catego-<br />
ries of <strong>the</strong> projected hydraulic structures, i.e. annual runoff,<br />
annual streamflow distribution, maximum discharges, Low (minimum)<br />
flow or periods of no flow in <strong>the</strong> river.<br />
The dependence of different runoff characteristics on basin<br />
area is different, In its general case it may be expressed by<br />
equation<br />
M.3<br />
where: M is specific runoff from basin area A; Q is parameter<br />
expressing runoff value independent of basin size;<br />
n is <strong>the</strong> index of runoff reduction with <strong>the</strong> change of basin<br />
area.<br />
For mean annual runoff within <strong>the</strong> limits of optimal areas<br />
ha? ; for <strong>the</strong> modulus of maximum discharge independent of<br />
basin size nLd ; for <strong>the</strong> mndulus of minimum discharge n>i<br />
Proceeding from <strong>the</strong> stated character of runoff reduction<br />
maps of mean annual runoff are plotted by <strong>the</strong> data related to<br />
<strong>the</strong> rivers with basin areas emtceeding; <strong>the</strong> lower limit of <strong>the</strong><br />
optimal area. These data are reduced to a long-term period on<br />
<strong>the</strong> basis of correlation with o<strong>the</strong>r points having long-term<br />
observation series and located in <strong>the</strong> given region or even<br />
beyond its boundaries.<br />
The duration of a long-term period is supposed to be<br />
sufficient if standard error of mean runoff does not exceed <strong>the</strong><br />
accuracy of measuremenets and annual runoff computation (in <strong>the</strong><br />
USSR it is accepted to be equal to !%).<br />
Data on large rivers are used only to control <strong>the</strong> correctness<br />
of plotting runoff isolines system. For this purpose runoff<br />
determined by <strong>the</strong> map as mean weighted value is compared with<br />
<strong>the</strong> actual mean runoff at <strong>the</strong> out<strong>le</strong>t obtained by measurements.<br />
Since <strong>the</strong> value of runoff on small rivers with basin areas<br />
<strong>le</strong>ss than <strong>the</strong> optimal value may be <strong>le</strong>ss because of <strong>the</strong> effect<br />
of prevailing non-climatic factor or it may exceed mean runoff<br />
value in <strong>the</strong> given Brea, a correction should be introduced to<br />
runoff determined by <strong>the</strong> map. The value of corrections is deter-<br />
mined by local graphs of relations between runoff and basin<br />
area.<br />
For <strong>the</strong> USSR area two types of mean runoff reduction from<br />
small basins have been established. In <strong>the</strong> zones of water<br />
surplus and variab<strong>le</strong> moistening river runoff from basin <strong>le</strong>as
than <strong>the</strong> optimal area tends to decrease due to incomp<strong>le</strong>te<br />
drainage of ground water within river basins. On <strong>the</strong> contrary<br />
in arid zones runoff tends to increase with <strong>the</strong> decrease of<br />
basin area due to decrease of losses by evaporation.<br />
Appropriate corrections have been determined for rivers<br />
in ùifferent geographic regions.<br />
To determine mean runoff of ungauged mountain rivers<br />
local graphs of relations between runoff and <strong>the</strong> altitude are<br />
usually usea. Mean basin e<strong>le</strong>vation essential for this purpose<br />
is obtained from topographic maps. As a ru<strong>le</strong>, within <strong>the</strong> limits<br />
of every geographic region <strong>the</strong>re are several local dependences<br />
of runoff change with <strong>the</strong> altitude. The number of <strong>the</strong>se graphs<br />
depends not only on <strong>the</strong> range of altitudes and mountain slopes<br />
exposure, but also on <strong>the</strong> number of observational points in <strong>the</strong><br />
given region. Their increase <strong>le</strong>ads to new local graphs. Thus,<br />
<strong>the</strong> availab<strong>le</strong> graphs are averaged for some territory.<br />
Normal runoff is <strong>the</strong> main water resources characteristic.<br />
But when planning water resources development it is essential<br />
to obtain data on runoff for wet and dry years with different<br />
frequency of occurrence. In <strong>the</strong> practice of hydrological computations<br />
in <strong>the</strong> USSR probab<strong>le</strong> runoff values are obtained by<br />
distribution curve of Pearsan III in its integral expression<br />
i.e. frequency curve. Normal runoff, coefficient of variation<br />
(C ) and coefficient of asymmetry (CS) are frequency curve<br />
pallameters. In case of observational data availab<strong>le</strong> <strong>the</strong> parameters<br />
are computed by ma<strong>the</strong>matical statistics methods. In case<br />
of data inadequacy <strong>the</strong>se parameters are established by geographicalinterpolation<br />
method.<br />
The computation of variation coefficient of annual runoff<br />
is based on <strong>the</strong> account of effect of climatic factors variability<br />
and factors of natural runoff control. Experimental data<br />
show that runoff variability tends to increase with <strong>the</strong> debrease<br />
of its value. Therefore maximum variations of runoff are observed<br />
in arid regions, whi<strong>le</strong> minimum ones - in <strong>the</strong> zone of water<br />
surplus. The normal runoff itself may serve as an index of nlimatic<br />
variability.<br />
141<br />
Among <strong>the</strong> factors of natural runoff control <strong>the</strong> capacity of<br />
river basin is of <strong>the</strong> greatest importance; it determines under-<br />
ground water storage. The basin area is an indirect index of<br />
basin capacity .<br />
An empirical formula has been obtained for <strong>the</strong> <strong>who<strong>le</strong></strong> USSR<br />
territory with <strong>the</strong> account of <strong>the</strong> two mentioned factors:<br />
here:bfo is normal runoff (l/sec per 1 sq.km); A is basin area<br />
?sq.km);B is parameter computed by substitution of She values<br />
known for <strong>the</strong> river-analogue in <strong>the</strong> given area into equation (3).
142<br />
The meaning; of coefficients of asymmetry in case of <strong>the</strong><br />
absence of observations is determined by <strong>the</strong> ratio of Cv and Cs<br />
established by <strong>the</strong> rivers-analogues in <strong>the</strong> given basin. If<br />
no analogues are availab<strong>le</strong> in <strong>the</strong> zones of water surplus or<br />
variab<strong>le</strong> moistening <strong>the</strong> follbwing ratio is accepted Ce = 2 C<br />
and for arid zones C, = 1.5 + 1.8 C,; for extremely arid re#&ns<br />
cs = 1.5 c,.<br />
When computing maximum discharges in case of no observations<br />
it should be taken into account that <strong>the</strong> flood character on<br />
rivers in any geographic region is mainly determined by clima-<br />
tic features and <strong>the</strong>refore data obtained from observations on<br />
some rivers are extended to all <strong>the</strong> rest of water courses of<br />
<strong>the</strong> same region.<br />
Different empirical formulae are used for maximum discharge<br />
computation, <strong>the</strong>ir parameters are determined by observational<br />
data on some rivers of <strong>the</strong> region under consideration.<br />
Rational formulae are widely used which are based on <strong>the</strong><br />
account of maximum or extreme rainfall intensity during flood<br />
concentration; in general <strong>the</strong>y may be given as follows:<br />
where: K is coefficient of dimensionality; h is maximum rate<br />
of rain or snow melt during lag-time ‘i ; dis coefficient of<br />
runoff during <strong>the</strong> same interval.<br />
The time of flood concentration is often determined by<br />
empirical relations between this value and river <strong>le</strong>ngth or<br />
basin area. Runoff coefficient is accepted by <strong>the</strong> analogy with<br />
flooãs on o<strong>the</strong>r rivers proceeding from <strong>the</strong> general nature of<br />
top cover and topography.<br />
For practical computations it is reasonab<strong>le</strong> to use reduction<br />
foriïiulae of a general type:<br />
where: maf-is maximum specific discharge;<br />
Je - is extreme specific discharge if A-0 and c= i<br />
C -is addition to basin area taking into account <strong>the</strong><br />
character of runoff maxima variations in case of<br />
small basin areas;<br />
n. -is <strong>the</strong> index of maximum runoff reduction.<br />
The parameters in <strong>the</strong> formula are established on <strong>the</strong> basis<br />
DQ processing of data on maximum discharges in <strong>the</strong> given region.<br />
Minimum (low) flow is determined by <strong>the</strong> rate of underground<br />
water drainage by rivers. The amount of underground water<br />
discharging into rivers depends on <strong>the</strong> number and capacity of<br />
aquifers cut through by river channel. The depth of erosion cut
143<br />
usually tends to increase with <strong>the</strong> increase of basin area. Therefore<br />
maximum runoff varies with <strong>the</strong> change of basin area.<br />
In this case <strong>the</strong> optimal basin area is supposed to be <strong>the</strong><br />
area when rivers cut through all <strong>the</strong> aquifers of <strong>the</strong> given<br />
region. It is possib<strong>le</strong> to plot a map of minimum flow for such<br />
rivers to be used for computations.<br />
For small water courses local graphs of relations between<br />
minimum flow and basin area are established.<br />
The rate of wa.ter resources development of some rivers<br />
is determined by <strong>the</strong> duration of no flow period. Such rivers<br />
occur in arid and permafrost zones. The duration of dry period<br />
is also determined by <strong>the</strong> basin area size.<br />
The experience of <strong>the</strong> use of indirect methods for <strong>the</strong> computation<br />
of main characteristics of water resources of <strong>the</strong> USSR<br />
rivers shows <strong>the</strong> eqediency of <strong>the</strong>ir use in countries with<br />
different climates and different physiographic features.<br />
1. Voskresenski K.P., Norma i izmenchivost godovogo atoka rek<br />
Sovetskogo Soyuza (Annual runoff norm and vari-<br />
ability for <strong>the</strong> USSR rivers), Hydrometeorological<br />
Publishing House, Leningrad, 1962, 545 p.<br />
2. Voskresenski K.P. Gidrologicheskie raschety pri proektiro-<br />
vanii sooruzheniy na malykh rekakh, ruchiakh i<br />
vrememykh vodotokakh (Hydrological computations<br />
for engineering projects on small rivers and<br />
temporary mater couraes), Hydrometeorological<br />
Publishing House, Leningrad, 1956, 468 p.
EVALUATION OF WATER RESOURCES OF MOUNTAIN AREAS IN CASE OF<br />
ABSTRACT<br />
ABSENCE OR INADEQUACY OF DATA ON RUNOFF<br />
Vuglinski V.S.<br />
State Hydrological Institute<br />
Leningrad, USSR<br />
V.A. Semenov<br />
Kazakn Research Hydrometeorological Institute<br />
Alma-Ata, USSR<br />
Normal annual runoff, as water resources indicator, may be<br />
determined for mountain areas on <strong>the</strong> basis of taking into account<br />
<strong>the</strong> laws of runoff distribQtion over territory and according to<br />
altitudinal zones established by <strong>the</strong> observational data from <strong>the</strong><br />
gauged rivers. These laws are connected with latitudinal and<br />
longtudinal zonalities, with differences in <strong>the</strong> nature of <strong>the</strong><br />
underlying surfaces and slopes exposure relative to moisture<br />
carrying air fluxes. These laws are quant'itatively expressed by<br />
regional dependences of normal runoff upon mean basin e<strong>le</strong>vation<br />
and <strong>the</strong> rate of its glacierization. Ano<strong>the</strong>r method, providing<br />
<strong>the</strong> determination of normal runoff also in case of comp<strong>le</strong>te<br />
absence of hydrometric data, is based on a combined solution of<br />
water and heat balance equations with <strong>the</strong> account of <strong>the</strong> energy<br />
component of <strong>the</strong> water cyc<strong>le</strong>. Data from standard meteorological<br />
network are used for computation.<br />
RES UME<br />
Le débit moyen annuel, en tant qu'indice des ressources en<br />
eau, peut être déterminé dans <strong>le</strong>s régions montagneuses en se<br />
basant sur <strong>le</strong>s lois de distribution établies pour l'ensemb<strong>le</strong> du<br />
territoire à partir des données obtenues aux stations de jau-<br />
geages, en tenant compte d'une división par zones d'altitude.<br />
Ces lois sont liées à la situation géographique (longitude et<br />
latitude), qui se traduit par des différences dans la nature du<br />
sous-sol et dans l'orientation des pentes par rapport 3 la di-<br />
rection des masses d'air humide. El<strong>le</strong>s se traduisent par des<br />
relations régiona<strong>le</strong>s quantitatives entre <strong>le</strong> ddbit moyen d'une<br />
part et l'altitude moyenne du bassin et <strong>le</strong> pourcentage de gla-<br />
ciers d'autre part. Une autre méthode, permettant d'évaluer <strong>le</strong><br />
débit moyen en l'absence tota<strong>le</strong> de données hydrométriques, met<br />
en jeu la résolution de deux équations, relatives l'une au bi-<br />
lan hydrologique, l'autre au bilan <strong>the</strong>rmique tenant compte des<br />
termes énergétiques du cyc<strong>le</strong> de l'eau. Les calculs sont effec-<br />
tués à partir des données fournies par <strong>le</strong> réseau m6téorologique.
146<br />
A hydrometric network is very scarce in mountain areas since<br />
<strong>the</strong>y are hardly accessib<strong>le</strong>. Methods for <strong>the</strong> evaluation of surface<br />
water resources in case of inadequacy or comp<strong>le</strong>te absence<br />
of observational data are based ei<strong>the</strong>r on <strong>the</strong> account of <strong>the</strong><br />
laws of space distribution of normal annual runoff @pical of<br />
<strong>the</strong> gauged regions or on <strong>the</strong> application of an appropriate<br />
design scheme.<br />
Space distribution of water resources (undisturbed by man's<br />
activities) in mountains and in plains is <strong>the</strong> result of hydrometeorological<br />
factors interactions (precipitation, air temperatue,<br />
evaporation) with underlying surfaces. But unlike plain areas<br />
where latitude and distance from <strong>the</strong> sea serve as main factors<br />
of heat and moisture ratio chasacterizing water resources, <strong>the</strong><br />
orography becomes <strong>the</strong> main factor of river runoff formation in<br />
mountains. The effect of topograpb on river runoff results in<br />
its direct influence on <strong>the</strong> flow velocity down <strong>the</strong> channels,<br />
depending on <strong>the</strong> slopes of watersheds and bqsins top cover. But<br />
<strong>the</strong> most important effect of topography on water resources is developed<br />
by its influence on water balance e<strong>le</strong>ments ( recipitation,<br />
evaporation, change of water storage in river basinsl;. This<br />
effect is of a particular importance in mountainous arid zones<br />
of Asia. For examp<strong>le</strong> gross precipitation in <strong>the</strong> mountains of<br />
Midd<strong>le</strong> Asia, Kazakhstan and Mongolis ranges from 150-lOO mm and<br />
<strong>le</strong>ss in areas protected from humid air masses (hollows, slopes<br />
of unfavourzb<strong>le</strong> orientation) up to 1500-2000 mm and more on<br />
favourably oriented slopes of periphery mountain ridges relative<br />
to air fluxes. The increase of precipitation according to e<strong>le</strong>vation<br />
and simultaneous losses by evaporation on high e<strong>le</strong>vations<br />
due to low air temperatures stipulate <strong>the</strong> improvement of conditions<br />
of river feeding characteristic for mountain areas as far<br />
as <strong>the</strong> basin e<strong>le</strong>vation increases.<br />
In connexion with <strong>the</strong> stated above, methods based on <strong>the</strong><br />
establishment of relations between runoff and orographic peculiarities<br />
of <strong>the</strong> location have been accepted in <strong>the</strong> USSR for <strong>the</strong><br />
evaluation of water resources in poorly gauged mountain areas.<br />
These orographic peculiarities are as follows: e<strong>le</strong>vation, slope<br />
and orientation of <strong>the</strong> region relative to <strong>the</strong> direction of moistw?e<br />
transfer.<br />
For <strong>the</strong> evaluation of mean annual runoff<br />
Q as an index of<br />
areal water resources <strong>the</strong> relations between specific discharge<br />
and e<strong>le</strong>vation of <strong>the</strong> watershed, which in majority of cases is<br />
expressed as mean weighted e<strong>le</strong>vation (H) are widely used.
147<br />
These relations are established for every region nn %hi? bapis<br />
of data obtain9d for gauged watersheds and are uwed ta /?cl;om-ine<br />
normal runoff of ungauged watersheds in <strong>the</strong> a2progriat;e repien.<br />
Sirice high mountain areas are very poorly gaiged tbs cvaluri-<br />
tion of water resources for such areas Is madß according im<br />
extrapolated portions of <strong>the</strong> dependences Cj E f (II). Data on<br />
precipitation, ablation and liquid glacia?. runoff are used %o<br />
make extrapolation more reliab<strong>le</strong> .<br />
In case of data on glacierization w&lab<strong>le</strong> <strong>the</strong>y are ursd<br />
both for <strong>the</strong> extrapolation of dependences Q. = f (HI an0 ?or<br />
direct conput;ation of mean annual runoff iron relat;ively smdl<br />
high mountain areas. According to ths investigations made by<br />
V.L. Schultz /1/ <strong>the</strong> rise of such empirical relations provi
14 8<br />
and subsoils- m is <strong>the</strong> exponent of runoff reduction; I is mean<br />
basin slope [ '/oo).<br />
When river basins are composed of karst rocks <strong>the</strong> effect of<br />
o<strong>the</strong>r azonal factors on river runoff may be neg<strong>le</strong>cted and only<br />
runoff changes caused by karst may be taken into account. Hence,<br />
for examp<strong>le</strong>, an appropriate correction (with negative sign) to<br />
zonal runoff for <strong>the</strong> Kazakh folded area is computed by empirical<br />
e quat ion :<br />
(2)<br />
where: Q is correction (l/sec per 1 km') due to karst effect.<br />
For <strong>the</strong> evaluation of zonal normal runoff <strong>the</strong> maps of isolines<br />
of normal runoff are used; <strong>the</strong>se maps are compi<strong>le</strong>d by<br />
observational data mainly from <strong>the</strong> basins fully located within<br />
one climatic zone. The method of isolines is usually preferab<strong>le</strong><br />
in case of natural water resources evaluation for large river<br />
basins and for poorly gauged mountain areas as a <strong>who<strong>le</strong></strong>.<br />
Very few maps of isolines of normal runoff plotted for<br />
particular mountain areas are availab<strong>le</strong> in <strong>the</strong> USSR. Since <strong>the</strong><br />
initial &ta are linited, <strong>the</strong>se maps are small-sca<strong>le</strong>d, mainly<br />
of 1 : 2 500 O00 sca<strong>le</strong> not more; <strong>the</strong>se maps are hardly suitab<strong>le</strong><br />
for <strong>the</strong> estimation of normal annual rynoff from small and midd<strong>le</strong>size<br />
watersheâs not exceeding 1000 km . Thus, <strong>the</strong> method of isolines<br />
provides a sufficiently accurate determination of normal annua3<br />
runoff mainly for large mountain watersheds (more than 1000 km 1.<br />
The method of col<strong>le</strong>ctive analom based on <strong>the</strong> graphs of relations<br />
between runoff and mean basin e<strong>le</strong>vation is used bn majoritg of<br />
cases for <strong>the</strong> computation of runoff from midd<strong>le</strong>-size basins, i.e.<br />
more than 500-600 km2. When computing runoff from small basins<br />
and often larger basins <strong>the</strong> use of <strong>the</strong> two mentioned methods<br />
is not always reasonab<strong>le</strong>. It is often explained by inadequacy of<br />
initial information on runoff and by a considerab<strong>le</strong> effect of<br />
azonal factors in mountains; in this connexion even basins with<br />
similar e<strong>le</strong>vation within <strong>the</strong> same mountain region may differ<br />
greatly in <strong>the</strong> conditions of runoff formation and its quantitative<br />
characteristics.<br />
In such cases <strong>the</strong> determination of normal annual runoff<br />
from mountain watersheds located in conditions of sufficient<br />
and excessive moistening is made by a combined solution of<br />
equations of water and heat balances. An indubitab<strong>le</strong> advantage<br />
of this method is in <strong>the</strong> fact it ensures a relatively accurate<br />
determination of normal annual runoff not only from large<br />
mountai basins but from watersheds with <strong>the</strong> areas not exceeding<br />
1000 km 9 .<br />
The computation is based on <strong>the</strong> equation of mean long-term<br />
annual water balance where normal annual runoff is determined
149<br />
by <strong>the</strong> difference between precipitation P and evaporation E:<br />
Q= P-E (3)<br />
When usin@; this equation it is essential to obtain a<br />
reliab<strong>le</strong> accuracy in determination of noml annual precipitation<br />
and evaporation. The method is applicab<strong>le</strong> for such mountain<br />
areas where <strong>the</strong> availab<strong>le</strong> hydrometeorological network provides an<br />
objective evaluation of precipitation distribution compared with<br />
runoff. In this case it should be kept in mind that evaporation<br />
is <strong>le</strong>ss variab<strong>le</strong> over area and altitudinal zones compared with<br />
runoff and it ILK' be computed for mountain watersheds with a<br />
sufficient accuracy .<br />
It should be noted that in equation (3) underground water<br />
exchange with adjacent watersheds is not taken into account1 As<br />
a ru<strong>le</strong>, this component is not big in mountains especially in<br />
permafrost zone. But in cases when its vali<strong>le</strong>s are commensurab<strong>le</strong><br />
with <strong>the</strong> o<strong>the</strong>r values of equation (3) <strong>the</strong> account of this<br />
component is essential.<br />
The determination of one of <strong>the</strong> parameters in equation (3)<br />
i.e. normal annual precipitation, is made with <strong>the</strong> use of<br />
graphs of precipitation and e<strong>le</strong>vation with <strong>the</strong> account of local<br />
orographic peculiarities. In this case correction should be introduced<br />
for <strong>the</strong> initial data which take into account <strong>the</strong> underestimation<br />
of precipitation by standard precipitation gauges.<br />
Computation of normal annual evaporation is made by equation:<br />
where: W is radiation balance of <strong>the</strong> moistened surface; Wa is<br />
tubu<strong>le</strong>& heat exchange; L is latent heat of evaporation; e<br />
is base of natural logarithms; th is hyperbolic tangent.<br />
Equation (4) is a precised version of M.I. Budyko's equation<br />
/3/ due to Wa value.<br />
In <strong>the</strong> right of equation (4) three unknown parameters are<br />
intrmduced: P, W and W .<br />
The way of de$ermina!ion of normal annual precipitation ie given<br />
above .<br />
The value6 of radiation balance of <strong>the</strong> moistened surface may<br />
be taken from appropriate maps or computed. For many areas within<br />
<strong>the</strong> USSR territory <strong>the</strong>re exist design formulae for W<br />
determina-<br />
tion according to latitude and e<strong>le</strong>vation of <strong>the</strong> loca%LQ. In<br />
particular, for Trans-Baikal area /4/ such formula may be<br />
presented as follows:
150<br />
where: '9" is mean watershed latitude; h = (H - SOO ) is <strong>the</strong><br />
exceedence of mean watershed latitude over 500 a.s.1. When<br />
computing radiation balance of <strong>the</strong> moistened surface of<br />
mountain watersheds its variations according to <strong>the</strong> exposure<br />
and steepness of slopes are taken into account.<br />
The uetermination of Wa as well as Wp is made ei<strong>the</strong>r according<br />
to appropriate maps or, in case of availab<strong>le</strong> data on mean<br />
long-term monthly air temperature, water vapour pressure and<br />
total cloudiness, by a combined solution of heat balance<br />
equation and <strong>the</strong> equation of Magnus. This method is presented<br />
in detail in some publications /5,6/. When using equation (4)<br />
it should be noted that mean long-term annual values of W<br />
and W are ra<strong>the</strong>r stab<strong>le</strong> characteristics slowly changing {ver<br />
territory a d altitudinal zones.<br />
After computing normal annual evaporation it is possib<strong>le</strong> to<br />
estimate mean long-term annual runoff by equation (3).<br />
It should be noted that <strong>the</strong> presented scheme of computation<br />
may be changed for watersheds located in low and midd<strong>le</strong>-height<br />
mountains. As to watersheds covering high mountain zone, this<br />
method of combined aolution of water and heat balances for<br />
normal annual runoff determination may be applied as well;<br />
<strong>the</strong> difference is that <strong>the</strong> number of terms in water balance<br />
equation increases ( it is essential to take into account <strong>the</strong><br />
ablation of glaciers, melting of snow fields, separate account<br />
of evasoration from different types of underïying surfaces<br />
in high aountains, i.e. ice, snow, talus and rocks).<br />
The evaluation of long-term variations of surface water<br />
resources of poorly gauged mountain areas is usually made &y an<br />
analytical frequency cume of annual river runoff. The values<br />
of runoff variation coefficient C, essential for its plotting<br />
are evaluated by <strong>the</strong>ir regional empirical relations be tween<br />
normal runoff, mean weighted e<strong>le</strong>vation of <strong>the</strong> watershed or <strong>the</strong><br />
glacierization area. These relations are established according<br />
to observational data from <strong>the</strong> gauged rivers, and <strong>the</strong> coefficient<br />
of asymmetry C is established by <strong>the</strong> ratio of this parameter<br />
and coefficient of variations for <strong>the</strong> gauged rivers of <strong>the</strong> region.<br />
Ìimpirical dependences of variation coefficient of annual<br />
runo?f and . <strong>the</strong> determining factors for ungauged mountain areas<br />
are usually<br />
expressed by equations:
where: a, b, c and r are regional parameters; H is mean<br />
weighted e<strong>le</strong>vation of watershed; is exponent of watershed<br />
glacierieation(percentage from <strong>the</strong> total draina@ area),<br />
The se<strong>le</strong>ction of <strong>the</strong> equation depends on <strong>the</strong> character of<br />
151<br />
river feeding. The dependence of variation ûoeff icient of annusl<br />
runoff and specific river discharge (equation 6) are used maim for areas with a considerab<strong>le</strong> portion of rainfalls in mountain<br />
river feeding.<br />
When estimating C, for ungauged mountain rivers of <strong>the</strong> arid<br />
zone where <strong>the</strong> effect of snow melt water is of particular<br />
importance, <strong>the</strong> preference is given to <strong>the</strong> relations of variation<br />
coefficient and mean weighted watershed e<strong>le</strong>vation (equation 7).<br />
For rivers located in basins where glaciers cover more than<br />
IQ% of <strong>the</strong> drainage area <strong>the</strong> dependence of C, upon mean weighted<br />
e<strong>le</strong>vation is usually broken and <strong>the</strong> preference is iven %o empirical<br />
relations between Cv and basin glacierization ? equation 8).<br />
If <strong>the</strong>re are no data availab<strong>le</strong> on <strong>the</strong> amount of glaciers <strong>the</strong>n<br />
instead of glacierization rate for C, determination of ungauged<br />
mountain rivers its indirect indices are sometimes used showing<br />
<strong>the</strong> relations between <strong>the</strong> area of altitudinal zone where glaciers<br />
are located and <strong>the</strong> area of <strong>the</strong> <strong>who<strong>le</strong></strong> basin.<br />
For <strong>the</strong> determination of variation coefficient of-annual run-<br />
off of mountain rivers <strong>the</strong> equation recommended by LP. Voskre-<br />
senski /7/ is used as well :<br />
where: X is regional parameter.<br />
The coefficient of asymmetry of mean annual runoff for ungauged<br />
mountain rivers is usually accepted as C, = 2Cv.<br />
RE F E R E N C E S<br />
1. Schultz V.L. Reki Srednei Asii (Midd<strong>le</strong> Asia rivers). Pt.1 and<br />
2, Hydrometeorol. Publ. House, Leningrad, 1965.<br />
2. Lavrentiev P.F., Semenov V.A., Khitrunova M.S. Uchet sredaei<br />
vysotg vodosborov, ikh orientatsii i azonalnykh faktorov<br />
podstila jushche i poverknosti gri rasc hetakh srednego<br />
godovogo stoka rek Severnogo Kazakhstana (The account of<br />
mean basins e<strong>le</strong>vation, <strong>the</strong>ir orientation and azonal facGor’P<br />
of <strong>the</strong> underlying surface when computing mean annual runoff<br />
of north Eazakhstan rivers), Trana. of Kaz. NIGBdI,<br />
VOI. 41, 1971.<br />
3. Budyko M.I. Teplovoi balans zemnoi poverkhnoeti (Heat ba<strong>le</strong>me<br />
of <strong>the</strong> Earth’s surface). Hydrometeorol. Publ. House,<br />
Leningrad, 1956.
152<br />
4. Vuglinski V.S. Yetodika rascheta radiatsionnogo balansa<br />
gornoi territorii i ee primenenie na primere<br />
basseina r. Vitini (Methods for <strong>the</strong> computation of<br />
radiation balance of mountain area and its applica-<br />
tion illustrated by <strong>the</strong> Vitim river basin). Trans.<br />
of GGI, 1972, ~01. 199.<br />
5. Vuglinski V.S. Raschet normy godovogo stoka neisuchennykh<br />
gornykh rek s primeneniem uravneniy vocino o i teplovo-<br />
go balansov (na priiaere basseina r. Vitimy. (Compu-<br />
tation of normal annual runoff of ungauged mountain<br />
rivers WI th <strong>the</strong> use of equations of water and heat<br />
balances !strated by <strong>the</strong> Vitfm river basin).<br />
Trans. of &(GI, 1972, vol. 200.<br />
6. Anàreyanov V.G. Vnutrigodovoe rasprede<strong>le</strong>nie rechnogo<br />
stoka (Annual stream flow distribution), Leningrad,<br />
Hyàrometeorol. Publ. House, 1961.<br />
as Voskresenski K,P. Horma i izmenchivost godovogo stoka rek<br />
Sovetskogo Sojusa (Normal annual runoff and its<br />
variations for <strong>the</strong> rivers of <strong>the</strong> Soviet Union).<br />
Leningrad, Hydrometeorol. Publ. House, 1962,
IMPROVEMENT OF HYDROLOGICAL INFORMATION FC?. FX.?:LCT<br />
DESIGN BY SHORT TERM MEASURES -<br />
1. Introduction<br />
GENERAL REPORT<br />
by<br />
Dr. John Rodda<br />
At <strong>the</strong> present time, when <strong>the</strong> amount of attention given to all asjxcta of<br />
th:: environnent is growing rapidly, <strong>the</strong> col<strong>le</strong>ct ion of enviromental<br />
information is increasing in importance, especicaìly for use as o m ~f <strong>the</strong><br />
bases Of measures for enviromental protection and combatting pollution.<br />
Kere <strong>the</strong> hydrologist and meteorologist are amongst <strong>the</strong> more fortam.tc of<br />
environmental scientists: <strong>the</strong>y probably have at <strong>the</strong>ir dislwaal a lager<br />
body of information re<strong>le</strong>vant to <strong>the</strong>ir needs than is availab<strong>le</strong> tc oihcr<br />
scientists in <strong>the</strong>ir particular fields. On <strong>the</strong> o<strong>the</strong>r hand it is true to c . ~<br />
that many watcr resources projects are designed with inaclc::imte data, indcd,<br />
sometimes with virtual1.y no date zt all.<br />
likely that wrong decisions will be taken, that wrone critcria will be<br />
se<strong>le</strong>cted and that inappropriate and uneconomic designs will be adopted.<br />
The end product can be a water resources system which pstly or entirely<br />
fails to meet <strong>the</strong> objectives that were foreseen for it, <strong>the</strong> bexr‘its it<br />
pro&uces,bcaring litt<strong>le</strong> relation to <strong>the</strong> capital invested.<br />
2. zata and Networks<br />
The classic response to this situation is to col<strong>le</strong>ct noce and m ro data<br />
for thc national archive. Aniassine; a large quaiitily or^ ìucirologi.:al<br />
information is even seen as an end in itself and virtus.lly mJ’ 5.ncrease<br />
in <strong>the</strong> total is considered of value.<br />
particularly where national data col<strong>le</strong>ction propmr,:es are nct plmned<br />
scieiitifically.<br />
types, generally rainfdl and streamflow records, o<strong>the</strong>r LW’C~S heina very<br />
largcly neg<strong>le</strong>cted; for examp<strong>le</strong>, sediment surveys and soil muistiire records.<br />
St&tions in <strong>the</strong> data network are often badly distributed, t5m-C arc<br />
differences in <strong>the</strong> <strong>le</strong>ngths of records and <strong>the</strong>ir quality is frcqixntly<br />
suspect. Such networks usually produce information inefficienti.;? and<br />
uneconomically - <strong>the</strong> oppoeite of <strong>the</strong> true objective of network &si@.<br />
Scientific design would produce a system whioh would add <strong>the</strong> moi;:<br />
know1ede;e for <strong>the</strong> <strong>le</strong>ast effort. This sytem would ncit only consïf;t of<br />
station-type time series observations, but also of surveys of various<br />
kinds, including questionnaires 2nd oensusee. It would not be ti riKi8 c;’stcrn,<br />
but one that would be altered and amended in response to needa and as<br />
objectives change.<br />
Lui if th.e purpose of a network can be stated clczrly .<strong>the</strong>n itn ?tasip is<br />
likely’to be fecilitated.<br />
Inadequste duta m ke it more<br />
This is not alwRjr:; <strong>the</strong> CasQ,(<br />
Usually <strong>the</strong> bulk of <strong>the</strong> information is of cno or two<br />
üf ooume networks iisly have various objectivce<br />
F’or project design purposes a network would have
I54<br />
a different form and composition from a network instal<strong>le</strong>d for researoh<br />
purposes, although both would be COmPOnCntO of and contributors to <strong>the</strong><br />
national network which would itself provide <strong>the</strong> overall information framework.<br />
in <strong>the</strong> oontext of this symposium it is important to consider firat<br />
<strong>the</strong> form of <strong>the</strong> national network and <strong>the</strong> attributes that would fit it best<br />
to <strong>the</strong> needs of project deoign end second <strong>the</strong> project nbtwoa itself.<br />
3. Data PrODertißS<br />
Langbein (1972) wpsted that water data, And thus <strong>the</strong> n dork for<br />
acquiring <strong>the</strong>m, have three intrinsio properties:<br />
and continuity. Imoartialitg relates to <strong>the</strong> aganoy or Bgenoiee that<br />
operate <strong>the</strong> network and archive <strong>the</strong> infomatttion from it. In its<br />
Perception of data prob<strong>le</strong>ms, <strong>the</strong> agency itself tends to introduce R bias<br />
in <strong>the</strong> data, a bias towards ita own speaialty. Por examp<strong>le</strong> M organieatioii<br />
concerned with water suppl.y wouiû tend to dieregard infomation about<br />
floods and <strong>the</strong> means of col<strong>le</strong>oting <strong>the</strong>se data. One solution to this<br />
prob<strong>le</strong>m is for basic data ool<strong>le</strong>ction to be <strong>the</strong> remit of en agenay without<br />
opcratiûïìal or cxccative ro<strong>le</strong>s, such ôs onc ifi-dve3 in reeearch.<br />
Re<strong>le</strong>vance of <strong>the</strong> data <strong>the</strong>n becmes important because thie type of data<br />
agency is one stage removed from <strong>the</strong> prob<strong>le</strong>ms to which <strong>the</strong> data arc applied.<br />
On <strong>the</strong> o<strong>the</strong>r hand, this avoids what Langbein calls <strong>the</strong> "squeeking wheel<br />
princip<strong>le</strong> whereby attention is continually uircctod toward8 mrrently urgcrì:<br />
prob<strong>le</strong>ma at <strong>the</strong> expense of <strong>the</strong> existing balance of <strong>the</strong> network and its<br />
oapacity to be employed for solving future as yet -own prob<strong>le</strong>ms,<br />
Continuity follows from <strong>the</strong> fact that hydrological datu are time-dependent,<br />
hence <strong>the</strong>ir col<strong>le</strong>ction needs to be oontinuous. Continuity ia at risk<br />
at times of national atresa euch as during tine of war, -turd disaster<br />
or finanoiai stringency.<br />
agencies are required to alter <strong>the</strong>ir progremmos and those not directed to<br />
o<strong>le</strong>ar and easily recornisab<strong>le</strong> objeotives tend to be curtai<strong>le</strong>d.<br />
4. The Fctwork<br />
impartiality, re<strong>le</strong>vance<br />
Organisationai change ia also a haed;<br />
Ideally <strong>the</strong> oountrywide hydrologiosl network should preoeed development,<br />
invariably <strong>the</strong> reverse is true. Most national networks have resultßd<br />
from ad hoc responses to particular prob<strong>le</strong>ms. Pow networks Beem to have<br />
reached <strong>the</strong> optimum in termo of distribution of stations, types of data<br />
and form of amhive. Perhaps <strong>the</strong> difficulty of deoiding what <strong>the</strong><br />
optimum is one reason for this, although Dswdy et al (1972) put forward <strong>the</strong><br />
idea of <strong>the</strong> <strong>le</strong>oel of information being optiwl when decisions involviq<br />
this infomation become insensitive to its m her inorease. !Phis ooucept<br />
has <strong>the</strong> difficulty that <strong>the</strong> optimum information <strong>le</strong>vel and thus tho optimum<br />
network differs for different objeotivee, so that it m w not be readily
applicab<strong>le</strong> to a malti-purpose countrywide network.<br />
prob<strong>le</strong>m of sca<strong>le</strong> and <strong>the</strong> fact that <strong>the</strong> component parts of <strong>the</strong> hydrolo&al<br />
network may have developed separatoly crnd to differing degrees.<br />
155<br />
<strong>the</strong> evaporation network and <strong>the</strong> water quality network are unlikely to have<br />
been co-ordinated and this may apply to <strong>the</strong> o<strong>the</strong>r variab<strong>le</strong>s.<br />
5. Sca<strong>le</strong> of Networks<br />
The factor of soa<strong>le</strong> ia pu1 important point to consider in project design for<br />
<strong>the</strong>re are differences between information needs on national and local<br />
sca<strong>le</strong>s. At <strong>the</strong> national <strong>le</strong>vel tho network would consist of long term,<br />
bench mark primary stations for sampling in <strong>the</strong> main, variations in time.<br />
The distribution of <strong>the</strong>so stations would relate to <strong>the</strong> degree of <strong>the</strong><br />
country's development and its hydrological heterogeneity. in o<strong>the</strong>r words<br />
a country with uniform climate, geolot;y and relief, a small number of<br />
inhzbitants utilizinl: few of <strong>the</strong> resources woulù most probtrbly possess<br />
a <strong>le</strong>ss developed network than oneaith diVerBe physical features, a<br />
large population and a strong industrial base.<br />
lhe twu couiitrias would present a siuiLr GGZ~IYLS:, bUt both crtrcrkz<br />
should necessarily be capab<strong>le</strong> of utilization, at <strong>the</strong> vem <strong>le</strong>ast, for<br />
accounting for <strong>the</strong> resource and for <strong>the</strong> warning of hazards.<br />
1Òcal sca<strong>le</strong> stations would tend to be of a short term, secondary tyae<br />
(Gandin 1967) established to samp<strong>le</strong> variability in space.<br />
network would be a major part of this secondary network.<br />
secondary network would mostly serve current information needs, <strong>the</strong> baoio<br />
countrywide network would satisfy future demands (Laagbein 1965).<br />
6.<br />
Use of Basic Network for Roject Desim Purposes<br />
Information from <strong>the</strong> basio network can be employed to provide estimates<br />
of hydrological variab<strong>le</strong>s for any Given point within a country and can<br />
thus be applied for project desi-, purposes. The estimâtion may be<br />
undertaken SbjJly bf interpolation between isop<strong>le</strong>ths on B countrywide<br />
map, constructed from <strong>the</strong> basic network. observations.<br />
data fram <strong>the</strong> network can be applied in a mapping technique suoh as<br />
<strong>the</strong> application of <strong>the</strong> grid system for storage and processing hydrological<br />
information from a large area and its use in relating <strong>the</strong> hydrolgical<br />
variab<strong>le</strong>s to <strong>the</strong> area's physical characteristics (Solomon et al 1968).<br />
Maps of mean annual precipitation, temperature and waporation were<br />
constructed by this method and <strong>the</strong>n employed with measures of <strong>the</strong><br />
topograpliy to develop a map of runoff.<br />
There io also <strong>the</strong><br />
importanoe of maps, maps also being important to that regionalisation<br />
type of approaoh. For examp<strong>le</strong>, measureß of <strong>the</strong> pertinent eurface or<br />
For exanp<strong>le</strong><br />
The information needs of<br />
At <strong>the</strong><br />
The project<br />
Whereas <strong>the</strong><br />
Alternatively<br />
This approach stresses <strong>the</strong>
156<br />
subsurface features of an area whioh can be mapped or measiired in <strong>the</strong><br />
field are related to a otatistic Of <strong>the</strong> hydrological vari;:b<strong>le</strong> in queetion.<br />
Relationships between mean annual rainfall amounts and niemures of -Lhe<br />
topo6.raph.y such ae a<strong>le</strong>vationplope and exposure have been widely de.lcrmined,<br />
likewise relations between <strong>the</strong> mean annual flood and catchment characteristics<br />
including area,channel slope anù drainé@ density.<br />
The paper 'sIrnprovement of runoff records in smal<strong>le</strong>r watershede, based on<br />
permeability of <strong>the</strong> geological subsurfacess by Dr Bala&Xun follows this<br />
2<br />
type of appl-oach. Records from basins <strong>le</strong>ss than 25OKu1 in area locaied<br />
in li!? USA and Central Europe were used in thio study. P~ak runoffs<br />
(m3 sec -' Km2) of 100 year return period were related to basin size, <strong>the</strong><br />
geological character of those basins assessed from permeability<br />
certain storm sires and intensities and also to <strong>the</strong> slope of <strong>the</strong> basin.<br />
Dr Halasi-Kun siicgests <strong>the</strong>re is evidence for a significant correlation<br />
betuem permeability and peak runoff but that <strong>the</strong> geological effect<br />
2<br />
"fades" for basins larger than 235h . This paper also ermines 50 year<br />
2<br />
low flows in <strong>the</strong> same bapjns (1 sec-' Km ) and again relates ihese flow6<br />
to p3OlOC;iC;rl characteristics. The author concludes that including<br />
permeability improves this type of approach and of course he is correct<br />
in <strong>the</strong> sense that <strong>the</strong> inclusion of any fur<strong>the</strong>r uncorrelated but<br />
quantifiab<strong>le</strong> catchment characteristics is a step forward.<br />
he hacl included as much of his basic data that it was possibla to<br />
publish.<br />
It does not deal with estimation from catchment characteristics but with<br />
reconstruction of records from a shorter period of more complote records<br />
applied to a longer period of more limited information. This is <strong>the</strong><br />
paper by Ih. Kovecs and Dr Kolnar: "Determination of snow water<br />
eqyivs<strong>le</strong>nt and enow melt water by thickness of snow cover data''<br />
One wishes<br />
Ano<strong>the</strong>r paper submitted for this section of <strong>the</strong> programme which (]:!ala<br />
uith information from <strong>the</strong> basic network is not strictly in <strong>the</strong> mine cztegorg.<br />
Snow<br />
depth has been observed at about 1000 stations in Hungary for 100 years<br />
and since 1960, water equiva<strong>le</strong>nt has also been measured at 60 stations.<br />
From studios of <strong>the</strong> bulk density of fresh snow ( Y min), mow saturatcd<br />
with capillmy water ( Y k) and melting mow ( 5' ma), snow depth and <strong>the</strong><br />
number of layers of enow developed during accumulation (R)o tanned <strong>the</strong><br />
critical bulk density, and R.<br />
between y max and R and dao a method for obtaining <strong>the</strong> duration of<br />
melting from air temperature records during <strong>the</strong> molting period. These<br />
relations are <strong>the</strong>n applied to <strong>the</strong> hindcasting of snow water equiva<strong>le</strong>nts<br />
from depth measurements and to forecasting duuration of <strong>the</strong> melt period<br />
and potential volume of melt water.<br />
Then <strong>the</strong>y arrive at a similar relation<br />
The predicted snow water e-iva<strong>le</strong>nt?:<br />
are oomparcd with meamred volumes at one site for part of 1963 and <strong>the</strong>
match between <strong>the</strong>m seems reasonably good.<br />
provided for <strong>the</strong> enow melt cdoulations.<br />
15 7<br />
The paper by Ur Beard œHydrological dzta fill in and Network Desi&* is<br />
similar to <strong>the</strong> previous one in that it deals with <strong>the</strong> extension of<br />
records from stream gaugingetations with long recoi.de to stations with<br />
only short ~%cordS. A stoolustic model, which can accept montly data, is<br />
based upon multip<strong>le</strong> linear regressions, using fransformed variab<strong>le</strong>s, and<br />
<strong>the</strong>se are derived from each station for eachGa<strong>le</strong>ndar month. To illustrate<br />
what happens as a result of chanoe variations in small samp<strong>le</strong>s, 10,000<br />
5-year samp<strong>le</strong>8 were drawn from a normal population and <strong>the</strong>ir means and<br />
standard deviations oalculated. For each samp<strong>le</strong> items were perated and<br />
<strong>the</strong>ir location in <strong>the</strong> parent population identified. It was found that in<br />
<strong>the</strong> oase of extremes tco many extreme vdues were generated indicating a<br />
bias in <strong>the</strong> estimates of extremes.mzde from small samp<strong>le</strong>s. To overcome<br />
this bias a transom hinotion was generated.<br />
(equation 2) showing <strong>the</strong> nucber ~f itefie 3: iiaedid in <strong>the</strong> shrt-tem<br />
recorà than can improve <strong>the</strong> accuracy of <strong>the</strong> short-term mean so that it ia<br />
reliab<strong>le</strong> as <strong>the</strong> mean obtained l'rom <strong>the</strong> lon&er reoord Values of I, are<br />
tabulated for various correlations and samplo ßizes against <strong>the</strong> longer<br />
record <strong>le</strong>ngth (tab<strong>le</strong> 2). Four different %year soquencee were se<strong>le</strong>cted<br />
from 40 years of reoord at one station and for each of tho four cases tho<br />
remaining 35 years were fil<strong>le</strong>d in.<br />
for <strong>the</strong> 40 years of reconstructed record ware compared with <strong>the</strong> actual<br />
40 year mean. The process wzs repeated Sor 3 o<strong>the</strong>r stations and a matrix<br />
is presented showing 8 comparison of statistics derived from <strong>the</strong>se recodo.<br />
The author concludes that for correlations above 0.95 short records need<br />
not be continued beyond 5 years but belar 0.8 short records should be<br />
oontinued. Between <strong>the</strong>se values a study of regional variations would<br />
reveal <strong>the</strong> relative advantage of continuing existing stations or starting<br />
new ones.<br />
A similar cornpanson is not<br />
An equation is given<br />
Then <strong>the</strong> mean flow for <strong>the</strong> %years and<br />
Arising from studies like <strong>the</strong>ee is <strong>the</strong> question of how estimateo compare<br />
with field measurements. Nash and Shaw (1966) in a study of United Kingdom<br />
floods, disoovered that even a sing<strong>le</strong> year of discharge records produced<br />
a more reliab<strong>le</strong> guida to <strong>the</strong> mean annual flood than <strong>the</strong> methods of<br />
estimation <strong>the</strong>n in current use. lore recently <strong>the</strong> UK Flood Study Team<br />
have found (Sutcliffe 1973) that estimated mean annual floods are within<br />
2 30$ of <strong>the</strong> mean of <strong>the</strong> measured annual maxima for <strong>the</strong> catchments<br />
studied.<br />
t
158<br />
7.<br />
Short Term Instrumental and’Observational kessurez<br />
There are a number Of constraints to <strong>the</strong> design of a project network,<br />
time probably being <strong>the</strong> most important. Usually <strong>the</strong>re are only 2 or 3<br />
years between <strong>the</strong> time a Project is conoeived and <strong>the</strong> time when <strong>the</strong><br />
design has to be finalised. The risks involved in employing <strong>the</strong>se<br />
2 or 3 years of information is <strong>the</strong>n a maximum but <strong>the</strong> risks diminish as<br />
<strong>the</strong> record <strong>le</strong>ngth inCrcaseS.<br />
more records may be COStly in terms of loss of benefit from <strong>the</strong> water<br />
resources system. At some Point a balance will be struck‘ between risk<br />
and benefit: this point will depend upon factors such a8 <strong>the</strong> type of<br />
project and <strong>the</strong> Proportion cf <strong>the</strong> resource to be develo2ed. Amongfit <strong>the</strong><br />
o<strong>the</strong>r constraints are those of finance, <strong>the</strong> skills availab<strong>le</strong> and <strong>the</strong><br />
location of <strong>the</strong> project. Uith adequate funds a well-equipped team can be<br />
brought toge<strong>the</strong>r and <strong>the</strong> project placed on a firm footing.<br />
location in terms of OlimatC and topopa& can be a very considerab<strong>le</strong><br />
handicap even with a well funded project.<br />
However, deferring a project to accumulate<br />
Depending on <strong>the</strong> nature of <strong>the</strong> projeot and <strong>the</strong> informstion it requires<br />
<strong>the</strong> defiign of <strong>the</strong> network hinges on, <strong>the</strong> answers to a nimber of<br />
An unfavourab<strong>le</strong><br />
questions:<br />
1. How is <strong>the</strong> information to be obtained?<br />
2. How many sites need it be obtpined from?<br />
3. Where are <strong>the</strong>se sites to be located?<br />
No papers were submitted describing an:! advances in inckwmentation or gouncï<br />
based survey techniques that might be applied to project design.<br />
are new instruments and new methods that could be employed to acquire<br />
information for pro jeot deoign. Batterj operated magnetic tape recording<br />
rain gauges and te<strong>le</strong>metering gauges proùuce more information more rapidly<br />
than conventional instruments, Automatic wea<strong>the</strong>r stations and automatically<br />
operated neutron probes do <strong>the</strong> same in <strong>the</strong> fields of evaporation aild soil<br />
moisture measuremcnt and <strong>the</strong>n <strong>the</strong>re are automatic dilution gauging devices<br />
for atream flow measurement to say nothing of <strong>the</strong> o<strong>the</strong>r methods of river<br />
gauging that do not require <strong>the</strong> conventional stilling well and structure<br />
in <strong>the</strong> channel.<br />
recent years, but <strong>the</strong>re was only one paper submitted to this Section in<br />
this category.<br />
Remote sensing techniques have dvaiced eiiormously in<br />
Hhat about <strong>the</strong> use of aerial photography,radar and <strong>the</strong><br />
various forms of imagery from satellites?<br />
exception is <strong>the</strong> paper by Dr <strong>le</strong>ijerink “Svaluation of local water resouses<br />
in a semi-arid hard rock region, by using photo-hydrological indices”.<br />
Yet <strong>the</strong>re<br />
They are not mentioned. The<br />
By interpreting aerial photographs an assessment was made of <strong>the</strong> local<br />
water resourcës in part of <strong>the</strong> Cuddapah Basin in south India. Following<br />
field surveys of <strong>the</strong> area‘s geologyssoils and land use <strong>the</strong> next stop was
to divide <strong>the</strong> basin into hydrologically homongeous 1andBczpes. The<br />
hyydrology of <strong>the</strong>se landscapes was deduced from <strong>the</strong> photographs from<br />
159<br />
features affected by surface flow and from <strong>the</strong> characteristics of <strong>the</strong><br />
superficiel deposits and solid geolow. For examp<strong>le</strong> within a particular<br />
landscapo <strong>the</strong> yield of a well is assmiod to be directly related to <strong>the</strong><br />
size of <strong>the</strong> irrigated area.<br />
for one landficape and also <strong>the</strong> recharge areas for those wells;<br />
relationship between <strong>the</strong>se factors giving a guide to yield &B a function<br />
of recharce area.<br />
and <strong>the</strong> results ct <strong>the</strong> interpretation were checked in <strong>the</strong> field for <strong>the</strong><br />
different relationships.<br />
The question of <strong>the</strong> number of sites to be samp<strong>le</strong>d is frequently answered<br />
in terms of <strong>the</strong> funds availab<strong>le</strong> for installing and oparating <strong>the</strong> netuork.<br />
For areas without records of any cort,arriving at a number is particularly<br />
difficillt for <strong>the</strong> number and location of stations hincos OA <strong>the</strong> distribution<br />
of <strong>the</strong> hydrologiczl variab<strong>le</strong>.<br />
according to a predermined grid or according to <strong>the</strong> distribution of e<strong>le</strong>vation.<br />
Ano<strong>the</strong>r method would be to delimit areas of homogeneous topograpbjand<br />
~eolocr and to site one station in each area.<br />
exist it is usually far simp<strong>le</strong>r to deterinine where tÒ site additional<br />
gauges.<br />
This is one of <strong>the</strong> topics discussed in <strong>the</strong> paper by Kessiers Delhomme and<br />
Delfiner: "Applicaton du Krigeage a' l'optimisation du'une coinpagne<br />
Pluviometrique en zone aride". The subject of this paper is <strong>the</strong><br />
2<br />
Kadjemeur Wadi in <strong>the</strong> east of Chad , a basin 245Km in area containing<br />
33 rain gauges. Here <strong>the</strong> technique of Kriging is employed to determine<br />
<strong>the</strong> o ptim weights of <strong>the</strong> gauges in thc network for <strong>the</strong> calculation of<br />
<strong>the</strong> mean basin rainfall.<br />
of <strong>the</strong> method and <strong>the</strong>n <strong>the</strong>y apply it to <strong>the</strong> description of a storm on<br />
6 AU~UQ~ 1966. Thc map obtained by ttìa technique of kriging niötohes <strong>the</strong><br />
hand drawn isobyetal map very well; in general it produces a broader<br />
smoo<strong>the</strong>r interpretation. A comparison of <strong>the</strong> estimates of <strong>the</strong> mean basin<br />
rainfalls is given for Krigingand three o<strong>the</strong>r methods, <strong>the</strong> "hiessen,<br />
mithmetic mean and planimetering methods. In general <strong>the</strong> results are sirnilar<br />
but <strong>the</strong> concentration of gauges on <strong>the</strong> western side of <strong>the</strong> basis distorts<br />
<strong>the</strong> arithmetic mean results ifi some storms.<br />
<strong>the</strong> prob<strong>le</strong>m of where to locate an extra page.<br />
gauges<br />
<strong>the</strong> barain or in <strong>the</strong> centre.<br />
The areas irrigated by wells were determined<br />
<strong>the</strong> basin, where <strong>the</strong> gain of information is a maaimum b? construoting<br />
isopeths of gain fiyre 8 shows where <strong>the</strong>se two points axe located - on<br />
<strong>the</strong><br />
A similar exercise was under-taken for surfzce wa.ter<br />
One method would be to site stations<br />
Where some stations already<br />
The authors provide a background to <strong>the</strong> <strong>the</strong>ory<br />
Finally <strong>the</strong> authorsooncider<br />
From <strong>the</strong> distribution of<br />
subjectively one would choose a site at <strong>the</strong> south eastern end of<br />
Kriging ~ 110~s determination of<strong>the</strong> point in
160<br />
<strong>the</strong> south eastern boundary and in <strong>the</strong> centre of <strong>the</strong> basin. There are<br />
various metliods for computing <strong>the</strong> mean basin .rainfall th8-t have been<br />
advocated recently - various forms Of surfaïri fittinc Sor examp<strong>le</strong>.<br />
problcm is that all <strong>the</strong>se methods rely on <strong>the</strong> accuracy of <strong>the</strong> point<br />
rainfall measurements which we kno~ as being far from accurato. The<br />
question of' what is <strong>the</strong> true mean basin rainfall remalns unanswered.<br />
CONCI Ir3IONS<br />
In <strong>the</strong> verj <strong>le</strong>ngthy tit<strong>le</strong> to thio section in <strong>the</strong> prop-mme for <strong>the</strong><br />
symposium, two separate topics were raised, first <strong>the</strong> improvement of<br />
hydrological information by short term measures and second <strong>the</strong> value<br />
of such measures, particularly as expressed by project economios. Whi<strong>le</strong><br />
one might argue that <strong>the</strong> first topic is covere? by <strong>the</strong> five papers<br />
reviewedobviously,<strong>the</strong> absence of any papers for <strong>the</strong> second is a<br />
significant pointer to tho need for work on thin topic. The reviewer<br />
proposes that UNESCO and Mi0 should consider streng<strong>the</strong>ning activities<br />
in this field by appointing a rapporteur to prepare a guide to methods<br />
that may be applied to this prob<strong>le</strong>m.<br />
References<br />
Langbein W B<br />
Dawdy D R<br />
Gandin L S<br />
Langbein W B<br />
Solomon S I<br />
Nash J E and<br />
Shaw B L<br />
1972 "Water Da-ta Today and in Prospect"<br />
flydroloaical Sciences Bul<strong>le</strong>tin<br />
Vol. 17 110 4 PP 369-385<br />
The<br />
Xoss !: E & Matalas N C 1972 9'Application of Systems Anulysis<br />
to Network Design"<br />
in Case<strong>book</strong> on yydrolopical Network Eesia Practice<br />
(Mitor U B Langbein)<br />
wE",O Chapter III - 4.1<br />
i967 "On <strong>the</strong> PlanninE of Metero1o:cicsl Networks<br />
ifil0 Commission for Clirnatoìoa 4i>p<br />
1965 "Nationzl Networks of Eydro1o;:ical Data"<br />
Denouvillies J P, Chart E J, klool<strong>le</strong>y J A Cadou C<br />
1968 "The Use of <strong>the</strong> Grid Square System for Computer Estimation<br />
of Precipitation, Tenperature and Runoff"<br />
Water Resources Reseerch<br />
VOI 4 NO 5 pp 919-926<br />
1966 "Mood Frequency as a Function of Catchment Characteristics'<br />
S.vm.sosium on River Flood Hydrolorn<br />
Institute of civil Engineers, London pp iiFi36<br />
Sutcliff J V 1973 Personal comniunication.<br />
I%ì JORN c RODDA Institute of IIydroloa present Department of <strong>the</strong> Dnvironment<br />
Wallingford, Berks. Address: 2 Karsham Street, LOiLO€T SW1<br />
May 1973 ~~~LUm>
ABSTRACT<br />
HYDROLOGIC DATA FILL-IN AND NETWORK DESIGN<br />
Leo R. Beard<br />
A study for <strong>the</strong> Texas Water Development Board in <strong>the</strong><br />
USA develops techniques for transferring streamflow data<br />
from locations of long record to locations of short record<br />
and uses such techniques to determine <strong>the</strong> relative value of<br />
continuing current records or establishing new stations.<br />
Multisite stochastic generation techniques are adapted to<br />
<strong>the</strong> prob<strong>le</strong>m of filling i,n missing data by use of recorded<br />
data at many o<strong>the</strong>r locations in <strong>the</strong> region. Several weaknes-<br />
ses of stochastic data analysis techniques are studied and<br />
new procedures are developed to overcome <strong>the</strong>se weaknesses.<br />
Results of <strong>the</strong> study are to be used for planning streamflow<br />
measurement programs.<br />
RESUMEN<br />
Un estudio hecho para el Texas Water Development Board<br />
en Los E.E.U.U. desarrolla t6cnicas para transferir datos de<br />
estaciones de largo período a estaciones de corto período y<br />
demuestra el valor relativo para continuar estaciones o esta<br />
b<strong>le</strong>cer nuevas estaciones. La generación de datos probabilís-<br />
ticos para reconstituir el periodo histôrico en varias esta-<br />
ciones en una regibn es demostrado por medio de otras esta-<br />
ciones en la región. Varias deficiencias en el uso del aná-<br />
lisis de datos probabillsticos son estudiadas y nuevos proce<br />
dimientos son desarrollados para sobreponerlas. Los resulta-<br />
dos del estudio serán usados para el planeamiento de progra-<br />
mas en el estudio del cauce en ríos.<br />
(1)<br />
Technical Director, Center for Research lin Water Resources,<br />
University of Texas, Austin, Texas, USA.<br />
(1)
162<br />
THE DATA FILL-IN PROBLEM<br />
In planning <strong>the</strong> design and operation of water resources projects, it is<br />
necessary to test <strong>the</strong> plans on <strong>the</strong> basis of at <strong>le</strong>ast 40 or 50 years of stream flow<br />
that can reasonably be expected to occur in <strong>the</strong> future. Many projects are<br />
influenced by stream flow and o<strong>the</strong>r hydrologic quantities that occur at several<br />
locations simultaneously. Accordingly, adequate testing of a design or operation<br />
plan requires 40 or more years of simultaneous hydrologic events at several<br />
locations. Usually it is desired to use for this purpose recorded past flows<br />
adjusted, if necessary, to future conditions. In many regions, even <strong>the</strong> best<br />
hydrologic records are very short, and in all regions <strong>the</strong>re are very short records<br />
that must be extended for planning purposes. Detai<strong>le</strong>d discussions of <strong>the</strong> use of<br />
syn<strong>the</strong>tic streamflows in addition to historical streamflows are contained in<br />
references 1 and 2.<br />
Also, in anticipation of future water resources studies, it is necessary to<br />
determine whe<strong>the</strong>r to continue records at existing hydrologic stations or to<br />
establish new stations with availab<strong>le</strong> resources. It is <strong>the</strong> purpose of this paper<br />
to describe a study made by The University of Texas for <strong>the</strong> Texas Water Development<br />
Board in <strong>the</strong> USA wherein techniques were developed for filling in missing data and<br />
for evaluating short records in relation to long records..<br />
THE DATA FILL-IN MODEL<br />
The computer model used in <strong>the</strong> study is one developed in <strong>the</strong> Hydrologic<br />
Engineering Center of <strong>the</strong> Corps of Engineers and described in reference 3. It<br />
accepts monthly stream flow, rainfall, evaporation or o<strong>the</strong>r hydrologic quantities<br />
as variab<strong>le</strong>s. The computation procedure consists of:<br />
a. Transforming all variab<strong>le</strong>s to logarithms<br />
b. Transforming all logarithms to form normal distributions<br />
c. Deriving, from <strong>the</strong> data, multip<strong>le</strong> linear regression equations for<br />
estimating missing quantities from <strong>the</strong> preceding quantity at <strong>the</strong> same station and<br />
<strong>the</strong> current or preceding quantity, depending on availability, at all o<strong>the</strong>r stations.<br />
d. Estimating missing quantities using <strong>the</strong> appropriate regression equation<br />
and a random component, and applying <strong>the</strong> reverse transform to obtain hydrologic<br />
quantities.<br />
In order to preserve <strong>the</strong> variance and <strong>the</strong> correlation matrix relating all<br />
variab<strong>le</strong>s, it is necessary to introduce a random component whose standard<br />
deviation is equal to <strong>the</strong> standard error of estimate of <strong>the</strong> regression equation.<br />
The model uses a different regression equation for each station and for each<br />
ca<strong>le</strong>ndar month at that station, and this regression equation can change every year<br />
depending on <strong>the</strong> availability of data at o<strong>the</strong>r stations during <strong>the</strong> current and
preceding months, Detai<strong>le</strong>d discussion of <strong>the</strong> data fill-in techniques and<br />
associated ma<strong>the</strong>matical prob<strong>le</strong>ms is contained in reference 4.<br />
S HORT-RECORD EFFECTS<br />
163<br />
When several hydrologic variab<strong>le</strong>s are analyzed simultaneously, it is<br />
usual that some records are very short and that records at some stations do not<br />
coincide in time with records at o<strong>the</strong>r stations. Because <strong>the</strong>se short records<br />
and <strong>the</strong>ir apparent interrelationships can be very mis<strong>le</strong>ading (due to unrepresenta-<br />
tive occurrences within <strong>the</strong> short time period), it is necessary to provide controls<br />
in <strong>the</strong> ma<strong>the</strong>matical model so that unreasonab<strong>le</strong> effects will not be generated.<br />
Also, it is necessary to devise estimates of intercorrelation for those pairs of<br />
variab<strong>le</strong>s where simultaneous data are not availab<strong>le</strong>.<br />
Each e<strong>le</strong>ment of <strong>the</strong> correlation matrix is computed using simultaneous<br />
values of each pair of variab<strong>le</strong>s after <strong>the</strong>y have been transformed to normal. For<br />
those stations where no simultaneous values exists, correlation coefficients<br />
are estimated by examining <strong>the</strong> common correlation coefficients that each of <strong>the</strong>se<br />
variab<strong>le</strong>s has with each of <strong>the</strong> o<strong>the</strong>r variab<strong>le</strong>s in <strong>the</strong> system. This yields information<br />
by which <strong>the</strong> maximum and minimum logical correlation coefficient between <strong>the</strong><br />
two variab<strong>le</strong>s can be established. After this has been done for all o<strong>the</strong>r variab<strong>le</strong>s,<br />
<strong>the</strong> correlation between <strong>the</strong>se 2 stations is established as an average of <strong>the</strong> logical<br />
maximum and minimum values. This is a necessary step for comp<strong>le</strong>ting <strong>the</strong> correla-<br />
tion matrix fromwhich regression equations must be computed.<br />
Ano<strong>the</strong>r short-period effect that can have serious consequences in planning<br />
is <strong>the</strong> instability of <strong>the</strong> mean and standard deviations of <strong>the</strong> logarithms of<br />
hydrologic quantities. It is possib<strong>le</strong> that, when records are as short as 4 or 5<br />
years, unusually extreme values can occur. When this happens, extrapolation to<br />
40 or 50 years can result in unreasonably extreme quantities being generated.<br />
Similarly, in such short records, it is possib<strong>le</strong> that no large or small events would<br />
occur, in which case extrapolation to long periods might not include events that<br />
would normally occur in such periods.<br />
Tab<strong>le</strong> 1 illustrates what happens as a result of <strong>the</strong>se small-samp<strong>le</strong> chance<br />
variations. Here, 10,000 5-year samp<strong>le</strong>s were drawn from a normal population,<br />
and <strong>the</strong> unbiased mean and standard deviation were computed for each. Then,<br />
for each samp<strong>le</strong>, 5 items were generated using <strong>the</strong>se samp<strong>le</strong> statistics, and <strong>the</strong>ir<br />
location in <strong>the</strong> true parent population identified. In <strong>the</strong> fourth line under ratios, it<br />
is shown that far too many extreme values were generated in this manner. Thus,<br />
<strong>the</strong>re is a significant bias in estimates made from small samp<strong>le</strong>s. In order to<br />
overcome this, <strong>the</strong> following empirical transform function to be applied to generated<br />
devia tes was developed:<br />
X I = x - . ~~x~/(N-U~'~<br />
in which I<br />
X<br />
X<br />
N<br />
=<br />
=<br />
=<br />
adjusted deviate (absolute value)<br />
generated deviate (absolute value)<br />
samp<strong>le</strong> size
164<br />
II)<br />
LI<br />
6<br />
H<br />
O<br />
O<br />
O<br />
O<br />
Ln<br />
Ln<br />
al<br />
N<br />
m<br />
al<br />
a<br />
-4<br />
.-4<br />
E<br />
cn<br />
LI<br />
O<br />
w<br />
v><br />
al<br />
m<br />
><br />
al<br />
U<br />
-4<br />
n<br />
ry<br />
O<br />
r:<br />
O<br />
-4<br />
rn<br />
LI<br />
al<br />
a<br />
II)<br />
-4<br />
n<br />
nl9mœI-<br />
V, ri19<br />
m<br />
8<br />
-I<br />
-<br />
b-<br />
O<br />
c b
The last line under ratios in Tab<strong>le</strong> 1 illustrates that this formula produces a<br />
very nearly normal distribution of values generated from a large number of<br />
small-samp<strong>le</strong> statistics.<br />
STABILITY PROVISIONS<br />
165<br />
The model used in this study for data fill-in includes a number of<br />
features that are necessary in arder to produce stab<strong>le</strong> projections when using<br />
short and intermittent records. When <strong>the</strong> correlation matrix that was derived<br />
as discussed above is used in constructing a regression equation, it is entirely<br />
possib<strong>le</strong> that <strong>the</strong> assemb<strong>le</strong>d Correlation coefficients will be mutually inconsistent,<br />
since <strong>the</strong>y are not based on simultaneous data. If this occurs, it simply means<br />
that <strong>the</strong> quantity to be estimated is over-defined and that some of <strong>the</strong> inconsistent<br />
data must be removed for <strong>the</strong> purpose of estimating that particular quantity. This<br />
is accomplished automatically in <strong>the</strong> computer by testing for consistency and,<br />
when <strong>the</strong> correlation coefficient exceeds unity, eliminating that variab<strong>le</strong> in <strong>the</strong><br />
equation which has <strong>the</strong> lowest direct correlation with <strong>the</strong> quantity to be estimated.<br />
This elimination process is continued automatically until <strong>the</strong> correlation<br />
coefficient becomes <strong>le</strong>ss than unity.<br />
Even though <strong>the</strong> correlation'matrix is consistent, it can still be highly<br />
unstab<strong>le</strong>. This occurs usually when 2 of <strong>the</strong> explanatory variab<strong>le</strong>s are highly<br />
interdependent. One indication of this condition is <strong>the</strong> occurrence of very<br />
high regression coefficients of opposite signs for those 2 variab<strong>le</strong>s. The test<br />
for this condition uses <strong>the</strong> beta coefficient, which is <strong>the</strong> regression coefficient<br />
that results when each of <strong>the</strong> variab<strong>le</strong>s is adjusted to unit variance (it thus<br />
measures <strong>the</strong> direct degree of impact of each variab<strong>le</strong> on <strong>the</strong> regression estimate) .<br />
If any beta coefficient exceeds 1.5 , variab<strong>le</strong>s are eliminated from <strong>the</strong> regression<br />
study until this condition no longer exists. In this manner, a primary cause of<br />
generating unreasonab<strong>le</strong> quantities is eliminated.<br />
REQUIREMENTS FOR MATHEMATICAL ACCURACY<br />
Experience with <strong>the</strong> use of this model for hydrologic data fill-in has<br />
indicated that ma<strong>the</strong>matical accuracy, integrity, and continuity are absolutely<br />
essential in order to avoid unreasonab<strong>le</strong> estimates. Many attempts have been<br />
made to smooth <strong>the</strong> statistics and correlation coefficients from month to month<br />
throughout <strong>the</strong> year in order to stabilize <strong>the</strong> estimates, but <strong>the</strong>se have usually<br />
resulted in ma<strong>the</strong>matical prob<strong>le</strong>ms that could not be readily overcome. Attempts<br />
have also been made to adjust coefficients within <strong>the</strong> correlation matrix in such<br />
a manner as to remove inconsistencies and increase stability, but <strong>the</strong>se also<br />
have resulted in erratic computation. It has become apparent that <strong>the</strong> regression<br />
equation for estimating a missing value must be used exactly as calculated<br />
from <strong>the</strong> observed or fil<strong>le</strong>d-in data.
166<br />
The transform function used for converting flows to normal has also<br />
been a source of serious ma<strong>the</strong>matical difficulty. If <strong>the</strong> data being transformed<br />
are highly skewed, transformed values can become highly erratic, particularly<br />
for small samp<strong>le</strong>s. In order to stabilize this transform, hydrologic quantities<br />
whose lower limit is zero are first transformed to logarithms (after adding a<br />
small increment). The size of this increment is <strong>the</strong>n adjusted so that <strong>the</strong><br />
skew of <strong>the</strong> logarithms does not differ much from zero. Then <strong>the</strong> approximate<br />
Pearson type III transform function appears to be comp<strong>le</strong>tely adequate for<br />
transforming <strong>the</strong> logarithms to normal. However, when <strong>the</strong> skew coefficient of<br />
<strong>the</strong> untransformed values differs from zero by more than a value of about 0.5,<br />
very serious transform prob<strong>le</strong>ms can occur.<br />
V&UE OF DATA FILL-IN<br />
It can be shown that adjustment of short-record statistics by use of<br />
long-record correlated data can result in improvement of <strong>the</strong> accuracy of <strong>the</strong><br />
mean value in accordance with <strong>the</strong> following equation:<br />
in which<br />
N1 = number of items in short record<br />
N2 =<br />
R , =<br />
number of items in long record<br />
cross correlation coefficient<br />
N1 = number of items that would be needed in <strong>the</strong> short record to<br />
obtain an accuracy of <strong>the</strong> mean that is equiva<strong>le</strong>nt to that<br />
obtainab<strong>le</strong> by <strong>the</strong> adjustment.<br />
Some values obtained with this equation are illustrated in tab<strong>le</strong> 2.<br />
In filling in missing values of monthly streamflows by correlation with<br />
long-record stations, correlation coefficients vary from month-to-month, so<br />
<strong>the</strong>re is not a simp<strong>le</strong> relationship that will show how much value is obtained<br />
by extending short records in this manner. However, a group of 4 stations having<br />
40 years of simultaneous data was used to estimate <strong>the</strong> increase in reliability of<br />
average-flow estimates based on 5 years of data correlated with 40 years of data<br />
at near-by locations. The experiment was conducted as follows:
Tab<strong>le</strong> 2<br />
Theoretically Equiva<strong>le</strong>nt Samp<strong>le</strong> Size<br />
for Computing Equally Reliab<strong>le</strong> Mean Value<br />
167<br />
Correlation Coefficient<br />
Samp<strong>le</strong><br />
Size .5 .8 .9 .95 .98<br />
Samp<strong>le</strong> Size of Related Variab<strong>le</strong> = 40<br />
5 6.4 11.4 17.2 23.8 31.3<br />
10 12.3 19.2 25.5 31 .O 35.8<br />
20 22.9 29.4 33.6 36.4 38.5<br />
40 40 .O 40 .O 40 .O 40 .O 40 .O<br />
Samp<strong>le</strong> Size of Related Variab<strong>le</strong> = 100<br />
5 6.6 12.8 21.7 35.1 57.1<br />
10 12.9 23.6 . 36.9 53.3 73.7<br />
20 25 .O 41 .O 56.8 71.9 86.3<br />
40 47.1 64.9 77.8 87.2 94.4<br />
Starting with one station, four different 5-year sequences were se<strong>le</strong>cted<br />
from <strong>the</strong> record. For each of <strong>the</strong>se, data were fil<strong>le</strong>d in for <strong>the</strong> remaining 35 years.<br />
For each of <strong>the</strong>se 5-year sequences, <strong>the</strong> mean flow for <strong>the</strong> 5 years and <strong>the</strong><br />
mean flow for <strong>the</strong> 40 years of fil<strong>le</strong>d in sequence were compared with <strong>the</strong> mean<br />
flow for <strong>the</strong> 40 years of actual record at <strong>the</strong> station. Standard errors from <strong>the</strong><br />
40-year recorded mean were computed.<br />
The ratios of <strong>the</strong> standard error of <strong>the</strong> 40-year fil<strong>le</strong>d-in data mean to<br />
<strong>the</strong> standard error of <strong>the</strong> 5-year data mean are shown in tab<strong>le</strong> 3 in <strong>the</strong> row<br />
designated as obsenred. This process was repeated for each of <strong>the</strong> 4 stations in<br />
order to obtain <strong>the</strong> 12 observed values of tab<strong>le</strong> 3.<br />
The expected ratios shown in tab<strong>le</strong> 3 were computed as <strong>the</strong> inverse<br />
ratios of <strong>the</strong> square root of effective record <strong>le</strong>ngths computed from equation 2.<br />
The standard-error ratios thus obtained are somewhat larger than expected,<br />
partly due to <strong>the</strong> fact that <strong>the</strong> 40-year record mean is not <strong>the</strong> true long-term<br />
mean and partly due to <strong>the</strong> variation of monthly correlation coefficients from <strong>the</strong><br />
correlation coefficients of annual flows shown in tab<strong>le</strong> 3.
168<br />
Short<br />
Record<br />
Sta tion<br />
1685<br />
Correl coef<br />
Observed<br />
Expected<br />
1675<br />
Correl coef<br />
Observed<br />
Expected<br />
1710<br />
Correl coef<br />
Observed<br />
Expect ed<br />
1730<br />
Correl coef<br />
Observed<br />
Expected<br />
Tab<strong>le</strong> 3<br />
Ratios of Standard Error of Fill-in<br />
Mean to Observed Mean for 5-year Records<br />
Correlated with 40-year Records<br />
1685<br />
.97<br />
.75<br />
.42<br />
.82 .79<br />
1 .o9 1 .O3<br />
.65 .68<br />
Long-Record Station<br />
1675 1710<br />
.97 .82<br />
.51 .82<br />
.42 .65<br />
.79<br />
.95<br />
.68<br />
.73 .75 .78<br />
.74 .92 .30<br />
.73 .72 .69<br />
1730<br />
.73<br />
.80<br />
.73<br />
.75<br />
.86<br />
.72<br />
.78<br />
1 .O7<br />
.69<br />
Although <strong>the</strong> results shown in tab<strong>le</strong> 3 are somewhat erratic due to <strong>the</strong> use<br />
of small samp<strong>le</strong>s and a small number of cases, it is apparent that <strong>the</strong> fill-in<br />
process described herein is generally valid and that tab<strong>le</strong> 2 can be used as a<br />
general guide in determining whe<strong>the</strong>r to continue short records or to start records<br />
at new locations where data are also needed. The advantage of <strong>the</strong> monthly fill-<br />
in model over a simp<strong>le</strong> adjustment of mean flows is that realistic variations of<br />
annual streamflow patterns for interrelated stations can be developed for use in<br />
simulation studies.<br />
Un<strong>le</strong>ss correlation coefficients between short-record and long-record values<br />
are well above 0.5, <strong>the</strong>re appears to be very litt<strong>le</strong> gain in reliability through<br />
correlation. Where <strong>the</strong>re is good correlation, <strong>the</strong> gain in reliability that can be<br />
expected through maintaining a short record for a longer period (such as continuing<br />
a 5-year record until it is 10 years long) is a function of <strong>the</strong> <strong>le</strong>ngth of near-by iong-<br />
record stations.
169<br />
Where correlation coefficients are well above .95, short records need<br />
not be continued much beyond 5 years ,but <strong>the</strong> near-by long record should<br />
be continued as long as greater reliability is needed. Where Correlation<br />
coefficients are much below .8, short records should be continued. Between<br />
<strong>the</strong>se limits, <strong>the</strong> relative value of continuing a short record or starting a new<br />
record depends on <strong>the</strong> unreliability of estimating flows at ungaged locations,<br />
which concerns an area of study beyond <strong>the</strong> scope of this paper.<br />
CONCLUSIONS<br />
The stochastic data fill-in model described can be used to estimate<br />
monthly values of missing hydrologic data at short-record locations where<br />
longer records exist in <strong>the</strong> region. The value of <strong>the</strong> fill-in procedure is a<br />
function of <strong>the</strong> correlation between <strong>the</strong> short-record and long-record data and<br />
<strong>the</strong> relative <strong>le</strong>ngths of record, generally as expressed in equation 2. This<br />
relation, as illustrated in tab<strong>le</strong> 2, can be used to determine whe<strong>the</strong>r to continue<br />
short records or establish new stations. It appears from tab<strong>le</strong> 2 that short<br />
records need not be continued beyond 5 years (un<strong>le</strong>ss hydrologic conditions<br />
change) where near-by records are continued that correlate at <strong>the</strong> .95 <strong>le</strong>vel<br />
or better. Where <strong>the</strong> correlation coefficient is below .8, records should generally<br />
be continued. Between <strong>the</strong>se two values, a study of regional variations would<br />
be needed to determine <strong>the</strong> relative advantage of continuing existing stations or<br />
starting new ones.<br />
ACKNOWLEDGMENT<br />
The study upon which this paper is based was supported by <strong>the</strong> Texas<br />
Water Development Board. Computation assistance was furnished by R.V.<br />
Juyal and J.W. Barron. Opinions and conclusions expressed are those of <strong>the</strong><br />
author.<br />
1.<br />
2.<br />
3.<br />
4.<br />
REFERENCES<br />
Beard, Leo R. (1965) Hydrologic Simulation Procedures in Water Yield<br />
Analysis, Sixth Congress, International Commission on Irrigation<br />
and Drainage, New Delhi, pp 22.103 - 22.116.<br />
Weiss, Arden O. and Beard, Leo R. (1971) A Multi-Basin Planning<br />
Strategy, Water Resources Bul<strong>le</strong>tin, Journal of <strong>the</strong> American Water<br />
Resources Association V.7, No.4, pp. 750-764.<br />
Beard, Leo R. (1965) Use of Interrelated Records to Simulate<br />
Streamflow, Journal of <strong>the</strong> Hydraulics Division, American Society of<br />
Civil Engineers, September 1965, pp. 13-22.<br />
Beard, Leo R., Fredrich, Augustine J. and Hawkins, Edward F. (1970)<br />
Estimating Monthly Streamflows within a Region, National Water Resources<br />
Engineering Meeting, American Society of Civil Engineers, Preprint 1125.
ABSTRACT<br />
APPLICATION DU KRIGEAGE A L'OPTIMISATION<br />
D'UNE CAMPAGNE PLUVIOMETRIQUE EN ZONE ARIDE<br />
-<br />
J.P. DELHOMME, P. DELFXNER<br />
In arid areas, hydraulic planning must often be performed in a<br />
few years: install a rain gauge network, streng<strong>the</strong>n it if necessary and<br />
determine <strong>the</strong> major features of <strong>the</strong> basin, mainly <strong>the</strong> volume of precipi-<br />
tation and its geographic distribution. It seems impossib<strong>le</strong> to utilize<br />
<strong>the</strong> usual elaborate statistical methods because <strong>the</strong>y appeal to time COT<br />
rrelations which can hardly be inferred, Indeed, after an initial pro-<br />
gram of precipitation measurements for a basin, data for only a sh-ort<br />
time interval are availab<strong>le</strong>, and regional climatological statiwn are<br />
commonly too far removed geograplìically to andd useful ingormqti'on. TO<br />
solve <strong>the</strong> interpolation prob<strong>le</strong>ms , only <strong>the</strong> spatial stxuctgre 08 preci-<br />
pitation on <strong>the</strong> basin itself can 6e considered. Kriging provides <strong>the</strong><br />
best linear estimates based on <strong>the</strong> experimental data, and this under<br />
very few assumption. In particular, it avoids <strong>the</strong> traditional assump-<br />
tion of second order stationarity, used in optimal filtering for exam-<br />
p<strong>le</strong>, and which is not justified in many cases. Moreover, Kriging per-<br />
mits quantification of precision of estimation and provides a solution<br />
to <strong>the</strong> prob<strong>le</strong>m of optimal location of new points of measurement, accor<br />
ding to a criterion of maximum gain of information,<br />
RESUME<br />
Lors d'une étude d'aménagement hydraulique en zone aride, on ne<br />
dispose souvent que de quelques années pour implanter un réseau pluvio-<br />
métrique, <strong>le</strong> renforcer si besoin est, et cerner <strong>le</strong>s caractéristiques ma<br />
jeures du bassin, principa<strong>le</strong>ment <strong>le</strong> volume d'eau tombé et sa réparti-<br />
tion. Les techniques statistiques élaborées traditionnel<strong>le</strong>ment semb<strong>le</strong>nt<br />
alors d'un emploi diffici<strong>le</strong> car el<strong>le</strong>s font intervenir des corrélations<br />
temporel<strong>le</strong>s dont l'inférence statistique est quasiment imposib<strong>le</strong>. En<br />
sffet, aprbs une première campagne de re<strong>le</strong>vés pluviométriques sur <strong>le</strong> b a<br />
ssin, on n'y possède que de tr8s courtes séries chronologiques et <strong>le</strong>s<br />
stations ciimatologiques régiona<strong>le</strong>s sont souvent trop élognées géogra-<br />
!hiquement pour apporter une information &el<strong>le</strong>ment valab<strong>le</strong>. Pour trai -<br />
ter <strong>le</strong>s problèmes d'interpolation, on ne peut donc prendre en considéra<br />
tion que la structure spatia<strong>le</strong> de la pluviométrie. Le Krigeage permet<br />
ie trouver <strong>le</strong>s meil<strong>le</strong>urs estimateurs linéaires construits sur <strong>le</strong>s va-<br />
<strong>le</strong>urs expérimenta<strong>le</strong>s, et ce, sous des hvpoth8ses tres larges: en parti-<br />
)ulier, l'hypothèse classique de la stationnaritg du second ordre, di-<br />
ffici<strong>le</strong>ment admissib<strong>le</strong> dans bien des cas, n'est pas nécessaire. Le Kri-<br />
Ceage permet en outre de quantifier la précision de notre estimation,<br />
?t appoiyte une sol-ution au problème de l'implantation optima<strong>le</strong> de nou-<br />
reaux points de mesure selon un critère de gain maximal d'information.
17 2<br />
Pour l'hydrogéologue, <strong>le</strong>s précipitations sont non seu<strong>le</strong>ment<br />
descriptif du climat, mais aussi, et surtout, l'élément constitutif<br />
du débit des cours d'eau.<br />
A ces deux aspects fondamentaux correspondent deus types<br />
d'approche différents d'une épisode pluvieux. I1 s'agit d'une part<br />
d'estimer en tout point du bassin la hauteur de précipitation pour<br />
avoir une vue d'ensemb<strong>le</strong> de la répartition spatia<strong>le</strong> de l'averse et<br />
pour en localiser <strong>le</strong>s épicentres, d'autre part, d'intégrer cette<br />
hauteur de précipitation sur toute la surface afin d'evaluer la qua2<br />
tité d'eau tombée sur <strong>le</strong> bassin durant ce laps de temps.<br />
Dans <strong>le</strong>s deux cas, on ne dispose au départ que des indications ponctuel<strong>le</strong>s<br />
recueillies aux stations pluviométriques. Si 1 'on veut obtenir des évaluations<br />
correctes ã partir de ces données en nombre limité, on doit attacher une<br />
grande importance au choix d'une méthode d'estimation qui soit adaptée aux<br />
buts poursuivis et présente <strong>le</strong> maximum de fiabilité. Que signifierait une quantité<br />
d'eau calculée avec 100 .d'erreur? Comme dans tout calcul physique, une<br />
va<strong>le</strong>ur numérique n'a de sens F qu'accompagnée d'un interval<strong>le</strong> d'incertitude. Si<br />
la précision n'est pas satisfaisante, il conviendra ã 1 'avenir d'instal<strong>le</strong>r de<br />
nouveaux pluviomètres. Quel<strong>le</strong> serait alors <strong>le</strong>ur implantation optiqa<strong>le</strong>? Ces<br />
questions trouvent une réponse satisfaisante dans <strong>le</strong> cadre de la théorie du<br />
krigeage de G. MATHERON (i), (Z), (3).<br />
I1 n'est pas place ici pour un long exposé théorique que <strong>le</strong> <strong>le</strong>cteur<br />
pourra trouver dans <strong>le</strong>s ouvrages de G. MATHERON cités en références.<br />
Aussi a-t-on préféré en montrer une application au cas concret d'une<br />
campagne pluviométrique en zone aride.<br />
PRESENTATION DU CADRE DE L'ETUDE<br />
Les données utilisées ont été empruntées ã une campagne de 1'ORSTOM<br />
dans la région Est du Tchad (4) en 1965-66. Durant la saison dds pluies a lieu<br />
la recharge de nappes souterraines de faib<strong>le</strong> importance qui fournissent 1 'essentiel<br />
des ressources pendant la saison sèche.<br />
Afin d'accroître cette recharge, un projet de construction de barrages<br />
de suralimentation sur certains ouadis a été décidé, <strong>le</strong>s études de reconnaissan-<br />
ce hydrologique devant s 'étendre sur deux années.<br />
On a retenu <strong>le</strong> cas du bassin de l'ouadi Kadjemeur d'une superficie de<br />
245 km2 et présentant de faib<strong>le</strong>s dénivellées (inférieures ã 100 m.).<br />
Les conditions climatiques sur ce bassin versant sont assez diffici<strong>le</strong>s<br />
?i estimer ã partir des stations climatologiques régiona<strong>le</strong>s (Fig.l), du fait de<br />
la rapidité des changements de régime climatique dans la région: en 400 km du<br />
Nord au Sud, on passe du régime sahélien sud d'Abeche au régime saharien de Fada<br />
Les périodes d'observation sont très inéga<strong>le</strong>s (Abeche: 31 ans, Guereda: 12 ans,<br />
Iriba: 8'ans, Biltine: 15 ans, Arada: 8 ans, Fada: 32 ans), et <strong>le</strong>s corrélations<br />
d'une station à l'autre ne sont pas satisfaisantes.
173<br />
On ne peut donc prendre en compte que <strong>le</strong>s données recueillies sur <strong>le</strong><br />
bassin lui-même, où l'on dispose de 33 points de mesures: 3 pluviographes, 19<br />
pluviomètres association et 11 totalisateurs (Fig.2).<br />
LES BASES CONCEPTUELLES DU KRIGEAGE<br />
Le phénomène étudié est considéré comme une fonction Z associant une<br />
va<strong>le</strong>ur numérique Z(x) à tout point x d'un certain domaine du plan ou de 1 'espace.<br />
On connaît <strong>le</strong>s va<strong>le</strong>urs prises par Z aux points expérimentaux xl, x2, ...., x<br />
Selon <strong>le</strong>s cas, on cherche ã estimer:<br />
N'<br />
i) la va<strong>le</strong>ur ponctuel<strong>le</strong> Z(xo) au point xo<br />
2) la va<strong>le</strong>ur moyenne sur un domaine S, soit i Is Z(x)dx<br />
3) la va<strong>le</strong>ur moyenne pondérée de Z, soit:<br />
Z, = j' Z(x)p(x)dx avec p(x)dx = 1<br />
Pour cela, on se'-donne un estimateur Z" de la va<strong>le</strong>ur exacte sous forme<br />
d'une combinaison linéaire des données disponib<strong>le</strong>s:<br />
N<br />
z* =J xi Z(Xi)<br />
1 =1<br />
I1 y a de multip<strong>le</strong>s facons de choisir <strong>le</strong>s coefficients de pondération<br />
xi: tout <strong>le</strong> problème est de déterminer <strong>le</strong>s meil<strong>le</strong>urs possib<strong>le</strong>s.<br />
A cet effet, on peut se laisser guider par des considérations physiques.<br />
La qualité de l'estimation doit dépendre de deux facteurs: <strong>le</strong> nombre et la disposition<br />
spatia<strong>le</strong> des points de mesure d'une part, la continuité, la régularité<br />
du phénomène étudié, de l'autre.<br />
Pour <strong>le</strong> premier point, il est clair que l'estimation est d'autant meil<strong>le</strong>ure<br />
qu'il y a plus de données expérimenta<strong>le</strong>s. Mais 1 'effectif du réseau de<br />
mesure n'est pas forcément déterminant. Interviennent éga<strong>le</strong>ment la disposition<br />
relative des points expérimentaux entre eux et <strong>le</strong>ur localisation par rapport au<br />
domaine a estimer (point ou surface). Par exemp<strong>le</strong>, pour estimer une quantité<br />
globa<strong>le</strong> sur une région, il est en général préférab<strong>le</strong> d'avoir moins de points mais<br />
disposés de façon uniforme que beaucoup de points agglutinés dans une seu<strong>le</strong> zone.<br />
La conclusion est inverse si 1 'on désire une estimation loca<strong>le</strong> au voisinage précisément<br />
de cette zone la mieux échantillonnée.<br />
Le second point est plus subtil et négligé dans la plupart des métho-<br />
des utilisées actuel<strong>le</strong>ment en hydrologie. Une fonction s'interpo<strong>le</strong> d'autant<br />
mieux qu'el<strong>le</strong> est plus régulière. S'agissant par exemp<strong>le</strong> d'estimer une va<strong>le</strong>ur<br />
ponctuel<strong>le</strong> Z(xo), il n'y a aucune raison d'utiliser la même formu<strong>le</strong> d'interpola-<br />
tion quand on travail<strong>le</strong> sur des pluies annuel<strong>le</strong>s ou des pluies journalières.<br />
Dans un cas la va<strong>le</strong>ur au point x diffère peu de cel<strong>le</strong>s des points voisins, dans<br />
1 'autre, <strong>le</strong> phénomène est plus ctaotique et <strong>le</strong>s points lointains apportent une<br />
information non négligeab<strong>le</strong>.
174<br />
Comment tenir compte de la régularité de la variab<strong>le</strong>?<br />
Les méthodes fonctionnel<strong>le</strong>s de 1 'analyse mathématique ordinaire ne sont<br />
guère utilisab<strong>le</strong>s pour <strong>le</strong>s fonctions traduisant un phénomène naturel. Cel<strong>le</strong>s-ci<br />
ont un comportement spatial bien trop comp<strong>le</strong>xe, trop erratique pour se laisser<br />
décrire ii 1 'aide d'expressions analytiques classiques. Pour souligner cette particularité,<br />
G. MATHERON (1) propose de donner a de tel<strong>le</strong>s fonctions <strong>le</strong> nom de<br />
"vari ab<strong>le</strong>s régionalisées".<br />
Une façon commode ii la fois sur <strong>le</strong> plan conceptuel et pratique de traiter<br />
une variab<strong>le</strong> régioiiziisée est de raisonner en termes probabilistes. On<br />
considère la variab<strong>le</strong> régionalisée comme une "réalisation de fonction aléatoire",<br />
c'est ii dire comme <strong>le</strong> résultat d'un tirage au sort dans un ensemb<strong>le</strong> de fonctions.<br />
Pour préciser cette idée, supposons qu'on range dans un même groupe un ensemb<strong>le</strong><br />
d'averses analogues, autrement dit, un ensemb<strong>le</strong> de fonctions Zi(X) associant à<br />
chaque point x la hauteur de précipitation en ce point. La fonction aléatoire<br />
Z est tel<strong>le</strong> que pour tout indice i et tout point x du domaine:<br />
z(x,i) = z.(x)<br />
1<br />
Au tirage au sort de l'indice i de l'averse correspond la fonction numérique or-<br />
dinaire Zi(X), c'est ii dire une réalisation de la fonction aléatoire Z. Ainsi<br />
sont fixées du même coup <strong>le</strong>s va<strong>le</strong>urs prises par la fonction en tous <strong>le</strong>s points<br />
de son domaine de définition, expérimentaux ou non.<br />
Dans <strong>le</strong> cadre de cette hypothèse, <strong>le</strong>s notions statistiques tel<strong>le</strong>s que<br />
moyenne, vari ance , covari ance ou auto-corrél ati on prennent un sens précis.<br />
E <strong>le</strong> symbo<strong>le</strong> "espérance mathématique", on a:<br />
Soit<br />
E CZ(x)l = m(x) moyenne<br />
E [Z(X)-m(x)]* = D2 [Z(X)]<br />
vari ance<br />
E [Z(x)-m(x)] [Z(y)-m(y)] = K(x,y) covariance<br />
K(X,Y)/=) .JK(y,y) = P(X,Y) auto-corrélation<br />
On voit que p(x,y) se déduit directement de K(x,y), la réciproque étant fausse.<br />
On utilisera donc plutôt K(x,y) qui contient plus d'information.<br />
Pour procéder valab<strong>le</strong>ment à 1 'inférence statistique de la moyenne et<br />
de la covariance aux différents points de l'espace, il faut disposer de chroni-<br />
ques suffisantes. Lorsque ce n'est pas <strong>le</strong> cas, come dans l'exemp<strong>le</strong> de Kadjemeur,<br />
des hypothèses supplémentai res sont nécessaires. Les méthodes optima<strong>le</strong>s du type<br />
de cel<strong>le</strong> du filtrage de WIENER (5), introduite en météorologie par L.S. GANDIN<br />
(6) se placent dans 1 'hypothèse où la variab<strong>le</strong> est "stationnaire d'ordre 2": la<br />
moyenne m(x) est constante et la covariance ne dépend pas séparément des points<br />
d'appui x et y, mais uniquement du vecteur x-y:<br />
E [~(x)] = m<br />
E [Z(x)-m] [Z(Y)-~] = K(x-Y)
175<br />
Ces hypothèses peuvent être trop restrictives. On sait par exemp<strong>le</strong> que<br />
<strong>le</strong>s précipitations sont plus abondantes en altitude qu'en plaine. Par conséquent,<br />
dans <strong>le</strong> cas général d'une région à relief varié, <strong>le</strong>ur moyenne m(x) présente une<br />
"dérive" et ne peut être considérée comme constante. Par ail<strong>le</strong>urs, il apparaît<br />
que <strong>le</strong>s calculs d'optimisation n'exigent pas que la variab<strong>le</strong> el<strong>le</strong>-même, mais<br />
uniquement ses accroissements y possède une covari ance stationnai re.<br />
Ceci étant, <strong>le</strong>s hypothèses du krigeage sont 1 es sui vantes :<br />
1) m(x) n'est pas forcément constante, mais est suffisamment régulière<br />
pour être représentée par une expression de la forme:<br />
k<br />
m(x) = 1 a, f'(x)<br />
1<br />
1 =o<br />
Les fonctions f fxì sont choisies à 1 'avance foolvnomes. fonctions<br />
trigonométriques; etc.. .) ; <strong>le</strong>s al sont des cÒef~cienG inconnus<br />
Une tel<strong>le</strong> formulation englobe <strong>le</strong> cas <strong>le</strong> plus simp<strong>le</strong> où la moyenne<br />
est constante. La "dérive" m(x) se réduit alors à:<br />
O<br />
m(x) = ao f (x)= ao<br />
fo(x) étant la fonction identiquemint éga<strong>le</strong> à 1.<br />
O<br />
On supposera toujours que f 1, car cela implique que l'erreur<br />
d'estimation Z-Z* est une combinaison linéaire d'accroissements<br />
de Z(x)<br />
2) Seconde hypothèse: 1 a variance des accroissements Z( x+h) -Z( x)<br />
ne dépend que du vecteur h. On pose:<br />
y(h) = i D2 [Z(x+h)-Z(x)]<br />
~ ( h ) est <strong>le</strong> vario ramme. Cette fonction du vecteur h renseigne<br />
sur 1 'isotropie +<br />
ou anisotropie de la variab<strong>le</strong> régionalisée.<br />
A direction fixée, el<strong>le</strong> indique comment varie, en moyenne quadratique<br />
l'écart de va<strong>le</strong>urs prises en deux points x et x+h<br />
lorsque la distance h augmente. A une variab<strong>le</strong> très régulière<br />
correspond un variogramme très continu, et inversement.<br />
Ces bases définies, il est possib<strong>le</strong> de résoudre tour à tour <strong>le</strong>s différents<br />
problèmes posés.<br />
KRIGEAGE DES ISOHYETES<br />
Soit Z(x) la hauteur de précipitation tombée sur un territoire pour une<br />
période déterminée. Afin d'estimer la va<strong>le</strong>ur ponctuel<strong>le</strong> Z(x,) , on cherche parmi<br />
<strong>le</strong>s estimateurs linéaires construits sur <strong>le</strong>s données expérimenta<strong>le</strong>s z*=xhiZ(xi)<br />
celui qui minimise 1 'erreur quadratique moyenne E[Z*-Z(xo)]2. Or: 1<br />
2<br />
E [Zf-Z(x0)] = D2 [Z*-Z(x0)] + [E[Z'-Z(x0)]]'
176<br />
Le premier terme D2 [Zf-Z(xo)] est la variance de l'erreur.<br />
fonction du variogrannne:<br />
El<strong>le</strong> s'explicite en<br />
D2 rhiZ(xi)-Z(x0)] = - 1 1 h-X.y(xi-x.) t 2 1 hiy(xi-xo)<br />
ij 1 J J<br />
i<br />
Le second terme [E[Z*-Z xQ)]I2 est <strong>le</strong> carré de l'erreur moyenne.<br />
moyenne représente un biais et il faut donc l'annu<strong>le</strong>r.<br />
E [fc-z(x0)] = E<br />
D'après <strong>le</strong>s hypothèses faites sur la dérive:<br />
d'où:<br />
Si l'on pose:<br />
m(xi) = 1 alf 1 (xi) et rn(xo) = 1 alf 1 (x,)<br />
1 1<br />
E [z*-z(xO)l = c al<br />
1<br />
'if 1 (xi) - f'(xO)]<br />
1 1<br />
1 hif (xi) = f (x,)<br />
1<br />
bc 1 = O, 1, ...., k<br />
Cette erreur<br />
him(xi) - m(xo<br />
l'erreur moyenne sera nul<strong>le</strong> quels que soient <strong>le</strong>s coefficients a qu'il ne sera<br />
1<br />
pas nécessaire de connaître.<br />
Minimisant 1 'erreur quadratique moyenne sous ces k+l conditions, on<br />
obtient <strong>le</strong> système de krigeage où figurent ktl paramètres de Lagrange ul:<br />
(SI)<br />
1<br />
1 h.y(xi-x.) t 1 ulf (xi) = y(xi-xo)<br />
j J J 1<br />
1 hjf<br />
1<br />
(Xj) = f<br />
1<br />
(x )<br />
O<br />
j<br />
(i=l, ..., N)<br />
(l=O,l,. ..,k)<br />
Ce système est régulier, donc admet une solution unique, pourvu que<br />
<strong>le</strong>s f (xi) soient linéairement indépendants sur 1 'ensemb<strong>le</strong> des points expérimen-<br />
taux (cf.(l) ou (2)).<br />
A l'optimum, la variance d'estimation a pour expression:<br />
D2[Z*-Z(Xo)] = 1 Xjy(x.-x ) t 1 plf 1 (x,)<br />
j 1<br />
On remarque que cette variance ne dépend que du variogramme et des<br />
solutions hi et pl du système de krigeage, c'est à dire uniquement de la struc-<br />
ture du phénomène et de la disposition des points de mesure.
177<br />
L'exemp<strong>le</strong> qui a été retenu est celui de 1 'averse du 6/8/66, la plus<br />
importante de l'année. I1 a été traité sur ordinateur à l'aide du programme<br />
BLUEPACK mis au point à Fontaineb<strong>le</strong>au(79,Le bassin de 1 'ouadi Kadjemeur ne présentant<br />
pas un relief très marqué, la pluviométrie n'y possède pas de dérive systématique.<br />
On a donc pris pour seu<strong>le</strong> fonction de base fo 5 1.<br />
Le variogramme est linéaire avec une discontinuité à 1 'origine.<br />
O pour h = o<br />
ríh) =<br />
en mm2. 20.4 t 11.23 h pour h # O en km<br />
I1 a été d'it plus haut que <strong>le</strong> variogramme est d'autant plus continu que<br />
la variab<strong>le</strong> est plus régulière.<br />
La discontinuité à l'origine du y(h) traduit une irrégularité à petite<br />
échel<strong>le</strong>. Ce phénomène a été observé depuis longtemps par <strong>le</strong>s hydrométéorologues<br />
qui 1 'expliquent par <strong>le</strong>s perturbations loca<strong>le</strong>s, l'instabilité du mouvement de<br />
l'air au voisinage du sol et l'arrivée de la pluie sur <strong>le</strong> pluviomètre par rafa<strong>le</strong>s<br />
irrégulières. A la limite, si on connaissait parfaitement <strong>le</strong>s hauteurs d'eau en<br />
tout point, il serait probab<strong>le</strong>ment impossib<strong>le</strong> d'en tracer la carte, <strong>le</strong>s fluctua-<br />
tions loca<strong>le</strong>s interdisant tout tracé continu.<br />
Prise entre la fidélité aux va<strong>le</strong>urs expérimenta<strong>le</strong>s et la nécessité de<br />
dégager des grands traits représentatifs du phénomène, la cartographie manuel<strong>le</strong><br />
exige en permanence des choix plus ou moins arbitraires. Ainsi sur 1 'averse du<br />
6 Août (Fig.4), il n'a été tenu aucun compte de la hauteur 27.6 mm mesurée au<br />
pluviometre n"26, alors que <strong>le</strong>s cotes extrêma<strong>le</strong>s 55.5 et 54.5 mm ont été scrupu-<br />
<strong>le</strong>usement respectées.<br />
Le krigeage, pour sa part, accorde aux va<strong>le</strong>urs expérimenta<strong>le</strong>s une importance<br />
directement 1 iée au degré de structuration du phénomène.<br />
La carte de la Fig.3 a été obtenue après estimation par krigeage aux<br />
noeuds d'une gril<strong>le</strong> régulière.Le pluviomètre n"29 (OU la hauteur mesurée est de<br />
55.5 mm) n'a ainsi contribué que pour environ 63% dans 1 'estimation du point de<br />
gril<strong>le</strong> <strong>le</strong> plus proche. L'influence, non négligeab<strong>le</strong>, des autres stations a ramené<br />
ce point de gril<strong>le</strong> à une va<strong>le</strong>ur de 48.7 mm.<br />
A cette estimation est attaché un écart-type de l'ordre de 6.25 mm, ce<br />
qui rend cette va<strong>le</strong>ur parfaitement compatib<strong>le</strong> avec la va<strong>le</strong>ur expérimenta<strong>le</strong> voisine.<br />
Sur 1 'ensemb<strong>le</strong> du bassin, <strong>le</strong>s écarts-types d'estimation sont compris entre 5.5<br />
et 14 mn, la zone la plus mal connue étant bien entendu la partie Sud-Est.<br />
ESTIMATION DE LA LAME D'EAU MOYENNE SUR UN BASSIN<br />
A l'heure actuel<strong>le</strong>, trois méthodes sont utilisées: <strong>le</strong> planimétrage<br />
des cartes tracées manuel<strong>le</strong>ment, la moyenne arithmétique simp<strong>le</strong>, la méthode des<br />
polygones de THIESSEN.
178<br />
La précision de la première méthode est directement liée à la qualité du<br />
tracé de la carte. Mais el<strong>le</strong> dépend aussi du soin de l'opérateur: de l'attention<br />
qu'il a portée au comptage des carreaux du papier mi1limétré.o~ à éviter <strong>le</strong>s a-<br />
coups dans <strong>le</strong> maniement du planimètre.<br />
La moyenne arithmétique est <strong>le</strong> procédé de calcul <strong>le</strong> plus simp<strong>le</strong>, pour ne<br />
pas dire <strong>le</strong> plus simpliste. Soit Q la quantité tota<strong>le</strong> d'eau qui s'est abattue<br />
sur <strong>le</strong> bassin be surface S. Si Z(x) est la hauteur d'eau au point x, la hauteur<br />
d'eau moyenne Z a pour expression:<br />
7 = Q/S soit 2 = i Is Z(x)dx<br />
Faire une moyenne arithmétique simp<strong>le</strong> sur <strong>le</strong>s Z(x):<br />
c'est perdre de vue que seu<strong>le</strong>s <strong>le</strong>s-quantités d'eau sont additives, et non <strong>le</strong>s<br />
hauteurs. Pour qu'une tel<strong>le</strong> moyenne ait un sens, il faudrait que tous <strong>le</strong>s pluvio-<br />
mètres soient équiva<strong>le</strong>nts, qu'ils représentent en quelque sorte chacun l/Nième du<br />
bassin ,<br />
Les polygones de THIESSEN (8) procèdent d'une analyse physique plus<br />
sérieuse. Ils reposent sur 1 'hypothèse qu'une station est représentative de 1 'ensemb<strong>le</strong><br />
des points du bassin pour <strong>le</strong>squels el<strong>le</strong> est la station la plus proche -<br />
voir Fig.2. L'idée de base est en fait plus généra<strong>le</strong> et ne repose pas réel<strong>le</strong>ment<br />
sur la forme géométrique des polygones. Soit en effet une partition quelconque<br />
du bassin en "zones d'influence'' de surface Si:<br />
s = s, + s, i- .... t SN<br />
Dire que la va<strong>le</strong>ur expérimenta<strong>le</strong> Z(xi) est représentative de la zone Si, c'est<br />
poser:<br />
L'estimation de la quantité d<br />
Q' =<br />
et la lame d'eau moyenne a pour va<strong>le</strong>ur:<br />
eau tota<strong>le</strong> est alors:<br />
Sur <strong>le</strong> plan formel, 1 'estimateur 2* n'est autre qu'une moyenne pondérée des Z(xi).<br />
Ce qui importe en vérité, ce sont <strong>le</strong>s poids Si/S et non la géométrie des zones<br />
d ' i n f 1 uence .<br />
On est ainsi tout naturel<strong>le</strong>ment amené à rechercher <strong>le</strong>s poids Xi qui<br />
opti mi sent 1 'es ti mateur :<br />
t*= 1 Xi Z(Xi)<br />
1
179<br />
En procédant de façon analogue au cas du krigeage ponctuel, on montre que <strong>le</strong>s hi<br />
sont solutions du système:<br />
Comme on choisit toujours pour fo(x) la fonctior constante identiquement<br />
éga<strong>le</strong> 5 1, la première des conditions sur <strong>le</strong>s fonctions f (pour 1=0) s'écrit<br />
si mpl emen t :<br />
I:xj=l<br />
j<br />
La variance d'estimation a pour expression, 5 1 'optimum:<br />
De même que pour <strong>le</strong> krigeage ponctuel, cette variance ne dépend que de<br />
la structure de la variab<strong>le</strong> et de la configuration des points expérimentaux. Par<br />
conséquent, si <strong>le</strong> variogramme est connu, on peut calcu<strong>le</strong>r la variance d'estimation<br />
sans avoir besoin de la va<strong>le</strong>ur des hauteurs d'eau. C'est cette propriété remarquab<strong>le</strong><br />
qu'on utilisera pour localiser un nouveau point de mesure par la "méthode<br />
du point fictif".<br />
Sur l'ouadi Kadjemeur, <strong>le</strong>s calculs ont été effectués pour <strong>le</strong>s 13 épisodes<br />
pluvieux de 1966. Vu <strong>le</strong> faib<strong>le</strong> nombre de points de mesure, il était diffici<strong>le</strong><br />
de procéder 5 1 'inférence statistique du variogramme averse par averse, d'autant<br />
plus que parfois certaines données étaient manquantes. Prendre bruta<strong>le</strong>ment<br />
<strong>le</strong> variogramme moyen sur l'ensemb<strong>le</strong> des averses eut été faire vio<strong>le</strong>nce à la nature<br />
car ces averses diffèrent par <strong>le</strong>ur intensité et <strong>le</strong>ur dispersion. Une hypothèse<br />
plus raisonnab<strong>le</strong> a été d'admettre que <strong>le</strong>s variogrammes des épisodes pluvieux<br />
sont proportionnels:<br />
Yk(h) = Wk<br />
oùyk(h) est <strong>le</strong> variogramme de la kième averse, y(h) <strong>le</strong> variogramme moyen et Wk<br />
<strong>le</strong> coefficient de proportionnalité. Cette relation équivaut 5 admettre qu'il y<br />
a conservation des corrélations spatia<strong>le</strong>s sur <strong>le</strong> bassin. En notant s la variance<br />
expérimenta<strong>le</strong> des hauteurs d'eau de la kiëme averse et 3 la moyenne !es sz, il<br />
en résulte que:<br />
Ok = s;/s;T<br />
-<br />
,
TABLEAU I<br />
xi (en a)<br />
Thiessen (9)<br />
averse<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
11<br />
12<br />
13<br />
effectif<br />
21<br />
28<br />
33<br />
33<br />
33<br />
33<br />
33<br />
33<br />
33<br />
33<br />
33<br />
31<br />
17<br />
moyenne<br />
mm<br />
13.81<br />
34.69<br />
5.34<br />
38.25<br />
1.06<br />
2.33<br />
21.81<br />
4.47<br />
32.29<br />
O .58<br />
O .40<br />
22.35<br />
6.35<br />
1.0 1.0 0.Y 1.2 1.1 1.b 1.1 5.3 3.3 5.U U.6<br />
1.1 1.1 0.8 1.1 1.5 1.5 8.3 3.4 3.8 2.1 1.1<br />
mm2<br />
19.58<br />
64.20<br />
31.90<br />
73.39<br />
1.31<br />
10.96<br />
225.61<br />
82.86<br />
152.21<br />
1.08<br />
1.13<br />
46.46<br />
45.75<br />
o .34<br />
1.10<br />
0.55<br />
1.26<br />
o .o2<br />
0.19<br />
3.88<br />
1.42<br />
2.62<br />
o .o2<br />
o .o2<br />
O .80<br />
O .79
181<br />
On constate une similitude assez nette qui justifierait a posteriori<br />
l'intuition de THIESSEN.<br />
De façon à permettre une comparaison plus complète des différentes méthodes<br />
évoquées, on a porté sur un même graphique (Fig.6) <strong>le</strong>s évaluations des lames<br />
d'eau pour <strong>le</strong>s 13 averses, par moyenne arithmétique, par planimétrage, par<br />
THIESSEN et par krigeage. De part et d'autre de la bissectrice des axes,sur<br />
laquel<strong>le</strong> figure la va<strong>le</strong>ur krigée, on a indiqué la fourchette à 2 écarts-types.<br />
I1 ressort que trois méthodes donnent des résultats à peu pres équiva<strong>le</strong>nts.<br />
Seu<strong>le</strong> la moyenne arithmétique se singularise, en particulier sur <strong>le</strong>s<br />
averses du 9 Août, du 13 au 14 Septembre, du 11 Août et du 23 Juil<strong>le</strong>t, où <strong>le</strong>s<br />
va<strong>le</strong>urs estimées se situent en dehors de la fourchette pourtant très large de 4<br />
écarts-types. Avec la moyenne arithmétique, tous <strong>le</strong>s pluviomètres ont même importance;<br />
<strong>le</strong> réseau étant plus fourni à l'Ouest, il y a systématiquement sousestimation<br />
ou sur-estimation de la lame d'eau selon que 1 'épicentre des averses<br />
se situe à l'Est ou à 1 'Ouest.<br />
Que retenir de ces comparaisons?<br />
Une fois écartée la moyenne arithmétique, il semb<strong>le</strong>rait,du moins sur<br />
l'exemp<strong>le</strong> traité, que l'avantage du krigeage ne soit pas très net. Pourtant,<br />
c'est la seu<strong>le</strong> méthode autour de laquel<strong>le</strong> a pu s'articu<strong>le</strong>r la comparaison, grâce<br />
au calcul d'erreur. En outre, <strong>le</strong>s auteurs ont pu remarquer que la méthode de<br />
THIESSEN, et dans une moindre mesure <strong>le</strong> planimétrage, s'avèrent en pratique longs<br />
et fastidieux. Le krigeage quant à lui ne nécessite que l'investissement d'un<br />
programme.<br />
OPTIMISATION DU RENFORCEMENT D'UN RESEAU PLUVIOMETRIQUE<br />
Remarquons tout de suite que cette question n'a de sens que si <strong>le</strong> but<br />
poursui vi a été cl ai rement défi ni .<br />
Dans <strong>le</strong> cas d'une reconnaissance en vue d'un aménagement hydraulique,<br />
1 'hydrologue a pour objectif l'étude de la relation pluie-débit: il s'intéresse<br />
donc en premier lieu à la quantité d'eau tombée journel<strong>le</strong>ment sur <strong>le</strong> bassin.<br />
La variance d'estimation par krigeage a permis de donner la fourchette d'incertitude<br />
avec laquel<strong>le</strong> cette quantité peut être calculée. El<strong>le</strong> fournit donc tout<br />
naturel<strong>le</strong>ment 1 'indicateur de précision nécessaire pour:<br />
1) décider de 1 'opportunité de renforcer <strong>le</strong> réseau,<br />
2) déterminer 1 'emplacement optimal d'un éventuel pluviomètre<br />
suppl émentai re.<br />
Pour ce faire, on utilisera la méthode du point fictif. A la question<br />
"comment implanter au mieux un nouveau point de mesure" peuvent être attachées<br />
certaines contraintes. Si ce choix ne se pose que parmi un certain nombre de<br />
points présé<strong>le</strong>ctionnés selon un autre critère (accès faci<strong>le</strong>, re<strong>le</strong>vé aisé,. . .),<br />
on implantera fictivement un pluviomètre en chacun d'eux et on déterminera <strong>le</strong><br />
gain dn précision correspondant.
182<br />
On procedera de même <strong>le</strong> long d'un cheminement si 1 'on a décidé a priori de .retenir<br />
une ligne caractéristique du terrain (piste d'accès, accident de terrain,. ..).<br />
Quand au contraire, 1 'on ne possêde aucun a priori , on tracera, toujours<br />
de la même manière, une carte d'isogain en précision sur l'estimation de<br />
la lame d'eau. C'est la solution qui a été adoptée pour l'étude du cas de<br />
1 'ouadi Kadjemeur.<br />
LOCALISATION OPTIMALE D'UN PLUVIOMETRE SUPPLEMENTAIRE<br />
Soit U; la variance d'estimation ã partir des 33 pluviomètres existants,<br />
de la hauteur d'eau tombée en moyenne sur <strong>le</strong> bassin pendant une averse.<br />
Si on implante fictivement un nouveau pluviomètre en un point M, cette<br />
variance d'estimation prend une nouvel<strong>le</strong> va<strong>le</strong>ur U ~M) < U:. Le gain en précision<br />
peut être défini comme:<br />
G(M) =<br />
uO<br />
Si 1 'on ne tenait aucun compte de 1 'existence de corrélations spatia<strong>le</strong>s, on donnerait<br />
comme gain correspondant à un 34e point de mesure: 1/34 -3%. Quant à la<br />
localisation, el<strong>le</strong> serait indifférente à 1 'intérieur du bassin.<br />
Considérant un domaines, <strong>le</strong> krigeage permet de déterminer <strong>le</strong> point M<br />
où ce gain est maximal.<br />
Pour Kadjemeur, <strong>le</strong> domaine retenu a été choisi de sorte qu'il englobe<br />
<strong>le</strong> bassin et ses abords immédiats. Le <strong>le</strong>cteur est invité à se reporter à la<br />
Fig.7 pour voir où il aurait lui-même implanté une nouvel<strong>le</strong> mesure.<br />
I1 suffit de comparer <strong>le</strong>s Fig.7 et 8 pour constater que sur de nombreux<br />
points, "1 'intuition" était insuffisante pour appréhender <strong>le</strong> problême d'une mani-<br />
ère globa<strong>le</strong>.<br />
Le gain maximum est de 13% au 'lieu des 3% donnés par une analyse sommaire.<br />
L'optimum absolu est situé en bordure du bassin, alors qu'au centre, une<br />
grande zone est dépourvue de point de mesure.<br />
Pourtant si l'on revient aux coefficients du krigeage ou ã ceux de<br />
THIESSEN, force est de constater que ces résultats vont dans <strong>le</strong> sens d'un<br />
"soulagement" des pluviomêtres de poids <strong>le</strong>s plus é<strong>le</strong>vés: on tend vers une égalisation<br />
de la contribution des différentes stations, ce qui satisfait <strong>le</strong> sens<br />
physique de 1 'hydrologue.<br />
Enfin, l'examen de la carte isogain (Fig.8) sur l'ensemb<strong>le</strong> du domaine<br />
est três instructif par lui-même. I1 apparaît qu'il vaut mieux implanter judicieusement<br />
un pluviomètre à l'extérieur du bassin plutôt que d'une manière<br />
redondante ã 1 'intérieur.
CONCLUSION<br />
183<br />
Dans une zone aride mal reconnue, 1 ' hydrométéorol ogue ne di spose pas<br />
de longues chroniques aux stations régiona<strong>le</strong>s; il se voit contraint de n'utiliser<br />
que <strong>le</strong>s données qu'il a pu recueillir pendant une ou deux campagnes<br />
annuel<strong>le</strong>s, pour calcu<strong>le</strong>r <strong>le</strong>s lames d'eau sur son bassin versant.<br />
Formalisant et généralisant la méthode des coefficients de THIESSEN, <strong>le</strong><br />
krigeage a permis, en fonction d'un objectif précis - estimation loca<strong>le</strong> ou glo-<br />
ba<strong>le</strong> - de trouver <strong>le</strong>s poids optimaux ii affecter aux différents pluviomètres.<br />
Un interval<strong>le</strong> de confiance a été associé à chaque estimation.<br />
Aussi <strong>le</strong> krigeage a-t-il permis de poser en termes de gain de précision<br />
<strong>le</strong> problème du renforcement d'un réseau: il donne objectivement <strong>le</strong> meil<strong>le</strong>ur<br />
endroit pour implanter un pluviomètre supplémentaire et la nouvel<strong>le</strong> précision avec<br />
laquel<strong>le</strong> pourra être estimée la grandeur étudiée.<br />
La méthode présentée est d'un emploi très soup<strong>le</strong>: <strong>le</strong> coût d'implantation<br />
peut varier selon 1 'emplacement, <strong>le</strong> bassin peut éga<strong>le</strong>ment être découpé en<br />
sous-bassins d'importance différente pour 1 'écou<strong>le</strong>ment.
184<br />
BIBLIOGRAPHIE<br />
Ma<strong>the</strong>ron, G. (1965). Les variab<strong>le</strong>s régionalisées et <strong>le</strong>ur estimation,<br />
Paris, Masson et Cie<br />
Ma<strong>the</strong>ron, G. (1969). Le krigeage universel, Fontaineb<strong>le</strong>au, Cahiers du<br />
Centre de Morphologie Mathématique , Fasc .1<br />
Ma<strong>the</strong>ron, G. (1970). La théorie des variab<strong>le</strong>s régionalisées et ses applications,<br />
Fontaineb<strong>le</strong>au, Cahiers du Centre de Morphologie Mathématique,<br />
Fasc .5<br />
Roche, M.A. (1968). Ecou<strong>le</strong>ment de surface, alimentation de nappe et<br />
transport sol ide des ouadis Fera, Kadjemeur et Sofoya, Fort-Lamy, ORSTOM<br />
Wiener, N. (1966). Extrapolation, interpolation and smoothing of stationnary<br />
time series, Cambridge, Mass., M.I.T. Press<br />
Gandin, L.S. (1963). Objective analysis of meteorological fields, Leningra<br />
Israël program for scientific translation<br />
Delfiner, P., Delhomme, J.P. (1973). Présentation du programme BLUEPACK,<br />
Fontaineb<strong>le</strong>au, Eco<strong>le</strong> des Mines de Paris, note interne<br />
Thiessen, A.H. (1911). Precipitation averages for large areas, Monthly<br />
Wea<strong>the</strong>r Rev. , vol. 39, n07, p. 1082<br />
Del finer, P. (1968). Cartographie et morphologie des précipitations<br />
considérées comme variab<strong>le</strong>s régionalisées , Université de Grenob<strong>le</strong><br />
(10) Delhomme, J.P. (1970). Présentation d'une méthode objective d'interpolatio<br />
pour la construction de cartes isopiézométriques , Douai , Communication au<br />
Groupe d'Etude de Bassins Versants Souterrains<br />
(11) Delhomme, J .P. (1971). Traitement géostatistique des données piézométrique<br />
<strong>le</strong> krigeage en hydrogéologie, Fontaineb<strong>le</strong>au, Recyclage en hydrogéologie<br />
mathématique<br />
(12) Delfiner, P. (1973). Analyse du géopotentiel et du vent géostrophique par<br />
krigeage universel, Paris, Note EERM - Météorologie Nationa<strong>le</strong>
Fig.1 - Situation du B.V de l'ouadi Kadjemeur<br />
185
186<br />
Fig.3- Carte obtenue par krigeage
2<br />
mm<br />
4<br />
100<br />
o 1 5 10<br />
c<br />
km<br />
Fig.5 -Variogramme moyen bp~vt~da~n&
188<br />
I<br />
Fig.6- Comparai.mi dei différ-iites méthodes d'estirnaticin globa<strong>le</strong>
189
P<br />
..
IMPROVEMENT OF RUNOFF RECORDS IN SMALLER WATERSHEDS BASED ON<br />
ABSTRACT<br />
PERMEABILITY OF THE GEOLOGICAL SUBSURFACE<br />
Dr. George J. Halasi-Kun<br />
Chairman, Columbia University Seminars<br />
on Pollution and Water Resources, New York, USA<br />
In smal<strong>le</strong>r watersheds with an area <strong>le</strong>ss than 250 km2, <strong>the</strong> recorded<br />
hydrologic data are scarce or non-existent because, Correlating<br />
<strong>the</strong> extreme runoff data with <strong>the</strong> characteristic permeability of<br />
<strong>the</strong> different geological formations can not only improve <strong>the</strong> va-<br />
rious methods for simulation or interpretation of hydrologic in-<br />
formation from o<strong>the</strong>r areas but also provides additional improve-<br />
ment in hydrological data ga<strong>the</strong>ring. An accoynt is given about<br />
tentative average values in millidarcys for permeability of diffe-<br />
rent geological formations based on se<strong>le</strong>cted bibliography and<br />
previous experience. Fur<strong>the</strong>r correlation of peak runoffs in Central<br />
Europe and in <strong>the</strong> Nor<strong>the</strong>astern coastal area of <strong>the</strong> United States<br />
with <strong>the</strong>ir specific geological subsurface is discussed.<br />
Finally, it is pointed out that <strong>the</strong> geological subsurface as a<br />
characteristic of <strong>the</strong> peak runoff does not apply at ail for water-<br />
sheds with an area over 300 km2. Similar but <strong>le</strong>ss c<strong>le</strong>arly defined<br />
correlation can be found between <strong>the</strong> lowest runoff and <strong>the</strong> storage<br />
capacity of <strong>the</strong> geological formations.<br />
RESUME<br />
Dans <strong>le</strong>s petits bassins, de surface infgrieure à 250 km2, <strong>le</strong>s<br />
données hydrologiques enregistrdes sont rares ou n<strong>le</strong>xistent %as.<br />
L'étude des corrélations entre <strong>le</strong>s mesures d'écou<strong>le</strong>ment extremes<br />
et la perméabilité caractéristique des différentes formations géo-<br />
logiques peut améliorer <strong>le</strong>s diverses méthodes de simulation ou<br />
d'interprétation des reqseignements hydrologiques provenant<br />
d'autres régions, ainsi que <strong>le</strong> rassemb<strong>le</strong>ment des donnees h.flrologi-<br />
ques. On a essayé de donner des va<strong>le</strong>urs moyennes de perméabilité,<br />
en l'millidarcyt', pour différentes formations géologiques en se<br />
basant sur une bibliographie sé<strong>le</strong>ctionnée et sur l'expérience anté-<br />
rieure. D'autres corr&latio?s, entre <strong>le</strong>s écou<strong>le</strong>ments de pointe en<br />
Europe Centra<strong>le</strong> et sur la Cote Nord-Est des Etats-Unis, et <strong>le</strong>urs<br />
sous-sols géologiques, sont aussi discutée:. Enfin on montre que,<br />
pour des bassins d'une surface supérieure a 300 km', la structure<br />
géologique souterraine ne constitue pas du tout une caractgristique<br />
des écou<strong>le</strong>ments de pointe. Des corrélations analogues, mais moi'ns<br />
clairement définies, peuvent se rencontrer entre <strong>le</strong>s écou<strong>le</strong>ments<br />
minimaux et la capacitd de stockage des formati'ons g@ologiques,
192<br />
(1) INTRODUCTION<br />
fi<br />
In smal<strong>le</strong>r watersheds with an area <strong>le</strong>ss than 250 lan", <strong>the</strong> recorded<br />
hydrologic data is scarce. Generally, no data for longer<br />
period is at hand, when <strong>the</strong> area is planned for development. In<br />
accordance with various studies conducted in moderate climatic<br />
conditions in Central lcurope and in <strong>the</strong> Nor<strong>the</strong>astern United States<br />
of !\merica, it seems that <strong>the</strong> peak runoff values of <strong>the</strong>se drainage<br />
basins are highly dependent on <strong>the</strong> geologic subsurface where <strong>the</strong><br />
permeability of <strong>the</strong> rock formations provides<strong>the</strong> ground water storage,<br />
or <strong>the</strong>ir impervious surface preconditions <strong>the</strong> extent of <strong>the</strong><br />
lake and swamp areas of <strong>the</strong> region cl]. Both <strong>the</strong>se characteristicr<br />
directly influence <strong>the</strong> drainage density in <strong>the</strong> areas with different<br />
permeability faotor of <strong>the</strong> geological formations [2], as can<br />
be demonstrated, for instance, by <strong>the</strong> hydrographioal map of Southweet<br />
Germany from <strong>the</strong> Upper-Danube region [Figure 1).<br />
Ano<strong>the</strong>r claeeio examp<strong>le</strong> can be <strong>the</strong> river training and flood<br />
control program of 1840-1950 in <strong>the</strong> Carpathian Basin in Central<br />
Europe where -- even in a large watershed like <strong>the</strong> Danube at<br />
Orshova, Romania (576,240 km- area of drainage basin with a yearly<br />
average rainfall of 900 mm) -- <strong>the</strong> diminishing of lake and swamp<br />
area by extensive drainage and flood control, <strong>the</strong> maximuin annual<br />
flood increased by 15~6 in <strong>the</strong> observed 110 year period (Figure 2)<br />
whi<strong>le</strong> <strong>the</strong> lake and swamp area decreased from llo7 to 3% of <strong>the</strong><br />
watershed. In <strong>the</strong> Carpathian Basin (318,030 1ans of <strong>the</strong> Danube<br />
drainage basin related to <strong>the</strong> observation station Orshova, Romania)<br />
39,000 km2 agricultural land was reclaimed in <strong>the</strong> flood and<br />
marshland region. (Figure 3 shows <strong>the</strong> reclaimed area for <strong>the</strong> 110<br />
year period including <strong>the</strong> two lakes of that basin.)<br />
A similar effect on peak runoff was observed in <strong>the</strong> State of<br />
New Jersey, U.S.A. in a territory of 20,295 h2, This obse ation<br />
is bas d on 67 gaging stations for watershed of from 25.6 2 to<br />
512 Inn 8 with an average yearly rainfall 1125 mm (Figure 4: Adjustment<br />
factor for effect of lakes and swamps on peak runoff in New<br />
Jersey 1897-1972 compared with data developed from <strong>the</strong> Danubian<br />
basin at gaging station Orshova, Romania 1840-1950).<br />
(2) hhXDdtma SURFACE FLOW<br />
Researohes conducted in <strong>the</strong> pa t decades concerned maximum<br />
flow in watersheds with area 250 km' or <strong>le</strong>ss, where <strong>the</strong> geological<br />
conditions, topographic characteristics and rock formations permit<br />
an evaluation of peak rates of runoff from smal<strong>le</strong>r watersheds.<br />
Such research revea<strong>le</strong>d a O<strong>le</strong>ar influence of <strong>the</strong>se factors on <strong>the</strong><br />
peak runoff. In accordance with <strong>the</strong> findings abroad (in Central<br />
Europe in an area of 49,008 h a) 13) and in <strong>the</strong> Nor<strong>the</strong>astern Uni-
193<br />
ted States (in an area of 20,295 km2) [23 <strong>the</strong> correlation of <strong>the</strong><br />
100 year peak flow with <strong>the</strong> geologic subsurface, <strong>the</strong> topographic<br />
conditions (slopes) and <strong>the</strong> size of <strong>the</strong> watershed can be put in<br />
<strong>the</strong> following equation:<br />
Q = C.A-e, where<br />
Q = 100 year peak runoff value in m3fsec.km2<br />
A = area of watershed in km2<br />
C = coefficient depending on <strong>the</strong> geological subsurface with value:<br />
In Central Europe [3] for 100-125 mm/day point rainfall<br />
intensity and 15 km by 50 km recorded storm patternfrom<br />
1 to 10.2 (Figure 51<br />
In Nor<strong>the</strong>astern United States of Amerlca [I] for 2OQ-250<br />
mm/day point rainfall intensity and 30 km by I05 km<br />
recorded storm pattern -<br />
from 7.3 to 147 (Tab<strong>le</strong> 1)<br />
e = exponent of <strong>the</strong> watershed area depending on <strong>the</strong> topographic<br />
character of <strong>the</strong> watershed (0.35 -plains; 0.37- slightly<br />
hilly plains; 0.44-0.46 -steeper hills and moderate mountains;<br />
0.50 - Alpine type mountains).<br />
Analyzing <strong>the</strong> figures of <strong>the</strong> geological runoff coefficient,,<br />
<strong>the</strong> result seems to be identical in both areas studied, if we<br />
assume that <strong>the</strong> point rainfall intensity and <strong>the</strong> size of recorded<br />
storm pattern have a direct influence on <strong>the</strong> peak runoff (Figure<br />
5). Fur<strong>the</strong>rmore <strong>the</strong> vegetative cover showed a 2 5% effect on <strong>the</strong><br />
peak flood. Similar influence was observed concerning <strong>the</strong> form of<br />
watershed Ct5% for fan-shaped and -30% elongated form of drainage<br />
basin). Urbanization alters <strong>the</strong> geological character of <strong>the</strong> surface<br />
and has a direct relation in increasing <strong>the</strong> surface peak flows<br />
41 *<br />
(3 1<br />
GROUND WATER STORAGE CAPACITY<br />
It is obvious that <strong>the</strong> rate of surface runoff must be related<br />
in an inverse way to <strong>the</strong> permeability of <strong>the</strong> geological subsurface<br />
of <strong>the</strong> watershed; and <strong>the</strong> quality and quantity of ground water<br />
storage is directly dependent on <strong>the</strong>se condltlons. Based on over<br />
70,ûOû well-record fi<strong>le</strong>s of domestic and industrial wells through-<br />
out <strong>the</strong> State of New Jersey, U.S.A.<br />
(area 20,295 km22 for <strong>the</strong><br />
period 1947-1972, <strong>the</strong> ground water availability in rock formations<br />
from Precambrian thorough Triassic in age and from unconsolidated<br />
sediments from <strong>the</strong> Cretaceous to <strong>the</strong> present, can be estimated.<br />
Comparison of large statlstical samp<strong>le</strong>s of well-records i’n <strong>the</strong><br />
rock formations to a depth of as much as 550 m has provided a means<br />
of estimating <strong>the</strong> ground water potential of areas underlain by
194<br />
specific rock types [5, 6, 7, 81. Several of <strong>the</strong>se estimates of<br />
ground water availability have been tested against <strong>the</strong> experience<br />
in areas of suburban development during times of drought 1961-1966.<br />
There us sufficient consistency in <strong>the</strong> results to indicate that<br />
underlying rock and sediment types may be determined from well<br />
data where <strong>the</strong>y are o<strong>the</strong>rwise concea<strong>le</strong>d by soil and overburden.<br />
The ga<strong>the</strong>red statistical data on ground water availability<br />
in New Jersey is based al60 on pumping tests and records of <strong>the</strong><br />
wells, and <strong>the</strong>ir figures can be accepted on <strong>the</strong> assumption of an<br />
average yearly rainfall 1125 mm -- which can drop for two consecu-<br />
tive years of' drought to 850 mm. Ground water is availab<strong>le</strong> in <strong>the</strong><br />
nor<strong>the</strong>rn half (rock country1 of <strong>the</strong> area studied only to a depth<br />
of 180 m below <strong>the</strong> surface. Boring tests proved that below that<br />
<strong>le</strong>vel, <strong>the</strong>re is a very marked decrease in fractures and fissures<br />
from which water can be obtained. There is also evidence, of<br />
course, that certain fracture zones may give abundant water at<br />
great depth, but <strong>the</strong>se fractures are those such as <strong>the</strong> Triassic<br />
border fault or o<strong>the</strong>rs that have been weil known. In <strong>the</strong> coastal<br />
half (coastal plains) of this examined region <strong>the</strong> limit, in<br />
accordance with <strong>the</strong> aquifer layers, is from 200 to 1000 m (from<br />
West to East) below <strong>the</strong> sea<strong>le</strong>vel 8 .<br />
Evapotranspiration and interception average 450-560 mm yearly,<br />
and <strong>the</strong> yearly average runoff is up to 550 mm from <strong>the</strong> annual<br />
precipitation. The ground water availability indicator has a value<br />
from O to 450 mrn yearly, depending on <strong>the</strong> permeability and storage<br />
capacity of <strong>the</strong> geological formations 191.<br />
Despite <strong>the</strong> fact that <strong>the</strong> estimate of <strong>the</strong> regional availabi-<br />
lity is complicated by factors such as recharge or transmissability<br />
frow adjacent areas, <strong>the</strong>re is c<strong>le</strong>ar evidence of correlation of<br />
permeability of geological formations with surface peak runoff and<br />
ground water availability. The comparison can be based only on<br />
average values of permeability because <strong>the</strong>y are measured under<br />
difficult conditions and in various geological formations which<br />
are similar to those formations used in establishing runoff for-<br />
mula coefficients and ground water availability indicators (Ta-<br />
b<strong>le</strong> 21. ït must be pointed out that <strong>the</strong> various formations are<br />
also, in general, already mixed or interwoven even in smal<strong>le</strong>r<br />
drainage areas. The uneven surface wea<strong>the</strong>ring, artificial imper-<br />
vious surface due to urbanization, <strong>the</strong> disintegrated underlying<br />
rock formations at various depths and possib<strong>le</strong> faults add to <strong>the</strong><br />
difficulty of establishing a practical average value of permeabi-<br />
lity, ground water availability or surface runoff even for a<br />
small waters he d.
195<br />
The various studies showed that <strong>the</strong> geological subsurface has<br />
an effect only on smal<strong>le</strong>r watersheds with an area of 250 km2 or<br />
<strong>le</strong>ss. The "geologic" character starts to "fade away" when <strong>the</strong><br />
size of <strong>the</strong> watershed is larger than 235 km2; and for a watershed<br />
of over 340 km2 <strong>the</strong> use of formulas based on geologic conditions<br />
is not recommended because o<strong>the</strong>r factors affect <strong>the</strong> flood flow,<br />
and <strong>the</strong> influence of geological factors is negligib<strong>le</strong>. Even more<br />
confined in area is <strong>the</strong> ground water availability estimate where<br />
<strong>the</strong> practical upper limit may be <strong>le</strong>ss than 200 km2, depending on<br />
<strong>the</strong> surface conditions and on <strong>the</strong> comp<strong>le</strong>xity of <strong>the</strong> subsurface<br />
rock formations.<br />
(4) LOWEST SURFACE FLOW<br />
The lowest flow in smal<strong>le</strong>r watersheds -- whose importance is<br />
to find out <strong>the</strong> pollution effect of various polluting sources<br />
depends , similarly, on <strong>the</strong> permeability of <strong>the</strong> geological subsur-<br />
face, <strong>the</strong> <strong>le</strong>ngth of drought, <strong>the</strong> frequency and'<strong>the</strong> amount of rain,<br />
<strong>the</strong> storage capacity of <strong>the</strong> aquifer layers, <strong>the</strong> evapotranspiration,<br />
<strong>the</strong> temperature of <strong>the</strong> atmosphere and of <strong>the</strong> soil, <strong>the</strong> retardation,<br />
effect of forests and of lakes and <strong>the</strong> e<strong>le</strong>vation of <strong>the</strong> watershed<br />
not to mention transfer of availab<strong>le</strong> water from one watershed to<br />
<strong>the</strong> o<strong>the</strong>r. From <strong>the</strong>se few factors is also evident that reliab<strong>le</strong><br />
data about lowest flow are even more scarce for smal<strong>le</strong>r watersheds<br />
than <strong>the</strong> peak flow. In general, <strong>the</strong> surface watey records contain<br />
prery li'mited amount of data pertaining to lowest flows.<br />
Despite <strong>the</strong>se circumstances, <strong>the</strong>re is sufficient evidence to<br />
develop also a formula for lowest runoff based on records from<br />
both areas col<strong>le</strong>cted especially from <strong>the</strong> drought periods 1921 and<br />
2947 in Czechoslovakia [IO, 11, 12) and 1961-1966 in New Jersey<br />
15. 6, 7, 81 as it follows:<br />
Q z CSA-~, where<br />
Q lowest runoff (50 years?) value in l/sec.km2<br />
A = area of watershed in km2<br />
C coefficient depending on <strong>the</strong> geological subsurface (Tab<strong>le</strong> 31<br />
In Central Europe from O to 5.58<br />
In New Jersey, U.S.A. from O to 5.75<br />
e 0,065; <strong>the</strong> exponent indicates an almost even distribution of<br />
<strong>the</strong> lowest runoff regard<strong>le</strong>ss of <strong>the</strong> size of watershed and <strong>the</strong><br />
availab<strong>le</strong> data and values were not sufficient evidence to<br />
evaluate <strong>the</strong> influence of slopes and topographic configuration<br />
of <strong>the</strong> drainage basins in fur<strong>the</strong>r details.
196<br />
In Both examined areas in <strong>the</strong> plain region, where <strong>the</strong> aquifer<br />
sediments prevail and reach considerab<strong>le</strong> depth as far as 1000 m<br />
below sea<strong>le</strong>vel, <strong>the</strong> runoff coefficients decrease in value because<br />
<strong>the</strong> surface runoff is absorBed by <strong>the</strong>se highly permeab<strong>le</strong> layers.<br />
As a contrast to this phenomena in ayeas such as <strong>the</strong>se along <strong>the</strong><br />
Atlantic Coast in New Jersey or <strong>the</strong> val<strong>le</strong>y of <strong>the</strong> Danube and <strong>the</strong><br />
Tisza in Sou<strong>the</strong>rn Czechoslavakia, <strong>the</strong> ground water col<strong>le</strong>cted, even<br />
from distant area, "spills over" fpon its subsurface storage and<br />
keeps <strong>the</strong> surface runoff values up to 10.4 lfsec,km2 in periods of<br />
drought. This outcropping of <strong>the</strong> gpound water can be observed also<br />
in <strong>the</strong> surface streams of <strong>the</strong> "Fine Barrens" area of Sou<strong>the</strong>rn New<br />
Jersey. Unfortunately, <strong>the</strong> data concerning low runoff is far <strong>le</strong>ss<br />
availab<strong>le</strong> than that for, peak runoff or well-records, This makes<br />
fur<strong>the</strong>r evaluation dad calculation extremely difficult, Therefore,<br />
this method of lowest runoff computation may be considered only as<br />
an estimate for planning and developing purposes.
197<br />
Brm I OGRAP IIY<br />
1. T-Ialasi-Kun, G.J. (1972). “Data Col<strong>le</strong>cting on Water Resources<br />
and Computations of MaximUm Flood for Smal<strong>le</strong>r Watersheds, If<br />
Simposio Internacional Yobre la Planifioacion de Recursos<br />
Ridraulicos - Ponenoias, Volume I, Mexico.<br />
2. rrerak, ñ!. , Stringfield, V.T. (1972). Karst, Amterdam-London-<br />
New York, Elsevier.<br />
3. Halasi-Kun, G.J. (1968). Die Ermittlu g von Hbchstabflüssen<br />
für Einzugsgebiete k<strong>le</strong>iner ala 300 b3 im Bereich der Slowakei,<br />
Braunsohweig, Leichtweiss-Inst itut.<br />
4. nalasi-Kun, G.J. (1969). ”Correlation Between Precipitation,<br />
Flood and Windbreak Phenomena o9 <strong>the</strong> Mountains, 1’ Proceedings<br />
of University Seminar on Pollution and Water Resources, Vol. I,<br />
~ e w York - Trenton, Columbia University - State of New Jersey,<br />
5. Mil<strong>le</strong>r, J. (1973). Geology and Ground Viater Resources of Sussex<br />
County ... , Geol. Bul<strong>le</strong>tin No. 73, Trenton, State of New Jersey.<br />
6. Widmer, K. (1905). Geology of <strong>the</strong> Ground Water Resources of<br />
Mercer County, Geoï. Report No. 7, Trenton, New Jersey ~eol.<br />
Surve y.<br />
7. Rhodehamel, E.C. (1970). A Hydrologic Analysis of <strong>the</strong> New Jersey<br />
Pine Barrens Region, Water Resources Circular No. 22, Trenton,<br />
State of New Jersey and U.S.G.S.<br />
8. Barksda<strong>le</strong>, H.C., Greenman, D.W., Land, S.M., Milton, O.S.,<br />
Outlaw, D.E. (1958). Groundwater Resources in <strong>the</strong> Tri-State<br />
Region Adjaoent to <strong>the</strong> Lower Delaware River, Special Report<br />
No. 13, Trenton, State of New Jersey.<br />
9. Halasi-Kun, G.J. (1971). llAspects hydrologiques de la pollution<br />
et des reBsources en eau, dans <strong>le</strong>e domaines urbains et industriels,lt<br />
Actes du Congres: Scienoes et Techniques An 2000, Paris,<br />
SICP.<br />
10. Dub, O. (1957). Hydrologia, hydrografia, hydrometria, Praha-<br />
Bratislava, SVTL.<br />
11. Halasi-Kun, G. J, (1949) Hydrologia, Kosice.<br />
12. Halasi-Kun, G.J. (1954). Voda v polnohospodárstve(Water in Agriculture),<br />
Bratislava, SPN.<br />
13. Davis, St.N., Dewiest, R.J.M. (1966). Hydrogeology, New Yorlc-<br />
London-Sydney, Je Wi<strong>le</strong>y 81 Sons .<br />
14. Lins<strong>le</strong>g, R.K.Jr., Koh<strong>le</strong>r, M.A., Paulhus, J.L.H. (1968). Hydrology<br />
for Engineers, New York - Toronto - London, hiCGraW-Hill.<br />
15. Water Resources Data for New Jersey: 1961-1971, Part 1: Surface<br />
Water Records (1962-1972). Treiiton, U.S.G.S.
198<br />
FIGURE I: HYDROGRAPHICAL MAP OF SOUTHWEST GERMANY FROM<br />
THE UPPER-DANUBE REGION.
YEARS<br />
FIGURE 2: MAXIMUM' ANNUAL FLOOD OF THE DANUBE RIVER AT<br />
ORSHOVA, ROMANIA, 1840-1950.<br />
0 SERIES OF PEAK FLOOD DISCHARGES.<br />
@ AVERAGE PEAK FLOOD.<br />
@TREND OF THE AVERAGE PEAK FLOOD.<br />
199
LAKE AND SWAMP AREAIIN PERCENT OF DRAINAGE AREA<br />
201<br />
FOR WATERSHEDS OF 25.6-512km2 (BASED ON 67GAGHG STATIONS IN NEW JERSEY,<br />
USA I 1897 - 1972)<br />
------- FOR IIWTERCHED OF 576.240 km2 (DANUBE AT ORSHOVA,ROMANIA, 1840-1950 WE<br />
TO LAND RECLAMATION AND FLOOD CONTROL)<br />
FIGURE 4: ADJUSTMENT FACTOR FOR EFFECT OF LAKES AND SWAMPS ON PEAK RUNOFF<br />
IN NEW JERSEY, U.S.A. AND DANUBE AT ORSHOVA, ROMANIA
202<br />
Tab<strong>le</strong> 1: Peak Runoff Coefficient in Various Hydrogeologic Regions:<br />
Hydrogeologic Regions: Peak Runof<br />
(formations) in Central Euroue:*<br />
(1) Kaolinite, Clay in-<br />
cluding argillaceous<br />
Triassic or Tertiary<br />
Pa<strong>le</strong>ogene Flysch<br />
(2) Pa<strong>le</strong>ozoic Sha<strong>le</strong>s,<br />
Schist and Mesozoic<br />
Mar 1<br />
(3) Igneous Rocks<br />
Tertiary Marl<br />
(5) Wea<strong>the</strong>red Igneous<br />
Rocks, Limestone, Tuff<br />
(6) Mesozoic Triassic<br />
Brunswick Pormat ions .<br />
(7) Mesozoic Cretaceous<br />
Clayey Sands, Tertiary<br />
Eogene Clayey Sands<br />
(8) Tertiary Miocene Sands<br />
and Quaternary Moraines<br />
(9) Tertiary Neogene<br />
! (for peak runoff<br />
17 5-18.2<br />
14<br />
10<br />
7<br />
6<br />
7<br />
7<br />
1 9-2 5<br />
1.9-2 5<br />
1<br />
Coeff icient<br />
* Ratio of combined effeot for point rainfall i tensity and size<br />
of storm pattern in <strong>the</strong> two observed regions 113:<br />
1 to 8.2 = Eastern Czechoslovakia to New Jersey, U.S.A.<br />
147<br />
100<br />
81<br />
70<br />
69<br />
37<br />
30<br />
26<br />
12<br />
7.3
Tab<strong>le</strong> 2: Ground Water Availability in Various Hydrogeologic<br />
Regions and Their Average Permeability:<br />
(format ions )<br />
(i) Kaolinite, Clay in<br />
eluding argillaceous<br />
Triassic or Tertiary<br />
Pa<strong>le</strong>ogene Flysch<br />
(2) Pa<strong>le</strong>ozoic Sha<strong>le</strong>s,<br />
Schist and Mesozoic<br />
hfarl<br />
(3) Igneous Rooks<br />
(except Basalt,Diabase<br />
Sandst ones y Meeozoio<br />
't'riassic Stockton Form<br />
(4) Dolomite, Besalt a<br />
Tertiary Marl<br />
(5) Wea<strong>the</strong>red lgneoae<br />
Rocks, Limestone a Tuff<br />
(6) MeeOZQiO Triassio<br />
Brunmiak Formations<br />
(7) Mesozoio Cretacteou<br />
Clayey Sands Tertiary<br />
Eogene Clayey Sands<br />
(8) Tertiary Miocene 3 nde<br />
and Quaternary Moraine 'I<br />
(10) Quaternary Beach<br />
Sands (cape May Form.)<br />
Woundwater Availability<br />
in New Jersey, U.S.A.<br />
(in mm/year [5,6,7,8)):<br />
Y<br />
d<br />
17 - 25 7<br />
<strong>le</strong>ss than 47<br />
63<br />
87 - 125<br />
150<br />
200-225<br />
2 50<br />
300<br />
3 50<br />
455<br />
203<br />
verage Permea-<br />
ility in milli-<br />
arcys 113,144 ) :<br />
1<br />
2<br />
1-1.9<br />
2<br />
2.5<br />
42<br />
3<br />
62<br />
7<br />
102-lP2<br />
102-142<br />
18.22<br />
Values are based on over 70,000 well-reoord fi<strong>le</strong>s of domestic and<br />
industrial wells of <strong>the</strong> State of New Jersey from <strong>the</strong> period of<br />
1947-1972. Fur<strong>the</strong>r information especially for regions (2),(3)y<br />
(6) and (71, is in references ba6,TY8,13,143. The form of data<br />
on average permeability makes easy comparison with Figure 5.
204<br />
Tab<strong>le</strong> 3: Lowest Runoff Coefficient in Various Hydrogeol. Regions:<br />
ifydrogeologic Regions:<br />
Lowest Runoff Coefficient<br />
( f orrnat ions)<br />
tn Central Europe: I in New<br />
I<br />
Jersey, U.S.A.;<br />
(for lowest runoff values in ï/seo.km2)<br />
(i) Tiaoïinite, Clay inc<br />
luding argillaceous<br />
Triassic or Tertiary<br />
Pa<strong>le</strong>ogene Flysch<br />
O *3-O 6<br />
0-0 26<br />
(2) Pa<strong>le</strong>ozoic Sha<strong>le</strong>s,<br />
Schist and h4e s oz oio<br />
Marl<br />
3) Igneous Rocks<br />
I except Basalt,Diabase)<br />
0-1 70<br />
Sands t ones , Mes oz oio<br />
Triassio Stockton Form.<br />
(4) Dolomite, Basalt an<br />
Tertiary Marl<br />
*<br />
1-2<br />
O 17-0 79<br />
(5) 'Nea<strong>the</strong>red Igneous<br />
Rocks, Limestone, Tuff<br />
(6) Mesozoio Triassi0<br />
Brunswick Formations<br />
(7) Mesozoic Cretaoeous<br />
Clayey Sands, Tertiary<br />
Eogene Clayey Sands<br />
(8) Tertiary Miooene Sands<br />
and Quaternary Moraines<br />
(9) Tertiary Neogene<br />
Sands, Mesozoic Cretaoeous<br />
Magothy-Raritan Formations,<br />
aternary River Drift<br />
O) Quaternary Beach<br />
ande (cape May Form.) -<br />
**<br />
4-5 68<br />
b<strong>le</strong>ss than 0.3"<br />
O. 62-0 91<br />
2 71-5 75
-- ABS TRACT<br />
-<br />
DETERMINATION OF SNOW WATER EQUIVALENT AND SNOWMELT WATER<br />
BY THICKNESS OF SNOW COVER DATAS<br />
George Kovács - George Molnár<br />
Res-earch Institute for Water Resources Develapment ,<br />
Budapest, Hungary'<br />
In studies on <strong>the</strong> accumulation- and melting process of snow bulk<br />
densities have been determined for fresh snow (Ymin), for snow saturated<br />
by capillarv water (y,> and for melting snow (ymax). Correlation<br />
studies have shown <strong>the</strong> magnitudes of y and Ymax to depend<br />
greatly on <strong>the</strong> number (RI Qf snow lagreps, Tke equations insolying <strong>the</strong><br />
bulk densities listed above form <strong>the</strong> Basi's of tEe computation charts<br />
prepared by <strong>the</strong> authors. These can be applied to two tñus far unsolved<br />
prob<strong>le</strong>ms: I. <strong>the</strong> reconstruction of past time serpes of water equi<br />
va<strong>le</strong>nt values for observing statìons w?tk data on tñe thi'ckness of <strong>the</strong><br />
snow cover only, and 2. forecasting <strong>the</strong> duration of <strong>the</strong> melting period<br />
and of <strong>the</strong> volume of snow-melt water form data on <strong>the</strong> thickness of cover<br />
and <strong>the</strong> air temperature predicted for <strong>the</strong> melting period.<br />
Au cours de l'analyse du phénomene d'accumulation et de fonte de<br />
e <strong>le</strong>s auteurs ont déterminé <strong>le</strong> poids volumétrique (y min,<br />
) initial de la neige fraiche, <strong>le</strong> poids volumétrique (y ,<br />
) de la neige a capìllaire-saturatton, <strong>le</strong> poids volum&tbique<br />
(y max, [g/cm3] 1 de la neige en fonte, Les analpee correllati'onal<strong>le</strong>s<br />
ont démontré que <strong>le</strong>s va<strong>le</strong>urs de y et y max dépendent d'une mesure con<br />
k<br />
sidérab<strong>le</strong> du nombre de couches de\la neige (7). Ce sont <strong>le</strong>s equationsexprimant<br />
<strong>le</strong>s poids volum~tri'qne prê=nt&ea pl.pe- EaPt qui servent de<br />
base au diagrama des auteurs, ce dernier permettant la solution de<br />
deux prob<strong>le</strong>mes jusqu'alors irrésolus: 1. Réalisation rétrospective des<br />
séries de données, dépendantes du temps, pour l'équiva<strong>le</strong>nt neige-eau<br />
dans <strong>le</strong> cas ou l'on n'a procédé qu'a l'observation de l'épaisseur de<br />
la neige, 2. Pronostics concernant la durse de la fonte et de la quantité<br />
d'eau qui s'y produit, a la Lase d'une épaisseur mesurée et d'une<br />
température prévue pour la période de fonte.
206<br />
ïNTPnODUCTIOIJ<br />
In water budget calculations, on <strong>the</strong> income side <strong>the</strong> snow or<br />
<strong>the</strong> meltage of snow is of great importance. The forecasting of<br />
spring floods and undrained runoff waters, <strong>the</strong> planning of reser-<br />
voir operation during <strong>the</strong> melting period and o<strong>the</strong>r prob<strong>le</strong>ms re-<br />
quire more accurate information on snow accumulation and melting,<br />
<strong>the</strong> continuous observation of <strong>the</strong> water content stored in <strong>the</strong> snow<br />
cover, $he recovery of past data and <strong>the</strong> reconstruction of snow<br />
water reoords with <strong>the</strong> help of observation data availab<strong>le</strong>. Above<br />
all <strong>the</strong> thickness of mow cover should be considered which has<br />
been continuously observed at more than 1000 stations for nearly<br />
100 years in Hungary. The network of water equiva<strong>le</strong>nt measuring<br />
stations, however, works only from 1960, comprising presently 60<br />
s tat i onse<br />
The paper summarizes <strong>the</strong> investigations'aiming at <strong>the</strong> dis-<br />
covery of <strong>the</strong> relation between snow cover thickness and snow-water<br />
equiva<strong>le</strong>nt on <strong>the</strong> basis of <strong>the</strong> numerous (about 200 O00 per eeason)<br />
data for <strong>the</strong> 12 years between 1960 and 1971.<br />
1. THE PROCESS OF TIB DEVELOPWT, ACCüMüUTION<br />
AND MELTING OF SNOW<br />
1.1 The development and accumulation of snow<br />
Snow crystals are formed by <strong>the</strong> hexagonal ice prisms deposit-<br />
ed around <strong>the</strong> concentration cores (soot, dust, etc.) overcoo<strong>le</strong>d to<br />
-15 - -25OC in <strong>the</strong> high regions of <strong>the</strong> atrnosphere.Lom temperatures<br />
(-10 - -15OC) and high moisture contents are conducive to <strong>the</strong> for-<br />
mation of crystals containing large, ramifying pore apace and to<br />
<strong>the</strong> deposition of <strong>the</strong>se crystals. Under reversed circumstances<br />
(temperature around O°C, low moisture content) so cal<strong>le</strong>d cylindric<br />
crystals and need<strong>le</strong>s of ice will develop and deposit densely pack-
ed, with high bulk density on <strong>the</strong> soil surface Il, 4, 103.<br />
The distribution examination performed using numerous data to<br />
determine <strong>the</strong> bulk density rg/cm31 of fresh snow having an<br />
intact crystal structure and containing no capillary water at all,<br />
yielded <strong>the</strong> following result:<br />
*min<br />
P 0.118 i 0.028<br />
3<br />
Ig/cm 3<br />
From <strong>the</strong> beginning of accumulation <strong>the</strong> snow cover becomes<br />
continuously more compact and its bulk density increases accord-<br />
ingly. This is <strong>the</strong> consequence on <strong>the</strong> one hand of <strong>the</strong> closer and<br />
closer agglomeration of snow crysta<strong>le</strong> and, on <strong>the</strong> o<strong>the</strong>r hand, of<br />
<strong>the</strong> increase of capillary water content stored in <strong>the</strong> pore spaces<br />
between <strong>the</strong> crystals. The increase in bulk density is caused by<br />
<strong>the</strong> combined effect of external (temperature, sunshine, wind,etc.)<br />
and internal (<strong>the</strong> weight of snow, etc.) factors to which <strong>the</strong> snow<br />
cover is exposed C3, 5, 6, 7, 8, 91.<br />
Obviously, <strong>the</strong> discharge of snow-melt water can only start<br />
when <strong>the</strong> capillary pores between <strong>the</strong> crystals have already been<br />
saturated with water.<br />
The bulk density of mow saturated with water is cal<strong>le</strong>d <strong>the</strong><br />
critical bulk density (a,)<br />
1.2 The process of mow melting<br />
According to <strong>the</strong> correlation examinations performed to de-<br />
termine <strong>the</strong> critical bulk density (rk), <strong>the</strong> development of rk con-<br />
siderably dependa on <strong>the</strong> number CR) of snow layers developed dur-<br />
ing accumulation. During a cold spell following a temporary melt-<br />
ing period with a duration of a few days, a lager of ice will de-<br />
velop on <strong>the</strong> surface which layer separates <strong>the</strong> old and <strong>the</strong> newly<br />
fal<strong>le</strong>n fresh mow. In <strong>the</strong> case of repeated recurrence of this<br />
phenomenon <strong>the</strong> snow layer is dissected into c<strong>le</strong>arly distinguish-<br />
ab<strong>le</strong> layers, <strong>the</strong> number(R) of which is a good indicator of <strong>the</strong><br />
207
208<br />
periodicity of accumulation and melting. The variations - i.e. <strong>the</strong><br />
periodic fluctuation - in <strong>the</strong> snow thickness time series is a good<br />
basis for estimating <strong>the</strong> number of layers. The reliability of es-<br />
timation remarkably increases, if, besides <strong>the</strong> snow thickness re-<br />
cord also <strong>the</strong> air temperature record is availab<strong>le</strong>.<br />
The relation between <strong>the</strong> numerous data of <strong>the</strong> critical bulk<br />
density (ak) and of <strong>the</strong> number of snow layers - both types of data<br />
obtained from <strong>the</strong> 12 years long period between 1960 and 1971 - is<br />
described by <strong>the</strong> following regression equation:<br />
rk = 0.153 +<br />
3<br />
0.050 R 2 0.025 Edcm 3<br />
Performing <strong>the</strong> correlation examination separately with <strong>the</strong><br />
data obtained from <strong>the</strong> mountaina and <strong>the</strong> lowlands, <strong>the</strong> following<br />
equations were obtained:<br />
'a<br />
km, t ain<br />
* 'lowland<br />
(2)<br />
= 0.160 + 0,042R 2 0.032 Ig/cm33 (3)<br />
t 0.146 + 0.052R 2 0,028 Cg/cm33<br />
It is interesting to note that <strong>the</strong> slope of <strong>the</strong> curve is some<br />
20 % flatter in <strong>the</strong> mountains than in <strong>the</strong> lowlands. This is prob-<br />
ably due to <strong>the</strong> fact that on <strong>the</strong> slopes in <strong>the</strong> mountain areas, <strong>the</strong><br />
water of a short melting period can immediately flow down, so that<br />
here <strong>the</strong> ice layers frozen subsequently will be relatively thinner<br />
than in <strong>the</strong> lowlands.<br />
Melting starts at <strong>the</strong> instant of capillary saturation,i.e. at<br />
<strong>the</strong> development of <strong>the</strong> critical bulk density. In <strong>the</strong> course of<br />
melting, <strong>the</strong> ice crystals are merged and destroyed progressively<br />
and become eventually comp<strong>le</strong>tely liquid. During this process <strong>the</strong><br />
bulk density of snow grows continuously until its maximum value is<br />
reached. The maximum bulk density (rma) is calculated using <strong>the</strong><br />
regression equation:<br />
max = 0.213 + 0.054R $; 0.035 Cg/cm33 c 5)
209<br />
The duration (mo) of meltinq - as verified by <strong>the</strong> examina-<br />
tions performed - depends primarily on <strong>the</strong> temperature conditions<br />
prevailing during <strong>the</strong> melting period and on <strong>the</strong> amount (A h) of<br />
snow-melt water.<br />
In <strong>the</strong> course of studies aimed at <strong>the</strong> discovery of <strong>the</strong> rela-<br />
tion between <strong>the</strong> different values of temperature and <strong>the</strong> amount of<br />
meltage numerous potential relations were tested, of which<br />
Ah = f(K) K æ tmax + tmin<br />
3<br />
( 6)<br />
has been se<strong>le</strong>cted as <strong>the</strong> beat, in which tmax and tDin are <strong>the</strong><br />
maximum and minimum temperatures, respectively, prevailing during<br />
<strong>the</strong> individual days of <strong>the</strong> melting period.<br />
The daily temperature value can be computed with Eq.(6) is<br />
cal<strong>le</strong>d rneltinR heat standard (K, ['CI).<br />
To calculate <strong>the</strong> duration of melting <strong>the</strong> relation<br />
can be used where is <strong>the</strong> daily average melting heat standard of<br />
<strong>the</strong> week after melting has started.<br />
It will readily be seen that <strong>the</strong> know<strong>le</strong>dge of <strong>the</strong> expected<br />
average maximum and minimum temperatures forecast in Hungary for a<br />
week by <strong>the</strong> National Meteorological Service is essential for pre-<br />
dicting <strong>the</strong> duration of <strong>the</strong> melting period.<br />
2. NEW CALCULATION METHODS<br />
Using <strong>the</strong> research results demonstrated above two as yet un-<br />
solved prob<strong>le</strong>ms can be approached:<br />
- The creation and reconstruction of mow-water equiva<strong>le</strong>nt time<br />
series - essential in hydrological practice - for snow measuring
21 o<br />
stations where only snow thickness observations are or were per-<br />
formed. In this way <strong>the</strong> water equiva<strong>le</strong>nt time series can be ex-<br />
tended in time for <strong>the</strong> duration of snow thickness observations.<br />
- Forecasting <strong>the</strong> volume of snow-melt water and <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong><br />
meltirg period on <strong>the</strong> basis of <strong>the</strong> snow thickness measured and<br />
of <strong>the</strong> air temperature forecast for <strong>the</strong> melting period.<br />
To solve <strong>the</strong>se two prob<strong>le</strong>ms <strong>the</strong> chart shown in Fia.1 has been<br />
constructed, <strong>the</strong> use of which and <strong>the</strong> course of calculation are<br />
demonstrated using <strong>the</strong> data obtained in 1963 at one of <strong>the</strong> snow-<br />
-water equiva<strong>le</strong>nt measuring stations - Dombori puszta - in <strong>the</strong><br />
lowlands. Since snow thickness and <strong>the</strong> water equiva<strong>le</strong>nt were ob-<br />
served simultaneously, <strong>the</strong> checking of <strong>the</strong> methods is also pos-<br />
sib<strong>le</strong>.<br />
2.1 Producing <strong>the</strong> snow-water eauiva<strong>le</strong>nt time series of <strong>the</strong> Deri0.d<br />
examined from of snow thickness data<br />
In <strong>the</strong> period examined - from <strong>the</strong> 11th of January to <strong>the</strong> 9th<br />
of March - snow was stored continuously. Within <strong>the</strong> period, = as<br />
revea<strong>le</strong>d by <strong>the</strong> snow thickness time series in <strong>the</strong> upper part of<br />
Fig. 2 - three greater intermediate and from <strong>the</strong> 2nd of March a<br />
final melting occured.<br />
a] To produce <strong>the</strong> snow-water equiva<strong>le</strong>nt time series <strong>the</strong> snow<br />
bulk density time series must be reproduced first from <strong>the</strong> snow<br />
thickness data availab<strong>le</strong>. in <strong>the</strong> calculation <strong>the</strong> bulk density of<br />
<strong>the</strong> freeh snow is supposed to be<br />
in all cases.<br />
rmin = 0.118<br />
3<br />
idcm 3<br />
During accumulation, when snow thickness is increasing, <strong>the</strong><br />
average bulk density of <strong>the</strong> <strong>who<strong>le</strong></strong> snow cover is calculated suppos-<br />
ing that <strong>the</strong> increment has also a bulk density of 0.118 g/cm3.
211<br />
E.g.: On <strong>the</strong> 19th of February <strong>the</strong> snow cover is 25.2 cm thick and<br />
has a bulk density of 0.320 g/cm 3 . Next day an additional<br />
amount of 1.6 cm snow cover fell. Thus <strong>the</strong> composition of<br />
<strong>the</strong> mow layer is calculated with <strong>the</strong> layers having<br />
and<br />
25.2 cm (94 %) thickness and 0.320 g/cm3 bulk density<br />
1.6 cm (6 %) thickness and 0.118 g/cm3 bulk density<br />
i.e. total 26,8 cm thickness and 0.308 g/cm3 bulk density.<br />
On <strong>the</strong> first day (nk) of <strong>the</strong> particular temporary and of <strong>the</strong><br />
final meltings <strong>the</strong> critical bulk density ( k) of <strong>the</strong> snow cover is<br />
taken into account with <strong>the</strong> corresponding number of layers.<br />
In <strong>the</strong> examp<strong>le</strong>, using part A) of <strong>the</strong> chart:<br />
rI.i5.<br />
ìr 11.12.<br />
r 11.21.<br />
r 111.2.<br />
= 0.202 g/cm3 because R = 1<br />
E 0.252 g/cm3 because R = 2<br />
t- 0,302 g/cm3 because A = 3<br />
E 0,351 g/cm3 because R = 4<br />
(These values are <strong>the</strong> solutions of Eq.(2) substituting<br />
R = 1, 2, 3 and 4, respectively.)<br />
On <strong>the</strong> last day of <strong>the</strong> particular temporary and of <strong>the</strong> final<br />
meltings <strong>the</strong> maximum bulk density of <strong>the</strong> snow cover is taken into<br />
account with <strong>the</strong> corresponding number of layers (see part A./ of<br />
<strong>the</strong> chart). Thus<br />
bx.is.<br />
r11.19.<br />
ìf 11.23.<br />
r111.9.<br />
= 0.267 g/03 because R 5 i<br />
= 0.320 g/cm3 because R = 2<br />
o 0.374 g/cm3 because R = 3<br />
= 0.428 g/cm3 because R = 4<br />
(These values are <strong>the</strong> solutions of Eq.(3) substituting<br />
B e 1, 2, 3 and 4, respectively.)<br />
Accordingly, <strong>the</strong> ske<strong>le</strong>ton of <strong>the</strong> bulk density time series is<br />
formed by <strong>the</strong> values rk and Y, chosen as <strong>the</strong> function of <strong>the</strong><br />
value min and of <strong>the</strong> number of layers. Intermediate values can
21 2<br />
only be estimated. In <strong>the</strong> period of accumulation <strong>the</strong> method al-<br />
ready demonstrated is used. in <strong>the</strong> melting period i.e. when <strong>the</strong><br />
thickness of snow cover decreases,a linear interpolation consider-<br />
ing <strong>the</strong> change of thicknees is made to choose values between<strong>the</strong><br />
critical and maximum bulk density.<br />
With <strong>the</strong>se considerations <strong>the</strong> <strong>who<strong>le</strong></strong> time series of bulk den-<br />
sity can be produced.<br />
b) Once <strong>the</strong> lime series of snow thickness (v) measured and of<br />
bulk density(J-) calculated are availab<strong>le</strong>, <strong>the</strong> time series of wa-<br />
ter equiva<strong>le</strong>nt (h) can be computed by <strong>the</strong> following equation:<br />
h Cmml = Cg/cm31 . 10 v Ccml (8)<br />
In Fig. 2, <strong>the</strong> data series of bulk density and of water equi-<br />
va<strong>le</strong>nt computed with <strong>the</strong> method demonstrated above are compared<br />
with <strong>the</strong> time aeries of data measured.<br />
2.2 Forecast of snow meltinq<br />
To demonstrate <strong>the</strong> method of forecasting <strong>the</strong> know<strong>le</strong>dge of<br />
only <strong>the</strong> snow cover thickness and of air temperature is suppoeed.<br />
a] First <strong>the</strong> initial water equiva<strong>le</strong>nt (ho) of <strong>the</strong> snow at <strong>the</strong><br />
start of melting is to be determined. This can be read directly<br />
from diagram B of <strong>the</strong> chart.<br />
In <strong>the</strong> examp<strong>le</strong>, at <strong>the</strong> start of final melting on <strong>the</strong> 2nd of<br />
March <strong>the</strong> thickness of mow cover containing 4 layers was 21.0 cm.<br />
A reading in <strong>the</strong> direction of <strong>the</strong> fat line on <strong>the</strong> chart yields an<br />
initial water equiva<strong>le</strong>nt of<br />
ho = 74.0 2 6 Cmm],<br />
i.e. melting is expected to produce<br />
74.0~6 rn of snow-melt water<br />
(<strong>the</strong> water equiva<strong>le</strong>nt actually measured was 71.6 mm, i.e. only 4 %
<strong>le</strong>ss than computed).<br />
b) With <strong>the</strong> water equiva<strong>le</strong>nt obtained as described above, and<br />
using <strong>the</strong> data on minimum and maxim air temperature forecast for<br />
<strong>the</strong> melting period, <strong>the</strong> duration (mo) of <strong>the</strong> melting period is<br />
estimated by means of part C of <strong>the</strong> chart.<br />
The meltina heat standard (K) needed for using <strong>the</strong> chart is<br />
calculated from Eq46).<br />
In <strong>the</strong> present examp<strong>le</strong>, for <strong>the</strong> week following <strong>the</strong> start of<br />
melting = 5.6OC was obtained.<br />
Consequently, by reading in <strong>the</strong> direction of <strong>the</strong> fat line, a<br />
melting period of<br />
rn =3 8 2 1 days<br />
O<br />
is predicted for <strong>the</strong> snow cover having a water equiva<strong>le</strong>nt of<br />
74.0 mm. (The actual melting period measured was 7 days).<br />
Note that <strong>the</strong> forecaet described should be repeat.ed daily<br />
with <strong>the</strong> latest data to make continuous allowance for changes in<br />
<strong>the</strong> wea<strong>the</strong>r.<br />
s x a t<br />
From <strong>the</strong> results of error analyses performed for checking <strong>the</strong><br />
two methods and from <strong>the</strong> first experiences gained with <strong>the</strong>ir ap-<br />
plication, <strong>the</strong> methods described for calculating <strong>the</strong> water equi-<br />
va<strong>le</strong>nt and forecasting snowmelt appear to be of practical interest.<br />
21 3
21 4<br />
REZERENCES<br />
Cil Karo1.B.P.: Snow cover (in Russian)<br />
Gidrometeorológitsheskoe izdátelstwo<br />
Leningrad, 1949.<br />
L23 Kovács.Gy.: The altitude system of snow conditions in <strong>the</strong><br />
winter 1968-69. (in Hungarian)<br />
Annual report on <strong>the</strong> work of VITTJKI 1972, Buda-<br />
pest<br />
L31 Kovács G The development, accumulation and melting of<br />
snow, <strong>the</strong> measurement and calculation of <strong>the</strong>se<br />
features ( in Hungarian)<br />
Bogdhffy, be Pályázat . 1973, Budapest<br />
141 Kuzmin.P.P.: Physics of <strong>the</strong> snow cover (in Russian)<br />
Gidrometeoizdat, 1957.<br />
151 Péczels,Gy.: The consideration of <strong>the</strong> accumulation and melt-<br />
ing of snow in <strong>the</strong> analysis of <strong>the</strong> precipitation<br />
system of catchments (in Hungarian)<br />
Idójárás 1969/1, Budapest<br />
161 Sa1amin.P.: The examination of snow melting in <strong>the</strong> Bükk<br />
mountains in Hungarian)<br />
Id6járás 1 4 56/5, Budapest<br />
173 Sa1amin.P.: The prob<strong>le</strong>ms of <strong>the</strong> examination of snow melting<br />
(in Hungarian)<br />
Discussion. Department of Agricultural Sciences,<br />
Hungarian Academy of Sciences, 1-3. IX. 1956.<br />
183 Sa1amin.P.: The influence of <strong>the</strong> relief on <strong>the</strong> accumulation<br />
and melting of snow (in Hungarian)<br />
Hidrológiai Köalöny 1960, Budapest<br />
191 To1lan.A.: Determination of Areal Values of <strong>the</strong> Water Equi-<br />
va<strong>le</strong>nt of Snow in a Representativ Basin<br />
(in English)<br />
Mordisk Hidrologisk Konferenc, Stockholm 1970.<br />
Li01 Yosida.2.: Physical Studies on Deposited Snow I-IV.<br />
(in Enalish)<br />
Gechanical Properties.<br />
Contributions from <strong>the</strong> Institute os Snow Tem-<br />
perature Suence, Sapporo, 1956.
RECONSTRUCTION OF THE WATER EQUIVALENT<br />
TIME SERIE8 OF JANUARY TO MARCH, 1963 AT<br />
DOHBORI PUSZTA.<br />
0,600<br />
0,500<br />
w 0,100<br />
- C0b:PU TED<br />
-_ OBSER VF D<br />
21 5
o<br />
216
EVALUATION OF LOCAL WATER RESOURCES IN AN SEMI-ARID,<br />
HARD ROCK REGION BY USING PHOTO-HYDROLOGICAL INDICES<br />
ABSTRACT<br />
By: A.M.J. MEIJERINK<br />
The local water resources of a semi-arid, hard rock<br />
area have been evaluated in a rapid, but approximate way,<br />
by using information derived from aerial photographs and<br />
from field observations.<br />
The sizes of irrigated areas of open, wide diameter<br />
wells, and of small reservoirs, have been taken as indices<br />
for <strong>the</strong> storage of groundwater and for <strong>the</strong> runoff generating<br />
capacity of small watersheds up -to a size of 40 kms2.<br />
Geological and geomorphological influences on <strong>the</strong><br />
groundwater storage and <strong>the</strong> recharge of <strong>the</strong> wells, could be<br />
established.<br />
By means of a judgment of <strong>the</strong> catchment characteris-<br />
tics, <strong>the</strong> relative runoff could be estimated.<br />
RESUMEN<br />
The limitations of <strong>the</strong> use of <strong>the</strong> indices are discussed.<br />
Los recursos hfdricos loca<strong>le</strong>s de una zona semi-árida,<br />
de rocas de baja permeabilidad, han sido evaluados de una<br />
forma rápida pero aproximada, mediante el uso de información<br />
obtenida de fotografías aéreas y de observaciones de campo.<br />
Los tamalios de las áreas irrigadas por pozos abiertos<br />
de amplio diámetro y por pequeños estanques, han sido toma-<br />
dos como indices de la reserva de agua subterránea y de la<br />
capacidad generadora de la escorrentia de pequefias cuencas,<br />
hasta de 40 km2. de extensión.<br />
Se pudo estab<strong>le</strong>cer las influencias geológicas y geo-<br />
morfológicas en las reservas y en la recarga de los pozos.<br />
Evaluando las características de las cuencas se pudo<br />
estimar la escorrentla relativa.<br />
Se discuten asimismo las limitaciones 'del uso de los<br />
indices.
218<br />
I. Introduction.<br />
The aim of this paper is to show <strong>the</strong> use of photo-interpretation for <strong>the</strong><br />
assessment of <strong>the</strong> local water resources in a semi-arid region, underlain by hard<br />
rocks with litt<strong>le</strong> storages.<br />
The region studied is a part of <strong>the</strong> Cuddapah Basin in south India (see figure 1 ),<br />
where <strong>the</strong> agriculture depend on <strong>the</strong> meager local water resources.<br />
By means of photo-interpretation <strong>the</strong> distribution and <strong>the</strong> relative quantities<br />
of <strong>the</strong> ground- and surface water resources could be studied.<br />
The study consisted of a simp<strong>le</strong> differentiation of <strong>the</strong> area in more or <strong>le</strong>ss<br />
hydrologically homogeneous units and of an analysis of <strong>the</strong> irrigated areas within<br />
<strong>the</strong> various terrain unit s.<br />
The sizes of <strong>the</strong> irrigated areas are in fact a control of <strong>the</strong> interpretation<br />
procedures and also an indicator of <strong>the</strong> hydrological situations.<br />
Approach.<br />
The following approach for <strong>the</strong> interpretation procedures has been adopted:<br />
____________________------_-_<br />
Differentiation of <strong>the</strong> region.<br />
The region has to be divided in certain 'landscapes'. Each landscape has<br />
its own comp<strong>le</strong>x of gross hydrological processes and gross water resources<br />
which differ from those in adjoining landscapes.<br />
Within each landscape <strong>the</strong>re are a number of smal<strong>le</strong>r land components which,<br />
on a reconnaissance sca<strong>le</strong>, may be considered to be hydrologically homogeneous.<br />
The land components may be of erosional or denudational nature such as a<br />
dissected alluvial fan y an inselberg comp<strong>le</strong>x with <strong>the</strong> surrounding<br />
embayments y etc.<br />
The land components may be sub-differentiated, if necessary, in land e<strong>le</strong>ments.<br />
The land e<strong>le</strong>ments are described here as terrain units which are closely<br />
associated with simp<strong>le</strong> hydrological processes.<br />
An examp<strong>le</strong> of a land e<strong>le</strong>ment in this study is: an area with well terraced<br />
agricultural fields, which are capab<strong>le</strong> of storing appreciab<strong>le</strong> quantities of<br />
surface runoff.<br />
Practical, ra<strong>the</strong>r than <strong>the</strong>oretical considerations are at <strong>the</strong> base of this scheme<br />
of land differentiation.<br />
However, <strong>the</strong> 'landscape' which has been described above, may be compared with<br />
Verstappen's 'Main geomorphological Unit (1 96fl), with <strong>the</strong> 'Land System' of <strong>the</strong><br />
Oxford Working Group (Brink et al. 1966) and with <strong>the</strong> 'Mesnosti' of <strong>the</strong> Eussian<br />
Authors (Vinogradow 1968).<br />
In <strong>the</strong> area of study, <strong>the</strong> boundaries of <strong>the</strong> landscapes follow closely <strong>the</strong><br />
boundaries of <strong>the</strong> main lithological units.<br />
The close association between <strong>the</strong> landscape and <strong>the</strong> geology in <strong>the</strong> Cuddapah Basin<br />
is caused by two facts:<br />
1)<br />
2)<br />
The lithologies are markedly different from each o<strong>the</strong>r and large outcrop<br />
areas are formed because of structural conditions.<br />
The denudational development of <strong>the</strong> area under predominantly semi-arid<br />
conditions has resulted in <strong>the</strong> adjustment of <strong>the</strong> geomorphology to <strong>the</strong><br />
pronounced geological fact ors.
- B. Hydrological evaluation.<br />
219<br />
Various landscapes occuring in <strong>the</strong> Basin, have been delineated and within <strong>the</strong><br />
landscapes a sub-differentiation was made of <strong>the</strong> land components and occasionally<br />
also of <strong>the</strong> land e<strong>le</strong>ments.<br />
The groundtruth of <strong>the</strong> interpretation, was in this particular case already<br />
availab<strong>le</strong>, because <strong>the</strong> author had carried out previously field checking of<br />
interpreted geology and geomorphology of <strong>the</strong> basin during four field seasons.<br />
Moreover, a soil survey of representative strips in each landscape had been<br />
made by a third party.<br />
The possib<strong>le</strong> hydrological significance has been estimated of <strong>the</strong> interpreted and<br />
mapped land components and land e<strong>le</strong>ments. The estimation is subjective.<br />
Details of <strong>the</strong> features, caused by overland flow and by concentrated sheet flow,<br />
as observed on <strong>the</strong> aerial photographs, have been used, as well as some o<strong>the</strong>r<br />
photographic characteristics. However, far more reliance was placed on <strong>the</strong><br />
possib<strong>le</strong> hydrological behaviour of <strong>the</strong> soils, wea<strong>the</strong>red zones, superficial<br />
deposits, particulars of <strong>the</strong> lithology, etc.<br />
The photo-interpretation is thus mainly useful as a means of rapid inventarization<br />
and for <strong>the</strong> study of <strong>the</strong> inter-relationships of <strong>the</strong> interpreted features.<br />
The results of <strong>the</strong> mapping and <strong>the</strong> hydrological evaluation have <strong>the</strong>n be compared<br />
with <strong>the</strong> outcome of <strong>the</strong> analysis of <strong>the</strong> index 'irrigated area'.<br />
In this text <strong>the</strong> emphasis is placed on <strong>the</strong> illustration of <strong>the</strong> use of <strong>the</strong> index<br />
in various terrain conditions, ra<strong>the</strong>r than on a description of <strong>the</strong> mapping<br />
procedures.<br />
The results obtained in three landscapes are discussed; first <strong>the</strong> groundwater<br />
occurrences, <strong>the</strong>n <strong>the</strong> surface water resources.<br />
Nature of <strong>the</strong> index.<br />
This index is actually a composite index, as <strong>the</strong> irrigated area is not only<br />
influenced by <strong>the</strong> availab<strong>le</strong> groundwater in <strong>the</strong> zone near <strong>the</strong> surface<br />
(<strong>the</strong> rocks are impermeab<strong>le</strong> in unwea<strong>the</strong>red conditions), but also by <strong>the</strong> irrigation<br />
practices. In <strong>the</strong> area, <strong>the</strong>re are thousands of open wells, out of which irrigation<br />
water is lifted by animal traction. The open, wide diameter wells pumped for a<br />
number of hours are <strong>the</strong>n <strong>le</strong>ft to recover for more than 24 hours.<br />
For <strong>the</strong> sake of briefness, a few examp<strong>le</strong>s are discussed.<br />
Because <strong>the</strong> agriculture is still carried out in a traditional manner, it has been<br />
assumed that <strong>the</strong> yield of <strong>the</strong> well is directly related to <strong>the</strong> size of <strong>the</strong> irrigated<br />
area.<br />
How accurate this relationship is, within a given landscape, has not been<br />
investigated.<br />
As is discussed fur<strong>the</strong>r on, it is not possib<strong>le</strong> to compare <strong>the</strong> acreage per well<br />
of one landscape with ano<strong>the</strong>r, because of differing soil conditions, crop<br />
rotation and water application.<br />
However, within a landscape <strong>the</strong> irrigation practices seem to be uniform.<br />
Therefore, <strong>the</strong> following discussion pertains only to <strong>the</strong> use of <strong>the</strong> index within<br />
a landscape.
220<br />
8-<br />
&<strong>le</strong> showing <strong>the</strong> effects of geomorphology on <strong>the</strong> occurence of<br />
groundwater.<br />
-<br />
Well clusters and <strong>the</strong> irrigated areas have been investigated on <strong>the</strong> Vempalli<br />
calcareous sha<strong>le</strong>s. A part of <strong>the</strong> area is shown in figure 2.<br />
It may be noted that <strong>the</strong> width of <strong>the</strong> recharge zone on <strong>the</strong> pediments, is<br />
related to <strong>the</strong> width of <strong>the</strong> irrigated areas. This relationship once established<br />
can be used to indicate potential irrigab<strong>le</strong> areas by means of mapping <strong>the</strong><br />
pediments in <strong>the</strong> area.<br />
Field observations have shown that <strong>the</strong> depth of wea<strong>the</strong>ring and of sheetwash<br />
deposits, increases on <strong>the</strong> pediments in downstream direction.<br />
On <strong>the</strong> upper parts, <strong>the</strong> unwea<strong>the</strong>red bedrock is close to <strong>the</strong> surface and litt<strong>le</strong><br />
infiltration can take place. However, fir<strong>the</strong>r downstream till <strong>the</strong> central<br />
drainage line is reached, <strong>the</strong> sheet flow may infiltrate partly, and recharge <strong>the</strong><br />
- limited - quantities of groundwater.<br />
The relationship between width and recharge zone and width of irrigated area has<br />
been found useful for locating 'under irrigated' areas in this particular<br />
landscape.<br />
- G.<br />
=amp<strong>le</strong> showing <strong>the</strong> use of statistical test for <strong>the</strong> evaluation of factors<br />
which influence <strong>the</strong> occurrence of groundwater.<br />
In a pediments landscape, <strong>the</strong> well indices have been used to compare <strong>the</strong> influences<br />
of <strong>the</strong> rock types on <strong>the</strong> well yields.<br />
Four samp<strong>le</strong> areas have been se<strong>le</strong>cted at <strong>the</strong> downstream parts of <strong>the</strong> pediments, in<br />
a ra<strong>the</strong>r narrow zone near ephemeral rivers, in order to minimize <strong>the</strong> influence of<br />
<strong>the</strong> morphological position.<br />
The four samp<strong>le</strong> areas are underlain by slates, by sericitic schists, by biotite<br />
schists and by gneisses.<br />
An analysis of variance shows that <strong>the</strong> differences in <strong>the</strong> samp<strong>le</strong> means are not<br />
significant at <strong>the</strong> 5% <strong>le</strong>vel. The samp<strong>le</strong> sizes varied from n = 12 to n = 24.<br />
Hence it is concluded that <strong>the</strong> influence of <strong>the</strong> lithology in this area on <strong>the</strong><br />
well yield is not significant. It should be noted however, that <strong>the</strong> rock types<br />
are ra<strong>the</strong>r impermeab<strong>le</strong> anyhow. The similarity of <strong>the</strong> well yields is attributed to<br />
<strong>the</strong> effects of wea<strong>the</strong>ring, type of soils,calcareous and siliceous crusts and to <strong>the</strong><br />
deposition of sheet wash deposits.<br />
A litt<strong>le</strong> south of <strong>the</strong> samp<strong>le</strong> areas, approximately 200 la2, fossil aeolian sands<br />
lare covering <strong>the</strong> pediment surfaces. The thickness of <strong>the</strong> sand cover varies from 1<br />
to over 20 meters.<br />
It can be expected that <strong>the</strong> well yields are higher than in <strong>the</strong> area without sands,<br />
because of <strong>the</strong> good infiltration possibilities in <strong>the</strong> sands and <strong>the</strong> higher specific<br />
yields of <strong>the</strong> sandy medium.<br />
This expectation is confirmed by <strong>the</strong> indeces. However, no significant differences<br />
in <strong>the</strong> well yields in this area could be detected, after <strong>the</strong> indices had been<br />
samp<strong>le</strong>d in four areas. The samp<strong>le</strong> areas have been se<strong>le</strong>cted on bxoad drainage<br />
divides and near ephemeral channels.
- D.<br />
Ekamp<strong>le</strong> showing <strong>the</strong> possibility of predicting approximate well yields<br />
f l o m _ s _ l l m p i e _ _ - ~ ~ - ~ ~ _______________<br />
~ - ~ ~ ~ ~ ~ - ~ ~ ~ ~<br />
In <strong>the</strong> three discussed examp<strong>le</strong>s, <strong>the</strong> nature of <strong>the</strong> recharge area was of<br />
interest. In <strong>the</strong> granite landscape along <strong>the</strong> western margin of <strong>the</strong> Cuddapah<br />
Basin, it was possib<strong>le</strong> to delineate <strong>the</strong> 'catchment' areas of individual wells,<br />
and thus compare <strong>the</strong> size of <strong>the</strong> 'catchment' or recharge area with <strong>the</strong> size<br />
of <strong>the</strong> area irrigated by <strong>the</strong> wells.<br />
The landscape in which three samp<strong>le</strong> areas have been se<strong>le</strong>cted, consists of<br />
convex interfluves and concave to flat val<strong>le</strong>y bottoms. On <strong>the</strong> interfluves<br />
<strong>the</strong> depth of wea<strong>the</strong>ring varies from O to 5 meters. The soils are red coloured,<br />
loamy sands to sandy loams with stone lines. The soils in <strong>the</strong> val<strong>le</strong>y bottom<br />
are grey coloured gritty clay loams to sandy clays.<br />
Althou& large outcrops may occur in or next to <strong>the</strong> val<strong>le</strong>y bottom, <strong>the</strong> average<br />
depth of wea<strong>the</strong>ring seems <strong>the</strong>re to be higher. Inselbergs are found scattered over<br />
<strong>the</strong> area.<br />
2 21<br />
The recharge area of individual wells have been samp<strong>le</strong>d wherever <strong>the</strong> local<br />
relief was sufficiently high to delineate <strong>the</strong> drainage divides on <strong>the</strong> interfluves<br />
and where <strong>the</strong> irrigated areas could be differentiated from <strong>the</strong> surrounding<br />
non-irrigated fields.<br />
The three samp<strong>le</strong> areas are shown in figure 1, from which it may be noted that<br />
<strong>the</strong> mean annual rainfall of <strong>the</strong> samp<strong>le</strong> areas is about <strong>the</strong> same.<br />
The three areas seem to be similar in geological and geomorphological respects.<br />
The areas irrigated by wells are shown in figure 3a, and have been plotted as<br />
a function of <strong>the</strong> recharge area in <strong>the</strong> graph of figure 3b.<br />
This graph shows <strong>the</strong> combined results of <strong>the</strong> three samp<strong>le</strong> areas.<br />
In all <strong>the</strong> three samp<strong>le</strong> areas <strong>the</strong> correlation coefficients are significant at<br />
<strong>the</strong> 5% <strong>le</strong>vel (Spearman rank correlation statistic), whi<strong>le</strong> no significant<br />
differences have been found between <strong>the</strong> three correlations (kskall and Wallis<br />
test of variance). For practical purposes, <strong>the</strong> line of best fit, shown in <strong>the</strong><br />
graph may be used as a guideline for <strong>the</strong> estimated yield of open wells as a<br />
function of <strong>the</strong> recharge area in <strong>the</strong> samp<strong>le</strong>d landscape.<br />
The scatter of <strong>the</strong> plotted points indicate approximately <strong>the</strong> degree of accuracy<br />
of <strong>the</strong> estimate.<br />
Remarks.<br />
These few examp<strong>le</strong>s show how <strong>the</strong> index irrigated area may serve as a check on <strong>the</strong><br />
expectations of <strong>the</strong> water occurrences and <strong>the</strong> approximate quantities, within<br />
well defined landscapes.<br />
It is not possib<strong>le</strong> to compare <strong>the</strong> indices of one landscape with ano<strong>the</strong>r, because<br />
<strong>the</strong> index is very sensitive to variations in irrigation practice, type of irrigated<br />
soils and <strong>the</strong> type of crops.<br />
If <strong>the</strong> yields in <strong>the</strong> various landscape have to be compared, <strong>the</strong> index has to be<br />
transposed in actual well yields.<br />
However, by employing this index as a control on <strong>the</strong> expectations, <strong>the</strong> value of<br />
<strong>the</strong> photo-interpretation is increased and <strong>the</strong> amount of field works is greatly<br />
reduced.
222<br />
1x1. Surface water.<br />
- ____________________________I___________-------------------------<br />
A. The use of <strong>the</strong> index 'area irrigated by water stored in small reservoirs'.<br />
Whi<strong>le</strong> scanning <strong>the</strong> aerial photographs, an apparent relationship was noted<br />
between <strong>the</strong> size of <strong>the</strong> catchment and <strong>the</strong> size of <strong>the</strong> irrigated areas below<br />
reservoirs. The reservoirs are usually constructed across <strong>the</strong> main drainage<br />
line, and are thus in a position to store <strong>the</strong> full runoff, provided of course,<br />
that <strong>the</strong> capacity of <strong>the</strong> tanks is sufficiently high.<br />
The reservoir capacities cannot be estimated on <strong>the</strong> aerial photographs<br />
accurately, because <strong>the</strong>y are very shallow, usually <strong>le</strong>ss than 2 meters deep.<br />
However, it may be supposed that <strong>the</strong> irrigated areas are closely adjusted to<br />
<strong>the</strong> average availab<strong>le</strong> quantities of water in <strong>the</strong> reservoirs.<br />
Thus, <strong>the</strong> irrigated areas are used as an index, or measure, for <strong>the</strong> runoff<br />
from <strong>the</strong> catchments. The se<strong>le</strong>cted catchments are smal<strong>le</strong>r than 40 h2.<br />
Comparisien of <strong>the</strong> indices are only possib<strong>le</strong> in a well defined landscape.<br />
Differences in irrigation practices, type of irrigated soil, etc., prohibit<br />
<strong>the</strong> comparision of <strong>the</strong> indices from two or more different landscapes with<br />
each o<strong>the</strong>r.<br />
Therefore, <strong>the</strong> index is mainly used to investigate whe<strong>the</strong>r variations of <strong>the</strong><br />
catchment characteristics, within a landscape, are associated with variations<br />
in <strong>the</strong> index.<br />
Before embarking on <strong>the</strong> discussion of <strong>the</strong> analysis, it may be useful to<br />
illustrate briefly <strong>the</strong> rainfall factors, <strong>the</strong> magnitude of <strong>the</strong> evaporation and<br />
<strong>the</strong> type of runoff.<br />
- -------I------_-_-------------<br />
B. Rainfall, runoff and evaporation.<br />
Inspection of <strong>the</strong> daily rainfall records of a few stations in <strong>the</strong> area shows<br />
that most of <strong>the</strong> rainfall occurs during <strong>the</strong> summer monsoon (SW monsoon).<br />
The maximum daily rainfall in <strong>the</strong> winter monsoon (NE monsoon) is much lower,<br />
usually <strong>le</strong>ss than 20 mms. per day. Mean monthly rainfall varies from 150 mms.<br />
during <strong>the</strong> SW monsoon to 5 ms. during <strong>the</strong> dry months.<br />
The frequency of <strong>the</strong> maximum daily rainfall, based on <strong>the</strong> partial seriesis<br />
shown in figure 4. The days with more than 30 mms rainfall (arbitrary standard)<br />
have been used for <strong>the</strong> compilation.<br />
Field observations show that isolated showers tend to produce flashy runoff in<br />
this semi-arid area, where <strong>the</strong> catchments have litt<strong>le</strong> storage possibilities.<br />
However, according to <strong>the</strong> local population, <strong>the</strong> tanks get fil<strong>le</strong>d up mainly by<br />
prolonged rainfall. It is not uncommon that in <strong>the</strong> SW monsoon, during three<br />
consecutive days with rainfall, more than 100 mms are recorded.<br />
Such occasions cause overflow of <strong>the</strong> reservoirs.<br />
On <strong>the</strong> o<strong>the</strong>r hand, during dry years <strong>the</strong> tanks may not get fil<strong>le</strong>d up, or may not<br />
be rep<strong>le</strong>nished for <strong>the</strong> irrigation of <strong>the</strong> rfpening crops.<br />
Analysis of <strong>the</strong> irrigdted areas in some catchments, as measured on <strong>the</strong> aerial<br />
photographs, indicated that <strong>the</strong> tank capacities are not capab<strong>le</strong> of storing <strong>the</strong><br />
most important runoff events, This was found by comparing <strong>the</strong> irrigated areas,<br />
expressed per unit of catchment area, of two or more tanks in sing<strong>le</strong> catchments.<br />
The downstream irrigated area was larger in 10 out of 11 cases.
Although <strong>the</strong> runoff may be prolonged when successive rainy days with high<br />
rainfall amounts occur, <strong>the</strong> runoff decreases rapidly after <strong>the</strong> cessation of <strong>the</strong><br />
rainfall. Most of <strong>the</strong> runoff seems to occur in <strong>the</strong> form of direct runoff with<br />
very litt<strong>le</strong> interflow (throughflow) and base flow.<br />
2<br />
The ephemeral rivers of <strong>the</strong> small catchments (up to 40 lan ) have dried up<br />
practically within a few days.<br />
The evaporation of <strong>the</strong> area is high. The highest mean monthly evapotranspiration<br />
in <strong>the</strong> area is 200 to 220 ms, and <strong>the</strong> minimum mean monthly value is 11 cms<br />
(at <strong>the</strong> time of <strong>the</strong> winterrains).<br />
bhen<strong>the</strong>meanyearly evapotranspiration figures, which are based on <strong>the</strong> Modified<br />
Penman fomla, are compared with <strong>the</strong> rainfall figures, <strong>the</strong> water shortages in <strong>the</strong><br />
area become obvious.<br />
The average depth of ihe tanks is usually very small (< 1.5 meters), but <strong>the</strong><br />
size of <strong>the</strong> tanks is comparatively very large (5 - 50 hectares).<br />
Monthly evaporation rates of more than 10 cms reduce <strong>the</strong>refore <strong>the</strong> effective<br />
storage of <strong>the</strong> tanks appreciably.<br />
Factors that influence <strong>the</strong> size of <strong>the</strong> irrigated area.<br />
From <strong>the</strong> above discussion, it is obvious that <strong>the</strong> index 'size of irrigated<br />
area' is an inaccurate measure for <strong>the</strong> runoff of <strong>the</strong> catchments.<br />
It is difficult to say for examp<strong>le</strong>, to what duration and frequency of <strong>the</strong><br />
discharges, <strong>the</strong> index is related.<br />
For <strong>the</strong> evaluation of <strong>the</strong> use of <strong>the</strong> index <strong>the</strong> following argument has been<br />
used:<br />
If <strong>the</strong> index is a perfect measure for <strong>the</strong> runoff production of <strong>the</strong> watersheds,<br />
a perfect correlation between <strong>the</strong> index and <strong>the</strong> size of <strong>the</strong> catchments can be<br />
expected. Variations in <strong>the</strong> relationship should be attributab<strong>le</strong> to variations<br />
of <strong>the</strong> hydrological effects of <strong>the</strong> landcomporients.<br />
Actually, it is <strong>the</strong> variation caused by <strong>the</strong> land components within a landscape,<br />
that is of interest in this study.<br />
However, <strong>the</strong> imperfectness of <strong>the</strong> index may be demonstrated by pointing out<br />
<strong>the</strong> following sources of error:<br />
1. Original differences in <strong>the</strong> capacities of <strong>the</strong> reservoirs, because of<br />
topographical differences, construction of <strong>the</strong> overflow, etc.<br />
2. Reduction of <strong>the</strong> original reservoir capacities by sedimentation.<br />
Age and history (breakages, desilting operations) of <strong>the</strong> tanks may differ<br />
within a landscape, also <strong>the</strong> sedimentation rates per unit of catchment area.<br />
3. Minor differences in <strong>the</strong> irrigation practices, water management.<br />
4. Interpretation and measuring errors on <strong>the</strong> aerial photographs.<br />
The variation caused by <strong>the</strong>se 4 factors cannot be evaluated without detai<strong>le</strong>d<br />
measurements in <strong>the</strong> field.<br />
Despite <strong>the</strong> fact that <strong>the</strong> mentioned factors are an important source of error,<br />
in all <strong>the</strong> four samp<strong>le</strong> areas, a significant correlation has been found between<br />
<strong>the</strong> parameters 'sise of catchment area' and 'sise of irrigated area'.<br />
223
224<br />
The influence of <strong>the</strong> land components on <strong>the</strong> index is discussed here for one<br />
large landscape, <strong>the</strong> 'Cumbum Landscape'.<br />
In aiio<strong>the</strong>r landscape, <strong>the</strong> 'landscape on <strong>the</strong> eastern basement comp<strong>le</strong>x', ra<strong>the</strong>r<br />
similar results have been obtained, and need <strong>the</strong>refore litt<strong>le</strong> elaboration.<br />
However, on <strong>the</strong> third landscape, <strong>the</strong> one on <strong>the</strong> granites in <strong>the</strong> west, where<br />
<strong>the</strong> land components seem to be equally distributed in <strong>the</strong> landscape, significant<br />
differences have been found in two samp<strong>le</strong> areas.<br />
In <strong>the</strong> o<strong>the</strong>r landscapes of <strong>the</strong> Cuddapah Basin, no sufficiently reliab<strong>le</strong><br />
measurements could be made for a proper analysis.<br />
--______________________________________----_--------------------<br />
The Cumbum landscape, an examp<strong>le</strong> of <strong>the</strong> runoff variation within a landscape.<br />
Descript ion.<br />
The landscape on <strong>the</strong> sha<strong>le</strong>s, siltstones and phyllites with occasional<br />
limestone beds of <strong>the</strong> Cumbums, forms a separate landscape in <strong>the</strong> Cuddapah Basi<br />
although <strong>the</strong> geomorphology of <strong>the</strong> landscape is not uniform.<br />
In some parts of <strong>the</strong> area, wea<strong>the</strong>red remnants of large but thin alluvial<br />
fans are found, supporting a dense vegetation of grassland and dense shrub.<br />
O<strong>the</strong>r parts may consist of eroded terrain of varying relief and ske<strong>le</strong>tical<br />
soils. South of this area extensive remnants of an old wea<strong>the</strong>red planation<br />
<strong>le</strong>vel are found.<br />
Estimation of <strong>the</strong> relative runoff from <strong>the</strong> aerial photographs.<br />
The runoff producing and runoff-storing land components in <strong>the</strong> watersheds<br />
have been interpreted and mapped.<br />
Land use features and geomorphological e<strong>le</strong>ments have been mapped separately,<br />
but have been plotted on a sing<strong>le</strong> map.<br />
The joint effects of <strong>the</strong> land use and geomorphology on <strong>the</strong> runoff has been<br />
evaluated by means of arbitrary standards:<br />
Land use features such as fields surrounded by ear<strong>the</strong>rn walls, behind small<br />
retention structures, etc. are capab<strong>le</strong> of storing runoff and <strong>the</strong> areas with<br />
many of such features have been classified as 'areas with low runoff'.<br />
Overgrazed, poorly cultivated fields on sloping land, fall in <strong>the</strong> class<br />
'high runoff'. The o<strong>the</strong>r areas have simply been classified as medium runoff.<br />
Similarly, geomorphological e<strong>le</strong>ments, such as old wea<strong>the</strong>red fans, thick<br />
slope deposits, buried pediments and e<strong>le</strong>ments like heavily eroded soil<br />
bare outcrops, true pediment slopes etc, fall in two opposing classes; high<br />
and low runoff. The remaining e<strong>le</strong>ments with no evident extreme hydrological<br />
behaviour fall in <strong>the</strong> medium class.<br />
The percentages of <strong>the</strong> areas falling in <strong>the</strong> three land use and in <strong>the</strong><br />
three geomorpho12gical classes are <strong>the</strong>n determined, and a final judgement<br />
puts <strong>the</strong> catchment in one of <strong>the</strong> three categories.<br />
The hydrological evaluation of <strong>the</strong> land components cannot be done without<br />
sufficient field how<strong>le</strong>dge. The procedure is arbitrary, and <strong>the</strong> results will<br />
<strong>the</strong>refore vary from one observer to ano<strong>the</strong>r.<br />
Analysis.<br />
The graph of figure 5a shows <strong>the</strong> index 'irrigated area' as a function of <strong>the</strong><br />
catchment area. The line of best fit has been established by graphical<br />
correlation (Lins<strong>le</strong>y, Koh<strong>le</strong>r, Paulhus 1949).<br />
For all <strong>the</strong> watersheds, shown on <strong>the</strong> graph, <strong>the</strong> runoff class has been<br />
estimated.
2 25<br />
The runoff class is now compared with <strong>the</strong> deviation of <strong>the</strong> plotted position<br />
of <strong>the</strong> points on <strong>the</strong> graph of figure 5a with <strong>the</strong> line of best fit.<br />
Points which are below <strong>the</strong> line of best fit are cal<strong>le</strong>d negative deviations,<br />
those above <strong>the</strong> line, positive deviations.<br />
Theoretically, possitive deviations should be associated with high or medium<br />
runoff classes, negative deviations with medium or low runoff classes.<br />
The results of <strong>the</strong> comparision are shown in figure 5b.<br />
Ideally, in <strong>the</strong> case of high runoff, all points should fall in <strong>the</strong> positive<br />
range and <strong>the</strong> cases of low runoff should fall in <strong>the</strong> negative range.<br />
The cases of medium runoff should have a symetrical distribution and <strong>the</strong><br />
magnitude of <strong>the</strong> deviation should be limited.<br />
As can be judged from <strong>the</strong> graph of figure 5b, no ideal result has been<br />
obtained, although <strong>the</strong> mediam values (see <strong>the</strong> graph) of <strong>the</strong> three runoff<br />
classes are at three different <strong>le</strong>vels ( + 36, - 38 and - 96 ).<br />
It should be remembered that <strong>the</strong> procedure is arbitrary and that <strong>the</strong>re are<br />
many causes of variation. The procedure followed, i.e. <strong>the</strong> author's<br />
judgment, <strong>le</strong>ads to an underestimation of <strong>the</strong> storages and losses in <strong>the</strong><br />
catchments. However, <strong>the</strong> method is not ment for an estimation of <strong>the</strong><br />
absolute runoff production, but for an estimation of <strong>the</strong> relative differences<br />
of <strong>the</strong> runoff in <strong>the</strong> various catchments within a landscape.<br />
For this purpose, we feel that <strong>the</strong> results of <strong>the</strong> comparision of <strong>the</strong> runoff<br />
estimates with <strong>the</strong> deviation of <strong>the</strong> index, are satisfactory, so that <strong>the</strong><br />
runoff classification may be applied in this landscape with some confidence.<br />
The landscape on <strong>the</strong> granites and examp<strong>le</strong> of unexplained differences in <strong>the</strong><br />
index.<br />
Catchments and irrigated areas have been measured in two samp<strong>le</strong>s areas on<br />
<strong>the</strong> granites, along <strong>the</strong> western margin of <strong>the</strong> Cuddapah Basin.<br />
!Che areas, denoted here with <strong>the</strong> nor<strong>the</strong>rn and <strong>the</strong> sou<strong>the</strong>rn area, are 250 lans<br />
apart. The mean annual rainfall in <strong>the</strong> two areas is to same, between 600 and<br />
700 mms. Only small differences, if any are expected in <strong>the</strong> frequencies of <strong>the</strong><br />
partial series.<br />
As has been discussed earlier, in <strong>the</strong> section on groundwater, both areas seem<br />
to be highly similar in geological and geomorphological aspects.<br />
The relationships between <strong>the</strong> catchment area and <strong>the</strong> size of <strong>the</strong> irrigated areas,<br />
is shown in figure 6.<br />
The Kendall rank correlation coefficient for <strong>the</strong> nor<strong>the</strong>rn samp<strong>le</strong> area is 0.78,<br />
for <strong>the</strong> sou<strong>the</strong>rn area 0.90. Both <strong>the</strong> correlations are significant at <strong>the</strong> 5% <strong>le</strong>vel.<br />
However, <strong>the</strong> two samp<strong>le</strong> areas show a marked difference between <strong>the</strong> indices<br />
'irrigated area' as a function of catchment size.<br />
The Muskall and Wallis test showed that <strong>the</strong> difference of <strong>the</strong> two relationships<br />
is significant at <strong>the</strong> 5% <strong>le</strong>vel.<br />
The explanation of <strong>the</strong> difference is difficult.<br />
Its has been tried to explain <strong>the</strong> difference by some additional photo-measurements<br />
of factors, that might influence <strong>the</strong> size of <strong>the</strong> irrigated area.<br />
Within <strong>the</strong> catchments, <strong>the</strong> percentage of outcrop areas in <strong>the</strong> two samp<strong>le</strong> areas<br />
have been compared, but no significant difference has been found.<br />
It was reasoned that <strong>the</strong> percentage of outcrop area would influence <strong>the</strong> runoff<br />
(and thus <strong>the</strong> index), because of <strong>the</strong> very low storage possibilities on <strong>the</strong><br />
outcrops. The number of wells in <strong>the</strong> irrigated area have also been compared with<br />
<strong>the</strong> irrigated area. It was thought that <strong>the</strong> wells, which re-use <strong>the</strong> water from<br />
<strong>the</strong> reservoirs, could be of influenoe. However, no significant correlation has bt.,:.?<br />
found.
226<br />
The explanation of <strong>the</strong> difference in <strong>the</strong> relationships may be hidden in<br />
factors, which are not measurab<strong>le</strong> on <strong>the</strong> aerial photographs.<br />
In this particular area, it is suggested that perhaps, <strong>the</strong> age of <strong>the</strong> reservoirs<br />
is an important causative factor.<br />
In <strong>the</strong> sou<strong>the</strong>rn area <strong>the</strong> reservoirs may be older than those in <strong>the</strong> nor<strong>the</strong>rn<br />
area, so that <strong>the</strong> smal<strong>le</strong>r irrigated areas in <strong>the</strong> sou<strong>the</strong>rn area, may be explained<br />
by a reduction of <strong>the</strong> reservoir capacity by sedimentation.<br />
Discussion of <strong>the</strong> results.<br />
In <strong>the</strong> area of study, <strong>the</strong> index 'area irrigated by open wells and by small<br />
reservoirs' provides a useful means of control of <strong>the</strong> hydrological significance<br />
of <strong>the</strong> photographic interpretation procedures.<br />
However, <strong>the</strong> use of <strong>the</strong> index requires a good deal of local know<strong>le</strong>dge of <strong>the</strong><br />
terrain characteristics, irrigation practices, etc.<br />
The index should be used in a careful way and <strong>the</strong> purely empirical character of<br />
<strong>the</strong> index restricts its use to well defined landscapes.<br />
However, a number of practical applications of some tested relationships have<br />
been found for some landscapes in <strong>the</strong> Cuddapah Basin.<br />
The experience and confidence gained by <strong>the</strong> study of <strong>the</strong> landscapes and by <strong>the</strong><br />
analysis of <strong>the</strong> index has been used for <strong>the</strong> hydrological evaluation of those<br />
landscapes, for which no sufficient indices could be samp<strong>le</strong>d.<br />
It is believed by <strong>the</strong> author, that for similar semi-arid,areas on hard rocks<br />
an approach along <strong>the</strong> same lines could give useful results, particularly when<br />
no appropriate hydrological data are existing.<br />
The indices are ra<strong>the</strong>r typical for <strong>the</strong> area investigated, but have been used<br />
for o<strong>the</strong>r areas in india as well. In regions where such indices are not<br />
existing, or are not very meaninghl, <strong>the</strong> evaluation of <strong>the</strong> landcomponents has<br />
to rely on o<strong>the</strong>r field observations. The field observations and measurements<br />
may consist of oral information of water <strong>le</strong>vel fluctuations in wells,<br />
determination of approximate yields, measurement of <strong>the</strong> base flow discharges,<br />
perhaps <strong>the</strong> estimation of <strong>the</strong> bankful discharges, and so on.<br />
The approach consists, in short, of:<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
Interpretation of geology, geomorphology and of aspects of soils,<br />
land use and vegetation.<br />
Differentiation of <strong>the</strong> regior. in 'homogeneous hydrological landscapes'.<br />
Provisional hydrological evaluation of <strong>the</strong> land components.<br />
Field checking of <strong>the</strong> interpretations and <strong>the</strong> col<strong>le</strong>ction of approximate<br />
hydrological data. The field work and <strong>the</strong> col<strong>le</strong>ction of data should have<br />
been planned on <strong>the</strong> basis of <strong>the</strong> interpretation results.<br />
Comparision of <strong>the</strong> results and final, approximate evaluation of <strong>the</strong><br />
influences of <strong>the</strong> land components for <strong>the</strong> development of <strong>the</strong> local water<br />
resources.<br />
-0-o-o-
Ref e re n ce s<br />
2 21<br />
Brink A.B., Mabutt J.A., Webster R. and Beckett P.H.T. (1966)<br />
Report of <strong>the</strong> working group on land classification<br />
and data storage. Milit. Engng. Exp. Establ.<br />
Christchurch, England. Engng. Report no. 940.<br />
Verstappen H.Th. and Zuidam R.A. (1969) I.T.C. system of<br />
geomorphological surveys. I.T.C. text<strong>book</strong> of P.I.<br />
VII, 2. 49 pp.<br />
Vinogradov B.V. (1968) Airphoto methods in geographical research<br />
in <strong>the</strong> U.S.S.R. Photogrammetria 23 pp 17-94<br />
Lins<strong>le</strong>y R.K., Koh<strong>le</strong>r M.A. and Paulhus J.L.H. (1949) Applied<br />
Hydrology Mc. Graw Hill, New York
228<br />
I = GRANITE LANDSCAPE , II = CUMBUM LANDSCAPE ,<br />
III = LANDSCAPE ON EASTERN BASEMENT COMPLEX -<br />
A = ANANTAPUR , K = KURNOOL , C = CUDDAPAH<br />
700 = ISOHYET<br />
Figure 1 - Map showing location of landscapes and <strong>the</strong> outline of <strong>the</strong><br />
Cuddapah Basin.<br />
Fiyre 2 - Irrigated area in relation to <strong>the</strong> width of <strong>the</strong> recharge area.<br />
( dots = open wells, white = pediments, hatches = hills ).
Figure 3a - Samp<strong>le</strong> area on <strong>the</strong> landscape on <strong>the</strong> granites, showing drainage<br />
divides, thalwegs, wells (open circ<strong>le</strong>s), samp<strong>le</strong>d wells (dots)<br />
and <strong>the</strong> corresponding recharge and irrigated areas.<br />
The schematical section indicates <strong>the</strong> depth of wea<strong>the</strong>ring.<br />
R E C H A R G E A R E A<br />
Figure 3b - Irrigated area as a function of recharge area, for three samp<strong>le</strong><br />
areas in <strong>the</strong> landscape on <strong>the</strong> granites.<br />
229
;"O1<br />
1.2<br />
1 .o<br />
4<br />
w<br />
m<br />
4.8<br />
FI<br />
W<br />
U<br />
H<br />
œ<br />
œ .4<br />
U<br />
.2<br />
2 3 o.<br />
,? -<br />
+<br />
(sca<strong>le</strong> for III )<br />
I I I i ,<br />
0.4 1 .o 2 4 6<br />
I I I I I L I , , I<br />
0.4 0.6 0.8 1.0 2 4 6 ô 10 years<br />
B E C U R R E N C E I N T E R V A L<br />
Figure 4 - Frequency curves of high daily rainfall (partial series), for<br />
three stations:<br />
I = Cuddapah, II = Kurnool, III = Anantapur.<br />
4 a 10 16 20 24 28 ~m2.<br />
D R A I N A G E A R E A<br />
Figure 5a - Area irrigated by reservoirs as a function of <strong>the</strong> catchent<br />
area for <strong>the</strong> Cumbum landscape.
Km2,<br />
2.8<br />
2.4<br />
2.0<br />
w<br />
m<br />
4<br />
1 .6.<br />
!a<br />
W<br />
I 3<br />
-4<br />
o 1.2<br />
H<br />
a<br />
e:<br />
H .a<br />
.4<br />
0<br />
rd<br />
5<br />
+<br />
200' -<br />
2 120<br />
3<br />
$ 40-<br />
3<br />
w o-<br />
M 40 -<br />
.rl C<br />
E! 120<br />
.rl<br />
*<br />
rl<br />
200<br />
-<br />
-<br />
-<br />
- IIII.I...IIL<br />
no. of catchments Y<br />
ii<br />
MEDIUA<br />
LOW<br />
Figure 5b - Results of comparison (see figure 5a and text).<br />
S A M P<br />
I. N.E.<br />
II. R A<br />
L E A R E A S :<br />
of K U R N O O L<br />
Y A C H O T I<br />
i/,-/ . ..<br />
/ +<br />
12 16 20 24 28<br />
D R A I N A G E A R E A<br />
231<br />
Figure 6 - Areas irrigated by reservoirs as a function of <strong>the</strong> catchment<br />
area. Two samp<strong>le</strong> areas are shown of <strong>the</strong> landscape on <strong>the</strong> granites.
ABSTRACT<br />
APPLICATION OF SATELLTTE CLOUD PICTURES<br />
IN SNOW HYDROLOGY OF TKE AIMALAYAS AND<br />
IN THE ESTIMATION OP RAINFALL OVER INDIA<br />
DURING SOUTKWEST MONSOON SEASON.<br />
P.S. PANT and N.G. GUPTA<br />
Many of <strong>the</strong> rivers in <strong>the</strong> Indi'an suB-continent bave <strong>the</strong>ir ori-<br />
gins in <strong>the</strong> Himalayas. An important souTce of water supply for <strong>the</strong>se<br />
rivers is, <strong>the</strong>refore, from <strong>the</strong> melting of <strong>the</strong> snow in <strong>the</strong> upper cat-<br />
chment o€ <strong>the</strong>se rivers. As much of this region is inaccessib<strong>le</strong>, <strong>the</strong><br />
conventional observations are not availaB<strong>le</strong>, Examination of <strong>the</strong> sate-<br />
llite Te<strong>le</strong>vision Pictures has revea<strong>le</strong>d tñe possibility of estimating<br />
<strong>the</strong> snow coverage over <strong>the</strong> river basi'na in tke Hihalayas and <strong>the</strong>reby<br />
to estimate tlìe contriBution of snowmelt to tñe flow in tEese rivers.<br />
An attempt has also been made to estìmate <strong>the</strong> 24 hour rainfall amounts<br />
over <strong>the</strong> plains of India duTing tñe soutñwest monsoon seanson, using<br />
cloud imageries taken By wa<strong>the</strong>r satelli'tes. Tiìese estimates are found<br />
to be in reasona6<strong>le</strong> agreement witñ tñe values obtaihed by isohyetal<br />
analysis of actual Tai'nfall 8ata. Tfie iesnlts'oBtaihed look promising.<br />
If confirmed from an analysis of several rainstorms, <strong>the</strong>se will find<br />
wide application in estimation of 24 hour average areal rainfall over<br />
data sparse river catcñments.<br />
RESUME<br />
Plus de rivieres en <strong>le</strong> SuB-continent de l'Inde ont <strong>le</strong>ur origi-<br />
ne dans <strong>le</strong>s Himalayas. Une importante source de l'approvisionment<br />
d'eau pour ces rivieres est, donc, la fusion de neige dans la captation<br />
supérieure de ces rhi'eaes. Comme Seaancoup ae cette .pêg?on eat Pnacce-<br />
ssib<strong>le</strong>, <strong>le</strong>s observations de convention ne sont pas disponib<strong>le</strong>s. Un ex5<br />
men de l'images têlévision du satellite a rdv'?lE! la possibilité pour<br />
estimer la enneigement sur <strong>le</strong>s garages de rivières dans <strong>le</strong>s Himalayas<br />
et ainsi pour évaluer <strong>le</strong> contribution pap la fusion de neilge an cou<strong>le</strong><br />
ment de ces rivilres. Un effort a &!té aussi Sai't pon? estihex la préci<br />
pitation pendant 24 heures sur <strong>le</strong>s plaines ae l'Inde pendant la sud<br />
-ouest mousson par 'nauge photograpñ2es de satelliye du temps, Ces eya-<br />
luations sont d'accord raisonag<strong>le</strong> avec <strong>le</strong>s va<strong>le</strong>rs obtenues par <strong>le</strong>s ana<br />
lyses isohyeta<strong>le</strong>s de actuels prScipitation donnees. Les resultants se<br />
présentent bien, S'il est confirmé d'une analyse de plus de tempete de<br />
pluie, ces trouverent large application pour entilmer la précipitation<br />
asrienne pendant 24 heures sur la captation de rivières oh <strong>le</strong>s données<br />
sont rares.
1. Ii@RDDUCTIDW<br />
1.1 &UIY rivers in India originate in <strong>the</strong> IIimalayas. The important<br />
some of water supply for <strong>the</strong>se rivers, is <strong>the</strong> snow over <strong>the</strong> upper catchment<br />
amm. Wrefore mapping <strong>the</strong> sncw cover and its variation mer <strong>the</strong> Himslayas<br />
is vital to forecast <strong>the</strong> stream flow in <strong>the</strong>ae rivers, which in turn is of groat<br />
inportance for generation OP power and irrigation through <strong>the</strong>se rivers.<br />
1.2 Large p&s of <strong>the</strong> upper catchent of <strong>the</strong>se rivera are inace8sibi.e<br />
mas and monitoring tho snow cover ard precipitation in <strong>the</strong>se areas by comentional<br />
methods is difficult: With <strong>the</strong> advent of Polar Orbitting Satellites <strong>the</strong><br />
possibïìity of mdtoring <strong>the</strong> abme-mentioned hylromerteorological parameters by<br />
reaiute sensing techniques has arisen. In <strong>the</strong> cloud imagery obtained through<br />
<strong>the</strong> satellites anow aver <strong>the</strong> Himalaya can be c<strong>le</strong>arly recognbed.<br />
R is also<br />
reiativalg easy to identify individual river val<strong>le</strong>ys on <strong>the</strong>se satellite picturm<br />
d e r cloud f'roe conditions.<br />
1.3 Ih case of rivers whose stream flow ia mainly dependeat on precipitation<br />
it is inportant to evaluate a~ acourately as p0~sibI.e <strong>the</strong> distribution<br />
of precipitation with area and duration k? order to obtain run-off from<br />
precipitation. 51 case of @lood forecasting <strong>the</strong> adàitional prob<strong>le</strong>m involved is<br />
that <strong>the</strong> 24 hour raFtiEaU data should be availab<strong>le</strong> eqmdi.tiousiy at <strong>the</strong> forecasting<br />
centre,( Since owll established conmsUnicatian links are nut availab<strong>le</strong><br />
for ail <strong>the</strong> river catchment areas, it WU be advantageous if at<strong>le</strong>ast a rough<br />
t38thIl&e of aerial precipitatiun (average) can be obtained frm in8tan'tBneoUS<br />
cloud imagery for calculating run-off.<br />
1.4 Before one can make an attempt to derive <strong>the</strong> 24. hour precipitation<br />
m <strong>the</strong> basis of m instantaneous satellite cloud picture, one bas to knm<br />
<strong>the</strong> characteristics of rainfall ono ia trying to estimate. Over large parts<br />
of Pidia more than 75 per cent of <strong>the</strong> annual rainfall is received during <strong>the</strong><br />
southwest mansoon season (June to September). During this seasong <strong>the</strong> rdnfall<br />
is not corrtinuou~ but ~ C W S in spells laethg for about 5 to 7 days. This Is<br />
uauolly associated with <strong>the</strong> occurrence of depressiona and <strong>the</strong>ir movement roughly<br />
dong <strong>the</strong> mowoon trough of lw pressure. These depressions which irsueliy<br />
fonn near <strong>the</strong> he4 of <strong>the</strong> Bay of Bengal cause locally heavy faUs varyiiig fra<br />
7 to 20 cm in 24. hour, The rainfall associated with <strong>the</strong>m extends mer areas<br />
aa ïarge as 100,ûûû to 200,ooO sq. hs; In hilly areas rainfalls as high as<br />
25 to 35 cm in a day are recorded in assmiation with <strong>the</strong>se depressions.<br />
Mansoon rabif& in general sbowe two diurnal peaks one in <strong>the</strong> afternoon d<br />
ano<strong>the</strong>r in <strong>the</strong> early morning hours. It ia also noticed that <strong>the</strong> monsoon clod<br />
pattern, p&iCixlarly tbme comected with situations af monsoon depressions,<br />
do not show much äiffereme between <strong>the</strong> afternoon and <strong>the</strong> morning. It is <strong>the</strong>refore<br />
felt that <strong>the</strong> sing<strong>le</strong> satdite cloud pictures from <strong>the</strong> orbitting wea<strong>the</strong>r<br />
satellites can provide a fairly reasonab<strong>le</strong> estimate of <strong>the</strong> +hour rainfall,
235<br />
1.5 in ader to attain reasombh auccees in our attempt to relate<br />
<strong>the</strong>se satellite cloud bagery with aerial distribirticn of precipitation, we &ve<br />
choeen only cccaaim Of raSn storms durhg <strong>the</strong> p1~oon season. Mher ue have<br />
&o restricted our attention to <strong>the</strong> p<strong>le</strong>jm thua avo%dhg <strong>the</strong> complications that<br />
VU set in hilly areas, With <strong>the</strong>se restrictions it is hopd that a reasonab<strong>le</strong><br />
degree of aucc888 can be achiwed,<br />
2. SNOW KyDRDi&GY OF HIULAYAS<br />
2.1 Monitoring of <strong>the</strong> snow cover over Hbhyaa far hy&dogical piurpases<br />
ha gained hportance in Tecent yoara in connection with cmstruction and<br />
operation of <strong>the</strong> dams cmatructed mer <strong>the</strong> rivers originating from <strong>the</strong> Himelayas.<br />
!i'he techniques wed in <strong>the</strong> application of satellite data for snaw mapping are<br />
basically those of Simp<strong>le</strong> photo-htarpretaticm i.e., detai<strong>le</strong>d vieu inapetion<br />
of individual photogmpha to identify <strong>the</strong> river comes and large val<strong>le</strong>ys to<br />
detenuine <strong>the</strong> aerial extent of 8now cover and to make an estimate of <strong>the</strong> height<br />
of snowline.<br />
2.2 A duly au <strong>the</strong> aasessment of water flow ki River Sut<strong>le</strong>j by<br />
Satellite picturea was conducted by Gupta and abbi (1 ) . The aver Sut<strong>le</strong>j<br />
originate from Lakes B$Las d Mansarovar in Tibet and foUons a aourse of about<br />
@û hm towards west-north-wemt through mountainm terrain before ananating in<br />
<strong>the</strong> plains of Punjab.4 The monthly werage discharge data of <strong>the</strong> Bempur stream<br />
gauge site, located near <strong>the</strong> Hhalagaa, ahow that <strong>the</strong> water discharge which ia<br />
2000 - 3ûûO c1i8808 during <strong>the</strong> vinter mcmtha @acember-Febninry) reaches <strong>the</strong><br />
peak velue during July when It b more than ten tkes <strong>the</strong> winter -off.<br />
w u be seen frcm <strong>the</strong> mdhly average discharge data for <strong>the</strong> mar 19669 given<br />
in T&ie ï, that whi<strong>le</strong> tbe monthly average of water dischare;@ valu@ during<br />
wMer months do not vary much from year to year, <strong>the</strong>re are large variati- in<br />
t b e for <strong>the</strong> anow melting period.<br />
Tab<strong>le</strong> 1<br />
MûNIHLY A m WER DBCXAWE SN CUSiES DIFERE3CE; 33 THE VfiUEs<br />
MONPH<br />
196F3<br />
1 %9<br />
OF WATER nmcmìm<br />
1968u1969<br />
JanUary<br />
31 88<br />
2359<br />
829<br />
February<br />
3068<br />
2665 43<br />
Mmh<br />
394<br />
3658<br />
3 O5<br />
d g d<br />
5944-<br />
5063<br />
881<br />
Hay<br />
11 6a3<br />
13868 -2260<br />
June<br />
31 893<br />
4r5892<br />
-1 2999<br />
July<br />
32/62 47081 -14319<br />
Bug;&<br />
2153 2<br />
N63 2<br />
-131<br />
-<br />
00<br />
September 1m<br />
17869<br />
-5592<br />
October 5w<br />
7357<br />
1450<br />
Naoeniber<br />
3760<br />
45@<br />
-788<br />
Decrember<br />
2800<br />
3500<br />
-700
236<br />
-tion of <strong>the</strong> satellite pictures for <strong>the</strong> above two years showed that BLICUdation<br />
of snow aver <strong>the</strong> western ELimalaya~ OCCW during <strong>the</strong> maths of<br />
Novembe-F'ebruary. The snow-malt b- spring i.e., from <strong>the</strong> m&ha of<br />
m h when <strong>the</strong> val<strong>le</strong>ys and river courses start becorning viaib<strong>le</strong> distinctly.<br />
The minimum snow cover ocam in <strong>the</strong> post-eionsoon period after <strong>the</strong> nœrth of<br />
Septeaiber. Figure 1 shows sane of <strong>the</strong> river courses in <strong>the</strong> aateïïite picture<br />
of 9 June 1969 cover- nor<strong>the</strong>rn and western Himalayas. Examination of day to<br />
day satellite Pictures revea<strong>le</strong>d that well sustained precipitation activity mer<br />
<strong>the</strong> Sut<strong>le</strong>j basin during <strong>the</strong> spring and pre-monsoon aeason of 1969, cdributed<br />
to higher values of water discharges from <strong>the</strong> month of May aL1WBPds. Th basin<br />
was also affected by <strong>the</strong> recurdng monsoon depressionsin <strong>the</strong> month of September<br />
I969 whereas <strong>the</strong> year 1968 was marked by <strong>the</strong> rar3y withdrawal of mmsoon from<br />
<strong>the</strong> region.'<br />
2.3 Snow cover stdy of nor<strong>the</strong>rn snd western Himalayas conducted by<br />
Srinivasan and Raman (2) with <strong>the</strong> help of se<strong>le</strong>cted NnIIBLE3 ïV and ESA4<br />
Pictures of 1969-70 bas confirmed that accmulatian of snow starte from<br />
<strong>the</strong> month of November and reaches <strong>the</strong> reixhum during Janu;iry-February wilh<br />
snow line generally at 213 l0n.b The snow melt begins in 4ril ad <strong>the</strong> mlnimimi<br />
snow cover was found to occur ki this stdy in Augmt-October with anow line<br />
rising upto about 5 km. The tearporai. sequence of aIIMBu9 III Image Diesector<br />
Camera mea (IDCS) pictures of Nor<strong>the</strong>rn ñhiìayas for <strong>the</strong> period of April<br />
196Waaueuy 1970 given in a report of NASA an NïMEiiJS (3) dematrates t b<br />
minimum of <strong>the</strong> snow cover aver <strong>the</strong> &dus river basin in <strong>the</strong> Himalaya dipring<br />
Augwt-September. The B~QV accmùlaticm mer <strong>the</strong> area cormences <strong>the</strong>reafter<br />
till <strong>the</strong> month of Bpril when mdting of snow begins.<br />
2.4 maph for <strong>the</strong> river Brahmputra, rivers originating fmn <strong>the</strong><br />
Centrai and W e r n Himalayas do nut have lmg courses along <strong>the</strong> mountab.<br />
These are <strong>the</strong>refore not 80 distinctly vieib<strong>le</strong> in satellite *dea a9 <strong>the</strong><br />
rivera in nor<strong>the</strong>rn end western Himalayas. Since <strong>the</strong> winter precipitatiar over<br />
<strong>the</strong>se regions is much <strong>le</strong>ss canpared to nor<strong>the</strong>rn anl western Hiinalayaa t b<br />
magnitude of snw cover difference during <strong>the</strong> winter and summep semons ie not<br />
significant, especially so over central Hhaiayas. Examinatiosi of snow cover<br />
over <strong>the</strong> val<strong>le</strong>ys in <strong>the</strong>se regions, <strong>the</strong>refore shows that <strong>the</strong> height of <strong>the</strong><br />
snowline is generally between 3.5 -5.0 Kms. The contribution of anw-mdt to<br />
<strong>the</strong> water discharge of rivers in <strong>the</strong>se regions e m <strong>the</strong>refore be expected to<br />
be <strong>le</strong>ss than that in <strong>the</strong> western Himalafraa.'<br />
3. EsTmIDN OF RAINFALL DURmG THE SOUl!ME3T MNSOON SEASON<br />
3.1 In recent pars <strong>the</strong> satellite iimgeries bave been increasi~gly<br />
used to derive <strong>the</strong> relatianship Wween <strong>the</strong> cloud pattelas and <strong>the</strong> WcipittLtion<br />
distribution over <strong>the</strong> data sparse areas of tropics. In India an early<br />
attempt was made by Hulshrestha and Gupta (4) to st* <strong>the</strong> rainfall. and o<strong>le</strong>d
patten associated with <strong>the</strong> mansoon depression. The nephandpis prepared by<br />
U.S. Wea<strong>the</strong>r Bureau were &tilb& by B-tt (5) to estimate <strong>the</strong> maithly rainfa<br />
in <strong>the</strong> Australian Region. The radiation data of TïBIS III waa utilised<br />
by &inbird (6) to derive <strong>the</strong> relatimmkip betuetm <strong>the</strong> height of clorid topa and<br />
precipitation depths. Similar studies for rainfall estimates have elso been<br />
conducted in &her parks of <strong>the</strong> globe.<br />
3.2 Tn order to establish a reasonably valid relation between <strong>the</strong><br />
instantaneous eateììite cloud picture and <strong>the</strong> averxe areal precipitation over<br />
a particular area, it ha~ to be ensured that <strong>the</strong> particular clouds toge<strong>the</strong>r with<br />
<strong>the</strong>ir pattern will have an influence an <strong>the</strong> precipitation which we are trying to<br />
estimate and <strong>the</strong>re will be no o<strong>the</strong>r significant development or changes which will<br />
&e our inference invaìid. B is also necessary to avoid at<strong>le</strong>ast at <strong>the</strong> first<br />
instance local peculiarities like <strong>the</strong> existence of marked features of orography,<br />
so that <strong>the</strong> important factor that influences precipitation over <strong>the</strong> area is<br />
mostly <strong>the</strong> clouds and <strong>the</strong>ir patterns. As already explaincd at paras 1.4 and<br />
1.5 ahove, we have <strong>the</strong>refore chosen occasions of rainstom only during <strong>the</strong><br />
monsoon seaon.<br />
3.3 The =A-9 cloud pictures (taken around 0900 Carr) covering our<br />
regimi for <strong>the</strong> 1969 and 1770 mansoon seasons were examined in conjmction with<br />
<strong>the</strong> hrs rainfall record at 0300 GMP of next day. This has lad to develop<br />
ment of <strong>the</strong> foilowing relations between cloud characteristics and mera<br />
average precipitation range.<br />
Tab<strong>le</strong> 2<br />
S.luo. Cloud cover Organisation and1 Aerial distribu- Probab<strong>le</strong> range of<br />
or appearance tion of rainfa 2L+ hr rainfall in cms.<br />
7. Overcast<br />
2. -ao-<br />
3. Broken to<br />
overcast<br />
4. ao-<br />
5. Scattered<br />
NO organisation<br />
smooth stratiform<br />
appearance.<br />
widespread<br />
ûrganised spiralling<br />
bands, convect ive<br />
appearance.<br />
&+<br />
Convect ive Fairly widebands<br />
spread along<br />
<strong>the</strong> bands<br />
Mainly stratiform<br />
with embedded bright<br />
convective patches<br />
Fairly widespread<br />
a) convective<br />
appearance<br />
Scattered<br />
b) stratiform scattered<br />
appearance.<br />
1-3 cma<br />
237<br />
7-12 C ~ S<br />
with scattered<br />
falls 712 C ~ S<br />
7-12 C ~ S<br />
1-3 cme with<br />
scattered falls<br />
of 6 6 cas.<br />
4-6 c m<br />
1-3 ciü~
238<br />
The probability range of 26 hour rainfall given in <strong>the</strong> abme tab<strong>le</strong> are in accordance<br />
with <strong>the</strong> criteria followed k? <strong>the</strong> Mia Meteorological Department to<br />
define <strong>the</strong> rainfall oharacteristics as moderate, ra<strong>the</strong>r heavy, heavy wd very<br />
heavy with precipitation ~IIIOUWLS k? <strong>the</strong> range of 1-3, &6, 7-12 and more than<br />
12 cms respectively. I3 will be seen from <strong>the</strong> taDie that whi<strong>le</strong> <strong>the</strong> organised<br />
convective clod bands can cause <strong>the</strong> rainstonas, <strong>the</strong> stratiform clou3s give<br />
widespread moderate rains .<br />
3.4 M a r a Aypm et al (7) Sttdied two rainStroma which 00curr.d<br />
during <strong>the</strong> south-west mcmaoon season of 1970. One of <strong>the</strong>se waa over <strong>the</strong><br />
eastern ottar Pradesh and was of two day8 duration i.e. 14-15 September. Tt<br />
caused floods in <strong>the</strong> River Ganges and its tribuLaries. The o<strong>the</strong>r rainstorm<br />
atxurred an 5-7 September 190 and cawed severe flocrds in <strong>the</strong> Nmda basin.<br />
Abbi et ai also studied (8) <strong>the</strong> I&- storm. These rainstom were aastxfated<br />
with well marked monsoon depressions, fn <strong>the</strong> fomer case <strong>the</strong> depression w m<br />
more or <strong>le</strong>ss dation- wer <strong>the</strong> area during <strong>the</strong>se two dap before recurving ki<br />
a nor<strong>the</strong>rly direction. The later depression followed a track which was aïmg<br />
<strong>the</strong> river basin, The ESSA-9 Satellite pictures correspondhg to rainstmms<br />
recorded on <strong>the</strong> above dates am shown at f- 2 to 6.<br />
3.5 The fht step in <strong>the</strong> process of estimation of 24. hour rainfell<br />
mer any partbular area ia an &hatian of <strong>the</strong> relative wupied by<br />
differed typa of clomis, mhly brlgkt commative and i3ttratifonn. For thie<br />
p ~ o san e overlay consisting of a fine mesh grid was prepared. By placing<br />
this mer <strong>the</strong> clod pictures and couabhg <strong>the</strong> nmber of squi3.e~ occupied by <strong>the</strong><br />
&ove types of CloudEl ia <strong>the</strong> overall area under consideration, <strong>the</strong> relative<br />
areas occupied by different typa of clouds were obtained. These relative<br />
areas were muïtipl<strong>le</strong>d by a factor of 10 in <strong>the</strong> caere of bright convective and<br />
by 0 in <strong>the</strong> case of stratiform. Tbe total number thus Mved for <strong>the</strong> area<br />
der consideration represents <strong>the</strong> average 2&hr rainfall over <strong>the</strong> area.<br />
Estimates of 24. hr. rainfail derived from satellite cloud pictures<br />
and corresponding values obtained from isohyetal anaiysis of actual rainfail<br />
recorded are given bdw at Tab<strong>le</strong>s 3 and 4.<br />
Date of<br />
satellite<br />
picture<br />
13-6-70<br />
1&%70<br />
Tab<strong>le</strong> 3<br />
Bright Stratiform<br />
cìouä type cio&<br />
coverage coverage<br />
Dtiniated -rial Precipitation<br />
average 24. hour in cms an tim<br />
rainfa in cms basis of isofor<br />
1,8O,ooO sq.hs hyetal maw<br />
for 1,4D,ooO eq.<br />
a$<br />
8%<br />
rsra<br />
75%<br />
43 5mo& = 50.9<br />
100<br />
-0<br />
L.0<br />
7 05
Tab<strong>le</strong> 4<br />
239<br />
Date of Total Bright Stratiform &sthated aerial Precipitaticm in<br />
satellite cloid cioud type COVB- average 24. hoar cmie on <strong>the</strong> besi8<br />
Pictun, aoverage ccmer.!ì%e rage, rainfall in cm of isokyetal mapa<br />
for 2,30,0ClOaq.Kmc1 for 2,00,000<br />
sq. Elms,<br />
4-Cp70 33% 2s %<br />
5-9-70 93%<br />
6-9-70 65%<br />
75% 15%<br />
Z3QMZZ= 100 2.9 483<br />
389[10*27112 100 4.3<br />
3s Bi can be seen from <strong>the</strong> above tab<strong>le</strong> that <strong>the</strong> average aerid rain-<br />
fall estimiatea from <strong>the</strong> satallite cloud pf.ctms apee fairly well with <strong>the</strong><br />
precipitatia depth8 b@ed 011 isohyetal anSlyeiaJ The value6 esthter3 in <strong>the</strong><br />
case of rainstrom over Utta Pradesh are hadever found on <strong>the</strong> higher e2de where-<br />
as estimates in cae of Narmeda basin ~z<strong>le</strong> lai=,' Thia will sugg8st that <strong>the</strong><br />
multiplloation factors appïieä for esthatuig <strong>the</strong> & hr rainfall. will differ<br />
from catchment to catchment,' Thia ie understandab<strong>le</strong> because local factors<br />
pïey an import& roli in <strong>the</strong> amount of precipita'cion that c m actuall;g be<br />
redised from a particular type of cloud.<br />
Whi<strong>le</strong> <strong>the</strong> resdts indicate that thi8 approach is prOimiahg we have yet<br />
to establish by applying <strong>the</strong> above criteria to many more s fma th& <strong>the</strong> fac-<br />
tors are valid,'<br />
4. CûNCLUS1DMS<br />
This preliminary study has shown that Satellite cloud pictures can<br />
be utilised for estimating <strong>the</strong> 2+hr rainfall which can be utilised for<br />
arriving at at<strong>le</strong>ast preliminary estimate of river discharge for initial<br />
decision making in flood foreoasting.<br />
cloud pictures @trimI reasonab<strong>le</strong> estimates of smw cover and snow line for<br />
utiilisation in t h esti.mates of snow-melt contribution to river<br />
dia charge ,<br />
8.0<br />
337<br />
It is also encouraging that satellite
240<br />
1, -ta, M.G, and Abbi, S.D.S(l9"l). ABsessmmt of water flow in<br />
river Sut<strong>le</strong>j by Satellite Pictures, Vayu kdal, V0l.l No.3,<br />
1 13-1 17.<br />
2.1 Srhivasan, U and Fiaman,S. (1 972). Satellite Pictures in <strong>the</strong><br />
etudy of sntm hydrology mer Western Himalayas, Indian ~ J.bt.<br />
&phSeiCs., vo1.23 No.3, Pp.335-3.44e<br />
3, The beet of Nimbus (1 VI) %P. Prepared for W4, Goddard<br />
Space Flight Center, lularyland, contract No. NAS 5-10343<br />
u Kubhreetha, S.M. and Gupta, M.G. (1 964). SataUite &My of<br />
an in&&wnsoon depression, Indian J .Met .Geoph'rs. , Vol.15,No0.2,<br />
pp. 175-182.<br />
5. Barrett, &C(1%0). The estimation of monthïy rainfail from<br />
satellite data, Mon.üeath.Rev., Vol.%, No.4, pp.322-327,<br />
6. Rainbird, A.F. (1 969). Some poterrtial amlicatiane of meteorological<br />
satellites in flood forecasting, Hydrological Forecasting,<br />
W.M,O. Tech. Note No.92, pp.73-80.'<br />
7;Harihara mar, P.S., Abbi, S.D.S. and Hem ñaj (1971)<br />
Rainfall and floods during 1970 southwest monsoon period,<br />
mdim J,M&.GeO&yS., Vol228 k.1, m.141-1@<br />
8. Abbi, S.D.S et al (1972). RainfaU study of <strong>the</strong> unprecidented<br />
fïoods of September 1970 in <strong>the</strong> Marmaäa basin, Mekeomlcytical<br />
Monograph, Hydrolowfio,-2/1 972,
FIGURE-1<br />
ESSA-9 PICTURE OF JUNE9,1969 SHOWS THE 5NOW<br />
COVERED MOUNTAIN RANGES OF NORTHERN AND<br />
WESTERN HIMALAYAS. DUE TO THE MELTING 8F SNOW<br />
FROM THE LOWER VALCEYSj MANY RIVER COURSES<br />
ARE VISIBLE IN THE PICTURE.<br />
241
24 2<br />
F IGURE-2<br />
ESSA-9 PICTURE OF SEPTEMBER 4,1969. AREA<br />
CONSIDERED FOR ESTIMATION OF AERIAL 24-<br />
HOUR RAINFALL IS SHOWN BY BLACK LINE.
FIGURE -3<br />
ESSA-9 PICTURE OF SEPTEMBER 5,1969. AREA<br />
CONSIDERED FOR ESTIMATION OF AERIAL 24-<br />
HOUR RAINFALL IS SHOWN BY BLACK LINE.<br />
243
244<br />
FIGURE - 4<br />
ESSA-9 PICTURE OF SEPTEMBER 6,1969. AREA<br />
CONSIDERED FOR ESTIMATION OF AERIAL 24-<br />
HOUR RAINFALL IS SHOWN BY BLACK LINE.
FIGURE- 5<br />
ESSA-9 PICTURE OF SEPTEMBER 13, 1969. AREA<br />
CONSIDERED FOR ESTIMATION OF AERIAL 24-<br />
HOUR RAINFALL IS SHOWN BY BLACK LINE.<br />
245
24 6<br />
FIGURE - 6<br />
ESSA-9 PICTURE OF SEPTEMBER 14, 1969.<br />
AREA CONSIDERED FOR ESTIMATION OF<br />
AERIAL 24-HOUR RAINFALL 15 SHOWN BY<br />
BLACK LINE.
THE USE OF SIMULATION TECHNIQUES, ESPECIALLY DESIGNED FOR<br />
DATA-SCARCE AREAS STATISTICAL METHODS AND DATA OPERATIONS<br />
Introduction<br />
General Report<br />
by<br />
Ivan C. James, II<br />
U.S. Geological Survey<br />
Simulation is not new to <strong>the</strong> field of water resources<br />
system design. The mass-curve analysis devised by W. Rippl<br />
ninety years ago continues in use as a graphical-simulation<br />
methodology for reservoir sizing. Al<strong>le</strong>n Hazen made a lasting<br />
contribution to reservoir design techniques sixty years ago by<br />
introducing <strong>the</strong> concept of a probability distribution of annual<br />
within-year storage requirements. Following this, <strong>the</strong>re were<br />
few substantial changes in water resources design techniques<br />
until <strong>the</strong> potential of <strong>the</strong> syn<strong>the</strong>sis of operations research,<br />
and <strong>the</strong> <strong>the</strong>n newly developing digital computers were recognizea<br />
in water resources planning and design. Within this last twenty<br />
years following this syn<strong>the</strong>sis <strong>the</strong>re has been an explosion in<br />
<strong>the</strong> size and number of directions of water resources research.<br />
Simulation has continued to be a widely used planning and<br />
design tool. The advent of high <strong>le</strong>vel programming languages<br />
and <strong>the</strong> continuing increases in processing rates with each new<br />
computer'generation has made it feasib<strong>le</strong> to simulate systems of<br />
an incredib<strong>le</strong> comp<strong>le</strong>xity. Simulations have been performed to<br />
test <strong>the</strong> responses of large sca<strong>le</strong> river basin developments,<br />
salinity control projects, aquifers, estuaries, and stream-<br />
aquifer systems to changes in design and operating variab<strong>le</strong>s,<br />
just to name a few applications. Current efforts to simulate<br />
world-wide wea<strong>the</strong>r systems will dwarf <strong>the</strong>se aforementioned<br />
simulation studies in terms of computations and data require-<br />
ments.<br />
Indeed, maybe we should stop to question this growth in<br />
comp1exit.y of simulation models. Have <strong>the</strong> requirements of our<br />
models outstripped <strong>the</strong> growth of our data base? Has <strong>the</strong> ability<br />
to build comp<strong>le</strong>xity and "realism" into our model$ exceeded our<br />
ability to interpret <strong>the</strong> results and make useful decisions from<br />
<strong>the</strong>m? The answers to <strong>the</strong>se questions depend upon one's objec-<br />
tive framework. I would argue that from <strong>the</strong> viewpoint Of economic<br />
efficiency, <strong>the</strong> first question presents a well posed, though not<br />
necessarily ma<strong>the</strong>matically trivial prob<strong>le</strong>m. Some of <strong>the</strong> papers<br />
of this very symposium are providing encouraging, though somewhat<br />
limited, results on <strong>the</strong> question of optimal amounts of informa-<br />
tion for decision prob<strong>le</strong>ms. The second question has much <strong>le</strong>ss<br />
of an analytical foundation. Marginal benefits from increasing<br />
<strong>the</strong> comp<strong>le</strong>xity of a model cannot be estimated if it is not known<br />
that <strong>the</strong> increase in comp<strong>le</strong>xity is converging to <strong>the</strong> "true<br />
nature" of <strong>the</strong> process being mode<strong>le</strong>d. Perspective on this point<br />
might be increased by recalling <strong>the</strong> tit<strong>le</strong> of Tocher's <strong>book</strong>,<br />
The Art of Simulation. Un<strong>le</strong>ss <strong>the</strong> field of general systems <strong>the</strong>ory<br />
develops some applied branches, <strong>the</strong> construction and evaluation<br />
of simulation mdoels will remain an art.
248<br />
Large sca<strong>le</strong> rivex bssin simulation models require hydrologic<br />
input traces at many points. Additionally, <strong>the</strong>re may also be<br />
requirements for o<strong>the</strong>r hydro-metrological input traces such as<br />
temperature, salinity, precipitation, solar insolation, and wind<br />
speed. In order for <strong>the</strong> response of <strong>the</strong> simulation model to be<br />
similar to that of <strong>the</strong> real system, generated input traces must<br />
maintain statistical relationships among <strong>the</strong>mselves as are found<br />
in <strong>the</strong> natural data.<br />
Long comp<strong>le</strong>te natural records would be ideal, but are not<br />
often availab<strong>le</strong>. In <strong>the</strong> more typical case <strong>the</strong>re is a mixture<br />
of record <strong>le</strong>ngths and record quality, and not unusually <strong>the</strong><br />
entire absence of a needed record. Even where all records cover<br />
a concurrent base period, <strong>the</strong> realization of <strong>the</strong> process during<br />
that period may exhibit such a pathologically singular behavior<br />
that a deterministic design using those data would be unwise.<br />
The prob<strong>le</strong>m, <strong>the</strong>n, is to go from short records of varying<br />
<strong>le</strong>ngths to long records. In doing so, one must establish a<br />
criterion for comparison among alternative techniques for infill-<br />
ing and generation of records. Philosophically we might use as<br />
a criteria <strong>the</strong> requirement that <strong>the</strong> decisions that are based on<br />
<strong>the</strong> simulation be <strong>the</strong> same as if long ,natural records were avail-<br />
ab<strong>le</strong>. This criterion is not measurab<strong>le</strong> and hence <strong>the</strong> usually<br />
accepted proxy ha y/b5?n3jhe maintenance of low order moments<br />
and correlations.- - - More recently it has been suggested<br />
that o<strong>the</strong>r statistics might be pertinent to some design situa-<br />
tions.41 ?/ 61 :/ The nurst coefficient is one of <strong>the</strong>se which<br />
may have importance for <strong>the</strong> design of long term storage carry-<br />
overs .g/<br />
There are a large number of uncertainties to be considered<br />
in <strong>the</strong> planning and design process. Uncertainties of <strong>the</strong> future,<br />
such as population, demand, technology, personal preferences,<br />
political choice, and hydrologic outcome plague us. As hydrolo-<br />
gists, we have tended to concentrate upon this latter source of<br />
uncertainty without a good perspective of our limited input<br />
into <strong>the</strong> total decision making process. Even in dealing within<br />
our domain of hydrologic uncertainty, we can fur<strong>the</strong>r subdivide<br />
this into <strong>the</strong> inherent stochastic uncertainty of <strong>the</strong> future<br />
events and our misspecification error in modeling <strong>the</strong> process.<br />
Making optimal decisions in <strong>the</strong> face of <strong>the</strong> inherent<br />
stochastic nature of <strong>the</strong> process is <strong>the</strong> justification for our<br />
detai<strong>le</strong>d analysis and study of <strong>the</strong>se processes; however, <strong>the</strong>re<br />
are numerous opportunities for <strong>the</strong> introduction of <strong>the</strong> misspeci-<br />
fication error in this process. Let us list a few:
1.<br />
2.<br />
3.<br />
4.<br />
Failure of <strong>the</strong> simulation (design) model to capture<br />
<strong>the</strong> re<strong>le</strong>vant characteristics of <strong>the</strong> real-world system.<br />
Failure of <strong>the</strong> decision process to optimize <strong>the</strong> objec-<br />
tive.<br />
Se<strong>le</strong>ction of an inappropriate or incorrect model for<br />
generating <strong>the</strong> input to <strong>the</strong> simulation.<br />
Sampling errors for <strong>the</strong> parameters of <strong>the</strong> flow<br />
generating models.<br />
The papers of this session must be evaluated primarily with<br />
respect to <strong>the</strong>se last two sources of error. The o<strong>the</strong>r sources<br />
of uncertainty should still be kept in mind.<br />
Review and Summary of Papers<br />
S. H. Charania Extension of Runoff Records for Small Catchments<br />
in Semi-arid Regions.<br />
249<br />
The Thomas-Fiering model is used for generation of syn<strong>the</strong>tic<br />
monthly streamflow traces for two small catchments, one <strong>the</strong><br />
Wakefield River in Australia, and <strong>the</strong> o<strong>the</strong>r <strong>the</strong> Kongoni River<br />
in Kenya. Transforms are applied to <strong>the</strong> streamflow data until<br />
<strong>the</strong> resulting values are approximately normally distributed.<br />
For <strong>the</strong> Wakefield River, <strong>the</strong> transform is <strong>the</strong> log of <strong>the</strong> square<br />
root of <strong>the</strong> flow.<br />
The generation of normally distributed random number6 is<br />
accomplished by a ra<strong>the</strong>r unusual technique. The area under <strong>the</strong><br />
normal distribution is divided into 100 equal sub-areas by ver-<br />
tical lines. The average distances to each of <strong>the</strong>se two bound-<br />
aries on each sub-area are tabulated for se<strong>le</strong>ction by use of <strong>the</strong><br />
computer generated uniformly distributed random number. More<br />
commonly used methods include averaging a number of uniformly<br />
distributed random numbers to approximate normalcy, or normaliz-<br />
ing transforms such as <strong>the</strong> sine-cosine and Haddamard matrix<br />
transformations.<br />
Statistics of generated flows are checked. On <strong>the</strong> two<br />
streams tested, 23 of <strong>the</strong> 24 monthly means and 19 of <strong>the</strong> 24<br />
monthly standard deviations of <strong>the</strong> generated data fall within<br />
<strong>the</strong> 95% confidence intervals. Skewness and kurtosis are appar-<br />
ently <strong>le</strong>ss weìl preserved.
250<br />
M. J. Ilamlin and N. T. Kotteyoda The Preparation OZ a Data Set<br />
for Hydrologic System Analysis<br />
Development of <strong>the</strong> water resources of <strong>the</strong> Wye and Severn<br />
River basins required a large sca<strong>le</strong> simulation model. Genera-<br />
tion of input data for <strong>the</strong> simulation model was difficult due<br />
to widely varying record <strong>le</strong>ngths and <strong>the</strong> necessity of adjusting<br />
records from <strong>the</strong> gaging site to <strong>the</strong> sites of potential interest.<br />
Additionally, it was felt necessary to generate daily flows.<br />
This was accomplished by first generating five day average flows<br />
and <strong>the</strong>n disaggregating this into <strong>the</strong> five daily flows which<br />
would approximately maintain <strong>the</strong> re<strong>le</strong>vant statistics.<br />
Development of records for <strong>the</strong> base period required:<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
adjustments to natural conditions of regulated<br />
records.<br />
adjustments of short records to a base period.<br />
adjustments to ano<strong>the</strong>r point on a stream based on<br />
drainage area and effective rainfall ratios.<br />
combinations of <strong>the</strong> above methods.<br />
construction of entire records at ungaged sites based<br />
on drainage and effective rainfall ratios applied to<br />
nearby streams. (Note: No stochastic component<br />
was added.)<br />
construction of records by summing lagged upstream<br />
records and making <strong>the</strong> usual ratio adjustments.<br />
(Note: No attenuation was used.)<br />
These adjusted base period records consisted of a long sequence<br />
of pentad (5-day average) data at all points of interest.<br />
All extensions of records to <strong>the</strong> base period are based<br />
upon a bivariate syn<strong>the</strong>sis using one of <strong>the</strong> two long term sta-<br />
tions as <strong>the</strong> independent variab<strong>le</strong>. The model is designed to<br />
maintain <strong>the</strong> seasonal means and standard deviations and <strong>the</strong><br />
serial and cross correlation coefficients. It might be noted<br />
that this method does not maintain cross correlations between<br />
sets of extended records. For a simp<strong>le</strong> examp<strong>le</strong>, note what<br />
happend when two stations X and Y are extended based on a<br />
station 2. Without loss of generality <strong>le</strong>t all means be zero<br />
and all variances unity. Then:
Y = p 2 4. (1 - P&) l/Z 6<br />
Y=<br />
where E, 6, are NIID(0,l). The cross correlation between X and<br />
Y is <strong>the</strong>n:<br />
The cross correlations generated by <strong>the</strong> authors is represented<br />
by <strong>the</strong> first term on <strong>the</strong> right hand side, whi<strong>le</strong> <strong>the</strong> physically<br />
possib<strong>le</strong> values are defined by <strong>the</strong> equation for all -1 p 6 5 ~ 1.<br />
I doubt that <strong>the</strong> last word is in on <strong>the</strong> data infilling question,<br />
but <strong>the</strong> method of Crosby and Maddock?/ looks promising.<br />
251<br />
A nbmber of o<strong>the</strong>r ad-hoc procedures were used to maintain<br />
certain characteristics. The higher correlation of lowflows<br />
was approximated by se<strong>le</strong>cting a threshold value below which <strong>the</strong><br />
correlation was increased. Crossing properties were maintained<br />
by adjustment of <strong>the</strong> skew coefficient to higher values than<br />
found in <strong>the</strong> historical data. Daily data was obtained by<br />
interpolation and noise addition on <strong>the</strong> pentad data.<br />
The authors propose generating syn<strong>the</strong>tic records by first<br />
generating records for <strong>the</strong> two major long term stations and<br />
<strong>the</strong>n infilling <strong>the</strong> o<strong>the</strong>r records using <strong>the</strong> base period statis-<br />
tics.<br />
Roberto L. Lenton and John C. Schaake, Jr. Potential Application<br />
of Bayesian Techniques for Parameter Estimation with Limited Data<br />
The authors review <strong>the</strong> use of Bayesian techniques for parameter<br />
estimation. Bayes <strong>the</strong>orem is a formalism for incorporating a prior<br />
probability distribution with samp<strong>le</strong> information to achieve a posterior<br />
probability distribution which gives appropriate weights to<br />
both <strong>the</strong> prior and samp<strong>le</strong> information. Prior distributions aan üe<br />
constructed from subjective judgments, information transfer, or<br />
a combination of <strong>the</strong>se.<br />
Bayesian decision making requirea <strong>the</strong> se<strong>le</strong>ction of an action<br />
such that <strong>the</strong> expected loss of utility is minimized. Thus, loss<br />
functions must be constructed for <strong>the</strong> parameters with probability<br />
distributions.<br />
An examp<strong>le</strong> of reservoir sizing using u first-order autore-<br />
gressive model is given. A beta distribution was fitted to serial<br />
correlations ùerived from 140 rivera of <strong>the</strong> world. Diffuse prior
252<br />
probability distributions were assumed for <strong>the</strong> two parameters whici<br />
contained information on <strong>the</strong> first two moments of flow. The Bayes<br />
estimator is compared to maximun likelihood estimators for several<br />
samp<strong>le</strong> record <strong>le</strong>ngths under an assumed quadratic loss function.<br />
As Bayesian techniques come into more use in hydraulic design<br />
<strong>the</strong>re seem to be some remaining questions of <strong>the</strong> method of <strong>the</strong>ir<br />
use. Se<strong>le</strong>cting a data base prior as <strong>the</strong> authors did should con-<br />
sider more of <strong>the</strong> physical makeup of <strong>the</strong> basin because invariably<br />
<strong>the</strong> size, shape, and geology should tell one that <strong>the</strong>re is more<br />
to be known about <strong>the</strong> basin correlation structure than that given<br />
by <strong>the</strong> worldwide distribution. If working in a smal<strong>le</strong>r region,<br />
one must also consider that his samp<strong>le</strong> and <strong>the</strong> data upon which<br />
<strong>the</strong> prior was based suffer from similar time sampling biases due<br />
to interstation correlation.<br />
M. E. Moss and D. R. Dawdy Stochastic Simulation for Basins<br />
with Short or no Records of Streamflow<br />
The authors show <strong>the</strong> application of a first-order auto-<br />
regressive-moving-average (ARMA) model to <strong>the</strong> generation of<br />
streamflow record for reservoir design. The method is partic-<br />
ularly applicab<strong>le</strong> where no records exist, but regioiial rela-<br />
tionships can define <strong>the</strong> mean5 and variances of monthly flow,<br />
and <strong>the</strong> means and variances of monthly effective basin precip-<br />
itation. The mean design size as determined by <strong>the</strong> use of <strong>the</strong><br />
sequent-peak algorithm on fifty syn<strong>the</strong>tic records of 58 years<br />
<strong>le</strong>ngth is found to be essentially <strong>the</strong> same as that determined<br />
from <strong>the</strong> historical record of <strong>the</strong> same <strong>le</strong>ngth.<br />
The paper also demonstrates an examp<strong>le</strong> of a seemingly<br />
growing area of research in hydrologic model building. This<br />
area is characterized by a syn<strong>the</strong>sis of ideas from determinis-<br />
tic model builders on how <strong>the</strong> components of a basin's hydrol-<br />
ogy should operate with stochastic modeling techniques. Note<br />
how <strong>the</strong> assumption that <strong>the</strong> basin re<strong>le</strong>ases base flow as a<br />
linear reservoir allows for <strong>the</strong> model parameters t.> be estimated<br />
as functions of precipitation parameters.<br />
One difficulty in using <strong>the</strong> model comes from its requirement<br />
for means and variances of effective monthly basin precipitation.<br />
These data are not among <strong>the</strong> commonly availab<strong>le</strong> wea<strong>the</strong>r records.<br />
Mean and variance of total monthly point precipitation are, or<br />
could be, mapped for many reqions; however, <strong>the</strong> reduction of<br />
<strong>the</strong>se values to <strong>the</strong> model input parameters would require adjust-<br />
ments for basin size and probably also basin shape and orienta-<br />
tion with predominant st.orm paths. This obstac<strong>le</strong> could be<br />
economically surmounted if <strong>the</strong> model was to be used extensively<br />
in one region.
T. A. McMahen and R. G. Xein Storage Yield Estimated with<br />
Inadequate Streamflow Data<br />
A seventeen year streamflow record is extended using a<br />
modified Boughton rainfall-runoff model and an 84 year daily<br />
rainfall record. Gould's stochastic model is applied to <strong>the</strong><br />
extended record to determine storage requirements.<br />
Boughton's model is similar to several o<strong>the</strong>r rainfall-<br />
runoff models, being of <strong>the</strong> conceptual-component type. Infil-<br />
tration, evapotranspiration, surface runoff and groundwater<br />
components are computed as functions of storage in <strong>the</strong> three<br />
conceptual zones of interception storage, uppersoil storage and<br />
lower soil storage. The lower soil storage zone is subdivided<br />
into two subzones, each with baseflow discharges to obtain a<br />
base flow with a doub<strong>le</strong> recession constant. The nine model<br />
parameters are estimated by a standard function minimization<br />
procedure using <strong>the</strong> split samp<strong>le</strong> technique such that one-half<br />
of <strong>the</strong> rscord is used for calibration and <strong>the</strong> o<strong>the</strong>r one-half<br />
for estimation of <strong>the</strong> fitting error. The criterion to be<br />
minimized in <strong>the</strong> calibration procedure is not stated.<br />
The relative information content is checked to show that<br />
<strong>the</strong>re is a gain of information about <strong>the</strong> mean due to <strong>the</strong> exten-<br />
sion. It would seem that storage requirements are also sensi-<br />
tive to <strong>the</strong> variance and serial correlation. The information<br />
content of <strong>the</strong>se statistics were apparently not checked.<br />
253<br />
Reservoir capacity for 50% and 90% drafts with 5% chance<br />
of failure were made with Gould'c stochastic storage model.<br />
A comparison of <strong>the</strong>se results with those obtained by behavioral<br />
analysis (reservoir routing) of <strong>the</strong> syn<strong>the</strong>sized flow record<br />
gives similar results at <strong>the</strong> 50% <strong>le</strong>vel of development without<br />
correction for <strong>the</strong> effect of serial correlation, but a much<br />
smal<strong>le</strong>r storage requirement for <strong>the</strong> Gould model (30% of behav-<br />
ioral value) at <strong>the</strong> 90% <strong>le</strong>vel of development. Correcting <strong>the</strong><br />
Gould model for serial correlation increases <strong>the</strong> storage require-<br />
ment to 86% of <strong>the</strong> behavioral value.<br />
The authors attribute <strong>the</strong> remaining discrepancy to being<br />
beyond <strong>the</strong> range of <strong>the</strong> serial correlation correction procedure.<br />
At such a high <strong>le</strong>vel of development, some hydrologists might<br />
argue for models with higher persistence than <strong>the</strong> lag-one<br />
Markov model.
254<br />
Pedro Porras G. and Alfredo Flores E. Stochastic Application in<br />
Ungaged Basins for Planning Purposes<br />
Water resources planning is described as being dynamic.<br />
Feedback from each iteration can be used to define requirements<br />
for more detai<strong>le</strong>d information. The first version of <strong>the</strong> National<br />
Plan of Development of Water Resources required an inventory of<br />
surface runoff. The approach for <strong>the</strong> second version of <strong>the</strong> plan<br />
was stymied due to inadequate data, hence <strong>the</strong> application of<br />
stochastic methods were tried.<br />
In designing <strong>the</strong>se methods, several generalizations from<br />
<strong>the</strong> data were helpful. Some of <strong>the</strong> physiographic factors affect-<br />
ing precipitation were more influential on rainfall quantity;<br />
o<strong>the</strong>rs were more influential in affecting <strong>the</strong> distribution of<br />
precipitation throughout <strong>the</strong> year. Ratios of monthly to annual<br />
precipitation were often similar in different zones. Data<br />
deficiencies made <strong>the</strong> construction of monthly isohyetal maps<br />
a difficult task.<br />
All data were reduced to percent of <strong>the</strong> 10-year average at<br />
that station and grouped into four sets each with a 500 mm/yr.<br />
range in average precipitation. It was found in low rainfall<br />
areas that <strong>the</strong> variance increased with <strong>the</strong> mean rainfall, whi<strong>le</strong><br />
<strong>the</strong>re was no relationship at high rainfalls. In ei<strong>the</strong>r case<br />
<strong>the</strong> Gumbel distribution was found to fit <strong>the</strong> data best.<br />
When <strong>the</strong> data were resca<strong>le</strong>d to common minimum and maximum<br />
values for <strong>the</strong> 10-year historical record, it was found that <strong>the</strong><br />
cumulative marginal distributions were essentially <strong>the</strong> same.<br />
Thus monthly rainfall at ungaged sites was generated by use of<br />
a transition process and rescaling with <strong>the</strong> 12 sets of maps of<br />
monthly maximum and minimum values and map of average rainfall.<br />
For a given precipitation, <strong>the</strong> marginal distribution of evapo-<br />
transpiration was found to be normal. This provided a mechanism<br />
for generating irrigation requirements from <strong>the</strong> generated rain-<br />
fall. A two parameter model was used to compute monthly runoff<br />
from monthly rainfall. A computer program was written to carry<br />
out <strong>the</strong>se computations for 2 minute-of-ang<strong>le</strong> grid points on<br />
<strong>the</strong> maps.<br />
One point to which <strong>the</strong> authors may want to address some of<br />
<strong>the</strong>ir comments is <strong>the</strong> use to which <strong>the</strong>ir results will be applied.<br />
The procedure <strong>the</strong>y have described generates results indepen-<br />
dently at neighboring grid points. Thus some possibly important<br />
properties of <strong>the</strong> generated irrigation requirements such as <strong>the</strong><br />
covariance among <strong>the</strong> grid points in a region are lost. This<br />
information may be of considerab<strong>le</strong> importance when estimating<br />
<strong>the</strong> distribution of regional water demands.
Marcel Roche Standardization and Interpolation of Data for a<br />
Simulation Model<br />
255<br />
The author finds that it is necessary to review and<br />
recompute all records when constructing a base period set of<br />
data for input to a sìmulation model. This involves obtain-<br />
ing <strong>the</strong> original gage-heights and applying <strong>the</strong> rechecked shift<br />
and datum corrections. Inspection of discharge rating curves<br />
and hydrograph cumparison with nearby stations provide addi-<br />
tional subjective checks on <strong>the</strong> quality of <strong>the</strong> records. Examp<strong>le</strong>s<br />
of substantial errors have been found using <strong>the</strong>se methods.<br />
Water quality computations also require checking, although<br />
of a different nature. Often one must combine records derived<br />
from conductivity measurements as well as partial and comp<strong>le</strong>te<br />
chemical analyses. Precipitation records should be checked by<br />
doub<strong>le</strong> mass curve analysis for any systematic errors. Original<br />
records should be checked for random transcription errors.<br />
These operations are necessary to obtain monthly values of<br />
<strong>the</strong>se parameters for <strong>the</strong> period of record.<br />
The extension of <strong>the</strong>se records to cover <strong>the</strong> base period may<br />
be accomplished hy using regression analysis. To preserve <strong>the</strong><br />
variance, a random component must be added back onto <strong>the</strong> regres-<br />
sion estimate. This process, however, can produce negative<br />
flows and o<strong>the</strong>r prob<strong>le</strong>ms. The author suggests instead estab-<br />
lishing a line of relation through <strong>the</strong> origin of <strong>the</strong> form y = Ax<br />
such that <strong>the</strong> variance is preserved.<br />
For some basins it is possib<strong>le</strong> to improve <strong>the</strong> prediction<br />
of <strong>the</strong> missing data by including an index of local rainfall as<br />
a factor in <strong>the</strong> multip<strong>le</strong> regression. An intuitively reasonab<strong>le</strong><br />
rainfall index is a weighted sum of previous rainfall amounts<br />
where <strong>the</strong> weights decrease in some geometric progression with<br />
time since <strong>the</strong> event.<br />
Variances of estimated salinities are preserved by empir-<br />
ically estimating <strong>the</strong> marginal distribution of salinities for a<br />
number of flow classes, and generating from <strong>the</strong> marginal distri-<br />
bution appropriate to <strong>the</strong> flow class.<br />
Finally, <strong>the</strong> prob<strong>le</strong>ms of adjusting records from points of<br />
col<strong>le</strong>ction to point of need, such as Hamlin and Kottegoda faced,<br />
must be solved. This is complicated by <strong>the</strong> necessity of main-<br />
taining an additional continuity relationship for <strong>the</strong> total<br />
dissolved load.
256<br />
Several of <strong>the</strong> pgints that <strong>the</strong> author brings up deserve<br />
some discussion. In <strong>the</strong> U.S., <strong>the</strong> annual computations of surface<br />
water records are rechecked and compi<strong>le</strong>d after a five year accum-<br />
ulation. Thereafter it would be unusual to recover any remain-<br />
ing errors. Statistical interpretation of historical water-<br />
quality records in <strong>the</strong> U.S. has sometimes been difficult because<br />
<strong>the</strong> chemical analyses were done on composited samp<strong>le</strong>s. The exist-<br />
ence of several methods of compositing added to this difficulty.<br />
Have <strong>the</strong> hydrological services of o<strong>the</strong>r countries had this<br />
prob<strong>le</strong>m? The autiior admits to <strong>the</strong> ine<strong>le</strong>gance of his practical<br />
techniques for maintaining variance, but one might also wonder<br />
if o<strong>the</strong>r unmaintained parameters such as <strong>the</strong> covariance prop-<br />
erties might be important to <strong>the</strong> decisions resulting from <strong>the</strong><br />
simulation model.<br />
H. D. Charma, A. P. Bhattacharya, and S. R. Jindal The use of<br />
Simulation Techniques for Sequential Generation of Short-Sized<br />
Rainfall Data and-its Application in <strong>the</strong> Estimation of Design<br />
Flood<br />
The authors attack <strong>the</strong> prob<strong>le</strong>m of, syn<strong>the</strong>tic generation of<br />
<strong>the</strong> 6 one-hourly rainfall values for <strong>the</strong> maximum annual storms.<br />
These were <strong>the</strong>n used for computing flood peaks which would<br />
presumably include worse conditions in <strong>the</strong> catchment than those<br />
experienced in <strong>the</strong> typically 10-20 years of record availab<strong>le</strong>.<br />
The historical data used were <strong>the</strong> 6-hour annual storms recorded<br />
at New Delhi in <strong>the</strong> 1956-1965 period.<br />
The rainfalls of an annual storm are assumed to result from<br />
an autoregressive process of <strong>the</strong> form:<br />
Xt = r<br />
+ t,t-1 Xt-l<br />
This model was used on <strong>the</strong> 10 years of data shown in tab<strong>le</strong> I to<br />
obtain <strong>the</strong> statistics shown in tab<strong>le</strong> II. Unfortunately, an<br />
error, possibly in programming, seems to occur in <strong>the</strong> generating<br />
model such that <strong>the</strong> process takes <strong>the</strong> form:<br />
X = E (E generated from a uniform distribution)<br />
1 1<br />
and<br />
- X = X t + r l < t L 6<br />
t,t-1 Xt-l<br />
hence, <strong>the</strong> only variation of <strong>the</strong> hourly rainfall increments is<br />
introduced by way of El and <strong>the</strong> variance of any hourly increment<br />
is <strong>the</strong>n:<br />
Et
where r<br />
1,Q<br />
E l<br />
and <strong>the</strong> variance of <strong>the</strong> total storm rainfall is<br />
This gives a variance of <strong>the</strong> totals of annual storms of about<br />
37 compared to <strong>the</strong> samp<strong>le</strong> variance in <strong>the</strong> historical data of<br />
about 430. This would seem a sufficient reason for <strong>the</strong> dis-<br />
crepancies between <strong>the</strong> historical and generated data shown in<br />
figures 2 and 3.<br />
Perhaps <strong>the</strong> authors could respond to <strong>the</strong> questions:<br />
1. Why was a uniform distribution se<strong>le</strong>ction for E ?<br />
1<br />
2. Why was no random component added on to each hourly<br />
value, independent of <strong>the</strong> o<strong>the</strong>r hourly values?<br />
J. H. Visser The Use of Stochastic Models in a Hydro-Agricul-<br />
tura1 Development Project in Lebanon<br />
257<br />
Consistent monthly temperature, rainfall, and streamfiow<br />
data were needed for a model used to simulate <strong>the</strong> operation of<br />
an irrigation project. The purpose of <strong>the</strong> simulation model was<br />
to provide an economic evaluation of <strong>the</strong> project and a design<br />
sizing of <strong>the</strong> reservoir.<br />
The historical data consisted of several 30-year rainfall<br />
records, some 15 year temperature series, two 14-year and 13<br />
three-to-five-year streamflow series.<br />
The data generating mechanism has several features peculiar<br />
to <strong>the</strong> <strong>le</strong>ngth of record and type of data being generated. Square<br />
root of precipitation and log of discharge were <strong>the</strong> transforma-<br />
tions chosen to approximately normalize <strong>the</strong> distribution of<br />
<strong>the</strong>se data. Strong annual but weak monthly correlations between<br />
precipitation and streamflow <strong>le</strong>d <strong>the</strong> authors to <strong>the</strong> following<br />
method of monthly streamflow generation. Annual streamflows<br />
were first generated based on a regression with annual precipi-<br />
tation. An autocorrelated series of monthly flows is <strong>the</strong>n
258<br />
generated and adjusted so that its sum is eque1 to Lhe previously<br />
generated annual flow. Temperatures are generated to maintain<br />
<strong>the</strong>ir serial correlation and a cross correlation with precipita-<br />
tion.<br />
For <strong>the</strong> short streamflow records, not enough data were<br />
availab<strong>le</strong> for estimation of <strong>the</strong> mean, variance, serial anã cross<br />
cross correlations for each ca<strong>le</strong>ndar month. The monthly means<br />
were removed and this series extended on <strong>the</strong> basis of oqe of<br />
<strong>the</strong> long term flow records which had been similarly transformed.<br />
J. R. Wallis and N. C. Matalas Relative Importance of Decision<br />
Variab<strong>le</strong>s in Flood Frequency Analysis<br />
The authors present interim results of a Monte Carlo simu-<br />
lation of <strong>the</strong> process of fitting flood frequency curves to data<br />
generated from known distributions. The ultimate objective,^ of<br />
<strong>the</strong> study is <strong>the</strong> development of strategies for optimal se<strong>le</strong>ction<br />
of flood frequency analysis techniques given <strong>the</strong> loss function,<br />
<strong>le</strong>ngth of record, samp<strong>le</strong> flood statistics, and a prior distri-<br />
bution over possib<strong>le</strong> frequency distributions for floods.<br />
The results presented by <strong>the</strong> authors are <strong>the</strong> probabilities<br />
of best fit of ei<strong>the</strong>r <strong>the</strong> normal, log-normal, or Gumbel dis-<br />
tribution to data generated in every point in <strong>the</strong> experimental<br />
hyperspace:<br />
distribution: normal, Gumbel;/S;gTmal with<br />
skew = 1/4, 1/2, 1, 1.14, 2, 2<br />
record <strong>le</strong>ngth : 10, 30, SO, 70, 90 years<br />
plotting position: Weibull, Hazen<br />
fitting criteria: minimum sum of squares, minimum sum of<br />
absolute deviations<br />
It should be noted here that probability of best fit is a measure<br />
of <strong>the</strong> f<strong>le</strong>xibility of a distribution in fitting a set of data and<br />
gives nei<strong>the</strong>r a connotation of better fit to <strong>the</strong> distribution<br />
that generated <strong>the</strong> data nor any measure of how well <strong>the</strong> fitted<br />
distribution estimates <strong>the</strong> T-year flood.<br />
A quick glance at <strong>the</strong> results allows for some possibly<br />
interesting interpretations. The maximum probability of se<strong>le</strong>ct-<br />
ing <strong>the</strong> correct distribution where <strong>the</strong> real world is normal<br />
comes from <strong>the</strong> use of <strong>the</strong> Weibull (W) distribution and <strong>the</strong> mini-<br />
mum sum of absolute deviations (MSAD) fitting criterion. Simi-<br />
larly if <strong>the</strong> real world is Gumb<strong>le</strong> <strong>the</strong>n se<strong>le</strong>ction of Hazen and
259<br />
MSAD for short records and Woibull-MSS (minimum sum of squares)<br />
for longer records gives <strong>the</strong> maximum probability of <strong>the</strong> under-<br />
lying distribution being of best fit. For all of <strong>the</strong> log-normal<br />
distributions, <strong>the</strong> MSS criteria with Weibull for short and Hazen<br />
for long records maximized this probability.<br />
What is apparent is that <strong>the</strong>re is no dominant strategy for<br />
se<strong>le</strong>ction of plotting position and criteria. The se<strong>le</strong>ction of<br />
<strong>the</strong>se two factors <strong>the</strong>n has an effect on <strong>the</strong> analysis to deter-<br />
mine <strong>the</strong> "best-fitting'' distribution. Perhaps <strong>the</strong> U.S. Water<br />
Resources Council should wonder how <strong>the</strong> acceptance of <strong>the</strong><br />
Weibull plotting position and <strong>the</strong> MSS criteria influenced<br />
<strong>the</strong>ir decision to use <strong>the</strong> log-Pearson type III distribution<br />
in flood frequency analysis.<br />
Discussions of <strong>the</strong> <strong>the</strong>oretical issues involved in <strong>the</strong><br />
se<strong>le</strong>ction of a plotting position formula can be found in<br />
LangbeinlO/, Benson=/, and Appel=/.<br />
G.Weiss Shot Noise Models for Syn<strong>the</strong>tic Generation of Multi-<br />
site Daily Streamflow Data<br />
This paper is ano<strong>the</strong>r examp<strong>le</strong> of a syn<strong>the</strong>tic gcnerating<br />
mechanism which has a physical interpretation. The shot noise<br />
process is a particular linear filtered Poisson process. For<br />
those familiar with unit hydrograph <strong>the</strong>ory, <strong>the</strong> psocess may be<br />
described as <strong>the</strong> convolution of a negative exponential shaped<br />
hydrograph with a time series of rainfall events that have a<br />
Poisson occurrence and an exponential distribution of magnitude.<br />
This generating mechanism was se<strong>le</strong>cted to give a first-order<br />
autoregressive process which would reproduce recessions.<br />
Analytical resolutions of prob<strong>le</strong>ms in parameter estimation<br />
and conversion from a continuous to a discrete-averaged time<br />
series are obtained. A generalization to two site generation<br />
is presented which maintains a cross-correlation. The general-<br />
ization to multip<strong>le</strong> sites is not given but could possibly be<br />
derived.<br />
Some shortcomings in <strong>the</strong> generated data required adjustments<br />
in <strong>the</strong> process. The skews were found to be too high, and <strong>the</strong><br />
monthly variances too low. The suspected reason for <strong>the</strong>se<br />
results was because <strong>the</strong> model did not consider <strong>the</strong> base flow<br />
component. A doub<strong>le</strong> shot noise process was developed which was<br />
<strong>the</strong> sum of two independent shot noise processes with different<br />
sets of parameters. One might imagine that this physically
260<br />
represents a surface runoff model superimposed on a base flow<br />
runoff model. The break in this line of physical interpretation<br />
comes because each process has a separate time series of pulses<br />
or rainfall. A more intuitive physical model might be one in<br />
which a fraction of <strong>the</strong> rainfall went into <strong>the</strong> surface runoff<br />
mechanism and its comp<strong>le</strong>ment into <strong>the</strong> baseflow mechanism.<br />
I realize that this may complicate <strong>the</strong> parameter estimation<br />
prob<strong>le</strong>m. The author is invited to give his assessment of <strong>the</strong><br />
prob<strong>le</strong>ms and benefits from extending <strong>the</strong> model in this manner.<br />
Eric F. Wood Flood Control Design with Limited Data - A Compar-<br />
ison of <strong>the</strong> Classical and Bayesian Approaches<br />
Classical and Bayesian techniques are compared in <strong>the</strong> design<br />
of a flood control structure. The author makes two reasonab<strong>le</strong><br />
assumptions about <strong>the</strong> distribution of floods: (1) Floods above<br />
a base <strong>le</strong>vel can be assumed to occur as a Poisson process; and<br />
(2) The upper tail of many right-side unbounded frequency dis-<br />
tributions is approximately exponential. From <strong>the</strong>se assumptions<br />
is derived an approximate cumulative probability function for<br />
<strong>the</strong> floods above <strong>the</strong> base <strong>le</strong>vel:<br />
where<br />
z = flood magnitude above <strong>the</strong> base <strong>le</strong>vel<br />
v = arrival rate of floods above <strong>the</strong> base <strong>le</strong>vel<br />
a = reciprocal of mean of floods above <strong>the</strong> base <strong>le</strong>vel<br />
t = time horizon<br />
This model is <strong>the</strong> basis for estimation by both <strong>the</strong> classical and<br />
Bayesian techniques.<br />
ln <strong>the</strong> classical technique, <strong>the</strong> parameters V and a are<br />
estimated by maximum likelihood techniques. This uses only<br />
<strong>the</strong> site record and no o<strong>the</strong>r information.<br />
In <strong>the</strong> Bayesian technique, <strong>the</strong> parameters are estimated<br />
by first forming prior probability distributions on <strong>the</strong> param-<br />
eters based on regional studies and subjective judgement. Bayes<br />
equation is used to incorporate <strong>the</strong> samp<strong>le</strong> information into <strong>the</strong><br />
prior distribution to obtain a posterior distribution on <strong>the</strong>se<br />
Parameters. Iii <strong>the</strong> examp<strong>le</strong> results of a regression analysis
261<br />
are used for estimating <strong>the</strong> parameters of <strong>the</strong> gamma-l prior<br />
distribution of (Y. Since large flood events are correlated,<br />
this method may underestimate <strong>the</strong> variance of (Y. Subjective<br />
judgement based on personal experience is assumed to provide<br />
<strong>the</strong> information for <strong>the</strong> parameters of <strong>the</strong> gamma-1 prior distri-<br />
bution of v.<br />
Caution should be exercised when interpreting <strong>the</strong> economics<br />
of <strong>the</strong> design application examp<strong>le</strong>. Note that <strong>the</strong>se costs assume<br />
<strong>the</strong> particular model correct, and are not measures of efficiency.<br />
For examp<strong>le</strong>, at <strong>the</strong> optimum <strong>le</strong>vel of protection <strong>the</strong>re is an<br />
equal marginal trade-off between protection costs and damage<br />
costs; <strong>the</strong>refore, <strong>the</strong> evaluation of <strong>the</strong> design based on <strong>the</strong><br />
classical model using <strong>the</strong> Bayesian model to estimate flood<br />
damages would give expected flood damages of <strong>le</strong>ss than <strong>the</strong><br />
$7 x lo5 value resulting from <strong>the</strong> <strong>le</strong>ss expensive protection<br />
work designed on <strong>the</strong> basis of <strong>the</strong> Bayesian model.<br />
Bayesian decision <strong>the</strong>ory as demonstrated in this examp<strong>le</strong><br />
may have much merit as a tool for incorporating information<br />
from regional studies and small samp<strong>le</strong>s for decision making in<br />
data scarce areas.<br />
Summary<br />
Syn<strong>the</strong>tic data generation for infilling and extension of<br />
records is a difficult task when <strong>the</strong> analyst hac a mixture of<br />
types, <strong>le</strong>ngths, and quality of availab<strong>le</strong> historical data.<br />
The approaches developed by <strong>the</strong>se authors attest to this<br />
variety of availab<strong>le</strong> data and to <strong>the</strong> various particular require-<br />
ments for input data of <strong>the</strong>ir simulation models and planning<br />
procedures. The literature is rep<strong>le</strong>te with examp<strong>le</strong>s of tech-<br />
niques developed for special prob<strong>le</strong>m applications.=/ =/ E/<br />
It was previously noted that in <strong>the</strong>ir data infilling and<br />
extension procedures several authors used methods which main-<br />
tained only one of <strong>the</strong> re<strong>le</strong>vane cross-correlations. Multisite<br />
syn<strong>the</strong>tic data generation also has prob<strong>le</strong>ms. Fierings/ dis-<br />
cusses some of <strong>the</strong> earlier attempts at overcoming <strong>the</strong> prob<strong>le</strong>m<br />
of inconsistent correlation matrices. More recent investiga-<br />
tions of this prob<strong>le</strong>m?/ =/ have <strong>le</strong>d to serious questions<br />
about <strong>the</strong> feasibility of consistent parameter estimation for<br />
<strong>the</strong> more complicated flow generating models=/. F@r practical<br />
reasons, one must achieve a compromise between e<strong>le</strong>gance and<br />
feasibility in <strong>the</strong>se extension procedures.<br />
Difficulties remain in <strong>the</strong> prob<strong>le</strong>m of how much and of<br />
what type of data are really needed for models used in decision
262<br />
processes. Decision <strong>the</strong>ory tools have only provided answers<br />
for simp<strong>le</strong> and often analytic models. The extension of <strong>the</strong>se<br />
tools into <strong>the</strong> pre-posterior 'analysis of data requirements for<br />
simulation models may be computationally prohibitive An examp<strong>le</strong><br />
of an approach is given by Young, Tseng, and Taylore/-<br />
Moss and Dawdy, and Weiss have proposed essentially new<br />
statistic models which use some physical interpretation from<br />
<strong>the</strong> basin in parameter estimation. Is this to be a new emphasis<br />
in model research? Wallis and Matalas, McMahon and Meir, and<br />
Wood are interested in <strong>the</strong> sensitivity of model and analytic<br />
se<strong>le</strong>ction on design results. Does this question have any poten-<br />
tial for being answered? These and o<strong>the</strong>r questions deserve some<br />
discussion.<br />
In this short time, I have attempted to cover a few of <strong>the</strong><br />
main points which <strong>the</strong> authors of <strong>the</strong> 12 papers have documented.<br />
These short synopses cannot do justice to <strong>the</strong> research and<br />
intel<strong>le</strong>ctual effort that was necessary in approaching <strong>the</strong>se<br />
very pressing and practical prob<strong>le</strong>ms. I urge each of you to<br />
read <strong>the</strong> papers. Perhaps this discussion can provide some<br />
insights that will be helpful in that task.<br />
of<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
To <strong>the</strong> authors I offer my apology for any mistakes or errors<br />
ei<strong>the</strong>r emphasis or interpretation.<br />
References<br />
Matalas, N, C., 1967, Ma<strong>the</strong>matical assessment of syn<strong>the</strong>tic<br />
hydrology, Water Resources Research, v. 3, no. 4, pp. 937-945.<br />
Fiering, M. B., 1965, Streamflow Syn<strong>the</strong>sis, Harvard University<br />
Press, Cambridge, Mass. 139 p.<br />
Beard, Leo R., 1965, Use of interrelated records to simulate<br />
streamflow, J. Hydraul. Div., Amer. Soc. Civil Eng., 91,<br />
pp 13-22.<br />
O'Connell, P. E., 1971, A simp<strong>le</strong> stochastic modelling of<br />
Hurst's law: Proceedings of <strong>the</strong> International Symposium on<br />
Ma<strong>the</strong>matical Models in Hydrology, Warsaw.<br />
Rodriguez-Iturbe, Ignacio, Jose M. Mejia, and David R. Dawdy,<br />
1972, Streamflow simulation 1. A new look at Markovian<br />
models, fractional Gaussian noise, and crossing <strong>the</strong>ory:<br />
Water Resources Research, v. 8, no. 4, pp. 921-930.
263<br />
6. Mejia, Jose M., Ignacio Rodriguez-Iturbe, and David R. Dawdy,<br />
1972, Streamflow simulation 2. The broken line process as<br />
a potential model for hydrologic simulation: Water Resources<br />
Research, v. 8, no. 4, pp. 931-941.<br />
7. Carlson, R. F., A. J. A. MacCormick, and D. G. Watts, 1970,<br />
Application of linear random models to four annual stream-<br />
flow series, Water Resources Research, V. 6, no. 4, pp.<br />
1070-1078.<br />
8. Wallis, J. R. and N. C. Matalas, 1972, Sensitivity of res-<br />
ervoir design to <strong>the</strong> generating mechanism of inflows:<br />
Water Resources Research, V. 8, no. 3, pp. 634-641.<br />
9. Crosby, D. S., and Thomas Maddock, III, 1970, Estimating<br />
coefficients of a flow generator for monotone samp<strong>le</strong>s of<br />
data: Water Resources Research, v. 6, no. 4, pp. 1079-1086.<br />
10. Langbein, W. B., 1960, Plotting positions in frequency<br />
analysis in Dalrymp<strong>le</strong>, Tate, Flood-frequency analyses:<br />
U.S. Geological Survey Water Supply Paper 1543-A.<br />
11. Benson, Manuel A., 1967, Average probability of extreme<br />
events: Water Resources Research, v. 3, no. 1, 225 p.<br />
12. Appel, Char<strong>le</strong>s A., 1968, A note on <strong>the</strong> average probability<br />
of extreme events: Water Resources Research, v. 4, no. 6,<br />
1359 p.<br />
13. Moreau, David H., and Edwin E. Pyatt, 1970, Weekly and<br />
monthly flows in syn<strong>the</strong>tic hydrology: Water Resources<br />
Research, v. 6, no. 1, pp. 53-61.<br />
14. Pentland, R. L., and D. R. Cuthbert, 1971, Operational<br />
hydrology for ungaged streams by <strong>the</strong> grid square techniqumr<br />
Water Resources Research, v. 7, no. 2, pp. 283-291.<br />
15. Benson, M. A., and N. C. Matalas, 1967, Syn<strong>the</strong>tic hydrology<br />
based on regional statistical parameters: Water Resources<br />
Research, v. 3, no. 4, pp. 931-935.<br />
16. Fiering, M. B., 1968, Schemes for handling inconsistent<br />
matrices: Water Resources Research, v. 4, no. 2, pp. 291-297.<br />
17. Matalas, N. C., and J. R. Wallis, 1971, Correlation con-<br />
straints for generating processes: Proceedings of <strong>the</strong> Inter-<br />
national Symposium on Ma<strong>the</strong>matical Models in Hydrology,<br />
Warsaw.
264<br />
18. Slack, J. R., 1972, Bias, illusion, and denial as data<br />
uncertainties: Proceedings of <strong>the</strong> International Symposium<br />
on Uncertainties in Hydrologic and Water Resources Systems,<br />
Tucson, Arizona.<br />
19. Young, G. K., M. T. Tseng, and R. S. Taylor, 1972, Data<br />
se<strong>le</strong>ction for environmental simulations - A water tempera-<br />
ture examp<strong>le</strong>: Water Resources Research, v. 8, no. 5,<br />
pp. 1226-1233.
ABSTRACT<br />
ETUDE DES RELATIONS PLUIE-DEBIT<br />
SUR TROIS BASSINS VERSANTS D'INVESTIGATION.<br />
Y. C OWRY - A. GUI LBOT<br />
Research basins are useful way to study hydrologic cyc<strong>le</strong><br />
On three of <strong>the</strong>m,with areas between 109 and 250 km2,<strong>the</strong><br />
authors have stablished a model relating rain and runoff<br />
This model is ab<strong>le</strong> to simulate mesured runoff series<br />
from rain series and provide a better understanding of hydrologic<br />
mechanisms at <strong>the</strong> sca<strong>le</strong> of this basins.<br />
ïhis paper suggests a méthodology which,applied to many<br />
ba.sins,should permit <strong>the</strong> identification of <strong>the</strong> relations between<br />
<strong>the</strong> physical caracteristics of a basin and <strong>the</strong> model parameters<br />
identification which is necessary in order to apply this model<br />
to ungaged basins.<br />
RES UME<br />
Les bassins versants d'invectigatioh constituent en ecx<br />
mêmes un outil de recherche privilégié en ce qui concerne <strong>le</strong>s<br />
mécanismes mis en jeu par <strong>le</strong> cyc<strong>le</strong> hydrologique naturel.<br />
Sur trois d'entre eux.de superficie comprise entre<br />
100 et 250 km2,<strong>le</strong>s auteurs ont établi un modè<strong>le</strong> de liaison pluie<br />
débits permettant la reconstitution des séries de débits observés<br />
à partir des séries concomitantes de pluie et autorisant une<br />
mell<strong>le</strong>ure connaissance des mécanismes hydrologiques considérés<br />
A l'échel<strong>le</strong> de ces bassins<br />
L'approche du cyc<strong>le</strong> hydrologique à nécessité diverses<br />
opérations tel<strong>le</strong>s que:<br />
-choix du schéma hydrologique et mise au point du mode<strong>le</strong><br />
-réglage du mode<strong>le</strong> et mise au point d'an processus de<br />
determination numérique des parametres ,<br />
-vérification de la validité du modè<strong>le</strong> par comparaison<br />
aux séries obserdes tau niveau des caractéristiques statistiques<br />
des principa<strong>le</strong>s grandeurs hydrologiques'<br />
-étude de la convergence des méthodes d'optimisation en<br />
présence d'erreurs aléatoires sur <strong>le</strong>s données d'entrées<br />
-analyse spatia<strong>le</strong> et temporel<strong>le</strong> des séries entrée-sortie<br />
(choix du pas de temps des entrées et determination du décalage<br />
pluie-débit par analyses spectra<strong>le</strong>s)<br />
Cette étude définit une méthodologie généra<strong>le</strong> d'utilisa-<br />
tion qui devrait permettre,à long terme,l'identification des<br />
relations liant <strong>le</strong>s caractéristiques physiques d'un bassin et <strong>le</strong>s<br />
parametres du mode<strong>le</strong>,identification nécpssaire dans <strong>le</strong> ras d'ap-<br />
plication du modè<strong>le</strong> à des bassins non contrôlés<br />
COWRY Yves - Ingénieur<br />
Agronome - Laboratoire National<br />
d'Hydraulique E.D.F. - Professeur Aesocié à l'Université des<br />
Sciences et Techniques du Languedoc - Montpellier (France)<br />
GUILBOT Alain - Ingénieur - Laboratoire d'Hydrologie -<br />
Université des Sciences et Techniques du Languedoc - Montpellier<br />
(France)
2 66 I. GENERALITES :<br />
Dans l'étude de la liaison pluie - débit,il s'agit<br />
d'élaborer généra<strong>le</strong>ment un mode<strong>le</strong>, type "boite noire" qui:<br />
considérant la séries des pluies comme"entrée",permet d'obtenir<br />
une"s0rtie"concordant sensib<strong>le</strong>ment avec la série chronologique<br />
concomitante des débits observés<br />
On peut alors envisager plusieurs types de mode<strong>le</strong>S.tels<br />
que <strong>le</strong>s mode<strong>le</strong>s linéaires classiques obtenus par corrélation,<br />
analyse multivariab<strong>le</strong>s,analyse factoriel<strong>le</strong>..,,<strong>le</strong>s mode<strong>le</strong>s à<br />
élément central linéaire (basé sur l'hypothèse de l'hydrogramme<br />
unitaire) ou <strong>le</strong>s mode<strong>le</strong>s conceptuels qui,en quelque sorte,font<br />
la synthèse généra<strong>le</strong>.<br />
La stucture d'un modè<strong>le</strong> conceptuel est fondée sur la<br />
connaissance ou la pseudo-connaissance des phénomenes en jeu<br />
dans <strong>le</strong> cyc<strong>le</strong> hydrologique. .<br />
On suppose,par exemp<strong>le</strong>,que <strong>le</strong>s taux et <strong>le</strong>s vitesses de<br />
transfert de l'eau de pluie par tel<strong>le</strong> ou tel<strong>le</strong> partie de cyc<strong>le</strong><br />
hydrologique sont asservis à l'état de remplissage de la zone<br />
correspondante par des fonctions à un ou deux parametres.<br />
La sortie résultante,en l'occurence la dérie des débits<br />
calculés,est comparée à la sortie observée daps <strong>le</strong> systeme réel,<br />
c'est à dire la série des débits observés à l'exutoire du bassin.<br />
Si la concordance ne semb<strong>le</strong> pas satisfaisante,on modifia<br />
<strong>le</strong>s psrametres des foictions des divers sous-systemes,jusqu'à<br />
obtenir une corredpondance satisfaisante entre <strong>le</strong>s séries observées<br />
et calculées<br />
Ceci ne devrait etre fait,non pas dans <strong>le</strong> but d'un calage<br />
spécifique permettant d'obtenir l'hydrogramme d'un bassin<br />
particulier,mais dans l'optique d'une recherche de liens entre<br />
<strong>le</strong>s va<strong>le</strong>urs des parametres du mode<strong>le</strong> et <strong>le</strong>s caractéristiques du<br />
bass in.<br />
I1 est donc nécessaire d'une part d'appliquer <strong>le</strong> même<br />
modè<strong>le</strong> à de nombreux bassins,d'autre part que tout mode<strong>le</strong> conceptuel<br />
soit,au départ,aussi simp<strong>le</strong> que possib<strong>le</strong> et que des modifications<br />
ne lui soient apportées que sf la nécessité absolue apparaisse.(raisons<br />
physiques ou amélioration évidente de la reproduction)<br />
Un systeme simp<strong>le</strong>,parceque dans un schéma élaboré,il<br />
y aura de fortes chances que <strong>le</strong> mode<strong>le</strong> comporte deux sous-systeme<br />
tout à fait équiva<strong>le</strong>nts et il sera extremement délicat de <strong>le</strong>ver<br />
l'indétermination sur l'attribution de la va<strong>le</strong>ur des parametres à<br />
l'un ou l'autre de ces sous-systèmes,ensuite parceque seul un<br />
mode<strong>le</strong> simp<strong>le</strong> permettra l'identification parometres-caractéris-<br />
tiques du bassin et donc son utilisation sur des bassins non<br />
jaugés.
267<br />
II.LES BASSINS ET LES DONNEES:<br />
L'étude porte sur trois bassins expérimentaux présentant<br />
des caracteres morphologiques,géologiques et pédologiques<br />
bien différenciés.<br />
-<strong>le</strong> bassin de ia DIEGE,affluent de la DORDOGNE,<br />
d'une superficie de 225 km2.Géré par EDF depuis 1960 puis par<br />
<strong>le</strong> Laboratoire d'Hydrologie de l'université des Sciences et<br />
Techniques du Languedoc,c'est un bassin montagneux.,cristsllin,<br />
bien boisé et soumis à des influences océaniques et méditerranéennes.<br />
-<strong>le</strong> bassin de l'ORGEVAL,affluent du GRAND MORIN<br />
d'une superficie de 104 km2 Géré par <strong>le</strong> C.T.G.R.E.F(Ministere de<br />
l'Agriculture) depuis 1962,c'est un vaste plateau limoneux,coJver<br />
dans sa majeure partie de culture et soumis à des influences<br />
océaniques et continenta<strong>le</strong>s.<br />
-<strong>le</strong> bassin de l'HALLUE,affluent de la SOW-,<br />
d'une superficie de 219 km2.Géré depuis 1966 par <strong>le</strong> B.R.G.F,<br />
c'est un bassin de relief modér6,formé de craie recouverte de<br />
limon et principa<strong>le</strong>ment mis en culture.11 est soumis essantiel-<br />
<strong>le</strong>ment à des influences océaniques.<br />
Ces trois bassins étant des bassins expérimentaur,<strong>le</strong>s<br />
données étaient caractérisées d'uns part par un volume important<br />
d'informations disponib<strong>le</strong>s,d'autre part par une qualité de l'enregistrement<br />
et du dépouil<strong>le</strong>ment (à de rares exceptions pres)<br />
Le choix d'une pluviométrie représentative fut fait,<br />
soit en fonction de nos propres connaissances du bassin(B.V de<br />
la DIEGE),soit en fonction des conseils de l'organisme de gestion<br />
(BV de i'OXGEV?,L),soit apres une analyse spatia<strong>le</strong> de ia pïuviom6triecB.V<br />
de 1'HALLUE).<br />
Le choix de l'indice d'ETP a,par contre,été mené de m2-<br />
niere quelque peu arbitraire et de façon indépendante pour <strong>le</strong>s<br />
tois bassins ce qui semb<strong>le</strong> une erreur,compte tenu de l'importance<br />
effective de sa variance interannuel<strong>le</strong> et de son niveau moyen<br />
Remarqueune méthode systématique de dépouil<strong>le</strong>ment a<br />
été mise au point et utilisée dans <strong>le</strong> cadre de cette étude.<br />
I1 s'agit de.traduire l'enregistrement pluviométrique<br />
ou limnimétrique dans un systeme (X,Y) sur machine D.MAC puis<br />
de transformer ces données"digita1isées en donées de pas de<br />
temps voulu (2h,10 mn..)<br />
III.LES MODELES:<br />
Dans <strong>le</strong> cadre d'une précédente étude,plusieurs mcde<strong>le</strong>s<br />
avaient été élaborés et testés par <strong>le</strong> Laboratoire.<br />
Trois d'entre eux ont été'retenus et rendus opérationnel<br />
Ce sont <strong>le</strong>s mode<strong>le</strong>s DIEGE.MER0 et CREC
268<br />
---- IV.LES METHODES EMPLOYEES:<br />
4.1.1:Méthodes des composantes principa<strong>le</strong>s appliquée à<br />
la détermination de la représentativité de l'.information pluviométrique<br />
:<br />
La méthode d'analyse en composantes principa<strong>le</strong>s permet<br />
de substituer k vecteurs X de n composantes corrélées entres el<strong>le</strong>s<br />
à k vecteurs Y de p composantes indépendantes avec p
269<br />
4 . 2 s ~ :<br />
Les parametres des fonctions des divers sous-systemes<br />
des mode<strong>le</strong>s fproduction,transfert) sont rarement déterminés a<br />
priori de façon précise.Nous avons accompli un effort tout particulier<br />
pour mettre au point une technique de détermination numérique<br />
de ces parametres,technique devant etre assez généra<strong>le</strong><br />
pour etre appliquée systématiquement 3 n'importe quel bassin en<br />
assurant une convergence réel<strong>le</strong> et rapide vers un optimum objectif.<br />
La méthodologie que nous proposons,testées initia<strong>le</strong>ment<br />
sur séries fictives,donc currespondsnt à une structure de mode<strong>le</strong><br />
et un jeu de parametres définis,semb<strong>le</strong> particulièrement intéressante:<br />
1.Définition de la zone de variation de chacun<br />
des parametres (en fonction de la nature du bassin et des result<br />
ats des analyses préalab<strong>le</strong>s des séries d'entrées)<br />
2.Tirage au hasard,d'abord dans une loi uniforme<br />
puis dans une loi norma<strong>le</strong> avec diminution de la variance<br />
de ia loi en cas de succes(ceci afin d'éviter Loe recherche systématique<br />
à partir d'un faux minimum)<br />
3.Recherche "direcre",avec rotation des axes<br />
de co~rdonnéos("nûCENaRû~~:.~.~c~erche séquentieiie effectuée successivefiient<br />
sur chacun des axes de coordonnées(correspondant<br />
chacun à un parametre) suivant un pas d'exploration modifié selon<br />
<strong>le</strong>s échécs et <strong>le</strong>s succes rencontrés.Si,dans toutes <strong>le</strong>s directions<br />
ont a enregistré au moins un succes suivi d'un échec,on défini<br />
alors la nouvel<strong>le</strong> direction du premier axe comme étant cel<strong>le</strong><br />
joignant !e point initial et <strong>le</strong> point final.La direction des<br />
autres axes est obtenue par la méthode d'ortogonalisation de<br />
SCHMIDT.<br />
4.Recherche fine par ta méthode du gradient<br />
conjuguée (Powell) lorsque la précédente méthode ne converge<br />
que tres 1entement.La méthode de POWELL utilise la méthode des<br />
directions conjuguée mais modifie <strong>le</strong> procédé de base afin d'accélérer<br />
la vitesse de convergence vers l'optimum tout en définissant<br />
un processus de recherc!:e <strong>le</strong> long d'un axe.<br />
Ce dtverses méthodes appliquées en cascade permettent<br />
la réduction d'un critere d'écart cho.si afin d'assurer une<br />
reconstitution satisfaisante et homogene sur une période déterminée,(<strong>le</strong><br />
critere choisi est de la forme<br />
1 IQobs-<br />
F = -5<br />
Qca$ IOobs - OmovPd<br />
9<br />
N Qobs moyen<br />
N étant <strong>le</strong> nombre de mois de la période,>de calage et Qmoyen <strong>le</strong><br />
modu<strong>le</strong> de la période de calage<br />
Ce choix a été fait dans <strong>le</strong> hut de rendre préférentiels<br />
<strong>le</strong>s écarts SUU <strong>le</strong>s va<strong>le</strong>urs extrêmes.En effet,dans cette expressio<br />
plus on s'écarte duidébit moyen,plus l'écart relatif est pondéré<br />
par une va<strong>le</strong>ur importante,et cela,aussi bien pour <strong>le</strong>s faib<strong>le</strong>s<br />
débits.que pour <strong>le</strong>s crues.
270<br />
Exemp<strong>le</strong> d'application de la méthode d'optimisation proposée<br />
En appliquant un jeu de paramètres à une série de données pluviométriques<br />
journalières, nous avons généré une série de débits fictifs journaliers<br />
à l'aide du modè<strong>le</strong> CREC.<br />
Nous nous sommes ensuite proposé de reconstituer cette série de<br />
débits en utilisant la méthode précédemment décrite.<br />
La réel<strong>le</strong> convergence de la méthode, tant au niveau de la fonction<br />
critère qu'au niveau des paramètres, semb<strong>le</strong> montrer son efficacit6 dans <strong>le</strong><br />
cas de séries parfaitement adéquates, sans erreúrs de mesure et avec un<br />
modè<strong>le</strong> vrai.<br />
x1<br />
x2<br />
x3<br />
x4<br />
x5<br />
X6<br />
x7<br />
Résultats de la recherche des paramètres<br />
du modè<strong>le</strong> (;I:EC par la méthode préconisée<br />
Erreur<br />
Va<strong>le</strong>ur Etape 1 Etape 2 Etape 3 relative %<br />
vraie<br />
O. 069 O. 0597 0.0765 O. 0693 O. 4<br />
o.. a43 O. 7322 o. 5871 o. a435 O. 06<br />
o. 0212 0.3922 0.0361 9.0221 4<br />
O. 0344 o. ooao O. 0272 O. 0343 O. 3<br />
3.902 6.5784 4.5951 3. a290 1.9<br />
7.992 6. a303 5.5790 a. 020 O. 4<br />
15.146 27.7554 4.6670 13. aaao 8.3<br />
O 636 58 1.5<br />
[Interprétation des résultat4<br />
L'adéquation du modè<strong>le</strong> CREC à l'étude de la liaison pluie-débit<br />
sur <strong>le</strong>s trois bassins étudiés peut s'accompagner d'une tentative de jus-,<br />
tification.<br />
Le schéma proposé par ce modè<strong>le</strong> présente au niveau du transfert<br />
une zone que l'on peut qualifier d'hypodermique et une zone souterraine.<br />
I1 apparaît que, pour <strong>le</strong> bassin de l'HALLUE, l'écou<strong>le</strong>ment calculé<br />
provient pour une part essentiel<strong>le</strong> de la zone souterraine, ce qui est en<br />
accord avec l'influence prépondérante des variati.ons de la nappe phréatique<br />
sur <strong>le</strong>s débits observés sur ce bassin.<br />
De même pour <strong>le</strong> bassin de l'ORGEVAL, drainé artificiel<strong>le</strong>ment<br />
(drainage agrico<strong>le</strong>) et ne présentant pas de réserves souterraines importantes,<br />
la majeure partie de l'écou<strong>le</strong>ment calculé provient de la zone<br />
définie comme "hypodermique'' (l'alimentation de la zone souterraine<br />
semblant être une constante du bassin).<br />
Enfin, sur <strong>le</strong> bassin de la DIEGE, l'écou<strong>le</strong>ment hypodermique est<br />
là aussi essentiel. De plus, deux remarques sont à faire : d'une part ce<br />
bassin peut présenter dans <strong>le</strong> cas d'une saturation importante du sol<br />
accompagnée de pluies intenses, du ruissel<strong>le</strong>ment "superficiel" (crue historique<br />
de 19601, ce que l'on retrouve au niveau du schéma du modè<strong>le</strong> CREC,<br />
d'autre part, il semb<strong>le</strong>rait que l'alimentation des réserves souterraines
(faib<strong>le</strong>s dans cette région) ne se<br />
seuil de teneur en eau de la zone<br />
I1 Y a donc une cohérence<br />
produisent qu'à partir d'un certain<br />
hypodermique.<br />
certaine entre la nature des divers<br />
271<br />
bassins et <strong>le</strong> comportement hydrologique du modè<strong>le</strong> proposé.<br />
Le manque de politique homogène au niveau du choix de l'indice<br />
d'ETP ne peut malheureusement pas permettre la comparaison de la fonction<br />
de production sur <strong>le</strong>s trois bassins et un effort reste à faire quant à<br />
ce choix.<br />
Conclusion<br />
Si, sur <strong>le</strong> plan opérationnel, <strong>le</strong>s modè<strong>le</strong>s utilisés se moritrent<br />
,différents au niveau de l'application (performance, sensibilité,....), ils<br />
restent tous discutab<strong>le</strong>s sur <strong>le</strong> plan conceptuel, puisqu'ils fixent a<br />
priori, en l'absence de foute veritab<strong>le</strong> information intermédiaire entre<br />
la pluie et <strong>le</strong> débit, <strong>le</strong> schéma du cyc<strong>le</strong> hydrologique.<br />
Néammoins, cette approche a permis de mettre en évidence l'ad+"-<br />
tion de certains schémas du cyc<strong>le</strong> hydrologique pour représenter plusieurs<br />
bassins, en autorisant une extrapolation temporel<strong>le</strong> (35 ans sur la DIEGE).<br />
Dans un esprit d'application de ces méthodes 2. des projet; d'aménage-<br />
ment des ressources en eau sans données suffisantes, il resterait à :<br />
- dégager des groupes de bassins justiciab<strong>le</strong>s de chaque schéma<br />
- définir des critères d'adéquation a priori d'un bassin à un<br />
schéma déterminé<br />
- caractériser chaque bassin par des index mesurab<strong>le</strong>smou analy-<br />
sab<strong>le</strong>s en absence de longues séries de données, et dont la détermination<br />
déboucherait sur l'appréciation quantitative des paramètres d'un modè<strong>le</strong><br />
global,<br />
Ceci permettrait <strong>le</strong> choix d'un modè<strong>le</strong> (schéma et va<strong>le</strong>urs des<br />
parametres) capab<strong>le</strong> de i-zprésenter <strong>le</strong> comportement d'un bassin non jaugé,<br />
dont on pourrait, à partir des séries climatologiques dispoqib<strong>le</strong>s, simu<strong>le</strong>r<br />
1 'écou<strong>le</strong>ment.
272<br />
R E F E R E N C E S<br />
(1) F. AUBIN - A. GUILBOT (note HYD 13/72 et 14/72)<br />
- Application de l'analyse spectra<strong>le</strong> - bassin de la DIEGE<br />
- Influence d'erreurs aléatoires sur la convergence d'une méthode<br />
d'opthisation. Tentatives de filtrage des séries chronologiques<br />
(2) Y. CORIíARY - A. GUILBOT (HYD 6/71)<br />
Etude généra<strong>le</strong> de quelques modè<strong>le</strong>s détermises de relations<br />
pluie-débit<br />
(3) Y. CORMARY - A. GUILBOT (HY3 44/70 - SHF - NOV. 1970)<br />
Méthodes d'optimisation des paramètres des modè<strong>le</strong>s déterdnistes<br />
(4) Y. CORIlARY - A. GUILBOT (HYD 16/71)<br />
Processus d'optimisation en quatre étapes applicab<strong>le</strong> B la recherche<br />
des paramètrss des modè<strong>le</strong>s déterministes<br />
(5) Y. CORMARY - S. RAMBAL (HYD 32/71)<br />
Relations pluie-débiG, bassin versant de 1'HALLUE B l'échel<strong>le</strong><br />
journalisre et à l'échel<strong>le</strong> bi-horairE<br />
(6) Y. CORMARY - G. GALEA (HYD 27/71)<br />
Relations pluie-débit, bassin versant de l'ORGEVAL, à l'échel<strong>le</strong><br />
j ourna 1 i gr e<br />
(7) Y. CORMARY - M. ANGLES (HYD 7/71)<br />
Relations pluie-débit sur <strong>le</strong> bassin de la DIEGE à l'échel<strong>le</strong><br />
journalii2te et à l'échel<strong>le</strong> bi-horaire<br />
(8) Y. CORMARY - M. LARINIER (HYD 34/71)<br />
Etudes théoriques des processus d'infiltration, d'évaporation<br />
et de drain8ge. Bibiiovraphie. Schemas d'approche du cyc<strong>le</strong><br />
hydrologique<br />
(9) Y. CORMARY - M. LARINIER ( HYD 33/71) .<br />
Utilisation du catalogue des sols pour la prédétermination des<br />
parametres dans <strong>le</strong>s modè<strong>le</strong>s HûLTAN et HANON<br />
(10) G, GALEA (thèse de 3ème cyc<strong>le</strong> - 1972)<br />
Etude des relations pluie-débit sur <strong>le</strong> bassin de 1'ORT;EVAL<br />
(11) M. ANGLES (thèse de.3Eme cyc<strong>le</strong>,- 1972)<br />
Etude des relacions pluie-déhit sur <strong>le</strong> bassin de la DïEGE<br />
000
Eva po ro. t ion<br />
Inférieures aux<br />
po<br />
&<br />
s s ib i 1 i t 6 s à I nbu:, rp 1 ion<br />
i(épartition de ia<br />
pluia absorbée dnns'<br />
los rdsorvoira S ut 1.1<br />
7 PARAMETRES<br />
MODELE CWEC 273<br />
RESERVOIR<br />
Percolation<br />
x, ct x2 tarissrment dos r6servoirs II et G<br />
x oL x4<br />
3<br />
x" et x4<br />
3<br />
x7<br />
rél~artition do In pluio entre <strong>le</strong>s réservoirs S et H<br />
Supérieures aux<br />
po s s ibil i t é s d I ab sorption<br />
'I<br />
rui s s e 11 ein en t<br />
suparficiol<br />
aliaiontntion du r6servoir G (A partir du résorvoir H)<br />
rEductj on do 1lIc.T. P.<br />
2 po~ciinbt.rab détoriiiinaiit 1 'atsoi'ption DETA at CAM>IA.<br />
7<br />
1<br />
I<br />
EX UT01 RE
274<br />
U<br />
a<br />
10 PARAMETRES<br />
MODELE MERO<br />
-<br />
Débordement 1 rcolation dan<br />
. (Ministore de 1 lAgriculture ferahlion<br />
xl, x2, ‘x3 , x4 répartition des surplus antre G, Q et L flu&dBt$c)<br />
x58 “61 x7 , x8<br />
décrue des réservoirs G et Q<br />
X 9 1 xl0 rcmplissage maxirnum de U et L
MODELE DIEGE<br />
STOCK MOYEN<br />
des IO JOURS<br />
PRECEDENTS<br />
i<br />
4 I<br />
I<br />
7 PARAMETRES 8'<br />
. . XI ,x~,xj parambtroa de 'ddcruo<br />
' . I<br />
S?OCI< JOURNALIER<br />
275<br />
I ,<br />
),:" &&&cg<br />
%?brX5,X7 parambtroo de rolotion stock-ddbit r & '3.<br />
x6 .' ~arambtro de rdduotion do l(ETP<br />
' ' ' _".
276<br />
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c<br />
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rn<br />
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w<br />
U<br />
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8<br />
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W<br />
CI<br />
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wu1<br />
Ual<br />
ww<br />
c<br />
U<br />
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o<br />
w<br />
m<br />
4<br />
rl<br />
-4<br />
U<br />
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rn<br />
o<br />
w<br />
e<br />
m<br />
m<br />
o<br />
cl<br />
U<br />
e<br />
5<br />
ri<br />
ri<br />
al<br />
m<br />
u)<br />
rl<br />
1<br />
s<br />
c<br />
m<br />
m m<br />
W U<br />
P C<br />
o<br />
.-E<br />
mal<br />
U+<br />
o a<br />
H O<br />
U<br />
w<br />
w
MODELE CAEC<br />
CRITIOUE DCS DEBITS GENIRES<br />
imk1<br />
4 3 9<br />
B""Y0I journo1ior<br />
MODELE rmRo xviii<br />
critiquo dos rcsultats<br />
1<br />
ANNUEL JOURNALIER<br />
I<br />
DEDIT MINIMUM '<br />
ANNUEL JOURNALIER I i<br />
I<br />
MODELE CR@C CHIIIOUE i suits)<br />
a<br />
i m;<br />
2<br />
1<br />
M3DCLE PSCRO<br />
I . b<br />
67 68. I 69 70<br />
I<br />
C.I.,. extrapolation<br />
Y, P<br />
BASSIN DE L'HALLUE<br />
critique (suite) XIX<br />
MDDCLE ANNUEL<br />
"7s "f ;<br />
2 1<br />
1-<br />
L<br />
67 s'a 39<br />
I<br />
~<br />
ca!aqe<br />
I<br />
_j<br />
8<br />
.<br />
extrepo~etion<br />
I ! .& id<br />
t . y:-<br />
-<br />
I<br />
!<br />
..i, - .,<br />
b<br />
0 Qcalc ....<br />
2 77
zo T-<br />
278<br />
I . . . .<br />
I C M I W I I A I O N O<br />
c
0.9<br />
0.6 -<br />
0.14<br />
MODELE CREC METHODE DE POWELL<br />
MODULES ANNUELS<br />
PIRIWE 11-61<br />
&er=<br />
.* CakkJn<br />
DEBITS MOYENS MINIMUM$<br />
. SUR IO JOURS CONSECUTIFS<br />
BASSIN DE L'ORGEVAL<br />
;-<br />
8<br />
6 -<br />
4 -<br />
2 -<br />
o<br />
279<br />
7 -<br />
Io<br />
m'l 1 Periode 63-69<br />
16 -<br />
14 - ob:erves<br />
- CO caicu<strong>le</strong>s<br />
e----+--*<br />
Période de calage : 1964 - 1966<br />
MODELE CREC<br />
METHODE DE POWELL<br />
d3 s'4 6: s's i7 s'0 619 -<br />
b<br />
AMEß
EXTENSION OF RUNOFF RECORDS FOR SMALL CATCHMENTS IN SEMI-ARID REGIONS<br />
ABS TRACT<br />
S. H. Charania<br />
A ma<strong>the</strong>matical model developed by Thomas and Fiering is used<br />
for <strong>the</strong> purpose. This model is based on statistical princip<strong>le</strong>s to<br />
produce an unlimited record which has <strong>the</strong> same statistical properties<br />
as <strong>the</strong> original record. The statistical characteristics of data used<br />
are <strong>the</strong> means, standard deviations, volumes, skewness, variances and<br />
kurtosis of flows. Briefly, <strong>the</strong> main steps for <strong>the</strong> syn<strong>the</strong>sls of data<br />
availab<strong>le</strong> or <strong>the</strong>ir transformations are as follows:<br />
(al determina'ion of statistical properties;<br />
(b) determination of frequency distributions;<br />
(c> regression and correlation analysis of data (or <strong>the</strong>ir<br />
transformati0ns)with a detai<strong>le</strong>d study of <strong>the</strong> confidence<br />
limits of lines of regression and statistical parameters;<br />
(dl generation of random numbers;<br />
(el syn<strong>the</strong>sis of flows using <strong>the</strong> above model.<br />
Tn order to speed up <strong>the</strong> work and to get more accurate results,<br />
computers are used for almost all <strong>the</strong> calculatlons; various computer<br />
programmes (Fortran IV), required for <strong>the</strong> development of syn<strong>the</strong>sized<br />
data, are developed. The data tested on <strong>the</strong> model are from small<br />
catchments in semi-arid regions of Kenya and Australia.<br />
RESUME<br />
Un modè<strong>le</strong> mathématique a été mis au point par Thomas et Fier-<br />
nig. I1 est conçu pour produire une série illimitée ayant <strong>le</strong>s mêmes<br />
propriétés statistiques que la série d'observations disponib<strong>le</strong>. Les<br />
caractéristiques statistiques prises en compte sont: <strong>le</strong>s moyennes,<br />
<strong>le</strong>s dcarts types, <strong>le</strong>s variances, <strong>le</strong>s coefficients d'assymétrie et<br />
d'aplatissement des distributions de dlbits. Les principa<strong>le</strong>s étapes<br />
pour la syn<strong>the</strong>se ou la transformation des données disponib<strong>le</strong>s sont<br />
résumées ci-dessous:<br />
(a) Détermination des propriétés statistiques.<br />
(b) Détermination des distributions de fréquence.<br />
(c) Etude des regressions et corrélations concernant <strong>le</strong>s données<br />
ou <strong>le</strong>urs transformées; étude détaillée des limites de confiance<br />
des courbes de régression et des paramètres statistiques.<br />
Cdl Production de nombres au hasard.<br />
(e) Synthèse de séries de débits ä partir du modkïe élaboré.<br />
Presque toutes <strong>le</strong>s opkrations sont faites sur ordinateur afin<br />
d'obtenir des rgsultats rapides et précis. Plusieurs programmes écrits<br />
en fortran IY ont dtd mis au point pour l'obtention des donnees syn-<br />
thétiques. Le modè<strong>le</strong> a 6te appliqué 3 des données recueillies SUT de<br />
petits bassins des rbgions semi-arides du Kenya et dfAustralie.
282<br />
INTRODUCTION<br />
Hew and untapped souroes of water need to be developed and<br />
control<strong>le</strong>d to meet <strong>the</strong> inoreasing demand for water. For this we<br />
require long records of flows with exact sequence of hydrological<br />
events. Such long records are, unfortunately, not availab<strong>le</strong>, es-<br />
pecially in <strong>the</strong> semi-arid regions. In semi-arid regions of Africa<br />
and Australia usually only 20 to 30 years of records are availab<strong>le</strong>.<br />
Short reoords can be extended to any required period of time<br />
by simulation techniques whioh involve certain types of ma<strong>the</strong>matical<br />
or logical models that describe <strong>the</strong> behaviour of <strong>the</strong> system over ex-<br />
tended periods of real time.<br />
There are two variants of simulation, operational gaming<br />
and <strong>the</strong> Monte Carlo techniques. The Monte Carlo technique is <strong>the</strong><br />
one used in this paper.<br />
PRINCIPLES OF THE MODEL BY TñOMAS AND FIERING<br />
Data used<br />
The streamflows considered should be independent values<br />
and also of a long period. Therefore annual, monthly and daily<br />
streamflows can be oonsidered; annual flows are definitely in-<br />
dependent flows, but <strong>the</strong>y are often not used when <strong>the</strong> period of<br />
record is too short. Daily flows are not used if <strong>the</strong>y are not<br />
independent flow values. Monthly flow values generally satisfy<br />
<strong>the</strong> requirements of independence and adequate periods of record and<br />
are used.<br />
The model rewiremsnts<br />
The data used should be serially correlated and normally<br />
distributed. If <strong>the</strong> historical flows are not normally distributed<br />
appropriate transformations are or should be used to make <strong>the</strong>m so.<br />
In such cases, <strong>the</strong> syn<strong>the</strong>sized flows should be converted to <strong>the</strong>ir<br />
original form.<br />
The model<br />
The method assume8 that <strong>the</strong> streamflows are made up of two<br />
components, <strong>the</strong> deterministic and <strong>the</strong> random. Therefore <strong>the</strong> model<br />
(a reoursive equation) for unit time interval (one month) to generate<br />
flows is:
I _- QJ =mean flows of consecutiva months.<br />
Qí~+i) AND<br />
Q; AND Q (i+,) =flows during <strong>the</strong> months i and ;+f.<br />
BJ =regression coefficient.<br />
ri =norma2 random variate.<br />
aj+- =standard deviation of flaws.<br />
In computations j runs cyclically from 1 to 12 and <strong>the</strong><br />
index i from O to 12 times <strong>the</strong> number of record years to be<br />
generated minus 1.<br />
Determination of Ti<br />
283<br />
The area under <strong>the</strong> normal distribution curve is divided<br />
into 100 equal areas, each bounded by a vertical line paral<strong>le</strong>l to<br />
y-axis. The values of X at <strong>the</strong> boundary lines repreeent <strong>the</strong> bound-<br />
ary values of Ti for each area with equal probability. A typical<br />
vglue of Ti for each area is determined by calculating <strong>the</strong> mean of<br />
<strong>the</strong> boundary values. Out of 100 typical values of Ti one is se<strong>le</strong>cted<br />
for use in <strong>the</strong> model using <strong>the</strong> random numbers.<br />
CRITERIA FOR SELECTION OF THE CATCHEIEñTS<br />
Two important phrases in <strong>the</strong> tit<strong>le</strong> need interpretation as<br />
follows r<br />
(a)<br />
(b)<br />
'for a small reservoir' - The size of <strong>the</strong> catchment is restricted<br />
to between i30 to 1300 square kilometres.<br />
'a eemi-arid region' - The rainfall in <strong>the</strong> catchment is<br />
limited to <strong>le</strong>ss than 508 milimetres per annum.<br />
The oatchments se<strong>le</strong>cted for <strong>the</strong> paper are Wakefield River<br />
catchment in Australia and Kongoni River catchment in Kenya.<br />
NOTES ON THE SELECTED CATCHMENTS<br />
Wakefield river catchment<br />
The catchment with an area of 420 square kilometres lies<br />
in south Australia. The general relief of <strong>the</strong> catchment consists<br />
of undulating plains with ridges ranging from 150 to 305 metres.<br />
The highest point in <strong>the</strong> catchment is 600 metres.
284<br />
The stream, 40 kilometres long, drops Prom 500 to 220<br />
metres and has two major tributaries, <strong>the</strong> Pine creek and Skillolga<strong>le</strong><br />
creek. It is considered perennial but flashy.<br />
500 mm.<br />
Average rainfall on <strong>the</strong> catchment is between 400 to<br />
Temperatures are high in summer with normal annual range<br />
from 40 to 50 degrees farenheit (4.5OC to IOOC).<br />
Relative humidity is also high in summer with a peak of<br />
80 per cent.<br />
Soils in <strong>the</strong> catchment are normal ma<strong>le</strong>e soils of a pinkish<br />
brown colour and light sandy texture. These are rich in lime but<br />
poor in humus and phosphate. Rocks in <strong>the</strong> area are mainly lime-<br />
stone.<br />
Monthly flows and <strong>the</strong>ir totals from 1953 to 1968 are<br />
availab<strong>le</strong> as are <strong>the</strong> monthly rainfalls for <strong>the</strong> five stations in<br />
and around <strong>the</strong> catchment.<br />
Eongoni river catchment<br />
A catchment with an area of approximately i5 square kilo-<br />
metres on <strong>the</strong> north west slope of Mount Kenya. The source of <strong>the</strong><br />
river is at an altitude of 2600 metres and <strong>the</strong> station at 2000<br />
metres. The stream passes from <strong>the</strong> forest area into grasslands.<br />
The stream, with no major tributaries, is about 13 kilometres<br />
long and perennial.<br />
Average rainfall in <strong>the</strong> catchment is between 500 to 600<br />
milimetres, with heavy falls in April and November.<br />
It is hot in <strong>the</strong> catohment throughout <strong>the</strong> year with mean<br />
temperatures around 60 degrees farenheit (i5.5OC).<br />
Daily and monthly flovs with <strong>the</strong>ir totals from i932 to i969<br />
excluding 1954, 1955, and 1956 are availab<strong>le</strong> for <strong>the</strong> catchment and<br />
also <strong>the</strong> monthly rainfalls of three stations in and around <strong>the</strong> area.<br />
ANALYSIS<br />
Statistics of flows<br />
These assist finally in establishing <strong>the</strong> success of <strong>the</strong><br />
method used. The following properties are required for each<br />
month of <strong>the</strong> year in recordt
(a) total volumes of flows;<br />
(a)<br />
(c)<br />
(d)<br />
(e)<br />
(f)<br />
(g)<br />
means of <strong>the</strong> flows and/or <strong>the</strong>ir transformed values;<br />
confidence limits of <strong>the</strong> means;<br />
variance and standard deviations of flows and <strong>the</strong>ir<br />
confidence limits;<br />
skewness of <strong>the</strong> flows - <strong>the</strong>se show <strong>the</strong> degree of departure<br />
of <strong>the</strong> distribution of flows from symmetry. There is a<br />
dimension<strong>le</strong>es measure but that i8 not used. The absolute<br />
and relative skewness are caloulated;<br />
kurtosis of flows - this measures <strong>the</strong> degree of spread of<br />
<strong>the</strong> data and is calculated using moments;<br />
285<br />
trends of flows - <strong>the</strong> calculation is based on <strong>the</strong> princip<strong>le</strong><br />
of <strong>le</strong>ast squares.<br />
Frequency distribution<br />
This is a very important aspect of <strong>the</strong> procedure os <strong>the</strong>re<br />
is a condition that <strong>the</strong> data used should be normally distributed.<br />
There are various methods to check whe<strong>the</strong>r <strong>the</strong> data is normally<br />
distributed or not. Por <strong>the</strong> paper <strong>the</strong> following procedure was<br />
followed:<br />
(a><br />
(b)<br />
(o)<br />
check <strong>the</strong> skewness and kurtosis of <strong>the</strong> flow data;<br />
if <strong>the</strong> skewness is not equal to zero and kurtosis is not equal<br />
to 3.0 <strong>the</strong> data is transformed and <strong>the</strong> statistics rechecked.<br />
The txansformations are carried until <strong>the</strong> values of skewness<br />
and kurtosis are 0.0 and 3.0 respectively;<br />
<strong>the</strong> data or <strong>the</strong> transformed values are plotted on <strong>the</strong><br />
probability paper using plotting position ia/a. If <strong>the</strong><br />
data give a straight line it is aesumed that <strong>the</strong> values<br />
are normally distributed.<br />
Bemession and correlation analysis<br />
A statistical relationship between <strong>the</strong> flows in different<br />
months is determined. A linear regression equation Y - A + Bx is<br />
developed for each month; <strong>the</strong> constants A and B are calculated as<br />
follows t
286<br />
Confidence limits of <strong>the</strong> regression constant B and <strong>the</strong><br />
regression line are also determined.<br />
Coefficient of correlation (or covarianoe), an expression<br />
for <strong>the</strong> degree of scatter in regression is calculated using <strong>the</strong><br />
following formula:<br />
Generation of random numbers<br />
The random numbers are used to obtain <strong>the</strong> random component<br />
of <strong>the</strong> model. There are alternative methods to generate a sequence<br />
of random numbers. The latest technique is <strong>the</strong> generation of psuedorandom<br />
numbers by digital computer methods. This produoes numbers<br />
that are<br />
(a) uniformly distributedl<br />
(b) statistically independent;<br />
(c) reproduaib<strong>le</strong>;<br />
(a)<br />
non-repeating for any desired <strong>le</strong>ngth.<br />
The validity of random numbers is tested by different<br />
methods but frequency test method is <strong>the</strong> most convenient.<br />
For <strong>the</strong> model we require random numbers that are normally<br />
distributed. The uniformly distributed random numbers are <strong>the</strong>re-<br />
fore transformed to be normally distributed. These are <strong>the</strong>n ad-<br />
justed for <strong>the</strong> mean and standard deviation.<br />
Calculations for <strong>the</strong> generation of streamflows for <strong>the</strong><br />
two catchments are illustrated in Figures 1, 2 and 3.
DESCRIPTION OF RESULTS<br />
287<br />
Wakefield river catchment: Monthly flow values are used<br />
as historical records availab<strong>le</strong> for <strong>the</strong> simulation of records.<br />
These values are not normally distributed so <strong>the</strong>y have to be trans-<br />
formed. It is found that <strong>the</strong> logs of square roots of flows are<br />
normally distributed.<br />
The total flows show that in i4 years of records availab<strong>le</strong>,<br />
<strong>the</strong> maximum annual runoff is 10.25 a lo3 cum/D. The years 1954,<br />
1957, 1959, 1962, 1965 and 1966 have very low runoffs compared to<br />
o<strong>the</strong>r years. The lowest runoff observed is 2.30 x lo3 cum/D.<br />
August has <strong>the</strong> highest total monthly flow of 88.5 x lo3 cum/D,<br />
<strong>the</strong> lowest flow is in February of 2.98 x 103 cum/^.<br />
The means, variances and <strong>the</strong> standard deviations follow a<br />
pattern which is similar to that for <strong>the</strong> volumes; on <strong>the</strong> o<strong>the</strong>r<br />
hand, skewness and kurtosis have comp<strong>le</strong>tely different patterns:<br />
this could be due to accumulated errors and hence <strong>the</strong>se values<br />
are only taken as a guide. The confidence limits of <strong>the</strong> statistiical<br />
parameters are very useful in <strong>the</strong> final check.<br />
It is found that <strong>the</strong>reis very litt<strong>le</strong> scatter of values<br />
in regression and correlation analysis; <strong>the</strong> correlations are<br />
poor for a coup<strong>le</strong> of months but for <strong>the</strong> rest <strong>the</strong>y are very high<br />
indeed.<br />
Konsïoni river catchmentr In this oase also, <strong>the</strong> monthly<br />
flows are used. in <strong>the</strong> records of 30 years, <strong>the</strong> highest flow is<br />
recorded in Aprilr 41.5 x lo6 cum/D; <strong>the</strong> lowest is in February:<br />
zero flow.<br />
All statistical parameters, except for skewness and kurtosis,<br />
follow <strong>the</strong> same pattern as that for volumes; again, <strong>the</strong> skewness and<br />
kurtosis seem to have accumulated errors and cannot be relied upon.<br />
COBCLUSIOIPS<br />
The correlation of flows appear to be extremely good.<br />
The usefulness of <strong>the</strong> stochastic model for low flows, by<br />
Thomas and Fiering, is well demonstrated. 39 thie paper <strong>the</strong> model<br />
was used to simulate flows from <strong>the</strong> small experimental catchments<br />
of Kongoni river in Kenya and Wakefield river in Australia, and for<br />
both catchments, <strong>the</strong> parameters of syn<strong>the</strong>tic flows lie within <strong>the</strong><br />
95 per cent confidence limits of <strong>the</strong> historical flows.
288<br />
The results <strong>the</strong>refore indicate that <strong>the</strong> model can be<br />
successfully applied to <strong>the</strong> flows from small catchments in<br />
semi-arid regions. Howevex, when applying <strong>the</strong> above <strong>the</strong>ory, <strong>the</strong><br />
following points should be consideredt<br />
(a><br />
<strong>the</strong> initial flow values should be reliab<strong>le</strong> to avoid any<br />
accumulation errors in <strong>the</strong> final results#<br />
(b) <strong>the</strong> frequency distribution of th+ flaüdg if <strong>the</strong> initial<br />
flows are not normally distributed, appropriate transform-<br />
ations have to be used to normalise <strong>the</strong>m;<br />
(c 1<br />
<strong>the</strong> se<strong>le</strong>ction of a value of 't'r this is an important<br />
part of <strong>the</strong> model; random numbers are used to achieve <strong>the</strong><br />
purpose as explained in <strong>the</strong> paper;<br />
(a) negative flows: on syn<strong>the</strong>sis, negative flows are obtained.<br />
If <strong>the</strong> transformed values used initially are in <strong>the</strong> log form,<br />
<strong>the</strong>n no changes have to be made; if <strong>the</strong> initial transformed<br />
values are not in <strong>the</strong> log forra, <strong>the</strong>n <strong>the</strong> negative values have<br />
to be removed by replacing <strong>the</strong>m with zero values.<br />
References<br />
Fiering, M.B. (1961). Queing <strong>the</strong>ory and simulation in reservoir<br />
design. Journal of Jïyd. Div. B.S.C.E., Vol. 87, pp. 36-59.<br />
Thomas, H.A. and Fiering, M.B. (1962). Ma<strong>the</strong>matical syn<strong>the</strong>sis<br />
of streamflow sequences for <strong>the</strong> analysis of river basin by<br />
simulation. Chapter 12 in 'Design of water reaourceE systems'<br />
by Maas et al, Harvard.<br />
Beard, L.R. (1967). Hydrologic simulation in water analysis.<br />
Journal of Irrigation and Drainage Div. A.S.C.E.<br />
Smty, T.L. (1961). E<strong>le</strong>ments of Queing <strong>the</strong>ory. McGraw Hill <strong>book</strong><br />
company Inc. New York.
Figure i - Wakefield river catchment<br />
289
290<br />
,-<br />
Figure 2 - Analysis of Kongoni river flows.<br />
I J A S O N D -<br />
Fm-3<br />
no<br />
.O.,”.
I,<br />
f =(SKU(NGl)+ SKUiNG)<br />
12) (-10)<br />
I READ STREAXí1.J) I=it d.14 I<br />
1 READ ARE4 U:.3)E? hORHPL CURVE 1<br />
__-<br />
I<br />
DETERVINE VPLXS U-: STAri?AkD FiORtJAL LARIATE FOR<br />
AREA UNGE P.ORMAL CUR\.€ FRCM 0.01 TO O5<br />
I STCAAil) = VkRAAil)+r0.5 I<br />
'<br />
..<br />
ss- xxxx ciz,rr,<br />
CSLV(l2,l) =xxxxi1z,141<br />
Ir11<br />
P= 1 A<br />
CALL THE RCNWM iiü:A;ER GNERATCG<br />
..<br />
S = 16.70<br />
NG2=100-NG+l<br />
/21 (-10) NO=NG2- 1<br />
T =CU(UíNG2) t SKL'ih'GJ!)<br />
/2a<br />
Figure 3 - Flow ohart for <strong>the</strong> syn<strong>the</strong>sis of flows<br />
291<br />
-<br />
flG2=lGC - D<br />
T ;(i353 t SSL!;N13!:<br />
/z O)
ABS TRACT<br />
SIMULATION OF HYDROLOGICAL SAMPLES BY NATURAL WATER<br />
FLOW CHARACTER1 S TI CT ICs<br />
A.I.Davydova, G.P.Kaliniii<br />
The paper is concerned with long hidrological series which have<br />
a specified distribution and are characterized by basin annual flows<br />
ra<strong>the</strong>r than by random number sensors. The <strong>the</strong>oretical basis for cons<br />
truction of a numerical sequence is a combined analysis of mean an-<br />
nual flow probabilities for groups of basins (incomp<strong>le</strong>tely homoge-<br />
neous) se<strong>le</strong>cted to suit certain correlational estimates. By using va<br />
rious techniques <strong>the</strong> basic statistical characteristics of initial t i<br />
me distributions are taken into account, The <strong>le</strong>ngth of such series<br />
depends on <strong>the</strong> amount of data on flows of rivers under study which<br />
are grouped by certain criteria.<br />
Simulation of hydrological series by natural water flow characteristics<br />
does not require any method to allow for <strong>the</strong> effect of a<br />
preceding value on <strong>the</strong> law whereby a subsequent one is distributed.<br />
Error is not accumulated in simulated series as <strong>the</strong>y grow,<br />
On a examiné dans <strong>le</strong> rapport en question la technique de cons-,<br />
truction de longues séries hydrologiques 2 diktrìbùtion calculée par<br />
caractéristiques de l'écou<strong>le</strong>ment annuel de bassins isolés, et non<br />
pas au moyen de capteurs de nombres occasionnels. C'est bien l'ana-<br />
lyse réunie de probabilités de va<strong>le</strong>urs annuel<strong>le</strong>s moyennes de l'écou-<br />
<strong>le</strong>ment par groupes de bassins (pas tout 2 fait homogJnes1, choisis<br />
suivant <strong>le</strong>s estimations corrélatives determinées, qui sert de base à<br />
la construction de succession numériques. Les différents procédés<br />
mis en oeuvre permettent de tenir compte des é<strong>le</strong>ments statistiques<br />
principaux des distributions de départ tempore<strong>le</strong>s. La longueur de t e<br />
l<strong>le</strong>s séries est fonction du volume de l'écou<strong>le</strong>ment des f<strong>le</strong>uves du<br />
monde, attirés au calcul et choisis selon <strong>le</strong>s critères définis.<br />
En simulant <strong>le</strong>s réalisations hydrologiques par caractéristiques<br />
naturel<strong>le</strong>s de l'écou<strong>le</strong>ment fluvial, point n'est besoin de recourir à<br />
une tel<strong>le</strong> ou tel<strong>le</strong> méthode de prise en considération de probabilités<br />
de la va<strong>le</strong>ur précédente sur la loi de distrifiti'on de prohEY2litds<br />
de la va<strong>le</strong>u: suivante, Aucune atcumulation de l'erreur nia 12eu dans<br />
<strong>le</strong>s séries a simu<strong>le</strong>r au fur et a mesure du prolongement de cel<strong>le</strong>s-ci'.
2 94<br />
Variou8 ways to extend <strong>the</strong> initial hydrological data are<br />
used h water flow calculation and forecasting. Short time<br />
serie8 of hydrological observations of ten fail to give adequate<br />
c harac teristic 8 of river flow and ensure reliab<strong>le</strong> calculations.<br />
Hydrological data are exbended by probabilistic techni-<br />
ques, among sbich <strong>the</strong> Monte-Carlo method ia <strong>the</strong> most widely<br />
U8ed. The esseqfe of <strong>the</strong> latter is that artificial curves of<br />
probabilistic processes can be obtained by generation of random<br />
numbers distributed by a certûin law. Note that a model of <strong>the</strong><br />
flow process thus obtained should have hundreds or thousands<br />
of terms to include all basic features of <strong>the</strong> process probabili-<br />
ty distribution f mtions. The applichtion of this technique<br />
for hydrological calculations was thoroughly developed by<br />
G.G.Svanidee [9J . The range of application of <strong>the</strong> Monte-Carlo<br />
insthod was considerably extended by o<strong>the</strong>r soviet researchers[2,6].<br />
This paper deals with construction of long hydrological<br />
series with discrete time by using annual flow dharacteristics<br />
ra<strong>the</strong>r than rarrlom number aensors.<br />
Water flow Q at a certain cross-section of a river is<br />
regarded as a function of time t. Eet be <strong>the</strong> number of ar-<br />
bitrary time instants t, .. m , t, for an arbitrary number oî<br />
basins x alia n values of flow. Any specified value can be<br />
expressed in terms of <strong>the</strong> probabiliby that it d1L not be ex-<br />
ceeded. Probability distribution func tions for non-excees of<br />
annual flow values, ma<strong>the</strong>matical expectation and o<strong>the</strong>r statis-<br />
tical characteristics of each series x are given as<br />
B,(t,* P, , ... t, Pn)* dere P i8 <strong>the</strong> probability distribu-<br />
tion density associated with <strong>the</strong> fumtion Fx.<br />
The approach consists in consecutive combination of river<br />
flow probability distributions for individual basins. in dohg<br />
so various techniques are employed to inClde basic statistical<br />
characteristics (such as ma<strong>the</strong>matical expectation, coefficients<br />
variation, asymmetry aid correlation) of <strong>the</strong> initial time dist-<br />
ribUtiOM of <strong>the</strong> flow.<br />
Conditional probability distxibutions of a combined space<br />
am time sequerice are effectively used in *at is known in<br />
hydrology as <strong>the</strong> year-point- method in which effbiercy criteria<br />
for combination of time series have been developed and<br />
used in plotting a faired empirical distribution curve.<br />
Combined analysis m thods for incomp<strong>le</strong>tely homogeneous<br />
hydrolo&ical characteristics used in calculation of aiaximum<br />
flow hawe been developed br S.V.fulitsky and ?uí.B.bnkel' 181.<br />
In this case <strong>the</strong> <strong>the</strong>oretical scheme for <strong>the</strong> construction<br />
of a numerical sequence is a combined analysis of probabilities<br />
of mean annual flows that do not substantially vary over <strong>the</strong><br />
period covered for groups of basins (incomp<strong>le</strong>tely homogeneous)
se<strong>le</strong>cted by certain correlation estimates.<br />
295<br />
Time series Oi river flow characteristics are grouped by<br />
values of <strong>the</strong> coefficients of correlation (r) betaen annual<br />
flows ob~erved and calculated for pabs of successive years.<br />
A certain relation between flows of Mfvidual years is established.<br />
The differences in correlation coefficients of river<br />
flows indthin a basin depend on <strong>the</strong> physical nnn gec,rapnical<br />
conditions &er which <strong>the</strong> flow vas furmå. When rivers are<br />
grouped by intra-series c osrelation indices, <strong>the</strong> physical and<br />
etatistical homogeneity of <strong>the</strong> flow series se<strong>le</strong>cted is to<br />
some extent alloued for.<br />
A group may izlude basins differing in <strong>the</strong> water content.<br />
Modular coefficients are employed to make flow indices of large<br />
and small basins conmeasurab<strong>le</strong>.<br />
Three groups covering <strong>the</strong> r range from O through 0.45<br />
have been se<strong>le</strong>cted (Tab<strong>le</strong> I) with intra-series correlation as<br />
a criterion, Calculations for o<strong>the</strong>r values are equally possib<strong>le</strong>,<br />
!Bib<strong>le</strong> 1<br />
Rivers Grouped,<br />
by Averaging Intervals of Bhst Self-Correlation Coeff ic <strong>le</strong>nts<br />
Averaged-<br />
values, r<br />
+0.10 +o. 25 +O.W<br />
The first group CO rises rivers whose flow intra-series<br />
correlation is Or r ~'3.220 and iarludes 40 basins, chbfly<br />
in Europe and North America, with coeff icients of variation<br />
F ranging from 0.20 to 0.60,<br />
The second group (+0.21 C r 5 +0,35) includes 45 qivers,<br />
chiefly in Ada and &8t Europe , with variation coefficients<br />
ranging from 0.10 to 0.40, i.e. below <strong>the</strong> range for <strong>the</strong> first<br />
group.<br />
The third group is characterized by coefficients of eor-<br />
relation between annual flows of successive years ranghg<br />
from 0.36 through 0.45 and ircludes 28 flow series with coefw<br />
ficients of variation from 0.20 to 0.40. The flow series in<br />
tius interval and duration, The latter varies from 40 to 150<br />
years.<br />
Over recent years <strong>the</strong> statistical analysis of correlation<br />
coefficients between neighboring terms of a series as a fu- tion of tinis intervals has revea<strong>le</strong>d that %he de4pendeDC8 does<br />
exist in most cases 5,6] . Fur<strong>the</strong>rmore, studies [ 21 of inherent
296<br />
and random errors in calculating this coefficient by standard<br />
formulae show that <strong>the</strong> error8 may erneed 0.07. Values of first<br />
self-correlation coefficieubs r in grouping flow seriee are<br />
se<strong>le</strong>cted in a certain variation range , Tab<strong>le</strong> I.<br />
To prove or disprove interdependence of <strong>the</strong>se series in<br />
each group, interseries-correlation coefficients R were calculated<br />
for 1921-1955. For some series, <strong>the</strong> comelation was<br />
found to be as low as iO.30. In order to meet <strong>the</strong> criteriw<br />
of indepeideme of samp<strong>le</strong>s, a hydro&ogicAl LIiqueme generated<br />
should consist of flow series from a certain grouping, <strong>the</strong><br />
correlation coefficients of which are close to zero. This is<br />
one of <strong>the</strong> conditions for lack of simltaneity in flow variationa<br />
of rivers analysed in groups, which results in an illcrease<br />
of <strong>the</strong> overall data contained in combined hydrological series.<br />
It should be noted, however, that <strong>the</strong> application of nor-<br />
mal correlation techniques to flow variation etudies may pro-<br />
ve an improper practice because of possib<strong>le</strong> nonlinear rela-<br />
tions. The depiidemes between values observed in initial hyd-<br />
rological series can be curvilinear. Therefore for some hyärolo-<br />
@cal series Cl 1 <strong>the</strong> initial characteristics ofQthad to be<br />
normalized.<br />
krmalized series thus obtained vm-e used to calculate<br />
correlation coefficients R. These -re compared with <strong>the</strong> coef-<br />
f icients obtainsd from actual flow characteristics. Formulae<br />
of normal correlation were used to find <strong>the</strong> proximity betwgen<br />
<strong>the</strong> associated correlation coefficients. This compromise can<br />
be justified by <strong>the</strong> lack of more refined techniques of estimati<br />
ing relations betaieen gamma-distributed random values. The am-<br />
lysis has shown that correlation coefficients obtained directly<br />
from series of observations and normalised series are close;<br />
for this reason first values of R were used in <strong>the</strong> calculatfoas.<br />
Flow series with inter-series correlation coefficients<br />
below 0.3, were tabulated h each group. These data <strong>le</strong>ad to<br />
<strong>the</strong> assumption that relations between flow characteristics of<br />
<strong>the</strong>se basins are immaterial. !he averahed coefficients B and<br />
<strong>the</strong> standard values of <strong>the</strong> totali- of series for each group<br />
analysed are shown in Tab<strong>le</strong> 2.<br />
Tab<strong>le</strong> 2<br />
---_--------<br />
--_-- I<br />
Average Values of B and dR for <strong>the</strong> Groups of Rivers<br />
---<br />
-------- - ___- .-_- - ~<br />
>-- __-<br />
--.-<br />
-<br />
fiange of self-correlation coefficient variation<br />
--- I"_ -II _-_-_ - ---- --__ --------- --------<br />
O 5 r L +0.20 +0.21 5 r 5 +0.35 0.36LrC+O.45<br />
""_ . ----_-<br />
k2 0.129 0.134 0.157<br />
¿fi 0.062 O. 081 O. 076<br />
----__--------------_________II__ -<br />
_* .- . . ---- --- --_ - -- ----
297<br />
This tab<strong>le</strong> proves <strong>the</strong> absence of any substantial relation-<br />
ship betwen <strong>the</strong> flow series analysed, Now,in each group <strong>the</strong><br />
flow series are combined in simulated sequ~11ce8. Thus from <strong>the</strong><br />
first group (O L, r 6 +0.20) a sequerce of 841 terms m s formed,<br />
<strong>the</strong> second group 40.21 5 r5t0.35) gave a sequeme of 620 terms,<br />
and <strong>the</strong> third group (+0.365 r-L+0.45) yielded a sequerice of<br />
530 terms. These flow characteristics are transformed using<br />
<strong>the</strong> coordinates of <strong>the</strong> se<strong>le</strong>cted type of distributions into <strong>the</strong><br />
curves of <strong>the</strong> event probability security P. The ordinates of<br />
securie cumes are computed wieh an allowme for coefficients<br />
of variation and asymmetry of each flow series. In this case<br />
<strong>the</strong> structure of <strong>the</strong> sequence of segueities comguted for <strong>the</strong><br />
entire set of series is indeperdent of <strong>the</strong> flow variation and<br />
normal flow in individual basins. If flow series included in<br />
one sequeme are regarded as S&@pl0S of indepenient random va-<br />
lues, <strong>the</strong>n <strong>the</strong> corresponding values of flow security are also<br />
i ndep e nde nt rand om value s.<br />
For each series x security curves were computed for taro<br />
types of distributions<br />
1. Values of securities in a tuee-pararnter Kritsky-<br />
&&e1 gamma-distribution P . This distribution was obtained<br />
b replm ing <strong>the</strong> variab<strong>le</strong> x'%# <strong>the</strong> gama-distribution equation<br />
2'71. %e va iab<strong>le</strong> is related to <strong>the</strong> initial value by <strong>the</strong> equa-<br />
lity Z E a r6 , where a and b are parameters to be deter-<br />
mined on <strong>the</strong> basis of experimrnial eviüeme (corresponding to<br />
C and C ). The equation of <strong>the</strong> distribution cume for y is<br />
ix this tase :<br />
y(x) = -- aa a8 b<br />
b ,<br />
f (4<br />
where a s and r(a) is <strong>the</strong> symbol of gamma-fulirtion.<br />
-k<br />
The ordinates of <strong>the</strong> security curve expressed by this<br />
equation are always positive when y = O and P = 10%. The shape<br />
behaviour of this distribution permits aqy relations betraeen<br />
a and b, i,e.between <strong>the</strong> variation coefficient Cv and <strong>the</strong> asymetry<br />
coefficient Cs<br />
cE3 (--- = I, 1.5, 2.0, 2.5, ... 6 )<br />
Three-parameter gamm-distribution curves fit I@ 11 <strong>the</strong><br />
flow series with high values of Cv.<br />
2. P, obtained from generalized curves proposed by<br />
Kaliain pahose studies substantiated <strong>the</strong> generality of <strong>the</strong> probabilitp<strong>the</strong>oretical<br />
schematics dereby various samp<strong>le</strong>s of<br />
flow are formed1 41 . Using <strong>the</strong> formulae K = f (P,C,) ,<br />
%- and <strong>the</strong> flow data for many rivers, <strong>the</strong> depedemies<br />
K=%v<br />
I( S(Cv) , K5% = f (Cv> , etc. for annual and mximum flows were<br />
1%obtained<br />
separately for each value of secul$.ty (P=l%; P=5%, etc.
298<br />
The tab<strong>le</strong>s of ordfnates of generalized curves for distribution<br />
of annual flow esess probabilities were <strong>the</strong>n compi<strong>le</strong>d.<br />
3. Fapirical values of mcurities for <strong>the</strong> entire sequerce<br />
were obtained by <strong>the</strong> saression<br />
Pem = m<br />
-e-. qoos ,<br />
&tI<br />
where m is <strong>the</strong> point in a sequence of n numbers.<br />
Thus, me have two <strong>the</strong>oretical aqd one empifica1 distributions<br />
&ich make up <strong>the</strong> long hydrological series constructed.<br />
Bor lack ~î npace and large sizes of <strong>the</strong> tab<strong>le</strong>s, <strong>the</strong> values<br />
of simulated samplings cannot be shown this paper.<br />
Let us now proceed to comparison of simulated empifica1<br />
and <strong>the</strong>ore tical distributions of securiQ probabilities.<br />
For a series of 841 ternis, <strong>the</strong> representativi of <strong>the</strong><br />
security probability distributions obtairied mas es 3 mated for<br />
Pea (empirical), PLM (Kritsky-bnkel) and Paen (generalized)<br />
because a choice of <strong>the</strong> distribution curve type may considerably<br />
affect <strong>the</strong> distribution of flow values varyi- in <strong>the</strong><br />
..<br />
probability of excess.<br />
The following versions of estimates -re consideredt<br />
1) Uniform quanti<strong>le</strong> distribution of Pe,, PK-M, Pge, .<br />
2) Alternation of series of increased anfi decreased water<br />
contents in simulated series.<br />
3) Convergence of <strong>the</strong> distributions obtaiiigd with respect<br />
to standard deviation.<br />
4) Comparison of samp<strong>le</strong>d spectra by <strong>the</strong> distribution types.<br />
The first estimace was to reveal <strong>the</strong> homogeneity of strutture<br />
ad uniformity of security distribution of simulated flow<br />
series. !The number of hits of security curves in distributions<br />
Pen, PK-BiI, Pgep ws compared in terms of arbitrary quanti<strong>le</strong><br />
security probability distributions with respect to a fixed<br />
quanti<strong>le</strong> do not coim:i.de for <strong>the</strong> three types. In most cases,<br />
kïowver, <strong>the</strong> deviation from <strong>the</strong> average velue does not exceed<br />
10%. The greatest variations are associated dCii quanti<strong>le</strong>s of<br />
15-2O% security.<br />
The statistical mthod of serial tests 103 was used in<br />
<strong>the</strong> analgcsls of <strong>the</strong> probability of an event t periods of higher<br />
or lower wter content as against <strong>the</strong> normal one) by <strong>the</strong> types<br />
of distribution. In this method, for several. samp<strong>le</strong>s of 100<br />
terms each in this case, <strong>the</strong> number of years (e<strong>le</strong>ments) m, with<br />
flow values above <strong>the</strong> average om ard % with values below <strong>the</strong>average<br />
ani! calculated. E<strong>le</strong>msnts of <strong>the</strong> same kind bounded on<br />
both sides by e<strong>le</strong>ments of ano<strong>the</strong>r kind form series of U.
299<br />
Lf <strong>the</strong> values of U and <strong>the</strong> ma<strong>the</strong>matical expectation E differ<br />
substantially, <strong>the</strong>n <strong>the</strong> no-radom nature of <strong>the</strong> event is pro-<br />
ved. In this case <strong>the</strong> number of series should be greater or<br />
smal<strong>le</strong>r than E by a value exceeding 33, (dispersions) of <strong>the</strong><br />
samp<strong>le</strong>.<br />
With <strong>the</strong> probability of higher and lower mter content<br />
studied in this way, ma obtained <strong>the</strong> relations of series in-<br />
dices, expectation and dispersion which are given in Tab<strong>le</strong> 3.<br />
The simulated series were analysed by both <strong>the</strong> technique dec-<br />
cribed in th& pa er and <strong>the</strong> Monte+Darlo method applied to<br />
<strong>the</strong> Krìtsky-Menke? distribution curve.<br />
Tab<strong>le</strong> 3<br />
Serial Test Method Parameters for Different Types of<br />
Becurity Distribution of a Simulated Sequeme of 841 Terms<br />
9<br />
41 8 405 w5 4û6<br />
?i! 423 43 6 43 6 435<br />
U 3w 370 3 67 371<br />
E 421 421 42 1 421<br />
D 14.5 44.4<br />
CI------------------------------..<br />
14.4 14.4<br />
. -- -L _. . ---_-u--------<br />
The numerical values of U ani E are seen to differ for ail<br />
<strong>the</strong> three typs of diskibution by more than 3D an&%imilar<br />
ratios. The qualitative indices obtaimd may prove, firstly,<br />
that <strong>the</strong> distribution of e<strong>le</strong>ments above or below <strong>the</strong> norm tn<br />
series analysed is not random anã, secondly, that <strong>the</strong> parameter<br />
values I%, , 9, U, E, D obtained for samp<strong>le</strong>s of 841 terms<br />
indicate uniform conditions for <strong>the</strong> event probability in all<br />
<strong>the</strong> distribution types analysed; in o<strong>the</strong>r words, we have essentially<br />
several representations of <strong>the</strong> sanie process.<br />
In estimating <strong>the</strong> representativity of <strong>the</strong> hydro10 'cal<br />
series obtained we analyse <strong>the</strong> convergence of eqlrica8Land<br />
<strong>the</strong>oretical securities for <strong>the</strong> entire series. This estimate<br />
was obtained from <strong>the</strong> r.m.s. deviation between <strong>the</strong> <strong>the</strong>oretical<br />
and empirical distributions of securities, in per cent, for<br />
various quanki<strong>le</strong> intervals aad for <strong>the</strong> entire series. In all<br />
<strong>the</strong> intervals <strong>the</strong> r.m.6. deviation does not exceed 1.5.<br />
The security of securities curves of <strong>the</strong> simulated sequernes<br />
(Pig.1) obtaimd for distributions Pgen have a similar<br />
trend with oniy minor differences.
300<br />
On <strong>the</strong> <strong>who<strong>le</strong></strong>, comparison of empirical and <strong>the</strong>oretical cur-<br />
ves for tne entire sequsnce proves <strong>the</strong> representativeness of<br />
<strong>the</strong> hydrological sequence obtained.<br />
Spectral analysis proved useful in studying <strong>the</strong> structure<br />
of <strong>the</strong> simulated sequemes. For comparison of quantirbative<br />
indices, <strong>the</strong> initial iflormation was furnished by <strong>the</strong> same<br />
samp<strong>le</strong> of 841 terms obtained for Various modifications of<br />
annual flow distsibution. Mote that in calculation of spectral<br />
density sequences of securiw probability in per cent, were<br />
used. Computations were made for nonfeired aIid faired estimates<br />
of a spectrum using <strong>the</strong> eqression [3] I<br />
values of <strong>the</strong> self-correlation fumtion, HI - maximum<br />
where shift, sé ordinal number of <strong>the</strong> shift, K = 1,2, ... , m.<br />
'phis equation makes it possib<strong>le</strong> to obtain a dispersion<br />
spectrum for each frequency bad as percentage of tho total<br />
dispersion of <strong>the</strong> time series under etudy. A fairea estimate<br />
of dispersion spectral denSity was obtained by using <strong>the</strong> Hanning<br />
fairing weight fu= tionr -<br />
i= o, I, 2, ... , m<br />
D1 = O<br />
- Big.2 represents plots of non-faired , S@), and faired,<br />
S(p), spectra computed for <strong>the</strong> 841 terms aad frequencies<br />
f = O 0.1, ... , 0.2 &. If <strong>the</strong> delay of m includes <strong>le</strong>ss<br />
than i- of <strong>the</strong> samp e (80 terms), <strong>the</strong> shift of estimates of<br />
S(p), S(p) being small.<br />
Because <strong>the</strong> simulated series contain compositional space<br />
and time information on <strong>the</strong> flow, <strong>the</strong> spectrum of <strong>the</strong>se series<br />
follow9 <strong>the</strong> pattern of white noise and is scattered among all<br />
f requemies.<br />
Fluctuations of S(p) and s(p) in individual samp<strong>le</strong>s can be<br />
found by computing <strong>the</strong> nean value of <strong>the</strong> spectrum, <strong>the</strong> dispersion<br />
ad <strong>the</strong> r.m.8. error when frequencies are changed. The<br />
latter charac teristic is computed from thsore tical spectrum<br />
for which is taken a samp<strong>le</strong>d spectrum computed with <strong>the</strong> use<br />
of Tab<strong>le</strong> 4.<br />
Tab<strong>le</strong> 4<br />
Samp<strong>le</strong>d Spec tra Estimates<br />
--c-_---_---------------------- - -L---._*_-_-_---------<br />
Non-f aired Baked<br />
IO. 48 10.00 10.45 10.80 10.00<br />
!%$ersion 5.81<br />
6.02 4.18 5.01 4.72<br />
%ba* 0.660 0.782 O 0.660 0.782 0
These estimates are evideme of homogeneous structures<br />
of <strong>the</strong> simulated sequences for various groupings by <strong>the</strong> m e s<br />
oî distributions.<br />
301<br />
The analysis <strong>le</strong>ads to <strong>the</strong> coralusion that hydrological<br />
series constructed from natural flow characteristics contain<br />
a vast body information on combination and duration of periods<br />
differing in water content and may prove useful in estimating<br />
possib<strong>le</strong> ranges of flow variations for certain rivers* The<br />
proposed assessmnt of laws governing hydrological variations<br />
may prove helpful in tackling various *ter industry prob<strong>le</strong>ms.
302<br />
REFERENCES<br />
1. A<strong>le</strong>kseev, G.A. Ob'ektivq'e metoäy vyraklnivaniya i normali-<br />
zatdi c orrelyataionnykh myazey. Gidrometeoixdat ,<br />
Leningrad, 1971.<br />
2. Vodnoenergethheskiye raachyoty metodom Monte-Carlo.<br />
Ed. by Resnikovsky A.Sh. "Energiya" , Moscow,<br />
19690<br />
otts 7l<br />
3. Jenkins, %---<br />
G. Spectral Analysis and its Application.<br />
4. Halinin, G.P. Prob<strong>le</strong>my glogalnoy gidrologii. Gidrometeo-<br />
iedat, Leningrad, 7968.<br />
5. Kalinin, G.P., Davydova, A. I. Pro~tramtvenno-meiaeMoy<br />
analiz tsiklichmsti atoka rek. Vestnik MGU, ser.<br />
Geographiya, No.4, 1967.<br />
6. Kilasoniya, A.I. IC voprosu vybsra nachala gidrologicheskogo<br />
goda pri vodokhozyastven4ykh i vodnoenergetioheskikh<br />
raschyotakh. Trudy GrmMIZ, XVIII, 1969.<br />
7. Eritsky, S.N., &&el, LF. Vybor krivykh rasprede<strong>le</strong>niya<br />
veroyatnostey dl raschyotov rechnogo stoka.<br />
IZV. AN SSSR, OTg No. 6, 1948.<br />
8. Kritsky, S.N., Unkel, M.F. O mtodike sovmestnogo analiza<br />
nabluàenig ea stokom gidrologicheskikh skhodnykh<br />
basseynov. Trudy GGI, Issue 180, Gldrometeoizdat,<br />
Leningrad, 1970.<br />
9. Svanidze, G.G. Osnovy raschyota regulirovaniya rechnogo<br />
stoka metodom &nl;e-Carlo. Metsniereba, Tbilisi,<br />
1964.<br />
10. Yanko, Y. Bdstematiko-statisticheakiye Tablitsi,<br />
Gosstroyizdat, Mosc OW, 1961
303
304<br />
200 - I<br />
1<br />
I<br />
I . - 7- :*u<br />
0.1 0.t 0.8 0.4 0.6<br />
I 1 I I I I ' m<br />
10 N) 30 40 bD 60 Po 80<br />
7 I I -<br />
Y i .<br />
Fig.2. Simulated flow samp<strong>le</strong> spectra for a generalized cume:<br />
(a) non-faired spectrum S(p)i (b> faired spectrum i@),
"THE PREPARATION OF A DATA SET FOR HYDROLOGIC SYSTEM ANALYSIS"<br />
ABSTRACT<br />
M.J. Hamlin, B.Sc. D.I.C. M.ASCE. M.I.W.E.<br />
N.T. Kottegoda, B.Sc. Ph.D. M.I.C.E. M.I.W.E.<br />
The potential of a comp<strong>le</strong>x water resource system can often be<br />
determined only by <strong>the</strong> construction of a ma<strong>the</strong>matical model which is<br />
<strong>the</strong>n used to simulate <strong>the</strong> operation of <strong>the</strong> system. Where flow data<br />
is inadequate <strong>the</strong> input data for <strong>the</strong> model may present <strong>the</strong> most<br />
chal<strong>le</strong>nging aspect of <strong>the</strong> design. The use of data generation techniques<br />
to supp<strong>le</strong>ment historic records to obtain a comp<strong>le</strong>te data set, pseudo-<br />
historic in character, provides a possib<strong>le</strong> solution. Totally syn<strong>the</strong>tic<br />
data sets, based only on <strong>the</strong> statistics of existing flow data, can be<br />
produced to provide an alternative approach. The operation of <strong>the</strong><br />
computed model must be based on a re<strong>le</strong>vant time unit. In most schemes<br />
studied in <strong>the</strong> United Kingdom decisions need to be taken daily and it<br />
is <strong>the</strong>refore appropiate for <strong>the</strong> data to be in daily form. In spite of<br />
considerab<strong>le</strong> research effort methods for <strong>the</strong> generation of sequences<br />
of daily records are still inadequate and pentads have been commonly<br />
used. These represent <strong>the</strong> aggregated flows over a five day period and<br />
are used for generation purposes. The five daily totals are subsequently<br />
sub-divided to give daily flow values.<br />
RES UME<br />
On peut souvent déterminer <strong>le</strong> potentiel d'un système comp<strong>le</strong>xe<br />
de ressources en eau en se contentant de construire et d'utiliser un<br />
modè<strong>le</strong> mathématique de simulation. Quand <strong>le</strong>s données sur <strong>le</strong>s apports<br />
sont insuffisantes, l'établissement des données d'entrées peut repré-<br />
senter l'aspect <strong>le</strong> plus ardu du calcul. L'emploi des techniques de<br />
génération de données pour suppléer aux observations historiques et<br />
obtenir une série de données complète, méthode de caractère pseudo-<br />
historique, fournit une solution possib<strong>le</strong>. Des series tota<strong>le</strong>ment syn-<br />
thétiques établies uniquement à partir de la statistique des données<br />
existantes, peuvent être elaborées et fournir une autre solution. Le<br />
fonctionnement du modè<strong>le</strong> exige <strong>le</strong> choix d'une unité de temps appropriée.<br />
Dans la plupart des cas étudiés dans <strong>le</strong> Royaume Uni, des décisions<br />
doivent être prises journel<strong>le</strong>ment; il convient donc que <strong>le</strong>s données<br />
soient journalières. Malgré un effort de recherche considérab<strong>le</strong>, <strong>le</strong>s<br />
méthodes destinées à créer des séries d'observations journalières<br />
restent insuffisantes et on doit fréquemment utiliser des données<br />
pentadaires. On génère ainsi des écou<strong>le</strong>ments pentadaires que l'on<br />
subdivise ensuite pour obtenir des va<strong>le</strong>urs journalières.
306<br />
Introduction<br />
For <strong>the</strong> development of <strong>the</strong> potential resources of <strong>the</strong> Wye and<br />
Severn river basins it is necessary to investigate both <strong>the</strong> design of<br />
individual components of <strong>the</strong> system and <strong>the</strong> operation of <strong>the</strong>se individual<br />
componenils in an integrated system. The operational study requires <strong>the</strong><br />
construction of a comp<strong>le</strong>x simulation model.<br />
Thig paper describes only one<br />
aspect of <strong>the</strong> prob<strong>le</strong>m namely <strong>the</strong> provision of a set of compatib<strong>le</strong> data as<br />
input to <strong>the</strong> simulation model. The work was undertaken for <strong>the</strong> Water<br />
Resources Board who were! responsib<strong>le</strong> for <strong>the</strong> main operational study.<br />
Theoretically a multisite generation model represents <strong>the</strong> most<br />
attractive solution. In practice for daily or even five daily flows this<br />
presents prob<strong>le</strong>ms for which <strong>the</strong>re are no immediate solutions and alternative<br />
possibilities had to be sought. The first of <strong>the</strong>se involves <strong>the</strong> use of<br />
historic and pseudo historic flow records to build up a set of data for each<br />
of <strong>the</strong> nodal points of <strong>the</strong> system and this was <strong>the</strong> course which was adopted.<br />
A second possibility is <strong>the</strong> generation of wholly syn<strong>the</strong>tic data based on <strong>the</strong><br />
statistics of existing gauging stations where a structure is created of<br />
primary and secondary sites which are linked using a standard bi-variate model.<br />
The choice of time unit needs considerab<strong>le</strong> care and units of five<br />
days were agreed as being appropriate. However <strong>the</strong>se tend to over estimate<br />
<strong>the</strong> resources and for a detai<strong>le</strong>d consideration of critical periods it is<br />
necessary to sub-divide <strong>the</strong>se into five daily values preserving, in as far as<br />
is possib<strong>le</strong>,,all <strong>the</strong> re<strong>le</strong>vant statistics of <strong>the</strong> daily time series.<br />
The Physical System<br />
A diagramatic sketch of <strong>the</strong> two river basins is shown in figure 1.<br />
Both-rivers rise in mid Wa<strong>le</strong>s and flow South into <strong>the</strong> Bristol Channel. They<br />
are both used for water supply purposes but <strong>the</strong>re are substantial untapped<br />
resources and <strong>the</strong> possibilities of inter-basin transfers both between <strong>the</strong><br />
Wye and Severn but more importantly from <strong>the</strong> Severn towards South East<br />
England are of considerab<strong>le</strong> national interest. In <strong>the</strong> first instance <strong>the</strong><br />
ma<strong>the</strong>matical model was to be operated using data for <strong>the</strong> 38 years from<br />
1932-1969 inclusive. These contain a number of well known low flow sequences<br />
and in particular <strong>the</strong> periods 193314 and 1949. For this purpose it was<br />
necessary to produce compatib<strong>le</strong> sets of data for all <strong>the</strong> nodal points in <strong>the</strong><br />
system. Some of this data was availab<strong>le</strong> from historical records for <strong>the</strong> full<br />
period. At o<strong>the</strong>r points only partial records were availab<strong>le</strong> and <strong>the</strong>re were<br />
a number of points devoid of any flow records. The sketch identifies a number<br />
of typical points within <strong>the</strong> system although <strong>the</strong>se do not represent <strong>the</strong> total<br />
number for which data was obtained. The points are classified from A to F<br />
as follows.<br />
A)<br />
Records at <strong>the</strong>se stations had been col<strong>le</strong>cted for a number of years and<br />
existed for <strong>the</strong> full period 1932-1969.
B) Gauged records exist for <strong>the</strong> full period 1932-1969 but required<br />
adjustment to allow for <strong>the</strong> effect of reservoirs in upland sub-<br />
catchments.<br />
C)<br />
Records exist for only part of <strong>the</strong> period and need infilling to<br />
comp<strong>le</strong>te <strong>the</strong> 1932-1969 sequence.<br />
307<br />
D) Gauged records exist for only part of <strong>the</strong> period and require both<br />
infilling and adjustment to allow for reservoirs in <strong>the</strong> upland subcatchments.<br />
E) No records are availab<strong>le</strong> for <strong>the</strong>se catchments but <strong>the</strong>y can be deduced<br />
from neighbouring catchments using <strong>the</strong> relationship<br />
where % and A are <strong>the</strong> areas of catchments E and A<br />
A<br />
respectively I<br />
and RIE and RIA are <strong>the</strong> effective rainfalls of<br />
catchments E and A respectively<br />
This relationship is purely deterministic and suffers from lack of a<br />
stochastic component. However since <strong>the</strong> stochastic component cannot<br />
be evaluated <strong>the</strong>re are no means for including it.<br />
F) No record is availab<strong>le</strong> for this catchment and flow values can only be<br />
deduced from upstream catchments using <strong>the</strong> following relationship<br />
is <strong>the</strong> total catchment area down to point F<br />
Al, A2, A are <strong>the</strong> areas down to points, 1,<br />
2 and 3 respectively<br />
RIF is <strong>the</strong> effective rainfall on Area AF<br />
RI1, RI2, RI3 are <strong>the</strong> effective rainfalls on Areas<br />
Al, A2 and A respectively<br />
3<br />
Q,, Q2, Q, are <strong>the</strong> flows at points, 1, 2 and 3 respectively<br />
“19 n are <strong>the</strong> times of travel from points 1, 2 and 3<br />
2’ “3 to point F respectively.<br />
The adjustments necessary to allow for reservoirs in <strong>the</strong> upland<br />
sub-catchments were calculated using a simp<strong>le</strong> accounting technique which made
308<br />
allowance for <strong>the</strong> flow times from reservoir to <strong>the</strong> nodal point. The choice<br />
of generation model and method of infilling data required considerably more<br />
attention.<br />
The bivariate model<br />
As all records extend up to December 1969, bivariate syn<strong>the</strong>sis was<br />
used to infill <strong>the</strong> earlier parts of records where this was necessary1. For<br />
this purpose pentad data at a satellite station, S, where <strong>the</strong> flow record is<br />
short is linked to <strong>the</strong> data at a key station, Ky which has a long and<br />
reliab<strong>le</strong> record,through a bivariate model. If such a model is to be<br />
acceptab<strong>le</strong> statistically it should maintain <strong>the</strong> coefficient of cross<br />
correlation, rks, between <strong>the</strong> stations, <strong>the</strong> lag one serial correlation<br />
coefficients, rk and rs of <strong>the</strong> two stations and <strong>the</strong> five day seasonal means<br />
Mkj+l and Msj+l, and seasonal standard deviations, Skj+l and SSj+ly at <strong>the</strong> two<br />
stationS.in season j+i, 1 i j 5 73, corresponding to time t+i. A bivariate<br />
model can be expressed by<br />
In this particular application as syn<strong>the</strong>sized data was required at<br />
station S only, Kt+l represents <strong>the</strong> historical flow at station K at time t+l<br />
in pentad units and St+l represents <strong>the</strong> concurrent syn<strong>the</strong>tic flow at station<br />
S with St as its antecedent value. The variab)e Xt+l at time t+l is given by<br />
1<br />
xt+l = (B/Ssj) (St - Ms.) + i1 (1 - B2) ........................... .(2)<br />
3 t+l<br />
where qt+l is a series of non-autocorrelated numbers with zero mean and unit<br />
variance and a distribution which is estimated from <strong>the</strong> distribution of <strong>the</strong><br />
historical data at <strong>the</strong> satellite station, S. As shown by Fiering2, <strong>the</strong> three<br />
correlation coefficients, rk, rs, rks, should be incorporated in <strong>the</strong> constant<br />
B as follows:-<br />
2 -1<br />
B = (1 - rks) (rs - rk Xs2) ...................................... .(3)<br />
In <strong>the</strong> first instance <strong>the</strong> Clywedog data at Llanidloes (station 1 in<br />
Tab<strong>le</strong> 1) was extended using <strong>the</strong> bivariate model and <strong>the</strong> data from <strong>the</strong> Elan<br />
Val<strong>le</strong>y Key station. The cross correlation coefficient is 0.89 and it was<br />
found that a three parametric gamma distribution (Pearson type 3) fits <strong>the</strong><br />
Clywedog data. These parameters were estimated and read into <strong>the</strong> main program.<br />
When evaluating reservoir storages from syn<strong>the</strong>tic data at <strong>the</strong><br />
satellite station certain defects in <strong>the</strong> model were found. A visual comparison<br />
of concurrent historic flows at <strong>the</strong> two stations in dry years such as 1959<br />
showed similarities in <strong>the</strong> low flows which were not reproduced by <strong>the</strong><br />
bivariate model in its original form. This discrepancy was observed in <strong>the</strong>
pattern of low flows in <strong>the</strong> syn<strong>the</strong>tic data prior to 1959,e.&,in <strong>the</strong> critical<br />
dry period of 1933 and 1934. In Fig 2 a comparison is made between 6 years of<br />
historical data at a Key Station, Kt, and concurrent data at a satellite<br />
station, St, which is partly historical (years 4, 5 and U) and partly<br />
syn<strong>the</strong>sised (years 1, 2 and 3) without adjustment. The patterns of concurrent<br />
sets of high flows are random in both parts but <strong>the</strong> long runs of low flows<br />
in historical dry years were not maintained ina<strong>the</strong> syn<strong>the</strong>sised data. In a<br />
separate analysis seasonal values of serial correlation were computed but no<br />
significant differences were found. This may be attributed to <strong>the</strong> fact that<br />
in this climatological zone <strong>the</strong> times of commencement of <strong>the</strong> dry and wet<br />
seasons and <strong>the</strong> <strong>le</strong>ngths of seasons are highly stochastic variab<strong>le</strong>s.<br />
However, it nas found that when <strong>the</strong> flows are below a certain<br />
threshold value?defined with respect to <strong>the</strong> data at <strong>the</strong> key station, which is<br />
TT1 in Fig 2, <strong>the</strong> standardized values of flows at <strong>the</strong> two stations are<br />
highly correlated so that <strong>the</strong>se are nearly equal. The <strong>le</strong>vel TT1 was found by<br />
trial and error and <strong>the</strong> data generation was repeated with <strong>the</strong> new criterion<br />
that when <strong>the</strong> flow in <strong>the</strong> key station is below <strong>the</strong> threshold value, <strong>the</strong><br />
standardized flows at both stations are equal or alternatively <strong>the</strong>y are<br />
different by a very small random component.<br />
A fur<strong>the</strong>r refinement was included to establish a recession curve on<br />
runs of low flows. This was estimated empirically and an average value was<br />
read into <strong>the</strong> program so that <strong>the</strong> droughts just resemb<strong>le</strong> <strong>the</strong> historical<br />
droughts. An examination of <strong>the</strong> adjusted syn<strong>the</strong>sised data showed that<br />
differences in concurrent low flows such as those illustrated in Fig 2 were<br />
eliminated.<br />
Choice of probability distribution<br />
309<br />
In studies dealing with extreme flows and <strong>the</strong> persistence of high or<br />
low flows <strong>the</strong> probability distribution of <strong>the</strong> data is fundamental and except in<br />
<strong>the</strong> case of certain annual series <strong>the</strong> distributions of historical data are<br />
significantly different from <strong>the</strong> Gaussian or normal type. This is because <strong>the</strong><br />
distribution of river flows in a historical samp<strong>le</strong> is bounded by zero or a<br />
positive value at its <strong>le</strong>ft extremity and has a long tail on <strong>the</strong> side of<br />
increasing flows.<br />
For this reason, <strong>the</strong> distributions are said to be positively<br />
skewed. Fur<strong>the</strong>rmore, <strong>the</strong> coefficient of skewness tends to increase inversely<br />
with <strong>the</strong> time unit on which <strong>the</strong> series of data is based, which means that <strong>the</strong><br />
skewness in pentad data is more than,, in, say, monthly data.<br />
Kottegoda3 has shown that <strong>the</strong> incorporation of a gamma distribution<br />
in a monthly data generation model could yield realistic flow sequences of<br />
syn<strong>the</strong>tic data. In particular, reservoir storage requirements evaluated from<br />
some of <strong>the</strong>se sequences surpass that from <strong>the</strong> historical records. In <strong>the</strong> case<br />
of pentad data, a wider range of distributions to include Pearson's Type I,<br />
-111 and VI functions are necessary in order to model <strong>the</strong> empirical<br />
distributions4.
310<br />
The underlying generating process in <strong>the</strong> bivariate model used in<br />
this study is autoregressive of <strong>the</strong> type<br />
k 1<br />
xt = -1 a.X t-j + nt (1 - R2) ...................................... (4)<br />
J -1<br />
in which Xt, an autocorrelated cyc<strong>le</strong>-free seriespand nt, a random series,<br />
have zero mean, unit variance and non-identical distributions, aj are <strong>the</strong><br />
autoregressive parameters, k is <strong>the</strong> order of <strong>the</strong> process, R is <strong>the</strong> coefficient<br />
of multip<strong>le</strong> correlation and t is a point on <strong>the</strong> time sca<strong>le</strong>.<br />
In an unpublished study,a comparison is made between crossing and<br />
o<strong>the</strong>r properties in historical pentad data and data syn<strong>the</strong>sised using an<br />
autoregressive model and o<strong>the</strong>r models of recent origin in all of which <strong>the</strong><br />
skewness in <strong>the</strong> random component, rit, is varied. The crossing properties of<br />
particular interest in hydrology are shown in Fig 3. A sequence of 5 day<br />
river flows, R, which varies with time t is intersected at two <strong>le</strong>vels, viz.,<br />
RU, an upcrossing <strong>le</strong>vel above which <strong>the</strong> flow is higher than <strong>the</strong> mean flow<br />
and RD, a downcrossing <strong>le</strong>vel below which <strong>the</strong> flow is lower than <strong>the</strong> mean.<br />
Because with increasing time R rises above <strong>the</strong> <strong>le</strong>vel Ru at 4 points, <strong>the</strong>re<br />
are 4 upcrossings with respect to Ru. Similarly <strong>the</strong>re are 2 downcrossings<br />
with respect to RD. The mean <strong>le</strong>ngth of <strong>the</strong> horizontal bases of <strong>the</strong> 4 shaded<br />
areas above <strong>the</strong> Ru line is cal<strong>le</strong>d <strong>the</strong> mean surplus run <strong>le</strong>ngth. The total<br />
area of <strong>the</strong> shaded parts or sums of ordinates within <strong>the</strong>m is <strong>the</strong> total<br />
surplus run sum.<br />
In <strong>the</strong> same way <strong>the</strong> mean <strong>le</strong>ngth of <strong>the</strong> intercepts at <strong>the</strong><br />
RD <strong>le</strong>vel is <strong>the</strong> mean deficit run <strong>le</strong>ngth and <strong>the</strong> total area below <strong>the</strong> RD <strong>le</strong>vel<br />
is <strong>the</strong> total deficit run sum.<br />
The crossing properties of historical pentad data of <strong>the</strong> river Wye<br />
at Rhyader and syn<strong>the</strong>sised data based on an autoregressive model are shown<br />
in Fig 4. For this analysis <strong>the</strong> 33 year historical record is divided into<br />
3 equal samp<strong>le</strong>s of 11 years so that sampling variations, that are commonly<br />
found in data of this type could be seen.<br />
The properties of Syn<strong>the</strong>sised data<br />
are based on <strong>the</strong> means of results from ten 11 year non-historical sequences.<br />
If a Gaussian distribution is used in <strong>the</strong> model <strong>the</strong> numbers of downcrossings<br />
are far in excess of <strong>the</strong> numbers expected from <strong>the</strong> historical data. The<br />
ratio of skewness applied to <strong>the</strong> nt series to that estimated from <strong>the</strong><br />
historical data ought to be between 1.0 and 2.0 if a realistic number of<br />
downcrossings is to be obtained. A fur<strong>the</strong>r increase in skewness results in an<br />
undesirab<strong>le</strong> reduction in downcrossings. The necessity of providing f r<br />
greater skewness in <strong>the</strong> nt series than in <strong>the</strong> historical data was shown by<br />
Thomas and Fiering5 , who investigated <strong>the</strong> storage-yield relationship.<br />
approximation to <strong>the</strong> optimum skewness as obtained by analysing <strong>the</strong> independent<br />
residuals, Zt, where<br />
k<br />
Zt = Xt -j=l I: Xt-j .................................................... (5)<br />
3,4<br />
has been shown in o<strong>the</strong>r studies<br />
An
Ano<strong>the</strong>r point in favour of using <strong>the</strong> appropriate value of skewness<br />
is <strong>the</strong> large number of negative values generated when <strong>the</strong> skewness is too low<br />
as is <strong>the</strong> case if <strong>the</strong> normal distribution is used. On <strong>the</strong> o<strong>the</strong>r :-:A Li che<br />
skewness applied is excessive, not only ari. Fcg-tive values totally eliminated<br />
but <strong>the</strong> lowest flows are too high as can be seen in <strong>the</strong> top <strong>le</strong>ft hand diagram<br />
in Fig 4. In addition <strong>the</strong> numbers of downcrossings at various <strong>le</strong>vels are<br />
much fewer than those in <strong>the</strong> historical records.<br />
31 1<br />
The deficit run <strong>le</strong>ngths and run sums are also shown to be dependent<br />
on <strong>the</strong> skewness but when compared to <strong>the</strong> numbers of downcrossings, <strong>the</strong> change<br />
with respect to skewness is in <strong>the</strong> opposite manner. With regard to upcrossings<br />
at <strong>le</strong>vels above 50 mms., an increase in skewness tends to 'correct' <strong>the</strong><br />
syn<strong>the</strong>sizeddata but more skewness is required than for <strong>the</strong> low flows. This<br />
may be achieved by incorporating two distributions in <strong>the</strong> model, one for high<br />
flows and <strong>the</strong> o<strong>the</strong>r for low flows6.<br />
There seems to be no basic difference in <strong>the</strong> results if.<strong>the</strong> underlying<br />
model is changed from <strong>the</strong> autoregressive type. Skewness is <strong>the</strong> more<br />
important criterion. This is generally true of all pentad flow series from<br />
this climatological zone. Full results will be published in <strong>the</strong> near future.<br />
Infilling of data<br />
'<br />
The ten stations at which pentad data was extended are listed in<br />
Tab<strong>le</strong> 1. The number of years of syn<strong>the</strong>sised data range from 5 to 29 years.<br />
The two key stations used in <strong>the</strong> syn<strong>the</strong>sis are given. In certain cases <strong>the</strong><br />
choice of key station for <strong>the</strong> syn<strong>the</strong>sis is obvious because of close proximity,<br />
e.g., <strong>the</strong> Wye at ñhyader and Elan at Caban Coch. In o<strong>the</strong>r cases, <strong>the</strong> key<br />
station was se<strong>le</strong>cted on <strong>the</strong> basis of <strong>the</strong> best cross correlation coefficient.<br />
The type of distribution adopted was ascertained from a preliminary programme<br />
and it is seen from <strong>the</strong> tab<strong>le</strong> that Pearson's Type 3 or 1 functions provide a<br />
good fit to <strong>the</strong> data. The best fitting distribution is determined by <strong>the</strong><br />
Kolmogorov-Smirnov two samp<strong>le</strong> tests between syn<strong>the</strong>sised and historical data<br />
at <strong>the</strong> satellite stations. The cross correlation coefficien.ts between<br />
concurrent records at <strong>the</strong> key stations and satellite stations range from 0.79<br />
to 0.96 for <strong>the</strong> historical periods at <strong>the</strong> satellite stations. Comparative<br />
values were obtained in respect of <strong>the</strong> syn<strong>the</strong>sised data at <strong>the</strong> satellite<br />
stations and concurrent historical data at <strong>the</strong> key stations. The tab<strong>le</strong> also<br />
indicates that <strong>the</strong> lag one serial correlation coefficient is naintained by<br />
<strong>the</strong> model.<br />
The flow diagram in Fig 5 shows <strong>the</strong> basic structure of <strong>the</strong> programe<br />
"Two station 5 Daily" which was used for <strong>the</strong> infilling of <strong>the</strong> pentad data at<br />
<strong>the</strong> ten stations. The programme contains twelve subroutines. Subroutine Fxy<br />
ascertains <strong>the</strong> threshold value, below which it is desirab<strong>le</strong> to incorporate<br />
higher cross correlation in <strong>the</strong> standardised data in order to maintain<br />
realistic low flow sequences in <strong>the</strong> syn<strong>the</strong>sised data.
312<br />
Subdivision to daily data<br />
Any system which is subject to daily changesin <strong>the</strong> operating<br />
strategy must have daily inputs. Initial studies can be undertaken using<br />
sets of data having a pentad or monthly time unit but ultimately this time<br />
unit has to be reduced.<br />
The method used was developed by Green7, a Ph.D student at <strong>the</strong><br />
University of Birmingham, who was interested in a river pollution model for<br />
which daily river flow data was essential. Pentad data is broken down into<br />
daily flows by interpolation and to <strong>the</strong> interpolated values is added an error<br />
term whose purpose is to maintain <strong>the</strong> statistical characteristics of <strong>the</strong><br />
actual daily flow data. The long term characteristics including floods and<br />
drought sequences are preserved in <strong>the</strong> five daily model.<br />
The syn<strong>the</strong>sis is carried out in two stages. Actual daily data is<br />
accumulated into five-day averages which are <strong>the</strong>n subdivided into syn<strong>the</strong>tic<br />
daily values. The syn<strong>the</strong>tic values are compared with <strong>the</strong> actual daily values<br />
so that <strong>the</strong> success of <strong>the</strong> parameters used in <strong>the</strong> process and of <strong>the</strong> process<br />
itself can be measured. Once <strong>the</strong> composition of <strong>the</strong> error term can be<br />
adequately described <strong>the</strong> syn<strong>the</strong>tic five-day averages are broken down to yield<br />
syn<strong>the</strong>tic daily flows.<br />
Alternative data sets<br />
The use of an historic sequence of data enab<strong>le</strong>s <strong>the</strong> system to be<br />
operated so that a <strong>who<strong>le</strong></strong> range of possib<strong>le</strong> planning decisions can be<br />
investigated. Each plan is compared against alternative plans based on <strong>the</strong><br />
same set of input data. When an optimum plan has been identified it is<br />
desirab<strong>le</strong> that <strong>the</strong> operational decisions and <strong>the</strong>ir consequences should be<br />
tested agai.nst o<strong>the</strong>r possib<strong>le</strong> sequences of input data. The historic data can<br />
only be used to investigate what would have happened in <strong>the</strong> past. Since <strong>the</strong><br />
flows will never be repeated in an identical sequence <strong>the</strong>re is no possibility<br />
of <strong>the</strong> same set of decisions occurring in <strong>the</strong> future.<br />
For this purpose it is proposed that a number of syn<strong>the</strong>tic flow<br />
records should be produced. These will consist of compatib<strong>le</strong> sets of data<br />
for <strong>the</strong> two major stations in <strong>the</strong> system namely Bewd<strong>le</strong>y and Elan Val<strong>le</strong>y.<br />
When <strong>the</strong>se have been prepared, data at <strong>the</strong> o<strong>the</strong>r existing nodal points can be<br />
obtained using <strong>the</strong> basic statistics given in Tab<strong>le</strong> 1.<br />
Conclusion<br />
The procedure outlined in this paper shows how water resource systems<br />
may be designed in spite of <strong>the</strong> inadequacy of historic flow records. An<br />
extension of <strong>the</strong> method allows for wholly syn<strong>the</strong>tic sets of data to be<br />
prepared so that <strong>the</strong> future effect of possib<strong>le</strong> flow sequences can be studied.<br />
It is an essential feature of <strong>the</strong>se syn<strong>the</strong>tic sets that whilst on <strong>the</strong> one hand
<strong>the</strong>y reproduce <strong>the</strong> statistics of <strong>the</strong> historic data <strong>the</strong>y also, on <strong>the</strong> o<strong>the</strong>r<br />
band, contain sequences of rare events which are not disclosed in <strong>the</strong><br />
original record.<br />
Acknow<strong>le</strong>dgements<br />
The authors wish to acknow<strong>le</strong>dge <strong>the</strong>,financial assistance and<br />
encouragement of <strong>the</strong> Water Resources Board and also of <strong>the</strong>ir col<strong>le</strong>ague<br />
Dr. Kelway and Mrs. Ross who yere:responsib<strong>le</strong> for <strong>the</strong> major task of data<br />
handling.<br />
References<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
Hamlin, M.J. and Kottegoda, N.T., (1971) "Extending <strong>the</strong> record of <strong>the</strong><br />
Teme", Jour. Hydrology 12, pp 100-116.<br />
Fiering, M.B., (1964) "Multivariate technique for syn<strong>the</strong>tic hydrology"<br />
Jour. ASCE, 90, No HY5, pp 43-60.<br />
Kottegoda, N.T., (1970) "Statistical methods of river flow syn<strong>the</strong>sis<br />
for water resources assessment". Proc. Inst. Civ. Engrs., Supp<strong>le</strong>ment<br />
(xviii) , Paper 7339s.<br />
Kottegoda, N.T., (1972) "Stochastic five daily stream 'flow model",<br />
Jour. ASCE, 98, HY5, pp 1469-1485.<br />
313<br />
Thomas, H.A. dr. and Fiering, M.B., (1963) "The nature of <strong>the</strong> storage-<br />
yield relationship", Operations Research in Water Quality Management,<br />
Chapter 1 of Report of <strong>the</strong> Harvard Resources Group to <strong>the</strong> U.S. Pub. Health<br />
Service, Cambridge, Mass.<br />
Kottegoda, N.T., (1972) "Flood evaluation - can stochastic models provide<br />
an answer?" Int. Symp. on Uncertainties in Hydrologic and Water Resource<br />
Systems, Tucson, Arizona.<br />
Green, N.M., (1973) "A syn<strong>the</strong>tic model for daily streamflow", Jour. of<br />
Hydrol, (in press).
314
Lake<br />
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FIG. 1<br />
catchment 1<br />
Not to sca<strong>le</strong><br />
315<br />
Avon
3-<br />
31 6
L a<br />
t<br />
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317
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31 8
[EVALUATE HARMONIC FITTED MEANS AND STD. DEVS. Subroutine Frier P'<br />
t<br />
IREAD DATA AT SATELLITE STATION 1<br />
t<br />
319<br />
~- __.<br />
TATION I<br />
t<br />
[COMPUTE SERIAL CORRELOGRAM OF RAW DATA Subroutine Correl<br />
+ 4<br />
ICOMPUTE SKEWNESS COEFFICIENTS OF FIVE<br />
[COMPUTE CROSS CORRELATION COEFFICIENTS. Subroutine Sat I<br />
t<br />
[FIND RUNS BELOW AND ABOVE MEAN. Subroutine Runs I<br />
t<br />
[OPTIONAL - STALL ANALYSIS ON HISTORICAL DATA. Subroutine Stall 1<br />
[OPTIONAL - DURATION ANALYSIS. HISTORICAL. Subroutine Dur<br />
GENERATE & TRANSFORM RANDOM NUMBERS TO PEARSON TYPE OR LOGNORMAL<br />
CHI SQUARE TEST Subroutine Rannor<br />
IDURATION ANALYSIS ON HISTORICAL AND SYNTHESIZED DATA. Subroutine Dur. 1<br />
1<br />
t<br />
t '\<br />
t<br />
J<br />
GENERATE DATA USING CONCURRENT DATA AT KEY STN. SET THRESHOLD VALUE<br />
FOR REALISTIC LOWFLOWS.. Subroutine f XY<br />
I<br />
COMPUTE MEANS, STD. DEVS., SKEW COEFFICIENTS OF SYNTHESIZED DATA.<br />
FIND HIGHEST AND LOWEST VALUES. Subroutine Percen.<br />
t<br />
IPUNCH CARD AND LINEPRINTED OUTPUT OF SYNTHESIZED DATA I<br />
b<br />
[ CO>íF'üTE CROSS CORRELATION COEFF. OF SYbTHECIZED AND M W DATA. Subroutine Sat11<br />
[OPTIONAL - STALL ANALYSIS ON SYNTHESIZED DATA. Subroutine Stall<br />
ISERIAL CORRELOGRAM OF SYKTHESIZED DATA. Subroutine Correl<br />
t<br />
[COMPARE RUNS IN SYNTHESIZED AND HISTORICAL DATA. Subroutine Runs<br />
I<br />
1<br />
IKOLMOGOROV-SMIRNOV TWO SAMPLE TEST OF SYNTHESIZED AND HISTORICAL DATA.<br />
I Subroutine Smir<br />
+<br />
I<br />
'I<br />
I<br />
1<br />
*r
POTENTIAL APPLICATION OF BAYESIAN TECHNIQUES FOR PARAMETER<br />
ESTIMATION WITH LIMITED DATA<br />
ABSTRACT<br />
Roberto L. Lenton, John C. Schaake Jr.<br />
and Ignacio Rodriguez-Iturbe<br />
Department of Civil Engineering<br />
Massachusetts Institdte of Technology<br />
The use of Bayesian Techniques for parameter estimation can<br />
potentially improve <strong>the</strong> availab<strong>le</strong> limited hydrologic data by taking<br />
into account not only <strong>the</strong> information contained in <strong>the</strong> historical<br />
samp<strong>le</strong>, but also all <strong>the</strong> information coming from o<strong>the</strong>r sources, both<br />
objective and subjective. At <strong>the</strong> same time, project economics can be<br />
considered by <strong>the</strong> use of a loss function which specifies <strong>the</strong> serious<br />
ness of choosing an estimate which is not <strong>the</strong> true one. For examp<strong>le</strong>,<br />
<strong>the</strong>se techniques can be applied to <strong>the</strong> estimation of <strong>the</strong> parameters<br />
of a first order autoregressive model. Moreover, if <strong>the</strong> hydrologist<br />
is willing to make certain simplifying assumptions and limit his pro<br />
b<strong>le</strong>m to <strong>the</strong> estimation of only <strong>the</strong> autocorrelation coefficient, <strong>the</strong>n<br />
comparatively simp<strong>le</strong> estimators result. The comparison between Bayes<br />
and Classical estimators for p on <strong>the</strong> basis of <strong>the</strong> risk function and<br />
<strong>the</strong> expected risk shows that <strong>the</strong> Bayes estimator is considerably mo-<br />
re advantageous, especially when <strong>the</strong> samp<strong>le</strong> is of a limited duration.<br />
RESUMEN<br />
La utilización de técnicas Bayesianas para la estimación de<br />
parámetros puede mejorar la informacibn limStada existente, aï tener<br />
en cuenta no sólo la información contenida en la muestra histórica<br />
sino también toda la información proveniente de otras fuentes tanto<br />
objetivas como subjetivas, A la vez, se pueden considerar los aspec-<br />
tos económicos mediante la utilización de una función de pérdidas<br />
que especifica la seriedad de escojer un esti’mado que no es el verda<br />
dero. Por ejemplo, se pueden utilizar estas têcnicas para la estima-<br />
ci6n de los parámetros de un modelo autoregresivo de primer orden.<br />
Mas afin, si el hidrólogo está dispuesto a .realizar ciertos supuestos<br />
simplificadores, y a limitar su prob<strong>le</strong>ma a la estimación del coefi-<br />
ciente de autocorrelación solamente, entonces se pueden obtener esti<br />
madores relativamente simp<strong>le</strong>s. La comparación entre los estimadores<br />
Bayesianos y clbsicos para el parametro p, en Base a la función de<br />
riesgo y al valor esperado del riesgo, demuestra que el estimador Ba<br />
yesiano presenta considerab<strong>le</strong>s ventajas, especialmente cuando la mues<br />
tra es de una duración limitada.
322<br />
INTRODUCTION<br />
Various models have been proposed in <strong>the</strong> past for modelling <strong>the</strong><br />
stochastic nature of <strong>the</strong> hydrologic processes; <strong>the</strong> purpose of using <strong>the</strong>se<br />
models has been to aid in making better investment and management decisions<br />
regarding Water Resource projects.<br />
Two factors are decisive in <strong>the</strong> choice of a model: <strong>the</strong> availab<strong>le</strong> in-<br />
formation and <strong>the</strong> prob<strong>le</strong>m to be solved. Once <strong>the</strong> model has been chosen, how-<br />
ever, <strong>the</strong>se two factors must continue to be considered. The only control <strong>the</strong><br />
hydrologist has over his model is in <strong>the</strong> estimation of its parameters. Hence<br />
<strong>the</strong> estimation technique should both use <strong>the</strong>%vailab<strong>le</strong> information in <strong>the</strong><br />
most efficient way, and in some manner take into account <strong>the</strong> prob<strong>le</strong>m at hand,<br />
Unfortunately, <strong>the</strong> classical methods of estimation do nei<strong>the</strong>r, and in this con-<br />
text have two important defects:<br />
1. They can only take into account <strong>the</strong> information contained in <strong>the</strong><br />
historical samp<strong>le</strong>. Evidently <strong>the</strong> hydrologist is limiting himself by not in-<br />
troducing information from o<strong>the</strong>r sources which could reduce <strong>the</strong> uncertainties<br />
of estimation.<br />
2. They produce values that are independent of <strong>the</strong> economic consequences<br />
of erroneous estimates. It appears evident that it would be more rational<br />
to assess <strong>the</strong> opportunity losses to be undergone by estimating a parameter<br />
erroneously, and <strong>the</strong>n use as a criterion for estimation <strong>the</strong> minimization of<br />
those expected opportunity losses.<br />
Bayes Theorem seems to provide a framework for approaching <strong>the</strong> prob<strong>le</strong>ms<br />
that have been indicated. The first point is taken into account by providing<br />
an "a priori" distribution on <strong>the</strong> parameter of interest. This prior<br />
distribution encompasses all infomation that is not in <strong>the</strong> historical samp<strong>le</strong>,<br />
providing an assessment of both <strong>the</strong> most likely values and <strong>the</strong> degree of uncertainty<br />
of <strong>the</strong> parameter in question. The prior information enters <strong>the</strong><br />
estimation procedure via Bayes Theorem, which expresses simply that<br />
where<br />
P (Om a p (O) p (Y/@) (1)<br />
Y = Vector of samp<strong>le</strong> observations<br />
O = Parameter<br />
p(O/Y) = "Posterior" pdf of O I given Y<br />
p(O) = Prior pdf of O<br />
p(Y/O) = Likelihood function for <strong>the</strong> parameter O .<br />
The posterior pdf now replaces <strong>the</strong> likelihood function as <strong>the</strong> means<br />
for making inferences about <strong>the</strong> parameter, and as a means for taking into<br />
account <strong>the</strong> second prob<strong>le</strong>m that was noted - i.e., <strong>the</strong> economic consequences<br />
of erroneous estimates.<br />
There are iïitroduced by means of a loss function R (O,@), which specifies<br />
<strong>the</strong> opportunity loss which is undergqne when O, <strong>the</strong> true value of <strong>the</strong><br />
parameter, is erroneously estimated as O . Hence <strong>the</strong> Bayesian criterion<br />
A
consists of choosing <strong>the</strong> value<br />
losses :<br />
or<br />
o =<br />
A h<br />
h<br />
0 that minimizes <strong>the</strong> expected opportunity<br />
min [a(@,@)] (2)<br />
o<br />
6 = min a(;,@) p (@/Y) do<br />
u n<br />
where fi is <strong>the</strong> region of <strong>the</strong> parameter 0 .<br />
(3)<br />
323<br />
Since it is evident that <strong>the</strong> Bayes approach depends ra<strong>the</strong>r heavily<br />
on two factors, <strong>the</strong> prior distribution and <strong>the</strong> loss function, <strong>the</strong>se two points<br />
will be discussed below in greater detail.<br />
THE PRIOR INFORMATION<br />
If <strong>the</strong> prior pdf is to adequately represent all information o<strong>the</strong>r<br />
than that contained in <strong>the</strong> samp<strong>le</strong>, it must be assessed with great care. The<br />
first question that must be answered is <strong>the</strong> source of ,this non-samp<strong>le</strong> infor-<br />
mation.<br />
One reasonab<strong>le</strong> source of information could be a col<strong>le</strong>ction of past<br />
records from o<strong>the</strong>r river basins on <strong>the</strong> value of <strong>the</strong> parameter of interest.<br />
The analysis can be performed on <strong>the</strong> frequency histogram of observed values,<br />
by fitting a known distributional form to it. This, of course, is only valid<br />
for non-dimensional parameters (such as <strong>the</strong> coefficient of variation or <strong>the</strong><br />
first order autocorrelation coefficient), <strong>the</strong> basic assumption being that<br />
<strong>the</strong>re are physical reasons which tend to make some values of <strong>the</strong> parameter<br />
more likely than o<strong>the</strong>rs, as ref<strong>le</strong>cted in <strong>the</strong> col<strong>le</strong>ction of records. As a<br />
first approximation, <strong>the</strong> hydrologist could analyze world-wide data; if he<br />
is not satisfied, he could regionalize this information or classify it, taking<br />
into account only rivers of similar characteristics to <strong>the</strong> one he is studying.<br />
Ano<strong>the</strong>r source of information could be <strong>the</strong> analysis of physical cha-<br />
racteristics of river basins which are related to <strong>the</strong> parameter of interest.<br />
By regression on <strong>the</strong>se characteristics, a measure of <strong>the</strong> mean and variance<br />
of <strong>the</strong> parameter can be obtained, and a probability distribution fitted to<br />
it. This is <strong>the</strong> approach used by Wood (1973), in ano<strong>the</strong>r paper presented at<br />
this conference,to derive prior information on exceedance flows. It also<br />
might be possib<strong>le</strong> to derive a prior distribution on <strong>the</strong> basis of <strong>the</strong>oretical<br />
considerations if <strong>the</strong> relationship between <strong>the</strong> parameter and <strong>the</strong> physical<br />
characteristics of <strong>the</strong> basin can be model<strong>le</strong>d. The model would give a measure<br />
of <strong>the</strong> mean value to assign to <strong>the</strong> prior distribution, whilst <strong>the</strong> variance<br />
must be obtained by assessing <strong>the</strong> reliability of <strong>the</strong> hypo<strong>the</strong>sized model.<br />
Finally, <strong>the</strong> hydrologist's judgement and experience must necessarily<br />
enter <strong>the</strong> picture. When a "data-based'' prior is used, <strong>the</strong> exact form of <strong>the</strong>
324<br />
prior pdf is tempered by <strong>the</strong> hydrologistvs subjective assessment; when<br />
no data is availab<strong>le</strong>, <strong>the</strong> hydrologist can approximate a prior distribution<br />
on <strong>the</strong> parameter from "introspection, casual observation or <strong>the</strong>oretical ob-<br />
servations" (Zellner, 1971); he must be extremely careful, however, in<br />
determining that <strong>the</strong> dispersion in his prior pdf properly represents his<br />
true state of know<strong>le</strong>dge or ignorance,<br />
THE LOSS FUNCTION<br />
The loss function k(6,o) has been defined as <strong>the</strong> function that<br />
specifieg <strong>the</strong> opportunity loss that obtains when <strong>the</strong> hydrologist "acts" as<br />
though O were <strong>the</strong> real parameter value, when in fact O is. These<br />
opportunity losses represent <strong>the</strong> difference between <strong>the</strong> benefits actually<br />
to be obtained from a given Water-Resources project and <strong>the</strong> greater value<br />
that would have been realized had <strong>the</strong> true parameter value been known, It<br />
is seen from this definition that <strong>the</strong> economic losses are a consequence of<br />
<strong>the</strong> decisions or ('actions'' that <strong>the</strong> hydrologist recommends on <strong>the</strong> basis of<br />
his estimate; for examp<strong>le</strong>, <strong>the</strong>se decisions could consist of constructing a<br />
reservoir of a certain storage capacity or, in a more comp<strong>le</strong>x system, of<br />
constructing a series of reservoirs, irrigation sites, power stations and<br />
diversions of a certain size or capacity,<br />
Formally, <strong>the</strong> loss function can be obtained in <strong>the</strong> following manner.<br />
(Pratt, Raiffa and Schlaifer, 1965). LetA<strong>the</strong>re be a set A of acts a<br />
(which are a consequence of <strong>the</strong> estimate 8 ), a set fi of parameter<br />
values O , and a value function (e.g. National Income Net Benefits) Vt<br />
with values vt(a,O). For every parameter point, <strong>the</strong> greatest of <strong>the</strong>se<br />
values is mpx vt (ayo). Therefore <strong>the</strong> opportunity loss of any particular<br />
act a given that <strong>the</strong> parameter is O is<br />
!L(a,O) max vt(a,O) - vt(a,O)<br />
a<br />
where a, is <strong>the</strong> optimal act a for O .<br />
Finally, if<br />
be expressed<br />
as is <strong>the</strong> optimal act a for 6 , <strong>the</strong>n (5) can<br />
These ideas are illustrated in Figure 1.<br />
Thus, to determine <strong>the</strong> structure of his loss function, <strong>the</strong> hydro-.<br />
logist must first evaluate <strong>the</strong> value functions associates with his prob<strong>le</strong>m.<br />
Application to a Reservoir Sizing Prob<strong>le</strong>m<br />
A simp<strong>le</strong>, though common prob<strong>le</strong>m in Water Resources Engineering is <strong>the</strong><br />
determination of <strong>the</strong> optimal storage capacity of a reservoir to provide re-
325<br />
gulated flow to an irrigation area of a given size with a given target demand.<br />
The action to be taken in this case consists of constructing <strong>the</strong> reservoir of<br />
a certain capacity S; <strong>the</strong> value functions could consist of <strong>the</strong> net benefits<br />
derived from <strong>the</strong> irrigation system.<br />
These can be computed on <strong>the</strong> basis of<br />
<strong>the</strong> long-term benefits derived from <strong>the</strong> operation of <strong>the</strong> irrigation site, <strong>the</strong><br />
cost of <strong>the</strong> reservoir and of <strong>the</strong> irrigation system, and <strong>the</strong> short term losses<br />
which occur when <strong>the</strong> water supplied by <strong>the</strong> reservoir is insufficient to meet<br />
<strong>the</strong> irrigation target.<br />
As th2 reservoir size is increased, <strong>the</strong> short term losses decrease at<br />
<strong>the</strong> expense of reservoir Costs, and <strong>the</strong>refore <strong>the</strong> prob<strong>le</strong>m essentially consists<br />
of a trade-off between <strong>the</strong>se two fac'tors.<br />
A discrete set of value functions can be easily determined in this<br />
case through simulation for a discrete number of parameter values<br />
(oi, i = l,Z,...,n) and for a discrete number of design storage capacities,<br />
or actions, (aj, j = 1,2,...,n). Using <strong>the</strong> assumed flow mod21 with parameter<br />
oi, and setting <strong>the</strong> reservoir capacity at aj, <strong>the</strong> system net benefits<br />
Vt(Oiy aj) can be determined after simulating for an appropriate period of<br />
years.<br />
This technique was applied to determine <strong>the</strong> loss function for a hypo<strong>the</strong>tical<br />
prob<strong>le</strong>m of determining <strong>the</strong> optimal storage capacity of a reservoir<br />
to supply an irrigation system in <strong>the</strong> Rio Colorado Basin in Sou<strong>the</strong>rn Argentina<br />
(see Lenton, Rodriguez-Iturbe, and Schaake, 1973) ; <strong>the</strong> assumed model was <strong>the</strong><br />
first-order normal autoregressive model with parameters p ,a2 and p . The<br />
loss function for p , assuming and u' equal to <strong>the</strong> samp<strong>le</strong> values, is<br />
shown in Figure 2.<br />
It should be pointed out that <strong>the</strong> loss function on all 3 parameters<br />
of <strong>the</strong> model could be determined using exactly <strong>the</strong> same technique, although<br />
a 6-dimensional matrix would be required.<br />
EXAMPLE APPLICATION: THE FIRST ORDER AUTOREGRESSIVE MODEL<br />
The Bayesian methodology for parameter estimation has been applied<br />
quite successfully to <strong>the</strong> case of <strong>the</strong> first-order normal autoregressive model<br />
by Lenton, Rodriguez-Iturbe and Schaake, (1973), <strong>the</strong> basic results of which<br />
are summarized below.<br />
The first order normal autoregressive process may be expressed as<br />
where y, = annual flow at year t<br />
w = independent normally distributed random-variab<strong>le</strong> with zero<br />
mean and unit variance
326<br />
p = mean of <strong>the</strong> process<br />
u2 = variance of <strong>the</strong> process<br />
p = first-order autocorrelation coefficient of <strong>the</strong> process<br />
The Bayesian analysis begins with <strong>the</strong> assessment of <strong>the</strong> prior in-<br />
formation. In this case, in order to make <strong>the</strong> posterior analysis ma<strong>the</strong>ma-<br />
tically tractab<strong>le</strong>, <strong>the</strong> model was reformulated in terms of <strong>the</strong> parameters<br />
v', U , and p, where v' is <strong>the</strong> inverse.pf <strong>the</strong> coefficient of variation.<br />
This approach has <strong>the</strong> considerab<strong>le</strong> advantage of permitting <strong>the</strong> assumption<br />
of independence between <strong>the</strong> parameters at an "a priori" <strong>le</strong>vel, and hence<br />
<strong>the</strong> prior pdf could be expressed<br />
The prior pdf on <strong>the</strong> parameter p , p(p), was derived from a col<strong>le</strong>ction<br />
of past records from 140 rivers of <strong>the</strong> world, ga<strong>the</strong>red by Yevjevich (1964).<br />
A Beta distribution of <strong>the</strong> form<br />
was fitted to <strong>the</strong> histogram of values of p derived from that col<strong>le</strong>ction,<br />
resulting in <strong>the</strong> following values of kl and k2<br />
kl = 9.888<br />
k2 = 14.499<br />
Assuming prior ignorance about <strong>the</strong> values of v' and U , (8)<br />
was expressed as<br />
It should be pointed out that it is possib<strong>le</strong> to incorporate informa-<br />
tion on <strong>the</strong> parameter v' as well, utilizing <strong>the</strong> same col<strong>le</strong>ction of records<br />
from which <strong>the</strong> prior information on p was derived, and fitting a Beta pdf<br />
to <strong>the</strong> frequency histogram. However, this procedure has <strong>the</strong> disadvantage of<br />
not permitting a marginal analysis on <strong>the</strong> parameter p , which can be con-<br />
siderably useful, as will be seen fur<strong>the</strong>r on.<br />
The likelihood function for <strong>the</strong> parameters was derived in <strong>the</strong> usual<br />
manner, and multiplying <strong>the</strong> prior pdf by <strong>the</strong> likelihood function for <strong>the</strong> 3<br />
parameters of <strong>the</strong> process, <strong>the</strong> posterior pdf was shown to be
where<br />
Equation (11) is <strong>the</strong>refore <strong>the</strong> key equation for <strong>the</strong> optimum estima-<br />
tion of <strong>the</strong> parameters of <strong>the</strong> first order autoregressive model. To do this<br />
in a practical design prob<strong>le</strong>m <strong>the</strong> following two steps must be undertaken:<br />
327<br />
1.) Derive <strong>the</strong> loss function &(6,0), where O and 8 are now<br />
3x1 column vectors, by application of Equation (6). The value functions can<br />
be obtained through simulation, as shown in <strong>the</strong> Rio Colorado examp<strong>le</strong>.<br />
2.) Obtain <strong>the</strong> optimum parameter estimates 0 by solving Equation<br />
(3), by substitution of (11). In practical applications, this minimization<br />
procedure must be undertaken numerically. Note that <strong>the</strong> determination of <strong>the</strong><br />
optimal estimates immediately gives <strong>the</strong> optimal action a6 through <strong>the</strong><br />
value function Vt (a,O), as indicated in Figure 1.<br />
Some Simplifying Approaches<br />
The procedure outlined above may be considerab1:r simplified if <strong>the</strong><br />
hydrologist is willing to limit his prob<strong>le</strong>m to <strong>the</strong> optimum estimation of only<br />
<strong>the</strong> parameter P ,<strong>the</strong> o<strong>the</strong>r parameters being estimated by classical proce-<br />
dures. This approach may be justified by noting that <strong>the</strong> variance of <strong>the</strong>se<br />
estimates is usually quite small.<br />
The marginal posterior pdf for P can be obtained in this case by<br />
integration of Equation (11). However, a much simp<strong>le</strong>r expression, and one<br />
that produces almost identical estimates, can be derived by making <strong>the</strong> trans-<br />
f o mat ion<br />
The model can now be expressed as<br />
Xt = y, - lJ (12)
328<br />
The marginal posterior pdf for P for this process can be shorn<br />
to be (Thornber, 1967; Rodriguez-Iturbe et al., 1972)<br />
where<br />
T<br />
a = C x 2<br />
t<br />
t=o<br />
T<br />
a 1 = - 2 C xt x t-1<br />
t=l<br />
a2 = I xL<br />
t-1<br />
t=2<br />
Fur<strong>the</strong>r fundamental simplificatiom can be made if <strong>the</strong> hydrologist<br />
is willing to fit a simp<strong>le</strong> functional form to his loss function. In this<br />
case <strong>the</strong> minimization of Equation (3) can be determined analytically. Some<br />
distribution-free results are availab<strong>le</strong> in <strong>the</strong> literature (Raiffa and<br />
Schlaifer, 1960), for examp<strong>le</strong>, if <strong>the</strong> loss function can be assumed quadratic,<br />
<strong>the</strong>n <strong>the</strong> optimum estimate is <strong>the</strong> mean of <strong>the</strong> posterior<br />
function can be assumed linear, i.e.<br />
pdf. If <strong>the</strong> loss<br />
<strong>the</strong>n <strong>the</strong> optimum estimate is obtained from <strong>the</strong> value that satisfies<br />
kU<br />
P(P/X) = -<br />
%+ ko<br />
where P(p/X) is <strong>the</strong> posterior cdf for p .<br />
Lenton, Rodriguez-Iturbe and Schaake (1973) extensively studied<br />
<strong>the</strong> properties of <strong>the</strong> Bayes estimators under <strong>the</strong> quadratic and linear loss<br />
functions, with various degrees of asymmetry. Basically, interest centered<br />
around comparing <strong>the</strong> performance of Bayes estimators with that of some clas-<br />
sical estimators. The criteria for comparison were <strong>the</strong> risk functions
R(p) and <strong>the</strong> expected risk B.<br />
where<br />
By definition,<br />
R = Region of <strong>the</strong> samp<strong>le</strong> vector Y<br />
Y<br />
Hence <strong>the</strong> expected risk is<br />
The prior pdf P(P) used was that given by Equation (9) with <strong>the</strong><br />
Yevjevich data parameters.<br />
Some se<strong>le</strong>cted results are reprinted in Tab<strong>le</strong>s 1 and 2. They<br />
show <strong>the</strong> Bayes estimator to be considerably superior to <strong>the</strong> Maximum Liks-<br />
lihood (ML) estimator in all cases, especially in <strong>the</strong> presence of limited<br />
data.<br />
Tab<strong>le</strong> 1<br />
Comparison of Estimators under <strong>the</strong> Quadratic Loss Function<br />
Samp<strong>le</strong> Length<br />
329<br />
The sensitivity of <strong>the</strong> Bayes estimator to <strong>the</strong> form of <strong>the</strong> loss function<br />
was also studied. The Bayes estimator was found to be remarkably robust;<br />
for examp<strong>le</strong>, for samp<strong>le</strong> <strong>le</strong>ngths of 10 years, if <strong>the</strong> coefficient<br />
ko was<br />
erroneously determined as ko = 4 instead of ko = 0.25, <strong>the</strong> Bayes estimator<br />
still performed almost twice as well as <strong>the</strong> ML estimator,under <strong>the</strong> expected
330<br />
risk criterion.<br />
Tab<strong>le</strong> 2<br />
Comparison of Estimators under <strong>the</strong> Linear Loss Function<br />
Fur<strong>the</strong>rmore, practically no difference in <strong>the</strong> expected risk was<br />
found when <strong>the</strong> symmetric linear loss function was used instead of <strong>the</strong><br />
quadratic. In general, it was concluded that errors in <strong>the</strong> form of <strong>the</strong><br />
loss function are <strong>le</strong>ss important than errors in <strong>the</strong> degree of asymmetry<br />
of <strong>the</strong> loss function. This observation tends to justify <strong>the</strong> simplification<br />
of fitting <strong>the</strong> loss function to a given functional form, provided that <strong>the</strong><br />
correct degree of asymmetry is preserved.<br />
CONCLUSIONS<br />
The Bayesian framework for parameter estimation permits <strong>the</strong> hydro-<br />
logist to correct <strong>the</strong> defects of classical methods of estimation through<br />
<strong>the</strong> consideration of fi sources of information and through <strong>the</strong> considera-<br />
tion of <strong>the</strong> economic consequences of erroneous estimates.<br />
Infonation obtained from sources o<strong>the</strong>r than <strong>the</strong> historical samp<strong>le</strong><br />
is incorporated in <strong>the</strong> prior pdf; however, <strong>the</strong> source of information must<br />
be analyzed very carefully. The samp<strong>le</strong> information enters <strong>the</strong> estimation<br />
procedure via <strong>the</strong> likelihood function. The economic consequences of erroneous<br />
estimates are taken into account by means of a loss function; a general tech-<br />
nique for obtaining this function for a hydrologic design prob<strong>le</strong>m is presen-<br />
ted.<br />
The general Bayesian approach to parameter estimation can be applied<br />
to <strong>the</strong> first order autoregressive model. In general, this procedure permits
<strong>the</strong> optimum estimation of <strong>the</strong> 3 parameters of <strong>the</strong> model. However, if <strong>the</strong><br />
hydrologist is willing to make some simplifying assumptions and limit<br />
his prob<strong>le</strong>m to <strong>the</strong> estimation of P , relatively simp<strong>le</strong> estimators can<br />
be derived. These estimators present considerab<strong>le</strong> advantages over <strong>the</strong><br />
classical estimators.<br />
ACKNOWLEDGEMENTS<br />
The work was supported by <strong>the</strong> Office of Water Resources Research,<br />
Office of <strong>the</strong> Interior, United States Govprnment, under Grant No.<br />
14-31-0001-9021.<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
REFERENCES<br />
Wood, Eric F., (1973). Flood Control Design with Limited Data -<br />
,A Comparison of <strong>the</strong> Classical and Bayesian Approaches, Presented<br />
at <strong>the</strong> Symposium on <strong>the</strong> Design of Water Resource Projects with<br />
Inadequate Data, Madrid, June 1973.<br />
Zellner, A. (1971). An Introduction to Bayesian Inference in Eco-<br />
nometrics, J. Wi<strong>le</strong>y &Sons.<br />
Pratt, J.W., H. Raiffa, and R. Schlaifer (1965). Introduction to<br />
Statistical Decision Theory, McGraw-Hill.<br />
331<br />
Lenton, Roberto L., I. Rodriguez-Iturbe and John C. Schaake, (1973).<br />
A Bayesian Approach to Autocorrelation Estimation in Hydrologic<br />
Autoregressive Models, Ralph M. Parsons Laboratory, Report No. 163,<br />
M.I.T.<br />
Yevjevich, V. (1964). Fluctuations of Wet and Dry Years, Part II,<br />
Analysis by Serial Correlation, Colorado State University Hydrology<br />
Paper No. 4, Fort Collins, Colorado.<br />
Thornber, H. (1967). Finite Samp<strong>le</strong> Monte Carlo Studies: An Auto-<br />
regressive Illustration, J. Am. Statist. ASSOC., September 1967.<br />
Rodriguez-Iturbe, I., J. Valdes, R. Lenton and D. Va<strong>le</strong>ncia, (1972).<br />
Bayesian Hydrological Model Building, Proceedings of <strong>the</strong> ïnter-<br />
national Symposium on Uncertainties in Hydrologic and Water Resource<br />
Systems, Volume II, University of Arizona, Tucson, Arizona.<br />
Raiffa, H, and R. Schlaifer, (1961). Applied Statistical Decision<br />
Theory, M.I.T. Press.
332<br />
Figure 1 . The value functions and <strong>the</strong> determination<br />
of <strong>the</strong> loss function.<br />
\<br />
a
_c,<br />
-p" -0 6 -0 2 02<br />
R
STORAGE-Y1,ELD ESTTMATEC WITH INADEQUATE STREAMFLOW DATA<br />
T.A. McMahon and R.G. Mein<br />
Department of Civil Engineering, Monash University, Australia.<br />
ABSTRACT<br />
Inadequate streamflow data may result from short records.<br />
Various techniques can be used to deal with this paper <strong>the</strong> design<br />
prob<strong>le</strong>m of estimating <strong>the</strong> storage-yield relationship for a large<br />
reservoir on a stream having only seventeen years of data is conside-<br />
red, There are two parts to this prob<strong>le</strong>m; <strong>the</strong> extension of <strong>the</strong><br />
streamflow record and <strong>the</strong> estimation of storage capacity.<br />
For <strong>the</strong> examp<strong>le</strong> studied, <strong>the</strong> streamflow record was extended<br />
from daily rainfall using a simp<strong>le</strong> rainfall-runoff procedure<br />
(Boughton's digital computer model modified with a groundwater<br />
component). The model was fitted to half of <strong>the</strong> availab<strong>le</strong> record and<br />
validated against <strong>the</strong> remaini'ng half. The agreement between estimated<br />
and historical data was better than that resulting from a month by<br />
month regression analysis between <strong>the</strong> historical flows and <strong>the</strong> flows<br />
at an adjacent site (with much longer streamflow records).<br />
In <strong>the</strong> part Gould's stochastic model is used to determine<br />
storage capacity. The mothod is independent of <strong>the</strong> initial conditions<br />
and takes into account seasonality and monthly serial correlation.<br />
Results are compared with those obtained using a behaviour analysis.<br />
RESUME<br />
L'insuffisance des données portant sur <strong>le</strong> débit peut résulter<br />
de la trop courte durée des re<strong>le</strong>vés. L'on peut adopter différentes<br />
techniques pour y remédier. Dans <strong>le</strong> présent artic<strong>le</strong>, nous étudierons<br />
plus particulierement <strong>le</strong> prob<strong>le</strong>me de l'élaboration d'une méthode<br />
permettant d'estimer <strong>le</strong>s relations débit-accumulation dans <strong>le</strong> cas<br />
d'un grand réservoir situé sur un cours d'eau pour <strong>le</strong>quel lienregis-<br />
trement des données ne remonte qu'a dix-sept ans. Le problème peut<br />
être divisé en deux parties: ia prolongation des données concernant<br />
<strong>le</strong> débit d'eau et l'estimation de la capacité d'emmagasinage.<br />
Dans <strong>le</strong> cas précis, la série des données concernant <strong>le</strong> ddbit<br />
d'eau a été prolongé au moyen de données pluviométriques grace 3<br />
l'utilisation d'une simp<strong>le</strong> procédure pluviométrie-écou<strong>le</strong>ment (modè<strong>le</strong><br />
numdrique de Boughton modifié par l'addition d'un autre facteur,l<strong>le</strong>au<br />
souterraine). Le modè<strong>le</strong> a St6 ajusté en utilisant la moiti8 des<br />
données disponib<strong>le</strong>s, pour être ensuite contrô<strong>le</strong> par comparaison ayec<br />
l'autre moitié. L'accord entre <strong>le</strong>s données estimées et historiques<br />
s'est révelé très supérieur à la corrélation, établie grâce à une<br />
analyse par régressions mensuel<strong>le</strong>s entre <strong>le</strong>s débits observés et ceux<br />
qui ont été obtenus dans un site voisin pour <strong>le</strong>quel <strong>le</strong>s re<strong>le</strong>vés por-<br />
tent sur une période beaucoup plus longue.<br />
Dans la deuxième partie <strong>le</strong> modè<strong>le</strong> stochastique de Gould est<br />
utilisé pour déterminer la capacité d'emmagasinage. Cette méthode est<br />
indépendante des conditions initia<strong>le</strong>s et el<strong>le</strong> ti'ent compte des fac-<br />
teurs saisonniers aussi bien que de la correlation sériel<strong>le</strong> mensuel<strong>le</strong>.<br />
Une comparaison est faite entre nos résultats et ceux obtenus par la<br />
méthode d'analyse dite "de comportement'' (behaviour analysis 1.
336<br />
INTRODUCTION<br />
Inadequate streamflow data may result from measurement errors and<br />
shortness of record. In this paper we are concerned with <strong>the</strong> latter prob<strong>le</strong>m.<br />
Specifically, we take a seventeen year streamflow record, considered<br />
inadequate for storage estimation, and show how this can be extended using<br />
a relatively simp<strong>le</strong> deterministic rainfall-runoff model.<br />
In <strong>the</strong> second part a stochastic storage model is used to estimate <strong>the</strong><br />
capacity of a sing<strong>le</strong> reservoir €or various regulating conditions and<br />
probabilities in failure.<br />
The methodology is illustrated using <strong>the</strong> ïñomson River catchment at<br />
'Ihe Narrows (ref. no. 225210, iat 37O 53'S, long 146O 24'E) in Victoria,<br />
Australia. This is a 518 km2 forested catchment (Fig.1) with e<strong>le</strong>vations<br />
varying between 400m and 1560m.<br />
In <strong>the</strong> south west, granodioritecountry gives<br />
rise to deep loam soils whereas <strong>the</strong> remaining area is sedimentary rocks with<br />
shallow soils. Mean annual catchment rainfall is 1540mm which varies from<br />
below 1OOOmm to more than 2500m over <strong>the</strong> catchment.<br />
Three standard daily read rain gauges (Fig.1) are availab<strong>le</strong> with<br />
records extending from 1886 to 1970. A fourth gauge with data from 1942<br />
onwards is also availab<strong>le</strong>. Stream heights at The Narrows have been recorded<br />
automatically since 1954 and masured gaugings have been made up to 531n~s-l.<br />
However, discharges up to 150m3s-l have been estimated. No long term<br />
evaporation measurements are availab<strong>le</strong> within or near <strong>the</strong> catchment. The<br />
closest station is at E<strong>le</strong>lbourne, 130 km to <strong>the</strong> west.<br />
STREAMFLOW DATA ESTIMATION<br />
Basically <strong>the</strong>re are two methods availab<strong>le</strong> for extending streamflow<br />
data at a gauging station. The first method consists in correlating <strong>the</strong><br />
flows at that station with those at a nearby station with long records.<br />
From <strong>the</strong> additional records and <strong>the</strong> regression equation/s relating <strong>the</strong> two<br />
stations, flows at <strong>the</strong> station with <strong>the</strong> shorter record period are estimated.<br />
Searcy (1960) explains c<strong>le</strong>arly <strong>the</strong> use of simp<strong>le</strong> analytical and graphical<br />
regression procedures to do this. Multip<strong>le</strong> regression methods are sometimes<br />
used. Examp<strong>le</strong>s of <strong>the</strong>se are given by Brown (1961) who illustrates <strong>the</strong><br />
procedures with monthly data for <strong>the</strong> Snowy Mountains region of Australia.<br />
The second technique is based on deterministic rainfall-runoff models.<br />
Of <strong>the</strong> many digital computer models now availab<strong>le</strong>, only one - Boughton's<br />
model - has been used extensively in Australia. Because of this and because<br />
it utilizes daily inputs, we adopted <strong>the</strong> Boughton model for extending <strong>the</strong><br />
seventeen years of streamflow at The Narrows, in an attempt to improve upon<br />
<strong>the</strong> results obtained using <strong>the</strong> standard regression method.<br />
In relation to data estimation, one question which arises is whe<strong>the</strong>r<br />
data should or should not be extended. In circumstances where consistency<br />
between record <strong>le</strong>ngths is required, data extension is mandatory. However,<br />
a check on whe<strong>the</strong>r statistically one is better off extending data can be made
y computing <strong>the</strong> relative information content using <strong>the</strong> procedure given by<br />
Fiering (1962). Checks made for this study showed extension of <strong>the</strong> data is<br />
beneficial.<br />
337<br />
In this paper <strong>the</strong> terms data extension and data estimation are used<br />
synonymously to des cribe procedures in which equiva<strong>le</strong>nt historical estimates<br />
are made. Herein, we calculate historical monthly flws for <strong>the</strong> period 1886<br />
to 1353. On <strong>the</strong> o<strong>the</strong>r hand, data generation is an analytical tool in which<br />
a model, which represents <strong>the</strong> stochastic streamflow process, produces flow<br />
sequences that statistically are no different to <strong>the</strong> historical sequence,<br />
but cannot be adopted to represent <strong>the</strong> observed flow record.<br />
Boughton's Rainfall-Runoff Model (Boughton 1966, 1968)<br />
?he Boughton model simulates for a catchment daily surface runoff from<br />
daily rainfall inputs and is operated in three distinct cyc<strong>le</strong>s - wetting,<br />
drying and drainage. The wetting cyc<strong>le</strong> is only considered on rainfall days,<br />
but <strong>the</strong> drying and drainage cyc<strong>le</strong> operate every day.<br />
a) Model structure.<br />
The model consists of four storages representing interception, upper-<br />
soil, drainage and <strong>the</strong> lower-soil zones (Fig.2).<br />
The interception store represents water stored on vegetation during<br />
rain periods. It fills during <strong>the</strong> wetting cyc<strong>le</strong> and evaporates (at <strong>the</strong><br />
Potential rate) during <strong>the</strong> drying cyc<strong>le</strong>.<br />
When <strong>the</strong> interception store is full, excess rainfall is admitted to<br />
<strong>the</strong> upper soil store which represents <strong>the</strong> moisture holding capacity of <strong>the</strong> top<br />
soil. Water is lost from this store during <strong>the</strong> drying cyc<strong>le</strong> by<br />
evapotranspiration.<br />
The drainage store fills during <strong>the</strong> wetting cyc<strong>le</strong> only after <strong>the</strong> upper<br />
soil store is full. This is intended to represent water in <strong>the</strong> upper soil<br />
which can later drain under gravity to <strong>the</strong> lower soil zone. If <strong>the</strong> drainage<br />
store is fil<strong>le</strong>d (i.e. <strong>the</strong> soil is saturated), surface runoff occurs. During<br />
<strong>the</strong> drainage cyc<strong>le</strong> <strong>the</strong> drainage store is dep<strong>le</strong>ted by water transferring to<br />
<strong>the</strong> lower soil store. No evapotranspiration occurs from <strong>the</strong> drainage store.<br />
The lower soil store represents water held in <strong>the</strong> sub-soil zone.<br />
Drainage from <strong>the</strong> drainage store adds to <strong>the</strong> volume in storage, whilst<br />
evapotranspiration and deep percolation dep<strong>le</strong>te it.<br />
üp-dating of <strong>the</strong> moisture status of <strong>the</strong> stores occurs daily.<br />
b) Infiltration.<br />
The model utilizes a relation similar to Horton's infiltration<br />
equation:<br />
-k . SS<br />
f = fc + (fo - fc) e . (1)
338<br />
where f = daily loss rate,<br />
fo = loss rate when soil is at wilting point,<br />
fc = limiting value which <strong>the</strong> loss rate<br />
approaches at high soil-misture <strong>le</strong>vels,<br />
k = an exponent, and<br />
SS = lower-soil moisture <strong>le</strong>vel.<br />
Thus infiltration is a function only of <strong>the</strong> lower soil moisture status. When<br />
this is low, <strong>the</strong> rate of infiltration from <strong>the</strong> drainage to <strong>the</strong> lower soil<br />
store is high, and vice versa.<br />
c) Evap o t rans pi r a t i on.<br />
As well as evaporation from <strong>the</strong> interception store, evapotranspiration<br />
takes place from <strong>the</strong> upper and lower soil stores. Evaporation need is first<br />
met from <strong>the</strong> interception store, and if that need is not fil<strong>le</strong>d, evapotranspiratic<br />
<strong>the</strong>n takes place simultaneously from <strong>the</strong> upper and lower soil stores. The rate<br />
is a function of both <strong>the</strong> evaporation potential and <strong>the</strong> soil moisture status<br />
of each store. This approach follows <strong>the</strong> work of Denmead and Shaw (1962)<br />
and is shown schematically in Fig. 3.<br />
d) Surface runoff.<br />
In <strong>the</strong> model no attempt is made to simulate <strong>the</strong> time sequencing of<br />
surface runoff. Runoff occurs only on days of rain. The algorithm for<br />
estimating daily runoff volume is:<br />
Q = P - f tanh (P/f)<br />
where Q = daily surface runoff,<br />
P = daily rainfall <strong>le</strong>ss interception and upper soil<br />
store requirements, and<br />
f = daily loss rate.<br />
e) Groundwater.<br />
As proposed by Boughton, <strong>the</strong> model yields surface runoff only. Ground-<br />
water loss whiclr is a function of <strong>the</strong> moisture status of <strong>the</strong> lower store '<br />
accretes to deep seepage. No base flow occurs.<br />
To overc8me this deficiency in <strong>the</strong> model, <strong>the</strong> authors substituted for<br />
<strong>the</strong> lower soil store shown in Fig. 2, <strong>the</strong> modification shown in Fig. 4.<br />
That is, <strong>the</strong> lower soil store is divided into two parts, each contributing<br />
to base flow. nie lower section of <strong>the</strong> store must be full before <strong>the</strong> upper<br />
section can hold water.<br />
On <strong>the</strong> assumption that <strong>the</strong>re is no deep groundwater<br />
loss from <strong>the</strong> catchment, base flow was represented by <strong>the</strong> following<br />
equations and added to <strong>the</strong> surface runoff component.<br />
. . . (2)
Qb = kl S1 if SS E Slmax<br />
Qb = kl S1 + k2 S2<br />
Qb = Sp + kl S1 + k2 S2 if SS > Slmm + SZmax<br />
339<br />
. . . (3)<br />
. . . (4)<br />
. . . (5)<br />
where Qb = daily base flow,<br />
S1,S2 = soil moisture <strong>le</strong>vels of each section of <strong>the</strong><br />
lower soil store,<br />
S = maximum capacity of lower soil store sections,<br />
simax> 2max<br />
SS = lower soil moisture status,<br />
Sp = spill from lower soil store if <strong>the</strong> total capacity is exceeded,<br />
kl,k2 = base flow recession constants for <strong>the</strong> lower soil store<br />
sections determined from <strong>the</strong> streamflow data.<br />
This particular algorithm was chosen because it represents on a daily basis<br />
<strong>the</strong> doub<strong>le</strong> sloped hydrograph recession limbs observed for The Narrows flow<br />
data.<br />
f) Parameter estimates and optimization.<br />
Before <strong>the</strong> Boughton model can be used to predict runoff, values for<br />
several parameters must be determined. nie usual method for this is to estimate<br />
values, run <strong>the</strong> model, and compare <strong>the</strong> predicted and observed values of runoff.<br />
Then changes are made to <strong>the</strong> parameters to see if <strong>the</strong> agreement can be<br />
improved. There were nine parameters for which values were required, namely<br />
<strong>the</strong> capacities of <strong>the</strong> interception, upper soil, drainage and lower soil<br />
stores, S2, k2, fo, fc, and k. (kl was determined from base-flow recessions<br />
in <strong>the</strong> data). Systematic variation of <strong>the</strong> values of <strong>the</strong> parameters (optimization)<br />
is made to determine <strong>the</strong> values.<br />
The most common optimization procedure is <strong>the</strong> steepest descent method.<br />
Ano<strong>the</strong>r procedure is <strong>the</strong> simp<strong>le</strong>x method. Boughton (1968) and Ned<strong>le</strong>r and Mead<br />
(1965) discuss <strong>the</strong>se techniques. Model parameters are determined normally<br />
using half <strong>the</strong> availab<strong>le</strong> conmon rainfall and streamflw record period, <strong>the</strong><br />
remaining half being used to test <strong>the</strong> model parameters by comparing <strong>the</strong><br />
computed flows with <strong>the</strong> observed ones.<br />
Fur<strong>the</strong>r details of <strong>the</strong> model can be found in Boughton's papers (1966,<br />
1968a,b) and in Pattison and McMahon (1973).<br />
Results.<br />
The model was applied on a daily basis to <strong>the</strong> Thomson catchment. For fhe<br />
period 1886-1971, daily catchment rainfalls were estimated from <strong>the</strong> daily ram gauge readings using Theissen weightings. AS daily streamflow data at The<br />
Narrows were availab<strong>le</strong> from 1954 to 1970, <strong>the</strong> period 1954-1962 was used to<br />
define model parameters, <strong>the</strong> remaining eight years was used as an independent<br />
test of <strong>the</strong>ir adequacy.
340<br />
Daily catchment evaporation estimates for <strong>the</strong> study period were estimated<br />
by applying monthly sunken tank pan coefficients taken from Wiesner (1970,<br />
Tab<strong>le</strong> 18) to kïbourne pan data. In addition to Melbourne being 130 km from<br />
<strong>the</strong> catchment, data was measured initially using a sunken tank but after 1967<br />
by American class 'Al pan. No suitab<strong>le</strong> class 'A' pan coeffecients are<br />
availab<strong>le</strong> for <strong>the</strong> site. This results in <strong>the</strong> open surface evaporation estimates<br />
from 1967 to 1970 being uncertain.<br />
From Tab<strong>le</strong> I it is seen that <strong>the</strong> computed monthly flows compare<br />
favourably with <strong>the</strong> observed values. In making this assessment, <strong>the</strong> simplicity<br />
of <strong>the</strong> model, <strong>the</strong> difficulties in estimating catdiment evaporation and <strong>the</strong><br />
normal accuracy of rainfall and streamflow data were all considered. The year<br />
1954 is difficult to simulate because of <strong>the</strong> effect of <strong>the</strong> choice of initial<br />
conditions, whi<strong>le</strong> <strong>the</strong> uncertainty of <strong>the</strong> correct evaporation values for 1967<br />
onwards is certainly a factor in <strong>the</strong> estimates for that period. A fur<strong>the</strong>r<br />
comparison is shown in Tab<strong>le</strong> II for <strong>the</strong> period 1954-1970 between historical<br />
flows and those calculated using <strong>the</strong> model and those calculated from monthly<br />
regression analysis between The Narrows data and its most reliab<strong>le</strong> set of<br />
adjacent flows. The model results are better.<br />
RESERVOIR CAPACITY ESTIMATION<br />
Joy and McMahon (1972) have reviewed a large number of procedures for<br />
computing <strong>the</strong> capacity of a sing<strong>le</strong> reservoir and conclude that Gould's<br />
method is a satisfactory design tool.<br />
Gould's Stochastic Storage Model (Gould, 1961)<br />
Gould's technique is classi£ied as a stochastic approach and is based<br />
on <strong>the</strong> pioneering work of Moran (1959). Using discrete time units, Moran set<br />
up a simp<strong>le</strong> mass balance of water in storage as follows:<br />
at+l = Pt + Xt - Yt . . . (6)<br />
where Xt,Pt+l = reservoir contents at <strong>the</strong> beginning and end of<br />
tth discrete time period,<br />
= inflow during tth time period, and<br />
Xt<br />
Yt = re<strong>le</strong>ase during tth time period.<br />
By neg<strong>le</strong>cting seasonality and annual serial correlation and dividing <strong>the</strong><br />
reservoir and streamflow into a number of equally sized zones, Moran was ab<strong>le</strong><br />
to obtain a system of equations (<strong>the</strong> coefficients of which are equiva<strong>le</strong>nt to<br />
<strong>the</strong> transition matrix of stored contents) describing <strong>the</strong> cumulative probability<br />
of stored contents. The solution of <strong>the</strong>se equations is <strong>the</strong> steady state<br />
condition of stored water.<br />
In Gould's technique, <strong>the</strong> reservoir is also divided into a number of<br />
zones. The transition matrix - <strong>the</strong> relation of <strong>the</strong> volume of water in storage<br />
at time t to <strong>the</strong> volume stored at time (t+l) - is obtained by routing each
year of <strong>the</strong> historical flow record through a storage of specified size, a month<br />
at a time, beginning each year in each zone. (Twenty zones were used in this<br />
study). Thus seasonality and monthly serial correlation are automatically taken<br />
into account. Re<strong>le</strong>ases from <strong>the</strong> reservoir can be varied seasonally or in any<br />
o<strong>the</strong>r specified manner. However, annual flows are assumed independent. By<br />
recording <strong>the</strong> starting zone, finishing zone and <strong>the</strong> number of failures, <strong>the</strong><br />
transition matrix of stored contents and <strong>the</strong> conditional probabilities of<br />
failure within <strong>the</strong> year subject to <strong>the</strong> reservoir contents at <strong>the</strong> start of<br />
<strong>the</strong> year are built up. From <strong>the</strong> transition matrix, <strong>the</strong> steady state content<br />
is obtained.<br />
341<br />
As applied bj, Gould, <strong>the</strong> conditional probabilities of failure were<br />
based on annual failures determined from monthly flows. This results in an<br />
over-estimation of <strong>the</strong> required storage size. In <strong>the</strong> procedure used here,<br />
<strong>the</strong> method is modified so that <strong>the</strong> conditional probabilities of failure are<br />
determined by monthly failures from monthly flows (see Joy and McMahon, 1972).<br />
Space precludes an adequate description of Gould's procedure. It is<br />
set down c<strong>le</strong>arly in examp<strong>le</strong> form in Appendix I of <strong>the</strong> original paper (Gould,<br />
1961).<br />
In <strong>the</strong> procedure annual flows are assumed independent. However, Gould<br />
provides an equation to correct for this if <strong>the</strong> annual serial correlation is<br />
significant. A limitation of <strong>the</strong> technique is that it requires computer<br />
facilities for solution. On <strong>the</strong> o<strong>the</strong>r hand, an advantage is that <strong>the</strong><br />
probability of failure is independent of <strong>the</strong> initial starting conditions.<br />
Moreover, because of <strong>the</strong> assumption of annual independence, records with<br />
missing years of data can be utilized without recourse to data extension<br />
techniques.<br />
Results.<br />
Gould's procedure, modified as noted above, was applied to <strong>the</strong> 85 years<br />
of estimated streamflow data for conditions of 5% probability of failure and<br />
constant draft rates of 50% and 90% of <strong>the</strong> mean monthly flow. In this context,<br />
5% probability of failure implies that 5% of <strong>the</strong> time <strong>the</strong> reservoir is unab<strong>le</strong><br />
to maintain a specified constant draft (o<strong>the</strong>r equiva<strong>le</strong>nt terms are yield,<br />
re<strong>le</strong>ase, regulation) which is defined as a percentage of <strong>the</strong> mean flow.<br />
Storage estimates are given in Tab<strong>le</strong> III and are compared with<br />
estimates based on a behaviour analysis. In <strong>the</strong> behaviour analysis, changes<br />
in <strong>the</strong> volume of water stored were examined on a monthly basis by adding inflows<br />
to, and subtracting re<strong>le</strong>ases from <strong>the</strong> water stored in a reservoir of finite<br />
capacity. Probability of failure was defined as <strong>the</strong> proportion of time units<br />
that <strong>the</strong> storage is empty to <strong>the</strong> number of units of historical flow run<br />
through <strong>the</strong> storage. In <strong>the</strong> behaviour analysis, it is assumed that <strong>the</strong><br />
reservoir is initially full.<br />
At 50% draft, <strong>the</strong> storage estimates are similar. On <strong>the</strong> o<strong>the</strong>r hand, at<br />
<strong>the</strong> higher draft <strong>the</strong> Gould estimate is about 30% of <strong>the</strong> behaviour value. At<br />
low drafts such as 50% of mean flow, annual serial correlation is unimportant
34 2<br />
(Tab<strong>le</strong> III). Never<strong>the</strong><strong>le</strong>ss, at high drafts with long draw .down periods extending<br />
over years, annual serial correlation must be taken into account. On adjusting<br />
<strong>the</strong> 90% value for <strong>the</strong> annual serial correlation of 0.34 using Gould's correction<br />
(Gould, 1961), <strong>the</strong> value is increased to 86% of <strong>the</strong> behaviour estimate. It<br />
should be noted, however, that 0.34* is beyond <strong>the</strong> range of correlations<br />
(O - O. 25) used by Gould in deriving <strong>the</strong> correction procedures.<br />
Consequently<br />
<strong>the</strong> correction factor is based on an extrapolation and this probably accounts<br />
for <strong>the</strong> difference in storage estimates at <strong>the</strong> 90% probability <strong>le</strong>vel. It has<br />
been shown elsewhere (Joy and McMahon, 1972) that Gould's method is a<br />
satisfactory procedure for estimating storage capacity.<br />
Because of <strong>the</strong> long <strong>le</strong>ngth of record (historical plus estimated) and <strong>the</strong><br />
low variability of Thomson River flows (for examp<strong>le</strong> <strong>the</strong> annual coefficient of<br />
variation is 0.42 compared with <strong>the</strong> more variab<strong>le</strong> Australian rivers with values<br />
over l.O), <strong>the</strong> behaviour storage estimate is considered to provide a reasonab<strong>le</strong><br />
check on <strong>the</strong> Gould results. However, in normal situations where ei<strong>the</strong>r <strong>the</strong><br />
availab<strong>le</strong> record is shorter or <strong>the</strong> stream more variab<strong>le</strong>, <strong>the</strong> behaviour procedure<br />
is not necessarily a satisfactow analytical tool. Some comments on this<br />
aspect may be found in McMahon, Codner and Joy (1972).<br />
CONCLUSIONS<br />
In this paper we have endeavoured to illustrate <strong>the</strong> use of a<br />
relatively simp<strong>le</strong> deterministic computer model by Boughton to extend an<br />
inadequate data sequence to one more acceptab<strong>le</strong> in <strong>le</strong>ngth. A modification<br />
which allowed <strong>the</strong> model to account for base flow was described.<br />
Reservoir capacity estimates were computed using Gould' s procedure<br />
again with slight modifications. Results were compared with those found<br />
using a behaviour analysis.<br />
ACKNOWLEDGEMENT<br />
nie authors wish to thank <strong>the</strong> Melbourne Metropolitan Board of Works<br />
for providing <strong>the</strong> rainfall and streamflow data for this project.<br />
Mr, G. Codner, Department of Civil Engineering, Monash University, provided<br />
<strong>the</strong> authors with <strong>the</strong> regression equations used to obtain <strong>the</strong> comparative<br />
results in Tab<strong>le</strong> II.<br />
*The value of 0.34 annual serial correlation is much higher than usual for<br />
Australian streams. It is not possib<strong>le</strong> to determine whe<strong>the</strong>r <strong>the</strong> Boughton<br />
model itself contributed to this high value.
REFERENCES.<br />
1.<br />
2.<br />
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Boughton, W.C. (1966). A Ma<strong>the</strong>matical Model for Relating Runoff to<br />
Rainfall with Daily Data, I.E. Aust., Civil Engg. Trans., CE8(1),<br />
pp. 83-93.<br />
343<br />
Boughton, W.C. (1968a). Evaluating <strong>the</strong> Variab<strong>le</strong>s in a Ma<strong>the</strong>matical<br />
Catchment Model, I.E. Aust., Civil Engg. Trans., CElO(1) pp. 31-39.<br />
Boughton, W. C. (1968b). A Ma<strong>the</strong>matical Catchment Model for Estimating<br />
Runoff, Jour. Hydrology (N.Z.), 7(2), pp. 75-100.<br />
Brown, J.A.E. (1961). Streamflow Correlation in <strong>the</strong> Snowy Mountains<br />
Area, Jour I.E. Aust., 33(3), pp. 85-95.<br />
Denmead, O.T. & Shaw, R.H. (1962). Availability of Soil Water to<br />
Plants as Affected by Soil Moisture Content and Meteorological<br />
Conditions,Agron. Jour. 54(5), pp. 385-389.<br />
Fiering, M.B. (1962). On <strong>the</strong> Use of Correlation to Augment Data,<br />
Jour. Amer. Stat. ASSOC., 57, pp. 20-52.<br />
Gould, B.W. (1961). Statistical Methods for Estimating <strong>the</strong> Design<br />
Capacity of Dams, Jour. I.E. Aust., 33(12), pp. 405-416.<br />
Joy, C.S. 6 McMahon, T.A. (1972) Reservoir Yield Estimation Procedures,<br />
I.E. Aust., Civil Engg. Trans., CE14(1), pp. 28-36.<br />
McMahon, T.A., Codner, G.P. 6 Joy, C.S. (1972). Reservoir-Storage Yield<br />
Estimates Based on Historical and Generated Streamflows. I.E. Aust.,<br />
Civil Engg. Trans., CE14(2) (in press).<br />
Moran, P.A.P. (1959). The Theory of Storage, Methuen, London.<br />
Ned<strong>le</strong>r, J.A. & Mead, R. (1965). A Simp<strong>le</strong>x Method for Function<br />
Minimization, Comp. Jour., 7, pp. 308-313.<br />
Pattison, A. & McMahon, T.A. (1973). Rainfall-Runoff Models Using<br />
Digital Computers, I.E. Aust., Civil Engg. Trans., CE15 (in press).<br />
Searcy, J.K. (1960). Graphical Correlation of Gauging Station Records,<br />
Geological Survey Water -Supply Paper 1541-C, Washington.
344<br />
TABLE I : COMPARISON OF BOUCHTON MODEL RESULTS WITH HISTORICAL VALUES<br />
FOR OPTIMIZING PERIOD AND INDEPENDENT TEST PERIOD.<br />
Parame ter<br />
Standard deviation<br />
(w/month)<br />
Corre lati on<br />
coe f fi cient<br />
between historical<br />
and predicted<br />
Optimizing Period (1953-62) Test period<br />
Historical Boughton His torical<br />
Value Mode 1 Value<br />
1 4 8 I 49<br />
1 38<br />
40 42 32<br />
- .94<br />
(1963-1970)<br />
Bought on<br />
Model<br />
TABLE II : OBSERVED STREAMFLOW PARAMETERS (FOR PERIOD 1954-1970) COMPARED<br />
WIM "HOSE ESTIMATED USING REGRESSION ANALYSIS AND BOUGHTON MODEL<br />
Parameter His torical<br />
Value<br />
Me an<br />
(mm/mon th)<br />
Standard deviation<br />
(m/mon th)<br />
Corre lation coefficient<br />
between observed and<br />
predicted I<br />
- .92<br />
TABLE III : STORAGE ESTIMATES FOR 5% PROBABILIW OF FAILURE<br />
(Millions of cubic metres)<br />
I Draft I Gould Model I Behaviour Analysis I<br />
1<br />
* Values in brackets include Gould's correction for<br />
annual serial correlation.<br />
86<br />
40<br />
37<br />
43<br />
41<br />
.89
o km<br />
Legend<br />
Daily read rain<br />
A Gauging station<br />
U<br />
~~<br />
5<br />
Granodiorite<br />
Clay, silt stone, silty sandstone<br />
FIG. 1. 'IHOSON RIVER CATCHMENT<br />
34 5
346<br />
Lower Soil Store<br />
-<br />
FIG. 2. SCHEMATIC DIAGRAM OF BOUGITON MODEL<br />
-
Actual<br />
ET<br />
Rate<br />
Ip potential<br />
ET<br />
Rate<br />
t<br />
eld<br />
pacity<br />
FIG. 3. COMPUTATION OF ACTUAL ET RATE FROM POTENTIAL ET RATE<br />
IN BOUGHTON MODEL. (Figure shows graphically <strong>the</strong><br />
method of calculation for moisture <strong>le</strong>vel S<br />
for potential rate p)<br />
Lower Sub-store 2<br />
Soi 1<br />
Store ---<br />
Sub-store 1<br />
Infiltration Evapotranspiration<br />
Baseflow 2 (kzSz)<br />
Baseflow 1 (klSi)<br />
-<br />
FIG. 4. MODIFICATION OF 'IHE LOWER SOIL STORE TO OBTAIN A BASE<br />
FLOW WITH DOUBLE RECESSION CONSTANT.<br />
347
ABS TRACT<br />
ESTIMATION OF GUMBEL LAW PARAMETERS IN SMALL SAMPLES<br />
Va<strong>le</strong>ntin Martfn Jadraque<br />
Civil Engineer<br />
A comp<strong>le</strong>te study of <strong>the</strong> distribution law of Gumbel for ex-<br />
treme values is realized and <strong>the</strong> methodology of estimation of <strong>the</strong><br />
characteristical parameters, mean and typical desviation in <strong>the</strong><br />
case of samp<strong>le</strong>s of few extension, which serve to determine <strong>the</strong><br />
typical parameters u and u of this law,<br />
RES UM EN<br />
Se realiza un estudio comp<strong>le</strong>to de la <strong>le</strong>y de distribución de<br />
Gumbel para valores extremos y la metodología de estimación de los<br />
parametros caracterfsticos, media y desviación típica en el caso<br />
de muestras de poca extensión, los cua<strong>le</strong>s sirven para determinar<br />
los pardmetros u y u de esta <strong>le</strong>y.
350<br />
Iii hydrological studies, and especially when studying<br />
<strong>the</strong> maximum annual flood of a river, this a<strong>le</strong>atory variab<strong>le</strong><br />
is considered as distributed according to <strong>the</strong> Gumbel law.<br />
However, <strong>the</strong> Gumbel law has more general applications, and<br />
its use is considered satisfactory as distribution of a<strong>le</strong>atory<br />
variab<strong>le</strong>s which are extremes (maximum or minimums) of a certain<br />
phenomenum produced in time.<br />
Ti<strong>le</strong> study we are making is partly a reminder of <strong>the</strong> main<br />
properties of this variab<strong>le</strong>, such as its distribution function,<br />
density function, moment generating function, and estimation of<br />
<strong>the</strong> moments regarding <strong>the</strong> origin, estimating <strong>the</strong> mean, variance<br />
and typical deviation, and partly a development of <strong>the</strong> study on<br />
estimation of <strong>the</strong> characteristic mean and typical deviation<br />
parameters in <strong>the</strong> case of samp<strong>le</strong>s of small extension, which in<br />
turn help to find <strong>the</strong> typical iy and u paramcters of this law.<br />
The a<strong>le</strong>atory variab<strong>le</strong> 5 with Gumbel distribution is one<br />
whose distribution function F (x) is:<br />
F (x) = Prob (5‘”) = e<br />
-e-% (x-u)<br />
(1)<br />
wliereMand u are parameters to be determined in each case, and<br />
whose estimation is analysed later on.<br />
The distribution function (i), as all distribution functions,<br />
fulfils <strong>the</strong> properties :<br />
F (x,) 5 F (x2 1 if x1 x2<br />
F (-”) = O<br />
F (+p.) = 1<br />
The density function f (x) will be:<br />
The distribution method is made by making:<br />
f’ (x) = FI’ (x) = 0
Taking Neper logarithms in (i) and deriving, one obtains:<br />
F" (x) = @.e-=' IF' (x) -oc.F(x) ] whereby;<br />
Thus :<br />
FI' (x) = O implies F' (x) = O(, F(x)<br />
e = 1 and <strong>the</strong>refore, X mode<br />
The poment generating function<br />
(t) = E (e 5 * t, =<br />
Making<br />
-I?¿. (x-u)<br />
-00<br />
-- 1<br />
Y = - ?<br />
dy = -U. x. dz<br />
ex = eU.y *<br />
= u<br />
Thus :<br />
u. t t o<br />
ys(t) = eu*t* (5 ! = e .r (i - (3)<br />
'p5(O) = 1<br />
351<br />
- "(x. u)<br />
. d. x<br />
[<br />
To calculate <strong>the</strong> moments in respect of origin &k , we recall<br />
tiiat :<br />
% = 'f k (0) = -$'f5(t4<br />
5<br />
t=O<br />
and <strong>the</strong>refore:<br />
y&(t> = 1 - o(1 . t + -. t2+ ... +- o(k* tk+ ...;<br />
Taking Neper logarithms in (3) , we get:<br />
d2<br />
l ! 2! k!<br />
LnV5 (t) = u. t -+ Ln r (i - t ) (41<br />
On <strong>the</strong> o<strong>the</strong>r harid, it is known that:<br />
r(i+x)=x! =lim nx<br />
n-+n<br />
(i+&-)<br />
J<br />
j = l
352<br />
Where<br />
C = 0,5772156 ... P Eu<strong>le</strong>r constant .<br />
If we call<br />
O0<br />
We will get:<br />
and, <strong>the</strong> re for e :<br />
O0<br />
K1=r+, c<br />
Ks = - s2<br />
a2 '<br />
..............<br />
...........<br />
Sr.(r - i)!<br />
xr =<br />
ar<br />
.............<br />
ïim ( i .L i i 1 .L ... L I . L<br />
n-+Eo 2 3 n "
Where hr is <strong>the</strong> cumulant r-esimo.<br />
Since iíl = dl = ,cam<br />
The typ ical devi at i on<br />
tnus :<br />
R<br />
D(5) = + = ka^<br />
Having obtained a samp<strong>le</strong> of values xl, x2 .. . , xn, and<br />
A<br />
estimating <strong>the</strong> meanPx from this samp<strong>le</strong> and <strong>the</strong> typical<br />
A<br />
deviation o-x , <strong>the</strong> estimation of <strong>the</strong> parametersGy u is<br />
made in accordance with <strong>the</strong> above formulas:<br />
=fc. - 0,45QOS. a;C<br />
With this <strong>the</strong>ore tical reminder , we shall now analyse <strong>the</strong><br />
estimators study centred on <strong>the</strong> meany, and <strong>the</strong> typical<br />
deviation cxL<br />
353<br />
The samp<strong>le</strong> mean xn is, as we know, an estimator/ X centered,<br />
of <strong>the</strong> population meanPx . In fact:<br />
-<br />
x1 I xp 1 ... 1 x n<br />
With x =<br />
n n
354<br />
x1 1 x2 1 ... .! x<br />
E(^ ) - E(Xn) = E( "1 = 2 .[E(xl) 1 E(x2) .! ... 5<br />
TK - n n<br />
In <strong>the</strong> case of <strong>the</strong> samp<strong>le</strong> typical deviatioii Sxn, this<br />
A<br />
estimatorcx, is not centered in <strong>the</strong> typical population deviation<br />
vX and we will tkrefore try and find<br />
&, , which<br />
depending on n, make - 3<br />
5Lxn- 1 .<br />
En En En -- n n<br />
be an estimator centered on x.<br />
Accordingly, it must be verified that:<br />
Where 5 is <strong>the</strong> Gumbel generical a<strong>le</strong>atory variab<strong>le</strong> with<br />
dis tribut ion function<br />
-d.(X - u)<br />
-e<br />
F(x) = e<br />
(d7 O) (-QiX,C .!a)<br />
Let us consider <strong>the</strong> variab<strong>le</strong> 7=a.('5- u) In o<strong>the</strong>r words:y=o
with' Sxn =<br />
n<br />
- n<br />
Considering Yn as estimator of /"y , we get:<br />
E($,) =-O(.E(X ) - d. u = .(PX - 1.1) = c<br />
n<br />
To calculate E (S ) , samp<strong>le</strong>s (y1, y2 .. . y,) are<br />
Yn<br />
formed, with extension n of <strong>the</strong> a<strong>le</strong>atory variab<strong>le</strong> 0 , whose<br />
distribution function is +(y) = Proh( ,r y) = - e-Y<br />
e 7<br />
(reduced Gumbel distribution o(= 1 u = O).<br />
Thus, for <strong>the</strong> simulation procedure , an a<strong>le</strong>atory number<br />
zi of <strong>the</strong> rectangular distribution (0,l) is formed, and<br />
making<br />
-Yi<br />
z.= e-e or in o<strong>the</strong>r words:<br />
1<br />
Yi = - I+LZi) one obtains a value yi of <strong>the</strong> a<strong>le</strong>atory<br />
variab<strong>le</strong> 12 .<br />
Having fixed <strong>the</strong> value of n (extension of <strong>the</strong> samp<strong>le</strong>) and<br />
obtained k samp<strong>le</strong>s of extension n, with sufficiently large k,<br />
we will get:<br />
355
Number of<br />
samp<strong>le</strong><br />
356<br />
Number of<br />
extension n<br />
Samp<strong>le</strong> mean<br />
1 -<br />
€(y 1 = c = 0'5772 -. .% Yni - Y,* 1 ... Y,!;<br />
n li<br />
?'lie value of En will be:<br />
n<br />
-<br />
= Y,<br />
and a centered estimation ofpx and TX will be:<br />
ci ,XT 1 x2 .L ... L<br />
'n - -<br />
Px = Il - xn<br />
Typical samp<strong>le</strong> desviation
Let us see ano<strong>the</strong>r procedure to estimate <strong>the</strong> typical<br />
population deviation G X.<br />
x,).<br />
357<br />
Let us consider <strong>the</strong> extension samp<strong>le</strong> n : (xi x2 ... xi ...<br />
Let us take a smal<strong>le</strong>r to larger scheduling of <strong>the</strong> form:<br />
kl< "6 . . . ..(Xi& * * sk,<br />
Cons i der ing <strong>the</strong> "quas i - ranges" :<br />
ox =? n -xl=x rnáx - Xmin = range<br />
rix = 5<br />
n-1 - '2<br />
rZx = - >i3<br />
and in general:<br />
I r<br />
hx = %-h<br />
If we have obtained k samp<strong>le</strong>s of extension n, by<br />
simulation in <strong>the</strong> Gumbel reduced Taw, and in each of<br />
<strong>the</strong>m <strong>the</strong> "quasi-range" r , we will get:<br />
11Y
358<br />
Let ? nh be <strong>the</strong> coefficient - function n - such that:<br />
Therefore:E(rhy--)<br />
o( .?nh<br />
Arid thus<br />
Pnh =<br />
In this case a centered estimation of pr and G, would be:<br />
L As 1 ... -L x -<br />
* = x<br />
n n<br />
Concluding, we can say that, given a samp<strong>le</strong> of extension n<br />
(Xi x2 ... Xn ):<br />
1) A centered estimation ofrx is:<br />
I x2<br />
-<br />
1 ... 1 xn -<br />
= x<br />
n<br />
n<br />
2) Centered estimations of 6 y are:<br />
Precisely as both estimators ox, and b, are centered<br />
inKx , in each case it would be convenient to take <strong>the</strong> one<br />
whose variance is <strong>le</strong>ss, in o<strong>the</strong>r words, where:<br />
h
we get:<br />
or in o<strong>the</strong>r words: :<br />
<strong>the</strong> variation coefficients of S and r respectively,<br />
Yn hY<br />
<strong>the</strong> above expressions stands as follows:<br />
And <strong>the</strong>refore,<br />
Within <strong>the</strong> "quasi-ranges" rhx, as all <strong>the</strong> quotients &<br />
Prix<br />
are centered estimators of cx and since r = H.rhx,<br />
hY<br />
we get:<br />
359
360<br />
For two particular values hl and h2 of h, we get:<br />
and <strong>the</strong>refore:<br />
One concludes, in all cases, that <strong>the</strong> centered estimator<br />
of Q to be used between SX, I<br />
D Or, ____ 'hlx<br />
En qnh 1<br />
r<br />
or, e will be <strong>the</strong> one for which <strong>the</strong> corresponding<br />
variation coefficient V(S ) or V(rhly) or V (r h2Y ) is <strong>le</strong>ss.<br />
Yn<br />
With <strong>the</strong>se grounds and criteria, we have obtained <strong>the</strong> following<br />
results for k = 20.000 samp<strong>le</strong>s of extension n, by statistical<br />
simulation of samp<strong>le</strong>s of Gumbel reduced law values:
-,- i<br />
I<br />
.<br />
. . . .<br />
....... -1..<br />
. __<br />
i ¡ , . -4<br />
!<br />
. I /<br />
. .- .~ - __ .<br />
.<br />
361
U o<br />
o<br />
U<br />
VI<br />
l.3<br />
U O<br />
m<br />
E<br />
.rl<br />
U<br />
VI<br />
2<br />
a<br />
o<br />
VI<br />
O<br />
c) s ci<br />
E<br />
ci x<br />
> o<br />
TI<br />
,-i<br />
d<br />
u<br />
.FI<br />
.d d<br />
bX<br />
- .d O<br />
U<br />
u<br />
VI<br />
O<br />
3<br />
rl<br />
o<br />
a<br />
r(<br />
5<br />
I
According to <strong>the</strong> above, and bearing inmind that:<br />
+ 3 I?
fi<br />
F)<br />
C<br />
C<br />
a<br />
364
ABSTRACT<br />
STOCHASTIC SIMULATION FOR BASINS WITH SORT<br />
OR NO RECORDS OF STREAMFLOW<br />
by M. E. Moss and D. R. Dawdy<br />
U.S. Geological Survey, Washington, D.C., USA<br />
Stochastic modeling of streamflows is a powerful tool<br />
in water resources systems desing. Statistics for simulation<br />
which are based on short records may be highly uncertain. A<br />
method is presented for <strong>the</strong> development of a stochastic model<br />
for an ungaged site. The means and variances of <strong>the</strong> monthly<br />
streamflows can be based on regional estimates or on physical<br />
characteristics of <strong>the</strong> basin. The autocorrelation structure<br />
is based on rainfall records and physical characteristics of<br />
<strong>the</strong> basin alone. Application of <strong>the</strong> method to <strong>the</strong> design of a<br />
reservoir is presented. A comparison is made with a reservoir<br />
design based on <strong>the</strong> recorded flows at <strong>the</strong> site.<br />
RESUMEN<br />
Los modelos estocásticos son una buena herramienta para<br />
el estudio de los recursos hidráulicos pero los métodos de SL<br />
mulación basados en series de pequeña extensión son muy peli-<br />
grosos se presenta un método de modelo estocástico en el cual<br />
la media y la varianza de los valores mensua<strong>le</strong>s de caudal se<br />
obtienen por comparación a escala regional de las caracteris-<br />
ticas físicas de las cuencas.<br />
La estructura de autocorrelación se basa en los valores<br />
de precipitación y características físicas de una sola cuenca<br />
se aplica este método al dimensionamiento de un embalse y se<br />
compara con los valores que se obtienen a partir de los cauda<br />
<strong>le</strong>s medidos en el emplazamiento de la presa.
366<br />
Introduction<br />
In many parts of <strong>the</strong> world <strong>the</strong>re are litt<strong>le</strong> or no streamflow da<br />
ta. Even in areas where <strong>the</strong>re is a relatively good set of streamflow<br />
data, projects for water resources development are desired for sites<br />
where <strong>the</strong> data do not exist. Oftentimes regional relations are developed<br />
to interpolate or, more rarely, to extrapolate streamflow characteristics<br />
to ungaged sites. The <strong>le</strong>ss data <strong>the</strong>re are, <strong>the</strong> <strong>le</strong>ss<br />
accurate are <strong>the</strong> regional relations based on those data, However, re<br />
gional relations often are <strong>the</strong> only basis for design. Thus, <strong>the</strong> mean<br />
flow, variance of flow, and o<strong>the</strong>r statistical characteristics may be<br />
related to drainage area, mean rainfall, or o<strong>the</strong>r physical basin mea<br />
sures. For instance, in <strong>the</strong> United States of America, multip<strong>le</strong> regre<br />
ssion relations are developed i .e,, c1) fram which can be computed<br />
mean flows and variances of flows for each month in <strong>the</strong> year, as<br />
well as for <strong>the</strong> total year.<br />
Stochastic simulation of streamflow is coming into widespread<br />
use for project design (2). The statistics required for stochastic<br />
simulation usually are based upon streamflow records col<strong>le</strong>cted at or<br />
near <strong>the</strong> project site. The accuracy of <strong>the</strong> statistics estimated for<br />
a stochastic model are deteymined by <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> streamflow re<br />
cords, Therefore, <strong>the</strong> regional relations mentioned earlier are a tool<br />
for supp<strong>le</strong>menting <strong>the</strong> data base for stochastic simulation, In fact,<br />
a reional relation may be superior to records col<strong>le</strong>cted at <strong>the</strong> site<br />
for estimation of some statistical parameters (3). For data scarce<br />
areas regional relations may be <strong>the</strong> primary source for such estima-<br />
tes.<br />
Stochastic simulation models require a know<strong>le</strong>dge of and<br />
estimation of <strong>the</strong> persistence of streamflow. By persistence is<br />
meant <strong>the</strong> degree to which streamflow today affects streamflows<br />
in <strong>the</strong> future. This is sometimes cal<strong>le</strong>d a carry-over effect.<br />
For <strong>the</strong> first order autoregressive models often used for stream-<br />
flow simulation, <strong>the</strong> first order autoregressive coefficient is<br />
sufficient to describe persistence. For more comp<strong>le</strong>x models,<br />
o<strong>the</strong>r measures of persistence may be needed. However, experi-<br />
ence to date indicates that regionalization methods currently<br />
used are inadequate for <strong>the</strong> statistical estimation of persistence<br />
characteristics 141. This is partly because <strong>the</strong>y are strongly<br />
dependent upon subsurface geology, for which no simp<strong>le</strong> regionali-<br />
zation techniques have been developed. Therefore, a methodology<br />
is needed which can be used to estimate <strong>the</strong> correlation structure<br />
of streamflow sequences. This correlation structure could <strong>the</strong>n<br />
be combined with <strong>the</strong> regional relations to develop <strong>the</strong> parameters<br />
for streamflow simulation models for use for project design in<br />
data-scarce areas.<br />
-
Method of Approach<br />
367<br />
Model choice and parameter estimation are <strong>the</strong> keys to<br />
effective use of stochastic streamflow sequences in hydrologic<br />
designs such as that of sizing a reservoir. The mixed-autore-<br />
gressive-moving-average (ARMA) model described by Moss [5]<br />
provides a scheme by which parameter estimation may be performed<br />
with a minimum of hydrologic data. It is a model that preserves<br />
many of <strong>the</strong> statistical characteristics that are commonly associ-<br />
ated with streamflow sequences. ARMA models can be developed<br />
that are covariance stationary and that preserve <strong>the</strong> memory of<br />
<strong>the</strong> streamflow process for longer periods than do <strong>the</strong> more com-<br />
monly used autoregressive models 161. A scheme with such<br />
attributes would seem to <strong>le</strong>nd itself to <strong>the</strong> development of<br />
data for design decisions in those cases where actual hydrologic<br />
data are too few to provide adequate solutions.<br />
A first order ARMA model for streamflow may be defined as<br />
M n = a M + b Pn-l + c Pn<br />
n- 1<br />
where M and P are <strong>the</strong> streamflow and effective precipitation,<br />
n<br />
respectively, for <strong>the</strong> nth time interval and a, b, and c are<br />
coefficients that are related to <strong>the</strong> basin characteristics.<br />
Moss 151 has shown that, if <strong>the</strong> baseflow from a basin behaves<br />
as a linear reservoir, that is<br />
-kt<br />
Qt = e QO<br />
where Q is discharge at time, t, and k is a constant, related to<br />
<strong>the</strong> geoiogy and measuring <strong>the</strong> streamf low recession rate , <strong>the</strong> ARMA<br />
parameters can be evaluated as<br />
-k<br />
a = e (3) I<br />
k (Tn-l)<br />
c i l - r e<br />
n<br />
and where r is <strong>the</strong> ratio of infiltration to effective precipi-<br />
tation and qn is a measure of <strong>the</strong> time distribution of effective<br />
precipitation during <strong>the</strong> nth time interval. T is in reality a<br />
random variab<strong>le</strong>, so that <strong>the</strong> ARMA model does n8t strictly des-<br />
cribe <strong>the</strong> streamflow process. However because T is usually<br />
restricted in its variations, <strong>the</strong> ARMA approximaeion may still<br />
be useful.<br />
(5) ,
368<br />
In many instances meteorologic information ei<strong>the</strong>r in <strong>the</strong><br />
form of raw-data or regional relations, such as maps, is avail-<br />
ab<strong>le</strong> where hydrologic data are not. Under such circumstances<br />
<strong>the</strong> parameters b and c can be defined as <strong>the</strong> expected values of<br />
equations 4 and 5, and <strong>the</strong> meteorologic information can be used<br />
to fit parameters to <strong>the</strong> ARMA model in order to generate syn-<br />
<strong>the</strong>tic 5treamflow sequences. Parameters b and c were defined<br />
in this study by assuming <strong>the</strong> T for each month was a uniform<br />
random variab<strong>le</strong> with a range from zero to one. These sequences<br />
can be used in design procedures, such as <strong>the</strong> sequent-peak<br />
algorithm of Thomas [71, in <strong>the</strong> same manner as actual observed<br />
discharge records. The model is not a perfect transfer mechanism,<br />
however, and <strong>the</strong> resulting designs will contain modeling errors<br />
in addition to <strong>the</strong> time-sampling errors that are inherent in <strong>the</strong><br />
meteorologic data. Similar time-sampling errors would be inclu,ded<br />
in actual streamflow records were <strong>the</strong>y availab<strong>le</strong>. Judgement of<br />
<strong>the</strong> model as a design tool should be relative to <strong>the</strong> best avail-<br />
ab<strong>le</strong> alternative because of exact design methodology does not<br />
exist.<br />
If <strong>the</strong> ARMA model is used in <strong>the</strong> manner described above as<br />
a monthly streamflow generator, <strong>the</strong> statistical moments of<br />
streamflow that will be preserved in <strong>the</strong> long run can be extracted<br />
from equation 1. The resulting correlation structure has been<br />
described by Moss [51. For <strong>the</strong> mean monthly streamflows a series<br />
of twelve linear equations is required:<br />
EIMnI = aEIMn-ll + bEIPn-lJ + cEIPnl, n = 2,12;<br />
12 12<br />
E[Mn] = c E[Pn]<br />
n= 1 n= 1<br />
where E[-] is <strong>the</strong> expected value or average of <strong>the</strong> variab<strong>le</strong> con-<br />
tained within <strong>the</strong> brackets. Similarly, for <strong>the</strong> variance and<br />
covariances of monthly streamflow<br />
Var [M ] = a2 Var IEln,ll + (b2 + 2abc) [Var Pn-ll<br />
n<br />
where Var 1.1 is <strong>the</strong> variance of <strong>the</strong> variab<strong>le</strong> contained within<br />
<strong>the</strong> brackets, and
= 1,12 i y = k-2-ke-2k- 2e -2k + 4e-k ] / Sk2<br />
369<br />
The coefficient 0.13 that appears In equation 8 was defined<br />
empirically by Monte Carlo methods by MOSS i51. The solutions of<br />
<strong>the</strong>se three sets of equations, although not necessary for <strong>the</strong><br />
imp<strong>le</strong>mentation of <strong>the</strong> model, yield estimates of <strong>the</strong> streamflow<br />
characteristics that can be examined for reasonab<strong>le</strong>ness prior<br />
to <strong>the</strong> design step.<br />
Data requirements for testing <strong>the</strong> model<br />
In order to test <strong>the</strong> model in a realistic design procedure,<br />
an existing 58-year streamflow record for <strong>the</strong> Toccoa River ne'ar<br />
Dial, Georgia, USA, was routed through a sequent peak algorithm<br />
to determine <strong>the</strong> reservoir capacity that would be required to<br />
meet <strong>the</strong> monthly water demands shown in figure 1. The demands<br />
described in figure 1 are hypo<strong>the</strong>tical, but <strong>the</strong>y vary seasonally<br />
in a realistic manner. The average demand is about fifty percent<br />
of <strong>the</strong> average streamflow, estimated from <strong>the</strong> existing record,<br />
for <strong>the</strong> Toccoa-Dial site.<br />
The ARMA model was subsequently used to generate 50 equally<br />
likely sequences of 58 years of monthly streamflow. For each<br />
syn<strong>the</strong>tic sequence a reservoir capacity, which could be compared<br />
with that defined by <strong>the</strong> actual record, was determined. Precipi-<br />
tation records from an existing station, Blue Ridge Dam, that is<br />
approximately 5 mi<strong>le</strong>s downstream from <strong>the</strong> streamgaging station<br />
were used in conjunction with Thornthwaite 181 estimates of<br />
evapotranspiration to estimate <strong>the</strong> mean monthly effective precipi-<br />
tation for each month. The standard deviations of monthly pre-<br />
cipitation for each month were assumed to be equal to those of<br />
<strong>the</strong> measured precipitation at <strong>the</strong> Blue Ridge Dam site. The use<br />
of variance of point precipitation as a measure of variance of<br />
precipitation over <strong>the</strong> basin tends to ovesestimate this parameter;<br />
however, because effective precipitation, which is <strong>the</strong> difference<br />
between precipitation and evapotranspiration, probably has a<br />
higher variance than precipitation, <strong>the</strong> assumption of variance<br />
of point precipitation as a surrogate for variance of basin-wide<br />
effective precipitation should not be unreasonab<strong>le</strong>. The twelve<br />
estimates of mean effective precipitation and <strong>the</strong> twelve esti-<br />
mates of standard deviation of effective precipitation in<br />
conjunction with <strong>the</strong> assumption of log normality of effective<br />
precipitation were used to syn<strong>the</strong>size <strong>the</strong> 58-year records of<br />
effective precipitation, which are converted to syn<strong>the</strong>tic stream-<br />
flow records by <strong>the</strong> use of equation 1.<br />
Fitting of Parameters<br />
For <strong>the</strong> application of equations 6-8 to convert an effective<br />
rainfall to a runoff record, four inputs are necessary. First,<br />
of course, is <strong>the</strong> record of rainfall itself. Second, rainfall<br />
must be converted to rainfall excess by abstracting losses.
370<br />
This was done for <strong>the</strong> Toccoa basin through use of <strong>the</strong> Thornthwaite<br />
equation to estimate evapotranspiration. The Thornthwaite<br />
equation was chosen for reasons of simplicity. If adequate data<br />
were availab<strong>le</strong> a more accurate estimation might be made, such as<br />
by <strong>the</strong> Penman formula. Third, <strong>the</strong> separation of effective precipitation<br />
into direct runoff and infiltration must be performed<br />
by se<strong>le</strong>cting r . Studies by <strong>the</strong> Tennessee Val<strong>le</strong>y Authority<br />
n<br />
(Eklund, C. D., oral communication) indicate an average value<br />
of approximately 0.7 for r. This value was used for each month.<br />
Stochastic Simulation Results<br />
The utility of <strong>the</strong> methodology for deriving parameters for<br />
stochastic generation of streamflow was tested next. The parameters<br />
shown in figures 2-4 were used with an ARMA model to generate 50<br />
syn<strong>the</strong>tic sequences of 58 years in <strong>le</strong>ngth, <strong>the</strong> same <strong>le</strong>ngth as<br />
<strong>the</strong> historical streamflow record. The sequent peak algorithm<br />
was <strong>the</strong>n used with <strong>the</strong> demand curve of figure 1 to generate<br />
design reservoirs for each of <strong>the</strong> 50 sequences. The 50 design<br />
reservoirs for <strong>the</strong> syn<strong>the</strong>tic sequences were <strong>the</strong>n arrayed into<br />
a probability distriaition, as shown on figure 5. The distri-<br />
bution of design sizes is approyirnately normal. The mean value<br />
of <strong>the</strong> design size is fortuitously close to that size which was<br />
based upon <strong>the</strong> recorded flows.<br />
The apparent reason for <strong>the</strong> excel<strong>le</strong>nt agreement between<br />
average simulated and recorded design sizes probably results<br />
somewhat from compensating errors. The design storage is<br />
equiva<strong>le</strong>nt to about one-seventh of <strong>the</strong> mean annual flow. The<br />
excesses of demand over supply occur mainly during August to<br />
October. During that critical period <strong>the</strong> model overestimates<br />
both mean flow (figure 2), which tends to decrease storage<br />
requirement, and variability of flow, which tends to increase<br />
storage requirement (figure 3). The covariance structure is<br />
closely reproduced.<br />
Figure 5 indicates that <strong>the</strong> stochastic simulation is rela-<br />
tively realistic with respect to reservoir design size, which<br />
is related to <strong>the</strong> variance and <strong>the</strong> correlation structure of<br />
flows. The fact that <strong>the</strong> actual record and <strong>the</strong> average of <strong>the</strong><br />
syn<strong>the</strong>tic records yield about <strong>the</strong> same design size indicates<br />
that <strong>the</strong> structure of <strong>the</strong> runoff series is maintained adequately.<br />
Therefore, in data-scarce areas, this approach may be a tool for<br />
use in project design.
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
References<br />
371<br />
Carter, R.F., (1970). Evaluation of <strong>the</strong> surface water data<br />
program in Georgia, U.S. Geol. Survey open-fi<strong>le</strong> report.<br />
Fiering, M.B., and Jackson, B.B., (1971). Syn<strong>the</strong>tic<br />
Streamfiows, Water Resources Monograph No. 1, American<br />
Geophysical Union, 98 p.<br />
Ber:son, M.A. , and Matalas, N.C., (1967). Syn<strong>the</strong>tic Hydrology<br />
based on regional statistical paramerers, Water Resources<br />
Research, 3(4), pp. 931-935.<br />
Thomas, D.M., and Benson, M.A., (1970). Generalization of<br />
streamflow characteristics from drainage-basin character-<br />
istics, U.S. Geol. Survey Water Supply Paper 1975, 55 p.<br />
MOSS, M.E., (1972). Serial-Correlation Structure of Discre-<br />
tized Streamflow, U.S. Geol. Survey open-fi<strong>le</strong> report.<br />
O'Connell, P.E., (1971). A simp<strong>le</strong> stochastic modeling of<br />
Hurst's law, Symposium on Ma<strong>the</strong>matical Models in Hydrology,<br />
IASH/UNESCO, Warsaw, Poland.<br />
Fiering, M.B., (1967). Streamflow syn<strong>the</strong>sis, Cambridge,<br />
Harvard Univ. Press, p. 69-73.<br />
Veihmeyer, F.J., (1964). Evapotranspiration, in Hand<strong>book</strong> of<br />
applied hydrology (edited by V. T. Chow), New York, McGraw-<br />
Hill CO., p. 11-26.
372<br />
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ABSTRACT<br />
CHCICE OF GENERATING MECHANISM IN SYNTHETIC<br />
HYDROLOGY WITH INADEQUATE DATA<br />
by<br />
P.E. O'CONNELL<br />
Department of Civil Engineering, Imperial Col<strong>le</strong>ge,<br />
University of London<br />
and<br />
J.R. WALLIS<br />
IBM T.J. Watson Research Center, Yorktown Heights,<br />
New York 10598<br />
A formidab<strong>le</strong> prob<strong>le</strong>m in syn<strong>the</strong>tic hydrology is <strong>the</strong> choice of a<br />
model which will best represent <strong>the</strong> generating mechanism of<br />
streamflow, which is unknown. Heretofore, such a choice has primarily<br />
been based on a statistical matching between historic record parameters<br />
and <strong>the</strong> parameters of <strong>the</strong> generating mechanism. However, for<br />
short historic records, an equally good match may be obtained for a<br />
number of models and statistical tests are not powerful enough to<br />
determine <strong>the</strong> appropriate model. Alternative mechanisms, however, may<br />
yield quite different design results, resulting in ei<strong>the</strong>r overdesign<br />
or underdesign with economic regrets in ei<strong>the</strong>r case. It is suggested<br />
that an alternative approach to model choice would be a decision<br />
<strong>the</strong>oretic approach, where <strong>the</strong> choice of model is based on an economic<br />
regret function, and where <strong>the</strong> model which gives <strong>the</strong> minimum overall<br />
regrets would be <strong>the</strong> appropriate choice. An examp<strong>le</strong> of this approach<br />
is given where flows are generated by a lag-one Markoj and an ARI A<br />
(1, O, 11 process and <strong>the</strong> secuent peak algorithm is u ilised in a<br />
deterministic sense for reservoir design toge<strong>the</strong>r with simp<strong>le</strong> economic<br />
regret functions.<br />
RESUME<br />
Un problème trzs diffici<strong>le</strong> de lrhydrologie syn<strong>the</strong>tique crest<br />
ltadoption d'un mode<strong>le</strong> <strong>le</strong>quel reprgsente <strong>le</strong> mieux <strong>le</strong> mecanisme gênétatrice<br />
de l'écou<strong>le</strong>ment, <strong>le</strong>quel est inconnu. Jusqu'ici ce choix été<br />
fondé principa<strong>le</strong>ment sur un assortiment des parametres des données<br />
historiques avec des parametres du mécanisme génératrice, Toutefois,<br />
en cas des re<strong>le</strong>vés historiques courts on obtient peutêtre un assortiment<br />
aussi bien en employant plusieurs mode<strong>le</strong>s et <strong>le</strong>s épreuves statistiques<br />
n'onts pas assez fortes pour détermin'e la modè<strong>le</strong> propre.<br />
Toutefois, <strong>le</strong>s autres mécanismes donnent peutêtre <strong>le</strong>s touts autres<br />
résultats pour dessein, <strong>le</strong> rêsultat est <strong>le</strong> sur-dessein ou <strong>le</strong> sous-<br />
-dessein accompanid en tout cas pap des regrets economiques., On prop'ose<br />
que l'approche alternatif au choix de la modè<strong>le</strong> est llqpproche de<br />
la théorie des decisions par <strong>le</strong>quel <strong>le</strong> choix de la modè<strong>le</strong> est fond6<br />
sur une fonction des regrets economiques et par <strong>le</strong>quel la modê<strong>le</strong> qui<br />
fourni <strong>le</strong>s moins regrets totals sera <strong>le</strong> choix propre. Un exemp<strong>le</strong> de<br />
cette approche est fourni quand <strong>le</strong>s écou<strong>le</strong>ments sont produi't par un<br />
procéds Markovien de deu1 retard et un proc6dé ASEVA 0, O, 12 et<br />
l'algorithme des pics successifs est utilisé au sens déterministique<br />
pour <strong>le</strong> dessein du réservoir avec des fonctions de regrets economiques<br />
ega<strong>le</strong>s.<br />
Y
378<br />
Introduction<br />
The ootid desig, of a water resource EF:E~~Z reqzres 'slow<strong>le</strong>dge of<br />
future flows within t h elstem, which, i- tu--r., ITlies tkat :Be gcze-xting<br />
process of <strong>the</strong> tows is kofi. Eowever, <strong>the</strong> gtrerazirg Srocess of srreamflow<br />
is gererallg &om, 2.~5 lilirelv projectiors 05 %hre zlous, cal<strong>le</strong>c -T-<strong>the</strong>tic<br />
strezzflows, may be gererated using approxic-:iozs to tbe UrCsrlyirg gereratirg<br />
process.<br />
(a)<br />
(b)<br />
The apFros3ation procedure involves. -<br />
<strong>the</strong> postulation of t'ne underlybg generatirg Frocess ar6 its specification<br />
through a set of przeters,<br />
<strong>the</strong> estimtion of <strong>the</strong> parameter values from a historic sequence, or<br />
through some alterrative strategy.<br />
Some generating processes cvreztly availat<strong>le</strong>, proFrties of historic<br />
secperces, and techniques of paraneter estinztio: inU now be considered<br />
briefly.<br />
Gener2tir.g Trocosses<br />
The postulation of a generatirg process kas 'reretofore been b e d on its<br />
ability to generste sjr<strong>the</strong>tic stremÎlows resez5lirg historic streznflows in<br />
terms of pärameters whic'n are thougit to infli<strong>le</strong>rce <strong>the</strong> äesie OÏ <strong>the</strong> water<br />
resorce system, [I), which necessitates that tte 3rocess re3reserts a redistic<br />
model of streamflow. The lag-oze ?%rkov process 'caä 5ee3 ra<strong>the</strong>r widely<br />
accepted as beizg ca-pb<strong>le</strong> of fulfillizg this l2:ter ro<strong>le</strong> mtll discrete time<br />
fractional Gaussiul noise (dfGn) was aävocateä as s more resiistic moäel of<br />
strezÏlow [2]. The spent for dfG3 as a ereriting process oÏ streãmflow<br />
finds its roots in <strong>the</strong> work of Hurst [:I, [4f u50 foud that, for some 800<br />
geopkrsical tine series, including streanflow<br />
- - 1 1<br />
S<br />
h<br />
where Iz/S is temed <strong>the</strong> resca<strong>le</strong>d ra^ge a d n is th-e record <strong>le</strong>r-gth. The<br />
expone'it h in equation (1) was found to have a a7e:age value OÏ 0.73 uith a<br />
stank-d deviati02 of O.Ca. For <strong>the</strong> lag-oae 3k-207 process =fi o<strong>the</strong>r processes<br />
lying withiri <strong>the</strong> BroEis Comain of attractioz, h equals 0.5, uhi<strong>le</strong> for dfûn,<br />
h q v assume ayy value in <strong>the</strong> rGge C < h < 1, vit2 <strong>the</strong> excepriori of h = 0.5,<br />
and b C.5 of particular interesi. Values o? h > C.5 are<br />
s~-no;nous with long te,- srcistexe, vith <strong>the</strong> distat psr; exertizg mall<br />
but azab<strong>le</strong> effects on present behaviour.<br />
The gelieration of a Lszmp<strong>le</strong> of d-Gn req-aires infixite Ember of operatios,<br />
azd, con~equeztk~, a~~roxinaiio?s are reTïred ir, oräer to Io-date<br />
dfG2 =s an operational gsreratkg process. Tses? assro-xkatioïs w e quite<br />
:oces of szem3ou dictirct fron <strong>the</strong> agroe=z:ioz or' t're gezerz;irn - _<br />
refezred to Tr3rLouslr ; -&e a~~z-o.sirsziors 10 è3z äm -2$rss17es sx5sequeitly<br />
used to approxicsite <strong>the</strong> gererating srocess of sirsmflou.
To date, a =.aber of z_nproximtions to d3n ha7e been >-o?ored. Y%zdelbrot<br />
i53 kzs proposed B fast d% approximatiori, wLich requires <strong>le</strong>ss ore~atiozs ;CI<br />
elierzte t?mn <strong>the</strong> tyge I ard II fractiord noise ap?roxinntio-s Frooosed ;-;tially<br />
LJ , Cut he did rot extend <strong>the</strong> docuieentäïion to <strong>the</strong> <strong>le</strong>vel iecesszyy for syr-tetic<br />
i~y.gdrolo~. &,ser5 02 <strong>the</strong> tne II ayproxibtio: Tropsed initidly by Yzlldelkot<br />
aLd Ydis [o], Y'tzhs =à Y&is [7] L57e pro-osed a filtered ,'ractionzl<br />
noise asproximatioil, and have formulated <strong>the</strong> gezeratizg process for <strong>the</strong> puTses of sgz<strong>the</strong>tic hydrology in terns of <strong>the</strong> ppiation rea, vzriarce, lag-or-e exiocorrelation<br />
u-d h, <strong>the</strong> Hust coefficienz. The KXP! (l,O,l) process [S) IFS<br />
beeo found by O'CozeU [g] to offer a sirp<strong>le</strong> approfixtion to dSs. The a ~roxicatioi<br />
is sufficiect in <strong>the</strong> serse, that, for a certais rage of pûzameter Taues<br />
adecute agreemerc uith Burst's law (e-tion (1)) is obtaired uithin sequezces<br />
which are sufficiel;tlg lorg for <strong>the</strong> purooses of syctììetic hyckolog. More<br />
recently, <strong>the</strong> broke2 line orocess has been progosed by Kejia et zl [IGJ as e<br />
model of <strong>the</strong> gezorati-g process of strezflow ; houever, ì-kdelbrot [Il] hzs<br />
shorn that <strong>the</strong> broke2 line process may be regzzded nìerely as zz ¿?.;prOxiuEtiGZ<br />
to fractional Gaussian noise.<br />
Historic Seauences<br />
Frequently, a historic seauence of a-~uai streamflow has been relied upr<br />
to determine <strong>the</strong> existence 9. aon-existence of persistence <strong>the</strong>reir. For this<br />
pu-?pose, tests of significbnce for inde-zdence zre availab<strong>le</strong> bâsed on <strong>the</strong><br />
<strong>the</strong>ory of runs [12], LI31 ar-d <strong>the</strong> distrikation of <strong>the</strong> 12 -one serial correlation<br />
coefficient [1q. However, Wallis a d Patalas [75! have fomd that for<br />
a variety of such tests, <strong>the</strong> probability of type II error (i.e. <strong>the</strong> probability<br />
of accepting <strong>the</strong> rull hypo<strong>the</strong>sis of independeme when it is false) is extrerely<br />
high for <strong>the</strong> sequexe <strong>le</strong>ngths usually avzilab<strong>le</strong> in hydrology.<br />
of persistence, estimates oi <strong>the</strong> lag-one autocorrelation tend to be biased<br />
towards zero, with <strong>the</strong> bias increasing with <strong>the</strong> intensity of <strong>the</strong> persistence,<br />
and to this latter Îact may be ascribed o=e of <strong>the</strong> reasons for <strong>the</strong> low power<br />
of <strong>the</strong> Acderson test. Where storage design is to be considered, <strong>the</strong> econorcic<br />
cor:sequences of ty-e II errors may welì be costly. Assuming that evidence of<br />
persistexe has Leer established, <strong>the</strong> hisioric sequence h s generdly been<br />
relied upon to &-Ovide reliab<strong>le</strong> evidence as to uhich ge2eratiig Eechanism is<br />
<strong>the</strong> appropriate ore for <strong>the</strong> flows. A model of <strong>the</strong> mderlyicg geserating<br />
necliacism is fitted EO <strong>the</strong> historic sequence, and goodness of fit testS.De<br />
<strong>the</strong>n enployed to àetermine <strong>the</strong> adeauacy 05 <strong>the</strong> model. A set of procedures for<br />
model €itti% ami vdidatios &ve been set out by Box axd Je-&ir?s [SI; however,<br />
whi<strong>le</strong> s ~ch procedees may provide reliab<strong>le</strong> resdts for <strong>the</strong> locger recorded<br />
sequerces U S ~ Y<br />
In <strong>the</strong> presezce<br />
availab<strong>le</strong> in industry E d economics, <strong>the</strong>y are liab<strong>le</strong> to<br />
pyovide mis<strong>le</strong>adkg results for 'short' mual strearflow sequences. A question<br />
mises as to what <strong>le</strong>ngth of record pay be considered 'short' ; uithout digressing<br />
too much, i; suflices to say th-t as 1or;g-terrn persisterce increases,<br />
<strong>the</strong> hfornztion coztent of a record [IÓ] decre- =ses.<br />
Whi<strong>le</strong> <strong>the</strong> kg-oze autocorrelation has bee? used in <strong>the</strong> _past as a measure<br />
of Fereistence, it essentially measures ody &ort-te,ri -ersistexce, ad, to<br />
quzzitify long-term Frsistezce, o<strong>the</strong>r messures must be used. Whi<strong>le</strong> h, <strong>the</strong><br />
Emst coeîficierit, is a measure of long term -xrsistezce, estimates from sms-ll<br />
sans<strong>le</strong>s gererated a lag oie Karkov process c d arrolaoatiox zo dftrri havc<br />
beer sko~?: to be hi- biased and extrezrly vz-iab<strong>le</strong>, 1V++ a d <strong>the</strong>refore II-relisbib<strong>le</strong><br />
for chooaì=& betmees short memoLT processes for ubi@ h = 0.5 and 10%<br />
379
380<br />
memory processes, for which h > 0.5. Xore receLtly, Wallis ard O'Cozell k81<br />
have attempted to separate sequences genented by a lolig nezory LW! (l,O,l)<br />
process from seq-Jences generated by a short meao,ry lag ore Yzrirov process<br />
using <strong>the</strong> distribution ol US, <strong>the</strong> resca<strong>le</strong>d rmgo, deriveä t ~ough Xocte Carlo<br />
simulrtions. For <strong>the</strong> sequelice <strong>le</strong>-gths w d l y anilab<strong>le</strong> ic. hydrology, <strong>the</strong>y<br />
foud that reliab<strong>le</strong> sepration was not possib<strong>le</strong>.<br />
Historic sequences have bee:: suggested by Slack 1191 as sometimes providir4<br />
litt<strong>le</strong> more tha? ari illugoil of what <strong>the</strong> wderlyi-4 gezeratizg process is, 2c<br />
a sizg<strong>le</strong> realization of H stochastic process will rarely 'have Ezap<strong>le</strong> estimates<br />
of parameters e q d to <strong>the</strong>ir populatiozi counter-ts. Irdeed, Clack has also<br />
noted that a gereratirg crocess mg 6ezy itself ir <strong>the</strong> seXe that <strong>the</strong> process<br />
may generate firite sänylrs to wnich <strong>the</strong> gezeratirg process itself W o t be<br />
fitted. Slack 123J has illustrated this point more fully for a miltivaiate<br />
lag-ore Markov process, for which certain constraicts exist OP <strong>the</strong> raqe of<br />
serial and cross correlations that <strong>the</strong> process ca3 acceyt.<br />
Coxequectly, <strong>the</strong><br />
fact that a historic sequence yields prameter estimates macceptab<strong>le</strong> to a<br />
model cannot be readily taken as evidence that <strong>the</strong> model is an inappropriate<br />
one.<br />
For <strong>the</strong> univariate lag-one Markov process, <strong>the</strong> question of 'self denial'<br />
does rot arise as estiusates of pl, <strong>the</strong> lag one aitocorrelation, will always<br />
lie in <strong>the</strong> range -1 < e, < 1, for uhich values of <strong>the</strong> process is stationary,<br />
and flous will be real-vdued. However, for dfGn, waich for zero mean and unit<br />
variace, is characterized by <strong>the</strong> covariance structure<br />
where s is <strong>the</strong> lag and h is <strong>the</strong> Hurst coefficient, ssmp<strong>le</strong> statistics be<br />
in conflict with <strong>the</strong> covariance structure of <strong>the</strong> nodel. Approximations to dfGn<br />
serve to approximate <strong>the</strong> covariance structure C(s,h), where C(l,h) is uniquely<br />
specified by h. As <strong>the</strong> pocess is specified bhre unit varizce, <strong>the</strong>n C(l,h)<br />
is <strong>the</strong> lag one serial correlztion, Q,, which for li > 0.5 must be positive, as<br />
must C(s,h) for s > 1.<br />
Eowever, finite sequences from a gezerating proce?<br />
with a covariance structure approGnating C(s,k! rzq yield h > 0.5 uni<strong>le</strong> QI < O,<br />
or, alternatively, h < 0.5 whi<strong>le</strong> el > O, which are incopatib<strong>le</strong> with <strong>the</strong><br />
structure of <strong>the</strong> model. However, <strong>the</strong> dari model isI\employed primily to<br />
preserve 102% ruII effects 2.s evidenced by values of h > 9.5, a d <strong>the</strong> evidence<br />
supplied bx QI, which is a measure of short run effects, nag be ignored.<br />
Provided QI > C and h > û.5, <strong>the</strong> filtered type II ayproxioratioil proposed by<br />
Matalss and Wallis [7] zlloïs <strong>the</strong> sinultarieous :reservation oÎ estimates of<br />
QI azd h through <strong>the</strong> ixorporation of an extrs filterizg yameter into <strong>the</strong><br />
generating process. As a result, <strong>the</strong> form of <strong>the</strong> covariace Îwction for dfGn<br />
not be closely followeä for small s- whi<strong>le</strong> for large s, <strong>the</strong> differerce<br />
shoulà be negligib<strong>le</strong>.<br />
The possibility of 'self-denial' eests for <strong>the</strong> ARIEiA (1 ,O,l) process.<br />
In or6er to ensure that <strong>the</strong> process is statiore? ar6 invertib<strong>le</strong>, t)ïe Lwp<strong>le</strong><br />
space for <strong>the</strong> paneters of <strong>the</strong> pzocess, azd a, skown i2 figure (la) is<br />
defired as -1 < # < +I, -1 < 8 < +I. The corres-rciirg Lssrif<strong>le</strong> -ce for QI<br />
and e2 is sho.ir.11 iri figre (13). Hoïever, fizite -<strong>le</strong>s froa <strong>the</strong> process<br />
may well yield estimtes oÎ pl u.d e2 lyhg ouiside <strong>the</strong> zcceptabìe rage.
3 81<br />
As a remit, a historic sequence m o t be relied u-pn to -est <strong>the</strong><br />
correct generatkg process for <strong>the</strong> flous, 2nd ET well <strong>le</strong>zd to 2 moàel beicg<br />
se<strong>le</strong>cted uhich is not reyesentative of <strong>the</strong> gereratixg rec'mim of streamflou,<br />
Houever, parameter estication t hrow statistics zes-mred is. .zi historic<br />
sequerce is lugely relied upon for matching a gezeraticg process n th a his-<br />
toric streamflom sequence.<br />
Parezeter Estimtion<br />
hs already ooted, a geuerating process is geaerally specified by a set<br />
of population Faeters, denoted as {a] = (aq,u2, ... aE), estimtes of uhich<br />
must be obtained from a historic sequezce before tie generating process may be<br />
rendered operatioral. For <strong>the</strong> lag-ore Ykrkov process, <strong>the</strong> set { a 1 usu2i.l~<br />
comprises <strong>the</strong> me=, vziriaxe, skeuness a d lag-ore autocorrelation 2t each<br />
site, and in <strong>the</strong> ttiuìtisite case, lag-zero cross-correlatioss betveeri sites.<br />
For qproximatiors to dL%, sn additiorial parmeter, <strong>the</strong> Hurst coefficieat,<br />
is izcluded. For estimztioa purposes, <strong>the</strong> mettod of moments is gezErallg<br />
emploFed [I ] with <strong>the</strong> L.mall samp<strong>le</strong> moment estimate of a parameter, a$, obtained<br />
from a historic sequence or' <strong>le</strong>ngth n bei% equated to its correspondxg pop dation parameter in <strong>the</strong> generating process, ai. Such a procedure assumes<br />
that<br />
E [",3 = a<br />
n<br />
i.e. that <strong>the</strong> estimate ai is statistically unbiased. However, in <strong>the</strong> presence<br />
of persistence, recent sicdies have shown that this assumption is not justified<br />
with respect to estimates of <strong>the</strong> variance [21], <strong>le</strong>g-one autocorrelation [I51<br />
and <strong>the</strong> Hurst coefficient [I?].<br />
The bias affects <strong>the</strong> resemblance uhich is<br />
obtei2ed between historic axxi syn<strong>the</strong>tic sequences, and is likely to adversely<br />
influence system desigri Mess some small samp<strong>le</strong> bias corrections are applied<br />
to <strong>the</strong> estimates. Io order to illustrate this liztter point, <strong>the</strong> small samp<strong>le</strong><br />
properties of generatiw processes rnust be considered.<br />
Small Samp<strong>le</strong> Prouerties o0 Ge-e<strong>le</strong>ratirE Processes<br />
!The lag-one biarkov generating process m y be specified as<br />
where p 6 and p are <strong>the</strong> pophtion mean, variance and lag-one autocorrelation<br />
coefficient, and et is an edependectly distributed random nomal variab<strong>le</strong><br />
with zero mean ard unit vâriance. Estimation of <strong>the</strong> paraneters e, 6 and p<br />
will cou be considered.<br />
In equation (2), e may be estimated as <strong>the</strong> lag-one serial correlation<br />
coefficient uhence,
382<br />
ñouever, avaïL&<strong>le</strong> estkators of pl yield biased estimates of e [UJ. If<br />
<strong>the</strong> estimator oÏ QI suggested by Box a d Je_nkins is used, <strong>the</strong>n e approximately<br />
satisfies<br />
For n = 25 and e= 0.3, aen E[@] = 0.21. If equation (3) is rearranged <strong>the</strong>n<br />
E@] + l/n<br />
e =<br />
(4)<br />
1 - 4/n<br />
A<br />
If e is obtahed :rom a historic sequence of size n using <strong>the</strong> Box and Jenkins<br />
algorithm, and E{Q] is replaced by 6 in equation (4) <strong>the</strong>n <strong>the</strong> ensuirg<br />
Ii<br />
estizate, e *, hill be Enroximately urtbissed. If Q is used in equation (21,<br />
<strong>the</strong>n estimates of <strong>the</strong> lzg-o?e autocorrelation measured in syn<strong>the</strong>tic sequences<br />
of size u, e , will satisfy<br />
whi<strong>le</strong>, if <strong>the</strong> <strong>le</strong>ngth of a syn<strong>the</strong>tic sequence approaches infinity, <strong>the</strong>n<br />
i.e.<br />
with<br />
where<br />
<strong>the</strong> proper resemblance ia maintained between historic and sy<strong>the</strong>tic sequences<br />
respect to <strong>the</strong> lag oce autocorrelation.<br />
<strong>the</strong>n whi<strong>le</strong><br />
2<br />
If <strong>the</strong> small samp<strong>le</strong> estimate of <strong>the</strong> variance, 6 , is defined as<br />
'> n<br />
s2 = (Xt -1) 2<br />
n- 1<br />
t=l<br />
2<br />
If g = O, <strong>the</strong>n E {s 1 = 62 uhi<strong>le</strong> if Q > O <strong>the</strong>n fh, Q> is positive, uhereby<br />
s2 terds to underestimate 8, with th2 bias iilcreasing as e ixreases afd<br />
n decreases. For LL = 25 c d = 0.5, f(n,Q> = C.963, SO tkt <strong>the</strong> bias Kill<br />
ger-erdly not be too severe for m-ual streamflou sequences. 3 order to correct<br />
for <strong>the</strong> bias in E? mezs-zed in a historic sequence, a unbiased estimate oÎ <strong>the</strong><br />
populztion variszce 2 %,y be defined as<br />
a2 = s2/f(n,p (8)<br />
A<br />
(5)
vhereupon<br />
E{$] = E{s2J/f(n,,) = 6 2<br />
"2<br />
so ttat 6 wiii be unbised. Hoïever, <strong>the</strong> foregoing correction- procedure pre-<br />
m e s that e is Lcco~",, &de, k practice, od? ar estfate, 8 , rill be<br />
avzïilzb<strong>le</strong>. .In tkis sitti-tiori =y be corrected for tis ii6 &ea* outlked<br />
and ar: estimate of <strong>the</strong> vzrkce del'bed as<br />
^62 = s2/f(n, e*> (9)<br />
and that<br />
h<br />
may be evaluated at <strong>the</strong> expected &lue of Q' which yields<br />
<strong>the</strong>n<br />
Hvïever, nei<strong>the</strong>r of <strong>the</strong> above two assu+ions are liab<strong>le</strong> to hold, c d a2 a<br />
result, difficulty is eilcoutered in definirg irnbiaseci estimate of 6 .<br />
Never<strong>the</strong><strong>le</strong>ss, equation (9) is likely to yield FS estimzte of & which is more<br />
appoxinstely wbissed than <strong>the</strong> straightformd estimate yielded by eqwtiozi<br />
(51 9<br />
383<br />
A''<br />
Using p, 6 as defired in eauation (9) a d e* as defined in equation (41,<br />
a syn<strong>the</strong>tic seculice OZ size n nay be generated using eywitiol? (2). If g2<br />
denotes aa esthte of <strong>the</strong> variaxe mewed tkerein usi% equation (5) <strong>the</strong>n :-<br />
2<br />
E($) a s<br />
whi<strong>le</strong> if <strong>the</strong> <strong>le</strong>qth of a syn<strong>the</strong>tic sequexe aPyoaches kfinity, <strong>the</strong>n aporoxii_<br />
matelg :<br />
E@> -6<br />
2
3 84<br />
A2<br />
where IS is defired via equation (9). Hence tFe correction procedure allows<br />
<strong>the</strong> growr resenMance betueen historic and syn<strong>the</strong>tic sequences to be ubzirtzhed<br />
aporoemtely.<br />
A fur<strong>the</strong>r quentie uKch may be of interest in rece--voir desigil studies<br />
is <strong>the</strong> variance of <strong>the</strong> sanp<strong>le</strong> mea, 8, uhich for equatiori (2) is given as [2$] :-<br />
nhkh is <strong>the</strong> variace of <strong>the</strong> samp<strong>le</strong> mean for an indeperdont rmdom procefis.<br />
However, foz Q > O, <strong>the</strong> term in braces is positive and greater than unity,<br />
vhereupon 6 will be larger t h for a randon time series. Hence if <strong>the</strong> iaformation<br />
contert of a sequezce is defized as <strong>the</strong> reciproczi of <strong>the</strong> vdance of<br />
<strong>the</strong> -?<strong>le</strong> mean [16], <strong>the</strong>a, as persisteme increases, tke information coatent<br />
relative to <strong>the</strong> mean decreases. For e= 0.3 aLd n = 25<br />
6m<br />
= 0.0724 cr2<br />
Never<strong>the</strong><strong>le</strong>ss, it should be noted that <strong>the</strong> lag-one Y!kov process is<br />
esseritially a short memory process for which h = 0.5. Ir. <strong>the</strong> preseme of locg<br />
term persistence, when 0.5 < h < 1, smdì sam?<strong>le</strong> biases in estimates of <strong>the</strong><br />
variarce ar,d lag-one autocorrelation become more severe. and <strong>the</strong> vaziame of<br />
<strong>the</strong> samp<strong>le</strong> mean-tends to i-crease. For dan, <strong>the</strong> variarce of <strong>the</strong> szmp<strong>le</strong> mean<br />
is given as<br />
2<br />
2<br />
-- 6<br />
(12)<br />
- 2-2h<br />
n<br />
2<br />
where IS is <strong>the</strong> variance<br />
result for white noise.<br />
2<br />
of <strong>the</strong> process. For b = 0.5, 6 reduces<br />
For h = 0.7, for which el = 0.30, and n<br />
3.6246 2<br />
- - = 0.145 6 2<br />
- 25<br />
to <strong>the</strong><br />
= 25<br />
Comprison of equations (11) and (12) illustrstes that for dsz, and, consecuentlg,<br />
for agroximitio-s <strong>the</strong>reto, estimtes of -<strong>the</strong> -?<strong>le</strong> me= are auch more varizb<strong>le</strong><br />
and meliab<strong>le</strong>, than for <strong>the</strong> lag-one I.'!kov &ort memoq process.<br />
Unfortunately, litt<strong>le</strong> is b o m of <strong>the</strong> s-3u samp<strong>le</strong> u-operties of <strong>the</strong> aoproximatiocs<br />
to dan proposed bp Mandelbrot 151, ?!!'das ar0 Kdis [7] and Xejia<br />
et al; [IO], and equatiog (12) will only be apFoximatelg true. An -tic<br />
derivation of such proprties would apear to Se a extreselg &iÎficult tz&<br />
in <strong>the</strong> face of <strong>the</strong> comaex m<strong>the</strong>mtical prooerties of t<strong>le</strong> approdmations. €?QU-<br />
ever; tlie LRDiA (I,O,I) p-ocess is ma<strong>the</strong>mticdly more tractab<strong>le</strong> and some scull,<br />
e l e results s v be derived.
The ABMB (l,O,l) generaticg process is &fined as<br />
where p and 6 are <strong>the</strong> nean and sLa&d deviatioa, $ zrd Q are <strong>the</strong> parameters<br />
of t'ne process, E d qt is u irdeperc<strong>le</strong>nt rzdom varil<strong>le</strong>. ne varhce o? /7<br />
must be defined 2s<br />
var 3, = +%E&- I+&- L<br />
Estirates of <strong>the</strong> parameters jZf 2nd O ~7 be dertved from 2 historic, sequence<br />
throqh estinatirg el z d p2, <strong>the</strong> hg-0r.e =à lag-two zutocorrelation coefî-<br />
icierts, or tho;@ mir: <strong>the</strong> more efficient rethod of -&m likelihood. [8]<br />
Alterzztively, e!, and f my be used to defire estimtes of j? and 8. For<br />
aporoximatiors to dr%, estimates of h Lid e 1 Lime bee= &om to be bizsed,<br />
As h is not<br />
vith tke SiEs kcreaskg kith ircresLzg h 2zd el I [17, 151.<br />
385<br />
<strong>the</strong>oretically de5red ai diff3rer.t from 0.5 for <strong>the</strong> AXPA (l,O,l) process,<br />
which never<strong>the</strong><strong>le</strong>ss yieldsAE fh] in <strong>the</strong> raxge C.5 to 1 for moderateJO large<br />
values of n, <strong>the</strong> bias iil h is difficult to Ruztify. E e bias in el could<br />
possibly be defired ardflim3.ly.k aAsinily fâshion to tkat for <strong>the</strong> lag-ore<br />
Markov process. Even if <strong>the</strong> b is in h zid could be defired, <strong>the</strong> appropiate<br />
bias correctiors m y not be copatiblz [19]. xri alterzative aggroach is to<br />
approximately. rztch observed Qq a d h values iri a =><strong>le</strong> of size n with <strong>the</strong><br />
aprro3riate Z {el] ard E 17 values for <strong>the</strong> ARRIT2 (l,O,l) process pre-<br />
defiEed through exteEsive Monte Carlo simulatioris [25]. This approach<br />
obviates <strong>the</strong> necessity lor bias corrections, but <strong>the</strong> estiptor of h adopted<br />
for <strong>the</strong> simulatios, giyen ifi [4),<br />
n<br />
h = K = {hg. (R/S)] / [Log "/2 1 (14)<br />
does not allow sufficient variability in E{ ^h] between afferent ,sets of para-<br />
meter values to effect a reliab<strong>le</strong> match.<br />
If <strong>the</strong> snail samp<strong>le</strong> estimate of <strong>the</strong> variace is defined as io equation (51,<br />
<strong>the</strong>n O'Connell 12.1 !ES &om that<br />
2<br />
E(s,) = 6<br />
where pl is <strong>the</strong> lag-oze autocorrelation defined as<br />
= (1 - $QI($ - 6)<br />
(1 + g2 - a)<br />
(15)<br />
(16)<br />
0'CoI;i<strong>le</strong>I.l [g] 0-r noted tbt vdues of $ in <strong>the</strong> rmge 2.50 < j? < 0.95 are of<br />
interest i2 modelling lox te-ai._rersis<strong>le</strong>zìce ; for suc3 values of $ m d for<br />
values of el us-:âïïy ercourtered for amua1 szreaflo*', <strong>the</strong> bias in s2 is<br />
gererqy h-ge. for L = 25, Q<br />
= 0.3 and #.= 0.85, ?(E, e:, $1 = 0.877,<br />
whi<strong>le</strong>, if $ = C.95, f(r, p?,$.> = 0.789. For fixed QI, sz ixrease in j?!<br />
reFGesezts ax +crease ia <strong>the</strong> iitersity of 10%-term persisterce, as eviderceà
386<br />
by higi<strong>le</strong>r observed values of h, <strong>the</strong> Hurst coefficient [ 91.<br />
As for <strong>the</strong> kg-orie Yarkov orocess, zn =biased estinate of ci<br />
2<br />
,<br />
42<br />
6 , may be<br />
defined, a s d g that $ acd 8, or, equivdently, J?f azd QI, are boni<br />
However, in practice, o- estimates of e ard Ø will be availab<strong>le</strong>, and a<br />
sinilzr prob<strong>le</strong>m to that encountered for <strong>the</strong> la- one Ymkov process is met in<br />
attemptirg to defize an unbiased estimate of I?:<br />
-<br />
The variance of <strong>the</strong> samp<strong>le</strong> mean, X for <strong>the</strong> ARIFA (l,O,l) process i$ given<br />
1253<br />
For el = !¿f, for u ~ c h <strong>the</strong> ARPIA (l,O,l) process reduces to <strong>the</strong> lag-one Markov<br />
process, equatiori (18) reduces to equation (11). Values of corresponding<br />
to large values oÏ j! ref<strong>le</strong>ct <strong>the</strong> low frequencies inherent in an approxiFtion<br />
For $ = c.85, PI = 0.3 and n = 25, Um2 = 0.15862 which compares<br />
to dfûn.<br />
trith equation (12) for dfGn with h = 0.7.<br />
Impact of Choice of Generatiig Process on System Design<br />
In <strong>the</strong> abseme of sufficiently lo= streamflow sequences to determine<br />
whe<strong>the</strong>r or.not lozg term persistence exists for a particular stream, and,<br />
lacking =y sound -&ysical basis for <strong>the</strong> ctoice of a generating process, consideratior<br />
must be even to <strong>the</strong> influence of choice of generating process on<br />
water resource system design.<br />
Few studies to date have investigated <strong>the</strong> sersi-<br />
tivity of system äesip to choice of gereratirg process for zmud streanflow.<br />
Wallis and Matalas [: 211 have studied <strong>the</strong> effects of long tern persistence on<br />
reservoir desigri through assuming prior know<strong>le</strong>dge of p and 6 2nd assessing <strong>the</strong><br />
impact of h on <strong>the</strong> äesign, W n g accomc of mall samp<strong>le</strong> biases in estimates<br />
of p1 a d 18 in <strong>the</strong>' Lr analysis. The reservoir desigr: Y ~ S evolved using <strong>the</strong><br />
sequent Fe& algorith, which was used to determine <strong>the</strong> minimum reservoir size<br />
necessary to meet 2 specified <strong>le</strong>vel of demd a expressed as a proportioli o€<br />
<strong>the</strong> observed averzge ITOU over <strong>the</strong> desi= period. For t'ûe desip considereä,<br />
<strong>the</strong> required reservo-ir capcity was fourd to depend on <strong>the</strong> mwitudes of h azd<br />
For equd e-cted vaues of <strong>the</strong> variance, E id), and <strong>the</strong> lag-one<br />
el.<br />
autocorre<strong>le</strong>tion, E [e?), in desigr sequezces of <strong>le</strong>ngth I?, and €or a > 0.80,<br />
approxicitions to 12-m with h > 0.5 yieloed reservoir sizes coxiderably in<br />
excess of those yielded by <strong>the</strong> lapone E%-kov process, thus e nmising <strong>the</strong><br />
relative impact or^ long-term acd short-term persistence on <strong>the</strong> design.<br />
By consideri--6 <strong>the</strong> water resource -stem design process, a basis emerges<br />
for <strong>the</strong> ckoice of a ge3eratizg process. A ge-eratirg process cas be postulated<br />
as beiri; tbt of <strong>the</strong> real worlä,.ad an oFtimd desigz evolved on this basis.<br />
Assuri$iors may <strong>the</strong>i be made concerzing tìe ideitity of <strong>the</strong> real world, acd<br />
<strong>the</strong> ecected reFets accrui3g from each assmotion my be evaluated. The procedure<br />
w be repeated for each postulated gexeratiF4 process for <strong>the</strong> real
world, with <strong>the</strong> ssuaei? generztirg process yielding <strong>the</strong> ~~ici~~m overall regrets<br />
representkg <strong>the</strong> 251~ro~riite ckoice. A simil- strategr has Seen employed by<br />
P?t&s acd YaXis [26J for <strong>the</strong> se<strong>le</strong>ction of a frequency distribution for <strong>the</strong><br />
evalmtion of a design flood magnitude.<br />
3 07<br />
A critical fictor in <strong>the</strong> choice of a gere-rating process concerrs <strong>the</strong><br />
ercistexe azd <strong>the</strong> intersity of long-term persistence to be mocelìed in syn<strong>the</strong>tic<br />
sequerces. Co-zequentLv, a set of si&tioz eqerimeris were evolved 25<br />
folious to dete,-mir;e strategies n th respect to long-term persistence, z ~ d<br />
to cetermine wke-her or rot, in <strong>the</strong> presexe of long-te,- persistence, bias<br />
corrections reed to be ap-lied to estimtes of <strong>the</strong> varizce azd lag-one auto-<br />
correlatisn measured in historic sequerces in o-der to obtaic realistic àesis<br />
results.<br />
Tuo generatkg processes uere adogted for <strong>the</strong> simulstion experiments,<br />
<strong>the</strong> lâg-oxe Markov process represexting short-term persistence and <strong>the</strong> ARPA<br />
(l,O,l) process represeztiq lorg-term persistelice. For <strong>the</strong> Latter process,<br />
<strong>the</strong> izterisity of long-ten persisterce may be cottrol<strong>le</strong>d by <strong>the</strong> parameter 6,<br />
consequent-, a value OÏ Ø = 0.85 YES M e n as represeEzing a medium intensity<br />
of lo-g-term persistence uhi<strong>le</strong> a value of ff = C.95 was se<strong>le</strong>cted to model a<br />
strozg intensity of 10%-te-m persistecce. Heme, <strong>the</strong> real uorid was assumed<br />
to be lag-one hrkov or ARIKA (l,O,l) with $ = 0.85 or f = 0.95 yielding 3<br />
possib<strong>le</strong> choices for <strong>the</strong> real world, identified by indices r = l,2,3.<br />
An o ptid design is required for each world, and a proceckire had to be<br />
evolved for evaluating <strong>the</strong> desip, which was defined as <strong>the</strong> minimum reservoir<br />
size necessu'y to meet 2 set of tzrget demands over <strong>the</strong> i?esig period rid,<br />
which was taken as 100 gears. Ra<strong>the</strong>r than defire <strong>the</strong> tzrget demands relztive<br />
to <strong>the</strong> samo<strong>le</strong> me= of <strong>the</strong> design sequence, <strong>the</strong> demar,ds uere defined BS percect-<br />
ages of <strong>the</strong> population meax of <strong>the</strong> r ed world so 2s to dlow <strong>the</strong> design to<br />
ref<strong>le</strong>ct more fully <strong>the</strong> varizbility of <strong>the</strong> =?ïe mean mong different uorlds.<br />
To permit a com-ison between optimal äesigr-s €or diÎferent worlds, each<br />
world was defined to hâve population parameters p and 6 such that<br />
and<br />
SES2] ,* = 9<br />
&e appro-priate values of 6 to be used i?l <strong>the</strong> gene=tiag *processes %v be<br />
defired fron eoiutions (9) iLld (17) for <strong>the</strong> lG-one Nzkov azà ARDA (l,O,l)<br />
processes, respeciively. The sequent ~3s.k algorithm u s use6 to evalute <strong>the</strong><br />
reserroir size, xxch YS defired 2s <strong>the</strong> mir?iia size slick tbt <strong>the</strong> reservoir<br />
rum C-p at host oxe orer <strong>the</strong> Cesis, period 'cui; suc3 tkt =e target de-~ds are dwqs met [F]. %e de-d <strong>le</strong>vels iàentiiied by izcices k = 1,2,3 ïere<br />
defeea 2s 75%, 85$ ana 955 of <strong>the</strong> populatioz cean = 70, which, in przctice,
388<br />
may redt in overdevelopmeat relative to <strong>the</strong> samp<strong>le</strong> mean for a design sequexe.<br />
ñowever, <strong>the</strong> sequent algorithm 2s originzllg formillatea by Thomas<br />
and Errrdez 1281 czot hard<strong>le</strong> <strong>le</strong>vels of develo~ent greater t h unity. In<br />
this sitiztion, <strong>the</strong> reservoir size was Bc-i-ic defined as for <strong>the</strong> case of urder-<br />
develo-ert usiLg 2 computer zlgorithm, with <strong>the</strong> initiâl reservoir storage<br />
neceesr-yy to zvoFd deficiercies beilig assiuned zvailab<strong>le</strong>. For each world, o otid<br />
desi,--s sere defT2ed as <strong>the</strong> expected reoervoir size for design, sequesces of<br />
<strong>le</strong>zgth "C-3 for kg-one autocorrelatiors e1 = 0.1, 0.3, 0.5 idectified by iräices<br />
j = l,í!,3 and de-ds 7.5, 8.5 ard 9.5 s m e d uniform over <strong>the</strong> ciesign period.<br />
The &!orite Carlo eweriments to be perfoned =y now be defined 2s follows.<br />
For each choice of real world <strong>the</strong> eqected reservoir size is denoted as<br />
E { (r, j,k)) for real world r with lag-one zutocorrelations j arid for <strong>le</strong>vel<br />
of äe~elopent k, {r,j,k = l,2,3] , and is defized through repetitive smplizg<br />
of desip sequences of <strong>le</strong>rgth 100 for which equations (19) and (2û) hold for<br />
r,j = 1,2,3.<br />
In order to assess <strong>the</strong> effects of bias correctious on <strong>the</strong> regrets, a6 uelì<br />
as essmptions about long-term persistence, <strong>the</strong> following Monte Czrlo samplirg<br />
proceduzes were defked.<br />
(1) Generate a 'historic' sequence identified by index i and <strong>le</strong>ngth n =<br />
50, 100 for world r with kg-one autocorrelation 2, decoted as {X I i,n,r,<br />
(2) hssirme <strong>the</strong> hi,storic sequence is gezerated by an assumed world with index<br />
1 = l,2,3, where <strong>the</strong> values of <strong>the</strong> index 1 refer to <strong>the</strong> same worlds as identical<br />
values of <strong>the</strong> index r. Estimates of <strong>the</strong> meari D, variance aí!, ard lag-one zutocorrelation<br />
PI are obtaired, at zhich juncture bias corrections may or may not<br />
be applied to #(i,D,r,j,l) and e (i,n,r,j,l). For 1 = 2,3, know<strong>le</strong>dge of <strong>the</strong><br />
assumed world includes krow<strong>le</strong>dge o$ <strong>the</strong> paranezer Ø(1).<br />
A 4 fi<br />
(3) Using p(i,n,r,j,l), 6 (i,n,r,j,l) ard el(i,n,r, j,l) (and $(i) for 1 = 2,3)<br />
a desim sequence oÎ <strong>le</strong>ngth nd = 100 is generated, whereupon, for k = '1,2,3, a<br />
design reservoir size is evaluated, denoted as h (i,E,r,j,k,l), which is <strong>the</strong><br />
reservoir size for a <strong>le</strong>vel of develoyezt k for desiga sequence i gecerated<br />
by a? asmed world 1. The pzraneters of <strong>the</strong> assumeä world are estimated from<br />
a historic sequecce of <strong>le</strong>ngth n from a real world r with lag-one autocorrelation<br />
index j. An overciesigii or uaderdesign relative to <strong>the</strong> opti,d design for <strong>the</strong><br />
real uorld may <strong>the</strong>2 be defked as<br />
Ah(i,n,r,j,k,l) = h(i,n,r,j,k,l) - E{>\ (r,j,k)] (21)<br />
which represents a simp<strong>le</strong> lines loss fìzxtion. Different sca<strong>le</strong>s BI and B<br />
may be äo-ied to wsitive zzd neetive losses if necessary. A quadratic goss<br />
function nay be del'iried through scusrirg Ah.<br />
+ReFeat (I), (21, (3) a sufficiently lzrge number of times to enab<strong>le</strong><br />
A (i,n,r, j ,k,l) ] , <strong>the</strong> expected positive loss, axd E {A-(i,n,r, j ,k,1)3<br />
<strong>the</strong> absolute vdue of <strong>the</strong> eqected negaative loss, to be defined. The expected<br />
overall loss may tiien be defined as<br />
uhere (i,s,r,j,k,l) is abreviated to (*).
3 89<br />
ordzr to pro9erL;;- assess *&e effects 00 an =&?tion about long term<br />
persistence, desigs &oula be evo1;red nskg Cesign seyiecces for ïdch<br />
E f.2) :.g for all sained uorlc?~. Hocever, even i' <strong>the</strong> bias correctioï<br />
procedures orscussed eelier ere z3flied to +Le small -?<strong>le</strong> ectinzt-es of pl<br />
arid 8, it uLL1 gel<strong>le</strong>rally rot be ooasib<strong>le</strong> to roet <strong>the</strong> rosuLrezst E [s2} yy: = 9<br />
within desigz seGuences, if oper&50%s (1) - (3) are fozoveo. However, <strong>the</strong><br />
series of operztLo.oris outlir?ed .we pr-y designed to illust,rate <strong>the</strong> effect<br />
of as-lying bis correctiors, prticularly i- &e preserre of long term persiti<br />
terce, as wen as detercrg <strong>the</strong> impzct of Frsiatence itself on <strong>the</strong> resets.<br />
KO wob<strong>le</strong>m is encountered k geEerati=g sp<strong>the</strong>tic sequences of <strong>le</strong>ogth n<br />
for a fixed value of el in<br />
fixed í! a d QI ix <strong>the</strong> a e<br />
process, <strong>the</strong> value of bS
390<br />
satisfy<br />
However, <strong>the</strong> estiaate of <strong>the</strong> variance satisfies<br />
E {s2(i,n2))<br />
vary with i and must <strong>the</strong>refore be considered as a random<br />
vci25<strong>le</strong> iikich satisfies<br />
"2 A<br />
which if 6 (i,nl) end f(n2, e(ilnl) are assumed indepecdent,<br />
= E [$2(i,n,)) E [ f (n2, G(i,nlJ<br />
However, in genera,<br />
A<br />
E {I(%, ê(i,nl)) # f(n2, E[@ (i,n,)I I<br />
as f(n2, @i,n ) is non-l+ezr. Howeveft, if Eff(n2..e (i,nlu is evaïEted<br />
at <strong>the</strong> expected vdue of e(i,nq), E {e (ilni) , which, if e (i,nl) is an<br />
unbiased estimate, equals E> , equation (2'7) reduces to<br />
E{E [s2(i,n2d] = 6 2 f(n,f')<br />
A<br />
=i E(S~],~ = 9 (28)<br />
which is what is required. However, nei<strong>the</strong>r of <strong>the</strong> two assumptions necessary<br />
to arrive at equation (28) are liab<strong>le</strong> to hold ; never<strong>the</strong><strong>le</strong>ss <strong>the</strong> departure from<br />
<strong>the</strong> required result may not be very serious. If, however @(i,nl) is biases,<br />
a larger discreparcy will occur which should accordingly manifest itself in<br />
<strong>the</strong> overall regets.<br />
The set of oserations (I) - (3) outlined above may be modified to study<br />
<strong>the</strong> effects of aosLying bias correctiocs separztely to estimtes of <strong>the</strong> varizce<br />
and <strong>the</strong> lag-one autocorrelatio2. For emp<strong>le</strong>, a simplified set of experimezts<br />
may be cìefined where <strong>the</strong> variance for <strong>the</strong> desis seque-ces, E [s2] 1co is<br />
assumed *-OX?,. in this situation, for each es2imate of <strong>the</strong> lag-one autocorrelation<br />
@(i,nl), <strong>the</strong> estimate of <strong>the</strong> variace, c2(i,nq) used to drive <strong>the</strong> gererator<br />
is defined as<br />
4<br />
b(i,n,) = E Is2] ,oo/f(n2, p(i,n,)) (29) .<br />
r\<br />
where E {szj is nou a constant but f(n2, @i,nl) is a random variab<strong>le</strong> <strong>the</strong>n,<br />
A<br />
2<br />
E ( G2(i,?)] = E [$2(i,nl) f(n2,p(i,nl)j = E [s Loo= 9<br />
from (29)
Cozsequently, <strong>the</strong> reqired vcriúzce can be miztahed -2 <strong>the</strong> desiga sequemes,<br />
whi<strong>le</strong> t'<strong>le</strong> effects of bizs correctionzs ??flied io %-ore autocorrelztio~s q v<br />
more esiïy be zssessed; as caa <strong>the</strong> hpact of tde time de-perxielit structure of<br />
<strong>the</strong> floäs.<br />
Cozclusions<br />
The prob<strong>le</strong>ms associated with choosizg a gecerati=& ?rocees for generatixg<br />
syn<strong>the</strong>tic streafo-is &-re been outlked, aid some of zLe prob<strong>le</strong>ms miskg a<br />
parmeter estdtion discussed. k <strong>the</strong> preserce of lorg-tem oersistence,<br />
<strong>the</strong> sadl Foperrties of gererati-4 processes trey ~ffer markedy from<br />
corres?ordirg poplation guzrititi-es, a d a set of simuï.ztio.ori experime-ts &.ve<br />
bees desir-ed to illustrate <strong>the</strong> izfluerce of 55s correctiox on <strong>the</strong> ete er<br />
reso'xce system Cesign -roceSS, whi<strong>le</strong> also allowing <strong>the</strong> coase:xexes of <strong>the</strong><br />
ixorrect moiellirg of persisteme to be assessed. €?ro?<strong>le</strong>rns are enco-tered<br />
in tzaixtaici3g a cozstzrt expecteà varicce ir desigri se-ences between differezt<br />
worlds, uhe- <strong>the</strong> vzzce is estiozted fyom hisroric sequexes, a=d oriy<br />
approxirate bis correctiozs m2y be ap?lied. Zowever, %te results of <strong>the</strong> sim-<br />
lâtio2 experineats whez 2vWlab<strong>le</strong> &odd illustrate <strong>the</strong> effects of qplyirg<br />
bizs coxectiors to <strong>the</strong> variace ard lag-one zxocorrelfcion, ead provide a<br />
guide es to what choice of gezerati-g mecha5m 29pears to -se <strong>the</strong> o verd<br />
regrets accsilizg from 2 partich choice of ge'reraticg mec-sm. The o otid<br />
choice of gererzting mechanism wo-dd be conditiod on a particular design<br />
process.<br />
3 91
392<br />
-1 +I<br />
Q<br />
Figure la<br />
-1 +I<br />
QI<br />
Pipe Ib<br />
+I<br />
+I<br />
-1<br />
d<br />
e2
Reterir-ces<br />
I.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
II.<br />
12.<br />
13.<br />
14.<br />
15.<br />
16.<br />
17.<br />
&tBLas, W.C. (1967), Ha<strong>the</strong>naticel assesmerd of sy-<strong>the</strong>tic hyär~logy,<br />
&ter Resow. Res. f(4), 931-935.<br />
P!delbrot, %B., Udlis, J.R. (19681, No&,<br />
kydrology, Kzter Beso*c. Res. 4(5) , 909-918.<br />
Joseph, a d operatio-di.<br />
Pxst, H.E. (1951), Lozg-term storege capacity of reservoiI’S, Th.us. Am.<br />
SOC. Civ. Engrs., 116, 770-8080<br />
393<br />
”st, B.$. (19561, ?!ethods of using long term storzge h reservoirs,<br />
Proc. Inst. Civ. Fagrs., 1, 519-543.<br />
.<br />
Yadelbrot, 3.B. (1971 ), A fast fractiod G2ïssim noise generator,<br />
Kater Resoil-. Bes. 7(3), 543-5530<br />
Fmdelbrot, B.B., WiLlis, J.R. (1969 1, Computer experiments with fractional<br />
Gmssian noises. Part I - Averages m-d variaces, Viter Resoure Res. 5(l),<br />
228-241.<br />
Yatalas, Nec., Wallis, J.R. (191 1, Statistical properties of multivariate<br />
fractional Eoise processes, Water Resow. Res. 7(61 , 1460-1468.<br />
Box, G.E.P., Jenkins, 6.13. (‘JPO), Time series analysis : Forecasting and<br />
control, S a Francisco, Holden-Day Inc., pp.553=<br />
O’Connelì, P.E. (1971), A simp<strong>le</strong> stochastic modellirg of Burst’s law,<br />
Proc. Interrztionaì Syqmsium on Mz<strong>the</strong>matjcd Models in Bydrology, Varszw,<br />
voi. I(I), 327-358, ïnt. Ass. Sci. ñydrol.<br />
Kejia, J.M., Rodriguez-Iturbe, I., Dawdy, D.R. (19721, Streamflow simulation.<br />
2 - !Che broke2 lice yocess as a potential mociel for hydrologic simulation,<br />
Water Resow. Res. 8(4), 931-941.<br />
Fadelbrot, B.B. (1972), Broken line process derived as ai- approximation<br />
to fractional noise, Yater Resour. Res. 86). 1354-1356.<br />
Kerdall, M.G. (1946), The advanced <strong>the</strong>ory of statistics, v.2, London,<br />
Char<strong>le</strong>s GriIfin a d Co. Ltd,, ?p. :24-125.<br />
Fisz, M. (7?63), Probability <strong>the</strong>ory *and ma<strong>the</strong>matical statistics, New York,<br />
John Wi<strong>le</strong>y ard Som I~C., pp. 421-423.<br />
kderson, R.L. (1942), Distribution of <strong>the</strong> serial correlation coefficient,<br />
An. Math. Stat., 13, 1-13.<br />
Vdis, J.R. , Natds, N.C. (1971). Correlopm =absis revisited,<br />
Vzter Resow. Res. 7(6), 1w-1459.<br />
Y!talas, N.C. ‘Langbeir,? Y.B. (19621, Info-tion content of <strong>the</strong> mean,<br />
Join. Geofir. Res. 6 7(9), S I - H .<br />
Vdis, J.R., Natalas, S.C. (Ig”O>, Sm11 ==?<strong>le</strong> pro-perties of II a d K -<br />
Estimators oi <strong>the</strong> Ewst coefficient h, Uster Resour. Bes. 6(6), 1931594.
394<br />
18. Vais, Jog., O'CoTlieU., P.E. (1973), Fi-- reserroir yield - hou reli251e<br />
are historic ?;7drolop~c records? 1-ternatiod Sjmoosium on <strong>the</strong> ñydrolog7<br />
of Mes, Helsizki, Firland.<br />
19. Slack, J.R. (:972), Bi=, illusion arid derial as data mcerteicties,<br />
Interatiod Spoosium 01: Uzcert-ties fi Hydrologic er6 Vater Resource<br />
System, Tucsx, ArizoE.<br />
20.<br />
21.<br />
24.<br />
25<br />
26.<br />
a.<br />
28.<br />
Slack, J.R. (373), I uould if I could (Self-denid by conditiorial models),<br />
Vater Resour. 2es. 9(1), 247-249.<br />
Wallis, J.B., Ystdas, N.C. (19721, Secsitivitr of reservoir design to<br />
<strong>the</strong> generat- nechzcìisin of inflows, Yater Resou. Res. 8(3), 634-641.<br />
22. Uallis, J.R., O'Connell, P.E. (19721, Small samo<strong>le</strong> estimation of Q<br />
Yater Resour. Res. 8(3), 707-712.<br />
23- Hatalas, N.C. (19671, Some aspects of time series dgsis in hydrologic<br />
studies, Proc. ñydzology Sym~osiuip ?io. 5 - 'Statistical Kethodc in Hydrology' -<br />
held at McGilì Univi. Kontreal, Feb. 1966, National Research Ccmcil of<br />
Cmda, ppm 41-99.<br />
Brooks, C.E.P., Carru<strong>the</strong>r6, N.C. (1953),<br />
in meteorologi, London, EPSO, pp. 412.<br />
Hand<strong>book</strong> of statistical methoàs<br />
O'Corsell, P.E. (19'731, The use of ARIMA models in <strong>the</strong> stochastic modellkng<br />
of long-term persistence, R.D. <strong>the</strong>sis (in preparation).<br />
Katalzs, N.C., Wallis, J.R. (19721, An approach to formulntina strategies<br />
for flood frequency -sis, Interratiorid Symposium on Uncertainties<br />
in ñydrologic a d Water Resource Systems, Tucson, hizona.<br />
Fierizg, M.B. (1967), Streanflow syn<strong>the</strong>sis, London, NcMiUm, pp. 139.<br />
Thomss, H.A., &den, R.P. (19631, Statistical aadysis of <strong>the</strong> reservoir<br />
yield relatioo, report, chap. 1, pp. 1-21, Harvard Water Resour. Group,<br />
Cambridge, F!s.
AB ST RACT<br />
STOCHASTIC APPLICATION IN UNGAGED BASINS<br />
FOR PLANNING PURPOSES<br />
By Pedro Porras G. and Alfredo Flores E.<br />
Water resources development planning, being iterative and dy-<br />
namic, requires increasingly detai<strong>le</strong>d basic information as each pla-<br />
nning <strong>le</strong>vel is surmounted. Successful planning is closely linked to<br />
<strong>the</strong> quality and quantity of <strong>the</strong> basic data. But when short periods<br />
are involved, <strong>the</strong> field of information is increasingly limited, as<br />
when dealing with monthly values instead of annual values. Moreover,<br />
methods and techniques are not as readily availab<strong>le</strong>, and besides,<br />
<strong>the</strong>y are more laborious, as compared with those used in connexion<br />
with long periods. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> use of sophisticated te-<br />
chniques, such as hydrologic simulation, is adequate at <strong>the</strong> project<br />
<strong>le</strong>vel but not at <strong>the</strong> planning stage, since it involves a more care-<br />
ful preparation of incoming data and its attendant remarkab<strong>le</strong> effect<br />
on <strong>the</strong> cost structure. Besides, it is timeconsuming to <strong>the</strong> extent of<br />
likely jeopardizing <strong>the</strong> requirement of keeping planning up-to-date,<br />
The method herein expounded deals with <strong>the</strong> attainment of average mon_<br />
thly values, covering a standard period of years, for precipitation,<br />
evaporation, net irrigation demands, and runoff, in <strong>the</strong> various<br />
stretches of <strong>the</strong> different rivers in a region, using transition pro-<br />
babilities (stochastic techniques], beginning wrth observed annual<br />
precipitation values, The method has been designed for computer solu-<br />
tion.<br />
El proceso de planificación hidráulica, por su carácter itera-<br />
tive y dinámico, requiere de una información básica cada vez mas de-<br />
tallada a medida que se van superando distintos nive<strong>le</strong>s. El êxito de<br />
la planificación está íntimamente ligado a la calidad y cantidad de<br />
datos básicos; pero la consecución de esta información se hace mas<br />
limitada cuando se imponen consideraciones de períodos cada vez más<br />
cortos, como ocurre cuando se trata de valores mensua<strong>le</strong>s en lugar<br />
de los anua<strong>le</strong>s; a esta limitación habría que añadir la menor disponl<br />
bilidad de mêtodos y técnicas que a su vez son m’as laboriosas que<br />
las usadas en periodos largos. Por otra parte, el emp<strong>le</strong>o de técnicas<br />
sofisticadas, como las simulaciones hidrolagicas, son adecuadas a ni<br />
vel de proyecto y no de planificación e implican una preparacian más<br />
meticulosa de los datos de entrada incidiendo notab<strong>le</strong>mente en costos<br />
y consumiendo un tiempo que podria poner en peligro la actualizaci’h<br />
que requiere la planificación. El método que aqui se expone trata de<br />
la consecuciön de valores medios mensua<strong>le</strong>s, pa??a un perPodo tipico<br />
de años, de precipitación, evaporación, demandas netas de riego y es-<br />
currimiento, en los diversos tramos de los di‘stintos ribs ae una re-<br />
gión, haciendo uso de las probabilidades de transl’ción (96cnicas e ~ -<br />
tocásticas) partiendo de los valores anua<strong>le</strong>s observados de precipita<br />
ción. El método ha sido diseñado para resolución por computadora.
396<br />
JUSTIFICATION<br />
For th elaboration f <strong>the</strong> first version of <strong>the</strong> National Plan of<br />
Development of Water Resources it was necessary to make a national inventory<br />
of <strong>the</strong> surface runoff. The characteristics of this first version were sufficient<br />
to know <strong>the</strong> mean annual volumes which were estimated through runoff<br />
isolines given by Hydric Balance method. One of <strong>the</strong> factors which determined<br />
<strong>the</strong> simplicity of application of this method was <strong>the</strong> number of years of <strong>the</strong><br />
chosen period- which allowed to accept <strong>the</strong> hypo<strong>the</strong>sis that <strong>the</strong> lateral transmissibility<br />
is negligib<strong>le</strong>.<br />
For <strong>the</strong> second version of <strong>the</strong> plan it was necessary to make an inventory<br />
of <strong>the</strong> surface runoff n mean monthly periods trying to reach a desirab<strong>le</strong><br />
<strong>le</strong>vel of detail but it was not possib<strong>le</strong> to make it directly because<br />
of insufficient information ex sting. This situation compel<strong>le</strong>d to make an<br />
investigation but unsuccessfully. Then it was necessary to change and follow<br />
o<strong>the</strong>r methodologies. From this emerged <strong>the</strong> idea of' making investigations<br />
through <strong>the</strong> application of stochastic methods to attain better results.<br />
CONSIDERATIONS FOR THE MODEL<br />
The proper characteristics of a region determine its own pluvial<br />
cyc<strong>le</strong> specified by <strong>the</strong> quantity and distribution of <strong>the</strong> rainfall. Among <strong>the</strong>se<br />
characteristics must be considered <strong>the</strong> latitude, longitude, proximity to <strong>the</strong><br />
sea or lakes, <strong>the</strong> topography, land form and so forth. Some of <strong>the</strong>m have<br />
greater influence on <strong>the</strong> quantity and o<strong>the</strong>rs on <strong>the</strong> distribution in <strong>the</strong> year;<br />
determining dry and wet periods. In <strong>the</strong> annual rainfall distribution may be<br />
appreciated two essential particularities : one, <strong>the</strong> annual total percentages<br />
corresponding to each month (<strong>the</strong>y represent similar figures for <strong>the</strong> different<br />
zones though it does not mean that <strong>the</strong> quantities of rainfall must be <strong>the</strong> same)<br />
<strong>the</strong> o<strong>the</strong>r, <strong>the</strong> sequence or <strong>the</strong> order of <strong>the</strong>ir presentation. The observations<br />
made have proved that in relatively small zones <strong>the</strong> variations of <strong>the</strong> rainfall<br />
may be important in quantity but not in its distribution.<br />
Generally <strong>the</strong> characteristics of <strong>the</strong> availab<strong>le</strong> data offer <strong>the</strong> opportunity<br />
that a lot of factors allow <strong>the</strong> elaboration of mean annual isohyetal<br />
maps with better reliability than <strong>the</strong> monthly ones (which in some cases can<br />
not even be elaborated). Among <strong>the</strong>se factors it is important to include <strong>the</strong><br />
following: <strong>the</strong> zonal variations better determined from <strong>the</strong> precipitatione<strong>le</strong>vation<br />
relation (topographic considerations); <strong>the</strong> comp<strong>le</strong>te annual totals<br />
series; <strong>the</strong> totalizer records;<br />
<strong>the</strong> comparison with mean annual runoff in<br />
gaged watershed; <strong>the</strong> simplicity in <strong>the</strong> estimation of lacking data through<br />
technical procedures like doub<strong>le</strong> mass curve, etc.
The most frequent difficulties in <strong>the</strong> elaboration of isohyetal monthly<br />
maps are: (a) <strong>the</strong> cluster of several months in succession since in many cases<br />
even having <strong>the</strong> annual totals of <strong>the</strong> comp<strong>le</strong>te measurement series makes difficult<br />
<strong>the</strong> assessment of monthly averages; (b) it does not exist c<strong>le</strong>ar relations<br />
between <strong>the</strong> altitude of <strong>the</strong> station and <strong>the</strong> monthly rainfalls; (c) not always<br />
exist definite relations betwe.en <strong>the</strong> monthly rainfalls measuredat near stations;<br />
(d) <strong>the</strong> zones with scattered stations where <strong>the</strong> mentioned considerations hamper<br />
drawing <strong>the</strong> monthly isohyetals. Here is <strong>the</strong>n <strong>the</strong> need to generate <strong>the</strong> monthly<br />
data through adequate methods.<br />
ANALYSIS OF DATA<br />
397<br />
For <strong>the</strong> stochastic generation of monthly values of rainfall, Region 1<br />
was dected(Maracaibo Lake Watershed) where 141 stations with 10 years period<br />
recorded were estimated: 1961-70.<br />
In a practical manner, even if <strong>the</strong> monthly average could not be deter-<br />
mined directly from <strong>the</strong> records, due to clustering of monthly values, it was<br />
possib<strong>le</strong> to determine more or <strong>le</strong>ss easily <strong>the</strong> monthly maximum and minimum that<br />
<strong>le</strong>d to generate series with standard deviation and mean similar to registered<br />
series according to <strong>the</strong> checking made at stations with comp<strong>le</strong>te recording.<br />
The se<strong>le</strong>cted value for <strong>the</strong> analysis was <strong>the</strong> percentage of each month<br />
in connexion with <strong>the</strong> annual average for those 10 years period. A frequency<br />
analysis was made of <strong>the</strong> group of data ga<strong>the</strong>red each month and by mean annual<br />
ranges of precipitation.<br />
The ranges of precipitation se<strong>le</strong>cted were <strong>the</strong> following: (a) from.<br />
50 mm to 1000 mm; (b) from 1000 mm to 1500 mm; (c) from 1500 mm to 2000 mm<br />
and (d) greater than 2000 mm. Then <strong>the</strong> following particularities appeared:<br />
(a)<br />
(b)<br />
(c)<br />
(d)<br />
In months of light precipitation ‘<strong>the</strong> interval of <strong>the</strong> percentages<br />
variation with respect to <strong>the</strong> annual average varied with <strong>the</strong> annual<br />
average.<br />
In months of heavy precipitation no relation was noticed.<br />
In all cases <strong>the</strong> Gumbel distribution function was <strong>the</strong> one which fits<br />
better on polygon frequency.<br />
The interval of <strong>the</strong> percentages variation, each month, is characteristic<br />
of each location. For instance: <strong>the</strong> interval in January in<br />
Paraguaipoa is permanent and is different to Machiques in <strong>the</strong> same<br />
month.
398<br />
DESCRIPTION OF THE MODEL<br />
It was determined to construct a stochastic model to generate (from<br />
<strong>the</strong> average values of annual rainfall) monthly series of 10 values which re-<br />
present a typical cyc<strong>le</strong> of 10 years. This matter has <strong>the</strong> following purposes:<br />
(a)<br />
(b)<br />
(c)<br />
To use <strong>the</strong> averages of those series in <strong>the</strong> elaboration of <strong>the</strong> mean<br />
monthly isohyetic maps.<br />
To compute (from generated rainfall values) probab<strong>le</strong> values of<br />
evaporation which accompany each rainfall in order to obtain balances<br />
and determine <strong>the</strong> water requirements for irrigation.<br />
To determine mean monthly rainfall over <strong>the</strong> watersheds area in order<br />
to estimate mean monthly runoff and obtain average values.<br />
For <strong>the</strong> generation of rainfall values it was necessary to estimate<br />
<strong>the</strong> possibility to elaborate a matrix of transition probability to obtain<br />
monthly rainfall values for consecutive years. This was not possib<strong>le</strong> because<br />
<strong>the</strong> monthly series (when are comp<strong>le</strong>ted) only consist of 10 terms and just like<br />
it was mentioned previously <strong>the</strong> number of <strong>the</strong>m is very reduced - this being<br />
one of <strong>the</strong> reasons why <strong>the</strong> model was prepared.<br />
The matrix of probability of transition was elaborated <strong>the</strong>n considerin!<br />
previous states equally likely that is to say, without distinction in all<br />
months. It was found that among <strong>the</strong> polygons of "accumulated relative fre-<br />
quencies" corresponding to <strong>the</strong> same previous state, but of different matrix<br />
pertaining to nearby regions, <strong>the</strong>re exists proportionality. That is, if we<br />
denominate F(x) <strong>the</strong> function that describes <strong>the</strong> best fit to <strong>the</strong> polygon corres.<br />
ponding to a previous state xi of a matrix [A and F, (x) that corresponding<br />
to <strong>the</strong> same previous state xi in <strong>the</strong> matrix zû] (Fig. 1) and if <strong>the</strong> functions<br />
take <strong>the</strong> same value for values x equal to s and t respectively that is to say<br />
if<br />
F (s)=F( ( t ) (1)<br />
and calling m and M <strong>the</strong> lower and upper limits of <strong>the</strong> interval variation of<br />
<strong>the</strong> first function and m, and M,<br />
is verified:<br />
to <strong>the</strong> second one; an important relation<br />
-<br />
-<br />
s - m M - m<br />
t - m, M,- m l
This last expression gives a simp<strong>le</strong> form to estimate <strong>the</strong> value of <strong>the</strong><br />
rariab<strong>le</strong> corresponding to a place when is knownitslimits of variation and a<br />
wilt matrix is valid for <strong>the</strong> region since <strong>the</strong> real value of x will be:<br />
x =<br />
s - m<br />
M - m<br />
(Mi - mi) + ml (2 1<br />
TO generate a cyc<strong>le</strong> of 10 years of rainfall values, any value (XO) is<br />
:hosen among <strong>the</strong> interval of variation of <strong>the</strong> percentages observed for <strong>the</strong><br />
nonth of December so as to begin <strong>the</strong> process. Boundaries for each month are<br />
<strong>the</strong> maximum and minimum limits where <strong>the</strong> percentages of each month change and<br />
<strong>the</strong> value of mean year rainfall in that place is determined. A random number<br />
is se<strong>le</strong>cted from O to 100; with this random number and with <strong>the</strong> initial value<br />
(xo) we come into <strong>the</strong> matrix of <strong>the</strong> transition probabilites and we read <strong>the</strong><br />
dalue of x which is changed through <strong>the</strong> expression 2.<br />
This value of x obtained this way and multiplied by <strong>the</strong><br />
,f precipitation and divided by 100, generate <strong>the</strong> first monthly<br />
dumerically is as follows:<br />
I<br />
xu P<br />
LL\ = J-<br />
-<br />
dhere P: mean year rainfall<br />
100<br />
x;: percentage corresponding to ( 'month) state<br />
i l<br />
With <strong>the</strong> value x: obtained and a new random number, <strong>the</strong> proced.ure is<br />
repeated so as to generate x2 and continually untili = 120 (10 years). The<br />
first 12 values correspond to each month of <strong>the</strong> first year generated; <strong>the</strong> next<br />
L2 correspond to <strong>the</strong> months of <strong>the</strong> second year and continually until <strong>the</strong> tenth<br />
rear.<br />
lbtainment of <strong>the</strong> Monthly Rainfall Over Area in a Watershed<br />
399<br />
mean year value<br />
rainfall value.<br />
In a watershed, as soon as <strong>the</strong> isohyets have been drawn with <strong>the</strong> mean<br />
mnual values of <strong>the</strong> availab<strong>le</strong> stations, <strong>the</strong> mean annual rainfall over area<br />
)btained with <strong>the</strong> average of all <strong>the</strong> point values inferred will be very close<br />
to <strong>the</strong> vaiue of <strong>the</strong> mean annual rainfall which has been obtained through <strong>the</strong><br />
danimeter. Based on this it is possib<strong>le</strong> to get <strong>the</strong> mean annual rainfall on a<br />
datershed, using <strong>the</strong> known mean annual totals or <strong>the</strong> generated mean annual<br />
lalues. Never<strong>the</strong><strong>le</strong>ss, applying this procedure to <strong>the</strong> monthly value generated,<br />
<strong>le</strong>ads to poor results, average excepted, since in two adjacent points it is<br />
impassib<strong>le</strong> to guarantee beginning both series with <strong>the</strong> representative values<br />
if <strong>the</strong> same year.<br />
1 typical cyc<strong>le</strong> of 10 years, not only regarding <strong>the</strong> magnitude of <strong>the</strong> terms but<br />
regarding <strong>the</strong> order too.<br />
(3)<br />
In spite of this <strong>the</strong> values of <strong>the</strong> series obtained represent
400<br />
In order to attain <strong>the</strong> series of <strong>the</strong> monthly values corresponding to<br />
<strong>the</strong> rainfall over <strong>the</strong> area already mentioned, <strong>the</strong> same procedure has to be re-<br />
peated (as it has been explained) for getting <strong>the</strong> monthly values at a point,<br />
starting from an initial value which is <strong>the</strong> average value of <strong>the</strong> initio1<br />
volumes at each point extended uniformly from <strong>the</strong> mean annual rainfall obtained<br />
according to <strong>the</strong> precedent explanation and <strong>the</strong> monthly maximum and minimum<br />
limits also obtained just like averages at each point.<br />
Generation of Monthly Values of Evaporation<br />
The precipitation of <strong>the</strong> zone was divided in 10 mm intervals and in eoc<br />
one of <strong>the</strong>m where <strong>the</strong> stations existed, <strong>the</strong> evaporation was studied and <strong>the</strong><br />
polygon of frequence which correspond to each rainfall interval was determine<br />
The distribution of frequency of <strong>the</strong> evaporation was anolyhed in interv<br />
class of 10 mm. Without making a strict analysis it was observed that <strong>the</strong> distribution<br />
of evaporation for each rainfall interval is normal. The range of<br />
<strong>the</strong> variation of <strong>the</strong> evaporation decrease whi<strong>le</strong> <strong>the</strong> mean value of <strong>the</strong> class of<br />
rainfall increase. Thus, to generate anevaporation value, as soon as <strong>the</strong> rainfall<br />
value has been generated, a random number is chosen as well as <strong>the</strong> interva<br />
to which <strong>the</strong> rainfall generated pertains; <strong>the</strong> random number determines <strong>the</strong><br />
corresponding evaporation value.<br />
With <strong>the</strong> pair of series obtained at each point, <strong>the</strong> accounting balance<br />
is made in order to determine <strong>the</strong> water requirements for irrigation.<br />
RESULTS<br />
With <strong>the</strong> aim of getting some indicator of <strong>the</strong> goodness of <strong>the</strong> results,<br />
<strong>the</strong> monthly rainfall values pertaining to places with measurements were generat<br />
and <strong>the</strong> series were compared through means and standard deviations.<br />
The linear correlation was calculated for each place among <strong>the</strong> twelve<br />
means generated (one for each month) and <strong>the</strong> measurements; <strong>the</strong> same also was<br />
made with <strong>the</strong> standard deviations.<br />
For illustration here are some results:<br />
!- Coefficient of cor;ela:ion between<br />
The Means Standard Deviations<br />
I<br />
O ,98<br />
o ,?2<br />
San José Bolivar<br />
Boroto<br />
Mochiques Gia.<br />
0,90<br />
O ,96<br />
O ,?3<br />
0,91<br />
0,81<br />
O ,96<br />
0,94<br />
0,89
JNOFF<br />
401<br />
The runoff is generated by <strong>the</strong> following equations explained in <strong>the</strong><br />
irk "Analisis sobre las Relaciones Escurrimiento-Precipitation en Periodos<br />
! n s u a 1 e s 'I (An a 1 y s i s o f Run o f f -Pr e c i pit at i on Re 1 at i on s by Mont hl y Per i od s )<br />
Pedro Porras G.<br />
ere :<br />
Ei = Si-' -<br />
Y<br />
o<br />
a<br />
4- (Pi-si- 1 -)B.<br />
I-a I-a I<br />
si = s. - 1<br />
a<br />
i (Pi - - s. 1 -) (i-+)<br />
I I 1-a<br />
: Number of <strong>the</strong> period of time (month)<br />
: Runoff in <strong>the</strong> period<br />
: Storage to <strong>the</strong> period<br />
i<br />
,f Coefficients depending on <strong>the</strong> characteristic of <strong>the</strong> watershed.<br />
This method has been developed in order to be applied by means of <strong>the</strong><br />
e of digital computers and to obtain ranges of mean monthly values which do<br />
t differ more than 10 per cent in general terms.<br />
For <strong>the</strong> application of this method it is convenient to use maps at<br />
100.000 sca<strong>le</strong>s with topography; on <strong>the</strong>se maps previously squared (an adequate<br />
dth among <strong>the</strong> lines must be no more than 2 minutes) we draw <strong>the</strong> mean annual<br />
ohyets for a period of 10 years and we build <strong>the</strong> twelve pairs of maps<br />
rresponding to <strong>the</strong> isopercentsofmaximum and minimum for each month.<br />
In <strong>the</strong> region all watersheds have to be shown upto<strong>the</strong> places of interest;<br />
e data must be prepared for each watershed.<br />
At <strong>the</strong> intersection point of <strong>the</strong> grid previously identified, we must<br />
ad <strong>the</strong> mean annual precipitation value as well as <strong>the</strong> maximum and minimum<br />
lues corresponding. This information plus <strong>the</strong> area of <strong>the</strong> watershed, initial<br />
lues of precipitation in percentage, <strong>the</strong> regional matrix of precipitation<br />
d evaporation and <strong>the</strong> values of alp, S establish <strong>the</strong> input data to <strong>the</strong> pro-<br />
O<br />
ss.<br />
(4)<br />
(5)
402<br />
The values of a,/, S must be obtained from nearby watersheds with<br />
O<br />
measurements and similar characteristics. In <strong>the</strong> procedure to be applied to<br />
get this result and in <strong>the</strong> se<strong>le</strong>ction of <strong>the</strong>se values, we must estimate <strong>the</strong><br />
very important physiographic characteristics of <strong>the</strong> watershed.<br />
RESOLUTION BY COMPUTER<br />
The application of <strong>the</strong> model requires a lot of computing which is<br />
very time consuming by hand; <strong>the</strong>refore it was necessary to elaborate a progrc<br />
for digital computer.<br />
The program has been elaborated in <strong>the</strong> PL/1 Language and it consists<br />
of a main program and five subprograms whose functions are <strong>the</strong> following:<br />
MA1N PROGRAM<br />
This program consists of six main phases:<br />
It reads <strong>the</strong> data of <strong>the</strong> transition matrix of <strong>the</strong> precipitation and<br />
<strong>the</strong> evaporation, as well as <strong>the</strong> characteristic data of <strong>the</strong> points<br />
of <strong>the</strong> grid in which <strong>the</strong> watershed has been subdivided (mean annual<br />
precipitation, initial monthly precipitation, monthly maximum limit<br />
and monthly minimum limit).<br />
It generates stochastically for each point of <strong>the</strong> grid <strong>the</strong> precipita.<br />
tion month to month for <strong>the</strong> se<strong>le</strong>cted period through <strong>the</strong> transition<br />
matrix of <strong>the</strong> precipitation.<br />
From <strong>the</strong> monthly precipitation of <strong>the</strong> point and through <strong>the</strong> transitic<br />
matrix of <strong>the</strong> evaporation, it calculates <strong>the</strong> monthly evaporations at<br />
<strong>the</strong> point.<br />
If <strong>the</strong> water requirements for irrigation are required at <strong>the</strong> point,<br />
<strong>the</strong> program through a control apply <strong>the</strong> method "Accounting Balance<br />
of Thornthwaite" using <strong>the</strong> precipitations and evaporation calculated<br />
in <strong>the</strong> phases 2 and 3.<br />
It generates stochastically <strong>the</strong> monthly precipitation for <strong>the</strong> water-<br />
shed from <strong>the</strong> transition matrix of <strong>the</strong> precipitation and taking as<br />
mean annual value and initial value, <strong>the</strong> average of <strong>the</strong>m already<br />
calculated for <strong>the</strong> points of <strong>the</strong> grid of <strong>the</strong> watershed.<br />
It calculates <strong>the</strong> runoff and monthly volumes for <strong>the</strong> watershed<br />
according to <strong>the</strong> work "Analysis of Runoff-Precipitation Relations<br />
by Monthly Periods" by Pedro Porras, Engineer.
SUBPROGRAMS<br />
Subprogram LENGTH:<br />
It calculates <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> interval of <strong>the</strong> probability curves<br />
accumulated by <strong>the</strong> transition matrix of <strong>the</strong> precipitation.<br />
Subprogram INTER:<br />
It interpo<strong>le</strong>s values in <strong>the</strong> transition matrix<br />
Subprogram ESTADI:<br />
It computes <strong>the</strong> mean and <strong>the</strong> standard deviation<br />
Subprogram DENERI:<br />
403<br />
It determines <strong>the</strong> water requirements for irrigation starting from <strong>the</strong><br />
method of "Accounting Balance of Thornthwaite".<br />
Subprogram RANDU:<br />
It generates random numbers between O and 1 used in <strong>the</strong> stochastic<br />
process.<br />
In <strong>the</strong> Fig. No. 2 attached is presented <strong>the</strong> Flow Chart of <strong>the</strong> program.<br />
VERIFICATION OF THE MODEL<br />
For <strong>the</strong> verification of <strong>the</strong> model <strong>the</strong> Socuy River watershed was<br />
se<strong>le</strong>cted. The basin is located in <strong>the</strong> north-west region of <strong>the</strong> Maracaibo<br />
Lake in Venzuela.
404<br />
Fin) Former state: xi<br />
MATRIX [AI<br />
Fiix) Former state:Xi<br />
MATRIX [d<br />
F,(t)rF(i) ---------<br />
Comparison of distribution functions for equal prior<br />
states of <strong>the</strong> various matrices<br />
PIOURE A
0<br />
COMIENZO<br />
MATRICES DE<br />
PRECIPITACION Y<br />
EVAPORA CI ON<br />
NUMERO DE<br />
CUENCAS<br />
LECTURA DE<br />
DATOS DE LA<br />
CUENCA<br />
PUNTO<br />
LECTURA OE<br />
DATOS DE LO5<br />
PUNTOS DE LA<br />
CUENCA<br />
I A<br />
::::;-I<br />
PR EC I P ITACI ON<br />
EVAPORACION<br />
I<br />
FIGURA 2<br />
- 9<br />
FIN PUNTO<br />
DEMANDAS<br />
FIN<br />
1 CALCULA P R F<br />
PITACION MEDIA<br />
ANUALY MENSUAL<br />
PARA LACUENCA<br />
IMPRIME<br />
PRECIPI TACION<br />
LECTURA DE<br />
CALCULA<br />
VOLUMENEC<br />
MENSUALES<br />
0<br />
FINAL<br />
~~<br />
DIAGRAMA DE FLUJO
406<br />
6<br />
1<br />
8<br />
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5<br />
11.11<br />
41.63<br />
154.31<br />
114.65<br />
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41-61<br />
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145.IC<br />
Il.5C<br />
C.15<br />
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5f.13<br />
45.15<br />
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?(.i1<br />
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42.21<br />
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31.11<br />
41.25<br />
ii.12<br />
I.i< 1-61<br />
15.15<br />
12.15<br />
1e2.27<br />
il.0<br />
145.41<br />
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1.51<br />
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I
ABSTRACT<br />
HOMOGENEISATION ET INTERPOLATION DES DONNES POUR<br />
UN MODELE DE SIMULATION<br />
par Marcel ROCHE<br />
When <strong>the</strong> topological schema of a project has been defined, it<br />
is necessary to establish data sets (in quantity and possibly in<br />
quality) wich are to be used for operating <strong>the</strong> simulation. This is<br />
all <strong>the</strong> more difficult as basic data are more seldom and of <strong>le</strong>ss good<br />
quality. It must 5e begun to ga<strong>the</strong>i as comp<strong>le</strong>tely as possib<strong>le</strong> data<br />
directly observed at availab<strong>le</strong> stations and to severely criticize<br />
those data. The second operation consists ia choosing <strong>the</strong> period of<br />
reference to be used Cas long as possib<strong>le</strong>) and in establishing an<br />
homogeneous series, for this period, from <strong>the</strong> basic data Ccorrelations).<br />
Finally, from this homogeneous series at availab<strong>le</strong> statïons, an<br />
homogeneous series at <strong>the</strong> various input points of <strong>the</strong> model has to<br />
be computed (interpolation in space). The author taRes as an exemp<strong>le</strong><br />
monthly yields and <strong>the</strong>ir mean salinity.<br />
RESUME<br />
Lorsque <strong>le</strong> schema topologique d‘un aménqgement a étd arrêté,<br />
il convient d‘établir los séquences hj-drologTqnes Cqyantit& et<br />
éventuel<strong>le</strong>ment qualité] oui devront être utilisées pour proceder a<br />
la simulatlon. L‘opératinn est d’autant plus df3licate que <strong>le</strong>s donnees<br />
de base sont plus payes et de moïns bonne qualite. On doi’t commencer<br />
par faire un bilan aussi comp<strong>le</strong>t quo possib<strong>le</strong> des données directement<br />
observses aux stations disponib<strong>le</strong>s et scumettrc ces donnges à une<br />
analyse critique ~6v’ere. La seconde opérntion consiste 3. choisir las<br />
période de référoncn qu‘on utilisera Cia plus 1Qngue possib<strong>le</strong>), et a<br />
établir pour cette ?-riode une série homogène a partir des données de<br />
base (corréiatioris). Enfin, on doit calcu<strong>le</strong>r, à partir de cette série<br />
homogène aux stati~ris disponib<strong>le</strong>s, une s&rie homogene aux différents<br />
points d’entrée di1 nodèlo (interpolation spatia<strong>le</strong>). L’auteur prend<br />
comme exemp<strong>le</strong> <strong>le</strong>s apports mensuels et <strong>le</strong>ur salinité moyenne.
408<br />
Un modè<strong>le</strong> mathématique de simulation pour un aménagement intégré est<br />
construit 5 partir d'un plan topologique tel que celui de la figvre 1. Sur ce<br />
plan, <strong>le</strong>s débits à injecter pour faire fonctionner <strong>le</strong> modè<strong>le</strong> sont représentés<br />
par <strong>le</strong>s symbo<strong>le</strong>s An et ACn, suivant qu'ils sont produits en tête de bassin ou<br />
dans un bassin intermédiaire. On peut, s'il est besoin, <strong>le</strong>ur associer <strong>le</strong>s salures<br />
moyennes correspondantes SAn et SACn.<br />
I1 est donc nécessaire de fournir <strong>le</strong>s va<strong>le</strong>urs de ces apports et éventuel<strong>le</strong>ment<br />
de ces salures sur la plus longue période possib<strong>le</strong>. Les débits sont<br />
mesurés à des stations de jaugeage qui ne coïncident pas toujours et avec toutes<br />
<strong>le</strong>s limites des unités hydrauliques. Par ail<strong>le</strong>urs, <strong>le</strong>s périodes sur <strong>le</strong>squel<strong>le</strong>s<br />
portent <strong>le</strong>6 observations ne sont jamais <strong>le</strong>s mêmes aux différentes<br />
stations. I1 en est de même pour <strong>le</strong>s observations concernant la qualité des<br />
eaux, avec encore moins de stations, davantage de lacunes et des périodes<br />
plus courtes.<br />
Les opérations destinées à préparer l'échantillon des An et des ACn, et<br />
éventuel<strong>le</strong>ment celui des SAn et des SACn seront donc <strong>le</strong>s suivantes (on supposera<br />
dans tout ce qui suit qu'on travail<strong>le</strong> à un pas de temps mensuel) :<br />
- mise au point des débits moyens mensuels observés aux stations,<br />
- mise au point des salures moyennes mensuel<strong>le</strong>s observées aux stations,<br />
- choix d'une période de travail, dite historique, et homogénéisation<br />
des débits moyens mensuels sur cette période,<br />
- homogénéisation, sur la période historique, des données concernant<br />
la qualité,<br />
- calcul, sur la période historique, des An, ACn, SAn et SACn, par<br />
interpolation géographique, et quelquefois par analogie ; équilibrage<br />
des volumes et des poids de sei.<br />
1.- Débits et salures observds aux stations<br />
Les données provenant des réseaux sont traitdes par <strong>le</strong>s services<br />
qui <strong>le</strong>s contr&<strong>le</strong>nt, de plus en plus par <strong>le</strong>s procddds de l'informatique.<br />
Quand on commence une étude hydrologique pour un amdnagement,<br />
on doit, dans toute la mesure du possib<strong>le</strong>, repartir des<br />
données origina<strong>le</strong>s non dlabordes,<br />
<strong>le</strong>s critiquer et <strong>le</strong>s traiter à no:<br />
veau, Pour <strong>le</strong>s hauteurs limnimêtriques, on reprend tous <strong>le</strong>s originaux<br />
des observateurs (<strong>le</strong>cteurs d'échel<strong>le</strong>s) et <strong>le</strong>s limnigrammes s'ils<br />
existent, On vérifie <strong>le</strong>s calages des échel<strong>le</strong>s en s'appuyant sur <strong>le</strong>s<br />
comptes rendus, <strong>le</strong>s contrô<strong>le</strong>s de zéro, sur tout document disponib<strong>le</strong>,<br />
on se <strong>livre</strong> au besoin 3 des enquetes sur <strong>le</strong> terrain; on essaye d'@va<br />
luer la qualitd des re<strong>le</strong>v'es d'apr&s la forme des limnigrammes, la<br />
tenue des feuil<strong>le</strong>s d'observations, en faisant des comparahons avec<br />
d'autres stations etc,<br />
Si on dispose des minutes des jaugeages, il n'est pas sans<br />
intérbt de contro<strong>le</strong>r quelques dépouil<strong>le</strong>ments; si on re<strong>le</strong>ve un purcep<br />
tage important d'erreurs il ne faut pas hésiter 3 <strong>le</strong>s reprendre en<br />
totalité, I1 ne s'agit pas toujours 12 d'un polissage raffiné; nous<br />
pourrions citer un cas dans <strong>le</strong>quel un contrô<strong>le</strong> a montrd que 25% des<br />
jaugeages présentaient des erreurs de dépouil<strong>le</strong>ment supgrieures 3<br />
20%. I1 faut ensuite refaire la courbe d'étalonnage, ou <strong>le</strong>s courbes<br />
si l'étalonnage a varié au cours de la période d'observation, ce qui<br />
est presque toujours <strong>le</strong> cas pour <strong>le</strong>s basses eaux,<br />
On reprend alors <strong>le</strong> calcul des ddbits à l'ordinateur et on<br />
en sort la série chronologique des débits moyens mensuels observés<br />
chacune des stations dans <strong>le</strong> ou <strong>le</strong>s bassins intéressant <strong>le</strong> projet,
Si l'on doit tenir compte de la qualité des eaux, notamment de la<br />
salinité, il faut faire sur <strong>le</strong>s observations correspondantes une<br />
opération analogue 8 ,la précédente, mais avec une méthodologie de<br />
contrô<strong>le</strong> tr&s différente,<br />
409<br />
Les mesures de salinités, par exemp<strong>le</strong>, portent généra<strong>le</strong>ment SUT des périodes<br />
beaucoup 21us courtes que cel<strong>le</strong>s des observations hydrométriques. R<strong>le</strong>s sont<br />
souvent disparates dans <strong>le</strong>urs méthodes dféchantillonnage (techniques de prélèvement)<br />
aussi bien que dans <strong>le</strong>s méthodes d'analyses. Pour ces dernières, on procède<br />
soit par analyse chimique complète, en différenciant <strong>le</strong>s sels dissous, soit par<br />
analyse sommaire : teneur globa<strong>le</strong> en sels dissous mesurée <strong>le</strong> plus souvent par conductivimétrie,<br />
I1 faut homogénéiser tous ces résultats, après une étude critique<br />
aussi poussée que possib<strong>le</strong> portant notamment sur la confiance qu'on peut attri-<br />
buer aux méthodes clsanalyse pratiquées<br />
et aux conditions dans <strong>le</strong>squel<strong>le</strong>s el<strong>le</strong>s<br />
ont été appliquées .e lorsqu'on <strong>le</strong>s connaît. On produit ainsi, pour <strong>le</strong>s .besoins<br />
du modè<strong>le</strong>, un échantillon de salures moyennes mensuel<strong>le</strong>s à un certain nombre de<br />
stations (l).,<br />
Si on a l'intention d'utiliser la pluviométrie disponib<strong>le</strong> pour étendre la<br />
période d'observation des débits, il faudra procéder éga<strong>le</strong>ment à l'indispensab<strong>le</strong><br />
étude critique des précipitations. On s'attachera notamment à détecter et à cor-<br />
riger <strong>le</strong>s erreurs systématiques, causes d'hétérogénéité dans <strong>le</strong>s séries, en appli-<br />
quant la méthode des doub<strong>le</strong>s cumuls (2). La encore, il ne s'agit nul<strong>le</strong>ment d'un<br />
débat académique ; <strong>le</strong>s erreurs systématiques dans ce genre de re<strong>le</strong>vés ne sont pas<br />
occasionnel<strong>le</strong>s, el<strong>le</strong>s constituent la règ<strong>le</strong> généra<strong>le</strong>. Pour éviter <strong>le</strong>s erreurs de<br />
transcription qui risquent d'affecter <strong>le</strong>s publications officiel<strong>le</strong>s, on recomman-<br />
de 15 aussi, dans toute la mesure du possib<strong>le</strong>, de partir des re<strong>le</strong>vés originaux.<br />
2.- Homogénéisation et extension des données I1débitsff<br />
Les opérations précédentes ont permis de constituer un échantillon de de-<br />
bits mensuels portant pour chaque station SUI' une période inférieure ou éga<strong>le</strong> a<br />
n années. L'homogénéisation va consister à choisir une période de référence au<br />
plus éga<strong>le</strong> à n années et, p y l'utilisation des regressions, à étendre <strong>le</strong>s re<strong>le</strong>-<br />
vés de toutes <strong>le</strong>s stations a ces n années, Les corrélations sont estimées mois<br />
par mois pour chaque coup<strong>le</strong> de stations, afin d'éviter l'influence de l'effet<br />
saisonnier.<br />
I1 est très important, lors de ces estimations, de ne pas fausser <strong>le</strong>s va-<br />
riances des échantillons calculés en utilisant sans autre précaution <strong>le</strong>s vérita-<br />
b<strong>le</strong>s équations de régression. Considérons pax exemp<strong>le</strong> <strong>le</strong>s stations i et j pour<br />
<strong>le</strong>squel<strong>le</strong>s on dispose, au mois m, d'une série d'observations communes portant sur<br />
p années, soit, pow: une année k donnée, q.m (k) et qj,m (k). On sait que pour<br />
utiliser aisément <strong>le</strong>s corrélations, il faut que la régression de qj,* en qi,,,,<br />
par exemp<strong>le</strong>, soit linéaire et, dans toute la mesure du possib<strong>le</strong>, homoscédastique,<br />
-C-------C----------___I___L_LC_________-<br />
(7) On lira avec profit, à propos du traitement des mesures de salme, m artic<strong>le</strong><br />
de J. CLAUDE, intitulé "une chaîne de programmes pour <strong>le</strong> traitement des données<br />
sur la salinitéf1, et publié dans <strong>le</strong>s CAHIERS ORSTOM, série HYDROLOGIE, Vol.<br />
IX, no 2, 1972 -<br />
(2) Voir l'artic<strong>le</strong> de Y. BRUNET-MOm intitulé ffEtude de l'homogénéité des séries<br />
chronologiques de précipitations annuel<strong>le</strong>s par la méthode des doub<strong>le</strong>s masses",<br />
et publié dans <strong>le</strong>s CAHIERS ORSTOM, série KYDROLOGIE, Vol. VïII, no 4, 1971 -
410<br />
I1 importe donc au départ de faire en sorte que, par anamorphose ou changement<br />
de variab<strong>le</strong>, ces conditions soient réalisées ; soit x la transformée correspondant<br />
à ,(k)<br />
is<br />
et yk la transformée de q (k). On sait que la régression de y<br />
9 .i9m -.<br />
en x s'exprime par la relation :<br />
YX = Yp + r<br />
avec <strong>le</strong>s notations habituel<strong>le</strong>s.<br />
P Y<br />
p 7<br />
(x-Z)<br />
P<br />
Mais yx ainsi calculée correspond à la moyenne conditionnel<strong>le</strong> des va<strong>le</strong>urs<br />
possib<strong>le</strong>s de y pour x donné et non pas à une va<strong>le</strong>ur isolée. Une tel<strong>le</strong> va<strong>le</strong>ur se-<br />
rait donnée par la relation y = yx + € dans laquel<strong>le</strong> & est une variab<strong>le</strong> aléatoire<br />
qui est souvent considérée comme étant norma<strong>le</strong> de moyenne nul<strong>le</strong> ; el<strong>le</strong> est indé-<br />
pendante de x si la condition d'homoscédasticité est réalisée. Négliger E dans <strong>le</strong><br />
calcul des débits non observés de la station conduit & diminuer artificiel<strong>le</strong>ment<br />
la variance de l'échantillon qu'on aura constitué, d'autant plus que <strong>le</strong> coeffi-<br />
cient de corrélation est plus faib<strong>le</strong>.<br />
Pour atre correct, si on veut utiliser l'équation de régression à ces fins,<br />
il faudrait d'abord déterminer la distribution de E, puis, au moment de la reconstitution,<br />
calcu<strong>le</strong>r yx et lui ajouter une va<strong>le</strong>ur € tirée au hasard dans la loi de<br />
distribution ainsi établie. Cela pose en fait un certain nombre de problèmes pratiques<br />
(apparition de débits négatifs) provenant du fait que <strong>le</strong>s hypothèses de<br />
base ne sont pas vraiment respectées et que l'estimation de l'écart-type de E est<br />
peu précise par suite de la petite tail<strong>le</strong> de l'échantillon qui sert à l'établir.,<br />
POW toutes ces raisons, il est fina<strong>le</strong>ment préférab<strong>le</strong> de procéder d'une manière<br />
beaucoup plus simp<strong>le</strong>, certes peu conforme à l'esthétique mathématique, mais qui<br />
respecte assez bien la variance initia<strong>le</strong> : prendre une droite passant par l'origine<br />
: y = Ax,<br />
I1 est parfois possib<strong>le</strong> d'améliorer la corrélation en tenant compte de la<br />
pluviométrie loca<strong>le</strong> par l'application d'une régression multip<strong>le</strong> ('IIo Supposons,<br />
pour fixer <strong>le</strong>s idEtes, que la variab<strong>le</strong> dépendante (cel<strong>le</strong> qu'on veut estimer) soit<br />
(k). Le bassin de surface S. qui fournit ce débit peut<br />
'j,m J<br />
-&re inclus dans <strong>le</strong> bassin de surface Si qui fournit %,m(k), on a<br />
alors S. (si,<br />
J<br />
-Inclure si, on a aïors S. ) si,<br />
J<br />
-n'avoir pas de point commun avec S..<br />
Dans' <strong>le</strong> premier cas, <strong>le</strong>s pluies tombant sur S. alimentent tota<strong>le</strong>ment S. et<br />
3 1<br />
on a peu de chance d'améliorer la régression en <strong>le</strong>s prenant en compte. Par contre,<br />
il n'est pas impossib<strong>le</strong> que la pluie tombant sur <strong>le</strong> bassin intermédiaire S exi-j<br />
plique une partie non négligeab<strong>le</strong> de la variance de q (k). Dans <strong>le</strong> second cas,<br />
j,m<br />
<strong>le</strong>s pluies sur Si expliquent au moins partiel<strong>le</strong>ment q.m(k) et ne peuvent expli-<br />
quer qj,m(k) que par l'intermédiaire de q<br />
i,m<br />
(k) : il est donc à priori inuti<strong>le</strong> de<br />
--------------------_______c____________--<br />
(1) Pour <strong>le</strong> détail de l'application des régressions multip<strong>le</strong>s à 1-'hydrologie, on<br />
peut se reporter par exemp<strong>le</strong> à l'artic<strong>le</strong> de P. TOUCKEBEiJF DE LUSSIQ'E "Régressions<br />
et corrélations multip<strong>le</strong>s en hydrologie", publié dans <strong>le</strong>s CAHIERS ORSTOM, série<br />
HYDROLOGIE, Vol. VïII, ne 4, 1971 -
411<br />
<strong>le</strong>s introduire. Par contre, qj est la somme de qi et de qj-i, variab<strong>le</strong> expliquée<br />
au moins partiel<strong>le</strong>ment par <strong>le</strong>s pluies qui tombent sur <strong>le</strong> bassin intermédiaire j-i,<br />
l'introduction de ces pluies dans la régression peut donc améliorer l'estimation.<br />
Dans <strong>le</strong> dernier cas il est évident que, si on dispose de pluies sur S il peut<br />
&tre uti<strong>le</strong> de <strong>le</strong>s introduire.<br />
j'<br />
I1 ne sera donc intéressant d'utiliser des régressions multip<strong>le</strong>s portant<br />
sur <strong>le</strong>s pluies que si <strong>le</strong>s données s'appliquent à un bassin contrblé par i ou par<br />
j, mais pas par <strong>le</strong>s deux à la fois. Il faut toutefois noter que, <strong>le</strong>s bassins et<br />
sous-bassins étant voisins, <strong>le</strong>s pluies, surtout à l'échel<strong>le</strong> du mois, ont des<br />
chances d'être assez fortement liées et on risque de vouloir faire expliquer à la<br />
variab<strong>le</strong> pluviométrique choisie une partie de la variance déjà expliquée par sio<br />
Dans <strong>le</strong> temps, l'influence de la pluie tombée <strong>le</strong> mois m sera ßans doute<br />
prépondérante, mais <strong>le</strong>s pluies des mois antérieurs peuvent avoir une influence<br />
non négligeab<strong>le</strong>. On introduira donc, suivant <strong>le</strong>s circonstances, soit <strong>le</strong>s pluies<br />
du mois (pluie mensuel<strong>le</strong> à un pluviomètre ou moyenne des pluies mensuel<strong>le</strong>s à plusieurs<br />
pluviomètres), soit un indice pluviométrique défini come une somme<br />
Pm + a P + a2 Pm-2 + ooo décroît quand i augmente, par exem-<br />
1<br />
OU a<br />
m-1<br />
+ a. P<br />
i m-i i<br />
p<strong>le</strong> en progression géométrique de raison i/2. Eh fait la plupart du temps on se<br />
limitera à la pluie du mois.<br />
Les relations ainsi établies, employées avec <strong>le</strong>s précautions indiquées en<br />
ce qui concerne <strong>le</strong> respect de la variance, servent à établir une chronique de dé-<br />
bits mensuels sur une période de n années pour toutes <strong>le</strong>s stations* On peut alors<br />
chercher à augmenter la durée de cette période avec <strong>le</strong> seul secours des données<br />
pluviométriques. Cette opération est préparée lors d.e 11 étude précédente d'homogé-<br />
néisation, mais l'absence de variab<strong>le</strong>s explicatives fld6bitsfl peut modifier assez<br />
considérab<strong>le</strong>ment l'influence relative des autres variab<strong>le</strong>s explicativeso<br />
On peut commencer à rechercher, pour chaque bassin, la relation entre <strong>le</strong><br />
débit moyen annuel Qi (k) et la pluie moyenne annuel<strong>le</strong> Pi (k) estimée par la mé-<br />
thode de miessen si on a plusieurs pluviomètres. La relation Q (P) comporte un<br />
seuil physiquement explicab<strong>le</strong> qui correspond en gros à la précipitation minima<strong>le</strong><br />
annuel<strong>le</strong> nécessaire à l'apparition de llécou<strong>le</strong>ment ; on la désignera par Poo A<br />
ce seuil se superpose une constante qui traduit la diminution de variance due à<br />
l'application de la régression. Comme P n'est pas connu à priori, on nia plus la<br />
O<br />
ressource de faire passer la droite de régression Q (P-PO) par l'origine (on suppose<br />
en effet que la régression est linéaire ;si el<strong>le</strong> ne l'est pas, il convient<br />
de faire <strong>le</strong>s transformations convenab<strong>le</strong>s) o On peut appliquer l'équation de régression<br />
vraie Q = A (P-Po), rechercher la loi de distribution des résidus, et Procéder<br />
au calcul de l'échantillon étendu comme on l'a indiqué pour l'homogénéisation,<br />
avec <strong>le</strong>s mgmes avantages et <strong>le</strong>s mêmes inconvénients. ûn préfère souvent<br />
utiliser un expédient dénué, il faut <strong>le</strong> dire, de base statistique solide, Au lieu<br />
d'appliquer <strong>le</strong>s moindres carrés aux résidus Q<br />
on <strong>le</strong>s applique<br />
calculé - Qobservé'<br />
aux distances des points représentatifs des coup<strong>le</strong>s (&k,<br />
à la droite w(p-pO)<br />
qui ne sera plus alors une vraie droite de régression. On dit qu'on utilise une<br />
''pseudo-r égressionl' .<br />
brsqulon a ainsi mis au point un échantillon étendu de débits moyens annuels,<br />
on reprend la m&me opération à Iféchel<strong>le</strong> mensuel<strong>le</strong>, en tenant compte au<br />
besoin de l'influence des pluies des mois antérieurs, ainsi qu'on l'a indiqué
41 2<br />
pour l'opération d'homogénéisation. Ces nouvel<strong>le</strong>s régressions sont surtout des-<br />
tinées à fournir la forme de la répartition des débits dans l'année, car <strong>le</strong>s dé-<br />
bits annuels déduits des débits mensuels ainsi reconstitub sont souvent moins<br />
valab<strong>le</strong>s que ceux qui sont obtenus par une regression à l'échel<strong>le</strong> de l'année ;<br />
il convient cependant de s'en assurer.,<br />
I1 reste à vérifier que <strong>le</strong>s données mensuel<strong>le</strong>s retenues pour <strong>le</strong>s diffé-<br />
rentes stations sont compatib<strong>le</strong>s entre el<strong>le</strong>s, c'est-à-dire qu'en général un débit<br />
d'une station aval doit &tre supérieur ou au moins égal à celui de toute station<br />
amont, que si une station aval AV est placée SUT un cours principal alimenté par<br />
deux bras dont <strong>le</strong>s débits sont contr81és par deux stations AM1 et AM2, <strong>le</strong>s débits<br />
de AV doivent &tre au moins égaux aux sommes des débits de AM1 et AM2. Autrement<br />
dit, on ne doit pas admettre de débit négatif dans un bassin versant intermg-<br />
diaire, sauf éventuel<strong>le</strong>ment dans deux cas :<br />
- il y a des pertes physiquement reconnues, soit par infiltration,<br />
soit par évaporation (marais o.oetcooo),<br />
- il y a des stockages naturels importants (lacs .etc.).<br />
Ces cas particuliers mis .$ part, si on constate des débits négatifs ou ri-<br />
dicu<strong>le</strong>ment faib<strong>le</strong>s, et cela arrive malheureusement assez souvent, c'est que, mal-<br />
gré l'étude critique et la mise en ordre initia<strong>le</strong> des données, il y a des erreurs<br />
dans l'étalonnage et/ou des erreurs systématiques dans <strong>le</strong>s re<strong>le</strong>vés d'échel<strong>le</strong> et/<br />
ou une mauvaise répartition de ces re<strong>le</strong>vés dans <strong>le</strong> temps (observations trop es-<br />
pacées compte tenu du régime), I1 faut revenir sur 1'8tude critique et essayer<br />
de déterminer quel<strong>le</strong>s sont <strong>le</strong>s stations auxquel<strong>le</strong>s on peut faire <strong>le</strong> plus confian-<br />
ce ; il est nécessaire d'aboutir à un choix, meme si ,celui-ci est un peu arbi-<br />
traire. On considérera comme bons <strong>le</strong>s débits des stations sé<strong>le</strong>ctionnées, qui<br />
doivent bien entendu &re compatib<strong>le</strong>s entre el<strong>le</strong>s, et on retouchera <strong>le</strong>s débits<br />
incriminés des autres stations jusqu'à remplir <strong>le</strong>s conditions de compatibilité.<br />
3. - Homogénéisat ion et extension des données Ilsalinit ésf1<br />
Les observations directes sur la salve des eaux sont presque toujours<br />
plus rares, dans <strong>le</strong> temps et dans l'espace, que pour <strong>le</strong>s débits. On sera donc<br />
appelé à comb<strong>le</strong>r plus de lacunes que pour <strong>le</strong>s débits et à procéder à une exten-<br />
sion plus importante des périodes.,<br />
Si <strong>le</strong>s rivieres sont restées en l'état naturel, ou tout au moins au rn&me<br />
degré de rejets susceptib<strong>le</strong>s de modifier la salure, <strong>le</strong>s données recueillies récemment<br />
sont susceptib<strong>le</strong>s d'&tre transposées dans <strong>le</strong> passé. Sinon la transposition<br />
n'est pas Ithistoriquement1l possib<strong>le</strong>, mais c'est d'importance secondaire. Eh effet,<br />
on ne doit pas, pour la simulation, employer des échantillons lJévolutifslt, car<br />
<strong>le</strong>s résultats qu'on en tirerait n'auraient pas de sens, Au contraire, si, par des<br />
tests quelconques, on s'apercevait que <strong>le</strong>s conditions gbéralss de salure ont<br />
changé, on ne devrait conserver que <strong>le</strong>s résultats <strong>le</strong>s plus récents, meme si la<br />
tail<strong>le</strong> de l'échantillon devait passab<strong>le</strong>ment s'en ressentir.<br />
h situation se présente de la façon suivante. Pour tout mois de la période<br />
irhomogènelt et éventuel<strong>le</strong>ment llétendue't de l'échantillon historique, on<br />
peut disposer :<br />
- d'une série continue de re<strong>le</strong>vés de salure qui, passée dans une chaPne<br />
de traitements de salinité permet d'établir une série complète<br />
de va<strong>le</strong>urs des saliires journalières et mensuel<strong>le</strong>s,
413<br />
- d'une sbrie incomplète mais pamqttant <strong>le</strong> caLcul d'un certain nombre<br />
de salures moyennes jourzi.alj.èrm9<br />
- de re<strong>le</strong>vés sporadiques,<br />
- d'au.cm re<strong>le</strong>vé,<br />
Soit une station pour laquel<strong>le</strong> on s., &mart toute la pér:ode homogène, la<br />
distribution d'observations journalières mivantes (ûuivant <strong>le</strong>s mois de l'arde<br />
d'exploitation numérot8s Î à -121,<br />
pur <strong>le</strong>s salinités moyennes journdi&rcs :<br />
nsjl nsj2 nsjj nsjg nsj5 nsj6 nsj7 nsj8 nnjq nsjI0 nsj II ns%2 y<br />
poi1.r <strong>le</strong>s débits m,oyer,s journ.diers :<br />
pour <strong>le</strong>s dé5its moyen..^ mensuels :<br />
Chaque terme nqmi est égal au nombre d.fann6es que comporte la serie homo-<br />
gène, mais u2 terme nqji p.fast pas forcément égd à nqmi .multiplié par <strong>le</strong> nombre<br />
de jours du mois i, puisqu'icn certain nonbre do débits moyens mensuels Ont pu<br />
8tre reccnstitués lars de l'opération dfhomogénéisation, sans qu'on possède au-<br />
cun re<strong>le</strong>vé joiirmlier ,?our <strong>le</strong>s mois corres2ondants. On cherchera dans une pre-<br />
mière étape à reconstiti;er, pour chaque qji à.isponib<strong>le</strong>, <strong>le</strong> sji correspondant qui<br />
n.'auiait pas &té observ8. Dans une seconde Qtape, on fera la meme opération Szn"<br />
<strong>le</strong>s qmi et <strong>le</strong>s mio<br />
Pour un r6gtne hydr2roiogiqu.e donné, la concentration en sel dissous depend<br />
EU premier chef de la nEtu.re minéralogique du bassin concerné, de l'importance<br />
des nappes soutermines et de 13 vitesse GU transit de Ifeau dans ces nappes, Vi-<br />
.cesse qui intervient sux la durée du contact de cette eau avec la roche- On Peut<br />
aSouter comme paramètre l'agressivité des pr6cipitations mesurbe par <strong>le</strong>ur teneur<br />
en CO2 libre et 1ev.ï degr6 de pureté. Le ph&niimène est donc comp<strong>le</strong>xe et il ne<br />
faut pas s'attendre 5 p'iwoir <strong>le</strong> représenter par des relations simp<strong>le</strong>s.<br />
I1 est toutefois logique ds penser que <strong>le</strong>s eaux souterraines sont norma<strong>le</strong>-<br />
men;: plus chargées f3n. sols dissous que <strong>le</strong>s eaux de surface, par suite de <strong>le</strong>ur<br />
contact prolone;& 2.vei <strong>le</strong>s roches. I1 faut donc s'attendre 5 ce que <strong>le</strong>s basses<br />
eaux soient plus chrnqkà cge <strong>le</strong>s débits importants at il est logique qu'il exis-<br />
te m e relation, certes m n tcjnctiomei?-e, entre ia salurs et ïe débit, L'expé-<br />
riencs montre qu'il en est bien ainsi ; el<strong>le</strong> met. de pl.us en evidence une in-<br />
f?u.eiic? saisomiere sur cntte rels.tion.<br />
?oc? 1'6tabiir OB -pxèd.e mois par mois, ou tout au m.oins trimestre par<br />
trimestre. Pour chaque mais :<br />
- on raycrte tocs <strong>le</strong>s sji obserTr8s en regard des qji qui <strong>le</strong>m cor-<br />
rnsporifimt ?<br />
- on oowtzte m e grande difipxioy, rnwie avec tendance tres nette<br />
5 i:.-ax aroissmc~ des sj.; avec <strong>le</strong>s ?%$iq
La dispersion est souvent tel<strong>le</strong> que l'utilisation sans précaution d'une ré-<br />
gression poserait des problèmes importants de réduction de variance, davantage<br />
que pour <strong>le</strong>s débits. Pour éviter ces inconvénients, nous avons mis au point la<br />
technique suivante qui a au moins l'avantage de respecter intégra<strong>le</strong>ment <strong>le</strong>s pro-<br />
priétés statistiques de l'échantillon.<br />
On détermine un certain nombre de classes de débits,<br />
Classe 1 O à ,qj<br />
Classe 2 lqj à ,qj<br />
------------------<br />
Classe k k-,qj à ,qj<br />
Classe k+l > ,qj ,<br />
de tel<strong>le</strong> façon qu'à l'intérieur de chacune on puisse considérer que l'influence<br />
de la variation du débit sur la salinité est négligeab<strong>le</strong>. Eh associant, à chaque<br />
qj de l~échantillon, la va<strong>le</strong>ur s. correspondante, on constitue autant de 'Iréservoirstt<br />
de salures qu'il y a de classes de débits. Avec <strong>le</strong>s précautions prises,<br />
dans chaque réservoir s est indépendant de qo Les différentes salinités contenues<br />
dans un, réservoir sont identifiées par un numéro.<br />
-<br />
A l'ORSTOM, l'opération est effectuée au moyen du programe 703 pour un<br />
découpage mensuel (comme ici) et par <strong>le</strong> programme 703 bis si <strong>le</strong> découpage est trimestriel.<br />
Le résultat est une matrice des salures à trois dimensions dont <strong>le</strong>s<br />
indices représentent<br />
- <strong>le</strong> numéro d'ordre de la salure dans <strong>le</strong> réservoir,<br />
- <strong>le</strong> mois,<br />
- la classe de débit à laquel<strong>le</strong> appartient <strong>le</strong> débit associé à la<br />
salinit é.<br />
Cette matrice permet de reconstituer <strong>le</strong>s salues correspondant à tous <strong>le</strong>s<br />
débits moyens journaliers observés pour <strong>le</strong>squels il n'y a pas eu de mesure de salinité.<br />
Le programme 704, qui fait cette opération pour un découpage mensuel,<br />
procède de la façon suivante :<br />
a - Wegistrement de la matrice des salures.<br />
- Lecture des débits limites de classeso<br />
- Lecture de la matrice des salures (aanS L'ordre : classe, mois,<br />
numéro de série de la salure) : ECHASA (NOCL, MOIS, K).<br />
b - Lecture des débits journaliers.<br />
- On lit <strong>le</strong>s débits journaliers pour un mois et on <strong>le</strong>s met dans<br />
un veateur à 31 positions DEB (JIo<br />
- Au fur et à mesure de la <strong>le</strong>cture par carte de quinzaine, on<br />
reperfore <strong>le</strong>s données pour constituer un jeu définitif débits<br />
et salures.<br />
C - Lecture des salures moyennes journalières.<br />
- ~n lit <strong>le</strong>s saïures moyennes pour un mois (ie méme que celui<br />
des débits qu'on vient de traiter) et on <strong>le</strong>s range dans un vecteur<br />
SAL (J).<br />
d - Détermination des salures journalières manquantes.<br />
- Dans une bouc<strong>le</strong> J = l,3l, on teste d'abord DEB (JIo S'il est<br />
négatif, c'est qu'il n'y a pas de débit observé pour <strong>le</strong> jour
415<br />
J ; il n'est donc pas possib<strong>le</strong> de complèter la salinité et on<br />
passe. S'il est positif, on teste SAI; (J) ; si el<strong>le</strong> est positive,<br />
c'est qu'il y a observation de sdinité ; on passe,, S'il<br />
est négatif, on complète.<br />
- Pour compléter : on cherche dans quel<strong>le</strong> classe se trouve<br />
X æ DEE3 (j), soit NOCZ ; on tire au hasard un nombre inférieur<br />
ou égal à NC, nombre de salures classées dans NOCL, soit K,<br />
et on associe à DEB (J) une salure SAL (J) éga<strong>le</strong> 2 ECHASA<br />
(NOCL, MOIS, K).<br />
e - Perforation des salinités sous la m&me forme que <strong>le</strong>s débits observés.<br />
On revient alors à b- pour lire <strong>le</strong>s débits du mois suivant, et<br />
on continue airisi jusqu'à épuisement des données.<br />
L'opération a permis de constituer un échmtillon pour <strong>le</strong>quel à<br />
chaque débit moyen journalier correspond une salme moyenne journalière. Pour <strong>le</strong>s<br />
mois comp<strong>le</strong>ts en débits observés, on peut alors calcu<strong>le</strong>r <strong>le</strong>s salures moyennes mensuel<strong>le</strong>s,<br />
Restent <strong>le</strong>s mois pour <strong>le</strong>squels on a pu reconstituer <strong>le</strong>s débits moyenc<br />
mensuels, sans posséder <strong>le</strong>s débits journaliers (homogénéisation) Pour <strong>le</strong>ur attribuer<br />
une salme moyenne, on procede d'une façon analoge à ce qui précède,<br />
4,- Calcul de l'échantillon historique pour <strong>le</strong> modè<strong>le</strong><br />
Lors du découpage géographique, on s'arrange pou que <strong>le</strong>s stations du réseau<br />
tombent autant que possib<strong>le</strong> à des limites d'unités hydrauliques, Mais cela<br />
n'est pas toujours possib<strong>le</strong> dfune part, et d'autre part <strong>le</strong>s unités hydrauliques<br />
sont toujours plus nombreuses que <strong>le</strong>s stations de mesure, I1 est donc nécessaire<br />
de procéder à une interpolation géographique, et m&ne parfois d'utiliser l'analogie<br />
et la transposition pour calcu<strong>le</strong>r tous <strong>le</strong>s An, SAn, ACn et SACn,<br />
Pour <strong>le</strong> calcul des An et ACn (apports), on comnence par dresser un tab<strong>le</strong>au<br />
donnant, pour chaque unité n, sa superficie et la nat-.ne des apports, Lorsque<br />
l'unité est encadrée en amont et en aval, on fait simp<strong>le</strong>ment une répartition au<br />
prorata des superficies, au moins dans un premier stade (interpolation géographique).<br />
Lorsque l'unité est en dehors du réseau des stations, on cherche à lui<br />
attribuer un débit spécifique par comparaison avec d'autres parties mieux connues<br />
du bassin ou avec d'autres bassins que l'on suppose avoir <strong>le</strong> mbe régime (extrapolation<br />
ou transposition) Cette dernière opération provoque nécessairement une<br />
légère erreur systématique par défaut sur la variance de l'échantillon global<br />
constitué pour <strong>le</strong> modè<strong>le</strong>, mais cette influence est presque toujomnégligeab<strong>le</strong>.<br />
Si l'unité hydraulique est confondue avec <strong>le</strong> bassin versant d'une station<br />
de base, on identifie <strong>le</strong>s apports As à la station a u apports An sur l'unité.<br />
si la station de base est unique et son bassin versant différent de l'uni-<br />
té, <strong>le</strong>s apports An sur l'unité sont obtenus à partir de ceux de la station de<br />
base As par calcul au prorata des superficies des bassins : An = As * Sn/Sso<br />
Si 2 stations de base encadrent la limite de l'mité hydraulique, <strong>le</strong>s ap-<br />
ports An sur l'unité sont calculés par interpolation linéaire entre <strong>le</strong>s apports<br />
As7 et As2 aux stations 7 et 2 : A ds1 -t (As2 - Asl) * (Sn - Ssl)/(Ss2 - Ss7)-<br />
Pour calcu<strong>le</strong>r des apports intermédiaires ACn avec 2 stations de base encadrant<br />
l'unité, on applique la relation : ACn = (As2 - AsII * Sn/(Ss2 - SSI)~<br />
Lors des calculs relatifs au 4ème cas, on rencontre parfois quelques difficultés<br />
: au pas de temps mensuel, la différence entre <strong>le</strong>s apports observés à la<br />
station aval et ceux de la station amont peut &tre négative bien que <strong>le</strong> bilan
416<br />
annuel soit norma<strong>le</strong>ment positif. Ceci se produit en particulier lorsque <strong>le</strong>s deux<br />
stations de base sont assez éloignées ou séparées par un bassin intermédiaire<br />
de grande surface comportant des affluents dont <strong>le</strong> régime hydrologique diffère,<br />
de celui du cours d'eau SUT <strong>le</strong>quel est située la station la plus amont. On peut<br />
alors procéder comme suit, pour différentes unités hydrauliques situées entre<br />
deux stations de base.<br />
- On calcu<strong>le</strong> <strong>le</strong>s apports intermédiaires mensuels et annuels entre <strong>le</strong>s<br />
2 stations.<br />
- On détermine la distribution temporel<strong>le</strong> moyenne de cet écou<strong>le</strong>ment<br />
intermédiaire pour la totalit 6 de la période disponib<strong>le</strong> (période d'observations<br />
communes entre <strong>le</strong>s deux stations). On obtient ainsi des coefficients mensuels de<br />
distribution exprimés en $ du modu<strong>le</strong> interannuel.<br />
- Pour chaque année de la période de reconstitution, on utilise ces<br />
coefficients pour <strong>le</strong> calcul de l'apport intermédiaire mensuel à partir de l'ap-<br />
port annuel observé.<br />
- On répartit cet apport intermédiaire mensuel sur chaque unité au<br />
prorata de sa superficie et de cel<strong>le</strong> du bassin versant intermédiaire.<br />
Cette difficulté ne devrait du reste pas se présenter si <strong>le</strong>s apports aux<br />
stations de base ont été soigneusement préparés.<br />
I1 y a de nombreuses façons de calcu<strong>le</strong>r <strong>le</strong>s SAn et SACn. On pourrait par<br />
exemp<strong>le</strong> passer par l'intermédiaire des poids de sel transités mois par mois aux<br />
stati'ons de base, et opérer de façon analogue à ce qui a été fait pour <strong>le</strong>s ap-<br />
ports. Cela supposerait une certaine homogénéité dans la production de la salure<br />
pour l'ensemb<strong>le</strong> du bassin, ou tout au moins pour des parties importantes du<br />
bassin faci<strong>le</strong>ment délitnitab<strong>le</strong>s. Cette condition n'est pas toujours r8alisée et<br />
l'origine de la salure des eaux est souvent localisée,<br />
L'interpolation géographique peut &tre sérieusement améliorée si on dispose,<br />
pour chaque bassin d'alimentation d'une unité, des surfaces des formations<br />
salines, et si on peut établir une relation entre cette surface et l'apport de<br />
sel, Ceci revient à définir pour un mois m donné une relation de la forme :<br />
Sape = fs (Qspe, pS) OU Supe est l'apport ,spécifique en sel, Q l'apport spéspe<br />
cifique en eau et pS <strong>le</strong> pourcentage de formation saline.<br />
Avec une bonne carte lithographique, on peut assez faci<strong>le</strong>ment déterminer<br />
pS pour tous <strong>le</strong>s bassins fournisseurs des unités et pour <strong>le</strong>s bassins contrbiés<br />
par des stations de réseau. Le problème serait alors résolu s'il était possib<strong>le</strong><br />
de déteminer fs avec une approximation convenab<strong>le</strong>. Cette détermination ne peut<br />
se faire qu'en traçant un faisceau de courbes expérimenta<strong>le</strong>s à partir des résul-<br />
tats des stations du réseau. La précision dépend du nombre de mesures disponi-<br />
b<strong>le</strong>s à chaque station, du nombre de stations et de la variabilité de pS, ce der-<br />
nier facteur étant particulièrement important.<br />
Si l'information disponib<strong>le</strong> est insuffisante, ce qui est presque toujours<br />
<strong>le</strong> cas, il est préférab<strong>le</strong> de procéder par analogie. Nous indiquerons la méthode<br />
utilisée par H. DOSSEUR (0,R.S.T.O.M.). On part de séries d'apports et de salures<br />
déjà constituées pour <strong>le</strong>s stations de base.<br />
Si <strong>le</strong> bassin de l'unité est confondu avec celui d'une station, on iden-<br />
tifie <strong>le</strong>s concentrations. Sinon, on affecte 5 chaque unité une station choisie<br />
de tel<strong>le</strong> façon que son bassin soit <strong>le</strong> plus représentatif de celui de l'unité<br />
considérée, compte tenu de sa situation géographique et de sa nature galogique.<br />
Ce choix peut &tre précisé à partir de renseignements concernant la salinité<br />
dans un secteur déterminé (mesures ponctuel<strong>le</strong>s, indications d'ordre qualitatif...).
41 7<br />
On associe à l'unité considérée <strong>le</strong>s réservoirs de d ures élaborés pour la sta-<br />
tion @ lui a été affectée, Pour chaque débit ACn, on détermine une concentra-<br />
tion moyenne SACn par tirage au hasard dans ces réservoirs, suivant la méthode<br />
indiquée antérieurement, avec toutefois une transformation préalab<strong>le</strong> des clas-<br />
ses de débits en classes de débits spécifiques pour tenir compte du rapport des<br />
superficies entre l'unité et <strong>le</strong> bassin de la station associ&,<br />
Du fait m&me de la méthode utilisée, l1échantillon des SACn présentera<br />
2 peu près sûrement des incompatibilités analogues à cel<strong>le</strong>s qui ont été signalées<br />
pour <strong>le</strong>s débits liquides, I1 faudra donc contr8<strong>le</strong>r, au moyen d'un programme<br />
annexe, que <strong>le</strong>s poids de sel P% = AG * SAC, obtenus pour chaque mois de chaque<br />
année de la période historique sont tels que <strong>le</strong> PS d'un point quelconque<br />
du réseau hydropaphique est au moins égal au PS de tout point situé à son<br />
amont, et que, si un point i limite à ï'avaï une unité limitée à ll~ont par<br />
des points 1, m r, Psi doit &tre au moins égal à Psi I- PS, I- oooooop PS, ,<br />
Pour tous <strong>le</strong>s mois Ou ces conditions ne sont pas réalisées, il est indispensab<strong>le</strong><br />
de retoucher la répartition des salinités dans <strong>le</strong> bassin pour rétablir<br />
la compatibilit 6.<br />
* *<br />
*
41 8<br />
Fig:l - SCHEMA TOPOLOGIQUE
ABSTRACT<br />
THE USE OF SIMULATION TECHNIQUES FOR SEQUENTAL<br />
GENERATION OF SHORT-SIZED RAINFALL DATA AND ITS<br />
APPLICATION IN THE ESTIMATION OF DESIGN FLOOD<br />
H.D.Sharma*, Dr.A.P.Bhattacharya** and S.R.Jindal;t**<br />
The studies based on rainfall runoff data are considerably vi-<br />
tiated in <strong>the</strong> event of inadequate data, as <strong>the</strong> reliability o€ <strong>the</strong><br />
probabilities of occurrence is reduced, It is, however, possib<strong>le</strong> to<br />
get over <strong>the</strong> lacuna of inadequacy of data by creating bigger-sized<br />
artificial series of rainfall. The use of such a series gives grea-<br />
ter precision in <strong>the</strong> estimations or projections based on expected m a<br />
ximum rainfall with specified <strong>le</strong>vels of occurrence and also provides<br />
better insight into possib<strong>le</strong> patterns of behaviour. This is done by<br />
<strong>the</strong> procedure of sequential generation fo data by <strong>the</strong> use of simula-<br />
tion techniques. Making use of <strong>the</strong>se, <strong>the</strong> technique has been applied<br />
for generating rainfall series of 100 nombers on <strong>the</strong> basis of recor-<br />
ded rainfall data for a period of ten years, The generated rainfall<br />
series was compared with <strong>the</strong> historical data which showed strong co-<br />
rrelation.<br />
These results have been used for <strong>the</strong> estimation of design<br />
flood for Yamuna river at Okhla (Delhi).<br />
Les procédés qui consistent à déduire <strong>le</strong>s écou<strong>le</strong>ments des pr5<br />
cipitations voient <strong>le</strong>ur efficacité considérab<strong>le</strong>ment diminuée lorsque<br />
<strong>le</strong>s observations concecnant cel<strong>le</strong>s-ci sont insuffisantes, par suite<br />
de l'imprécision qui regne alors sur l'estimation des probabilités<br />
de ces précipitations. On peut essayer de tourner la difficulté en<br />
créant artificiel<strong>le</strong>ment de longues séries d'observations pluviométri<br />
ques. L'utilisation de tel<strong>le</strong>s séries conduit a une meil<strong>le</strong>ure préci-<br />
sion des estimations ou des prédéterminations basées sur la pluie ma<br />
xima<strong>le</strong> attendue avec une probabilité donnée; el<strong>le</strong> permet aussi une<br />
meil<strong>le</strong>ure vue suc <strong>le</strong>s schémas possib<strong>le</strong>s du comportement des précipi-<br />
tations. On procede par génération séquentiel<strong>le</strong> des données, en uti-<br />
lisant <strong>le</strong>s techniques de simulation, On donne comme exemp<strong>le</strong> la cons-<br />
titution d'une série de 100 ans à partir d'une période de 10 ans<br />
d'observations. La séries engendrée , comparée avec la sérìe histori<br />
que, met en 'evidence une forte corrélation.<br />
Ces résultats ont étd utilisés pour l'estimation d'une crue<br />
de projet à Okhla, sur <strong>le</strong> f<strong>le</strong>uve Yamuna [Delhi).<br />
* Director , Irrigation Research Institute , Roorkee, U .P,<br />
$:* Research Officer , Basic Research Division, Irrigation Research<br />
Institute, Roorkee, U.P.<br />
;'
420<br />
i. INTRODUCTION<br />
1.1 In all iqr-go<strong>the</strong>tical investigations, particularly in <strong>the</strong> estimation<br />
of design flood of river basins, it 1s essential to have an idea of <strong>the</strong><br />
distribution of rainfall as also <strong>the</strong> relationship between rainfall a d<br />
runoff. This is, however, not always possib<strong>le</strong> in case of small sized<br />
data, extending over 8ay 10 to 20 years as is usually met vit<br />
oractice, as <strong>the</strong>se may not be representative of <strong>the</strong> vorst possib<strong>le</strong><br />
conditions prevaillng in <strong>the</strong> catchment. On account of such shortcomings,<br />
it is likely that <strong>the</strong> findings based <strong>the</strong>reon may not be realistic. This<br />
difficulty may be overcome by resorting to <strong>the</strong> technique of sequential<br />
generation with <strong>the</strong> aid of which it is possib<strong>le</strong> to artificially create<br />
larger sized data series.<br />
2. CONCWT OF W?UENTIAL GEWERI'EION<br />
2.1 sequential generation is a statistical process usiag Monte Carlo<br />
methods to produce a random sequence of hydrologic or any o<strong>the</strong>r data<br />
on <strong>the</strong> basis of a stochastic model for <strong>the</strong> hydrologic process. Monte<br />
Carlo method is an experimental or merical probability method used<br />
for <strong>the</strong> statistical sampling of random variab<strong>le</strong>s. The sequence so<br />
generated makes possib<strong>le</strong> detai<strong>le</strong>d study of <strong>the</strong> performance of various<br />
hydrologic events, thus helping <strong>the</strong> development of well balanced hydo<br />
rologic designs.<br />
2.2 Un<strong>le</strong>ss <strong>the</strong> record is too meagre to be considered as a represento-<br />
tive samp<strong>le</strong>, <strong>the</strong> statistical parameters derived from It should enab<strong>le</strong><br />
<strong>the</strong> hydrologist to construct a suitab<strong>le</strong> model that wlll generate<br />
hydrologic information for as long a period of time as desired. Bnce<br />
<strong>the</strong> statistical parameters of <strong>the</strong> population of <strong>the</strong> generated data<br />
are necessarily <strong>the</strong> same as those estimated from <strong>the</strong> bistorical date,<br />
<strong>the</strong> new information is limited<br />
that are inherent in <strong>the</strong> observed record.<br />
3<br />
errors of measurement and sampling<br />
n
2.3 The procedure of sampling by shuffling; cards which waa among <strong>the</strong><br />
srl<strong>le</strong>st techniques can be simplified by <strong>the</strong> use of random number tab<strong>le</strong>s.<br />
naugh random number tab<strong>le</strong>s are availab<strong>le</strong> as punched cards, with Increase<br />
3g use of digital cornputor, ma<strong>the</strong>matical methods for generating pseu-<br />
Fndom numbers within <strong>the</strong> computing machine have been developed'in order<br />
2.4<br />
eliminate <strong>the</strong> need for extensive input of random numbers.<br />
3 <strong>the</strong> basis of required statistical <strong>le</strong>vels of errors and confidence,<br />
Lthough <strong>the</strong> optimal size may be determined more realistically by compar-<br />
<strong>the</strong> cost of <strong>the</strong> Increased samp<strong>le</strong> size with <strong>the</strong> benefits of <strong>the</strong> corres-<br />
4 21<br />
The size of <strong>the</strong> hydrological data to be generated may be estimated<br />
mding increase in accuracy, provided that <strong>the</strong> benefit and cost data<br />
:e availab<strong>le</strong>.<br />
, ANàLYSIS OF RAINFALL QATA<br />
3.1<br />
The rainfall data analyse8 herein pertain to 6 hour annual storms<br />
?corded at New Delhi for a period of 10 years from 1956 to 1965. They<br />
ive been arranged in such e manner that <strong>the</strong> storm starts with <strong>the</strong> first<br />
burly rainfall and ends at <strong>the</strong> 6th hourly rainfall, although in reality<br />
Le arrangement may be vitiated in some cases by <strong>the</strong> occurrence of a<br />
Bizz<strong>le</strong> before <strong>the</strong> recording of <strong>the</strong> main _portion of <strong>the</strong> storm or by<br />
beaks within <strong>the</strong> duration of <strong>the</strong> storm. The recorded data may be seen<br />
I Tab<strong>le</strong> I.<br />
FORMULBTION OF THE MATHEMûTICAL MODEL<br />
:.1 To develop a suitab<strong>le</strong> model to represent <strong>the</strong> time degendent<br />
ndom process of <strong>the</strong> hourly rginfalls, <strong>the</strong> following non-stationary<br />
rkov-chain niodel(l) was found to be consistently satisfactory.<br />
.) Ven Te Chow, Hand<strong>book</strong> of Applied Hydrology, pp. 8-93,<br />
McGraw Hill Book Co.
422<br />
......... (1)<br />
where xt x <strong>the</strong> hourly rainggll of any one of B annual<br />
storms at <strong>the</strong> t hour,<br />
xt-1 z <strong>the</strong> hgurly rainfall at <strong>the</strong> preceding or <strong>the</strong><br />
(t-i) h hour,<br />
t = time in hour ranging from 1 to m,<br />
r = Markov Chain Coefficient,<br />
6~ = random component due to hourly rainfall xt ,<br />
For <strong>the</strong> first hour when t = 1, <strong>the</strong> trend component r Xt,l become<br />
zero and X1 may be taken to be equal to €1 . The Markov Chah Coeffi-<br />
cient r and <strong>the</strong> random component €G may be determined from <strong>the</strong> give1<br />
rainfall data by <strong>the</strong> method of <strong>le</strong>ast squares by fitting a straight<br />
line between Xt and Xt-1.<br />
4.2 For <strong>the</strong> rainfall data recorded at New Delhi Station, <strong>the</strong> storm<br />
duration m =6 hours and number of annual storms, Ns10. The distribi<br />
tion parameters, mean and standard deviation of <strong>the</strong> historical rain.<br />
fall data were determined for each hour and are given in column 2 ai<br />
3 of Tab<strong>le</strong> II. The values of <strong>the</strong> random component et and <strong>the</strong> Markov<br />
Chain Coefficient r were worked out by <strong>the</strong> method of <strong>le</strong>ast squares<br />
and are shown in columns 4 end 5 in Tab<strong>le</strong> II.<br />
4.3 In <strong>the</strong> present analysis based on sequential generation, <strong>the</strong><br />
oractice followed has been to generate 100 pseudo-random numbers fo:<br />
uniform distribution of <strong>the</strong> first hourly rainfall by I.B.M. Compute:<br />
1401, whose programme is given in igpendix I. These 100 generated<br />
random mmbers of a uniform distribution have been taken as first<br />
hourly rainfalls of 100 storms and have been utilized for computing<br />
100 second hourly rainfalls by <strong>the</strong> Markov-chain model given in<br />
equation (1).
4.4 The rainfall data have been generated for each successive hour on<br />
<strong>the</strong> basis of <strong>the</strong> rainfall in tlx? previous hour according to <strong>the</strong> Markov<br />
chain model formulated. Knowing <strong>the</strong> Markov chain coefficient r and<br />
random component 6,for <strong>the</strong> second hour derived from <strong>the</strong> historical data<br />
(vide Tab<strong>le</strong> II) a random series of 100 second hourly rainfalls can be<br />
comouted by means of equation (i). These 100 generated second hourly<br />
rainfall were <strong>the</strong>n utilized to compute 100 third hourly rainfalls with<br />
<strong>the</strong> help of Markov chain coefficient r and random component (vide<br />
3<br />
Tab<strong>le</strong> II) by using equation (i). This procedure has been repeated for<br />
successive hourly rainfalls until ser<strong>le</strong>s of 100 hourly rainfalls for<br />
all <strong>the</strong> six hours were generated. The involved operations were carried<br />
out on IBM computer, 1620 as per programe given in Appendix II. The<br />
sequentially generated data has been shown in Tab<strong>le</strong> III.<br />
4.5 The cumulative probability function P(x) of <strong>the</strong> variate X may be<br />
obtained by <strong>the</strong> following equation;<br />
where ,.ho 5 Y & ,h~<br />
.o (2)<br />
fiois <strong>the</strong> lower limit of <strong>the</strong> variate X which may be assumed to be zero<br />
an8 is <strong>the</strong> upper limit of variaue X.<br />
4.5.1<br />
In <strong>the</strong> present analysis, <strong>the</strong> total hourly rainfall of annual<br />
storms have been worked out by adding all <strong>the</strong> six hourly rainfalls for<br />
each storm of historical data as well as generated data as per column 8<br />
of Tab<strong>le</strong>s I and III respectively. The cumulative probability per cents<br />
have been evaluated by <strong>the</strong> use of equation (a for ten storms of <strong>the</strong><br />
historical data as ?er column (9) of Tab<strong>le</strong> I as also for 100 storms of<br />
<strong>the</strong> generated data as per column (9) of Tab<strong>le</strong> III.<br />
423
424<br />
5. EsTIYVìTION OF DESIGN FLOOD WITH THE AID OF GENERATED RAINFALL S ~ I<br />
5.1<br />
It is possib<strong>le</strong> to derive a series of runoffs from <strong>the</strong> generated<br />
rainfall series provided that <strong>the</strong> relationship between rainfall and<br />
off for a particular basin is known. In <strong>the</strong> present Case, in which<br />
sequential generation techniques have been applied for only on rainfall<br />
station in <strong>the</strong> Yamuna catchment, vie. New Delhi and for wNch 110 rain-<br />
fall-runoff relationship was availab<strong>le</strong>, an assum3tion has been made tha<br />
surface runoff from rain storm is 80 per cent of rainfall during <strong>the</strong><br />
period of high floods when most of <strong>the</strong> catchment is saturated and in-<br />
filtration losses are of low order. Based on this preamb<strong>le</strong>, a series<br />
of runoffs may be assumed to be generated. The abovezentioned series<br />
can be utilised to compute <strong>the</strong> peak floods with <strong>the</strong> help of unit<br />
hydrograph developed at <strong>the</strong> gauge site and o<strong>the</strong>r methods.<br />
5.2 The series of 100 peak floods comguted for <strong>the</strong> river Yamuna at<br />
Okhla (catchment area = 6811 sq. Kms.) shown in column 10 of Tab<strong>le</strong><br />
III has been used to derive <strong>the</strong> following stochastic model on <strong>the</strong><br />
Dasis of princin<strong>le</strong>s of stochastic hydrology reported earlier for<br />
<strong>the</strong> estimation of design<br />
wnere yo is <strong>the</strong> design flood and Tk is <strong>the</strong> recurrence interval.<br />
5.3 From Mg. 4 based on above, <strong>the</strong> design flood with a recurrence<br />
interval of 500 years works out to 7794.5 cumec for <strong>the</strong> Yamuna river<br />
at Okhla (Delhi). It may however be 2ointed out that this should be<br />
ta<strong>le</strong>n to be more as an illustration of <strong>the</strong> application of <strong>the</strong> techn-<br />
ique of sequential generation for <strong>the</strong> estimation of <strong>the</strong> design flood<br />
in view of <strong>the</strong> limitations of <strong>the</strong> rainfall data for <strong>the</strong> entire catch-<br />
ent and - ilitv of a rainfall ru ela t i o =hi D<br />
(2) ,,ttZharya>A8P., Jindal, S.R. and RamJ%ff :Estimation of Design--<br />
Flood of <strong>the</strong> Ganga Fiver by processes of Stochastic hydrology",<br />
U. 2. Annual Besearch rieport, 1967 (Technical Memorandum No. 37) .<br />
1
5. DISCUSSION OF RLiSULTS<br />
6.1 Figure 1 gives a comparison between <strong>the</strong> worst possib<strong>le</strong> raiaall<br />
;tarm of <strong>the</strong> historical data and <strong>the</strong> generated series an <strong>the</strong> basis of<br />
I gra3hical plot between time in hours and hourly rainfall. It is<br />
.ndicated that <strong>the</strong>re is close Conformity for <strong>the</strong> entire storm dura-<br />
,ion comorising six hours.<br />
6.2<br />
425<br />
Gra^hical comparison has been made bbtwecn historical and generated<br />
Iata with respect to cumulative probability distributian of rainfall at<br />
he third hour, at which <strong>the</strong> peak rainfall was rècorded in <strong>the</strong> observa-<br />
ional as well as seqtientially generated data as per Figure 2. Close<br />
ionformity is indicated between <strong>the</strong> two distributions.<br />
6.3 similar comlarison has also been made for <strong>the</strong> two series for total<br />
ix-hourly rainfall for <strong>the</strong> annual storms as shown in Figure 3. Close<br />
ionformitg is observed in this case as well, both for ehird hourly rain-<br />
'all and total six-hourly rainfall, which provides added evidence regard-<br />
ng <strong>the</strong> representativeness of <strong>the</strong> sequentially generated series.<br />
6.4 mom <strong>the</strong> generated rainfall series, it has been ,possib<strong>le</strong> to derive<br />
cm?<br />
runoff ser<strong>le</strong>s which has been utilised toda series of 100 peak dischar-<br />
es. The latter orovide <strong>the</strong> background for <strong>the</strong> derivation of a stochastic<br />
ode1 wherefrom a hypo<strong>the</strong>tical 500-year design flood for <strong>the</strong> Yamuna<br />
iver at okhla (Delhi) may be estimated.<br />
7.1<br />
COI;CLU~IOMS<br />
he size of <strong>the</strong> historical data, particularly in such investigations<br />
herein this may be a limiting factor for analytical studies.<br />
7.2<br />
The technique of sequentiaï generation may be adopted for increasing<br />
storm rainfall is a time dependent raridom series and may be treated<br />
y El finite duration discrete non-stationary process that is ameneb<strong>le</strong> to<br />
a<strong>the</strong>matical formlation and analysis. For rainfall at New Delhi, <strong>the</strong><br />
istorical data of hourly rainfall in <strong>the</strong> annual storm has been regresented<br />
y nan-stationary Markov-chain model, <strong>the</strong> data consisting of ten .six-
426<br />
hourly storms.<br />
7.3 A compari n of <strong>the</strong> historical and generat LI 100 y ar data, both<br />
for third hourly rainfall and total six hourly rainfall, shows that <strong>the</strong><br />
sequentially generated series is fairly representative of <strong>the</strong> charactei<br />
istics of <strong>the</strong> historical data.<br />
7.4 The generated hydrologic series of rainf'all has been utilised to<br />
estimate <strong>the</strong> design flood of <strong>the</strong> Yamuna river at Okhla(De1hi) with a<br />
recurrence, interval of 500 years.<br />
The authors wish to acknow<strong>le</strong>dge <strong>the</strong> useful help extended by<br />
Messrs Ramjeet and D.C.Mltta1 in <strong>the</strong> analysis and computational work.<br />
APPENDIX I<br />
Fortran program for <strong>the</strong> generation of PseudÕrandom<br />
numbers in Uniform Mctribution. 4<br />
SE Q spm FORTUN STATENE2iT<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
11<br />
12<br />
13<br />
14<br />
15<br />
16<br />
17<br />
18<br />
19<br />
20<br />
C GEN-RATION OF 100 ?SEUDORANDUM Nuz.[BERS IN<br />
UNIFORM DI SJXtBUTION<br />
10 IALFA- 10**17 -C 3<br />
IRN1 = 10*(10**19-1) -b 7<br />
8% 0.0<br />
N=l<br />
91 READ 95,B<br />
95 FC-WAT (F4.2)<br />
Do 2 I = 1,100<br />
IRN IRNl*IALFA<br />
RSN= IRN<br />
RSJ!N= RUN* 10.0**(-20)<br />
SN = (B-A)* RSTN .) A<br />
R=N+1<br />
IRNI= Im<br />
2RIhT loo, SW<br />
100 FORMAT (2E 16 e 8)<br />
2 COICTIhUE<br />
IF (SENSE ShkTCS O) 92,3<br />
3 GO TO 91<br />
92 STOP 555<br />
END
10<br />
100<br />
APPENDIX II<br />
Fortran program for <strong>the</strong> generation of i00 slx hourly<br />
rainfall storms for New Delhi Station by Markov-chain<br />
Model.<br />
DIMEESIONS X(iOO),A(lOO) ,B(100) ,C(iOO) ,D(iOO) ,G(100),<br />
DIEIEKSI ONS Y ( 100 ,V ( 100 )<br />
READ 100, (X(I), I = 1,100)<br />
FORMAT (~oF7.4)<br />
cupi = 0.0<br />
SUMA = 0.0<br />
SUMB = 0.0<br />
SUlC = 0.0<br />
m!D = 0.0<br />
SUMG = 0.0<br />
smfl = 0.0<br />
DO 200 I = 1,100<br />
b(1)' 0.973 -k 1*551*X(I)<br />
B(1) 2 15.023 46.694*A(I)<br />
C(I) = 12.297- 0*036*B(I)<br />
D(1) z L.871 + O. 106*C(I)<br />
G(1) = 0.138 4 0.400*D(I)<br />
Y(1) = X(1) + A(1) t BU) t C(l) -k D(I)S G(1)<br />
V (I ) = 664.9 *Y (I )<br />
X(1)<br />
suMx= s w+<br />
SUMA = SUMA t A(1)<br />
SüMl3 = SUMB f B(I)<br />
swc SUMC -t C(I)<br />
SUMD = SUMD + D(1)<br />
SUMG = SUIVIG + G(1)<br />
SUMV = smn +V(I)<br />
PUNCH300, X(I ,A (I 1 , B(I 1 , C (I ) , D( I , G (I ,Y (1<br />
PUNCH350 ,V (I )<br />
350 FORMi1T (FS0.4)<br />
300 FORMAT (7F10.4)<br />
200 CGEJTI NUE<br />
PUNCH 400, SUMX, S W<br />
400 FORMAT (6F12.4)<br />
,"UI\+CH 500,SUMV<br />
500 FORMAT (F 25.4)<br />
STO?<br />
ENI)<br />
, SUME , SUMC, SUMD , SUMG<br />
42 7
428<br />
TABLE I<br />
Historical hourly rainfall data for annual storms for New Delhi Station<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
a<br />
9<br />
10<br />
20; 7.56<br />
13.9.57<br />
29-90 58<br />
6.9.59<br />
5.10.69<br />
24.9.61<br />
20.9062<br />
8.8.63<br />
14.7.64<br />
2.9.65<br />
0.25<br />
1. 80<br />
2.00<br />
‘O. 10<br />
O. #<br />
o. 10<br />
0.30<br />
1.50<br />
O. 40<br />
1.90<br />
O. 50<br />
2.10<br />
3.30<br />
4.60<br />
O. 80<br />
0.40<br />
O. 50<br />
<strong>le</strong> 80<br />
1.50<br />
7.80<br />
19.30<br />
22. so<br />
42.00<br />
54.20<br />
19.10<br />
8.50<br />
21.50<br />
new<br />
30.00<br />
61.20<br />
14.75<br />
8.10<br />
13. so<br />
3.50<br />
9000<br />
5.50<br />
9.10<br />
22.00<br />
17.20<br />
9.30<br />
4.06 1.02<br />
5030 3060<br />
11.90 5.60<br />
0.50 0.20<br />
2.00 0.40<br />
2.40 0.08<br />
0.10 0.10<br />
1.80 1.50<br />
1.80 0.90<br />
0.70 0.20<br />
39.88<br />
43.40<br />
78.30<br />
63.10<br />
31.70<br />
16.98<br />
31.60<br />
56. So<br />
51.80<br />
81.10<br />
49.2<br />
53.5<br />
96.5<br />
770 8<br />
39.1<br />
20.9<br />
39.0<br />
69.7<br />
63.9<br />
100 o<br />
TABLE II<br />
fa<br />
Parameters of <strong>the</strong> Markov-Chain Model(hour1y rainfall of annual storms<br />
of New mihi ‘Sation.<br />
Time í t 1 Mean (mm/hour) Stendard Random Markov-Chai n<br />
in hours devi at i on component coefficient<br />
(mmhour 1 et r<br />
1<br />
1 2 3 4<br />
-<br />
b<br />
o. 875 O 8122 .I<br />
2 2.330 2.3522 0.973 lo 551<br />
3 30.620 16o7600 15.023 6.694<br />
4<br />
5<br />
11.195 5.8190 12.297 -0 036<br />
3.056 3.4928 1.871 0.106<br />
6 l b 360 1.8311 0. 138 0.400<br />
II
429
Co<br />
I<br />
431
432<br />
FIG 1 - DISTRIBUTION OF WORST RAINFALL STORM<br />
FOR NEW DELHI<br />
10 30 50 80 90 95 99 99.8 999!<br />
CU M U L A T IV E PR OB AB ILtTY PERCE NT<br />
F IG.2 - CUMLILATIVE PROBABILITY DISTRIBUTION OF<br />
THIFiC 40URLY RAINFALL IN ANNUAL STORMS
E<br />
O<br />
I<br />
-I<br />
4<br />
I-<br />
O<br />
k- 20<br />
433<br />
40 60 80 90 95 98 99 99.8 99.99<br />
CU MU LAT IVE PROBABIL I TY PERCE NT<br />
RECURRENCE INTERVAL (Tk) IN YEARS<br />
FIG.4- STOCHASTIC MODEL FOR ESTIMATION OF<br />
DESIGN FLOOD (Yo1 FROM RECURRENCE<br />
INTERVAL ( TK)
ABSTRACT<br />
THE USE OF STOCHASTIC MODELS IN A HYDRO-AGRICULTURAL<br />
DEVELOPMENT PROJECT IN LEBANON"<br />
by<br />
J.H. Visser<br />
Stochastic modelling techniques were employed in order to<br />
provide long sequences of monthly streamflow and water demand needed<br />
for irrigation scheme design (<strong>the</strong> historic flow records being too<br />
short to serve this purpose).<br />
The water requirement calculations (with <strong>the</strong> Blaney Cridd<strong>le</strong><br />
formula) were based on generated series of monthly rainfall and<br />
monthly mean temperature. The generation "in phase" of <strong>the</strong>se variab<strong>le</strong>s<br />
with streamflow, ensured that a dry year was characterised by high<br />
demand and low flow, This strategy of Ilin phase" generation was<br />
preferred to <strong>the</strong> more usual treatment of assuming a fixed annual<br />
cyc<strong>le</strong> of demands and allowed for a better assessment of <strong>the</strong> design<br />
parameters and a better economic evaluation.<br />
The Ilin phaset1 series required <strong>the</strong> generation of<br />
- monthly rainfall (simp<strong>le</strong> model due to absence of<br />
persistance)<br />
- monthly mean temperature (mixed model with auto regression<br />
and linear regression on monthly rainfall)<br />
- annual streamflow (with linear regression on annual<br />
rainfall )<br />
- monthly streamflow, related to <strong>the</strong> annual flow, using auto<br />
regression (for stations having a historic record of more<br />
than 10 years)<br />
- monthly streamflow, related to <strong>the</strong> annual flow, using auto<br />
regression of a deseasonalized variab<strong>le</strong> and and linear<br />
regression on <strong>the</strong> same variab<strong>le</strong> of ano<strong>the</strong>r (better) station.<br />
(for stations having <strong>le</strong>ss than 10 years of record).<br />
Undertaken jointly by <strong>the</strong> Government of Lebanon and <strong>the</strong> Food and<br />
Agriculture Organ2sation of <strong>the</strong> United Nations.
436<br />
RESUME<br />
Des méthodes stochastiques ont été utilisées pour pouvoir<br />
disposer de séries longues dtapport et de demande mensuels, nscessaires<br />
pour Irétude dfun projet d'hrigation. CLes séries historiques<br />
d'apport étant trop courtes pour être utilisées).<br />
Les besoins en eau (calculés avec la formu<strong>le</strong> de Blaney-<br />
-Gridd<strong>le</strong>) ont été basés sur des séries générées de la pluie et de la<br />
température mensuel<strong>le</strong>s. La génération "en phase" de ces variab<strong>le</strong>s<br />
avec <strong>le</strong>s apports a fait que l'année sèche se caractérise par une<br />
demande é<strong>le</strong>vée et des apports faib<strong>le</strong>s.<br />
Cette stratégie de génération Ifen phase" a été préférge par<br />
rapport à la méthode plus habituel<strong>le</strong> dlun cyc<strong>le</strong> annuel fixe de la<br />
demande, et a permis une meil<strong>le</strong>ure &valuation de la gestion diun<br />
projet d'irrigation aïnsi quiune meil<strong>le</strong>ure évaluation économique.<br />
' -<br />
Les séries "en phase" ont nécessité la génération de<br />
- pluie mensuel<strong>le</strong> (modè<strong>le</strong> simp<strong>le</strong> vu l'absence de persistance)<br />
- température mensuel<strong>le</strong> (modè<strong>le</strong> mixte de régression sériel<strong>le</strong><br />
et de régression linéaire par rapport ?i la pluie mensuel<strong>le</strong><br />
- apport annuel (régression simpie par rapport à ia pluie<br />
annuel<strong>le</strong>)<br />
apport mensuel, par rapport à ifapport annuel, avec<br />
régression sériel<strong>le</strong> (pour <strong>le</strong>s stations ayant au moins 10<br />
années d'observations)<br />
- apport mensuel, par rapport à ifapport annuel, avec<br />
régression sériel<strong>le</strong> diune variab<strong>le</strong> désaisonnalisée et de<br />
régression linéaire par rapport à la même yariab<strong>le</strong> drune<br />
autre (meil<strong>le</strong>ure] station (pour <strong>le</strong>s stations ayant moins de<br />
10 annges d'observations).
1 - INTRODUCTION<br />
43 7<br />
1.1. One of <strong>the</strong> objectives of <strong>the</strong> UNDP/FAO Project LEBANON 13,<br />
concerning <strong>the</strong> hydro-agricultural development of North Lebanon, was to study<br />
an irrigation scheme of about 7 O00 ha in <strong>the</strong> KOURA-ZGHARTA region. For <strong>the</strong><br />
water supply of this scheme a dam has to be constructed on <strong>the</strong> Aasfour river.<br />
The reservoir inflow can be provided by <strong>the</strong> Aasfour discharges toge<strong>the</strong>r with<br />
part of <strong>the</strong> streamflow of an adjacent river.<br />
To assess <strong>the</strong> performance of <strong>the</strong> design reservoir long series of<br />
streamflow are needed to be routed through such a reservoir, Such long series<br />
of historic records were missing and <strong>the</strong> presence of outliers (very wet and<br />
several consecutive dry years), made it virtually impossib<strong>le</strong> to establish<br />
with any confidence a return period for <strong>the</strong>se outliers.<br />
1.2. The hydrologic information availab<strong>le</strong> in <strong>the</strong> Project area was<br />
based mainly on <strong>the</strong> following data :<br />
- -<br />
two streamflow series with 14 years of record<br />
-<br />
thirteen streamflow series with 3 to 5 years of record<br />
several rainfall series of about 30 years of record<br />
some temperature series of about 15 years of record.<br />
A good correlation exists between annual rainfall and streamflow<br />
but low values are found of <strong>the</strong> correlation coefficient between monthly rainfall<br />
and streamflow. This can be explained by <strong>the</strong> fact that <strong>the</strong> response of<br />
<strong>the</strong> catchments to rainfall has a delay factor of one to two months due to <strong>the</strong><br />
presence of snow and / or springs. It was thus impossib<strong>le</strong> to apply <strong>the</strong> conventional<br />
technique of extending <strong>the</strong> shorter streamflow records by correlating<br />
<strong>the</strong>m with <strong>the</strong> longer rainfall records, Unfortunately monthly streamflow data<br />
are needed for reservoir analysis studies,<br />
As <strong>the</strong> conventional techniques were unab<strong>le</strong> to provide <strong>the</strong>se data,<br />
<strong>the</strong> use of hydrological modelling techniques became necessary.<br />
1.3. Two main categories of ma<strong>the</strong>matical models can be used in<br />
princip<strong>le</strong> for this kind of prob<strong>le</strong>ms : Deterministic and stochastic. The first<br />
category permits to extend <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> historic streamflow series to<br />
<strong>the</strong> same <strong>le</strong>ngth as <strong>the</strong> (longer) historic rainfall record. The stochastic mo-<br />
delling however permits to generate syn<strong>the</strong>tic events of any <strong>le</strong>ngth adequate<br />
for certain design purposes.<br />
A mixt use of a stochastic input into a deterministic model could<br />
be useful in princip<strong>le</strong> but unfortunately no such valuab<strong>le</strong> generating models<br />
for daily rainfall existed.
43 8<br />
1.4. The statistics, such as <strong>the</strong> expected frequency of failure<br />
of <strong>the</strong> design system, depend largely on <strong>the</strong> variation of <strong>the</strong> streamflow, i.e.<br />
on <strong>the</strong> values of <strong>the</strong> variance of monthly and annual flow. The stochastic model-<br />
ling techniques can improve <strong>the</strong> estimate of <strong>the</strong> variances of <strong>the</strong> shorter<br />
records by using <strong>the</strong> infoption availab<strong>le</strong> in <strong>the</strong> longer series.<br />
It was for all <strong>the</strong>se reasons that <strong>the</strong> Lebanon 13 Project decided<br />
to apply stochastic modelling techniques.<br />
1.5. In order to apply generated series of streamflow in <strong>the</strong><br />
reservoir simulation studies it was necessary to generate also long series of<br />
rainfall and temperature in order to calculate long series of water demand. To<br />
avoid generating series of streamflow, rainfall and temperature that were un-<br />
correlated, a method of "in phase" generation was adopted, This phasing will<br />
ensure, for examp<strong>le</strong>, that during a dry year <strong>the</strong> values of streamflow and rain-<br />
fall are both low, toge<strong>the</strong>r with high temperature values resulting in high de-<br />
mand for <strong>the</strong> year.<br />
1.6. For <strong>the</strong> calculation of crop water needs <strong>the</strong> Blaney-Cridd<strong>le</strong><br />
formula was used in view of <strong>the</strong> insufficiency of data for <strong>the</strong> Penman method.<br />
However from <strong>the</strong> point of view of <strong>the</strong> methodology, <strong>the</strong>re 2s no objection to<br />
replace <strong>the</strong> Blaney-Cridd<strong>le</strong> formula by that of Penman or ano<strong>the</strong>r. The methodo-<br />
logy for adjusting water resources and water demand as applied to North Lebanon<br />
is explained schematically in <strong>the</strong> attached flow chart.<br />
2 - CHOICE OF TYPE OF STOCHASTIC MODELS<br />
2.1. The stochastic models, described hereafter, for <strong>the</strong> gene-<br />
ration of long time series for different variab<strong>le</strong>s which are mutually in phase,<br />
were proposed by Mr. J. Bernier, Chief of <strong>the</strong> Statistics Group at <strong>the</strong> Labora-<br />
toire National d'Hydraulique, Chatou (France) and consultant to <strong>the</strong> North<br />
Lebanon Project for stochastic hydrology. The models chosen were in response to<br />
<strong>the</strong> availability of data and o<strong>the</strong>r local conditions as well as to <strong>the</strong> objecti-<br />
ves of <strong>the</strong> study in particularly to provide input data for <strong>the</strong> simulation<br />
studies, which explains <strong>the</strong> use of monthly values of <strong>the</strong> different variab<strong>le</strong>s.<br />
-<br />
2.2. The e<strong>le</strong>ments on which this choice was based were :<br />
-<br />
a caracteristics of availab<strong>le</strong> data :<br />
streamflow : <strong>the</strong> presence of 2 series of 14 years of record
-<br />
The<br />
439<br />
- temperature : <strong>the</strong> presence of some series of about 15 years of<br />
record and a significant value for <strong>the</strong> corre-<br />
lation between monthly rainfall and temperature<br />
during spring and autumn.<br />
- rainfall : several series of 30 years of record and a good<br />
correlation between rainfall and streamflow on<br />
annual basis but a bad one an monthly basis<br />
(due to snowfall and / or karsticity)<br />
- perennial flow of <strong>the</strong> rivers.<br />
(<strong>the</strong> temperature and rainfall series toge<strong>the</strong>r permit <strong>the</strong> use of<br />
Blaney-Cridd<strong>le</strong>'s formula for <strong>the</strong> calculation of crop water needs)<br />
requirements imposed by <strong>the</strong> methodology used for adjusting water resour-<br />
ces and water demand :<br />
- <strong>the</strong> series of streamflow and demand have to be in phase<br />
- <strong>the</strong> "being in phase" of streamflow and demand requires automati-<br />
cally <strong>the</strong>' "in phase" generation of <strong>the</strong> series of rainfall, tem-<br />
perature and streamflow.<br />
2.3. The short streamflow series (3 to 5 years of record) can<br />
only be used after "deseasonalisation" of <strong>the</strong> variab<strong>le</strong> (3) resulting in series<br />
of 36 to 60 months of record in which <strong>the</strong> different characteristics of <strong>the</strong> par-<br />
ticular months have been neutralised,<br />
2.4. The following transformations of <strong>the</strong> variab<strong>le</strong>s were necessa-<br />
ry in order to be ab<strong>le</strong> to use <strong>the</strong> normal distribution :<br />
- <strong>the</strong> logarithm of <strong>the</strong> discharges instead of <strong>the</strong> discharges<br />
- <strong>the</strong> square root of <strong>the</strong> rainfall instead of <strong>the</strong> rainfall (<strong>the</strong><br />
temperature is taken without any transformation).<br />
2.5. The particular features described above <strong>le</strong>d to <strong>the</strong> applica-<br />
tion of <strong>the</strong> following generating models :<br />
1) A simp<strong>le</strong> model for monthly rainfall due to <strong>the</strong> absence of<br />
persistance in <strong>the</strong> monthly rainfall series. This model uses<br />
only mean and variance of <strong>the</strong> historic record toge<strong>the</strong>r with<br />
generated random numbers (eq.1).<br />
2) A mixed model for monthly temperature using autoregression<br />
plus regression on ano<strong>the</strong>r variab<strong>le</strong> (monthly rainfall) and<br />
a random number generator (eq.2).
44 O<br />
3 - DESCRIPTION OF MODELS USED<br />
3) A simp<strong>le</strong> regression model for annual streamflow using re-<br />
gression o,n annual rainfall plus a random number generator<br />
(for stations with at <strong>le</strong>ast 10 years of record) (eq.3).<br />
4) An autoregression model for monthly streamflow using <strong>the</strong><br />
relation monthly / annual streamflow to give monthly stream-<br />
flow plus a random number generator (eq.4).<br />
5) A mixed autoregression model for monthly streamflow of a sta-<br />
tionary ("deseasonalised") variab<strong>le</strong> with regression on <strong>the</strong><br />
same variab<strong>le</strong> of ano<strong>the</strong>r station plus random number generator<br />
(stations having <strong>le</strong>ss than 10 years of record) (eq.7).<br />
3.1. Generation of monthly rainfall<br />
Hypo<strong>the</strong>sis : normal distribution of<br />
(Pt = monthly rainfall).<br />
The statistical analysis of <strong>the</strong> historic series of monthly rainfall<br />
gives <strong>the</strong> values of mt = <strong>the</strong> mean of <strong>the</strong> fit of <strong>the</strong> month t (12 values)<br />
and <strong>the</strong> s2t = <strong>the</strong> variance of <strong>the</strong> fit of <strong>the</strong> month t (also 12 values),<br />
Generating model :<br />
fitemt -+ st . u (eq.1)<br />
( 6 = o for Pt = 0)<br />
where : u is <strong>the</strong> random normal deviate with zero mean and unit variance<br />
- u = o, s2(u) = 1<br />
The square root of <strong>the</strong> rainfall depends thus on mt, st and U.<br />
3.2. Generation of mean monthly temperatures<br />
Hypo<strong>the</strong>sis : normal distribution of Tt (Tt = mean monthly tempe-<br />
rature of month t).<br />
The generator model uses a correlation of temperature with
ainfall and is as follows :<br />
Tt,i = mean monthly temperature of month t (t runs from 1 to 12 and i<br />
represents <strong>the</strong> ith month after <strong>the</strong> start generation, i = 1,2 .... ><br />
Ut<br />
K i =<br />
mt<br />
U<br />
vt<br />
al,t<br />
= <strong>the</strong> mean of <strong>the</strong> mean monthly temperatures of month t, in period<br />
of record (12 values)<br />
square root of <strong>the</strong> rainfall of month t,i<br />
= <strong>the</strong> mean of <strong>the</strong> square root of <strong>the</strong> rainfall of month t, in period of<br />
record (12 \talues)<br />
= random nonual deviate with 3 = O and s2(u) = 1<br />
= <strong>the</strong> variance of <strong>the</strong> residuals of T in <strong>the</strong> month t, in period of<br />
record (12 values)<br />
441<br />
and d2,t = partial regression coefficients (for method of calculation<br />
see standard works, e.g. Ven Te Chow "Applied Hydrology", page 8:6û)<br />
For <strong>the</strong> first value of Tt,l, ill <strong>the</strong> following fonuula is used :<br />
Tt-1 = M t-1 + G u'<br />
where M and v are known and a sing<strong>le</strong> value for <strong>the</strong> random normal deviate u'<br />
is sufficient to define <strong>the</strong> value of Ttml.<br />
Once <strong>the</strong> coefficients of <strong>the</strong> generator model for mean monthly tem-<br />
perature are known for each month, a sequence can be generated which will be in<br />
phase, at <strong>the</strong> monthly <strong>le</strong>vel, with rainfall.<br />
3.3. Generation of monthly streamflows for stations having at<br />
<strong>le</strong>ast 10 years of record<br />
The poor degree of correlation found in <strong>the</strong> study area between<br />
monthly streamflows and monthly rainfall, due to <strong>the</strong> effect of snow and karsti-<br />
city of <strong>the</strong> basin, necessitated first a correlation of <strong>the</strong> annual streamflows<br />
(14 years of record) and <strong>the</strong> annual rainfall (<strong>the</strong> same station as used in para<br />
3.1. above). This was achieved through <strong>the</strong> following equation based on <strong>the</strong><br />
hypo<strong>the</strong>sis of a normal distribution of log Q :
log ~a = M +e( q- m> + K u<br />
M = <strong>the</strong> mean of <strong>the</strong> logarithm of <strong>the</strong> annual flow Qa (14 values of Qa)<br />
m =<br />
u = random normal deviate<br />
íeq.3)<br />
<strong>the</strong> mean of a (Pa = annual rainfall for <strong>the</strong> rainfall station 32 years)<br />
v = variance of <strong>the</strong> residuals of log Qa<br />
In order to adjust to <strong>the</strong> long series of rainfall (32 years)<br />
<strong>the</strong> following correction to M was prepared. This type of correction is only<br />
necessary when records of <strong>the</strong> period of M are not typical of <strong>the</strong> period of m.<br />
In fact, since <strong>the</strong> rainfall record (32 years) is longer than<br />
<strong>the</strong> streamflow record (14 years) and <strong>the</strong> two are correlated, <strong>the</strong> estimate M<br />
of <strong>the</strong> population mean of log Q and <strong>the</strong> estimate v of <strong>the</strong> variance of <strong>the</strong><br />
residuals can be improved, <strong>le</strong>adtng to revised estimates M;! and v2 as follows :<br />
2 2<br />
s2 (log Qa> = s1 (log Qa) - r<br />
2<br />
where <strong>the</strong> suffixes 1 and 2 refer to <strong>the</strong> 14 and 32 year records respectively.<br />
2<br />
The value of 62 (log Qa) so obtained can be used to derive an<br />
improved estimate v from <strong>the</strong> relation :<br />
a<br />
(eq. 3b)<br />
2 2<br />
v 2 = (1 r s2 (log Q,) (eq.3~)<br />
All <strong>the</strong> coefficients of equation 3 being known, a long series<br />
of logarithms of annual flow Q, can be generated and will be in phase with<br />
<strong>the</strong> annual rainfall.<br />
The next step is <strong>the</strong> generation of <strong>the</strong> variate Zt,i = log Qt,i - log Qa (eq.4)<br />
(eq.4)
443<br />
where :<br />
- - Zt = mean of (log Q<br />
t,i log Q a<br />
for period of recorii (12 values, t = 1,2,... 12)<br />
r = correlation coefficient of Zt on ZtWl (12 values)<br />
t<br />
s(Zt> = standard deviation of Zt (12 values)<br />
n<br />
vt = variance of <strong>the</strong> residuals = - . 2 2<br />
s<br />
n-2<br />
2<br />
u = random normal deviate with u = O, s (u) = 1.<br />
-<br />
(2,) . (1 C. rt)<br />
The application of equation (4) <strong>the</strong>n gives a generated series<br />
of Qtai following.<strong>the</strong> transformation<br />
-<br />
Qt,i - Qa e ('t,i'<br />
íeq.4a)<br />
for each month. The annual totals of <strong>the</strong>se Qtai should be equal to <strong>the</strong> values<br />
of Qa generated with equation (3) but this will not usually be so, in<br />
which case <strong>the</strong> following correction must be made for <strong>the</strong> generated Q<br />
. tai<br />
(eq. 4b)<br />
where :<br />
E 'a<br />
QL,i - ' Qt,i<br />
Q'a<br />
I<br />
Qt,i<br />
Qa<br />
QIa<br />
= revised value of Qtai (monthly generated flows)<br />
= annual flows generated by eq. (3)<br />
= annual sum of monthly flows Q generated by eq. (4)<br />
t,i<br />
The result of <strong>the</strong>se processes is a generated series of monthly<br />
streamflows which are in phase annually with rainfall which is in turn in<br />
phase monthly with <strong>the</strong> mean monthly temperatures.<br />
3.4. Generation of monthly streamflows for stations having <strong>le</strong>ss<br />
than 10 years of record<br />
For short series equations (3) and (4) cannot be used because<br />
a short series does not permit a sufficiently precise calculation, month by<br />
month, of <strong>the</strong> mean, variance and correlation coefficient. To avoid this diffi-<br />
culty <strong>the</strong> observed series of <strong>the</strong> monthly streamflow Qt i can be "deseasonalized"<br />
to produce a new series Y giving, in <strong>the</strong> case of 4 yeah of observations, 48<br />
values of Y. To "deseasonalize" one of two transforms was used :
444<br />
Y = log Q - a cos (-<br />
t,i<br />
12<br />
(eq.5)<br />
where (eq.5) : <strong>the</strong> origin was taken as <strong>the</strong> month of November (t = 1)<br />
<strong>the</strong> phase determined in such a way as to place <strong>the</strong> mean<br />
e = maximum flow in <strong>the</strong> required month, which also determines<br />
<strong>the</strong> month of minimum flow which will be six months later.<br />
a<br />
= <strong>the</strong> amplitude of <strong>the</strong> variate (<strong>the</strong> logarithm of <strong>the</strong> monthly<br />
flows) chosen in such a way as to give <strong>the</strong> best fit to<br />
<strong>the</strong> observed maximum and minimum monthly values.<br />
The choice of <strong>the</strong> transform depends on <strong>the</strong> shapc of <strong>the</strong> hydro-<br />
graph. In <strong>the</strong> Lebanon project equation (5) was found to give <strong>le</strong>ss good re-<br />
sults and <strong>the</strong>refore equation (6) was used.<br />
Each of <strong>the</strong>se equations gives a series of Y which combines all<br />
months. From this series <strong>the</strong> mean (i) can be calculated and also <strong>the</strong> variance<br />
(
-<br />
X<br />
=I mean of X (during period of record)<br />
u = randa normal deviate<br />
v = variance of <strong>the</strong> residuals of Y<br />
44 5<br />
When all <strong>the</strong> coefficients of equation (7) are known and data have<br />
been generated for <strong>the</strong> "long" record station as described in para. 3.3, <strong>the</strong> model<br />
can be used to generate <strong>the</strong> long series of Yi and so, using <strong>the</strong> transforms in<br />
equations (5) and (6) above, long series of monthly streamflows Q<br />
t,i'<br />
The resulting values of Qt,+ are related to <strong>the</strong> monthly generated<br />
flows of a station having a long record which are <strong>the</strong>mselves in phase annually<br />
with <strong>the</strong> annual rainfall. The annual rainfalls are <strong>the</strong> sum of <strong>the</strong> monthly rainfalls<br />
which <strong>the</strong>mselves are in phase with <strong>the</strong> mean monthly temperatures.<br />
- 4 CONCLUSIONS<br />
4.1. The checks used on <strong>the</strong> generated time series are <strong>the</strong> sta-<br />
tistical moments (mean, variance and coefficient of variation) and periodicity<br />
(winter - summer). It was found that <strong>the</strong> generated time series were in general<br />
of good quality but that <strong>the</strong> value for <strong>the</strong> coefficient of variation (i.c. <strong>the</strong><br />
variance) was too high. This did not matter in <strong>the</strong> Lebanon case because <strong>the</strong><br />
reservoir simulation studies done with <strong>the</strong>se time series kept us on <strong>the</strong> safe<br />
side, but for <strong>the</strong> sake of comp<strong>le</strong>teness some kind of correction should be intro-<br />
duced in future.<br />
4.2. All <strong>the</strong> programmes have been written in Fortran IV for <strong>the</strong><br />
IñM 1130 computer in such a way that <strong>the</strong>y can be used separately or in series.<br />
In this latter case <strong>the</strong> calculation time is about one hour per run. To obtain<br />
as output in one run <strong>the</strong> results of system simulation (reservoir size, irrigab<strong>le</strong><br />
area, failures and effects on o<strong>the</strong>r water users) <strong>the</strong> following input data are<br />
needed :<br />
(a) a historic record of rainfall (monthly)<br />
(b) a historic record of mean temperature (monthly)<br />
(c) a "long" (12-14 years in North Lebanon) historic record of streamflow<br />
(monthly<br />
(d)<br />
a "short" (3 to 5 years) historic record of streamflow (monthly)
446<br />
<strong>the</strong> duration of sunshine as a percentage p of <strong>the</strong> maximum possib<strong>le</strong><br />
<strong>the</strong> monthly crop coefficient K (by crop)<br />
<strong>the</strong> maximum usab<strong>le</strong> soil moisture storage (by ero;)<br />
<strong>the</strong> coefficient of growth of <strong>the</strong> plant (by crop)<br />
<strong>the</strong> phasing of irrigation development of <strong>the</strong> <strong>who<strong>le</strong></strong> area, and <strong>the</strong><br />
subdivision of <strong>the</strong> area by crop (in %)<br />
<strong>the</strong> rate of changedver from <strong>the</strong> present olive groves to <strong>the</strong> new crops<br />
coefficient of irrigation efficiency<br />
geometric characteristics of <strong>the</strong> reservoir<br />
P (e) and (f) are necessary for <strong>the</strong> application of <strong>the</strong> Blaneydridd<strong>le</strong> formula.<br />
e
I II I<br />
Il<br />
m<br />
P 00<br />
I I m<br />
FLOW CHART FOR DATA GENERATION AND SYSTEM ANALYSIS<br />
I 1. II ..... i ace -. 3.7.4 or t a North Ltibonon irriqaion scheme<br />
1-1 i<br />
o' indirti. Bencator<br />
Jaouory 1972 lidopied from Prqe~i Droir~ng AE-2513
RELATIVE IMPORTANCE OF DECISION VARIABLES<br />
IN FLOOD FREQUENCY ANALYSIS<br />
Wallis, J.R.<br />
IBM, Thomas J. Katson Research Center , Worktown Heights , N .Y, , USA<br />
ABSTRACT<br />
Matalas, N. C.<br />
U.S. Geological Survey, Washington, D. C., USA<br />
Monte Carlo simulations were used to assess flood and overde-<br />
sign losses that result from differing choices of assumed frequen-<br />
cy distribution, plotting position, criterion of best fit and<br />
<strong>le</strong>ngth of record. Probabilities of best fit for an assumed world<br />
distribution, given a real world distribution, are given.<br />
RESUMEN<br />
El método de simulación de Monte Carlo se utiliza para eva-<br />
luar los daños producidos por máximas crecidas en funci’on de las<br />
<strong>le</strong>yes de distribución de frecuencias, de las estaciones utilizadas<br />
y de la calidad y extensión de las series hidrológicas. De la mues<br />
tra puede obtenerse el valor minimo teórico de los daños estimados.
450<br />
Introduction<br />
In many cases, <strong>the</strong> design of multipurpose water resource<br />
systems includes flood control as one of <strong>the</strong> purposes. Whi<strong>le</strong> <strong>the</strong><br />
design process may specify <strong>the</strong> sizing of flood control structures<br />
as a function of <strong>the</strong> T-year flood, <strong>the</strong> design process must cope<br />
with <strong>the</strong> uncertainty as to <strong>the</strong> magnitude of <strong>the</strong> T-year flood.<br />
Given that floods are a random phenomenon, <strong>the</strong> magnitude of <strong>the</strong><br />
T-year flood depends upon <strong>the</strong> underlying probability distribution<br />
of flood events and <strong>the</strong> values of <strong>the</strong> distribution's parameters.<br />
Among <strong>the</strong> objectives of flood frequency analysis is that of<br />
determining <strong>the</strong> magnitude of <strong>the</strong> T-year flood, referred to as <strong>the</strong><br />
design flood. Whi<strong>le</strong> <strong>the</strong> underlying distribution of floods is<br />
unknown, an estimate of <strong>the</strong> design flood can be provided. A dis-<br />
tribution may be assumed or chosen in accordance with some criter-<br />
ion of best fit to observed flood sequences. Given an observed<br />
flood sequence of <strong>le</strong>ngth n, and an assumed or chosen distribution<br />
estimate of <strong>the</strong> distribution's parameter values, <strong>the</strong> design flood<br />
can be derived. These estimates are subject to sampling errors,<br />
<strong>the</strong> magnitudes of which depend upon n, and on uncertainties that<br />
are only partially a function of n. To reduce sampling errors,<br />
longer flood sequences are needed. To acquire longer sequences<br />
through direct observation might necessitate <strong>the</strong> delays in <strong>the</strong><br />
design of <strong>the</strong> water resource system. Delays would be economically<br />
feasib<strong>le</strong> if over <strong>the</strong> period of data col<strong>le</strong>ction no benefits were<br />
foregone. In those cases where benefits would be foregone effec-<br />
tively longer sequences might be obtained through regional analyses.<br />
However, even with a very large but finite flood sequence,<br />
uncertainty would still exist in <strong>the</strong> estimate of <strong>the</strong> T-year flood.<br />
The Uncertainty arises because <strong>the</strong> assumed or chosen distribution<br />
used to estimate <strong>the</strong> T-year flood does not necessarily have to<br />
be <strong>the</strong> correct real world distribution, and in fact if a criter-<br />
ion of best fit is used, <strong>the</strong> chosen distribution might vary given<br />
ano<strong>the</strong>r flood sequence of equal <strong>le</strong>ngth.<br />
By using an estimate of <strong>the</strong> T-year flood as <strong>the</strong> design flood,<br />
ei<strong>the</strong>r of two types of losses is likely to be incurred. The first<br />
type refers to overdesign costs of flood control structures which<br />
would be incurred if <strong>the</strong> estimated T-year flood exceeded <strong>the</strong> true<br />
value of <strong>the</strong> design flood. The second type refers to <strong>the</strong> down-<br />
stream damages which would be incurred from underdesign if <strong>the</strong><br />
true value of <strong>the</strong> design flood exceeded <strong>the</strong> estimate of <strong>the</strong> T-year<br />
flood. In <strong>the</strong> design process, what is of concern is not how well<br />
.a particular distribution fits an observed flood sequence par se,<br />
but to what extent <strong>the</strong> two types of design losses are affected by<br />
<strong>the</strong> choice of a particular distribution. To <strong>the</strong> designer, <strong>the</strong>
451<br />
criterion of best fit refers to choosing a distribution to minimize<br />
design losses.<br />
To gain some insight as to <strong>the</strong> magnitudes and sensitivities<br />
of <strong>the</strong> design losses to uncertaintfes in <strong>the</strong> choice of a flood<br />
frequency distribution and estimates of <strong>the</strong> distribution's param-<br />
eter values, several computer-based experiments, employing Monte<br />
Carlo techniques, are currently being performed. In this paper<br />
<strong>the</strong> nature of <strong>the</strong>se experiments is briefly discussed, and some<br />
experimental results as to <strong>the</strong> probabilities of fitting of<br />
observed flood sequences with particular distributions are<br />
presented.<br />
Monte Carlo Experiments<br />
Two sets of distribution functions are considered. The first<br />
set, referred to as <strong>the</strong> real world set, consists of several dis-<br />
tributions, any one of which may be <strong>the</strong> underlying distribution<br />
of floods. From <strong>the</strong> real world set, a distribution function is<br />
chosen where <strong>the</strong> distribution's parameters values are related to<br />
<strong>the</strong> mean, u, <strong>the</strong> standard deviation, o, and <strong>the</strong> coefficient of<br />
skewness, y. For this distribution, 18,000 flood sequences of<br />
<strong>le</strong>ngth n are generated.<br />
The second set of distribution functions, referred to as <strong>the</strong><br />
imagined or assumed set, contains several distributions, any one<br />
of which may be fitted to observed flood sequences. Each e<strong>le</strong>ment<br />
of <strong>the</strong> imagined set is fitted to each of <strong>the</strong> generated sequences,<br />
and on <strong>the</strong> basis of various methods for defining plotting positions<br />
and measuring goodness of fit, <strong>the</strong> particular distribution of best<br />
fit is determined for each generated sequence. From each of <strong>the</strong>se<br />
distributions, <strong>the</strong> flood having an exceedance probability of 1/T<br />
is determined. These floods are estimates of <strong>the</strong> real world flood<br />
of exceedance probability 1/T.<br />
Initially, overdesign and underdesign linear loss functions<br />
in terms of <strong>the</strong> difference between <strong>the</strong> real world T-year flood<br />
and its estimate are assumed. Nonlinear loss functions will be<br />
considered in subsequent experiments. Given <strong>the</strong> 18,000 values of<br />
<strong>the</strong> differences between <strong>the</strong> real world T-year flood and its<br />
estimates, <strong>the</strong> probabilities of incurring overdesign and under-<br />
design losses and <strong>the</strong> expected values of <strong>the</strong> losses are estimated.<br />
Similarly, <strong>the</strong>se values are estimated for each of <strong>the</strong> o<strong>the</strong>r dis-<br />
tributions belonging to <strong>the</strong> real world set.<br />
Three flood control design objectives are considered:<br />
1) minimizing <strong>the</strong> expected overdesign losses, 2) minimizing <strong>the</strong><br />
expected underdesign losses, and 3) minimizing a weighted sum<br />
of <strong>the</strong> expected overdesign and expected underdesign losses. For
452<br />
<strong>the</strong> third objective, <strong>the</strong> weights, say CI and ß, where a > O, ß > O,<br />
and o: + ß = 1, are varied. Among <strong>the</strong> methods for defìning plotting<br />
positions and measuring goodness of fit, <strong>the</strong> particular method and<br />
measure by which <strong>the</strong> various design objectives are met were deter-<br />
mined. The sensitivities of <strong>the</strong> design losses to <strong>le</strong>ss than optimal<br />
choices of <strong>the</strong> plotting position method and measure of goodness<br />
of fit were assessed.<br />
The experiments were carrìèd out for every feasib<strong>le</strong> point in<br />
<strong>the</strong> following experimental hyperspace:<br />
p = 2600<br />
u = $00<br />
y = O, 1/4, 1/2;*, 1, 1 . 1 4 K 2<br />
n = 10, 30, 50, 70, 90<br />
The results of <strong>the</strong>se experiments are conditional on <strong>the</strong> distribu-<br />
tions belonging to <strong>the</strong> real world set. Subsequent experiments<br />
will consider prior information on <strong>the</strong> real world distribution<br />
function and regional estimates of <strong>the</strong> distribution's parameter<br />
values.<br />
Probabilities of Best Fit:<br />
Some preliminary results of <strong>the</strong>se experiments are presented<br />
namely, <strong>the</strong> probabilities of best fit. Both <strong>the</strong> real world and<br />
imagined world sets consisted of three e<strong>le</strong>ments -- <strong>the</strong> normal<br />
distribution, <strong>the</strong> log-normal dìstribution, and <strong>the</strong> Type I extremai<br />
(Gumbel) distribution. For each distribution belonging- to <strong>the</strong> real<br />
woerfd sat) i8000 sequences were generated f o each ~ feasib<strong>le</strong><br />
point in <strong>the</strong> experimental hyperspace. Floods for each sequence<br />
of <strong>le</strong>ngth n were ranked in order of magnitude from <strong>the</strong> largest,<br />
having rank m = 1, to smal<strong>le</strong>st, having rank m = n. The flood of<br />
rank m was assigned an exceedance probability, P[m,nl , by both<br />
<strong>the</strong> "Weibul method," defined as<br />
and by <strong>the</strong> "Hazen method," defined as<br />
(See Chow: 1964).<br />
P [m,nl = m/(n+l) (1)<br />
W<br />
P,[m,nl = (2m-i) / 2n (2)<br />
For a given e<strong>le</strong>ment of <strong>the</strong> real world set and a given e<strong>le</strong>ment<br />
of <strong>the</strong> imagined world set, two sets of differences of flood magni-<br />
tudes were formed for each generated sequence. The first set con-<br />
sisted of <strong>the</strong> differences between <strong>the</strong> observed, that is generated,
453<br />
floods and <strong>the</strong> corresponding imagined world floods having exceed-<br />
ance probabilities defined by Pw(m,nJ. Similarly, <strong>the</strong> second set<br />
of differences was based on exceedance probabilities defined by<br />
PH (m,n).<br />
Two measures of goodness of fit were considered -- <strong>the</strong> sum of<br />
squares of <strong>the</strong> differences in flood magnitudes and <strong>the</strong> sum o€ <strong>the</strong><br />
absolute differences in flood magnitudes. For each sequence based<br />
on an e<strong>le</strong>ment of <strong>the</strong> real world set and relative to each method of<br />
assigning exceedance probabilities, <strong>the</strong> e<strong>le</strong>ment of <strong>the</strong> imagined<br />
set which provided <strong>the</strong> best fit to <strong>the</strong> sequence of floods was<br />
determined, where best fit was defined by each of two criteria --<br />
minimum sum of squares of differences in flood magnitudes and<br />
minimum sum of absolute differences in flood magnitudes.<br />
The probability of <strong>the</strong> event that a sequence, having a par-<br />
ticular e<strong>le</strong>ment of <strong>the</strong> real world set as its underlying distribu-<br />
tion, is best fitted by a particular e<strong>le</strong>ment of imagined world<br />
set, relative to a particular method of assigning exceedance<br />
probabilities and a particular criterion of best fit, was esti-<br />
mated by N/18,000, where N denotes <strong>the</strong> number of times <strong>the</strong> event<br />
occurred and 18,000, <strong>the</strong> total number of times <strong>the</strong> event could<br />
have occurred. The probabilities for each of <strong>the</strong> 36 points in<br />
<strong>the</strong> event space relative to each feasib<strong>le</strong> point in <strong>the</strong> experi-<br />
mental hyperspace were determined.<br />
Remarks<br />
For <strong>the</strong> experimental hyperspace, <strong>the</strong> estimates of <strong>the</strong> probabil-<br />
ities of best fit multiplied by 1000 over <strong>the</strong> event space are given<br />
in Lah<strong>le</strong>s 1 through 9, where MSS denotes minimum sum of squares,<br />
MSAD denotes minimum sum of absolute differences, N denotes <strong>the</strong><br />
normal distribution, G denotes <strong>the</strong> GuInbel distribution, and L denotes<br />
<strong>the</strong> log-normal distribution. These estimates were based on <strong>the</strong> follow-<br />
ing additional experimental operating ru<strong>le</strong> -- if <strong>the</strong> computed value<br />
of <strong>the</strong> coefficient of skewness, y, for a generated sequence was equal<br />
to or <strong>le</strong>ss than 0.007, <strong>the</strong>n <strong>the</strong> sequence was considered to have been<br />
drawn from a normal distribution. Whi<strong>le</strong> <strong>the</strong>se probabilities give<br />
some indication as to <strong>the</strong> power for identifying <strong>the</strong> real world flood<br />
distribution from an observed sequence, <strong>the</strong>y do not give any indica-<br />
tion of <strong>the</strong> optimum strategy to use for choosing a design distribu-<br />
tion. Interpretation of <strong>the</strong>se experimental results in terms of<br />
overdesign-underdesign strategies will be <strong>the</strong> subject of subsequent<br />
papers.<br />
Reference<br />
Chow, Ven T. (1964). Hand<strong>book</strong> of Applied Hydrology, McGraw-Hill.
454<br />
Tab<strong>le</strong> 1. -- Real World is normal<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
n N L G N L G N L G N L G<br />
~~ ~~<br />
10 527 468 5 602 314 84 523 289 188 556 234 211<br />
30 531 432 36 608 341 51 530 432 38 579 364 58<br />
50 528 455 17 614 360 26 526 465 9 489 387 24<br />
70 526 468 6 613 377 io 526 473 2 595 397 8<br />
90 532 466 2 620 374 5 529 470 1 603 394 4<br />
Tab<strong>le</strong> 2. -- Real World is Gumbel with skew = 1.14<br />
Weibuil Hazen<br />
MSS MSAD MSS MSAD<br />
n N L G N L G N L G N L G<br />
10 216 766 i8 287 523 190 212 416 371 245 358 398<br />
30 40 646 314 78 624 299 40 658 303 57 590 352<br />
50 7 612 380 27 642 331 8 717 275 19 629 352<br />
70 2 596 402 10 625 364 2 756 242 7 641 352<br />
90 0 591 409 6 622 372 O 767 233 3 635 361<br />
Tab<strong>le</strong> 3 . -- Real World is log-normal with skew = i/4<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
n N L G N L G N L G N L G<br />
10 441 551 8 521 370 109 436 323 241 474 261 265<br />
30 342 573 85 433 469 99 341 570 89 395 481 124<br />
50 279 659 62 382 550 68 279 683 38 347 583 70<br />
70 234 727 38 339 621 41 232 750 18 313 649 $8<br />
90 205 771 24 313 661 26 204 788 8 290 688 23
Tab<strong>le</strong> 4. -- Real World is log-normal with skew = 112<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
n N L G N L G N L G N L G<br />
455<br />
10 359 630 11 442 421 i37 356 352 292 393 290 317<br />
30 197 643 161 281 554 165 196 641 163 240 556 204<br />
50 116 721 i63 202 655 143 116 780 i04 169 678 153<br />
70 71 789 i40 147 735 117 70 859 71 124 758 118<br />
90 46 837 i17 118 789 93 46 901 53 97 812 91<br />
Tab<strong>le</strong> 5. -- Real World is log-normal with skew = v1/2<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
n N L G N L G N L G N L G<br />
10 302 684 i4 382 461 157 298 375 327 333 315 353<br />
30 i14 662 224 i86 597 218 i14 662 224 i50 582 268<br />
50 48 688 264 106 672 222 48 775 177 83 677 241<br />
70 22 712 266 64 719 217 21 832 i47 49 731 221<br />
90 11 737 253 43 756 201 11 865 125 32 765 203<br />
Tab<strong>le</strong> 6. -- Real World is log-normal with skew = 1<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
n N L G N L G N L G N L G<br />
10 235 750 i5 309 508 183 231 406 362 260 350 390<br />
30 50 658 292 96 631 273 51 663 286 72 594 334<br />
50 13 625 363 39 649 312 13 736 251 27 640 333<br />
70 5 607 388 20 651 330 4 767 229 i4 653 334<br />
90 1 592 406 io 657 333 1 787 212 6 667 327
n<br />
Tab<strong>le</strong> 7 .-- Real World is log-normal with skew = 1.14<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
N L G N L G N L G N L G<br />
10 209 775 17 281 528 192 205 418 376 234 364 403<br />
30 34 651 314 70 638 292 34 660 306 52 594 354<br />
50 7 606 386 24 638 338 7 721 272 i7 633 351<br />
70 2 586 412 9 635 356 2 755 243 6 641 353<br />
90 O 582 418 4 632 364 0 777 223 3 647 351<br />
n<br />
Tab<strong>le</strong> 8. -- Real World is log-normal with skew =<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
N L G N L G N L G N L G<br />
10 169 814 16 231 565 204 166 438 396 193 387 421<br />
30 17 655 328 38 658 304 16 667 317 27 615 358<br />
50 2 620 378 11 657 332 2 727 271 6 648 346<br />
70 0 618 381 3 658 339 O 772 228 2 671 327<br />
90 O 629 371 1 664 335 o 804 196 1 685 314<br />
Tab<strong>le</strong> 9. -- Red World is log-normal with skew = 2<br />
Weibull Hazen<br />
MSS MSAD MSS MSAD<br />
n N L G N L G N L G N L G<br />
10 113 865 21 163 614 223 112 473 415 135 427 438<br />
30 4 696 300 11 710 279 4 710 286 8 679 313<br />
50 0 723 277 1 751 247 O 798 202 1 754 245<br />
70 O 770 230 0 795 205 0 860 i40 O 807 193<br />
90 o 810 190 O 826 174 O 903 97 0 842 158
AB STRA CT<br />
SHOT NOISE MODELS FOR SYNTHETIC GENERATION<br />
OF MULTISITE DAILY STREAMFLOK DATA<br />
bY<br />
G, WEISS<br />
Department of Ma<strong>the</strong>matics Imperi'al Col<strong>le</strong>ge<br />
University of London<br />
Whi<strong>le</strong> multisite models for generating syn<strong>the</strong>tic streamflow da-<br />
ta on a monthly basis have been successfully used, adequate daily<br />
models are lacking, In particular, existing models based on Gau-<br />
ssian processes are unsuitab<strong>le</strong> in reproducing <strong>the</strong> recessions which<br />
are c<strong>le</strong>arly observab<strong>le</strong> in daily data. The models currently being de<br />
veloped at Imperial Col<strong>le</strong>ge, London, under contract for <strong>the</strong> Water<br />
Resources Board, England, are based on "Shot Noise" or filtered Poi<br />
sson processes. These processes consist of a series of Poisson<br />
events, each of which generates a pulse of random height and some<br />
fixed recession shape, In <strong>the</strong> simp<strong>le</strong>st of <strong>the</strong>se models <strong>the</strong> pulses<br />
consist of jumps which are exponentially distributed in magnitude,<br />
and which decay exponentially with a fixed decay rate. This is a<br />
continuous time first order autoregressive (Markovian) process, and<br />
its instantaneous values have a Gamma distribution. This model can<br />
be fitted to streamflow data so as to preserve <strong>the</strong> observed means,<br />
standard deviations, serial and cross correlations of daily data.<br />
Using a more complicated model consisting of two shot noise proce-<br />
sses, monthly statistics can be preserved in addition to <strong>the</strong> daily<br />
statistics. This model gave satisfactory results with data from SO-<br />
me East Anglia sites,<br />
-- RESUME<br />
Alors qu'on a réussi a construire des modg<strong>le</strong>s capab<strong>le</strong>s de €OU:<br />
nir artific2el<strong>le</strong>ment des sérri'es ae dé:its moyens mensuels en plusieurs<br />
sites, :n manque encore de mode<strong>le</strong>s satisfaisants pour <strong>le</strong>s.va<br />
<strong>le</strong>urs journalieres. I1 faut souligner en particulier que <strong>le</strong>s mode<strong>le</strong>s<br />
stochastiques actuels, basés sur des processus gaussiens, ne sont<br />
pas capab<strong>le</strong>s de reproduire <strong>le</strong>s décrues qui sont faci<strong>le</strong>s 2 mettre en<br />
évidence dans <strong>le</strong>s re<strong>le</strong>v6s.journaliers. Les modè<strong>le</strong>s qu'on est en<br />
train de mettre au point a l'Imperia1 Col<strong>le</strong>ge (Londres), pour <strong>le</strong><br />
compte du Water Resources Board (Ang<strong>le</strong>terre) sont basés sur <strong>le</strong><br />
"shot noise" ou processys de Poisson filtr'es. Ces processus se composent<br />
d'une série d'évenements obéissant a une loi de Poisson, dont<br />
chacun produit une impulsion d'amplitude aléatoire dêcroissant suivant<br />
une forme déterminée. Dans <strong>le</strong>s plus simp<strong>le</strong>s de ces modè<strong>le</strong>s,<br />
<strong>le</strong>s impulsions consistente en des sauts dont <strong>le</strong>s amplitudes aléatoA<br />
res sont distribuées de facon exponentiel<strong>le</strong> et sont affectées, une<br />
fois produites, d'une décroissa?ce exponentiel<strong>le</strong> dans <strong>le</strong> temps, la<br />
constante de temps étant fixée a l'avance. Ceci constitue un proce5<br />
sus (Markovien) autorégressif de premier ordre continu dans <strong>le</strong> temps,<br />
dont <strong>le</strong>s va<strong>le</strong>urs insta2tannées sont distribuées suivant une loi<br />
Gamma. Le modè<strong>le</strong> peut etre ajusté aux données disponib<strong>le</strong>s concernant<br />
l'écou<strong>le</strong>ment de facon a respecter <strong>le</strong>s moyennes, <strong>le</strong>s écarts-types et<br />
<strong>le</strong>s corrélations croisées des débits journaliers. En utilisant un<br />
modè<strong>le</strong> plus compliqué, formé de deux processus ltshot noi~e'~, il est<br />
possib<strong>le</strong>2 en plus des caractéristiques statistiques des va<strong>le</strong>urs<br />
journalieres , de conserver cel<strong>le</strong>s des va<strong>le</strong>urs mensuel<strong>le</strong>s. Ce-modè<strong>le</strong><br />
a donné des résultats satisfaisants pour un ensemb<strong>le</strong> de rivieres<br />
dans l'Est de l'Ang<strong>le</strong>terre.
Introduction<br />
The present work is aimed at supplying multisite daily syn<strong>the</strong>tic stream-<br />
flow data for <strong>the</strong> British Water Resources Board. The Water Resources Board<br />
is currently developing regional simulation programs to help in <strong>the</strong> planning<br />
and operation of water supply, and sensitivity tests have shown that at <strong>the</strong><br />
<strong>le</strong>vel of detail used in <strong>the</strong>se programs, monthly data are insufficient and<br />
daily data are indeed required.<br />
Generation of syn<strong>the</strong>tic monthly data in Hydrology was apparently first<br />
attempted in <strong>the</strong> Harvard Water Program [I], and has been developed as a useful<br />
tool since ([2], [ 3J,[ 41). Basically, in <strong>the</strong> methods previously used, <strong>the</strong><br />
data or a transformed series obtained from <strong>the</strong> data are assumed to follow a<br />
multivariate Gaussian 1st order autoregressive process. Some sophisticated<br />
transformations and some higher order regression models have also been used<br />
([5], [6])*<br />
Autoregressive models based on <strong>the</strong> Gaussian distribution have also been<br />
applied to daily data [7J, [SJ. Such models, however, are inadequate in <strong>the</strong><br />
following sense : one of <strong>the</strong> prominent features of daily streamflows is <strong>the</strong><br />
presence of peaks and recessions c<strong>le</strong>arly observed in <strong>the</strong> data. Yet no model<br />
based on <strong>the</strong> Gaussian distribution can reproduce <strong>the</strong>se recessions, no matter<br />
what transformation or what order of autoregression is used.<br />
In this paper a class of models which reproduce recessions is introduced.<br />
A simp<strong>le</strong> model from this class, of <strong>the</strong> same degree of comp<strong>le</strong>xity as <strong>the</strong><br />
Gaussian 1st order autoregressive process is developed, and its imp<strong>le</strong>mentation<br />
for data generation described. The <strong>the</strong>oretical aspects of reproducing recessions<br />
are discussed. Finally, <strong>the</strong> application of <strong>the</strong> model to some British<br />
streamflows is illustrated.<br />
Filtered Poisson Processes<br />
Let N(t) be a P&isson process, <strong>le</strong>t Y be a random variab<strong>le</strong>, and <strong>le</strong>t<br />
w(t,y) be some function. Let <strong>the</strong> sequence ..., 7-1, TO, 71, .O be <strong>the</strong> times<br />
of events of <strong>the</strong> process N(t), and <strong>le</strong>t ..., y-1, yo, YI, ... be mutually<br />
independent random values having <strong>the</strong> same distribution as Y, and all of which<br />
are independefit of N(t). A filtered Poipson process X(t) is defined by :<br />
For fur<strong>the</strong>r details of <strong>the</strong>se processes refer to CS].<br />
A physical interpretation in hydrological terms can be given to <strong>the</strong><br />
filtered Poisson process. The events at random times T ~, given by <strong>the</strong> Poisson<br />
process can be thought of as beginnings of rain storms. The random value ym<br />
associated with T; could correspond to <strong>the</strong> amount of water in <strong>the</strong> rainstorm.<br />
Finally, T and y , will produce a response in <strong>the</strong> flow given by w(t-.cm,ym)<br />
and thus w?t,y) represents <strong>the</strong> system transfer function.
459<br />
The foregoing interpretation is only approximate, since rainstorms are<br />
not independent as <strong>the</strong> Ym'S are required to be in <strong>the</strong> definition. One can<br />
however imagine that an independent series of climatic events exists initially,<br />
and that w(t,y) is <strong>the</strong> transfer function which transforms such climatic events<br />
into streamflow.<br />
The Shot Noise Process<br />
A particular linear filtered Poisson process was adopted for modelling<br />
streamflow, which is referred to as <strong>the</strong> shot noise process. In a filtered<br />
Poisson process, <strong>le</strong>t N(t) be a Poisson process with rate 3 . Let Y be a<br />
random variab<strong>le</strong> with an exponential distribution and with mean 8, and <strong>le</strong>t<br />
w(t,y) = ye-bt, for t > O. The shot noise process is defined as :<br />
The process has three parameters : 9 - <strong>the</strong> event rate, 8 - <strong>the</strong> average<br />
jump height, and b - <strong>the</strong> decay rate. !he process has <strong>the</strong> following properties<br />
(found by applying <strong>the</strong>orems in [9J) :<br />
X(t) has a Gamma (Pearson type 2) distribution with parameters (Q, q/b),<br />
and thus X(t) is nonnegative and positively skewed, and has probability density<br />
function :<br />
The moments of X(t) are given by :<br />
From eq. (2) <strong>the</strong> process at time t+s, X<br />
(s > o)<br />
t+s), can be written as :<br />
The two terms in (5) are independent. The first represents <strong>the</strong> effect<br />
of events previous to t, and is equal to e-bs X(t). The second includes <strong>the</strong><br />
events in (t, C+s) and is <strong>the</strong> innovation term.<br />
~~(t+s) one has<br />
Denoting <strong>the</strong> innovation by
460<br />
Thus, <strong>the</strong> shot noise process is in fact a 1st order autoregressive process<br />
in continuous time. However, it differs from <strong>the</strong> Gaussian 1st order autoregressive<br />
process in that ES(t+s), instead of being Gaussian, has a skewed distribution<br />
with a positive probability of being exactly zero. This arises when<br />
no events occur in (t, t+s).<br />
Some Aspects of Modelling Recessions<br />
When modelling a stochastic process in hydrology one often makes <strong>the</strong> in-<br />
exact but not unrealistic assumption of a linear system. This amounts to<br />
assuming that <strong>the</strong> process X(t) is of <strong>the</strong> form :<br />
where dY(t) is a comp<strong>le</strong>tely uncorrelated and independent process, which<br />
describes all <strong>the</strong> randomness in X(t), and h(t) is <strong>the</strong> system transfer function.<br />
A usual choice for dY(t) is a Gaussian white noise. The characteristic<br />
feature of <strong>the</strong> shot noise process is that dY(t) is chosen as zero almost every-<br />
where, except for a series of spikes. These spikes occur at random time<br />
instants determined by a Poisson process and each spike has some random mass.<br />
(Note that in eqns. (1) and (2) summation over <strong>the</strong> spikes replaces <strong>the</strong> integral<br />
in (7)).<br />
The choice of <strong>the</strong> transfer function h(t) determines <strong>the</strong> autocorrelation<br />
of X(t). In particular h(t) = e-bt gives a 1st order autoregressive process,<br />
and corresponds to a sing<strong>le</strong> linear reservoir. In addition h(t) must determine<br />
<strong>the</strong> shape of recessions in X(t).<br />
However, if dY(t) is chosen as Gaussian white noise, no recessions will<br />
appear in X(t). The absence of recessions may be explained intuitively by <strong>the</strong><br />
fact that Gaussian white noise is changing by minute quantities very quickly ;<br />
hence <strong>the</strong> recession shape of h(t) appears in minute form and is immediately<br />
swamped by <strong>the</strong> next change in dY (t) . Thus linear Guassian processes cannot<br />
reproduce recessions, irrespective of <strong>the</strong> form of h(t), and <strong>the</strong> same will be<br />
true even if a non-linear transformation is used pointwise on X(t).<br />
The ability of <strong>the</strong> shot noise process to reproduce recessions prompted<br />
its use in <strong>the</strong> model1ing;of daily streamflows.<br />
Averaged Sampling of <strong>the</strong> Shot Noise Process<br />
Natural streamflow and <strong>the</strong> stochastic shot noise process are continuous<br />
time processes. Recorded daily streamflows and <strong>the</strong> syn<strong>the</strong>tic data to be<br />
produced are on <strong>the</strong> o<strong>the</strong>r hand discrete time processes. The usual approach<br />
in <strong>the</strong> modelling of monthly data is to consider <strong>the</strong> data as a discrete samp<strong>le</strong><br />
of <strong>the</strong> process, i.e. <strong>the</strong> values of <strong>the</strong> continuous process at discrete time<br />
points. However, discrete sampling is inaccurate, since <strong>the</strong> data are actually<br />
obtained by averaging <strong>the</strong> flows over <strong>the</strong> period between <strong>the</strong> discrete sampling
time points.<br />
461<br />
The difference between <strong>the</strong> two approaches, of discrete sampling<br />
or of average sampling, is negligib<strong>le</strong> for serial correlations of up to P = 0.5<br />
which are typical for monthly data.<br />
which is typical for daily data.<br />
It is however substantial for e= 0-8<br />
It is assumed here that streamflow follows a continuous time shot noise<br />
process X(t), and that <strong>the</strong> observed data (and <strong>the</strong> generated data), are averages<br />
of this process over a period of T = 1 day. The data X,, X2, ... are thus<br />
defined as :<br />
The moments of X. are slightly different from those of X(t), and are given<br />
by : J<br />
SQ<br />
E(X.1 =<br />
J<br />
Var (x.) = - 9 Q2 2 [b-(l-e-b))<br />
3 b (9)<br />
b2<br />
(s 1)<br />
In addition, <strong>the</strong> averaging changes <strong>the</strong> shape of <strong>the</strong> recessions. Whereas<br />
in <strong>the</strong> process X(t) recessions start from a vertical rising limb, for <strong>the</strong><br />
averaged values X <strong>the</strong> transfer function is of <strong>the</strong> shape,<br />
j<br />
1 -bt)<br />
f; (1-e<br />
that is <strong>the</strong> rise is gradual over O \< t < 1.<br />
Fitting <strong>the</strong> Shot Noise Model to Daily Streamflow<br />
(o < t < 1)<br />
In fitting <strong>the</strong> shot noise model an approach similar to that employed by<br />
Matalas [3J is used. Values of 9 , 8, b are calculated which preserve <strong>the</strong><br />
values of 1.1, o2 and p(l) obsezved in <strong>the</strong> data. Thus <strong>the</strong> samp<strong>le</strong> mean, variance<br />
and first serial correlation, p, 82 and g(1) are calculated from <strong>the</strong> hietorica:<br />
data. These are substituted in equations (9), which are solved for 4,<br />
8, b. The estimated decay rate 6 is solved for from <strong>the</strong> thigd equation by<br />
numerical methods, and <strong>the</strong> o<strong>the</strong>r two equations yield 8 and 9.<br />
An alternative approach would be to estimate b directly from observed<br />
recessions or from <strong>the</strong> unit hydrograph of <strong>the</strong> basin, and to estimate 3 by
46 2<br />
observing times of peaks in <strong>the</strong> data. This latter approach was attempted for<br />
<strong>the</strong> British streamflow data at our disposal. However, preservation of observed<br />
p, o2 and (1) using this method did not ensue. A similar approach may<br />
however prove useful for different data, for instance, streams in semi-arid<br />
regions, where data may consist of short records of frequent observations.<br />
Syn<strong>the</strong>tic Generation of Shot Noise Data<br />
Let 3, 8, b be <strong>the</strong> parameters estimated from historical data. The algo-<br />
rithm for generating syn<strong>the</strong>tic data is as follows :<br />
Denoting by Xt, t = 1, 2, ..., <strong>the</strong> averaged shot noise to be generated,<br />
and by X(t), t = O, 1, 2, ..., <strong>the</strong> values of <strong>the</strong> continuous process, one<br />
obtains from (5) and (IO) :<br />
where <strong>the</strong> first term in eqns. 11 and 12 is <strong>the</strong> contribution from events<br />
preceding t , and <strong>the</strong> second is <strong>the</strong> contribution of events in (t, t+l).<br />
Starting with an initial value for X(O), XI and X(1) are generated.<br />
X(1) is <strong>the</strong>n used to generate X and X(2) and so on. Assuming XI, ..., Xt<br />
and X(t) have been generated, tge following steps <strong>le</strong>ad to Xt+l, X(t+l).<br />
1) The first terms of (11,121 are calculated, from X(t). X(t) can <strong>the</strong>n be<br />
discarded.<br />
2) Time of last event preceding (t,t+l) need not be remembered.<br />
are initiated by putting m = O, .cm = O.<br />
Event times<br />
3) The next event .cm+l is generated as zm+l = T~+I, where I is a random<br />
number generated from an exponential distribution with mean (1/3 1.<br />
4) If T ~ > + 1 ~ all events in (t,t+l) have been exhausted and so generation<br />
of Xt+l and X(t+l) is comp<strong>le</strong>te.<br />
5) For T ~+I < 1, ya+-, is generated as a random number from an exponential<br />
distribution with mean 8.<br />
6) The contribution of y to (11, 12) is calculated as :<br />
m+ 1<br />
1 (l-e-b(l-Tm+l 1 ) and e -b(l-.cm+l), and added to <strong>the</strong> values<br />
;<br />
of Xt+l and X(t+l) respectively.<br />
7) m is set to m+l and steps 3 to 7 are repeated.<br />
Thus <strong>the</strong> generation requires only random numbers from exponential distri-<br />
butions, which are easy to create.
Multisite Shot Noise Processes<br />
463<br />
The shot noise process already defined can be easily extended to a<br />
multisite process. Let Xl(t), ..., XM(t), be continuous shot noise processes<br />
at M sites, with parameters Sk, Qk, bk, k = 1, ..., M.<br />
A multisite process incorporating all of <strong>the</strong> parameters will be defined<br />
by assuming that some of <strong>the</strong> events occur simultaneously at several sites,<br />
and give rise to correlated jumps yk at <strong>the</strong>se sites.<br />
For two of <strong>the</strong> sites, k and 1, <strong>le</strong>t <strong>the</strong> events which occur simultaneously<br />
be at rate Jkl(with 3k1 < 3 k, 3 kl < 3 i), and <strong>le</strong>t <strong>the</strong> jumps associated<br />
with a simultaneous event, y , yl, have a correlation coefficient ckl. Then<br />
<strong>the</strong> correlation between \(ty and X,(t) is :<br />
c + I<br />
kl<br />
.& . 2<br />
In this expression <strong>the</strong> first term shows <strong>the</strong> effect of <strong>the</strong> different decay<br />
rates on <strong>the</strong> cross correlation, (i.e. <strong>the</strong> effect of <strong>the</strong> two different recession<br />
shapes), and <strong>the</strong> o<strong>the</strong>r two terms arise from <strong>the</strong> correlation between <strong>the</strong> two<br />
series of events and jumps.<br />
By (13) Ski and Ckl can be chosen for each pair k,l, to preserve <strong>the</strong><br />
observed cross correlation in <strong>the</strong> multisite data.<br />
The Doub<strong>le</strong> Shot Noise Process<br />
Some difficulties arose in fitting <strong>the</strong> shot noise process to average<br />
daily flows from some English streams. Whi<strong>le</strong> <strong>the</strong> model did preserve <strong>the</strong> mean,<br />
standard deviation and lag one serial correlation coefficient of <strong>the</strong> daily<br />
data, when <strong>the</strong> syn<strong>the</strong>tic data was averaged over months, <strong>the</strong> syn<strong>the</strong>tic monthly<br />
data had much smal<strong>le</strong>r standard deviations and lag one serial correlation<br />
coefficients than those observed in <strong>the</strong> historic data. Moreover, in <strong>the</strong> syn<strong>the</strong>tic<br />
data <strong>the</strong> recessions decayed too fast towards zero, and too many rises<br />
and recessions were generated.<br />
Inspection of <strong>the</strong> historic daily ctreamflow hydrographs showed that <strong>the</strong><br />
streams model<strong>le</strong>d have a pronounced base flow component which is not reproduced<br />
by <strong>the</strong> shot noise process.<br />
Therefore a more sophisticated model was proposed, which assumes X(t)<br />
to be <strong>the</strong> sum of two independent shot noise processes, Xq(t) with parameters<br />
+A, QI, bl and X2(t) with parameters $2, 82 and b2 (cf equation 2). In<br />
<strong>the</strong>se two process $1, QI, bl are assumed to be larger than $2, 82, b2, so<br />
that Xl(t) has more recessions, higher jumps and a faster decay rate than<br />
X2(t). In physical terms, Xl(t) may be thought of as representing a surface<br />
runoff mechanism and X2(t) as representing a baseflow mechanism.
464<br />
In fitting <strong>the</strong> model, <strong>the</strong> six parameters can be calculated so as to<br />
preserve <strong>the</strong> observed mean, standard deviation and lag one serial correlation<br />
of <strong>the</strong> observed daily flows, and <strong>the</strong> standard deviation and lag one serial<br />
correlation of <strong>the</strong> observed averaged monthly flows.<br />
An Application of <strong>the</strong> Doub<strong>le</strong> Shot Noise Model<br />
Data from <strong>the</strong> river Nene in East Anglia and some of its tributaries, and<br />
of one tributary of <strong>the</strong> neighbouring Great Ouse was used to generate syn<strong>the</strong>tic<br />
data. The Nene flows through East Anglia, and discharges into <strong>the</strong> Wash. It<br />
has a drainage area of 1630 km2, it receives an average annual rainfall of<br />
623 mm, and has an annual runoff of 157 mm.<br />
The historic data consists of 11 years of average daily streamflows<br />
concurrent at 8 sites. !Che doub<strong>le</strong> shot noise model was fitted to <strong>the</strong> data so<br />
as to preserve at each site <strong>the</strong> overall mean, <strong>the</strong> standard deviation of <strong>the</strong><br />
daily and of <strong>the</strong> monthly series, and <strong>the</strong> lag one serial correlation coefficient<br />
of <strong>the</strong> daily and <strong>the</strong> monthly series, and so as to preserve <strong>the</strong> daily cross<br />
correlations between <strong>the</strong> sites. Seasonality was accounted for through estima-<br />
ting <strong>the</strong> parameters separately for each ca<strong>le</strong>ndar month.<br />
Twelve series of syn<strong>the</strong>tic data, each of <strong>the</strong>m equal in <strong>le</strong>ngth to <strong>the</strong><br />
historic record, were generated. Tab<strong>le</strong> 1 summarises some of <strong>the</strong> results from<br />
<strong>the</strong> historic and generated data. The tab<strong>le</strong> includes quantities calculated for<br />
<strong>the</strong> River Nene at Orton, close to <strong>the</strong> outflow point, and for <strong>the</strong> ca<strong>le</strong>ndar month<br />
of January. Flows are listed in m3/s.<br />
The tab<strong>le</strong> gives a comparison between properties of <strong>the</strong> historic data,<br />
properties of <strong>the</strong> <strong>the</strong>oretical model calculated analytically, and properties<br />
of <strong>the</strong> syn<strong>the</strong>tic data. Column 1 refers to <strong>the</strong> historic data. Columns 2-4<br />
refer to <strong>the</strong> <strong>the</strong>oretical model. Column 2 contains quantities calculated for<br />
<strong>the</strong> doub<strong>le</strong> shot noise model, and <strong>the</strong> decomposition of <strong>the</strong>se quantities into<br />
<strong>the</strong> process modelling <strong>the</strong> surface runoff mechanism (fast process) and <strong>the</strong><br />
process modelling <strong>the</strong> baseflow mechanism (slow process) are given in columns<br />
4 and 3 respectively. Columns 5-7 refer to <strong>the</strong> syn<strong>the</strong>tic data. Column 5<br />
contains values which are averages of all <strong>the</strong> twelve syn<strong>the</strong>tic series whi<strong>le</strong><br />
columns 6 and 7 list <strong>the</strong> lowest and highest values obtained for each quantity<br />
out of <strong>the</strong> twelve series.<br />
The different rows of <strong>the</strong> tab<strong>le</strong> list <strong>the</strong> values of several quantities<br />
of interest. The quantities which are starred, are those used in fitting<br />
<strong>the</strong> model. For some of those quantities <strong>the</strong> model preserves <strong>the</strong> historical<br />
value exactly, whi<strong>le</strong> o<strong>the</strong>rs were only preserved approximately, due to numerical<br />
difficulties. e12 is <strong>the</strong> cross correlation between Nene at Orton and Great<br />
Ouse at Thornborough Mill.<br />
The rest of <strong>the</strong> quantities listed were not used in fitting <strong>the</strong> model,<br />
and success in preserving <strong>the</strong>m can serve as a measure of <strong>the</strong> adequacy of <strong>the</strong><br />
model. Of <strong>the</strong>se quantities which were not fitted, <strong>the</strong> lag two and lag three<br />
daily serial correlation coefficients are extremely well preserved. On <strong>the</strong><br />
o<strong>the</strong>r hand <strong>the</strong> skewness of <strong>the</strong> data was overestimated by <strong>the</strong> model. It is<br />
very encouraging that <strong>the</strong> model seems to yield reasonab<strong>le</strong> values of high and<br />
low flows.
465<br />
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466<br />
Conclusions<br />
The shot noise model has been developed as a physically more realistic<br />
model of daily streamflow data than has heretofore been proposed, and in<br />
particular models recession effects which are a prominent feature of daily<br />
streamflow data.<br />
In its basic form, <strong>the</strong> shot noise process in its conception, statistical<br />
properties and with <strong>the</strong> associated method of fitting and method of data gener-<br />
ation is as simp<strong>le</strong> and easy to hand<strong>le</strong> as <strong>the</strong> Gaussian 1st order autoregressive<br />
model.<br />
The use of <strong>the</strong> doub<strong>le</strong> shot noise model for some English streans gave<br />
satisfactory results, and illustrates <strong>the</strong> adaptibility of this class of models.<br />
It is felt that <strong>the</strong>se mdels, with <strong>the</strong>ir emphasis on events and recessions,<br />
could with fur<strong>the</strong>r research provide a link between deterministic and stochastic<br />
hydrology. Thus studies by deterministic methods of <strong>the</strong> instantaneous unit<br />
hydrograph and of <strong>the</strong> mechanism of base flow etc. could provide some of <strong>the</strong><br />
parameters needed for a stochastic model based on shot noise processes.<br />
Acknow<strong>le</strong>dgements<br />
This work is financed by <strong>the</strong> Water Resources Board of England and Wa<strong>le</strong>s,<br />
and is being carried out at Imperial Col<strong>le</strong>ge of Science and Technology in<br />
London under <strong>the</strong> supervision of Professor D.R. Cox of <strong>the</strong> Department of<br />
Ma<strong>the</strong>matics and Mr. T. OIDonnell and Mr. P.E. O'Connel1 of <strong>the</strong> Hydrology<br />
Section, Department of Civil Engineering, to whom I am deeply indebted for<br />
ideas and help.
References<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
467<br />
Thomas, H.A. and Fiering, M.B. (19621, Ma<strong>the</strong>matical syn<strong>the</strong>sis of stream-<br />
flow sequences for <strong>the</strong> analysis of river basins by simulation, Ch. 12 in<br />
'Design of Water Resources Systems', by Maass, A., et al, London, Macmillan,<br />
pp. 459-493.<br />
Fiering, M.B. (19641, Multivariate techniques for syn<strong>the</strong>tic hydrology,<br />
Proc. Am. Soc. Civ. Engrs., J. Hydraul. Div., Vol. 90, HY5, pp. 43-60.<br />
Matalas, N.C. (19671, Ma<strong>the</strong>matical assessment of syn<strong>the</strong>tic hydrology,<br />
Water Resources Research, Vol. 3, pp. 937-945.<br />
Young, G.K. and Pisano, W.C. (19681, Operational hydrology using residuals,<br />
Proc. Am. Soc. Civ. Engrs., J. Hydraul. Div. Vol. 94, HY4, ppe 909-923.<br />
Beard, L.R. (1965), Use of interrelated records to simulate streamflows,<br />
Proc. Am. Soc. Cive Engrs., J. Hydraul. Div., Vol. 91, Hy5, pp. 13-22.<br />
Moreau, D.H. and Fyatt, E.E. (19701, Weekly and monthly flows in syn<strong>the</strong>tic<br />
hydrology, Water Resources Research, Vol. 6, pp. 53-61.<br />
Quimpo, R.G. (19681, Stochastic analysis of daily river flows, Proc. Am.<br />
Soc. Civ. Engrs., J. Hydraul. Div., Vol. 94, HYI, pp. 43-58.<br />
Payne, K., Newman, W.R. and Kerri, K.D. (19691, Daily streamflow simulation,<br />
Proc. Am. Soc. Cive Engrs., J. Hydraul. Div., Vol. 95, HY4, pp. 1163-1180.<br />
Parzen, E. (19641, Stochastic processes, San Francisco, Holden-Day,<br />
pp- 144-159.
ABSTRACT<br />
FLOOD CONTROL DESIGN WITH LIMITED DATA - A COMPARISON<br />
OF THE CLASSICAL AND BAYESIAN APPROACHES<br />
Eric F. Wood<br />
Department of Civil Engineering<br />
MASSACHUSETTS INSTITUTE OF TECHNOLOGY<br />
Water Resource planners usually design flood control structures<br />
by choosing an extreme value model. The model's parameters are esti-<br />
mated from <strong>the</strong> availab<strong>le</strong> streamflow data and design decisions are ma<br />
de by finding <strong>the</strong> discharge related to a particular <strong>le</strong>vel of risk.<br />
In most design prob<strong>le</strong>ms <strong>the</strong> data on extreme events is severely limi-<br />
ted, making parameter estimation difficult. Two different parameter<br />
estimation approaches are investigated - classical and Bayesian -<br />
which are applied to a flood design prob<strong>le</strong>m for a nortkeastern U.S.<br />
river. The classical approach uses <strong>the</strong> maximum likelihood cri'terion<br />
for parameter estimation. The Bayesian approach is performed for 05-<br />
jective prior information based upon observations from o<strong>the</strong>r rivers<br />
and for subjective prior information derived by considering <strong>the</strong><br />
effect of river basin development upon flood discharges. The results<br />
indicate that <strong>the</strong> classical and Bayesian approaches <strong>le</strong>ad to diffe-<br />
rent design discharges for <strong>the</strong> same <strong>le</strong>vel of risk.<br />
Normalmente los Ingenieros Hidráulicos diseñan las estructuras<br />
para el control de crecidas eligiendo un modelo de valores extremos;<br />
los parámetros del mismo se estiman mediante los datos de aforo y<br />
se diseña mediante la determinación de la crecida asociada con un<br />
cierto nivel de riesgo. En la mayoria de los casos, la información<br />
disponib<strong>le</strong> sobre los valores extremos es muy escasa, haciendo muy<br />
difícil la estimación de los parámetros. Se investigan dos métodos<br />
de estimación de pardmetros - el clásico y el Bayesiano - que se<br />
aplican a un prob<strong>le</strong>ma de control de crecidas para un rio del nordes<br />
te de los Estados Unidos. El método cl'lsico utiliza el criterio de<br />
verosimilitud máxima para la estimación de parámetros. El enfoque<br />
Bayesiano se desarrolla con informaci8n priori" objetiva basada<br />
en observaciones realizadas en otros ríos, y con información "a<br />
priori" subjetiva obtenida al considerar el efecto del desarrollo<br />
de la cuenca sobre los cauda<strong>le</strong>s de crecida. Los resultados indican<br />
que los métodos clásico y Bayesiano l<strong>le</strong>van a diferentes valores de<br />
cauda<strong>le</strong>s de crecida para un mismo nivel de riesgo,
470<br />
INTRODUCTION<br />
In water resources planning <strong>the</strong> hydrologist's main function is analysis<br />
that <strong>le</strong>ad to engineering decisions. The decision variab<strong>le</strong> is not a hydrologic<br />
variab<strong>le</strong> but a general engineering variab<strong>le</strong> like <strong>the</strong> height of a dike or <strong>the</strong><br />
size of a spillway. The separation of hydrologic analysis and economic<br />
analysis can not occur if efficient designs are to be obtained. In <strong>the</strong><br />
decision process <strong>the</strong> design variab<strong>le</strong>s are related to <strong>the</strong> estimation of hydrologic<br />
variab<strong>le</strong>s through a loss function which ref<strong>le</strong>cts <strong>the</strong> different economic<br />
implications of <strong>the</strong> project.<br />
If one accepts this ro<strong>le</strong> for <strong>the</strong> hydrologist within <strong>the</strong> planning process<br />
<strong>the</strong>n a number of important qualitative implications follow.* It is useful to<br />
describe comp<strong>le</strong>x phenomena such as rainfall or runoff processes by <strong>the</strong> application<br />
of probability <strong>the</strong>ory only from <strong>the</strong> point of view that it produces a more<br />
economical design. If streamflows can be treated as random variab<strong>le</strong>s <strong>the</strong>p it<br />
is consistent to treat <strong>the</strong> unknown parameters of <strong>the</strong> distributions of streamflows<br />
as random variab<strong>le</strong>s. To treat <strong>the</strong> parameters of distribution of random<br />
variab<strong>le</strong>s as random variab<strong>le</strong>s is not "permitted" within <strong>the</strong> framework of<br />
classical statistics.<br />
The extension of this argument is that it is useful and professionally<br />
sound to treat any uncertain factor as a random variab<strong>le</strong> if it <strong>le</strong>ads to better<br />
decisions. This includes variab<strong>le</strong>s such as <strong>the</strong> quality of workmanship in construction,<br />
cost and benefit adjustments due to inflation as well as <strong>the</strong> more<br />
traditional hydrologic variab<strong>le</strong>s.<br />
The designer should consider three types of uncertainty in his analysis -<br />
uncertainty of a probabilistic nature (i.e. frequency of occurrence), statistical<br />
uncertainty due to <strong>the</strong> limited number of observations from which parameters<br />
are to be estimated, and professional uncertainty arising from incomp<strong>le</strong>te information<br />
concerning <strong>the</strong> underlying process and its probabilistic representation<br />
(Cornell, 1972).<br />
The methodology of <strong>the</strong> decision process must be ab<strong>le</strong> to con-<br />
sider <strong>the</strong>se forms of uncertainty as well as be ab<strong>le</strong> to utilize professional<br />
judgement obtained from related experience with similar projects.<br />
Bayesian analysis within <strong>the</strong> framework of statistical decision <strong>the</strong>ory<br />
(ñaiffa, 1968) prescribes a methodology for making decisions under uncertainty.<br />
Decision <strong>the</strong>ory allows <strong>the</strong> decision maker to consider toge<strong>the</strong>r both <strong>the</strong> uncertainty<br />
of <strong>the</strong> model<strong>le</strong>d process, <strong>the</strong> quantifying of <strong>the</strong> decision outcomes and<br />
<strong>the</strong> preferences for <strong>the</strong>se outcomes. Bayesian analysis is a probabilistic framework<br />
by which <strong>the</strong> uncertainty in any design variab<strong>le</strong> and <strong>the</strong> know<strong>le</strong>dge about<br />
that variab<strong>le</strong> can be considered. This paper is concerned with <strong>the</strong> latter aspect<br />
of <strong>the</strong> proposed methodology. The application of Bayesian analysis in water<br />
resources planning is gaining acceptance as decision makers recognize <strong>the</strong><br />
* Paral<strong>le</strong>l arguments have been used previously in support of a statistical<br />
decision approach to structural reliability analysis by Cornell (1972).
inherent advantages that combining information sources and treating uncertain<br />
parameters as random variab<strong>le</strong>s <strong>le</strong>ads to better designs. In recent years many<br />
researchers have made significant contributions to this area. These include<br />
<strong>the</strong> work of Bernier (1967), Shane and Gaver (1970), Davis et al (1972a)<br />
Bogardi and Szidarovszky (1972) amongst o<strong>the</strong>rs.*<br />
Probabilistic Model Formulation<br />
471<br />
One area of particular concern to water resource planners is <strong>the</strong> analysis<br />
of extreme events, mainly floods. This prob<strong>le</strong>m is especially applicab<strong>le</strong> to <strong>the</strong><br />
issues raised earlier since data is often scarce, with <strong>the</strong> consequences due to<br />
an inadequate design often severe.**<br />
The issues we wish to focus upon in this paper is a comparison of <strong>the</strong><br />
classical approach and <strong>the</strong> Bayesian approach to flood analysis and design when<br />
direct observation of extreme events are ei<strong>the</strong>r scarce, non-existant, or non-<br />
stationary. Substantial urbanization of a river basin introduces non-<br />
stationarity effects into <strong>the</strong> direct observations, thus decreasing <strong>the</strong>ir<br />
information content.<br />
The first step in any analysis, classical or Bayesian, is <strong>the</strong> construction<br />
(or assumption) of an underlying probabilistic model which represents <strong>the</strong><br />
physical process. Consider <strong>the</strong> hypo<strong>the</strong>tical streamflow trace presented in<br />
Figure 1. The flows of interest are those flows greater than Qo and it is<br />
assumed that flows larger than Qo can be described by a Poisson process (<strong>the</strong><br />
time between events are exponentially distributed) with an average annual<br />
arrival rate and <strong>the</strong> probability distribution of <strong>the</strong> flows of interest (flows<br />
greater than Qo) can be represented by <strong>the</strong> exponential distribution<br />
where<br />
This is a fairly general form since <strong>the</strong> upper tails of many distributions<br />
may be represented as exponential. The proposed model has been used for<br />
extreme flows by Shane and Lynn (1964) and Todorovic and Ze<strong>le</strong>nhasic (1970) and<br />
for rainfall events by (Davis et al (1972b)) and Grayman and Eag<strong>le</strong>son (1971).<br />
*The numerous papers at <strong>the</strong> International Symposium on Uncertainties in<br />
Hydrologic and Water Resource Systems, December 11-14, 1972, Tucson, Arizona,<br />
U.S.A. is proof of <strong>the</strong> growing interest in this field.<br />
**The decision makers may also consider besides economic consequences social<br />
and professional consequences due to failure of a flood control structure.<br />
These may be loss of life, disruption of community services and <strong>the</strong> loss of<br />
professional prestige. On <strong>the</strong> o<strong>the</strong>r hand, over design commits resources that<br />
could be used on o<strong>the</strong>r projects.
472<br />
A A<br />
The probability that, in any sing<strong>le</strong> occurrence, a discharge z (z = q -<br />
exceeds <strong>the</strong> discharge z is Pz, where<br />
-a2<br />
Pz = 1 - FZ(z) = e<br />
<strong>the</strong> process of <strong>the</strong>se occurrences is Poisson with an average arrival<br />
rate VP and <strong>the</strong> probability that in time t n exceedances of <strong>le</strong>vel z will<br />
Z<br />
occur is<br />
n -UP t<br />
P[N = n 1 = (vP,) e<br />
n !<br />
No exceedances of z in time t is just<br />
probability function of z. Substituting<br />
[ -az<br />
F (z) = fvte z>o<br />
Z<br />
z< o<br />
The probability <strong>the</strong> z = O is equal to <strong>the</strong> probability that a peak discharge q<br />
is <strong>le</strong>ss than Q,. I If z is such that <strong>the</strong> probability of exceeding z is small<br />
and <strong>the</strong> arrival rate of such events is small <strong>the</strong>n FZ (z) can be approximated by<br />
-a2<br />
FZ (z) 1 - vte<br />
(3)<br />
P [nz = O] = F (2); <strong>the</strong> cumulative<br />
P from (2) into<br />
Z FZ(z) gives<br />
The probabilistic model of <strong>the</strong> underlying physical process serves both <strong>the</strong><br />
classical and Bayesian analyst but in slightly different ways.<br />
Assume that <strong>the</strong>re is no uncertainty in <strong>the</strong> model itself but only in its<br />
parameters CY and V . The classical analyst <strong>the</strong>n obtains point estimators,<br />
V and a , (usually by <strong>the</strong> maximum likelihood criterion) from <strong>the</strong> observed<br />
streamflow record. His probabilistic model is<br />
The<br />
<strong>the</strong><br />
Bayesian analyst, meanwhi<strong>le</strong>, obtains probability density distributions on<br />
unknown parameters, v and a , from combining all sources of information.<br />
The Bayesian approach to <strong>the</strong> use of probabilistic methods recognizes that<br />
<strong>the</strong> subjective information of <strong>the</strong> analysis is inseparab<strong>le</strong> from <strong>the</strong> objective<br />
aspects. Subjective infomation is incorporated into <strong>the</strong> analysis through a<br />
prior probability distribution which ref<strong>le</strong>cts <strong>the</strong> information content. This<br />
prior information is combined with objective information - direct data observations<br />
- to provide <strong>the</strong> analyst with a posterior distribution. This ref<strong>le</strong>cts<br />
all of his information. If <strong>the</strong> prior information is vague and <strong>the</strong> samp<strong>le</strong> information<br />
is very good <strong>the</strong>n <strong>the</strong> posterior distribution of <strong>the</strong> information will be<br />
(5)<br />
QO)
4'1 3<br />
negligibly affected by <strong>the</strong> prior. The opposite also holds. The prior information<br />
may be looked upon as that information an analyst wovld use if he had no<br />
observab<strong>le</strong> data. In <strong>the</strong> design for floods a number of sources of information,<br />
are availab<strong>le</strong>. These include such sources as regression equations based on <strong>the</strong><br />
physical characteristics of <strong>the</strong> basin (Benson, 1962) and analytical derivation<br />
of extreme flow dynamics (Eag<strong>le</strong>son, 1972) as well as engineering experience and<br />
expertise. To ignore <strong>the</strong>se sources is to throw away potentially significant<br />
information which could <strong>le</strong>ad to better designs. The use of diffuse or noninformation<br />
prior is in most cases wrong since it side steps this important<br />
aspect of Bayesian <strong>the</strong>ory.<br />
The prior information and <strong>the</strong> direct observations are combined through<br />
Bayes <strong>the</strong>orem<br />
where<br />
f" (a) = t(aJUamp1e) f'(a) (7)<br />
f" (a) is <strong>the</strong> posterior probability distribution of parameter a<br />
&(alSampie) is <strong>the</strong> likelihood function of a given <strong>the</strong> observed<br />
samp<strong>le</strong>s.<br />
f'(a) is <strong>the</strong> prior probability distribution of parameter a.<br />
The posterior distribution, f'l(a), can be found analytically if <strong>the</strong> prior distribution<br />
is a natural conjugate. To obtain <strong>the</strong> posterior distribution from a<br />
prior which is not a natural conjugate usually requires <strong>the</strong> application of<br />
numerical methods.<br />
jugate for both parameters Y and a.<br />
obtained from:<br />
The gamma-1 probability density function is <strong>the</strong> natural con-<br />
1 -<br />
FZ (2) = FZ<br />
all Y all a<br />
The Bayesian distribution of z, FZ (21, is<br />
(zIv,a) f"(v) f"(a) dvda (8)<br />
where FZ (zIv,a) = Fz (z) of equation (5)<br />
By assuming <strong>the</strong> posterior distribution on parameter v to be gamma - 1 with<br />
parameters u" , SI' and parameter a to be gamma - 1 with parameters Y", E"<br />
(<strong>the</strong>se aze obtained with natural conjugate priors) permits analytical evaluation<br />
of 1 - FZ (2).<br />
v a
474<br />
where<br />
L ""+1 J<br />
1 - F (2) = Ut 1 +E<br />
z<br />
cI=vII+1<br />
E''<br />
3 = - u"+l<br />
S "<br />
This is <strong>the</strong> probabilistic model for <strong>the</strong> Bayesian analysis. It is interesting<br />
to note that <strong>the</strong> form is comp<strong>le</strong>tely different from classical analysis model<br />
(equation 6).<br />
Design Model Formulation<br />
The motivation for developing <strong>the</strong> probabilistic models of extreme events<br />
is to apply <strong>the</strong>m in making decisions. Suppose we are interested in <strong>the</strong> damage<br />
associated with <strong>the</strong> exceedance flow z which is larger than <strong>the</strong> flood pro-<br />
tection flow <strong>le</strong>vel r . The total cost is comprised of a damage cost C,(z)<br />
and a protection cost C (r). For our examp<strong>le</strong>, <strong>le</strong>t's assume that <strong>the</strong> damage<br />
cost can be expressed b!:<br />
C,(z) = C1 (z-r) (10)<br />
whi<strong>le</strong> <strong>the</strong> cost of protection can be expressed by:<br />
C,(r) = K + Co r (11)<br />
If <strong>the</strong> expected criterion is used to evaluate different protection <strong>le</strong>vels <strong>the</strong>n<br />
<strong>the</strong> expected cost, E[c], of protecting for a flow r is<br />
m<br />
z=r<br />
For <strong>the</strong> classical model f(z) is:<br />
from differentiating equation (6). Thus <strong>the</strong> annual exp,ected damage E[CZ] from<br />
flooding when flood protection r is provided is<br />
This assumes that r<br />
FZ(z) is valid.<br />
a<br />
is large enough that <strong>the</strong> upper tail approximation for
In <strong>the</strong> Bayesian framework equation (12) applies but with <strong>the</strong> Bayesian<br />
density function f(z) which is obtained from equation (9) as:<br />
The annual expected damages due to flooding is<br />
again assuming that<br />
Examp<strong>le</strong> Application<br />
<strong>the</strong> upper tail approximation of Fz (z) is valid.<br />
475<br />
The analytical formulations developed here are applied to a river in <strong>the</strong><br />
Nor<strong>the</strong>astern region of <strong>the</strong> United States. The mean of fhe maximum yearly flood<br />
is about 5800 cfs. Exceedance events were considered to be flows greater than<br />
10,500 cfs which is somewhere around <strong>the</strong> 10 year recurrence intervals. Only<br />
three flows in <strong>the</strong> 37 years of record (1929 through 1965) exceeded this base<br />
flow.<br />
Bayesian Parameter Estimation<br />
Estimation for a<br />
Prior information on a , <strong>the</strong> event magnitude distribution, was obtained<br />
from a regression on 36 o<strong>the</strong>r Nor<strong>the</strong>astern United States basins. The regression<br />
related exceedance flows to physical characteristics found within any drainage<br />
basin. The following regression was obtained:<br />
.153 o 2.87 A .81 .74 .54 .65<br />
Qm- St<br />
where*<br />
is mean exceedance flow, in cubic feet per second<br />
% is orographic factor<br />
A is drainage basin area, in square mi<strong>le</strong>s<br />
S<br />
T<br />
is main channel slope, in feet per mi<strong>le</strong><br />
is average January, degrees below freezing, in degrees Fahrenheit<br />
St<br />
is percent of surface storage area plus .5 percent<br />
Since partial duration series and annual flood series are virtually identi-<br />
cal above a frequency of about <strong>the</strong> 10 year flood (Langbein, 1949) <strong>the</strong> use of <strong>the</strong><br />
annual series for <strong>the</strong> prior was considered to be adequate for this examp<strong>le</strong>.<br />
Research is presently being conducted by <strong>the</strong> author to study <strong>the</strong> prob<strong>le</strong>ms of<br />
appropriate prior information.<br />
*The physical characteristics are from Benson (1962) and <strong>the</strong> streamflow data<br />
from <strong>the</strong> U.S. Geological Water Supply Papers. (1301-A, 1721-A, 1901-A).<br />
J
From <strong>the</strong> regression an estimate of <strong>the</strong> mean exceedance flood, Q and an<br />
estimate of <strong>the</strong> variance of <strong>the</strong> mean flood were found to be:<br />
P9<br />
= 734 cfs<br />
QP<br />
5 2<br />
V[Q ] = 4.9 x 10 (cfs)<br />
P<br />
Due to <strong>the</strong> assumed distribution of <strong>the</strong> magnitude of exceedance events, <strong>the</strong> mean<br />
exceedance flood can be related to <strong>the</strong> event magnitude distribution parameter<br />
by Q = l/a. If Q is assumed to be distributed as an inverted gamma -1 distribetion<br />
with pargrneters v' and R' <strong>the</strong>n a is distributed gamma -1 with parameters<br />
v', $' (biffa and Cchlaifer. 1961); that is<br />
with<br />
V[Qp] = v'>1<br />
(v')2 (VI-i)<br />
This gave parameters v' = 2, R' = 1468. Thus <strong>the</strong> prior distribution on O! ,<br />
f' (a) is<br />
-1468a<br />
fgYl(a) = e a2 (i468I3<br />
r (3)<br />
The posterior of , f"(a) can now be evaluated by equation (7). Thus<br />
5 -33668a<br />
f" (a) = Ka e<br />
YI<br />
where<br />
(20)<br />
n<br />
The posterior of a is gamma -1 with parameters E' = + zi ; VI' = V I+ n<br />
Estimation for v .<br />
The estimation of prior information on u, <strong>the</strong> average arrival rate,<br />
involves, as a first step, <strong>the</strong> estimation of <strong>the</strong> first two central moments of<br />
<strong>the</strong> distribution of <strong>the</strong> arrival rate of a peak flow that will exceed <strong>the</strong> base<br />
flow Qo. In our examp<strong>le</strong>, this base flow was 10,500 cfs. There are some<br />
probabilistic or statistical methods one may use to approach this prob<strong>le</strong>m or<br />
<strong>the</strong> engineer may have said simply "based upon my experience in <strong>the</strong> area, my<br />
best estimate of v<br />
minus .O25 of .l'I.<br />
is .1 and <strong>the</strong>re is a 50-50 chance that v could be plus or<br />
The implication of that statement is that <strong>the</strong> standard
deviation is about ,033. If that is accepted for our examp<strong>le</strong> and if a g a m - 1<br />
distribution for <strong>the</strong> prior<br />
-<br />
of V is assumed <strong>the</strong>n<br />
-0'V U'<br />
f' (v) e (s'v) s'<br />
YI<br />
-<br />
r (u'+i)<br />
with u' 8<br />
s' = 92<br />
(21)<br />
The posterior distribution of , fy1" (u) is just<br />
3 .-37 v8 .-92~<br />
f II (y) = v<br />
Y1<br />
-129~<br />
= v" e<br />
(22)<br />
- yhich is g a m - 1 distributed with parameters u" 11, si' = 129, and mean<br />
v= .O85 events per year.<br />
Substituting <strong>the</strong>se into <strong>the</strong> Bayesian design model of Equation (9) yields<br />
- 1 - FZ (z) 5 .085t<br />
-<br />
Thus <strong>the</strong> Bayesian model, 1 - F (z), of equation (23) is shown in<br />
2<br />
Figure 2. The effect of considering diffuse prior information (no observations<br />
in no years of data) is also shown in Figure 2.<br />
Classical Proceedures<br />
Application of <strong>the</strong> classical estimation proceedures is straight forward.<br />
Estimators for both <strong>the</strong> average arrival rate, v , and <strong>the</strong> parameter of <strong>the</strong><br />
event magnitude distr&bution, a can be obtained by applying <strong>the</strong> maximum likelihood<br />
criterion. For v , <strong>the</strong> estimator for v, <strong>the</strong> likelihood function is:<br />
and <strong>the</strong> maximum likelihood criterion; 2 = O yields<br />
Similarly for a ;<br />
a,<br />
j = 0 = 3 = .O81<br />
tr<br />
-<br />
37<br />
-5<br />
a = n = 9.3 x 10<br />
zi<br />
i=l
478<br />
A Kolmogorov-Smirnos test on <strong>the</strong> models using <strong>the</strong> derived estimators<br />
passed <strong>the</strong> .10 significance <strong>le</strong>vel with ease.<br />
Thus <strong>the</strong> classical estimator model for our examp<strong>le</strong> is<br />
-9.3 10-~~<br />
1 - FZ(z) .O811 e<br />
which is <strong>the</strong> probability of observing a peak flow z in <strong>the</strong> next interval of<br />
time. The classical model, 1 - F<br />
Z<br />
(z) represented by equation (25), is compared<br />
to <strong>the</strong> Bayesian model in Figure 2.<br />
Design Application<br />
Using cost coefficients for Equations (10) and (11) as being:<br />
4<br />
c1 = $10 Icfs<br />
K = $25 x lo4 equiva<strong>le</strong>nt annual cost over a<br />
= $102/cfs proposed 50-year project life.<br />
for <strong>the</strong> classical design proceedures in Equation (16) utilizing <strong>the</strong> Bayesian<br />
design model and in Equation (14) for <strong>the</strong> classical design model.<br />
The expected annual cost of providing protection against <strong>the</strong> 100 year<br />
flood is presented in Tab<strong>le</strong> I.<br />
100 yr flood Expected Flood Equiva<strong>le</strong>nt Annual<br />
discharge Damages ($1 Protection Cost-50 Yr Life($)<br />
Bayesian Model 17500 7 105 20 105<br />
Classical Model 22500 10.76 lo5 25 105<br />
Tab<strong>le</strong> I - Comparisons of Costs and Damages for Bayesian<br />
and Classical Models<br />
For each model <strong>the</strong> flood which had <strong>the</strong> lowest expected total cost also was <strong>the</strong><br />
100 year flood.<br />
Discussion<br />
Incorporating Non-Stationarity Effects<br />
The prob<strong>le</strong>ms of flood analysis when non-stationarity has been introduced<br />
into <strong>the</strong> streamflow records due to increased development of <strong>the</strong> drainage basin<br />
have not been comp<strong>le</strong>tely solved. Recent studies by Bras (1972) have shorn<br />
~~
479<br />
increases in flood peaks of developed catchments of between 30% to 115% depend-<br />
ing upon <strong>the</strong> particular size and shape of <strong>the</strong> storm. Basin development tends<br />
to remove natural stream storage areas as well as decrease impervious areas and<br />
holding ability of natural ground cover. The effects of changing <strong>the</strong>se charac-<br />
teristics can best be investigated by a deterministic catchment runoff model<br />
that utilizes a stochastic rainfall generator (Har<strong>le</strong>y, Wood and Schaake, 1973).<br />
The Bayesian analyst has a number of options open to him which include a<br />
rainfall analysis, a runoff analysis from <strong>the</strong> catchment analysis, and o<strong>the</strong>r<br />
approaches. He can ei<strong>the</strong>r utilize <strong>the</strong> streamflow data or ignore it, applying<br />
his engineering judgment in many ways.<br />
The classical analysis has few, if any, options open to him. The strict<br />
application of his <strong>the</strong>ory permits him only to consider <strong>the</strong> historical record<br />
which will not apply to <strong>the</strong> developed basin. If <strong>the</strong> amount of development is<br />
small, <strong>the</strong>n <strong>the</strong> historical record may still contain valuab<strong>le</strong> information but if<br />
extensive modifications have taken place and <strong>the</strong> classical analyst still uses<br />
his historical record <strong>the</strong>n he must be ab<strong>le</strong> to defend it.<br />
Conclusions<br />
The ro<strong>le</strong> of analysis is to aid decision making. The two approaches presented<br />
here <strong>le</strong>ad to quite different design decisions.<br />
The classical approach restricts <strong>the</strong> analyst to <strong>the</strong> observab<strong>le</strong> hydrologic<br />
data to which o<strong>the</strong>r information sources can not be added. Fur<strong>the</strong>rmore, it is<br />
not possib<strong>le</strong> to include within <strong>the</strong> analysis o<strong>the</strong>r uncertain parameters which<br />
may affect <strong>the</strong> design.<br />
Instead some o<strong>the</strong>r artifical mechanism is used such as<br />
adding a factor of safety to <strong>the</strong> design variab<strong>le</strong>, designing for <strong>the</strong> largest<br />
possib<strong>le</strong> event or using some o<strong>the</strong>r method which can not be related to a mean-<br />
ingful economic (or social) preference criterion.<br />
Too often too much weight is given to a few observab<strong>le</strong> data points and too<br />
litt<strong>le</strong> weight to o<strong>the</strong>r availab<strong>le</strong> information. Lhe Bayesian analysis is a methodology<br />
which enab<strong>le</strong>s <strong>the</strong> combination of information sources as well as allows<br />
<strong>the</strong> explicit evaluation of <strong>the</strong> effect of all sources of uncertainty upon <strong>the</strong><br />
decision variab<strong>le</strong>s. The application of <strong>the</strong> Bayesian approach will <strong>le</strong>ad to<br />
better design than will a classical analysis which is restricted to a few observations<br />
and whose conclusions are difficult to interpret.<br />
Acknow<strong>le</strong>dgments<br />
The work was supported by <strong>the</strong> Office of Water Resourc Research, Office<br />
of <strong>the</strong> Interior, United States Government under Grant No. 14-31-0001-9021.<br />
References<br />
1. Benson (1962). "Factors Influencing <strong>the</strong> Occurrence of Floods in a Humid<br />
Region of Diverse Terrain" U.S. Geological Survey Water Supply Paper 1580-B,<br />
Washington, D.C.
480<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13,<br />
Bernier (1967). "Les Methods Bayesiennes En Hydrologie Statistique"<br />
Proceedinvs of <strong>the</strong> Int. Hydro Symp., September 1967, Colorado State<br />
University, Fort Collins, Colorado, USA.<br />
Bogardi and Szidarovszky (1972)."The Margin of Safety for Compensating<br />
Losses due to Uncertainties in Hydrological Statistics" Proceedings of<br />
<strong>the</strong> Int. Symp on Uncertainties in Hydrologic and Water Resource Systems.<br />
Dec. 1972, University of Arizona, Tucson, Arizona, USA.<br />
Bras (1972)."Effects of Urbanization on Runoff Characteristics of Small<br />
Basins in Puerto Rico" Unpublished Bachelor of Science <strong>the</strong>sis, Department<br />
of Civil Engineering, Massachusetts Institute of Technology, Cambridge,<br />
Massachusetts, U.S.A.<br />
Cornell (1972). "Bayesian Statistical Decision Theory and Reliability-<br />
Based Design" Structural Safety and Reliability, (A. Freudenthal, ed.),<br />
Pergamon Press, New York.<br />
Davis, Kisiel, and Duckstein (1972a)."Bayesian Decision Theory Applied to<br />
Design in Hydrology" Water Resources Research, Vol. 8 No. 1.<br />
Davis, Duckstein and Kisiel (1972b)."Uncertainty in <strong>the</strong> Return Period of<br />
Maximum Events : A Bayesian Approach" Proceedings of <strong>the</strong> Int. Symp. on<br />
Uncertainties in Hydrologic and Water Resource Systems. December 1972,<br />
University of Arizona, Tucson, Arizona, U.S.A.<br />
Eag<strong>le</strong>son (1972). "Dynamics of Flood Frequency" Water Resource Research<br />
Vol. 8, No. 4.<br />
Grayman and Eag<strong>le</strong>son (1971). "Evaluation of Radar and Raingage Systems for<br />
Forcasting" Ralph M. Parsons for Water Resources and Hydrodynamics T.R. No.<br />
138, Department of Civil Engineering, M.I.T., Cambridge, Mass. U.S.A.<br />
Har<strong>le</strong>y, Wood, and Schaake (1973). "The Application of Hydrologic Models to<br />
Urban Planning" Presented at 54th Annual Meeting, American Geophysical<br />
Union, Washington, D.C., April, 1973.<br />
Langbein a949). "Annual Floods and <strong>the</strong> Partial Duration Flood Series"<br />
American Geophysical Union Transaction, V. 30, pp. 879-881.<br />
Raiffa (1968). Decision Analysis, Addison-Wes<strong>le</strong>y, Reading, Mass., U.S.A.<br />
Raiffa and Schlaijer (1961). Applied Statistical Decision Theory, M.I.T.<br />
Press, Cambridge, Mass., U.S.A.<br />
14. Shane and Gaver (1970). "Statistical Decision Theory Techniques for <strong>the</strong><br />
Revision of Mean Flow Regression Estimates" Water Resources Research,<br />
Vol. 6, No. 6.
16. Todorovic and Fe<strong>le</strong>nhasic (1970). "A Stochastic Model for Flood Analysis"<br />
Water Resources Research, Vol. 6, No. 6.<br />
481<br />
17. United States Department of <strong>the</strong> Interior. "Surface Water of North Atlantic<br />
Slope Basins, throu$-11950", U.S.G.S.<br />
D.C., 1957.<br />
Water Supply Paper f301, Washington,<br />
18. . "Surface Water of North Atlantic<br />
Slope Basins, 1950-60", U.S.G.S. Water Supply Paper 1721, Washington, D.C.,<br />
1969.<br />
19. . "Surface Water Supply of <strong>the</strong> U.S.<br />
1961-65Jater Supply Paper 1901, Washington,<br />
D.C., 1969.
482<br />
4<br />
Q<br />
W<br />
TIME<br />
O. DISCHARGE Q I EXCEEDANCE DISCHARGE 2 = Q-Qo<br />
Figure 1: Typical Discharge Record Showing Exceedance Events and<br />
Showing <strong>the</strong> Probability Density Functions for both Discharges<br />
and Exceedance Events.<br />
Qo
O O<br />
rn<br />
O<br />
2<br />
O<br />
O<br />
O U1<br />
O<br />
O<br />
o<br />
O<br />
i<br />
O<br />
Lrl<br />
ri<br />
O<br />
01<br />
O<br />
O<br />
ln<br />
N<br />
483
Authors and Tit<strong>le</strong>s;<br />
axe:<br />
THE USE OF MATHEMATICAL (DETERMINISTIC) MODELS<br />
General Report<br />
bY<br />
J. E. Nash<br />
University Col<strong>le</strong>ge, Galway. Ireland.<br />
At <strong>the</strong> time of writing four papers have been received. These<br />
(1) "A Rainfall-Runoff Model Based on <strong>the</strong> Watershed Stream Network" by<br />
J.W. De<strong>le</strong>ur and N.T. Lee, of <strong>the</strong> School of Civil Engineering, Purdue<br />
University, and <strong>the</strong> Department of Agricultural l3conomics of <strong>the</strong><br />
university of Illinois, respectively.<br />
(2) "Monthly Streamflow EstLnation from Limited Data" by C.T. Haan,<br />
of <strong>the</strong> Agricultural Engineering Department, University of Kentucky.<br />
(3) "Obtaining of Deficient Information by Solving Inverse Prob<strong>le</strong>ms<br />
from Ma<strong>the</strong>matical runoff models", by V.I. Koren and L.S. Kutchent,<br />
of <strong>the</strong> Hydrometeorological Centre of <strong>the</strong> U.S.S.R.<br />
(4) "The. Ma<strong>the</strong>matical Model of Water Balance for Data Scarce Areas" by<br />
Nabil Rofail, Water Resources Department, Desert Institute of Cairo.<br />
Introduction: In view of <strong>the</strong> relatively small number of papers it had been<br />
suggested to me by <strong>the</strong> OrgGisers that I should include some introductory comment<br />
of my own,on <strong>the</strong> subject of catchment modelling.<br />
However, whi<strong>le</strong> <strong>the</strong> number of<br />
Papers is indeed small, <strong>the</strong>y are all interesting and two of <strong>the</strong>m are of a
486<br />
ma<strong>the</strong>matical nature and will require time to elucidate.<br />
interesting,in that it presents a practical tool developed and used in <strong>the</strong><br />
Soviet Union but, as far as I am aware, not generally known in <strong>the</strong> West.<br />
paper also happens, understandably, to be very difficult to follow, and on<br />
<strong>the</strong>se two accounts, I propose to devote a somewhat disproportionate part of <strong>the</strong><br />
time availab<strong>le</strong> to its consideration. I feel sure that <strong>the</strong> o<strong>the</strong>r authors will<br />
not consider this in any way a slight and, as I am sure that this distinguished<br />
audience would prefer me to devote any time availab<strong>le</strong> to <strong>the</strong> consideration<br />
of khis interesting technique, I shall keep my own general comments on <strong>the</strong><br />
subject of modelling as brief as possib<strong>le</strong>.<br />
Hydrological Modelling:<br />
One of <strong>the</strong>se is particular3<br />
This<br />
The variety of tit<strong>le</strong>s among <strong>the</strong> papers we are considering<br />
ref<strong>le</strong>cts <strong>the</strong> widely different senses in which <strong>the</strong> term modelling is understood.<br />
Never<strong>the</strong><strong>le</strong>ss, <strong>the</strong>re is a strong common link between <strong>the</strong>m.<br />
that a natural phenomenon such as <strong>the</strong> conversion of rainfall into discharge, or<br />
<strong>the</strong> movement of water in a porous medium, is represented by an hypo<strong>the</strong>sis or<br />
model, expressed as a series of operations which are,performed<br />
This would seem to be,<br />
on one<br />
function of time (<strong>the</strong> input) to convert it to ano<strong>the</strong>r function of time (<strong>the</strong><br />
output), or as a sing<strong>le</strong> ma<strong>the</strong>matical relationship, a partial or ordinary<br />
differential equation which must be solved in terms of boundary CQnditions.<br />
The relationship between <strong>the</strong> three e<strong>le</strong>ments may be represented by<br />
Where <strong>the</strong> relationship is of a causal nature this may be indicated by<br />
6 f fe.c E-<br />
y ?)
I make this distinction because <strong>the</strong> ma<strong>the</strong>matical solution of such prob<strong>le</strong>ms is<br />
often against <strong>the</strong> direction ,of <strong>the</strong> arrow, which from <strong>the</strong> ma<strong>the</strong>matical point of<br />
view may,<strong>the</strong>refore,be considered irre<strong>le</strong>vant. I shall use <strong>the</strong> terms cause and<br />
487<br />
effect for emphasis, only when <strong>the</strong> direction of <strong>the</strong> arrow is physically re<strong>le</strong>vant<br />
Ei<strong>the</strong>r diagram represents a relationship between three quantities one only of<br />
which may be unknown in any realistic prob<strong>le</strong>m.<br />
The solution sought may be <strong>the</strong> output, <strong>the</strong> input, or a description of,<br />
(or parameters of) <strong>the</strong> operation itself (e.g. a unit hydrograph or <strong>the</strong> coefficients<br />
of a differential equation).<br />
Generally speakingrit is true in <strong>the</strong> hydrological context that <strong>the</strong> operations,<br />
viewed in <strong>the</strong> direction from cause to effect are stab<strong>le</strong>lin <strong>the</strong> sense that bounded<br />
causes produce bounded effects and small variations in <strong>the</strong> causes produce smal<strong>le</strong>r<br />
variation in <strong>the</strong> effects.<br />
Precisely because of this fact, <strong>the</strong> inverse operation<br />
discussed by Koren and Kutchment, of <strong>the</strong> discovery of <strong>the</strong> cause of an observed<br />
effect, or <strong>the</strong> discovery of an operation itself, tends to be unstab<strong>le</strong> and small<br />
variations oierrors in <strong>the</strong> observed output produce larger variations or errors<br />
in <strong>the</strong> computed cause,or <strong>the</strong> computed values of <strong>the</strong> parameters of <strong>the</strong> operation.<br />
Por this reason <strong>the</strong> solution of <strong>the</strong> inverse prob<strong>le</strong>m is usually very much more<br />
difficult than <strong>the</strong> solution of <strong>the</strong> direct prob<strong>le</strong>m.<br />
The Direct Prob<strong>le</strong>m:<br />
which arise usually involve questions of convergence of finite difference<br />
solutions of differential equations.<br />
This has at <strong>le</strong>ast a logical simplicity and <strong>the</strong> difficulties<br />
The paper by Nabill Rofail describes <strong>the</strong><br />
solution of one such prob<strong>le</strong>m which we shall discuss in some deail later.
488<br />
Among <strong>the</strong> inverse prob<strong>le</strong>ms it is useful to distinguish three types<br />
(a) The input is unknown<br />
(b) The values of <strong>the</strong> parameters of <strong>the</strong> operation are unknowr<br />
(c) The form and parameter values of <strong>the</strong> operation are unknowr<br />
In <strong>the</strong> particular case of a lumped linear system where <strong>the</strong> input, <strong>the</strong><br />
operation and <strong>the</strong> output may be represented by<br />
L<br />
where x(t) is <strong>the</strong> input, y(t) is <strong>the</strong> output and h(t) <strong>the</strong> impulse response, <strong>the</strong><br />
three classes collapse to one. For such systems <strong>the</strong> form of <strong>the</strong> operation may<br />
be described uniquely by <strong>the</strong> impulse response of <strong>the</strong> system and,<strong>the</strong>oretically at<br />
<strong>le</strong>ast,this may be found without prior specification of its form. Therefore <strong>the</strong><br />
second and third classes merge.<br />
Fur<strong>the</strong>rmore, because of <strong>the</strong> symmetry in h and<br />
x in <strong>the</strong> two equations, a symmetry which becomes more obvious when <strong>the</strong> relationshi1<br />
ià expressed in terms of Lapace transforms through <strong>the</strong> Faltung <strong>the</strong>orm,<br />
Ys) = X(sj HU) (3)<br />
<strong>the</strong> prob<strong>le</strong>ms of discovering h and x are ma<strong>the</strong>matically <strong>the</strong> same, so that all three<br />
distinctions vanish.<br />
in <strong>the</strong> nonlinear prob<strong>le</strong>m, however, or when recognition<br />
of <strong>the</strong> system implies discovery of <strong>the</strong> coefficients of a partial differential<br />
equation (a distributed system) <strong>the</strong> distinctions remain valid.<br />
The Lumped,Linear Model:<br />
For functions which are not simp<strong>le</strong> expressions, it is<br />
usually easiest to deal with <strong>the</strong>se models in discrete form.<br />
Eqs. 1 and 2 are<br />
replaced by €Y] = IhJ i4 (4 1<br />
and ius = PI €h3 c=>
where [x] and<br />
at equal time invervals.<br />
[y] are vectors of <strong>the</strong> input and output respectively, samp<strong>le</strong>d<br />
ordina tes of <strong>the</strong> impulse response as<br />
h, O 0:<br />
[hJ is a rectangular matrix formed from <strong>the</strong><br />
[h] =<br />
Similarly Lx] in eq. 5 is a rectangular matrix formed from <strong>the</strong> input<br />
ordinates in <strong>the</strong> same way.<br />
.<br />
The direct. prob<strong>le</strong>m of finding fy3 is trivial. The inverse prob<strong>le</strong>ms of<br />
finding 1.3 of eq. 4 or {XI from eq. 5 are ma<strong>the</strong>matically <strong>the</strong> same.<br />
Because of <strong>the</strong>,great stability of <strong>the</strong> operation in <strong>the</strong> direct direction,<br />
solution of <strong>the</strong> inverse prob<strong>le</strong>ms tends to be very unstab<strong>le</strong> and <strong>the</strong>refore<br />
489<br />
difficult. Traditional means usually involve a <strong>le</strong>ast squares solution;(Snyder)<br />
or <strong>the</strong> imposition of constraints on <strong>the</strong> impulse response e.g. harmonic analysis<br />
with a limited number of terms (O’D~nnell). A new method of obtaining a <strong>le</strong>ast<br />
squares solution under a constraint is described in <strong>the</strong> paper by Koren and<br />
Kutchment.<br />
Distributed Linear Models: If <strong>the</strong> input is distributed in space in a<br />
constant manner <strong>the</strong> direct prob<strong>le</strong>m is essentially <strong>the</strong> same as that of <strong>the</strong><br />
lumped linear system (Kraijenhoff van de Leur, Venetis). If however <strong>the</strong><br />
input is arbitrarily distributed in space,<strong>the</strong> differential equation must be<br />
solved numerically for <strong>the</strong> direct prob<strong>le</strong>m, usually by reducing <strong>the</strong> prob<strong>le</strong>m<br />
to linear difference equations, which are solved at <strong>the</strong> node points of an xt<br />
plane. An examp<strong>le</strong> is provided in <strong>the</strong> paper by Nabil Rofail.
490<br />
Threatment of <strong>the</strong> inverse prob<strong>le</strong>m to discover <strong>the</strong> input or <strong>the</strong> coefficients<br />
in a known linear system expressed as a partial differential equation,are rare.<br />
An attempt to discover <strong>the</strong> characteristics of an aquifer is described in ref. 6 of<br />
<strong>the</strong> paper by Koren and Kutchment, and <strong>the</strong> paper itself gives two examp<strong>le</strong>s of such ai<br />
attempt to determine values of <strong>the</strong> conveyances and cross sectional areas as<br />
functions of space and time in an open channel.<br />
The General Nonlinear Prob<strong>le</strong>m: The direct prob<strong>le</strong>ms are again relatively<br />
straightforward - <strong>the</strong> nonlinearity complicates <strong>the</strong> numerical solution of distribute(<br />
systems (partial differential equations) but .<strong>the</strong> lumped parameters systems are<br />
scarcely affected.<br />
The Inverse Prob<strong>le</strong>m involves, generally, postulation of <strong>the</strong> form of <strong>the</strong><br />
operation (i.e. a conceptual model) and estimation of <strong>the</strong> parameters<br />
successive approximations. The first approximations are inserted in <strong>the</strong> model<br />
and <strong>the</strong> output computed.<br />
This is compared with <strong>the</strong> observed output and a<br />
sing<strong>le</strong> expression of <strong>the</strong> observed errors (<strong>the</strong> objective function) is systematically<br />
reduced by subsequent trial and error adjustments of <strong>the</strong> parameters values.<br />
Examp<strong>le</strong>s are provided in <strong>the</strong> papers by De<strong>le</strong>ur and Lee and Haan. The major'<br />
difficulties with this method are that <strong>the</strong> set of solutions obtained may not<br />
be unique and, particularly when two or more parameters represent similar<br />
operations,<strong>the</strong> optimised values are subject to very .high sampling variance.<br />
It would be interesting to speculate whe<strong>the</strong>r such prob<strong>le</strong>ms could be made amenab<strong>le</strong><br />
to direct <strong>le</strong>ast squares approximations, as so often used in <strong>the</strong> corresponding<br />
linear case.<br />
Theorétically, this would seem possib<strong>le</strong>, but it may be, as seems<br />
to be generally assumed, that <strong>the</strong> comp<strong>le</strong>xity. of <strong>the</strong> equation representing <strong>the</strong><br />
by
dependence of <strong>the</strong> objective function On <strong>the</strong> parameters might render its formulation<br />
difficulty. I feel however that this possibility ought to be explored.<br />
Having thus classified <strong>the</strong> papers according to <strong>the</strong> nature of <strong>the</strong> prob<strong>le</strong>m discussed<br />
we corne to a consideration of <strong>the</strong> papers <strong>the</strong>mselves in some detail. These I<br />
would like to take in <strong>the</strong> order of <strong>the</strong>ir classification above.<br />
491
492<br />
A Ma<strong>the</strong>matical Model of Water Balance for Data Scarce Areas<br />
by Rofail<br />
The tit<strong>le</strong> of this paper is somewhat mis<strong>le</strong>ading.<br />
fact <strong>the</strong> numerical solution of <strong>the</strong> linearised equations of motion of groundwater<br />
in an unconfined aquifer.<br />
distributed linear system. Neg<strong>le</strong>cting any vertical component of velocity, <strong>the</strong><br />
horizontal components paral<strong>le</strong>l to <strong>the</strong> x and y axis in a homogeneous aquifer are<br />
proportional to <strong>the</strong> gradients of <strong>the</strong> piezometric head (h+z). The constant of<br />
proportionality (k) is known as <strong>the</strong> coefficient of permeability, (authors eqs.<br />
1 and 2).<br />
The subject matter is in<br />
It is <strong>the</strong>refore a case of a direct solution of a<br />
The continuity equation (aut4ors eq.3) expresses <strong>the</strong> fact that <strong>the</strong> rate<br />
of rise of <strong>the</strong> surface of saturation<br />
, at a’point in <strong>the</strong> aquifer, is<br />
proportional to <strong>the</strong> rate of percolation down to <strong>the</strong> aquifer at this point, plus<br />
<strong>the</strong> net rate of flow towards <strong>the</strong> point (<strong>the</strong> negative of <strong>the</strong> divergence). The<br />
constant of proportionality is known as <strong>the</strong> specific yield and is given <strong>the</strong><br />
symbol/h in authors eq. 3. These two equations are combined in <strong>the</strong> authors eq.4<br />
by replacing <strong>the</strong> velocity terms by <strong>the</strong> corresponding gradients of <strong>the</strong> piezometri<br />
head, yielding an equation in <strong>the</strong> head only.<br />
3Yhtz) k ukæl , a h<br />
3% b>c<br />
03 1
This equation contains first and second order derivatives of <strong>the</strong> depth h and<br />
<strong>the</strong> e<strong>le</strong>vation of <strong>the</strong> aquifer bed z, and is non-linear due to <strong>the</strong> occurrence of<br />
- By fur<strong>the</strong>r assuming that <strong>the</strong> gradient of h is small relative to that of z,<br />
terms such as (&become ah small relative to s. 9s and are dropped,thus<br />
linearibing <strong>the</strong> equation (authors'eq. 5)<br />
9 8 - h $:jL- WZ. ah<br />
11<br />
az- - arc- E<br />
bLh hg?, ah .k-- -<br />
-h--, -<br />
bu<br />
DY ay k<br />
493<br />
N =o (Id<br />
This assumption is attributed to"Boussenzq" and is stated to be that <strong>the</strong><br />
powers of derivatives of <strong>the</strong> first order are of a <strong>le</strong>sser order of magnitude<br />
than <strong>the</strong> derivatives <strong>the</strong>mselves. This would, of course, be an acceptab<strong>le</strong> assumption<br />
but it does not seem to be that which is in fact made by <strong>the</strong> author in obtaining<br />
his eq. 5 from eq. 4. Perhaps <strong>the</strong> authors would like to comment on this.<br />
To emphasise <strong>the</strong> linearity of eq. 5 in terms of <strong>the</strong> partial derivatives of h,<br />
<strong>the</strong> partial derivatives of z (assumed to be known) are written as 7, and fy<br />
and <strong>the</strong> second order derivatives (also assumed to be known as r, and fiv<br />
in eq. 10. (Note that in <strong>the</strong> text <strong>the</strong> quantities 7,, and Vy are incorrectly<br />
stated to be <strong>the</strong> partial derivatives of h; this in only a printing errer).<br />
In order to advance <strong>the</strong> solution from time n'to <strong>the</strong> n+l <strong>the</strong> author<br />
replaces <strong>the</strong> equation of motion with two distinct finite difference approximations.
494<br />
The first is applied to <strong>the</strong> first half of <strong>the</strong> time interval and <strong>the</strong> second to<br />
<strong>the</strong> second half. For ease of comparison I reproduce here <strong>the</strong> equiva<strong>le</strong>nces as<br />
used in <strong>the</strong> two successive steps.<br />
j and k refer to <strong>the</strong> node point location in <strong>the</strong> x and y directions and n<br />
to <strong>the</strong> number of <strong>the</strong> time interval. approximation<br />
1st half 2nd half<br />
derivative step.<br />
step<br />
-<br />
These approximations have a certain symmetry, which when <strong>the</strong> finite differenc<br />
approximations are added for <strong>the</strong> two half time steps,make <strong>the</strong> results consistent<br />
with <strong>the</strong> original equation up to <strong>the</strong> second order. They have <strong>the</strong> additional merit<br />
that <strong>the</strong> finite difference<br />
written implicity in terms<br />
equation for <strong>the</strong> first half step in time, can be<br />
n 42<br />
of linear combination of (h,-l ., h, and hJ+l)<br />
with <strong>the</strong> coefficients all known (author's eq. il). Similarly <strong>the</strong> equation<br />
applied to <strong>the</strong> second<br />
rt'<br />
half step yields an implicit equation linear in<br />
h, and hrti ail <strong>the</strong> coefficients again being known (authors eq. 12).<br />
(LI<br />
These implicit equations can be solved as a linear set when <strong>the</strong> boundary condition8<br />
are provided to yield h for all node points at a sing<strong>le</strong> time.<br />
<strong>the</strong> solution through time.<br />
Repetition extends
"A rainfall runoff model based on <strong>the</strong> watershed and stream network"<br />
ßy Del<strong>le</strong>ur and Lee<br />
The prob<strong>le</strong>m discussed, i.e. that of recognising a lumpedpon-linear model,<br />
is in <strong>the</strong> inverse category and <strong>the</strong> method used is that of postulating <strong>the</strong> form<br />
of <strong>the</strong> system and optimisation of <strong>the</strong> parameters.<br />
The authors begin by pointing out that even with forty years of record <strong>the</strong><br />
errors in estimating <strong>the</strong> parameters of a stochastic model of annual flows<br />
may be quite high.<br />
rainfall-runoff process,likewise,require long term series of both <strong>the</strong> rainfall<br />
and runoff for <strong>the</strong>ir calibration. They conclude, <strong>the</strong>refore, that for regions<br />
with inadequate data one may have to resort to deterministic models ei<strong>the</strong>r of<br />
<strong>the</strong> "black box" or of <strong>the</strong> "physical" type.<br />
to whe<strong>the</strong>r <strong>the</strong> model form attempts to mirror <strong>the</strong> physical processes or is merely<br />
a linear regression. The authors point to <strong>the</strong> obvious deficiency of black<br />
box models,that <strong>the</strong>y cannot be transferred from one location to ano<strong>the</strong>r because<br />
<strong>the</strong>re is an absense of a one to one relationship between <strong>the</strong> parameters of <strong>the</strong><br />
model and <strong>the</strong> parameters<br />
They mention also that stochastic linear models of <strong>the</strong><br />
This distinction is made according<br />
<strong>the</strong> watershed. They conclude, <strong>the</strong>refore, that a<br />
495<br />
physical model requiring only a small number of identifiab<strong>le</strong> parameters or a model<br />
based on data which can be obtained in a relatively short time is required.<br />
I am not sure what <strong>the</strong> authors mean by 'a stochastic linear model of <strong>the</strong><br />
rainfall runoff process" nor am I sure that <strong>the</strong>se several models are<br />
alternatives for <strong>the</strong> same piirpose.<br />
It seems to me that if one requires a<br />
stochastic model in order to generate a time series having <strong>the</strong> properties of<br />
<strong>the</strong> observed samp<strong>le</strong> it can scarcely be rubstituted for by a deterministic model<br />
(whe<strong>the</strong>r of a black box nature or o<strong>the</strong>rwise) relating rainfall to discharge.
496<br />
Before such a model could be used to produce a syn<strong>the</strong>tic discharge record <strong>the</strong><br />
stochastic properties of <strong>the</strong> rainfall input would have to be computed and a<br />
syn<strong>the</strong>tic rainfall record fed into <strong>the</strong> deterministic model.<br />
that <strong>the</strong> prob<strong>le</strong>m had merely been transferred ra<strong>the</strong>r than solved by <strong>the</strong> substitution<br />
of <strong>the</strong> deterministic model.<br />
Thus it would seem<br />
If of course a much longer ,rainfall record was<br />
availab<strong>le</strong> this might be useful. The authors intention is to provide a deterministi<br />
model with a small number of parameters preferably identifiab<strong>le</strong> from <strong>the</strong> catchment<br />
characteristics. I don't .think <strong>the</strong>re would be any argument about <strong>the</strong> usefulness<br />
of such a model even if it would not substitute for a stochastic model for a<br />
different purpose.<br />
The authors suggest that <strong>the</strong> availability of modern techniques of photography<br />
and general remote sensing technology make it possib<strong>le</strong> to observe re<strong>le</strong>vant<br />
catchment characteristics on a large sca<strong>le</strong> and <strong>the</strong>refore to include <strong>the</strong>se in <strong>the</strong><br />
deterministic model.<br />
In <strong>the</strong> authors'model use,is made of <strong>the</strong> following catchment<br />
characteristics obtained by aerial photography - <strong>the</strong> plan form of <strong>the</strong> stream<br />
network, <strong>the</strong> topography, and <strong>the</strong> soil type. The model attempts to relate <strong>the</strong><br />
areal mean of <strong>the</strong> rainfall to <strong>the</strong> discharge at <strong>the</strong> gauging site, both as functions<br />
of time.<br />
area which is <strong>the</strong> area contributing at any given instant to <strong>the</strong> flow at <strong>the</strong><br />
gauging station i.e. <strong>the</strong> function of time representing that portion of <strong>the</strong><br />
catchment from which runoff is currently passing <strong>the</strong> gauging station at <strong>the</strong> time<br />
in question.<br />
The structure of <strong>the</strong> model depends largely on <strong>the</strong> concept of a contributi<br />
Obviously <strong>the</strong> contributing area is a functi,on of <strong>the</strong> catchment<br />
wetness and <strong>the</strong> model for this quantity, described by <strong>the</strong> authors'eq. 1 is c<strong>le</strong>arly<br />
dependent upon <strong>the</strong> antecedent rainfall.
The contribut?.ng area at timeibt is A (fdk) and <strong>the</strong> total catchment area is A,.<br />
I assume that in eq.11 <strong>the</strong> second negative sign from <strong>the</strong> right in <strong>the</strong> numerator<br />
should in fact be positive, and with this interpretation I understand <strong>the</strong><br />
assumption of eq. 11 to be that <strong>the</strong> contributing area expressed as a proportion<br />
of <strong>the</strong> total catchment area varies with <strong>the</strong> Nth power of a wetness index, which<br />
is obtained by <strong>the</strong> sumation up to <strong>the</strong> time under consideration of <strong>the</strong> proportion<br />
of <strong>the</strong> net rainfall in each previous time interval, weighted in an exponential<br />
manner according to remoteness in time. Later it is stated that this equation<br />
is subject to a constraint of continuity, that is, that <strong>the</strong> total effective<br />
rainfall is equal to <strong>the</strong> total discharge. It is not explained how khis condition<br />
is fulfil<strong>le</strong>d but if eq. 11 is taken as a statement of proportionality ra<strong>the</strong>r than<br />
of equality, giving, <strong>the</strong>refore, <strong>the</strong> relative valuaat ail times of <strong>the</strong> contributing<br />
areas, <strong>the</strong> constant of proportionality may be chosen to satisfy <strong>the</strong> requirement<br />
of continuity.<br />
This is my interpretation of <strong>the</strong> authors'intention. The quantity<br />
B would seem to be a constant loss rate existing throughout <strong>the</strong> storm. I think,<br />
perhaps, <strong>the</strong> authors might like to clarify <strong>the</strong>se few points and explain what<br />
happens if <strong>the</strong> rainfall intensity is <strong>le</strong>ss han B.<br />
The next step is to distribute <strong>the</strong> total contributing area A(7At) along <strong>the</strong><br />
channel. of <strong>the</strong> catchment. The contributing area per unit <strong>le</strong>ngth of channel<br />
At> is obtained under <strong>the</strong> following assumptions.<br />
1. Constant velocity at a given time throughout <strong>the</strong> catchment.<br />
2. Uniform distribution of drainage density.<br />
3. Uniform distribution throughout <strong>the</strong> catchment of first order streams.<br />
497
498<br />
Under <strong>the</strong>se assumptions <strong>the</strong> total contributing area at any given time<br />
is distributed according to <strong>the</strong> distance (or time of flow) from <strong>the</strong> gauging site,<br />
in th? same way as tli? number of channels is distributed according to distance<br />
from <strong>the</strong> gauging site. Thus it is possib<strong>le</strong>,on an examination of <strong>the</strong> stream<br />
system,to define <strong>the</strong> function al2 6 k) in space and time.<br />
The input of each reach, in each time e<strong>le</strong>ment, is obtained by multiplying<br />
<strong>the</strong> appropriate contributing area by <strong>the</strong> net rainfall intensity (i.e. <strong>the</strong><br />
total rainfall intensity minus <strong>the</strong> loss rate) and this input is routed through<br />
<strong>the</strong> channel system by a 1inear.method (Dooge and Har<strong>le</strong>y) to give <strong>the</strong> output<br />
at <strong>the</strong> gauging station as a function of time.<br />
Because <strong>the</strong> routing is linear <strong>the</strong> output due to <strong>the</strong> input on a given reach<br />
could be represented by a convolution integral (but <strong>the</strong> kernel may vary from<br />
reach to reach). The total output as a function of time is obtained as <strong>the</strong><br />
spatial integral of this convolution integral. The kernels <strong>the</strong>mselves vary<br />
with stream slope, a reference discharge, and a roughness parameter.<br />
To reduce <strong>the</strong> comp<strong>le</strong>xity it is proposed to replace <strong>the</strong> actual stream network<br />
by a "folded up" one in which (I believe) all stream e<strong>le</strong>ments lying <strong>the</strong> same<br />
distance from <strong>the</strong> gauging site would be assumed equal to one ano<strong>the</strong>r in <strong>the</strong><br />
properties of <strong>le</strong>ngth, roughness, slope and reference discharge. They would<br />
also agree, of course, in <strong>the</strong> depth of contributing area. It is mentioned later<br />
that <strong>the</strong> roughness and slope parameters are obtained by actual observation but<br />
it is not c<strong>le</strong>ar to me how this can be done in <strong>the</strong> idealised or "folded up" model.
The parameters of <strong>the</strong> model are:<br />
A catchment area<br />
D and N numerical parameters in eq. 1.1<br />
B <strong>the</strong> constant loss rate<br />
CZ <strong>the</strong> roughness coefficient (one parameter only)<br />
QR <strong>the</strong> reference discharge (one parameter)<br />
SL <strong>the</strong> main charnel slope (one parameter)<br />
Subsequently in <strong>the</strong> work, <strong>the</strong> parameter B was set to zero and <strong>the</strong> area<br />
49 9<br />
and slope parameters were obtained by physical measurement (I think <strong>the</strong> authors<br />
might like to explain this a litt<strong>le</strong> fur<strong>the</strong>r) <strong>the</strong> remaining parameters D,N, CZ<br />
and QR were obtained by optimisation.<br />
Details of <strong>the</strong> method are not given,<br />
nor are we told what sampling variance of <strong>the</strong> optimum values was obtained.<br />
We are told,however,that <strong>the</strong> model was insensitive to D (when D exceeded 0.5)<br />
and a fixed value of D = 0.8 was chosen. QR was found to vary only slightly<br />
between 1.1 and 1.4 cubic meters per second for rainfall values ranging from<br />
2.5. to 14 milimeters. Thus only N and CZ are <strong>le</strong>ft as free parameters.<br />
The model was applied to 13 basins in <strong>the</strong> eastern haî€ of <strong>the</strong> United<br />
'States and when.<strong>the</strong> optimised values had been obtained, relations were sought<br />
between <strong>the</strong>se and <strong>the</strong> catchment characteristics, so that <strong>the</strong>se relations could<br />
be used to provide estima'tes of <strong>the</strong> parameters values for use subsequently<br />
on wigauged catchments.<br />
N was found to vary with <strong>the</strong> ratio of runoff to rainfall volumes for <strong>the</strong><br />
storm according to authors eq. 8.<br />
# = esp (0.464- R,)/O.24L
500<br />
This ratio itself was found to vary between storms in accordance with <strong>the</strong><br />
daily temperature, an index of soil permeability, rainfall volume, and maximum<br />
in t ens it y e<br />
CZ was significantly related to basin area, stream slope and <strong>the</strong> base<br />
flow value at <strong>the</strong> time of occurence of <strong>the</strong> storm.<br />
Using <strong>the</strong>se relations between <strong>the</strong> model and <strong>the</strong> catchment to estimate<br />
<strong>the</strong> parameters of <strong>the</strong> former for insertion in <strong>the</strong> model which was subsequently<br />
fed with'<strong>the</strong> observed rainfall, good results were obtained, hydrograph peaks<br />
being reproduced with an error of <strong>the</strong> order of 20% in magnitude, and 10% in<br />
timing .
Monthly streamflow estimation from limited data<br />
by Haan.<br />
501<br />
The purpose of <strong>the</strong> exercise described in this paper is very similar to that<br />
in <strong>the</strong> paper by De<strong>le</strong>ur and Lee.<br />
monthly runoff from daily rainfall and <strong>the</strong> parameters are related to catchment<br />
characteristics by regression equations. The model is of <strong>the</strong> physical ra<strong>the</strong>r<br />
than <strong>the</strong> "black box" type according to <strong>the</strong> distinction of De<strong>le</strong>ur and Lee<br />
and according to <strong>the</strong> classification I have suggested, <strong>the</strong> prob<strong>le</strong>m is inverse<br />
non-linear lumped.<br />
A four parameter model is used to compute<br />
The structure of <strong>the</strong> model is not described but we are told that <strong>the</strong>re<br />
are four parameters.<br />
fmax- maximum infiltration rate (cm-hr)<br />
-- maximum daily seepage loss (cm)<br />
'max<br />
c -- "<strong>the</strong> water holding capacity of that part of <strong>the</strong> soil, from<br />
which <strong>the</strong> evapo-transpiration rate is <strong>le</strong>ss than <strong>the</strong> potential<br />
rate, un<strong>le</strong>ss this portion of <strong>the</strong> soil is saturated".<br />
F -- fraction of seepage that becomes runoff.<br />
s<br />
The input to <strong>the</strong> model is a series of daily rainfall values and average<br />
monthly values of potential evaporation (evapo-transpirat'ion).<br />
The optimisation is obtained by comparing computed and observed values<br />
of monthly discharges and summing <strong>the</strong> squares of <strong>the</strong> errors to obtain <strong>the</strong><br />
objective function.<br />
in turn.<br />
The search is carried out along <strong>the</strong> axis of each parameter
502<br />
The model was.app1ied to 27 catchments in Kentucky and South Carolina and<br />
a four percent average error was found in <strong>the</strong> prediction of <strong>the</strong> annual discharge.<br />
This of course is not a very efficient test of <strong>the</strong> model. Details of <strong>the</strong> results<br />
obtained are not provided - in particular <strong>the</strong> estimates of <strong>the</strong> sampling variances<br />
of <strong>the</strong> optimum parameter values are not provided.<br />
To provide for ungauged catchments,regressions were sought for <strong>the</strong> optimum<br />
values of <strong>the</strong> parameters obtained from 17 catchments on certain characteristics<br />
of <strong>the</strong>se catchments. The independent variab<strong>le</strong>s were 12 in number (see tab<strong>le</strong> 1<br />
of <strong>the</strong> paper) <strong>le</strong>aving, it would seem, only 5 degrees of freedöm, though perhaps<br />
even this is an overestimate as covarience terms appear in <strong>the</strong> regression equations<br />
It would be interesting to <strong>le</strong>arn how significant <strong>the</strong> coefficients in <strong>the</strong>se<br />
equations appear to be.<br />
Having obtained <strong>the</strong> regression equation <strong>the</strong> model was applied to six<br />
catchments not used in obtaining <strong>the</strong> regressions. The model parameters were<br />
obtained from <strong>the</strong> regressions and <strong>the</strong> runoff simulated.<br />
<strong>the</strong> total runoff for <strong>the</strong> <strong>who<strong>le</strong></strong> period varied from 1.8 to 11.8.<br />
do not indicate how <strong>the</strong> model performed over shorter periods, for examp<strong>le</strong> of<br />
one year, one month, peak flows, etc.,etc..<br />
Percentage errors for<br />
These figures<br />
On a sing<strong>le</strong> catchment <strong>the</strong> effects of different methods of parameter<br />
estimation are explored.<br />
regression equations asd a percentage error (in <strong>the</strong> total flow?) of 8.64%<br />
observed.<br />
increased this figure to 10.13% and when 2 or 3 years of records were so used<br />
figures of 2.19 and 9.38 were found.<br />
one.<br />
Firstly, <strong>the</strong> parameters are estimated from <strong>the</strong><br />
Optimisation of <strong>the</strong> parameters in <strong>the</strong> first year of record surprisingly<br />
The last optimisation was a ra<strong>the</strong>r curious<br />
The parameters were first obtained through optimisation in <strong>the</strong> first years
ecord and <strong>the</strong> remaining 21 years Of Output simulated. Next <strong>the</strong> worst two of<br />
<strong>the</strong>se years, from <strong>the</strong> point Of View Of agreement between computed and observed<br />
outputs, were noted and <strong>the</strong> parameters optimised.again,independently in <strong>the</strong><br />
records of <strong>the</strong>se two years.<br />
averages of <strong>the</strong> two sets weighted according to <strong>the</strong> sum of deviations of<br />
observed and simulated flows.<br />
The final parameters were taken as weighted<br />
using <strong>the</strong>se final values of <strong>the</strong> model parameters was made <strong>the</strong> observed<br />
error in <strong>the</strong> total discharge was only 0.56%.<br />
503<br />
When <strong>the</strong> simulation for <strong>the</strong> full period of record
504<br />
"Obtaining of Deficient Information by solving inverse prob<strong>le</strong>ms for Ma<strong>the</strong>matics<br />
Runoff Models" by Koren and Kutchment.<br />
The authÒrs'definition of an inverse prob<strong>le</strong>m is in agreement with %hat which<br />
1 have been using.<br />
prob<strong>le</strong>ms and mention <strong>the</strong> lack of uniqueness of <strong>the</strong> solutions obtained by<br />
postulating <strong>the</strong> form of <strong>the</strong> operation and adjusting <strong>the</strong> coefficient or parameters<br />
by trial and error.<br />
Tikonev which restores <strong>the</strong> proper posing of <strong>the</strong> prob<strong>le</strong>m and limits <strong>the</strong> possib<strong>le</strong><br />
variation of <strong>the</strong> solution in accordance with "a priori" information on <strong>the</strong><br />
s o lu t ion.<br />
They explain <strong>the</strong> difficulty of obtaining solutions to such<br />
Instead <strong>the</strong>y propose <strong>the</strong> application of an algorithm due to<br />
Unfortunately, I am not familiar with <strong>the</strong> sources quoted and my interpretatio<br />
of <strong>the</strong> method derives so<strong>le</strong>ly from<strong>the</strong> present paper.<br />
I would hope <strong>the</strong> authors<br />
would forgive me if I misinterpret <strong>the</strong>ir intention and I would hope that, if<br />
at all possib<strong>le</strong>, time should be provided to allow <strong>the</strong>m to correct me and explain<br />
sezral points of difficulty which I stil1,only very imperfectly,understand.<br />
The method involves <strong>the</strong> algebraic minimisation of an objective funtion and<br />
is akin to Lagrange's method of undertermined multipliers.<br />
Consider a function F(h) where h is a vector .hl,h2...and suppose<br />
we wish to minimise F(h) subject to a constraint on h, e.g.T(h) = O.<br />
Lagrange's method states that <strong>the</strong> conditional minimum of F(h) occurs at <strong>the</strong><br />
same h ao <strong>the</strong> unconditional minimum of G(h,a) where
G(h,oc) a F(h) +- 9th) (is 1<br />
A formal algebraic proof is possib<strong>le</strong> but scarcely necessary.<br />
constraint cp(i-i) = O <strong>the</strong> functions G(h,*) and F(h)<br />
Along <strong>the</strong><br />
are identical and <strong>the</strong>refore<br />
<strong>the</strong>ir (conditional) minima agree. But <strong>the</strong> unconditional minimum of G(h,a)<br />
obtained by differentiating G(h,ch) with respect to h a ndu and simultaneously<br />
equating <strong>the</strong> derivatives to zero, implies q(h) = O or <strong>the</strong> general (unconditional)<br />
and conditional minima of G(h,a) agree.<br />
of C(h,5) gives <strong>the</strong> value of h which corresponds <strong>the</strong> conditional minimum of F(h).<br />
An optimum value for a is also found.<br />
to be an adaptation ra<strong>the</strong>r than <strong>the</strong> straightforward use of this method. A<br />
series of values of <strong>the</strong> vector h which minimise G(h,@) for a series of values<br />
of agradually increasing from zero toward <strong>the</strong> optimum (SC are found by<br />
opt<br />
differentiating. The first of <strong>the</strong>se vectors h (corresponding to = O) corresponds<br />
505<br />
to <strong>the</strong> unconditional minimum of F(h). The last (corresponding toca*<br />
opt<br />
corresponds to <strong>the</strong> conditional minimum (i.e. to <strong>the</strong> constraint fully imp<strong>le</strong>mented)<br />
and <strong>the</strong> intermediate solutions correspond to <strong>the</strong> partial imp<strong>le</strong>mentation of <strong>the</strong><br />
constraint.<br />
Consequently <strong>the</strong> unconditional minimum<br />
The method used by <strong>the</strong> authors would seem<br />
in this way <strong>the</strong> investigator is enab<strong>le</strong>d to seek about in <strong>the</strong> vicinity of <strong>the</strong><br />
optimum h for one which provides a reasonab<strong>le</strong> compromise between satisfying <strong>the</strong><br />
constraints and minimising <strong>the</strong> function.<br />
in <strong>the</strong> three examp<strong>le</strong>s quoted in <strong>the</strong> paper <strong>the</strong> physical prob<strong>le</strong>m is reduced,<br />
in one case after <strong>the</strong> application of much ingenuity, to <strong>the</strong> solution of a set of<br />
linear algebraic eqs.<br />
4<br />
Q =AZ (I 6)<br />
Where Q and h are vectors and A a rectangular matrix. Assuming redundancy among<br />
<strong>the</strong> equations a <strong>le</strong>ast squares solution could be found by minimising<br />
F(h) 5 /I Ph- all"<br />
(I 7J
506<br />
As shown by Snyder, <strong>the</strong> solution of this equation is<br />
A*Ah = A*Q (12)<br />
or h = (A*A)-l A*Q (1 9)<br />
In <strong>the</strong> inverse prob<strong>le</strong>m, in <strong>the</strong> hydrological context (e.g. h is <strong>the</strong> impulse<br />
response or <strong>the</strong> input to a linear system) this equation is often badly conditioned<br />
and <strong>the</strong> h obtained may be seriously distorted by small errors..in Q or A.<br />
In particular, h may fail to conform to some physical requirements (e.g. unit<br />
area or smoothness of <strong>the</strong> impulse response). The authors method is to minimise<br />
i.e., to find h from<br />
(A*A+ aE)h = A* Q (where i? is <strong>the</strong> unit matrix)<br />
with prese<strong>le</strong>cted values of e(presumab1y increasing from zero.<br />
Obviously <strong>the</strong> smal<strong>le</strong>r <strong>the</strong> value of q <strong>the</strong> more /lh112 is permitted to increase<br />
and <strong>the</strong>refore increasing q corresponds to increasing <strong>the</strong> permissib<strong>le</strong> fluctuation<br />
in h.<br />
Presumably a se<strong>le</strong>ction is <strong>the</strong>n made between <strong>the</strong> several h, bearing in<br />
mind that <strong>the</strong> nearer oi is to zero <strong>the</strong> nearer h is to <strong>the</strong> <strong>le</strong>ast squares solution<br />
of <strong>the</strong> equation.<br />
It is c<strong>le</strong>ar that <strong>the</strong> constraint need not be precisely stated. It is sufficiei<br />
that <strong>the</strong> coefficient of arepresents some quantity which increases with <strong>the</strong><br />
undesirab<strong>le</strong> property of h.<br />
<strong>the</strong> Lagrangian method would probably be <strong>the</strong> better.<br />
If <strong>the</strong> constraint can be precisely stated,e.g.rh = 1,<br />
Of <strong>the</strong> three examp<strong>le</strong>s quoted by <strong>the</strong> authors, <strong>the</strong> first involves finding <strong>the</strong><br />
effective rainfall input given <strong>the</strong> impulse response and <strong>the</strong> discharge.<br />
prob<strong>le</strong>m as we have seen is identical (even in <strong>the</strong> constraint) to that of finding<br />
<strong>the</strong> unit hydrograph given <strong>the</strong> input and output. The authors mentioned various<br />
The
constraints includkigrh = 1 which would yield<br />
9 =Ikh-QI\ + aZh<br />
and a smoothness constraint /h/ yielding<br />
=Ih-Q/I + ec )I hl<br />
Straightforward application of Lagrange s method would of course yield h = O<br />
which would be use<strong>le</strong>ss. The solutions for <strong>le</strong>sser values of &would permit h # O<br />
2<br />
whi<strong>le</strong> restraining /h// .<br />
The second prob<strong>le</strong>m discussed by <strong>the</strong> authors is that of discovering <strong>the</strong><br />
coefficients of <strong>the</strong> Saint-Venant eqs. for flow-in open channels<br />
- .II<br />
;; = $+$*&(e)<br />
as +Fk<br />
+i&($)<br />
507<br />
(d<br />
= 0 )<br />
This prob<strong>le</strong>m arises in, for examp<strong>le</strong>, flood routing, where it is impracticab<strong>le</strong><br />
to measure <strong>the</strong> conveyance and areal relations K(z,x) and F(z,x) for each<br />
cross section.<br />
from observations made on <strong>the</strong> discharges and water<strong>le</strong>vels as functions of space<br />
and time, Q=Q(x,t) and z=z(x,t), during <strong>the</strong> passage of a particular flood.<br />
ûnce.<strong>the</strong>se relations have been established <strong>the</strong>y may be used directly in subsequent<br />
routing operations.<br />
Instead, smoo<strong>the</strong>d values of <strong>the</strong>se functions may be obtained<br />
The continuity equation integrated with respect to x, provides<br />
Q(x, t >-Q(o, t 1 = & F (i,, t 1%<br />
where 7) is a dummy variab<strong>le</strong> along <strong>the</strong> <strong>le</strong>ngth x.<br />
In ?finite difference form,<br />
this equation applied to a reach of channel from K=O to K=i, becomes<br />
f$ (j+l, k 1 + F (j+l, k+l 1-F (j , k 1-F ( j , k+l)]<br />
I< :<br />
It.<br />
~~Q:q(j+l,o) + Q(j+i,i) - Q(j,il-Ki,od<br />
where j and k refer to time and space, respectively.
508<br />
The authors state that this eq. maybe arranged as<br />
-?+<br />
AF = Q<br />
one such equation existing for each discrete time and <strong>the</strong> vectors running, as it<br />
were, along <strong>the</strong> channel.<br />
It would seem that 2); has been omitted, but even allowing for this, I cannc<br />
express eq. 24 in this form. Nor does it seem to me that eq. 25 is redundant.<br />
It would certainly be interesting to have this point c<strong>le</strong>ared up, but I think we<br />
can all accept that <strong>the</strong> finite difference equation can somehow be reduced to a<br />
set of linear equations between <strong>the</strong> changes in time in F(x) and in distance iii<br />
Q(>r). Such a set of equations would apply for one instant only and would take<br />
<strong>the</strong> form of<br />
The authors apply <strong>the</strong> algorithm already expiained,with <strong>the</strong> constraint that<br />
Ik-FOIr, where F is an initial estimate of F<br />
0-<br />
0)<br />
is minimised for<br />
With regard<br />
chosena's by solving<br />
Q@,F,) = //AF-Q 11 +a IIP-gI<br />
(A*A+aE )F=A*Q+ a EP<br />
is kept small,<br />
(27<br />
to choosinga <strong>the</strong> authors mention <strong>the</strong> "method of discrepancy" whit<br />
I don't quite understand, nor can I see how <strong>the</strong> smooth variation of P with time<br />
can be insured, as P(x) seems to De found independently for each time step.'<br />
Perhaps in calculating F <strong>the</strong> values of F obtained .in <strong>the</strong> previous time step may bc<br />
used in <strong>the</strong> algorithm for F and thus, by constraining //F-Fo1\2 <strong>the</strong> change in F<br />
O<br />
is distributed regularly over all x's.<br />
Having found, P(x,t), thus, from <strong>the</strong> continuity equation, and having z(x,t)<br />
C=
already, F(x,z) can be found.<br />
The dynamic equation is similarly used to find K(x,t) and hence K(x,z).<br />
Details are not given by <strong>the</strong> authors but <strong>the</strong> computation would seem to be<br />
quite independent of <strong>the</strong> Computation of P(x,t,).<br />
in <strong>the</strong>ir thikd and final exmp<strong>le</strong> <strong>the</strong> authors deal with <strong>the</strong> same equations<br />
but Pith different boundary conditions.<br />
discharges are known as functions of time, only at <strong>the</strong> beginning and <strong>the</strong> end<br />
of <strong>the</strong> channel reach, i.e. Q(o,t) and Q(L,t).are known.<br />
z(x,t) is known.<br />
case.<br />
This time <strong>the</strong>y assume 'that <strong>the</strong><br />
509<br />
They assume also that<br />
These conditions are more parsimonious than in <strong>the</strong> former<br />
The difference between <strong>the</strong> discharges at <strong>the</strong> ends of <strong>the</strong> channel reach<br />
is related to <strong>the</strong> rate afincrease in storage in <strong>the</strong> channel by <strong>the</strong> continuity<br />
The right hand side is a sing<strong>le</strong> known quantity for sach t he step and may<br />
thus be expressed as a vector in time.<br />
The <strong>le</strong>ft hand side, because of <strong>the</strong><br />
integration with respect to x, is also a vector in th(unknown).<br />
Assume that P(x,t) can be expressed as a smooth function of space and<br />
time by a Chebishev polynomial.<br />
Ex;>anded, this would be an ordinary polynominal in x and z and in xz with<br />
constant coefficients depending on Aks.
510<br />
To evaluate <strong>the</strong> coefficients, P(x,z) could be inserted in eq.29 but as this<br />
is a difference equation Li P, <strong>the</strong> solution would be underdetermined at <strong>le</strong>ast<br />
to <strong>the</strong> extent of <strong>the</strong> arbitrary constant. Instead of using F, <strong>the</strong> authors use<br />
<strong>the</strong> top width B(x,z) and.expand this in x and z as<br />
I€ <strong>the</strong>re are m by n t e m in <strong>the</strong> expansion of B(x,z),<strong>the</strong>re will be m by n<br />
unknown coefficients A and,<strong>the</strong>refore,at <strong>le</strong>ast this number of equations in<br />
ks<br />
<strong>the</strong> form of eq.29 must be found. This can be done by taking sufficient time<br />
intervals in <strong>the</strong> rising and falling hydrograph and evaluating <strong>the</strong> right hand<br />
side of eq.29 accordingly to yield xl,x2,X3, .<br />
For every k and s <strong>the</strong> quantity<br />
is known for every.x and t and <strong>the</strong> integral with respect to x can <strong>the</strong>refore be<br />
found.<br />
Hence <strong>the</strong> coefficients of every A in eq.29 after substitution<br />
k.9<br />
can be written down yielding.<br />
%st 'ks<br />
(3 21<br />
= Xt and this can be arranged in matrix form, if (33)<br />
necessary, and I think it would be necessary, with Akis written in vector form<br />
A,l, Ak2, Ak3, ............. Thus <strong>the</strong> linear equation in <strong>the</strong> unknown A would<br />
ks<br />
be obtained as<br />
0 Q =x 9<br />
where $ is <strong>the</strong> matrix of <strong>the</strong> coefficients Cks in eq. 33 and <strong>the</strong> solution for<br />
9<br />
8 <strong>the</strong> vector of unknown A,s found by application of <strong>the</strong> algorithm.<br />
(Sf 1<br />
(Q* FBI0 = 9*x @a<br />
In this case, <strong>the</strong> constraint imposed is 191 small. The choice afa(and<br />
<strong>the</strong>refore of e) seems to be made at <strong>the</strong> value ofawhere a fur<strong>the</strong>r change<br />
inawould produce only a minhum change in 8 expressed by eq.36.
Y<br />
(c = 2 p(QP+,) -<br />
J- I<br />
Once F(x,z) has been determined, and remembering that we have already<br />
z(x,t), K(x,z) can be found from <strong>the</strong> dynamic equation simplified to<br />
and, thus, all <strong>the</strong> parameters of <strong>the</strong> equation are availab<strong>le</strong>. I cannot quite<br />
follow <strong>the</strong> authors explanation of this part of <strong>the</strong> project.<br />
511<br />
be some subt<strong>le</strong>ties here which I am missing,but I can see no particular difficulty<br />
in solving it along <strong>the</strong> lines I have indicated.<br />
There may<br />
This is an extremely interesting though difficult paper and I hope <strong>the</strong><br />
authors will be availab<strong>le</strong> to correct iy very inadequate exposition of it and<br />
perhaps resolve some of <strong>the</strong> difficulties which I have mentioned and o<strong>the</strong>rs<br />
which may be tròuhling o<strong>the</strong>r col<strong>le</strong>agues.<br />
References<br />
(1) Snyder, W., Tennessee Val<strong>le</strong>y Authority (1961)<br />
"Matrix operations in hydrograph computations"<br />
(2) O'Donnell T., (1960) 'Instantaneous unit hydrograph derivation by harmonic<br />
analysis'IASH (Helsinki) Pub No 51<br />
(3) Kraijenhof van de Leur, D.A., "A study of non-steady ground water flow with<br />
special reference dto a reservoir coefficient"<br />
De Ingenieur , 70(19 1 (1 93 8 1.<br />
(4) Venetis C. "Estimating infiltration and/or <strong>the</strong> parameters of unconfined<br />
aquifers from ground water <strong>le</strong>vel observations" Jour Hyd. 12(1971)
ABSTRACT<br />
DONNEES INADEQUATES ET MODELES MATHEMATIQUES<br />
DE LA POLLUTION EN RIVIERE<br />
Par J.BERNIER<br />
Laboratoire National d'Hydraulique<br />
CHATOU - France<br />
The value of I'inadequate1l information must be judged<br />
relatively to <strong>the</strong> ma<strong>the</strong>matical tools used and <strong>the</strong> practical prob<strong>le</strong>m<br />
to be solved, Concerning river pollution where <strong>the</strong> prob<strong>le</strong>m is to<br />
design projects as sewage treatment plants for instance, <strong>the</strong><br />
limitation of <strong>the</strong> usual Information col<strong>le</strong>cted in situ is shown,<br />
These limitations do no appear with <strong>the</strong> standard math.ematjca1 model.<br />
Taking in account of more realistic stochastic model allows us to<br />
measure <strong>the</strong> value of this information and to design experiment for<br />
col<strong>le</strong>cting acceptab<strong>le</strong> data,<br />
RESUME<br />
La va<strong>le</strong>ur de l'information inadequate doit être jugee en<br />
fonction des problèmes à rbsoudre et des outils mathématiques utili-<br />
sês pour cette résolution, En matière de pollution en rivière où il<br />
s'agit de dêfinir <strong>le</strong>s caractdristiques des moyens de lutte comme<br />
cel<strong>le</strong>s des stations dlspuration par exemp<strong>le</strong>, on montre <strong>le</strong>s limita-<br />
tions de l'information usuel<strong>le</strong>ment recueillie in situ, Ces limita-<br />
tions n'apparaissent pas avec <strong>le</strong>s modè<strong>le</strong>s mathgrnatiques standard,<br />
La prise en compte de modè<strong>le</strong>s stochastiques plus réalistes permet<br />
de mesurer la va<strong>le</strong>ur de cette information et de definir <strong>le</strong>s condi-<br />
tions de col<strong>le</strong>cte de données acceptab<strong>le</strong>s,
51 4<br />
I - INTRODUCTION<br />
En matière d'aménagement des ressources en eau, l'insuffisance<br />
des données est souvent <strong>le</strong> premier écueil auquel on se heurte. Cependant<br />
pour mieux apprécier la validité et la précision des réponses aux questions<br />
posées à l'hydrologue ou l'ingénieur, il faut noter que cette insuffisance<br />
de données n'est en fait que <strong>le</strong> ref<strong>le</strong>t de l'inadéquation des inhthodes uti-<br />
lisées pour résoudre <strong>le</strong>s problèmes. Ces méthodes doivent etre adaptées ><br />
la nature de l'information disponib<strong>le</strong> ou 2 recueillir. Certes dans bien<br />
des cas <strong>le</strong>s données disponib<strong>le</strong>s doivent être coiilplétées mais l'organisa-<br />
tion de la col<strong>le</strong>cte des données complémentaires, <strong>le</strong> choix des conditions<br />
opératoires de mesures ne peuvent valab<strong>le</strong>ment être &finis qu'en fonction<br />
des méthodes iiiobilisant l'inforination recueillie. Dans ce contexte, certai-<br />
nes iitéthodes usuel<strong>le</strong>s sont particulièreiiient inadéquates. On peut en trouver<br />
des exemp<strong>le</strong>s dans <strong>le</strong> domaine de la pollution en rivière notamment dans<br />
l'étude du inouverrient et des réactions auxquels sont soumises des matières<br />
polluantes en riviere 2 l'aval d'un point de rejet en vue d'apprécier la<br />
capacité d'autoépuration de la rivière compte tenu de ce rejet. Nous trai-<br />
terons ici du seul problème de la pollution biochiinique'caractbris6e par<br />
<strong>le</strong> bilan d'oxygène.<br />
II - LA METHODE USUELLE<br />
La méthode classique utilise <strong>le</strong> modè<strong>le</strong> de Streeter et Phelps<br />
décrivant <strong>le</strong> bilan dynamique d'oxygène sous la forme de deux équations<br />
différentiel<strong>le</strong>s (voir la liste de notations en fin de note).<br />
J
515<br />
Mises sous forme intégra<strong>le</strong> et en faisant apparaître l'abscisse<br />
longitudina<strong>le</strong> x prise <strong>le</strong> long de la rivière et liée au teiiips d'écouìe-<br />
2<br />
nient t et à la vitesse moyenne du courant u pa; t = - , <strong>le</strong>s équa-<br />
tions donnent :<br />
L(x) = Lo e<br />
X<br />
-K3 u<br />
X<br />
- K -<br />
B (x) =ao e ~ ~ ( 2 u -e e<br />
OÙd (x) est <strong>le</strong> déficit en oxygène : 3 = Cs - C.<br />
U<br />
'J - Kg U)<br />
Ainsi peut-on mesurer l'incidence d'un rejet (spécifiant <strong>le</strong>s<br />
concentrations initia<strong>le</strong>s d'oxygène Co et de demande biologique en<br />
oxygène L ) sur l'autoépuration à l'aval.<br />
La mise en oeuvre courante de ce iiiodè<strong>le</strong> demande l'estimation<br />
préalab<strong>le</strong> des coefficienb de reoxygénation K2 et de biodégradation K<br />
1<br />
et K3.<br />
On utilise généra<strong>le</strong>inent des formu<strong>le</strong>s empiriques ou quelques observations<br />
recueillies dans <strong>le</strong> tronçon de rivière étudié. Les multip<strong>le</strong>s formu<strong>le</strong>s<br />
empiriques disponib<strong>le</strong>s établies en laboratoire ou sur des rivières de<br />
caractéristiques biochimiques particulieres présentent des résultats<br />
extre'mement dispersés diffici<strong>le</strong>ment extrapolab<strong>le</strong>s hors des limites du<br />
doniaine où el<strong>le</strong>s ont été établies. I1 reste l'inforniation recueillie<br />
dans chaque cas d'espèce. A la limite Kg et K2 peuvent ëtre calculés<br />
par l'intermédiaire des formu<strong>le</strong>s (2) en fonction d'un seul coup<strong>le</strong> d'ob-<br />
servations amont (Lo, Co) et d'un seul coup<strong>le</strong> aval ( L(x), C(x)). Une<br />
tel<strong>le</strong> procédure, trop souvent utilisée pratiquement, est justifiée dans<br />
<strong>le</strong> contexte du modè<strong>le</strong> déterministe strict décrit par (2) mais ce carac-<br />
tère déterministe est extrêmement fallacieux come nous allons <strong>le</strong> voir.<br />
Par ail<strong>le</strong>urs il importe de se soucier de la cohérence spatia<strong>le</strong> des mesures,<br />
<strong>le</strong> décalage des époques d'observations a l'amont et à l'aval devraient<br />
tenir compte du temps d'écou<strong>le</strong>ment t . En pratique cette condition iinpé-<br />
rative est rarement respectée et la non prise en compte de la menie masse<br />
d'eau 2 l'amont et à l'aval entrarne une dispersion notab<strong>le</strong> des observations<br />
et des estimations erronées de KI, K2 et Ka.
516<br />
III - INSUFFISANCES DU MODELE DETERMINISTE DE STREETER ET PHELPS<br />
Le modè<strong>le</strong> (i), (2) schéiuatise l'hydraulique de l'écou<strong>le</strong>ment :<br />
celui-ci est supposé permanent, uniforme ; de plus on néglige la disper-<br />
sion de polluants dissous ou en suspension dans l'eau imputab<strong>le</strong> au phéno-<br />
mène de diffusion turbu<strong>le</strong>nte et aux fluctuations de vitesse dans chaque<br />
section. Cependant l'effet de cette dispersion est surtout notab<strong>le</strong> en<br />
régime transitoire et devient négligeab<strong>le</strong> en régime de pollution permanent<br />
ou lorsque cette pollution (caractérisée par L et CI présente une évo-<br />
lution <strong>le</strong>nte, d ms <strong>le</strong> cas de rivière à coefficient de dispersion faib<strong>le</strong><br />
(cf. [i] et [2] ).<br />
L'inadéquation en nature du modè<strong>le</strong> de Streeter et Phelps provient<br />
surtout des hypothèses caractérisant <strong>le</strong>s phénomènes biochimiques :<br />
- prise en compte de la seu<strong>le</strong> phase carbonée des réactions de<br />
dégradation des matières organiques (non prise en compte de<br />
la phase azotée) ;<br />
- non prise en compte des effets de la photosynthèse de la respi-<br />
ration des boues de fonds, de la sédimentation des matières<br />
polluantes etc ...<br />
Certaines tentatives [a] ont été faites pour inventorier et<br />
modeliser plus comp<strong>le</strong>tement <strong>le</strong>s phénomènes mais la multiplicité des para-<br />
. .<br />
mètres et <strong>le</strong>s difficultés pratiques d'estimation de ces paraniètres rendent<br />
illusoire l'apparente précision qui semb<strong>le</strong>rait résulter d'un inventaire<br />
exhaustif des mécanismes biologiques et physico-chimiques.<br />
Par ail<strong>le</strong>urs si <strong>le</strong>s concentrations en oxygène dissous peuvent<br />
être mesurées in situ avec une précision acceptab<strong>le</strong>, il n'en est pas de<br />
même de la demande biologique en oxygène qui n'est pas estimée directement<br />
mais seu<strong>le</strong>ment par 1 l intermédiaire d'un test chiriiique, la DB05, effectué<br />
au laboratoire sur des échantillons pré<strong>le</strong>vés en rivière. Des essais ont<br />
montré que l'imprécision de ce test est notab<strong>le</strong> et que la chaîne comp<strong>le</strong>xe<br />
des conditions opératoires, depuis <strong>le</strong> prélèvement en riviere jusqu'a<br />
1 'analyse chimique en laboratoire introduit des erreurs systématiques et<br />
aléatoires importantes. On ne peut donc considérer cette DBO comme une<br />
5<br />
mesure exacte de la consommation d'oxygène in situ mais comme un index<br />
représentatif de cette consommation en plus ou moins bonne comélation<br />
statistique avec el<strong>le</strong>.
IV - UN MODELE STOCHASTIQUE<br />
517<br />
I1 est classique en statistique de prendre en compte globa<strong>le</strong>ment<br />
l'ensemb<strong>le</strong> des phénomènes négligés dans un modè<strong>le</strong> déterministe schématique<br />
sous foriiie determes d'erreurs aléatoires. Cette conception permet d'intro-<br />
duire la soup<strong>le</strong>sse nécessaire à une bonne adéquation du modè<strong>le</strong> aux données<br />
d'observations en nature. Les équations (1) sont remplacées pour un sys-<br />
teme différentiel stochastique :<br />
Les paramètres pl et p2 représentent <strong>le</strong>s irioyennes, c'est-à-dire la<br />
2<br />
part systématique. des erreurs dues aux phénoiiiènes négligési o12 et u2<br />
représentent <strong>le</strong>s variances des termes d'erreurs. E, et .C2 sont alors<br />
<strong>le</strong>s erreurs centrées réduites (de moyenne nul<strong>le</strong> et de variances éga<strong>le</strong>s à 1).<br />
Dans ce contexte aléatoire, <strong>le</strong>s paramètres K1, K2, K3 peuvent<br />
être interpréter coime <strong>le</strong>s coefficients de la régression statistique des<br />
variations de concentrations en fonction des grandeurs<br />
L etd et ils<br />
ont une signification statistique plutôt que physique. On peut alors rem-<br />
placer l'index DB05 par tout autre index qui soit en corrélation avec<br />
la demande en oxygène (par exemp<strong>le</strong> : demande chiiiiique en oxygène, carbone<br />
organique total, etc... indexes qui sont plus aisément mesurab<strong>le</strong>s que la<br />
DB05). I1 est possib<strong>le</strong> d'intégrer <strong>le</strong> système ( 3) (cf. [ 41 1. En adinettant<br />
des conditions initia<strong>le</strong>s fixées et go au point origine x = O, on<br />
LO<br />
peut montrer que L(t) et3 (t) sont des variab<strong>le</strong>s aléatoires quasi-<br />
gaussiennes dont <strong>le</strong>s espérances niathémat iques et variances sont :<br />
E (L) =<br />
- P2 + (Lo - -) P2 e<br />
K2 Kg<br />
- K3 t<br />
E(d)=-t- Pi<br />
K1<br />
P2<br />
K2 Kg<br />
pci +- '1<br />
- K3-K2<br />
P2<br />
(Lo - -)I<br />
K2<br />
-Kpt<br />
e<br />
+-í-<br />
K1 P2<br />
- K3t<br />
-Lo) e<br />
K3-K2 K3<br />
(4)<br />
(5)
v -<br />
518<br />
Var (LI =<br />
- 2 K3t<br />
v22 (i - e 1<br />
2 2<br />
2 u2 t ui r,<br />
Var ($1 = ( u1 +<br />
2<br />
(K~-K~) K3-K2<br />
formu<strong>le</strong>s dans <strong>le</strong>squel<strong>le</strong>s<br />
erreurs Cl et E,.<br />
K3<br />
- 2 K t<br />
(i - e 3 )<br />
Kg<br />
- 2 K,t<br />
(i - e 1<br />
K2<br />
K1<br />
(u u r+-<br />
K3-K2 1 2 K3-K2<br />
(i - e 1<br />
Kg +<br />
K2<br />
(7)<br />
r est <strong>le</strong> coefficient de corrélation entre <strong>le</strong>s<br />
LE MODELE STOCHASTIQUE AVEC ERREURS DE MESURES SUR L et d<br />
Connie nous l'avons déjà souligné, 1' inadéquation du iiiodè<strong>le</strong> aux<br />
données in situ est liée éga<strong>le</strong>ment aux erreurs de mesures importantes sur<br />
ces données. Pour analyser plus complèteinent <strong>le</strong> problèiiie, il importe de<br />
préciser <strong>le</strong>s conditions d'observations. Nous ne traiterons ici que de la<br />
méthode usuel<strong>le</strong> OU l'on observe deux points amont x et aval distants<br />
de x = - xcI.<br />
écrit :<br />
Compte tenu des foririu<strong>le</strong>s (4) à (7) <strong>le</strong> modè<strong>le</strong> intégré peut être<br />
en regroupant sous forine des constantes a et b <strong>le</strong>s termes indépendants<br />
des conditions initia<strong>le</strong>s dans <strong>le</strong>s espérances inathématiques et sous forme<br />
des constantes p1, p2, p3 <strong>le</strong>s coefficients de Lo et do. Les variab<strong>le</strong>s<br />
aléatoires d'écart EL et I!& ont alors des variances u et u2 données<br />
L<br />
par <strong>le</strong>s formu<strong>le</strong>s (6) et (7) et qui sont donc indépendantes de ces conditions<br />
2<br />
=2
initia<strong>le</strong>s. En fait ce que l'on observe n'est pas directement<br />
mis deux grandeurs X et Y tel<strong>le</strong>s que :<br />
X=L+ 7)<br />
1<br />
Y =a + 7)<br />
2<br />
519<br />
L ou 3<br />
où Il e; 7)2 sont <strong>le</strong>s erreurs de mesures aléatoires de variances respec-<br />
2<br />
tives VL et Va . L'estiiriation in situ des parainetres du modè<strong>le</strong> stochas-<br />
tique et notanunent des vitesses de reoxygénation et de biodégradation<br />
K2 '<br />
K et K3 , demande la connaissance préalab<strong>le</strong> des variances d'erreurs de<br />
1<br />
mesures. De façon précise on supposera qu'ont été obtenus n ensemb<strong>le</strong>s<br />
de grandeurs observab<strong>le</strong>s (Xoy X1, Yo, Y<br />
1<br />
) aux deux points amont et aval<br />
du tronçon de rivière considérée. Les paramètres statistiques de ces cou-<br />
_ - - -<br />
2 2 2 2<br />
p<strong>le</strong>s (moyennes Xo, XI' Yo YI , variances S x0, S x13 S yo, S y1 , et<br />
covariances<br />
S&xl3 Sx0y1, etc ... ) permettent l'estimation des coefficients<br />
du inode<strong>le</strong> au moyen des relations :<br />
P, = sxlxo<br />
2 2<br />
SX, - VL<br />
- -<br />
a = X1 - p 3 Xo<br />
(10)<br />
(12)
520<br />
VI - LE PROBLEME DES ERREURS D'ECHANTILLONNAGE<br />
Dans la plupartdes cas pratiques l'estimation des paramètres du<br />
modè<strong>le</strong> d'oxygène ne peut être effectuée que sur un nombre<br />
n de répéti-<br />
tions d'observations assez liiriité. I1 importe alors de iiiesurer la précision<br />
de ces estimations par <strong>le</strong>urs variances d'échantillonnage. Nous ne pouvons<br />
ici développer l'ensemb<strong>le</strong> des formu<strong>le</strong>s, nous renvoyons <strong>le</strong> <strong>le</strong>cteur à [2]<br />
pour un aperçu sur <strong>le</strong> problème. En expriniant <strong>le</strong>s divers paramètres comme<br />
des fonctions des variances et covariances<br />
<strong>le</strong>s formu<strong>le</strong>s précédentes peuvent permettre <strong>le</strong> calcul approché de ces va-<br />
riances d'échantillonnages à partir de cel<strong>le</strong>s des variances et covariances<br />
estimées (cf. [ 51 ). En ce qui concerne notaniiiient <strong>le</strong>s paramètres pi qui<br />
déterminent <strong>le</strong>s vitesses des réactions biochiiriiques , on aura des formu<strong>le</strong>s<br />
de la foriiie :<br />
VI1 - VALEUR DE L'INFORMATION RECUEILLIE EN NATURE<br />
S2<br />
XO<br />
2<br />
S yl, Sxoxl, etc ...<br />
Pour la suite de la discussion il est conimode d'appe<strong>le</strong>r variances<br />
2<br />
d'erreurs d'adéquation du inode<strong>le</strong> <strong>le</strong>s paramètres uL , mg2 car <strong>le</strong>ur va<strong>le</strong>ur<br />
est liée à l'importance de l'explication des variations d'oxygène dissous<br />
par <strong>le</strong>s paramètres<br />
C, DB05 ... pris en compte. L'interprétation des<br />
variances d'échantillonnages, perinet de préciser <strong>le</strong> noiiibre d'observations n<br />
nécessaires à l'obtention d'une précision d'estiniation donnée ; on pourra<br />
observer généra<strong>le</strong>ment la loi généra<strong>le</strong> de l'augmentation de n en fonction :<br />
- des va<strong>le</strong>urs croissantes de l'erreur d'adéquation .<br />
- des va<strong>le</strong>urs croissantes de l'erreur de mesure.<br />
Quels que soient <strong>le</strong>s paramètres de pollution pris en compte, quel<strong>le</strong>s que<br />
soient <strong>le</strong>s procédures opératoires de mesures, il restera toujours des<br />
erreurs d'adéquation et de mesure irréductib<strong>le</strong>s. Dans de tel<strong>le</strong>s circons-<br />
tances, on ne peut utiliser <strong>le</strong>s procédures classiques d'estimation des<br />
modè<strong>le</strong>s d'oxygène supposés déterministes et qui n'utilisent que des infor-<br />
mations trop partiel<strong>le</strong>s. réduites trop souvent 2 un unique ensemb<strong>le</strong> des<br />
4 va<strong>le</strong>urs Xo, Yo, XI, Y1. I1 est absolument indispensab<strong>le</strong> de faire des<br />
mesures répétitives en nombre n suffisant. Le modè<strong>le</strong> stochastique est
alors un guide précieux pour la planification de la col<strong>le</strong>cte de cette<br />
information et des procédures opératoires de mesures.<br />
521<br />
Placé devant un problème de décision, <strong>le</strong> gestionnaire de la qua-<br />
lité de l'eau d'une rivière aura donc a sa disposition des observations<br />
cohérentes recueillies in situ ; mais ce n'est pas la seu<strong>le</strong> source d'in-<br />
formations disponib<strong>le</strong>. Le gestionnaire dispose éga<strong>le</strong>ment de données plus<br />
ou moins qualitatives sur <strong>le</strong>s vitesses de réactions, de forinu<strong>le</strong>s semiempiriques<br />
diverses [ 61 dont la dispersion des résultats est tel<strong>le</strong> qu'el<strong>le</strong><br />
ne peut. donner que des ordres de grandeurs assez grossiers. Une tel<strong>le</strong><br />
information est cependant précieuse si el<strong>le</strong> permet de réduire <strong>le</strong> nornbre<br />
d'observations in situ. La disposition d'un modè<strong>le</strong> stochastique permet<br />
l'utilisation des méthodes bayésiennes [7] , [a] dont <strong>le</strong> but est l'incor-<br />
poration des inforiiiations de diverses origines dans un modè<strong>le</strong> quantifié.<br />
Davis, Kisiel et Duckstein [E] ont montré tout l'intérêt de ces techniques<br />
appliquées ?i l'étude des risques,associées aux décisions en matière de<br />
gestion des ressources en eau et au calcul de la va<strong>le</strong>ur économique de<br />
l'information hydrologique qui tend à réduire ces risques. Ia prise en<br />
compte d'un modè<strong>le</strong> stochastique est la première étape de l'approche déci-<br />
sionnel<strong>le</strong> dans <strong>le</strong>s problèmes de qualité de l'eau.
522<br />
c :<br />
N O T A T I O N S<br />
concentration en oxygène dissous,<br />
concentration en oxygène dissous 2 la saturation<br />
Cs - C déficit en oxygène<br />
DB05 : demande biologique en oxygène mesurée au laboratoire sur 5 jours<br />
à la température de 2OoC sur un échantillon supposé représentatif<br />
de la rivière<br />
K1 : coefficient de consoinmation d'oxygène dans <strong>le</strong> modè<strong>le</strong> de Streeter -<br />
Phelps<br />
coefficient de réoxygénation<br />
K2 :<br />
K3 : coefficient de dégradation de la DBO restante<br />
L : deniande biologique en oxygène restante<br />
x : abscisse longitudina<strong>le</strong> de la rivière<br />
u : vitesse moyenne de l'écou<strong>le</strong>ment.<br />
moyenne par section de rivière<br />
L
B I B L I O G R A P H I E<br />
[i ] W.E. DOBBINS : BOD and Oxygen relationships in streams<br />
Froc. ASCE Sanit Div. S A 3 - 1964<br />
[2]<br />
523<br />
J. BERNIER - P. LENCIONI : Utilisation d'un modè<strong>le</strong> stochastique pour<br />
organiser 13 col<strong>le</strong>cte in situ des données de qualité<br />
de l'eau d'une rivière - 15ème Congrès de 1'A.I.R.H.<br />
Ictainboul 1973.<br />
[3] D. LEFORT : Modè<strong>le</strong>s mathérriatiques de pollution en riviere -<br />
La Houil<strong>le</strong> Blanche - nuiiiéro spécial 8/1971<br />
[4] D.R. COX - H.D. MILLER : The <strong>the</strong>ory of stochastic processes<br />
Methuen - 1965<br />
[ 5 ] T.W. ANDERSON : An introduction to multivariate statistical analysis<br />
Wi<strong>le</strong>y - 1958<br />
[6]<br />
M. NEGULESCU - V. ROJANSKI : Recent Research to determine reaeration<br />
[ 71 J. BERNIER<br />
[ 81<br />
coefficient - Water Research - Vol 3 no 3 - 1969<br />
: Les méthodes bayésiennes en hydrologie statistique.<br />
Froc. Intern. Hydrology Symp. Colorado State Univer-<br />
sity - Fort Collins - 1967<br />
D.R. DAVIS - C.C. KISIEL - L. DUCKSTEIN : Bayesian Decision Theory<br />
applied to design in Hydrology - Water Resources<br />
Research - Vol 8 no 1 - 1972.
"REGIONAL GROUNDWATER RECHARGE ESTIMATES VIA METEOROLOGICAL DATA"<br />
ABSTRACT<br />
SAMUEL P.COOK AND SAMUEL G,MBURU<br />
In arid regions <strong>the</strong> planning of agricultural development<br />
requires estimates of <strong>the</strong> availability of groundwater, Adequate<br />
detai<strong>le</strong>d data bases are unlikely to exist in most areas of interest<br />
in developing countries, We have attempted to compute <strong>the</strong><br />
groundwater recharge potential for East Africa using primarily <strong>the</strong><br />
availab<strong>le</strong> meteorological data, The procedure has been to generate<br />
a syn<strong>the</strong>tic year by averaging <strong>the</strong> meteorological data for each<br />
month at each meteorological site over a period of years, Then from<br />
<strong>the</strong> monthly precipitation, <strong>the</strong> estimated evapotranspiration and <strong>the</strong><br />
estimated run off is subtracted, This computation proceeds according<br />
to an assumed soil moisture storage and transport model whose<br />
throughput constitutes <strong>the</strong> potential groundwater recharge, Contour<br />
lines of this cuantity are plotted and <strong>the</strong>se can serve as a guide<br />
to rural planners for optimizing <strong>the</strong> se<strong>le</strong>ction of sites for new<br />
development,<br />
RESUME<br />
Aux regions arides quand on fait un plan du development agri-<br />
co<strong>le</strong> il faut evaluer l'eau souterraine desponib<strong>le</strong>, I1 est tres peu<br />
probably qu'on trouve <strong>le</strong>s donnees de base assez detail<strong>le</strong>es dans la<br />
plupart des regions en observation aux pays que son en train de se<br />
developper. On a essayé de computer <strong>le</strong> potentiel de la recharge des<br />
?aux souterraines pour <strong>le</strong>s pays en Afrique de l'Est en utilisont,<br />
?rincipa<strong>le</strong>ment, <strong>le</strong>s données meteorologique disponib<strong>le</strong>s, On a produit<br />
sne ann&e des donnges syn<strong>the</strong>tìques en faìsant la moyenne <strong>le</strong>s<br />
ionnêes meteorologiqueo pour chaque mois a chaque hstallatlon meteo-<br />
nologique pendant une periode des annêes, Puis on a soustrait de la<br />
>recipitation mensuel<strong>le</strong>, l'evaluation de ltevapotranspirat2on et de<br />
L'ecou<strong>le</strong>ment total, Cette computation continue suivant un mode<strong>le</strong><br />
;uppose de la capacité de l'eau et du transport de l'eau dans <strong>le</strong> sol<br />
lui evalue <strong>le</strong> potentiel de la recharge de l'eau souterraine. Les<br />
:ourbes de niveau de cette quantite sont tracées et <strong>le</strong>s organisateurs<br />
lu developpement rural peuvent s'en reg<strong>le</strong>r pour optimiser la<br />
;e<strong>le</strong>ction des situations pour des projets neufs.
526<br />
At <strong>the</strong> East African Agriculture and Forestry Research<br />
Organization we are interested in estimating regional groundwater<br />
recharge rates to provide basia information for agricultural<br />
planning. Our approach is based on a water balance calculation. The<br />
gmundwater recharge is <strong>the</strong> residual which reinains after subtracting<br />
from <strong>the</strong> precipitation <strong>the</strong> losses due to evaporation and surface and<br />
subsurface run-off. The total precipitation can be computed reasonably<br />
well from <strong>the</strong> rainfall records. The remaining terms can be estimated.<br />
T. Woodhead, 1. Dagg and D.A. Rijks (1, 2, 3, 4) have<br />
studied extensively <strong>the</strong> computation of <strong>the</strong> Penman potential<br />
evapotranspiration from <strong>the</strong> data sources in Eaet Africa. The actual<br />
evaporative losses may be estimated with <strong>the</strong> use of a physioal model<br />
of <strong>the</strong> soil moisture storage and transport system. Por a regional<br />
computation in a predominently arid region <strong>the</strong> net surface and<br />
subsurface run-off may be taken a8 aero to yield an upper bound on<br />
<strong>the</strong> potential groundwater recharge. The accuraoy of <strong>the</strong> final result<br />
will depend on <strong>the</strong> ohoice of <strong>the</strong> soil moisture storage and transport<br />
model. This must represent <strong>the</strong> average regional response to <strong>the</strong><br />
stimuli of rain, wind and sun.<br />
Water Balanoe Oalculation 0.f Groundwater Beoharge<br />
GWFli Potential Groundwater Resharge<br />
Pa Precipitation<br />
Eo: Penman Potential Evapotranspiration<br />
E : Actual Evapotranspiration<br />
Q: Soil Moisture Storage<br />
AQ; Change in Soil Moisture Storage<br />
RO: Rn-off<br />
GWR = P - E -.U) - RO<br />
In order to compute <strong>the</strong> actual evapotranspiration from <strong>the</strong><br />
Penman potential evapotranepiration <strong>the</strong> dynamics of a soil moisture<br />
model m e invoked. Many such models arc possib<strong>le</strong> and future studies<br />
may improve this phase of <strong>the</strong> work. For <strong>the</strong> present computation a<br />
nbucket model" has been chosen. This model al<strong>le</strong>mes that moisture<br />
stored in <strong>the</strong> soil root sone is freely availab<strong>le</strong> for transpiration<br />
up to <strong>the</strong> Penman potential Eo demand. If <strong>the</strong> root zone soil moisture<br />
is exhausted no fur<strong>the</strong>r evapotranspiration takes place regard<strong>le</strong>ss of<br />
<strong>the</strong> Penman Eo demand. Fur<strong>the</strong>rmore, lhe root abne has a finite<br />
capacity, Qo, for moisture storage. In <strong>the</strong> event <strong>the</strong> monthly<br />
precipitation minus <strong>the</strong> monthly Penman Eo exceeds Qo, downward<br />
percolation of <strong>the</strong> excess moisture takes place and constitutes <strong>the</strong><br />
potential groundwater recharge.
"Bucket Model" of Soil Moisture Stcrage ana Transport<br />
1. Root zone has a maximum moisture storage capacity, Qo.<br />
2. If QSQo, groundwater recharge i? cil.<br />
3. If monthly moisture input plus storage exceeds Qo, <strong>the</strong><br />
excess is potential groundwater rechzrge.<br />
We have made use of meteorological data col<strong>le</strong>cted by<br />
T. Woodhead, M. Dagg, and D.A. Rijks (iq 2, 3, 4). This data is<br />
based on 80 stations in Kenya, 50 in Tanzaniz and 17 in Uganda, at<br />
ozly a few stations were oomp<strong>le</strong>te records of precipitation, wind run,<br />
and insclation availab<strong>le</strong>. Several methods were devised by Wlcndhead to<br />
fill in <strong>the</strong> blanks. In order to eompute <strong>the</strong> Penman potential<br />
evapotranspiration, Eo, according to <strong>the</strong> method of McCullooh (5) <strong>the</strong><br />
following inputs are needed:<br />
2<br />
R: insclation in cakories/cm /day<br />
n/N: =?io 3f observed to maximum possib<strong>le</strong> number of daily<br />
smshine hours.<br />
Ta:<br />
zverage screened ambient temperature 'C.<br />
o<br />
Td: mean deily temperature of den point C.<br />
U: win& run in mi<strong>le</strong>s per day at 2 meter e<strong>le</strong>vation.<br />
Eased cn fifteen stations a linear regression between R and<br />
n/N has been derived (6, 7). This regressior, was used to derive one<br />
of <strong>the</strong>ne quantities when <strong>the</strong> o<strong>the</strong>r was availab<strong>le</strong> from <strong>the</strong> meteorological<br />
recnrds. When nei<strong>the</strong>r R nor n/N were recorded use was made of a,<br />
rehtionship esteblished (1, 2) between <strong>the</strong> monthly mean of daily<br />
sucshim duraticn and <strong>the</strong> tot21 cloud amount. Estimates of total<br />
cloud smount are made at most civil airfields. When records of wind<br />
run were not availab<strong>le</strong> use was made of e relation established<br />
betaeen Beaufort sca<strong>le</strong> assessments of wind velocity and wind run (3).<br />
After <strong>the</strong>se procedures had been applied to comp<strong>le</strong>te <strong>the</strong><br />
recr-ds <strong>the</strong>y were processed to construct a syn<strong>the</strong>tic year. Por each<br />
site sverages over <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> record were made for each given<br />
mocth of <strong>the</strong> year. Monthly values of precipitation and Penman<br />
potential evapotranspiration were obtained. At each site an estimate<br />
was also made of <strong>the</strong> model parameter, Qo, <strong>the</strong> maximum m il moisture<br />
storage capacity in <strong>the</strong> root zone.<br />
The soil moisture storage and transport system is conceived<br />
of E? a dynamic system whose output is <strong>the</strong> downward percolating groundwater<br />
recharge which is determined jointly by <strong>the</strong> system driving funation<br />
3n3 <strong>the</strong> system physical parameters. The present output depends on <strong>the</strong><br />
whc<strong>le</strong> past behavior of <strong>the</strong> input. If <strong>the</strong> syetem ie 3inesr, <strong>the</strong> system<br />
oiityut is <strong>the</strong> convolution of <strong>the</strong> system unit impulse rnspcnse and<br />
tke system c?ri.ving function. If <strong>the</strong> system is non-licesr, given <strong>the</strong><br />
527
528<br />
input we can compute <strong>the</strong> output ueing <strong>the</strong> system parameters.<br />
For our bucket model system we proceeded as follows.<br />
If <strong>the</strong> Penman Eo exceeded <strong>the</strong> precipitation for several months we<br />
tentatively assumed that <strong>the</strong> soil moisture was totally dep<strong>le</strong>ted at <strong>the</strong><br />
end of <strong>the</strong> dry season. Starting at that month <strong>the</strong> <strong>book</strong>eeping was<br />
begun arid carried out each month for <strong>the</strong> duration of <strong>the</strong> syn<strong>the</strong>tic<br />
year. On <strong>the</strong> o<strong>the</strong>r hand, if <strong>the</strong> precipitation exceeded <strong>the</strong> Penman<br />
Eo for most of <strong>the</strong> year <strong>the</strong> assumption was made that <strong>the</strong> soil was<br />
saturated with moisture at <strong>the</strong> end of <strong>the</strong> wettest period. Then <strong>the</strong><br />
oookeeping was comp<strong>le</strong>ted for <strong>the</strong> syn<strong>the</strong>tic year. One of <strong>the</strong>se<br />
assumptions always <strong>le</strong>d to a consiatent sat of <strong>book</strong>eeping entries.<br />
The monthly groundwater recharge was summed at each site over all<br />
mrnths of <strong>the</strong> syn<strong>the</strong>tic year. These totals were <strong>the</strong>n noted on a<br />
map and contour lines of equal groundwater recharge were interpolated.<br />
After obtaining <strong>the</strong> meteorological data <strong>the</strong> sing<strong>le</strong> model<br />
payameter 20 determines <strong>the</strong> reault. Therefore, <strong>the</strong> sensitivity of<br />
<strong>the</strong> cont Ars to a variation of Qo ia of interest. It is evident that<br />
If <strong>the</strong> soil moisture is not comp<strong>le</strong>tely dep<strong>le</strong>ted at some time during<br />
7-e gear, a change in Qo alone will not affect <strong>the</strong> groundwater recharge.<br />
For this situation, only <strong>the</strong> moisture in permanent storage will be<br />
changed. On <strong>the</strong> o<strong>the</strong>r hand, if <strong>the</strong> sril moiature is comp<strong>le</strong>tely<br />
exhausted at one time during <strong>the</strong> year, a change in Qo will change<br />
<strong>the</strong> throughput by an equal and opposite amount. Therefore, in arid<br />
regions a low soil moisture storage capacity favors groundwater<br />
recharge. Low storage capacity implies that a short intense rain<br />
will rapidly fill up <strong>the</strong> soil moisture reservoir and <strong>the</strong> excess will<br />
quickly percolate downward beyond <strong>the</strong> root zone to <strong>the</strong> subsurface<br />
storage aquifer. In arid regions vegetative cover is generally <strong>le</strong>ss,<br />
which re8'uces <strong>the</strong> losses due to tranepiration. Deep rooted<br />
vegetation wnuld negata this advantage.<br />
In order to investigate <strong>the</strong> eensitivity of <strong>the</strong> potential<br />
groundwater reoharge contours to a change in <strong>the</strong> value of QG, <strong>the</strong><br />
ac.mputations were carried out for two choices of this parameter at<br />
each site.<br />
Oritiaue of <strong>the</strong> Method<br />
<strong>le</strong> hare attempted to obtain same idea of <strong>the</strong> regional<br />
potential groundwater recharge ratea using <strong>the</strong> availab<strong>le</strong> data, mainly<br />
meteorological records. By <strong>the</strong> use of a simp<strong>le</strong> aoil moisture<br />
storage and transport model we estimate <strong>the</strong> actual evapotranspiration<br />
from <strong>the</strong> Penman potential evapotranspiration. The residue from <strong>the</strong><br />
precipitat irn after subtracting <strong>the</strong> evapotranspiration and <strong>the</strong><br />
increment tc soil moisture storage we identify as <strong>the</strong> deep percelation<br />
or potential groundwater recharge. The model we used for <strong>the</strong> soil<br />
mt-isture is a one parameter model and we have studied <strong>the</strong> effect on<br />
tSe throughpiit of <strong>the</strong> choice of this parameter.
Several improvements are poserib<strong>le</strong> in <strong>the</strong> treatment. A study<br />
co-ild be carried out to improve <strong>the</strong> accurary of <strong>the</strong> model. It is easy<br />
tri devise multi-psrameter models. These could be compared to field<br />
me-rsurements. Ano<strong>the</strong>r refinement would incorporate sdditional input<br />
data. Neutron soil moisture probes are now fairly widely availab<strong>le</strong>.<br />
Fer future work such data should be incorporated in <strong>the</strong> computation.<br />
The use of this additional input can lighten <strong>the</strong> burden of <strong>the</strong> model<br />
in <strong>the</strong> determination of <strong>the</strong> actual evapotranspiration.<br />
The neutron moisture probe data could be used in <strong>the</strong><br />
following way which modifies <strong>the</strong> present soil moisture model. Soil<br />
misture profi<strong>le</strong>s could be taken at monthly or weekly intervals to a<br />
depth of five meters. The total soil moisture to a fixed depth will<br />
??e tota<strong>le</strong>d fnr each measurement. During <strong>the</strong> interval between<br />
measurements <strong>the</strong> precipitation and <strong>the</strong> Penman potential<br />
zvspotranspiration will be tota<strong>le</strong>d. If during that interval <strong>the</strong>re<br />
i? soil moist*ire storage exceeding <strong>the</strong> wilting point in <strong>the</strong> roet zone<br />
than <strong>the</strong> amel evapctranspiration will be taken as <strong>the</strong> Penman Eo. If<br />
wt, <strong>the</strong> actual evap6transpiration will be taken as zero.<br />
With this accounting procedure we could compute <strong>the</strong> sum of<br />
<strong>the</strong> deep percolation and <strong>the</strong> difference between surface and subsurface<br />
run on and runoff. If <strong>the</strong>se last categories are in approximate<br />
halance, <strong>the</strong>n we have <strong>the</strong> deep percolation or potential groundwater<br />
recharge as <strong>the</strong> residual, From measurements at a network of sites <strong>the</strong><br />
regional maps msy be constructed.<br />
The authors wish to thank <strong>the</strong> Director of EBBFRO for<br />
aermission to present this paper at <strong>the</strong> Symposium on <strong>the</strong> Design 00<br />
Tater Resources Projects with Inadequate Data, Madrid, June 1973.<br />
REFERENCES<br />
I. . Woodhead, T. (1966). Empirical relations between cloud<br />
amount, insolation and sunshine duration in East Africa:<br />
i, E. Afr. Agric. For. J., 2, pp211.<br />
2.<br />
3.<br />
Woodhead, T. (1967) Empirical relations between cloud<br />
amount, insolation and sunshine duration jr? East Africa:<br />
II, E. Afr. Agric. For. J., 2, pp474.<br />
Woodhead, T. (1970) mapping potential evaporaticn for<br />
tropical East Africa; <strong>the</strong> accuracy of Penmen estimates irem<br />
indirect assessments of radiation and wind speed, Proc.<br />
Reuding Symposium World Wg<strong>le</strong>tgFi,B&lanCe I. A.C.H.<br />
. I._ I -<br />
Dagq, M. and Woodhead, T. and Rijks, D.A., Evapcmtirr<br />
jn Enst Africa, nul. I.A.S.H., XV, 1, pp61.<br />
529
530<br />
c .. .<br />
6.<br />
7.<br />
McCulloch, J .S .O. (1965), Tab<strong>le</strong>s for <strong>the</strong> rapid i:omputE+,i.on<br />
of <strong>the</strong> Penman estimate of evaporation, E. Aï'?. AgTjc. For.<br />
J., 22, pp.286.<br />
Woodhead, T. (1968), Studies of Potential Evaporaticn in<br />
Kenya, Government nf Kenp, Nairobi.<br />
Woodhead, T. (196R), Studies of Potertial Evapuration i.c<br />
Tarizania, Dar es Saham.
A RAINFALL-RUNOFF MODEL BASED ON THE WATERHED STREAM NETWORK<br />
A BS TRAC T<br />
by J.W. Del<strong>le</strong>ur and M.T. Lee*<br />
School of Civil Engineering<br />
Purdue University '<br />
West Lafayette, Indiana 47907, USA<br />
Physical models of <strong>the</strong> rainfall-runoff process are better<br />
suited than ei<strong>the</strong>r stochastic or black box models for areas with<br />
limited data, The model parameters must have a physical signifi-<br />
cance, be convenient to obtain and <strong>the</strong>ir number should be small,<br />
The framework of a model meeting <strong>the</strong>se objectives is proposed and<br />
is based primarily on <strong>the</strong> geomorphologic characteristics of <strong>the</strong><br />
stream network obtainab<strong>le</strong> from maps or from aerial photographs.<br />
There is analytical and experimental evidence that hydrographs are<br />
dominated by direct runoff from very short overland flow paths<br />
from precipitation on transient, near channel wetlands. This<br />
wetland area is dynamic in <strong>the</strong> sense that it varies in terms of <strong>the</strong><br />
history of <strong>the</strong> excess of <strong>the</strong> precipitation over <strong>the</strong> "B" horizon<br />
permeability, The distribution of <strong>the</strong> dynamic contributing area<br />
along <strong>the</strong> main stream is obtained under <strong>the</strong> assumptions that <strong>the</strong><br />
velocity of flow along <strong>the</strong> stream network is uniform, that <strong>the</strong><br />
drainage density is a constant within a given watershed and that <strong>the</strong><br />
first order streams are uniformly distributed in <strong>the</strong> basin, The<br />
runoff from <strong>the</strong> dynamic contributing area is <strong>the</strong>n routed through <strong>the</strong><br />
syn<strong>the</strong>sized stream network to obtain <strong>the</strong> direct runoff at <strong>the</strong> basin<br />
out<strong>le</strong>t,<br />
RESUME<br />
Les modè<strong>le</strong>s physiques des transferts pluies-débits s'adaptent<br />
mieux que <strong>le</strong>s modè<strong>le</strong>s stochatiques ou que <strong>le</strong>s "boîtes noires" aux<br />
régions où <strong>le</strong>s donndes sont limitées, Les paramêtres de ces mode<strong>le</strong>s<br />
doivent être pourvus d'un sens physique, faci<strong>le</strong>s 2 obtenir et <strong>le</strong>ur<br />
nombre doit être petit, Le cad~e du modè<strong>le</strong> proposé se conforme 2 ces<br />
objectifs et est basé principa<strong>le</strong>ment sur <strong>le</strong>s caractéristiques gêomorphologiques<br />
du rlseaa fluvial que l'on peut obtenir de cartes ou<br />
de photographies aériennes, I1 a dtb démontrê analytiquement et<br />
expérimenta<strong>le</strong>ment que <strong>le</strong>s hydrogrammes sont en general dominbs par<br />
<strong>le</strong> ruisel<strong>le</strong>ment sur de petits parcours situês dans <strong>le</strong>s zones mouiliges<br />
près des cours d'eau, Ces zones mouillêes son dynamiques dans<br />
<strong>le</strong> sens qu'el<strong>le</strong>s varient pendant la pluie et avec la saison, Un modè<strong>le</strong><br />
mathêmat2que de ces zones est formu<strong>le</strong> en fonction de Ifhistoire<br />
de la précipitation excédant la permsabilité de l'horizon iiB't. La<br />
distribution de ces zones <strong>le</strong> long du rdseau fluvial est obtenue en<br />
supposant que la vitesse de l'écou<strong>le</strong>ment est uniforme dans <strong>le</strong> réseau<br />
fluvial, que la densitd de drainage est constante pour <strong>le</strong> bassin<br />
donne, et que <strong>le</strong>s cours d'eau du premier ordre sont uniformdment<br />
distrlbuh dans <strong>le</strong> bassi'n, Les ruissel<strong>le</strong>ments des zones dynamiques<br />
sont achemint% au travers d'une synthèse du rdseau fluvial pour obtenir<br />
i'ëcouïement à l'exutoire.<br />
* Current Addres,s: Dept, of Agricultural Economics, Univ. of<br />
Illinois, Urbana, 1llinoi.s y 61801 y USA,
532<br />
Stochastic models for <strong>the</strong> generation Of river flow sequences require long<br />
historical time series for <strong>the</strong> appropriate calibration of<strong>the</strong> parameters. In<br />
many parts of <strong>the</strong> world, actual streamflow records are not sufflciently long to<br />
attempt to deflne an e<strong>le</strong>mentary model such as a flrst order Markov process for<br />
annual streamflow series. According to Rodriguez-Xturbe [i] for ser<strong>le</strong>s shorter<br />
than 40 years <strong>the</strong> error in estimating <strong>the</strong> annual man might run from 2% to 20%.<br />
for <strong>the</strong> variance from 15% to 60%. and for <strong>the</strong> rank one serial correlation it<br />
might be as high as 200%. It may be fìati<strong>le</strong> to attempt to develop generating models<br />
which preserve parameters, <strong>the</strong> estimation of which carries such an uncertainty.<br />
The formulation of mnthly models requires shorter records, but with a<br />
record of i5 years, <strong>the</strong> error in <strong>the</strong> rank one serial correlation coefficient is<br />
still of <strong>the</strong> order of 40%. Btochastic linear models of <strong>the</strong> rainfall-runoff process<br />
likewise require long time ser<strong>le</strong>s of both <strong>the</strong> rainfall and runoff for <strong>the</strong>ir<br />
calibration. It would, <strong>the</strong>refore, appear that for regions with inadequate data,<br />
one may have to resort to deterministic models. At this point, <strong>the</strong> choice may<br />
be between a %lack box" type of model and a physical model. Black box models<br />
ceanot be transferred from one location to ano<strong>the</strong>r as <strong>the</strong> meaning of <strong>the</strong> peu-8meters<br />
in terma of <strong>the</strong>ir representation of <strong>the</strong> components of <strong>the</strong> hydrologic<br />
cyc<strong>le</strong> is usually undeflned. The proper choice appears to be a physical model<br />
which requires a small number of easily identifiab<strong>le</strong> parameters, or a model<br />
based on data w-hich can be obtained in a relatively short time, perhaps by new<br />
techniques.<br />
These physical models could conceivably be formulate&, by making use of information<br />
that is becoming availab<strong>le</strong> through remote sensing from airCrart and<br />
from satellites. By means of <strong>the</strong>se new techniques, large areas CBP be observed<br />
and analyzed in a short time, end require a small amount of observations on <strong>the</strong><br />
ground. The recent developments in remote seneing technology thue seem to point<br />
to a new direction for hydrologic investigations in -(LB with inadequate data.<br />
Remote sensing from aircraft or from satellite is best applied to observing<br />
or monitoring fairly large area and thus <strong>le</strong>nde itself to <strong>the</strong> hydrologic studies<br />
of comp<strong>le</strong>te watersheds. Images taken at different times cm show changes In <strong>the</strong><br />
watershed, such as variations ia <strong>the</strong> land we.<br />
The potential of remote sensing<br />
for water resources stubies has been discussed by Kiefer and Scherz [2], but <strong>the</strong><br />
principal application of remote sensing to hydrology has been through aerial<br />
photography.<br />
The eye can see light from about .4 to .7 microns, but photographscan sense<br />
from about 0.3 to 1.0 microns, thus extending <strong>the</strong> range to lower and higher wave<br />
<strong>le</strong>ngths. Color end color infrared photography have been used with great success<br />
in forestry .and in agricultural crop identification. [3] Themai scanning op-<br />
erates in <strong>the</strong> heat emission part o? <strong>the</strong> energy spectrum in <strong>the</strong> wan <strong>le</strong>ngth riPr<br />
3 to 20 mincrons. Pluhowski [4] shoved that with Infra-red Imagery in <strong>the</strong> 8 to<br />
14 micron range, it is possib<strong>le</strong> to discern <strong>the</strong>rmal contrasts of 1' or 2'C. Thin<br />
technique enab<strong>le</strong>s <strong>the</strong> hyarolo,5lSt to detect areas of &K>inid water dlacharge mad<br />
to identify circulation patternr in large vater bodies.
533<br />
More advanced techniques include mUitiSpeCtra1 scanning and side looking<br />
radar. Multispectral scanners produce as many as 20 separate images in wave<br />
<strong>le</strong>ngths ranging from <strong>the</strong> ref<strong>le</strong>cted infrared region to <strong>the</strong> ultravio<strong>le</strong>t region.<br />
These images may <strong>the</strong>n be analyzed by means of computer data processing programs<br />
which classi* <strong>the</strong> surface materials. This classification is accomplished by<br />
separating materials in a known area according to <strong>the</strong>ir spectral response characteristics<br />
and <strong>the</strong>n applying <strong>the</strong>se criteria to unknown areas. [3] The side<br />
looking airborne radar can operate through dense cloud covers. It has been used<br />
in mapping sou<strong>the</strong>astern Pan- and northwestern Columbia, which could not be<br />
mapped by conventional aerial photography becauee of <strong>the</strong> cloud cover. As an examp<strong>le</strong>,<br />
Weaver [5] cites that <strong>the</strong> meandering pattern of <strong>the</strong> hiira river was revea<strong>le</strong>d<br />
by this technique.<br />
Black and white, color and color infrared photography combined can be used<br />
to delineate water bodies, rivers and streams, <strong>the</strong> drainage structure of watersheds,<br />
to give indications on <strong>the</strong> underlying geology and on <strong>the</strong> soil types of<br />
<strong>the</strong> region. Waltz and Myers [6] have shown that <strong>the</strong>re exists a significant correlation<br />
between <strong>the</strong> optical density measured from an aerial film and soil water<br />
content measured by neutron probes and also between ground water temperatures as<br />
measured through infrared <strong>the</strong>rmal scanner and <strong>the</strong> soil water content of fallow<br />
or bare soil. Zachary et.al. [7] has applied multispectral. remote sensing to<br />
soil survey research in Indiana.<br />
These techniques may also be used for enalysie<br />
of water quality and for monitoring water pollution. [a] A general review of<br />
<strong>the</strong> application of remote sensing in <strong>the</strong> management of earth resources has been<br />
prepared by Colwell [9].<br />
It appears that at present, black and white, color and infrared aerial<br />
color photography, can be used to obtain <strong>the</strong> basic information regarding stream<br />
networks, water bodies, main geologic and soil features needed in hydrologic investigations.<br />
It also appears that in <strong>the</strong> near future, more dependab<strong>le</strong> informa-<br />
tion on soil water will become availab<strong>le</strong> through remote sensing.<br />
The remote<br />
sensing techniques thus appear to be of particular interest in areas with inade-<br />
quate data, as a substantial area can be mapped in a relatively short time with<br />
a minimum of ground observation.<br />
MODEL FRAMEWORK<br />
It is <strong>the</strong> purpose of this paper to explore <strong>the</strong> feasibility of developing<br />
rainfall-ninofi models based primarily on information that can be obtained from<br />
remote sensing aerial photography and to establish a framework for such models.<br />
The simp<strong>le</strong>r observations obtainab<strong>le</strong> f rm <strong>the</strong> aerial photography being <strong>the</strong> plan<br />
form of <strong>the</strong> stream network. <strong>the</strong> topography and <strong>the</strong> soil type. <strong>the</strong> proposed model<br />
is based on <strong>the</strong>se three types of information and particularly on <strong>the</strong> stream net-<br />
work s<br />
Geomorphologists have developed parameters which describe <strong>the</strong> topology, <strong>the</strong><br />
structure, <strong>the</strong> planform and <strong>the</strong> relief of stream networks. [lo] Some of <strong>the</strong>se<br />
parameters can be used ae indices of <strong>the</strong> hydrologic behavior of <strong>the</strong> basins since<br />
scmral characterietics of <strong>the</strong> hydrograph depend upon <strong>the</strong> efficiency of <strong>the</strong>
534<br />
drainage networks. For examp<strong>le</strong>, <strong>the</strong> bifurcation ratio (ratio of number of streem<br />
segments of one order to number of stream segments of next higher order) is an<br />
important control over <strong>the</strong> peakedness of <strong>the</strong> runoff hydrograph. Ano<strong>the</strong>r geomorphologic<br />
parameter which affects <strong>the</strong> moff pattern is <strong>the</strong> drainage density (smmation<br />
of stream <strong>le</strong>ngths divided by basin area) which is approximately one halr<br />
of <strong>the</strong> reciprocal of <strong>the</strong> overland flow <strong>le</strong>ngth. A high drainage density indicates<br />
a rapid removal of <strong>the</strong> surface runoff, a decrease in <strong>the</strong> lag time and an<br />
increase in <strong>the</strong> peak of <strong>the</strong> hydrograph.<br />
A model based on <strong>the</strong> stream network also <strong>le</strong>nds itself to <strong>the</strong> application or<br />
<strong>the</strong> dynamic source area concept ra<strong>the</strong>r than <strong>the</strong> application of classical Horton<br />
infiltration <strong>the</strong>ory for <strong>the</strong> purpose of estimating <strong>the</strong> runoff-producing-rainfall.<br />
Freeze [li] has shown <strong>the</strong>oretically that on concave slopes with lower permeabilities<br />
and on all convex slopes, hydrographs are dominated by direct runoff with<br />
a very short overland flow path from precipitation on transient, near channel<br />
wetlands which form <strong>the</strong> variab<strong>le</strong> response area.<br />
DuMe and Black [12] reported<br />
that <strong>the</strong> area contributing to <strong>the</strong> overland flow ie dynamic in <strong>the</strong> sense that it<br />
varies seasonally and throughout a storm. Nutter and Hew<strong>le</strong>tt 1131 have depicted<br />
<strong>the</strong> growth of <strong>the</strong> source area during a storm from areas adjacent to <strong>the</strong> lover<br />
order streems and gradual4 expanding to <strong>the</strong> main stream in one direction and to<br />
efflmeral stream in <strong>the</strong> o<strong>the</strong>r. It seems logical to assume that <strong>the</strong> response<br />
area will depend on <strong>the</strong> soil type adjacent to <strong>the</strong> stream and on <strong>the</strong> antecedent<br />
rainfall.<br />
In view of <strong>the</strong> comp<strong>le</strong>xity that would result from estimating and routing <strong>the</strong><br />
runoff in each tributary, it is proposed to syn<strong>the</strong>size <strong>the</strong> stream network by<br />
folding it along <strong>the</strong> main stream in a manner similar to that mea by Lkwge [i41<br />
with <strong>the</strong> time-area diagram.<br />
Several routing procedures could be used, <strong>the</strong> %om-<br />
p<strong>le</strong>te linear routing" method of Dooge and Har<strong>le</strong>y [i51 was used because of its<br />
superior accuracy among o<strong>the</strong>r linearand emperical methods. It is in <strong>the</strong> appli-<br />
cation of <strong>the</strong> routing procedure that <strong>the</strong> slope of <strong>the</strong> main stream plays a major<br />
ro<strong>le</strong>. For full details <strong>the</strong> reader is directed to ref. 17.<br />
iVRMULàTION OF THE MDDEL<br />
The model of <strong>the</strong> contributing area A(iAt) is expressed by <strong>the</strong> relationship<br />
i;,<br />
i-1<br />
[R(kAt) - B]At + [R(iAt) - B]At<br />
A(iAt) = Ao I<br />
i<br />
T<br />
[R(kAt) - 1<br />
B]At<br />
k=O<br />
where A(iAt) is <strong>the</strong> contributing (response) ana at time iAt<br />
R(kAt) is <strong>the</strong> rainfall intensity at time kAt<br />
B is <strong>the</strong> "B" horizon permeability<br />
D is <strong>the</strong> fraction of <strong>the</strong> antecedent rainfall contributing to <strong>the</strong><br />
response area<br />
N is a parameter<br />
T is <strong>the</strong> total nuniber of sampling points of <strong>the</strong> runoff -&-ogreph<br />
(1)
k<br />
i<br />
is an index to count <strong>the</strong> time of antecedent rainfall excess,<br />
k*i<br />
is an index indicating <strong>the</strong> current time<br />
Equation (1) is subject to <strong>the</strong> conetraint that <strong>the</strong> continuity equation must<br />
be satisfied, namely <strong>the</strong> volume of rainfall excess muet be equal to <strong>the</strong> volume<br />
of direct runoff:<br />
- T T<br />
1 QO(lAt)At 1 A(iAt) [R(iAt) - B]At<br />
i =o i=o<br />
where Q,(iAt) is <strong>the</strong> direct runoff at <strong>the</strong> out<strong>le</strong>t at time Ata<br />
535<br />
The syn<strong>the</strong>sis of <strong>the</strong> stream network is based, in part, on <strong>the</strong> observation<br />
made,by Leopold [i61 that <strong>the</strong>re is no definite tendency îor <strong>the</strong> flow velocity to<br />
have a great change along <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> stream for a given retuni period or<br />
fkequency. It may thus be assumed that locations having equal distances measured<br />
along <strong>the</strong> stream network to <strong>the</strong> out<strong>le</strong>t, have <strong>the</strong> same runoff travel time to<br />
<strong>the</strong> out<strong>le</strong>t. If it may be fur<strong>the</strong>r assumed that <strong>the</strong> drainage density is approximately<br />
uniform within a watersheä, <strong>the</strong>n <strong>the</strong> total stream <strong>le</strong>ngth upstream of a<br />
particular point on a stream ia proportional to <strong>the</strong> tributary drainage area at<br />
that point. Thus <strong>the</strong> distribution of <strong>the</strong> travel times is proportional to <strong>the</strong><br />
distribution of <strong>the</strong> drainage areas along <strong>the</strong> stream reaches, and only <strong>the</strong> latter<br />
need to be considered. Fig. 1 shows <strong>the</strong> method of estimation of <strong>the</strong> distribution<br />
of <strong>the</strong> drainage arem s(JAL) along <strong>the</strong> main stream reaches for a idealized<br />
waters he d.<br />
The vol^ of runoff may be obtained by adding <strong>the</strong> runoffs from each of <strong>the</strong><br />
e<strong>le</strong>mentary contributing areas. Calling a(jAL, IAt) <strong>the</strong> dynamic response area<br />
at stream reach jAL and at time iAt, <strong>the</strong> continuity equation may be written<br />
T T S<br />
1 Qo(iAt)At 1 1 a(JAL, iAt) [R(iAt) - BIAL At (3)<br />
is0 i=O, j=o<br />
where S is <strong>the</strong> total nimiber of stream reaches.<br />
Assuming fur<strong>the</strong>r that <strong>the</strong> first order atreams or <strong>the</strong> atream Bources are plaiformly<br />
distributed over <strong>the</strong> watershed, <strong>the</strong>n, at a given time iAts <strong>the</strong> ratio 0%<br />
<strong>the</strong> dynamic response area a(jAL, iAt) at stream reach jAL to <strong>the</strong> tributaPy<br />
drainage area at <strong>the</strong> same stream reach is equal to <strong>the</strong> ratio of <strong>the</strong> total pesponse<br />
area A(iAt) to <strong>the</strong> total watershed area Ao- Thus<br />
The continuity equation (3) thua becows
536<br />
The direct runoffs from <strong>the</strong> individual stream reaches are <strong>the</strong>n routed<br />
through <strong>the</strong> stream network by means of a linear routing procedure. li- 2<br />
shows schematically <strong>the</strong> routing procedure for a stream reach.<br />
X(JAL, kAt), in reach j at time<br />
-<br />
The input,<br />
kAt is <strong>the</strong> direct runoff given by<br />
X(JAL, kAt) ao(jAL) A(kAt) [R(kAt) - BI (5)<br />
AO<br />
and <strong>the</strong> routed outflow from reach,j, at time iAt is Y(JAL, iAt), given by <strong>the</strong><br />
convolution integral shown in pig. 2 which Is approximated by <strong>the</strong> convolution<br />
sum<br />
i<br />
Y(jAL, iAt) 2: Hu(JAL, (i-k)At) X(jAL, kAt)At (6)<br />
k=O<br />
where H (JAL, (i-k)At) is <strong>the</strong> kernel function or instantsneous unit hydrograph<br />
of <strong>the</strong> bear routing procedure used. The runoff at <strong>the</strong> out<strong>le</strong>t, Bo, is obtained<br />
by summation over <strong>the</strong> stream reaches<br />
S i<br />
2 (iAt) = 1 1 H,(jAL, (i-k)At) * Ia0(jAL) A(kAt) [R(kAt)-BI - At '<br />
C<br />
(7)<br />
j=O k=O AO<br />
where A(kAt) is given by equation (1). The kernel fbctions for 10 of <strong>the</strong> most<br />
common linear routing models have been listed by Toebes and Chang [la]. In this<br />
particuiar study <strong>the</strong> linear channel routing kernel function used is based on a<br />
linearization of <strong>the</strong> Saint Venant equations developed by Rwge and Har<strong>le</strong>y [15].<br />
This kernel function has three parameters: <strong>the</strong> stream slope, a reference discharge,<br />
and a roughness parameter.<br />
IMPLEMENTATION OF THE MODEL<br />
The watersheds se<strong>le</strong>cted for testing <strong>the</strong> model are located in <strong>the</strong> state of<br />
Indiana, near <strong>the</strong> center of <strong>the</strong> esstem half of <strong>the</strong> United States. Thirteen basins<br />
were used with areas ranging from 8 to 400 square kilometers. The drainage<br />
maps for <strong>the</strong>se watersheds were prepared from aerial photographs at <strong>the</strong> sca<strong>le</strong> of<br />
1:20,000 by <strong>the</strong> staff of <strong>the</strong> Airphoto Interpretation Laboratory, School of Civil<br />
Engineering at -due University.<br />
The maps used were at <strong>the</strong> sca<strong>le</strong> of 1:63,360<br />
(one inch equals one mi<strong>le</strong>). The longitude and latitude of all stream junctions<br />
and stream sources within <strong>the</strong> basins were digitized and stored on punched cards<br />
by means of an automatic digitizer. The details of assembly and of <strong>the</strong> storage<br />
of <strong>the</strong> hydrologic and geomorphologic data on magnetic tapes have been reported<br />
by Lee, Blank and Del<strong>le</strong>ur [is]. Fig. 3 presents a CALCOMP restitution of <strong>the</strong><br />
drainage network of a watershed from <strong>the</strong> data stored on magnetic tape. Also<br />
ahawn on Fig. 3 are <strong>the</strong> etream link m d <strong>the</strong> drainage eue8 distributions along
537<br />
<strong>the</strong> main stream. The rainfall imposed on <strong>the</strong> dynamic contributing areas is <strong>the</strong>n<br />
used as <strong>the</strong> input into <strong>the</strong> linear routing procedure for each main stream link<br />
and <strong>the</strong>n summed over all <strong>the</strong> stream links. Fig. 4 (right) shows <strong>the</strong> outflow hy-<br />
drograph obtained by <strong>the</strong> comp<strong>le</strong>te linear routing method using <strong>the</strong> parameter val-<br />
ue shown (QI3 = reference discharge in d/Sec, CZ = roughness coefficient in<br />
d2/sec, SL = main stream slope). With <strong>the</strong> exception of <strong>the</strong> slope, <strong>the</strong> parame-<br />
ter velues were obtained by an optimization procedure which minimized <strong>the</strong> differ-<br />
ence between observed and calculated peak discharges and observed and calculated<br />
timesto <strong>the</strong> peak discharge. The parameters so obtained were correlated with<br />
climatological and geomorphological characteristics of <strong>the</strong> watersheds with <strong>the</strong><br />
following results.<br />
For <strong>the</strong> watersheds used in this study it was found that <strong>the</strong> outflow hydrographs<br />
were not sensitive to <strong>the</strong> choice of D for D > 0.5. A value of D = 0.8<br />
was used. The value of B was taken as zero as <strong>the</strong> soils were generally impervious<br />
because of <strong>the</strong>ir clayey type and high permanent water tab<strong>le</strong> in <strong>the</strong> contributing<br />
areas adjacent to <strong>the</strong> streams. The value of N was found to be related t6<br />
<strong>the</strong> moPf ratio, R (ratio of measured runoff to measured rainfall):<br />
r'<br />
0.464 - Rr<br />
for = D = 0.8, B = O<br />
0.242<br />
(8)<br />
The runoff ratio was in turn related to <strong>the</strong> storm characteristics, <strong>the</strong> tempera-<br />
ture and an average soil permeability index of <strong>the</strong> basin by a regression equa-<br />
tion of <strong>the</strong> type<br />
a 8 6 ~<br />
Rr = Tmin 'I 'max<br />
where T+n is <strong>the</strong> minimum daily temperature when <strong>the</strong> storm occurs, Sf =: soil<br />
permeability index determined by assigning soil permeability VaEues to major<br />
soil types occurring in <strong>the</strong> basin, and calculating <strong>the</strong> weighted average for each<br />
basin, Pt is <strong>the</strong> rainfa-ll. volume and P- is <strong>the</strong> maximum rainfall intensity. The<br />
independent variab<strong>le</strong>s in <strong>the</strong> right hand side of Eq. (9) are listed from <strong>le</strong>ft to<br />
right in order of decrearkig significance. Al1 <strong>the</strong> exponente were negative and<br />
<strong>le</strong>ss than one (a = -0.42, f3 = -0.15, y = -0.18, 6 = -0.25). The multip<strong>le</strong> correlation<br />
coefficient was 0.91.<br />
The roughness coefficient C, was significantly correlated to <strong>the</strong> basin area,<br />
<strong>the</strong> stream slope, and <strong>the</strong> b me flow per unit area by a regression equation of<br />
<strong>the</strong> t yp<br />
where Bf is <strong>the</strong> base flow per unit area when <strong>the</strong> storm occurs, 4 is <strong>the</strong> drain-<br />
age area and So is <strong>the</strong> slope of <strong>the</strong> main stream. The independent variab<strong>le</strong>s are<br />
listed in order of decreasing aignificance. The exponents p and v were positive<br />
(1.0 and 1.4 respectively) but A was negative (-0.21)* The multip<strong>le</strong> correlation<br />
coefficient was 0.64. The value of <strong>the</strong> reference discharge varied between nar-<br />
row limits, 1.1to 1.4 cubic meter per second for atom wlumas ranging from 2.5<br />
(9)
538<br />
to i4 mm. Making use o? equations 8 throua 10, <strong>the</strong> model regenerated well <strong>the</strong><br />
shapes of <strong>the</strong> hydrographs; <strong>the</strong> peak discharges were in general, reproduced within<br />
20% and <strong>the</strong> times to peak within 10% of <strong>the</strong> observed values. It should be<br />
remembered that equations 8, 9 and 10 are unlikely to be vali8 outside of <strong>the</strong><br />
geographical arca ?or which <strong>the</strong>y were obtained. They indicate, however, <strong>the</strong><br />
type of variab<strong>le</strong>s which influence <strong>the</strong> model parameters and <strong>the</strong>ir corresponding<br />
sensitivity.<br />
DISCUSSION AND CoNCurSIONS<br />
The framework has been developed for a model which makes it possib<strong>le</strong> to es-<br />
timate <strong>the</strong> runoff from rainfall and from data obtainab<strong>le</strong> from aerial photography<br />
and from remote sensing. As presented, aerial photograph is needed for <strong>the</strong> de-<br />
termination of <strong>the</strong> stream network, <strong>the</strong> main stream slope and <strong>the</strong> watershed a r e a n<br />
In addition infrared color photography and/or ground observations are needed to<br />
obtain <strong>the</strong> soil permeability index, <strong>the</strong> soil types for <strong>the</strong> estimation o? <strong>the</strong> 'B"<br />
horizon permeability and <strong>the</strong> base flow.<br />
with <strong>the</strong> rapid progress of <strong>the</strong> remote sensing technology, it is expected<br />
that, in <strong>the</strong> near future, <strong>the</strong> amount of field work necessary may be greatly re-<br />
duced and lidted to calibration areas to obtain <strong>the</strong> spectral response charac-<br />
teristics needed for <strong>the</strong> interpretation of <strong>the</strong> remote sensing scanning.<br />
The rainfall-runoff process in a watershed was simulated by three basic<br />
components: a dynamic contributing area model, a contributing area distribution<br />
curve which integrates <strong>the</strong> contributing areas along <strong>the</strong> stream network, and a<br />
linear routing technique. In <strong>the</strong> proposed dynamic contribution area model <strong>the</strong><br />
exponent N, which quantifies <strong>the</strong> rate of expansion of <strong>the</strong> response area, Was<br />
found to be <strong>the</strong> dominant parameter, and was found to be correlated to <strong>the</strong> -off<br />
ratio. The "B" horizon infiltration and <strong>the</strong> weight of <strong>the</strong> antecedent rainfd1 D<br />
were not <strong>the</strong> primary parameters. In <strong>the</strong> linear routing, <strong>the</strong> roughness Parameter<br />
was found to be correlated to geomorphologic parameters and to <strong>the</strong> baseflow Wr<br />
unit area. The reference discharge did not change significantly from Storm to<br />
storm or from watershed to watershed.<br />
ACKNOh'LEIKMENT<br />
The Work presented herein was supported by Lhe Office o? Water Resources Re search, U.S. Department of <strong>the</strong> Interior under grant OWRR-B-008-IliD, by <strong>the</strong> Purdue<br />
Research Foundation under grant XR 5869 and by PiPrdue University. m-<br />
thors wish to exprese <strong>the</strong>ir thanks to <strong>the</strong> sponsors,<br />
REFERENCES<br />
1. Rodriguez-Iturbe, I. (1969) Estimation of Statistical Parmeters for Annual<br />
River Flows, Water Resources Research, 5, pp. 1418-1421-<br />
2. Keifer, R. W. and J. A. Sherz (1971) Aerial photograph for Water reSomceS<br />
studies, Jour. o? Surveying and Mapping Div. , Am. Soc. civil Enva. vo<strong>le</strong> 97,<br />
No- SU29 PP. 321-333.
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
539<br />
Laboratory for Agricultural Remote Sensing, Purdue University, Lafayette,<br />
Indiana (i968 and 1971) Remote Multispectral sensing in Agriculture, Vol.<br />
3 (Annual Rept., 1968) Res. Bull. No. 844, Agr. Exp. Station, also Vol. 4<br />
(Annual Rept., 1971) Res. Bull. NO. 873, Agr. Exp. Station.<br />
Pluhowski, E. J. (1972) Hydrologic interpretations based on infrared ima-<br />
gery of Long Island, New York, Geel. Surv. Water Supply Paper 2009, U.S.<br />
Govt. Print. Off.<br />
Weaver, K. F. (1969) Remote sensing: new eyes to see <strong>the</strong> world, Natl. Geo-<br />
graphic, Vol. 135, NO. 1, pp. 47-73.<br />
Waltz, F. A. and Y. I. mers (1970)<br />
Remote sensing of hydrologic resources<br />
in <strong>the</strong> Great Plains, Rept. #I, Remote Sensing Inst., Univ. of South Dakota.<br />
(Availab<strong>le</strong> from IVTIS, NO. PB 195 451)<br />
Zachary, A. L., J. E. Cipra, R. J. Diderickson, S. J. Kristof, and M. F.<br />
Baumgardner (1972) Application of multispectral remote sensing to soil<br />
survey research in Indiana, Lab. for Application of Remote Sensing, Purdue<br />
Univ., Lafayette, Indiana, Print 110972. '<br />
8. Scherz, J. P. (1971) Monitoring water pollution by remote sensing, Jour. ai'<br />
Survy. and Map. Div., Am. Soc. Civil Engr., Vol. 97, No. SU2, pp. 307-320.<br />
9. Colwell, R. N. (1973) Remote sensing in <strong>the</strong> management of earth resources,<br />
American Scientist, Vol. 61, NO. 2.<br />
10. Strah<strong>le</strong>r, A. N. (1964) Quantitative geomorphology of drainage b&ns and<br />
channel networks, in Hand<strong>book</strong> of Applied Hydrology, V. T. Chow, Ed., McGraw-<br />
Hill Book Co., pp. 4-40, pp. 44-74.<br />
11. Freeze, R. A. (1972) Ro<strong>le</strong> of subsurface flow in generating surface runoff,<br />
2, Upstream Source Areas, Water Resources Res., 8. pp. 1272-1283.<br />
12. Dunne, T., Bnd Black, R. D. (1970) Partial area contributions to storm runoff<br />
in a s-1 New Englua watershed, Water Resources Res., 6, pp. 1296-1311.<br />
13. Nutter, W. J., and Hew<strong>le</strong>tt (1971) Stream flow production from permeab<strong>le</strong> upland<br />
basin, paper presented to <strong>the</strong> Third Internatl. Seminar for Hydrology<br />
Professors, Furdue Univ., Lafayette, Ind., USA, July 1971.<br />
14. Dooge, J. C. I. (1959) A general <strong>the</strong>ory of <strong>the</strong> unit hydrograph, Jour. o?<br />
Geophys. Res., Vol. 64, NO. 2, pp. 241-256.<br />
15 Dooge, J. C. I. and B. M. Har<strong>le</strong>y (1967) Linear routing in uniform chmela,<br />
Proc. inti. wdroïogy Symp., Sept. 1967, Fort Collins, Colorado, USA, 1, pp.<br />
57-63.<br />
16. kopold, L. E. (1953) Downstream change of velocity in rivers, Am. Jour. of<br />
Science. 25. PP. 606-624.<br />
17. Lee, M.-T. t&d-J. W. Del<strong>le</strong>ur (1972) A program for estimating muoff from<br />
Indiana Watersheds, Part III, Analysis of geomorphologic data snd a dyndc contributing area model for runoff estimation, Purdue Univ. Water Resources<br />
Res. Center, Lafayette, Ind. Tech. Rept. No. 24.<br />
18. Toebes, G. H. and T. P. Chang (1972) Simulation model for <strong>the</strong> Upper Wabash<br />
surface water system, Purdue Univ. Water Resources Res. Center, Lafayette,<br />
Ind. Tech. Rept. NO. 27.<br />
19. Lee, M. T., D. ~lank, J. W. Del<strong>le</strong>ur<br />
(1972) A program for eetimting mori from Indiana watersheds , Part II , Assembly of hydroloaic euid geomorphologic<br />
data for small watershed0 in Indiana, Purdue Unio. Water Resource9 Ra#. Ccnter,<br />
Lafayette, Indiens, Tech. Rept. No. 23.
540<br />
a<br />
U<br />
O<br />
O<br />
STREAM REACH<br />
2 4 6 8<br />
STREAM REACH i<br />
FIGURE I DRAINAGE AREA DISTRIBUTION ALON<br />
THE STREAM REACHES
INPUT 5 q (0,t)<br />
INPUT - 1 SYSTEM1 OUTPUT -<br />
INPUT<br />
DE LTA<br />
FUNCTION<br />
q(0.t) = Xf0,t)<br />
-<br />
t<br />
t<br />
OUTPUT = q (L,t)<br />
OUTPUT<br />
2 PHYSICAL D?AGRAM OF UPSTREAM INFLOW<br />
INSTANTANEOUS UNIT HYDROGRAPH<br />
FOR SINGLE STREAM REACH<br />
t<br />
541
m (D U N<br />
542
8 N<br />
f d-<br />
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I-<br />
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W<br />
rr n<br />
âLL<br />
=IA<br />
o- 0<br />
"w z<br />
z3<br />
I-o-<br />
CJ<br />
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543
MONTHLY STREAMFLOW ESTIMATION FROM LIMITED DATA<br />
C.T. Haan<br />
-- ABSTRACT<br />
A four parameter, monthly water yield model has been developed<br />
and tested that makes it possib<strong>le</strong> to estimate monthly streamflow<br />
volumes from daily rainfall information. The four parameters of <strong>the</strong><br />
model can be determined from as litt<strong>le</strong> as two years of observed<br />
monthly streamflow data. This makes it possib<strong>le</strong> to install<br />
temporary, short term stream gaging stations to col<strong>le</strong>ct two or<br />
three years of monthly streamflow data and from this data determine<br />
<strong>the</strong> four parameters of <strong>the</strong> water yield model. The model uses a<br />
self-optimizing procedure so that <strong>the</strong> user is not necessarily involved<br />
in <strong>the</strong> parameter estimation process. A study of 24 watersheds<br />
has also shown that <strong>the</strong> four model parameters can be related<br />
to soil, geomorphic, and geologic characteristics of <strong>the</strong> basin.<br />
In this way <strong>the</strong> parameters for an ungaged basin can be estimated<br />
without requiring any data from that particular basin. Once <strong>the</strong><br />
model parameters are determined, long traces of monthlv streamflow<br />
data cán be simulated us ng only daiiy rainfa 1 as a model input.<br />
RESUME<br />
_ _ mis au point un mode<br />
On a développé et<br />
e à quatre paramètres,<br />
pour l'évaluation du rendement mensuel en eau, qui rend possib<strong>le</strong><br />
une estimation du volume de l'écou<strong>le</strong>ment mensuel à partir de<br />
l'information que constituent <strong>le</strong>s pluies journalières. Les quatre<br />
nnram5tres - - . - - - - - di1 - - mod31e . - - - - - n~iivent - - . - -- - P+PP - - - - d8tPvrninLs - - - - , , -. . - - > - na-etiv r-.- --- ~tin~nmrna-<br />
- ....I"L...tions<br />
aussi restreintes que <strong>le</strong>s résultats de deux ans d'observation<br />
des débits. Cela permet d'instal<strong>le</strong>r des stations de jaugeage des<br />
débits, temporaires, à court terme, afin de réunir des informations<br />
portant sur deux ou trois ans, sur <strong>le</strong>s débits mensuels, et, 2<br />
partir de ces informations deduire <strong>le</strong>s quatre paramètres du modè<strong>le</strong><br />
de rendement d'eau. Le modè<strong>le</strong> fait usage d'un procédé qui évalue<br />
automatiquement <strong>le</strong>s informations et <strong>le</strong>s dispose de la meil<strong>le</strong>ure<br />
facon possib<strong>le</strong> (self-optimizing procedure) de sorte que l'usager<br />
n'a pas nécessairement à se préoccuper de l'estimation des paramètres.<br />
Une étude de 24 bassins (watersheds) a aussi montré que<br />
<strong>le</strong>s quatre paramètres du modè<strong>le</strong> peuvent être reliés aux caracteristiques<br />
géomorphiques et géologiques, ainsi qu'a cel<strong>le</strong>s du sol, pour<br />
un bassin (basin) donné. De cette facon, <strong>le</strong>s paramètres pour un<br />
bassin non jauge peuvent être évalués sans que l'on ait besoin de<br />
faire appel a des renseignements sur ce bassin en question. Une<br />
fois que <strong>le</strong>s paramètres du modè<strong>le</strong> sont déterminés, on peut produire<br />
artificiei<strong>le</strong>ment un tracé important (long traces) des débits<br />
mensuels en se servant seu<strong>le</strong>ment, pour alimenter <strong>le</strong> modè<strong>le</strong>, des<br />
précipitations j ournaïières .
546<br />
The design of surface water supply systems requires an<br />
estimate of streamflow volume characteristics. Ideally, <strong>the</strong><br />
designer would like to have availab<strong>le</strong> long streamflow records.<br />
Generally, for <strong>the</strong> design of water supply reservoirs, a long<br />
record of monthly runoff volumes would be sufficient. Unfortunate-<br />
ly <strong>the</strong>se streamflow records are not availab<strong>le</strong> for a vast majority<br />
of <strong>the</strong> streams draining watersheds of up to 1500 square kilometers.<br />
Thus, methods of estimating monthly runoff volumes for <strong>the</strong>se un-<br />
gaged basins are needed.<br />
Watershed models that simulate continuous streamflow records<br />
of fer promise in this area. Several comprehensive watershed<br />
models are presently availab<strong>le</strong> [1,2,3]. These models may be<br />
categorized as parametric models in that <strong>the</strong>y contain several<br />
parameters that must be estimated before <strong>the</strong>y can be applied to a<br />
particular watershed. The parameters must be determined manually<br />
and are based on a comparison of observed streamflows with<br />
simulated streamflows.<br />
Realizing <strong>the</strong> ability of computers to find parameter sets that<br />
satisfy certain objectives and <strong>the</strong> differences that occur when<br />
different peop<strong>le</strong> determine <strong>the</strong> values for model parameters, Liou<br />
141 attempted to develop a self-optimizing procedure for <strong>the</strong><br />
Stanford Watershed Model [i]. Fur<strong>the</strong>r work on <strong>the</strong> Stanford Model<br />
was done by Ross [SI in an attempt to relate <strong>the</strong> optimum model<br />
parameters to watershed characteristics.<br />
Haan [6] has developed a relatively simp<strong>le</strong> runoff model that<br />
enab<strong>le</strong>s one to estimate monthly streamflow from daily precipitation<br />
and estimated average daily potential evapotranspiration. The<br />
model contains four parameters that must be determined for each<br />
basin. These parameters are :<br />
f<br />
Smax<br />
Cmax<br />
= maximum infiltration rate (cm/hr).<br />
= maximum daily seepage loss (cm) .<br />
= water holding capacity of that part of<br />
<strong>the</strong> soil from which <strong>the</strong> evapotranspiration<br />
rate is <strong>le</strong>ss than <strong>the</strong> potential rate<br />
un<strong>le</strong>ss this portion of <strong>the</strong> soil is<br />
saturated (cm) .<br />
'Associate Professor, Agricultural Engineering Department ,<br />
University of Kentucky , Lexington , Kentucky, 40 506 , USA.
Fs = fraction of <strong>the</strong> seepage that becomes<br />
runoff (-1.<br />
The optimum values for <strong>the</strong>se parameters are defined to be<br />
those that minimize <strong>the</strong> sum of squares between <strong>the</strong> observed and<br />
simulated monthly runoff volumes. Thus, to get <strong>the</strong> optimum<br />
parameter values , some observed streamflow data is required.<br />
Work with <strong>the</strong> model has shown that two years of monthly streamflow<br />
data are usually sufficient to obtain a satisfactory set of<br />
optimum parameter values.<br />
The model has <strong>the</strong> capability of determining <strong>the</strong> optimum<br />
parameter values for a particular basin when provided with daily<br />
rainfall, average daily potential evapotranspiration by months,<br />
some observed monthly streamflowc for <strong>the</strong> basin, and a set of<br />
initial estimates for <strong>the</strong> parameter values.<br />
The optimization procedure that is<br />
univariate technique. The value of <strong>the</strong><br />
n<br />
c (Vo - Vs. ) is computed using <strong>the</strong><br />
i=l i 1<br />
parameter value. in this expression, n<br />
flow and Vo is <strong>the</strong> observed volume and<br />
streamflow $or <strong>the</strong> ith month. Next <strong>the</strong><br />
presently used is a simp<strong>le</strong><br />
objective function<br />
initial estimates for <strong>the</strong><br />
is <strong>the</strong> number of monaths of<br />
Vs. <strong>the</strong> simulated volume of<br />
vatue of one of <strong>the</strong> parameters<br />
is chanued bv a fixed amount and <strong>the</strong> obiective function is<br />
recomputed. &e vaiue of this parameter contiAues to be changed<br />
as long as <strong>the</strong> objective function is improving (getting smal<strong>le</strong>r).<br />
The o<strong>the</strong>r three parameters are adjusted in <strong>the</strong> same manner one at<br />
a time. Since <strong>the</strong>se parameters are not independent, <strong>the</strong> entire<br />
process is <strong>the</strong>n repeated one or two times. The result of this<br />
iterative process is taken as <strong>the</strong> optimum set of parameters.<br />
This model has been tested on 24 watersheds in Kentucky<br />
ranging in area from 1.74 to 1225 square kilometers and on 3 water-<br />
sheds in South Carolina ranging in area from O .ll to 2.27 square<br />
kilometers. The results of <strong>the</strong>se evaluations are given in 161 ,<br />
[71 and [8]. Defining <strong>the</strong> average prediction error (%) as 100<br />
times <strong>the</strong> absolute value of <strong>the</strong> difference between <strong>the</strong> observed<br />
and simulated average annual streamf low divided by <strong>the</strong> observed<br />
average annual streamflow, <strong>the</strong> average prediction error for <strong>the</strong>se<br />
27 watersheds is 4.0 percent. For <strong>the</strong>se watersheds, <strong>the</strong> average<br />
annual runoff varied from 18.7 to 48.6 cm.<br />
For streams on which <strong>the</strong>re are no records availab<strong>le</strong>, at <strong>le</strong>ast<br />
two procedures can be used to estimate <strong>the</strong> optimum parameter values.
548<br />
If sufficient time is availab<strong>le</strong>, a temporary stream gaging station<br />
can be established and operated for two or more years. This<br />
station would only have to provide information on <strong>the</strong> monthly<br />
flows. The data from this short-term gaging program could <strong>the</strong>n<br />
be used in <strong>the</strong> optimization scheme described earlier.<br />
Jarboe and Haan [7] have used a second technique for estimat-<br />
ing <strong>the</strong> four model parameters for ungaged basins. This method uses<br />
streamflow information from gaged basins in <strong>the</strong> vicinity of <strong>the</strong><br />
ungaged basin of interest. The optimum model parameters for <strong>the</strong><br />
gaged basins are determined and related to measureab<strong>le</strong> character-<br />
istics of <strong>the</strong> gaged basins. These relationships are <strong>the</strong>n used<br />
to estimate <strong>the</strong> model parameters for <strong>the</strong> ungaged basin.<br />
The basin characteristics used by Jarboe and Haan [7] are<br />
shown in Tab<strong>le</strong> 1. The four model parameters were related to <strong>the</strong>se<br />
factors using multip<strong>le</strong> linear regression. Twenty-three watersheds<br />
were included in <strong>the</strong> study. Six of <strong>the</strong> watersheds were se<strong>le</strong>cted<br />
at random and treated as ungaged basins.<br />
The remaining 17 basins<br />
were used in developing <strong>the</strong> following prediction equations for <strong>the</strong><br />
model parameters :<br />
fmx = 11.83 - 11.51 Smax - 0.0147 SdSb - 0.030 A H<br />
g<br />
- 0.334 PkFc + 0.692 VrPR<br />
= 0.073 + 0.0031 Wc + 0.00075 Iw L - 0.0021 P H<br />
nax a g<br />
+ 0.00011 FcL - 0.0057 V H<br />
r g<br />
C = 7.69 + 0.739 IwSb + 0.011 S H + 0.0243 FcIw<br />
d 9<br />
Fs = 0.325 + 0.0068 L + 0.444 PkSb + 0.00027 PsSd<br />
- 0.018 WcPk<br />
These equations should not be used on watersheds (1) greater<br />
than 100 square kilometers in area, (2) on urban watersheds, or<br />
(3) on watersheds that differ greatly in <strong>the</strong>ir hydrologic char-<br />
acteristics from <strong>the</strong> watersheds used to derive <strong>the</strong> equations.<br />
(1)<br />
(2)<br />
(3)<br />
(4)
Tab<strong>le</strong> 1. Watershed characteristics used by Jarboe and<br />
Haan [7] to estimate <strong>the</strong> water yield model<br />
parameters.<br />
Geomorphic Factors<br />
A Basin area (km')<br />
percent of basin under forest cover (%)<br />
Percent of basin in lakes and ponds (%)<br />
Slope of <strong>the</strong> main stream (%)<br />
Length of <strong>the</strong> main stream (km)<br />
%<br />
Soil Factors<br />
W Average availab<strong>le</strong> soil water capacity (cm)<br />
HC U. S. Departnuint of Agriculture, Soil<br />
Conservation Service hydrologic soil<br />
group converted to a numerical index<br />
from 1 to 4 (-1<br />
Sd Average soil depth (cm)<br />
P Average soil permeability (cm/hr)<br />
p:<br />
Average permeability of upper soil horizon<br />
(cm/hr)<br />
549<br />
Geologic Factors<br />
vr<br />
"Rock" volume = mean basin e<strong>le</strong>vation above<br />
basin out<strong>le</strong>t times <strong>the</strong> basin area (krn3)<br />
I,<br />
Water availability index (an index ranging<br />
from 1 to 4 depending on <strong>the</strong> ability of <strong>the</strong><br />
material underlying <strong>the</strong> basin to yield water<br />
to wells) (-)
5 50<br />
Tab<strong>le</strong> 2 presents a s-ry of <strong>the</strong> results of using <strong>the</strong> above<br />
4 equations to estimate <strong>the</strong> parameters of <strong>the</strong> model on <strong>the</strong> 6 basins<br />
that were taken as unqaged. The simulated runoff values were<br />
obtained by estimating <strong>the</strong> model parameters from equations 1<br />
through 4 and <strong>the</strong>n using <strong>the</strong>se estimated parameters in <strong>the</strong> water<br />
yield model to simulate monthly streamflows. The six watersheds<br />
listed in tab<strong>le</strong> 2, although actually gaged, were considered as<br />
ungaged and not used in developing equations 1 through 4. These<br />
results indicate that reasonably good estimates of runoff volumes<br />
can be made on watersheds for which no streamflow records are<br />
avai lab <strong>le</strong>.<br />
EXAMPLE APPLICATION<br />
The South Fork of <strong>the</strong> Litt<strong>le</strong> Barren River in Kentucky was<br />
used to illustrate <strong>the</strong> application of this model under various<br />
conditions. Streamflow records have been maintained by <strong>the</strong> U.S.<br />
Geological Survey for this watershed for <strong>the</strong> period October of<br />
1948 through September of 1970. A U.S. Wea<strong>the</strong>r Bureau rain gage<br />
at Edmonton, Kentucky, about 6 1/2 kilometers from <strong>the</strong> watershed,<br />
was used to provide <strong>the</strong> needed precipitation input. Some of <strong>the</strong><br />
watershed physical characteristics are given in tab<strong>le</strong> 3. This<br />
watershed was not se<strong>le</strong>cted because of <strong>the</strong> ability of <strong>the</strong> model to<br />
simulate its monthly flow, but because of <strong>the</strong> long gaging record<br />
for <strong>the</strong> stream that could be used to check <strong>the</strong> simulated results.<br />
Tab<strong>le</strong> 4 summarizes <strong>the</strong> various simulations made on <strong>the</strong> South<br />
Fork of <strong>the</strong> Litt<strong>le</strong> Barren River watershed. Methods a through d<br />
are examp<strong>le</strong>s of how <strong>the</strong> model might be used to simulate streamflow<br />
from a previously ungaqed area. Method a required no streamflow<br />
records in that <strong>the</strong> parameters were estimated from equation 1<br />
through 4. Methods b, c, and d illustrate how <strong>the</strong> model can be<br />
used if it is possib<strong>le</strong> to initiate a stream gaging program and<br />
col<strong>le</strong>ct 1, 2, or 3 years of data respectively on monthly runoff<br />
volumes. In method e <strong>the</strong> entire 22 years were used in a procedure<br />
described by Haan 161 to obtain <strong>the</strong> parameter values. In tab<strong>le</strong> 4<br />
<strong>the</strong> percent error is as previously defined, <strong>the</strong> correlation<br />
coefficient is <strong>the</strong> simp<strong>le</strong> correlation between <strong>the</strong> observed and<br />
simulated monthly flows for <strong>the</strong> entire 22 year period of record,<br />
and <strong>the</strong> slope is <strong>the</strong> slope of a simp<strong>le</strong> regression line relating<br />
<strong>the</strong> observed and simulated flows.
Tab<strong>le</strong> 2. Comparison of observed and simulated average<br />
annual runoff for six Kentucky watersheds when<br />
<strong>the</strong> mode1 parameters are estimated by equations<br />
(1-4).<br />
Observed<br />
Average<br />
Annual<br />
Watershed Runoff<br />
Helton Branch 43.59 crn<br />
McGills Creek 41.50<br />
Perry Creek 34.16<br />
Stillwater Creek 48.59<br />
Litt<strong>le</strong> Plum Creek 46.74<br />
N. F. Nolin River 39.90<br />
Simulated<br />
Average<br />
Annual<br />
Runoff<br />
44.63 cm<br />
46.41<br />
33.55<br />
42.98<br />
48.01<br />
43.46<br />
551<br />
Percent Watershed<br />
Error Area<br />
2<br />
2.4 2.20 km<br />
11.8 5.54<br />
1.8 4.45<br />
11.5 62.16<br />
2.7 13.33<br />
8.9 94.28<br />
Tab<strong>le</strong> 3. Physical characteristics of South Fork of <strong>the</strong><br />
Litt<strong>le</strong> Barren River watershed, Kentucky.<br />
Area<br />
Forest cover<br />
Lakes and ponds<br />
Slope of main stream<br />
Length of main stream<br />
Availab<strong>le</strong> soil water capacity<br />
Index of USDA hydrologic soil group<br />
Average soil depth<br />
Average soil permeability<br />
Average permeability of upper soil horizon<br />
II Rock I' volume<br />
Water availability index<br />
47.4 km2<br />
62 %<br />
0.11 %<br />
0.32 %<br />
15.96 km<br />
17.68 cm<br />
2.30<br />
84.84 cm<br />
3.02 cm/hr<br />
3.35 cm/hr<br />
2.25 km3<br />
2.0
552<br />
The mean annual runoff for <strong>the</strong> South Fork of <strong>the</strong> Litt<strong>le</strong><br />
Barren River is 50.17 cm. Thus an error of 1 percent represents<br />
an average annual error in <strong>the</strong> simulated runoff of O .5 cm. When<br />
<strong>the</strong> model parameters were calculated from equations 1 through 4,<br />
<strong>the</strong> error in <strong>the</strong> average annual runoff was 4.3 cm. Figure 1 shows<br />
a portion of <strong>the</strong> simulated and observed streamflows for <strong>the</strong> watersheds.<br />
The simulated monthly runoff shown in this figure were<br />
obtained using parameters calculated from equations 1 through 4<br />
in Haan's [6] wateryield model.<br />
Again it is cautioned that <strong>the</strong>se<br />
equations may not produce reliab<strong>le</strong> parameter estimates for regions<br />
hydrologically different than Kentucky. The technique of deriving<br />
parameter prediction equations should, however, be valid else-<br />
where.<br />
Methods b, c, and d of tab<strong>le</strong> 4 illustrate how a few years of<br />
streamflow data can be used to estimate model parameters which in<br />
turn can be used to simulate long traces of monthly flows. The<br />
variab<strong>le</strong> nature of streamflow from year to year is apparent in<br />
<strong>the</strong> runoff records from this watershed. As an examp<strong>le</strong> <strong>the</strong> first<br />
three years of <strong>the</strong> 22 year record produced <strong>the</strong> highest, third<br />
highest, and sixth highest annual runoff. The average annual run-<br />
off for <strong>the</strong> first three years was 75.79 cm as compared to 50.17 cm<br />
for <strong>the</strong> entire period of record. It was <strong>the</strong>se three wet years<br />
that were used in determining <strong>the</strong> model parameters indicated in<br />
tab<strong>le</strong> 4 under methods b, c, and d. This indicates that even<br />
though <strong>the</strong> years used in obtaining <strong>the</strong> model parameters may not be<br />
representative, reasonab<strong>le</strong> estimates of streamflow can still be<br />
obtained.<br />
Tab<strong>le</strong> 4 also indicates that <strong>the</strong> accuracy of <strong>the</strong> simulation<br />
depends on <strong>the</strong> years used in determining <strong>the</strong> model parameters.<br />
The fact that using two years of flow data to obtain <strong>the</strong><br />
parameter values produced better simulated results for <strong>the</strong> entire<br />
22 year period than did <strong>the</strong> parameters obtained from three years<br />
of data is not unusual; however, in general <strong>the</strong> more years used<br />
to obtain <strong>the</strong> parameters, <strong>the</strong> better will be <strong>the</strong> simulated<br />
results.<br />
Method e consisted of (1) optimizing <strong>the</strong> model on <strong>the</strong> first<br />
year of record, (2) simulating <strong>the</strong> entire 22. years of flow with<br />
<strong>the</strong>se parameters, (3) reoptimizing <strong>the</strong> model on <strong>the</strong> 2 years from<br />
<strong>the</strong> entire 22 year record that produced <strong>the</strong> poorest fit, and<br />
(4) finally determining <strong>the</strong> final parameters as a weighted<br />
average of <strong>the</strong> resulting two optimum sets of parameters where <strong>the</strong><br />
weighting factors are <strong>the</strong> sum of <strong>the</strong> deviations of observed flows<br />
from simulated flows. The parameters obtained in this manner
Tab<strong>le</strong> 4. Methods used to optimize parameters on S.F.L. Barren<br />
River and summary of simulation results.<br />
Method<br />
~<br />
Des cri D ti on<br />
(a)<br />
íb 1<br />
Parameters calculated from equations 1-4.<br />
Parameters determined by optimization on first year of<br />
data.<br />
(Cl Parameters determined by optimization on first two years<br />
of data.<br />
(dl<br />
(e)<br />
Parameters determined by optimization on first three<br />
years of data.<br />
Parameters optimized by Jarboe 181.<br />
Percent Correlation<br />
C<br />
fmax 'ma,<br />
Method Error Coefficient Slope cm/hr cm/day cm<br />
(a) 8.64 0.91 0.93 3.58 0.21 13.33 0.49<br />
(b) 10.13 0.87 0.96 3.45 0.25 16.38 0.54<br />
(c) 2.19 O .92 0.91 3.58 0.20 . . 12.57 0.54<br />
(d) 9.38 O .92 0.89 5.61 0.22 11.81 0.69<br />
(e) O .56 o .91 0.92 3.30 0.22 12.45 0.58<br />
when used with <strong>the</strong> watershed model were ab<strong>le</strong> to simulate <strong>the</strong> 22<br />
years of record with an average annual error of only 0.56 percent<br />
or 0.28 cm. Obviously this technique cannot be used on a data<br />
scarce watershed. It is included here only to provide an<br />
indication of <strong>the</strong> ability of <strong>the</strong> model to simulate monthly stream-<br />
flows.<br />
This model like most parametric hydrologic models, is<br />
in a constant state of change as improvements are incorporated to<br />
make <strong>the</strong> model easier to use, to reduce computer processing time<br />
and to increase <strong>the</strong> accuracy of <strong>the</strong> simulations.<br />
FS<br />
553
554<br />
SUMMARY<br />
Two procedures for using a four parameter water yield<br />
model for simulating traces of monthly streamflaw from watersheds<br />
with ei<strong>the</strong>r no or very limited streamflow information are<br />
presented. The two procedures are (1) to relate <strong>the</strong> model<br />
parameters to watershed physical characteristics using stream-<br />
flow data from watersheds located near <strong>the</strong> watershed of interest<br />
or (2) to establish a short term gaging program on <strong>the</strong> stream<br />
draining <strong>the</strong> watershed and use <strong>the</strong>se streamflow records to<br />
determine <strong>the</strong> model parameters. Once <strong>the</strong> model parameters are<br />
determined, long streamflow traces can be generated using ei<strong>the</strong>r<br />
measured or syn<strong>the</strong>tic daily rainfall. These two procedures were<br />
illustrated on a watershed in Kentucky and demonstrated that<br />
reasonably accurate estimates of monthly streamflow can be<br />
obtained.<br />
Acknow<strong>le</strong>dgements: The work in which this report is based was<br />
supported in part by <strong>the</strong> Kentucky Division of Water and in part<br />
by <strong>the</strong> Kentucky Agricultural Experiment Station as a contribution<br />
to Sou<strong>the</strong>rn Regional Project S-53 "Factors Affecting Water Yields<br />
from Small Watersheds and Shallow Ground Aquifers". The paper<br />
is published with <strong>the</strong> approval of <strong>the</strong> Director of <strong>the</strong> Kentucky<br />
Agricultural Experiment Station.
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
BIBLIOGRAPHY<br />
555<br />
Crawford, N. H. and R. K. Lins<strong>le</strong>y. (1966). Digital simul-<br />
ation in hydrology: Stanford watershed model IV. Technical<br />
Report 39 , Stanford University , Department of Civil<br />
Engineering, Stanford, California.<br />
Holtan, H. N. and N. C. Lopez. (1971). USDAHL-70 Model of<br />
Watershed Hydrology. Technical Bul<strong>le</strong>tin No. 1435, Agricultural<br />
Research Service, U. S. Department of Agriculture, Washington,<br />
D.C. 84 pp.<br />
Tennessee Val<strong>le</strong>y Authority. (1972). Upper Bear Creek<br />
Experimental Project: A continuous daily streamflow model.<br />
Division of Water Control Planning, Hydraulic Data Branch,<br />
Knoxvil<strong>le</strong>, Tennessee. 99 pp.<br />
Liou, E. Y. (1970). OPSET: Program for computerized<br />
se<strong>le</strong>ction of watershed parameter values for <strong>the</strong> Stanford<br />
Watershed Model. University of Kentucky Water Resources<br />
Research Report 34, Lexington, Kentucky.<br />
Ross, G. A. (1970). The Stanford Watershed Model: The<br />
correlation of parameter values se<strong>le</strong>cted by a computerized<br />
procedure with measureab<strong>le</strong> physical characteristics of <strong>the</strong><br />
watershed. University of Kentucky Water Resources Institute<br />
Research Report 35 , Lexington, Kentucky.<br />
Haan, C. T. (1972). A water yield model for small watersheds.<br />
Water Resources Research 8 (No. 1) , pp 58-69.<br />
Jarboe, J. E. and C. T. Haan. (1972). Calibration of a four-<br />
parameter water yield model to small ungaged watersheds in<br />
Kentucky. Paper No. 73-207 for presentation at <strong>the</strong> 1973<br />
Annual Meeting of <strong>the</strong> American Society of Agricultural<br />
Engineers, Lexington, Kentucky, June 17-20, 1973.<br />
Jarboe, J. E. (1972). Calibration of a four-parameter water<br />
yield model for use on small, ungaged watersheds in Kentucky.<br />
Unpublished M. S. Thesis in Civil Engineering Library , University<br />
of Kentucky, Lexington, Kentucky.
556<br />
20[ PART OF<br />
- S.F.L. BARREN R.<br />
- RUNOFF RECORD<br />
- -<br />
OBSERVED RO O<br />
SIMULATED RO -<br />
TIME<br />
Figure 1. Examp<strong>le</strong> of monthly streamflaw simulation<br />
results using calculated model parameters.
AB ST RACT<br />
OBTAINING DEFICIENT INFORMATION BY SOLVING<br />
INVERSE PROilLEMS FOR MATHEMATICAL RUNOFF MODELS<br />
V.I. Koren and L.S. Kutchment?:<br />
Possibilities are considered for increase of deficient<br />
information for extending observation series by solving <strong>the</strong><br />
"inverse prob<strong>le</strong>mt1 for ma<strong>the</strong>matical runoff models. The results<br />
of applying <strong>the</strong> <strong>the</strong>ory of "improperly posed prob<strong>le</strong>ms'' are<br />
presented. Examp<strong>le</strong>s are given for representing hydrological,<br />
geometrical and hydraulic characteristics of <strong>the</strong> basin by<br />
lumped and distributed parameter runoff models.<br />
RESUME<br />
Les auteurs examinent <strong>le</strong>s possibilités de la resolution<br />
du problème inverse appliquée aux modè<strong>le</strong>s mathématiques d'eco!<br />
<strong>le</strong>ment, en vue de compléter <strong>le</strong>s lacunes des séries d'observa-<br />
tions et d'étendre la période couverte par ces séries. Ils ex-<br />
posent <strong>le</strong>s résultats qui ont été obtenus par l'application de<br />
la théorie des problèmes posés incorrectement. Ils citent des<br />
exemp<strong>le</strong>s de détermination des caractéristiques hydrologiques,<br />
topographiques et hydrauliques a l'aide de modè<strong>le</strong>s d'ecou<strong>le</strong>ment<br />
globaux ou matriciels.<br />
:k Hidrometeorological Centre of <strong>the</strong> USSR.
558<br />
Ma<strong>the</strong>matical modelling of hydrological processes is increas-<br />
ingly used to provide for missing information and to extend hydrolo-<br />
gical time series. Ma<strong>the</strong>matical models are predominantly used for<br />
<strong>the</strong> solution of <strong>the</strong> so-cal<strong>le</strong>d 'direct prob<strong>le</strong>m', consisting of deriv-<br />
ation of unknown hydrological variab<strong>le</strong>s by solving respective differ-<br />
ential equations with known coefficients and known initial and bound-<br />
ary conditions, In a large number of cases it is necessary to solve<br />
<strong>the</strong> 'inverse prob<strong>le</strong>m' namely to find <strong>the</strong> coefficients and establish<br />
<strong>the</strong> initial and boundary conditions using observed values of <strong>the</strong><br />
hydrological variab<strong>le</strong>s included in <strong>the</strong> equations. This approach has<br />
as yet gained relatively rare use due to <strong>the</strong> fact that <strong>the</strong> solution<br />
of <strong>the</strong> 'inverse prob<strong>le</strong>m' is more difficult than that of <strong>the</strong>ldirect<br />
prob<strong>le</strong>m'. The solution of <strong>the</strong> 'inverse prob<strong>le</strong>m' may be circumvented<br />
by multip<strong>le</strong> solutions of <strong>the</strong> 'direct prob<strong>le</strong>m' for examp<strong>le</strong> by <strong>the</strong><br />
methods of trial and error and subsequent optimization. Thia may<br />
<strong>le</strong>ad however to a non-unique or inferior solution. The principal<br />
difficulty in <strong>the</strong> solution of <strong>the</strong> inverse prob<strong>le</strong>m consists in <strong>the</strong><br />
fact that it may be incorrectly posed and thus <strong>le</strong>ads to <strong>the</strong> non-<br />
existence of some or any initial conditions or <strong>le</strong>adsto a solution<br />
in which a small change of initial conditions (data) due for examp<strong>le</strong><br />
to observational errors, results in major changes in <strong>the</strong> results.<br />
This has caused in <strong>the</strong> past a reluctance toward <strong>the</strong> use of this<br />
method, since <strong>the</strong> solution being of very low accuracy and high un-<br />
certainty casts doubt on its physical significance.<br />
A number of studies were made in recent years (particularly<br />
by A.N. Tikhonov and his school) aiming at <strong>the</strong> correct posing of<br />
<strong>the</strong> prob<strong>le</strong>m by establishing <strong>the</strong> necessary conditions for it.<br />
A.N. Tikhonov has shown that it is possib<strong>le</strong> to u13e a priori inform-<br />
ation on <strong>the</strong> solution to ensure a continuous dependance of <strong>the</strong><br />
solution of an incorrectly posed prob<strong>le</strong>m on its initial conditions<br />
and to derive special algorithm:: which prevent bringing out <strong>the</strong> solution<br />
outside <strong>the</strong> limits of its uniqueness and of <strong>the</strong> existence of its initial<br />
conditions. In particular it made possib<strong>le</strong> to solve with sufficient<br />
stability such classical incorrectly-posed prob<strong>le</strong>ms as <strong>the</strong> integral<br />
equation of <strong>the</strong> first type, algebraic systems with improper initial<br />
conditions, <strong>the</strong> Cauchy prob<strong>le</strong>m c?f <strong>the</strong> Laplace equation and o<strong>the</strong>rs.<br />
The <strong>the</strong>ory of <strong>the</strong> 'inverse prob<strong>le</strong>m' has thus stimulated <strong>the</strong> formu-<br />
lation of algorithms used in many scientific and technical fields.<br />
The method was particularly useful in geophysics, where it permitted<br />
<strong>the</strong> solving, for examp<strong>le</strong>, of prob<strong>le</strong>ms of determination of rock charac-<br />
teristics not accessib<strong>le</strong> for direct measurement as well as restora-<br />
tion of missing information, to cite only <strong>the</strong> most important points.<br />
The use of this method in hydrology appears also as most promising.<br />
Examp<strong>le</strong>s of such studies, used in hydrological practice, are given<br />
below. They illustrate also <strong>the</strong> princip<strong>le</strong>s and possibilities of <strong>the</strong><br />
<strong>the</strong>ory of incorrectly posed prob<strong>le</strong>ms.<br />
1. Determination of <strong>the</strong> input functions of <strong>the</strong> models<br />
with lump parameters<br />
Let us suppose that <strong>the</strong> process of transforming an input h(t)<br />
in <strong>the</strong> catohment (effective rainfall or an inflow) into an output<br />
Q(t) can be described by <strong>the</strong> Duhamel integral:
559<br />
where P(t) is some known function of influence. Then having <strong>the</strong> observations<br />
on Q(t) and knowing <strong>the</strong> function P( t) (by historic observations or<br />
from physiographic and hydraulic data) it is possib<strong>le</strong> using (1) to derive<br />
h(t). Thus an improperly posed prob<strong>le</strong>m is solved - consisting of an integral<br />
equation of <strong>the</strong> first type. It is possib<strong>le</strong> to solve this prob<strong>le</strong>m<br />
on <strong>the</strong> basis of A.N. Tikhonov's algorithm. Integral (1) is replaced by<br />
a summation according to <strong>the</strong> method of rectang<strong>le</strong>s and a smoo<strong>the</strong>d functional<br />
curve is constructed:<br />
-b<br />
3<br />
where Q = a vector, designating <strong>the</strong> Ordinates of <strong>the</strong> given hydrograph Q(t);<br />
h = a vector of <strong>the</strong> unknown ordinates h A = a matrix with e<strong>le</strong>ments<br />
5'<br />
P ; d= a positive Constant. Finding <strong>the</strong> minimum of this functional<br />
mk&'it possib<strong>le</strong> to receive a sequence of stab<strong>le</strong> solutions %, which<br />
converge to <strong>the</strong> accurate solution providing <strong>the</strong>re are no errors in <strong>the</strong><br />
given data. However since <strong>the</strong>re are always errors in <strong>the</strong>se, changing<br />
<strong>the</strong> parameter &(cal<strong>le</strong>d parameter of regularization) we se<strong>le</strong>ct such<br />
solution which corresponds best to <strong>the</strong> a priori information about <strong>the</strong><br />
function h(t). For exam e good results are obtained with <strong>the</strong> aid of<br />
<strong>the</strong> condition Th(t)dt=$(t)dt.<br />
o<br />
O<strong>the</strong>r kinds of a priori information, allowing <strong>the</strong> narrowing<br />
of <strong>the</strong> interval of unknown solutions, may be a suggestion on <strong>the</strong> smoothness<br />
of <strong>the</strong> solution, <strong>the</strong> non-negativeness of <strong>the</strong> ordinates, <strong>the</strong> closeness<br />
to some known function and so on. Naturally, <strong>the</strong> narrower <strong>the</strong> interval<br />
of <strong>the</strong> solution, <strong>the</strong> higher accuraoy will be obtained.<br />
Results in using<br />
functional (2) to determine <strong>the</strong> input functions of <strong>the</strong> runoff models,<br />
described by <strong>the</strong> Duhamel integral, are presented in greater detail<br />
in (3)' where examp<strong>le</strong>s of constnicting effective rainfall, hydropower<br />
station re<strong>le</strong>ases and snowmelt intensity are treated. Ano<strong>the</strong>r approach<br />
to <strong>the</strong> solution of <strong>the</strong> inverse prob<strong>le</strong>ms for models described by <strong>the</strong><br />
Duhamel integral (linear models with lump parameters) are indicated in<br />
(6).<br />
2. Determination of geometric and hydraulic charactexistics<br />
of river channels using observations of flow<br />
To describe unstea9flow in a river channel Saint Venant<br />
equations may be used:<br />
(3)
560<br />
where 2 (x,t) = stage at point x at time t, Q(x,t) = discharge, K(x,z)<br />
forces of resistance1 g= acce<strong>le</strong>ration of gravity. Because of great<br />
variability of geometry and roughness of <strong>the</strong> river channels <strong>the</strong><br />
functions F(x,z) and K(x,z) determined by <strong>the</strong> observations in<br />
separate points are not quite representative for <strong>the</strong> <strong>who<strong>le</strong></strong> river reach,<br />
even with large frequency of observations. Thus a prob<strong>le</strong>m of determining<br />
<strong>the</strong> averaged relations P(x,z) or B(x,z) = aF/aZ and K(x,z) by observations<br />
of flow (<strong>the</strong> determination of coefficients of <strong>the</strong> system (3))<br />
is of great significance for <strong>the</strong> establishment of <strong>the</strong> most characteristic<br />
geometry and hydraulic properties of <strong>the</strong> river channel as well as<br />
for ensuring sufficient accuracy of <strong>the</strong> calculationa. It can be shown<br />
that this prob<strong>le</strong>m is improperly posed and for its solution it is<br />
necessary to derive special calculating algorithms. We shall discuss<br />
below two of <strong>the</strong> approaches tried by us in solving this prob<strong>le</strong>m.<br />
(A) The discharges and Water <strong>le</strong>vels are known in a ra<strong>the</strong>r large<br />
number of sites.<br />
Integration of <strong>the</strong> continuity equation (3) with respect to x,<br />
<strong>le</strong>ads to:<br />
Finite differences are substituted for <strong>the</strong> derivatives and<br />
instead of an integral it is possib<strong>le</strong> to construct for every time moment j<br />
<strong>the</strong> following system of equationst<br />
In order to solve this system it is necessary to have Q(x,t) F(x,o) and<br />
F(o,t). As <strong>the</strong> prob<strong>le</strong>m is improperly posed <strong>the</strong> solution of <strong>the</strong><br />
system (5) is unstab<strong>le</strong>. For its regularization <strong>the</strong> solution of<br />
A.N. Tikhonov's functional is with introducing initial approximation.<br />
As a result for every time suchFarefoanä which correspond to <strong>the</strong><br />
minimum of <strong>the</strong> functional.
561<br />
where 2 is <strong>the</strong> given initial approximation, d= positive parameters,<br />
thus a golution is found which not only secures <strong>the</strong> minimum of square<br />
deviation of <strong>the</strong> right part of <strong>the</strong> system (5) from <strong>the</strong> <strong>le</strong>ft part, but<br />
at <strong>the</strong> same time it is <strong>le</strong>ast deviated from <strong>the</strong> initial approximation.<br />
The condition of functional extreme gives:<br />
To se<strong>le</strong>ct <strong>the</strong> quantitydmethod of discrepancy has been used.<br />
The idea of this methori COnSiStS in conforming <strong>the</strong> accuracy of <strong>the</strong><br />
prob<strong>le</strong>m's solution to <strong>the</strong> accuracy of observed data.<br />
It is supposed that <strong>the</strong> error0 of <strong>the</strong> given information forming<br />
discrepancy of <strong>the</strong> system (5) are known and an d is found which<br />
secures this discrepancy 8'. It is possib<strong>le</strong> to prove that if <strong>the</strong><br />
functional (6) is used <strong>the</strong> parameterdsecuring <strong>the</strong> given discrepancy<br />
is unique. The initial pproximation can be made in a ra<strong>the</strong>r crude<br />
mannerbarticularly for 9 = O), however giving a good initial approximation<br />
contributes toam-aocurate optimum d. Use of <strong>the</strong> initial<br />
approximations provides great possibilities for improvement of <strong>the</strong><br />
solution by introduction of a priori information. Such a priori<br />
information can be an empirical relationship between geometrical and<br />
hydraulic characteristics, observed in separate sites, and different<br />
<strong>the</strong>oretical formulas (for examp<strong>le</strong>, we have used <strong>the</strong> equation of <strong>the</strong><br />
typical form of river Otrinnel derived from <strong>the</strong> princip<strong>le</strong> of minimum<br />
dissipation of energy).<br />
The values of F (x,t) found according to equation (4) have<br />
been used for determining <strong>the</strong> characteristics of <strong>the</strong> resistant forces.<br />
For this purpose <strong>the</strong> momentum equation has transcribed:<br />
Derivatives with respect to t have been replaced by forward directed<br />
finite differences and <strong>the</strong> integrals have been replaced by sums de-<br />
rived by <strong>the</strong> method of rectang<strong>le</strong>s. The resulting algebraical systems<br />
have been solved for all time intervals with <strong>the</strong> help of <strong>the</strong> same<br />
algorithm as <strong>the</strong> system (5) (without <strong>the</strong> initial approximation).<br />
Aa for determining F(x,t) and K(x,t) <strong>the</strong> discrepancy has been<br />
taken equal to 5 per cent of <strong>the</strong> average modu<strong>le</strong> from <strong>the</strong> <strong>le</strong>ft integral<br />
equation's part.<br />
This method has been tested on data obtained by special<br />
observations of unsteady movement in <strong>the</strong> merca river and it has<br />
given satiafactory results (a comparison ha8 been made between <strong>the</strong><br />
relations F(x,z) and K(x,z) which have been derived by different<br />
floods by measurements in separate sites) (see figure 1).
562<br />
(B) The stages are known in a ra<strong>the</strong>r great number of sites and<br />
<strong>the</strong> discharges only in <strong>the</strong> first and <strong>the</strong> last site.<br />
Let us integrate <strong>the</strong> continuity equation with respect to<br />
time (in <strong>the</strong> interval (Ti, Ti+l)) and to distance (in <strong>the</strong> interval<br />
(0, LI):<br />
to solve it in <strong>the</strong> form:<br />
where ‘y- <strong>the</strong> Chebishev polinomials. Let us put (10) in (9):<br />
No terms with zero polinomial are in <strong>the</strong> <strong>le</strong>ft part of <strong>the</strong> equation (li),<br />
because in this case <strong>the</strong> integral would be equal to zero. The equation<br />
(li) is <strong>the</strong>refore not sufficient for <strong>the</strong> full determination of <strong>the</strong><br />
function (10). However it can be used for determining <strong>the</strong> function<br />
B(x,z), which can be presented:<br />
To find <strong>the</strong> coeffiaients we shall construct a system of equations<br />
(<strong>the</strong>ir number must not b e h w than Y=(n +l)m , and change <strong>the</strong> limita<br />
of <strong>the</strong> integration with respect to time in (dl so as to embrace <strong>the</strong><br />
<strong>who<strong>le</strong></strong> amplitude of variation of discharges and of stages on <strong>the</strong> rising<br />
as well as on <strong>the</strong> falling, part of <strong>the</strong> hydrograph. Let us write this<br />
system in <strong>the</strong> matrix forms<br />
Bere is <strong>the</strong> matrix of +th order, its e<strong>le</strong>ments are equal
563<br />
id- vector of <strong>the</strong> unknown coefficients $(B. x - right part with e<strong>le</strong>ments:<br />
Since <strong>the</strong> system (13) is unstab<strong>le</strong>, ita solution is possib<strong>le</strong><br />
with A.N. Tikhonov's functional. AS a result <strong>the</strong> following system is<br />
found :<br />
where%'* - matrix transformed with relation top, E - <strong>the</strong> unit matrix.<br />
The parameter of regularization d has been determined from <strong>the</strong><br />
conditions of minimum of <strong>the</strong> function<br />
where A.PP,,), A (dp) - j-th e<strong>le</strong>ments for <strong>the</strong> two successive<br />
values JO(. +or determining (ni + i) coefficients entering in (10)<br />
we shall replace in (9) discharge with <strong>the</strong> product of a cross section<br />
area and <strong>the</strong> velocity of <strong>the</strong> current U(x,t) and shall make <strong>the</strong> proper<br />
integration with respect to time and to distance. Putting in <strong>the</strong><br />
resulting equation <strong>the</strong> relation (10) we shall find:<br />
Here C - matrix of (nI + I) x N-th order with e<strong>le</strong>ments<br />
The rest of symbols are <strong>the</strong> same. lystem (17) is solved by analogy<br />
with system (13). Having determined 3 and it is possib<strong>le</strong>, asing<br />
relations (10) and (12) to find function B(x,z).<br />
This approach has<br />
been tested on <strong>the</strong> data of special observations in <strong>the</strong> Svir river.<br />
In figure 2 functions B(x,z) for some sites, calculated by relation (10)<br />
are shown: fur<strong>the</strong>rmore widths were determined aocording to topographic<br />
data. For controlling <strong>the</strong> results of <strong>the</strong>se calculations <strong>the</strong> discharges<br />
in-<strong>the</strong> intermediate sites have been determined with <strong>the</strong> help of equation:
564<br />
These discharges have been found as very close to those observed.<br />
The coefficients received from <strong>the</strong> different floods have turned out<br />
to be quite simila % and this fact indicates <strong>the</strong>ir sufficient stability.<br />
Let vs see now a scheme for determining <strong>the</strong> hydraulic'characteristics<br />
of river channels. We use <strong>the</strong> dynamic St. Venant equation, assuming<br />
that <strong>the</strong> ineLtial terms are equal to zero<br />
Putting (18) into (19) and integrating with respect to distance in<br />
<strong>the</strong> interval (0,ï) we gett<br />
às earlier we shall find <strong>the</strong> solution in <strong>the</strong> form:<br />
Putting (21) into (20) and using Tikhonov's functional by analogy with<br />
<strong>the</strong> previous one we construct <strong>the</strong> system of equations for determining<br />
<strong>the</strong> coefficients Dks:<br />
-b<br />
where D - vector of unknown coefficients, 3- vector with e<strong>le</strong>ments<br />
&=Z(t) - Z(t), - matrix of (n2+1).(m2+l) N-th order with e<strong>le</strong>ments<br />
The function B(t, t) hae been calculated according to relation<br />
(12) including <strong>the</strong> earlier determined coefficients Ilks. The found<br />
functions have been oompared with <strong>the</strong> functions determined by <strong>the</strong> method<br />
of optimization. It was found that a strong smoothing is observed.<br />
This can be eliminated by taking logarithms in equation (19).
References<br />
565
566<br />
I50<br />
SO<br />
/Iff<br />
96<br />
a 0<br />
e<br />
E<br />
a *'<br />
0<br />
8<br />
e<br />
O<br />
e<br />
8<br />
b
x r acm<br />
I l<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
15.0 250 2.50<br />
I I I<br />
200 390 4uu<br />
.x= 3.9cm<br />
I I I<br />
i50 250 350<br />
i<br />
/<br />
567
ABS TRACT<br />
THE MATHEMATICAL MODEL OF<br />
WATER BALANCE FOR DATA-SCARCE AREAS<br />
by<br />
Nabil Rofail<br />
The ma<strong>the</strong>matical model of water balance for data-scarce areas<br />
is designed. The solution of this prob<strong>le</strong>m is considered of general<br />
type of boundary conditions.<br />
The equations of motion and mass conservation <strong>le</strong>ad to <strong>the</strong> lin<br />
ear parabolic partial differential equation. The equation is solved<br />
by implicit scheme and <strong>the</strong> alternating direction procedure is appli<br />
ed for computation. The numerical procedure has second order accu-<br />
rracy and is unconditionally stab<strong>le</strong>. As it contains no iterative --<br />
routines, it is also exceptionally economical in computing time and<br />
memory requiments. Therefore <strong>the</strong> procedure is recommended for areas<br />
of inadequate hydrological data.<br />
RESUME<br />
L'auteur décrit un modè<strong>le</strong> mathématique de bilan hydrologique<br />
destiné aux régions pour <strong>le</strong>squel<strong>le</strong>s on dispose de peu de données. -<br />
La solution du prob<strong>le</strong>me est envisagée pour des conditions aux limi-<br />
tes généra<strong>le</strong>s.<br />
Les équations du mouvement et de la conservation de masse con<br />
duisent à une équation aux différentiel<strong>le</strong>s partiel<strong>le</strong>s linéaire para<br />
bolique. Cette équation constitue un système implicite qu'on résoud<br />
par approximations successives. Le mode de calcul numérique est in-<br />
conditionnel<strong>le</strong>ment stab<strong>le</strong> et permet une précision du second ordre.<br />
Comme il ne contient pas de procédé itératif, il est aussi exception<br />
nel<strong>le</strong>ment économique en temps de calcul e; en dimension de mémoire.<br />
I1 est donc recommande pour <strong>le</strong>s régions ou <strong>le</strong>s données hydrologiques<br />
sont insuffisantes.<br />
~~ ~ ~<br />
* Water Resources Dept. Desert Institute, Mataria, Cairo, Egypt.<br />
-
570<br />
Introduction<br />
The system of equations of motion and equations of mass con-<br />
servation for <strong>the</strong> ground water flow <strong>le</strong>ads to linear pzrabolic di-<br />
fferential equations. In <strong>the</strong> present work <strong>the</strong> impervious bed has<br />
been considered of any configuration and <strong>the</strong> recharge or dischar-<br />
ge from <strong>the</strong> aquifer has been introduced. Moreover <strong>the</strong> effect of<br />
boundnries are considered ei<strong>the</strong>r of a river or of a continuation<br />
of <strong>the</strong> aquifer of different parameters. The present aaterial deals<br />
with <strong>the</strong> solution of <strong>the</strong> balance equation that can be easily app-<br />
lied for areas of inadequate hydrologicd. data.<br />
Formulation of muation<br />
The aquifer is considered homogenous and <strong>the</strong> effect of <strong>the</strong><br />
impervious bed is &%Ven. According to Dupuit assumption, <strong>the</strong> equ-<br />
ation of motion cam be written as followsj ( see Fig. i) ,<br />
v = - k 2 i h - t ~ ) = - k a h - k dz (2)<br />
"Y "Y<br />
The equation of mass conservation can be considered as follows;
!he dischuge or <strong>the</strong> recharge to <strong>the</strong> aquifer is considered as a<br />
:unction of time to <strong>the</strong> exposed area i.e. N = f ( x,y,t) equat-<br />
tons (1),(2) and (3) proYide <strong>the</strong> following equation system,<br />
- k d(h+z) ah<br />
- k h QQ(lZz'<br />
- kh --<br />
9X ax<br />
Q'íh+z) - k a(h+z) ah - N =a (4)<br />
9 Y* ay al/<br />
?or simplifying <strong>the</strong> solution of equation (4), Boussensq acsump-<br />
:ion ( <strong>the</strong> powers of derivatives of one order, SLTB of lower ma-<br />
pitude than <strong>the</strong> derivatives <strong>the</strong>mselves ) is applied, <strong>the</strong>refore<br />
<strong>the</strong> system will be,<br />
kherefore equation (5) can be written in <strong>the</strong> alternate farm,<br />
571
512<br />
\A consistent imdicit difference scheme<br />
The three implicit difference scheme has been applied for equ-<br />
ations (6) and (7), considering <strong>the</strong> grid spacing of A5 and time<br />
intervai At, ( see fig. 2 >. Thus <strong>le</strong>ads to;<br />
AS<br />
and (g), such that;<br />
Equation (10) is found by Taylor's series expansion, to be term<br />
by term consistent with equation (5). The von Neuinann method is<br />
used for examining <strong>the</strong> stability of <strong>the</strong> finite deference scheme.<br />
It has been found out that <strong>the</strong> amplification factor 5 1, that<br />
1mAS<br />
<strong>the</strong> original component e will not increase with time.<br />
Therefore <strong>the</strong> scheme can be considered absolutely consistent to<br />
second order accuracy, i.e. to O ( Asz, A tz), and absolutely
unconditionally stab<strong>le</strong>. Thus it is not unexpected as <strong>the</strong> applied<br />
scheme is implicit.<br />
Alternating Direction Algorithon<br />
Equations (8) and (9) can be written respectiviely in <strong>the</strong><br />
following form,<br />
@here, A,B,C and D are coefficients of known values,i.e.,<br />
573
574<br />
Each of equations of equation5 (11) a d (12) forms a tridiago1<br />
al vector system that may be solved using <strong>the</strong> aìternating directior<br />
algorithm (e.g. Richtmyer and Mooton,1967 1, by introducing auxili-<br />
ary variab<strong>le</strong> E and P in <strong>the</strong> x sweep as follows:<br />
Introducing (11) in (131, that provides recurrence relations;<br />
)-I , ,$= Cq- A, $+,)(A,%+,+%)<br />
5<br />
= - CJ ( AJ El,, +B,<br />
Method of Computation:<br />
Thensgion of interest in <strong>the</strong> x-y plane in which <strong>the</strong> numurical<br />
solution of equation (5) is carried out, is divided by a mesh of g~<br />
id lines. The distance between grid lines need not be <strong>the</strong> same, ths<br />
is considered one of <strong>the</strong> reasons for recommending this metnod of cc<br />
putption for areas of inadequate hydrological data? Equation (8) is<br />
solved for <strong>the</strong> x sweep for <strong>the</strong> time <strong>le</strong>vel (9) to <strong>the</strong> time <strong>le</strong>vel<br />
( n + j$ 2, and equation (9) for y-sweep for <strong>the</strong> time <strong>le</strong>vel ( n +<br />
to <strong>the</strong> time <strong>le</strong>vel ( n 4 1 ). The recurrence can be initiated from<br />
boundpry conditions of any two - point type; two velocities ( con-<br />
tinuiation of <strong>the</strong> aquifer ), two depths ( bounded by rivers ), one<br />
velocity and one depth.<br />
If <strong>the</strong> boundary is a continuiation of <strong>the</strong> aquifer, this could<br />
be illustrated as follows;
.s <strong>the</strong> boundary point is situated at JJ A x, <strong>the</strong> equation (14) can<br />
expressed as follows,<br />
r <strong>the</strong>, case where <strong>the</strong> boundary is bounded by a river, <strong>the</strong>refore;<br />
575<br />
The comput.afbncan be applied for <strong>the</strong> x-sweep by determining <strong>the</strong><br />
zfficients P and P from one bounary to <strong>the</strong> o<strong>the</strong>r bounduy and bet-<br />
3n <strong>the</strong>m for all grid points. Thus <strong>the</strong> new values of <strong>the</strong> potentials<br />
<strong>the</strong> time <strong>le</strong>vel ( n + % ) can be determined from <strong>the</strong> recurrence re-<br />
-<br />
;ion (13). These new values of <strong>the</strong> time <strong>le</strong>vel ( n + M ) are used<br />
? <strong>the</strong> computation of <strong>the</strong> coefficients of y-sweep and <strong>the</strong> values of<br />
;ential at <strong>the</strong> end of time <strong>le</strong>vel ( n + 1 ) have been found out ( see<br />
IW chart diagram ). This method is known as <strong>the</strong> multi - sweep method.
576<br />
Conclusions<br />
A program in AIGOL - 60 has been executed on ICL - 1900 machine<br />
for <strong>the</strong> water balance equation. The program was tested for different<br />
boundaries ( e.g. river, dyke, .*) of different parameters.<br />
As <strong>the</strong> method of solution is based mainly on <strong>the</strong> three implicit<br />
difference scheme that <strong>the</strong>re is no conditions for choosing <strong>the</strong> dis-<br />
tance between <strong>the</strong> grid points and <strong>the</strong> time interval A t. Use this<br />
procedure contains no iterative routines and it has been found out<br />
that this method is exceptionally ecomomical ln computing time and<br />
memory requifements and <strong>the</strong> solution is considered of high accuracy<br />
Thus this method of computation is recommended for areas of in-<br />
adequate hydrological data.
BOR n = 1 STEP 1 UNTIL nn<br />
I<br />
[COMPUTATION i)F h AT n + $ 1<br />
1 -[h<br />
.y<br />
= Km h<br />
J.<br />
,COMPUTATION OF SWBE2 IN Y-DIHECTION<br />
FOR J = 1 STEP 1 UNTIL JJ<br />
\'<br />
i<br />
COU'UTATION OF SwIEE;p IN X-DIWTIÙN<br />
c<br />
[COMPUTATIUN FOR E & F FOB kk - 1 to i]<br />
1<br />
ICOMPUTAIION OF h AT n+l<br />
I<br />
FLOW CRART DIAGRAM<br />
577
570<br />
Symbo 1 e<br />
h : heigtit of water tab<strong>le</strong> above <strong>the</strong> impervious bed.<br />
u,v: <strong>the</strong> flow components. in x and y directions respectiv;aly.<br />
k : Coefficient of permeability.<br />
: Coefficient of specific yield.<br />
ïV t intensi- of infiltration to <strong>the</strong> ground water tab<strong>le</strong>.<br />
AS : <strong>the</strong> grid spacing.<br />
At : <strong>the</strong> grid spacing on t-axis.<br />
JJ : <strong>the</strong> number of grid points on x-axis.<br />
kk : <strong>the</strong> number of grid pointe on y-axis<br />
nn : <strong>the</strong> number of grid points on t-sis.<br />
3 : any grid point on <strong>the</strong> x-axis.<br />
k : any grid point on <strong>the</strong> y-axis.<br />
n : any grid point on <strong>the</strong> t-axis.<br />
AOKNOWLEDGWT<br />
This work is sponsored in part by <strong>the</strong> Water Resources Depart-<br />
ment of <strong>the</strong> Desert Institute, Cairo, Egypt, to which <strong>the</strong> author is<br />
gratef uì .
Literature<br />
1. Abbott M.B. ( 1967 ). Difference methods, Lecture note, Inter-<br />
national course of Hydraulic Engineering, Delft, Holland.<br />
2. Mitchell A.R. ( 1969 ). Computational methods in partial diff-<br />
erential equations, John Wi<strong>le</strong>y.<br />
3. Nabil Rofail ( 1972 ). The numerical computation of parabolic<br />
equation using inplicit difference scheme and alternating<br />
direction methods, gth Conference on statistics and computat-<br />
ional science, Cairo, Egypt. pp. 572~- 593.<br />
4. Richtmyer R.D. and Mooton K.P. ( 1967 ). Difference mothods for<br />
ijlitial value prob<strong>le</strong>ms, Interscience.<br />
5. Uri Shamir, (1967). The use Of computers in ground water hydrology,<br />
hydro dynamics Laboratory, Beport NQS. 105, Yasaachusetts.<br />
579
580<br />
t<br />
cr><br />
4<br />
t<br />
tn<br />
0<br />
r:<br />
3ig.l Diagramatic representation of unconfined aquifer<br />
Y<br />
a R U<br />
J+1 J J-1<br />
k-1 n+l<br />
k<br />
4 k-1 n<br />
a<br />
Fig.2 The three <strong>le</strong>vel Scheme<br />
,<br />
'COS- AS 3
ABS TRACT<br />
DATA ACQUISITION AND METHODOLOGY FOR A SIMULATION MODEL<br />
OF THE LLOBREGAT DELTA (BARCELONA, SPAIN)<br />
Francisco VilarÓ Rigo1 y Emilio Custodio Gimena<br />
The Llobregat Delta (Barcelona) is a 80 sqKm., area supplying up to<br />
150 million cubic meters per year of water for industrial, urban and<br />
agricultural uses, in order of decreasing importance. The construction<br />
of a exploitation simulation model has been necessary in order to study<br />
carefully <strong>the</strong> new prob<strong>le</strong>ms concequence of a increasing rate of<br />
abstraction, <strong>the</strong> conversion of extense irrigation lands in industrial<br />
areas, <strong>the</strong> dredging of a new harbor and <strong>the</strong> forcoming river regulation<br />
with dams. Historical data were initialy scarce. In one hand <strong>the</strong>y were<br />
restricted to <strong>the</strong> rainfall and main river discharge know<strong>le</strong>dge and in<br />
<strong>the</strong> o<strong>the</strong>r hand to some disperse ground water <strong>le</strong>vel data and fi<strong>le</strong> of<br />
well dril<strong>le</strong>rs logs without interpretation. After <strong>the</strong> classification of<br />
<strong>the</strong> existing data, some specific studies were iniciated in order to<br />
comp<strong>le</strong>te <strong>the</strong> know<strong>le</strong>dge of <strong>the</strong> system and finally <strong>the</strong> model was<br />
constructed, followed with an important stage of value adjustment,<br />
specially those related to intermediate aquitard properties. The<br />
ajusted model has been used in several stages to forecast <strong>the</strong> response<br />
to preestablished possib<strong>le</strong> future situations.<br />
Key words: scarce data, model, adjustment, exploration.<br />
RESUMEN<br />
El delta del Llobregat (Barcelona) constituye una zona de 80 km2,<br />
que l<strong>le</strong>ga a proporcionar hasta 150 millones de m3 anua<strong>le</strong>s de agua para<br />
usos industria<strong>le</strong>s, urbanos y agrícolas, por orden decreciente de impor<br />
tancia. Ha sido necesaria la construcci6n de un modelo de simulacibn -<br />
de la explotaci6n a fin de estudiar con detal<strong>le</strong> los prob<strong>le</strong>mas apareci-<br />
dos a causa de la cada vez más intensa explotación, transformación de<br />
áreas agrícolas extensas en industria<strong>le</strong>s, apagado de un nuevo puerto y<br />
próxima regulación del río mediante embalses. tos datos histbricos --<br />
existentes inicialmente eran escasos. Por un lado se reducían al cono-<br />
cimiento de la pluviometrla y del caudal del rio principal y por otro<br />
lado a algunos datos esporadicos de nive<strong>le</strong>s del agua y un archivo de -<br />
perfi<strong>le</strong>s de pozos sin elaborar. Se han realizado estudios detallados -<br />
orientados a comp<strong>le</strong>mentar el conocimiento del sistema y finalmente se<br />
ha construido el modelo con una importante fase de ajuste de los valo-<br />
res estimados, en especial a los referentes al acuitardo intermedio. -<br />
El modelo ajustado ha sido utilizado en varias fases de previsión de -<br />
respuesta ante determinadas situaciones futuras posib<strong>le</strong>s.<br />
Palabras clave: datos escasos, modelo, ajuste, explotación.<br />
( ) Comisaría de Aguas del Pirineo Oriental y Curso Internacional de -<br />
Hidrología Subterránea. Barcelona.<br />
I
582<br />
1.- LOCATION AND BACKGROUND<br />
The Bajo Llobregat is an area spreading from Barcelona<br />
Eastwards and <strong>the</strong> Garraf Limestone Massive Westwards and SW<br />
(Fig. 1). It is largly occupied by <strong>the</strong> val<strong>le</strong>y of <strong>the</strong> Llobregat<br />
river and its delta, whose alluvial formations occupy around<br />
80 Km2., of which slightly over 50 Km2. correspond to <strong>the</strong> delta<br />
itself.<br />
The proximity to <strong>the</strong> important urban nuc<strong>le</strong>us of Barcelona,<br />
<strong>the</strong> fertility of <strong>the</strong> land, <strong>the</strong> easy availability of water and<br />
<strong>the</strong> existence of a big market for its products, have given rise<br />
to and important agricultural and industrial development. Today,<br />
<strong>the</strong> industry is tending to take <strong>the</strong> place of farming and both<br />
are rejected by <strong>the</strong> expanding urban area of <strong>the</strong> town of Barce-<br />
lona. On <strong>the</strong> o<strong>the</strong>r hand,<strong>the</strong> current expansion of <strong>the</strong> Barcelona<br />
harbour, needs new service areas to be prepared, which <strong>the</strong> Bajo<br />
Llobregat easily offers.<br />
The prob<strong>le</strong>ms of important water extractions, of increasing<br />
interest in <strong>the</strong> sands and gravels of <strong>the</strong> delta and val<strong>le</strong>y for<br />
construction, <strong>the</strong> additional communication lines, <strong>the</strong> prolifer-<br />
ation of discharges and tipping of all classes, etc., create a<br />
harmful and apprehensive climate, and <strong>le</strong>ads to <strong>the</strong> destruction<br />
of <strong>the</strong> aquifers by emptying and contamination and it may produce<br />
a deep sea intrusion. Its rational administration requires a<br />
good know<strong>le</strong>dge of <strong>the</strong> characteristics and hydraulic operation<br />
of <strong>the</strong> aquifers in <strong>the</strong> area.<br />
In 1909 a detai<strong>le</strong>d study was made on <strong>the</strong> groundwater<br />
hydrology of <strong>the</strong> delta (61, but <strong>the</strong> systematic and detai<strong>le</strong>d<br />
studies started in 1964, which is <strong>the</strong> inicial point of a series<br />
of ‘mportant works and reports which are partly listed in <strong>the</strong><br />
references. They have mostly been prepared by personnel of <strong>the</strong><br />
General Hydraulic Works Board, through <strong>the</strong> East Pyrenees Water<br />
Committee and <strong>the</strong> De<strong>le</strong>gation in Barcelona of <strong>the</strong> Public Works<br />
Geological Service.<br />
The complicated factors raised <strong>the</strong> need to have a simulation<br />
model of <strong>the</strong> aquifer systems availab<strong>le</strong>. The Public Works<br />
Geological Service built a R-C (resistors and capacities) model<br />
in 1970 as a first approximation, and almost simultaneously,<br />
<strong>the</strong> East Pyrenees Nater Board and <strong>the</strong> Public Works Geological<br />
Service prepared a digital ma<strong>the</strong>matical model of <strong>the</strong> explotation,<br />
capab<strong>le</strong> of fur<strong>the</strong>r details and more f<strong>le</strong>xib<strong>le</strong> use (i). The main<br />
prob<strong>le</strong>m when building such models lies in <strong>the</strong> scanty historical<br />
data availab<strong>le</strong>, since <strong>the</strong> systematic control studies of <strong>the</strong> area<br />
were initiated sistematicaly after 1966.
2.- AQUIFERS IN THE AREA<br />
583<br />
Figure 2, shows <strong>the</strong> general features of <strong>the</strong> aquifers in<br />
<strong>the</strong> area, by means of three cross-sections. In <strong>the</strong> Llobregat<br />
val<strong>le</strong>y, <strong>the</strong>re is a sing<strong>le</strong> aquifer of coarse gravel which divides<br />
up in <strong>the</strong> delta entrance, into two superposed ones, separated<br />
by a silt-clayey intercalation, which increases in thickness<br />
towards <strong>the</strong> sea. Thus an upper aquifer, which is mostly a water<br />
tab<strong>le</strong> one,and a deep confined aquifer with a weakly semi-<br />
pervious roof are separated. The silt intercalation narrows<br />
and becomes sandy towards <strong>the</strong> delta margins, and finally<br />
disappears, thus allowing both aquifers to lie directly above<br />
one ano<strong>the</strong>r, and in easy hydraulic relation (8) (9) (14).<br />
The aquifer of <strong>the</strong> val<strong>le</strong>y and <strong>the</strong> deep aquifer of <strong>the</strong> delta<br />
present areas of high transmissivity where <strong>the</strong>re are important<br />
pumpings, whereas <strong>the</strong> upper aquifer of <strong>the</strong> delta support only<br />
reduced explotation.<br />
Both <strong>the</strong> delta and <strong>the</strong> val<strong>le</strong>y ape marginated by materials<br />
which may be considered as impervious.<br />
3.- EXTRACTIONS AND HYDRAULIC OPERATION<br />
In figure I, <strong>the</strong> main extractions and hydraulic conditions<br />
of <strong>the</strong> model area were shown. In <strong>the</strong> delta, <strong>the</strong> two largest<br />
pumping centres are found in Prat de Llobregat and <strong>the</strong> Free Port,<br />
and <strong>the</strong>y gravitate on <strong>the</strong> deep aquifer; in <strong>the</strong> val<strong>le</strong>y <strong>the</strong>y lie<br />
alongside a lower end (Cornella-Sant Joan D'Espi) and neighbour-<br />
hood of Sant Feliu de Llobregat. O<strong>the</strong>r extraction nuc<strong>le</strong>i are<br />
found along <strong>the</strong> SU edge of <strong>the</strong> delta, besides o<strong>the</strong>r isolated<br />
points, served from both aquifers. The upper aquifer of <strong>the</strong> delta<br />
receives an excel<strong>le</strong>nt recharge from irrigation return flow and<br />
waste water discharge, and it is drained by <strong>the</strong> sea, <strong>the</strong> final<br />
stretch of <strong>the</strong> river, <strong>the</strong> drains of <strong>the</strong> airport and <strong>the</strong> marginal<br />
pumping areas. The aquifer of <strong>the</strong> val<strong>le</strong>y receives its main<br />
recharge through river water infiltration and from <strong>the</strong> irrigation<br />
canals, but <strong>the</strong> permeability of <strong>the</strong> beds impedes <strong>the</strong> maintenance<br />
of a direct hydraulic connection, and a non-saturated mediun<br />
exists between water tab<strong>le</strong> and <strong>the</strong> bottom of <strong>the</strong> surface water.<br />
The deep aquifer of <strong>the</strong> delta receives <strong>the</strong> recharge direct from<br />
<strong>the</strong> val<strong>le</strong>y or from <strong>the</strong> upper aquifer in <strong>the</strong> marginal areas or by<br />
vertical infiltration through <strong>the</strong> silt <strong>le</strong>ns. These relations<br />
and actions can be seen in <strong>the</strong> doub<strong>le</strong> piezometric surface of<br />
figure 3, and are ref<strong>le</strong>cted in some detai<strong>le</strong>d studies based on<br />
balance criteria. (9) (10) (141, hydraulic computations (18) (19)<br />
and geohydrochemical evaluation (2) (3) (5) (8).<br />
4.- MOTIVATION OF THE MODEL<br />
Delta groundwater explotation for industrial uses has been<br />
increasing at a rapid pace during <strong>the</strong> last ten years, at <strong>the</strong>
5 84<br />
same time as normal extractions for <strong>the</strong> Barcelona supply have<br />
been dropping as a result of <strong>the</strong> direct utilization of <strong>the</strong><br />
river water, after a suitab<strong>le</strong> treatment. Total extraction<br />
however has gradually increased and it will rise possibly in<br />
<strong>the</strong> immediate future when it is necessary to reactivate <strong>the</strong><br />
urban supply wells to meet growing âemand. The total capacity<br />
of water stored in <strong>the</strong> aquifer system and easily mobilizab<strong>le</strong>,<br />
is between 100 and 150 million m3., a small figure compared<br />
with <strong>the</strong> annual extraction which non exceeds 140 million m3.,<br />
and can reach 200 with <strong>the</strong> present existing pumping capacity.<br />
This means that in <strong>the</strong> absence of recharge, in a few months,<br />
certain parts of <strong>the</strong> aquifer dry up or are <strong>le</strong>ft with an insuf-<br />
ficient saturated thickness to maintain vel1 discharges. The<br />
river infiltration does not increase when <strong>the</strong> extractions rise,<br />
as a result of its disconnection with <strong>the</strong> water tab<strong>le</strong> in <strong>the</strong> main<br />
recharge area (corresponds to <strong>the</strong> val<strong>le</strong>y), and <strong>the</strong>re is no o<strong>the</strong>r<br />
important recharge source except <strong>the</strong> sea, this inducing a<br />
steadily advancing sea water intrusion. (5) (20).<br />
The study of <strong>the</strong> effect of new extractions or of different<br />
natural or artificial hydrological, river situations, as a<br />
result of its dam regulation or water transportation to o<strong>the</strong>r<br />
areas and also <strong>the</strong> conversion of farming areas into industrial<br />
zones, is comp<strong>le</strong>x. Por this reason it was decided to built a<br />
simulation model to analyse <strong>the</strong> explotation of <strong>the</strong> ground waters,<br />
which would also help to assess <strong>the</strong> river recharge, <strong>the</strong> sea<br />
intrusion (by indirect evaluation) and <strong>the</strong> interferences.<br />
The different objectives and variab<strong>le</strong>s to be estimated may<br />
be summed up as follows: (4)<br />
Study of <strong>the</strong> effects of <strong>the</strong> explotation in certain places,<br />
with or without disappearance of some of <strong>the</strong> present pumpings.<br />
Study of <strong>the</strong> artificial recharge effects by spreading and<br />
by wells, and analysis of <strong>the</strong>ir technical, economic and<br />
<strong>le</strong>gal feasibility.<br />
Study of <strong>the</strong> effects and suitability of a recharge litoral<br />
barrier to reduce sea intrusion, in <strong>the</strong> upper aquifer, in<br />
<strong>the</strong> deep one, or in both, im se<strong>le</strong>cted areas.<br />
Study of <strong>the</strong> Llobregat river regulation effects and/or<br />
derivation of larger discharges for supply.<br />
Study of <strong>the</strong> suppression effects of irrigated areas or <strong>the</strong><br />
modulation of <strong>the</strong> irrigation discharges.<br />
Study of <strong>the</strong> geotechnical prob<strong>le</strong>ms derived from abandonment<br />
of <strong>the</strong> main current pumpings.<br />
Study of <strong>the</strong> operation of <strong>the</strong> aquifers as local reservoirs<br />
for <strong>the</strong> most adequate service of demand.
The study of <strong>the</strong>se possibilities includes:<br />
1) Determination of <strong>the</strong> external balance,<br />
2) Determination of <strong>the</strong> internal balance.<br />
585<br />
3) Estimation of <strong>the</strong> fresh water discharges into <strong>the</strong> sea and<br />
river.<br />
4) Estimation of <strong>the</strong> sea water encroachment areas and <strong>the</strong>ir<br />
possib<strong>le</strong> evolution.<br />
5) Estimation of <strong>the</strong> deficits which may turn up in <strong>the</strong><br />
different zones.<br />
5.- MODEL NETWORK<br />
The shape of <strong>the</strong> piezometric surface, <strong>the</strong> distribution of<br />
<strong>the</strong> pumpings, <strong>the</strong> plant of <strong>the</strong> aquifer system and present<br />
know<strong>le</strong>dge, advised an assymetric network, digital ma<strong>the</strong>matical<br />
model, similar to <strong>the</strong> one established by <strong>the</strong> California Water<br />
Resources Department (17) as being <strong>the</strong> best suited. In accordance<br />
with <strong>the</strong> already known estimation princip<strong>le</strong>s (12) were made <strong>the</strong><br />
necessary adaptations for its programming and handling on <strong>the</strong><br />
doub<strong>le</strong> memory IBM 1630 computer at <strong>the</strong> Computation Office of <strong>the</strong><br />
Public Works Ministry in Madrid, and a series of special<br />
modifications in <strong>the</strong> boundary conditions. Its constructive and<br />
network details have been published on several occasions (1) (4)<br />
(15). In <strong>the</strong> delta, <strong>the</strong> two aquifers are simulated by means of<br />
a doub<strong>le</strong> network of polygons (fig. 4) connected by a vertical<br />
conductor branch. The sea condition is established as a constant<br />
<strong>le</strong>vel directly for <strong>the</strong> upper aquifer and by means of a resistent<br />
e<strong>le</strong>ment (<strong>the</strong> aquitard) for <strong>the</strong> deep aquifer. The condition of <strong>the</strong><br />
draining river is imposed as om ano<strong>the</strong>r constant <strong>le</strong>vel, and <strong>the</strong><br />
river condition in recharge area is established giving a recharge-<br />
discharge set of figures by polygon in terms of <strong>the</strong> river discharge.<br />
6.- RESOLUTION OF INSUFFICIENCY OF DATA. ADJUSTMENT.<br />
At <strong>the</strong> time when <strong>the</strong> need for <strong>the</strong> model came about, <strong>the</strong><br />
number of availab<strong>le</strong> data were small, especially regarding <strong>the</strong><br />
<strong>le</strong>ngth of <strong>the</strong> observation period.<br />
The number of data figures needed is very varied and com-<br />
prises those referring to <strong>the</strong> geometric form of <strong>the</strong> aquifers<br />
and <strong>the</strong>ir hydraulic parameters, up to those referring to <strong>the</strong><br />
temporary and spacial distribution of <strong>the</strong> extractions, passing<br />
by <strong>the</strong> infiltration of <strong>the</strong> rainwater, <strong>the</strong> river and <strong>the</strong> irrigations<br />
(6) (7) and <strong>the</strong>y should have a sufficient precision and represen-<br />
tative nature in accordance with <strong>the</strong> model network. A set of data<br />
should be availab<strong>le</strong> in each node and branch of <strong>the</strong> projected<br />
model.
586<br />
In this case, <strong>the</strong> geological structure was well known,<br />
owing to a high number of bore-ho<strong>le</strong>s (fig. 3) and wells with<br />
fi<strong>le</strong>d lithological log, but not SO <strong>the</strong> hydraulic characteris-<br />
tics of <strong>the</strong> different formations. These values were fractionary<br />
and corresponded to some precise data of tests in piezometers<br />
and wells made very ofer under difficult conditions, and some<br />
few prolonged pumping tests, with complicated interpretation<br />
due to <strong>the</strong> notab<strong>le</strong> piezometric fluctuations that are produced,<br />
wich sometimes exceed a metre throughout <strong>the</strong> day.<br />
With <strong>the</strong> availab<strong>le</strong> data, a plan of isotransmissivities was<br />
comp<strong>le</strong>ted and a distribution of <strong>the</strong> seepage coefficient of <strong>the</strong><br />
aquitard (intermediary silts) was estimated, based on a few<br />
granulometric tests and geohydrochemical considerations, which<br />
only gave <strong>the</strong> approximate magnitude.<br />
C<strong>le</strong>arly a model built under <strong>the</strong>se circunstances,with a<br />
poorly known connection with <strong>the</strong> river, and for which <strong>the</strong>re was<br />
only a few partial semi-quantitive figures, mostly obtained by<br />
statistical analysis of <strong>the</strong> river discharges, supply extractions<br />
and <strong>le</strong>vels in val<strong>le</strong>y (131, is a long way from reproducing <strong>the</strong><br />
reality, An adjustment process is necessary, based on comparison<br />
of its response to actions taken from <strong>the</strong> historic series and<br />
comparison with <strong>the</strong> effects observed in <strong>the</strong> aquifer. These<br />
actions are <strong>the</strong> extractions and recharges and <strong>the</strong> effects are<br />
<strong>the</strong> piezometric <strong>le</strong>vels.<br />
The adjustment process requires a sufficiently long and<br />
comp<strong>le</strong>te set of historic data, in order to comp<strong>le</strong>te, correct and<br />
suit <strong>the</strong> imprecise data, or <strong>the</strong> estimated or non-existent data.<br />
This adjustment process allows some data to be corrected if<br />
o<strong>the</strong>r can be taken as sufficiently precise. O<strong>the</strong>rwise, no<br />
so<strong>le</strong> situation is reached, or <strong>the</strong>re is no satisfactory solution<br />
nor one which responds to <strong>the</strong> real conditions of <strong>the</strong> prototype<br />
or real system. The set of historic data should be for each<br />
polygon, and this is difficult even in well known areas, and<br />
with a good systematic of measurements. In <strong>the</strong> case of <strong>the</strong> Llo-<br />
bregat delta, it was decided to take as "exact" data, despite<br />
certain uncertainties in <strong>the</strong>ir determination:<br />
a)<br />
b)<br />
The extractions by pumping anã <strong>the</strong> recharges by wells and<br />
drains, using as contrast criterion: for agricultural uses,<br />
<strong>the</strong> irrigated surface anã calculated water needs; for indus-<br />
trial uses, <strong>the</strong> type of production, number of workers and<br />
real production in those cases where it was known; and for<br />
supply uses, <strong>the</strong> urbanistic <strong>le</strong>vel and population served.<br />
The infiltrations of <strong>the</strong> rainwater, <strong>the</strong> irrigation water<br />
and runoff of <strong>the</strong> nearby areas obtained from water balances<br />
in <strong>the</strong> soil, and <strong>the</strong>refore of <strong>the</strong>oretic type. Never<strong>the</strong><strong>le</strong>ss,<br />
as this is a mild climate area, flat and with classic<br />
irrigation crops, a small error is expected.
507<br />
c) The water losses of <strong>the</strong> canals by infiltration based on <strong>the</strong><br />
in<strong>le</strong>t and out<strong>le</strong>t measurements and <strong>the</strong> irrigation quantities.<br />
In winter, <strong>the</strong>se irrigation quantities are almost non-existent,<br />
wich permits an acceptab<strong>le</strong> estimation.<br />
d) The piezometric surfaces and hydrograms of availab<strong>le</strong> ground<br />
water. Most hydrograms have been obtained with eight water<br />
<strong>le</strong>vel recorders, plus daily measurements on ano<strong>the</strong>r six<br />
piezometers, plus monthly readings on a few more points. The<br />
piezometric surfaces correspond to intense and periodical<br />
measurement campaigns of one or two days duration, but <strong>the</strong>se<br />
may have errors due to variations in <strong>the</strong> measurement hour,<br />
or introduction of some dynamic data or tridimensional flow<br />
areas; never<strong>the</strong><strong>le</strong>ss <strong>the</strong>y are sufficiently valid.<br />
e) River discharges, obtained with certain guarantee at <strong>the</strong><br />
upper val<strong>le</strong>y in<strong>le</strong>t, in Martorell.<br />
f) Geometric dimensions of <strong>the</strong> model<strong>le</strong>d units.<br />
The data to be adjusted are:<br />
1. on <strong>the</strong> model in itself, based on already mentioned<br />
previous values, and with a pre-established variation<br />
margin, taken from existing know<strong>le</strong>dge.<br />
- Transmissivities of <strong>the</strong> surface and deep aquifer, with<br />
reduced variations.<br />
- Vertical permeability of <strong>the</strong> aquitard (intermediary<br />
silts) for which <strong>the</strong> previous data could be notably<br />
erroneous.<br />
- Porosity of <strong>the</strong> water tab<strong>le</strong> aquifer, only admitting<br />
slight variations in accordance with <strong>the</strong> lithology.<br />
- Coefficient of elastic storage of <strong>the</strong> captive aquifers<br />
within a logical margin according to <strong>the</strong> existing<br />
structure and figures on <strong>the</strong> interpretation of pumping<br />
tests and <strong>the</strong> response to sea tide in some water <strong>le</strong>vel<br />
recorders of ground waters.<br />
2. on <strong>the</strong> actions impossed on <strong>the</strong> aquifer in <strong>the</strong> adjustment<br />
period, not directly known.<br />
- River recharge, estimated previously from balances,<br />
simplified analysis of <strong>the</strong> piezometric surfaces of <strong>the</strong><br />
val<strong>le</strong>y and a statistical correlation betwen discharges<br />
of <strong>the</strong> river-supply extractions and <strong>le</strong>vels in <strong>the</strong><br />
val<strong>le</strong>y (13).<br />
- Discharge to <strong>the</strong> river in <strong>the</strong> final stretch, estimated<br />
by partial balances and sumplified analysis of <strong>the</strong><br />
piezometric surfaces. This is a relatively small value.
588<br />
- Discharge to <strong>the</strong> sea and sea water encroachment values,<br />
according to <strong>the</strong> aquifer and <strong>the</strong> coastline area con-<br />
sidered. Measured very roughly due to estimation dif-<br />
ficulties, excepting <strong>the</strong> central coastal stretch of<br />
<strong>the</strong> water tab<strong>le</strong> aquifer.<br />
The distribution of <strong>the</strong> recharge between <strong>the</strong> upper and deep<br />
aquifer of <strong>the</strong> delta is a result of <strong>the</strong> adjustment, and also is<br />
<strong>the</strong> water discharge circulating through <strong>the</strong> aquitard.<br />
The chief difficulties regarding <strong>the</strong> adjustment are derived<br />
from insufficient data on <strong>le</strong>vels and a need to account on <strong>the</strong><br />
seasonal variations, owing to <strong>the</strong> great importance of extractions<br />
in relation with <strong>the</strong> quickly mobilizab<strong>le</strong> ground storage volume<br />
of water. The first piezometric surface useab<strong>le</strong> is at <strong>the</strong> begin-<br />
ning of 1966, and although ano<strong>the</strong>r six comp<strong>le</strong>te ones and one<br />
partial one were availab<strong>le</strong>, <strong>the</strong>ir distribution was nei<strong>the</strong>r regular<br />
in time, nor covered each of <strong>the</strong> quarterly periods into which<br />
<strong>the</strong> year was to be divided up. It was <strong>the</strong>refore decided to use<br />
<strong>the</strong> availab<strong>le</strong> piezometric surfaces, with minor corrections to<br />
adopt <strong>the</strong>m to <strong>the</strong> final moment of each quarterly interval,<br />
forming new interpolated piezometric surfaces, based on <strong>the</strong> data<br />
of <strong>the</strong> continuous piezometric measurements in some points, already<br />
discussed, trying to maintain <strong>the</strong> flow shape character.<br />
To comp<strong>le</strong>te <strong>the</strong> quarter figures, <strong>the</strong> water balance estimations<br />
were made in <strong>the</strong> soil, and <strong>the</strong> extractions were calculated from<br />
<strong>the</strong> inventory according to <strong>the</strong> annual rate of use and moment <strong>the</strong><br />
wells went into operation in some cases, or based on <strong>the</strong> demand<br />
curves given by some users.<br />
In figure 5, a samp<strong>le</strong> of <strong>the</strong> result of <strong>the</strong> final adjustment<br />
process can be seen, taking <strong>the</strong> 4 years of figures, distributed<br />
into 16 quarterly terms. This final adjustment need 13 stages<br />
with <strong>the</strong> definitive network. Some prior trials were made with a<br />
simplified network, to know <strong>the</strong> magnitude and convergence rates<br />
of <strong>the</strong> estimation process. This adjustment stage incorporated<br />
<strong>the</strong> modifications suggested by <strong>the</strong> previous one, mainly modifying<br />
<strong>the</strong> hydraulic characteristics of <strong>the</strong> units and <strong>the</strong> recharge of <strong>the</strong><br />
river. Before making a modification, <strong>the</strong> results obtained were<br />
carefully studied, taking into account previous results with early<br />
stages, and in order to be in accordance with <strong>the</strong> physical charac-<br />
teristics of <strong>the</strong> system.<br />
An important result of <strong>the</strong> adjustment process is not only <strong>the</strong><br />
correction of <strong>the</strong> imprecise data, but obtaining o<strong>the</strong>r necessary<br />
data for <strong>the</strong> model explotation process, previously unknown. Such<br />
is <strong>the</strong> relation Qr (river discharge) versus IR, thus allowing<br />
<strong>the</strong> (river infiltration) computation of IR (not measurab<strong>le</strong>) with<br />
availab<strong>le</strong> data on QR. The adjustment obtained shows <strong>the</strong>re is this<br />
relation with a sufficient statistical degree of significance.
589<br />
Figure 6 shows <strong>the</strong> inicial map of transmissivities of <strong>the</strong><br />
deep aquifer and <strong>the</strong> val<strong>le</strong>y gravels and <strong>the</strong> one obtained after<br />
<strong>the</strong> adjustment. The differences are not important and in many<br />
cases, <strong>the</strong> variations are not merely a correction of an erroneous<br />
value, but <strong>the</strong> adaptation of a precise value (test in bore ho<strong>le</strong><br />
or well) or of a regional value (pumping test or analysis of<br />
piezometric oscillations) to <strong>the</strong> dimensions and forms of each<br />
polygon.<br />
7.- UTILIZATION OF THE MODEL<br />
The model has been built for use under different conditions<br />
as those prevailing during <strong>the</strong>adjustmen process. This creates<br />
various prob<strong>le</strong>ms. For examp<strong>le</strong>, <strong>the</strong> validity of <strong>the</strong> model for<br />
o<strong>the</strong>r distributions of <strong>the</strong> pumping or recharge-disoharge, or<br />
those corresponding to piezometric surfaces notably different.<br />
Also, one must consider <strong>the</strong> validity of <strong>the</strong> Qr - 1, (river<br />
discharge - river recharge) relation, under different circumstances<br />
of <strong>the</strong> river system or after conditioning works in <strong>the</strong> bed. The<br />
adjustment period is ra<strong>the</strong>r short, but sufficient to insure<br />
credib<strong>le</strong> results under conditions similar to <strong>the</strong> adjustment<br />
ones and in time periods not much greater. If one attempt to<br />
simulate 20 years or under pumping conditions with o<strong>the</strong>r centres<br />
of extraction, noticeably different as those existing now, <strong>the</strong><br />
results would possibly only be semi-quantitative.<br />
One of <strong>the</strong> recent processes of using <strong>the</strong> model arose to study<br />
<strong>the</strong> possibility of temporarily increasing <strong>the</strong> groundwater<br />
extractions for supply, in <strong>the</strong> event of a succession of dominatly<br />
dry year combined with a delay in <strong>the</strong> first service of <strong>the</strong> new<br />
surface water regulation works of <strong>the</strong> Llobregatriver (ll), taking<br />
into account <strong>the</strong> normal pumping increase forecastsfor o<strong>the</strong>r pur-<br />
poses. The injuries and needs of redistribution and conditioning<br />
of <strong>the</strong> pumpings under various foreseen hypo<strong>the</strong>sis have been<br />
assessed, and <strong>the</strong> sea water encroachment and <strong>the</strong> later return to<br />
a "normal" situation after <strong>the</strong>se regulation works have been<br />
finished. Some of <strong>the</strong> possib<strong>le</strong> extraction situations have not<br />
been made as <strong>the</strong>y produce excessive drops which prevent <strong>the</strong><br />
pumping capacity of <strong>the</strong> wells to be maintained.<br />
The use of <strong>the</strong> model permits <strong>the</strong> aquifer system of <strong>the</strong> Bajo<br />
Llobregat to be hand<strong>le</strong>d as a regulating reservoir, analysing<br />
<strong>the</strong> guarantees of <strong>the</strong> different ground water demands in different<br />
natural or man-made hydrological situations, and a know<strong>le</strong>dge of<br />
<strong>the</strong> rate and location of <strong>the</strong> progressive salinization process or<br />
<strong>the</strong> effectiveness of <strong>the</strong> measures adopted to reduce it. These<br />
eventualities were analysed by seven different hypo<strong>the</strong>sis<br />
following <strong>the</strong> adjustment process (1) (15), including <strong>the</strong> analysis<br />
of <strong>the</strong> possib<strong>le</strong> artificial recharge. The model, in its explotation<br />
phase, works with six monthly intervals instead of <strong>the</strong> quarterly<br />
intervals of <strong>the</strong> adjustment.
590<br />
8.- CONCLUSION<br />
The careful1 modelling of an aquifer permits a very useful<br />
work tool to be obtained, even though <strong>the</strong> initiai data is<br />
incomp<strong>le</strong>te or non-existent in certain aspects, provided ano<strong>the</strong>r<br />
series of sufficiently precise data, or with known error is<br />
availab<strong>le</strong>, and which is such that it permits an adjustment<br />
process with a sufficient number of steps.<br />
9. - REFERENCES<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
Cuena, J. and Custodio, E. (1971) - Construction and adjustment<br />
of a two layer ma<strong>the</strong>matical model of <strong>the</strong> Llobregat Delta.-<br />
International Symposium on Ma<strong>the</strong>matical Models in Hydrology.-<br />
International Association of Scientific Hydrology.- Varsovia.<br />
Custodio, E. (1967) - Etudes geohydrochimiques dans <strong>le</strong> delta<br />
du Llobregat, Barcelone (Espagne) - Geochimie, Precipitations,<br />
Humidité du Sol, Hydrometrie. Assemblée Généra<strong>le</strong> de Berne,<br />
1967. Association International d’Hydrologie Scientifique.-<br />
Gentbrugge - pp. 134/155.<br />
Custodio, E. (1969) - Ground water entries in <strong>the</strong> Llobregat<br />
river delta.- Hydrological Reasearch Documents No. 6, Barce-<br />
lona Water REsearch, Applications and Studies Centre. Speech<br />
in Pamplona 1967 - pp 205/237.<br />
Custodio, E., Cuena, J. and BayÓ, A. (1971) - Prob<strong>le</strong>m,<br />
execution and use of a two layer ma<strong>the</strong>matical model for <strong>the</strong><br />
Llobregat delta aquifers (Barcelona).- First Spanish - Por-<br />
tuguese - American Congress on Economic Geology.- Madrid-<br />
Lisbon. sep 1971. Section 3. pp 171/198.<br />
Custodio, E., Bayo, A. and Pelaez, M.D. (1971) - Geochemistry<br />
and water entries for study of <strong>the</strong> movement of ground water<br />
in <strong>the</strong> Llobregat river delta (Barcelona) - First Spanish-<br />
Portuguese-American Congress on Economic Geology.- Madrid-<br />
Lisbon. sep. 1971. Section 6 pp 51/80.<br />
Custodio, E. and López-Garcia, L. (1972) - Construction and<br />
utilization process of a model. Chapter 5 Basic Theory on<br />
Analogical and Digital Models of Aquifers. Informations and<br />
Sutides, Bul<strong>le</strong>tin, 37, Geologicql Service of Public Works.<br />
Madrid.<br />
Custodio, E. (1973) - Basic data for building an aquifer<br />
simultation model. Chapter 16.1. on Subterranean Hydrology<br />
Omega Editorial. Barcelona (at press).
591<br />
8. Custodio, E. and o<strong>the</strong>rs (1973) - Compiling of works made<br />
during <strong>the</strong> period 1966/1972 in <strong>the</strong> Bajo Llobregat. Water<br />
Board of <strong>the</strong> East Pyrenees and Public Works Geological<br />
Service. Barcelona (in preparation).<br />
9. Llamas, M.R. and Molist, J. (1967) - Hydrology of <strong>the</strong> Besos<br />
and Llobregat River deltas.- Hydrological Investigation<br />
Documents NQ 2 - Water Research, Applications and Studies<br />
Centre. Barcelona. Speech in Barcelona (1966).<br />
10. Llamas, M.R. and VilarÓ, F. (1967) - Die Rol<strong>le</strong> der Grund-<br />
wasserspeicher bei der Wasservorsorgung von Barcelona.-<br />
Das Gas-und-Wasserfach, Wasser Abwasser, Vol. 34. No. 15,<br />
August 1967. pp. 945/953.<br />
11. Martin-Arnaiz, M. (1972) - Report on <strong>the</strong> explotation<br />
possibilities of <strong>the</strong> Llobregat river delta aquifers. General<br />
Board of Hydraulic Works. East Pyrenees Water Board. Barce-<br />
lona (prior report).<br />
12. Mc Neal, R.M. (1958) - An asymetrical finite difference<br />
network - Quarterly of Applied Ma<strong>the</strong>matics. Vol. XI. No. 3<br />
1958.<br />
13. Montalbán, F. (1969) - Factorial analysis of <strong>the</strong> oscillations<br />
of <strong>the</strong> deep aquifer of <strong>the</strong> Llobregat river. Hydrological<br />
Investigation Documents No. 6. Water Research, Applications,<br />
and Studies Centre. Barcelona, Pamplona speech (1967).<br />
14. Ministry of Public Works (1965).- Study of <strong>the</strong> Total Hydraulic<br />
resourcs of <strong>the</strong> East Pyrenees - Second Report East Pyrenees<br />
Water Board and Public Works Geological Service. Barcelona.<br />
15. Ministry of Public Works - Report on <strong>the</strong> construction and<br />
application of a ma<strong>the</strong>matical simulation model of <strong>the</strong> Llo-<br />
bregat delta aquifers.- Study of <strong>the</strong> Total Hydraulic Resources<br />
of <strong>the</strong> East Pyrenees. Central Area. Report CE-111.- East<br />
Pyrenees Water Board and Public Works Geological Service.<br />
Barcelona.<br />
16. Santa Maria, L. and Marin A. (1909) - Hydrological studies<br />
on <strong>the</strong> Llobregat river basin.- Bul<strong>le</strong>tin of <strong>the</strong> Commission<br />
of <strong>the</strong> Geological Map of Spain LX 2nd Series.<br />
17. Tyson, H.N. and Weber, E.M. (1964).- Ground water management<br />
for <strong>the</strong> nations future computer simulation of ground-water<br />
basins - proceedings of <strong>the</strong> ASCE, Journal of <strong>the</strong> Hydraulics<br />
Division. New York. Jyly 1964.<br />
18. VilarÓ, F. (1967) - Balance of <strong>the</strong> present use of <strong>the</strong> Bajo<br />
Llobregat. Hydrological Investigation Papers No. 2. Water<br />
Investigations, Applications and Studies Centre. Barcelona<br />
Speech in Barcelona, (1966) - pp 155/169.
592<br />
19. VilarÓ, F. and Martin Arnbiz, M. (1968) - Hydric Balance<br />
of <strong>the</strong> Bajo Llobregat - Hydric Balance Seminar - F.A.O. -<br />
Geology and Mining Institute of Spain. Madrid.<br />
20. VilarÓ, F. Custodio, E., and Bruington, A.E. (1970) - Sea<br />
Water intrusion and water pollution in <strong>the</strong> Pirineo Oriental<br />
(Spain) - ASCE National Water Resources Engineering Meeting,<br />
Memphis, Tennence. - Meeting Preprint 1122.
Fi g. 1 .- Plano general de situaci& y de extracciones.<br />
General location and pumping map.
O<br />
O 00 O O<br />
Yi<br />
sariaw - soiiaui S~JI~UI- soiiaw<br />
s '"" r<br />
O'' . I ' U<br />
594
-2-<br />
..2-.-<br />
595<br />
-<br />
Escal o-Sca<strong>le</strong><br />
O 1 2 3 L SKm.<br />
L ogunas pantanosas natural es<br />
Limite de los zonas permeab<strong>le</strong>s<br />
Limite del ocuifero<br />
profundo<br />
Isopieza del acuifero profundo I ml.<br />
Isapieza del acuifero suprficial [m 1.<br />
e Sondeo piezomctrico<br />
-.-.-<br />
Natural marshy lagoons<br />
Boundary of <strong>the</strong> permeab<strong>le</strong> oreas<br />
Boundary of <strong>the</strong> deep aquifer<br />
A- Isopiestic line Of <strong>the</strong> drepoquifer(ml<br />
,-2--- Isopiestic line of <strong>the</strong> upperoquiferIm)<br />
Observation bore- ho<strong>le</strong><br />
Fi g. 3 - Superficies PiezornCtricas en Abril de 1.967 (s& Custodio) y<br />
situaci& de los sandeos.<br />
Piezometric surfaces in April 1.967 (after Custodio) and 10Cb<br />
tion of <strong>the</strong> boreho<strong>le</strong>s.
I<br />
596
in<br />
c<br />
I in .e<br />
I - a<br />
I u<br />
I o 4<br />
I C<br />
/* :<br />
Y --<br />
597
598<br />
Tranrmirividad del aeuifero del<br />
val<strong>le</strong> y profundo del delta en m2/dia<br />
Transmissivity of val<strong>le</strong>y and delta<br />
deep aquifers in sqml day<br />
-- id<br />
- - - - Dato inicial Preliminary figure<br />
1000 Valor ajustadocon Value mstchcd with <strong>the</strong><br />
el modelo model<br />
Fig. 6.- Valores de la tranrrtirividaà del acuftero del val<strong>le</strong> y prohrado &l dal-<br />
ta del Llobregat.<br />
Values of <strong>the</strong> val<strong>le</strong>y aad delta upper aquifers OP <strong>the</strong> Llobregat delta.
and in Etudes et rapports d ‘hydrataptie 16<br />
gn of<br />
r- resou rces<br />
inadeq uate<br />
projects<br />
data<br />
Proceedings of <strong>the</strong> Madrid Syinposiurn<br />
June 1973<br />
Elaboration des projets<br />
d’utilisation des ressources en eau<br />
sans données suffisantes<br />
Volume 2<br />
Unesco - WMO - <strong>IAHS</strong><br />
Unesco - OMM - AISH<br />
Actes du colloque de Madrid<br />
Juin 1973
Studies and reports in hydrology/Etudes et rapports d’hydrologie 16
TITLES IN THIS SERIES / DANS CETTE COLLECTION<br />
1.<br />
2.<br />
3.<br />
4.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13.<br />
14.<br />
1s.<br />
16.<br />
The use of analog and digital computers in hydrology: Proceedings of <strong>the</strong> Tucson Symposium.<br />
June 1966 / L'utilisation des calculatrices analogiques et des ordinateurs en hydrologie: Actes du<br />
colloque de Tucson, juin 1966. Vol. 1 & 2. Co-edition <strong>IAHS</strong>-Unesco / Coédition AISU-Unesco.<br />
Water in <strong>the</strong> unsaturated zone: Proceedings of <strong>the</strong> Wageningen Symposium, June I967 / L'eau dans<br />
la zone non saturée: Actes du symposium de Wageningen, juin 1967. Edited by / Edité par P. E.<br />
Rijtema & H. Wassink. Vol. 1 & 2. Co-edition <strong>IAHS</strong>-Unesco / Coédition AISH-Unesco.<br />
Floods and <strong>the</strong>ir computation: Proceedings of <strong>the</strong> Leningrad Symposium, August 1967 / Les crues<br />
et <strong>le</strong>ur évaluation: Actes du colloque de Leningrad, août 1967. Vol. 1 & 2. Co-edition IARS-Unesco-<br />
WMO / Coédition AISH-Unesco-OMM.<br />
Representative and experimental basins: An international guide for research and practice. Edited<br />
by C. Toebes and Y. Ouryvaev. Published by Unesco.<br />
Les bassins représentatifs et expérimentaux: Guide international des pratiques en matière de re-<br />
cherche. Publié sous la direction de C. Toebes et V. Ou-vaey. Publié par l'Unesco.<br />
'Discharge of se<strong>le</strong>cted rivers of <strong>the</strong> world / Débit de certain cours d'eau du monde. Published by<br />
Unesco / Publié par l'Unesco.<br />
Vol. I : General and régime characteristics of stations se<strong>le</strong>cted 1 Caractéristiques généra<strong>le</strong>s et<br />
caractéristiques du régime des stations choisies.<br />
Vol. II: Monthly and annual discharges recorded at various se<strong>le</strong>cted stations (from start of obser.<br />
vations up to 1964) / Débits mensuels et annuels enregistrés en diverses stations sé<strong>le</strong>ctionnées<br />
(de l'origine des observations à l'année 1964).<br />
'Vol. III: Mean monthly and extreme dlscharges (1%5-1969) / Débits mensuels moyens et débits<br />
extrêmes (19651969).<br />
List of International Hydrological Decade Stations of <strong>the</strong> world / Liste des stations de la Décennie<br />
hydrologique internationa<strong>le</strong> existant dans <strong>le</strong> mmde. Published by Unesco 1 Publié par l'Unesco.<br />
Ground-water studies: An international guide for practice. Edited by R. Brown, I. Ineson. V. KO-<br />
noplyantsev and V. Kova<strong>le</strong>vski. (Will also appear in French, Russian gnd Spanish / Paraitrg<br />
éga<strong>le</strong>ment en espagnol, en français et en russe.)<br />
Land subsidence: Proceedings of <strong>the</strong> Tokyo Symposium, September 1969 / Affaisement du sol:<br />
Actes du colloque de Tokyo, septembre 1969. 'Vol. 1 & 2. Co-edition <strong>IAHS</strong>-Unesco / Coédition<br />
AISH-Unesco.<br />
Hydrology of deltas: Proceedings of <strong>the</strong> Bucharest Symposium, May 1969 / Hydrolaße des deltas:<br />
Actes du colloque de Bucarest, mai 1969. Vol. 1 & 2. Co-edition <strong>IAHS</strong>-Unesco / Coédirion AISH-<br />
Unesco.<br />
Status and trends of research in hydrology / Bilan et tendances de la recherche en hydrologic.<br />
Published by Unesco 1 Publié par l'Unesco.<br />
World water balance: Proceedings of <strong>the</strong> Reading Symposium, July 1970 / Bilan hydrique mondial:<br />
Actes du colloque de Reading, juil<strong>le</strong>t 1970. Vol. 1-3. Co-edition ZAHS-Unesco-WhfO 1 Coédirion<br />
AISH-Unesco-OMM.<br />
Results of research on representative and experimental basins: Proceedings of <strong>the</strong> Wel1inp;ton<br />
Symposium, December 1970 / Résultats de recherches sur <strong>le</strong>s bassins représentatifs et ex érimen-<br />
taux: Actes du cowoque de Wellington, décembre 1970. 'Vol. 1 & 2. Coedition <strong>IAHS</strong>-Jnesco /<br />
Coédition AISH-Unesco.<br />
Hydrometry: Proceedings of <strong>the</strong> Kob<strong>le</strong>nz Symposium, September 1970 / Hydrométrie: Actes du<br />
colloque de Cob<strong>le</strong>nce, septembre 1970. Co-edition ZAHS-Unesco-WMO / Coédition AISH-Unesco-<br />
OMM.<br />
Hydrologic information systems. Co-edition Unesco-WMO.<br />
Ma<strong>the</strong>matical models in hydrology: Proceedings of <strong>the</strong> Warsaw Symposium, July 1971 / Les mc-<br />
de<strong>le</strong>s mathématiques en hydrologie Actes du colloque de Varsovie, juil<strong>le</strong>t 1971. Vol. 1-3. Co-<br />
edition <strong>IAHS</strong>-Unesco-WMO / Coédit on AISH-Unesco-OMM.<br />
Design of water resources projects with inadequate data: Proceedings of <strong>the</strong> Madrid sym ,<br />
June 1973 / Elaboration des projets d'utilisation des ressources en eau sans données sufp:z:<br />
Actes du colloque de Madrid, juin 1973. Vol. 1-3. Co-edition Unesco-WMO-<strong>IAHS</strong> / Coéditiori Unesco-<br />
OMM-AISH.
Design of<br />
water resources projects<br />
with inadequate data<br />
Proceedings of <strong>the</strong> Madrid Symposium<br />
June 1973<br />
Elaboration des projets<br />
d’utilisation des ressources en eau<br />
sans données suffisantes<br />
A contribution to <strong>the</strong> Lnternationat Hydrological Decade<br />
Une contribution a la Ecennie hydrologique internationa<strong>le</strong><br />
Con reshmenes en csuañol<br />
Volume 2<br />
Actes du colloque de Madrid<br />
Juin 1973<br />
Unesco - WMO - <strong>IAHS</strong> 1974<br />
Uiiesco - OMM - AISH
Published jointly by<br />
<strong>the</strong> United Nations Educational, Scientific<br />
and Cultural Organization,<br />
7, Place de Fontenoy, 75700 Paris, 3) ><br />
World Meteorological Organization,<br />
41 av. Giuseppe-Motta, Geneva, and<br />
<strong>the</strong> International Association of Hydrological Sciences (President: J.-A. Rodier),<br />
19, rue Eugène-Carrière, 75018 Paris<br />
Publié conjointement par<br />
l’organisation des Nations Unies pour<br />
l‘éducation, la science et la culture.<br />
7, place de Fontenoy, 75700 Paris.<br />
l’organisation météorologique mondia<strong>le</strong>,<br />
41, av. Giuseppe-Motta, Genève, et<br />
l’Association internationa<strong>le</strong> des sciences hydrologiques (président: 3.-A. Rodier),<br />
19, rue Eugène-Carrière. 75018 Paris<br />
Impreso por el Centro de Estudios Hidrográficos, Madrid<br />
The se<strong>le</strong>ction and presentation of material and <strong>the</strong>’ opinions expressed in this publication<br />
are <strong>the</strong> responsibility of <strong>the</strong> authors concerned and do not necessarily ref<strong>le</strong>ct <strong>the</strong><br />
views of <strong>the</strong> publishers.<br />
The designations employed and <strong>the</strong> presentation of <strong>the</strong> material do not imply <strong>the</strong><br />
expression of any opinion whatsoever on <strong>the</strong> part of <strong>the</strong> publishers concerning <strong>the</strong> <strong>le</strong>gal<br />
status of any country or territory, or of its authorities, or concerning <strong>the</strong> frontiers<br />
of any country or territory.<br />
Le choix et la présentation du contenu de cet ouvrage et <strong>le</strong>s opinions qui s‘y<br />
expriment n’engagent que la responsabilité des auteurs et ne correspondent pas<br />
nécessairement aux vues des éditeurs.<br />
Les dénominations employées et la présentation des divers éléments n’impliquent<br />
de la part des éditeurs aucune prise de position à l’égard du statut juridique de l’un<br />
quelconque des pays et territoires en cause, de son régime politique ou du tracé<br />
de ses frontières.<br />
ISBN 92-3-001137-1<br />
0 Unescc-WMO-<strong>IAHS</strong>-1974<br />
Printed in Spain
PREFACE<br />
The International Hydrological Decade (IHD) 1965-74 was launched by<br />
<strong>the</strong> General Conference of Unesco at its thirteenth session to promote<br />
international co-operation in research and studies and <strong>the</strong> training of spe-<br />
cialists and technicians in scientific hydrology. Its purpose is to enab<strong>le</strong><br />
all countries to make a ful<strong>le</strong>r assessment of <strong>the</strong>ir water resources and a<br />
more rational use of <strong>the</strong>m as man’s demands for water constantly increase<br />
in face of developments in population, industry and agriculture. In 1974<br />
National Committees for <strong>the</strong> Decade had been formed in 108 of Unesco’s<br />
131 Member States to carry out national activities within <strong>the</strong> programme<br />
of <strong>the</strong> Decade. The imp<strong>le</strong>mentation of <strong>the</strong> programme is supervised by a<br />
Co-ordinating Council, composed of 30 Member States se<strong>le</strong>cted by thc Ge-<br />
neral Conference of Unesco, which studies proposals for developments<br />
of <strong>the</strong> programme, recommends projects of interest to all or a large<br />
number of countries, assists in <strong>the</strong> development of national and regional<br />
projects and co-ordinates international co-operation.<br />
Promotion of collaboration in developing hydrological research techni-<br />
ques, diffusing hydrological data and planning hydrological installations<br />
is a major feature of <strong>the</strong> programme of <strong>the</strong> IHD which encompasses all<br />
aspects of hydrological studies and research. Hydrological investigations<br />
are encouraged at <strong>the</strong> national, regional and international <strong>le</strong>vel to streng-<br />
<strong>the</strong>n and to improve <strong>the</strong> u6e of natural resources from a local and a global<br />
perspective. The programme provides a means for countries well advanced<br />
in hydrological research to exchange scientific views and for developing<br />
countries to benefit from this exchange of information in elaborating re-<br />
search projects and in imp<strong>le</strong>menting recent developments in <strong>the</strong> planning<br />
of hydrological installations.<br />
As part of Unesco’s contribution to <strong>the</strong> achievement of <strong>the</strong> objectives<br />
of <strong>the</strong> IHD <strong>the</strong> General Conference authorized <strong>the</strong> Director-General to<br />
col<strong>le</strong>ct, exchange and disseminate information concerning research on<br />
scientific hydrology and to facilitate contacts between research workers<br />
in this field. To this end Unesco initiated two series of publications: Studies<br />
and Reports in Hydrology and Technical Papers in Hydrology.<br />
The Studies and Reports in Hydrology series, in which <strong>the</strong> present<br />
volume is published, is aimed at recording data col<strong>le</strong>cted and <strong>the</strong> main<br />
results of hydrological studies undertaken within <strong>the</strong> framework of <strong>the</strong><br />
Decade, as well as providing information on research techniques. Also<br />
included in <strong>the</strong> series are proceedings of symposia. Thus, <strong>the</strong> series com-<br />
prises <strong>the</strong> compilation of data, discussions of hydrological research techni-<br />
ques and findings, and guidance material for future scientific investigations.<br />
It is hopped that <strong>the</strong> volumes wil furnish material of both practical and<br />
<strong>the</strong>oretical interest to hydrologists and governments participating in <strong>the</strong><br />
IHD and respond to <strong>the</strong> needs of technicians and scientists concerned<br />
with prob<strong>le</strong>ms of water in all countries.<br />
A number of <strong>the</strong>se volumes have been published jointly with <strong>the</strong> In-<br />
ternational Association of Hydrological Sciences and <strong>the</strong> World Meteoro-<br />
logical Organization which have co-operated with Unesco in <strong>the</strong> imp<strong>le</strong>-<br />
mentation of several important projects of <strong>the</strong> IHD.
PRÉFACE<br />
La Conférence généra<strong>le</strong> de l’Unesco, à sa treizième session, a décidé<br />
de lancer, pour la période s’étendant de 1965 à 1974, la Décennie hydrologique<br />
internationa<strong>le</strong> (DHI), entreprise mondia<strong>le</strong> visant a faire progresser la con-<br />
naissance en matière d’hydrologie scientifique par un développement de<br />
la coopération internationa<strong>le</strong> et par la formation de spécialistes et de<br />
techniciens. Au moment où l’expansion démographique et <strong>le</strong> développement<br />
industriel et agrico<strong>le</strong> provoquent un accroissement constant des besoins<br />
en eau, la DHI permet à tous <strong>le</strong>s pays de mieux évaluer <strong>le</strong>urs ressources<br />
hydrauliques et de <strong>le</strong>s exploiter plus rationnel<strong>le</strong>ment.<br />
I1 existe actuel<strong>le</strong>ment dans i08 des 131 Etats membres de l’Unesco un<br />
comité national qui, pour tout ce qui a tratit au programme de la Décen-<br />
nie, impulse <strong>le</strong>s activités nationa<strong>le</strong>s et assure la participation de son pays<br />
aux entreprises régiona<strong>le</strong>s et internationa<strong>le</strong>s. L’exécution du programme<br />
de la DHI se fait sous la direction d’un Conseil de coordination composé<br />
de 30 Etats membres désignés par la Conférence généra<strong>le</strong> de l’Unesco; ce<br />
conseil étudie <strong>le</strong>s propositions concernant <strong>le</strong> programme, recommande<br />
l’adoption de projets intéressant l’ensemb<strong>le</strong> des pays ou un grand nombre<br />
d’entre eux, aide à la mise sur pied de projets nationaux et régionaux, et<br />
coordonne la coopération à l’échelon international.<br />
Le programme de la DHI qui porte sur tous <strong>le</strong>s aspects des études et<br />
des recherches hydrologiques, vise essentiel<strong>le</strong>ment à développer la col-<br />
laboration dans la mise au point des techniques de recherches, dans la<br />
diffusion des données hydrologiques, dans l’organisation des installations<br />
hydrologiques. I1 encourage <strong>le</strong>s enquêtes nationa<strong>le</strong>s, régiona<strong>le</strong>s et interna-<br />
tiona<strong>le</strong>s tendant à accroître et à améliorer l’utilisation des resources na-<br />
turel<strong>le</strong>s, dans une perspective loca<strong>le</strong> et généra<strong>le</strong>. I1 permet aux pays avancés<br />
en matière de recherches hydrologiques d’échanger des informations; aux<br />
pays en voie de développement, il offre la possibilité de profiter de ces<br />
échanges pour élaborer <strong>le</strong>urs projets de recherches et pour planifier <strong>le</strong>urs<br />
installations hydrologiques en tirant parti des acquisitions <strong>le</strong>s plus récentes<br />
de l’hydrologie scientifique.<br />
Pour permettre a l’Unesco de contribuer au succès de la DHI, la Con-<br />
férence généra<strong>le</strong> a autorisé <strong>le</strong> Directeur généra<strong>le</strong> à rassemb<strong>le</strong>r, à échanger<br />
et à diffuser des informations sur <strong>le</strong>s recherches d’hydrologie scientifique<br />
et à faciliter <strong>le</strong>s contacts entre <strong>le</strong>s chercheurs dans ce domaine. A cette<br />
fin, l’Unesco fait paraître deux nouvel<strong>le</strong>s col<strong>le</strong>ctions de publications:
tique que théorique, et qu’el<strong>le</strong> répondra aux besoins des techniciens et<br />
des hommes de science de tous pays qui s’occupent des problèmes de l’eau.<br />
Certains de ces ouvrages sont publiés en coopération avec l’Association<br />
internationa<strong>le</strong> des sciences hydrologiques ou I’Organisatioii mMorologique<br />
mondia<strong>le</strong> dans <strong>le</strong> cadre de projets réalisés conjointement par ces orga-<br />
nisations et l’Unesco.
Design d water resources projecis with inadequate data: P-dings d <strong>the</strong> Madrid aympoaium,<br />
June 1973 / Elaboration des projeta d'utilisation dei resswTas en eau aona d onka auffluntes:<br />
Actea du wlloque de Madrid. juin 1973<br />
Volume II Contents Tab<strong>le</strong> des matidres<br />
Foreword/Avant-propos<br />
TOPIC II.1A . METHODS FOR STUDIES LN DATA-SCARCE AREAS AND<br />
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF<br />
DATA FOR PURPOSES OF PLANIFICATION OF WATER<br />
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).<br />
METHODOLOGY FOR ASSESSING HYDROLOGICAL CHA-<br />
RACTERISTICS IN DATASCARCE AREAS.<br />
POINT II.1A . METHODES D'ETUDES UTILISEES DANS LES REGIONS OU<br />
LES DONNEES SûNT INSUFFISANTES ET INFLUENCE SUR<br />
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR<br />
L'ELABORATION DES PROJETS DE L'UTILISATION DES<br />
RESSOURCES EN EAU (A L'EXCLUSION DES CRUES ET<br />
DES DEBITS DE BASSES EAUX).<br />
METHODOLOGIE POUR L'EVALUATION DES CARACTE-<br />
RISTIQUES HYDROLOGIQUES DANS LES REGIONS OU<br />
LES DONNEES SûNT RARES.<br />
BASSO, EDUARDO. (UNDPMIMO) GENERAL REPORT<br />
ABIODUM, ADIGUN ADE. (NIGERIA)<br />
Water resources projects in Nigeria and <strong>the</strong> hydrological data employed in<br />
<strong>the</strong>ir planning and development ................................<br />
BASSO, E., ARRIAGADA, A., NEIRA, H., PEREZ DELGADO, M. (COSTA<br />
RICA)<br />
An examp<strong>le</strong> of regional co-operation for improving <strong>the</strong> hydrological and<br />
meteorological information ...................................<br />
CUBAS GRANADO, FRANCISCO. (SPAIN)<br />
Existing methodology for estimating free water surface evaporation ....<br />
CUSTODIO, EMILIO. (SPAIN)<br />
Geohydrological studies in small areas without systematic data ........<br />
DALINSKY, JOSEPH S. (ISRAEL)<br />
Methods of analysing deficient discharge data in arid and semi-arid zones<br />
for <strong>the</strong> design of surface water utilization .......................<br />
D'OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)<br />
Mapai river hydrological study (Limpopo's river) ...................<br />
D'OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)<br />
Application of Coutagne's and Turc's formulas to sou<strong>the</strong>rn Mozambique<br />
rivers ...................................................<br />
HERAS, R. (SPAIN)<br />
Report hydrological programa of <strong>the</strong> Center for Hydrographic Studies for<br />
<strong>the</strong> investigation of hydraulic resources with insufficient data .........<br />
1<br />
21<br />
35<br />
59<br />
77<br />
95<br />
141<br />
121<br />
155
KARAUSHEV, A.V., BOGOLIUBOVA, I.V. (U.S.S.R.)<br />
Computation of reservoin wdLnrntition .........................<br />
KLIGUE, R.K., MECHDI EL SACHOB (U.S.S.R.)<br />
CilnilritionofrunoffinIraq ..................................<br />
KUZMIN, P.P., VERSHININ, A.P. (U.S.S.R.)<br />
Determination of evaporation in caw of <strong>the</strong> abmnce or inadequacy of<br />
data .....................................................<br />
PENTA, A., ROSSI, F. (ITALY)<br />
Objective criteria to daclare a aerier of data sufficient for technical pur-<br />
poses ....................................................<br />
QUINTELA GOIS, CARLOS. (PORTUGAL)<br />
Objective criteria used in hydrology with inadequate data ............<br />
SMITH, ROBERT L. (U.S.A.)<br />
Utilizing climatic data to appraise potentiai water yields .............<br />
STANESCU, SILVIU. (COLOMBIA)<br />
Determination of hydrological characteristics in points without direct<br />
hydrometricdata ...........................................<br />
TEMEZ, J.R. (SPAIN)<br />
New models of frequency law of runoff starting from precipitations ....<br />
TRENDEL, R., DER MEGREDITCHIAN, G., RULLIERE, MARIE CLAIRE.<br />
(FRANCE)<br />
Traitement opérationnel des données pluviométriques entachées d'erreurs<br />
ouinsuffisantes ............................................<br />
TOPIC II.1B . METHODS FOR STUDIES IN DATA SCARCE AREAS AND<br />
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF<br />
DATA FOR PURPOSES OF PLANIFICATION OF WATER<br />
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).<br />
INFLUENCE OF INADEQUACY OF HYDROLOGICAL DATA<br />
ON PROJECT DESIGN AND FORMULATION.<br />
POINT II.1B - METHODES D'ETUDES UTILISEES DANS LES REGIONS OU<br />
LES DONNEES SONT INSUFFISANTES ET INFLUENCE SUR<br />
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR<br />
L'ELABORATION DES PROJETS DE L'UTILISATION DES<br />
RESSOURCES EN EAU (A L'EXCLUSION DES CRUES ET<br />
DES DEBITS DE BASSES EAUX). INFLUENCE DU MANQUE<br />
DE DONNEES HYDROLOGIQUES SUR LE CALCUL DU<br />
PROJET ET SA FORMULATION.<br />
BEARD, L.R. (U.S.A.) GENERAL REPORT<br />
BANERJI, S., LAL, V.B. (INDIA)<br />
Design of water resources projects with inadequate data in India. General<br />
& Particular Case Studies ................................... 323<br />
199<br />
207<br />
217<br />
221<br />
24 1<br />
253<br />
265<br />
287<br />
30 1<br />
315
JAMB, IVAN C. (U.S.A.)<br />
Data requirements for <strong>the</strong> optimization of reservoir dengn and operating<br />
dedetermination ..........................................<br />
REID, GEORGE W. (U.S.A.)<br />
The design of water quality management projecta with inadequate data<br />
SABHERWAL, R.K. (INDIA)<br />
Designing projects for <strong>the</strong> development of ground water resources in <strong>the</strong><br />
alluvial plains of nor<strong>the</strong>rn India on <strong>the</strong> basis of inadequate data .......<br />
SEXTON, J.R., JAMIESON, D.G. (U.K.)<br />
Improved techniques for water resource systems design .........<br />
WEBER, J., KISIEL, CHESTER C., DUCKSTEIN, LUCIEN (U.S.A.)<br />
Maximum information obtainab<strong>le</strong> from inadequate design data: from<br />
multivariate to Bayesian methods ..............................<br />
TOPIC 11.2 - CURRENT PRACTICES FOR ASSESSING DESIGN FLOODS<br />
AND DESIGN LOW FLOWS, INCLUDING THE USE OF<br />
SYNTHETIC UNIT HYDROGRAPH, WITH PARTICULAR<br />
EMPHASIS ON MAXIMALISATION AND MINIMALISATION.<br />
POINT 11.2 - PRATIQUES COURANTES POUR L'EVALUATION DES<br />
CRUES ET DES DEBITS D'ETIAGES PRIS EN COMPTE DANS<br />
LE PROJET, COMPRENANT L'EMPLOI D'HYDROGRAMMES<br />
UNITAIRES DE SYNTHESE, AVEC ETUDE PARTICULIERE<br />
DE LA MAXIMALISATION ET DE LA MINIMALISATION.<br />
ROCHE, MARCEL. (FRANCE) GENERAL REPORT<br />
BATLLE GIRONA, MODESTO. (SPAIN)<br />
Estimation of floods by means of <strong>the</strong>ir silt loads .................<br />
BERAN, M.A. (U.K.)<br />
Estimation of design floods and <strong>the</strong> prob<strong>le</strong>m of equating <strong>the</strong> probability<br />
ofrainfailandrunoff ........................................<br />
DAVIS, DONALD R., DUCKSTEIN, L., KISIEL, CHESTER C., FOGEL, MAR-<br />
TIN M. (U.S.A.)<br />
A decision-<strong>the</strong>oretic approach to uncertainty in <strong>the</strong> return period of<br />
maximum flow volumes using rainfall data .......................<br />
HALL, M.J. (U.K.)<br />
Syn<strong>the</strong>tic unit hydrograph technique for <strong>the</strong> design of flood al<strong>le</strong>viation<br />
works in urban areas ......................................<br />
HELLIWELL, P.R., CHEN, T.Y. (U.K.)<br />
A dimension<strong>le</strong>ss unitgaph for Hong Kong ........................<br />
HERAS, R., LARA, A. (SPAIN)<br />
Study of maximum floods in small basins of torrential type ..........<br />
335<br />
349<br />
365<br />
383<br />
40 1<br />
419<br />
439<br />
459<br />
473<br />
485<br />
501<br />
517
HERBST, P.H., VAN BIWON, S., OLIVIER, J.P.J., HALL, J.M. (SOUTH<br />
AFRICA)<br />
Flood estimation by determination of regional parameten from limited<br />
data ....................................................<br />
JARASWATHANA, DAMRONG., PINKAYAN, SUBIN. (THAILAND)<br />
Practices of design flood frequency for small watersheds in Thailand ...<br />
KINOSITA, TAKEO., HASHIMOTO, TAKESHI. (JAPAN)<br />
Design discharge derived from design rainfall ..................<br />
LEESE, MORVEN N. (U.K.)<br />
The use of censored data in estimating t-year floods .........<br />
POGGI PEREIRA, PAULO. (BRAZIL)<br />
Assessment of design floods in Brazil ........................<br />
RENDON-HERRERO, OSWALD. (U.S.A.)<br />
A method for <strong>the</strong> prediction of washload in certain small watersheds ....<br />
RODIER, J.A. (FRANCE)<br />
Méthodes utilisées pour l'évaluation des débita de m e des petits com<br />
d'eau en régions tropica<strong>le</strong>s ....................................<br />
SOKOLOV, A.A. (U.S.S.R.)<br />
Methods for <strong>the</strong> estimation of maximum dischargea of snow melt and<br />
rainfall water with inadequate observational data ..................<br />
VLADIMIROV,A.M.,CHEBOTAREV, A.I. (U.S.S.R.)<br />
Computation of probabiustic valuea of low flow for ungauged riven .<br />
WON, TAE SANG. (U.S.A.)<br />
A study on maximum flood discharge formulas ....................<br />
TOPIC III - RELATION BETWEEN PROJECT ECONOMICS AND HYDROLO-<br />
GICAL DATA<br />
POINT 111 - RELATION ENTRE LES DONNEES ECONOMIQUES DU PRO-<br />
JET ET LES DONNEES HYDROLOGIQUES<br />
BURAS, NATHAN. (ISRAEL)<br />
The cost-effectiveness of water resources systems considering inadequate<br />
hydrologiddata ...........................................<br />
FILOTTI, A., FRANK, G., PARVULESCU, C. (ROMANIA)<br />
Optimization of water resources development projects in case of inade-<br />
quate hydrologic data ....................................<br />
POBEDIMSKY, A. (ECE)<br />
Relation between project economics and hydrologicai data ...........<br />
54 1<br />
553<br />
551<br />
563<br />
517<br />
5 89<br />
603<br />
615<br />
625<br />
635<br />
649<br />
66 1<br />
683
INTRODUCTION<br />
The Symposium on <strong>the</strong> Development of Water Resources Projects with<br />
Inadequate Data was held in Madrid from 4 to 8 June 1973 for <strong>the</strong> purpose<br />
of focusing on <strong>the</strong> methodology for hydrologic studies for water resources<br />
projects with inadequate data and on current practices for <strong>the</strong> assessment<br />
of design parameters.<br />
The Symposium was opened at <strong>the</strong> Palacio de Exposiciones on <strong>the</strong><br />
morning of 4 June by Miniester of Public Workes of Spain Addresses were<br />
<strong>the</strong>n given by Dr. Dumitrescu on behalf of <strong>the</strong> Director General of Unesco,<br />
Professor Nevmec on behalf of <strong>the</strong> Secretary-General of WMO, Dr. Rodier<br />
as President of <strong>IAHS</strong> and by Dr. Briones, on behalf of <strong>the</strong> Spanish Na-<br />
tional Committee for <strong>the</strong> IHD.<br />
The Symposium was attended by 480 participants from 77 countries.<br />
The technical programme, detal<strong>le</strong>d in <strong>the</strong> Tab<strong>le</strong> of Contents, included<br />
consideration of 3 major areas:<br />
1. Methodology for hydrological studies with inadequate data,<br />
2. Current practices in different countries,<br />
3. Relation between project economics and hydrological data.<br />
Each area was fur<strong>the</strong>r sub-divided into topics for each of which <strong>the</strong><br />
individually contributed papers were abstracted into a general report, orally<br />
presented by an invited expert, and followed by discussion.<br />
Since <strong>the</strong> individual papers were not presented at <strong>the</strong> Symposium orally<br />
by <strong>the</strong> authors, <strong>the</strong>ry are reproduced here in <strong>the</strong> orden in which<br />
<strong>the</strong>y were reported in each general report under each topic.
Contents<br />
Tab<strong>le</strong> des matières<br />
Volume II<br />
Foreword/Avant-propos ................................<br />
TOPIC II.1A - METHODS FOR STUDIES IN DATA-SCARCE AREAS AND<br />
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF<br />
DATA FOR PURPOSES OF PLANIFICATION OF WATER<br />
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).<br />
METHODOLOGY FOR ASSESSING HYDROLOGICAL CHA-<br />
RACTERISTICS IN DATA-SCARCE AREAS.<br />
POINT II.1A - METHODES D’ETUDES UTILISEES DANS LES REGIONS OU<br />
LES DONNEES SONT INSUFFISANTES ET INFLUENCE SUR<br />
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR<br />
L’ELABORATION DES PROJETS DE L’UTILISATION DES<br />
RESSOURCES EN EAU (A L’EXCLUSION DES CRUES ET<br />
DES DEBITS DE BASSES EAUX).<br />
METHODOLOGIE POUR L’EVALUATION DES CARACTE-<br />
RISTIQUES HYDROLOGIQUES DANS LES REGIONS OU<br />
LES DONNEES SONT RARES.<br />
BASSO, EDUARDO. (UNDP/WMO) GENERAL REPORT<br />
ABIODUM, ADIGUN ADE. (NIGERIA)<br />
Water resources projects in Nigeria and <strong>the</strong> hydrological data employed in<br />
<strong>the</strong>ir planning and development ................................<br />
BASSO, E., ARRIAGADA, A., NEIRA, H., PEREZ DELGADO, M. (COSTA<br />
RICA)<br />
An examp<strong>le</strong> of regional co-operation for improving <strong>the</strong> hydrological and<br />
meteorological information ...................................<br />
CUBAS GRANADO, FRANCISCO. (SPAIN)<br />
Existing methodology for estimating free water surface evaporation ....<br />
CUSTODIO, EMILIO. (SPAIN)<br />
Geohydrological studies in small areas without systematic data ........<br />
DALINSKY, JOSEPH S. (ISRAEL)<br />
Methods of analysing deficient discharge data in arid and semi-arid zones<br />
for <strong>the</strong> design of surface water utilization .......................
II<br />
D’OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)<br />
Mapai river hydrological study (Limpopo’s river) ...................<br />
D’OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)<br />
Application of Coutagne’s and Turc’s formulas to sou<strong>the</strong>rn Mozambique<br />
rivers ....................................................<br />
HERAS, R. (SPAIN)<br />
Report hydrological programs of <strong>the</strong> Center for Hydrographic Studies for<br />
<strong>the</strong> investigation of hydraulic resources with insufficient data .........<br />
KARAUSHEV,A.V., BOGOLIUBOVA, I.V. (U.S.S.R.)<br />
Computation of reservoirs sedimentation .......................<br />
KLIGUE, R.K., MECHDI EL SACHOB (U.S.S.R.)<br />
Calculation of runoff in Iraq ..................................<br />
KUZMIN, P.P., VERSHININ, A.P. (U.S.S.R.)<br />
Determination of evaporation in case of <strong>the</strong> absence or inadequacy of<br />
data .....................................................<br />
PENTA, A., ROSSI, F. (ITALY)<br />
Objective criteria to declare a series of data sufficient for technical pur-<br />
poses ....................................................<br />
QUINTELA GOIS, CARLOS. (PORTUGAL)<br />
Objective criteria used in hydrology with inadequate data ............<br />
SMITH, ROBERT L. (U.S.A.)<br />
Utilizing climatic data to appraise potential water yields .............<br />
STANESCU, SILVIU. (COLOMBIA)<br />
Determination of hydrological characteristics in points without direct<br />
hydrometric data ...........................................<br />
TEMEZ, J.R. (SPAIN)<br />
New models of frequency law of runoff starting from precipitations ....<br />
TRENDEL, R., DER MEGREDITCHIAN, G., RULLIERE, MARIE CLAIRE.<br />
(FRANCE)<br />
Traitement opérationnel des données pluviornetriques entachées d’erreurs<br />
ou insuffisantes ............................................
TOPIC II.1B - METHODS FOR STUDIES IN DATA SCARCE AREAS AND<br />
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF<br />
DATA FOR PURPOSES OF PLANIFICATION OF WATER<br />
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).<br />
INFLUENCE OF INADEQUACY OF HYDROLOGICAL DATA<br />
ON PROJECT DESIGN AND FORMULATION.<br />
POINT II.1B - METHODES D’ETUDES UTILISEES DANS LES REGIONS OU<br />
LES DONNEES SONT INSUFFISANTES ET INFLUENCE SUR<br />
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR<br />
L’ELABORATION DES PROJETS DE L’UTILISATION DES<br />
RESSOURCES EN EAU (A L’EXCLUSION DES CRUES ET<br />
DES DEBITS DE BASSES EAUX). INFLUENCE DU MANQUE<br />
DE DONNEES HYDROLOGIQUES SUR LE CALCUL DU<br />
PROJET ET SA FORMULATION.<br />
BEARD, L.R. (U.S.A.) GENERAL REPORT<br />
BANERJI, S., LAL, V.B. (INDIA)<br />
Design of water resources projects with inadequate data in India. General<br />
& Particular Case Studies ...................................<br />
JAMES, IVAN C. (U.S.A.)<br />
Data requirements for <strong>the</strong> optimization of reservoir design and operating<br />
ru<strong>le</strong> determination ..........................................<br />
REID, GEORGE W. (U.S.A.)<br />
The design of water quality management projects with inadequate data .<br />
SABHERWAL, R.K. (INDIA)<br />
Designing projects for <strong>the</strong> development of ground water resources in <strong>the</strong><br />
alluvial plains of nor<strong>the</strong>rn India on <strong>the</strong> basis of inadequate data .......<br />
SEXTON, J.R., JAMIESON, D.G. (U.K.)<br />
Improved techniques for water resource systems design ..............<br />
WEBER, J., KISIEL, CHESTER C., DUCKSTEIN, LUCIEN (U.S.A.)<br />
Maximum information obtainab<strong>le</strong> from inadequate design data: from<br />
multivariate to Bayesian methods ..............................<br />
TOPIC 11.2 - CURRENT PRACTICES FOR ASSESSING DESIGN FLOODS<br />
AND DESIGN LOW FLOWS, INCLUDING THE USE OF<br />
SYNTHETIC UNIT HYDROGRAPH, WITH PARTICULAR<br />
EMPHASIS ON MAXIMALISATION AND MINIMALISATION.
IV<br />
POINT 11.2 - PRATIQUES COUFUNI"'I'S POUR L'EVALUATION DES<br />
CRUES ET DES DEBITS D'ETIAGES PRIS EN COMPTE DANS<br />
LE PROJET, COMPRENANT L'EMPLOI D'HYDROGRAMMES<br />
UNITAIRES DE SYNTHESE, AVEC ETUDE PARTICULIERE<br />
DE LA MAXIMALISATION ET DE LA MINIMALISATION.<br />
ROCHE, MARCEL. (FRANCE) GENERAL REPORT<br />
BATLLE GIRONA, MODESTO. (SPAIN)<br />
Estimation of floods by means of <strong>the</strong>ir silt loads ..............<br />
BERAN, M.A. (U.K.)<br />
Estimation of design floods and <strong>the</strong> prob<strong>le</strong>m of equating <strong>the</strong> probability<br />
of rainfall and runoff ........................................<br />
DAVIS, DONALD R., DUCKSTEIN, L., KISIEL, CHESTER C., FOGEL, MAR-<br />
TIN M. (U.S.A.)<br />
A decision-<strong>the</strong>oretic approach to uncertainty in <strong>the</strong> return period of<br />
maximum flow volumes using rainfall data .......................<br />
HALL, M.J. (U.K.)<br />
Syn<strong>the</strong>tic unit hydrograph technique for <strong>the</strong> design of flood al<strong>le</strong>viation<br />
works in urban areas ........................................<br />
HELLIWELL, P.R.,CHEN, T.Y. (U.K.)<br />
A dimension<strong>le</strong>ss unitgraph for Hong Kong ........................<br />
HERAS, R., LARA, A. (SPAIN)<br />
Study of maximum floods in small basins of torrential type ..........<br />
HERBST, P.H., VAN BILJON, S., OLIVIER, J.P.J., HALL, J.M. (SOUTH<br />
AFRICA)<br />
Flood estimation by determination of regional parameters from limited<br />
data .....................................................<br />
JARASWATHANA, DAMRONG., PINKAYAN, SUBIN. (THAILAND)<br />
Practices of design flood frequency for small watersheds in Thailand ...<br />
KINOSITA, TAKEO., HASHIMOTO, TAKESHI. (JAPAN)<br />
Design discharge derived from design rainfall ......................<br />
LEESE, MORVEN N. (U.K.)<br />
The use of censored data in estimating t-year floods ................
POGGI PEREIRA, PAULO. (BRAZIL)<br />
Assessment of design floods in Brazil .............................<br />
RENDON-HERRERO, OSWALD. (U.S.A.)<br />
A method for <strong>the</strong> prediction of washload in certain small watersheds ...<br />
RODIER, J.A. (FRANCE)<br />
Méthodes utilisées pour l’évaluation des débits de crue des petits cours<br />
d’eau eri régions tropica<strong>le</strong>s ....................................<br />
SOKOLOV, A.A. (U.S.S.R.)<br />
Methods for <strong>the</strong> estimation of maximum discharges of snow melt and<br />
rainfall water with inadequate observational data ..................<br />
VLADIMIROV, A.M., CHEBOTAREV, A.I. (U.S.S.R.)<br />
Computation of probabilistic values of low flow for ungauged rivers ....<br />
WON, TAE SANG. (U.S.A.)<br />
A study on maximum flood discharge formulas ....................<br />
TOPIC III - RELATION BETWEEN PROJECT ECONOMICS AND HYDROLO-<br />
GICAL DATA<br />
POINT III - RELATION ENTRE LES DONNEES ECONOMIQUES DU PRO-<br />
JET ET LES DONNEES HYDROLOGIQUES<br />
BURAS, NATHAN. (ISRAEL)<br />
The cost-effectiveness of water resources systems considering inadequate<br />
hydrological data ...........................................<br />
FILOTTI, A., FRANK, G., PARVULESCU, C. (ROMANIA)<br />
Optimization of water resources development projects in case of inade-<br />
quate hydrologic data ....................................<br />
POBEDIMSKY, A. (ECE)<br />
Relation between project economics and hydrological data ...........
Foreword<br />
Whi<strong>le</strong> <strong>the</strong> need for hydrological and meteorological data of many types<br />
for <strong>the</strong> design of water resources projects is obvious, it is often found,<br />
especially in many developing countries, that such data are ei<strong>the</strong>r lacking<br />
or inadequate.<br />
Recognizing <strong>the</strong> existence of this prob<strong>le</strong>m, <strong>the</strong> Co-ordinating Counci.1 of<br />
<strong>the</strong> IHD appointed a group of experts (third session, Paris, June 1967) to<br />
study <strong>the</strong> prob<strong>le</strong>m of design of water resources projects with inadequate<br />
data.<br />
Similarly, <strong>the</strong> Commission for Hydrology of WMO (third session, Geneva,<br />
September 1968) established a Working Group on Hydrological Design<br />
Data for Water Resources Projects to prepare guidance material on this<br />
subject for <strong>the</strong> WMO Guide to Hydrological Practices and to maintain<br />
liaison with <strong>the</strong> IHD group of experts appointed by <strong>the</strong> Co-ordinating<br />
Council.<br />
As a means of taking stock of <strong>the</strong> work carried out by <strong>the</strong> hydrological<br />
community in coping with project design with scarce data, Unesco and<br />
WMO jointly convened a symposium on this subject. The Symposium was<br />
organized with <strong>the</strong> co-operation of <strong>the</strong> <strong>IAHS</strong> and <strong>the</strong> Spanish National<br />
Committee for <strong>the</strong> IHD and was held in Madrid from 4 to 8 June 1973 at<br />
<strong>the</strong> invitation of <strong>the</strong> Government of Spain.<br />
The Madrid Symposium concentrated on <strong>the</strong> methodology of hydro-<br />
logical studies for water resources projects with inadequate data and on<br />
current practices for <strong>the</strong> assessment of design parameters.<br />
The Minister of Public Works of Spain opened <strong>the</strong> Symposium at <strong>the</strong><br />
Palacio de Exposiciones on <strong>the</strong> morning of 4 June. Addresses were given<br />
by Dr. Dumitrescu on behalf of <strong>the</strong> Director-General of Unesco, Professor<br />
Nemec on benalf of <strong>the</strong> Secretary-General of WMO, Dr. Rodier as President<br />
of <strong>IAHS</strong> and by Dr. Briones, on behalf of <strong>the</strong> Spanish National Committee<br />
for <strong>the</strong> IHD.<br />
The Symposium was atteneded by 480 participants from 77 countries.<br />
The technical programme, detai<strong>le</strong>d in <strong>the</strong> Tab<strong>le</strong> of Contents, included<br />
consideration of 3 major areas:<br />
1. Methodology for hydrological studies with inadequate data;<br />
2. Current practices in different countries;<br />
3. Relation between project economics and hydrological data.<br />
Each area was fur<strong>the</strong>r sub-divided into topics for each of which <strong>the</strong><br />
indivi,dually contributed papers were abstracted into a general report,<br />
orally presented by an invited expert, and followed by discussion.
This volume of proceedings was compi<strong>le</strong>d by <strong>the</strong> Spanish National Com-<br />
mittee for <strong>the</strong> IHD; it includes all <strong>the</strong> general reports and individual<br />
papers presented at <strong>the</strong> Symposium, as well as <strong>the</strong> discussions. It is issued<br />
as a joint Unesco/WMO/<strong>IAHS</strong> pub,lication in <strong>the</strong> spirit in which <strong>the</strong> three<br />
Organizations have collaborated during <strong>the</strong> IHD.<br />
Since <strong>the</strong> individual authors did not present <strong>the</strong>ir papers orally at <strong>the</strong><br />
Symposium, <strong>the</strong> papers are reproduced here in <strong>the</strong> order in which <strong>the</strong>y<br />
are discussed in <strong>the</strong> general report for each topic.<br />
Unesco, WMO and <strong>IAHS</strong> wish to record <strong>the</strong>ir thanks to <strong>the</strong> Spanish<br />
National Committee for <strong>the</strong> IHD for <strong>the</strong> many contributions of its members<br />
towards <strong>the</strong> organization of <strong>the</strong> Symposium, and for <strong>the</strong> Committee’s as-<br />
sistance in <strong>the</strong> publication of <strong>the</strong>se proceedings.
AVANT-PROPOS<br />
I1 est évident que, pour élaborer des projets d’utilisation des ressources<br />
en eau il est nécessaire de disposer de données hydrologiques et météoro-<br />
logiques de types très divers; or il apparaît que ces données sont souvent<br />
inexistantes ou insuffisantes, notamment dans beaucoup de pays en voie<br />
de développement.<br />
Conscient de ce problème, <strong>le</strong> Conseil de coordination de la DHI a créé,<br />
lors de sa troisième session (Paris, juin 1967) un groupe d’experts chargé<br />
d’étudier <strong>le</strong>s moyens d’elaborer des projets d’utilisation des ressources<br />
en eau sans disposer de données suffisantes.<br />
De son côté, la Commission d’hydrologie de l’OMM a constitué à sa<br />
troisième session (Genève, septembre 1968) un groupe de travail sur <strong>le</strong>s<br />
données hydrologiques nécessaires à l’élaboration des projets d’arnénagement<br />
des ressources hydrauliques; ce groupe de travail a été chargé de<br />
formu<strong>le</strong>r des recommandations destinées à figurer dans <strong>le</strong> Guide OMM des<br />
pratiques hydrologiques, et d’assurer la liaison avec <strong>le</strong> groupe d’experts<br />
de la DHI créé par <strong>le</strong> Conseil de coordination.<br />
Afin de faire <strong>le</strong> point des travaux accomplis par la communité hydrologique<br />
en ce qui concerne l’élaboration de projets pour <strong>le</strong>squels on ne<br />
dispose pas de données suffisantes, l’Unesco et l’OMM ont décidé de réunir<br />
conjointement un colloque consacré à cette question. Ce colloque, organisé<br />
avec la collaboration de I’AISH et du Comité national espagnol pour la<br />
DHI, s’est tenu à Madrid en juin 1973, à l’invitation du gouvernement espagnol.<br />
Le colloque de Madrid a traité en particulier de la méthodologie des<br />
études hydrologiques sans données suffisantes et des pratiques courantes<br />
utilisées pour l’évaluation des paramètres de calcul.<br />
Le colloque a été ouvert par <strong>le</strong> ministre espagnol des travaux publics,<br />
<strong>le</strong> matin du 4 juin, dans <strong>le</strong> cadre du Palais des expositions. Des allocutions<br />
furent prononcées par M. Dumitriscu, au nom du Directeur général de<br />
l’Unesco, par <strong>le</strong> professeur Nemec, au nom du Secrétaire général de l’OMM,<br />
par M. Rodier, président de l’AISH, et par M. Briones, au nom du Comité<br />
national espagnol pour la DHI.<br />
480 participants, venant de 77 pays, participèrent au colloque.<br />
Le programme technique, dont <strong>le</strong> contenu détaillé figure dans la tab<strong>le</strong><br />
des matières, portait sur trois domaines principaux:<br />
1. Méthodologie des études hydrologiques sans données suffisantes;<br />
2. Les pratiques courantes utilisées dans différents pays;<br />
3. Relation entre <strong>le</strong>s données économiques du projet et <strong>le</strong>s données<br />
hydrologiques.
Chacun de ces domaines était subdivisé en thèmes, et sur chaque thème<br />
un rapport général synthétisant <strong>le</strong>s communications individuell<strong>le</strong>s était pré-<br />
senté par un expert, puis suivi d’une discussion.<br />
Les Actes du colloque, établis par <strong>le</strong> Comité national espagnol pour<br />
la DHI, comprennent l’ensemb<strong>le</strong> des communications individuel<strong>le</strong>s et des<br />
rapports généraux, ainsi que <strong>le</strong> compte rendu des débats auxquels ils ont<br />
donné lieu. Ils constituent une publication conjointe de l’Unesco, de l’OMM<br />
et de I’AISH, reflétant l’esprit dans <strong>le</strong>quel <strong>le</strong>s trois organisations ont col-<br />
laboré pendant la DHI.<br />
Comme <strong>le</strong>s communications individuel<strong>le</strong>s n’ont pas été présentées ora-<br />
<strong>le</strong>ment par <strong>le</strong>urs auteurs, el<strong>le</strong>s sont reproduites dans l’ordre où el<strong>le</strong>s sont<br />
apparues dans <strong>le</strong> rapport <strong>le</strong>s concernant.<br />
Unesco, l’OMM et I’AISH tiennent à remercier <strong>le</strong> Comité national es-<br />
pagnol pour la DHI du concours qu’il a apporté à l’organisation du colloque<br />
et à la publication de ses Actes.
METHODOLOGY FOR ASSESSING HYDROLOGICAL CHARACTERISTICS<br />
IN DATA SCARCE AREAS<br />
General Report<br />
bY<br />
Eduardo Basso*<br />
INTRODUCTION<br />
Three of <strong>the</strong> eight papers reviewed in this Report describe in general <strong>the</strong> form<br />
of assessing hydrological characteristics in data-scarce areas (Nigeria, Ango-<br />
la and <strong>the</strong> Central American Isthmus). The o<strong>the</strong>r five papers deal with <strong>the</strong> ap-<br />
Plication of certain particular methods, covering estimation of runoff, evapo-<br />
ration, sedimentation and o<strong>the</strong>r hydrological parameters. Therefore, <strong>the</strong> re-<br />
vision will be made in this order.<br />
The procedure to be followed in this summarizing report consists of present-<br />
ing summaries of <strong>the</strong> papers followed by a discussion of <strong>the</strong> main subjects and<br />
by some general comments on <strong>the</strong> <strong>who<strong>le</strong></strong> subject.<br />
REVIEW OF THE PAPERS<br />
Okavango Basin in Angola. - The paper by Mr. Quintelag** presents <strong>the</strong><br />
studies made in Angola and in particular those of <strong>the</strong> Okavango Basin, which<br />
is one of <strong>the</strong> big international rivers of <strong>the</strong> South of Angola. The basin is shown<br />
in Figure 1 of <strong>the</strong> paper. Its drainage area is about 150 O00 Km2 and most of<br />
<strong>the</strong> rainfall occurs from October to April. Altitudes range from 1000 to 1800<br />
meters. For <strong>the</strong> rainfall studies, 28 stations could provide 20 years of records<br />
after comp<strong>le</strong>ting some shortages by correlation. With <strong>the</strong>se annual values, an<br />
isohyetical map was drawn taking into account altitudes and some climatical<br />
factors. From <strong>the</strong>re, <strong>the</strong> mean annual rainfall was computed and analysed by<br />
applying <strong>the</strong> Foster-Hazen method. The result is shown in Figure 2 of <strong>the</strong><br />
paper, from which a mean annual precipitation of 950 mm was debermined.<br />
19 flow measuring stations operate in <strong>the</strong> basin, but only records for 7 years<br />
were availab<strong>le</strong>. As <strong>the</strong> mean rainfall of <strong>the</strong>se seven years is near <strong>the</strong>average<br />
<strong>the</strong> author concludes that <strong>the</strong> mean annual flow can be estimated by averaging<br />
<strong>the</strong> flows of those seven years for every station.<br />
One station operatdby <strong>the</strong><br />
South African Services had longer records (25 years) and for it <strong>the</strong> Foster-<br />
Hazen method was used (Figure 3 of <strong>the</strong> paper). Finally, Figure 4 shows a<br />
curve indicating <strong>the</strong> variation of <strong>the</strong> specific annual flow within <strong>the</strong> drainage<br />
area.<br />
* Project Manager, Central American Hydrometeorological Project, (UNDP/<br />
WMO). Managua, Nicaragua.<br />
** See list of references at <strong>the</strong> end of this Report,
2<br />
Niveria's Case. Abiodun's paperg deals first with Nigeria's water policy and<br />
with <strong>the</strong> institutional arrangements in relation 90 water resources studies.<br />
It later presents some examp<strong>le</strong>s of utilization of hydrological data in existing<br />
projects.<br />
The Kainji multipurpose scheme is located on River Niger (Figure 1 of <strong>the</strong><br />
paper). Although construction was started in 1964, no water <strong>le</strong>vels were observed<br />
prior to 1959. The precipitation network was also insufficient until in<br />
1953 when new stations were instal<strong>le</strong>d allowing a seven year record (1953-59)<br />
from which <strong>the</strong> rainfall over <strong>the</strong> catchment area was calculated for this period.<br />
A relatively long record at Jebba, upstream of <strong>the</strong> dam, could not be used<br />
because of lack of adequate datum information. A correlation between monthly<br />
rainfall and runnoff at Jebba was obtained using <strong>the</strong> newly observated discharges<br />
at Jebba for seven years and w e -ed for ccmp&ing <strong>the</strong> discharge from <strong>the</strong> basin<br />
between Niamey (upstream of <strong>the</strong> dam site at Kianzi) and Jebba Cbserved and<br />
computed flows for two years are shown in Figure 2 of <strong>the</strong> paper.<br />
The &ect of lage up to one month were considered in <strong>the</strong> correlation (equation<br />
1 of <strong>the</strong> paper). Finally, <strong>the</strong> discharge at <strong>the</strong> damsite was obtained substracting<br />
<strong>the</strong> eetimated runoff on two areas, estimating runoff coefficients of O. 1 and<br />
O. 2 (equation 2 of <strong>the</strong> paper).<br />
The paper refers <strong>the</strong>m to <strong>the</strong> prob<strong>le</strong>ms produced by <strong>the</strong> lack<br />
cal data in Midwestern Nigeria.<br />
of hydrogeologi-<br />
in Western Nigeria many long and reliab<strong>le</strong> evaporation and rainfall data are<br />
availab<strong>le</strong>, but river discharges are very scarce. According to <strong>the</strong> author, <strong>the</strong><br />
standard practice is to base <strong>the</strong> water scheme design in a conservative form<br />
using a monthly evaporation of 127 mm and computing runoff with <strong>the</strong> formula<br />
in which Q, is <strong>the</strong> catchment annual runoff, A <strong>the</strong> basin drainage area, Rpsn<br />
<strong>the</strong> basin rainfall value corresponding to correspondjng <strong>the</strong> probability of -Urtaace.in<br />
5ûy~are Co;'Coefficient of runoff for <strong>the</strong> basin, estimated at 4%. Abig<br />
dun indicates that variations of this formula are widely used in western Nigeria.<br />
and <strong>the</strong> uee of a form of it was not used for computing <strong>the</strong> flood for <strong>the</strong><br />
spillway for Asejire Project because of <strong>the</strong> advice of a foreign consultant. Ins<br />
tead, a runoff of 490 l/sec was used. basedan similar occurrences in o<strong>the</strong>r<br />
West African dtreams.<br />
The paper refers also briefly to <strong>the</strong> Lake Chad basin studies which count with<br />
cooperation from <strong>the</strong> United States Geological Survey, FAO and UNESCO. In<br />
this case FAO'e efforts have been directed to <strong>the</strong> harmonization and evaluation<br />
of <strong>the</strong> data. infra-red aerial photography has been used in connection with<br />
<strong>the</strong>e e tas ka.<br />
The paper concludes with an appraisal of <strong>the</strong> studies used and with a brief des-<br />
cription of <strong>the</strong> future activities in <strong>the</strong> field of water resources investigations in<br />
Nigeria. Here, <strong>the</strong> use of new techniques such as rsmote sensing is recom-<br />
mended.<br />
The Central American Hydrometeorological Proiec<br />
The paper by Basso. Arriagada, Neira and Pérez 13. describes <strong>the</strong> activities<br />
of <strong>the</strong> Central American Hydrometeorological Project, a co-operative effort<br />
-
etween <strong>the</strong> countries of <strong>the</strong> Central American Istbmis and <strong>the</strong> United Nations<br />
Development Programme acting as executive agency <strong>the</strong> World Meteorological<br />
Organization. The main objectives of <strong>the</strong> Project are: (i) Installation of a basic<br />
network of meteorological and hydrological station; (ii) Col<strong>le</strong>ction, processing<br />
and publication of <strong>the</strong> data; (iii) training of personnel by means of courses,<br />
fellowships or through technical publication and manuals; and (iv) The institutional<br />
streng<strong>the</strong>ning of <strong>the</strong> meteorological and hydrological services of <strong>the</strong><br />
area.<br />
At <strong>the</strong> beginning of <strong>the</strong> Project (1966) <strong>the</strong> conditions in <strong>the</strong> area varied widely<br />
from country to country. In <strong>the</strong> average <strong>the</strong> few river discharge measuring<br />
stations had short and sometimes unreliab<strong>le</strong> data, <strong>the</strong> meteorological network<br />
was poorly distributed and bore no connection with <strong>the</strong> hydrological network, a<br />
defect that has been also reported in o<strong>the</strong>r of <strong>the</strong> papers under review; ra-<br />
diation, evaporation and rainfall intensity information was comp<strong>le</strong>tely insuf-<br />
ficient, and --except for one or two countries-- no sediment or water quality<br />
measurements were made at all. A few capab<strong>le</strong> technicians were availab<strong>le</strong>,<br />
but extensive training was a pressing need.<br />
The paper describes in some detail <strong>the</strong> steps taken by <strong>the</strong> Project, as result<br />
of which <strong>the</strong> present situation is quite satisfactory for developing conditions.<br />
Of particular interest for <strong>the</strong> subject of this meeting is <strong>the</strong> description of some<br />
methods proposed by <strong>the</strong> project for assessing hydrological characteristics<br />
with insufficient data.<br />
The use of <strong>the</strong> sediment rating curve, Figure 7 of <strong>the</strong> paper. has been used for<br />
computing sediment transportation. The remarks on <strong>the</strong> variation of <strong>the</strong> coe&<br />
ficient n of <strong>the</strong> equation C SA Qn with annual precipitation (G:sediment dis-<br />
charge, Q: Discharge; A, n coefficients) are of interest in aMlyZing scarce se-<br />
diment information. Figure 8 of <strong>the</strong> paper shows <strong>the</strong> results of some measure-<br />
ments made by <strong>the</strong> Project, indicating <strong>the</strong> effect of rainfall and vegetation cover<br />
in <strong>the</strong> sediment yield. The effect of <strong>the</strong> destruction of <strong>the</strong> vegetab<strong>le</strong> cover by a<br />
volcanic eruption should be noticed as a quite particular case.<br />
Flood and rainfall envelopes (Figure 9 and 10 of <strong>the</strong> paper) have been used as<br />
a first estimate of maximum discharges and precipitation studies. Studies of<br />
regionalized flood frequency analysis are now under way.<br />
O<strong>the</strong>r achievements of <strong>the</strong> Project include etudies for determining evapotrans -<br />
piration and water requirements for irrigation, studies on runoff forecasting<br />
groundwater studies using a regional analog computer, etc.<br />
The report refers also to <strong>the</strong> prob<strong>le</strong>m of network imp<strong>le</strong>mentation in areas with<br />
access prob<strong>le</strong>ms, and <strong>the</strong> use of prefabricated e<strong>le</strong>ments Éhould be noted<br />
(Figures 3 and 4 of <strong>the</strong> Report). Figures 5 and 6 shows <strong>the</strong> change in areal<br />
coverage as result of <strong>the</strong> action of <strong>the</strong> project. The successful use of modern<br />
mechanical methods for processing meteorological and hydrological information<br />
should encourage o<strong>the</strong>r developing countries in <strong>the</strong> use of <strong>the</strong>se methods.<br />
The report concludes with a remark on <strong>the</strong> importance of adequate institu-<br />
tional support for <strong>the</strong>se activities, which< imitia1Ly requi res <strong>the</strong> creation of<br />
concern of <strong>the</strong> Governments on <strong>the</strong> importance of meteorology and hydrology.<br />
3
4<br />
Est i mat i ng Water Yiel ds<br />
Smith's9 paper presents an interesting examp<strong>le</strong> of estimating water yields<br />
using only precipitation and temperature measurements.<br />
The basic water balance equation applied to a catchment area may by expressed<br />
as:<br />
P R+E+ AS<br />
P precipitation; R total basin outflow, E evapotranspiration and AS<br />
change in storage. For a long period AS becomes negligib<strong>le</strong>, and making<br />
some transformations in equation (1) it is possib<strong>le</strong> to rewrite it as:<br />
Thus, in <strong>the</strong> long term, <strong>the</strong> runoff oefficient C is governed by climatic consider-<br />
ations. ln 1967 Guisti and López$ proposed that <strong>the</strong> mean stream discharge<br />
could be determined as a function of <strong>the</strong> mean annual precipitation and <strong>the</strong> basin<br />
climatic index, BCI, defined as:<br />
where P: average monthly precipitation in centimeters and T: average monthly<br />
temperature in degrees centigrade. A relation between C and BCI based in 250<br />
catchments in <strong>the</strong> United States and Puerto Rico is shown in Figure 1 of <strong>the</strong> paper.<br />
The use of regional relations between BCI and P as those shown in Figure 2 of<br />
<strong>the</strong> paper allows to derive C only from precipitation data. The basic C versus<br />
BCI relationship was tested with satisfactory results as those shown in Figure 3<br />
of <strong>the</strong> paper.<br />
The basic relationships can also be used to appraise <strong>the</strong> effect of changes of <strong>the</strong><br />
precipitation, If subscript 1 represents natural conditions and 2 represented<br />
augmented conditions (in <strong>the</strong> case of an increase in rainfall) <strong>the</strong>n <strong>the</strong> gain in runoff<br />
can be written as:<br />
Where, PM L P2/Pi<br />
Jn tab<strong>le</strong> 1 <strong>the</strong> author compares <strong>the</strong> results of using this method with <strong>the</strong> results<br />
of ueing hydrologic simulation as reported by several investigators with good<br />
agreement..<br />
Using a reasonab<strong>le</strong> amount of judgment it is possib<strong>le</strong> to determine flow characteristics<br />
o<strong>the</strong>r than <strong>the</strong> mean. Figure 4 shows a comparison of calculated and<br />
observed annual runoff distributions for <strong>the</strong> Marias de Cygnes River, Kansas,<br />
USA. However, <strong>the</strong> limitation of this method, as c<strong>le</strong>arly indicated in <strong>the</strong> text<br />
of <strong>the</strong> paper, should be considered before using it.<br />
estimation of monthly yields allocating <strong>the</strong>m in proportion to <strong>the</strong>ir contribution<br />
to <strong>the</strong> BCI (a two month running average should be used due to tag prob<strong>le</strong>ms).<br />
(1)<br />
This also applies to <strong>the</strong>
The use of <strong>the</strong> basic relation can also be extended with <strong>the</strong> help of certain flow<br />
and miscellaneous field measurements.<br />
The paper closes showing <strong>the</strong> application of <strong>the</strong> method for appraising <strong>the</strong> po-<br />
tential yield characteristics of coastal aquifers in sou<strong>the</strong>rn Puerto Rico and<br />
presenting one examp<strong>le</strong> of <strong>the</strong> adjustments required when <strong>the</strong> natural conditions<br />
have been changed by man's activities.<br />
Application of Coutagne's and Turc's Formulas<br />
The paper by D'Oliveira and Mip~so6J applies Coutagne's and Turc formulas<br />
for <strong>the</strong> sou<strong>the</strong>rn Mozambique rivers.<br />
Coutagne's general ru<strong>le</strong> states:<br />
D-H-KH2<br />
D: Runoff deficit = H - E; H: Mean rainfall height; K! Coutagne's constant<br />
Also C = KH where C: Runoff Coefficient ; &<br />
H<br />
The most probab<strong>le</strong> value of K is obtained by equating to zero <strong>the</strong> first derivative<br />
of E(C - KH)2, which results in:<br />
Turc's general ru<strong>le</strong> can be expressed as:<br />
P.<br />
H<br />
/.z-g-<br />
Where L: Turc's constant = A t 25T t O. 05T3,<br />
P: Evaporation plus percolation looses ( runoff deficit), H: Precipitation,<br />
A: Constant; T: Mean temperature (In degrees centigrade)<br />
Turc applied his ru<strong>le</strong> for 254 basins, using A 300, finding that in 53% of <strong>the</strong><br />
cases <strong>the</strong> difference between <strong>the</strong> real and computed D was <strong>le</strong>es than 40 mm; in<br />
43% of <strong>the</strong> cases this difference was <strong>le</strong>ss than O. 1 of measured D and in 65% <strong>the</strong><br />
difference was <strong>le</strong>ss than O. 2 measured D.<br />
The application of both formulas to seven basins was divided nto two groups; <strong>the</strong><br />
Limpopo River group (Rainfall 450-650 nun; temperatures 18' C-20' C) and <strong>the</strong><br />
Incomati, Sabie, Umbeluzi and Usoto Group (Rainfall 800 mm; temperatures<br />
higher than 20').<br />
Detai<strong>le</strong>d results are presented, which can be sumarized as follows:<br />
5
6<br />
Limpopo area<br />
E<strong>le</strong>phants River<br />
Beit Bridge<br />
Trigo de Morais<br />
All group<br />
C ontanne' s Turc relation<br />
K A = 300<br />
Per cent of D -Dcalc<br />
greater than O. 1 Dcalc<br />
o. O00055<br />
O. 000031<br />
O. 000047<br />
o. 000050<br />
Incomati, Sabie. Umbeluzi and Usoto area<br />
Incomati River O. 0001 50<br />
Sabie River<br />
O. 000131<br />
Umbeluzi River<br />
O. 000145<br />
Usuto River<br />
O. 000162<br />
All group<br />
O. 000140<br />
-<strong>the</strong> use of Turc's relation with A 300 produces poor results, <strong>the</strong> authors<br />
present a nomograph (Figure 2 of <strong>the</strong> paper) to compute <strong>the</strong> value of A. Using<br />
<strong>the</strong>se new values of <strong>the</strong> constants <strong>the</strong> difference between calculated and measured<br />
D is reduced to acceptab<strong>le</strong>s <strong>le</strong>vels.<br />
Estimati on of Lvapotranspiration<br />
The paper by Kuzmin and Vershininu deals with <strong>the</strong> determination of evapora-<br />
tion in case of <strong>the</strong> absence or inadequacy of data.<br />
Since methods for direct evaporation measurements are still being developed,<br />
computations are <strong>the</strong> main source of information. These can be divided into<br />
three groups: (i) methods based in <strong>the</strong> physical analysis of <strong>the</strong> process, (ii)<br />
methods combining <strong>the</strong> physical analysis with semi-empirical constants deter -<br />
mined from actual evaporation in representative regions and (iii) purely statistical<br />
methods.<br />
The first group includes methods using heat balance equation. water balance<br />
equation and turbu<strong>le</strong>nt diffusion. In <strong>the</strong> USSR equation (1) which has been<br />
deduced from <strong>the</strong> simplified equation of <strong>the</strong> heat balance of <strong>the</strong> land surface with<br />
<strong>the</strong> account of Bowen ratio is widely applied.<br />
where: E: Evapotranspiration, R is <strong>the</strong> measured value of <strong>the</strong> radiation balance<br />
of <strong>the</strong> surface, B is <strong>the</strong> heat income into <strong>the</strong> soil, L is <strong>the</strong> latent heat of evapor-<br />
ation, Cp is <strong>the</strong> heat capacity under constant pressure, H is <strong>the</strong> atmospheric<br />
pressure, t and e are respectively <strong>the</strong> differences in temperature and water<br />
pressure measured at two <strong>le</strong>vels above <strong>the</strong> ground.<br />
Equation (1) should ra<strong>the</strong>r belong to <strong>the</strong> second group than to <strong>the</strong> first one, since<br />
it does not represent all physical factors that affect <strong>the</strong> phenomena.<br />
Full water balance is not applied in <strong>the</strong> practice but in <strong>the</strong> case of deep water<br />
tab<strong>le</strong>. in this case, <strong>the</strong> fobwing equation is used in <strong>the</strong> USSR for estimating-<br />
evapotranspiration from non-irrigated fields;<br />
64%<br />
16%
E = X +(W1 . W2) (2)<br />
X: Precipitation; W1 and W2 are <strong>the</strong> moieture storage in soil at <strong>the</strong> beginning<br />
and at <strong>the</strong> end of <strong>the</strong> design period. Some conditions for using this relation<br />
are indicated by <strong>the</strong> author. Ano<strong>the</strong>r partial solution of water balance equation<br />
is <strong>the</strong> estimation of mean annual sums of evapotranspiration as <strong>the</strong> dif -<br />
ference between precipitation and runoff. After indicating <strong>the</strong> possibilities of<br />
methods based on turbu<strong>le</strong>nt diffusion prob<strong>le</strong>ms in deriving a universal equation<br />
are also stated. Therefore, <strong>the</strong> convenience of equations using non-specialized<br />
observations is evident. One of <strong>the</strong>se for regions - of natural moistening is due<br />
to Budyko:<br />
-Ro XL<br />
;h Ro (cm year-1 ) (3)<br />
4<br />
equation (3) includes only one observational parameter, X: long term average<br />
precipitation (cm/year) R o is <strong>the</strong> average annual radiation balan e of <strong>the</strong> undey<br />
lying surface which can be obtained from <strong>the</strong> m ap of Referenced. L is <strong>the</strong><br />
latent heat of evaporation, A method for distributing <strong>the</strong> mean annual sums<br />
estimated from equation (3) is explained by <strong>the</strong> ,authors.<br />
Monthly evapotranspiration from irrigated fields are estimated with <strong>the</strong> help<br />
of simplified heat balance equations. The standard error is about 15% when<br />
special observations are availab<strong>le</strong> or about 30% with standard observations.<br />
The use of em+ical relations similar to that of Blaney and Cridd<strong>le</strong> can be<br />
used only if <strong>the</strong> empirical coefficients are tested and corrected for each point<br />
of <strong>the</strong>ir application.<br />
The most simp<strong>le</strong> equations allowing <strong>the</strong> estimation of evaporation from water,<br />
snow and ice surfaces by means of standard observational data, are <strong>the</strong><br />
following binomial and monomial equations:<br />
and<br />
E (a t ab&) (es - e2)<br />
E = A U2 (es - e2)<br />
being: E; evaporation -/day; U, wind speed at <strong>the</strong> height z above <strong>the</strong> surface<br />
in misec; es and e2 are <strong>the</strong> maximum water vapor pressure estimated<br />
from surface temperature and water pressure at 2 meters in mb; A, a and b<br />
are experimental constants. For estimating evaporation from snow <strong>the</strong> values<br />
of a -0.18 ab=O. 098 z=lOm should be used in equation (8). For lake evaporation<br />
a=O. 14 b=O. 72 and 9m should be used in equation (8). O<strong>the</strong>r cases are also<br />
discussed in <strong>the</strong> paper.<br />
RESERVOIR SEDIMENTATION<br />
The paper by Karaushev and Bogeliubevag presents a method for estimating<br />
reservoir sedimentation based on <strong>the</strong> equation of sediment balance as applied to<br />
<strong>the</strong> <strong>who<strong>le</strong></strong> reservoir or its parts.<br />
The inflow of sediments is computed by observational data or by indirect<br />
methods. The outflow of sediments is computed based in hydraulic and sediment<br />
characteristics. The determination of sedimentation during one year is<br />
reduced to estimating tha portion of <strong>the</strong> sediment inflow that is accumulated in<br />
<strong>the</strong> reservoir.<br />
7
8<br />
Equation (1) of <strong>the</strong> paper shows <strong>the</strong> computation of sedimentation for any si ze<br />
fraction in a design interval:<br />
m -6<br />
Paj = Z Pi in j - Qter j A tj 10 ils si terj (1)<br />
is1<br />
P aj is <strong>the</strong> amount of sediments of all size fractions trapped by <strong>the</strong> reservoir<br />
during k tj; Pi in j is <strong>the</strong> inflow of sediment for each i-th fraction; Q ter j is<br />
<strong>the</strong> mean water outflow (m3/s) and Si ter j is <strong>the</strong> mean turbidity (concentration)<br />
for <strong>the</strong> time 4 tj and for <strong>the</strong> i-th fraction of size. Equation (2) to (9) are used<br />
for computing si ter j and are based in hydrod namic considerations and <strong>the</strong><br />
reservoir characteristics such as <strong>le</strong>ngth and depth. The amount of bed load in<br />
<strong>the</strong> reservoir is computed by equation (10):<br />
Pa bed j = lom3 (R bed in j - R bed ter j) A tj (1 0)<br />
Pa bed j is <strong>the</strong> weight of bed load in <strong>the</strong> reservoir (Tons), R bed in j and R bed<br />
ter ' indicate bed load discharge at <strong>the</strong> initial and terminal discharge sites<br />
(Kgjsec), A tj is <strong>the</strong> time interval, Bed load, R bed, is computed with Shamovls<br />
equation (equations 11 to 14 <strong>the</strong> text).<br />
The annual accumulation of all sediment fractions for <strong>the</strong> first year of reser-<br />
voir operation is obtained by adding <strong>the</strong> suspended and bed sediments as indicated<br />
in equation (15) of <strong>the</strong> paper. The value Pai (tons) so obtained is transformed<br />
into volumetric units Wai:<br />
- Ys is <strong>the</strong> specific weight of <strong>the</strong> sediment (T/m3). After <strong>the</strong> first<br />
duced volume W Wa is used for <strong>the</strong> computations of next year.<br />
For <strong>the</strong> computation of <strong>the</strong> chronological variations of sedimentation <strong>the</strong> Shamov<br />
method is recommended:<br />
Where Wat is <strong>the</strong> sediment volume in t years; Wal is <strong>the</strong> sedimentation volume<br />
during <strong>the</strong> first year, computed as explained before, W a ext is <strong>the</strong> extreme<br />
volume of sediments in <strong>the</strong> reservoir, approximately computed by:<br />
Where W is <strong>the</strong> initial volume of <strong>the</strong> reservoir, Ur is <strong>the</strong> area of river cross<br />
section when discharge is close to maximum and up is <strong>the</strong> maximum cross<br />
section area of <strong>the</strong> upper pool near <strong>the</strong> dam.<br />
Surface Water Utilization in Arid and Semi Arid Zones. - The paper by Dalinskyly<br />
shows *e experience of Tahal-Water Planning for Israel Ltd. in various methods<br />
of analyzing stream flows. For planning of utilization <strong>the</strong> following information<br />
is required: (a) <strong>the</strong> average volume of annual flows (P.ave),. representing <strong>the</strong><br />
stream water resources potential; <strong>the</strong> average annual feasib<strong>le</strong> utilizab<strong>le</strong> flows<br />
is a portion of this value; (b) <strong>the</strong> stream's flow regime including flood frequency;<br />
(c) <strong>the</strong> stream variability within a season, a year, or from one year to ano<strong>the</strong>r.
For determining <strong>the</strong> annual flood return periods <strong>the</strong> author proposes <strong>the</strong> use of<br />
<strong>the</strong> well known T = formula; for longer return periods <strong>the</strong> estimates of<br />
m<br />
order of magnitude of annual flows for longer return periods can be obtained<br />
by extrapolation on probability paper.<br />
The next section deals with <strong>the</strong> well known flow-durati on curves,<br />
The concept of Ilhorizontal cut" of <strong>the</strong> stream hydrograph is useful in <strong>the</strong> case<br />
of a diversion of a stream, as indicated in sketchs 2 and 3 of <strong>the</strong> paper. The<br />
"horizontal cut1' can be expressed ma<strong>the</strong>matically as:<br />
Qd = Qi when<br />
Qd (ad) max when<br />
mix<br />
Qi 3 Qd) m ax<br />
Where: Q: atreamflow discharge<br />
Qd diverted discharge<br />
(Qd) max: maximum diverted discharge<br />
... (2)<br />
For a period of n years, a series of n annual diverted volumes can be obtained<br />
and <strong>the</strong> average diverte&annual flow (va) can be calculated for each value of<br />
(Qd) max. The funtion Ud = f (ad) max has <strong>the</strong> form indicat& in sketch 4.<br />
Three zones can be distinguished in this curve; in zone I Ud is<br />
'mmax<br />
relatively large and almost constant; in zone II <strong>the</strong> derivative decrease quickly<br />
as (Qd) max increases; in zone III <strong>the</strong> derivative trends to eeru, when (Qd)max+ Q<br />
Most of diversions will be economically justified in_zone i, and unfeasib<strong>le</strong> in<br />
Zone IU. Formula (3) can be used for calculating Ud from <strong>the</strong> flow-duration<br />
curve.<br />
Adjustments for baseflows or minimum diverted discharges can be made easily<br />
changing <strong>the</strong> origin of coordinates.<br />
When <strong>the</strong>re are limitations to <strong>the</strong> diversion of baseflow discharges, a "doub<strong>le</strong><br />
cuttt is required as indicated in sketch 5. This case will arise when baseflow<br />
is undesirab<strong>le</strong> due to high salinity or o<strong>the</strong>r reasons for diversion. A maximum<br />
desirab<strong>le</strong>discharge is determined generally by sedimentation conditions. The<br />
value of UA, average diverted flow can be computed as:<br />
where ÜB is established by means of a horizontal cut and E by means of a<br />
vertical cut. Funtions Ug and Uc can be easily calculatfi by computer, An<br />
examp<strong>le</strong> is shown in Fig. 2, App A. Using equation (5) UG can be calculated<br />
from <strong>the</strong> flow-duration curve without use of a computer.<br />
Ano<strong>the</strong>r uae of <strong>the</strong> vertical cut is presented for planning of diversiomwith limitations<br />
of maximum discharges due to sedimentation:<br />
Qd m Q<br />
for Q 4 Qdmax Qd = O for Q > IQdmax l<br />
The resulting discharge curve is combined with <strong>the</strong> sediment concentration<br />
flow discharge curve shown in Figure 4, App. A for computing <strong>the</strong> sediment<br />
transport as detai<strong>le</strong>d in App. B. of <strong>the</strong> paper.<br />
9
10<br />
Next section deals with <strong>the</strong> determination of annual storab<strong>le</strong> flows as a .function<br />
of reservoir capacity. Assuming that losses during <strong>the</strong> rainy season can be<br />
neg<strong>le</strong>cted, <strong>the</strong> following equation applies:<br />
- UR: is <strong>the</strong> n years' averageannual amount of water stored in <strong>the</strong> reservoir<br />
(Net average capacity'RN)<br />
UR)^ is <strong>the</strong> amount of water stored in <strong>the</strong> i th year;<br />
UR)i Ui when Ui 4 RN<br />
Iud.<br />
i: (RN)i when Ui P (RN)i<br />
Ui: is <strong>the</strong> annual streamflow<br />
(Rn)i represents <strong>the</strong> net reservoir capacity in <strong>the</strong> ith year<br />
Limitations of <strong>the</strong>se relations arc indicated in <strong>the</strong> text (Neg<strong>le</strong>cting losses, etc. )<br />
Relations between K a n d <strong>the</strong> reservoir efficiencyhs function of average<br />
Uave<br />
net reservoir capacity are shown schematically in sketch 6. (The meaning of<br />
<strong>the</strong> three zones is <strong>the</strong> same as in sketch 4). This analysis is important for<br />
preliminary estimates and/or feasibility calculations. Recent investigations<br />
reported in <strong>the</strong> paper prove that <strong>the</strong>se relations can be approximately estimated<br />
on a regional basis using as a parameter <strong>the</strong> dimension<strong>le</strong>ss standard deviatio-<br />
Formulae for computing RN and RN. based in <strong>the</strong> decrease of <strong>the</strong> capacity of<br />
<strong>the</strong> reservoir due to sedimentation are given in o<strong>the</strong>r section of <strong>the</strong> paper. The<br />
paper concludes recommending hydrological investigations to find, on a regional<br />
basis,parameters allowing to represent <strong>the</strong> main functions discus sed in <strong>the</strong><br />
artic<strong>le</strong>.<br />
ASSESSING mROLOCICAL CHARACTERISTICS IN DATA -SCARCE AREAS<br />
Estimating flow regime when no data are availab<strong>le</strong>. - The first prob<strong>le</strong>m with<br />
which <strong>the</strong> hydrologist has to deal in data-scarce areas consists of obtaining<br />
<strong>the</strong> hydrological characteristics of <strong>the</strong> region. First of all, <strong>the</strong> average discharge<br />
has to be estimated. For this, a wide variation of methods can be used<br />
depending in <strong>the</strong> availability of information<br />
The worst case consists in a comp<strong>le</strong>te lack of information Here <strong>the</strong> estimates<br />
should be based in observations in similar gauged zones. The estimations made<br />
for Asejire Project seem to belong to this case. However, even in this extreme<br />
case, <strong>the</strong> scarce availab<strong>le</strong> information should not be neg<strong>le</strong>cted. Topography,<br />
altitude, shape, orientation, geology and ve@aHecover of <strong>the</strong> basin are easily<br />
obtained and should be always used.<br />
The most e<strong>le</strong>mentary equation is <strong>the</strong> surface relation:<br />
A<br />
Q=- Qb<br />
Ab<br />
where: Q : Flow at <strong>the</strong> site under study; Qb : Flow at a base station; A: Surface<br />
of <strong>the</strong> basin under study and Ab: Surface of <strong>the</strong> basin of <strong>the</strong> base station.<br />
This equation, ii obviaidya wrypoor representation of <strong>the</strong> +noniena, and sbculd be used
only for gross estimates.<br />
When precipitation data is availab<strong>le</strong>,this mdhod can be improved by introducing <strong>the</strong><br />
precipitation data for <strong>the</strong> basin under study. (P) and for <strong>the</strong> basin of <strong>the</strong> -base<br />
station (Pb). The relation in this case is:<br />
A P<br />
Q a-<br />
A b pb Q b (b)<br />
If <strong>the</strong> yield of <strong>the</strong> basin is defined as K e 2, equation (b) becomes:<br />
PA<br />
Q-KbPA (4<br />
A variation of this method, used sucessfully in Chi<strong>le</strong> and Central America" J<br />
consists 8f analyzing <strong>the</strong> variation of K with <strong>the</strong> basin conditions, topography,<br />
e<strong>le</strong>vation, vegetation, geology, orientation, etc.. . Figure 1 shows an examp<strong>le</strong><br />
of this method.<br />
Abiodun uses this method in his paper. No explanations, however, are given<br />
for <strong>the</strong> criteria in se<strong>le</strong>cting K = O. 04 and for <strong>the</strong> use of a 1:50 years precipitation.<br />
This seems an exaggerately pesimistic estimation and should result<br />
in underestimation of <strong>the</strong> water resources. However, since as <strong>the</strong> paper explains<br />
that efforts are been made for enlarging <strong>the</strong> scope of <strong>the</strong> hydrological investigations<br />
in Nigeria, it is hoped that soon it will be possib<strong>le</strong> to revise <strong>the</strong>se computations<br />
with more accurata methods.<br />
When <strong>the</strong> Economic Comission for Latin America decided to make a preliminary<br />
survey of <strong>the</strong> water resources in <strong>the</strong> Central American Isthmus, <strong>the</strong> UNbP/<br />
@Mo project prepared <strong>the</strong> maps of curves of runoff deficit shown in Figure 2,<br />
wnich allowed first estimates for ungauged areas. The trace of <strong>the</strong> curves<br />
should take into account <strong>the</strong> already mentioned physical factors.<br />
A fur<strong>the</strong>r improvement consists of <strong>the</strong> introduction of climatic factors. such as<br />
temperature. Examp<strong>le</strong>s of <strong>the</strong>se methods are those of Khosla, Langbein,<br />
Coutagne, Turc and <strong>the</strong> one propeed by Smith in his paper. Application of<br />
<strong>the</strong>se methods with universal constants produce, sometimes, large errors, so<br />
<strong>the</strong>y should be limited to regional use, previously determining <strong>the</strong>ir constants<br />
in gauged areas of similar characteristics.<br />
in D'Olivieria's case, <strong>the</strong> use of Coutagne's ru<strong>le</strong> with <strong>the</strong> original constants<br />
would hata introduced very large errors in <strong>the</strong> estimates. The same occures<br />
when A = 300 Le used in Turc's formula.<br />
As Smith shows in his pa er a good relation between precipitation and tempe-<br />
rature (or Flimatic indexf can be found in a Fegionalized basis. The reporter<br />
has added entra1 American values to Smiths relations for Puerto Rico and<br />
Kansas with good results (Figure 3). However, <strong>the</strong> Constants used in <strong>the</strong>sb<br />
rnethds elaould be verified-on a regional basis.<br />
A check has been made to all <strong>the</strong>se methods using <strong>the</strong>m to estimate <strong>the</strong> mean<br />
annual discharge (in mm) of e<strong>le</strong>ven CentralAmerican streams of quite different<br />
conditions. The following methods have been used: Equations a. b and c,<br />
Coutagne, Turc and Smith, and <strong>the</strong> results are summarized in tab<strong>le</strong> i.<br />
11
12<br />
Tab<strong>le</strong> L - Comparison of <strong>the</strong> use of several methods for estimating mean annual<br />
runoff (mm) of Central American streams.<br />
Country Drainage Estimates of mean annual runoff Obserx<br />
and Basin ed<br />
Station sq Km. Eq.a Eq, b Eq, c Coutagne Turc Smith runoff<br />
Guatemala<br />
Candelaria<br />
Honduras<br />
Re. Pimienta<br />
El Salvador<br />
Bande ras<br />
San Marcos<br />
Nicaragua<br />
Dar $0<br />
Tamarindo<br />
Costa Rica<br />
Cachi<br />
El Humo<br />
Palmar<br />
Panamá<br />
David<br />
Majk<br />
849. 5<br />
883.8<br />
432.8<br />
180. O<br />
91 5<br />
165<br />
904.1<br />
135<br />
486 3<br />
1392<br />
321 8<br />
470<br />
720<br />
89 0<br />
560<br />
160<br />
280<br />
2280<br />
26 50<br />
1590<br />
1970<br />
1370<br />
480 550<br />
720 750<br />
840 800<br />
590 620<br />
160 200<br />
250 480<br />
2000 2160<br />
5700 6000<br />
2300 2350<br />
2150 2200<br />
1520 1550<br />
1550<br />
540<br />
7 50<br />
81 O<br />
50<br />
340<br />
21 O0<br />
(6 500)<br />
2270<br />
2880<br />
880<br />
1830 1190 440<br />
1250 320 740<br />
1500 660 500<br />
1500 1000 590<br />
670 50 110<br />
1040 230 500<br />
2250 1700 2000<br />
(5240) (6100) 6270<br />
2420 1800 1970<br />
2620 2150 2650<br />
1670 800 1560<br />
Average<br />
error ‘já 35 21 15 40 39 60<br />
( ) Extrapolations.<br />
Equation (c) gives <strong>the</strong> best results, followed by <strong>the</strong> simp<strong>le</strong> areal relation corrected<br />
for taking into account <strong>the</strong> change in precipitation (Equation b). However, <strong>the</strong>se<br />
are <strong>the</strong> results for a particular area, Central America, and <strong>the</strong>re is no assurance<br />
that similar results should apply to o<strong>the</strong>r regions of <strong>the</strong> world. The best advice<br />
could be to try several of <strong>the</strong>se methods and check as soon as possib<strong>le</strong> <strong>the</strong> results<br />
with measurements at <strong>the</strong> site under study.<br />
Extending short or incomp<strong>le</strong>te records, - The most commonly used method for<br />
extending short or incomp<strong>le</strong>te records is to correlate <strong>the</strong> records of <strong>the</strong> station<br />
with <strong>the</strong> records of a station with longer records. The correlation can be done<br />
with mean annual, mean monthly, mean daily or instantaneous discharges: <strong>the</strong><br />
quality of <strong>the</strong> correlation decreasing in this order. For daily or instantaneous<br />
dischargestlag effects have to be taken into account. in larger basins --as in<br />
<strong>the</strong> case reported by Abiodun-- lag effects apply also to monthly discharges.<br />
The quality of <strong>the</strong> correlation can be determined easily by means of simp<strong>le</strong><br />
statistical tests. This quality depende on <strong>the</strong> physical and meteorological<br />
characteristics of <strong>the</strong> basins being compared. in general, correiations between<br />
two stations nearly located over <strong>the</strong> same river give good results. The fol1 w-<br />
ing results should be expected when <strong>the</strong> basins of Figure 4 are comparedld:
Basins compared Quality of correlation<br />
1 and 2<br />
2 and 3<br />
3 and 4<br />
2 and 5<br />
4 and 6<br />
5 and 6<br />
Good : Basins of similar form,<br />
size and orientat i on .<br />
Fair : The orientation of <strong>the</strong><br />
val<strong>le</strong>y is different.<br />
Poor: Different altitude and<br />
orientation.<br />
Fair : Same form but different<br />
a It it ude.<br />
Poor : Different characteristics<br />
Poor : Different orientation and<br />
altitude.<br />
When no hydrometric information is availab<strong>le</strong>, correlation can be tried with<br />
longer precipitation series.<br />
A long time average can be obtained assuming a constant yield of <strong>the</strong> basin, or:<br />
In this case, long (n year) precipitation records are availab<strong>le</strong>, Pt and Qt are<br />
<strong>the</strong> rainfall and discharge averages over <strong>the</strong> t years for which discharge data<br />
are availab<strong>le</strong>. This method, after checking that Pn = P was used by Quintela<br />
in his paper. However, an interesting verification in that case would had been<br />
correlating <strong>the</strong> seven year's records with <strong>the</strong> South African station<br />
Studies made in Chi<strong>le</strong> and in <strong>the</strong> Central American Isthmus show that <strong>the</strong> results<br />
of correlation studies are far more reliab<strong>le</strong> than <strong>the</strong> methods explained<br />
in <strong>the</strong> preceeding section. However, extreme caution has to be exercised when<br />
records are too short, carefully avoiding to be too influenced by some statistical<br />
indicators. In this case comparison with o<strong>the</strong>r methods is an useful auxiliary<br />
tool. Comp<strong>le</strong>te verification of <strong>the</strong> base information should be <strong>the</strong> starting point<br />
of any extension of hydrological records.<br />
Estimating evaporation and evapotranspiration. - The estimation of evaporation<br />
and evapotranspiration has several important implications in hydrological<br />
studies, such as computations of reservoir evaporation, water balances and of<br />
requirements for agriculture.<br />
Direct measurements are difficult; <strong>the</strong> U. S. Wea<strong>the</strong>r Bureau type A pan, <strong>the</strong><br />
mQst frequently used instrument in developing countri es, is not always correctly<br />
read and <strong>the</strong> relation from pan to lake evaporation remains in doubt . The<br />
development of a simp<strong>le</strong> formula for computing potential evaporation, is <strong>the</strong>refore<br />
of great importance.<br />
Kuzmin and Vershinin give an excel<strong>le</strong>nt summary of formulas used in <strong>the</strong> USSR..<br />
For data-scarce areas, however, formulas based in <strong>the</strong> physical interpretation<br />
of tFe fenomena are quite difficult to apply. Equations (2) and (3) are certainly<br />
promising and it would be interesting to have more details on <strong>the</strong>m Binomial<br />
formulas are widely used. Equation (9) lightly different coefficients has<br />
been used in Chi<strong>le</strong> and in Central America wit&<br />
with unsatisfactory results.<br />
13
14<br />
Equation (9), as reported by Kuemin and Vershinin, ly been compared with<br />
Blaney -Gridd<strong>le</strong>, Penman, Hargreaves -Christiansen and Meyer formulas<br />
for five locations in <strong>the</strong> Central American I sthmus with <strong>the</strong> following results:<br />
Tab<strong>le</strong> II<br />
Evaporation computed with several formulas<br />
Station Madden San José Chorrera Guija G ua t emala<br />
Panamá Costa El El Guatemala Average<br />
Formula Rica Salvador Salvador<br />
USSR, Binomial<br />
lake evaporation 79 8 488 1293 1133 765 89 5<br />
Blaney-Cridd<strong>le</strong> 2062 1749 2041 1910 1647 1882<br />
Penman 1328 1077 1345 1530 1350 1326<br />
Hargreaves-<br />
Christians en 1400 1 O00 1340 1280 1070 1218<br />
Meyer 1394 1227 2147 1577 1628 1594<br />
Potential<br />
Evaporation<br />
(Measured in<br />
pan x O. 77) 1020 1145 1760 1460 1050 1287<br />
in average, <strong>the</strong> best agreement ie reached Penman formula. However, as <strong>the</strong><br />
Hargreaves-Chrintiansen equation was a plied using only temperature, humidity<br />
and precipitation (wind was estimated7 it provides a simp<strong>le</strong> alternative, Blaney-<br />
Cridd<strong>le</strong> and Meyer (a simp<strong>le</strong> binomial formula) give excessive values. The USSR<br />
binomial formula gives very low values, which is probably due to <strong>the</strong> obvious<br />
differences in climate with respect to <strong>the</strong> conditions from which <strong>the</strong> formula was<br />
de rived.<br />
Sediment studies. - Three of <strong>the</strong> papers show examp<strong>le</strong>s of sediment determinations,<br />
which quite frequently have to be made with insufficient information,<br />
The first prob<strong>le</strong>m refers to estimating sediment yields from streams without<br />
sediment measurements. Figure 8 of <strong>the</strong> paper on <strong>the</strong> Central American Hydro<br />
meteorological Project shows <strong>the</strong> wide variation, within a regiun, of sediment<br />
yields. Thus, determining sediment transportation loads without field measur -<br />
menta is quite unreliab<strong>le</strong>. Hydrologic and hydraulic information from <strong>the</strong><br />
gauging stations "somehow improves <strong>the</strong>se estimates. However, very simp<strong>le</strong> sq<br />
diment measurements allow a relatively acc rat estimate of suspended sediment<br />
loads. A good correlation has been found18)1 between <strong>the</strong> concentration of a<br />
samp<strong>le</strong> taken with a bott<strong>le</strong> by U n d of unskil<strong>le</strong>d obskrvers and <strong>the</strong> mean concen-<br />
tration obtained with conventional sampling.<br />
The sediment rating curves (two examp<strong>le</strong>s presented in <strong>the</strong> papers) allow to<br />
compute <strong>the</strong> total suspended load in a quite simp<strong>le</strong> form, The points of <strong>the</strong> curve<br />
relating <strong>the</strong> solid and liquid discharges have a large dispersion (due to errors
in measurements, differences in <strong>the</strong> raising and decreasing stages of a flood,<br />
variations in <strong>the</strong> availability of sediment supply, etc. ), but its mean trend has<br />
been found to be relatively stab<strong>le</strong>, which allows estimates of suspended loads<br />
with series of observations as short as one year.<br />
Determination of bed load presents just <strong>the</strong> opposite prob<strong>le</strong>m. Direct measurements<br />
are difficult and provide in most of <strong>the</strong> cases non-meaningful results.<br />
Use of well-known formulas is thus encouraged, in spite of <strong>the</strong> fact that <strong>the</strong>y<br />
give enormous differences. Therefore, <strong>the</strong> use of several methods is suggested<br />
including, if possib<strong>le</strong>, methods, such as modified Einstein, which use <strong>the</strong> availab<strong>le</strong><br />
suspended sediment measurements. It is also quite useful to observe <strong>the</strong><br />
'!critical discharge", i. e. discharge at which <strong>the</strong> bed movement starts, which<br />
in some cases can be determined by detection. of stone noise by <strong>the</strong> stream<br />
gaugers.<br />
Karaushev and Bogeliuva's paper deals with <strong>the</strong> important prob<strong>le</strong>m of predicting<br />
<strong>the</strong> chronology of <strong>the</strong> filling of a dam, and esents a new interesting approach<br />
to <strong>the</strong> prob<strong>le</strong>m also studied by Brownlv. However, <strong>the</strong> main difficulty<br />
as seen by this reporter, is <strong>the</strong> estimation of "in situ" specific weight of <strong>the</strong><br />
sett<strong>le</strong>d suspended sediment.<br />
very fine its sett<strong>le</strong>ment is very slow and subject to relatively complicate laws.<br />
For this <strong>the</strong> formula of Reference 14/ can be used:<br />
yT : -k k( T-l T Log T - 1 )<br />
15<br />
The prob<strong>le</strong>m here is that when <strong>the</strong> sediment is<br />
k is a constant depending on <strong>the</strong> size and mechanical distribution of <strong>the</strong> material;<br />
YT <strong>the</strong> specific weight after T years and y1 <strong>the</strong> s ecific weight of <strong>the</strong> sediment<br />
("in situ") after one year of settling. Referencelggives values of k and Y<br />
but to <strong>the</strong> reporteis know<strong>le</strong>dge, no check of <strong>the</strong>se values have been made for<br />
1,<br />
most of <strong>the</strong> world. These "in situ" determinations are difficult, since it is<br />
practically impossib<strong>le</strong> to obtain indisturbed samp<strong>le</strong>s of submerged clay or lime.<br />
The use of y Ray diffusion probes can be usefull, but <strong>the</strong>y require careful<br />
laboratory calibrations and expensive equipment.<br />
Water Resources Studies. - Dalinsky's paper present an interesting and simp<strong>le</strong><br />
method for preliminary studies of water resources projects, and should be con-<br />
sidered a preliminary approach to those exposed in o<strong>the</strong>r sections of this Sym-<br />
posium.<br />
CONCLUSIONS<br />
Assessing hydrological characteristics in data-scarce areas is indeed a difficult<br />
prob<strong>le</strong>m The difficulties in <strong>the</strong> studies increase inversely with <strong>the</strong> amount of<br />
information. availab<strong>le</strong>, not because of <strong>the</strong> intrins ic ma<strong>the</strong>matical and operational<br />
prob<strong>le</strong>ms, but because extremely good judgement is required. Unfortunately<br />
good hydrological judgement depends on <strong>the</strong> know<strong>le</strong>dge of <strong>the</strong> meteorological,<br />
physical and hydrological characteristics of <strong>the</strong> region under study.<br />
Several excel<strong>le</strong>nt examp<strong>le</strong>s have been shown of what can be done with scarce<br />
information, but <strong>the</strong> possibilities of big mistakes appeared also evident. These<br />
can be avoided ei<strong>the</strong>r with excel<strong>le</strong>nt judgement or with <strong>the</strong> help of a few, but<br />
adequate data. These data do not need to be long term series or sophisticated<br />
measurements, thus can be col<strong>le</strong>cted at a relatively low cost. This cost represents<br />
only a small fraction of <strong>the</strong> eventual overexpenditures or losses from<br />
poorly designed schemes.
16<br />
The ideal, obviously, would be to undertake in each data-scarce area a com-<br />
prehensive meteorological and hydrological survey, as <strong>the</strong> U DP/WMO projects<br />
in several parts of <strong>the</strong> world. Evaluation of <strong>the</strong>se projectslg allows to show<br />
several concrete examp<strong>le</strong>s where a small investment in <strong>the</strong>se basic surveys has<br />
resulted in economic benefits several times larger than <strong>the</strong> expenditures in me-<br />
teorology and hydrology.<br />
REFERENCES<br />
Quintela Gois, C. - Some Criteria Used in Hydrologic Studies with Inadequate<br />
Data. Symposium on <strong>the</strong> Design of Water Resources Projects<br />
with Inadequate Data. Madrid 1973.<br />
Abiodun, A. A. - Water Resources Projects in Nigeria and <strong>the</strong> Hydrological<br />
Data Employed in <strong>the</strong>ir Planning and Development. Symposium on <strong>the</strong><br />
Design of Water Resources Projects with Inadequate Data. Madrid 1973.<br />
Basso, E., Arriagada, A., Neira H. and Pérez Delgado, M. - An Examp<strong>le</strong><br />
of Co-operation for Improving <strong>the</strong> Hydrological and Meteorological Information.<br />
Symposium on <strong>the</strong> Design of Water Resources Projects with Inadequate<br />
Data. Madrid 1973.<br />
Smith, R. - Utilizing Climatic Data to appraise Potential Water Yields.<br />
Simposium on <strong>the</strong> Design of Water Resources Projects with Inadequate<br />
Data. Madrid 1973.<br />
Giusti, E. V. and López M. A. - Climate and streamflow of Puerto Rico,<br />
Caribbean Journal of Science, Vol. 7, pp 87-93, 1967.<br />
D'Oliveira Martens, E. E. and Mimoso Loureira, J. J. - Application of<br />
Coutagne's and Turc formulas to <strong>the</strong> Sou<strong>the</strong>rn Mozambique rivers. Sym-<br />
posium on <strong>the</strong> Design of Water Resources Projects with-Inadequate Data.<br />
Madrid 1973.<br />
Kuzmin, P. P. and Vershinin, A. P. - Determination of Evaporation in<br />
case of <strong>the</strong> Absence or Inadequacy of Data. Symposium on <strong>the</strong> Design<br />
of Water Resources Projects-with Inadequate Daia. Madrid 1973.<br />
Materialy Mezhduvedomstvennogo Sovetchchania PO prob<strong>le</strong>me Izuchenia<br />
i Obosnovania Metodov Rasheta Isparenia s vodnoi Poverkhnosti i Suchi.<br />
(Materials of Interagency Meetings on <strong>the</strong> Prob<strong>le</strong>m of Study and Substantiation<br />
of Methods for <strong>the</strong> Computation of Evaporation from Water and<br />
Land Surfaces). Edited by CGI, Valdai 1966.<br />
Karaushev, A. V. and Bogeliulova L V. - Computation of Reservoir Sedi-<br />
mentation. Symposium on <strong>the</strong> Design of Water Resources Projects with<br />
Inadequate Data. Madrid 1973.<br />
Dalinsky, J S. - Methods of Analysing Defficient Discharge Data in Arid<br />
and Semi-arid zones for <strong>the</strong> Design of Surface Water Utilization Symposium<br />
on <strong>the</strong> Design of Water Resources Projects with Inadequate Data.<br />
Madrid 1973.<br />
Central American Hydrometeorological Pro.iect. - Manual de Instrucciones:<br />
Estudios Hidrológicos (Manual of Instructions: Hydrological -<br />
Studies)<br />
Publicación No, 70, San José, Costa Rica 1972.
Central American Hydrometeorological Proiect. - Medida de la Evaporación<br />
(Measurement of Evaporation) Publicación No. 19, San José, Costa<br />
Rica,. 19 68.<br />
Brown, C. B. - Discussion of "Sedimentation in Reservoirs" by B. J.<br />
Witzig" transactions ASCE Vol. 109, 1944, pp 1080-1086.<br />
Office of Indian Affairs, Bureau of Reclamation, Tennessee VaBey Authority<br />
Corps of Engineers, Geological Survey, Department of Agriculture and<br />
Iowa institute of Hydraulic Research. - A Study of Methods Used in Measurment<br />
and Analysis of Sediment Loads in Streams. Report 9 "Density of<br />
Sediments Deposited in Reservoirs", St. PBul District Sub-Office, Corps<br />
of Engineers, Hydraulic Laboratory University of Iowa, Iowa City, Iowa<br />
194.<br />
Central American Hydrometeorological Project. - Estimación Preliminar<br />
del Balance de Aguas en el Istmo Centroamericano (Preliminary estimat-<br />
ion of <strong>the</strong> water Balance in <strong>the</strong> Central American Isthmus) Pubiicación<br />
No. 18, San José, Costa Rica 1968.<br />
World Meteorological Organization. - Twenty Years of WMO Assistance.<br />
WMO-No. 338, Geneva, Switzerland 1972.<br />
Central American Hydrometeorological Project. - Deficiendas de Agua<br />
en Centro América B Panamá (Water Defficiencies in Central America<br />
and Panad) Repodprepared by G. Hargreaves as a consultant to <strong>the</strong><br />
Central American Hydrometeorological Project. Publication No. 88,<br />
Managua, Nicaragua, 1973.<br />
Central American Hydrometeorological Project. - Emp<strong>le</strong>o de la Muestra<br />
Puntual para la Determinación del Sedimento en Suspensión (Use of <strong>the</strong><br />
Puricbial Samp<strong>le</strong> for <strong>the</strong> determination of Suspended Sediment) Publica -<br />
ciÓn No. 1, San José, Costa Rica, 1967.<br />
Central American Hydrometeorological Project. - Manuel de Instruccio-<br />
nes: Hidrometría (Manual of Instructions: Hydrometry) Publicación No.<br />
47, Segunda Edición, San José, Costa Rica, 1972.<br />
17
18<br />
Figure 1. -<br />
Method for estimating<br />
hydrologic yield of<br />
u ngauged areas<br />
(From Reference lu)<br />
Figure 4. -<br />
Basins used for<br />
checking results of<br />
correlation<br />
(From Reference ,l#
x<br />
20<br />
300<br />
200<br />
Kansas<br />
L<br />
5u<br />
tra1 America<br />
1 I I<br />
1 O0 200 500<br />
P MEAN ANNUAL PRECIPITATION CENTIMETERS<br />
Figure 3. - Central American Values plotted into Smith's BCI f(P)
ABSTRACT<br />
WATER RESOURCES PROJECTS IN NIGERIA AND<br />
THE HYDROLOGICAL DATA EMPLOYED IN THEIR<br />
PLANNING AND DEVELOPMENT<br />
Adigun Ade Abioduna<br />
The need for adequate water supply to meet <strong>the</strong> demands of<br />
Nigeria's growing population is well known. However, <strong>the</strong> technical<br />
adviser is seriously handicapped in his planning efforts by <strong>the</strong><br />
lack of sufficient information. As a result, different kinds of<br />
data and different <strong>le</strong>vels of efficiency have been employed by <strong>the</strong><br />
various agencies which have planned <strong>the</strong> existing major water related<br />
projects un Nigeria. This investigation shows and intensity of<br />
rainfalls and <strong>the</strong> attendant floods, small sca<strong>le</strong> project modelling,<br />
projections based on hydrologic data from o<strong>the</strong>r but climatologically<br />
similar places, provision of missing data by statistical correlation,<br />
and intensive surveys over short periods to obtain rapid and exten-<br />
sive information. These schemes have been reviewed and <strong>the</strong> hydrologic<br />
information employed in designing <strong>the</strong>m has been appraised. This study<br />
also shows that Nigeria must intensify her efforts to provide exten-<br />
sive basic data on her surface and groundwater resources if costly<br />
mistakes are to be avoided in <strong>the</strong> future. A case is also made for <strong>the</strong><br />
use of new techniques such as Remote Sensing for rapid identification<br />
and appraisal of <strong>the</strong>se resources.<br />
RES UME<br />
On sait quels sont <strong>le</strong>s besoins du Nigbria pour un approvision-<br />
nement en eau capab<strong>le</strong> de satisfaire <strong>le</strong>s demandes de sa population<br />
croissante. Or il se trouve que <strong>le</strong> conseil<strong>le</strong>r technique y est sérieu-<br />
sement handicapé, dans son effort de planification, par l'insuffi-<br />
sance de l'information. Les diverses agences qui sont chargées, au<br />
Nigeria, des grands projets d'aménagement des eaux, doivent utiliser<br />
des données disparates ayant des niveaux d'efficacité différents.<br />
L'analyse des problèmes montre que l'gtude de ces projets doit faire<br />
appel -3 l'information loca<strong>le</strong> sur la fréquence et l'intensité des<br />
pluies, et <strong>le</strong>s crues qui en sont la conséquence (petits aménagements),-<br />
aux évaluations tirées des donnés hydrologiques recueillies dans de<br />
régions climatiques semblab<strong>le</strong>s, -à l'utilisation des corrélations<br />
pour boucher <strong>le</strong>s lacunes,- a l'observation intensive sur de courtes<br />
périodes pour étendre rapidement l'observation. Des efforts ont été<br />
faits dans ce sens, mais il reste que <strong>le</strong> Nigeria doit <strong>le</strong>s intensifier<br />
pour rassemb<strong>le</strong>r une masse importante de données de bases sur <strong>le</strong>s<br />
ressources en eaux de surfaces et en eaux souterraines, afin d'éviter<br />
dans l'avenir de coûteuses erreurs. On ne néglige pas non plus<br />
l'utilisation des techniques nouvel<strong>le</strong>s, tel<strong>le</strong>s que la détection 'a<br />
distance, pour améliorer lainventaire de ces ressources.<br />
* Lecturer, Dept. of Agric. Engineering, University of Ife, I<strong>le</strong>-Ife,<br />
Nigeria.
22<br />
1. IBTRODUCTIOH<br />
The developat of water reaiources miithin <strong>the</strong> pat deeade, in Nigeria,<br />
has concentrated moetly on <strong>the</strong> prorieion of adequate pipe-borne water for<br />
domeetic and institutional supplies. The trend, however, ia changiiig, and<br />
it is now realieed that water resource8 developent, a8 a ipa$Or economlo<br />
revolutionary tool, ihould Qiphaeise ita total harnesaing, control and<br />
utilization to provide in addition to watar supply, such o<strong>the</strong>r benefits 88<br />
hydro-power, irrigation water, flood control, water transportation into and<br />
from <strong>the</strong> hinter<strong>le</strong>d, fish and wild-life, recreation and pollution abatarsient.<br />
The awaxenees of <strong>the</strong>se needa has provoked riome dee Unking and has,<br />
in part, precipitated <strong>the</strong> putting toge<strong>the</strong>r of <strong>the</strong> Färat ]81962-68) and <strong>the</strong><br />
Second (1970-74) National Developent Plans. The objective of <strong>the</strong> latter,<br />
according to <strong>the</strong> National Economic Counail, being<br />
"<strong>the</strong> achievement and mainteamce of <strong>the</strong> highest poeaib<strong>le</strong> rate<br />
of increase in <strong>the</strong> standard of living and <strong>the</strong> creation of <strong>the</strong><br />
neceieary conditions to this end, inoluding public support<br />
and awareness of both <strong>the</strong> potaiti&<strong>le</strong> that exist and <strong>the</strong> sac-<br />
rificee that will be required."<br />
The imp<strong>le</strong>mentation of <strong>the</strong> variow schemes coatained in <strong>the</strong>m pl- have<br />
experienced sime hardship especially where technical man-power and information<br />
were needed. In many inatances, <strong>the</strong> technPlogist is often cal<strong>le</strong>d upon to<br />
make far reaahing professional deaisiona, and quite often, he ia seriouelg<br />
handicapped in hie $Lanning efforts by <strong>the</strong> lack of scientific information.<br />
This prob<strong>le</strong>m of planning dthout facta waa amply stated by Andu (1) about<br />
bore ho<strong>le</strong> drilling (for water) in Yeatern Nigeria:<br />
"1 have emphasised <strong>the</strong> handicap due to ecantinesa of hydrolo-<br />
gical data; and eince <strong>the</strong> gigantia Five Year Developnent<br />
Programme cannot be held up becaune of this, <strong>the</strong> practice<br />
now is to confine drilling to areas with favourab<strong>le</strong> geolo-<br />
gical formations. Time facbr has made any exploratory<br />
test drilling virtually impoaaib<strong>le</strong>. The location of a bore<br />
ho<strong>le</strong> wen in a geologically favourab<strong>le</strong> area is chancy -<br />
and <strong>the</strong>re ia no sufficient guarantee that water of adequate<br />
quantity ehall be etruck. It ie not uncomon to drill far<br />
deeper than expected where <strong>the</strong> exhibited geological patterme<br />
indieate o<strong>the</strong>rwise...."
In order to achieve thd goals spel<strong>le</strong>d out in <strong>the</strong> National Dwelopent<br />
Plane, expertise are often imported to analyae our local data or to use<br />
<strong>the</strong>ir "ingenuityn to generate needed scientific information on which our<br />
planning and development programmes could rely.<br />
rical data are ei<strong>the</strong>r scanty, unreliab<strong>le</strong> or absent, and <strong>the</strong> syn<strong>the</strong>sized<br />
data can only be 88 reliab<strong>le</strong> as <strong>the</strong> historical but scanty data availab<strong>le</strong>.<br />
For many foreign experts, handling <strong>the</strong> problwie of <strong>the</strong> tropics is a new<br />
educational experience and most of <strong>the</strong>se techniml consultants, who are<br />
often from temperate climates can only draw on <strong>the</strong>ir bow<strong>le</strong>dge and ewe-<br />
rience of <strong>the</strong>ir own temperate environment and adapt <strong>the</strong>m to plan for <strong>the</strong><br />
needs of <strong>the</strong> tropical zones.<br />
23<br />
More often than not, histo-<br />
Most of <strong>the</strong> existing water resources sehemes have been hand<strong>le</strong>d in <strong>the</strong><br />
nanner enunciated above, and sane of <strong>the</strong>se techniques can in some caees be<br />
referred to as "educated guesen work by <strong>the</strong> experts. Hence, this paper<br />
examinea, in closer details, a few of <strong>the</strong> existing water resources projecte<br />
in Higeria with a view to high-lighting <strong>the</strong> kinds of hydrologic data, analysis,<br />
and <strong>the</strong> different <strong>le</strong>vels of efficiency that have characterized <strong>the</strong>ir planning<br />
and developeat. Such an evaluation should offes some guide-lines for<br />
systematic planning in <strong>the</strong> future.<br />
2. HYDROLOGICAL DATA COLLECTION<br />
The hydrological data needed to effeat adequate study of water resou~.ces<br />
inolude data on precipitation, evaporation, stream-flou and groundwater.<br />
In Nigeria, <strong>the</strong> so<strong>le</strong> responsibility for col<strong>le</strong>cting rainfall data<br />
reste on <strong>the</strong> Federal Meteorological Service (IPPS).<br />
over lux) rain gauging stations throughout <strong>the</strong> country, utilizing <strong>the</strong><br />
recorded data Rom <strong>the</strong>se statione for water resources planning would require<br />
fur<strong>the</strong>r interpretation and analyeis. This is so because <strong>the</strong>se stations<br />
uere not established in relation to river basins.<br />
Although M!CS maintains<br />
Fur<strong>the</strong>rmore, those co-<br />
l<strong>le</strong>cting <strong>the</strong> rainfall data such as <strong>the</strong> local school teachers and looal post<br />
office personnel owe no al<strong>le</strong>giance to <strong>the</strong> WITS since <strong>the</strong> latter never rewards<br />
<strong>the</strong>m in any way or form for <strong>the</strong>ir services. Hence, <strong>the</strong> accuracy and relia-<br />
bility of data col<strong>le</strong>cted under <strong>the</strong> aforeaentioned condition are often in<br />
grave doubt.<br />
The measurement of evaporation data acroas <strong>the</strong> nation is also done by<br />
<strong>the</strong> employing some 68 clans A evaporation pans in an area almost 590,000<br />
square kilometres. In additiop, <strong>the</strong>re are three lysimeter statione in Nigeria<br />
- two at Ibadan and one in Zaria. Although reservoirs are being built on 8<br />
continuing baais, and evaporation acrose <strong>the</strong> land v miw between 102 to 204<br />
eentimetrea a year, <strong>the</strong> impact of evaporation on <strong>the</strong> yields of <strong>the</strong>se reservoirs<br />
is probably still not fully realised.
24<br />
Stream flow data are col<strong>le</strong>cted by auch ageucies as <strong>the</strong> IpLand Waterways<br />
Department (IWD) and <strong>the</strong> Ministries of Work. The former maintains over 100<br />
gauging stations along <strong>the</strong> major rivers of Higeria for <strong>the</strong> expresseu purpose<br />
of recardiiig stage heights which are used to determine navigab<strong>le</strong> waterways.<br />
The Ministries of Work on <strong>the</strong> o<strong>the</strong>r hand are mom interested in potential<br />
areas for <strong>the</strong> location of highway bridges, hence, most of <strong>the</strong>ir hydrological<br />
stations are non-self recording. The unavailability of diacharge measurements<br />
or proper rating curves that could be used to interprete <strong>the</strong> recorded<br />
stage heights has rendered most of <strong>the</strong> date availab<strong>le</strong> unworkab<strong>le</strong>.<br />
In <strong>the</strong><br />
Nor<strong>the</strong>rn States, where <strong>the</strong>re were ZIO reel hydrological net-rrodf until after<br />
1960, most of <strong>the</strong> rivers are non-perennial and shifting; <strong>the</strong> latter situatinn<br />
makes it mandatory to provide more than one ratirig curve per Station per<br />
seaBon thus rendering most of <strong>the</strong> availab<strong>le</strong> record difficult to interprete.<br />
The Geological Survey of Nigeria (GSN) is totally responsib<strong>le</strong> for col<strong>le</strong>cting<br />
data on <strong>the</strong> groundwater resourceB of <strong>the</strong> nation. Most of GSl's<br />
efforts have been concentrated in <strong>the</strong> Nor<strong>the</strong>rn States where <strong>the</strong>re is abundant<br />
supply of grounawater and very litt<strong>le</strong> surface water supply. The GSü<br />
in collaboration with <strong>the</strong> United States Geological Survey, haa carried out<br />
some investigation in <strong>the</strong> Chad Basin comp<strong>le</strong>x, and estimates have been made<br />
of <strong>the</strong> life of <strong>the</strong> aquifers in <strong>the</strong> basin as a result of groundwater mining.<br />
However, no attempt has been made to quantify <strong>the</strong> annual natural groundwater<br />
recharge or <strong>the</strong> contribution of <strong>the</strong> groundwater to river discharge. Information<br />
is also not availab<strong>le</strong> on <strong>the</strong> groundwater flou conditions.<br />
Although <strong>the</strong> agencies cited above col<strong>le</strong>ct vast quantities of data<br />
annually, <strong>the</strong> fact ia that until very recently, <strong>the</strong> data col<strong>le</strong>cted have been<br />
piecemeal and <strong>the</strong> hgdrologicd records were never checked nor analyred. In<br />
many cases, <strong>the</strong> records have no duplicate8 and hence distribution is often<br />
impomsib<strong>le</strong>. This state of affairs is often due to two major factors - lack<br />
of funds and lack of badly needed technical mawpower. This dearth of<br />
adequate hydrological information has not however precluded <strong>the</strong> planning and<br />
<strong>the</strong> actual developent of a -ber of major water schemes in Nigeria euch as<br />
<strong>the</strong> Kainji Dam and Lake Project on River Niger.
30 HYDBOLOCICAL DATA USED IN EXISTING PROJECTS<br />
Sequential generation of hydrological data has been a tool <strong>the</strong> hydro-<br />
logist haa often used to create syn<strong>the</strong>tic records, in <strong>the</strong> absence of very<br />
long hiebrical records, that could be used in hia water reaourcea planning<br />
efforts. Since <strong>the</strong> generated set of data is only as good as <strong>the</strong> historiaal<br />
set employed in such a syn<strong>the</strong>sis, <strong>the</strong> historical set should not be too ahort.<br />
In <strong>the</strong> absenoe of such a hiatorical information, <strong>the</strong> planning anã derelopent<br />
of <strong>the</strong> existing water resources schemes in Nigeria have relied on such tech-<br />
niques of hydrological data derivation as - local information, eduaateä<br />
gueas method, projection based on hydcological data from o<strong>the</strong>r but climatolo-<br />
gically similar places, provision of missing data by correlation and intenaive<br />
surveya over short periods. A few significant schemes are examined below.<br />
A. KAINJI W B ND D q<br />
The moat wide-ranging water reeourcee project undertaken to date in<br />
Nigeria is <strong>the</strong> Kainji Dam and kke Saheie on River Niger (Fig.1.). The scheme<br />
was conceived to provide hydro-e<strong>le</strong>ckic pwer, flood control, regulated water<br />
for navigation, and fishery benefits. Although actìml construction started<br />
at <strong>the</strong> Kainji site in 1964, water <strong>le</strong>vels were never observed <strong>the</strong>re prior to<br />
1959. The two nearest statitma where ologic data were observed on <strong>the</strong><br />
Niger prior to 1959 vere Jebba (Nigeria Y and Niamey (Niger), both of which<br />
sandwich <strong>the</strong> Kainji site and are 906 anä 1630 kilometres respectively amy<br />
from <strong>the</strong> Atlantic mouth of Xivex Niger.<br />
The pre-construction density of rainfall net work within <strong>the</strong> catchment<br />
area of <strong>the</strong> Keinji project was too low to serve as <strong>the</strong> baais for any reliab<strong>le</strong><br />
hydrological interpretation. Conaequently, new rainfall uging stations<br />
were established for <strong>the</strong> project and a seva year record K955f959) was<br />
obtained by <strong>the</strong> consulting f im (2).<br />
method, <strong>the</strong> total amount of rainfall on <strong>the</strong> catchment area was calculated<br />
for <strong>the</strong> seven year period.<br />
25<br />
Through <strong>the</strong> application of <strong>the</strong> Thieasen<br />
Although records of water <strong>le</strong>vels at Jebba were availab<strong>le</strong> for <strong>the</strong> years<br />
1915-24 and 1947-64, <strong>the</strong> ahiftirig positions of <strong>the</strong> gauges during those years<br />
&e it impossib<strong>le</strong> to oorrelate <strong>the</strong> datum points of all <strong>the</strong> gauges used.<br />
Consequently, a decieim was made to correlate <strong>the</strong> rainfall with <strong>the</strong> nia-off<br />
within <strong>the</strong> Niamey-Jebba catchment, using <strong>the</strong> newly observϊ atage discharges<br />
at Jebba for seven years, and to employ this correlation curve with <strong>the</strong> cal-<br />
culated rainfall data to establish a 1939-59 discharge reaord for Jebba.<br />
Owing to <strong>the</strong> relative insignificant average value of <strong>the</strong> inflou between<br />
Jebba and KainJi, <strong>the</strong> obaei-red and <strong>the</strong> generated discharge data for Jebba<br />
were aaaumed to be <strong>the</strong> same for linin31 - which ie upstream of Jebba - and<br />
were analyzed accordingly. In establishing a satisfactory correlation between<br />
<strong>the</strong> rainfall and runoff data, two steps were taken:
26<br />
Because of <strong>the</strong> wide variation in both <strong>the</strong> rainfall and <strong>the</strong> discharge<br />
data betueen gauging stations, only monthly tota<strong>le</strong> were uaed in <strong>the</strong><br />
dYSi8. This approach was found to produce smoo<strong>the</strong>r oorrelation<br />
curves than thore obtained from daily or 54ay records.<br />
The Jebba-Niamey catchment area i8 extensive, and <strong>the</strong> runoff contri-<br />
bution to <strong>the</strong> Niger flow from <strong>the</strong> Dahomey catchment area takes a<br />
longer time to reach Jebba than <strong>the</strong> othr catchent6 downstream.<br />
Hence, a sequence of lag time wa8 introduced into <strong>the</strong> data analysis<br />
to yield an expression heeeby derived as<br />
where<br />
QA,P+= CU~,.O + CP Re&- Ah),$ + C3Rr.n ( 1)<br />
BA,,.,, = Runoff of <strong>the</strong> month of August from <strong>the</strong> Jebba-Biamey<br />
catchment srea;<br />
R4.~ U Bainiail of <strong>the</strong> month of July on <strong>the</strong> Dahomey catchment<br />
area;<br />
R($-k),s= Bainfall of second-half of July plus that of firsthalf<br />
of August on <strong>the</strong> Sokoto basin;<br />
e,,. = Rainfall of <strong>the</strong> month of August for <strong>the</strong> rest cf <strong>the</strong><br />
Jebba-Niamey catchent area; and<br />
'<br />
C,, C2, C are nuioff coefficients far Dahomey, Sokoto and <strong>the</strong><br />
rest of Jebba-Niamey catchment area respectively.<br />
The introduction of <strong>the</strong> coefficients of runoff in <strong>the</strong> above equation<br />
became necessary as a result of <strong>the</strong> wide variation in <strong>the</strong> geographioal<br />
nature of <strong>the</strong> catchment area.<br />
Through <strong>the</strong> step enumerated above, <strong>the</strong> m 4 f f data from debba-iainey<br />
catchment area were deduced for <strong>the</strong> period 193959, and th6Se w8re added to<br />
<strong>the</strong> Niamey observed record. The net result is w.2, <strong>the</strong> hydrograph of <strong>the</strong><br />
Niger discharge at Jebba. This figure was in turn used to develop <strong>the</strong> maas<br />
inflow curve info bke I[a;Lnji. The most important atreaai between gaiaji and<br />
Jebba is River Oïi with an estimted aiktchment correlation ooefficient of<br />
0.2 The remainder of <strong>the</strong> drainage basin had an estimated runoff coefficient<br />
of 0.1 These coefficients were used by <strong>the</strong> oonaultants in <strong>the</strong> equation<br />
wwe<br />
Q,,,,,<br />
= QJebh - -e,, - O.'F?, (2)<br />
Q E river discharge in eubic metres/ulrit of time<br />
P = rainfall in cubia metres/unit of time<br />
to arrive at <strong>the</strong> mass inflow curve for Lake Kainji.
The daily discharges used in <strong>the</strong> hydrologfcal analysis are very<br />
interrelated and <strong>the</strong> peaks are interdependent. Sime <strong>the</strong>se daily discharges<br />
exhibited a tendency towards persistence in succeasive stream flows, <strong>the</strong><br />
Goodrich dietributiona were used in <strong>the</strong> frequency calculations; <strong>the</strong> latter<br />
were of <strong>the</strong> exponential type and were similar to <strong>the</strong> exponential Gabel<br />
distributions.<br />
B. WATER SUPPLY II MIDWESTERN NIGERIA<br />
Water resources activities in <strong>the</strong> lid-West are centred mostly on<br />
water supply. The latter is tapped, in general, from <strong>the</strong> various aquifere<br />
which underlie 9% of <strong>the</strong> State. The Benin sand aquifer has <strong>the</strong> greatest<br />
potential - about 333 metres thick extending laterally to an appreciab<strong>le</strong><br />
distance - but <strong>the</strong> hydrological studies from which <strong>the</strong> aquifer chacterietics<br />
could be obtained are etill in <strong>the</strong> planning stage. m y of <strong>the</strong> ;aquifers,<br />
such as <strong>the</strong> Benin sand (3) and <strong>the</strong> Coastal Plain aquifers oan be described<br />
only in <strong>the</strong> moat general tarma because of <strong>the</strong> lack of recorded data. There<br />
i8 also no information on <strong>the</strong> hundreds of veils tht tap water daily from<br />
<strong>the</strong>se aquifers.<br />
C. UTER SUPPLY II WESTERN NIGWIA<br />
The Western Nigeria Water Corporation is entirely responsib<strong>le</strong> for <strong>the</strong><br />
planning and <strong>the</strong> developinent of Water supply in <strong>the</strong> State. The Corporation<br />
obtains <strong>the</strong> necessary evaporation and rainfall data, m y of which are very<br />
long and reliab<strong>le</strong>, from <strong>the</strong> Federal Baeteerological Service in Lagoa. However,<br />
because of <strong>the</strong> scantiness of data on river discharges, <strong>the</strong> standard praotice<br />
in those parts of <strong>the</strong> West, where surface uater has been developed, is to<br />
base <strong>the</strong> rater scheme design on <strong>the</strong> following hydrological assumptions in<br />
addition to a very liberal monthly evaporation of 127 mm:<br />
(i) A conservative runoff coefficient of 4s<br />
(ii) A once-in-50 years recurrence probability in rainfall with<br />
where P = Percentage probability W rainfall being equal to or<br />
<strong>le</strong>sa than a given talue;<br />
m = rank of <strong>the</strong> year; and<br />
n = number of years of record<br />
The catchment annual ruaoff, Q, which is based on <strong>the</strong>se assumptiom<br />
can be computed from <strong>the</strong> expreseion<br />
where A = Basing drainage mea:<br />
27
28<br />
= Baein rainfall value correspnâing to <strong>the</strong> probability of<br />
orne-in-% yeare oocurenae.<br />
Co = Coefficient of m f f for <strong>the</strong> basin.<br />
]Equation (3) or a forin of it hae been widely applied on <strong>the</strong> numerous eurfaae<br />
water apply eohetmee in <strong>the</strong> West. Eowvwr, bey%$ of <strong>the</strong> vast arai- area<br />
of 7,500 sq. kilometres that is governed by <strong>the</strong>,project, a form of <strong>the</strong> equation<br />
(3) shown above was not employed to predict <strong>the</strong> maximm probab<strong>le</strong> flood.<br />
Instead, Professor M. Parde of <strong>the</strong> University of Grenob<strong>le</strong> in France, a<br />
speciaìiet in <strong>the</strong> field of flood studies, advised <strong>the</strong> consultants that a<br />
runoff in <strong>the</strong> order of 490 litres per second per square kilometre is known<br />
to hava occured within West African strema of similar importance ae <strong>the</strong><br />
Oehun river on which <strong>the</strong> achenie is estsb1ished.b Sapply Ibadan aith water.<br />
Hence thb value was used in caltu<strong>le</strong>ting <strong>the</strong> project's spillway<br />
deeign flood of 3680 cmbic metres per second.<br />
The developaenf of groundwater resonroes in <strong>the</strong> West has encountered<br />
a number of difficulties. When <strong>the</strong> existing wells were been developed, <strong>the</strong><br />
areal extent of <strong>the</strong> bed-rock formation was not fully known, and <strong>the</strong> lithographic<br />
characterietics of <strong>the</strong> water bearing formations, in many cases, were<br />
etill to be studied. Absence of perfonaance data on <strong>the</strong> wells has only<br />
aggravated <strong>the</strong> situation, and in most cases, local informtion on dug weU, was often obtained from <strong>the</strong> inhabitanto.<br />
The Geological Survey of Nigeria (WN) in collaboration with <strong>the</strong> United<br />
States Geological Survey haa undertaken aome imestigativ6 work which had <strong>le</strong>d<br />
<strong>the</strong>m to edict a 30 year yielding life for <strong>the</strong> Lake Chad Baain midd<strong>le</strong> eone<br />
aquifer Kg.3) at a withdrawal rate of 5000 gph with wells placed at 16 kilometres<br />
apart. Huiidred8 of bore ho<strong>le</strong>s have been dril<strong>le</strong>d but co-ordinated dayto-day<br />
performance data on <strong>the</strong>se bore-ho<strong>le</strong>s are lacking. Most of <strong>the</strong> information<br />
that can be readily obtained on <strong>the</strong> basin's aquifere are availab<strong>le</strong> only<br />
in special reports. It is also irapossib<strong>le</strong> to undertake a meaninghil study<br />
-<br />
of <strong>the</strong> basin's aquifere within Bigeria along since four countries share <strong>the</strong><br />
bain area. The FAQarpd' -@are assisting <strong>the</strong> Lake Chad Basin Commiedon<br />
a regional organization of <strong>the</strong> countries that have territorial claims over<br />
parts of <strong>the</strong> basin, to<br />
(i) Compi<strong>le</strong> all <strong>the</strong> availab<strong>le</strong> date in <strong>the</strong> baain;<br />
(ii) melop an analme compu4er model that would miwilate au. <strong>the</strong><br />
activities that affect <strong>the</strong> quantity of water in <strong>the</strong> basin; and<br />
(iii) Define <strong>the</strong> various aquifers in <strong>the</strong> Chad bydro-geological basin and<br />
erdeavour to arrim at a syn<strong>the</strong>sie or composite picture covering<br />
<strong>the</strong> correlations between <strong>the</strong> atmospheric, emrface and groundwater<br />
ae well as between individual aquifera.
From hydrological stand point, <strong>the</strong> Kainji project has been more intenaively<br />
studied than any o<strong>the</strong>r water scheme in Nigeria. A number of houn<br />
standard methods were used to develop some reasonab<strong>le</strong> results such as <strong>the</strong><br />
correlation established between runoff and rainfall of <strong>the</strong> Jebba-Niamey<br />
catchment area. Since <strong>the</strong> project design flows were in prt derived from <strong>the</strong><br />
discharge data obtaineä at Niamey, <strong>the</strong> accuracy of <strong>the</strong> rating curve used to<br />
determine <strong>the</strong> Niamey discharges shonld have been verified. The importance<br />
of this project also warranted a longer hydrological record than <strong>the</strong> 20 year<br />
reconstituted record used. This could have been sequentially generated.<br />
In order to emure enough water supply, coneervative estimates of rainfall,<br />
runoff and liberal estimater of evaporation have been <strong>the</strong> 8taRdard<br />
practice in <strong>the</strong> West. But such educated guesaes have not prevented water<br />
shortages resulting froa both drought and under-design for <strong>the</strong> needs of <strong>the</strong><br />
communities served. It appears that <strong>the</strong>se educated guess teahniques haye<br />
never taken into consideration that moat of <strong>the</strong>se watershetie would be opened<br />
up in <strong>the</strong> near future ae a result of extensive and medhanised farming practices.<br />
The occurence of %aximam possib<strong>le</strong> rain-stOmR vouìd also yield higher peak<br />
flow than most of <strong>the</strong> existing achemea, except <strong>the</strong> Aaejire project, have<br />
been designed to hand<strong>le</strong>. And <strong>the</strong> eubsequent floafiing resulting from <strong>the</strong><br />
faracing practices or <strong>the</strong> maximum rainstorm would not only wipe aut <strong>the</strong>se<br />
water schemes but would also endangaz lives and property.<br />
Groundwater develogaent based on inadequate or scanty data has been<br />
found to be both unecoaoioicnl and frustrating. Typical examp<strong>le</strong>s include<br />
bore ho<strong>le</strong> failures at Agbor and Warri owing to <strong>the</strong> mollapse of cui-,<br />
and poor yield as a result of drilling for water in granite aone at Ijebu-<br />
Ode. The problame of grodwater developaent in <strong>the</strong> Hid-Vest are eubatan-<br />
tial, and only a through analysis can fore-stall more prob<strong>le</strong>ma in <strong>the</strong><br />
future. And <strong>the</strong> efforts of Mo within <strong>the</strong> chad Baain has aided and aace<strong>le</strong>-<br />
rated not only <strong>the</strong> haraonizatim of <strong>the</strong> existing data, but aiso <strong>the</strong> evaluation<br />
of data concerning evaporation and temperature meaeurenent by infra-red<br />
aerie1 photography.<br />
5. HPDBOLOOIcbL RESS(XIRCE<br />
Planning without facts ehould na longer plague <strong>the</strong> water resource8<br />
program of <strong>the</strong> nation, mre especially, as we shift our emphiusis from<br />
sing<strong>le</strong> giirpose water supply to multi-purpose schemes such aa <strong>the</strong> Kainji and<br />
Kano river schemes. The latter, also a victim of hydrological data scarcity<br />
ia enviaaged to pride barrefita in such areas ae irrigation, hydro-pouer and<br />
flood control.<br />
In order to enmare systematic planning in <strong>the</strong> future, <strong>the</strong><br />
newly created Water Resoaaraoa btitute.8haeld eatablieh a Hydrological<br />
resource^ Centre whoee primary hction uill firet be <strong>the</strong> aol<strong>le</strong>cting and<br />
compiling of <strong>the</strong> existing piecemeal data and nates that are scattered all over<br />
29
30<br />
<strong>the</strong> nation. This responsibility should be a continuow one and <strong>the</strong> inforaur-<br />
tion so col<strong>le</strong>cted ahould be published annually and made availab<strong>le</strong> to <strong>the</strong><br />
public on ea<strong>le</strong>. The centre should also standardice, nation wide, <strong>the</strong> inetru-<br />
mentation and hyàrological data col<strong>le</strong>cting and recording procedures.<br />
The information availab<strong>le</strong> to <strong>the</strong> centre can be upgraded both in quality<br />
and in quantity through <strong>the</strong> application of Remote Sensing Technique (RST).<br />
The latter ia currently availab<strong>le</strong> through participation in <strong>the</strong> United Stateet<br />
urth ~esourcea ~echnology Satellites program (EE~TSP). The data ana imegery<br />
obtaiasd though such a pmg~am can be utilised in sweral ways including<br />
<strong>the</strong> rapid identification and appraisal of our water resource^. The immediate<br />
hydrological investigation required in Nigeria, uuder <strong>the</strong> EELTSP, includes:<br />
The overall delienation of <strong>the</strong> aquifers of <strong>the</strong> Chad Basin and <strong>the</strong><br />
pattern of <strong>the</strong> groundwater movsment in <strong>the</strong> basin. Such information<br />
can be integrated into <strong>the</strong> basin'e existing analogne model;<br />
<strong>the</strong> monitoring of changes in reservoira' <strong>le</strong>vels resulting from<br />
evaporation and changes in rater courses resulting Rom erosibn and<br />
siltation;<br />
%he identification and quantification of <strong>the</strong> groundwater resources<br />
of Sou<strong>the</strong>rn Nigeria including <strong>the</strong> location of <strong>the</strong> position and<br />
evaluation of <strong>the</strong> extent of salbwater intrusion along <strong>the</strong> coastal<br />
aquifers. Data obtained from ERTSP would generate more awareness<br />
of <strong>the</strong> prob<strong>le</strong>m in surface water developeat and would provide<br />
needed information on which conjunctive ground and surface water<br />
programa oould be baed.<br />
6. CoáIcLusIOB<br />
Within <strong>the</strong> past decade, a =ber of water resources schemes have been<br />
developed, ard in genSral, <strong>the</strong>se schemes hare been plaaned with very limited<br />
hydrological data tbat were often extended through <strong>the</strong> applicaticm of<br />
statistical techiqueo to provide rational design parameters. In o<strong>the</strong>rs,<br />
"educated guess" technique wan subatitnted. The net resuit of auch methode<br />
haa been <strong>the</strong> failure of many water supply schemes to meet demands eapeoially<br />
aurfng <strong>the</strong> dry seaeon ana <strong>the</strong> location of unproductive bore-hoiea which had<br />
to b abaadoned.<br />
The future of io~iiy of <strong>the</strong>se a ches osnnot be accurately preàicted at<br />
this point. bwever, <strong>the</strong>re is <strong>the</strong> urgent need to upgrade <strong>the</strong> scanty data,<br />
through a contirmoire qioiritoring proc688, on which <strong>the</strong>m schemee were built.<br />
such a step WOUM provide a sound baeie for ature schemer, and would e mme<br />
<strong>the</strong> propat execution of any modification on existing sc@nms when warranted.
The scarcity of reliab<strong>le</strong> data is mostly due to <strong>the</strong> acute shortage of<br />
hydrologists and midd<strong>le</strong>-<strong>le</strong>vel techniciana in this diecipline. Herne, it<br />
will be necessary for <strong>the</strong> Water Resources IMtitUte in collaboration dth<br />
some of <strong>the</strong> eriating Universities to develop and execute achenes uhereby<br />
a large number of such teohnologiate and techniciana might be trained<br />
locally to meet <strong>the</strong> urgent needs of <strong>the</strong> nation.<br />
In order to eneure systematic planning in <strong>the</strong> futare, <strong>the</strong> power to<br />
col<strong>le</strong>ct, compi<strong>le</strong> and hairnonise all <strong>the</strong> hydrological data in <strong>the</strong> country<br />
should be vested in a Hydrological Resource Centre. The hydrological<br />
information availab<strong>le</strong> to such a centre could include data and imagery<br />
obtained through <strong>the</strong> use of Remote Sensing Technique.<br />
The information<br />
obtained will enab<strong>le</strong> <strong>the</strong> nation to acce<strong>le</strong>rate <strong>the</strong> pace of ita natural<br />
resources developneat.<br />
31
32<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
BIBLIOGWHY<br />
Andu, J. A. (1965). Exploitation and Developent of Groundvater in<br />
Western Nigeriay Ministry of Works and Trsmeport, Ibadan, Nigaria.<br />
IJEDECO (1961). Niger Dame Project, Vol. 2, Hydrology and Bei<strong>le</strong>rvoir<br />
Operation, Report auhitted to <strong>the</strong> Fedaral Ooverment of Nigeria,<br />
Lagoa<br />
Tahal (Water Planning) Ltd. (1965). Master Plan for Urban and Rural<br />
Water Supply, Report submitted to <strong>the</strong> Hid-West Ministry of Works and<br />
l!rsnsport, Benin, Nigeria.<br />
Tahal Consulting Ebgineers Ltd. (1969). Akungba-Shapureka-Ido&<br />
Water Supply Sahane, Plauning Report suinuitte8 to <strong>the</strong> Western Nigeria<br />
Water Corporation, Ibadan.<br />
Tahal and Motor Columbus Ltd. (1961). Ibadan Water Supply - hejire<br />
Daia, Final Design Report submitted to <strong>the</strong> Western Nigeria Hinietry of<br />
Works and Tranpport.<br />
Mil<strong>le</strong>r, B. E., R. E. Johnaton, J. A. Oloni, and J. A. Umma (1968).<br />
Groundwater ñyärology of <strong>the</strong> Chad Baein in B om and Dikwa Emiratem,<br />
N.E. Mgeria, with special Eaphasis on <strong>the</strong> Flow Life of <strong>the</strong> Artmian<br />
System. USGS Water Supply Paper 1757-1, U.S. Govt. Printing Oîîïce,<br />
Washington, D.C.<br />
üHESC0 (1970). Study of Water Resources in <strong>the</strong> Chad Basin, Report<br />
on <strong>the</strong> resulta of <strong>the</strong> hoject, Conclueionis and Recamendatione, PSriS,<br />
fiFrance.<br />
HEDECO (1970). Feasibility Study-gan0 River Project: Report eubdtted<br />
to <strong>the</strong> %o State Winistriea of Agriculture sad Iaatursl Besotarcea, and<br />
Worka and Sumeye, b o , Isigeria.<br />
Federal Ministry of Infonuation, Iagos (1970). Second Hationel<br />
Developent Plan, Fed. Govt. Printer, Lagos, Hgeria.
3G1 Ma2 of W Africa showing R Niger and its tributaries<br />
---_ -~~drograph<br />
I<br />
I L -<br />
derived Smm rom<br />
4dL<br />
- N+ogTh<br />
derived From &sei<br />
. ved. w* -.rat5<br />
FIG 2 Hydrogragh of R Niger at bbba (1955-571 (Reference 2)<br />
33<br />
..
ABSTRACT<br />
AN EXAMPLE OF REGIONAL CO-OPERATION FOR IMPROVING<br />
THE HYDROLOGICAL AND METEOROLOGICAL INFORMATION<br />
Eduardo Basso*<br />
Andrés Arriagadann<br />
Hcsrnando Neira**<br />
Manuel PBrez Delgado***<br />
T&e Centra8 A ~s~ican Hydrometeorological Project initiat ed<br />
in September 1961 rapresents & co-operative effort among <strong>the</strong> countries<br />
of <strong>the</strong> Central Amsricea Isthmus (Costa Rica. El Salvador, Guatemala,<br />
Honduras, Hiceragua and Panam%) and <strong>the</strong> United Nations Development<br />
Programme, acting aar executivo agency <strong>the</strong> World Meteorological<br />
Organizatioa. Its objectives are <strong>the</strong> following: (i) installation of a<br />
basic netW6Pk of meteorological and hydrological stations, (ii)<br />
col<strong>le</strong>ction, preceesing and publication of <strong>the</strong> data, (iii) training<br />
of personnel by neans of course.%, fellowships or through technical<br />
publications end manuals and (iv) <strong>the</strong> institutional strenghtening of<br />
<strong>the</strong> meteorological and hydpalogical services in <strong>the</strong> area. Important<br />
Project activities have been <strong>the</strong> Paeting of new equipment used in<br />
developed countries in order to study <strong>the</strong>ir application to <strong>the</strong><br />
characteristics and tropical climate of <strong>the</strong> area, and <strong>the</strong> development<br />
and application of methods for meteorology, hydrology and sediment<br />
studies with limited information. It is concluded that <strong>the</strong>ir use in<br />
o<strong>the</strong>r areas with similar conditions can be useful and that regional<br />
cooperation can be one effective means for coping with inadequate<br />
data through <strong>the</strong> pooling of individual countries efforts.<br />
RES UMEW<br />
El Proyecto Hidrometeorológico Centroamericano iniciado en -<br />
Setiembre de 1967 constituye un esfuerzo cooperativo entre los paí-<br />
ses del Istmo Centroamericano (Costa Rica, El Salvador, Guatemala,<br />
Honduras, Nicaragua y Panamá) y el Programa de las Naciones Unidas<br />
para el Desarrollo, actuando como agencia ejecutora la Organización<br />
Meteorolögica Mundial. Sus objetivoe principa<strong>le</strong>s los constituyen: -<br />
(i) la instalación de una red básica de estaciones meteorológicas e<br />
hidrolbgicas, (ii) la reco<strong>le</strong>cciön, proceso y publicación de los da-<br />
tos, (iii) el adiestramiento del personal ya sea con becas y cursos<br />
o mediante publicaciones y manua<strong>le</strong>s tgcnicos, y (iv) el robusteci--<br />
miento institucioaal de los servicios meteorolbgiccs e hidrológicos<br />
en el área. Actividades importantes del Proyecto han sido la prueba<br />
de nuevos equipos utilizados an países desarrollados para estudiar<br />
su adaptación a lae características y clima tropical del area, y el<br />
desarrollo y aplicacids de métodos para la ejecución de estudios mo<br />
teorológicos, hidrol6gicas y de sedimentación con información limi-<br />
tada. Se concluye estimando que su uso en otras áreas COQ condicio-<br />
nes similares puede ser de utilidad.<br />
* Project Manager, Central American Hydrometeorological Project<br />
** Hydrologist Expert., Central American Hydrometeorological Project<br />
*** Hydrometeorological Expert, Central American Hydrometsorological<br />
Pro j ect
36<br />
INTRODUCTION<br />
The Central American Hydrometeorological Project is a joint effort between<strong>the</strong><br />
Governments of <strong>the</strong> Central American Isthmus ( Costa Rica, El Salvador, Guate<br />
mala, Honduras, Nicaragua and Panamá) and <strong>the</strong> United Nations Development<br />
Programme. The World Meteorological Organization acts as Executing Agency<br />
The Project started in September 1967, at a cost of 9. 2 millions dollars (3.3<br />
millions UNDP and 5.9 millions Governments), which makes this Project one<br />
of <strong>the</strong> largest in this field. in March 1973 a second phase of <strong>the</strong> Project was<br />
started devoted mostly to <strong>the</strong> Coordination and Consolidation of <strong>the</strong> activities in<br />
Meteorology and Hydrology. This second phase has a duration of three years<br />
and <strong>the</strong> global contribution of UNDP adds to 1.3 millions dollars.<br />
PROJECT OBJECTIVES<br />
The main objectives of <strong>the</strong> first phase of <strong>the</strong> Project, already comp<strong>le</strong>ted were<br />
<strong>the</strong> following:<br />
Installation of 290 hydrometric stations in <strong>the</strong> six countries.<br />
Installation of 830 climatological stations (60 main, 240 secondary and 530<br />
pluviomet ric).<br />
c) Institutional strenghtening of <strong>the</strong> Meteorological and or Hydrological Services<br />
and <strong>the</strong> col<strong>le</strong>ction, preparation and publication of <strong>the</strong> data in both <strong>the</strong> new<br />
and old stations.<br />
d) Training of <strong>the</strong> personnel, by means of fellowships, seminars, courses,<br />
publications and on-<strong>the</strong>-job training.<br />
At <strong>the</strong> end of <strong>the</strong> project <strong>the</strong> number of stations constructed surpassed by far <strong>the</strong><br />
goals; more than 350 hydrological and more than 950 of all kinds<br />
of meteorological stations were comp<strong>le</strong>ted. The achievements in <strong>the</strong> activities<br />
of data processing and publication as in personnel training were most remarkab<strong>le</strong>.<br />
In some countries, meaningful results were obtained in <strong>the</strong> important task of<br />
institutional building. In o<strong>the</strong>rs <strong>the</strong> present condition is not yet adequate for <strong>the</strong><br />
needs of <strong>the</strong>ir development, but it is expected that during <strong>the</strong> Second Phase it<br />
will be possib<strong>le</strong> to comp<strong>le</strong>te <strong>the</strong> necessary arrangements for this. For Co-ordinating<br />
at a regional <strong>le</strong>vel <strong>the</strong> activities in Meteorology and Water Resources Investigations,<br />
a Regional Committee was created. This Cornmiittee, formed by<br />
<strong>the</strong> presidents of <strong>the</strong> National Coordinating Committees of <strong>the</strong> six countries, has<br />
proved to be an excel<strong>le</strong>nt arrangement and can be considered a good examp<strong>le</strong> of<br />
regional Co-or dination.<br />
PRE-PROJECT CONDITIONS<br />
The conditions before <strong>the</strong> beginning of <strong>the</strong> project varied widely from country to<br />
country. However, in some countries it was practically inexistent. Although<br />
about 180 hydrometric stations were in operation in 1966, only a few provided<br />
reliab<strong>le</strong> data. Even <strong>the</strong>se had a very short period of records, normally <strong>le</strong>ss.<br />
than five years. In some countries <strong>the</strong> stations consisted only of a staff gauge,<br />
without bridge or cab<strong>le</strong> for flood measurements, in o<strong>the</strong>r cases limnigraplis were<br />
instal<strong>le</strong>d without a device for checking <strong>the</strong> river <strong>le</strong>vels, Sediment measurements<br />
were made in only one country and water quality determinations were made. How<br />
ever, <strong>the</strong> main deficiencies arised from <strong>the</strong> methods of col<strong>le</strong>cting and processing
<strong>the</strong> data. The O. 6 depth method of velocity measurement was used in some cases<br />
introducing errors in <strong>the</strong> stream gauging data. The discharge rating curves were,<br />
generally extrapolated graphically, and no checks were made for <strong>the</strong> consistency<br />
of <strong>the</strong> resulting information. Even though most of <strong>the</strong>se defects were recognized<br />
<strong>the</strong> counterpart pcrsonnel lacked <strong>the</strong> means and <strong>the</strong> influence for improving <strong>the</strong><br />
situation.<br />
The situation in Meteorology was similar. Even considering that some countries<br />
had fairly well organized cervices, <strong>the</strong> network was absolutely insufficient for<br />
<strong>the</strong> needs of <strong>the</strong> region. Several Services had only one main meteorological<br />
station, in <strong>the</strong> principal airport. The number of secondary stations in good work<br />
ing standard were <strong>le</strong>ss than 20. The rainfall observation network comprised only<br />
<strong>the</strong>.main inhabited areas, and even <strong>the</strong>re, only a few recording instruments were<br />
availab<strong>le</strong>. Some countries showed a comp<strong>le</strong>te lack of rain recorders and o<strong>the</strong>rs<br />
of evaporation stations. The processing of <strong>the</strong> data was quite rudimentary, and<br />
<strong>the</strong>ir publication with a few exceptions, inexistent. Only five meteorologist with<br />
university degree were availab<strong>le</strong>, and all five worked in one country. Practically,<br />
no co-ordination between meteorological and hydrological services existed.<br />
TRAINING<br />
The activities of personnel training at all <strong>le</strong>vels were considered fundamental<br />
and received a preferential treatment from <strong>the</strong> project.<br />
Training in <strong>the</strong> Region. Training in <strong>the</strong> region was done with courses --in-<br />
cluding cour s e s by correspondence - -, seminar s, on-<strong>the</strong> - job training, confer ence s<br />
and publications. Without including on <strong>the</strong> job training, approximately 500 peop<strong>le</strong><br />
received formal or informal courses. This does not include <strong>the</strong> personnel<br />
trained by o<strong>the</strong>r WMO projects, such as <strong>the</strong> Chair of Meteorology at <strong>the</strong> University<br />
of Costa Rica or <strong>the</strong> Mobi<strong>le</strong> Center for Training of Meteorological Personnel.<br />
Practically all <strong>the</strong> graduates of <strong>the</strong>se courses are engaged in activities connected<br />
with <strong>the</strong> Project.<br />
Fellowships. 37 fellowships with a total of 324 men-month were made availab<strong>le</strong><br />
to <strong>the</strong> Project, for <strong>the</strong> preparation of new personnel or for <strong>the</strong> improvement<br />
of <strong>the</strong> training of <strong>the</strong> existing ones. These fellowships were a fundameotal<br />
comp<strong>le</strong>tement for <strong>the</strong> local training which was devoted mainly to a large number<br />
of low <strong>le</strong>vel technicians. Most of <strong>the</strong> fellows comp<strong>le</strong>ted successfully <strong>the</strong>ir<br />
studies, and some obtained higher degrees in well known Universities. The importance<br />
given to <strong>the</strong> practical training must be noted; <strong>the</strong> course for preparing<br />
technicians in meteorological instruments--Buenos Aires-- must be specially<br />
remarked. Unfortunately, some of <strong>the</strong> fellows ieft <strong>the</strong>ir jobs with <strong>the</strong> Government<br />
sometime after <strong>the</strong> comp<strong>le</strong>tion of <strong>the</strong>ir studies, which means that <strong>the</strong>ir<br />
and UNDP's effort was wasted. However, <strong>the</strong> percentage of fellows in this<br />
case was relatively low, 13%. in addition, <strong>the</strong> Project co-operated actively<br />
to obtain fellowships from national and multilateral sources. in such a way,<br />
64 more fellowships were obtained, without including <strong>the</strong> ones used for <strong>the</strong><br />
courses already mentioned. As a consequence, <strong>the</strong> total of trained personnel<br />
has been significantly higher than <strong>the</strong> quantity that should have resulted only<br />
from <strong>the</strong> fellowships assigned to <strong>the</strong> Project. Even so, <strong>the</strong> shortage of capab<strong>le</strong><br />
personnel can still be noticed, specially in <strong>the</strong> Meteorological Services.<br />
31
38<br />
Publications. The Project considered that one <strong>the</strong> most effective forms of<br />
training in a dispersed regional project was <strong>the</strong> intensive use of technical<br />
publications. in general, <strong>the</strong> reaction to <strong>the</strong>se publications were encouraging.<br />
and resulted in a large demand of <strong>the</strong>m, both from <strong>the</strong> countries of <strong>the</strong> area<br />
and from outside. Their main merit aves to <strong>the</strong> fact that bibliography in<br />
Spanish was made a.vailab<strong>le</strong> to all counterpart <strong>le</strong>vels.<br />
pared, about 100 technical publications and 60 reports had been re<strong>le</strong>ased, .To<br />
make <strong>the</strong> diffusion of Project activities more availab<strong>le</strong> a by-mounthly news<strong>le</strong>tter<br />
was edited, Over 500 copies of each issue were printed, making possib<strong>le</strong> for<br />
all members of <strong>the</strong> Committee to know <strong>the</strong> activities of <strong>the</strong> o<strong>the</strong>rs. The editorial<br />
activity of <strong>the</strong> Project stimulated also <strong>the</strong> publications of <strong>the</strong> counterpart, increasing<br />
<strong>the</strong>ir technical reports and data publication. By far <strong>the</strong> most important<br />
publication of <strong>the</strong> Project is <strong>the</strong> "Manual of Instructions". which has been plan-<br />
ned in four volumes.<br />
three chapters: (1) Field measurements and installations, (2) Data processing<br />
and (3) Sediments. Standards are set for <strong>the</strong> installations, field measurement<br />
and methods for processiqg <strong>the</strong> information. The second volume is devoted to<br />
IIHydrological Studies" comprising: (1) Verification and Correction of Hydrolo -<br />
gical Records, (2) Extension of Hydrological Records, (3) Duration and Variation<br />
Studies, (4) Hydrometeorological Studies, (5) Floods. (6) Draughts, (7) Hydrological<br />
Forecasts (8) Hydrological Studies for Power Developments (9) Agricuitural<br />
Hydrology (10) Economic Aspects in Hydrology (11) Use of Mechanical Data<br />
Processing. The third volume refers to "Meteorological Observations" and has<br />
been edited only in a preliminary form. The last volume "Ground Water Hydro-<br />
logy" will be prepared -h <strong>the</strong> future.<br />
When this paper was prg<br />
The first one deals with "Hydrometry" and comprises<br />
The Manual is aimed to <strong>the</strong> medium <strong>le</strong>vel<br />
technicians and includes several numerican examp<strong>le</strong>s, with information of <strong>the</strong><br />
area. Special emphasis has been given to <strong>the</strong> specific prob<strong>le</strong>ms arising from<br />
<strong>the</strong> lack of long and reliab<strong>le</strong> records. (1) (2).<br />
EQUIPMENT<br />
. The equipment component, formed <strong>the</strong> major part of UNDP's contribution,<br />
adding to a total of about 1,9 million dollars.<br />
Meteorological Equipment. The main meteorological stations (type A) were<br />
in general provided with universal wind recorder, mercury barometer, microbarograph,<br />
psychrometer, maximum and minimum <strong>the</strong>rmometers. <strong>the</strong>rmohydrograph,<br />
set of geo<strong>the</strong>rmometers, Robitzch actinograph, Campbell-Stokers heliograph,<br />
Piche and tank evaporimeter, tank <strong>le</strong>vel anemometer, water <strong>the</strong>rmometer,<br />
raingauge and rain recorder (Figure 1).<br />
The ordinary climatological stations were provided with psychrometer, maximum<br />
and minimum <strong>the</strong>rmometers, raingauge, rain recorder and Piche evaporimeter.<br />
in most of <strong>the</strong> station of this type evaporation tank, pan <strong>le</strong>vel anemometer, and<br />
<strong>the</strong>rmohydrograph were instal<strong>le</strong>d and in several, anemograph, heliograph, actinograph<br />
and or soil <strong>the</strong>rmometers were also included. (Figure 2). Some stations<br />
included also agrometeorological instruments. such as soil-moisture gauges,<br />
dew recorders, lysimeters, extrasoil <strong>the</strong>rmometers. etc. Several pre-Project<br />
stations were reinstal<strong>le</strong>d and or comp<strong>le</strong>ted with new instruments.<br />
The meteoro<br />
logical equipment included also six standard barometers, which were included<br />
in <strong>the</strong> principal station of each country, replacement parts for <strong>the</strong> period of <strong>the</strong><br />
project and for some time after its comp<strong>le</strong>tion and equipment for inspection and<br />
maintenance. Also included were equipment for part of a regional laboratory<br />
for calibration of meteorological equipment. The equipment provided was, in
general, of good quality, and adequate for <strong>the</strong> needs of <strong>the</strong> Project. As far as<br />
possib<strong>le</strong> equipment of complicated operation or maintenance was avoided, preferring<br />
simp<strong>le</strong> and sturdy ones suitab<strong>le</strong> for tropical conditions. In some cases<br />
'defects were detected, but <strong>the</strong>y were satisfactorily corrected by <strong>the</strong> manufacturers,<br />
by introducing several changes in <strong>the</strong> design of <strong>the</strong> instruments.<br />
Hydrological Equipment. The stations instal<strong>le</strong>d included <strong>the</strong> total or part<br />
of <strong>the</strong> following e<strong>le</strong>ments provided by UNDP: limnigraph --of <strong>the</strong> float type or<br />
bubb<strong>le</strong> gauge type (manometric) --damping pipe, housings, connections and pack<br />
ings of <strong>the</strong> limnigraph, sets of staff gauges, cab<strong>le</strong>s and accessories (Cab<strong>le</strong> caq for <strong>the</strong> cab<strong>le</strong>way instailation plus a reasonab<strong>le</strong> quantity of spare parts.<br />
portant to note <strong>the</strong> fact that <strong>the</strong> standardized prefabrication of <strong>the</strong> construction<br />
e<strong>le</strong>ments, especially <strong>the</strong> cab<strong>le</strong>way towers, which were designed for 3, 6 and 9<br />
meters height, allowed a simplification of <strong>the</strong> construction of <strong>the</strong> stations, making<br />
easier its transportation and mounting;a fact that was of fundamental importance<br />
to reach isolated and difficult zones (Figure 3). The equipment was designed in<br />
order to ensure a maximum of safety during construction and operation, providing<br />
<strong>the</strong> cab<strong>le</strong> cars with safety brakes, <strong>the</strong> towers with stairways protected with<br />
safety rings, etc.. . In addition <strong>the</strong> publication of standards of construction and<br />
operation aimed to ensure this objective. As a consequence of this, and reversing<br />
<strong>the</strong> pre-project conditions, serious accidents happened nei<strong>the</strong>r during <strong>the</strong><br />
constructionnor during <strong>the</strong> operation of <strong>the</strong> stations. (Figure 4) The equipment<br />
included a current-meter calibrating tan!!, which was instal<strong>le</strong>d at <strong>the</strong> Universidad<br />
Centroamericana in Managua. Probably due to <strong>the</strong> careful supervision of<br />
<strong>the</strong> design and construction of <strong>the</strong> building, <strong>the</strong> installations remained undamaged<br />
by <strong>the</strong> earthquake of December 1972.<br />
instruments for <strong>le</strong>vel recordings, three digital limnigraphs were qperated expe-<br />
rimentally for some years. The results of <strong>the</strong>ir operation was, in general, LUISA<br />
tisfactory, because of extreme humidity and lack 'of adequate maintenance. This lias<br />
proved that <strong>the</strong> se<strong>le</strong>ction of mechanical equipment was a wise one, and that <strong>the</strong><br />
gradual introduction of digital equipment should wait for more development to<br />
solve <strong>the</strong> observed defects and to allow training of specialized personnel.<br />
39<br />
It is im-<br />
in order to gain experience with modern<br />
Equipment for flow and Sediment Measurement. The Hydrological services<br />
were provided with flow meters, counterweights, winches and cranes, etc. to<br />
ensure <strong>the</strong> adequate operation of <strong>the</strong> hydrometric network. Se<strong>le</strong>cting <strong>the</strong> type of<br />
current meters was also subject of detai<strong>le</strong>d studies, and it was decided to use<br />
both <strong>the</strong> axial and <strong>the</strong> Price current meter. after a consideration of <strong>the</strong>ir relative<br />
merits. The manual of Instructions (1) of <strong>the</strong> Project contains instructions regarding<br />
<strong>the</strong> criteria to be used in se<strong>le</strong>cting one or o<strong>the</strong>r instrument. in addition,<br />
measurements made in Costa Rica and El Salvador proved that <strong>the</strong> difference<br />
between <strong>the</strong> measurements made with both kinds of current meters is very small.<br />
The Project started and intensive programme of sediment sampling, for which<br />
<strong>the</strong> acquisition of standardized D49 and DH-48 samp<strong>le</strong>rs has been fundamental<br />
for <strong>the</strong> successful achievement of this goal. In addition <strong>the</strong> Project provided<br />
construction,laboratory, navigation and transportation equipment.<br />
Data Processing Equipment. This comprises fundamentally two groups: e-<br />
<strong>le</strong>ctronic calculators and peripherical comnutation equipment. The first group<br />
cornprises conventional and programmab<strong>le</strong> calculators, which have been used<br />
preferentialy in <strong>the</strong> computation of streamflow measurements, hydrograms and
discharge rating curves. The second group includes mainly card perforators<br />
for imput to conventional e<strong>le</strong>ctronic computers. This equipment will be used as<br />
a base for <strong>the</strong> future data processing centers planned in <strong>the</strong> second stage of <strong>the</strong><br />
Project.<br />
DESIGN AND CONSTRUCTION OF THE NETWORK<br />
The design of <strong>the</strong> climatological network was based upon <strong>the</strong> following crite<br />
ria:<br />
a. The first priority for <strong>the</strong> main mateorological stations was given to <strong>the</strong> irrip<strong>le</strong>mentation<br />
of <strong>the</strong> basic synoptic network, which had been planned before <strong>the</strong><br />
Project. The remaining main stations were located in intermediate points,<br />
trying to obtain a relative uniform density and a good representation of <strong>the</strong> different<br />
climates of <strong>the</strong> area. Preference was given to installation in <strong>the</strong> main<br />
airports.<br />
b. When possib<strong>le</strong>, an ordinary station was instal<strong>le</strong>d in each mayor agricultural<br />
area. In isolated val<strong>le</strong>ys with characteristic microclimates, an effort was made<br />
to asign a station to each of <strong>the</strong>m.<br />
c. To obtain <strong>the</strong> necessary interrelation between <strong>the</strong> hydrological and <strong>the</strong> me- teorological network, at <strong>le</strong>ast an ordinary st,ation was assigned to each mayor basin<br />
or sub-basin with co-ordinated operation of <strong>the</strong> meteorological and hydrdogical<br />
networks.<br />
d. in scarcely populated areas <strong>the</strong> main consideration was <strong>the</strong> availability of<br />
observers.<br />
e. Finally, <strong>the</strong> availability of air, land or water access was a limiting factor<br />
in some jung<strong>le</strong>, mountainous or isolated areas. The pluviometric network was<br />
planned following <strong>the</strong> recommendations of <strong>the</strong> Guide for Hidrometeorological<br />
Practices of WO, with <strong>the</strong> limitations imposed iy <strong>the</strong> lack of observers and<br />
<strong>the</strong> inaccessibility of some regions. Regarding <strong>the</strong> design of <strong>the</strong> hydrological<br />
stations, <strong>the</strong>se were located in <strong>the</strong> following places:<br />
i Near <strong>the</strong> mouth of <strong>the</strong> principal rivers and or <strong>the</strong>ir main tributaries. ii. in<br />
each main lake. iii. At <strong>the</strong> out<strong>le</strong>t of each main lake. iv. Where dams of major<br />
hydraulic works were planned. v. At <strong>the</strong> entrance of a river to a mayor<br />
val<strong>le</strong>y.<br />
vi. At <strong>the</strong> crossing of a major river of an international boundary. in<br />
addition, some stations were located in urban and minor basins, based mainly<br />
on utility criteria or, in some cases, for use as representative basins. it<br />
was planned to make sediment measurements in part of <strong>the</strong> network mainly at<br />
<strong>the</strong> stations listed under i and iv. The comp<strong>le</strong>te plan was co-ordinated at a<br />
regional <strong>le</strong>vel and approved by <strong>the</strong> Regional Committees (3).<br />
The impact of <strong>the</strong><br />
Project in <strong>the</strong> meteorological network coverage can be appreciated in Figure 5<br />
which shows <strong>the</strong> situation before and after of <strong>the</strong> Project. A similar compari-<br />
son has been made for <strong>the</strong> hydrological network in Figure 6.<br />
Sediment Measurements. (4) One of <strong>the</strong> subjects of main interest for <strong>the</strong><br />
Project was measurement of <strong>the</strong> sediment loads of <strong>the</strong> rivers, since --with <strong>the</strong><br />
exception of Costa Rica-- practically no information was availab<strong>le</strong> at <strong>the</strong> begin.<br />
ning of <strong>the</strong> Project.<br />
At present., systematic samplings are made in 136 of <strong>the</strong><br />
gauging stations in <strong>the</strong> area. Samplings are made in accordance with <strong>the</strong> usual<br />
techniques and are later analized in <strong>the</strong> laboratories established in <strong>the</strong> six
countries to derive <strong>the</strong> sediment load. When access prob<strong>le</strong>ms limit <strong>the</strong> number<br />
of measurements, some local observers take a point daily samp<strong>le</strong> it was found<br />
that in most of <strong>the</strong> cases <strong>the</strong> concentration of this samp<strong>le</strong> correlate well with <strong>the</strong><br />
average of <strong>the</strong> compusite samp<strong>le</strong>s. The use of <strong>the</strong> sedbent rating curve, relating<br />
<strong>the</strong> solid discharge (G) with <strong>the</strong> liquid discharge (Q) has been used for comp<strong>le</strong>ting<br />
<strong>the</strong> records. Figure 7 shows one typical sediment rating curve and<br />
Figure 8 summarizes some of <strong>the</strong> first results obtained by <strong>the</strong> project.<br />
The bed load is computed using several of <strong>the</strong> usual formulas, and several examp<strong>le</strong>s<br />
have been published in arder ta explain <strong>the</strong> procedure to <strong>the</strong> counterpart<br />
technicians (12). An interesting result concerning <strong>the</strong> sediment rating curve is<br />
that <strong>the</strong> coefficient of <strong>the</strong> equation G = A an, varies between 1.4 ad 4. O. The<br />
lower values of n (1.4 to 2. O) are associated with rivers crossing arid areas,<br />
and <strong>the</strong> .value of n in general increases as <strong>the</strong> rainfall also increases. A <strong>the</strong>ory<br />
for explaining this has been developed by <strong>the</strong> project (20) and will be <strong>the</strong> object<br />
of fur<strong>the</strong>r publications.<br />
'<br />
Water Quality. Although this objective was not originally though of, <strong>the</strong><br />
Project has started a minimum programme of measurements of water quality.<br />
At present systematic samplings are made in only 48 stations of Costa Rica,<br />
El Salvador and Guatemala, but it is expected that in <strong>the</strong> future this programme<br />
will be expanded.<br />
STUDIES AND APPLIED RESEARCH<br />
Most of <strong>the</strong> prob<strong>le</strong>ms in hydrology and meteorology in Central America<br />
arise from <strong>the</strong> lack of appropriate information, <strong>the</strong>refore this subject falls<br />
directly in <strong>the</strong> main <strong>the</strong>me of this Seminar. Although in <strong>the</strong> area of <strong>the</strong> Project<br />
a few, very few, meteorological stations existed with information up to <strong>the</strong> begin<br />
ning of <strong>the</strong> century, this fact did not help much in <strong>the</strong> evaluation of water re-<br />
sources and much <strong>le</strong>ss for <strong>the</strong> feasibility studies.<br />
<strong>the</strong> Project provides <strong>the</strong> necessary coverage so in most of <strong>the</strong> cases <strong>the</strong> prob<strong>le</strong>m<br />
is nowof "insufficient data" and not of comp<strong>le</strong>te "lack of Information1I. In Central<br />
America it is now possib<strong>le</strong> to undertake <strong>the</strong> study of <strong>the</strong> potential resources of a<br />
basin or for estimating <strong>the</strong> maximum design flow, even considering that <strong>the</strong><br />
stations giving an adequate coverage have only two or three years record. With<br />
this information, a model of <strong>the</strong> wea<strong>the</strong>r responsib<strong>le</strong> for <strong>the</strong> major floods can be<br />
prepared.<br />
model which CaA%e transposed in time to <strong>the</strong> most intensive storms, knowing<br />
only data at a few rainfall stations and very iimited hydrological information; <strong>the</strong><br />
maximum historical gauge <strong>le</strong>vels par examp<strong>le</strong>. The above mentioned method is<br />
now being used for <strong>the</strong> design flood of a large hydroe<strong>le</strong>ctrical dam in sou<strong>the</strong>rn<br />
Costa Rica, and will be published in a future report of <strong>the</strong> Project.<br />
wind, present wea<strong>the</strong>r, meteorological phenomena, temperature and humidit y<br />
obtained at two possib<strong>le</strong> sites for <strong>the</strong> new airport for Tegucigalpa for short<br />
periods of observation, have established <strong>the</strong> need of fur<strong>the</strong>r information for a<br />
meaninful decision. in this case <strong>the</strong> lack of information on cloud cover and vi-<br />
sibility made impossib<strong>le</strong> a decision as in <strong>the</strong> previous case. Therefore, it can<br />
be seen that <strong>the</strong> prob<strong>le</strong>ms of evaluation with insufficient data differ substantially<br />
from one case to ano<strong>the</strong>r, and it is impossib<strong>le</strong> to propose fixed solution methods.<br />
The first case shows how <strong>the</strong> action of <strong>the</strong> Central American Hydrometeorological<br />
Project has made possib<strong>le</strong> <strong>the</strong> evaluation of water resources with insufficient in-<br />
formation by means of a closed and co-ordinated work between <strong>the</strong> meteorologist<br />
41<br />
The network established by<br />
Based in this wea<strong>the</strong>r model it is possib<strong>le</strong> to develop an isohyetical<br />
Data on
42<br />
and <strong>the</strong> hydrologist. in <strong>the</strong> Symposium, it would be important to recognize this<br />
fact. Special enphasis has to be placed in <strong>the</strong> fact that in <strong>the</strong> area of evaluation<br />
of natural resources with limited information, <strong>the</strong> prob<strong>le</strong>ms will be solved best<br />
wtth a close collaboration between hydrologists and meteorologists, since it is<br />
impossib<strong>le</strong> to separate <strong>the</strong> aerial and terrestial phase of <strong>the</strong> hydrologic cyc<strong>le</strong>.<br />
The lack of hydrological information can be compensated with meteorological<br />
information and viceversa. "Elastic relations" which allow to extrapolate <strong>the</strong><br />
few observed data, based on some know<strong>le</strong>dge of <strong>the</strong> mechanics of <strong>the</strong> phenomena,<br />
should be used as far as possib<strong>le</strong>. The fact that <strong>the</strong> hydrometric data are<br />
based on pluviometric information must not be forgotten, since it provides <strong>the</strong><br />
most effective tool for <strong>the</strong> evaluation of water resources.<br />
The scope of this<br />
paper makes impossib<strong>le</strong> to detail all <strong>the</strong> studies of <strong>the</strong> Project. A list of some<br />
of <strong>the</strong>m of which most were published is <strong>the</strong> following: - Studies for determining<br />
water requirements for irrigation (5) (6) (7) (8). - Studies on runoff forecasting<br />
(9) (10). (Already being used for forecasting <strong>the</strong> operation of several reservoirs<br />
in <strong>the</strong> area). Effect of <strong>the</strong> eruptions of <strong>the</strong> Irazú Volcano on <strong>the</strong> sediment discharge<br />
of <strong>the</strong> Reventazón River (11) (12). - Sediment computations, specially<br />
bed load, for several projects.<br />
for several projects. -<br />
- Assistance for <strong>the</strong> computation of design flood<br />
Development of methods for estimating floods in <strong>the</strong><br />
area. Figure 9 shows some flood envelopes for all <strong>the</strong> Central American area.<br />
Figure 1 O shows some rainfall envelopes for <strong>the</strong> area (1 3) (14). -Groundwater<br />
studies with <strong>the</strong> analog computer were made for <strong>the</strong> Project at El Salvador (1 5).<br />
- - Water balance studies (16) (17) (18). Figure 11 shows schematically <strong>the</strong> results<br />
of a preliminary study for all <strong>the</strong> Central American area. Effect of <strong>the</strong><br />
temperature on <strong>the</strong> sediment load (19) Figure 12 summarizes <strong>the</strong> result of this<br />
study.<br />
STUDIES WITH INADEQUATE DATA<br />
The inadequacies of data arise from (i) incorrect data and (ii) short or insuf<br />
ficient records. Although coping with this is one of <strong>the</strong> tasks of <strong>the</strong> Second<br />
Phase of <strong>the</strong> Project, efforts for correcting and extending <strong>the</strong> availab<strong>le</strong> data have<br />
been made up to now. The Manual ofInstructions of <strong>the</strong> Project (2) details <strong>the</strong><br />
techniques suggested for this.<br />
Doub<strong>le</strong> mass curves are used for a first check of <strong>the</strong> quality of <strong>the</strong> data.<br />
When errors are found in <strong>the</strong> hydrological records <strong>the</strong>y are generally due to incorrect<br />
extrapolation of <strong>the</strong> stage-discharge curve. Jn this case several methods<br />
for determining this curve are proposed, some based in hydraulic relations and<br />
o<strong>the</strong>r in <strong>the</strong> hydrological balance of <strong>the</strong> basis.<br />
The filling or extension of <strong>the</strong>se records is made ei<strong>the</strong>r using simp<strong>le</strong> or mgl<br />
tip<strong>le</strong> corelation andfor estimating <strong>the</strong> runoff based in <strong>the</strong> meteorological data<br />
and basic characteristics.<br />
Up to now, <strong>the</strong> checking and extension of meteorological and hydrological<br />
records has been made following specific needs, but it is planned to undertake<br />
thin task in a co-ordinated and comprehensive form for all Central American<br />
Isthmus during <strong>the</strong> Second Pahse of <strong>the</strong> Project.<br />
DATA PROCESSING AND PUBLICATION<br />
One of <strong>the</strong> main Project activities has been to ensure <strong>the</strong> prompt and adequate
processing of <strong>the</strong> information. This has been achieved, both in meteorology and<br />
hydrology, by means of modern systems based in <strong>the</strong> use of e<strong>le</strong>ctronic computers.<br />
Meteorology. The data col<strong>le</strong>cted at <strong>the</strong> stations are directly written in <strong>the</strong><br />
computer entrance forms, except where, due to limitations of <strong>the</strong> observer, this<br />
has to be done in <strong>the</strong> central office of <strong>the</strong> meteorological services. The detail of<br />
<strong>the</strong> forms and instructions for filling <strong>the</strong>m are indicated in Publication No 84 of<br />
<strong>the</strong> Project (20). The results of reading <strong>the</strong> graphs of <strong>the</strong> recording instruments<br />
are also fi<strong>le</strong>d on <strong>the</strong> form. At this stage, <strong>the</strong> adjustment of <strong>the</strong> graphs by cornpa<br />
rison with <strong>the</strong> direct reading instruments has to be made. Finally, before<br />
punching <strong>the</strong>se data on IBM cards, <strong>the</strong> consistency of <strong>the</strong> data is checked. This<br />
system allowed <strong>the</strong> publication of <strong>the</strong> first meteorological year<strong>book</strong> (21) using<br />
services of a rented computer. In <strong>the</strong> future, this system will be changed for<br />
one that.wil1 requiere a minimum of services of commercial firms. Plans for<br />
mechanizing <strong>the</strong> reading of bands and for preparing some secondary processing<br />
are also being taken into account. Each country will prepare its part of <strong>the</strong><br />
year<strong>book</strong> on uniform format, so that <strong>the</strong> preparation of a regional year<strong>book</strong> wili<br />
consist of joining <strong>the</strong> national parts only.<br />
Hydrology. The action of <strong>the</strong> Project has allowed <strong>the</strong> standarization of data<br />
processing, following <strong>the</strong> usual recommendations in this kind of work Therefore<br />
it is now possib<strong>le</strong> to ensure <strong>the</strong> reliability of most of <strong>the</strong> records that are<br />
published. At <strong>the</strong> same time <strong>the</strong> deficiencies of <strong>the</strong> previous data are now evident.<br />
Therefore, <strong>the</strong> revision of <strong>the</strong>se old data constitutes a fundamental activity<br />
of <strong>the</strong> second phase of <strong>the</strong> Project. The Project has proposed a comp<strong>le</strong>te niechanized<br />
processing, as indicated in <strong>the</strong> instructions (2) (22), but for <strong>the</strong> lack of<br />
computing facilities this objective could be achieved only partially. In practice,<br />
<strong>the</strong> computation of stream gauging is made mechanically, ei<strong>the</strong>r by means of<br />
programmab<strong>le</strong> calculators or by conventional computers. The use of small programmab<strong>le</strong><br />
calculâtors or mini-computers will be extended to <strong>the</strong> second phase<br />
of <strong>the</strong> Project. The translation of <strong>the</strong> graphs of <strong>the</strong> limnigraphs has been made<br />
up to now by manually, but <strong>the</strong> rest of <strong>the</strong> process from <strong>the</strong>re on is more or <strong>le</strong>ss<br />
mechanized up to <strong>the</strong> tab<strong>le</strong>s for publication. Mechanization of all this process<br />
is contemplated in <strong>the</strong> second phase of <strong>the</strong> Pr'oject. The rest of <strong>the</strong> processes,<br />
i. e. : rating curves, sediment computations, duration curves, etc.. . , is made<br />
manually or with <strong>the</strong> use of <strong>the</strong> few programmab<strong>le</strong> calculators provided up to<br />
date, but <strong>the</strong> trend is towards to a comp<strong>le</strong>te mechanizations of <strong>the</strong>se computations.<br />
The Project has published four regional year<strong>book</strong>s (23). of which <strong>the</strong> last three<br />
have been prepared with <strong>the</strong> help of e<strong>le</strong>ctronic computers. These publications<br />
have received excel<strong>le</strong>nt comments by <strong>the</strong> users of <strong>the</strong> information. The year<strong>book</strong>s<br />
contain in addition to streamflow records, lake <strong>le</strong>vels, sediment discharges,<br />
water quality, duration curves and flood envelopes.<br />
OUTLOOKFORTHEFUTURE<br />
The impact of <strong>the</strong> Project on <strong>the</strong> meteorological and hydrological activities<br />
in <strong>the</strong> Central American Isthmus has been impressive not only in <strong>the</strong> amount of<br />
availab<strong>le</strong> information, but in <strong>the</strong> increase of <strong>the</strong> public concern with <strong>the</strong> importance<br />
*<br />
of <strong>the</strong>se. The second phase of <strong>the</strong> project is aimed maidy to comp<strong>le</strong>ting <strong>the</strong><br />
institutional strenghtening necessary to ensure <strong>the</strong> continuity of <strong>the</strong> activities re -<br />
quired for providing <strong>the</strong> basic information needed for <strong>the</strong> social and economical<br />
43
44<br />
development of <strong>the</strong> Central American Isthmus. The Project will have at that<br />
time prepared <strong>the</strong> local Services for providing all necessary information for pr2<br />
ject design Where this information is insufficient, tools will be availab<strong>le</strong> for<br />
mbking a reasonab<strong>le</strong> good estimate which will avoid delaying <strong>the</strong> imp<strong>le</strong>mentation<br />
of, <strong>the</strong> Project. When <strong>the</strong> information is inexistent, criteria for obtaining a mini<br />
mum set of data will be well known to <strong>the</strong> local technicians. Finally, <strong>the</strong> mete2<br />
rological and hydrological services will be in a good position for influencing national<br />
polices on natural resources, ensuring a rational and efficient use of <strong>the</strong>m.<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
1 o.<br />
11.<br />
12.<br />
13.<br />
14.<br />
-<br />
REFERENCES<br />
PHCA. Manual de Instrucciones; Hidrometría, Publicación No 49<br />
PHCA. Manual de Instrucciones; Estaciones Meteorológicas, Publication<br />
No 70.<br />
-<br />
PHCA. Programa Regional de Instalaciones; Publication No 20<br />
- PHCA. Medida de<br />
Publication N"79.<br />
Sedimento s en Algunos Ríos del Istmo Centroamericano<br />
PHCA. El cálculo de los requerimientos de agua en Costa Rica.<br />
tion No 39.<br />
Publica-<br />
Hargreaves, G. Requerimientos de Irrigación y Balance de Agua; Proyec-<br />
to propuesto Arenal, Costa Rica, Publication No 87 del PHCA.<br />
Hargreaves, G. Necesidades y Requerimientos para Irrigación; Comayagua<br />
y Vecindades, Hondruas. Publication No 86 del PHCA.<br />
Hargreaves, G. Deficiencias de Agua en Centroamérica y Panamá. Publication<br />
No 88 del PHCA.<br />
- PHCA. Previsiones de Escorrentía. Publication N"46.<br />
PHCA. Pronósticos Hidrológicos para la Operación de Plantas Hidroeléctr-<br />
'=&tas del Seminario de Managua) Publication N091.<br />
Basso, E. Sediment measurements in several rivers of <strong>the</strong> Central Ameri-<br />
can Isthmus. Fall meeting of <strong>the</strong> American Geophysical Unnion, San Fran-<br />
cisco 1971.<br />
- Se dimento en algunos rfos del istmo Centroamericano,<br />
PHCA. Medidas de<br />
Publication No 79.<br />
Basso, E. Some Methods for Estimation of Floods with Limited Information<br />
in One Tropical Area. Second international Hydrology Symposium Fort<br />
Collins, Colorado 1972.<br />
-<br />
PHCA. Envolvente de Precipitaciones en el Istmo Centroamericano, Publication<br />
No 81.
15. PHCA. Factibilidad del Riego con pozos en el Proyecto Usulután<br />
dor, Publication No 25.<br />
16. PHCA. Estimación Preliminar del Balance de Aguas en el Istmo Centroa-<br />
mericano; Publication No 18.<br />
45<br />
El Salva<br />
17. Alghren, L. ; Basso, E; Jovel R. Preliminary Evaluation of <strong>the</strong> Water<br />
Balance in <strong>the</strong> Central American Isthmus. Symposium on <strong>the</strong> Water Balance<br />
in North America; Banff 1970.<br />
18. PHCA. Estimación preliminar gel Balance de Agdas del Lago de Managua.<br />
Publication No 7 5.<br />
19. PHCA. Efecto de la Temperatura en el Transporte de Sedimentos. Publi-<br />
cation No 6 1.<br />
20. PHCA. Curva de Descarga de Sedimentos. Publication No 8.
46<br />
Figure 1<br />
Main Met e or olog ica 1 Stat i on.
Figure 2<br />
O r dina ry Meteor olog ical Sta tion.<br />
A_....<br />
47
48<br />
Figure 3<br />
Typical Hydrometric Installation.
Figure 4<br />
Prefabricated Cab<strong>le</strong>way Tower.<br />
49
50<br />
Figure 5<br />
Meteorological coverage, before and after<br />
<strong>the</strong> Project.
Figure 6<br />
Hydrological coverage after <strong>the</strong> PFoject.<br />
51
52<br />
Figure 7<br />
Sediment rating curve.
AVERAGE ANNUAL PREC/P/TAT/ON MM<br />
Figure 8<br />
Results of <strong>the</strong> Sediment measurements<br />
53
54<br />
Figure 9<br />
Flood Envelopes.
Figure 10<br />
Maximum Rainfall Envelopes.<br />
55
56<br />
Figure 11<br />
Water Balance in Central America
StZE OF PARTlCLES MM<br />
Figure 12<br />
Effect of Temperature in sediment transportation.<br />
57
ABS TRACT<br />
METHODOLOGY EXISTING FOR ESTIMATING<br />
FREE SURFACE WATER EVAPORATION<br />
by<br />
Francisco Cubas Granado<br />
The purpose of this paper is to recount <strong>the</strong> metodology<br />
for estimating free surface water evaporation and particulary<br />
in <strong>the</strong> case of a reservoir when studing <strong>the</strong> regulation curves<br />
<strong>the</strong>ceof or <strong>the</strong> regulation-exploitation system, for estatistics<br />
and empirical methods.<br />
RESUMEN<br />
El objetivo de este artículo es recopilar los distintos<br />
métodos para estimar la evaporación en lámina libre y particu-<br />
larmente en el caso de un embalse en función de la regulación<br />
que efectue y del sistema regulación-explotación utilizando m5<br />
todos empíricos y estadlsticos.
60<br />
Al1 water returning to <strong>the</strong> atmospherd due so<strong>le</strong>ly to evaporation<br />
procosoen is an important e<strong>le</strong>ment in <strong>the</strong> hydrologic cyo<strong>le</strong>. Moreover,<br />
it io a limitin;? factor for <strong>the</strong> effecient utilisation o9 free surface<br />
mtor (reservoira, lnken, rivers, etc.).<br />
In vie# of its big influonce in <strong>the</strong> water cyc<strong>le</strong>, wvapqration has<br />
beon <strong>the</strong> subject of innumerab<strong>le</strong> surveys which, because of <strong>the</strong> diversity<br />
of sndc pursued in enmh one <strong>the</strong>reof, have not given rise to R homopmaour<br />
<strong>the</strong>ory that could be accepted unanirnoualy.<br />
pronent papar io to recount <strong>the</strong> methoùology exinting for estimating<br />
I'rso rJurfaoo MttCr evaporo.tion.<br />
The so<strong>le</strong> purpose of this<br />
l~nrticularly in <strong>the</strong> oose of a reßervoir,<br />
vhen studgin,? <strong>the</strong> re$Tlation curve8 <strong>the</strong>reof or <strong>the</strong> re~lation-exploitation<br />
riyntern, formulan are required which nay enab<strong>le</strong> <strong>the</strong> mapomtion ocourring<br />
to be cotimnted ox evaluated when it occurs on J. large sc¿i<strong>le</strong> or must<br />
lie taken into account for r.orkinL; out <strong>the</strong>se calculations.<br />
oithcr empirical or on n phgBica.1 basis, may often mitipte <strong>the</strong> lack of<br />
"in nitu" data.<br />
1.2. %ctoro affectinp: <strong>the</strong> atmomhere's evaporating Dower<br />
Suah formulas,<br />
The a.tmoephore's evaporating power in tho avapoxation rate, ex-<br />
prossed in millimetres of water, for tho period determined (mm. of<br />
uistor per day, for examp<strong>le</strong>).<br />
The atmoaphere'o evaporating povier faotors are: <strong>the</strong> hypometrio<br />
deficit, temperaature of <strong>the</strong> water, temepraturo of <strong>the</strong> air, insolation,<br />
fiperd and turbu<strong>le</strong>nce of <strong>the</strong> wind, barornotrio pressure, <strong>the</strong> quality Of<br />
Lhc irntar and u1ti tude.<br />
In fact, moct of <strong>the</strong>se po.ra,metere are corelatecl to each o<strong>the</strong>r and<br />
th@ nractica.1 formulae used Tor eva.luntinf: evaporation gfily use8 those<br />
:mmmnotera which ame <strong>the</strong> moot important or easiest to measure.<br />
1.3. b'actorr, afrectinn wa.-aorcr.tion of Tree viater surfacea.<br />
¡?ret? wa.tc,r r:iirfn.cr rv&porntion, ,Tiven that th'e atmosphere's<br />
~wl)n~*it~tj.n,~~ :>oi:e:r ir: cox:t:.i.nt, dencnds on ita ri.rea..l+and dapth. The I?I~.SS
OF imter acto as P. regqlator so that if it is not R large area and<br />
in aliallow, t!ir ternpcrn.tnre of thp <strong>who<strong>le</strong></strong> eanilg follovrs <strong>the</strong> law of<br />
<strong>the</strong>rmal variation as np1)lied to itr, surfo.ce.<br />
Iprne imter nurfri.ce evaporation in all <strong>the</strong> <strong>le</strong>ss in hot seaaona m a<br />
pentrr in cold ven<strong>the</strong>r, <strong>the</strong> bi,:ger in area and in 'depth <strong>the</strong> water<br />
napne ie.<br />
It can bo considered that, ba.nically, free surface water evaporation<br />
dc:)enda<br />
-<br />
on:<br />
'ih enorgy availabla from uo1n.r rr.diation.<br />
- The perceptib<strong>le</strong> heat traiiomitted throunh <strong>the</strong> air.<br />
- 'l'ho airari cnpncity to trianoport imter vapour.<br />
Tlir diff~rent methods propozcd for estimating waporation may be<br />
poiipod into tiro ccitr:;orieo:<br />
EL) Empirioal methods giving rise to Yormulae baaed, mostly, on<br />
BEI lton,a law with modi í.'icationo to Che fmtors affeating evaporntion.<br />
b) VathodB with n rational basis of physicaï <strong>the</strong>ories that may ha<br />
aummrd up ant<br />
- Ilcthotln baned on vater evaluation, consisting in performing<br />
EL water input and output LaLance with evaporation being<br />
calculated as an unknown in <strong>the</strong> balnnoe equation.<br />
- i<strong>le</strong>thocin harad on <strong>the</strong> evnluation of energy where <strong>the</strong> bn<strong>le</strong>nce<br />
mnde io a,n energy enterinc and <strong>le</strong>aving brzlanoe. The cslculation<br />
of evaporation io aimilar to <strong>the</strong> foregoing g~oup.<br />
- Methodo bnaad on <strong>the</strong> rnnßri trnnuaort <strong>the</strong>ory vihere cvaporation<br />
in evaluated from <strong>the</strong> wind cpeed and <strong>the</strong> vapour preoouro<br />
61<br />
,ymdient between lhe surface i!ntPr and <strong>the</strong> supwinounbent<br />
lnyerc of air.<br />
The irater or cner.3 bo.l:i.nce methods are <strong>the</strong>oretically suitab<strong>le</strong> for u88<br />
in cn1culntin:y evaporation in 1a.k~~ and reservoirs. Never<strong>the</strong><strong>le</strong>ss, it is
62<br />
difficult to a.pnly tham in prnotice because of <strong>the</strong> error committed<br />
in msaaurin(: some terms in <strong>the</strong> balance.<br />
Nore rocent renenrch hno drmonstratod that over relatively long<br />
prriodo, >t Icaat one month, <strong>the</strong> potential ovapo-transp'iration is<br />
conota,nt Pnd only depends on climatic factors. This has <strong>le</strong>d <strong>the</strong><br />
researchers to Geek empiric formulae depending on <strong>the</strong>se factors.<br />
In pmrcr!.l, empiric rormnlae Iinvc. been nought after by oo-relating<br />
eva n o r-, t i on wit li t ti e Po 1 lowi n,n met oo ro logica 1 fa0 t c w i<br />
-<br />
The temperature of tho air<br />
Incident SOIR.T ro.dia,tion<br />
- Air humidity<br />
- A combination of tho foresing<br />
Howover, many of <strong>the</strong>se îormulno hnvo to be checked in practice<br />
b(?îore using <strong>the</strong>m on uurfnces or areas which are not thoee where<br />
<strong>the</strong>y were first obtained. Their contrast is obtained by oelibrating <strong>the</strong>m<br />
by actual measurements of evapora.tion on <strong>the</strong> basis of ihatruments al-<br />
ren4dy exintin,? (rafte, tanice, evmorimeters, ato.). Xn fact, <strong>the</strong> only<br />
prccodure fox direot meanurement of evaporation lies in solving a water<br />
bn lance.<br />
Let uö romnmber that:<br />
. The hycrornetric deficit or atmosphere saturation deficit,<br />
obtained ao n difference between saturating vapour tenaioh Fe lo <strong>the</strong><br />
irater nurfme tompera,turr T and <strong>the</strong> actual vapour tension pa in <strong>the</strong><br />
ri.mbisnt air, ir, <strong>the</strong> main ;,pa.rameter of <strong>the</strong> atmoophere avnpomting QOWBF.<br />
- 'The hyqrometric condition or r3eiFee E of thc air<br />
rcTrm5.n:: to tho viater nurfoce temperature T is <strong>the</strong> quotient betmen <strong>the</strong>
<strong>the</strong> tencione 1%. 8.nd Pe ( f.<br />
= Fan/fe) and represents <strong>the</strong> relative<br />
humidity or -tha air.<br />
- The psiahrnmetric diffarence, obta.ined an a difference<br />
b
64<br />
it in not p017::ii~i.e to expect a,nythin!y rno:t*e Lhm an a.pproririo,tioii froin<br />
this typo OP rrtimnte, a.ü IJC? have u.lreedy mid.<br />
The rooults obta.ir.eci in <strong>the</strong> evapora.tion tanka must be multiplied<br />
by tho trin!c coefficient, which will be peculiar to ench type, in order<br />
to oritir!ia.-tc! <strong>the</strong> actual evawration.<br />
ïri addition, <strong>the</strong> pa.rn.meter va.luec wc h3.w to consider are those<br />
exia Lin:: in <strong>the</strong> air-water surface intoqhase which<br />
to rdiwniire wit1 iro Iin,vo to observo <strong>the</strong>m at <strong>the</strong> inost ncceosibla points where<br />
i.1; il: niipponcd that <strong>the</strong>ir values Co-rrlíite well with those which would<br />
h:i.vr beon obt:iiiiccl in tho micl intarphase.<br />
2.3.1.-n0s forrnula<br />
are generally impoccib<strong>le</strong><br />
In 1802, I):i.lton deduced that , just likc! o<strong>the</strong>r parameters, evaporation<br />
on a. free mter surface is proportioria.1 to <strong>the</strong> hygrometric deficit.<br />
hi2 cvo.por:i t ion forrnula:<br />
E = (Pc? - Fa) = o( (Fe - Fa.), depending on <strong>the</strong> hygrometric<br />
or, what ir: tho ao.rne8<br />
II<br />
deficit<br />
= d :pe (i-€), drprndin,y on tho saturatinC;<br />
vapour t enci i on and hjr$romet ri c<br />
de I;r ea,<br />
IIence,
In thc Pirat expression of Uaiton’s forinuls, II represents <strong>the</strong> total<br />
:)resnurc! (,pm plus water vapour) above <strong>the</strong> evaporatine ourface. H’s<br />
inl.‘lirence only intervenes ac a corrective term in evaporation prob<strong>le</strong>m0<br />
riml m;ry br tiiccanxlsd in R. rirst approximation.<br />
‘i’hc cocPPioicnt 4 in cha.racteri::tic of <strong>the</strong> meterological station under<br />
o o iin i rl erat i on.<br />
‘J!hi:i formula civon very vRria.b<strong>le</strong> evo.poration valuos Prom one plme to<br />
i’i.notJior, uhioh limite iti; une.<br />
in irhiolri:<br />
- i3 i: <strong>the</strong> water evaporated in o. month of n days.<br />
- Po (in mrn. of r.lg.) ir: tho nvera,p saturz,tin,s water vapour<br />
tension nt temnoraturs T. This in obtained from hygrometric<br />
tnb<strong>le</strong>s.<br />
- Fn. (in min. of JIg) in <strong>the</strong> actual a.verage monthly water vapour<br />
ttrnoion of‘ tho fiar nt <strong>the</strong> tima of tho T readings. Thin ia<br />
obtained by multiplying Fe by <strong>the</strong> hypometrio deGree.<br />
- I3 (in min. o.€ !!go) in <strong>the</strong> Lx1,roniotric axerage monthly pressure.<br />
- T (in QG) ia <strong>the</strong> a,vora.ge monthly tomporature.<br />
65
66<br />
vhore:<br />
- E (in mrn.) io tho evn7oration in 24 houm.<br />
- U, <strong>the</strong> :i.riiìnesn. 'Phi:: is calculated by tha equation D=lO@-humidity<br />
at u ntmocphereo.<br />
- V (in mi<strong>le</strong>s/hour) io <strong>the</strong> :i,vera,ye wincl opeed over 24 hours.<br />
- 'I' (in QB), .<strong>the</strong> ::.vcrnp temperature over 24 hours.<br />
3) Tiorton's eqiiation:<br />
i - p + r Y - i<br />
W-h<br />
with P i o for n inrater ahcet irith 2. crflall ßurfaoe.<br />
4) Rohwer' a equation:<br />
E=O.7'(1 (1./16s-0.01:36 n1.Y. (Fda)
D = averaee barometric pressure.<br />
C numerioal coefficient<br />
1% = prenent vapour pressure<br />
Be sotimating vapour pressure<br />
h m lblative humidity of tho air<br />
P = fraction of time during which <strong>the</strong> wind is turbu<strong>le</strong>nt.<br />
t = nurnber cf days.<br />
Ta = average tempwature cf <strong>the</strong> air<br />
Tw = average temporature of surfaoe water.<br />
N P monthly wind speed average<br />
'y wind factor,<br />
&3.6. ûbnarvation<br />
The dií'ïiculty in applying <strong>the</strong>se formulae lies in that <strong>the</strong><br />
mnjoiity of <strong>the</strong> vnrisb<strong>le</strong>s appear as nn average vsluo and it is possib<strong>le</strong><br />
io2 <strong>the</strong>ir valueci not to represent <strong>the</strong>ir total VQlUe well.<br />
'1.1. Critime on <strong>the</strong> mothods<br />
Tho methods based on <strong>the</strong> enerm balance enter more into <strong>the</strong><br />
field of research than in that of praoti-1 usage.<br />
contrant or chock purposes.<br />
or tima which are nufficiently long.<br />
67<br />
Yhey can be usied for<br />
They can give evaporation Values over periods<br />
Tho methodo based on <strong>the</strong> water balonce or OR ,th0 mass twnspcrt<br />
ihoory 1i.m likawins mur@ suitab<strong>le</strong> for uoinc 81 ohecke than fa2 graatioal<br />
utla~a. ThePie are, however, more recomendab<strong>le</strong> for invsetigating svapomticn<br />
over ohort periorln of timo ( a few hours).<br />
'Pheee nothaaa also always hitve to be chaoked &na contrasted, as<br />
hnppene with tho ampirio mothods, on <strong>the</strong> basis of direot eveporntion mean-<br />
iiromonto, bccnuee without <strong>the</strong>se contTaStB and <strong>the</strong> modifications resulting<br />
thorofrom, oounte-active resulto may be given.
68<br />
In view oi <strong>the</strong> faet thRt in <strong>the</strong> imter or energy balance<br />
rn(>thods it is difficult to measure <strong>the</strong> terms appearing for evaporation<br />
ctiidy on free inter eurfaccs, <strong>the</strong> use of methods based on <strong>the</strong> mass<br />
transport <strong>the</strong>ory is more otxongly recomnendad.<br />
3.2. :mos trmmort methods<br />
In masE tmnsport methods, <strong>the</strong> value of evaporation is entimated<br />
ïroin <strong>the</strong> wind opeod aiid <strong>the</strong> vapour pressure ,?radient between <strong>the</strong> surface<br />
water cind <strong>the</strong> laysra of air, bu wing a formula of <strong>the</strong> typez<br />
R = ri fl (u) f 2 (ao - 1%)<br />
iilioro II ir. <strong>the</strong> pronortionality constant commonly known as <strong>the</strong> 'ImaSB<br />
tmmiport coePí'iciontlt, 21 and f2 nre knovm functions of <strong>the</strong> wind apeed<br />
anti vapour prop nure {yadi snt res? cct ively.<br />
The í'o.re{:oin,y formu1t-c ir: rrritten, in its more uma1 forrnt<br />
iihioh enn.b<strong>le</strong>ii E to be calculnted if i:e know <strong>the</strong> value of H beforehand.<br />
3.7.1. Cp.lcula.tin,s !I<br />
N'E: value in obtained in two waynr<br />
1) Dy catirnntiny: <strong>the</strong> evaporation by o<strong>the</strong>r methods and dividing<br />
i I; by ths product U.(Fe-Fa)<br />
2) Dy obtaining n linear c.clyreocion arqmtion betwnn <strong>the</strong> chango<br />
[)I' tlio r:tnto OP <strong>the</strong> wa.tcrl\lI md. thr product U(l%-%), in <strong>the</strong> following<br />
i.!ayg<br />
A 11 fl.IT. (F'c?-~) 5 C<br />
iiherr bhc coii:;to,rit C giver. <strong>the</strong> avcrn,Te loss from filtration in <strong>the</strong> free<br />
1I:l:LPr ::iii..l'n.cc.
69<br />
Thr ficrr t pro.-,edure for i”ind,inf; <strong>the</strong> mass transport coefficient<br />
mtriiirec n. qrncise water balance crhich forces exact measuring o€ <strong>the</strong> input<br />
and output or <strong>the</strong> surface wa.t,or unùer study to be ca,rried out.<br />
not; bc prn.ctica.l in till CRDBB.<br />
This may<br />
The nocond. procedure in cntisfmtory if <strong>the</strong> losses not due to<br />
rva.pora,tion do not vary to any ,great extent or are not <strong>the</strong> most part of<br />
tho mtnr loet from thc surface in question.<br />
It ia interesting to look into <strong>the</strong> aeroAynarnic formula, here,<br />
that cnnbloo T7 to bc calouliited RB n function of <strong>the</strong> shape (perimeter) and<br />
six0 (nroa) of <strong>the</strong> free irater nurface, i+moncFt o<strong>the</strong>r variab<strong>le</strong>s.<br />
3.%.2. The rnntooroloaianl .station rrcruirod Eor viorking out <strong>the</strong><br />
a ero dynnmi c met hod<br />
The fo11owin:y rnateriii.1 iu necesnsry for <strong>the</strong> typical station:<br />
- A Ch:::: A raft on <strong>the</strong> edge of <strong>the</strong> water surface.<br />
- ‘l‘wo CUD type wind Buceo<br />
- %io limnigrnphs<br />
- A water tempersture recordar.<br />
- A hy,iro<strong>the</strong>rmograph and a pyrmometro.<br />
- Several plqvioTra.phs cet out all around <strong>the</strong> perimeter of <strong>the</strong><br />
ourfnco in question.<br />
J.?. 3. kiorkinrt out <strong>the</strong> evanore,tion esuation<br />
Ilvnpomtion ia, with rajytrd to i’aorkinc out <strong>the</strong> formula mentioned<br />
in 3.2.1., ri. diffunion proceso in which <strong>the</strong> water vapour is transported<br />
froin <strong>the</strong> tinter nurface to thn Fi.tmocnherc.<br />
Vrrtichl trn.nni)orta.tlon of tho vo.nour depends on thn effective-<br />
naon of <strong>the</strong> tur.hii<strong>le</strong>nt mixture in <strong>the</strong> lor:er layers of air, r.nd <strong>the</strong> main<br />
inl.‘liisnce tharcin in tho wind npcred and ro1ir;hnaon OP <strong>the</strong> surfa.ce. A turb<br />
ii1an.t coefficinnt ban be found which varies rrith <strong>the</strong> vind cpeeù Por each
70<br />
tlotcrininotl niirfnco, ir1 thc caso whcrrby <strong>the</strong> narodynarnio characteristics of<br />
tlic 1::tter romain conntant.<br />
The tramsport of <strong>the</strong> vapour takes place under <strong>the</strong> vapour<br />
~)recsure {:radient cet up between thc vapour saturation pressure' at <strong>the</strong><br />
curlno
) It iu accci>tcri thnt <strong>the</strong> laminar sublnyer ia of a negligib<strong>le</strong><br />
thicknsnn nntl thnt tho turbu<strong>le</strong>nt boundary layer extondB below <strong>the</strong><br />
wrfaca water.<br />
c) The riincl cpeed profi<strong>le</strong> is given by <strong>the</strong> law:<br />
m<br />
u(z) = aZ<br />
tihere t h conntc?.nto 2 and fi depend on <strong>the</strong> etability nnd roghness of <strong>the</strong><br />
: : iirf ace.<br />
Combini'iig <strong>the</strong> equations given for E ana T, we havez<br />
From $ho wind profi<strong>le</strong> i: found:<br />
irliioh a.lloi~n us to irrite:<br />
m-1<br />
dz. E. z<br />
Integrating for<br />
E = <<br />
u2<br />
Throou,yh a,nalap;y trith <strong>the</strong> fluid flow through & unifom tube,<br />
it c m be shotrn thnt:<br />
(2m+l) (di)<br />
O,:! Q.2 0.4, 1,8<br />
U<br />
nliich civer: rice to tho followinc expreesion for E:<br />
71
72<br />
where: - -<br />
Tho a.vern.go speed value U is equa.1 to <strong>the</strong> product<br />
(K2. IJ2)<br />
- K1 and 1 2 aro numerical constants.<br />
- V ic <strong>the</strong> kinetic viscosity.<br />
- e,<br />
X in riven by X with A and P being <strong>the</strong> area and <strong>the</strong><br />
perimeter of <strong>the</strong> surface under study.<br />
139 expresein,y <strong>the</strong> aaecific humidities as a function of <strong>the</strong><br />
vapour t en0 iam, <strong>the</strong> evnpora.tion equation becomes t<br />
if 6 is tho thickncco of <strong>the</strong> turbu<strong>le</strong>nt layer, it is easy to show that:<br />
and thon:<br />
ii,o-rotlyrinmic rnet hod.<br />
Kl=m+l;K2= 1 .<br />
m 4 3<br />
Tho ciluationn (1) ; ml (7’) aro <strong>the</strong> formuin proposed by thio<br />
In order to facililate <strong>the</strong> calculation of evaporation in small<br />
. ux’hcrs, lhe Col loirinl: hypoihcocs can Uc rnnde:
3.2.5. Conclusiono<br />
a) Give <strong>the</strong> wind profilo exponent <strong>the</strong> valus<br />
1<br />
m=s<br />
b) Take a, value of 6 metren a6 <strong>the</strong> thickness of <strong>the</strong> turbu<strong>le</strong>nt<br />
boundary layerr<br />
Thon, tho mass tranoport coeffioient ie given by2<br />
-4 p<br />
N = (2.62 x 10 ) (A)<br />
o1 2<br />
Thena fortnulno ara vary useful when, st al1 times, tho area<br />
nnd perimeter of <strong>the</strong> \ranter surface under survey are known. They are very<br />
important, <strong>the</strong>n, for applying to tho study of <strong>the</strong> evaporation change that<br />
would OCOUT in n re~ervoir in every situation <strong>the</strong>reof.<br />
In order to oalculate <strong>the</strong> value of N, th4 values for U, Fe and Fa<br />
obtained at R meteorological station near <strong>the</strong> zone under survey can be<br />
UCoa.<br />
The inclusion of <strong>the</strong> perimeter and area in <strong>the</strong> formula for<br />
cnlcutnting I compenses <strong>the</strong> variability OS this mass transport cooffioient.<br />
The application of this formula to very irregular ehapad water<br />
ririrfiloao may <strong>le</strong>ad to an excaesiva ca.lculation of evaporation because of<br />
<strong>the</strong> effect of tho perimeter in <strong>the</strong> formula. In such caees, to mitigate<br />
.thio exoesn, we c ~ maker n<br />
u. Thort hw&t WIo3 t zmanOo formula<br />
‘Phis expraeoeo evaporation by<br />
(in inches/hour)<br />
73
74<br />
EJ h ere:<br />
- F1 ana B'2 are <strong>the</strong> vapour pressures (in inchee of Hg) at<br />
nltitudeo hl and hp.<br />
- U1 and U2 are <strong>the</strong> winü speeds (in m/h) at <strong>the</strong> said altitudes.<br />
- iri thc nvertrn,Te temperature (inop) of <strong>the</strong> air between altitudee<br />
hl n.nd h2<br />
3.4. PItnmnn's formula<br />
'Chio haCj tho expresoion:<br />
E I 0.4 (1 4 0.17 U) (Fe - Fa)<br />
whore E io civen in mm/day and <strong>the</strong> wind veloaity U, at 2 m. height, in<br />
mi<strong>le</strong>5 per hour.
UT i3LIOGRAPlIY<br />
tl;dodon en uso y DU emp<strong>le</strong>o para cdlculo de la eva,potranspira~idn~~,<br />
by Paustino Lonnno Cnrcfa.- February 1964.- Publication no. 23 of <strong>the</strong><br />
C.E.11. OP tiio Xiriistry of Public Blorks.<br />
" L'hytl xo loei e d R 1 ' ingeni eur", by C. ii6rn6ni &$as. -Publishad by Eyro 1 l es.<br />
'llI:indbaok of applied i!:;ciro1.ogyW, by Van Te Chon.-Published by E<strong>le</strong>i.c-Graw Hill.<br />
"Netodos prdcticoo PRM. el estudio hidrologico comp<strong>le</strong>to de una cuenca",<br />
by R. fieran.- Published by <strong>the</strong> C.F.H. of <strong>the</strong> Ministry of Publio Works.<br />
75
ABSTRACT<br />
GEOHYDROLOGICAL STUDIES IN<br />
SMALL AREAS WITHOUT SYSTEMATIC DATA<br />
Emilio Custodio Gimenan<br />
Frequently are needed studies to profit ground water resources by<br />
means of wells or gal<strong>le</strong>ries in areas with non existing data on river<br />
and spring flows and on recharge, but in which injuries may be imposed<br />
on pre-existing water uses. One begins looking for availab<strong>le</strong> data in<br />
several kinds of fi<strong>le</strong>s and inquiring local peop<strong>le</strong>. Moreover, <strong>the</strong> size<br />
of <strong>the</strong> existing water concessions and <strong>the</strong>ir specific use allows <strong>the</strong><br />
appraisal of mean and base discharge. The pluviometry is obtained though<br />
<strong>the</strong> closest stations, and some corrections on judgement. The key prob<strong>le</strong>m<br />
is <strong>the</strong> effective ground water recharge calculation, beeing solved<br />
through <strong>the</strong> consideration of three independent points of view:<br />
1) modified hydrometeorological balance<br />
2) ground water flow calculation based on existing or estimated data<br />
3) salt balance, specially chloride, based on water - tab<strong>le</strong> chemical<br />
analysis and rain water composition<br />
Generally is possib<strong>le</strong> to get coherent results. As an illustration,<br />
three cases are presented:<br />
a) Montroig Area (Tarragona). It is a coastal plain<br />
b) Riera de Carme Basin (Barcelona). It is a limestone formation<br />
c) Famara Massive (Lanzarote, Canary Islands). It is a basaltic<br />
formation in an arid clima<br />
Key words: scarce data, ground water, chemical balance, perameter<br />
estimation, subterranean flow, case histories.<br />
RESUMEN<br />
Con frecuencia deben realizarse estudios para aprovechamiento de -<br />
aguas subterráneas mediante pozos o ga<strong>le</strong>rías en zonas en las que exis-<br />
ten datos sobre cauda<strong>le</strong>s de ríos y fuentes, ni sobre la recarga, pero<br />
en las que se esperan afecciones a usos ya estab<strong>le</strong>cidos. Se procede a<br />
la búsqueda de los posib<strong>le</strong>s datos en los archivos y al interrogatorio<br />
de los habitantes. Por otro lado la importancia de las concesiones --<br />
existentes y su destino permite apreciar los cauda<strong>le</strong>s y los cauda<strong>le</strong>s -<br />
de base. La pluviometría se interpola a partir de las estaciones más -<br />
próximas efectuando correciones estimativas. El prob<strong>le</strong>ma clave es el -<br />
cálculo de la recarga eficaz a los acuíferos y se ataca bajo tres pun-<br />
tos de vista:<br />
1) balance hidrometeorológico modificado<br />
2) cálculo del flujo de agua subterránea a partir de datos disponib<strong>le</strong>s<br />
o estimativos<br />
3) balance en sa<strong>le</strong>s, en especial en cloruros, a partir de los análisis<br />
del agua freática y de la composición del agua de lluvia.<br />
En general se obtienen resultados coherentes. A título de ilustración<br />
se comentan tres casos prácticos:<br />
1) área de Montroig (Tarragona). Es un llano costero<br />
b) cuenca de la Riera de Carme (Barcelona). Es un macizo calcareo<br />
c) macizo de Famara (Lanzarote, Islas Canarias). Es un macizo basálti-<br />
CO en clima arido.<br />
Palabras clave: datos escasos, agua subterránea, balance químico, esti<br />
mación de parámetros, flujo su,bterráneo, casos rea<strong>le</strong>s.<br />
9; Comisaria de Aguas del Pirineo Oriental y Curso Internacional de Hidrología<br />
Subterránea. Barcelona.
78<br />
1. INTRODUCTION -<br />
Frequently, geohydrological studies are made in small basins<br />
where prob<strong>le</strong>ms of water use exist or are foreseen. To solve<br />
<strong>the</strong>se prob<strong>le</strong>ms, data is required which has not usually been<br />
compi<strong>le</strong>d or taken down. Usually, <strong>the</strong>re are only a few<br />
pluviometers in <strong>the</strong> area, and are of dubious reliability; <strong>the</strong>re<br />
are have no measurements of <strong>the</strong> water courses as <strong>the</strong>y are small<br />
or ephernerous, and <strong>the</strong> springs or sources have not been<br />
control<strong>le</strong>d. On <strong>the</strong> contrary, <strong>the</strong> exploitations established may<br />
be of a <strong>the</strong> same order of magnitude of <strong>the</strong> total availab<strong>le</strong> water<br />
resources.<br />
It is not possib<strong>le</strong> to give general working norms, since<br />
<strong>the</strong>re is a very wide range of climatic, geological structural<br />
conditions etc. After setting out some general ru<strong>le</strong>s, three<br />
cases will consequently be discussed, showing notab<strong>le</strong> differ-<br />
ences in conditions, discussing <strong>the</strong> form of operation and <strong>the</strong><br />
guarantee of <strong>the</strong> estimations made.<br />
The main objectives of <strong>the</strong> work to be carried out may be<br />
summarized as follows:<br />
a) Know<strong>le</strong>dge of <strong>the</strong> groundwater flow pattern, including<br />
recharge, circulation and discharge. The identification<br />
of <strong>the</strong> main aquifers is one of <strong>the</strong> stages to be covered.<br />
b) Obtain a reasonab<strong>le</strong> hydraulic balance, if possib<strong>le</strong><br />
coherent with <strong>the</strong> results of various independent<br />
estimation processes.<br />
c) Analysis of <strong>the</strong> existing and projected water up-taking<br />
ernphasing <strong>the</strong> possib<strong>le</strong> interferences between <strong>the</strong>m and<br />
with <strong>the</strong> water courses and springs, and also, if possib<strong>le</strong>,<br />
obtaining <strong>the</strong> foreseen user's extraction programme.<br />
It is important to remember that one is obliged to cany ont<br />
<strong>the</strong>se studies should be during certain months along or at <strong>the</strong><br />
most within a year; consequently <strong>the</strong> only data availab<strong>le</strong> to<br />
compute <strong>the</strong> components of <strong>the</strong> mean hydrological cyc<strong>le</strong> are<br />
those existing at <strong>the</strong> time. The hydrological data taken during<br />
<strong>the</strong> study are not mean values, but depend on <strong>the</strong> climatic<br />
conditions during <strong>the</strong> study and past actions, and <strong>the</strong>y should<br />
consequently be corrected to obtain a mean or pre-established<br />
situation.<br />
One needs to solve <strong>the</strong> prob<strong>le</strong>m by various channels as<br />
independently as possib<strong>le</strong>. In princip<strong>le</strong>, <strong>the</strong>y may be included<br />
in any of <strong>the</strong> following three large groups:<br />
a) Hydrometeorological methods.<br />
b) Geohydrochemical methods.<br />
c) Hydrodynamic methods.
Details of <strong>the</strong>se methods will not be discussed in this<br />
paper as <strong>the</strong>ir general lines are well known. For fur<strong>the</strong>r<br />
details, <strong>the</strong> reader may consult <strong>the</strong> two volume text:<br />
"Hidrologia Subterránea" coordinated by M. R. Llamas and E.<br />
Custodio, at present being printed by Ediciones Omega, Barce-<br />
lona.<br />
The investigation and special methods are expressly<br />
excluded, since this is not <strong>the</strong> right place to discuss <strong>the</strong>m,<br />
but <strong>the</strong> studies to solve <strong>the</strong> real prob<strong>le</strong>ms raised and which<br />
require a prompt answer and an order of magnitude of <strong>the</strong>ir<br />
confidence. More delicate works can later be set up to find<br />
or affirm <strong>the</strong> basic estimations and hypo<strong>the</strong>sis.<br />
2.- DATA COMPILATION AND SYNTHESIS<br />
The data compilation and syn<strong>the</strong>sis work is necessary in<br />
any hydrological study, but in small basins with insufficient<br />
data, it assumes peculiar features, since it is frequently<br />
necessary to test all possibilities in various aspects.<br />
First, it should be defined <strong>the</strong> sort of is necessary data,<br />
to later define <strong>the</strong> search places where data can be found and<br />
finally establish <strong>the</strong> methodology of compilation and elaboration.<br />
When discussing <strong>the</strong> three factors defined in <strong>the</strong> introduction,<br />
will be specified what data is necessary, if <strong>the</strong>y already exist.<br />
The places where <strong>the</strong> data can be found vary from one country to<br />
ano<strong>the</strong>r, and from one place to ano<strong>the</strong>r and a list of <strong>the</strong>m would<br />
prove very tedious. The official centres officially in charge of<br />
filing and compiling certain types of data, and <strong>the</strong>ir publications<br />
should be permanently in mind. The consultation and help of local<br />
experts may prove essential, and also <strong>the</strong> water-well companies;<br />
fur<strong>the</strong>rmore one should not forget <strong>the</strong> local peop<strong>le</strong> as well,<br />
without whose collaboration many important aspects may pass<br />
unnoticed, and even essential data or also some time <strong>the</strong> main<br />
profited sources and wells.<br />
Except perhaps for very litt<strong>le</strong> developed areas, with abundant<br />
water resources, <strong>the</strong> local peop<strong>le</strong> have a noticeab<strong>le</strong>, often<br />
unconcious, know<strong>le</strong>dge of <strong>the</strong> local hydrology, of a qualitative<br />
nature, but which may be quantized and built up with adequate<br />
surveys. This method of obtaining data not only saves a lot of<br />
work and time, but perhaps is <strong>the</strong> only way of obtaining historic<br />
know<strong>le</strong>dge and erroneous conclusions, by building a logical<br />
structure on not well foundes basis.<br />
A good know<strong>le</strong>dge of <strong>the</strong> local idiosyncrasy and peop<strong>le</strong><br />
customs is needed for <strong>the</strong>se tasks, and <strong>the</strong>y should not be given to<br />
under-qualified peop<strong>le</strong> who raise suspicions and are not capab<strong>le</strong><br />
of handling, screening and correcting <strong>the</strong> information received.<br />
Generally speaking, <strong>the</strong> local inhabitants are not very<br />
willing in princip<strong>le</strong> to reveal <strong>the</strong>ir know<strong>le</strong>dge, out of fear<br />
79
80<br />
it may prejudice <strong>the</strong>m. The interviewer should be prepared<br />
to “waste time” in winning over <strong>the</strong>ir confidence and present<br />
<strong>the</strong> survey without <strong>the</strong>m noticing it, making notes discreetly.<br />
One should try to get <strong>the</strong> information to flow out on its<br />
own, just channelling it and loocking for <strong>the</strong> interesting<br />
details.<br />
It is generally difficult to pass judgement on <strong>the</strong> data<br />
obtained in this way and it requires a great critical sense,<br />
a good know<strong>le</strong>dge of <strong>the</strong> area and a continuous contrasting.<br />
The collaboration of <strong>the</strong> local inhabitants is most<br />
important in locating springs, bore-ho<strong>le</strong>s, wells, etc., and<br />
to establish <strong>the</strong> most important characteristics of <strong>the</strong>m. On<br />
<strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> local corporations and Town Councils are<br />
usually important sources of information.<br />
3.- - OBTAINING THE OBJECTIVES<br />
To obtain <strong>the</strong> objectives listed in <strong>the</strong> introduction, it<br />
is frequently necessary to set up a general water balance in<br />
homogeneous part ia1 areas , bearing in mind <strong>the</strong> limitations<br />
and inevitab<strong>le</strong> errors <strong>the</strong>y contain. One should not only see<br />
an equation between mean values in <strong>the</strong> word ”balance”, but<br />
also <strong>the</strong> possib<strong>le</strong> variations in <strong>the</strong> different values<br />
intervening and <strong>the</strong>ir interconnection (SC). This is specially<br />
important when <strong>the</strong> ground-water reservoir capacities availab<strong>le</strong><br />
are small in relation with <strong>the</strong> water volumes to be exploited<br />
annually, giving rise to accentuated seasonal effects.<br />
This raises <strong>the</strong> prob<strong>le</strong>m that in one of <strong>the</strong>se small areas<br />
where data is scarce and not very reliab<strong>le</strong>, an elaboration<br />
and definition is necessary, with a depth not common in <strong>the</strong><br />
case of large basins.<br />
One of <strong>the</strong> greatest unknown factors is usually <strong>the</strong><br />
infiltration and recharge to <strong>the</strong> aquifers , which should be<br />
estimated using <strong>the</strong> best methods availab<strong>le</strong>.<br />
4.- -- HYDROMETEOROLOGICAL METHODS<br />
The hydrometerological methods to establish water balance<br />
and define deep infiltration are <strong>the</strong> classical ones, except<br />
in arid or semi-arid areas, where a daily computation (10)<br />
must be made to avoid excessive errors in monthly data handling.<br />
(*) The autor is conscious of <strong>the</strong> limitations of <strong>the</strong> water<br />
balance but feels it is a very useful tool if <strong>the</strong> person<br />
handling it is aware of its restrictions and errors, and<br />
<strong>the</strong> variability of <strong>the</strong> involved magnitudes.
The pluviometry must be obtained through <strong>the</strong> usually<br />
scant stations availab<strong>le</strong>, which generally do not cover <strong>the</strong><br />
mountainous parts where <strong>the</strong> pluviometry is usually greater<br />
than in <strong>the</strong> lowlands.<br />
A first measure is to correlate <strong>the</strong> different stations<br />
and comp<strong>le</strong>te <strong>the</strong> series, trying to obtain a definition of<br />
areas with <strong>the</strong> same rainfall (quantity, distribution and<br />
intensity), making estimated altimetric and topographic<br />
corrections.<br />
Next, <strong>the</strong> graphs of accumulated deviations of <strong>the</strong><br />
pluviometry should be drawn, and <strong>the</strong>se will be <strong>the</strong> basis of<br />
<strong>the</strong> study on <strong>the</strong> springs and water courses discharge and<br />
water-<strong>le</strong>vel in <strong>the</strong> wells. With <strong>the</strong>se relations hips, <strong>the</strong><br />
conditions observed during <strong>the</strong> study will be changed into<br />
mean conditions or those conditions of particular interest<br />
in order to ascertain extent <strong>the</strong> pluviometric variations<br />
influence <strong>the</strong> ground waters.<br />
When estimating <strong>the</strong> surface runoff, <strong>the</strong> know<strong>le</strong>dge of<br />
<strong>the</strong> local peop<strong>le</strong> may provide interesting data helping <strong>the</strong><br />
morphological appreciations made. Frequently, local peop<strong>le</strong><br />
can tell <strong>the</strong> heights and frequences of water in <strong>the</strong> river<br />
beds under various circumstances, and thus draw an initial<br />
scheme of <strong>the</strong> system. When <strong>the</strong>re are permanent waters is<br />
frequent <strong>the</strong> presence of manufacturing or irrigation<br />
installations which use <strong>the</strong>m, and in this case <strong>the</strong>y are usually<br />
dimensioned for <strong>the</strong> base discharge or some figure slightly<br />
higher. A know<strong>le</strong>dge of this discharge and <strong>the</strong> user's remarks<br />
are of great importance, as it permits <strong>the</strong> characteristics<br />
of <strong>the</strong> surface hydrology to be reconstructed approximately,<br />
based on one or various river flow measurement campaigns in<br />
se<strong>le</strong>cted points. The absence of noticeab<strong>le</strong> surface uses by<br />
means of simp<strong>le</strong> derivations, may be a c<strong>le</strong>ar sign of temporary<br />
discharges.<br />
Rarely are <strong>the</strong>re homogeneous and well defined crops in<br />
<strong>the</strong>se basins, and frequently <strong>the</strong>re is forest, brush, bare<br />
rock areas and great slopes, and consequently <strong>the</strong> classic<br />
evapotranspiration estimations are not applicab<strong>le</strong>. Added to<br />
this is <strong>the</strong> rare availability of <strong>the</strong> necessary meteorological<br />
data, excepting some <strong>the</strong>rmometric station. Thorntwaite's<br />
method (9) may give an initial idea of <strong>the</strong> potential value.<br />
Successive balances based on an estimated field capacity,<br />
enab<strong>le</strong> <strong>the</strong> real evapotranspiration to be calculated by<br />
difference. In low pluviosity areas, with a high evapotrans-<br />
piration capacity, <strong>the</strong> errors may be very important and a<br />
value of <strong>the</strong> infiltration plus surface runoff below 10 or 20<br />
per cent of <strong>the</strong> annual mean pluviometry, may only give a mere<br />
indicative figure. In this case, o<strong>the</strong>r balance methods should<br />
be established.<br />
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82<br />
5. GEOCHEMICAL METHODS<br />
In studies where <strong>the</strong>re is insufficient data, <strong>the</strong> chernical<br />
characteristics of <strong>the</strong> ground water may be extremely useful,<br />
if <strong>the</strong>y are correctly interpreted.0ne of <strong>the</strong> chief advantages<br />
of <strong>the</strong> geochemical methods lies in <strong>the</strong> low variability of <strong>the</strong><br />
chemical composition of <strong>the</strong> ground waters, averaging <strong>the</strong><br />
annual and seasonal variations, and in <strong>the</strong> low cost of an<br />
overall indicative analysis if <strong>the</strong>re are sufficient points for<br />
<strong>the</strong> sampling. The interpretation however, is a delicate affair<br />
and should be made by an experienced person with sufficient<br />
know<strong>le</strong>dge of <strong>the</strong> local hydrogeology and geology.<br />
On <strong>the</strong> one hand, <strong>the</strong> geohydrochemical methods may help<br />
to establish <strong>the</strong> paternof <strong>the</strong> groundwater .flow, comparing <strong>the</strong><br />
analysis of various points of water, using graphs (mainly<br />
those of logarithmic vertical columns or Schoel<strong>le</strong>r's;<br />
triangular with three fields, or Piper's; and Stiff's modified<br />
polygonals) (8) and ionic indexes, helped if necessary by dis-<br />
persion diagrams (correlation between two chemical charac-<br />
teristics (8).<br />
From ano<strong>the</strong>r point of view, <strong>the</strong> chemical composition of<br />
<strong>the</strong> ground water may fur<strong>the</strong>r information on <strong>the</strong> recharge. For<br />
this, it should be admitted that <strong>the</strong> aquifer does not notably<br />
modify <strong>the</strong> salt contents of <strong>the</strong> infiltered water. To remove<br />
<strong>the</strong> possib<strong>le</strong> influence of solution and modifying phenomena<br />
such as ionic exchange, redox reaction aggressiveness to<br />
carbonates, precipitation etc., <strong>the</strong> chloride ion is taken as<br />
reference, which can only be changed by an addition of a new<br />
chloride ion by <strong>the</strong> aquifer. In alluvial aquifers, limestone,<br />
dolomite, etc., no important additions are expected if for<br />
from <strong>the</strong> sea.<br />
In this case, all <strong>the</strong> chloride of <strong>the</strong> ground water would<br />
come from rain, and <strong>the</strong>refore we can state: (2) (4) (5) (8):<br />
(P - E) . Ca I . Cs<br />
Where:<br />
lr<br />
P annual mean pluviometry<br />
E = annual mean surface runoff<br />
I = annual mean deep infiltration<br />
Ca mean concentration in rain water chloride<br />
C, = concentration in ground water chloride<br />
It can be easily deduced that:<br />
I (F - E) cs
Some care is required when applying <strong>the</strong> method. One is<br />
that <strong>the</strong> activities on ground surface should not modify <strong>the</strong><br />
chloride contribution. The method <strong>the</strong>refore has a dubious<br />
application in intensive crop areas with irrigation or in<br />
areas with disposal and infiltration of important amounts of<br />
direc.t residual waters or though tippers, etc., and also in<br />
immediate coastal areas with direct sea influence.<br />
The Cs value may easily be obtained from <strong>the</strong> ground<br />
water analysis provided this is stab<strong>le</strong>. If not, <strong>the</strong> method<br />
should not be applied. The value of Ca is not normally known,<br />
as it is rare to find systematic analysis of <strong>the</strong> rainwater;<br />
during <strong>the</strong> study some analysis of this rainwater may be made,<br />
but before taking a content as <strong>the</strong> mean value, various deter-<br />
minations must be compared, since this content varies each<br />
season according to <strong>the</strong> origin of <strong>the</strong> clouds and even within<br />
a same rainfall. During initial estimation attempts, it may<br />
be considered that in areas several dozen km. away from <strong>the</strong><br />
sea, <strong>the</strong> chloride content is generally <strong>le</strong>ss than 10 ppm, and<br />
that in areas some hundred kms from <strong>the</strong> sea, it is <strong>le</strong>ss than<br />
1 ppm (8).<br />
In areas near <strong>the</strong> coast, <strong>the</strong> variations and contents may<br />
be higher. Close to populated areas, in particular if <strong>the</strong>se<br />
are industrial, high values may also occur.<br />
The solub<strong>le</strong> salts brought along by <strong>the</strong> infiltrated water,<br />
may not only come from <strong>the</strong> rain, but also from <strong>the</strong> atmospheric<br />
dust and this is ano<strong>the</strong>r reason for doubt. In princip<strong>le</strong>, <strong>the</strong><br />
rainwater col<strong>le</strong>ctor units should also col<strong>le</strong>ct <strong>the</strong> atmospheric<br />
dust, but at a sufficient height to avoid <strong>the</strong> local partic<strong>le</strong><br />
movement at low <strong>le</strong>vel.<br />
Experience shows that <strong>the</strong> method is good in arid areas,<br />
where <strong>the</strong> rain concentration due to evaporation is high and<br />
<strong>the</strong> surface runoff is scarce. The result is more prob<strong>le</strong>matic<br />
in <strong>the</strong> more humid areas where <strong>the</strong> infiltration is an important<br />
fraction of <strong>the</strong> pluviometry and where <strong>the</strong> surface runoff is<br />
notab<strong>le</strong> and errors in estimation greatly influence <strong>the</strong> P - E<br />
value. However, in <strong>the</strong>se areas it is possib<strong>le</strong> to apply <strong>the</strong><br />
salt balance in order to separate <strong>the</strong> components of <strong>the</strong><br />
hydrogram of a gauging station, if a sufficient chemical<br />
analysis series is availab<strong>le</strong> during a rainfall and <strong>the</strong> later<br />
period, but <strong>the</strong> author has no direct experience in such cases<br />
(12) (15).<br />
6. HYDRODYNAMIC METHODS<br />
The hydrodynamic methods try to determine <strong>the</strong> infiltration<br />
based on <strong>the</strong> hydraulic characteristics of <strong>the</strong> aquifer and <strong>the</strong><br />
piezometric surface. The most correct way of making <strong>the</strong> balance<br />
is using a simulation model, but this is generally a detai<strong>le</strong>d<br />
study phase and requires a notab<strong>le</strong> amount of data (13).<br />
83
The methods given herein refer to simp<strong>le</strong> situations within<br />
<strong>the</strong> study area with a well defined piezometric surface and<br />
with a pattern and slope which scarcely varies throughout <strong>the</strong><br />
year, so that an almost stationary situation can be imagined,<br />
with a well differentiated recharge and drainage area. The<br />
method means that <strong>the</strong> flow of water per unit of transversal<br />
width is equal to <strong>the</strong> mean recharge upwards. The application<br />
means having <strong>the</strong> mean transmissivity of <strong>the</strong> aquifer in <strong>the</strong><br />
analysis area obtained by means of some pumping tests and<br />
bore-ho<strong>le</strong>s and that <strong>the</strong> piezometric surface has been observed<br />
in a sufficient number of points to precisely know <strong>the</strong> mean<br />
gradients. The estimation is a mere application of Darcy's law.<br />
q T. i.<br />
where q = discharge per unit width<br />
T = transmissivity<br />
i = piezometric gradient<br />
Darcy's law is generally valid in most normal circumstances.<br />
7. EXAMPLES<br />
To illustrate <strong>the</strong> above, three examp<strong>le</strong>s have been chosen,<br />
corresponding to studies in areas of <strong>le</strong>ss than 100 Km2, one in<br />
a semi-humid area, ano<strong>the</strong>r in a semi-dry area and <strong>the</strong> o<strong>the</strong>r in<br />
a sub-desert climate. To better locate <strong>the</strong> data, <strong>the</strong> examp<strong>le</strong><br />
has been broken down into multip<strong>le</strong> paragraphs:<br />
7.1 Montroig Area<br />
Location.- S.W. of Tarragona, in <strong>the</strong> Baix Camp (fig. 1)<br />
Physiographic characteristics.- Flat coastal strip, 4 km<br />
wide, bordered by <strong>the</strong> mountain range. The water divide line<br />
is 12 km from <strong>the</strong> sea (1) (3) (5).<br />
Geological characteristics.- Plain of detritic materials<br />
resting on clay formations. Mountain range materials are of<br />
low permeability (1) (11).<br />
Water exploitation.- Traditional use for irrigation. Near<br />
<strong>the</strong> coast, pumping of 10 m3/year approximately, for supply of<br />
<strong>the</strong> Vandellos Nuc<strong>le</strong>ar Station. New extractions for Tarragona<br />
are ready to start in a short time (3) (5).<br />
Basic prob<strong>le</strong>m.- Find out <strong>the</strong> resources and sea intrusion<br />
process when new well will start pumping.<br />
Existing data.- Scarce, reduced and partial hydro-<br />
meteorological data. Mean pluviometry 400-500 mm in <strong>the</strong> plain,<br />
higher in <strong>the</strong> mountain range (1). Almost non-existant hydrological<br />
data; <strong>the</strong>re are no permanent water courses. The existing ones
are short-lived dry creeks. Contributions are estimated be<br />
means of local surveys. Almost non-existant hydrogeological data<br />
prior to <strong>the</strong> studies for <strong>the</strong> Vandellós Nuc<strong>le</strong>ar Station; later,<br />
values of <strong>the</strong> transmissivity of <strong>the</strong> piezometric gradients in a<br />
reduced area were availab<strong>le</strong>. The aquifers drain directly into<br />
<strong>the</strong> sea (1) (5) (11).<br />
Hydrometeorological balance.- Of dubious value owing to<br />
incertitude of data and low infiltration (2).<br />
Geohydrochemical balance.- Good application conditions in<br />
non-irrigab<strong>le</strong> land areas. There is no direct data on <strong>the</strong><br />
chloride content of <strong>the</strong> rain water, but this can be obtained<br />
by comparison with similar areas with data (2).<br />
Hydrodynamic balance.- Ideal conditions for application,<br />
but <strong>the</strong> interpretation of <strong>the</strong> pumping tests becomes comp<strong>le</strong>x (5).<br />
Results.- The mean recharge obtained by each of <strong>the</strong> three<br />
methods was as follows, in thousands of m3 per year, per km<br />
of coastline$: ( 2).<br />
a) Hydrometeorological method ....... 600<br />
b) Geohydrochemical method .......... 900<br />
c) Hydrodynamic method .............. 1.100<br />
Method a) foresees a progressive marine intrusion; method<br />
b) foresees a critical situation and method c) a certain<br />
residual flow to <strong>the</strong> sea, which would stabilize <strong>the</strong> salt water-<br />
fresh water interfacies in a new position.<br />
Check-ups.- Two lines of piezometers were instal<strong>le</strong>d to<br />
control <strong>the</strong> sea intrusion and three water tab<strong>le</strong> e<strong>le</strong>vation<br />
recorders to control <strong>the</strong> <strong>le</strong>vels were instal<strong>le</strong>d. After over<br />
three years exploitation, <strong>the</strong> interphase movement is more in<br />
accordance with result c) than with <strong>the</strong> o<strong>the</strong>r two. The<br />
hydrometeorological method is excessively pessimistic. The<br />
geohydrochemical method is acceptab<strong>le</strong> in a first approximation,<br />
and <strong>the</strong> hydrodynamic method is <strong>the</strong> closest to reality (5).<br />
7.2. Riera de Carme Basin<br />
Location.- S. of <strong>the</strong> town of Igualada (Barcelona) (fig. 2).<br />
Physiographic characteristics.- It spreads over 100 km2<br />
betwen 250 and 900 m of altitude. The <strong>le</strong>ngth of <strong>the</strong> small river<br />
is 25 km.<br />
Geological characteristics.- The materials are clay, silt<br />
and limestone Eocene formations, resting on clay and chalk of<br />
<strong>the</strong> Keuper formation. The tectonic alteration is important.<br />
There are important travertine and calctuff formations (6).<br />
* In reality, a maximum and a minimum value was calculated.<br />
85
86<br />
Water exploitation.- The discharge of <strong>the</strong> Riera de Carme<br />
is notably regular. The main ground regulating reservoir are<br />
<strong>the</strong> alveoline limestones, discharging relatively important<br />
springs. The main spring discharges in Capellades, outside <strong>the</strong><br />
basin. There is an intense industrial use.<br />
Basic prob<strong>le</strong>m.- Some wells have been chil<strong>le</strong>d from which<br />
100 l/sec., are going to be pumped continuously. Find out<br />
whe<strong>the</strong>r it is possib<strong>le</strong> to obtain this discharge in dry seasons<br />
and what type of injuries will be produced to springs and water<br />
courses. The tests have been made in an extraordinarily humid<br />
season, and it is <strong>the</strong>refore essential to obtain <strong>the</strong> probab<strong>le</strong><br />
situation under o<strong>the</strong>r conditions.<br />
Existing data.- Various peripherical pluviometric stations,<br />
and only one interior one, at present out of service. Spacial<br />
distribution of <strong>the</strong> pluviometry relatively regular, around<br />
600 mm/year.<br />
The hydrological data is very scarce. With only one gauging<br />
station operating since 3 years ago. The runoff characteristics<br />
have been reconstructed, based on a inventory, survey of <strong>the</strong><br />
canals, seried gauging and comparison with o<strong>the</strong>r,basins. The<br />
normal basic discharge of <strong>the</strong> river is 400 l/sec., which should<br />
rise to 500 ì/sec., if <strong>the</strong> ground discharge to a nearby basin<br />
is considered. The hydrogeological data are almost non-existant,<br />
except a prolonged pumping test lasting for two months, and '<br />
various tests on bore-ho<strong>le</strong>s (6). Various springs have been<br />
regularly gauged and <strong>the</strong> data has been apparently satisfactorily<br />
comp<strong>le</strong>ted, by means of local surveys on <strong>the</strong> field and in<br />
factories.<br />
Hydrometeorological balance.- Not very reliab<strong>le</strong> since most<br />
of <strong>the</strong> basin has high slopes, with wood or brush, and with<br />
sometimes very permeab<strong>le</strong> materials.<br />
Geohydrochemical balance.-In <strong>the</strong> main springs area, ground<br />
water has 15 to 24 ppm in Cl-. The scarce rain water samp<strong>le</strong>s<br />
show chloride content betwen 5 and 10 ppm. The accuracy is<br />
very low. Direct surface runoff is not known with an adequate<br />
degree of confidence. Data is only indicative. Possibly <strong>the</strong><br />
chemical conditions for hydrogram components separation by<br />
means of salt balance discharges are optimum, but has not been<br />
made as <strong>the</strong> influence of <strong>the</strong> industrial discharges is not very<br />
well known. .<br />
Hydrodynamic balance.- The important variability of<br />
transmissivity conditions, of <strong>the</strong> main fractured aquifer and<br />
its comp<strong>le</strong>x arrangement, make estimations difficult. The best<br />
way is by a study of discharge recession curves in se<strong>le</strong>cted<br />
points.<br />
Results.- Estimation of <strong>the</strong> total infiltration in millions<br />
of m3/year, including <strong>the</strong> groundwater discharge outside <strong>the</strong><br />
basin (6):<br />
a) Hydrometeorological method ........ 10<br />
b) Geohydrochemical method ........... 18
c) Hydrodynamic method ........ ?<br />
d) Separation of hydrogram<br />
components ................. 15<br />
Check-ups.- No direct check-ups are made, but <strong>the</strong>y will be<br />
obtained after comp<strong>le</strong>tion of <strong>the</strong> study wiht <strong>the</strong> 2-month pumping<br />
test, and related observations.<br />
7.3. Famara M,assive<br />
Location.- N. of <strong>the</strong> Island of Lanzarote, Canary Islands<br />
(fig. -3).<br />
Physiographic characteristics.- Massive of over 600 m. in<br />
altitude, which forms a notab<strong>le</strong> cliff over <strong>the</strong> W. coastline.<br />
Spreads over 80 km2. Very scanty vegetation, almost sub-desert<br />
climat e.<br />
Geological charact.eristics.- Tahular hsalts, of more than<br />
1.000 m. thickness, buried cinder cones, very continuous and very<br />
litt<strong>le</strong> permeab<strong>le</strong> subhorizontal clay-like <strong>le</strong>vels (almagre].<br />
Water exploitation.- Reserve area of ground waters for<br />
supplying <strong>the</strong> capital. Col<strong>le</strong>ctions by means of gal<strong>le</strong>ries which<br />
penetrate deep into <strong>the</strong> massive, with horizontal drills and at<br />
present also some vertical ones, to increase <strong>the</strong> drainage. Discharge<br />
obtained 2 to 3 l/sec., at present temporarily increased to 20<br />
l/sec. (4)<br />
Basic prob<strong>le</strong>m.- Get to know <strong>the</strong> reserves of <strong>the</strong> massive and<br />
determine <strong>the</strong> exploitab<strong>le</strong> discharges, <strong>the</strong>ir rate and recession<br />
curves, Assess <strong>the</strong> possib<strong>le</strong> resources.<br />
Hydrometeorological data.- Sufficient rainfall network, except<br />
in <strong>the</strong> highest areas, where most of <strong>the</strong> low infiltration mus-t be<br />
produced. Compiling of data and detai<strong>le</strong>d elaboration by <strong>the</strong><br />
Hydrographic Study Centre In Madrid c"]. Mean rainfall below 200 &year.<br />
Hydrological data.- Absence of surface runoff except in<br />
strong storms. There are no direct availab<strong>le</strong>.<br />
Hydrogeological data.- Almost non-existent, except in <strong>the</strong><br />
gal<strong>le</strong>ries where <strong>the</strong>re is a data record and several deep exploration<br />
bore-ho<strong>le</strong>s. There are various small springs and oozes, of very<br />
fine or inappreciab<strong>le</strong> discharge, inventoried by <strong>the</strong> Public Works<br />
Geological Service, and with some chemical analysis.<br />
Hydrometeorological balance.- Of dubious Worth, due to <strong>the</strong><br />
highly arid climate and because many suppositions have had to be<br />
made. A daily balance calculates a mean infiltration of 3 mmLyear.<br />
There is possibly no recharge if <strong>the</strong> daily rainfall does not exceed<br />
20 to 40 mm., which only occurs a few times in a period of several<br />
years.<br />
By reason of <strong>the</strong> Scientific Study of Water Resources of <strong>the</strong> Canary<br />
Islands, made by <strong>the</strong> General Board of Hydraulic Florks of <strong>the</strong><br />
Spanish Government, and UNESCO.<br />
a7
88<br />
Hydrogeochemical balance.- Chloride content of <strong>the</strong><br />
infiltration water obtained from <strong>the</strong> analysis of <strong>the</strong> small water<br />
oozes on <strong>the</strong> almagres at high altitude (around 300 to 700 ppm.)<br />
and surface wells in Haría (900 ppm). The water of <strong>the</strong> gal<strong>le</strong>ries<br />
is more salty possibly due to basalt contributions by <strong>the</strong> high<br />
holding time. There are no direct data on chloride content in <strong>the</strong><br />
rainwater, but it can be estimated from <strong>the</strong> data of <strong>the</strong> island of<br />
Gran Canaria (4).<br />
Hydrodynamic balance.- Made with precautions from <strong>the</strong><br />
freatic surface obtained by a careful study of <strong>the</strong> data on <strong>the</strong><br />
bore-ho<strong>le</strong>s and gal<strong>le</strong>ries, and <strong>the</strong> hydraulic characteristics of<br />
<strong>the</strong> basalts obtained by various procedures (4). The hydraulic<br />
gradients at times exceed 10% per cent in a plane at O e<strong>le</strong>vation.<br />
Results.- The rechar e obtained by each of <strong>the</strong> three<br />
methods in thousands of m 5 /year, are (4):<br />
- Mean<br />
Min.<br />
Max.<br />
-<br />
-<br />
Famara Heights (19 Km2) .... 225 28 5 190<br />
Famara Lows (28 Km2) .... 140 224 84<br />
Marginal plains (25 Km2) . . , 50 75 25<br />
Total (72 Km2) ,. .. 415 584 299<br />
In this case, <strong>the</strong> best method would appear to be <strong>the</strong><br />
geohydrochemical one. There is no direct verification, but<br />
additional information is obtained by means of isotopes and<br />
ambient radioisotopes, apart from a study of <strong>the</strong> salinity of <strong>the</strong><br />
soil and dust, in elaboration,<br />
Co'p<strong>le</strong>mentary,- Since <strong>the</strong> explotation is mainly of<br />
reserves, it has been computed by hidrodynamic study methods of<br />
<strong>the</strong> recession curves of <strong>the</strong> gal<strong>le</strong>ry discharges, that <strong>the</strong> water<br />
yield should vary between 0,03 and 0,05. This figures, jointly<br />
with <strong>the</strong> o<strong>the</strong>r data allow to estimate exploitab<strong>le</strong> reserves in <strong>the</strong><br />
gal<strong>le</strong>ry area, betwen 20 to 60 million m3, The overall transmissi-<br />
vity is 100 m'/day, with a thickness between 200 and 500 m.<br />
8. CONCLUSIONS<br />
In areas with small infiltration in relation to <strong>the</strong><br />
pluviometry, <strong>the</strong> geohydrochemical method applied with<br />
precautions, is a very useful tool which can improve <strong>the</strong><br />
hydrometeorological balance method. In more humid areas, <strong>the</strong><br />
results are not so c<strong>le</strong>ar. The hydrodynamic balance is <strong>the</strong> best<br />
method but in some cases it needs appropriate conditions for<br />
application, and in any case, it requires numerous reconnoisance<br />
tests to determine <strong>the</strong> hidrodynamic characteristics of <strong>the</strong> aquifer.<br />
9. REFERENCES
1. Custodio, E., Molist J., and Martin Arnaiz, M. (1968).<br />
First Report on <strong>the</strong> works for supply of <strong>the</strong> Vandellós<br />
Nuc<strong>le</strong>ar Station. Geoteclinics Geokgists Consultants.<br />
Barcelona.<br />
2. Custodio, E. (1969). Report on <strong>the</strong> present state of<br />
<strong>the</strong> possibilities of <strong>the</strong> Montroig col<strong>le</strong>ctions to- supply<br />
water to <strong>the</strong> Vandellós Nuc<strong>le</strong>ar Station (internal report).<br />
3. Custodio, E., Bayo, A., and Orti, F., (1971). Geological,<br />
Hydrogeological and geochemical characteristics of <strong>the</strong><br />
coastal aquifers between Cambrils and L'Ametlla de Mar<br />
(Tarrag ona)i. -<br />
on Economic Geology, Madrid-Lisbon. Section 3. pp 1471170.<br />
4. Custodio, E., and Saénz de Oiza, J., (1972) Geohydrological<br />
study on <strong>the</strong> Famara Massive, Lanzarote. General Board of<br />
Ilydraulic Works. Las Palmas - Barcelona. 2U4 pp.<br />
5,<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13.<br />
89<br />
Fi rs t Span i s h - Portuguese -Amer i can Congr es s<br />
Custodio, E., and o<strong>the</strong>rs (1973). Final Report on <strong>the</strong><br />
works to supply <strong>the</strong> Vandellós Nuc<strong>le</strong>ar Station. Geotechnics<br />
Geologist Consultants, Barcelona (being elaborated).<br />
Custodio, E., and o<strong>the</strong>rs (1973). Geohydrological study of<br />
<strong>the</strong> Carme Basin. Barcelona. East Pyrenees Water Board<br />
and Public Works Geological Service. Barcelona.<br />
Custodio, E. (1973). Hydraulics of water col<strong>le</strong>ctions.<br />
Section 9 of Subterranean liydrology. Vol, 1.- Omega<br />
Editorial. Uarcelona (at printers).<br />
Custodio, E. (1973). Geo1iydrocliemistry.- Section 10 of<br />
Sub terruiieanz liydrology , Vol, 2, - Omega Editorial, Barcelona<br />
(at printers),<br />
Martin Arnaiz, M. (1973). Components of <strong>the</strong> Hydrological<br />
Cyc<strong>le</strong>, Section 6 of SUBterranean Iiydrology, Vol. 1.- Omega<br />
Editorial. Barcelona (at printers).<br />
Mero F. (1969). Approach to daily hydro-meteorological water<br />
balance computations for surface and groundwater basins.-<br />
Proceedings ITC - UNESCO Seminar on Integrated Surveys for<br />
River Basin Development.- Delft. pp 89/116.<br />
Orti , F. (1970). Notes on <strong>the</strong> hydrogeological prospecting<br />
made for supply of <strong>the</strong> Vandellós Nuc<strong>le</strong>ar Station (Tarragona)<br />
Geological Research Institute of <strong>the</strong> Provincial De<strong>le</strong>gation.<br />
Vol. XXIV, pp 75/88 Barcelona.<br />
Pinder, G.F. and Jones, J.F. (1909) Determination of <strong>the</strong><br />
ground water component of peak discharge from <strong>the</strong> chemistry<br />
of total runoff.- Water Resources Research, Vol 5. No 2<br />
April 1969 pp 438/445.<br />
Public Works Geological Service (1972). Basic Tiieory on<br />
analogical digital models of aquifers. (See especially<br />
chapter 5, Process of construction and use of a model<br />
by E. Custodio and L. Lopez-Garcia), Information and<br />
Studies, Bul<strong>le</strong>tin No. 37 - Public Works Geological Service,<br />
Madrid, October 1972. 178 pp.
90<br />
14. VilarÓ, F., Custodio, E., aiid Bruington, A.E., (1970)<br />
Sea Water intrusion and water pollution in <strong>the</strong> Pirineo<br />
Oriental (Spain). ASCE National Water Kesources Engineering<br />
Meeting, Memphis, Tennesse. Meeting Preprint 112.<br />
15. Visocky, A.P. (1970). Estimating <strong>the</strong> groundwater con-<br />
tribution to storm runoff by <strong>the</strong> e<strong>le</strong>ctrical Conductance<br />
method.- Ground water, Vo. 8. No. 2, March-April 1970.<br />
pp 5/10.
Fig.1 - Situdción del Area de Montroig<br />
Locat i on map of Fdontroi g Area<br />
91
92<br />
Fig. 2 - Sitiiacibri de la Cuenca del Carme<br />
Locat i on of Carme Has i n
Q Al cgranza<br />
93
METHODS OF ANALYSING DEFICIENT DISCHARGE DATA<br />
IN ARID AND SEMI-ARID ZONES FOR THE DESIGN OF SURFACE WATER UTILIZATION<br />
-____ ABSTRACT<br />
bY<br />
Joseph S. Dalinsky<br />
TAHAL - Water Planning for Israel Ltd., Tel Aviv, Israel<br />
Technion, Israel Institute of Technology, Haifa, Israel<br />
This paper surveys various methods of analysing stream flows:<br />
frequency of annual volumes, discharge-volume relationship with<br />
horizontal, vertical and doub<strong>le</strong> hydrograph cutting, and calcula-<br />
tion of <strong>the</strong> storage volumes availab<strong>le</strong> as a function of reservoir<br />
capacity.<br />
Application of <strong>the</strong>se methods, which have been successfully<br />
applied by Tahal-Water Planning for Israel Ltd. over <strong>the</strong> past '<br />
ten years, can generate data for <strong>the</strong> design of surface water<br />
utilization schemes when flow records are availab<strong>le</strong> for only<br />
a few years.<br />
The understanding and application of <strong>the</strong> general design<br />
aspects, even if only qualitative, enab<strong>le</strong>s <strong>the</strong> planning engineer<br />
to reduce his basic hydrological requirements to <strong>le</strong>ss than 10<br />
years duration.<br />
It is proposed that applied hydrological research be di<br />
rected towards evaluation of a number of important hydrologi-<br />
cal design parameters on a regional basis to enab<strong>le</strong> nondimen-<br />
sional curves to be established.<br />
RESUME<br />
L'auteur examine différentes méthodes pour l'analyse du<br />
débit des riviére: fréquence des volumes annuels-relation entre<br />
ces volumes et <strong>le</strong>s débits dérivés pour l'utilisation par tronca<br />
ture'des hydrogrammes, cette troncature pouvant etre vertica<strong>le</strong>,<br />
horizonta<strong>le</strong>, ou <strong>le</strong>s deux à la fois- calcul des volumes stockés<br />
disponib<strong>le</strong>s en fonction de la capacité du réservoir.<br />
Ces méthodes ont été utilisées avec SUCC~S par Tahal-Water<br />
Planning, pour Israël Ltd, au cours des dix derni2res années.<br />
Leur application permet de fournir des donnérs pour l'aménagement<br />
des eaux, lorsqu'on ne dispose de données di'écu<strong>le</strong>ment que<br />
pour un'petit nombre d'années.<br />
Une mise en oeuvre intelligenye des aspects généraux d'un<br />
projet, même sous une forme purement qualitative, permet à l'i2<br />
génieur de planification de se contenter, pour <strong>le</strong>s données hy-<br />
drologiques de base, de moins de 10 ans d'observation.<br />
L'auteur propose que la recherche hydrologique appliquée<br />
soit orientée vers l'estimation des parametres hydrologiques -<br />
important à la réalisation des projets. Une tel<strong>le</strong> étude doit<br />
être menée sur une base régiona<strong>le</strong> et déboucher sur l'établisse-<br />
ment d'abaques adimensionnels.
96<br />
INTRODUCTION<br />
The need for water in <strong>the</strong> arid and semi-arid zones is in most cases<br />
greater than <strong>the</strong> water resource potential, since <strong>the</strong>re are generally<br />
large areas of good soils that cannot be cultivated as a result of <strong>the</strong><br />
scarcity of water for irrigation.<br />
utilization is directed toward maximum exploitation of <strong>the</strong> lfmited re-<br />
sources at reasonab<strong>le</strong> cost.<br />
Hence, <strong>the</strong> planning of surface water<br />
The annual yield of surface water resources varies considerably as<br />
a result of <strong>the</strong> extremely non-uniform climatic conditions that prevail in<br />
arid and semi-arid zones. In many cases, <strong>the</strong> availab<strong>le</strong> source cannot by<br />
itself provide an adequate supply and o<strong>the</strong>r solution& must be found,<br />
Possib<strong>le</strong> solutioqs are as follows:<br />
(1) To recharge surface water to suitab<strong>le</strong> groundwater aquifers which<br />
would serve as long-term reservoirs.<br />
will be <strong>the</strong> varying annual volumes of surface water, in addition<br />
to <strong>the</strong> natural rep<strong>le</strong>nishment, whi<strong>le</strong> <strong>the</strong> output will be an ap-<br />
proximately constant annual rate.<br />
In this case <strong>the</strong> input<br />
(2) In <strong>the</strong> case of supply from surface reservoirs fed from intercep-<br />
tion of stream flows, this source can be integrated with some<br />
o<strong>the</strong>r certain or steady but limited source. In this case, in<br />
years of adequate flow from <strong>the</strong> <strong>le</strong>ss reliab<strong>le</strong> or variab<strong>le</strong><br />
source, <strong>the</strong> yield of <strong>the</strong> steady source is retained €or use in<br />
those years in which <strong>the</strong> yield of <strong>the</strong> variab<strong>le</strong> source is in-<br />
adequate or non-existent.
In such cases, planning should be based on <strong>the</strong> average flow which<br />
can be diverted and recharged, or stored in <strong>the</strong> surface reservoir under<br />
dif fereniconditions.<br />
The techniques proposed in <strong>the</strong> following are aimed at calculating<br />
<strong>the</strong>se average values as a function of planning parameters such as maximum<br />
diverted flow or maximum net capacity of <strong>the</strong> surface reservoir.<br />
Acquaintance with streamflow regimes is best acquired by study of<br />
hydrographs.<br />
<strong>le</strong>dge it provides.<br />
although detai<strong>le</strong>d time dependence of discharges (instantaneous discharges)<br />
are <strong>the</strong> subject of greatest interest; where daily discharges do not under-<br />
go rapid changes (e.g. in rivers, springs, and baseflows), monthly data<br />
may be sufficient.<br />
The more detai<strong>le</strong>d <strong>the</strong> information, <strong>the</strong> more exact <strong>the</strong> ùnow-<br />
Data of hourly or daily flow rates are important,<br />
Hydrograph study provides valuab<strong>le</strong> information on <strong>the</strong> flow regime<br />
and can <strong>le</strong>ad to diversified techniques of analysis.<br />
The following information on streamflows is essential for <strong>the</strong> plan-<br />
ning of utilization:<br />
- The average volume of annual flows (U 1, which represent <strong>the</strong><br />
ave<br />
stream water resources potential; <strong>the</strong> average annual feasib<strong>le</strong><br />
utilizab<strong>le</strong> flows is a portion of this value.<br />
- The stream's flow regime: Is <strong>the</strong> stream perennial, intermittent,<br />
or ephemeral?<br />
Does it have a significant base flow or only dis-<br />
continuous floods? What is <strong>the</strong> duration of flow or floods, <strong>the</strong><br />
yearly number of floods, and <strong>the</strong> interval between successive<br />
floods?<br />
97
98<br />
- Streamf low variability, which comprises variability within a season<br />
or a year and variability from one year to ano<strong>the</strong>r.<br />
Hydrograph analysis can provide <strong>the</strong> required information s&h as:<br />
monthly and annual flows and <strong>the</strong>ir frequencies; flow-duration curves;<br />
annual average flows in relation to diverted discharges - represented by<br />
horizontal and vertical hydrograph cuts; and annual average flows in rela-<br />
tion to possib<strong>le</strong> reservoir capacities.<br />
In many cases <strong>the</strong> planning has to be done whi<strong>le</strong> insufficient hydro-<br />
logical data are availab<strong>le</strong>.<br />
<strong>the</strong> limited data availab<strong>le</strong> to be used efficiently and <strong>the</strong>refore reduce to<br />
a minimum <strong>the</strong> period of records needed for planning purposes - very often<br />
to <strong>le</strong>ss than 10 years, if <strong>the</strong> period for which records are availab<strong>le</strong> can<br />
be taken as representative of climatic conditions.<br />
A. ANNUAL FLOOD BETURN PERIODS<br />
The techniques presented in this paper enab<strong>le</strong><br />
For practical purposes of surface water utilization planning, annual<br />
flood return periods can be computed by using <strong>the</strong> established formula:<br />
n + l<br />
TE- m<br />
where: T is <strong>the</strong> return period (in years);<br />
n is <strong>the</strong> number of annual flow data;<br />
... (1)<br />
m is <strong>the</strong> serial number of annual flow data arranged in descending<br />
order, by size.<br />
By using <strong>the</strong> above formula, return periods approximately equal to<br />
<strong>the</strong> period for which data are availab<strong>le</strong> can be reasonably evaluated.<br />
estimates of annual flows (order of magnitude) for longer return periode<br />
can be obtained by extrapolation on probability paper by using <strong>the</strong> points<br />
The
which were calculated according to formula (1) as plotting points; though,<br />
for practical purposes, rare annual flows are of litt<strong>le</strong> importance, if any,<br />
since in most cases, a project based on rare flows will not be economic,<br />
B. FLOW-DURATION CURVES<br />
Flow-duration curves express <strong>the</strong> average duration of occurring die-<br />
charges equal to or greater than given values (Q > Q ); or dischargea<br />
i- 1<br />
equal to or smal<strong>le</strong>r than given values (Qi 5 Q21e Schematic representation<br />
of flow-duration curv<br />
is given in Sketch 1.<br />
Sketch 1: Flow-Duration Curve - Schematic Representation<br />
The duration can be expressed as <strong>the</strong> average number of days per year<br />
on which <strong>the</strong> said discharges occur, or as <strong>the</strong> total number of days in n<br />
years (see illustration of flow-duration curves ‘for <strong>the</strong> Qishon stream in<br />
Fig. 1 in App. A), or as <strong>the</strong> relative duration (which is similar to<br />
relative frequency ) .<br />
The computed discharges can be hourly or daily averages, or averages<br />
for any period - in accordance with <strong>the</strong> aims of <strong>the</strong> analysis and <strong>the</strong><br />
nature of <strong>the</strong> data, In general, average daily discharges will be used<br />
when <strong>the</strong> daily discharge fluctuations are not appreciab<strong>le</strong>, or wnen <strong>the</strong><br />
representative changes are daily changes. For streams characterized by<br />
99<br />
,
100<br />
a flood flow regime - where <strong>the</strong> flows are of short duration and <strong>the</strong>re is<br />
no significant baseflow - average hourly discharge or averages for even<br />
shorter periods can be chosen. It is customary to express <strong>the</strong> relative<br />
duration in percentages (p).<br />
The area delimited by Lhe curve Q = f(p), when /Qdp or Z(Q x Ap), is<br />
equal to <strong>the</strong> average discharge of <strong>the</strong> stream.<br />
The relations can be easily and economically established when calcu-<br />
lations are made by computer, in many cases as by-product of <strong>the</strong> computer<br />
analysis of streamflow data. For planning purposes, <strong>the</strong> direct use of<br />
flow-duration relations or curves is not convenient and <strong>the</strong>ir use is limi-<br />
ted to assisting <strong>the</strong> computation of data needed for drawing<br />
represent <strong>the</strong> horizontal and vertical hydrograph cuts (as shown in <strong>the</strong><br />
following sections of this paper). It should be stressed that: (1) The<br />
flow-duration relations can represent <strong>the</strong> streamflow character; (2) These<br />
relations can be achieved with satisfactory accuracy in <strong>the</strong> zone of <strong>the</strong><br />
practical importance (<strong>the</strong> zone where relatively small or medium size dis-<br />
charges occur) using data of a relatively short period (few years, mostly<br />
<strong>le</strong>ss than 10 years).<br />
curves which<br />
C. AVERAGE ANIUAL FLOW IN RELATION TO MAXIMUM DIVERTED DISC-GE -<br />
HORIZONTAL CUT<br />
When stream diversion i8 considered, whe<strong>the</strong>r by gravity flow or<br />
pumping, <strong>the</strong> dependence of annual diverted flows on <strong>the</strong> maximum diverted<br />
discharge is computed using <strong>the</strong> historical data. The curve representing<br />
<strong>the</strong> dependence of <strong>the</strong> average annual diverted flows on <strong>the</strong> maximum diver-<br />
ted discharges can be considered as <strong>the</strong> stream's "visiting card".<br />
meaning of <strong>the</strong> diversion, from <strong>the</strong> hydraulic aspect, is that all <strong>the</strong><br />
The
discharges which are equal to or smal<strong>le</strong>r than a certain magnitude, (Q,),<br />
are being diverted (see Skbtch 2).<br />
<strong>the</strong> diffeTence [(Q,) - (Qd)max I will overspill, whi<strong>le</strong> <strong>the</strong> diverted dis-<br />
charge, Q, will be approximately constant, at <strong>the</strong> magnitude of about<br />
(Qd)maxg<br />
Qd i Diversion<br />
discharge<br />
When <strong>the</strong> discharge (Q,) exceeds (Q,),<br />
Sketch 2: Schematic Layout of Diverdion Sketch 3: Hydrograph Horiaontal Cut<br />
Q<br />
I<br />
From a hydrological point of view this means a "horizontal cut"'of<br />
<strong>the</strong> streamflow hydrographs (see Sketch 3).<br />
where:<br />
The "horizontai cut" can be expressed ma<strong>the</strong>matically as:<br />
Qi<br />
Qd<br />
(Qä'rnax<br />
is <strong>the</strong> streamflow discharge;<br />
is <strong>the</strong> diverted discharge;<br />
is <strong>the</strong> maximum diverted discharge.<br />
101
102<br />
For every maximum diverted discharge a certain volume can be diverted<br />
every year; for a period of n years - a series of n annual diverted volmes<br />
can be obtained, out of which <strong>the</strong> average annual diverted flow (Ud) can be<br />
calculated for each value of (Q )<br />
d max'<br />
Sketch 4.<br />
The function cd = f (Qdlmx has <strong>the</strong> shape illustrated schematically in<br />
When Qd -+ œ, <strong>the</strong>n * U e where U represents <strong>the</strong> stream poten-<br />
d ave' ave<br />
tia1 (average annual flows).<br />
The curve representing <strong>the</strong> dependence of Ü on (Q )<br />
d d max<br />
into three m in zones according to <strong>the</strong> tangent slopes:<br />
Zone I: AÜd/A(Qd)mx is relatively<br />
large and almost constant. The<br />
diversion will be most justified<br />
economically<br />
-<br />
Zone II: AUd/A(Qd)max quickly de-<br />
creases as (Q ) increases.<br />
d max<br />
This is a transition zone.between<br />
Zones I and III.<br />
-<br />
Zone III: AUd/A(Qd),, diminishes<br />
as (Qd)mx increases, and tends<br />
towards zero when (Q ) * m, In<br />
d max<br />
this zone, diversions are usually<br />
not worthwhi<strong>le</strong> o<br />
A<br />
5<br />
CI<br />
v<br />
-<br />
Sketch 4: Ud -<br />
can be divided<br />
I<br />
-H I<br />
Uave<br />
f [(Qd)max]
- Note:<br />
(i) The above relations can be easily obtained from <strong>the</strong> computer<br />
at small cost.<br />
They can also be calculated without a computer,<br />
in some cases easily (depending on <strong>the</strong> data).<br />
(ii) If <strong>the</strong> flow-duration curve is used, expressed by relative<br />
durations (p), and <strong>the</strong> average nupiber of flow days per year<br />
(ta) are given, <strong>the</strong>n üd = f [ (Qd),] can be calculated by<br />
<strong>the</strong> formula:<br />
-<br />
- -<br />
where:<br />
t is <strong>the</strong> average number of flow days per year;<br />
. . . (3)<br />
p is <strong>the</strong> relative duration (expressed as a fraction) of<br />
discharges equal to or greater than <strong>the</strong> appropriate Q.<br />
(iii) Althmgh <strong>the</strong> historical streamflow data will not be repeated<br />
in <strong>the</strong> future, <strong>the</strong> calculated diverted volumes represent<br />
fairi,y satisfactorily <strong>the</strong> amounts and distribution of <strong>the</strong><br />
expected volumes for practical purposes of planning.<br />
In planning streamflow utilization by diversion of flows up to a<br />
certain maximum diversion discharge, <strong>the</strong> above relations, based on a<br />
simp<strong>le</strong> "horizontal cut" of hydrographs, are widely used (see illustration<br />
of horizontal, vertical and doub<strong>le</strong> cuts of hydrographs in App. A, Fig. 2).<br />
The maximum diverted discharge is determined according to direct or in-<br />
direct economic considerations (for instance - <strong>the</strong> limitation of <strong>the</strong><br />
diverted discharges in order to minimize <strong>the</strong> inflow of sedimentary<br />
materials).<br />
When limitations exist with regard to small discharges,<br />
this method cannot be applied (see Section D).<br />
103
1 o4<br />
When water rignts refer to baseflows and/or discharges up to a minimm<br />
diverted discharge, Qo, (when Q > Q only Q is utilized), <strong>the</strong> computation<br />
O<br />
can be made by translating <strong>the</strong> pivot of <strong>the</strong> axes to a new starting point<br />
- - *<br />
(Q0; U >. In this case, new sca<strong>le</strong>s will Óe used: Qd (Qd - Qo) and<br />
O - U: = Üd - -<br />
Üo. If <strong>the</strong> point (Qo; U ) lies outside of Zone I, or at its<br />
edge - <strong>the</strong> best flows are already utilized, .even though this does not mean<br />
a priori that <strong>the</strong> proposed scheme will not De feasib<strong>le</strong>.<br />
O<br />
D. AVELUGE ANHUAI., FLOW IN RELATION TO DIVERTED DISCURGES - DOUBLE CUT<br />
When <strong>the</strong>re are limitations to <strong>the</strong> diversion of baseflow discharges,<br />
or any definite discharges <strong>le</strong>ss than a certain magnitude, <strong>the</strong> relation of<br />
average annual flows to diverted discharges cannot be computed by means of<br />
a simp<strong>le</strong> "horizontal cut",<br />
and horizontal cut, is required (see Sketch 5).<br />
In <strong>the</strong>se casesp a "doub<strong>le</strong> cut", i.e. a vertical<br />
Doub<strong>le</strong> Cut Horizontal Cut Vertical Cut<br />
Sketch 5: Schematic Representation of Doub<strong>le</strong> Cut<br />
Such a case will arise when baseflow is of undesirab<strong>le</strong> quality for<br />
diversion purposes - generally too saline; conversely, during flood flows<br />
tne water is of good quality and can be diverted up to a certain maximum<br />
value, (Qd)max*<br />
In this case <strong>the</strong> doub<strong>le</strong> cut - shown in Sketch 5-A - ie
105<br />
used, When from a certain discharge onwards tiie sediment concentration is<br />
undesirab<strong>le</strong> for artificial recharge or from <strong>the</strong> aspect of reservoir capa-<br />
city losses, and it is decided not to divert those discharges - a vertical<br />
cut for Q = (Qd)mx is used (see Sketch 5-C).<br />
Since during planning it is still unknown which Q and which (Q )<br />
d max<br />
will be se<strong>le</strong>cted, different combinations have to be examined. Tiiis can be<br />
done by establishing a series of curves which describe <strong>the</strong> relations be-<br />
tween annual average flows to maximum diverted discharges for different<br />
values of Q<br />
[Qo = constant]. This analysis has <strong>the</strong> disadvantage of being<br />
related to discontinuous values of Qo and <strong>the</strong> need for repeating <strong>the</strong> calcu-<br />
lations for each value of Q<br />
involved, many sets of curves'will be required).<br />
O<br />
(when <strong>the</strong> storage capacity of a reservoir is<br />
Such work is superfluous<br />
and can be limited to <strong>the</strong> calculation of only two curves - <strong>the</strong> horizontal<br />
cut curve and <strong>the</strong> vertical cut curve - if <strong>the</strong> following equation is used:<br />
- - -<br />
UA = UB - uc<br />
where:<br />
-<br />
UA = fl<br />
- UB = f2<br />
-<br />
... (4)<br />
Qo; (Qd)mxl, represent <strong>the</strong> doub<strong>le</strong> cut';<br />
(Q,),] o is established by means of a horizontal cut;<br />
U = f [Q ] , is established by means of a vertical cut.<br />
c 3 0<br />
It is possib<strong>le</strong> to calculate Ü<br />
A<br />
for any desired combination of (Q<br />
d<br />
)<br />
max<br />
and Qo by <strong>the</strong> use of <strong>the</strong> two curves (see Sketch 5-B and C).<br />
Each of <strong>the</strong><br />
functions Ü and Û can easily be calculated by computer. These functions,<br />
B C<br />
as calculated for <strong>the</strong> Qishon stream in Israel, are illustrated in App.&<br />
'pig. 2.
1 O b<br />
-<br />
Function Ü can easily be calculated from U*, without use of a compu-<br />
C<br />
ter, on <strong>the</strong> basis of a flow-duration curve, when <strong>the</strong> duration indicates<br />
<strong>the</strong> average number of days per year of any given discharge or discharges<br />
exceeding <strong>the</strong> given value.<br />
be expressed by:<br />
-<br />
where:<br />
In this case <strong>the</strong> relation between <strong>the</strong> two will<br />
ta and p as in equation (3)<br />
t* is <strong>the</strong> average number of days per year of a discharge of Qo or<br />
-<br />
more (see Sketch 3); t* = ta x (P)~,.<br />
E. THE USE OF THE VERTICAZ. CUT<br />
In addition to <strong>the</strong> contribution of <strong>the</strong> vertical cut curve for<br />
simplifying <strong>the</strong> doub<strong>le</strong> cut technique and its use for planning of di-<br />
versions with constraints of maximum discharges owing to sedimentation<br />
IQ0 (QdImxi Qd Q for Q 5 (Qd1-i Qd = 0 for Q ’ (Qd)maxla <strong>the</strong><br />
curves are used for calculating <strong>the</strong> average annual sediment concentra-<br />
tion or load.<br />
The average annual sediment load can be calculated using simul-<br />
taneously <strong>the</strong> vertical cut curve and <strong>the</strong> curve describing <strong>the</strong> relation<br />
of <strong>the</strong> sediment concentrations to flow discharges (mostly log-log rela-<br />
tions). The computation is carried out as demonstrated in Appendix B.<br />
The average annual volume of sediment load, (Us)ave, of <strong>the</strong> stream<br />
is calculated as:<br />
m<br />
... (6)
Accordingly, <strong>the</strong> average annual sediment concentration, ave(Cv), is<br />
calculated as:<br />
where :<br />
(AUVIj<br />
-<br />
i- (Uslave<br />
ave('v) 'ave<br />
.. (7)<br />
107<br />
-<br />
(ÜVli - (Uv)i-l, indicates <strong>the</strong> contribution of <strong>the</strong> discharges<br />
within <strong>the</strong> limits of Qi-i to Q, to <strong>the</strong> average annual flow;<br />
-<br />
(U<br />
v<br />
)<br />
i<br />
indicates <strong>the</strong> average annual flow from <strong>the</strong> vertical cut<br />
curve [for Q 2 Qil.<br />
[(Cv)avel, is <strong>the</strong> mean volumetric sediment concentration of <strong>the</strong> dis-<br />
charges within <strong>the</strong> limits of Qi-i to Q,.<br />
j indicates <strong>the</strong> intervals of <strong>the</strong> discharges (AQ), chosen for<br />
<strong>the</strong> calculation.<br />
It should be noted that since <strong>the</strong> significant reduction in reservoir<br />
storage capacity resulting from sedimentation in arid and semi-arid zones<br />
is mostly due to rare high rate floods, <strong>the</strong>re is a need for data of a<br />
relatively long period. In such cases, it is <strong>the</strong>refore recommended to<br />
use probability analysis in <strong>the</strong> evaluation of <strong>the</strong> frequencies. However,<br />
regional analysis supported by analysis of historical flood water marks<br />
for rough estimates of <strong>the</strong> "maximum historical floods" enab<strong>le</strong>s relatively<br />
short period data to be used for evaluating <strong>the</strong> expected average annual<br />
sediment load order of magnitude (in this case - channel sections with<br />
a stab<strong>le</strong> bed should be chosen; o<strong>the</strong>rwise large mistakes may occur as a<br />
result of marked changes in <strong>the</strong> channel bed).<br />
F. ANNUAL STORABLE FLOW AS A FUNCTION OF RESERVOIR CAPACITY<br />
The quantities of reservoir-stored streamflows which can be<br />
utilized depend on flows, net reservoir capacity, reservoir operation,<br />
and seepage and evaporation losses.<br />
When <strong>the</strong> reservoir is to be emptied every year (e.g. <strong>the</strong>re is<br />
a rainy season in which <strong>the</strong> flows are stored and a dry season when <strong>the</strong><br />
stored water is used), it is possib<strong>le</strong> to estimate <strong>the</strong> annual stored
108<br />
quantities according to annual flows and net reservoir capacity, as long<br />
as losses, at <strong>le</strong>ast during <strong>the</strong> rainy season, are small. When <strong>the</strong> expected<br />
losses are large, <strong>the</strong> possib<strong>le</strong> losses must be known or estimated before<br />
calculations can be made; however,this information is often not availab<strong>le</strong>.<br />
When losses during <strong>the</strong> rainy season can be disregarded, <strong>the</strong> reservoir<br />
can every year store quantities smal<strong>le</strong>r than or equal to its net capacity:<br />
in which<br />
Here:<br />
iL<br />
UR = - n<br />
i-1<br />
(URIi = ui when Ui 2 RN<br />
(UR)i = (%li when Ui - ' (%)i<br />
Ui<br />
indicates <strong>the</strong> annual streamflow in <strong>the</strong> ith year - when <strong>the</strong><br />
reservoir is on <strong>the</strong> channe1,and annual diverted flow - when<br />
<strong>the</strong> reservoir is off <strong>the</strong> channel;<br />
(%)i representing net reservoir capacity in <strong>the</strong> ith year;<br />
(U<br />
R<br />
)<br />
i<br />
is <strong>the</strong> amount of water stored in <strong>the</strong> ith year;<br />
-<br />
UR<br />
is <strong>the</strong> n years' average annual amount of water stored in <strong>the</strong><br />
-<br />
reservoir (whose average net capacity is %)<br />
n is <strong>the</strong> number of annual data (calculated by <strong>the</strong> use of ei<strong>the</strong>r<br />
observed, historical, or of reconstructed syn<strong>the</strong>tic data).<br />
The quantity ÜR is always smal<strong>le</strong>r than Uave, when Uave designates<br />
<strong>the</strong> average possib<strong>le</strong> annual inflows into <strong>the</strong> reservoir, such as average<br />
diverted streamflow or average streamflow.
Annual net reservoir capacity is defined as freel,,annual capacity<br />
up to maximum operational height (e.g. up to <strong>the</strong> spillway crest). When<br />
<strong>the</strong> reservoir is operated in consideration of a planned dead storage, net<br />
operational capacity is constant until <strong>the</strong> dead storage is rep<strong>le</strong>te with<br />
-<br />
sediment; i.e. (%Ii = RN = constant. Storage losses are dependent on<br />
two major factors: <strong>the</strong> volume of annually deposited sediment, and <strong>the</strong><br />
volume of water remaining in <strong>the</strong> reservoir at <strong>the</strong> end of each year (<strong>the</strong><br />
"remaining volume" is generally constant, owing to <strong>the</strong> reluctance to pump<br />
mud, except in reservoirs which store water for more than one year; <strong>the</strong><br />
case of such reservoirs 'is not dealt with in this artic<strong>le</strong>).<br />
The following should be noted:<br />
(a) The computations as described give approximate solutions.<br />
If losses (by seepage and/or evaporation) are relatively<br />
large, at <strong>le</strong>ast monthly water balances are required in order<br />
to calculate <strong>the</strong> stored inflows with reasonab<strong>le</strong> accuracy.<br />
(b) The amount,,of water which can be annually utilized also depends<br />
in each case on <strong>the</strong> operational regime of <strong>the</strong> reservoir and on<br />
<strong>the</strong> losses (<strong>the</strong>re is a difference between <strong>the</strong> utilized and <strong>the</strong><br />
stored amounts, since losses occur whi<strong>le</strong> <strong>the</strong> stored water is<br />
being utilized).<br />
(c) In arid and semi-arid zones - in many cases, due to <strong>the</strong> limited<br />
potential of <strong>the</strong> stream and <strong>the</strong> considerab<strong>le</strong> seepage and evapora-<br />
tion losses - surface water utilization is based on artificial<br />
recharge of aquifers. In such cases <strong>the</strong> reservoirs are used<br />
for regulating and silting purposes; <strong>the</strong>refore (UR)i can exceed<br />
<strong>the</strong> net reservoir storage capacity, as it is a product of a<br />
109
number of floods which entered <strong>the</strong> reservoir after it was<br />
emptied or partly emptied (after every flood <strong>the</strong> stored water<br />
is transferred to spreading grounds for artificial recharge).<br />
Principally, <strong>the</strong> calculations, <strong>the</strong> character, and <strong>the</strong> analysis<br />
of <strong>the</strong> relations between FR and RN are <strong>the</strong> same as dealt with<br />
in this Section.<br />
The average annual volume of stored water (uR) and <strong>the</strong> reservoir<br />
efficiency (ÜR/uaVe) as functions of average net reservoir capacity (i$)<br />
are shown schematically in Sketch 6. (The meaning of <strong>the</strong> different zones<br />
is as explained in Section C for Sketch 4)<br />
d<br />
Zond Zonelzone (UR/Uave><br />
A Zonalzone I Zone<br />
2-<br />
Rd -<br />
Sketch 6: Schematic Representation of U R = f (s) and (ÜR/UaVe) = F (s)<br />
The curves, which summarize <strong>the</strong> aforementioned influences, illus-<br />
trate <strong>the</strong> contribution of average net reservoir capacity (Q).<br />
-<br />
Here too,<br />
as in <strong>the</strong> analysis of <strong>the</strong> relations illustrated in Sketch 4, different<br />
zones of <strong>the</strong> curves can be discerned, characterized by <strong>the</strong> magnitude of<br />
<strong>the</strong> slopes of <strong>the</strong> tangents to <strong>the</strong> curves (AÜR/A% or A(ÛR/Uave)/A$).<br />
These slopes represent <strong>the</strong> marginal additions of <strong>the</strong> average annual<br />
quantity of storab<strong>le</strong> water for <strong>the</strong> addition of a unit of net reservoir<br />
capacity .
-<br />
It is characteristic that as % increases, AÜR/AQ decreases. For<br />
111<br />
high values of RN <strong>the</strong> value AÜR/A$ is small, as it represents rare flood<br />
flows; its reliability is <strong>the</strong>refore limited.<br />
An analysis of this kind is of great importance for preliminary<br />
estimates and/or feasibility calculations, since it makes it possib<strong>le</strong><br />
to find easily <strong>the</strong> approximate economic solution.<br />
Recent investigations made by <strong>the</strong> Surface Water Utilization Depart-<br />
ment of Tahal - Water Planning for Israel Ltd. prove that <strong>the</strong> relationship<br />
- -<br />
between UR and % can be approximately estimated on a regional basis using<br />
as a parameter <strong>the</strong> dimension<strong>le</strong>ss standard deviation (<strong>the</strong> ratio u /U<br />
u ave’<br />
where U is <strong>the</strong> standard deviation, and U is <strong>the</strong> average annual flow).<br />
U ave<br />
It was found that <strong>the</strong> ratio ouIU is, in many cases, of regional<br />
ave<br />
character. (The above-mentioned investigations have not yet been con-<br />
cluded and hence cannot yet be summarized).<br />
The function described above is illustrated in App. A, Fig. 3.<br />
G. THE COMPUTATION OF RN AND RN<br />
where :<br />
Annual net reservoir capacity can be computed from <strong>the</strong> equation:<br />
(%li = - ... (9)<br />
($>i<br />
is <strong>the</strong> net reservoir capacity at <strong>the</strong> end of <strong>the</strong><br />
ith year;<br />
(%)i-1 is <strong>the</strong> net reservoir capacity at <strong>the</strong> end of <strong>the</strong><br />
(i-i) th year;<br />
(Rs)i is <strong>the</strong> volume of <strong>the</strong> sediment trapped in <strong>the</strong><br />
reservoir during <strong>the</strong> ith year.
112<br />
The volume of <strong>the</strong> sediment deposits trapped in <strong>the</strong> reservoir during<br />
a certain year can be calculated from <strong>the</strong> equation:<br />
where :<br />
(CS) i<br />
(RsIi = (EVIi x Uix(T.E.Ii = - x Ui x (T.Ea)i ... (10)<br />
YS<br />
is <strong>the</strong> average concentration of <strong>the</strong> transported sediment<br />
during <strong>the</strong> ith year, by volume;<br />
(C ) is <strong>the</strong> average concentration of <strong>the</strong> transported sediment<br />
s i<br />
th<br />
during <strong>the</strong> i year, by weight (e.g. in p.p.m);<br />
YS<br />
ui<br />
is <strong>the</strong> average specific weight of <strong>the</strong> trapped sediment<br />
(generally approximately constant, depending on sediment<br />
qualities and reservoir operation);<br />
represents <strong>the</strong> reservoir inflow in <strong>the</strong> ith year;<br />
(T.E.Ii is <strong>the</strong> trap efficiency in <strong>the</strong> ith year - <strong>the</strong> portion of <strong>the</strong><br />
sediment which remains in <strong>the</strong> reservoir (if <strong>the</strong>re is any<br />
overspill, part of <strong>the</strong> sediment <strong>le</strong>aves <strong>the</strong> reservoir with<br />
<strong>the</strong> overspill).<br />
For a design period of n years, especially when <strong>the</strong> value of n is<br />
high (tens of Years), <strong>the</strong> total loss in reservoir storage capacity re-<br />
sulting from sedimentation can be calculated as:<br />
will be<br />
Therefore, <strong>the</strong> average annual loss in reservoir storage capacity<br />
-<br />
Rs = - (Rs)i = Uave x (T.E.1 x [ave(Cv)l ."a (11)<br />
n i=l
where :<br />
ave('v)<br />
is <strong>the</strong> average concentration, by volume,of <strong>the</strong> sediment<br />
deposited by <strong>the</strong> transported water at <strong>the</strong> reservoir<br />
location;<br />
(T.E.) ,<strong>the</strong> average trap efficiency.<br />
The average net reservoir capacity for a period of n years will be<br />
estimated as:<br />
where :<br />
Ro<br />
is <strong>the</strong> initial reservoir capacity.<br />
... (12)<br />
11 3<br />
It should be noted that since it is impossib<strong>le</strong> to predict <strong>the</strong> future<br />
annual flows, <strong>the</strong>re is no o<strong>the</strong>r practical possibility of evaluating <strong>the</strong><br />
net reservoir storage capacity. For practical purposes, <strong>the</strong> use of average<br />
net storage capacity (%) is sufficient.<br />
H. RECOMMENDED HYDROLOGICAL INVESTIGATIONS<br />
Hydrological investigations directed towards finding parameters<br />
which enab<strong>le</strong> non-dimensional curves to be established which represent<br />
<strong>the</strong> main functions discussed in this artic<strong>le</strong>, are recommended, especially<br />
on a regional basis.<br />
The reconstruction of such a regional syn<strong>the</strong>tic curve, even though<br />
not "scientifically accurate", will be of great assistance in planning<br />
surface water utilization schemes, and especially in planning <strong>the</strong> first<br />
stage of such schemes.
114<br />
BIBLIOGRAPHY<br />
This artic<strong>le</strong> is based on <strong>the</strong> experience gained in Tahal - Water<br />
Planning for Israel Ltd., in <strong>the</strong> last 20 years, &.on <strong>the</strong> Technical<br />
Reports published by Tahal in Hebr’ew, as also on <strong>the</strong> foliowing works.<br />
1. Kuiper, E., Water Resources Development, Buttexworths,<br />
London, 1965<br />
2. Lins<strong>le</strong>y, Ray K. and J. B. Franzini, Water Resources<br />
Engineering, McGraw-Hill Book Co., London, 1964<br />
3. Searcy, J. K., Flow-Duration Curves, Manual of Hydrology,<br />
Geological Survey Water Supply Paper 1542-A, Washington D.C.,<br />
1959
APPENDIX 13: CALCULATION OF AVERAGE ANNUAL SEDIMENT. VOLUME<br />
TRANSPORTED BY LOWER QISHON FLOWS<br />
li 5<br />
1. Streamflow hydrographs were used for preparing a %cirtical cut curve"<br />
representing <strong>the</strong> average annual values of flow (Uc) contributed by<br />
discharges up to any value of QI as explained in Sections D and E,<br />
and illustrated in Fig. 2 of App. A.<br />
2. Simultaneous data of sediment concentrations and instantaneous dis-<br />
charges, supplied by <strong>the</strong> Israel Jydrological Service, drawn on a<br />
log-log paper enab<strong>le</strong> <strong>the</strong> construction of a correlation line between<br />
<strong>the</strong> average sediment concentration and <strong>the</strong> instantaneous discharges.<br />
In order to be on <strong>the</strong> safe side, <strong>the</strong> line was removed toward <strong>the</strong><br />
higher concentrations (of each discharge) - see Fig. 4 of App. A.<br />
3.. The calculations are shown in detail in <strong>the</strong> following tab<strong>le</strong>.<br />
-
11 6.<br />
CALCULATION OF AVERAGE ANNUAL SEDIMENT LOAD<br />
EXAMPLE: LOWER QISHON STREAM (ISRAEL)<br />
-<br />
uc<br />
cs<br />
L CU. mf s ec<br />
-<br />
5<br />
O<br />
1<br />
3<br />
5<br />
1 .20<br />
10<br />
15<br />
--<br />
Total<br />
LEGElID:<br />
-<br />
MCMIY r<br />
3.0<br />
6.0<br />
8. O<br />
10.4<br />
II..<br />
12.0<br />
13. O<br />
Q = diccnarge<br />
3,O<br />
3.0<br />
2. o<br />
2.4<br />
1.0<br />
O. 6<br />
1.0<br />
13.0<br />
PPm<br />
400<br />
800<br />
1 , 200<br />
1,700<br />
2,300<br />
2,700<br />
-<br />
300<br />
600<br />
1 , O00<br />
1 , 450<br />
2,000<br />
2,500<br />
4,000<br />
5 x AÜc<br />
ave<br />
tonlyear<br />
APP.<br />
Sneet 2<br />
900<br />
1 , 800<br />
2,000<br />
3 , 480<br />
2,000<br />
1,500<br />
4,000<br />
15,680<br />
U2 = average annual flow volume related to Q calculated by<br />
vertical cut of hydrographc (from Fig. 2 of App. A)<br />
I -<br />
AUc = <strong>the</strong> interval of Uc contributed by discharge interval<br />
-<br />
Cs = average sediment concentration, by weight (from Fig. 4<br />
of App. A) , high values<br />
- -<br />
(Cs)ave = average CS for discharge interval<br />
4. The result obtained from <strong>the</strong> calculations shown in <strong>the</strong> above tab<strong>le</strong>,<br />
is that average annual sediment load transported by <strong>the</strong> Qishon stream-<br />
ilows amounts to about 16,000 ton. Assuming an average trap efficiency<br />
of 90 percent and sediment deposits specific weight of 1.5 ton per cu.m -<br />
<strong>the</strong> average annual value of sediment trapped and deposited in <strong>the</strong> planned<br />
reservoir will be about 10,000 cu.m per year ( 16~000x0'9 9,600<br />
1.5<br />
i0,OOO cu.m per year).
Rainy yerre - average: t* > 40 daye I<br />
---Ueriium reinfall years - averaec:<br />
20 1 t* 40 daye<br />
Dry year# - average: t* < 20 daye<br />
dischargee exceeding Q<br />
t* - Number OP daye with discharges cxceeding<br />
1.5 m3Jeec<br />
.,...u. Average for 1940141 to 1964165<br />
FIG. 1: FLOW-DURATION CURVES FOR QISHON<br />
- STREAM (ISUAEL)<br />
FIG. 2: HORIZûNTAL, VERTICAL AND DOUBLE<br />
CUTS OF HYDROGRAPHS OF QISHON<br />
STREAM (ISRAEL)<br />
117<br />
Appendìx A
118<br />
FLG. 3: THE DEPENDENCE OF THE AVERAGE STORABLE<br />
FLOWS AND THE STORAGE EFFICIENCY ON<br />
THE NET AVERAGE RESERVOIR CAPACITY AT<br />
THE UPPER QISHON (UPSTREAM THE HYDRO-<br />
METRIC STATION TO WHICH THE DATA OF<br />
FIG. 1 AND 2 REFER), ISML<br />
Amendix A
Appendix A<br />
11 9
ABSTRACT<br />
APPLICATION OF COUTAGNE'S AND TURC FORMULAS<br />
TO THE SOUTHERN MOZAMBIQUE RIVERS<br />
Emilio Eugénio D'Oliveira Mertens<br />
Joäo José Mimoso Loureiro<br />
Checking of Coutagne's and Turc formulas, was purposed to<br />
obtain values, though approximated, for <strong>the</strong> annual mean runoff<br />
of <strong>the</strong> several rivers at sou<strong>the</strong>rn Mozambique where few gauging<br />
stations exist. Therefore, measured rainfall and temperature<br />
values were col<strong>le</strong>cted from <strong>the</strong> meteorological and gauging stations,<br />
as well as <strong>the</strong> runoff values observed in <strong>the</strong> locations.<br />
We conclude from <strong>the</strong> results obtained that <strong>the</strong> application<br />
of <strong>the</strong>se ru<strong>le</strong>s has given us, with relative guarantee, <strong>the</strong> annual<br />
mean runoff values, with deviations inferior to 10% which can be<br />
considered as satisfactory.<br />
RESUME<br />
Les formu<strong>le</strong>s de Coutagne et Turc ont été utilisées pour<br />
obtenir des va<strong>le</strong>urs, même approximatives, de l'écou<strong>le</strong>ment moyen<br />
annuel pour <strong>le</strong>s différents f<strong>le</strong>uves de la région sud de Mozambique<br />
dans laquel<strong>le</strong> on ne dispose que d'un nombre très limité de<br />
stations de jaugeage.<br />
Les calculs ont St6 effectués à partir des va<strong>le</strong>urs des<br />
précipitations et des températures mesurées aux stations météorologiques<br />
et pluviométriques, ainsi que des va<strong>le</strong>urs des écou<strong>le</strong>ments<br />
observées à différentes stations.<br />
Les résultats obtenus montrent que l'application de ces<br />
deux formu<strong>le</strong>s donne, avec une précision relative, des va<strong>le</strong>urs de<br />
l'écou<strong>le</strong>ment moyen annuel. Les écarts sont inférieurs à lo%, ce<br />
qui peut être considéré comme satisfaisant.
The hidrological phenomenons o- greater interest, relating to <strong>the</strong> hidro-<br />
logical studies of a catchment area under consideration, are namely:-<br />
Rainfall<br />
Air temperature<br />
Relative humidity<br />
Evaporation<br />
Hidrometical records<br />
Flow discharges of streams and runoff<br />
Sediment discharges<br />
The main purpose of a certain hidrological study, consists on <strong>the</strong> deter-<br />
mination for each one of <strong>the</strong> observed actions, of <strong>the</strong> variability princip<strong>le</strong>s<br />
<strong>the</strong>reof at distinguished intervals, analogy princip<strong>le</strong>s of <strong>the</strong> phenomenon itself<br />
from site to site and of <strong>the</strong> correlation princip<strong>le</strong>s amongst <strong>the</strong> several pheno-<br />
menons.<br />
One of <strong>the</strong> basilar e<strong>le</strong>ments necessary for <strong>the</strong> planning of an economical<br />
development program is <strong>the</strong> know<strong>le</strong>dge of <strong>the</strong> value and distribution of its hidrg<br />
logical resources.<br />
In Mozambique, registration of <strong>the</strong> hydric resources has been facing<br />
great difficulties not only in what refers to <strong>the</strong> extension of <strong>the</strong> territory but<br />
also, and essentially, by lack of observations of <strong>the</strong> hidrological phenomenons,<br />
namely <strong>the</strong> runoff and flow discharges of rivers and water-sources.<br />
From a report presented by Dr. L. Turc on <strong>the</strong> 3rd.iiidrological Ehgeneer-<br />
ing Congress organized by <strong>the</strong> 'Societé iiidrologique de France', which took<br />
place in Argel, in 1954, we were suggested to follow <strong>the</strong> idea of verifying <strong>the</strong><br />
possibility in <strong>the</strong> application of Coutagne's and Rirc general ru<strong>le</strong>s, related to<br />
<strong>the</strong> Sou<strong>the</strong>rn Mozambique water-sources.<br />
2 - COUTAGNE'S AND TURC GENERAL RULES<br />
These general ru<strong>le</strong>s allow us to estimate, by simp<strong>le</strong> calculation, <strong>the</strong><br />
value of a catchment area's runoff deficit, provided that rainfall and tempe-<br />
rature are known.
2.<br />
Runoff deficit - is <strong>the</strong> difference between mean rainfall height<br />
123<br />
pertinent to a certain site in <strong>the</strong> water-source and <strong>the</strong> corresponding height<br />
to <strong>the</strong> flow discharge estimated at <strong>the</strong> referred site.<br />
2.1 - COUTAGNE'S GENERAL RULE<br />
Being :<br />
H = Mean rainfall height<br />
general ru<strong>le</strong> is<br />
E = Hficient rainfall height, that is, <strong>the</strong> height which transform<br />
itself <strong>the</strong>oretically, in <strong>the</strong> <strong>who<strong>le</strong></strong>, to runoff.<br />
D = Runoff deficit = H - E<br />
C = Runoff coefficient = 2<br />
H<br />
K = Coutape's constant<br />
2<br />
D = H - KH2 being E = KH<br />
and since C = E and D=H-E<br />
H<br />
Now as:<br />
C,D-H or C = KH<br />
H<br />
2<br />
(C - KH)2 = (c 1 - KH + (c 2 - KH 2)2 + (c 3 - KH 3) + ....**<br />
to minimize this sum, it will do equalizing zero to <strong>the</strong> first derivative:<br />
wherefore:<br />
(C 1 - KH 1) H 1 + (C 2 - KH 2) H 2 + .......= O<br />
CH<br />
CH-KH2 = O K = -<br />
H2<br />
From <strong>the</strong> above determination it is given <strong>the</strong> most probab<strong>le</strong> value for K.
124<br />
/3.<br />
2.2 - TURC'S GENERAL RULE<br />
L = Turc's constant<br />
P = Evaporation plus lost by percolation<br />
T =Mean temperature<br />
A = Constant<br />
being<br />
wherefore<br />
H<br />
'dFtJ2<br />
L2<br />
L = A + 25 T + 0,05 T3<br />
The author still precises that applying his general ru<strong>le</strong> in 254<br />
Catchment areas, considering A = 300, distributed towards every climate in <strong>the</strong><br />
world, it has been reckmed that values of 0 observed and calculated from <strong>the</strong><br />
referred general ru<strong>le</strong>, came out as to <strong>the</strong> undermentioned results:-<br />
or<br />
or ra<strong>the</strong>r<br />
in 53% <strong>the</strong> cal. D - me86 D < 40 mni<br />
in 43% <strong>the</strong> cal. D - meas. D < 0,l meas. D<br />
i,n 65% <strong>the</strong> cal. D - meas.D < 0,2 meas. D
4.<br />
3- REPORT ûF THE CONSIDERED LOCATIONS<br />
125<br />
Described hereyder are <strong>the</strong> considered locations at <strong>the</strong> Limpopo's<br />
(incl. E<strong>le</strong>phant's River), Incomati, Umbeluzi, Sabie and Usuto Rivers (D. 1)<br />
3.1 - ELEPHANT'S RIVER<br />
Location: Maçuço - Mozambique<br />
2<br />
Catchment area: 66.600 Km<br />
Mean rainfall height: 636 mm<br />
Mean temperature: 18,9OC.<br />
Observation years : 1944/45 to 1970/71.<br />
3.2 - LIMPOPO'S RIVER<br />
Location:Beit bridge - R.A.S.<br />
2<br />
Catchment area: i88.000 Km<br />
Mean rainfall height: 481 mm<br />
Mean temperature : 2OoC.<br />
Observation years : 1955/56 to 1963/64<br />
3.3 - LIMPOPO'S AND ELEPHANTS RIVERS<br />
Location: Vila Trigo de Morais - Mozambique<br />
2<br />
Catchment area: 340.000 Km<br />
Mean rainfall height: 541 mm<br />
Mean temperature: 20,2OC<br />
Observation years: 1951/52 to 1969/70<br />
3.4 - INCOMATI'S RIVER<br />
Location: Ressano Garcia - Mozambique<br />
2<br />
Catchment area: 21.600 Km<br />
Mean rainfall height: 832,2 mm<br />
Mean temperature: 18,8OC<br />
Observation years : 1955/56 to 1969/70<br />
./.
126<br />
3.5 - SABIfl'S RIVER<br />
Location: Machatuine - Mozambique<br />
Catchment area: 6.200 Km2<br />
Mean rainfall height: 766,4 mm<br />
Mean temperature: 20,7OC<br />
Observation years: 1955/56 to 1969/70<br />
3.6 - UMBELUZI 1 s RIVER<br />
Location: Goba - Mozambique<br />
Catchment area: 3.100 Km<br />
2<br />
Mean rainfall height: 820,3 mm<br />
Mean temperature: 21,6OC<br />
Observation years : 1955/56 to 1970/7i<br />
3.7 - MAPUTO'S RIVER<br />
Location: Sipofaneni - Swaziland<br />
2<br />
Catchment area: 12.903 Km<br />
Mean rainfall height: 83i,7 mm.<br />
Mean temperature : 22OC<br />
Observation years : 1958/59 to 1964/65<br />
3.8 - Therefore we get two distinguished groups in regarding to pluviosity<br />
and temperature:-<br />
- Limpopofs River Groue - with mean rainfalls between 450 and 650 mm<br />
and temperatures from 18OC to 2OoC;<br />
- Incomati's, Sabie, Umbeluziaid Usuto Group with mean rainfall<br />
values of 800 m. and temperatures higher than 2OoC.<br />
4 - APPLICATION OF COUTAGW'S GENERAL RULE<br />
We have tried Coutagne's general ru<strong>le</strong> for each one of <strong>the</strong> above<br />
groups and locations <strong>the</strong>rein.
4.1 - LIMPOPO'S RIVER CATCHMENT AREA<br />
127<br />
2<br />
The 340.000 Km of <strong>the</strong> Limpopo's River Catchment Area relating to<br />
2<br />
Vila Trigo de Morais' gauging station, include <strong>the</strong> 66.600 Km of Maçuço's<br />
gauging station at E<strong>le</strong>phants' River and <strong>the</strong> 188.000<br />
2<br />
Km pertinent to Beit<br />
Bridge location.<br />
4.1.1 - For <strong>the</strong> 27 observation years at E<strong>le</strong>phants' River we have reached to<br />
<strong>the</strong> following type of Coutagne's ru<strong>le</strong>:-<br />
D = H - 0,000055 H (1)<br />
The most probab<strong>le</strong> values for <strong>the</strong> measured runoff deficits (612,9 mm)<br />
and calculated ones (613,3 mm) differ in 4 mm to a mean deviation of _+ 8,7 mm<br />
and a mean observation error of 7,4 mm.<br />
Extension of this ru<strong>le</strong> for <strong>the</strong> availab<strong>le</strong> 66 rainfall observation<br />
years would be plainly acceptab<strong>le</strong> in view of <strong>the</strong> fact that for a period of 34<br />
years of which we own <strong>the</strong> closest possib<strong>le</strong> runoff estimatives, difference is<br />
kept for <strong>the</strong> measured and calculated deficit.<br />
4.1.2 - For <strong>the</strong> 9 observation years in Limpopo's River area at Beit Bridge we<br />
reached to <strong>the</strong> results hereunder, to Coutagne's ru<strong>le</strong>:<br />
2<br />
D = H - OJ00O031 H<br />
(II)<br />
recording <strong>the</strong> most probab<strong>le</strong> values of <strong>the</strong> measured runoff deficits (472,7 mm)<br />
and calculated ones (473,7 mm) being <strong>the</strong> mean deviation and each observation<br />
error of _+ 6,2 mm and 3,7 mm respectively.<br />
4.1.3 - Finally for <strong>the</strong> i9 observation years in Vila Trigo de Morais, situated<br />
after <strong>the</strong> confluence with Limpopo's and E<strong>le</strong>phants Rivers, Coutagne's ru<strong>le</strong><br />
presents us <strong>the</strong> following result:-<br />
2<br />
D = H - 0,000047 H<br />
(III)<br />
Measured and calculated runoff deficits have a similar probab<strong>le</strong>st<br />
value, but <strong>the</strong> mean deviation is increased in ,+ 10,l mm and mean observation<br />
error amounts to f 7,6 mm.<br />
./.
128<br />
/7.<br />
4.1.4 - It is <strong>le</strong>ft to determinate now an availab<strong>le</strong> Coutagne's general ru<strong>le</strong><br />
to <strong>the</strong> entire group of 55 observations, as <strong>the</strong> principal e<strong>le</strong>ments taken to<br />
its calculation - runoff coefficient (C) and mean rainfall (H) - are not<br />
dependent values on those of <strong>the</strong> referring catchment areas, same being consi-<br />
dered to <strong>the</strong> runoff deficit.<br />
Ordering <strong>the</strong> values of <strong>the</strong> observed mean rainfall, <strong>the</strong> undermentioned<br />
ru<strong>le</strong> is calculated (Q1):-<br />
2<br />
D = H - 0,000050 H (IV)<br />
Through <strong>the</strong> same comparative system, it was obtained to <strong>the</strong> measured<br />
runoff deficits, <strong>the</strong> value of 560,3 mm and for <strong>the</strong> calculated ones through<br />
<strong>the</strong> same ru<strong>le</strong> (IV) 56O,6 mm being 0,3 mm <strong>the</strong> difference <strong>the</strong>reof.<br />
Mean deviation of <strong>the</strong> measured and calculated values amounts to<br />
+ 9,l mm with a mean observation error for each one of 7,2 mm.<br />
-<br />
Seing that <strong>the</strong> values of medium rainfalls, relating to <strong>the</strong> observa-<br />
tion periods, are respectively of 636, 481 and 541 mm with a short difference<br />
from <strong>the</strong> medium normal rainfall, we may conclude that Coutagne's ru<strong>le</strong> of<br />
which coefficient is equal to 0,000050, can be applied to every catchment<br />
area of which medium rainfall is comprehended between 450 and 650 mm. However<br />
it is necessary to point out that its application, in view of <strong>the</strong> great ex-<br />
tensions that <strong>the</strong>y comprehend, cannot be considered as absolutely precise,<br />
except for mean values.<br />
Application of this ru<strong>le</strong> every year may <strong>le</strong>ad us to<br />
mistake, since it calculates a regular correlation amongst runoff and rainfall<br />
which is not precised in <strong>the</strong> practice because of <strong>the</strong> powerful stream of Lim-<br />
popo's River, namely before <strong>the</strong> confluence with E<strong>le</strong>phants' River.<br />
4.2 - CATCHMENT AREA'S GROUP OF INCOMATIIS, SABIE, UMBELUZI AND USUTO RIVERS<br />
2<br />
The 43,803 Km of this group comprehend all <strong>the</strong> rivers which drain<br />
off in Lourenqo Marques' Bay and are located in an area, mean altitudes of<br />
which excede <strong>the</strong> 800 m. and mean rainfall is estimated between 750 mm and<br />
850 mm.<br />
All <strong>the</strong> above catchment areas are neighbouring.<br />
4.2.1 - For <strong>the</strong> 15 observation years of <strong>the</strong> Incomati's River at Ressano<br />
Garcia rainfall station, we conclude from Coutagne's general ru<strong>le</strong> <strong>the</strong> next:<br />
2<br />
D = H - 0,000150 H (V)<br />
./.
129<br />
The mean deviation value amounts to 19,7 mm and <strong>the</strong> mean observation<br />
error to i5 mm for <strong>the</strong> measured and calculated runoff deficits of 737,3 and<br />
739,O mm respectively.<br />
4.2.2 - For <strong>the</strong> Machatuinels rainfall station of Sabie's River and Goba's<br />
rainfall station of Umbeluzi's River, respectively with 15 and 16 observation<br />
years pertinent to an identical period, Coutagne's general ru<strong>le</strong> figures like:<br />
D = H - 0,000131 H<br />
2<br />
2 (VI)<br />
D = H - 0,000145 H<br />
To <strong>the</strong> first location, measured and calculated runoff deficits are<br />
similar (684,7 mm) and mean deviation amounts to _+ 20,4 mm.<br />
At Umbeluzi's River, mean deviation amounts to <strong>the</strong> decreasing value of<br />
+ 181 mm and difference between <strong>the</strong> measured and calculated deficits amounts<br />
-<br />
to 718,s and 719,O.<br />
4.3.3 - For <strong>the</strong> 7 observation years at <strong>the</strong> Sipofaneni's rainfall station in<br />
Swaziland, <strong>the</strong> main confluent of Maputo's River, Coutagne's general ru<strong>le</strong> is<br />
as follows:-<br />
D = H - 0,000162 (VI1 )<br />
Measured values (717,3 mm) and calculated ones (717,O) differ from<br />
O,3 mm and mean deviation amounts to &17,3.<br />
4.3.4 - Similary to what has been done in Limpopo, it was arranged <strong>the</strong> group<br />
of 53 observation years (Q2) in regard to mean rainfall, which alters from<br />
500 mm to 1.540 mm and <strong>the</strong> exposed Coukagne's ru<strong>le</strong> is:-<br />
D = H - 0,000140 H2 (VIII)<br />
Difference from <strong>the</strong> results determined by measuring and obtained from<br />
this ru1.e (VIII) is of 1,2 mm with a mean deviation of f 19,3 mm and _+ 15,8 mm<br />
for <strong>the</strong> medium error of each observation.<br />
To this catchment area's group mean rainfall wherein exceeding 800 mm<br />
and mean small deviations qualifying same as of minor torrentiality, and,<br />
consequently, higher stream regularity, application of Coutagne's general ru<strong>le</strong><br />
more than granting us accurate values for <strong>the</strong> annual runoff deficits yet allow<br />
./.
130<br />
/9.<br />
us its application year after year.<br />
5. - APPLICATION OF TURC'S GENERAL RUlE<br />
Application of this ru<strong>le</strong>, such as formed by Turc, that is, considering<br />
A=300, could not be used but running <strong>the</strong> risk of forming gross estimate errors<br />
seing that in Mozambique, <strong>the</strong> catchment areas in study have great extensions,<br />
usually.<br />
5.1 - We have tried to both of <strong>the</strong> groups <strong>the</strong> application of <strong>the</strong> general ru<strong>le</strong><br />
formed by Turc, and have found <strong>the</strong> next following results:-<br />
5.1.1 - IJMPOW'S CATCHMENT AREA<br />
Difference calc. D - meas. D L<br />
II II L<br />
II II A<br />
II II L<br />
II 11 <<br />
II II L<br />
II II<br />
II II<br />
1<br />
Ls<br />
20 mm -<br />
40 mm -<br />
40 ~UW -<br />
0,Ol m.D -<br />
0,05 m.D -<br />
0,l m. D -<br />
0,l m. D -<br />
0,2 m. D -<br />
5.1.2 - INCOMATI'S, SABIE, UMBELUZI AND USUTO GROUP-(Q2)<br />
Difference calc. D - meas. D c 20 m -<br />
11 Il < 40mm -<br />
II II<br />
> 40 ìüiìì -<br />
11 II < 0,Ol m.D -<br />
II II < 0,05 m.D -<br />
11 II < 0,l m.D -<br />
II 11 > 0,l m.D -<br />
11 Il > 0,2 m.D -<br />
53%<br />
76%<br />
24%<br />
14%<br />
22%<br />
36%<br />
64%<br />
O<br />
3%<br />
66%<br />
34%<br />
1 5%<br />
50%<br />
84%<br />
16%<br />
5.2 - Percentages differ from <strong>the</strong> obtained values by Turc for his Group of<br />
254 catchment areas and considering <strong>the</strong> 64% (Limpopols Group) of events<br />
superior to 0,l of measured D, we are not ab<strong>le</strong>d to consider <strong>the</strong> ru<strong>le</strong> as<br />
applicab<strong>le</strong>.<br />
10%
lo.<br />
131<br />
Therefore it was tried to find a rectifying solution of <strong>the</strong> constants<br />
in function of <strong>the</strong> mean rainfall and annual mean temperature for every catch-<br />
ment area.<br />
Thus we have traced <strong>the</strong> graphic shown in D.2, consequence of<br />
succeeding considerations on <strong>the</strong> measured values.<br />
5.3 - Upon <strong>the</strong> above application of Turc's general ru<strong>le</strong>, established <strong>the</strong><br />
constant A from <strong>the</strong> graphic, we reached to <strong>the</strong> following results:<br />
5.3.1 - LIMPOPO'S CATCHMENT AREA (Q3)<br />
Difference calc. D - meas. D < 20 mm<br />
II 11 < 40 mm<br />
II II<br />
40mm<br />
11 II<br />
d 0,Ol m.D<br />
II II < 0,05 m.D<br />
11<br />
II<br />
11<br />
II<br />
II<br />
11<br />
< 0,l m. D<br />
> o,1 m. D<br />
> 0,2 m. D<br />
88%<br />
95%<br />
5%<br />
33%<br />
io%<br />
5.3.2 - INCOMATI'S, SABIE, UMBELUZI AND USUTO CATCHMENT AREAS (a)<br />
Difference calc. D -<br />
II II<br />
II<br />
II<br />
II<br />
II<br />
Il<br />
11<br />
II<br />
11<br />
II<br />
11<br />
11<br />
II<br />
10%<br />
O<br />
O<br />
meas. D < 20 mm - 43%<br />
< 40 iiìüi - 87%<br />
> 40 ïìüü - 13%<br />
¿ 0,Ol m.D - 25%<br />
< 0,05 m.D - 8%<br />
< 0,l m.D - 96%<br />
> 0,l m.D - 4%<br />
> 0,2 m.D - O<br />
5.4 - From <strong>the</strong> <strong>who<strong>le</strong></strong> of <strong>the</strong> 108 compared values, we conclude that in 87% of<br />
<strong>the</strong> cases <strong>the</strong> difference among <strong>the</strong> calculated and observed values does not<br />
exceed 40 mm and in 96% <strong>the</strong> difference does not exceed 0,l from <strong>the</strong> measured<br />
runoff deficit. Never<strong>the</strong><strong>le</strong>ss, <strong>the</strong> most re<strong>le</strong>vant results are that in 2% of <strong>the</strong><br />
cases <strong>the</strong> deviation does not exceed 0,Ol meas.D and 53% does not amount to<br />
O,O5 of <strong>the</strong> observed runoff deficit.<br />
. /*
132<br />
/li.<br />
6 - CONCLUSIONS<br />
In <strong>the</strong> water-sources situated at <strong>the</strong> South of Save's River, <strong>the</strong>re<br />
might be applied <strong>the</strong> Coutagne's and Turc general ru<strong>le</strong>s on <strong>the</strong> following way:<br />
and 700 nun.<br />
COüTAGNE'S GENERAL RULE:<br />
Catchment areas with annual medium rainfalls coaiprehend between 450<br />
2<br />
D = H - 0,000050 H<br />
for mean rainfall superior to 700 nun.<br />
TURC'S GENERAL RULE<br />
2<br />
D = H - o,000140 H<br />
Determining constant A from <strong>the</strong> graphic<br />
Utility in <strong>the</strong> application of <strong>the</strong>se ru<strong>le</strong>s becomes evident in view<br />
of <strong>the</strong> non-existence of gauging stations along <strong>the</strong> multip<strong>le</strong> water-sources in<br />
<strong>the</strong> area under consideration, meteorologic and rainfall stations taking <strong>the</strong>ir<br />
place instead.<br />
But, seing that for <strong>the</strong> inventorying of <strong>the</strong> hidrological resources<br />
is matter of extreme necessity <strong>the</strong> know<strong>le</strong>dge, though approximated, of <strong>the</strong><br />
annual mean rainfall, and yet because it has become evident through <strong>the</strong> appli-<br />
cation of <strong>the</strong> aforesaid ru<strong>le</strong>s that precise values amount to <strong>le</strong>ss than i%, we<br />
may say that have succeed with <strong>the</strong> reaching of our main purposes.
133<br />
D1
850<br />
800<br />
750<br />
700<br />
650<br />
6 O0<br />
55 o<br />
500<br />
450<br />
\<br />
\<br />
\<br />
\<br />
\<br />
\<br />
\ '<br />
\ i<br />
\ v<br />
35 O<br />
400<br />
C5 O<br />
500<br />
ln<br />
U<br />
L<br />
C<br />
m .-<br />
U - W<br />
u<br />
O<br />
P<br />
a<br />
.- E<br />
-I<br />
VI<br />
O<br />
.-<br />
a<br />
A (TURC2<br />
ABACO PARA A DEIERMINAFAO<br />
DA CON5fANTE DE TURC
DETERMINAÇ~O<br />
DO COEFICIENTE DE COUTAGNE<br />
Rios Limpopo e E<strong>le</strong>fantes
DEIERMIN4ÇÁO DO COEFICIENTE DE COUTPGNE<br />
Rios Limpopo e E<strong>le</strong>fantes<br />
1
DETERMINPÇÃO DO COEFICIENTE O€ COüTAGNE<br />
Rios Incomati, Sabié, ümùeltízi e Maputo<br />
OEFIC E NI<br />
e E SCGAIAI<br />
C<br />
-__<br />
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83,46<br />
448.900 107.20<br />
A98 329<br />
462.40Q<br />
74,47<br />
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463,761 48 ,LO<br />
467,856<br />
485,809<br />
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101.27<br />
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109.85<br />
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79 120<br />
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142.40<br />
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+?5 625<br />
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+12 __ 144<br />
+11 121<br />
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-22 484<br />
+.4 16<br />
- -<br />
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38<br />
DETERMIN4C$O DO COEFICIENTE DE COUT4GNE<br />
2
QUADRO COMPARATIVO DOS RESULTADOS OBTIDOS<br />
PELA FORMULA DE TURC E DA CONSTANTE TI,<br />
RADA DOÁBACO<br />
ZONA Bios Limpopo e E<strong>le</strong>fantes mor<br />
3
140<br />
QUADRO COMPARATIVO DOS RESULTADOS OBTIDOS<br />
PELA FORMULA DE TURC E DACONSTANTE TI-<br />
RADA DO ÁBACO<br />
ZONA Incomati, Sabi6, Umùelilzi e Naputo<br />
4
ABSTRACT<br />
MAPA1 HI DROLOGI CAL STUDY ( LIMPOPO ' S RI VER)<br />
EMILIO EUGENIO D'OLIVEIRA MERTENS<br />
JOÃO JOSE MIMOSO LOUREIRO<br />
Lack of observations in <strong>the</strong> flow discharges and runoff,<br />
taken at <strong>the</strong> future location of mapai's dam, have compel<strong>le</strong>d<br />
us to <strong>the</strong> essaying of diversed methodology viewing its<br />
obtention.<br />
It was se<strong>le</strong>cted <strong>the</strong> method of <strong>the</strong> specifica1 runoff<br />
technic which has conducted us to most consistant and<br />
significant results in conjunction with those observed and<br />
calculated for o<strong>the</strong>r locations at <strong>the</strong> catchment area.<br />
RESUME<br />
L'abscence des observations relatives aux débits et<br />
écou<strong>le</strong>ments measurab<strong>le</strong>s au futur lieu du barrage du mapai,<br />
nous a forcé d'essayer diverse méthodologie pour en obtenir.<br />
I1 a ;te choisie la method: de la technique dés débits<br />
specifiques que nous a conduit a des resultats tres concordants<br />
et significatives en conjunction avec ceux observés<br />
et calculés pour <strong>le</strong>s autres lieu du bassin versant.
142<br />
1 - CATCHMENT AREA<br />
1.1 - Site, area, relief and hydrography:<br />
The hidrographic basin of <strong>the</strong> Limpopo River has its major part in <strong>the</strong><br />
territories of South Africa, Rhodesia and Botswana, its area of 412 O00 km2<br />
being devided in <strong>the</strong> following manner (Drawing 1):<br />
South African Republic .................... 193 500 km2<br />
Rhodesia .................................. 66 O00 km2<br />
Botswana .................................. 73 O00 km2<br />
Mozambique ................................ 79 500 km2<br />
Rounded off, <strong>the</strong> catchment area is situated beyween 220 and 260<br />
South and 269 and 350 East, its highest altitude being 2.300 metres near <strong>the</strong><br />
city of Lydenburg.<br />
In National territory, situated between paral<strong>le</strong>ls 210 and 250 South and<br />
meridians 310 and 359 East, <strong>the</strong> basin has to <strong>the</strong> North, that of <strong>the</strong> River Save,<br />
to <strong>the</strong> South, that of <strong>the</strong> River Incomati and to <strong>the</strong> East, that of <strong>the</strong> River<br />
Govuro, and a coastal strip where a few closed catchment areas are found from<br />
which <strong>the</strong> water-sources accumulate in lakes.<br />
In Mozambique <strong>the</strong>re is no noticeab<strong>le</strong> irregularity, this occurring only<br />
in <strong>the</strong> limiting zone to <strong>the</strong> south of <strong>the</strong> Limpopo, in a reduced area with e<strong>le</strong>va-<br />
tions of 400 metres.<br />
In its total <strong>le</strong>ngth <strong>the</strong> average height is of 840 metres, its being 977,<br />
964 and 950 metres, respectively in Beitbridge, Mapai and Trigo de Morais.<br />
The average slopes of <strong>the</strong> course of <strong>the</strong> water are:<br />
Upper stream ....................................... 2, 50 dlan<br />
Central stream ..................................... 1,80 m/km<br />
Lower stream ....................................... 0,Og m/km<br />
The Limpopo is one of <strong>the</strong> most important rivers of South Africa and<br />
Mozambique and, as happens with <strong>the</strong> Incomati River, it is contained in <strong>the</strong><br />
lower part of <strong>the</strong> great drainage area, which includes more than half of <strong>the</strong><br />
Transvaal and a considerab<strong>le</strong> part of South Rhodesia.<br />
./.
143<br />
The Limpopo is a strange river, very changeab<strong>le</strong> and capricious, perhaps<br />
due to <strong>the</strong> influence of <strong>the</strong> dissimilarity of its hydrographical basin; its vol5<br />
me of water is extremely variab<strong>le</strong> as in dry wea<strong>the</strong>r it is very reduced and during<br />
<strong>the</strong> rainy season, reaches heights of 7 metres which flood large areas of ground<br />
in <strong>the</strong> central and lower courses. The Limpopo River, when it enters our territory,<br />
has already a definite bed, where it has three large tributaires: on <strong>the</strong><br />
right bank, <strong>the</strong> E<strong>le</strong>phants River, and on <strong>the</strong> <strong>le</strong>ft bank, <strong>the</strong> Nuanetzi and <strong>the</strong><br />
Changane; it is to <strong>the</strong>se that it owes its permanent volume of water for <strong>the</strong><br />
flow from those joining it, its principal supplier being <strong>the</strong> E<strong>le</strong>phants River, a<br />
water-source which crosses a region of high rains, its hydrographic basin having<br />
a somewhat impermeab<strong>le</strong> geological configuration.<br />
It belongs to <strong>the</strong> hydrographical system of <strong>the</strong> African Continent and it<br />
is of <strong>the</strong> torrential rate of permanent volume.<br />
The course of <strong>the</strong> water which takes <strong>the</strong> name of Limpopo River, is formed<br />
by <strong>the</strong> junction of <strong>the</strong> Marico and Crocodi<strong>le</strong> Rivers which have <strong>the</strong>ir sources at<br />
an altitude of 1.500 metres to <strong>the</strong> west of <strong>the</strong> city of Pretoria.<br />
The principal tributaries of <strong>the</strong> right bank, all with <strong>the</strong>ir sources in<br />
<strong>the</strong> Transvaal, from <strong>the</strong> source to <strong>the</strong> mouth of <strong>the</strong> Limpopo River are as follow:<br />
River Matablas , Pongola, Palala, Sand, Pafuri (flowing in close to Pafuri,<br />
already in Portuguese territory) and <strong>the</strong> E<strong>le</strong>phants River, <strong>the</strong> largest and most<br />
important which joins it within Mozambique after some 110 kms. On its <strong>le</strong>ft bank,<br />
<strong>the</strong> Limpopo receives large courses of water all with <strong>the</strong>ir sources in Rhodesia,<br />
<strong>the</strong> principal ones being:<br />
River Notwani, Macloutsie, Tuli, Umtzingwane, Bubye, Nuanetzi (which has<br />
already flown about 50 kms in Mozambique) and <strong>the</strong> River Changane.<br />
1.2 - Geological Aspect, Soils and Vegetation:<br />
In <strong>the</strong> Limpopo basin, formations are found which belong to different<br />
systems, such as Karroo, Waterberg, Primitive System.<br />
The basin in South African and Rhodesian territory seems to be constituted<br />
of basaltic lava, Serie Ecca, siliceous detrital rocks (sandstone) of brown<br />
red and purp<strong>le</strong> colours, formations of conglomerates, graphite and gneiss.<br />
In Mozambique, <strong>the</strong> basin is mainly constituted of sedimentary formations.<br />
In a narrow area near <strong>the</strong> border, volcanic rocks are found, in <strong>the</strong> upper<br />
course of <strong>the</strong> Limpopo River and E<strong>le</strong>phants River formations of <strong>the</strong> Cretaceous Era,<br />
in <strong>the</strong> rest, Quaternary formations with alluvium, sandstone, calcarium and sand<br />
deposits.<br />
./.
144<br />
The vegetation in foreign territory is mainly constituted of bush and<br />
grass, of great density in <strong>the</strong> highlands, and mixed bush and grass plains. In<br />
Mozambique, <strong>the</strong> vegetation is of <strong>the</strong> bushy type and plains with some trees,<br />
<strong>le</strong>vel grass plains and large stretches of grassy land.<br />
The predominant soils in our territory are: sandy in <strong>the</strong> coastal area,<br />
salty in <strong>the</strong> river va<strong>le</strong>s, soils of mananga in <strong>the</strong> lower Changane and conglome-<br />
rates.<br />
1.3 - Climate:<br />
In respect to <strong>the</strong> area situated in Mozambique, it appears that <strong>the</strong><br />
average annual temperatures are practically <strong>the</strong> same in almost all <strong>the</strong> basin,<br />
being 240 C with <strong>the</strong> exception of <strong>the</strong> north eastern side, where it goes as low<br />
as 220 C.<br />
On <strong>the</strong> coastal and north-eastern areas, <strong>the</strong> average maximum daily tem-<br />
peratures are 300 and 320. C and in <strong>the</strong> central area 34Q.C.<br />
The average temperature in <strong>the</strong> hottest month is 280 C. and <strong>the</strong> lowest<br />
260.c., <strong>the</strong> annual variation of <strong>the</strong>se averages being between 60 and 9Q.C.<br />
The average temperature in <strong>the</strong> coldest month is 20% in <strong>the</strong> central<br />
area, and 18% in thê rest, whi<strong>le</strong> <strong>the</strong> coastal area has an average minimum in<br />
<strong>the</strong> coldest month of 12%.<br />
The annual average relative humidity in <strong>the</strong> central area is 65%, increa2<br />
ing to <strong>the</strong> north and south to reach <strong>the</strong> highest rate of 75%.<br />
According to <strong>the</strong> classification of Koppen, <strong>the</strong> climate of <strong>the</strong> basin is<br />
in general <strong>the</strong> dryness of steppes with a dry season in winter, dryness of <strong>the</strong><br />
desert in <strong>the</strong> area of Pafuri, dryness of <strong>the</strong> steppes in <strong>the</strong> south of <strong>the</strong> basin,<br />
and in <strong>the</strong> coastal area, tropical raininess of a savanna.<br />
The predominant winds in <strong>the</strong> months of September to February are those<br />
from <strong>the</strong> East and, during <strong>the</strong> o<strong>the</strong>r months, almost entirely with predominance<br />
from <strong>the</strong> West.<br />
In <strong>the</strong> <strong>who<strong>le</strong></strong> basin, one finds that it is situated between <strong>the</strong> iso<strong>the</strong>rmics<br />
of 240 and 170 with <strong>the</strong> average temperatures of 200, 2003 and 2002 respectively<br />
for <strong>the</strong> areas of Beitbridge, Mapai and Trigo de Morais.<br />
1.4 - Hydrological Occupation:<br />
Both <strong>the</strong> South African Republic and Rhodesia have a network of udometric<br />
and hydrometric stations which, for <strong>the</strong> African Continent, can be considered<br />
dense: one pluviometer for 200 km2 and one hydrometric station for 4 o00 h2.
145<br />
There are readings from 31 Rhodesian udometric stations and from 90<br />
South African posts, <strong>the</strong> majority of which with more than 30 years of existence.<br />
The more significant hydrometric stations in Rhodesia and South African<br />
not only for <strong>the</strong> area <strong>the</strong>y cover and <strong>the</strong>ir locality, but for <strong>the</strong> extension of<br />
<strong>the</strong>ir records, are:<br />
Rhodes ia :<br />
South Africa:<br />
- River Tuli - 4 144 km2<br />
- Unzimgwane River - 2 533 km2<br />
- Bubye River - 8 029 lan2<br />
A3 MO7 - Eerste Poor - Groot Marico Rivier<br />
A2 M25 - Hardekool Bulti - Crocodi<strong>le</strong> River<br />
A5 MO2 - Vischgat - Palala River<br />
A5 MO3 - Oxenham Ranch - Limpopo River<br />
A7 MO4 - Beitbridge - Limpopo River<br />
A7 MO3 - Zamenkomst - Sand River<br />
A9 MO1 - Schuinshoogte - Luvuhu River<br />
- Liverpool - Olifants Rivier<br />
- (354) - Olifants Rivier<br />
- Manorv<strong>le</strong>i - Letaba River<br />
- Letaba Ranch - LetkaRiver<br />
- Driehoek - Blyde River<br />
- 8 588 h2 - 21 i09 lan2<br />
- 2 341 h2 - 97 850 km2<br />
-180 O00 h 2<br />
- 6 900 km2<br />
- 912 lan2<br />
- 42 352 km2<br />
- 27 928 km2<br />
- 668 km2<br />
- 4 716 h2 - 2 199 km2<br />
In Mozambique, <strong>the</strong>re are 33 udometric stations, some with a significant<br />
period of regular readings, and 12 stations for <strong>the</strong> measurement of water volu-<br />
me, some equipped with linmographs, <strong>the</strong> most significant of those with regular<br />
readings being those of Maçuço (66 O00 km2) and Tiobine (68 450 km2) on <strong>the</strong><br />
E<strong>le</strong>phants River; Vila Trigo de Morais (340 O00 km2), Pafuri (235 930 km2),<br />
Mapai (246 O00 km2), Mohambe (342 780 h2), João Belo (407 970 km2) on <strong>the</strong><br />
Limpopo River, and Chibuto (43 200 km2) on <strong>the</strong> Changane River.<br />
As regards evaporation, <strong>the</strong>re are 6 U.S. Class A Tina evaporemeters in<br />
Rhodesia, 20 Standard Symons in South Africa, and 5 U.S. Class A Tina evapore-<br />
meters and 7 Piche atmometers in Mozambique.<br />
./.
146<br />
Also in <strong>the</strong> Portuguese part of <strong>the</strong> basin are 3 lysimetric stations, 3 cli<br />
matological stations, 2 agronomic-climatological posts and 4 climatological posts.<br />
2 - RAINS<br />
2.1 - Introduction:<br />
From <strong>the</strong> analysis of <strong>the</strong> normal isohyetal map, it is noted that <strong>the</strong> basin<br />
is within <strong>the</strong> ishoyetal extremes of 400 and 1.500 mm., <strong>the</strong> monthly distribution<br />
of rainfall being divided in a deficient way throughout <strong>the</strong> year,with a concentrg<br />
tion of about 85% of <strong>the</strong> total during <strong>the</strong> months from October to March inclusive.<br />
To determine <strong>the</strong> average annual rainfall in <strong>the</strong> Limpopo basin as far as<br />
Mapai, three methods were used:- of <strong>the</strong> rainy districts; of <strong>the</strong> area of influes<br />
ce and <strong>the</strong> isohyetal figures.<br />
2.2 - The method of <strong>the</strong> rainy districts<br />
The South African Republic is divided into restricted zones by rains of<br />
average equality, certain rainy districts under <strong>the</strong> same princip<strong>le</strong> also dividing<br />
<strong>the</strong> areas of Rhodesia.<br />
In accordance with <strong>the</strong> udometric records for <strong>the</strong> period 1914/15 to 1963/<br />
/64 (50 years), <strong>the</strong> average annual rainfall figures were determined in respect<br />
to <strong>the</strong> basin, taking into account <strong>the</strong> fall in each district.<br />
From <strong>the</strong> period of 50 years, <strong>the</strong> average annual rainfall arrived at was<br />
579 mm, its having been 582, 566 and 560 mm respectively during <strong>the</strong> past 10, 15<br />
and 25 years.<br />
2.3 - Method of Area of Influence<br />
In accordance with <strong>the</strong> records existing for <strong>the</strong> period 1954/55 to 1963/<br />
/64 (10 years) and using <strong>the</strong> 71 udometric posts, <strong>the</strong> average annual rainfall was<br />
determined, <strong>the</strong> figure obtained being 514 mm.<br />
2.4 - Isohyetal Method<br />
With <strong>the</strong> normal figures - 30 years - recorded at <strong>the</strong> udometric posts,.<br />
isohyetal curves were interpolated, <strong>the</strong> areas between <strong>the</strong> adjacent isohyetal<br />
figures being <strong>the</strong>reafter measured.<br />
The average figure for <strong>the</strong> rainfall in <strong>the</strong> basin thus obtained was 504m.
2.5 - Correlation of <strong>the</strong> methods:<br />
147<br />
For <strong>the</strong> common period of 1954/55 to 1963/64 and 1955/56 to 1963/64 of<br />
which <strong>the</strong> average annual rainfall figures are availab<strong>le</strong>, <strong>the</strong> correlation between<br />
<strong>the</strong> two methods was determined, <strong>the</strong> following equations having been obtained:<br />
1954/55 to 1963/64 - x = 0,94y + 99<br />
1955/56 to 1963/64 - x = 1,02y + 62<br />
which give a lineal relation between <strong>the</strong>m, <strong>the</strong> rates of <strong>the</strong> correlation being<br />
very significant (0,98 and 0,96) <strong>the</strong> figures arrived at showing only small dif-<br />
f erences .<br />
2.6 - Resumé:<br />
ANNUAL RAINFfiLL<br />
Averages :<br />
3 - RUNOFF<br />
Period of 50 years ........... 579 mm<br />
Period of 25 years ........... 560 mm<br />
Period of 15 years ........... 566 mm<br />
Period of 10 years ........... 582 mm<br />
Rainiest year ................ 971 mm (1924/25)<br />
Rainiest year w/lOO year occrence<br />
........................ 1 101 mm<br />
Driest year .................. 355 mm (1963/64)<br />
Driest year w/lOO year OCCUT-<br />
rence ........................ 290 mm<br />
3.1 - E<strong>le</strong>mentary Princip<strong>le</strong>s:<br />
In view of <strong>the</strong>re not being any measurements of <strong>the</strong> volume of <strong>the</strong> Limpopo<br />
River in Portuguese territory, it was necessary to resort to comparative studies,<br />
taking as a basis <strong>the</strong> specific flowage in <strong>the</strong> various hydrometric stations,<br />
existing upstream in <strong>the</strong> region of Mapai.<br />
./*
148<br />
3.2 - Details of <strong>the</strong> Study<br />
Barrows, in his <strong>book</strong> "Water Power Ehgeneering" stated that "it is often<br />
possib<strong>le</strong> to consider, without serious error, that <strong>the</strong> specific volume of a river<br />
is similar to <strong>the</strong> successive contours along <strong>the</strong> same river".<br />
also advises that, whenever possib<strong>le</strong>, comparisons and corrections should be esta<br />
blished, not only in respect to <strong>the</strong> rainfall, but also to <strong>the</strong> altitude, slopes,<br />
constitution and <strong>the</strong> rock formations of <strong>the</strong> soil.<br />
The same author<br />
Based on <strong>the</strong> measurements made at <strong>the</strong> Hydrometric stations of <strong>the</strong> Repu-<br />
blic of South Africa and Rhodesia which cover 195 841 km2, 79,6% of <strong>the</strong> respective<br />
basin in <strong>the</strong> area of <strong>the</strong> barrage of Mapai, we took into account <strong>the</strong> specific run-<br />
off year by year and we calculated <strong>the</strong> specific runoff of <strong>the</strong> locality under<br />
study.<br />
Since 1963, efforts have been made to estimate <strong>the</strong> average annual runoff<br />
always using <strong>the</strong> methods of <strong>the</strong> specific volume but adopting various criteria.<br />
The average annual figure obtained for <strong>the</strong> period of 12 years was 3 095<br />
million cubic metres, which corresponds to <strong>the</strong> specific runoff of 12 580 m3<br />
k2 - 1 (chart attached).<br />
Thus we have:<br />
Study in 1963 ( 6 years) specific runoff 13 700 m<br />
3<br />
(k2)-1<br />
Study in 1965 ( 7 years) specific runoff 13 658 m3 (k2)-1<br />
Present study (12 years) specific runoff 12 583 m 3 (k2)-1<br />
If we observe <strong>the</strong> sequence of <strong>the</strong> years in which measurements existed<br />
for <strong>the</strong> 3 studies realised, it will be noted that <strong>the</strong> period of 12 years includes<br />
an excessively dry year (1963-1964) and one high runoff (1966-67) whi<strong>le</strong> <strong>the</strong>re<br />
were no measurements taken in 1965-66 at <strong>the</strong> fundamental station of study (Beit-<br />
bridge) due to <strong>the</strong> hydrometric station having been under water (floods in Februa-<br />
ry, 1966) and as a result of which, <strong>the</strong> figure now obtained is necessarily defec-<br />
tive.<br />
The 1970 study covering a period of 15 years, places <strong>the</strong> specific average<br />
3 2<br />
runoff as 13 O00 m (k )-l.<br />
For <strong>the</strong> 246 O00 km2 of <strong>the</strong> river basin, <strong>the</strong> figures obtained for <strong>the</strong><br />
average annual runoff in <strong>the</strong> region of Mapai are respectively as follow:<br />
1963 study .............. 3 370 million m 3<br />
1965 study .............. 3 360 million m 3<br />
1970 study .............. 3 200 million m 3<br />
Present study ........... 3 095 million m<br />
3<br />
./.
149<br />
Any of <strong>the</strong>se figures fall within <strong>the</strong> admissib<strong>le</strong> limits based on <strong>the</strong> rate<br />
of <strong>the</strong> runoff observed at <strong>the</strong> stations of Beitbridge and Vila Trigo de Morais,<br />
<strong>the</strong> average of which is respectively 0,015 and 0,026.<br />
Thus for a figure of 3 O00 million m3 and for <strong>the</strong> average rainfall of<br />
579 nun., runoff coefficient is 0,021, which is within <strong>the</strong> observed limits.<br />
Using <strong>the</strong> method of Coutagne only for <strong>the</strong> average annual figures, for<br />
a rainfall of 579 mm as <strong>the</strong> average over 50 years, we arrive at a runoff of<br />
2 969 10 6 m 3 and for 582 nun as <strong>the</strong> average for <strong>the</strong> past 10 years, <strong>the</strong> figure<br />
of 2 999,7 10<br />
6<br />
m<br />
3 .<br />
Observing and trying all <strong>the</strong>se ways and means, we shall adopt chart<br />
attached hereto for <strong>the</strong> annual runoff because, as <strong>the</strong>y arise from direct mea-<br />
surements, <strong>the</strong>y fall within all <strong>the</strong> estimated figures.<br />
3.3 - Monthly Distribution<br />
The monthly distribution is based on a hydrometric station in <strong>the</strong> Repu-<br />
blic of South Africa (Beitbridge), which already has years of sufficient read-<br />
ings, its area being much like that of Mapai.<br />
Thus we haïe:<br />
4 - FLOODS<br />
October ................ 0,5%<br />
November ................ 0,6%<br />
December ................ 4,6%<br />
January ................ 25,%<br />
February ................ 33,8%<br />
March ................ i8,i%<br />
April ................ 8,s<br />
May ................ 4,3%<br />
June ................ i,%<br />
July ................ 1,1%<br />
August ................ o, 9%<br />
-<br />
September ................ 0,3% 10%<br />
The River Limpopo is typically torrential and as such, not only dries<br />
during consecutive months as it is susceptib<strong>le</strong> to exceptional floods.<br />
Many records of <strong>the</strong> volume of floods have been compi<strong>le</strong>d in <strong>the</strong> Republic<br />
of South Africa since 1915, always based on specific volumes and <strong>the</strong>y obtained<br />
. /.
150<br />
measured details which extended to <strong>the</strong> area of Mapai, gave us <strong>the</strong> figures of<br />
16 925 m3/s (1933) and 12 792 m3/s (1966).<br />
Thus, using various formulas, one can estimate <strong>the</strong> volume of floods for<br />
return periods of 100 and 200 years.<br />
100 years 200 years<br />
Water affairs formula ............ 13 243 14 963 m3/s<br />
Mimoso Loureiro formula ........ 14 304<br />
18 375 II<br />
Ful<strong>le</strong>r formula ................. 19 763 21 284 11<br />
(Period 34 years and specific<br />
volumes measured)<br />
Larivail<strong>le</strong> formula ............... - 15 375 'I<br />
The estimate is thus very difficult and depends a great deal on <strong>the</strong><br />
type of barrage to be adopted.<br />
The hydrograph of a maximum flood was also determined, based on <strong>the</strong><br />
formula of Giandotti in <strong>the</strong> calculation of <strong>the</strong> times of concentration of <strong>the</strong><br />
peak and of <strong>the</strong> swell of <strong>the</strong> flood, and on <strong>the</strong> hydrographs of <strong>the</strong> floods record-<br />
ed at <strong>the</strong> border (Pafuri) during <strong>the</strong> years 1955, 1958, 1959, 1966 and 1967.<br />
It was verified that, in <strong>the</strong> flood of 1966, <strong>the</strong>re was agreement in <strong>the</strong><br />
calculated and observed times, because <strong>the</strong> calculation placed <strong>the</strong> figure at<br />
153 hours and from observation, at 150 hours for <strong>the</strong> time of concentration,<br />
<strong>the</strong>re being, however, a difference in <strong>the</strong> time of <strong>the</strong> swell of 600 with 724<br />
hours (sketch attached).<br />
5- Evaporation and Solid flows<br />
Using <strong>the</strong> details measured in Rhodesia, South Africa and Mozambique,<br />
we can place <strong>the</strong> resulting evaporation at Mapi as 1 344 mm. As to solid volumes,<br />
<strong>the</strong> figures are few and vary greatly; as for <strong>the</strong> rest, definitely, confix<br />
med by <strong>the</strong> comp<strong>le</strong>x composition of <strong>the</strong> hydrographic basin, as for <strong>the</strong> same volu -<br />
me one obtain 21,8 kg/s and 97,l kg/s, without any meaning.
0<br />
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151
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1
Relation of hydrological programs of <strong>the</strong> Center of Hydrographic Studies<br />
for comp<strong>le</strong>te studies of hydraulic resources with insufficient data<br />
Dr. Rafael HERAS<br />
1 LIS TA Lista las ochenta columnas de las fichas.<br />
The programme lists <strong>the</strong> eighty columns of <strong>the</strong> cards.<br />
2 LIS 65 Lista las ochenta columnas de las fichas, poniendo en la<br />
cabecera de la pagina las columnas numeradas y saltan-<br />
do página cada 65 fichas.<br />
It lists eighty columns of <strong>the</strong> card, putting on <strong>the</strong> heading<br />
of <strong>the</strong> page <strong>the</strong> numered colum and skipping a page after<br />
every 65 cards.<br />
3 LIFLA Lista fichas con flags.<br />
It lists carde with flags.
156<br />
4 LIMO1 Lista cinta de papel del limni'grafo.<br />
It lists ribbon of <strong>the</strong> limnigraphe.<br />
5 LICAN Listado de longitudina<strong>le</strong>s.<br />
Longitudinal listing.<br />
6 LIGUI Listado de transversa<strong>le</strong>s.<br />
Transversal listing.<br />
7 LGMET Listado de datos geográficos.<br />
Listing of geographical data.<br />
8 LPMET Lista la cuenca, número de estación, aflo y datos de prg<br />
cipitaciones mensua<strong>le</strong>s.<br />
It lists <strong>the</strong> basin, number of station, year and monthly<br />
rainfall data.<br />
9 LFRNT Lista cinta de papel perforada en código FERRANTI.<br />
It lists perforated ribbon in FERRANTI code.
10 LI-C-CA Lista cabeceras de los cana<strong>le</strong>s de aforos.<br />
It lists <strong>the</strong> headings of <strong>the</strong> channels of valuation.<br />
11 L-NI-24H Lista precipitaciones máximas en 24 horas.<br />
It lists <strong>the</strong> maximum rainfall in 24 hours.<br />
12 LTVP Lista tablas de vertidos probab<strong>le</strong>s,<br />
It lists tab<strong>le</strong>s of probab<strong>le</strong> downpour.<br />
13 LTAC Lista las tablas de alturas-cauda<strong>le</strong>s.<br />
It lists tab<strong>le</strong>s of altitudes-flows.<br />
157<br />
14 TAVAL Lista los valores de las curvas alturas-cauda<strong>le</strong>s y está<br />
preparado para obtener valores que no figuren en las ta<br />
blas, interpolando linealmente entre los dos puntos más<br />
próximos.<br />
It lists <strong>the</strong> values of altitudes-flows lines and it is<br />
prepared to obtain values not appearing in <strong>the</strong> tab<strong>le</strong>s,<br />
by linear interpolation between <strong>the</strong> closest points.
158<br />
15<br />
16<br />
TAB- 1<br />
TAB-2<br />
17 TAB- 3<br />
18 TAB- 4<br />
Este programa lista, de las cinco fichas de que consta<br />
la información de cada pozo, lo siguiente: número de pg<br />
zo, fechas de muestreo, número de muestras (laborato-<br />
rio), latitud, longitud, nivel de agua/l, columna de agua,<br />
horas de bombeo, caudal en Ils., procedencia, número<br />
de laboratorio, fecha del análisis, temperatura del aire,<br />
temperatura del agua e indicador.<br />
This programme lists <strong>the</strong> five cards which contains <strong>the</strong><br />
information of each well, as <strong>the</strong> following: number of<br />
well, dates of sampling, number of samp<strong>le</strong>s (laboratory),<br />
latitude, longitude, <strong>le</strong>vel of water/l, column of water,<br />
hours of pumping, flow in Ils., origin, laboratory number,<br />
date of analysis, temperature of <strong>the</strong> air, temperature of<br />
water and indicator.<br />
Lista: número de pozo, fecha de muestreo, calcio, mag-<br />
nesio, manganeso, sodio, potasio, cloruro, sulfato, fluo<br />
ruro, silice, fosfato y carbonato.<br />
It lists: number of well, date of sampling, calcium,<br />
manganese, sodium, potassium, chloride, sulphate,<br />
fluoride, silica, phosphate and carbonate.<br />
Lista: número de pozo, fecha de muestreo, bicarbonato,<br />
nitrito, nitrato, amoniaco, boro, hierro, pH, resistivi-<br />
dad, gravedad especifica, sólidos disueltos, litio, estrog<br />
cio, ni’quel.<br />
It lists: number of well, date of sampling, bicarbonate,<br />
nitrite, nitrate, boron, ammonia, iron, pH, resistivity,<br />
gravity/m, solid in dissolution, lithium, strontium, -<br />
nickel.<br />
Lista: nimero de pozo, fecha de muestreo, cobalto, iodo,<br />
bromo, molibdeno, zinc, plomo, cromo, cobre, vanadio,<br />
mer curio, ar sé nico.
159<br />
It lists: number of well, date of sampling, cobalt, iodine,<br />
bromine, molibdenum, zinc, plumbum, chromium, cop-<br />
per, vanadium, mercury and arsenic.<br />
19 TAB-5 Lista: número de pozo, fecha de muestreo, pH, resisti-<br />
vidad, gas carbónico libre, oxigeno disuelto, dureza, dg<br />
reza (sin carbonatos), alcalinidad, T. A., HC03.<br />
It lists: number of well, date of sampling, pH, resistivity,<br />
free carbonic gas, oxygen disolved, hardness (without<br />
carbonates), alkalinity, T. A., HC03.<br />
20 TNFAG Tablas de interpolación polinómica de cuarto grado.<br />
Tab<strong>le</strong>s of polynomical interpolation of <strong>the</strong> fourth grade.<br />
21 TANG Tablas de senos, cosenos y tangentes.<br />
Tab<strong>le</strong>s of sine, cosine, tangents.<br />
22 TNUA Mete en disco tablas de números a<strong>le</strong>atorios.<br />
It puts in <strong>the</strong> disk tab<strong>le</strong>s of fortuitous numbers.<br />
23 T-V-P Mete en disco tablas de datos hidrológicos.<br />
It puts in <strong>the</strong> disk tab<strong>le</strong>s of hydrological data.
160<br />
24 T-EVA<br />
25 EN-PRT<br />
26 SAPRI<br />
27 PDPRC<br />
28 PC 128<br />
Mete en disco tablas de evaporaciones.<br />
It puts in <strong>the</strong> disk tab<strong>le</strong>s of evaporation.<br />
Dada una serie de estaciones, almacena en disco dichas<br />
estaciones.<br />
It stores in <strong>the</strong> disk stations which are previously given.<br />
Obtiene un listado de las estaciones almacenadas en disco.<br />
It obtains a listing of stations stored in <strong>the</strong> disk.<br />
Perfora datos con anos consecutivos para los diversos<br />
programas de regulaciones.<br />
It perforates data with consecutive years for <strong>the</strong> different<br />
programs of regulations.<br />
Dada una serie de cauda<strong>le</strong>s diarios con formato 12 F 6.2,<br />
los perfora con formato 9 F 8.3, para ser utilizados por<br />
el programa I~TCDAP".<br />
Given a series of daily flows with format 12 F 6. 2, it<br />
perforates <strong>the</strong>m with format 9 F 8. 3, to be used by <strong>the</strong><br />
program I'TDFAF~~.
29 PRD-I Ordena un máximo de 1.000 datos, de mayor a menor,<br />
con formato 12 F 6.2.<br />
161<br />
It puts in order a maximum of 1.000 data, in descendent<br />
order, with format 12 F 6. 2.<br />
30 PRD-2 Igual que el anterior, pero con formato 9 F 8.3.<br />
The same as above, but with format 9 F 8.3.<br />
31 PRD-3 Ordena números en coma fija, de mayor a menor.<br />
It puts in order numbers with fixed point, in descendent<br />
order.<br />
32 DUPLP Duplica las ochenta columnas de las fichas.<br />
It duplicates <strong>the</strong> eighty columns of <strong>the</strong> cards.<br />
33 DUP-Mp Duplica las ochenta columnas, modificando la columna<br />
que se desee.<br />
It duplicates <strong>the</strong> eighty columns, modifying <strong>the</strong> column<br />
that is wished.<br />
34 SUMNU Suma o resta un número a una serie de datos mensua<strong>le</strong>s.<br />
It adds or deducts a number from a series of monthly data.
162<br />
35 SUMRE Dadas dos series de datos mensua<strong>le</strong>s, las suma o las<br />
resta.<br />
it adds or deducts two series of monthly data which are<br />
given.<br />
36 SUMAR Suma series de datos hidrológicos.<br />
37 MULTI<br />
38 MULTA<br />
39 MUL 12<br />
It adds series of hydrological data.<br />
Multiplica series de datos por un número fijo y obtiene<br />
el listado, así como las fichas perforadas, con estos nue<br />
vos valores.<br />
It multiplies series of data by a fixed number and it<br />
obtains <strong>the</strong> listing, in <strong>the</strong> same way as <strong>the</strong> perforated<br />
cards, with this new values.<br />
Multiplica los 12 números de la primera fila por cada<br />
uno de los datos, los pone en orden decreciente y los lis<br />
ta en 12 columnas.<br />
It multiplies <strong>the</strong> 12 numbers of <strong>the</strong> first row by each one<br />
of <strong>the</strong> data, it puts <strong>the</strong>m decreasing order and it lists -<br />
<strong>the</strong>m in 12 columns.<br />
Multiplica los datos mensua<strong>le</strong>s por el número que ocupa<br />
el lugar correspondiente a ese mes en la primera ficha,<br />
lista y perfora.<br />
It multiplies <strong>the</strong> monthly data by <strong>the</strong> number that occupies<br />
<strong>the</strong> corresponding place of that month in <strong>the</strong> first card, it<br />
lists and Derforates.
40 MULAN<br />
41 MRS 12<br />
42 MA TEN<br />
43 SECUA<br />
44 DIS -KM<br />
163<br />
Multiplica los datos mensua<strong>le</strong>s de cada ano por el &me-<br />
ro que ocupa el lugar correspondiente a ese ano en las<br />
primeras fichas que <strong>le</strong>e.<br />
It multiplies <strong>the</strong> monthly data of each year by <strong>the</strong> number<br />
that occupies <strong>the</strong> corresponding place to that year in <strong>the</strong><br />
first cards that is read.<br />
Multiplica, suma o resta dos o una series de datos sin<br />
limitación.<br />
It multiplies, adds or deducts two or one series of data<br />
without limitation.<br />
E<strong>le</strong>va una matriz a la potencia enésima.<br />
It e<strong>le</strong>vates a matrix to <strong>the</strong> n power.<br />
Resuelve un sistema de ecuaciones (40 como máximo).<br />
It solves a system of equations (40 as maximum).<br />
Dada la situación geográfica de una serie de estaciones<br />
por su latitud y longitud, este programa se<strong>le</strong>cciona los<br />
grupos de estaciones a comparar con el criterio de que<br />
las distancias entre las estaciones sean menores de una<br />
cantidad fija.<br />
Given <strong>the</strong> geographical position of a series Of stations by<br />
<strong>the</strong>ir latitude and longitude, this programme chooses <strong>the</strong><br />
group of stations to be compared with <strong>the</strong> criterium that<br />
<strong>the</strong> distance among <strong>the</strong> stations be smal<strong>le</strong>r than a fixed<br />
quantity.
164<br />
45 LISDA Lista series de datos mensua<strong>le</strong>s (sin limitación de exten<br />
sión), imprime cabecera y calcula la suma anual y las -<br />
medias mensua<strong>le</strong>s.<br />
It lists series of monthly data (without limitation in its<br />
scope), prints heading and computes <strong>the</strong> yearly sum and<br />
<strong>the</strong> monthly averages.<br />
46 MAXLL Dados los valores de precipitación total mensual, dias<br />
de lluvia y los valores máximos en 24 horas, obtiene los<br />
máximos en 24 horas y los dilas de lluvia a escala anual.<br />
Given <strong>the</strong> values of total monthly rainfall, days of rain<br />
and <strong>the</strong> maximum values in 24 hours, it obtains <strong>the</strong> -<br />
maximum in 24 hours and <strong>the</strong> days of rain in a yearly<br />
s ca<strong>le</strong>.<br />
47 INT-ES Dados los cauda<strong>le</strong>s medios mensua<strong>le</strong>s y anua<strong>le</strong>s, la apoy<br />
tación media de los años precedentes y el caudal máximo<br />
(medios diarios e instantáneo y la fecha), obtiene cauda-<br />
<strong>le</strong>s medios anual y mensual, caudal y aportación mensual.<br />
Deduce la aportación y caudal anual, caudal medio y apor<br />
tación media de la serie anual.<br />
Given <strong>the</strong> monthly and annual average flows, <strong>the</strong> average<br />
afford of <strong>the</strong> preceding years and <strong>the</strong> maximlim flow (daily<br />
and instantaneous averages and <strong>the</strong> date), it obtains year<br />
ly and monthly average flows, monthly flow and afford. It<br />
deducts yearly afford and flow, average flow and <strong>the</strong> aver<br />
age afford of <strong>the</strong> yearly series.<br />
48 INT-EM Dado el volumen embalsado, aportación de salida y media<br />
precedente, obtiene entradas y salidas, reserva, aporta-<br />
ción del año, aportación de entrada y salida, media de e;<br />
trada y salida (aportación y caudal).
49<br />
50<br />
51<br />
INT -CA<br />
ADfDB- 1<br />
ADfDB-2<br />
165<br />
Given <strong>the</strong> stored volume, afford of exit and <strong>the</strong> preceding<br />
average, it obtains entries and exits, reserve, yearly -<br />
afford, afford of entry and exit, average of entry and -<br />
exit (afford and flow).<br />
Realiza la misma función, pero sin el caudal máximo<br />
instantáneo.<br />
It performs <strong>the</strong> same function, but without <strong>the</strong> maximum<br />
instantaneous flow.<br />
Dadas las series de datos hidrológicos de un conjunto de<br />
estaciones a escala anual, mensual, etc., forma "esta-<br />
ción tipo" (media aritmética de las estaciones de cada<br />
grupo) y a continuación compara cada una de las estacio<br />
nes con su "estación tipo", acumulando las series y dag<br />
do, además de las sumas acumuladas, la relación entre<br />
las acumulaciones de cada estación con las acumulacio-<br />
nes de la "estación tipo".<br />
Given <strong>the</strong> series of hydrological data of an assembly of<br />
stations at yearly sca<strong>le</strong>, monthly sca<strong>le</strong>, etc., it forms<br />
''type station!' (arithmetical mean of <strong>the</strong> stations in each<br />
group) and afterwards compares each stations with its<br />
'hype station", accumulating <strong>the</strong> series and giving besides<br />
<strong>the</strong> accumulated sums, <strong>the</strong> relation among <strong>the</strong> accumula-<br />
tion of each station with those of "type station".<br />
Este programa es análogo al anterior, pero no utiliza -<br />
estación tipo, haciendo todas las comparaciones posib<strong>le</strong>s<br />
en cada grupo.<br />
This programme is analogous to <strong>the</strong> preceding one, but<br />
it does not use type station, performing all <strong>the</strong> possib<strong>le</strong><br />
comparisons in each group.
166<br />
52 A-AC-96 Acumula datos hidrológicos mensua<strong>le</strong>s (de 1 en 1 hasta<br />
96 en 96 meses). Imprime 96 cuadros.<br />
It accumulates hydrological monthly data (from 1 to 1 up<br />
to 96 in 96 months). It prints 96 charts.<br />
53 AC-96-P Este programa es análogo al anterior, pero perfora los<br />
resultados en ficha.<br />
This programme is analogous to <strong>the</strong> preceding one, but<br />
it perforates <strong>the</strong> results in card.<br />
54 AM-AC 1 Acumula aportaciones mensua<strong>le</strong>s y ordena de menor a<br />
mayor .<br />
It accumulates monthly affords putting in ascendent order.<br />
55 AM-C 96 Dada una serie de datos hidrológicos mensua<strong>le</strong>s, acumu-<br />
la de 1 a 96 meses consecutivos, y los ordena de menor<br />
a mayor.<br />
Given a series of hydrological monthly data, it accum;I-<br />
lates from 1 to 96 consecutive months, and putting <strong>the</strong>m<br />
in ascendent order.<br />
56 AC-T~D Acumula todos los valores de una serie.<br />
It accumulates all <strong>the</strong> values of a series.
57 AD~BP Dibuja en el Plotter los diagramas de las acumulaciones<br />
dob<strong>le</strong>s.<br />
167<br />
It designs in <strong>the</strong> Plotter <strong>the</strong> diagrams of <strong>the</strong> doub<strong>le</strong> - -<br />
accumula tion.<br />
58 ALAOA Dadas las aportaciones diarias, las lista, ordena de mg<br />
nor a mayor y las acumula.<br />
Given <strong>the</strong> daily affords, it lists, puts in ascendent order<br />
and accumulates <strong>the</strong>m.<br />
59 TCDAP Dada una serie de cauda<strong>le</strong>s diarios, la transforma en -<br />
aportaciones diarias.<br />
Given a series of daily flow, this programme transforms<br />
<strong>the</strong>m in daily affords.<br />
60 TCAP~ Dada una serie de cauda<strong>le</strong>s mensua<strong>le</strong>s, la transforma en<br />
aportaciones mensua<strong>le</strong>s.<br />
Given a series of monthly flows, this series is transform<br />
ed in monthly affords.<br />
61 TAPPC Dada una serie de aportaciones mensua<strong>le</strong>s, la transforma<br />
en cauda<strong>le</strong>s mensua<strong>le</strong>s.<br />
Given a series of monthly affords, it transforms <strong>the</strong>m in<br />
monthly flows.
168<br />
62 TANAD Dada una serie de aportaciones natura<strong>le</strong>s diarias, la -<br />
transforma en aportaciones derivab<strong>le</strong>s diarias.<br />
Given a ser<strong>le</strong>s of daily natural affords, <strong>the</strong> programme<br />
transforms <strong>the</strong>m into daily derivab<strong>le</strong> affords.<br />
63 C-C-D-ES Dada una serie de cauda<strong>le</strong>s diarios, obtiene los cauda<strong>le</strong>s<br />
derivados diarios en m3/s., medias y aportaciones men_<br />
sua<strong>le</strong>s, cauda<strong>le</strong>s clasificados, aportación y caudal total<br />
del año.<br />
Given a series of daily flows, <strong>the</strong> programme obtains <strong>the</strong><br />
daily derivab<strong>le</strong> flows in CU. m/s., averages and monthly<br />
affords, classified flows, afford and total flow of <strong>the</strong> year.<br />
64 C -C -D-CA Realiza la misma función que el programa anterior, pero<br />
con cauda<strong>le</strong>s mensua<strong>le</strong>s.<br />
It performs <strong>the</strong> same function as <strong>the</strong> preceding programme,<br />
but with monthly flows.<br />
65 C-C-D-EM Dados los volúmenes y salidas diarias de un embalse, ob<br />
tiene las reservas diarias (Hm3), cauda<strong>le</strong>s diarios (sali-<br />
das en m3/s. ), media mensual, salida y entrada mensual,<br />
evaporacion y un resumen anual (cauda<strong>le</strong>s medios, salida,<br />
entrada, evaporación).<br />
Given <strong>the</strong> volumes and <strong>the</strong> daily exits of a reservoir, <strong>the</strong><br />
programme obtains <strong>the</strong> daily reserves (CU. Hm), daily<br />
flows (exits in CU. m/s), monthly average, monthly entry<br />
and exit, evaporation and an annual summary (averages<br />
flows, entry, exit, evaporation).
169<br />
66 C-C-D-E1 Dados los datos de alturas de escala diaria y los valores<br />
de las curvas alturas-cauda<strong>le</strong>s de una estación, calcula<br />
los cauda<strong>le</strong>s diarios de dicha estación. Los cauda<strong>le</strong>s que<br />
no figuran en las tablas se calculan interpolando lineal-<br />
mente entre los dos más próximos.<br />
Además del caudal diario, este programa obtiene los cag<br />
da<strong>le</strong>s máximos y mínimos, los cauda<strong>le</strong>s medios mensua-<br />
<strong>le</strong>s y las aportaciones mensua<strong>le</strong>s.<br />
Given <strong>the</strong> daily sca<strong>le</strong> heights and <strong>the</strong> values of <strong>the</strong> height-<br />
flow charts of a station, <strong>the</strong> programme computes <strong>the</strong> --<br />
daily flows of said station. The flows that are not appear-<br />
ing in <strong>the</strong> tab<strong>le</strong>s, are calculated by linear interpolation -<br />
between <strong>the</strong> closest points.<br />
In addition to <strong>the</strong> daily flow, this programme obtains <strong>the</strong><br />
maximum and minimum flows, <strong>the</strong> monthly average flows<br />
and <strong>the</strong> monthly affords.<br />
67 C-C-D-E2 Dados los datos de alturas de escala diaria y los valores<br />
de las curvas alturas-cauda<strong>le</strong>s de una estación, calcula<br />
los nive<strong>le</strong>s diarios en metros, los cauda<strong>le</strong>s diarios en<br />
m3 /s. , medias mensua<strong>le</strong>s, máxima instantánea, aporta<br />
ci& mensual en Hm3, cauda<strong>le</strong>s clasificados y un resu--<br />
men de los datos del año (aportación y caudal total y es-<br />
pecifico y cauda<strong>le</strong>s caracteristicos).<br />
Given <strong>the</strong> daily sca<strong>le</strong> heights data and <strong>the</strong> values of height-<br />
flow lines of a station, <strong>the</strong> programme computes <strong>the</strong> daily<br />
<strong>le</strong>vels in meters, <strong>the</strong> daily flows in CU. m/s., monthly -<br />
averages, instantaneous maximum, monthly afford in --<br />
CU. Hm, classified flows and an annual summary data --<br />
(afford and total flow, specific flow and caracteristic --<br />
flows).<br />
68 C-C-D-ES Dados los datos de alturas de escala diaria y los valores<br />
de alturas-cauda<strong>le</strong>s de una estación, calcula los cauda<strong>le</strong>s<br />
diarios y el caudal medio anual.<br />
Los datos son los obtenidos en limnigrafo.
170<br />
69<br />
71<br />
72<br />
CURGA<br />
Given <strong>the</strong> daily sca<strong>le</strong> heights data and <strong>the</strong> values of height-<br />
flow lines of a station, <strong>the</strong> programme computes <strong>the</strong> daily<br />
flows and <strong>the</strong> yearly average flow.<br />
Data are obtained by <strong>the</strong> limnigraphe.<br />
A partir de unos puntos base tabula una tabla de gastos.<br />
Lista y dibuja los diagramas de las curvas de gastos.<br />
From a basic point, it tabulates a tab<strong>le</strong>s of expenses.<br />
It lists and designes <strong>the</strong> diagram of <strong>the</strong> expense lines.<br />
ME Y ME Dadas las aportaciones mensua<strong>le</strong>s de una serie de esta-<br />
ciones de una cuenca, calcula e imprime las medias mec<br />
sua<strong>le</strong>s de cada estación y la media de las medias de todas.<br />
MEDIP<br />
Given <strong>the</strong> monthly affords of a series of stations of a basin,<br />
it computes and prints <strong>the</strong> monthly average of each station<br />
and <strong>the</strong> average of all means.<br />
Dados los datos diarios de años de una estación, los<br />
imprime y calcula las sumas y medias mensua<strong>le</strong>s de ca<br />
da ailo y las medias de las medias (mediorum).<br />
Given <strong>the</strong> daily data of c years of a station, <strong>the</strong> programme<br />
prints <strong>the</strong>m and computes <strong>the</strong> sums and monthly averages<br />
of each year and <strong>the</strong> average of <strong>the</strong> n_ means (mediorum).<br />
MET, ME Dada una serie de datos mensua<strong>le</strong>s, calcula ias medias<br />
para cualquier periodo.<br />
It calculates <strong>the</strong> average for any period of a series of<br />
monthly data, which are given.
73<br />
74<br />
75<br />
76<br />
CIC LJD<br />
MED -AN<br />
A-ESP<br />
INF - 1<br />
171<br />
Calcula las medias acumuladas en periodos de n años y<br />
sus relaciones con la media del periodo total.<br />
It calculates <strong>the</strong> accumulated averages in periods of E<br />
years and <strong>the</strong>ir relations with <strong>the</strong> average of <strong>the</strong> total<br />
period.<br />
Dada una serie de datos anua<strong>le</strong>s, calculala media para<br />
cualquier pedodo.<br />
Given a series of yearly data, <strong>the</strong> programme calculates<br />
<strong>the</strong> average for any period.<br />
Dadas unas series pluviométricas rea<strong>le</strong>s, obtiene una se<br />
rie real, media de las anteriores, a partir de la cual ob-<br />
tiene otra de precipitaciones efectivas, de la que se con-<br />
siguen las aportaciones especificas y los cauda<strong>le</strong>s en una<br />
cuenca.<br />
Given an actual pluviometrical series, <strong>the</strong> programme<br />
obtains an actual series, average of <strong>the</strong> preceding, from<br />
which it obtains ano<strong>the</strong>r series of effective rainfalls, -<br />
from which it gets specific affords and <strong>the</strong> flows in a basin.<br />
Dadas las precipitaciones mensua<strong>le</strong>s de una cueBca y los<br />
coeficientes de capacidad de infiltración mensual en mm.,<br />
de humedad inicial del suelo y de superficie en Has., ob-<br />
tiene las aportaciones especificas, infiltración y evapora<br />
cion.<br />
Given <strong>the</strong> monthly rainfall of a basin and <strong>the</strong> coefficients<br />
of monthly infiltration capacity in mm., initial humidity<br />
from <strong>the</strong> earth and from surface in Has., it gets <strong>the</strong> --<br />
specific affords, infiltration and evaporation.
172<br />
77 INF - 2 Realiza la misma función que el programa anterior, pe-<br />
ro a nivel diario.<br />
It performs <strong>the</strong> same function as <strong>the</strong> preceding one, but<br />
in a daily <strong>le</strong>vel.<br />
78 EVAP- 1 Dados los volúmenes, superficies y reserva, halla las<br />
evaporaciones.<br />
The programme computes <strong>the</strong> evaporations, knowing <strong>the</strong><br />
volume, area and stock.<br />
79 EVAPT Dado el número de estaciones, superficie de la cuenca<br />
(kmz), aportación media (media de una serie) y precip'<br />
tación media, obtiene la aportación en Hm3, coeficiente<br />
de escorrentia y déficit de escorrentía.<br />
It obtains <strong>the</strong> afford in CU. Hm., runoff coefficient and<br />
deficit of runoff, knowing <strong>the</strong> number of stations, area<br />
of <strong>the</strong> basin (sq. Km. ), average of <strong>the</strong> afford (average<br />
of a serie) and <strong>the</strong> average rainfall.<br />
80 REST-l Lista: número total de muestras, media aritmética, va-<br />
lor máximo, valor minimo, desviación tipica, tempera-<br />
tura del agua.<br />
The programme lists: total number of samp<strong>le</strong>s, arithmeg<br />
cal mean, maximum and minimum value, standard desvia<br />
tion. temperatura of <strong>the</strong> water.<br />
81 REST-2 Lista los mismos parámetros para calcio, magnesio, mall<br />
ganeso, sodio, potasio, cloruro, sulfato, fluoruro, silice,<br />
fosfato y carbonato.
82 REST-3<br />
83 REST-4<br />
84 D-FLSI<br />
85 GE~NE<br />
It lists <strong>the</strong> same parameters for calcium, magnesium,<br />
manganese, sodium, potassium, chloride, sulphate, -<br />
fluoride, silica, phosphate and carbonate.<br />
173<br />
Lista los mismos parámetros para bicarbonato, nitrito,<br />
nitrato, amoniaco, hierro, resistividad y sólidos disuel<br />
tos.<br />
It lists <strong>the</strong> same parameters for bicarbonate, nitrite,<br />
nitrate, ammonia, iron, resistivity, solid in dissolution.<br />
Lista los mismos parámetros para pH, resistividad, gas<br />
carbónico libre, oxígeno disuelto, dureza, dureza (sin -<br />
carbonatos), alcalinidad, T. A. , HC03.<br />
It lists <strong>the</strong> same parameters for pH, resistivity, free<br />
carbonic gas, oxygen dissolved, hardness, hardness -<br />
(without carbonates), alkalinity, T. A., HC03.<br />
Dado el perimetro y la superficie por encima de cada co-<br />
ta de una cuenca, obtiene el rectángulo equiva<strong>le</strong>nt e, coe-<br />
ficiente de Gravelius, indice de pendiente, pendiente m e-<br />
dia y altitud media de la cuenca.<br />
Given <strong>the</strong> perimeter and <strong>the</strong> area above each e<strong>le</strong>vation of<br />
a basin, <strong>the</strong> programme obtains <strong>the</strong> equiva<strong>le</strong>nt rectang<strong>le</strong>,<br />
Gravelius factor, pendant index, average of <strong>the</strong> pendant<br />
and average altitude of <strong>the</strong> basin.<br />
A partir de coordenadas de puntos fijos dados, de medi-<br />
ciones, direcciones y distancias para una conexión de pu9<br />
tos aislados o múltip<strong>le</strong>s, el programa obtiene las coorde-<br />
nadas de los nuevos puntos.
86<br />
87<br />
88<br />
DIS-2P<br />
HELME<br />
REPPU<br />
89 ARCES<br />
From coordinates of fixed given points, of measuring,<br />
directions and distances to a connection of isolated or<br />
multip<strong>le</strong>s points, <strong>the</strong> programme gets <strong>the</strong> coordinates<br />
of <strong>the</strong> new points.<br />
Dadas las coordenadas de dos puntos, calcula su dis-<br />
tancia.<br />
Given <strong>the</strong> coordinates of two points, it calculates <strong>the</strong>ir<br />
distance.<br />
Convierte coordenadas instrumenta<strong>le</strong>s en terrestres<br />
(Helme rt ).<br />
It converts from instrumentais coordinates to terrestrial<br />
(Helmert).<br />
En función de las coordenadas de los puntos calcula azi-<br />
mutes y distancias para replanteo.<br />
In function of coordinates of <strong>the</strong> points, it calculates azi-<br />
muths and distances to be reconsidered.<br />
Calcula la superficie en planta que abarca cada curva en<br />
un embalse.<br />
It calculates <strong>the</strong> area which comprises each line in a<br />
reservoir.
90 EAKIN Dada la superficie del tramo máximo de un embalse, su<br />
longitud y la superficie del perfil de sondeo, obtiene el<br />
volumen de esas superficies.<br />
Given <strong>the</strong> area of <strong>the</strong> maximum stretch of a reservoir,<br />
<strong>the</strong>ir longitude and <strong>the</strong> area of <strong>the</strong> profi<strong>le</strong> of sounding,<br />
it obtains <strong>the</strong> volumes of those surfaces.<br />
91 RAPEM Cubica un embalse según la fórmula RAPEM.<br />
It cubes a reservoir according to formula RAPEM.<br />
92 C~R-AM Dada una serie mensual de datos hidrológicos, calcula<br />
la correlación ortogonal, los momentos de la serie reg<br />
pecto al origen y el coeficiente de correlación.<br />
175<br />
Given a monthly series of hydrological data, it calculates<br />
<strong>the</strong> ortogonal correlation, <strong>the</strong> moments of <strong>the</strong> series with<br />
regards to <strong>the</strong> origin and <strong>the</strong> correlation factor.<br />
93 CPR-LM Calcula la correlación lineal mensual, obteniendo el cog<br />
ficiente de correlación, las medias, varianzas, disper--<br />
si&, coeficiente angular y ordenada en el origen de la<br />
recta de regresión.<br />
The programme calculates <strong>the</strong> monthly linear correlation,<br />
obtaining <strong>the</strong> correlation factor, <strong>the</strong> averages, variances,<br />
dispersion, grade ang<strong>le</strong> coefficient and ordinate at <strong>the</strong> -<br />
origin of <strong>the</strong> regression straight.<br />
94 CPR-AN Este programa se diferencia del anterior Únicamente en<br />
que, partiendo de los datos mensua<strong>le</strong>s, utiliza solamente
176<br />
los anua<strong>le</strong>s, efectuando la correlación entre ellos con -<br />
arreglo al esquema ya reseñado.<br />
This programme differs from <strong>the</strong> previous one only in<br />
that starting from <strong>the</strong> monthly data, it uses only <strong>the</strong> -<br />
annual ones and performing <strong>the</strong> correlation among <strong>the</strong>m,<br />
in accordante with <strong>the</strong> indicated scheme.<br />
95 C~R-DM Con los datos hidrológicos mensua<strong>le</strong>s de tres estaciones,<br />
z, x e y, realiza la correlación dob<strong>le</strong> de x e y con z, ob-<br />
teniendo la ecuación del plano de correlación.<br />
With <strong>the</strong> monthly hydrological data of <strong>the</strong> three stations,<br />
z, c and y, it performs <strong>the</strong> doub<strong>le</strong> correlation of x and y<br />
with z, obtaining <strong>the</strong> equation of <strong>the</strong> plane of correlation.<br />
96 CPR-PA El esquema es igual al de los anteriores que realizan cg<br />
rrelaciones, dando éste la ecuación de la parábola de re<br />
gresión de y sobre x.<br />
The scheme is identical to <strong>the</strong> previous one which performs<br />
correlations, , <strong>the</strong> equation of <strong>the</strong> parabola of regression<br />
of y on x is given by <strong>the</strong> scheme.<br />
97 C ~ R - ~ R Dadas dos series mensua<strong>le</strong>s de datos hidrológicos, calcg<br />
la la correlación ortogonal obteniendo la recta cuya suma<br />
de cuadrados de distancias a los puntos es minima.<br />
Given two monthly series of hydrological data, <strong>the</strong> pro-<br />
gramme calculates <strong>the</strong> ortogonal correlation, obtaining<br />
<strong>the</strong> straight whose sum of squares to <strong>the</strong> points in min-<br />
imum.
98 CfbR-A-p Dada una serie mensual de datos hidrológicos, realiza<br />
la correlación anual ortogonal.<br />
Given a monthly series of hydrological data, it performs<br />
<strong>the</strong> yearly ortogonal correlation.<br />
99 CPR-48 Realiza la correlación ortogonal para los meses de estia<br />
je (4) y el resto de los meses (8).<br />
It performs <strong>the</strong> ortogonal correlation for <strong>the</strong> summer<br />
months (4) and to <strong>the</strong> balance of <strong>the</strong> months (8).<br />
100 CPR-LP Dadas dos series de datos hidrológicos, obtiene la recta<br />
de correlación ortogonal entre sus logaritmos.<br />
Given two series of hydrological data, it gets <strong>the</strong> straight<br />
of ortogonal correlation among <strong>the</strong>ir logarithms.<br />
10 1 COR-O-P El esquema es igual al del propama "CqR-qR" y, ade-<br />
más, dibuja la nube de puntos en el Plotter.<br />
The scheme is identical to "COR-OR" programme, in<br />
addition it draws <strong>the</strong> clouds of points in <strong>the</strong> Plotter.<br />
102 cpycfb Comp<strong>le</strong>ta y corrige una serie de datos hidrológicos dan-<br />
do las ecuaciones de las rectas de regresión (y=ai x - bi)<br />
entre las dos series (se puede hacer con una ecuación o<br />
con dos simultáneamente).<br />
It comp<strong>le</strong>tes and corrects a series of hydrological data,<br />
giving <strong>the</strong> equation of <strong>the</strong> regression straights (x=ai x - bi)<br />
between <strong>the</strong> two series (it can be done with an equation or<br />
two simultaneously).
178<br />
io3 ~p-DRR Comp<strong>le</strong>ta los datos hidrológicos según la recta de regre-<br />
sión.<br />
It comp<strong>le</strong>tes <strong>the</strong> hydrological data according to <strong>the</strong> reg-<br />
sion straight.<br />
104 IN-D-M1 Inventa datos mensua<strong>le</strong>s de una estación con datos anua-<br />
<strong>le</strong>s, a partir de los datos mensua<strong>le</strong>s de otras dos esta--<br />
ciones.<br />
It creates monthly data of a station with yearly data, star<br />
ting from <strong>the</strong> monthly data of o<strong>the</strong>r two .stations.<br />
105 IN-D-M2 Inventa datos mensua<strong>le</strong>s de una estación con datos anua-<br />
<strong>le</strong>s, a partir de los datos mensua<strong>le</strong>s de otra según la fÓ'<br />
mula B (I) = [A (I) * (SUMB (I) / SUMA (I) 1.<br />
It creates monthly data of a station with yearly data,<br />
starting from monthly data of o<strong>the</strong>r station, according<br />
to formula: B (I) = I A (I) * (SUMB (I) / SUMA (I) 1 .<br />
106 COQUI Calcula 20 correlaciones ortogona<strong>le</strong>s, entre e<strong>le</strong>mentos<br />
quhnicos, pintando por Plotter los puntos y la recta de<br />
regresión y dos para<strong>le</strong>las a una distancia igual a la dis-<br />
persion.<br />
The programme calculates 20 ortogonal correlations,<br />
among chemical e<strong>le</strong>ments, drawing in Plotter <strong>the</strong> points<br />
and <strong>the</strong> regression straight and two paral<strong>le</strong>l lines to a<br />
distance identical to <strong>the</strong> dispersion.
179<br />
107 BERKA Dibuja el diagrama de Berkaloff-Schol<strong>le</strong>r por el Plotter,<br />
uno por cada pozo, en una escala logaritmica. Pinta los<br />
puntos de los siguientes e<strong>le</strong>mentos: CA, MG, ALC, CL,<br />
SO4, HCOQ + CO3, NO3. y une con segmentos dichos -<br />
puntos.<br />
It designs <strong>the</strong> diagram of Berkaloff-Schol<strong>le</strong>r by means of<br />
Plotter, one diagram to each well, in logarithmical sca<strong>le</strong>.<br />
It draws <strong>the</strong> points of <strong>the</strong> following e<strong>le</strong>ments: CA, MG, -<br />
ALC, CL, SO4, HC03 t COQ, NO3, and it joins <strong>the</strong> - -<br />
mentioned points with <strong>the</strong> segments.<br />
108 STIF Dibuja el diagrama de Stif.<br />
The programme draws <strong>the</strong> Stif diagram.<br />
109 PIPER Dibuja el diagrama de Piper. Consta de dos triángulos<br />
equiva<strong>le</strong>ntes; en la base del primero se marca en 70 el va<br />
lor CA, en mgl/l, y en los otros dos lados, MG y NA+K,<br />
y mediante para<strong>le</strong>las a 10s lados opuestos obtenemos un<br />
punt o.<br />
De la misma forma, en el segundo triángulo pintan en el<br />
lado base, CL, y en los otros dos CO3 t HCH03 y so4 t<br />
t NOP en 70 y mediante para<strong>le</strong>las obtenemos otro punto.<br />
Esta operación se repite para cada pozo y para las ocho<br />
zonas.<br />
It draws <strong>the</strong> Piper diagram. It is composed of two equiva<br />
<strong>le</strong>nt triang<strong>le</strong>s; in <strong>the</strong> base of <strong>the</strong> first one, it marks in 70<br />
<strong>the</strong> value CA, in mgl/l, and in <strong>the</strong> o<strong>the</strong>r two sides, MG<br />
and NA t K, and by means of paral<strong>le</strong>l lines to <strong>the</strong> opposite<br />
sides obtaining a point.<br />
In <strong>the</strong> same way, in <strong>the</strong> second triang<strong>le</strong>, it marks in <strong>the</strong><br />
sidebase, CL and in <strong>the</strong> o<strong>the</strong>r two sides, Co3 + HCHO<br />
and SO4 + NOP in 70 and by paral<strong>le</strong>l lines obtaining ano<strong>the</strong>r<br />
point. This execution is repeated for each well and for <strong>the</strong><br />
eight bands.
180<br />
i10 G ~ K A los valores de una serie de datos hidrológicos, <strong>le</strong> ajug<br />
ta una <strong>le</strong>y de Goodrich y contrasta la bondad del ajuste -<br />
mediante el test de Kolmogoroff.<br />
Given a series of values of hydrological data, it fits with<br />
Goodrich’s law and contrasts <strong>the</strong> perfection of <strong>the</strong> fitting<br />
by Kolmogoroff’ test.<br />
111 GPK~L A partir de una serie de datos hidrológicos anua<strong>le</strong>s, se<br />
ajusta una <strong>le</strong>y de Goodrich y se contrasta la bondad del<br />
ajuste mediante el test de Kolmogoroff.<br />
Los datos de entrada son series mensua<strong>le</strong>s. El programa<br />
obtiene las series anua<strong>le</strong>s mensua<strong>le</strong>s, la media, los mo-<br />
mentos respecto al origen de orden 2 y 3, los momentos<br />
centra<strong>le</strong>s de segundo (varianza) y de tercer orden, as( -<br />
como los parámetros de la <strong>le</strong>y de distribución de Goodrich.<br />
Starting from a series of hydrological annual data, Good-<br />
rich’s law is adjusted and <strong>the</strong> perfection is contrasted by<br />
Kolmogoroff’test.<br />
The entry data are monthly series. The programme - -<br />
obtains, annual series, monthly series, <strong>the</strong> average, <strong>the</strong><br />
moments with regard to <strong>the</strong> origin of order 2 and 3, <strong>the</strong><br />
central moments of second (variance) and third order, -<br />
<strong>the</strong> programme obtains also <strong>the</strong> parameters of <strong>the</strong> Good-<br />
rich distribution law.<br />
112 GPKAF Dada una serie de datos hidrológicos, ajusta la <strong>le</strong>y de dig<br />
tribución de Goodrich.<br />
Given a series of hydrological data, <strong>the</strong> Goodrich distri-<br />
bution law is adjusted by <strong>the</strong> programme.
113 GPKPL Realiza la misma función que el programa IIGQKOLII y<br />
dibuja las curvas.<br />
It performs <strong>the</strong> same function that <strong>the</strong> "GOKOL"<br />
programme and draws <strong>the</strong> curves.<br />
114 GPK 70 Dada una serie de datos hidrológicos, ajusta una <strong>le</strong>y de<br />
Goodrich y contrasta la bondad del ajuste mediante el<br />
test de Kolmogoroff.<br />
181<br />
Given a series of hydrological data, it fits with Goodrich's<br />
law and contrasts <strong>the</strong> perfection of <strong>the</strong> fitting by Kolmogo<br />
roff ' s tes t.<br />
115 GUMB 1 Ajusta una <strong>le</strong>y de Gumbel a una serie de datos hidrológi-<br />
cos. El programa nos da los diversos valores que resul-<br />
tan de la <strong>le</strong>y de Gumbel, ajustada para tiempos de recu-<br />
rrencia de 5, 10, 25, 50, 100, 500 y 1000 anos ylista --<br />
los datos origina<strong>le</strong>s clasificados de menor a mayor, asig<br />
nándo<strong>le</strong>s a cada uno la frecuencia 2n-1 / 2 N, donde n es<br />
el número de orden y N el total de datos.<br />
Datos de entrada (12 F. 6. 2). Anua<strong>le</strong>s.<br />
This programme fits <strong>the</strong> Gumbel's law to a series of --<br />
hydrological data. The programme gives us several values<br />
according to <strong>the</strong> Gumbel's adjusted law, for time of --<br />
recurrences 5, 10, 25, 50, 100, 500, 1000 years and it<br />
lists <strong>the</strong> original data classified in a crecent order assign-<br />
ing to each one a frecuency equal to 2n- 1 / 2 N, where n is<br />
<strong>the</strong> number of order and N <strong>the</strong> total number of data.<br />
Entry data (12 F 6. 2). Yearly.<br />
116 GUMB 2 Realiza la misma función que el programa "GUMB l", pg<br />
ro los datos de entrada son (9 F 8.3).<br />
It performs <strong>the</strong> same function as "GUMB l", but <strong>the</strong> entry<br />
data are (9 F 8. 3).
182<br />
117 GUMB 3<br />
118 GUMB 4<br />
119 GUMB-P<br />
Realiza la misma función que el programa "GUMB 1".<br />
pero los datos de entrada son mensua<strong>le</strong>s.<br />
It performs <strong>the</strong> same function as <strong>the</strong> "GUMB l", but <strong>the</strong><br />
entry data are monthly data.<br />
Ajusta una <strong>le</strong>y de Gumbel a una serie de datos mensua-<br />
<strong>le</strong>s. Obtiene mensua<strong>le</strong>s y máximos anua<strong>le</strong>s.<br />
This programme fits <strong>the</strong> Gumbel's law to a series of<br />
monthly data. It obtains also monthly and maximum<br />
annual series.<br />
Realiza la misma función que el programa "GUMB 1"<br />
y además dibuja la nube en el Plotter.<br />
It performs <strong>the</strong> same function as <strong>the</strong> programme "GUMB 1"<br />
and draws <strong>the</strong> clouds of points in <strong>the</strong> Plotter.<br />
120 C-C-D Dados los datos diarios de aportaciones natura<strong>le</strong>s, se ob<br />
tienen datos diarios de aportaciones derivadas para dis-<br />
tintos cauda<strong>le</strong>s de derivación, con los que se obtienen ds<br />
tos mensua<strong>le</strong>s de aportaciones natura<strong>le</strong>s y derivadas. Con<br />
estas parejas de datos se ajustan unas curvas, que sirven<br />
para obtener datos mensua<strong>le</strong>s de aportaciones derivadas<br />
cuando sólo se tengan datos mensua<strong>le</strong>s de aportaciones -<br />
natura<strong>le</strong>s.<br />
Given <strong>the</strong> daily data of natural affords, <strong>the</strong> programme -<br />
obtains daily data of derived affords for different flows -<br />
of derivation by which are obtained monthly data of natural<br />
and derived affords; with <strong>the</strong>se pairs of data, some lines<br />
are adjusted, which are used to obtain monthly data of<br />
derived affords, when only monthly data of natural affords<br />
are had.
183<br />
1 .21 GAMMA Dada una se<strong>le</strong>cción de 10 valores equidistantes de la fun_<br />
ciÓn g amma de X (con 16 cifras significativas) y sus seis<br />
primeras diferencias en el intervalo (1, 2), obtiene por<br />
interpelación cualquier g amma de X.<br />
Given a se<strong>le</strong>ction of 10 equidistant values with <strong>the</strong> func-<br />
tion g amma of X (with 16 significative digits) and <strong>the</strong>ir<br />
six first differences in <strong>the</strong> interval (1. 2), <strong>the</strong> programme<br />
obtains by interpolation any gamma of X.<br />
122 NUMRE Dada una se<strong>le</strong>cción de 10 valores equidistantes de n y<br />
sus seis primeras diferencias en el intervalo (-0. 75,<br />
4.25), obtiene la función inversa de la función de Good-<br />
rich.<br />
Given a se<strong>le</strong>ction of 10 equidistant values of n and <strong>the</strong>ir<br />
six first differences in <strong>the</strong> interval (-0. 75, .4. 25), <strong>the</strong><br />
programme obtains <strong>the</strong> inverse function of <strong>the</strong> Goodrich<br />
function.<br />
123 AM-C 4P Ajusta una <strong>le</strong>y de frecuencias parabólicas a los diez m e-<br />
nores valores de una serie de aportaciones clasificadas,<br />
sacando el valor de la aportación correspondiente a una<br />
garantía dada.<br />
The programme fits a law of parabolic frecuency to <strong>the</strong><br />
ten <strong>le</strong>ast values of a series of classified affords, obtain-<br />
ing <strong>the</strong> value of <strong>the</strong> afford corresponding to a given - -<br />
guarantee.<br />
124 CD SOO Calcula las aportaciones acumuladas correspondientes a<br />
una garantía dada, obteniendo la curva de seguridad.<br />
It evaluates <strong>the</strong> accumulated affords corresponding to a<br />
given guarantee, obtaining <strong>the</strong> safety lines.
184<br />
125 CD SO1<br />
126 CD 502<br />
Calcula las aportaciones acumuladas durante un afio,<br />
obteniendo la curva de seguridad.<br />
It calculates <strong>the</strong> accumulated affords during a year,<br />
obtaining <strong>the</strong> safety line.<br />
Calcula las aportaciones acumuladas, después las clasi-<br />
fica considerando los siguientes periodos:<br />
Oct Nov Set<br />
Oct + 1 Nov + 1 Set + 1<br />
_ _ _ _ - - - - - - - - - - -<br />
Ott + k NO~ + k Set + k<br />
y obtiene después las aportaciones acumuladas corres -<br />
pondientes a una garantia determinada. Posteriormente,<br />
calcula las demandas acumuladas para los mismos peri2<br />
dos y seguidamente las superficies evaporantes corres-<br />
pondientes a un volumen cualquiera vi, tomando como sg<br />
perficie evaporante en un mes la media aritmética de las<br />
correspondientes al estado inicial y final, y aplicándolo<br />
a la evaporación unitaria mensual.<br />
A partir de estas pérdidas obtiene, para la curva de ga-<br />
rantia dada, las pérdidas tota<strong>le</strong>s para cada periodo.<br />
Finalmente, calcula por iteración la curva de seguridad<br />
por meses, según la ecuación<br />
siendo :<br />
Ei (G) = volumen embalsado al principio del mes (i)<br />
para las curvas de garantia G.<br />
Dik<br />
Pik<br />
= demanda real acumulada desde el principio<br />
del mes (i) durante k meses sucesivos.<br />
= pérdidas del embalse acumuladas desde el<br />
principio del mes (i) durante k meses suce-<br />
sivos.<br />
Aik (G) = aportación del embalse acumulada desde el<br />
principio del mes (i) durante k meses suce-<br />
sivos, que tiene una probabilidad G de ser<br />
superada.
185<br />
It evaluates <strong>the</strong> accumulated affords, and afterwards it<br />
classifies <strong>the</strong>m considering <strong>the</strong> following periods:<br />
Oct Nov Set<br />
Oct t 1 Nov + 1 Set + 1<br />
_ _ _ _ _ _ _ - - - - - - - -<br />
Ott t k NO~ t k Set + k<br />
computing afterwards <strong>the</strong> accumulated affords corresponcj<br />
ing to a determined guarantee. Lately, it evaluates <strong>the</strong> -<br />
accumulated demands for <strong>the</strong> same periods and <strong>the</strong>reafter<br />
<strong>the</strong> evaporating areas corresponding to a given volume vi,<br />
taking as <strong>the</strong> evaporating areas in one month, <strong>the</strong> arithm-<br />
etical mean corresponding to <strong>the</strong> initial and final state -<br />
and applying this to <strong>the</strong> unitary monthly evaporation.<br />
Starting from this losses, <strong>the</strong> programme obtains for <strong>the</strong><br />
given line of guarantee, <strong>the</strong> total losses to each period.<br />
Finally, it evaluates iteratively <strong>the</strong> safety line by months,<br />
according to <strong>the</strong> equation<br />
where:<br />
Ei (G) = stored volume at <strong>the</strong> beginning of <strong>the</strong> month (i)<br />
for <strong>the</strong> guarantee line G.<br />
Dik<br />
'ik<br />
= actual accumulated demand since <strong>the</strong> begin-<br />
ning of <strong>the</strong> month (i) during k consecutive -<br />
months.<br />
losses in <strong>the</strong> reservoir accumulated, since<br />
<strong>the</strong> beginning of <strong>the</strong> month (i) during k conse-<br />
cutive months.<br />
Aik (G) = accumulated afford in <strong>the</strong> reservoir since <strong>the</strong><br />
beginning of <strong>the</strong> month (i) during k consecutive<br />
months, that has a G probability of being over<br />
pas sed.
186<br />
127 CDSSE<br />
128 REGCV<br />
Calcula las curvas de seguridad de un embalse, para -<br />
cualquier nivel de garantía de suministro de una deman<br />
da dada en función de la serie histórica de aportaciones<br />
de hasta 60 anos de duración.<br />
It evaluates <strong>the</strong> safety lines of a reservoir, to any <strong>le</strong>vel<br />
of giiarantee of supply of a given demand in function of<br />
<strong>the</strong> teorica1 series of affords up to 60 years of duration.<br />
A partir de la serie de aportaciones mensua<strong>le</strong>s en un pun<br />
to, calcula las capacidades de embalse estricto mediante<br />
el método de las diferencias acumuladas, por la expresión<br />
c = q - Aki.<br />
El mismo programa distingue dos casos:<br />
a) Regulación a caudal constante<br />
Mediante el programa obtenemos el principio y la<br />
duración del período de vaciado del embalse, del<br />
intervalo de meses sucesivos que da el máximo VE<br />
lor positivo a la suma de las diferencias ni q - Aki,<br />
siendo qi el caudal minimo continuo garantizado -<br />
durante el periodo considerado. Calcula también el<br />
volumen medio regulado en % de la aportación m e-<br />
dia y en Hm3, el caudal regulado y las capacidades<br />
de embalse estrictas para asegurar estos cauda<strong>le</strong>s<br />
en tanto por ciento de la aportación media y en Hm3.<br />
b) Regulación con caudal variab<strong>le</strong><br />
En este caso el caudal para hacer el cálculo de la -<br />
regulación es variab<strong>le</strong> en cada uno de los meses. -<br />
El programa nos da el principio y la duración del -<br />
periodo de vaciado del embalse, volumen medio re<br />
gulado en 70 de A m y en Hm3, capacidades de embF2<br />
se estricto en 7' de A m y en Hm3 y, además, el nu-<br />
mero de Has. regab<strong>le</strong>s con los volúmenes medios -<br />
regulab<strong>le</strong>s.<br />
Starting from <strong>the</strong> series of monthly affords in a point, <strong>the</strong><br />
programme calculates <strong>the</strong> capacities of strict reservoir<br />
according to <strong>the</strong> method of <strong>the</strong> accumulated differences, -<br />
by <strong>the</strong> expression C z ni q - Aki.<br />
The same programme distinguishes two cases :
a)<br />
Regulation at a constant flow<br />
187<br />
By means of <strong>the</strong> programme we obtain <strong>the</strong> beginning<br />
and <strong>the</strong> duration of <strong>the</strong> period of emptying <strong>the</strong> resec<br />
voir, <strong>the</strong> interval of following months which gives<br />
<strong>the</strong> maximum positive value of <strong>the</strong> sum od differen_<br />
ces ni qi - Aki, being qi <strong>the</strong> minimum continuos -<br />
flow guaranteed during <strong>the</strong> period under consider-<br />
ation. It calculates also <strong>the</strong> average regulated - -<br />
volume in 70 of <strong>the</strong> average afford and in CU. Hm,<br />
<strong>the</strong> regulated flow and <strong>the</strong> strict capacities of --<br />
reservoir to assure <strong>the</strong>se flows in percentage of<br />
<strong>the</strong> average afford and in CU. Hm.<br />
b) Regulation with variab<strong>le</strong> flow<br />
In this case <strong>the</strong> flow to perform <strong>the</strong> calculation of<br />
<strong>the</strong> regulation is variab<strong>le</strong> in each month. The pro-<br />
gramme gives us <strong>the</strong> beginning and <strong>the</strong> duration of<br />
<strong>the</strong> emptying period of <strong>the</strong> reservoir, averages -<br />
regulated volume in 70 of A m and in CU. Hm, capa<br />
cities of <strong>the</strong> strict reservoir in 70 of Am and in CU.<br />
Hm in addition, <strong>the</strong> number of Has irrigab<strong>le</strong>s with<br />
<strong>the</strong> average regulab<strong>le</strong> volumes.<br />
129 REG25 Realiza la misma función que el programa "REGCV",<br />
pero la entrada de datos está calculada para que regu<strong>le</strong><br />
estaciones durante 25 horas.<br />
The programme performs <strong>the</strong> same function as <strong>the</strong> pro-<br />
gramme "REGVF", but <strong>the</strong> entry data is evaluated to<br />
regulate stations during 25 hours.<br />
130 REGVA Dada una serie histórica de aportaciones mensua<strong>le</strong>s, unos<br />
consumos, una serie de precipitaciones mensua<strong>le</strong>s sobre<br />
el cultivo, unas capacidades de embalse máximo muerto<br />
y dando distintos porcentajes del consumo, calcula las -<br />
variaciones de volumen embalsado, los dé€icits y verti-<br />
dos, después de abastecer los regadios con unos consu-<br />
mos determinados, a los que descuenta la precipitación<br />
sobre el cultivo. El programa tiene en cuenta la evapora<br />
ciÓn mensual del embalse.
188<br />
Given an historical series of monthly affords, some -<br />
consumptions, a series of monthly rainfall over <strong>the</strong> crop,<br />
a capacities of maximum dead reservoir and giving dif-<br />
ferent porcentages of <strong>the</strong> consumption, this programme<br />
calculates <strong>the</strong> variations of volume of <strong>the</strong> reservoir, <strong>the</strong><br />
deficits and emptying, <strong>the</strong>m to supply <strong>the</strong> irrigated land<br />
with a determined consumptions, deducting <strong>the</strong> rainfall<br />
on <strong>the</strong> cultivation. It has in consideration also <strong>the</strong> month-<br />
ly evaporation of <strong>the</strong> reservoir.<br />
131 REG-RA Estudia la regulación para riegos y abastecimientos de<br />
forma análoga al REGVA.<br />
This programme studies <strong>the</strong> regulation for irrigation<br />
and supply in <strong>the</strong> same way to <strong>the</strong> REGVA programme.<br />
132 REG-K2 Estudia la regulación conjunta de un sistema de embal-<br />
ses considerando evaporación, para lo que utiliza los -<br />
siguientes datos:<br />
a)<br />
Las series de aportaciones en uno, dos tres o c u ~<br />
tro embalses de los que se trata de efectuar una -<br />
regulación conjunta.<br />
3<br />
b) Los consumos mensua<strong>le</strong>s en Hm , suponiendo que<br />
se consume anualmente el 100% de la aportación -<br />
media de cada embalse.<br />
c)<br />
d)<br />
La evaporación mensual en cms.<br />
Las caracteristicas de los embalses, ecuaciones<br />
de las curvas alturas -volúmenes, superficies -voli<br />
menes, capacidad total y volumen de embalse - -<br />
muer to.<br />
El programa realiza entre las aportaciones de cada em-<br />
balse sorteos equiprobab<strong>le</strong>s de 5 en 5 años, hasta 1000<br />
años, y a las series de 50, 100, 150 - 1000 <strong>le</strong>s aplica el<br />
proceso de regulación conjunta, en hipótesis de consumo<br />
de diversos % de la aportación media, dando como res-<br />
tado el 70 de fallos en cada serie de anos para cada uno -<br />
de los embalses considerados.
133 REG-SU<br />
189<br />
It studies <strong>the</strong> compound regulation of a system of reser-<br />
voirs considering evaporation, usind <strong>the</strong> following data:<br />
a)<br />
The series of affords in one, two, three or four<br />
reservoirs of which it treats to realize a compound<br />
regulation.<br />
b) The monthly consumption in CU. Hm, supposing -<br />
that one hundred per cent of <strong>the</strong> average afford is<br />
used yearly in each reservoir.<br />
c) Monthly evaporation in cms.<br />
d)<br />
The characteristics of <strong>the</strong> reservoirs, equations<br />
of <strong>the</strong> height-volume lines, area-volumes, total<br />
capacity and volume of <strong>the</strong> dead reservoir.<br />
The programme performs among <strong>the</strong> affords of each<br />
reservoirs equi-probab<strong>le</strong> casting lots every 5 years<br />
until 1000 years, and to <strong>the</strong> series.50, 100, 150, 1000<br />
<strong>the</strong> programme uses <strong>the</strong> compound regulation process,<br />
in <strong>the</strong> hypo<strong>the</strong>sis of different 70 of consumption of <strong>the</strong><br />
average afford, given as a result <strong>the</strong> percentage of -<br />
failures in each series of years for each reservoir under<br />
consideration.<br />
Este programa estudia la regulación sucesiva de una se-<br />
rie de embalses, sin limitación de número, utilizando las<br />
curvas de regulación del programa anterior y en la hipó-<br />
tesis de que la capacidad de embalse se utiliza para reg5<br />
lar la aportación de la cuenca propia y los cauda<strong>le</strong>s no re<br />
gulados aguas arriba, obteniéndose como resultado los -<br />
volúmenes regulados por cuencas parcia<strong>le</strong>s y tota<strong>le</strong>s.<br />
This programme studies <strong>the</strong> sequential regulation of a<br />
series of reservoir, without limitation of number, using<br />
<strong>the</strong> regulation lines of <strong>the</strong> preceding programme and -<br />
under <strong>the</strong> hypo<strong>the</strong>sis that <strong>the</strong> capacity of reservoir is -<br />
used to regulate <strong>the</strong> afford of its own basin and <strong>the</strong> non<br />
regulated upstream flows, obtaining as a result <strong>the</strong> regu<br />
lated volumes by partial and total basins.
190<br />
134 REG-KI<br />
135 REG-K3<br />
EAM<br />
RYPJU<br />
RELLO<br />
RESE<br />
EBBE<br />
136 CMAR<br />
137 EAM<br />
Igual que el REG-KI sin embalse muerto.<br />
The same as REG-KI, but without dead reservoir.<br />
Realiza la misma función que el programa "REG-KI",<br />
calculando directamente varias hipótesis.<br />
It performs <strong>the</strong> same function as <strong>the</strong> programme "REG-<br />
K2", evaluating directly several hypo<strong>the</strong>sis.<br />
Dados los valores de abastecimiento, riegos en valor<br />
absoluto y 70, calcula los consumos mensua<strong>le</strong>s.<br />
Given <strong>the</strong> values of supply, irrigations in absolute value<br />
and percentage, it calculates <strong>the</strong> monthly consumption.<br />
Estudia la explotación de hasta seis embalses, interco-<br />
nectados entre ellos por una red principal de cana<strong>le</strong>s de<br />
conducción, que se simula mediante una malla de 24 nu<br />
dos.<br />
Utiliza una serie de aportaciones generadas por sorteo<br />
a<strong>le</strong>atorio teniendo en cuenta o no la autocorrelación de<br />
las aportaciones anua<strong>le</strong>s.<br />
Los Órdenes de desembalse se estab<strong>le</strong>cen en función de<br />
los vertidos probab<strong>le</strong>s de cada uno de los embalses y de<br />
la demanda a satisfacer. Se tiene en cuenta las pérdidas<br />
por evaporación en los embalses y la capacidad de las -<br />
conducciones.
191<br />
EAM It studies <strong>the</strong> development of up to six reservoirs, inter<br />
connected among <strong>the</strong>m by a principal net of channels of<br />
conduction, which are simulated by a mesh of 24 knots.<br />
The programme uses a series of generated affords by<br />
fortuitous casting lots having present or no <strong>the</strong> self-cor<br />
relation of yearly affords.<br />
The orders of emptying are established in relation to <strong>the</strong><br />
probab<strong>le</strong> emptying of each reservoir and of <strong>the</strong> demand<br />
to satisfy. Having present <strong>the</strong> losses by evaporation in<br />
<strong>the</strong> reservoir and <strong>the</strong> capacity of <strong>the</strong> conductions.<br />
138 RYPJU Este modelo simula la explotación y la producción ener-<br />
gética de un conjunto de aprovechamientos.<br />
Se aplica a un sistema de aprovechamientos (embalses y<br />
saltos hidroeléctricos) situados sobre dos rios en forma<br />
de Y, al cual se pueden añadir cauda<strong>le</strong>s regulados en --<br />
otras cuencas o detraer cauda<strong>le</strong>s regulados por el sist:<br />
ma.<br />
A partir de una serie de aportaciones generada por sor-<br />
teo a<strong>le</strong>atorio y teniendo en cuenta la autocorrelación, se<br />
estab<strong>le</strong>cen los Órdenes de desembalse en función de las<br />
demandas y de los vertidos probab<strong>le</strong>s en cada embalse.<br />
El modelo calcula las producciones en todos los saltos<br />
y la garantia de suministro de la demanda prevista.<br />
This model pretends <strong>the</strong> exploitation and energetic pro-<br />
duction of an assembly of utilizations.<br />
It is applied to a system of utilizations (reservoirs and<br />
hydroe<strong>le</strong>ctric waterfall) located on two rivers in <strong>the</strong> form<br />
of Y, to which it could be added regulated flows in o<strong>the</strong>r<br />
basins or take away flows regulated by a system.<br />
Starting from a series of affords, generated by fortuitous<br />
casting lots and having present <strong>the</strong> self-correlation, <strong>the</strong><br />
order of emptying in relation to <strong>the</strong> demand and <strong>the</strong> pro-<br />
bab<strong>le</strong> emptying in each reservoir is established.<br />
The model calculates <strong>the</strong> productions in all waterfalls<br />
and <strong>the</strong> guarantee of supply of <strong>the</strong> calculated request.
192<br />
139 RELLO<br />
140 RESE<br />
Este modelo simula la explotación coordinada de los re-<br />
cursos superficia<strong>le</strong>s y subterráneos.<br />
Supone la existencia de un embalse subterráneo del que<br />
se puede extraer un caudal uniforme prefijado, en fun-<br />
ción de los estados de los embalses del sistema.<br />
Utiliza una serie de aportaciones generadas por sorteo<br />
a<strong>le</strong>atorio, teniendo en cuenta la autocorrelación de las<br />
aportaciones anua<strong>le</strong>s y las curvas de seguridad de un -<br />
embalse equiva<strong>le</strong>nte a la suma de los embalses del sis-<br />
tema, determinadas mediante el programa CDSSE.<br />
La explotación se simula teniendo en cuenta las pérdidas<br />
por evaporación en los embalses y obtiene la garanda de<br />
suministro de la demanda de abastecimiento junto con -<br />
los valores de la extracción anual media del acuifero y<br />
del periodo de máxima duración de la extracción máxi-<br />
ma prevista.<br />
This model pretends <strong>the</strong> coordinated exploitation of su-<br />
perficial and underground resources.<br />
It assumes <strong>the</strong> existence of an underground reservoir<br />
from which a uniform flow can be extracted fixed in ad-<br />
vance, depending on <strong>the</strong> state of <strong>the</strong> system of <strong>the</strong> resec<br />
voir.<br />
It uses a series of generated affords by fortuitous casting<br />
lots, having present <strong>the</strong> self-correlation of <strong>the</strong> yearly<br />
affords and <strong>the</strong> safety lines of a reservoir equiva<strong>le</strong>nt to<br />
<strong>the</strong> sum of <strong>the</strong> system, which is determined by <strong>the</strong> pro-<br />
gramme SAFLI.<br />
The exploitation is simulated, having present <strong>the</strong> losses<br />
by evaporation in <strong>the</strong> reservoir, <strong>the</strong> guarantee of supply<br />
of <strong>the</strong> demand, toge<strong>the</strong>r with <strong>the</strong> values of <strong>the</strong> annual -<br />
average extraction and of <strong>the</strong> period of maximum calcu-<br />
lated extraction.<br />
Este modelo simula un sistema de explotación con varios<br />
embalses situados sobre una misma corriente, uno de -<br />
los cua<strong>le</strong>s puede ser el origen de un aprovechamiento -<br />
hidroeléctrico.<br />
La explotación se estab<strong>le</strong>ce a partir de una serie de apor<br />
taciones generadas por sorteo a<strong>le</strong>atorio y teniendo en -
141 EBBE<br />
193<br />
cuenta la autocorrelación de las aportaciones anua<strong>le</strong>s,<br />
en función de las curvas de seguridad de un embalse tg<br />
tal equiva<strong>le</strong>nte para atender a una demanda de usos cog<br />
suntivos. Al mismo tiempo que calcula la garantía de su<br />
ministro de la demanda prevista obtiene las produccio-<br />
nes históricas en todos los saltos, distinguiendo la enec<br />
gi’a de puntas y la energía producib<strong>le</strong> en horas l<strong>le</strong>nas en<br />
el periodo critico (nov. a feb. ) de máxima demanda ener<br />
gética. Ordena los valores de la energía optenidos y as{<br />
puede suministrar los valores de la energia de distinta<br />
calidad garantizada en el per
194<br />
of <strong>the</strong> calculated reservoirs. It calculates <strong>the</strong> values of<br />
<strong>the</strong> possib<strong>le</strong> overflows at <strong>the</strong> same time that it calculates<br />
<strong>the</strong> guarantee of supply of <strong>the</strong> calculated demands and <strong>the</strong><br />
energetic productions and consumptions in <strong>the</strong> hydroe<strong>le</strong>c<br />
trical utilizations of <strong>the</strong> basin.<br />
142 LAMI 1 Estudia la laminación de un embalse, supuesto un nivel<br />
inicial determinado y pudiendo utilizar uno o varios si-<br />
temas de desagües, en función de las caracteristicas -<br />
del embalse y de la crecida.<br />
It studies <strong>the</strong> lamination of a reservoir, supposing an<br />
initial <strong>le</strong>vel determined in advance and being possib<strong>le</strong><br />
<strong>the</strong> use of one or several drainage systems, in function<br />
of <strong>the</strong> characteristics of <strong>the</strong> reservoir and <strong>the</strong> flood.<br />
143 HIDR 1 Calcula el hidrograma para diversas hipótesis de inten-<br />
sidad horaria de precipitación, coeficiente de escorren-<br />
tia y duración de la tormenta.<br />
It evaluates <strong>the</strong> hydrogram for different hypo<strong>the</strong>sis of<br />
hourly intensity of rainfall, runoff coefficient and <strong>the</strong><br />
duration of <strong>the</strong> storm.<br />
144 HIDR 2 Calcula el hidrograma con intensidad y coeficiente de<br />
escorrenti’a corriente.<br />
It computes <strong>the</strong> hydrogram with normal intensity and<br />
usual runoff coefficient.<br />
145 ABC Realiza el estudio económico (análisis, beneficio y cos -<br />
to), expresándolo en forme de corrientes monetarias as
tualizadas en función de la tasa de descuento y obtiene<br />
la ratio<br />
Beneficio - Gastos<br />
costos<br />
195<br />
It performs <strong>the</strong> economic study (analysis, profit and<br />
price) expressing it in actual monetary currency in func-<br />
tion of <strong>the</strong> standard rate of deduction and it obtains <strong>the</strong><br />
ratio<br />
Profit - Expenses<br />
Prices<br />
146 ABC 10 Programa ABC para estudio económico de varias cen-<br />
tra<strong>le</strong>s.<br />
It programmes ABC for an economical study of several<br />
centrals.<br />
147 ABC TV Programa ABC para estudio económico, con tasa varia-<br />
b<strong>le</strong>.<br />
It programmes ABC for an economical study, with varia<br />
b<strong>le</strong> standard rate.<br />
148 BNZ Dado el número de muestra, tiempo de <strong>le</strong>ctura y número<br />
de desintegración, obtiene la concentración de Tritio pa-<br />
ra muestras de agua.<br />
Given <strong>the</strong> number of <strong>the</strong> samp<strong>le</strong>, <strong>le</strong>cture time and num-<br />
ber of disintegration, it obatains <strong>the</strong> concentration of -<br />
Tritium for samp<strong>le</strong> of water.
196<br />
149 PARAM Calcula los siguientes indices : RMG/RCA, RNA/RK,<br />
RNA/RCA, ANA/RMG, (RCA-RMG), RALC, BR/CL,<br />
RCL/RHCO3, RHCO3/RCL, RS04/RCL, (RCA + RHCOQ)/<br />
/ (RCH + RS04), RHC03/RHC03 t RS04 t RCL, RALC/RCL,<br />
RCA/RALC, SAR, ICB, ID, FI, Tipo de agua. Además,<br />
lista los e<strong>le</strong>mentos en meq/l. y en 70.<br />
It calculates <strong>the</strong> following indexes: RMG/RCA, RNA/RK,<br />
RNA/RCA, ANA/RMG, (RCA-RMG), RALC, BR/CL,<br />
RCL/RHC03, RHC03/RCL, RSOq/RCL, (RCA t RHCOS)/<br />
/ (RCH t RS04), RHC03/RHC03 + RSO4 + RCL, RALC/<br />
/ RCL, RCA/RALC, SAR, ICB, ID, FI, Type of water.<br />
Besides, it lists <strong>the</strong> e<strong>le</strong>ments in meq/l. and in %.<br />
150 HISTO Calcula los histogramas de las siguientes relaciones,<br />
clasificándolos en clases y valores fuera de clase:<br />
RCL/RS04, RCL/RHCO3, RALC/RCL, RNAIRCA, RCL,<br />
RSO4, RHC03, RN03, RALC, Res. seco, T.D.S., Dureza<br />
total. El histograma lo dibuja por impresora.<br />
It calculates <strong>the</strong> hystogram of <strong>the</strong> following relations,<br />
classifying <strong>the</strong>m in classes and values out of class:<br />
RCL/RS04, RCL/RHCO3, RALC/RCL, RNA/RCA, RCL,<br />
RSO4, RHC03, RN03, RALC, Res. (dry), T.D.S., total<br />
hardness. The hystogram is designed by <strong>the</strong> printer.<br />
151 TUBEC Dado un muestrario de tubedas de diferentes diámetros,<br />
con sus precios y caracteristicas hidráulicas,, determi-<br />
na para una configuración topológica y topografica de la<br />
red y para diversas hipótesis, la combinación de distri-<br />
bución de tubos más económica que permita el suminis-<br />
tro solicitado con la minima pérdida de carga.<br />
Given a samp<strong>le</strong> <strong>book</strong> of pipes of differents diameters,<br />
with <strong>the</strong>ir prices and hydraulical characteristics, it<br />
determines for a topological and topographical form of<br />
<strong>the</strong> system and for several hypo<strong>the</strong>sis, <strong>the</strong> combination<br />
of distribution of pipes more economical, that allow <strong>the</strong><br />
solicited supply with <strong>the</strong> minimum loss of loading.
197<br />
152 CANAL Definido un canal por sus secciones y pendientes en difg<br />
rentes tramos, as: como por distintos tipos de cornpuer_<br />
tas, el programa determina la evolución de los cauda<strong>le</strong>s<br />
transportados en el curso del tiempo, as: como los cala<br />
dos alcanzados en los distintos tramos del canal.<br />
Permite estudiar las maniobras de apertura y cierre de<br />
compuertas más convenientes para la explotación del cg<br />
na1 .<br />
Defined a canal by its sections and pendants in different<br />
stretchs, as well as by different types of floodgates, -<br />
this programme determines <strong>the</strong> evolution of <strong>the</strong> flows<br />
carried in <strong>the</strong> course of time, as well as soakage reach-<br />
ed in <strong>the</strong> different stretchs of <strong>the</strong> canal.<br />
It allows also to study <strong>the</strong> process of opening and closing<br />
of floodgates more convenient to exploitation to <strong>the</strong> canal.<br />
153 SER-EL Depura los datos suministrados por las empresas hidro-<br />
eléctricas relativos a la producción mensual de energi'a<br />
de los diferentes saltos de cada una, recogidos en tarje<br />
tas perforadas. La depuración se hace verificando la -<br />
concordancia de los datos geográficos, número de horas<br />
de utilización de los contro<strong>le</strong>s en función de la potencia<br />
instalada y producción.<br />
Una vez corregidos todos los errores detectados, pre-<br />
para unos cuadros resúmenes estadisticos de producción<br />
de energTa, clasificados por diferentes conceptos :<br />
Producciones globa<strong>le</strong>s por cuencas hidrográficas en ca-<br />
da mes.<br />
Producciones globa<strong>le</strong>s UNESA, IN1 y otr os en cada mes.<br />
Producciones anua<strong>le</strong>s clasificadas por:<br />
Empresa o concesionario.<br />
Centra<strong>le</strong>s por magnitud de su producción.<br />
Centra<strong>le</strong>s y cuencas por magnitud de su produc-<br />
cion.<br />
Centra<strong>le</strong>s y rios por magnitud de su producción.<br />
Centra<strong>le</strong>s y provincias por magnitud de su produc-<br />
ción.
198<br />
It purifies <strong>the</strong> supplied data by hydroe<strong>le</strong>ctrical companies<br />
relatives to monthly productions of energy of <strong>the</strong> diffe--<br />
rents waterfalls of each one, col<strong>le</strong>cted in perforated cards.<br />
The c<strong>le</strong>ansing is done verifying <strong>the</strong> harmony of geographi-<br />
cal data, number of hours of utilization of controls in -<br />
function of instal<strong>le</strong>d power and production.<br />
Once corrected all <strong>the</strong> detected errors, it prepares a<br />
summary of statistical charts of production of energy,<br />
classified by different ideas :<br />
Total productions for hydrographical basin in each month.<br />
Total productions UNESA, INI, and o<strong>the</strong>rs in each month.<br />
Annual production classified by:<br />
Company or dea<strong>le</strong>r.<br />
Centrals by magnitude of production.<br />
Centrals and basins by magnitude of production.<br />
Centrals and rivers by magnitude of production.<br />
Centrals and provinces by magnitude of production.
COMPUTATION OF RESERVOIRS SEDIMENTATION<br />
A.V. Karaushev, I.V. Bogoliubova<br />
State Hydrologic a 1 Institut e<br />
Leningrad, USSR<br />
-- ABSTRACT<br />
Methods for <strong>the</strong> computation of sedimentation by suspended<br />
sediments and bed load of <strong>the</strong> projected reservoirs are given,<br />
or <strong>the</strong> first year Of <strong>the</strong> reservoir operation computation is<br />
made according to <strong>the</strong> balance of sediments computed by <strong>the</strong><br />
difference between <strong>the</strong> transport capacity and <strong>the</strong> hydraulic<br />
parameters of <strong>the</strong> current at <strong>the</strong> upper pool (transient region)<br />
and at <strong>the</strong> dan of <strong>the</strong> reservoir. The subsequent attenuation of<br />
<strong>the</strong> process as well as <strong>the</strong> total duration of sedimentation is<br />
evaluated by empirical relations obtained from <strong>the</strong> observational<br />
data on reservoirs under operation.<br />
RESUME<br />
Les auteurs exposent des méthodes pour <strong>le</strong> calcul de<br />
l'envasement des barrages par <strong>le</strong>s matériaux transportés en<br />
charriage ou en suspensión. On fait, pour la première année<br />
d'exploitation, <strong>le</strong> bilan des matériaux déposés dans la retenue<br />
par différence entre ce qui entre à l'amont (station de mesure<br />
dar,s la zo~e du remous) et ce qui sort par <strong>le</strong> barrage; ces<br />
mesures sort reliées aux paramètres hydrauliques du cours d'eau.<br />
u', extrapo<strong>le</strong> <strong>le</strong>s résultats dans <strong>le</strong> futur en utilisant des<br />
relations empiriques obtenues pour d'autres réservoirs en cours<br />
! 1 exploit at ion.
200<br />
The construction of reservoirs in mountain areas and at <strong>the</strong><br />
foothills on rivers with a considerab<strong>le</strong> sediment concentration<br />
inevitably faces with <strong>the</strong> necessitg to remove or to impede sediments<br />
transported by <strong>the</strong> river to keep <strong>the</strong> projected capacity<br />
of <strong>the</strong> reservoir. The present paper gives methods accepted in<br />
<strong>the</strong> USSR providing <strong>the</strong> evaluation of possib<strong>le</strong> sedimentation rate<br />
for <strong>the</strong> <strong>who<strong>le</strong></strong> reservoir or for its individual parts during <strong>the</strong><br />
first year of its operation and for subsequent years.<br />
Methods for <strong>the</strong> computation of re semoirs sedimentat ion are<br />
based on <strong>the</strong> equation of sediments balance applied to <strong>the</strong> <strong>who<strong>le</strong></strong><br />
reservoir or its parts, to <strong>the</strong> gross composition of <strong>the</strong> transported<br />
sediments or its particular fractions. The use of this equation<br />
makes it possib<strong>le</strong> to compute <strong>the</strong> difference between sediments inflow<br />
and its discharge out of <strong>the</strong> reservoir i.e. sediments<br />
accumulation. The inflow of sediments is computed by observational<br />
out of <strong>the</strong> reservoir<br />
data or by indirect methods. The discharge of sediments is estimated<br />
by equations of <strong>the</strong> transporting capacitg of <strong>the</strong> current at<br />
<strong>the</strong> specified values of water discharge Q, mean depth Hm, mean<br />
current velocity Vm, and granulometaAc sediment composition.<br />
The computation of sedimentation during one year is reduced<br />
by <strong>the</strong> determination of that portion of sediment discharge<br />
which is accumulated in <strong>the</strong> reservoir. When starting computation<br />
it is essential to establish design values of annual water<br />
discharge, of suspended sediments and bed load, as well as typical<br />
chronological graphs of <strong>the</strong>se values for <strong>the</strong> inflow site of<br />
<strong>the</strong> reservoir. It is recommended to divide <strong>the</strong> hydrograph of <strong>the</strong><br />
typical year into 3 or 4 design time intervals& and to compute<br />
sediments accumulation according to <strong>the</strong> values o $ Q, Vm Hm, etc.,<br />
averaged for every time interval. The computation of se Aimentation<br />
rate is made by individual fractions i, in this case it is<br />
sufficient to subdivide all <strong>the</strong> transported fractions into 3 or<br />
5 categories. Then sediments are summarized according to all<br />
categories of <strong>the</strong> fractions.<br />
The computation of sedimentation by suspended fractions for<br />
a design interval A tj is niade by equation:<br />
where: Pa j is <strong>the</strong> amount of sediments of all <strong>the</strong> fractions<br />
(tons) in <strong>the</strong> reservoir or in <strong>the</strong> design area dur A t ;<br />
*i in J is inflow of sediments of <strong>the</strong> i-th fractio3tonsg<br />
durmg time ~t through <strong>the</strong> initial (upper) discharge site of<br />
<strong>the</strong> reservoir $ or its part)determined by <strong>the</strong> chronological<br />
graph or by computation das made for <strong>the</strong> upstream area; Qter<br />
is mean water discharge (<br />
sec) for time at through <strong>the</strong><br />
terminal (downstream) discharge site of <strong>the</strong> Jeservoir ( at <strong>the</strong><br />
dam) or <strong>the</strong> design area; Si t is mean particular turbidity<br />
for time At. of <strong>the</strong> i-th fractfoi at <strong>the</strong> terminal discharge site<br />
of <strong>the</strong> resehoir (certain area) (g/m3); ~ t is j time interval<br />
(sec).
202<br />
Turbidity of <strong>the</strong> i-th fraction at <strong>the</strong> terminal discharge<br />
site Si ter j is computed by equation of A.T. Karaushev:<br />
- G"AL<br />
(2)<br />
where: Si 4" J Ois particular turbidity at <strong>the</strong> initial discharge<br />
site mean or time interval B ta; S? is <strong>the</strong> turbidity<br />
corresponding to a particular thins$of%idg capacity of <strong>the</strong> current<br />
computed by equation (6) given below; e is <strong>the</strong> base of natural<br />
l2gar ith;<br />
G is dimension<strong>le</strong>ss value determined by equation<br />
where: ui is fall velocity of fraction i under consideration;<br />
kg is a parameter having a dimensionality of velocity and which<br />
is computed by equation<br />
LL; ri<br />
(4)<br />
The value of r is <strong>the</strong> value of hydroneclnanic param?.t;er<br />
of sediments whichi= be obtained by graphs according to <strong>the</strong><br />
Chezygs coefficient C and ratio of *i (Fig. 1 ). In equation (3)<br />
AL indicates <strong>the</strong> <strong>le</strong>ngth<br />
-<br />
of <strong>the</strong> reservoir (or it8 part) given<br />
In relative units:<br />
A L<br />
AL =----<br />
Hm (5)<br />
where: AL is <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> reservoir (or design area (m);<br />
IL, is mean depth of <strong>the</strong> reservoir ( or some area), (m) for time<br />
iIltel?ral A t*o<br />
A particulAr transporting aapacj. of <strong>the</strong> current S: tr -<br />
(for <strong>the</strong> i-th fraction of i8 CO uted with <strong>the</strong> %se of<br />
data on bed load composition. The value of 8 tr j is computed<br />
with <strong>the</strong> use of hydraulic e<strong>le</strong>ments of <strong>the</strong> current mean for time<br />
interval A tj related to <strong>the</strong> <strong>who<strong>le</strong></strong> reservoir or its desim area:<br />
Here a<br />
actual<br />
indicates a correcting factor estimated by <strong>the</strong> ratio of<br />
and computed turbidity at <strong>the</strong> initial discharge site:<br />
SS4. r a= --<br />
s cokvlvip (7)
2 02<br />
d is composition ( in per cent) of <strong>the</strong> i-th weighted<br />
f&&&o& in <strong>the</strong> roiling portion of bed load.<br />
The value 0% droil i is determined by <strong>the</strong> ratio<br />
--<br />
IC0 , (8)<br />
roil i - t CL bed i<br />
where A is <strong>the</strong> portion (per cent) of <strong>the</strong> i-th fraction in<br />
bed load &&&sition; r is gross portion ( in per cent) of <strong>the</strong><br />
weighted fraction in bed load composition. In this case sediments<br />
wPth fall velocity corresponding to <strong>the</strong> condition u 4 1- are<br />
regarded as weighted fractions, and Vi indicates maxmum value<br />
of <strong>the</strong> vertical component of <strong>the</strong> puls%on velocitg. The latter<br />
salue is computed by a special equation according to mean velocity<br />
of <strong>the</strong> current and Chew's coefficient. Gross turbidity of roiling<br />
(Sroil) is obtained by equation:<br />
where: N is characteristic dimension<strong>le</strong>ss number depending on<br />
Chew's eoefficient C; is <strong>the</strong> ratio of velocity at <strong>the</strong> bottom<br />
go mean veloci%y; <strong>the</strong> ! est of <strong>the</strong> symbols are given in previous<br />
equations, *<br />
When comguting.Si tr for <strong>the</strong> first time interval <strong>the</strong> composition<br />
of bed load xn <strong>the</strong> reservoir is accepted according to <strong>the</strong><br />
averaged conposition of river alluvium in <strong>the</strong> cñannel and in <strong>the</strong><br />
f lood-plain; for subsequent intervals it is essential to consider<br />
<strong>the</strong> composition of sediments obtained by computations.<br />
If sedimentation is computed for certaih areas, <strong>the</strong>n Si 8' j<br />
estimated by equation (2) for <strong>the</strong> first (upper) area 1s use as<br />
Si in j for <strong>the</strong> second area domstream, etc.<br />
The computation of reservoir sedimentation by bed load is made<br />
according to <strong>the</strong> same design intemals a8 by suspended sediments.<br />
For an approximate evaluation it i8 possib<strong>le</strong> to be confined to<br />
<strong>the</strong> computation for flood periods when <strong>the</strong> major portion of coarse<br />
fractions flows into <strong>the</strong> reservoir.<br />
The amount of bed load in <strong>the</strong> reservoir is determined by <strong>the</strong><br />
difference:<br />
(10)<br />
where: Pa bed is <strong>the</strong> weight of bed load in <strong>the</strong> reservoir (tons);<br />
n t. is time interval (sec); Rbed in<br />
J<br />
and Rbed ter jindicate<br />
bed load discharge at <strong>the</strong> initial and termi.mil discharge sites<br />
(kg/sec) nean for design time intervalat.. For bed load discharge<br />
computation it is reasonab<strong>le</strong> to recommed $he equationsof G.I.<br />
Shamov, VaNe Gomharov, I.V. Egiaearov, K.I. Rossinski, et al. The<br />
equation of G.I. Shamov is <strong>the</strong> most simp<strong>le</strong> one providing a suffici-
203<br />
ent coincidence of computation results with <strong>the</strong> data of measure-<br />
ments at a wide range of fractions dimensions:<br />
where: Rbed is bed load discharge (kg/sec); B is current width<br />
(m); Hm is mean depth (m); a, is mean diameter of mobi<strong>le</strong> partic<strong>le</strong>s<br />
of bed load (m); Vm is mean velocity of <strong>the</strong> current (m/sec); vsed<br />
is mean velocitg of <strong>the</strong> current (m/sec) when fractionswith Q<br />
in diameter stop moving; K is coefficient consider- non-homo-<br />
geneitg of bed load composition.<br />
The value of is computed by equation:<br />
WL (12)<br />
d =c~oíz.i.ul!<br />
i=/ L '<br />
WL<br />
where: d, and d. res ectively indicate percentage and mean<br />
diameter o) a certain ?i-th) fraction; summation is made accord-<br />
ing to all <strong>the</strong> mobi<strong>le</strong> fractions, <strong>the</strong> number of <strong>the</strong>se fractions<br />
is-indicated as<br />
Separation of<br />
equation:<br />
m.<br />
immobi<strong>le</strong> (coarse) fraction is made by<br />
3<br />
9,o 12 'yl<br />
-<br />
4,t -<br />
i-<br />
L H,<br />
-<br />
The value of Vsed is obtained by equation:<br />
fis<br />
= 3,) d, H ~<br />
(13)<br />
( 14)<br />
All <strong>the</strong> computations of bed load transport are made accord-<br />
ing to hydraulic e<strong>le</strong>ments mean for time interval A tj.<br />
Annual accumulation of all <strong>the</strong> sediment fractions for <strong>the</strong><br />
first year of reservoir operation is determined by equation:<br />
where: Pai is gross sediment weight for <strong>the</strong> 1st year (tons);<br />
and Pa bed j respeCtiVely indicate <strong>the</strong> Wight of suspend-<br />
88 g%hde.ents and bed load for <strong>the</strong> 1st year (tons)j j is <strong>the</strong> number<br />
of design internali n is <strong>the</strong> number of intervals during a year.<br />
If computation 1s made according to certain areas, <strong>the</strong>n<br />
summation is made for all <strong>the</strong> areas to obtain gross sedimentation<br />
of <strong>the</strong> reservoir. The obtained value for <strong>the</strong> <strong>who<strong>le</strong></strong> of <strong>the</strong> reservoir<br />
is transformed into volumetric units:
204<br />
where: Wai is <strong>the</strong> volume of sediments during <strong>the</strong> 1st ear (m 3 );<br />
P is <strong>the</strong> weight of sediments for <strong>the</strong> 1 t year (tons5; fs is<br />
&e volumetric weight of sediments ( t/m 3 ). After <strong>the</strong> computation<br />
being done <strong>the</strong> initial volume of <strong>the</strong> reservoir W is corrected.<br />
The obtained volume of <strong>the</strong> reservoir at <strong>the</strong> end of <strong>the</strong> 1st year<br />
W, = W - Wa<br />
is used for <strong>the</strong> computation of sedimentation for <strong>the</strong><br />
next year. $he computation of sedimentation for subsequent years<br />
may be performed in <strong>the</strong> same way as for <strong>the</strong> 1st year or by<br />
extrapolation equations considering time attenuation of sedimenta-<br />
tion.<br />
It is recommended to use <strong>the</strong> equation of G.I. Shamov for <strong>the</strong><br />
computation of chronological variations of sedimentation:<br />
where: Wa t is <strong>the</strong> volume of Sedimentation (m3) in t years;<br />
M'ad is sedimentation volume during <strong>the</strong> 1st year (m3) computed<br />
by <strong>the</strong> methods described above; Wa ext is <strong>the</strong> extreme volume<br />
of sediments in reservoir (m3) approximately computed by<br />
e quat - ion:<br />
3<br />
where: W is <strong>the</strong> initial volume of <strong>the</strong> reservoir (m ); Wtis<br />
area of river cross section (m2) when dischar e is close to<br />
maximm;Ldp is <strong>the</strong> maximum cross section area fm2) of <strong>the</strong> upper<br />
pool near <strong>the</strong> dam.<br />
The method is applicab<strong>le</strong> to <strong>the</strong> reservoir as a <strong>who<strong>le</strong></strong>.<br />
One year should be accepted as a design time interval. In<br />
case of correct initial values <strong>the</strong> method provides variations<br />
of reservoir sedimentation close to actual.<br />
REFERENCES<br />
1. Bogoliubova I.V. Resultam po<strong>le</strong>vykh iss<strong>le</strong>dovaniy i rascheta<br />
stoka v<strong>le</strong>komykh nanosov r. Mzymty (The results of field<br />
investigations and bed load discharge computation for <strong>the</strong><br />
Mzymta river) Trans. of <strong>the</strong> State Hydrological Inst.,<br />
1968, ~01. 156, PP* 3943.<br />
2. Karauschev A .V . Rechnaya gidravlika (River hydraulic 8) .<br />
Leningrad, ñydrometeorological Publishing House, 1969,<br />
PP. 303-778.<br />
36 Razumikhina K.V . Primenenie formuly transportiruyushchei<br />
sposobnosti dlia rascheta godovogo stoka vzveshennykh<br />
nanosov (Application of transporting capacity equation<br />
for <strong>the</strong> computation of annual discharge of suspended<br />
sediments). Trans. of <strong>the</strong> State Hydrological Inst., 1969,<br />
vol. 175, ppe 137-154.<br />
4. Ukasania PO raschetu eai<strong>le</strong>nia vodokhranilishch pri strof.t;elnom<br />
proektirovanii (Instructions for <strong>the</strong> computation<br />
of reservoirs sedimentation for engineering projects).<br />
Leningrad, Qdrometeorological Publishing House, 1968,<br />
54 PP.<br />
5. Shamov G.I. Recbnye nanoqy (River sediments). Leningrad,<br />
Hydrometeorological Publishing House, 1959, pp. 2-282.
c<br />
Figure 1. - Computation of reservoirs sedimentation<br />
205
ABSTRACT<br />
CALCULATION OF RUNOFF IN IRAQ<br />
R.K. KLIGUE, MECHDI EL SACHOB<br />
There given a general characteristic of runoff in<br />
Iraq and its distribution within <strong>the</strong> area of <strong>the</strong> country.<br />
The authots analyse correlation of runoff with e<strong>le</strong>vation<br />
of <strong>the</strong> territory, river basin area and o<strong>the</strong>r factors with<br />
<strong>the</strong> aim of using <strong>the</strong>se relationships for regions without<br />
runoff data. Hydrologic time series analysis of runoff<br />
and analysis of runoff fluctuations through <strong>the</strong> territory<br />
are cited.<br />
RES UME<br />
On donne la caractéristique généra<strong>le</strong> d'écou<strong>le</strong>ment<br />
fluvial en Irak et sa distribution par la territoire. Sont<br />
analisdes <strong>le</strong>s relations entre écou<strong>le</strong>ment fluvial, l'altitude<br />
de lieu, l'aire de surface réceptrice et d'autres facteurs.<br />
L'objectif de cette analyse d'est utilisation de ces<br />
relations pour <strong>le</strong>s régions avec l'absence des données<br />
d'écou<strong>le</strong>ment. On fait l'analyse des séries hydrologiques<br />
et des variations de débits par la territoire.
208<br />
The investigation of river flows in Iraq is of great<br />
importance keeping in view <strong>the</strong> constantly increasing water<br />
balance stress in <strong>the</strong> country that arouses <strong>the</strong> necessity to<br />
design special rnultipurpose projects as well as projects<br />
aimed at more comp<strong>le</strong>te use of water resources by construct-<br />
ing irrigation systems, overyear storage reservoirs, by im-<br />
proving crop management under irrigation and by wider use<br />
of groundwaters.<br />
The main rivemof Iraq - <strong>the</strong> Tigris and Euphrates -<br />
cross <strong>the</strong> oountry by <strong>the</strong>ir midd<strong>le</strong> and lower reaches. Con-<br />
fluencing <strong>the</strong>y form a river cal<strong>le</strong>d Shatt-al-Arab flowing<br />
into <strong>the</strong> Perbian Gulf. The main tributaries of <strong>the</strong> Tigris<br />
within Iraq are <strong>the</strong> Greater Zab, Lesser Zab, Adhaim and<br />
Diyala. The Euphrates river have no tributaries on <strong>the</strong> ter-<br />
ritory of <strong>the</strong> country. The arid regions are characterized<br />
by <strong>the</strong> existence of "wadi".<br />
Water resources of Iraq are mainly determined by <strong>the</strong><br />
flow of <strong>the</strong> Tigris and Euphrates rivers making about 77.7 km 3<br />
3 3<br />
a year - about 22.2 km flows into <strong>the</strong> sea and 55.5 km<br />
(71.4%) is used for irrigation, municipal and industrial<br />
water supply and power generation. A considerab<strong>le</strong> part of<br />
flow is lost due to evaporation, transpiration and filtra-<br />
tion.<br />
The mean annual flow of <strong>the</strong> Euphrates on entering <strong>the</strong><br />
territory of Iraq is 928 cumecs decreasing downstream (Na -<br />
\I<br />
siriya) to 454 cumecs. Thus, <strong>the</strong> rate of flow changes along<br />
<strong>the</strong> river <strong>le</strong>ngth from 3.52 to 1.57 1/s. km'.<br />
The Tigris river on <strong>the</strong> territory of Iraq has several
ig tributaries, which increase its flow from 587 cumecs<br />
209<br />
at Tusan to 1534 cumecs at Salman-Pak. Downstream <strong>the</strong> flow<br />
decreases due to intensive withdrawal for irrigation and<br />
makes 49.6 cumecs.at Qalat-Sa<strong>le</strong>h. The mean annual rate of<br />
flow in <strong>the</strong> Tigris basin changes from 12.7 l/s.km<br />
2<br />
in <strong>the</strong><br />
upper part to 0.26 l/s.km<br />
2<br />
(Qalat-Sa<strong>le</strong>h) in <strong>the</strong> lower<br />
part .<br />
The coefficient of annual flow variation (C,) for <strong>the</strong><br />
Tigris and Euphrates changes from 0.26 to 0.31 decreasing<br />
with <strong>the</strong> altitude of watershed (Haam, m). It can be ex -<br />
pressed by a correlation:<br />
The flow of <strong>the</strong> Tigris and Euphrates is distributed<br />
very uneven through a yew' - <strong>the</strong> greater part of it falls on<br />
<strong>the</strong> flood period (April-May), making about eo"/4 in <strong>the</strong> upper<br />
reaches and 5% - in <strong>the</strong> lower reaches.<br />
The beginning of flood period is closely dependent on<br />
<strong>the</strong> mean altitude of watershed m) and may occur in<br />
January-April. The mean duration of flood period in <strong>the</strong><br />
piedmont regions is 45 days, in <strong>the</strong> midd<strong>le</strong> mountain regions -<br />
90 days and in high mountain - 135 days. The more is <strong>the</strong><br />
water availability through <strong>the</strong> year, <strong>the</strong> longer is <strong>the</strong> flood<br />
period. Por <strong>the</strong> year with mean water availability <strong>the</strong><br />
duration of <strong>the</strong> flood period (I, days) may be calculated<br />
by <strong>the</strong> equation:<br />
T = Hmean - 20.6<br />
14.4
210<br />
In Icaq <strong>the</strong> maximum flow of rivers occurs due to snow<br />
melting and rain. The rates of maximum daily flows of <strong>the</strong><br />
Tigris river and its tributaries vary from 220 l/s.km2 (<strong>the</strong><br />
2<br />
Khaair) and 135 l/s.km (<strong>the</strong> Greater Zab) in <strong>the</strong> mountain<br />
2<br />
regions to 1.6 l/s.hm2 (<strong>the</strong> Tigris-Amara) and 0.71 l/s.hm<br />
(<strong>the</strong> Qalat-Sa<strong>le</strong>h). in <strong>the</strong> south of <strong>the</strong> Mesopotamian lowland.<br />
The rates of maximum daily flow of <strong>the</strong> Euphrates river vary<br />
from 13.3 l/s.km2 (Hit) to 8.5 l/s.km<br />
2<br />
(downstream <strong>the</strong> dam<br />
Hindiya). The rates of maximum monthly flow of <strong>the</strong> Tigris ri-<br />
2<br />
ver and its tributaries vary from 48 l/a.km (<strong>the</strong> Greater<br />
2<br />
Zab river at Eski-Ka<strong>le</strong>k) in mountain regions to 0.6 l/s,km<br />
(<strong>the</strong> Tigris river at Qalat-Sa<strong>le</strong>h) in <strong>the</strong> south lowlands. For<br />
<strong>the</strong> Euphrates river <strong>the</strong> rate of maximum monthly flow fluc -<br />
tuate from 8.6 l/s.km<br />
2<br />
(Hit) to 4.0 l/s.km<br />
2<br />
(Nasiriya). As<br />
I ru<strong>le</strong> <strong>the</strong>re is observed an increase in <strong>the</strong> maximum rates of<br />
flow throughout <strong>the</strong> mountain regions a<br />
In <strong>the</strong> mountain regions of Iraq,above 2000 m (<strong>the</strong> Tigris,<br />
Greater Zab and Euphzlates river basins) <strong>the</strong> rate of maximum<br />
daily river flow increase from 8.5 to 136 l/s.km2, making<br />
<strong>the</strong> mean about 35 l/e.km<br />
2<br />
for every 100 m. For this region<br />
a certain dependence between maximum rates of flow (%ax)<br />
and mean annual rates (Mme= an. ) is traced.
211<br />
In <strong>the</strong> midd<strong>le</strong> mountain region with <strong>the</strong> altitudes 1000-<br />
2000 m (<strong>the</strong> Lesser Zab and Diyala basins) <strong>the</strong> rates of maxi-<br />
2<br />
mum daily flow fluctuate within <strong>the</strong> limits 48-112 l/s.km ,<br />
increasing averagely by 38 l/c,km2 on every 100 m of alti-<br />
tude* The dependence of maximum and mean annual flows for<br />
this region is expressed by <strong>the</strong> correlation<br />
- 45.8<br />
%ax äaiïy 0.138<br />
In piedmont regions, lower than 1000 m (AdhaFm and<br />
Khazir river basins) <strong>the</strong> rates of maximum daily flow being<br />
inversely proportional to <strong>the</strong> altitude of watershed increase<br />
from 48 to 220 l/s.km2. The phenomenon is observed in <strong>the</strong><br />
case of <strong>the</strong> maximum monthly flow. This can be explained by<br />
<strong>the</strong> fact that <strong>the</strong> Khazir river has <strong>the</strong> <strong>le</strong>sser, compared to<br />
<strong>the</strong> Adhaim basin, area of watershed but receives greater<br />
rainfall. The dependence between <strong>the</strong> maximum and mean annual<br />
flows is expressed by <strong>the</strong> correlation.<br />
%ax monthly = 2.5 %ea* an.<br />
The coefficients of maximum daily flow variation for<br />
<strong>the</strong> rivers of Iraq fluctuate within <strong>the</strong> limits from 0.14<br />
(<strong>the</strong> Tigris-Amara) to 0.74 (<strong>the</strong> Adhaim-Injana). The coeffi-<br />
cients of maximum monthly flow are limited by 0.18-0.29. There<br />
observed <strong>the</strong> inverse proportion between coefficients of va-<br />
riation and <strong>the</strong> altitude of watershed. The correlation of<br />
coefficients of maximum monthly flow variations (Cv max 1 and<br />
<strong>the</strong> coefficients of annual flow variations (Cv an. ) can be<br />
put as:<br />
'v max monthly =(2*1 'v an. ) - 0.21
212<br />
For <strong>the</strong> rivers of Iraq it is characteristic <strong>the</strong> increase<br />
of maximum flow with <strong>the</strong> decrease of <strong>the</strong> watershed area<br />
(F b2),<br />
Barnax daily = 62 - 0.2 F + -9<br />
Minimum flow usually occurs in autumn (September-Octo-<br />
ber) and is caused by <strong>the</strong> groundwater dep<strong>le</strong>tion by <strong>the</strong> end<br />
of <strong>the</strong> hot and dry period. Small rivers in piedmont regions<br />
dried up as early as <strong>the</strong> beginning of summer and till <strong>the</strong><br />
winter rains <strong>the</strong>y have no flow.<br />
One of <strong>the</strong> most important factors of <strong>the</strong> Iraq rivers<br />
regime is <strong>the</strong> low-water period, when <strong>the</strong> river flow is cha-<br />
racterized by stab<strong>le</strong> low <strong>le</strong>vels and discharges and when <strong>the</strong><br />
rivers under <strong>the</strong> condition of great reduction of surface flow<br />
or its comp<strong>le</strong>te cessation are recharged through groundwaters.<br />
Low-water period usually occur in summer or autumn (oftener<br />
from June to December). Its beginning (t days from <strong>the</strong> begin-<br />
ning of <strong>the</strong> year) has a certain dependence on altitude,<br />
OSI75 +<br />
days = 64.6 (jy,ean-lOOO) - K<br />
where, K - coefficient of water availability of <strong>the</strong> designed<br />
period. It can vary from + 20 (for high-water period) to -20<br />
(for low-water period). At <strong>the</strong> mean <strong>le</strong>vel it approaches zero.<br />
As a ru<strong>le</strong> <strong>the</strong> greater is <strong>the</strong> altitude of a watersheii, <strong>the</strong><br />
shorter is <strong>the</strong> period of low water. In piedmont zones it<br />
makes averagely 171 days, in midd<strong>le</strong> mountain zones - 159 days<br />
and in high mountain - 153 days. The latter is mainly due<br />
to more favourab<strong>le</strong> conditions of humidification in <strong>the</strong> moun-<br />
tains where <strong>the</strong> rainfall reaches 1000 mm than in piedmont re-<br />
gions where <strong>the</strong> rainfall makes 200-300 m.
213<br />
The rates of minimum flows of Iraq rivers have a wide<br />
range of variations from 0.04 (daily) and O.Oí'(monthly) for<br />
<strong>the</strong> river Adhaim (Injana) to 5,56 (daily) and 5.71 (monthly)<br />
l/s.km2 for <strong>the</strong> Greater Zab river (Bekhma). Usually <strong>the</strong><br />
rates of minimum daily and monthly flows increase with <strong>the</strong><br />
altitude and this relationship can be expressed ass<br />
10 3 J<br />
'L$in = 5.8 IO-.<br />
On <strong>the</strong> territory of Iraq <strong>the</strong>re observed quite a defi-<br />
nite reduction of <strong>the</strong> minimum flow rates with <strong>the</strong> increase<br />
of a watershed area. For examp<strong>le</strong>, on <strong>the</strong> Tigris river at<br />
2<br />
Al-Fatha <strong>the</strong> minimum monthly flow equals 2.95 l/s.km , <strong>the</strong><br />
2<br />
watershed area being 1076b0 km and downstream <strong>the</strong> Kut bar-<br />
2<br />
rage <strong>the</strong> flow - 1.35 l/s.km and <strong>the</strong> watershed area -<br />
177540 b2. For <strong>the</strong> Greater Zab and Tigris river basins<br />
(high mountain zone) <strong>the</strong> relationship will be similar,<br />
%in monthly = 6.35 - 3.14 IÕ7F<br />
%in daily = 6.14 - 3.08<br />
The decrease of <strong>the</strong> minimum flow rates with <strong>the</strong> in -<br />
crease of a watershed area can be explained mainly by great<br />
withdrawals of water for irrigation and due to considerab<strong>le</strong><br />
evaporation.<br />
The mean annual flow of Iraq being studied better than<br />
<strong>the</strong> minimum one <strong>the</strong>ir interrelation may arouse certain in-<br />
terest<br />
'min monthly = 'Os4 'mean an.<br />
%in daily = O*** mean an. 0.15
214<br />
These relationships do not ref<strong>le</strong>ct <strong>the</strong> conditions<br />
in <strong>the</strong> lower reaches where <strong>the</strong> regime of <strong>the</strong> rivers is great-<br />
ly distorted under <strong>the</strong> influence of anthropogenic factors.<br />
The variations in <strong>the</strong> flow of Iraq rivers (mean, mini-<br />
mum and maximum) occur principally at <strong>the</strong> same time. For <strong>the</strong><br />
investigation of <strong>the</strong> minimum flow variation of <strong>the</strong> Tigris<br />
river (at Mosul, 1920-1970), <strong>the</strong> Euphrates river (at Hit,<br />
1932-1970) and <strong>the</strong> Greater Zab river (1938-1970) <strong>the</strong>y used<br />
<strong>the</strong> method of <strong>the</strong> differential integral curves permitting<br />
to bring out <strong>the</strong> succession of low-water and high-water<br />
groups of years in <strong>the</strong> considered period. The duration of<br />
low-water periods with minimum flow (mean coefficient of wa-<br />
ter availability 0.78) can change within 1-18 years (<strong>the</strong> Tig-<br />
ris river at Mosul) and high-water periods (mean coefficient<br />
of water availability 1.24) - within 1-11 years. The coeffi-<br />
cients of variation (C,) for minimum monthly and daily river<br />
flows fluctuate from 0.18 (monthly, <strong>the</strong> Tigris river at Al-<br />
Fatha) and 0.17 (daily, <strong>the</strong> Euphrates river at Hit) to 1.27<br />
(monthly,<strong>the</strong> Aähaim river at Injana) and 2.03 (daily, <strong>the</strong><br />
Adhaim river at Injana). The value of <strong>the</strong> coefficient of va-<br />
riation is inversely proportional to <strong>the</strong> altitude which is<br />
explainab<strong>le</strong> by a considerab<strong>le</strong> aridity of lowland territories<br />
in Iraq.<br />
The values of Cv for minimum flow, definitely correlate<br />
to Cv for mean annual flow. For <strong>the</strong> watersheds where <strong>the</strong><br />
withdrawal of water for irrigation is low this relationship<br />
can be expressed by <strong>the</strong> following empirical equation,<br />
'v min daily = 1 0 ~ 'v ~ mean 5 an.<br />
- 0.16
In Iraq because of <strong>the</strong> drying up many rivers have no<br />
215<br />
flow for a considerab<strong>le</strong> period (<strong>the</strong> Adhaim, Al-Wend, Galal-<br />
Bedrah, Wadi-river, etc.). The watershed areas of drying-up<br />
2<br />
rivers may reach 13000 km (<strong>the</strong> Aàhaim river) and <strong>the</strong> dura-<br />
tion of a drain<strong>le</strong>ss period exceed 250 days. In Iraq, especialare<br />
ly, in its south-western part <strong>the</strong>re a numerous strema of<br />
temporal nature, "wadi", which have flow only several days<br />
a year. The <strong>le</strong>ngth of some of <strong>the</strong>m reach many tens of kilo-<br />
meters. This phenomena is a result of <strong>the</strong> extreme aridiky of<br />
<strong>the</strong> region where <strong>the</strong>y are met (<strong>the</strong> rainfall is <strong>le</strong>ss than<br />
100 mm and evaporation - over 2500 mm).<br />
It should be noted that <strong>the</strong> given relationships of &if-<br />
ferent flow characteristics on <strong>the</strong> territory of Iraq despite<br />
<strong>the</strong>ir approximate nature and <strong>the</strong> necessity of fur<strong>the</strong>r preci-<br />
sion, permit to give duly evaluation of a number of flow pa-<br />
rameters for <strong>the</strong> insufficiently studied regions of <strong>the</strong> consi-<br />
dered territory.
ABSTRACT<br />
DETERMINATION OF EVAPORATION IN CASE OF THE<br />
ABSENCE OR INADEQUACY OF DATA<br />
P.P. Kuzmin, A.P. Vershinin<br />
State Hydrological Institute<br />
Leningrad, USSR<br />
The possibilities for <strong>the</strong> determination of evaporation<br />
from water surface and land are given in case of <strong>the</strong> absence<br />
of data of direct evaporation measurements. The analysis and<br />
classificatiqn of methods for <strong>the</strong> computation of evaporation<br />
are presented. Practical recommendation for <strong>the</strong> determination<br />
of evaporation by means of standard observational data from<br />
hydrometeorological stations are given.<br />
Les auteurs examinent <strong>le</strong>s possibilités d'évaluation de<br />
l'évaporation des surfaces d'eau libre lorsqu'il n'existe pas<br />
d'observation directe. Ils analysent <strong>le</strong>s différentes méthodes<br />
utilisées et en proposent une classification. Ils font des<br />
recommandations pour l'évaluation de l'évaporation 'a partir<br />
des observations standards effectuées dans <strong>le</strong>s stations<br />
hydrométéorologiques.
218<br />
The determination of evaporation under natural conditions<br />
is of great importance for <strong>the</strong> ewtimation of <strong>the</strong> present and<br />
future water resources, Por water resources management and<br />
for <strong>the</strong> solution of various <strong>the</strong>oretical prob<strong>le</strong>ms in <strong>the</strong> field<br />
of hydrology and meteorology. Methods of direct evaporation<br />
measurements under natural conditions are still being developed,<br />
<strong>the</strong>refore computations are <strong>the</strong> main source of information.<br />
The existing computation methods might be subdivided into<br />
three groups. The first group comprises <strong>the</strong> methods based on<br />
<strong>the</strong> physical analysis of <strong>the</strong> evaporation process. The second<br />
group (combined ox comp<strong>le</strong>x methods) includes methods based on<br />
physical princip<strong>le</strong>s combined with semi-empirical constants which<br />
can be determined with <strong>the</strong> help of accurate measurements of<br />
actual evaporation in representative regions.<br />
Methods based on <strong>the</strong> statistical analysis using only<br />
empirical relations, where empirical constants and coefficients<br />
are h igw variab<strong>le</strong> and depend on meteorological conditions,<br />
make <strong>the</strong> third group.<br />
Besides, according to <strong>the</strong> basic data (factors) included i-nto<br />
<strong>the</strong> design schemes, it should be noted that computation methods<br />
may be comp<strong>le</strong>x and simp<strong>le</strong> as well as difficult and easy to be<br />
applied in practice. In this respect <strong>the</strong> most simp<strong>le</strong> and<br />
practicab<strong>le</strong> methods are those of <strong>the</strong> third and some of <strong>the</strong> second<br />
groups, whi<strong>le</strong> <strong>the</strong> methods of <strong>the</strong> first group which are based<br />
on <strong>the</strong> physical analysis, are most inconvenient in practice.<br />
The first group includes <strong>the</strong> well-known methods of estima-<br />
tion of evaporation from heat balance eqqtion, water balance<br />
equation and turbu<strong>le</strong>nt diffusion method /ll/; <strong>the</strong> accurate<br />
solution of <strong>the</strong>se equations cannot be obtained because it is<br />
impossib<strong>le</strong> to estimate with sufficient degree of accuracy some<br />
individual components of <strong>the</strong> above equations.<br />
The estimation of <strong>the</strong> turbu<strong>le</strong>nt heat exchange between <strong>the</strong><br />
underlying surface (water or land) and <strong>the</strong> atmosphere is one<br />
of <strong>the</strong> difficultiee of <strong>the</strong> solution of heat balance equation.<br />
This component can be estimated approximately without conside-<br />
ration of temperature stratification and horizontal gradients<br />
of turbu<strong>le</strong>nt heat exchange (advection).<br />
In particular, in <strong>the</strong> course of estimating evaporation from<br />
<strong>the</strong> reservoir surface it is difficult to determine time<br />
variation8 of <strong>the</strong> heat accumulated by <strong>the</strong> reservoir (heat<br />
content) as well as heat income and losses due to all kind8<br />
of water inflow and outflow (both surface and subsurface).<br />
Therefore this method is applied only in research studies.<br />
Heat b8Lance equation of <strong>the</strong> land surface is more complicated<br />
than that of <strong>the</strong> water surface /U/#<br />
In <strong>the</strong> 'USSR, however, 8 method of estimating evapotranspira-<br />
tion has been developed and is widely applied,from <strong>the</strong>following<br />
equation /14/:<br />
ß-B<br />
( 1)
which is deduced of <strong>the</strong> heat<br />
balance of <strong>the</strong> land with <strong>the</strong><br />
account of Bowen ratio:<br />
219<br />
Here: E is evapotranspiration, R is <strong>the</strong> measured value of <strong>the</strong><br />
radiation balance of <strong>the</strong> surface, B is heat income into %he<br />
soil, II is <strong>the</strong> atmospheric pressure, P is turbu<strong>le</strong>nt heat<br />
exchange with <strong>the</strong> atmosphere, C is heat capacity under<br />
constat pressure, L is <strong>the</strong> latgnt heat of evaporation;<br />
at and ai are respectively <strong>the</strong> differences in temperature<br />
and water vapour pressure measured at two <strong>le</strong>vels above <strong>the</strong><br />
ground.<br />
Equation (l), naturally, would ra<strong>the</strong>r belong to <strong>the</strong> second<br />
group of methods than to <strong>the</strong> first one; it does not include<br />
horizontal gradients of turbu<strong>le</strong>nt heat exchange (advection)<br />
and temperature stratification. It can be applied for homogeneous<br />
areas large enough to ensure wind r u over homogeneous<br />
top cover over plain area at <strong>the</strong> distance of 300-400 m.<br />
The relative standard error of 10-day and monthly evapotranspiration<br />
sums estimated from equation (1) for <strong>the</strong> re ions of<br />
natural moistening and for irrigated fields makes f 1%.<br />
Equation (1) cannot be recommended for <strong>the</strong> estimation of<br />
evapotranspiration in very dry regions (semi-deserts, deserts) .<br />
Full water balance equation is not applied in practice<br />
since it is both difficult to determine water exchange with<br />
<strong>the</strong> bed of reservoir (<strong>the</strong> difference between underground<br />
water inflow and outflow in a reservoir) whi<strong>le</strong> estimating<br />
evaporation from <strong>the</strong> water surface and to determine water<br />
exchange between <strong>the</strong> upper layer of <strong>the</strong> aeration zone and <strong>the</strong><br />
underlying ground (upward and downward streams of moisture in<br />
<strong>the</strong> ground) whi<strong>le</strong> estimating evapotranspiration from <strong>the</strong> land<br />
surface in a river basin.<br />
In case of deep water tab<strong>le</strong> ( no <strong>le</strong>sa than 3-5 m) <strong>the</strong><br />
simplified water balance equation is used in <strong>the</strong> USSR to<br />
estimate evapotranspiration from non-irrigated agricultural<br />
fields; according to this equation evaporation is estimated<br />
from precipitation (x) and <strong>the</strong> change of moisture storage in<br />
<strong>the</strong> upper soil layer:<br />
E =K+(L4pW.) (2)<br />
where8 W and W are moisture storage in soil at <strong>the</strong><br />
beginnid and a? <strong>the</strong> end of <strong>the</strong> design period.<br />
Equation (2) can be used only under <strong>the</strong> condition that all<br />
precipitation is absorbed by <strong>the</strong> soil and no surface runoff<br />
is formed and besides that <strong>the</strong> depth of rainfall water per-<br />
colation should not exceed <strong>the</strong> depth up to which soil moisture<br />
content was measured and moisture content was determined. Such<br />
conditions usually exist during <strong>the</strong> vegetation period. The
220<br />
depth of <strong>the</strong> upper layer of soil in which soil moisture storage<br />
should be determined is 1 m in wet areas and up to 3 m in<br />
arid zones. In case of reliab<strong>le</strong> estimation of precipitation<br />
and moisture storage <strong>the</strong> standard error of <strong>the</strong> estimation of<br />
monthly sums of evapotranspiration by this method makes<br />
approximately l5-2m. Ano<strong>the</strong>r examp<strong>le</strong> of a partial solution<br />
of water balance equation is <strong>the</strong> estimation of mean annual<br />
sum8 of evapotranspiration as <strong>the</strong> difference between precipitation<br />
and runoff /3/.<br />
Theore tical and experimental development of <strong>the</strong> turbu<strong>le</strong>nt<br />
diffusion method which is also known as <strong>the</strong> gradient or<br />
aerodynamic method, has not yet reached <strong>the</strong> stage which would<br />
allow its wide application in practice /1,2,9,12,15,16/. However,<br />
this method :is promising. Being universal, this method is<br />
based on gradient measurements of wind speed and air humidity<br />
and provides estimation of evaporation from any land or sea<br />
surface irrespective of <strong>the</strong> state and character of <strong>the</strong> latter.<br />
Accurate enough (universal) solutions of <strong>the</strong> equations<br />
of <strong>the</strong> first group require a consideration of a great number<br />
of factors and special observations to be made. The development<br />
of simplified semi-empirical and empirical design schemes<br />
will provide possibilities for <strong>the</strong> estimation of evaporation<br />
without <strong>the</strong> data of specialized observations. At present it<br />
seems possib<strong>le</strong> that standard observational data from hydrometeorological<br />
stations are enough far <strong>the</strong> estimation of<br />
long term average annual and monthly evaporation sums for<br />
a given territory and monthly evaporation sums for individual<br />
years, as well as for <strong>the</strong> estimation of evaporation from<br />
different surfaces - snow cover, swamps, forests, irrigated<br />
and non-irrigated agricultural fields /1,4,5,10,17/.<br />
Most convenient are <strong>the</strong> methods of computation of mean<br />
annual evapotranspiration sums for <strong>the</strong> regions of natural<br />
moistening based on <strong>the</strong> equation developed by Y.I. Puàyko /IO/.<br />
The right part of equation /3/ includes only one parameter<br />
taken from standard observational data, i.e. long term average<br />
precipitation X ( cm year'l) which ref<strong>le</strong>cts <strong>the</strong> natural moisture<br />
content of a region. Ano<strong>the</strong>r parameter, ref<strong>le</strong>cting <strong>the</strong> heat<br />
regime and <strong>the</strong> character of <strong>the</strong> underlying surface - averag<br />
annual adiation balance of moistened surface Ro (kcal cm -2<br />
year - $ is $&en from <strong>the</strong> map prepared by N . Efimova /IO/.<br />
L is <strong>the</strong> latent heat of evaporation (kcal, $*). The standard<br />
error of evaporation estimated from equation (3) makes about<br />
17% /6/0<br />
Mean annual sums estimated from equation /3/ can be<br />
easily distributed by months because long term mean monthly<br />
'
221<br />
evaporation sums given as percenta e of <strong>the</strong> annual sum, change<br />
regularly according to geobotanic ?soil-climatic) zones. This<br />
method is cal<strong>le</strong>d <strong>the</strong> method of percentage ratios /7/. The<br />
percentage ratios by months are given in tab<strong>le</strong>s, developed<br />
by experimental or design ways. Tab<strong>le</strong> I, for Instance, presents<br />
monthly evapotranspiration values as percentage of annual sums<br />
for <strong>the</strong> main geobotanic zones of <strong>the</strong> European territory of <strong>the</strong><br />
USSR .<br />
Tab<strong>le</strong> I<br />
C onif mous<br />
forests O 0,5 2 6 17 25 22 15 8 4 O,5 C<br />
Mixed and<br />
decideous forests,<br />
fore st-steppe s 0,5 I 3 9 18 20 ia i3 9 5 3 05<br />
Steppes 1 I 3 1 1 1 9 2 0 1 6 12 8 5 3 I<br />
M.I. Budyko suggested a combined method for <strong>the</strong> computation<br />
of monthly evapotranspiration sums with <strong>the</strong> use of <strong>the</strong> main<br />
e<strong>le</strong>ments of heat and water balances /1/. This method can be<br />
applied in practice since it is based on <strong>the</strong> use of <strong>the</strong> standard<br />
observational data, i.e. precipitation (x), runoff c y), air<br />
temperature and humiditg.<br />
In this case it is assumed that, when soil moisture content<br />
is <strong>le</strong>ss than its water holding capacitg, monthly evapotranspira-<br />
tion ( E ) is proportional to th8 monthly sum of potential<br />
evapora8on ( E ) and to <strong>the</strong> average monthly storage of<br />
productive moisgure in 1-metre layer of soil WI + W2<br />
2<br />
( WI and W2 are moisture storage at <strong>the</strong> beginning and at <strong>the</strong><br />
end of <strong>the</strong> month), that is:
222<br />
where: W is critical storage of productive moisture in<br />
soil lay& 1 m deep at which and above which<br />
to Eo. Equations (4) and (5) are applied to<br />
<strong>the</strong> year. WI at <strong>the</strong> beginning of <strong>the</strong> first warn month ( Fa<br />
spring) is estimated approximately, and later it is assumed<br />
to be equal to W estimated for <strong>the</strong> end of each previous<br />
month from equatzon:<br />
or from equation:<br />
where: y indicated runoff.<br />
Eo and WO are taken from graphs and tab<strong>le</strong>s included in<br />
publication /IO/. The value of E depends on <strong>the</strong> conventional<br />
humkdkty deficit of <strong>the</strong> air whicf: is determined as <strong>the</strong><br />
difference between maximum water vapour pressure estimated<br />
from mean monthly air temperature, and vapour pressure of <strong>the</strong><br />
air at <strong>the</strong> altitude of 2 metres.<br />
This method was developed in two variants and is applied<br />
for <strong>the</strong> estimation of long term average monthly evapotranspiration<br />
sums of individual months of certqin years /4/.<br />
Aver<br />
ed areal evapotranspiration values (areas to 1000 -<br />
3000 km2y are determined from equations (3) and (4) - (7).<br />
Design schemes for <strong>the</strong> determination of evapotranspiration<br />
from different kinds of surfaces include parameters which are<br />
seldom measured at observational stations. For instance, in<br />
<strong>the</strong> scheme developed by V.V. Romanov /13/ evapotranspiration<br />
from a swamp is assumed to be proportional to <strong>the</strong> radiation<br />
balance of <strong>the</strong> swamp surface ( Q =dR ); in <strong>the</strong> scheme<br />
developed by S.F. Fedorov /18/ evapotranspiration from forests<br />
is proportional to <strong>the</strong> potential evaporation and <strong>the</strong> proportion<br />
coefficient is presented as <strong>the</strong> function of <strong>the</strong> radiation<br />
index of dryness ( R/LX).<br />
Monthly sums of evapotranspiration from irrigated fields are<br />
estimated with <strong>the</strong> help of simplified heat balance equation /i/
223<br />
using special observational data, <strong>the</strong> standard error bei%<br />
l5%, or with <strong>the</strong> help of modified formulae of <strong>the</strong> comp<strong>le</strong>x<br />
method /ïg/ usiq standard observational data, <strong>the</strong> standard<br />
error being 3%. To estimate evapotranspiration from irrigated<br />
agricultural fields empirical design schemes similar to that<br />
of Blaney and Cridd<strong>le</strong> /li/ can be used if only empirical<br />
coefficients are tested and corrected for each point of <strong>the</strong>ir<br />
application,<br />
Most simp<strong>le</strong> design schemes allowing estimation of evapora-<br />
tion from water, snow and ice surfaces by means of standard<br />
observational data, are <strong>the</strong> following binomial and monomial<br />
equations :<br />
and<br />
(9)<br />
where E is evaporation in mm/day, U, is <strong>the</strong> wind speed at<br />
<strong>the</strong> height 2, above <strong>the</strong> surface in m/sec; es and e2 axe th<br />
maximum water vapour pressure estimated from <strong>the</strong> surface<br />
temperature and water vapour pressure at <strong>the</strong> height of 2 m<br />
in mb; A, a and b are coefficients estimated from experiments.<br />
Substituting a = 0.18 ab = 0.098 in equation (8) and<br />
assuming z = IO m one can obtain <strong>the</strong> formula for <strong>the</strong> estima-<br />
tion of evaporation from <strong>the</strong> snow surface /8,11/,<br />
When a L 0.14, b = 0.72 and z = 2 m, equation /8/ can<br />
be used for <strong>the</strong> estimation of evaporation from lake (reservoir)<br />
surface. In this case in equation (8) parameters uz , e and<br />
e2 shouid be substituted by correspondent values measure8 at<br />
different points above <strong>the</strong> reservoir and averaged for a month<br />
with respect to <strong>the</strong> <strong>who<strong>le</strong></strong> water area of <strong>the</strong> reservoir,<br />
In case of <strong>the</strong> absence of such obsemrational data one can<br />
use <strong>the</strong> data from land meteorological stations situated in <strong>the</strong><br />
same climatic zone. T9e transition of <strong>the</strong> obeained above-land<br />
coefficients &z/ ef and 4; to <strong>the</strong> corresponding above<br />
reservoir values h ou d be carried out with respect to <strong>the</strong><br />
transformation of <strong>the</strong> air flux affected by <strong>the</strong> underlying SUT-<br />
face, <strong>the</strong> topography of <strong>the</strong> environment, <strong>the</strong> rate of wind<br />
protection of <strong>the</strong> reservoir and <strong>the</strong> average <strong>le</strong>ngth of wind<br />
run above <strong>the</strong> reservoir /l7/.<br />
In conclusion it should be mentioned that <strong>the</strong> present paper
224<br />
deals with <strong>the</strong> methods which can be used in practice and produce<br />
relatively reliab<strong>le</strong> estimates of evaporation on <strong>the</strong> basis of<br />
standard observational data from meteorological stations.<br />
Therefore more complicated methods of <strong>the</strong> first group<br />
which cause difficulties being applied in practice, and<br />
numerous empirical design schemes which produce unreliab<strong>le</strong><br />
results, are not treated here. Penman and Turc methods are<br />
not mentioned since <strong>the</strong>y are known well enough. It should be<br />
mentioned as well that <strong>the</strong> above classification of methods<br />
is conventional.<br />
All methods are closely interrelated, and <strong>the</strong>ir develop-<br />
ment, particularly <strong>the</strong> improvement of methods of computation<br />
in case of inadequate data, depends greatly on fur<strong>the</strong>r<br />
experimental and <strong>the</strong>ore tical research on <strong>the</strong> evaporation<br />
prob<strong>le</strong>m .<br />
R E F E R E N C E S<br />
1. Qudyko Y.I., 1956. Teplovoi balans zemnoi poverkhnosti<br />
(Heat balance of <strong>the</strong> Earth's surface). Hydrometeorological<br />
Publishing House, Leningrad.<br />
2, Buãyko M.I., 1948, Isparenie v estestwenoykh usloviakh<br />
(Evaporation under natural conditions). Hydrometeorological<br />
Publishing House, Leningrad.<br />
3. Wodnye resuray i wodny balans territorii Sovetskogo Sojuza<br />
(Water resources and water budget of <strong>the</strong> USSR area).<br />
Hydrometeorological Publishing House, Leningrad, 1967.<br />
4. Zubenok L.I., 1968, Ob oprede<strong>le</strong>nii sumaiarnogo isparenia za<br />
otdelnye godg (On estimation of evapotranspiration<br />
during particular years). Trans. of GGO, vol. 233,<br />
Leningrad.<br />
5. Konstantinov A.R., Astakhova N.I., Levenko B.A., 1971,<br />
Metoày rascheta isparenia s selskokhoziaystvennykh<br />
po<strong>le</strong>i (Methods for <strong>the</strong> computation of evaporation<br />
from agricultural fields), Hydrometeorological<br />
Publishing House, Leningrad.<br />
6. Kuzmin P.P., 1966. Teoreticheskaya skhema otsenki oshibok<br />
rascheta isparenia s poverkhnosti sushi (Theoretical<br />
scheme of evaluation of estimation errors of<br />
evaporation from land). Materials of Interagency<br />
meeting on <strong>the</strong> prob<strong>le</strong>m of study and substantiation<br />
of methods of evaporation computations from water<br />
and land. Ed. GGI, Valdai.<br />
7. Kuzmin P.P., Zubenok L.I. Konstantinov A.R., Astakhova N.I.,<br />
Vinogradov V .V . , 1968 . Vnutrigidivie rasprede <strong>le</strong>nie sumsuschi<br />
na territorii SSSII (Annual distribution of<br />
evapotranspiration from land over <strong>the</strong> USSR territon),<br />
Trans. of GGI, vol. 151.
225<br />
8. Kuzmin P.P., 1953. K metodike iss<strong>le</strong>dovania i zascheta<br />
isparenia s poverkhnosti snezhnogo pokrova. (On<br />
methodology of research and computation of evaporation<br />
from snow pack surface) Trana. of GI, vol.<br />
41 (95).<br />
9. Leichtmap D.L., l9W. Profil vetra i obmen v prizemnom<br />
sloe atmosfery (Wind profi<strong>le</strong> and exchange in <strong>the</strong><br />
lowest atmosphere). Izv. AN SSSR, ser. geofis.,<br />
No.1.<br />
IO . Materialy mezhduvedomstvennogo sovetchchania PO prob<strong>le</strong>me<br />
izuchenia i obosnovania metodov rascheta isparenia s<br />
vodnoi poverkhnosti i suchi. (Materials of Interagency<br />
meeting on <strong>the</strong> prob<strong>le</strong>m of study and substantiation<br />
of methods for <strong>the</strong> computation of<br />
evaporation from water and land surfaces). Ed.<br />
by GGI, Valdai, 1966.<br />
11. Measurement and estimation of evaporation and evapotranspiration.<br />
Technical Note No. 83, WO-N0.201. TP.<br />
105, 1966, Geneva.<br />
12. Monin A.S., Obukhov A.M., 1954. Osnovnye xakonomernosti<br />
turbu<strong>le</strong>ntno o peremeshivania v prizemnom sloe<br />
atmospgery $Principal laws of turbu<strong>le</strong>nt mixing in<br />
<strong>the</strong> lowest atmosphere) . Trans. of Geophysical Inst.,<br />
AN SSR, vol. 24 (151).<br />
13. Romanov V.V., 1962. Isparenie s bolot Xvropeiakoi territorii<br />
SSSR (Evaporation from swamps from <strong>the</strong> USSR<br />
European territory). Hydromet. Publ. House, Leningrad;<br />
14. Rukovodstvo PO gradientnym nabliudeniam i oprede<strong>le</strong>niu sostavlia<br />
jushchikh teplovogo balansa (Guide on gradient<br />
observations and determination of heat balance components)<br />
* Hydromet. Publ. House, Leningrad, 1962.<br />
15. Rusin N.P., 1959, Gradientny metod oprede<strong>le</strong>nia isparenia<br />
s sushi i ego ispolzovanoe na seti stantsiy (Gradient<br />
method of estimation of evaporation from land and its<br />
use on <strong>the</strong> network of stations). Trans. of III-rd<br />
All-ünion Hydrological Congress, vol. III, Hydromet.<br />
Publ. House , Leningrad.<br />
16. Tbornthwaite C.W. and Holtzman B., 1942. Measurements<br />
of evaporation from land and water surfaces. U.S.<br />
Dept. Agr. Technical Bul. 817.<br />
17. Ukazania PO raschetu isparenia s poverkhnosti vodoemov<br />
(Instructions for <strong>the</strong> computation of evaporation from<br />
reservoir surface). Hydromet. Publ. House, Leningrad,<br />
1969<br />
18. Fedorov S.F., 1969. O reaultatakh iss<strong>le</strong>dovania digrologicheskoi<br />
roli <strong>le</strong>sa. (On <strong>the</strong> research results of hydrological<br />
ro<strong>le</strong> of forest). Trans. of GGI, vol. 176.<br />
19. Kharchenko S.I., 1968. Gidrologia oroshaemykh zemel<br />
(Hydroìogy of irrigated areas). Hydromet. Publ.<br />
House, Leningrad.
ABSTRACT<br />
OBJECTIVE CRITERIA TO DECLARE A SERIES OF<br />
DATA SUFFICIENT FOR TECHNICAL PURPOSES<br />
by<br />
Penta A., Rossi F.<br />
It is supposed: that for technical purposes it is<br />
necessary to estimate <strong>the</strong> values xo that an hydrological<br />
variab<strong>le</strong> x may assume with a given probability 6; that x can<br />
be measured directly and that its n values have been recorded.<br />
The series of <strong>the</strong> n values of x is'defined sufficient<br />
if it consents to estimate xo with a reliability adequate for<br />
technical purposes.<br />
By referring to <strong>the</strong> usual statistical methodologies,<br />
<strong>the</strong> authors present objective criteria to recognize whe<strong>the</strong>r<br />
<strong>the</strong> series of n values is sufficient. The authors furnish some<br />
diagrams that indicate which minimum values of n are necessary<br />
for <strong>the</strong> series to be considered sufficient.<br />
From <strong>the</strong> diagrams it is evident that for <strong>the</strong> same values<br />
of n <strong>the</strong> series sufficiency is strictly linked to <strong>the</strong> variability<br />
of x.<br />
Particularly, <strong>the</strong> authors considere <strong>the</strong> normal, <strong>the</strong> lognormal<br />
and <strong>the</strong> doub<strong>le</strong> exponential (Gumbel) distributions, <strong>the</strong><br />
m.ost applied laws of hydrology.<br />
-- RESUME<br />
On suppose qu'à l'égard du problème technique il faut<br />
estimer <strong>le</strong>s va<strong>le</strong>urs XQ qu'une variab<strong>le</strong> hydrologique x peut<br />
assumer avec la probabilité 0, que x peut être mesurée et que<br />
n va<strong>le</strong>urs de x ont été enregistrées.<br />
La série des n va<strong>le</strong>urs de x est définie suffisante si<br />
par el<strong>le</strong> on peut estimer xa avec une confiance adéquate au but<br />
du technicien.<br />
En se rapportant aux méthodologies statistiques usuel<strong>le</strong>s<br />
on donne des criteriums objectifs pour reconnaitre si la série<br />
des n va<strong>le</strong>urs est suffisante.<br />
On donne des diagrammes par <strong>le</strong>squel<strong>le</strong>s on indique <strong>le</strong>s<br />
va<strong>le</strong>urs minima du nombre n qui son necessaires afin que la série<br />
soit suffisante.<br />
D'après <strong>le</strong>s diagrammes il apparait évident que, n ayant<br />
la même va<strong>le</strong>ur, la suffisance de la série dépend de la variabilité<br />
de x.<br />
En particulier, <strong>le</strong>s auteurs considèrent la loi norma<strong>le</strong>,<br />
la loi log-norma<strong>le</strong> e la loi de Gumbel, qui sont plus fréquemment<br />
employées en hydrologie.
228<br />
Symbols and definitions<br />
1: Let us indicate by I<br />
- x , a generic hydrological variab<strong>le</strong>3<br />
-E , ax and y, respectively <strong>the</strong> mean, <strong>the</strong> standard deviation<br />
and <strong>the</strong> coefficient of variation of <strong>the</strong> x population;<br />
- @(XI, <strong>the</strong> distribution function of x ;<br />
- xQ, , <strong>the</strong> value of x corresponding to <strong>the</strong> cumulated probability<br />
@ e<br />
Moreover, <strong>le</strong>t us also indicate by :<br />
- x , with 14isn , <strong>the</strong> n values of x registered, in each<br />
sing<strong>le</strong> year,iduring <strong>the</strong> observation period i<br />
- - x and ax, respectively <strong>the</strong> estimates of [ and a ;<br />
- Pix) , <strong>the</strong> estimate of <strong>the</strong> distribution function (Dix);<br />
- xppa,, <strong>the</strong> estimate of<br />
xa ;<br />
- y(xP,@), <strong>the</strong> sampling coefficient of variation of 5 E@<br />
2: If x is normally distributed,<strong>the</strong> best estimate xpsio, of x<br />
is obtained [ 1 1 by t<br />
X pn(D = Z + u<br />
@<br />
where u is <strong>the</strong> value of <strong>the</strong> variab<strong>le</strong> u, that in equation:<br />
@<br />
1 2<br />
2<br />
1<br />
@(U) E -<br />
du (2)<br />
corresponds to <strong>the</strong> fixed value of @ .<br />
The sampling coefficient of variation of xp could be obtained<br />
approximately [2] by<br />
-@<br />
I<br />
or whenever n is sufficiently high, <strong>the</strong> equation (3) becomes :<br />
1 +u$/*<br />
i i n<br />
(1)<br />
(3')<br />
(D
3: If x is distributed according to <strong>the</strong> log-normal function,<br />
having established that y = log x , we indicate by :<br />
- Yi 9 <strong>the</strong> value of y corresponding to <strong>the</strong> generic value xi ;<br />
I - y and s respectively <strong>the</strong> mean and <strong>the</strong> standard .deviation of<br />
<strong>the</strong> n values of Yi.<br />
Y'<br />
Therefore, <strong>the</strong> best estimate x of x is obtained [a] by<br />
P=UJ 0<br />
(4)<br />
log xp' E y. + U@ 8<br />
=a><br />
Y<br />
229<br />
where <strong>the</strong> value of uUJ is deduced by means of equation (2) whi<strong>le</strong> <strong>the</strong> values<br />
of y and s are deduced respectively by <strong>the</strong> equations :<br />
' n<br />
log xi<br />
- 13.1<br />
Y" n<br />
and<br />
r n<br />
s =<br />
Y<br />
n-1<br />
The sampling coefficient of variation of xPIUJ could be obtained<br />
approximately [ 21 by :<br />
2<br />
1 + u0/2<br />
or, whenever y* is sufficiently mall, <strong>the</strong><br />
I 2<br />
- 1 (7)<br />
equation (7) becomes :<br />
4: If x is distributed according to <strong>the</strong> doub<strong>le</strong> exponentlal,namely<br />
@umbel function, an almost correct and efficient estimate xP=@ of x UJ is<br />
obtained [4] by :<br />
xppUJ= + K UJ<br />
(8)<br />
where Ka, is <strong>the</strong> value of <strong>the</strong> variab<strong>le</strong> K that'in <strong>the</strong> equation :<br />
-- 6<br />
1<br />
K = (0,5772 + In In -<br />
(9)<br />
x a,<br />
corresponde to <strong>the</strong> fixed vaïue of
230<br />
The sampling coefficient of variation of x could be obtained<br />
approximately [ 21 by :<br />
5: When <strong>the</strong> variab<strong>le</strong> x is measured in k gaging stations, lying<br />
in a detertnined zone, <strong>the</strong>re exists an hydroloRica1 similitude between <strong>the</strong> k<br />
stations if <strong>the</strong> parameters, or some parameters al , z2 , ..... of <strong>the</strong> x<br />
diatribution assume <strong>the</strong> same value or if <strong>the</strong>y vary from one to ano<strong>the</strong>r with a<br />
known regression relation in function of a certain number of parameters<br />
y1<br />
y2 ..... '<br />
[5].<br />
The inter-station correlation is <strong>the</strong> correlation which exists, in<br />
such cases, among <strong>the</strong> values of x registered in <strong>the</strong>m contemporaneously (e.g.<br />
in <strong>the</strong> same year if maximum and minimum annual values are considered).<br />
Therefore, <strong>the</strong> information that can be derived from <strong>the</strong> k stations,<br />
considered all toge<strong>the</strong>r, in regard to <strong>the</strong> x distribution parameters al, a2,..<br />
..... is <strong>the</strong> same as <strong>the</strong> information furnished by a number k of independent<br />
stations. Such a number, known as <strong>the</strong> equiva<strong>le</strong>nt number, depegds both on k<br />
and on <strong>the</strong> mean interstation correlation coefficient F , thus becoming so<br />
smal<strong>le</strong>r than k, <strong>the</strong> higher <strong>the</strong> value of F is.<br />
So, e.g. if in <strong>the</strong> k stations <strong>the</strong> mean 6 of x would assume <strong>the</strong><br />
same value, <strong>the</strong> information that <strong>the</strong> comp<strong>le</strong>x of <strong>the</strong> data registered in <strong>the</strong> k<br />
etatiomwould furnish in regard to would. be equal 16) to <strong>the</strong> information<br />
furnished by an equiva<strong>le</strong>nt number of independent stationsequal to :<br />
k<br />
keE l+F(k-1)<br />
Basic Risk. Uncertainty and Effective Risk<br />
6: Normally, for design purposes, by referring to a given hydrolog&<br />
cal variab<strong>le</strong> x,we indicate by :<br />
xd , <strong>the</strong> value of<br />
(deBiRn Value) ;<br />
N , <strong>the</strong> desinn duration.<br />
x that Is assumed as <strong>the</strong> basis for <strong>the</strong> design<br />
Particularly, in a flood prob<strong>le</strong>m, we se<strong>le</strong>ct a value Of so that<br />
<strong>the</strong>re exists a probability of failure W that xd will be exceeded<br />
xd<br />
at <strong>le</strong>ast<br />
once in N years.<br />
Consêquently, xd coincides with <strong>the</strong> value xD of x whlch corre<br />
sponda to a value of D of <strong>the</strong> cumulated probability furnished by :
E.g. when N = 25 years and W=0,025, @ is equal to 0,999.<br />
Likewise, in a drought prob<strong>le</strong>m, we se<strong>le</strong>ct a value of so<br />
xd<br />
that<br />
<strong>the</strong>re exists a probability of failure W that xd will not be exceeded at <strong>le</strong>ast<br />
once in N years.<br />
Consequently, instead of using equation (121, we must apply <strong>the</strong><br />
following equation :<br />
E.g.<br />
1<br />
@ = l -<br />
/N<br />
(1 - w)<br />
when N = 25 years and W = 0,025, @ is equal to 0,001 .<br />
The basic risk is defined [ 71 as <strong>the</strong> risk that would be encountered<br />
if, by knowing <strong>the</strong> probability distribution of x,we would assume x EX@ .Such<br />
d<br />
risk is measured by means of <strong>the</strong> probability of failure<br />
In reality, however, <strong>the</strong> distribution of x is not known. Consequently,<br />
having fixed <strong>the</strong> basic risk W and having calculated @ by means of eqpations<br />
(12) or (131, with <strong>the</strong> use of a series of n values of x,only an estimate<br />
x of x could be had, aiid, <strong>the</strong>refore, to assume x = xQ an error<br />
equal toP =?x - xD ) would be made. In reality <strong>the</strong> effective risk that is<br />
encountered ?;treater than <strong>the</strong> basic risk due to <strong>the</strong> uncertainty with which<br />
<strong>the</strong> value of x could be estimated.<br />
0<br />
Sufficiency of a Sing<strong>le</strong> Series of Data<br />
7: Once <strong>the</strong> basic risk W has been determined, to judge whe<strong>the</strong>r a<br />
sing<strong>le</strong> series of data is sufficient for technical purpose4,i.t is necessary to<br />
take into account <strong>the</strong> uncertainty with which x@ could be estimated.<br />
Generally, by considering also <strong>the</strong> observation periods which are usually<br />
availab<strong>le</strong>, a series of at <strong>le</strong>ast 30+40 data is defined IIlonP and it is<br />
implicitly retained sufficient; a series with <strong>le</strong>ss than 30+40 data is defined - Vshort1I and is considered insufficient.<br />
In reality, however, such criterion might be erroneous. In fact, if<br />
<strong>the</strong> uncertainty, with which x0 could be estimate, is measured by means of<br />
y{xp } , from<br />
eq. (31, or eq. (7) or eq. (101, we recognize Immediately that<br />
<strong>the</strong> sad uncertainty, beside n , depends also on :<br />
i) <strong>the</strong> variability of <strong>the</strong> hydrological magnitude x being considered,<br />
which can be measured by y ;<br />
w.<br />
231<br />
ii) <strong>the</strong> probability Q of <strong>the</strong> design value xd , which is a function<br />
of <strong>the</strong> basic risk W and <strong>the</strong> design duration N.<br />
In particular, <strong>le</strong>t us consider e.g. <strong>the</strong> annual rainfall depth xuh distributed generally according to <strong>the</strong> log-normal function [ 81, with a coefficient<br />
of variation y, which varies from 0,l to 0,9 as we progressively move from<br />
sub-humid zones to semi-arid and arid zones, <strong>the</strong> mean annual rainfall changes<br />
from vaïues of circa i 500 mm to values of circa 50 mm [ 9 J .
232<br />
As it could be noticed from <strong>the</strong> diagram (a) of fig. 1, if it were nec<br />
essary to estimate <strong>the</strong> median value x 5o of x,a long series could be retained<br />
sufficient from a technical point of v hw for each of <strong>the</strong> possib<strong>le</strong> values of yx,<br />
since in no case y{xp, would be greater than 15%.<br />
However, wheh we fix <strong>the</strong> duration N equal to 25 years and <strong>the</strong> basic<br />
risk equal to 2,5%, by applying eq. (12) or eq. (13) we notice that we must refer<br />
to values of @ equal to 0,999 or 0,001. In this case, from <strong>the</strong> diagram (b)<br />
of fig. 1 it ie evident that a long series of data would be sufficient from a<br />
technical point of view only if y were ra<strong>the</strong>r low.<br />
In fact, even for values of y, greater than 0,5, Y{X~,~) could<br />
s be greater than 20%.<br />
On <strong>the</strong> o<strong>the</strong>r hand, from <strong>the</strong> same diagrams (a) and (b) of fig. 1 ,it<br />
can be derived that a short series of data, which is certainly insufficient<br />
for values greater than y , could be sufficient if y, would assume too<br />
small values.<br />
Analagoue considerations could be made if x follows <strong>the</strong> doub<strong>le</strong><br />
exponential distribution by examining <strong>the</strong> diagrams (a) and (b) of fig. 2 in<br />
which are represented <strong>the</strong> function of y{xp, 1 as n and<br />
(corresponding to <strong>the</strong> distribution mode) -and fi? CD = Y,<br />
0,999.<br />
<strong>the</strong> Data Registered in O<strong>the</strong>r Stations<br />
for @ P 0,368<br />
9: The regions where regular hydrological measurements have been<br />
taken for a short period of observation, have often arid or semi-arid climate,<br />
<strong>the</strong>refore, it becomes practically impossib<strong>le</strong> to estimate from a sing<strong>le</strong> series<br />
of data <strong>the</strong> values that, with a given probability, those magnitudes might assume.<br />
It becomes <strong>the</strong>refore necessary to recognize if it is possib<strong>le</strong> to improve <strong>the</strong><br />
estimate of xrp in a given station by using <strong>the</strong> data obtained in o<strong>the</strong>rs. As<br />
it is known, to render this possib<strong>le</strong>, it is necessary that <strong>the</strong> different stations<br />
considered be hydrologically similar (see pgr.5). For this to happen, it is<br />
necessary that <strong>the</strong> values taken by x in <strong>the</strong> &d stations depend on common<br />
meteorological and hydrological factors. Consequently, this implies that <strong>the</strong>re<br />
exists an inter-station correlation.<br />
It is udeful to point out that from this point of view it is very<br />
important to consider ei<strong>the</strong>r one of <strong>the</strong> hydrological magnitude. In fact, <strong>the</strong><br />
mean inter-station correlation coefficient P is amal<strong>le</strong>r when <strong>the</strong> daily or<br />
weekly rainfall is considered, whi<strong>le</strong> it is greater when we take into account<br />
annual rainfall [ 101 . In <strong>the</strong> case of annual rainfall, in a research conducted<br />
from <strong>the</strong> information furnished by 1141 pluviometers instal<strong>le</strong>d in <strong>the</strong> Western<br />
D.S. and in <strong>the</strong> South-West California [lOl, Caffey has shown that <strong>the</strong> mean<br />
inter-station correlation coefficient F situated in a zone meteorologically<br />
homogeneous varies from Q,30 to 0,SO. In a recent research on pluviometers<br />
instal<strong>le</strong>d in Basilicata and in Sou<strong>the</strong>rn Italy, we have found fn0,5 t 0,6 and
in a research on <strong>the</strong> Morocco pluviometers, being conducted at <strong>the</strong> time of this<br />
report, r = 0,90 which is still higher.<br />
233<br />
10: By referring to <strong>the</strong> mean value 6 of x, in <strong>the</strong> diagram of.<br />
fig.3, ne have repreeented equation (11) which formulates <strong>the</strong> law according which<br />
<strong>the</strong> equiva<strong>le</strong>nt number ke of independent stations, defined in pgr.5, varies as<br />
a function of r' and <strong>the</strong> number k , of statione instal<strong>le</strong>d in <strong>the</strong> zone.<br />
As it can be observed from fig.3, for each value of F, ke increasel<br />
at each increase in k ,tending asymptotically toward a maximum value<br />
kernax= F<br />
Consequently, <strong>the</strong> maximum increase of information that is obtained in<br />
regard to 5 by applying <strong>the</strong> hydrological similitude criteria is inversely<br />
proportional to F . E.g. when F = 0,5 , <strong>the</strong> information, at <strong>the</strong> most, could be<br />
doub<strong>le</strong>d; for still greater values of ? , which are often encountered in hydrology,<br />
<strong>the</strong> advantage obtained could be almost negligib<strong>le</strong>.<br />
On <strong>the</strong> o<strong>the</strong>r hand, no real benefit is obtained by increasing <strong>the</strong> number<br />
of k stations above a certain limit strictly connected to F . To prove this,<br />
we have represented in <strong>the</strong> diaáram of fig.4 <strong>the</strong> law with which - ke<br />
varies as<br />
k<br />
a function of k for different values of r. As it can be noti$eyxif we are<br />
satisfied with <strong>the</strong> 90% of <strong>the</strong> maximum information that can be obtained, by<br />
ke<br />
accepting that - = 0,9, this objective could be reached with only 9<br />
k<br />
stations, for F = with only 4 stations for r 5 0,7.<br />
CONCLUSIONS<br />
11: In eome countries, systematic, reliab<strong>le</strong>, homogenous measurements<br />
have been taken for only few years and in few stations. Moreover, to render <strong>the</strong><br />
prob<strong>le</strong>m more severe, such regions have an arid, or semi-arid climate. Therefore,<br />
due to <strong>the</strong> extreme variability of <strong>the</strong> hydrological magnitudes, with <strong>the</strong> same<br />
number of data, <strong>the</strong> uncertainty with which <strong>the</strong> probability distribution of <strong>the</strong>m<br />
could be estimated, is greater.<br />
Consequently, in <strong>the</strong> said regions it is particularly important to<br />
utilize all <strong>the</strong> information that <strong>the</strong> few availab<strong>le</strong> data could furnish, by applying<br />
ei<strong>the</strong>r correct statistical methods to interpret each sing<strong>le</strong> series of data and/or<br />
by defining objectively some hydrological similitude criteria that would consent<br />
<strong>the</strong> interpretation on how <strong>the</strong> magnitude varies from one station to ano<strong>the</strong>r.<br />
Particularly, for a reference magnitude x , by applying <strong>the</strong> hydrolog-<br />
ical similitude criteria, it is possib<strong>le</strong> :<br />
a) to obtain a reliab<strong>le</strong> estimate of xo even for points where no<br />
direct measurements of x were even taken ;
2 34<br />
b) to improve <strong>the</strong> estimate of x in points where only few data<br />
are availab<strong>le</strong>.<br />
Q,<br />
The advantages obtained in regard to point b) could be noticeably<br />
limited by <strong>the</strong> inter-station correlation located within an hydrologically<br />
homogeneous zone.<br />
In any case, only when all <strong>the</strong> information availab<strong>le</strong> has been uti-<br />
lized, it is possib<strong>le</strong> to establish whe<strong>the</strong>r <strong>the</strong> data availab<strong>le</strong> are sufficient<br />
or not to be used in practical applications.<br />
12: If <strong>the</strong> availab<strong>le</strong> data in <strong>the</strong> region should be insufficient, a<br />
supp<strong>le</strong>mentary research program would be necessary. Even in this case, it is<br />
absolutely necessary to take into account <strong>the</strong> information furnished by all <strong>the</strong><br />
data availab<strong>le</strong> so that <strong>the</strong> research program is carried out in an adequate manner.<br />
On <strong>the</strong> o<strong>the</strong>r hand, we must be well aware of <strong>the</strong> results of a short<br />
research program.<br />
In fact, if an appropriate localization of <strong>the</strong> stations is made, it<br />
is useful :<br />
i) to individualize and improve <strong>the</strong> delimitation of <strong>the</strong> region in<br />
hydrologically homogeneous zones ;<br />
ii) to determine <strong>the</strong> regression law of a variab<strong>le</strong> x as function of<br />
some parameters which characterize <strong>the</strong> point or <strong>the</strong> basin (e.g. <strong>the</strong> regression<br />
relation of <strong>the</strong> mean rainfall depth vs <strong>the</strong> <strong>le</strong>vel of <strong>the</strong> point or <strong>the</strong> regression<br />
relation of <strong>the</strong> mean annual runoff vs mean annual rainfall).<br />
On <strong>the</strong> o<strong>the</strong>r hand, when both aims have been attained, <strong>the</strong> research<br />
program could be useful also to estimate <strong>the</strong> probability distribution of x<br />
in different points (or basins) only if in <strong>the</strong> region <strong>the</strong>re are one or more<br />
gaging stations functioning for a long time.
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
KENDALL M.G. and STUART A., (1967). The Advanced Theory of Statistics.<br />
London, Griffin, Vol. 2, 2nd Ed., p.54.<br />
ROSSI F., (1 972) Distribuzione di campionatura di alcune grandezze stg<br />
tistiche. Fac. di Ing. delltllniv. di Napoli, 1st. di Costr. Idr.,<br />
Quad. no 5 .<br />
AITCUISQM J. and BROWN J.A.C., (1957). The Lognormal Distribution.<br />
Cambridge University heas.<br />
235<br />
LOWRaY N.D. and NASH J.E., (1970). Methods of Fitting Doub<strong>le</strong> Exponential<br />
Distribution. Journal of Hydrology, 10, pp. 259 - 275.<br />
VIPARELLI C., (1 965). Idrologia applicata all 1 Ingegneria. Parte II, Fond.<br />
Politecnica del Mezzogiorno d'Italia, Napoli.<br />
MATALAS N.C. and BENSON M.A., (1961). Effect of Interstation Correlation<br />
on Regression Analysis. Journal of Geophysical Research, Vo1.66, nolo.<br />
YEVJEVICH V., (1972). Probability and Statistics in Hydrology. Water<br />
Resources Publ., Fort, Collins, Colorado.<br />
MARKQVIC R.D., (1965). Probability Functions of Best Fit Distributions of<br />
Annual Precipitation and Runoff. Hydrology Papers no 8, Fort Collins, Co-<br />
lorado.<br />
GARCIA-AGREDA R., RASULO G., and VIPARELLI R., (1 973) Pluviometric Zones<br />
and <strong>the</strong> Criteria to Define <strong>the</strong>ir Boundaries for Regions with Scarce Data.<br />
Simposio sobre proyectos de recursos hidraùlicos con datos insuficientes,<br />
Madrid.<br />
CAFFEY J.E., (1965). Inter-station Correlations in Annual Precipitation<br />
and in Annual Effective Precipitation. Hydrology Papers no 6. Fort Collins,<br />
Colorado.
20 40 60<br />
OBJECTIVE CRITERIA TO DECLARE A SERIES OF DAPA SUFFICIENT I'OR<br />
TECHNICAL PURPOSES<br />
*O n
OBJECTIVE CRITERIA TO DECLARE A SERIES OF DATA SUFTICICNT FOX<br />
TECHNICAL PURPOSES
10<br />
Ke<br />
8<br />
6<br />
4 I<br />
2<br />
O<br />
6 - /<br />
- I<br />
O, 50<br />
Y<br />
O, 70<br />
I<br />
?=%O<br />
10 20 30<br />
1<br />
Fig. 3<br />
OBJECTIVE CRITERIA TO DECLARE A SERIES OF DATA SUFFICIENT FOR<br />
TECHNICAL PURPOSES<br />
I I<br />
I I<br />
1 K. 50
l,oo<br />
0,8 O<br />
0,60<br />
0,4 O<br />
0,20<br />
- 1,oo-<br />
/<br />
Fig. 4<br />
10 20 30 50<br />
40 K<br />
OBJECTIVE CRITERIA TO DECLARE A SERIES OF DATA SUFFICIENT FOR<br />
TECHNICAL PURPOSES
ABSTRACT<br />
SOME CRITERIA USED IN HYDROLOGIC STUDIES<br />
WITH INADEQUATE DATA<br />
Carlos Quintela Góis<br />
In territories where <strong>the</strong> hydrologic networks are<br />
still scarce, it is necessary to adopt simplified designing<br />
criteria which might <strong>le</strong>ad to sufficiently reliab<strong>le</strong> results.<br />
In this paper those which are normally used for <strong>the</strong> hydro-<br />
logic characterization of <strong>the</strong> drainage basins under <strong>the</strong>se<br />
conditions are presented and examp<strong>le</strong> of <strong>the</strong>ir application<br />
to Okavango Basin in Angola is given.<br />
RESUME<br />
Dans <strong>le</strong>s territoires où <strong>le</strong>s réseaux hydrologiques<br />
sont encore insuffisants, il faut recourir a des procédés<br />
de c alcul simplifiés qui puissent conduire a des résultats<br />
dignes de confiance. L'auteur expose des méthodes norma<strong>le</strong>ment<br />
utilisées pour évaluer. dans de tel<strong>le</strong>s conditions<br />
<strong>le</strong>s caractéristiques hydrologiques des bassins, en prenant<br />
pour examp<strong>le</strong> du bassin du Cubango, en Angola.<br />
* Civil Engineer - Member of <strong>the</strong> Working Group<br />
of <strong>the</strong> Overseas Ministry (Portugal) for <strong>the</strong> I.H.D.
242<br />
1. Introduction<br />
In <strong>the</strong> framework of <strong>the</strong> hydraulic policy which has b en followed for he<br />
last years in <strong>the</strong> Portuguese Overseas Provinces, <strong>the</strong> study of <strong>the</strong> general plans<br />
for <strong>the</strong> development of <strong>the</strong> water resources plays a very important ro<strong>le</strong>. In <strong>the</strong><br />
two main provinces of Africa - Angola and Mozambique - this action <strong>le</strong>d to <strong>the</strong> fact<br />
that <strong>the</strong> main drainage basins are already covered by such studies; that enab<strong>le</strong>s<br />
an adequate hydroe<strong>le</strong>ctric and hydro-agricultural overall planning to be made,<br />
Hydrological studies are obviously <strong>the</strong> fundamental basis of such general plans<br />
because <strong>the</strong>y. determine <strong>the</strong> hydrologic characterization of <strong>the</strong> basin and from <strong>the</strong>-<br />
re <strong>the</strong> preliminary design of <strong>the</strong> several schemes and estimate of <strong>the</strong>ir potentia -<br />
lities. In this field international cooperation which was achieved with <strong>the</strong> o<strong>the</strong>r<br />
territories of Sou<strong>the</strong>rn Africa, as a result of established agreements, is also of<br />
a great importance and it gives an idea of <strong>the</strong> value that water has got for <strong>the</strong><br />
common development on that part of <strong>the</strong> world.<br />
The inhospitab<strong>le</strong> characteristics of <strong>the</strong>se areas toge<strong>the</strong>r with <strong>the</strong> communica-<br />
tion difficulties and low human occupation result usually in very scarce and<br />
recent hydrologic networks so that on carrying out hydrologic studies one faces<br />
<strong>the</strong> difficulty of applying <strong>the</strong> classic methods or those used for more developed<br />
areas.<br />
Therefore it is necessary to adopt approximative methods and special crite -<br />
ria enabling to arrive at sufficiently correct and reliab<strong>le</strong> results for <strong>the</strong> ai -<br />
med purposes.<br />
In this paper <strong>the</strong> methods which have been followed for carrying out <strong>the</strong> abo-<br />
ve mentioned hydrologic studies are presented and <strong>the</strong> approximate criteria that<br />
have been adopted as a result of inadequacy of data are pointed out; at <strong>the</strong> end<br />
a practical examp<strong>le</strong> is given for <strong>the</strong> case of a drainage basin in Angola. Only<br />
<strong>the</strong> aspects of rainfall and run-off in average terms are stressed because <strong>the</strong>y<br />
are of most interest for <strong>the</strong> hydrologic studies of general plans.<br />
2. Rainfall<br />
Among <strong>the</strong> hydrologic data, rainfall is commonly measured for a longer pe-<br />
riod, even in developing territories. Although networks do not cover satisfacto-
ily <strong>the</strong> areas to be studied, <strong>the</strong>y enab<strong>le</strong> <strong>the</strong> characterization of <strong>the</strong> phenomenon<br />
with enough accuracy to be achiewed.<br />
243<br />
Usually <strong>the</strong> daily precipitation data measured in raingauges normally loca-<br />
ted at villages or townships are availab<strong>le</strong>. Record periods of twenty years ormore,<br />
at <strong>le</strong>ast in some of <strong>the</strong> stations, are frequent and <strong>the</strong> use of correlation techni-<br />
ques enab<strong>le</strong>s to obtain monthly rainfall all over <strong>the</strong> stations of <strong>the</strong> network, On<br />
<strong>the</strong> o<strong>the</strong>r hand, uniform rainfall regime of <strong>the</strong> African subtropical regions with a<br />
long period of four months without precipitation is well known, which makes it ea-<br />
sier to fulfil some failures in <strong>the</strong> records.<br />
The study of that regime is usually done by taking <strong>the</strong> annual weighed pre-<br />
cipitations obtained from <strong>the</strong> isohyet maps drawn for <strong>the</strong> basin. The isohyet me -<br />
thod is considered to be <strong>the</strong> most adequate when dealing with incomp<strong>le</strong>te informa-<br />
tion, because local surveys, topography, etc. may help to introduce corrections<br />
or indicate <strong>the</strong> best drawing of <strong>the</strong> curves of equal precipitation so that a pat-<br />
tern, as close as possib<strong>le</strong> with reality, can be obtained. Once <strong>the</strong> basins have<br />
usually a drainage area of tens of thousands of square kilometers, <strong>the</strong> used sca-<br />
<strong>le</strong> for drawing isohyet maps is normally 1:l O00 000.<br />
After those maps are obtained, some characteristic sections are chosen and<br />
<strong>the</strong> weighed values are determined. These are <strong>the</strong> bases for <strong>the</strong> study of <strong>the</strong> rain-<br />
fall regime and periods of about 20 years permit <strong>the</strong> application of stochastic me-<br />
thods. Among <strong>the</strong>se, <strong>the</strong> method of Hazen-Foster has been considered to be <strong>the</strong> most<br />
adequate to interpretate <strong>the</strong> phenomenon. After graphical and analytical confirma-<br />
tion of its applicability, it is possib<strong>le</strong> to obtain <strong>the</strong> mean annual value and tho-<br />
se corresponding to characteristic return periods. The probability relating to each<br />
one of <strong>the</strong> years of <strong>the</strong> period can be obtained as well.<br />
This analysis gives a first idea of <strong>the</strong> natural sequence of <strong>the</strong> years and<br />
principally <strong>the</strong> occurence of dry periods and <strong>the</strong>ir degree of drought so that fur-<br />
<strong>the</strong>r studies for comparison with <strong>the</strong> run-off can be done.<br />
The study of rainfall is usually comp<strong>le</strong>ted with a short analysis of dry and<br />
wet seasons and mainly of <strong>the</strong> frequency with which longer dry seasons may occur.<br />
3. Run-off<br />
As far as flow measurements are concerned, data is always very scarce and only
few flow stations in Portuguese Africa have records availab<strong>le</strong> for more than 5 to<br />
10 years. Besides, it has been verified that <strong>the</strong> study of general plans normally<br />
shows <strong>the</strong> need and <strong>le</strong>ad to <strong>the</strong> best choice and establishment of <strong>the</strong> hydrometric<br />
networks.<br />
Stochastic methods cannot be applied safely with such short periods and<br />
<strong>the</strong>refore <strong>the</strong> first approximative criterium to be used is trying to characterize<br />
<strong>the</strong> availab<strong>le</strong> flow record period by relating it with <strong>the</strong> similar period of <strong>the</strong><br />
rainfall studies. Hence it is possib<strong>le</strong>, as a first approximation, to consider <strong>the</strong><br />
same probability of occurence for <strong>the</strong> annual flow and rainfall of a certain year.<br />
From this it is often possib<strong>le</strong> to chose certain years which can be considered<br />
as average or with a given degree of dryness. Therefore a critical period<br />
corresponding to an unfavourab<strong>le</strong> sequence of years can be chosen in order to fix<br />
<strong>the</strong> storage capacity of interannual reservoirs and to obtain a comp<strong>le</strong>te regulation<br />
of <strong>the</strong> flows. This sequence is normally formed by an average year followed<br />
by two or more dry years with fixed characteristics. Undoubtedly this is an approximate<br />
approach, but experience has shown that for studies at <strong>the</strong> <strong>le</strong>vel of general<br />
plans this analysis is quite acceptab<strong>le</strong> and safe because <strong>the</strong> pessimism in<br />
<strong>the</strong> reasoning compensates <strong>the</strong>.uncertainties resulting from <strong>the</strong> inadequacy of data.<br />
Sometimes, as an exception, <strong>the</strong>re exists in <strong>the</strong> basin a measuring section<br />
with a longer period of records and for which stochastic methods can be applied.<br />
Two ways can <strong>the</strong>n be followed, (1) correlation analysis with o<strong>the</strong>r stations of<br />
<strong>the</strong> basin, trying to obtain more data for those which have shorter records or<br />
(2) characterization of <strong>the</strong> shorter period by relating it with <strong>the</strong> longer one<br />
of that station in a similar way as mentioned in <strong>the</strong> previous paragraph for <strong>the</strong><br />
rainfall.<br />
The first method is not always easy to apply, because <strong>the</strong> rivers might<br />
show a change of regime along <strong>the</strong>ir course as a result of <strong>the</strong> phisiography and<br />
correlations are no more valid.<br />
The second one is more reliab<strong>le</strong> and on applying it, it is possib<strong>le</strong> to ar-<br />
rive at safe and easily interpretab<strong>le</strong> results. Normally one can obtain not only<br />
<strong>the</strong> annual flow but also <strong>the</strong> monthly ones of <strong>the</strong> average and dry )ears of <strong>the</strong> cho-<br />
sen critical period and <strong>the</strong>refore carry out more reliab<strong>le</strong> regulation studies.
245<br />
The study of rainfall/run-off relations has not been, as far as our expe-<br />
rience is concerned, successful for large drainage basins as a method of enlar -<br />
ging <strong>the</strong> availab<strong>le</strong> flow record period. This is probably <strong>the</strong> result of <strong>the</strong> speci-<br />
al type of <strong>the</strong> rainfall regime of those regions - short, heavy and localized storms-<br />
toge<strong>the</strong>r with high temperatures and evaporation rates which affect <strong>the</strong> usual me-<br />
chanism of transforming rainfall into run-off. Besides, this method would only<br />
<strong>le</strong>ad to global annual values and its distribution along <strong>the</strong> year is not possib<strong>le</strong><br />
to obtain.<br />
4. Application examp<strong>le</strong><br />
4.1 - General characterization of <strong>the</strong> prob<strong>le</strong>m<br />
The Okavango is one of <strong>the</strong> three big international rivers of <strong>the</strong><br />
South of Angola. It springs on <strong>the</strong> central plateau of <strong>the</strong> territory and flows<br />
more or <strong>le</strong>ss North-South down to <strong>the</strong> border with Southwest Africa where it<br />
shifts eastwards, forming <strong>the</strong> border, crossing Kaprivi Strip and spreads in-<br />
to a wide swampy area ( Figure 1).<br />
Its drainage basin in Angola is about 150 O00 km2 from which 61 O00<br />
km2 belong to its main tributary Cuito.<br />
The nor<strong>the</strong>rn part of <strong>the</strong> basin is <strong>the</strong> most rainy one and <strong>the</strong>re <strong>the</strong><br />
altitudes reach 1 800 m, decreasing gradually southwards to 1 O00 m.Here <strong>the</strong><br />
climaté is semi-arid. Rainfall occur in <strong>the</strong> wet season from October to April;<br />
<strong>the</strong> o<strong>the</strong>r months are dry.<br />
From <strong>the</strong> geological standpoint, <strong>the</strong> northwest part of <strong>the</strong> basin is<br />
formed by igneous and metamorphic rocks; sedimentary formations occur in <strong>the</strong><br />
rest of <strong>the</strong> basin.<br />
The hydrographic pattern is characteristical as well, <strong>the</strong> tributa -<br />
ries being normally paral<strong>le</strong>l to each o<strong>the</strong>r and flowing North-South. The sha-<br />
pe of <strong>the</strong> beds is ru<strong>le</strong>d by <strong>the</strong> local geological and topographical conditions.<br />
As far as <strong>the</strong> vegetation is concerned, it changes from <strong>the</strong> more or<br />
<strong>le</strong>ss dense forest in <strong>the</strong> North into <strong>the</strong> savana in <strong>the</strong> South.<br />
The prob<strong>le</strong>m was to carry out <strong>the</strong> general plan for <strong>the</strong> development of
246<br />
<strong>the</strong> water resources and obviously <strong>the</strong> first step was <strong>the</strong> hydrological study.<br />
In <strong>the</strong> following chapters a summary will be presented of <strong>the</strong> analy-<br />
sis made for <strong>the</strong> study of <strong>the</strong> rainfall and run-off, according to <strong>the</strong> methods<br />
and criteria mentioned above in this paper, once <strong>the</strong> availab<strong>le</strong> data was ina-<br />
dequate.<br />
4.2 - Rainfall studies<br />
For <strong>the</strong> rainfall studies, <strong>the</strong> records of 28 stations for <strong>the</strong> period<br />
1943/1970 were availab<strong>le</strong>. However, only from 1950/51 onwards, <strong>the</strong> number of<br />
stations with comp<strong>le</strong>te records was sufficient and <strong>the</strong>refore <strong>the</strong> basical study<br />
period considered was 20 years, from 1950 to 1970. Some shortage of monthly<br />
records necessary for <strong>the</strong> evaluation of <strong>the</strong> annual values was easily overcome<br />
by correlation with more comp<strong>le</strong>te and reliab<strong>le</strong> stations.<br />
With <strong>the</strong>se annual values, <strong>the</strong> isohyet maps were drawn on a sca<strong>le</strong><br />
1:l O00 O00 introducting <strong>the</strong> influence of altitude and o<strong>the</strong>r known climatical<br />
factors and avoiding a cold interpretation of <strong>the</strong> plotted values.<br />
After chosing some characteristic sections, <strong>the</strong> weighed annual rain-<br />
fall was determined and analysed by applying <strong>the</strong> Foster-Hazen method. Figure<br />
2 shows a diagram with <strong>the</strong> sequence of annual precipitation and <strong>the</strong> correspon-<br />
ding probability graph for a section of <strong>the</strong> main course of <strong>the</strong> river where <strong>the</strong><br />
international border starts.<br />
From <strong>the</strong> joint study of <strong>the</strong>se graphs, some conclusions can be drawn.<br />
First of all, <strong>the</strong> applied stochastic method can be considered adequate to interpretate<br />
<strong>the</strong> phenomenon and <strong>the</strong>refore it is possib<strong>le</strong> to determine a mean annual<br />
precipitation of 950 mm as well as precipitations corresponding to certain<br />
return periods. One can note <strong>the</strong> occurence of a sequence of four dry<br />
years which might be considered as <strong>the</strong> basis of <strong>the</strong> critical period for regulation<br />
purposes.<br />
4.3 - Run-off studies<br />
The basin has 19 flow measuring stations and <strong>the</strong> records started to<br />
be obtained early in 1963. Before that date, <strong>the</strong>re were some random measure-
247<br />
ments made with floating device> but <strong>the</strong>ir reliability was doubtful. The net-<br />
work is nowadays equipped with automatic <strong>le</strong>vel gaugings and flows are measu -<br />
red with current meters suspended from steel cab<strong>le</strong>s crossing <strong>the</strong> river from<br />
one bank to <strong>the</strong> o<strong>the</strong>r.<br />
It was <strong>the</strong>n possib<strong>le</strong> to have flow records for a period of 7 years<br />
consisting of maximum and minimum flows, average daily flows, and consequent-<br />
ly monthly and annual values.<br />
For such a short period stochastic methods are not applicab<strong>le</strong> with<br />
reliability; never<strong>the</strong><strong>le</strong>ss <strong>the</strong> analyses made for <strong>the</strong> rainfall showed that such<br />
period has average characteristics and <strong>the</strong>refore <strong>the</strong> mean annual flow can be<br />
estimated by averaging <strong>the</strong> flows of those seven years for every station.<br />
The same criteria cannot be applied to determine <strong>the</strong> dry year flow,<br />
because in this seven years period (1963/1970) any of <strong>the</strong> years of <strong>the</strong> cri-<br />
tical period obtained from <strong>the</strong> rainfall study is not included.<br />
Fortunately, <strong>the</strong>re is a station in <strong>the</strong> international strech measu-<br />
red by <strong>the</strong> South African Services which has got records for a longer period<br />
(25 years) from 1945 on, although some of its valueshave been obtained by cor-<br />
relation. It was <strong>the</strong>n possib<strong>le</strong> for this station to apply <strong>the</strong> Foster-Hazen rne-<br />
thod which showed a ra<strong>the</strong>r well interpretation of <strong>the</strong> phenomenon.<br />
Figure 3 shows in <strong>the</strong> same way as for <strong>the</strong> rainfall <strong>the</strong> diagram of<br />
annual flow sequence and <strong>the</strong> probability graph.<br />
The former indicates a notorious resemblance with <strong>the</strong> one of <strong>the</strong><br />
rainfall, being characteristical <strong>the</strong> four dry year period 1966/1970. It skws<br />
as well that <strong>the</strong> period 1963/1970 is an average one and that 1966/67 can re-<br />
present <strong>the</strong> dry year of <strong>the</strong> critical period.<br />
In order to obtain <strong>the</strong> annual flows in any section of <strong>the</strong> river,tk<br />
curves showing <strong>the</strong> variation of <strong>the</strong> specific annual flow with <strong>the</strong> drainage ba-<br />
sin for <strong>the</strong> average and dry year, were drawn ( Figure 4); <strong>the</strong>se curves show<br />
bi uniform pattern and thus one can consider <strong>the</strong>m sufficiently reliab<strong>le</strong> for<br />
obtainment of <strong>the</strong> desired values.<br />
The regulation of flows can be studied by considering <strong>the</strong> sequence
248<br />
of an average year followed by four dry years as determined above.<br />
5. Conclusions<br />
Some criteria normally utilized for hydrological studies of <strong>the</strong> general<br />
plans for <strong>the</strong> development of <strong>the</strong> water resources of rivers in semi-arid areas of<br />
Portuguese African territories were presented and an examp<strong>le</strong> of <strong>the</strong>ir application<br />
given. The obtained results are obviously approximate but <strong>the</strong>y can be considered<br />
sufficiently safe for <strong>the</strong> purpose and moreover when decisions would be taken for<br />
<strong>the</strong> design of specific projects fur<strong>the</strong>r data will be availab<strong>le</strong> and <strong>the</strong>n a more re-<br />
liab<strong>le</strong> analysis can be made.<br />
.........................
W<br />
I<br />
I-<br />
249
nm<br />
6 O0<br />
LOO<br />
200<br />
O00<br />
BOO<br />
600<br />
LOO<br />
200<br />
O<br />
Pmm<br />
500<br />
LOO<br />
300<br />
200<br />
100<br />
O00<br />
900<br />
SEQUENCE GRAPH<br />
FOSTER- HAZEN ADJUSTMENT<br />
- a% m m<br />
0 0 0 - N Y i 0 O O 0 0 0 0 O O y> m O b m 6<br />
- N<br />
- < m u > c m m m m a m r n m<br />
FiGQRE 2 - STUDY OF ANNUAL RAINFALL<br />
PROEABILIT Y<br />
YEAR
IL LL<br />
O<br />
io6,'<br />
10 O00<br />
z 9000<br />
I 3<br />
< 8000<br />
z<br />
7000<br />
6 O00<br />
5000<br />
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O<br />
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1000<br />
3 000<br />
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a O00<br />
7 O00<br />
6000<br />
5 O00<br />
4000<br />
3000<br />
2 O00<br />
1 O00<br />
SEQUENCE GRAPH 251<br />
FOSTER- HAZEN ADJUSTMENT<br />
-ri yl In "01<br />
0 0 0 - h i - 0 0 0 0 0 0 0 0 o ~n m m o i m m<br />
- N<br />
m - m w p i m 01 m 0 . m m m m<br />
PROBABILITY<br />
FIGURE 3 - STUDY OF ANNUAL RUNOFF<br />
i<br />
1
252<br />
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0
AB S TRA CT<br />
UTILIZING CLIMATIC DATA TO APPRAISE POTENTIAL WATER YIELDS<br />
Robert L. Smith"<br />
Precipitation and temperature measurements often represent <strong>the</strong><br />
only significant hydrologic data availab<strong>le</strong> in developing areas. Initial<br />
assessments of potential surface and ground water supplies must build<br />
on this limited climatic base. Early in <strong>the</strong> planning studies <strong>the</strong>re is<br />
need for an accurate estimate of mean annual streamflow, and of <strong>the</strong><br />
probab<strong>le</strong> variance in annual flows. These determinations can be made<br />
utilizing an empirical function relating <strong>the</strong> mean annual runoff<br />
coefficient to <strong>the</strong> aforementioned climatic parameters. The relationships<br />
have been tested in a wide range of environments, and <strong>the</strong>ir general<br />
utility can be extended appreciably with limited surface and subsurface<br />
observations. Applicability of <strong>the</strong> recommended relationships is<br />
demonstrated by se<strong>le</strong>cted case studies involving a variety of prob<strong>le</strong>ms.<br />
Included are examp<strong>le</strong>s illustrating <strong>the</strong> calculation of: (a) mean yields<br />
for ungaged areas, (b) <strong>the</strong> probability distribution of annual flows for<br />
ungaged areas, (c) daily flow duration curves, (d) potential yield of<br />
se<strong>le</strong>cted groundwater areas, and (e) <strong>the</strong> potential impact of precipita-<br />
tion augmentation on surface water supplies.<br />
RESUMEN<br />
A menudo las medidas de precipitación y temperatura son los Úni<br />
cos datos hidrolbgicos disponib<strong>le</strong>s para áreas en desarrollo. Los esti-<br />
mados inicia<strong>le</strong>s sobre abastecimientos potencia<strong>le</strong>s de aguas superficia-<br />
<strong>le</strong>s y subterráneas deben partir de esta limitada base climática. Muy -<br />
pronto en el curso de la planificación se hace necesario un estimado -<br />
preciso del caudal promedio anual y de la variación probab<strong>le</strong> en flujos<br />
anua<strong>le</strong>s. Estas determinaciones pueden hacerse mediante la utilización<br />
de una función empírica relacionando el coeficiente de escorrentía me-<br />
dia anual con los antes mencionados parbmetros climáticos. Este tipo -<br />
de relación ha sido puesto a prueba en una amplia serie de medio am--<br />
bientes y su utilidad general puede extenderse apreciab<strong>le</strong>mente con li-<br />
mitadas observaciones sobre y bajo tierra. El éxito con que se han --<br />
aplicado las relaciones recomendadas se demuestra por medio de casos -<br />
escogidos que cubren una variedad de prob<strong>le</strong>mas. Se incluyen ejemplos -<br />
que ilustran el cálculo de: (a) rendimientos promedios para áreas Ca--<br />
rentes de medidas, (b) la distribución probabilística de cauda<strong>le</strong>s anua<br />
<strong>le</strong>s en áreas carentes de medidas, (c) curvas diarias de caudal-dura---<br />
ción, (d) rendimiento potencial de áreas de agua subterránea escogidas<br />
y, (e) el impacto potencial de la incrementación de precipitación so--<br />
bre abastecimientos de agua superficial.<br />
-<br />
JI Deane Ackers Professor of Civil Engineering, University of Kansas,<br />
Lawrence, Kansas, USA.
254<br />
The water resources planner is often required to appraise <strong>the</strong> water yield<br />
characteristics of streams for which flow data is unavailab<strong>le</strong>. In <strong>the</strong>se situ-<br />
ations <strong>the</strong> initial appraisal has to be based on climatic factors supp<strong>le</strong>mented<br />
by prior experience in similar terrains. This paper presents an empirical<br />
relationship designed to fur<strong>the</strong>r this appraisal, and which <strong>the</strong> author has found<br />
useful on a number of occasions.<br />
The basic water balance equation applied to a catchment area may be<br />
expressed as<br />
P = R + E + AS (1)<br />
where all temo represent units of depth over <strong>the</strong> catchment area, and P = precipitation,<br />
R = basin outflow, E = evapotranspiration and AS = change in<br />
storage.<br />
For <strong>the</strong> condition of an extended time interval <strong>the</strong> AS term becomes<br />
negligib<strong>le</strong>. In this case, and after dividing all terms by P, <strong>the</strong> equation may<br />
be rewritten as<br />
RIP = c = 1 - E/P (2)<br />
Thus, in <strong>the</strong> long term <strong>the</strong> runoff coefficient C is governed by climatic considerations.<br />
Geographers and agricultural scientists have long utilized climatic<br />
parameters in appraisinz water balance questions relating to management of soil<br />
moisture. In 1967 Guisti and Lopez [i] proposed that <strong>the</strong> mean stream discharge<br />
could be determined as a €unction of (a) mean annual precipitation and (b) <strong>the</strong><br />
basin climatic index. BCI. The latter is based on <strong>the</strong> work of Thornthwaite i21<br />
Jan *'"I<br />
where P is <strong>the</strong> average monthly precipitation in centimeters and T is <strong>the</strong> aver-<br />
age monthly temperature in degrees Centigrade.<br />
If <strong>the</strong> hypo<strong>the</strong>sis presented by Guisti and Lopez has merit, it should be<br />
possib<strong>le</strong> to develop a relationship between BCI and <strong>the</strong> deviations from <strong>the</strong> mean<br />
line dram on a sc.atter diagram of average precipitation versus average runoff.<br />
Their initial efforts to develop such a relationship were limited to examination<br />
of relatively short term data in Puerto Rico. Smith 131 subsequently extended<br />
this approach by examining data from approximately 250 ca.tchments in <strong>the</strong><br />
United States and Puerto Rico. The resulting empirical relationship between<br />
<strong>the</strong> coefficient C in equation (2) and <strong>the</strong> BCI is graphed in Figure 1. It dif-<br />
fers appreciably from t.he curve initially presented by Guisti and Lopez.<br />
availab<strong>le</strong> data provided firm definition of <strong>the</strong> relationship for BCI values<br />
ranging froiri 35 to 150. Currently, extension beyond <strong>the</strong>se limits is most ten-<br />
tati.ve and i.s Eased on <strong>the</strong> following. The lower end was extended to <strong>the</strong> obvious<br />
terminal al: <strong>the</strong> origin. Extension of <strong>the</strong> upper end of <strong>the</strong> curve was based un<br />
concurrer.t appraisal of <strong>the</strong> nature of <strong>the</strong> 13CI vs P relationship in high rainfall<br />
zrens, and on recognition that <strong>the</strong> change in C with X I should be such that <strong>the</strong><br />
i.iic.rment.al percent of precipitation which becomes runoff is constantljr<br />
The
increasing bur never exceeds unity. One word of caution. Data utilized in<br />
developing <strong>the</strong> re1 onship was obtained €rom catchments for which <strong>the</strong> sub-surface<br />
outflow was negligib<strong>le</strong>. Thus <strong>the</strong> ruhoff calculated by Figure 1 represents<br />
tot91 runoff and cannot be directly equated to streamflow in those instances<br />
wtie're a significant percentage of <strong>the</strong> yield' is discharged as sub-surface flow.<br />
Utilization of <strong>the</strong> relationship is enhanced by conversion of existing<br />
climatic data into a basic P vs Bdi relationship for <strong>the</strong> area in question.<br />
Worldwide <strong>the</strong> relatioriship between BCI' and P varies markedly. Regionally It<br />
preciably with topographie considerations. However, for a given<br />
he relationship between BCï and P is weJ.1 defined. Figure 2 illustrates<br />
a typ'ical relationship for a basin in <strong>the</strong> State of Kansas in <strong>the</strong> central<br />
United States Qhere e<strong>le</strong>vation changes are negqigib<strong>le</strong>, and similar relations fcr<br />
Puerto Rico where e<strong>le</strong>vation is a significant factor. Note that <strong>the</strong> slope of <strong>the</strong><br />
relationship also varies slightly with location. Figures 1 and 2 caq be used<br />
conjunctively to develop <strong>the</strong> mean annual rainfall-runoff relationship for <strong>the</strong><br />
catchment. Experience has shown that actual data will scatter about <strong>the</strong> curve<br />
so determined because AS is seldom negligib<strong>le</strong> on an annual basis. The individ-<br />
ual curves tend to approach a 45" asymptote as evapotranspiration tends EO<br />
255<br />
become fully satisfied and <strong>the</strong>reby constant. For examp<strong>le</strong>, in Puerto Rico <strong>the</strong><br />
evapotranspiration demand is satisfied at all e<strong>le</strong>vations when <strong>the</strong> rainfall<br />
exceeds SOO centimeters, but <strong>the</strong> magnitude of this consumptive loss is a function<br />
of e<strong>le</strong>vation.<br />
The basic C vs BCI relationship has been tested in several ways with<br />
satisfactosy results. Figure 3 will seme LO illustrate. Figure 3(a) presents<br />
a coiqparisbii of calculated versus observed discharge for thirty streams in<br />
Puerto Rico [4]. The calculated values were determined via conjunctive use of<br />
<strong>the</strong> appropriate curve from Figure 2 and Figure 1.<br />
Since <strong>the</strong> qbserved records<br />
wexe relatively short, many no longer than three years in <strong>le</strong>ngth, <strong>the</strong> applicab<strong>le</strong><br />
BCI was based on <strong>the</strong> average precipitation during <strong>the</strong> period of observed stream-<br />
flow. BCI values for <strong>the</strong>se streams range from 49 to 178. Figure 3(b) presents<br />
<strong>the</strong> mean annual precipitation versus mean aqnual runoff relationship €or <strong>the</strong><br />
State of Kansas. The solid curve <strong>the</strong>reon was based on observed data from 122<br />
basins [5]. The dashed cuí-ve was calculated using <strong>the</strong> Kansas curve of Figure 2<br />
and Lhe basic coefficient chart of Figure 1. Basin BCI values for <strong>the</strong> ctndition<br />
of mean precipitation range from 25 to 70.<br />
The basic relationships can also be utilized to appraise possib<strong>le</strong> stream<br />
response under several yeats of above or below nomal precipitation. For examp<strong>le</strong>,<br />
in recent years appretiab<strong>le</strong> attention has been directed to <strong>the</strong> potential<br />
application of wea<strong>the</strong>r modification tachniqves in improving water supply con-<br />
dtionc.<br />
Although <strong>the</strong> bulk of <strong>the</strong> research effoxt has been directed toward<br />
seeding techniques and understanding <strong>the</strong> mechanisms of cloud physics, several<br />
investigators in <strong>the</strong> 1Jnited States, via <strong>the</strong> use of hydrologic simulation techniques,<br />
have attempted LO explore how streams would respond to a given increase<br />
in precipitation. The relationships in Figures 1 and 2 can be utilized to<br />
estimate <strong>the</strong> percent gain in runoff thaL will oc €or a given increase in<br />
average precipi tatjon. Let <strong>the</strong> subscript I represent natural conaitioi-s, subscript<br />
2 represent augmented conditions, and <strong>the</strong> symbol PM equal P2/P,. Then
256<br />
Percent gain in runoff = 100 - = 100<br />
Ri<br />
PC-PC (PM1 c*-cl<br />
22 113100 (4 1<br />
plcl cl<br />
Tab<strong>le</strong> 1 summarizes <strong>the</strong> impact of precipitation augmentation on water yield<br />
as determined by hydrologic simulation and as reported by Lins<strong>le</strong>y and Crawford<br />
[6], Crawford [7], Lumb [8] and Smith [3]. The first three authors utilized<br />
<strong>the</strong> Stanford Watershed Model and <strong>the</strong> latter utilized <strong>the</strong> Kansas Watershed Model.<br />
In aggregate, <strong>the</strong>se investigators conducted simulations on 14 separate watersheds,<br />
13 in <strong>the</strong> United States and one in New South Wa<strong>le</strong>s. The last two columns<br />
provide a comparison of <strong>the</strong> average increase in yield as determined by<br />
simulation and as estimated by use of Figure 1.<br />
The calculations assumed that<br />
<strong>the</strong> slope of <strong>the</strong> BCI vs P relationship was equiva<strong>le</strong>nt to <strong>the</strong> typical Kansas<br />
curve. This approximation introduces some error because <strong>the</strong> slope of this<br />
relationship does vary slightly from watershed to watershed, None<strong>the</strong><strong>le</strong>ss, <strong>the</strong><br />
calculated and simulated values are -most comparab<strong>le</strong>. Examination of <strong>the</strong> computer<br />
simulations again reveals that year to year increases scatter about <strong>the</strong><br />
mean value listed in <strong>the</strong> tab<strong>le</strong>. - Tab<strong>le</strong> 1 - Comparative evaluation of <strong>the</strong> impact of precipitation aiigmentatiun<br />
on mean yield.<br />
-<br />
-<br />
,ength lbserved<br />
of Period<br />
'eriod ainfall -- Runoff<br />
'ears cm/year iainf all PM<br />
One HundredlTen Mi<strong>le</strong> Creek,<br />
Kansas -<br />
Stranger Creek, Kansas - 11<br />
Doniphan Creek, Kansas - i/<br />
Black Vermi lion River,<br />
Kansas A<br />
Salt Creek, Kansas - 11<br />
17<br />
17<br />
17<br />
20<br />
8<br />
14<br />
14<br />
14<br />
8<br />
88.4<br />
88.4<br />
88.4<br />
90.5<br />
86.4<br />
77.5<br />
77.5<br />
77.5<br />
58.0<br />
.205<br />
.205<br />
,205<br />
.200<br />
.261<br />
.125<br />
.125<br />
.125<br />
,066<br />
1.05<br />
1.10<br />
1.20<br />
1.10<br />
1.10<br />
1.05<br />
1.10<br />
1.20<br />
1.05<br />
S. Fk. Solomon River, Ks - 11<br />
Beaver Creek, Kansas<br />
Cottonwood Creek, Calif. - 21<br />
8<br />
8<br />
20<br />
21<br />
2<br />
58.0<br />
58.0<br />
53.1<br />
45.6<br />
40.9<br />
.O66<br />
.O66<br />
.O57<br />
.O16<br />
,080<br />
1.10<br />
1.20<br />
1.10<br />
1.10<br />
1.15<br />
Wollombi Brook, 3l<br />
New South Wa<strong>le</strong>s -31<br />
5 107.7 .141 1.10<br />
Beargrass Creek, Ky T~ 5 110.6 .403 1.10<br />
Arroyo Seco, Calif. -<br />
5 68.2 .386 1.10<br />
LaBrea Creek, Calif. -4/<br />
41<br />
18 28.6 .O84 1.10<br />
Dry Creek, California - 22 130.0 .472 1.10<br />
Saxtons River, Vermont - 41<br />
16 111.6 .499 1.10<br />
- i/ From data presented by rn i70)<br />
- 2/ From data presented by Lins<strong>le</strong>y and Crawford (1962)<br />
- 3/ From data presented by Crawford (1965)<br />
- 4/ From data presented by Lumb (1969)<br />
5/ Not calculated<br />
-<br />
-<br />
-<br />
% Gain i<br />
Computer<br />
#irnulatiori<br />
16<br />
33<br />
74<br />
35<br />
30<br />
21<br />
41<br />
94<br />
23<br />
49<br />
107<br />
41<br />
62<br />
82<br />
35<br />
20<br />
19<br />
41<br />
18<br />
19<br />
Runoff<br />
:alculated<br />
17<br />
34<br />
70<br />
32<br />
31<br />
22<br />
40<br />
87<br />
26<br />
52<br />
117<br />
53<br />
- 51<br />
78<br />
40<br />
25<br />
24<br />
44<br />
20<br />
20
257<br />
Earlier reference was made to <strong>the</strong> fact that a plotting .of annual precipitation-runoff<br />
values for a given basin will scatter about <strong>the</strong> mean annual relationship<br />
one develops with Figure 1 and <strong>the</strong> basin applicab<strong>le</strong> Figure 2. Also,<br />
it was noted that year to year percentage gains in flow from precipitation<br />
augmentation, and as determined by computer simulation, would scatter about <strong>the</strong><br />
average gain observed for <strong>the</strong> entire period of record. This scattering is due<br />
to <strong>the</strong> well established phenomenon of hydrologic persistence and ref<strong>le</strong>cts shortterm<br />
storage changes. Question arises, <strong>the</strong>refore, as to whe<strong>the</strong>r <strong>the</strong> relation-<br />
ship can be used to determine flow characteristics o<strong>the</strong>r than <strong>the</strong> mean.<br />
answer is yes but a reasonab<strong>le</strong> amount of judgment is required. Determination<br />
of <strong>the</strong> distribution of annual flows will serve to illustrate.<br />
Availab<strong>le</strong> climatic data can be utilized to develop <strong>the</strong> probability<br />
distribution of basinwide annual precipitation, and <strong>the</strong> basin average curve<br />
for Figure 2. When working with a basin whose geologic structure is not conducive<br />
to <strong>the</strong> development of significant baseflow components, i.e., a basin<br />
with minimum persistence characteristi.cs, an estimate of <strong>the</strong> prcbability distribution<br />
of annual flows can be developed by direct application of <strong>the</strong> pre-<br />
cipitation probability function to Figures 2 and 1.<br />
Figure 4 provides a<br />
comparison of calculated and observed annual runoff distributions for <strong>the</strong><br />
Marias des Cygnes River, Kansas, USA. This basin has litt<strong>le</strong> natural storage<br />
and experiences a wide range. in annual precipitation, from <strong>le</strong>ss than 50 an to<br />
more than 150 a.<br />
Experience has shown that <strong>the</strong> foregoing approach is generally applicab<strong>le</strong><br />
to <strong>the</strong> above average years. However, where lag or persistence is expected to<br />
be a significant factor <strong>the</strong> lower portion of <strong>the</strong> distribution function should<br />
be hand<strong>le</strong>d differently. In this case, replotting of <strong>the</strong> precipitation proba-<br />
bility function using a two year running average will provide a more appropriate<br />
solution. The effect, of course, is to convert <strong>the</strong> naturally skewed distribu-<br />
tion which results from direct application of <strong>the</strong> basic coefficient relation-<br />
ship to a more normal distribution so often encountered in <strong>the</strong> annual flow<br />
relationship. Exercise of <strong>the</strong> judgment option inherent in <strong>the</strong> alternative<br />
approaches outlined above requires that <strong>the</strong> planner be cognizant of <strong>the</strong> nature<br />
of typical distribution functions in basins of similar geologic character.<br />
For areas where freeze is of minor concern mean monthly yields can be<br />
estimated by allocating monthly values in proportion to <strong>the</strong>ir contribution to<br />
<strong>the</strong> BCI as defined in equation (3). However, this calculation should be made<br />
using <strong>the</strong> average two month running total due, again, to <strong>the</strong> prob<strong>le</strong>m of lag.<br />
Extension of this concept as a means of developing a stochastic generator of<br />
monthly yield needed for preliminary appraisal of storage-yield relations is<br />
currently underway.<br />
That is, monthly BCI values based on two month running<br />
averages are being utilized to determine <strong>the</strong> regression, correlation, and<br />
standard deviation parameters required for stochastic generation of long term<br />
monthly yield 191.<br />
Utility of <strong>the</strong> basic relationships can be extended to <strong>the</strong> determination<br />
of additional flow characteristics with <strong>the</strong> acquisition of certain short-term<br />
The
258<br />
and miscellaneous field measurements. For examp<strong>le</strong>, experience has shown that a<br />
daily flow duration curve obtained from a short-term record acquired over a pe-<br />
riod of two to three years can be adjusted to a long-term appraisal if <strong>the</strong> ordi-<br />
nates of <strong>the</strong> short-term record are expressed as a dimension<strong>le</strong>ss ratio to <strong>the</strong><br />
average flow observed during <strong>the</strong> short record period. Subsequent mul.tiplication<br />
of <strong>the</strong>se ratios by <strong>the</strong> long-term mean as determined from Figures 1 and 2 will<br />
provide a reasonab<strong>le</strong> approximation of <strong>the</strong> long-term flow duration curve,<br />
The relations described herein have also proven useful in appraising <strong>the</strong><br />
potential yield characteristics of coastal aquifers in sou<strong>the</strong>rn Puerto Rico [IO'.<br />
Historic groundwater use from <strong>the</strong>se aquifers far exceeds <strong>the</strong> possib1.e direct<br />
recharge assuming all <strong>the</strong> locally generated flow, as determined from Figure 1,<br />
is di.verted to <strong>the</strong> groundwater aquifer. In this case <strong>the</strong> princip<strong>le</strong> recharge<br />
mechanism, excluding <strong>the</strong> recirculation effect of well irrigation, is infi.l.tration<br />
of surface water as it flows across <strong>the</strong> alluvial plain. Figures 1 and 2<br />
were utilized to determine <strong>the</strong> mean surface inflow from <strong>the</strong> mountainous central<br />
core at Lne point where <strong>the</strong> water entered <strong>the</strong> coastal plain. Following <strong>the</strong><br />
analysis of various short-term flow duration records which were availab<strong>le</strong>, this<br />
mean yield was converted to a daily flow duration curve as described above.<br />
Local stream seepage measurements , availab<strong>le</strong> from <strong>the</strong> U. S. Geological Survey,<br />
were coup<strong>le</strong>d with o<strong>the</strong>r similar information from prior studies to develop a<br />
channel infiltration rate as a function of channel width and slope.<br />
Applica-<br />
tion of <strong>the</strong> potential loss capacity of each channel to its flow duration curve<br />
allowed subdivision of <strong>the</strong> surface flow into two components; <strong>the</strong> portion which<br />
was infiltrated into <strong>the</strong> subsurface and <strong>the</strong> portion which escaped tcj th* sea.<br />
An areal mass balance was <strong>the</strong>n performed to determine <strong>the</strong> magnitude of trie<br />
subsurface discharge to <strong>the</strong> sea (precipitation on <strong>the</strong> plain plus streamflow<br />
from <strong>the</strong> mountains minus <strong>the</strong> sum of direct local runoff plus evapotranspiration<br />
plus surface flow escaping to <strong>the</strong> sea). The recharge due to infiltration and<br />
<strong>the</strong> subsurface discharge to <strong>the</strong> sea, both as determined above, were <strong>the</strong>n incor-<br />
porated in a subsequent mass balance of <strong>the</strong> subsurtace aquifer in which ïwliarge<br />
was equated to i<strong>le</strong>t pumping (gross p-mpirig minus recirculation or return flow)<br />
plu: subsurface discharge tu <strong>the</strong> sea. The calculations were repeated '01 con-<br />
dition? o<strong>the</strong>r than <strong>the</strong> mean, e.g., <strong>the</strong> vondition of protracted drouth. Results<br />
of <strong>the</strong>se calculations provided a satisfactory explanation of <strong>the</strong> respons< in<br />
aquiftr water <strong>le</strong>vels that has been expeiienced during both noimal and subnormd<br />
climatic condi t I .,ns.<br />
One additional word of caution beforc losjxg this discussion. The<br />
estimates obtained from Figures 1 and 2 as2uine natural catchments, and wtural<br />
climatic ,oriditLons. Whenever man's activities have materially altered he<br />
naturai environment, e.g., by applicatio? of irrigation water or <strong>the</strong> I UI~ ruc-<br />
tion of appreciab<strong>le</strong> impervious areas, adjustments must be niade. The fc,l 3ing<br />
will serve L'J illustrate.<br />
Thr lower 6500 hectares of Cherry Creek, Colorado is partially iirbanized.<br />
Approxinial eli' 15 percent of <strong>the</strong> area is impervious surface for which hie run..ff<br />
coefficient appioxiniates 0.9, and an addit,( nai 10 perrent is in urban awns<br />
which are heavily irrigated (an average of ~uciut t.7 cm per year). 'Hie mem<br />
anmal rainfall appri,. hates 38 m . nd ';c tesponding BCI Is 32. The ad-jii. ;. I:
BCI for <strong>the</strong> irrigated area approximates 87. An estimate of annual yield, with<br />
and without consideration of <strong>the</strong> man induced changes, is summarized below.<br />
-<br />
Percent Moisture Applied Weighted Runoff<br />
Area (a.) C cm.<br />
Natural Conditions<br />
Modified Conditions<br />
100 38 .O25 .97<br />
Natural<br />
Impervious<br />
Irrigated<br />
55<br />
15<br />
30<br />
38<br />
38<br />
104<br />
.O25<br />
.goo<br />
.320<br />
.52<br />
5.12<br />
- 10.00<br />
15.64<br />
The observed mean annual discharge from this 6500 hectare portion of <strong>the</strong> basin<br />
for <strong>the</strong> four ca<strong>le</strong>ndar years 1966-69 was approximately 16 centimeters.<br />
The foregoing examp<strong>le</strong> is of interest on two counts. First, it illustrates<br />
<strong>the</strong> procedure required for adjusting <strong>the</strong> appraisal of mean yield to accommodate<br />
significant man induced changes. Second, it provides a relatively unique exam-<br />
p<strong>le</strong> of <strong>the</strong> impact of urbanization on flow response.. In many areas <strong>the</strong> effect of<br />
urbanization is to reduce <strong>the</strong> opportunity for recharge and thus diminish baseflow<br />
contributions. The reverse is true for <strong>the</strong> examp<strong>le</strong> cited above. .Here, <strong>the</strong> im-<br />
pact of lawn irrigation in a relatively dry climate has created a substantial<br />
baseflow contribution to <strong>the</strong> stream and a significant increase in overall yield.<br />
In summary, precipitation and temperature measurements often represent <strong>the</strong><br />
only significant hydrologic data availab<strong>le</strong> to <strong>the</strong> water resources planner. AE<br />
empirical function relating <strong>the</strong> mean runoff coefficient to <strong>the</strong> aforementioned<br />
parameters hac been developed. The relationship has been tested in a wide<br />
range of environments, and has proven most useful in undertaking preliminary<br />
assessments of water availability. The general utility of <strong>the</strong> relationship can<br />
be extended appreciably with limited field data and <strong>the</strong> application of basic<br />
hydrologic concepts. Continued exploration of <strong>the</strong> utilization of climatic data<br />
in <strong>the</strong> preliminary appraisal of water yield characteristics must be encouraged.<br />
Acknow<strong>le</strong>dgements<br />
A portion of <strong>the</strong> material described herein was deve.loped during conduct of<br />
a research project sponsored by <strong>the</strong> Kansas Water Resources Research Institute<br />
and <strong>the</strong> Office of Water Resources, U. â. Department of <strong>the</strong> Interior. The<br />
author is also indebted to Black & Veatch, Consulting Engineers, Kansas City,<br />
Missouri; R. A. Domenech & Associates, Hato Rey, Puerto Rico; and <strong>the</strong> Puerto<br />
Rico Aqueduct and Sewer Authority for permission to cite information developed<br />
by <strong>the</strong>se several offices in <strong>the</strong>ir analysis of water availability in Puerto Rico.<br />
References Cited<br />
1. Guisti, E.V. and Lopez, M.A., (1967). Climate and streamflow of Puerto<br />
Rico, Carribbean Journal of Science, Vol. 7, pp 87-93.<br />
259
260<br />
2. Thornthwalte, C. W., (1931). The climates of North America according to a<br />
qew classification, Geographic Review, Vol. 21, pp 633-55.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
Smith, RL., (1970). Water utilization aspects of wea<strong>the</strong>r modification in<br />
Kansas, Contribution No. 46, Kansas Water Resources Research Institute,<br />
Lawrence, Kansas.<br />
Black & Veatch - R. A. Domenech & Assoc., (1971). Water Resources of<br />
Puerto Rico, phase 2, surface water appraisal, Puerto Rico Aqueduct<br />
and Sewer Authority, San Juan, Puerto Rico.<br />
Furness, L.W., (1959). Kansas streamflow characteristics, part 1, flow<br />
duration, Kansas Water Resources Board Technical Report No. 1, Topeka,<br />
Kansas.<br />
Lins<strong>le</strong>y, B.K. and Crawford, N.H., (1963). Estimate of <strong>the</strong> hydrologic<br />
results of rainfall augmentation, Journal of Applied Meteorology,<br />
Vol. 2, NO. 3, pp 426-427.<br />
Crawford, N.H., (1965). Hydrologic consequences of wea<strong>the</strong>r modification:<br />
case studies, Human Dimensions of <strong>the</strong> Atmosphere, University of Chicago<br />
Press, Chicago, Illinois, pp 41-57.<br />
Lumb, A.M., (1969). Hydrologic effects of rainfall augmentation, Tech.<br />
Report 116, Dept. of Civil Engineering, Stanford University, Palo<br />
Alto, Calif ornia.<br />
Thomas, KA., Jr. and Fiering, M., (1962). Ma<strong>the</strong>matical syn<strong>the</strong>sis of<br />
streamflow sequences for <strong>the</strong> analysis of river basins by simulation,<br />
Design of Water Resource Systems, Chapter 12, Harvard Press, Cambridge,<br />
Mass achuse t t s.<br />
Black & Veatch - R. A. Domenech & Assoc., (1970). Water Resources of<br />
Puerto Rico, phase 1, ground water appraisal, Puerto Rico Aqueduct<br />
and Sewer Authority, San Juan, Puerto Rico.
z<br />
9<br />
t- U<br />
o -7<br />
O .6<br />
o -5<br />
k<br />
0<br />
o<br />
E 0.4<br />
n.<br />
z<br />
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w<br />
2<br />
I*<br />
k 0.3<br />
O<br />
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K<br />
z<br />
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w<br />
E 0.2<br />
0.1<br />
o .o<br />
COORDINATES<br />
COEFFICIENT<br />
110 .430<br />
I20 .470<br />
140 .5 35<br />
I GO ,583<br />
I RO .624<br />
20 o .655<br />
I I I<br />
O 40 80 120 I60 200<br />
BASIN CLIMATIC INDEX<br />
Figure 1 - Basic climatic index related to <strong>the</strong> ratio of mean runoff<br />
divided by mean precipitation<br />
261
262<br />
300<br />
200 -<br />
PUERTO RICO CURVES<br />
ME& BASIN ELEVATION<br />
1500 METERS<br />
1000 METERS<br />
x<br />
Li1<br />
O<br />
z<br />
o100 -<br />
i=<br />
a<br />
z - J<br />
o<br />
5<br />
cn<br />
a 50m<br />
TYPICAL<br />
CURVE<br />
SOO METERS<br />
-<br />
20 20 50 100 200 500<br />
MEAN ANNUAL PRECIPITATION - CENTIMETERS<br />
Figure 2 - Se<strong>le</strong>cted examp<strong>le</strong>s of <strong>the</strong> relationship between precipitation<br />
and basin climatic index
‘5 1.0<br />
w<br />
E<br />
w<br />
2 0.5<br />
J<br />
3<br />
o -1<br />
a<br />
O<br />
o. 2<br />
OBSERVED MEAN DfSCHARGE IN CMS<br />
MEAFJ ANNUAL RAINFAL.LA<br />
RUNOFF RELATION FOR<br />
STATE OF KANSAS, USA.<br />
BASED ON 122 DATA POINTS<br />
0<br />
(FURNES)<br />
CALCULATED USING STANDARD COEFFI-<br />
CIENT CHART<br />
I I I<br />
40 60 80 100 I20<br />
MEAN ANNUAL PRECIPITATION - CENTIMETERS<br />
Figure 3 - Sone canparative rec4ults obtaiiied with <strong>the</strong> basic BCI vs C<br />
relationship<br />
-<br />
263<br />
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\ \<br />
-++H-OBSERVED ANNUAL RAINFALL<br />
O000 OBSERVED ANNUAL STREAMFLOW<br />
CURVE FITTED TO RAINFALL DATA<br />
---- CALCULATED STREAMFLOVJ CURVE<br />
\<br />
O ooo\<br />
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o \<br />
O\<br />
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O L<br />
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.O5 .I5 .30 .50 .70 .85 .95<br />
PROBABILITY OF EXCEEDANCE<br />
Figure 4 - An examp<strong>le</strong> plot of calculated versus observed probability<br />
distribution of annual flows<br />
O
ABSTRACT<br />
DETERMINATION OF HYDROLOGICAL CHARACTERISTICS<br />
IN POINTS WITHOUT DIRECT HYDROMETRIC DATA<br />
S i 1 v i u S t an e s c u Jc<br />
In Colombia, hydrological information is very scarce.<br />
Consequently no direct hydrometric data are availab<strong>le</strong> for most<br />
of <strong>the</strong> sites of projected hydrotechnical works and exploitation<br />
of water. Therefore one must generally apply methods of generalization,<br />
transfer of direct information from observed points<br />
to points of interest, and indirect estimation of <strong>the</strong> hydrological<br />
characteristics. In relation with this, <strong>the</strong>re are several<br />
proceedings of indirect determination of mean, maximum and<br />
minimum runoff, as well as of o<strong>the</strong>r characteristics of <strong>the</strong><br />
hydrological regime which are applied to <strong>the</strong> concrete conditions<br />
of Colombia. The examp<strong>le</strong>s, which are included to illustrate <strong>the</strong><br />
application of <strong>the</strong> methods pointed out, are se<strong>le</strong>cted from comp<strong>le</strong>x<br />
hydrological studies, elaborated or in <strong>the</strong> process of elaboration,<br />
within <strong>the</strong> frame of <strong>the</strong> activities of interpretation and hydrological<br />
calculations worked out in <strong>the</strong> Colombian Service of Meteoro -<br />
logy and Hydrology.<br />
RESUMEN<br />
En Colombia, la información hidrológica es muy escasa. Co-<br />
mo consecuencia, la mayoría de los sitios de proyectos de obras -<br />
hidrotécnicas y aprovechamientos de agua no disponen de datos hi-<br />
drométricos directos. De tal manera, se deben aplicar ampliamente<br />
métodos de generalización, de transferencia de información direc-<br />
ta desde puntos en que se dispone de observaciones y mediciones -<br />
hacia puntos de interés práctico, así como métodos de calculo in-<br />
directo de las características hidrológicas. Relacionado con esto,<br />
se presentan varios procedimientos de determinación indirecta de<br />
cauda<strong>le</strong>s medios, máximos y minimos, así como de otras caracteris-<br />
ticas del régimen hidrológico, aplicados a las condiciones concre<br />
tas de Colombia. Los ejemplos que se incluyen para ilustrar la -<br />
aplicación de los métodos indicados son se<strong>le</strong>ccionados de estudios<br />
hidrológicos comp<strong>le</strong>jos, elaborados o en curso de elaboración den-<br />
tro del marco de labores de interpretación y cálculos hidrológi--<br />
cos desarrollados en el Servicio Colombiano de Meteorología e Hi-<br />
drología.<br />
* Hydrology Expert - World Meteorological Organization and United<br />
Nations Development - Program. Bogotá, Colombia - Servicio Co--<br />
lombiano de Meteorología e Hidrología.<br />
-
266<br />
Introduction and generali ties<br />
In Colombia, owing to <strong>the</strong> scarcity of direct hydrometric<br />
information, in most of ti<strong>le</strong> specific cases, <strong>the</strong> sites of hydro-<br />
technical works, water uses or o<strong>the</strong>r works which come in direct<br />
or indirect contact with <strong>the</strong> rivers, do not coincide with <strong>the</strong><br />
sites of <strong>the</strong> hydrometric stations. In such cases, <strong>the</strong> necessary<br />
hydrological parameters must be determined by methods of indirect<br />
estimation. The indirect nydrologic estimations are not however<br />
considered as final values, but only used in an approximate way<br />
or for guidance. The magnitude of <strong>the</strong>se estimations is verified<br />
by means of hydrological field activities, which are generally<br />
grouped into two categories:<br />
a) Temporary hydrometric stations.<br />
b) Expeditional hydrological activities.<br />
The temporary hydrometric stations with intensive and comp<strong>le</strong>x<br />
programs of observations and measurements have been widely used<br />
in Colombia, and in particular for dam and reservoir projects<br />
for hydroenergetic purposes, drinking and industrial water uses,<br />
for <strong>the</strong> most important towns in <strong>the</strong> country, irrigation and<br />
drainage districts of national importance, and, to a <strong>le</strong>sser<br />
extent, for navegation, construction arid protection of bridges,<br />
and also for preservation of hydrographic basins.<br />
The expeditional hydrological activities have also encountered<br />
a very wide field of application, especially in aqueduct projects<br />
for medium and small towns, road bridges and sewers, land use<br />
programming, reforestation programmes,.prefeasibility studies for<br />
work projects which come in direct or indirect contact with <strong>the</strong><br />
water currents, and also when planning <strong>the</strong> use of hydric resources.<br />
Regarding this latter aspect, <strong>the</strong> expeditional hydrological<br />
activities not only act as a means of verifying indirect Iiydrological<br />
estimations, but tiicy moreover constitute almost UE only<br />
really acceptab<strong>le</strong> and reliab<strong>le</strong> way of assessing <strong>the</strong> scope of <strong>the</strong><br />
hydrological parameters in over half <strong>the</strong> national territory (in<br />
o<strong>the</strong>r words 60ú- 000 km2) , where <strong>the</strong> stationary hydrological<br />
activities are almost comp<strong>le</strong>tely missing.<br />
The use of temporary hydrometric stations is very similar to<br />
<strong>the</strong> use of what are normally cal<strong>le</strong>d secondary stations. As this<br />
is generally a known method, no details on this question will be<br />
given. Only <strong>the</strong> most important features of <strong>the</strong> prob<strong>le</strong>m will be<br />
presented.<br />
More emphasis however will be made when describing<br />
<strong>the</strong> most usual methods of <strong>the</strong> expeditional hydrological<br />
activities, since in Colombia, <strong>the</strong>se vast fields of application<br />
are not only found in <strong>the</strong> past, but also in <strong>the</strong> present and <strong>the</strong><br />
future.
Verification of hydrological estimations by means of<br />
temp o r a ry hy d rometric stations,<br />
The hydrometric activity of one or several temporary<br />
stations with comp<strong>le</strong>x and intensive observations and measurements<br />
programmes, verifies and comp<strong>le</strong>tes <strong>the</strong> hydrological estimations,<br />
riot only by means of direct data it provides during its operation,<br />
but also through <strong>the</strong> possibility of spreading its series of direct<br />
data, by means of correlatioris with data of o<strong>the</strong>r reference<br />
stations, which have been working for over 15 years, in zones<br />
of similar hydrological characteristics.<br />
In most cases, preference is to install one of <strong>the</strong> temporary<br />
hydrometric stations exactly in or near <strong>the</strong> sites of <strong>the</strong> projected<br />
works. The stretch of river corresponding to <strong>the</strong> work site does<br />
not always however meet satisfactory conditions to install a<br />
hydrometric station. In such cases, various hydrometric stations<br />
must be instal<strong>le</strong>d in <strong>the</strong> hydrographic basin where <strong>the</strong> site of <strong>the</strong><br />
works is found, and later deduct <strong>the</strong> hydrological parameters<br />
by interpolation, balance of discharges or relation with physiographic<br />
or morphometric characteristics of <strong>the</strong> basin,<br />
In many cases, although a temporary hydrometric station can<br />
be instal<strong>le</strong>d in <strong>the</strong> very site of <strong>the</strong> projected works, it proves<br />
preferib<strong>le</strong> to install several more stations in <strong>the</strong> corresponding<br />
hydrographic basin, in order to have possibilities of<br />
controlling <strong>the</strong> activity developed in <strong>the</strong> station of <strong>the</strong> works<br />
site, comp<strong>le</strong>te <strong>the</strong> eventual gaps in observations and measurements,<br />
avoid errors and confirm <strong>the</strong> results,<br />
The working duration of <strong>the</strong> temporary hydrometric stations<br />
is variab<strong>le</strong>, pursuant to <strong>the</strong> specific conditions. When <strong>the</strong><br />
temporary station is right in <strong>the</strong> river as .<strong>the</strong> reference station,<br />
and <strong>the</strong> areas of its hydrographic basins differ by <strong>le</strong>ss than lo%,<br />
<strong>the</strong> correlation is generally established in one year alone.<br />
\<strong>the</strong>n <strong>the</strong> two stations are in <strong>the</strong> same river, but <strong>the</strong> areas of<br />
<strong>the</strong>ir basins differ by over lo%, <strong>the</strong> time needed to establish<br />
a reliab<strong>le</strong> correlation takes various years, In short, when <strong>the</strong><br />
temporary station is not in <strong>the</strong> same river as <strong>the</strong> reference one,<br />
<strong>the</strong> operation of <strong>the</strong> first one does not end when an acceptab<strong>le</strong><br />
ma<strong>the</strong>matical correlation is obtained, but as soon as this is<br />
physically verified, during a period which contains sufficient<br />
humid and dry average years, in o<strong>the</strong>r words, ra<strong>the</strong>r representative<br />
for <strong>the</strong> average nultiannual situation, O<strong>the</strong>rwise <strong>the</strong> correlation<br />
obtained, although good from a ma<strong>the</strong>matical point of view, may<br />
only express a temporary situation, which would <strong>le</strong>ad to great errors<br />
Besides <strong>the</strong> above cases, situations have occasionally been<br />
found where <strong>the</strong> reference station was too far from <strong>the</strong> site of<br />
<strong>the</strong> projected works. Hence a direct correlation was practically<br />
impossib<strong>le</strong> to establish. The prob<strong>le</strong>m could however be solved<br />
with various intermediary stations, which facilitated <strong>the</strong> transfer<br />
of data, by means of chain correlations.<br />
267
268<br />
The decision to suspend <strong>the</strong> running of a temporary hydro-<br />
metric station has always constituted a great difficulty.<br />
Generally, only in a very few cases can <strong>the</strong> duration of operation<br />
of <strong>the</strong>se stations be really considered sufficient. Therefore,<br />
even after <strong>the</strong> works execution has commenced, it is considered<br />
preferib<strong>le</strong> to continue running temporary stations near <strong>the</strong>se<br />
work sites, and <strong>the</strong>se stations sometimes remain in operation<br />
even after <strong>the</strong> corresponding exploitation has started. The<br />
supp<strong>le</strong>mentary data supplied by <strong>the</strong>se stations prove highly useful<br />
to comp<strong>le</strong>te and confirm <strong>the</strong> hydrological estimations, and also<br />
as guidance for eventual improvements in both <strong>the</strong> works and in<br />
<strong>the</strong> water uses programmes,<br />
Verification of indirect hydrological estimations by<br />
cxpeditional methods<br />
The verification of hydrological estimations by means of<br />
temporary hydrometric stations with intensive and comp<strong>le</strong>x<br />
programmes of observations and measurements, constitutes a superior<br />
method, from a qualitative point of view, in relation with <strong>the</strong><br />
verification of estimations by means of expeditional hydrological<br />
activities. It is not always however possib<strong>le</strong> to install and<br />
operate stations in <strong>the</strong> sites of <strong>the</strong> projected works and water<br />
uses, or in <strong>the</strong>ir hydrographic basins. In scarcely populated<br />
regions, it is difficult to operate hydrometric stations, but<br />
<strong>the</strong> estimation of <strong>the</strong> main hydrological parameters of <strong>the</strong>se<br />
areas is essential to plan <strong>the</strong> uses of hydric resources on a<br />
long term basis and also for prefeasibility studies. This<br />
situation is due to certain specific conditions, In Colombia,<br />
in more than half <strong>the</strong> national territory, <strong>the</strong> land communication<br />
lines are comp<strong>le</strong>tely or partly missing, or are in a deficient<br />
state, such that in rainy periods, penetration is rarely possib<strong>le</strong>.<br />
In <strong>the</strong>se areas moreover, it is very difficult to find<br />
satisfactorily qualified peop<strong>le</strong> to act as hydrometric observers,<br />
and <strong>the</strong> installation of limnigraphs, in areas which have no<br />
watchkeepers, generally proves a hazard and failure.<br />
The lack of hydrometric networks, and <strong>the</strong> difficulty of<br />
organizing stationary hydrological activities in almost half<br />
<strong>the</strong> national territory, constitute conditions which favour <strong>the</strong><br />
wide use of expeditional hydrological activities. Although<br />
<strong>the</strong>se cannot give well defined determinations of <strong>the</strong> hydrological<br />
system characteristics, as <strong>the</strong> Stationary systematic hydrometry<br />
cannot be substituted, <strong>the</strong>y constitute essential work to verify<br />
hydrological estimations or to obtain approximate or guide<br />
indications in isolated regions of difficult access, where <strong>the</strong><br />
installation and operation of hydrometric stations fail to<br />
encounter satisfactory conditions.<br />
The hydrological measurements in campaigns are frequently<br />
applied in Colombia to determine <strong>the</strong> following factors:
a) Maximum discharges, duration of floods and<br />
time of wave propagation;<br />
b) Minimum flows and duration of low waters;<br />
c) Sediment charges;<br />
d) Water temperatures;<br />
e) Physical, chemical and biological<br />
characteristics of <strong>the</strong> water;<br />
f) Overall hydrological characteristics of<br />
<strong>the</strong> currents.<br />
üetermination of maximum discharges and flood characteristics<br />
by means of hyd rological expeditions<br />
269<br />
In most of <strong>the</strong> concrete situations (except <strong>the</strong> case of<br />
flood sweeping), <strong>the</strong> aim is to determine <strong>the</strong> maximum runoff,<br />
pursuant to information on maximum historic <strong>le</strong>vels and maximum<br />
floods known in <strong>the</strong> region, which supposes <strong>the</strong> almost total<br />
absence of evident traces of maximum waters in <strong>the</strong> beds of <strong>the</strong><br />
currents.<br />
Thus, <strong>the</strong> information obtained on <strong>the</strong> field, acquires<br />
decisive importance. It can be classified into two categories:<br />
a) Information supplied by <strong>the</strong> river bank dwel<strong>le</strong>rs,<br />
b) Microphysiographic analysis in <strong>the</strong> largest beds<br />
of <strong>the</strong> rivers and on <strong>the</strong>ir banks.<br />
The information solicited from <strong>the</strong> river-bank dwel<strong>le</strong>rs<br />
refers to <strong>the</strong> following factors:<br />
a) The maximum <strong>le</strong>vel of <strong>the</strong> greatest flood known,<br />
b) The year and eventually <strong>the</strong> date when <strong>the</strong> flood<br />
came about,<br />
c) The time <strong>the</strong> flood waters took to reach <strong>the</strong>ir<br />
maximum <strong>le</strong>vel,<br />
d) The time <strong>the</strong> waters took in dropping to <strong>the</strong>ir<br />
normal <strong>le</strong>vels,<br />
e) Eventual artificial influences on <strong>the</strong> maximum<br />
runoff system,<br />
Pursuant to <strong>the</strong> specific possibilities, <strong>the</strong> information<br />
on maximum waters is solicited from river-bank dwel<strong>le</strong>rs who<br />
have lived on <strong>the</strong> premises for over 30 years and <strong>the</strong> questions<br />
are put to various peop<strong>le</strong>, in order to have a chance to compare<br />
replies,<br />
The microphysiographic analysis in <strong>the</strong> largest beds of<br />
<strong>the</strong> currents and on <strong>the</strong>ir banks, consider <strong>the</strong> possibility<br />
of finding certain traces about <strong>the</strong> maximum water <strong>le</strong>vels,<br />
These analysis generally refer to <strong>the</strong> following factors:<br />
a) Geomorphological aspect,<br />
b) Alluvial material and grounds;<br />
c) Vegetation and organic vegetab<strong>le</strong> material
270<br />
The geomorphological or morphohydrographic aspect of <strong>the</strong> bed<br />
constitutes <strong>the</strong> first sign on <strong>the</strong> possibility of <strong>the</strong> river waters<br />
overflowing. The indicative details are micromorphological<br />
aspects of very recent age, traces of erosion processes, alluvial<br />
formations scarcely fixed by <strong>the</strong> vegetation etc, Sometimes<br />
one can even determine lines of separation between <strong>the</strong> lowest parts<br />
of <strong>the</strong> banks, characterized by very recent morphogenetic<br />
processes, and <strong>the</strong> upper parts of same, relatively fixed and of<br />
more advanced evolution. All <strong>the</strong> information resulting from<br />
<strong>the</strong> detai<strong>le</strong>d analysis of <strong>the</strong> bed micromorphology, cannot <strong>le</strong>ad<br />
to an exact determination of <strong>the</strong> maximum <strong>le</strong>vel of <strong>the</strong> waters, but<br />
it does offer a first and very useful general guidance, on <strong>the</strong><br />
extension of <strong>the</strong> maximum flood and its possib<strong>le</strong> lines of demark-<br />
ation on <strong>the</strong> banks or in <strong>the</strong> largest bed.<br />
The micromorphological analysis is comp<strong>le</strong>ted with observ-<br />
ations on <strong>the</strong> alluvial materials and <strong>the</strong> soil. The fine sediments,<br />
coming from <strong>the</strong> smal<strong>le</strong>r bed, found on <strong>the</strong> banks, are a sure sign<br />
of flooding. The secondary soil, discontinuous on surface, and<br />
those which are scarcely at <strong>the</strong> beginning of <strong>the</strong> formation<br />
processes, also indicate <strong>the</strong> overflow of <strong>the</strong> waters, Finally,<br />
<strong>the</strong> mineralogical analysis of <strong>the</strong> fine sepry recent sediments<br />
of <strong>the</strong> largest bed, may indicate <strong>the</strong> presence of materials which<br />
are not of that place but come from upstream in <strong>the</strong> section under<br />
study, which indicates <strong>the</strong> flooding of <strong>the</strong> larger bed. The<br />
vegetation can also indicate <strong>the</strong> overflow of <strong>the</strong> waters, On <strong>the</strong><br />
one hand, <strong>the</strong> discontinuity of <strong>the</strong> vegetab<strong>le</strong> formations indicates<br />
<strong>the</strong> approximate limit of <strong>the</strong> flood. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong><br />
detail<strong>le</strong>d inventary of <strong>the</strong> vegetab<strong>le</strong> species of <strong>the</strong> area can<br />
constitute a highly important piece of information, because all<br />
vegetab<strong>le</strong> material which is different, in <strong>the</strong> larger bed, may<br />
have been brought by floods from upstream. The detai<strong>le</strong>d<br />
laboratory analysis of <strong>the</strong> vegetab<strong>le</strong> content of <strong>the</strong> samp<strong>le</strong><br />
sediments aiid soils may <strong>le</strong>ad to decisive results, when pol<strong>le</strong>n<br />
partic<strong>le</strong>s are found in <strong>the</strong>m which do not belong to <strong>the</strong> vegetab<strong>le</strong><br />
species of <strong>the</strong> area under study, but to o<strong>the</strong>rs from upstream zones.<br />
During <strong>the</strong> .land activities, <strong>the</strong> river-bank dwel<strong>le</strong>rs'<br />
information is always comp<strong>le</strong>ted with microphysiographic analysis<br />
made in <strong>the</strong> largest beds of .<strong>the</strong> currents and on <strong>the</strong> banks of same.<br />
Without <strong>the</strong>se analysis, <strong>the</strong> riversiders' information cannot be<br />
verified, and can consequently i d to very great mistakes, The<br />
errors arise from subjective reasons which make <strong>the</strong> riversiders<br />
hide <strong>the</strong> truth or merely offer information on unknown events.<br />
The microphysiographic information, although unab<strong>le</strong> to fix <strong>the</strong><br />
maximum <strong>le</strong>vel of <strong>the</strong> waters, indicates essential approximations,<br />
as general guidance and verification factors.<br />
Besides finding information on <strong>the</strong> characteristics of <strong>the</strong><br />
maximum runoff (iiiformation frorii river-side dwel<strong>le</strong>rs and micro-<br />
physiographic information), <strong>the</strong> following main operations are<br />
carried out in each section studied:
271<br />
Survey of three cross-sectional profi<strong>le</strong>s, spaced at<br />
equal distances or more, of <strong>the</strong> river width, and<br />
continuing for no <strong>le</strong>ss than 1 m, above <strong>the</strong> maximum<br />
historic <strong>le</strong>vel of <strong>the</strong> waters. During <strong>the</strong> survey, <strong>the</strong><br />
maximum <strong>le</strong>vels are markeù on <strong>the</strong> profi<strong>le</strong>s, and any<br />
lithological sign of soil or vegetab<strong>le</strong> removed from<br />
<strong>the</strong> place;<br />
Survey of <strong>the</strong> longitudinal profi<strong>le</strong> of <strong>the</strong> current,<br />
with a <strong>le</strong>ngth equal to or at <strong>le</strong>ast 5 times <strong>the</strong> width<br />
of <strong>the</strong> river.<br />
Execution of at <strong>le</strong>ast one gaging (if <strong>the</strong> natural<br />
conditions so permit)<br />
Approximate drawing of <strong>the</strong> river span, including <strong>the</strong><br />
largest bed, <strong>the</strong> marking of <strong>the</strong> cross-sectional profi<strong>le</strong>s;<br />
indications on <strong>the</strong> types and sizes of lithological<br />
materials and vegetation of <strong>the</strong> largest and smal<strong>le</strong>st bed,<br />
<strong>the</strong> lines defining <strong>the</strong> maximum flood, certain reference<br />
e<strong>le</strong>ments, etc.);<br />
Sampling of sediments of <strong>the</strong> smal<strong>le</strong>r bed and of alluvial<br />
material, and eventually soils from <strong>the</strong> larger bed and banks,<br />
along <strong>the</strong> cross-sectional profi<strong>le</strong>s made;<br />
Inventary of <strong>the</strong> vegetab<strong>le</strong> species of <strong>the</strong> area and eompiling<br />
of vegetab<strong>le</strong> remains differing to <strong>the</strong> local species,<br />
The litliological samp<strong>le</strong>s of soil or vegetab<strong>le</strong>s are suitably<br />
packed, and all <strong>the</strong> necessary references are marked on <strong>the</strong><br />
packages to establish <strong>the</strong> site from which <strong>the</strong>y have been taken.<br />
Afterwards, pursuant to possibilities, <strong>the</strong> samp<strong>le</strong>s are analysed<br />
in <strong>the</strong> laboratory.<br />
The above activities are made in various representative<br />
sections of <strong>the</strong> hydrographic basin studied, in order to have<br />
sufficient data availab<strong>le</strong> to permit a comparison of values,<br />
an analysis of <strong>the</strong> territorial variation of same and generalization<br />
of runoff maximum. During <strong>the</strong> field work, <strong>the</strong> information from<br />
different sections are permanently compared, bearing in mind <strong>the</strong><br />
territorial continuation of <strong>the</strong> processes, <strong>the</strong> variation of<br />
<strong>the</strong> magnitudes, <strong>the</strong> periods and dates on which <strong>the</strong> events have<br />
come about, etc,<br />
Once <strong>the</strong> field and laboratory activities have beencomp<strong>le</strong>ted,<br />
<strong>the</strong> following factors are determineo during office work:<br />
a) Maximum discharges of homogeneous probability (generally 1%).<br />
b) Main flood characteristics;<br />
c) Eventually, time of wave propagation.<br />
The discharges corresponding to <strong>the</strong> maximum historic <strong>le</strong>vels<br />
are calculated by hydraulic methods. The measurements made<br />
during <strong>the</strong> expeditions help to determine <strong>the</strong> hydraulic formula<br />
factors which contain <strong>the</strong> rugosity coefficient. These values<br />
are not used directly when estimating <strong>the</strong> maximum discharges,<br />
but merely offer comparison criteria. Once <strong>the</strong> maximum historic
2 72<br />
discharges corresponding to a certain frequency have been<br />
calculated (for examp<strong>le</strong> 3% if <strong>the</strong>y have been produced in<br />
30 yars), <strong>the</strong> values should be increased, in accordance with<br />
<strong>the</strong> coefficients, which permit one to pass from larger<br />
frequencies to rare occurrences. Thus a homogeneization of data<br />
is made (<strong>the</strong> 1% probability is convenient), essential for<br />
comparisons and generalization. In order to change values<br />
of various probabilities into 1% probability values, it is<br />
preferib<strong>le</strong> to use coefficients established with base on <strong>the</strong><br />
direct hydrometric data availab<strong>le</strong> in <strong>the</strong> same zones or in<br />
regions which are hydrologically similar. If <strong>the</strong>se comp<strong>le</strong>tely<br />
fail, coefficients will <strong>the</strong>n be used estimated with base on <strong>the</strong><br />
<strong>the</strong>oretic frequency curves, considered adequate for <strong>the</strong> region<br />
under survey,<br />
A final verification of <strong>the</strong> 1% maximum probability discharges<br />
- is made through generalizations of various forms. The most<br />
comfipn are <strong>the</strong> type: Qmax = f (A); lg qmax = f(1gA); qmax =<br />
f ( ); etc, where Qmas = maximum discharge, in m3/s; A area<br />
of R e basin in km2; qmax = maximum yield, in l/s/km2, or mm;<br />
Hm = average e<strong>le</strong>vation of <strong>the</strong> basin, in m; n = a subunit exponent,<br />
specific for <strong>the</strong> natural conditions of a given zone; f = a<br />
different function for each zone.<br />
The chief flood characteristics (swelling time and total<br />
duration of same) are also verified by comparison of data and<br />
generalizations. These latter are determined by reason of<br />
various morphometric and physiographic factors of <strong>the</strong> hydro-<br />
graphic basins (<strong>le</strong>ngth of currents, gradients of same, etc,)<br />
Likewise, <strong>the</strong> time of wave propagation is also verified and<br />
defined. The generalizations are generally determined in<br />
relation with <strong>the</strong> <strong>le</strong>ngths and gradients of <strong>the</strong> currents,<br />
The final verification of <strong>the</strong> results is made by comparison<br />
with <strong>the</strong> direct hydrometric data availab<strong>le</strong> in <strong>the</strong> region. Thus,<br />
it is not acceptab<strong>le</strong> that <strong>the</strong> 1% probability maximum discharges<br />
estimate? by expeditional methods, be <strong>le</strong>ss than <strong>the</strong> discharges<br />
measured in hydrometric stations, during short intervals,<br />
Determination of minimum discharges and duration of low waters<br />
by expeditional methods.<br />
In most of <strong>the</strong> concrete situations, <strong>the</strong> minimum low water<br />
characteristics are determined during <strong>the</strong> expeditions made to<br />
find <strong>the</strong> maximum runoff, subject to <strong>the</strong> condition that <strong>the</strong>se be<br />
made during low waters.<br />
The activities developed on <strong>the</strong> field have three categories:<br />
a) Compiling of information froin <strong>the</strong> riversiders.<br />
b) Hydrological and topographic work in <strong>the</strong> bed of <strong>the</strong> current.<br />
c) Observations on <strong>the</strong> lithology and freatic layers of <strong>the</strong> region,<br />
The reports from <strong>the</strong> riversiders refer to <strong>the</strong> following aspects
273<br />
a) Eventual interruption of <strong>the</strong> runoff,<br />
b) Minimum historic <strong>le</strong>vels;<br />
c) Year and month when <strong>the</strong> runoff was interrupted or when<br />
<strong>the</strong> minimum <strong>le</strong>vel came about,<br />
d) Low waters and duration of same,<br />
e) Eventual artificial influences on <strong>the</strong> minimum runoff system.<br />
Preferibly <strong>the</strong> information is requested from various peop<strong>le</strong><br />
who have lived near <strong>the</strong> river, for ovcr 30 years. The most<br />
marked sections €or analysis are those which pertain to spans<br />
of current where ancient floodgate openings are found, and also<br />
<strong>the</strong> sections near to irrigation land. The existence of<br />
derivations, upstream from <strong>the</strong> section under survey, must be<br />
considered, to avoid considering <strong>the</strong> minimum discharges in<br />
influenced state as minimums in natural state.<br />
Ti<strong>le</strong> hydrological aiid topographic work in <strong>the</strong> bed refer to<br />
<strong>the</strong> following:<br />
a) Execution of measurements;<br />
b) Topographic survey of loiigi tudinal prof i<strong>le</strong>s,<br />
The measurements are generally made by wading. After making<br />
<strong>the</strong> measurements, <strong>the</strong> wet section is drawn and on this, <strong>the</strong> line<br />
of <strong>the</strong> surface of <strong>the</strong> water corrcsponding to <strong>the</strong> lowest water<br />
(in accordance with <strong>the</strong> information on minimum historic <strong>le</strong>vels).<br />
The topographic survey of longitudinal profi<strong>le</strong>s is made on<br />
<strong>the</strong> water surface, and spreads for at <strong>le</strong>ast three times <strong>the</strong><br />
width of <strong>the</strong> lower bed. These operations are made in various<br />
sectioiis representing <strong>the</strong> basin or area under survey, which<br />
are generally assimilated in <strong>the</strong> main confluences, The information<br />
is permanently compared, bearing in mind <strong>the</strong> territorial<br />
continuity of <strong>the</strong> hydrological phenomena.<br />
Throughout <strong>the</strong> hydrological expeditions , <strong>the</strong> lithology of<br />
<strong>the</strong> region is continually observed. If geological maps are<br />
availab<strong>le</strong>, <strong>the</strong>y are taken to <strong>the</strong> field, to have prior indications<br />
on <strong>the</strong> areas where <strong>the</strong>re are permeab<strong>le</strong> rocks Any discontinuity<br />
in <strong>the</strong> runoff during low waters should be explained ei<strong>the</strong>r as<br />
a result of human activities or due to lithological influences,<br />
Research on <strong>the</strong> depth of <strong>the</strong> freatic layers, in existing wells,<br />
is also made, and also on possib<strong>le</strong> contacts of <strong>the</strong>se layers with<br />
<strong>the</strong> flows, which could constitute an important additional inform-<br />
ation for estimating <strong>the</strong> minimum runoff characteristics.<br />
The estimations, interpretation, verification and generalization<br />
of data are made at a later stage, at <strong>the</strong> office. The minimum<br />
discharges are estimated by means of hydraulic methods, For <strong>the</strong><br />
factor containing <strong>the</strong> rugosity coefficient, <strong>the</strong> values are used<br />
which result from <strong>the</strong> measurements made, The discharges of<br />
diverse statistical probabilities are transformed into 97%<br />
probability discharges (three times in 100 years) to obtain liomogeneous<br />
values which can be compared. The coefficients used to
274<br />
change values of greater frequency into values of <strong>le</strong>sser<br />
probability shauld be determined based on direct hydrometric<br />
data of <strong>the</strong> zone or regions with similar hydrological system,<br />
If <strong>the</strong>se fail, determinate coefficients may be used based on<br />
<strong>the</strong> <strong>the</strong>oretic frequency curves, considered adequate for <strong>the</strong><br />
region studied.<br />
In <strong>the</strong> event of intermittent runoff flows once in 30 years,<br />
all <strong>the</strong> minimum discharges with probabilities above 959. may<br />
be considered <strong>the</strong> same or zero.<br />
The 97% probability minimum discharges are firstly analysed<br />
in relation with <strong>the</strong> areas of basins and by means of balances of<br />
discharges. The yields are analyzed by means of generalization<br />
relations, which may be of type qmin = f (iim) for mountainous<br />
areas and qmin = f (B%) or qmin = f (Ud) for flat areas, In<br />
<strong>the</strong>se relations qmin = minimurn yield, in l/s/km2, or mm; Iim =<br />
average e<strong>le</strong>vation of <strong>the</strong> basin, in m; B% = forestal covering<br />
coefficient of <strong>the</strong> basin, in %; Ud = drainage density or density<br />
of <strong>the</strong> hydrographic network in km/km2; and f = a different function<br />
in each zone.<br />
If maps are availab<strong>le</strong> with monthly mean isohyets, <strong>the</strong> minimum<br />
yields may be compared with <strong>the</strong> monthly mean precipitations of<br />
<strong>the</strong> driest month, by means of relations of type qmin = f(Pm),<br />
where Pm = mean precipitation of <strong>the</strong> driest month, in <strong>the</strong> basin<br />
correspoiiding to each section, in mm.<br />
The duration of <strong>the</strong> low waters is analysed from <strong>the</strong> point of<br />
view of territorial continuation of <strong>the</strong> hydrological phenomena<br />
aiid moreover, in relation with <strong>the</strong> distribution of <strong>the</strong> precipit-<br />
ations within <strong>the</strong> year, When special meteorological maps are<br />
availab<strong>le</strong>, indicating <strong>the</strong> average duration of <strong>the</strong> drought periods,<br />
this data can be used to verify <strong>the</strong> maximum low water durations,<br />
by means of relations of type Te = f(Ts), where Te = time or<br />
duration of <strong>the</strong> drougnt period, in days; and f = a different<br />
function in each zone.<br />
Determination of sediment charges by expeditional methods<br />
The hydrological campaigns to determine sediment charges<br />
are made during high waters periods and after floods of certain<br />
importance have occurred, The expeditions organized during high<br />
waters periods try to determine sediment charges in suspension,<br />
whereas <strong>the</strong> o<strong>the</strong>rs refer to haulage volumes.<br />
The main activities to determine sediment charges in suspension<br />
are as follows:
a) Sampling of waters with suspensions;<br />
b) Execution of measurements;<br />
c) Geomorphological observation of <strong>the</strong> land.<br />
275<br />
The water samp<strong>le</strong>s are taken during tlie execution of<br />
measurements, in tlie same points where tlie speeds of <strong>the</strong><br />
water are measured. The measurements and water sampling<br />
are made in various characteristic sections of <strong>the</strong> hydrographic<br />
basin or study zone, where bridges or o<strong>the</strong>r facilities<br />
are availab<strong>le</strong> to execute <strong>the</strong> gagirigs.<br />
iluring <strong>the</strong>se runs, permanent geomorphological observations<br />
are made on <strong>the</strong> existence of erosion processes, land degradation,<br />
lithological conditions ardvegetation, and also <strong>the</strong>ir relation<br />
with washing of <strong>the</strong> soils, etc. All <strong>the</strong>se factors help to<br />
explain <strong>the</strong> abrupt changes in <strong>the</strong> territorial variation of <strong>the</strong><br />
sediment concentration. To make our work easier and as general<br />
guidance, it is convenient to take to <strong>the</strong> field <strong>the</strong> lithological<br />
or geological, general geomorphological and special geomorphological<br />
maps (gradient, fragmentation of <strong>the</strong> relief, erosion, etc) if<br />
<strong>the</strong>y exist.<br />
Once <strong>the</strong>y have been estimated, <strong>the</strong> sediment charges in<br />
suspension are analysed, bearing in mind <strong>the</strong> territorial<br />
continuity of <strong>the</strong> hydrological processes, Any discontinuity<br />
ihould be explained by tlie different contribution of any of<br />
<strong>the</strong> affluents, or by evident morpholithological changes in <strong>the</strong><br />
basin, which determine changes in <strong>the</strong> erosion and <strong>the</strong> transport<br />
of sediments. The concentration of sediments in suspension may<br />
moreover be analysed in function of <strong>the</strong> territorial variation of<br />
tlie corresponding runoff, within <strong>the</strong> hydrographic basin or area<br />
studied.<br />
In order to establish <strong>the</strong> magnitude of <strong>the</strong> averages of sedi-<br />
ment concentration in suspension, various hydrological campaigns<br />
are made, until relations between discharges and sediment<br />
charges, in various characteristic sections can be determined.<br />
Thus, <strong>the</strong> verification and generalization of concentrations or<br />
sediment charges in suspension is made through <strong>the</strong> flow and run-<br />
off magnitudes,<br />
Finally, <strong>the</strong> verification of <strong>the</strong> magnitude of <strong>the</strong> average<br />
values of eoncentretion of suspensions is made, by morpholitho-<br />
logical zones, in function of <strong>the</strong> variation of <strong>the</strong> gradients,<br />
coefficients of covering with forestal vegetation, etc.<br />
To find voluniEs of dragged sediments, certain activities are<br />
carried out, during low water periods, and <strong>the</strong> following are <strong>the</strong><br />
most important among <strong>the</strong>se:<br />
a) Set up marks and fixed reference points.<br />
b) Topographic surveys of alluvial accumulations in flow beds;<br />
c) Set up and recover traps for sediments and measure accumula<br />
t i on s .
276<br />
These operations are performed in various characteristic<br />
sections , generally downstream of important confluences regard-<br />
ing <strong>the</strong> drags contribution. To obtain a general idea on tlie<br />
size of <strong>the</strong> sediment charges dragged along, various campaigns<br />
are made. In <strong>the</strong> first, <strong>the</strong> marks and fixed reference points<br />
are set up on <strong>the</strong> banks, in <strong>the</strong> larger lied, and sometimes even<br />
iii ti<strong>le</strong> smal<strong>le</strong>r bed of <strong>the</strong> currents, and also <strong>the</strong> sediment traps<br />
in <strong>the</strong> smal<strong>le</strong>r and larger river beds. In later campaigns,<br />
topographic surveys are made of <strong>the</strong> alluvial accumulations;<br />
<strong>the</strong> sediments accumulated in <strong>the</strong> traps are removed; <strong>the</strong> marks<br />
are repaired and also <strong>the</strong> reference points that have been<br />
damaged during floods, and <strong>the</strong> traps are again set up for bottom<br />
sediments.<br />
lhe dragged sediment charges are analysed in relation with<br />
<strong>the</strong> magnitude of <strong>the</strong> discharges and suspension charges, and also<br />
in function of <strong>the</strong> morpliolitliolpgical local conditions (litho-<br />
logical comp<strong>le</strong>xes, erosion and gradient processes, etc.)<br />
Finally, coeffients may be established which, for each zone<br />
of specific morpnolithological conditions, indicate <strong>the</strong> magnitude<br />
of tlie proportion that <strong>the</strong> sediment charges dragged along<br />
represent , in relation with <strong>the</strong> suspension charges.<br />
Determination of water temperatures by expeditional methods<br />
The hydrological expeditions which determine <strong>the</strong> water<br />
temperatures, refer to <strong>the</strong> following operations:<br />
a) Measurement of air temperatures.<br />
b) Measurement of water temperatures,<br />
c) Observations on <strong>the</strong> land lithology.<br />
The air temperatures are measured in order to have values<br />
availab<strong>le</strong> to determine correlations between <strong>the</strong>se and <strong>the</strong> water<br />
temperatures. Once <strong>the</strong> correlations have been established,<br />
characteristic values and <strong>the</strong> variation in space and time of<br />
<strong>the</strong> water temperatures can be determined, based on <strong>the</strong> values<br />
of <strong>the</strong> former, Naturally, in such cases, maps with iso<strong>the</strong>rms<br />
of <strong>the</strong> air in tlie surveyance regions are availab<strong>le</strong>,<br />
‘She water temperatures are measured in various characteristic<br />
sections, paral<strong>le</strong>l with those of <strong>the</strong> air temperatures. The<br />
variation in <strong>the</strong>ir values is analysed bearing in mind <strong>the</strong> territorial<br />
continuation of <strong>the</strong> hydrological processes, Aiiy jump in<br />
<strong>the</strong> water temperatures, throughout a flow, should be explained<br />
ei<strong>the</strong>r by confluences with different temperature flows, or by<br />
imp0 r t ant prouiid water contributions,<br />
Observations oii <strong>the</strong> lithology of <strong>the</strong> region are made to<br />
detect possib<strong>le</strong> substantial ground water contributions, and related<br />
to this, explain <strong>the</strong> sharp changes in temperature experienced by<br />
<strong>the</strong> waters throughout <strong>the</strong> flows,
277<br />
in order to compi<strong>le</strong> representative data not only from<br />
<strong>the</strong> territorial variation point of view, but also regarding<br />
tiic temporary variation, expeditions are made tliroughout all<br />
<strong>the</strong> seasons of <strong>the</strong> year.<br />
Determination of physical, chemical and biological character-<br />
istics of <strong>the</strong> water by means of expeditional methods<br />
The campaigns to determine <strong>the</strong> quality of <strong>the</strong> waters are<br />
organized during low water periods, when <strong>the</strong> physical , chemical<br />
and biological characteristics of <strong>the</strong> flow waters are most<br />
stab<strong>le</strong>.<br />
Iii cases of waters whose quality is unchanged by human<br />
activities, <strong>the</strong> characteristic sections for expeditional work<br />
are <strong>the</strong> confluences. To <strong>the</strong> contrary , <strong>the</strong> conf luences with<br />
drainage and sewerage are also taken into account,<br />
The main activities carried out during <strong>the</strong> campaigns arc<br />
as follows:<br />
a) Compiling of water samp<strong>le</strong>s;<br />
b) Execution of measurements;<br />
c) Analysis of water samp<strong>le</strong>s , and eventually, preservation<br />
and packing of same;<br />
d) Geological observations.<br />
The water samp<strong>le</strong>s are analysed on <strong>the</strong> field, if mobi<strong>le</strong><br />
laboratories are availab<strong>le</strong> (<strong>the</strong> most suitab<strong>le</strong>]. When <strong>the</strong>re<br />
are no possibilities of making comp<strong>le</strong>te analysis on <strong>the</strong> field,<br />
<strong>the</strong> samp<strong>le</strong>s are preserved, and at <strong>le</strong>ast <strong>the</strong> analysis of <strong>the</strong> easily<br />
changeab<strong>le</strong> characteristics are made, anù which can be only<br />
determined in fresh tests. ‘Iiie samp<strong>le</strong>s sent to <strong>the</strong> laboratory<br />
are suitably packed, and all tiic indications regarding <strong>the</strong> site,<br />
and date of col<strong>le</strong>ction are noted on <strong>the</strong> packets.<br />
The measurements are made to find <strong>the</strong> discharges to which<br />
<strong>the</strong> characteristics measured correspond, anù also <strong>the</strong> amounts<br />
of waters availab<strong>le</strong> for dilution of chemical coiicentrations,<br />
of vital use, especially in cases of pollution tipping.<br />
The geological observations are similar to those made during<br />
expeditions to find <strong>the</strong> miiiimum runoff characteristics. In <strong>the</strong><br />
case of physical, chemical and biological qualities of <strong>the</strong> flow<br />
waters, any sharp change should be explained ei<strong>the</strong>r by artificial<br />
influence (tipping of pollutions), or by natural influence, due<br />
to confluences with flows of different biological, chemical and<br />
physical characteristics, or due to an abundant food of ground<br />
waters from different lithological zones,<br />
As soon as thc physical, cliemral and biological character-<br />
istics of <strong>the</strong> water have been determined, <strong>the</strong>y are analysed,<br />
bearing in mind <strong>the</strong> territorial continuity of <strong>the</strong> hydrological
278<br />
processes, and <strong>the</strong>y are verified, according to litliokgical<br />
zones, in relation with <strong>the</strong> discharge and runoff magnitudes.<br />
Determination of <strong>who<strong>le</strong></strong> hydrological characteristics of <strong>the</strong><br />
Flows by means of observations and measuremeiits on campaigns<br />
In practice, <strong>the</strong> caso very frequently turns up of <strong>the</strong>re<br />
being no direct hydrometric data availab<strong>le</strong> in certain Iiydro-<br />
graphic basins, or estimations of various hydrological<br />
characteristics must be verified.<br />
In such situations, comp<strong>le</strong>x expeditional hydrological<br />
activities are developed, based essentially on tiie following<br />
p r i nc ipk s :<br />
a) iixccution of simultaneous measurements hydrologically,<br />
in various sections;<br />
b) Periodicity of campaigns, in accordance wi tli <strong>the</strong> hydro-<br />
logical method phases;<br />
c) Installation of recorder apparatus, with long duration,<br />
autonomous operation;<br />
Ci) General Observations o11 <strong>the</strong> genetic characteristics of<br />
<strong>the</strong> hydrological sys tem.<br />
The measurements may refer to most of <strong>the</strong> hydrological<br />
characteristics (<strong>le</strong>vels, discharges, sediment charges, temperature<br />
and physical, chemical arid biological characteristics of <strong>the</strong><br />
waters, etc,) and <strong>the</strong>y are made in various representative<br />
sections of <strong>the</strong> basins under survey, and also in nearby<br />
hydrometric stations, located iii areas with similar hydric<br />
system.<br />
The princip<strong>le</strong> of Iiydrological simultaneity should be strictly<br />
respected during <strong>the</strong> measurements, in order to compare and<br />
correlate <strong>the</strong> results. From an operational point of view, this<br />
supposes a need to execute work by means of various teams of<br />
hydrologists working paral<strong>le</strong>l, in accordance wi tli strictly<br />
established programs regarding <strong>the</strong> sections and measurement hours,<br />
The periodicity of <strong>the</strong> campaigns, iri functinn of <strong>the</strong> hydro-<br />
logical system, is irnposcd as a compulsory condition, in order<br />
to establish <strong>the</strong> variation ranges of <strong>the</strong> characteristics measured<br />
and <strong>the</strong> correct correlations between <strong>the</strong> data of nrious sections<br />
arid those of <strong>the</strong> nearby liydrometric stations.<br />
"lie installation of recording apparatus, of long duration<br />
autonomous operation, is convenient when <strong>the</strong> periodicity and<br />
frequency of <strong>the</strong> campaigns cannot be assured on a satisfactory<br />
<strong>le</strong>vel, and also when oiie is trying to comp<strong>le</strong>te <strong>the</strong> correlations<br />
between <strong>the</strong> data of <strong>the</strong> sections studied and <strong>the</strong> reference<br />
hydrometric stations. tiowever, in most specific cases, <strong>the</strong> land<br />
difficulties prevent execution of works for installation of<br />
recorder apparatus (lack of roads and labour; <strong>the</strong> maintenance and
279<br />
periodic inspection of <strong>the</strong> installations and apparatus cannot<br />
be assured, etc.)<br />
The measurement of tlie hydrological characteristics is<br />
organized in accordance with <strong>the</strong> indications given in <strong>the</strong> '<br />
above paragraphs.<br />
The data analysis is made bearing in mind <strong>the</strong> territorial<br />
continuity of <strong>the</strong> hydrological processes and <strong>the</strong> correlations<br />
between various sections and <strong>the</strong> reference hydrometric stations,<br />
arid also in terms of <strong>the</strong> local physiographic conditions<br />
influencing <strong>the</strong> variation of <strong>the</strong> hydrological system factors,<br />
Naturally, <strong>the</strong> most important thing is to properly determine<br />
ti<strong>le</strong> correlations so as to extend <strong>the</strong> series of data of <strong>the</strong><br />
sections studied, in terms of <strong>the</strong> long series of data availab<strong>le</strong><br />
in <strong>the</strong> reference hydrometric stations. It is <strong>the</strong>refore<br />
convenient for <strong>the</strong> expeditioiiai hydrological activities to be<br />
developed in each zone, at <strong>le</strong>ast during two comp<strong>le</strong>te years.<br />
The verification, analysis and interpretation of <strong>the</strong> data<br />
is made before suspending <strong>the</strong> field activities. Pursuant to<br />
tlie results, <strong>the</strong> initial programmes can be changed and <strong>the</strong><br />
work intensified, to define <strong>the</strong> processes which have not yet<br />
been satisfactorily determined. The total suspension of <strong>the</strong><br />
expeditionai hydrological activities in <strong>the</strong> study area can only<br />
be mac<strong>le</strong> after conclusive results have been obtained, or,<br />
exceptionally, when <strong>the</strong> sure conclusion is reached that <strong>the</strong><br />
methods used are sufficient to deterriiine or verify <strong>the</strong> hydro-<br />
logical characteristics which must be known.<br />
bibliography<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
Uiaconu, C., Lazarescu D, (1965). Iiidrologie, Bucuresti.<br />
Irjorld Meteorological Organization (1970). Hydrometeorological<br />
Practices Guide, OMM-No. 168 TP. 82, Geneva.<br />
Roche, M. (1963). llydrologie de suÏface, Paris,<br />
Stanescu, S. (1969). Chief prcsent day prob<strong>le</strong>ms of <strong>the</strong><br />
national network organization of hydrological stations<br />
in Colombia, Aperiodic Publication 1 , SCMII, Bogota,<br />
Stanescu, S. (1971). Expeditional Iiydrological Activities.<br />
Aperiodic Publication 22, SCMiI, Bogota.<br />
Vircol, Al. (1960). Calculul debitelor maxime folosind<br />
cercetari<strong>le</strong> expeditionare, Studii de liidrologie 1, Bucuresti.<br />
World Meteorological Organization (1972). Case<strong>book</strong> on<br />
Hydrological Network Design Practice, WMO- No. 324 , Geneva.
280<br />
1100<br />
260<br />
E STAC I ON ME TE ORO LOG IC A I NGE N I O MAN U EL I TA<br />
PROMEDIO 1901 -1970<br />
ESTACION HIDROMETRICA CAUCA - JUANCHITO<br />
PROMEDIO 1934-1970<br />
24 O<br />
1935 1940 1945 1950 1955 1960 1%5 1970<br />
COMPARACION DE PROMEDIOS MULTIANUALES SUCESIVOS (GLISANTES) DE PRECIPITA -<br />
CION CON EL PROMEDIO DEL PERIODO 1901-1970 EN LA ESTACION METEOROLOGICA<br />
INGENIO MANUELITA (A) Y DE CAUDAL CON EL PROMEDIO DEL PERIODO 1934.1970<br />
EN LA ESTACION HIDROMETRICA CAUCA- JUANCHITO (8)<br />
FIGURA I<br />
I GRAFICO PARA CURVA DE FRECUENCIA I<br />
COMPARACION DE CURVAS DE FRECUENCIA DE CAUDALES MEDIOS ANUALES<br />
DEL PERIODO 1934-1970 Y i951 -1970 EN LA ESTACION HIDROMETRICA<br />
CAUCA - JUANCHITO<br />
FIGURA 2<br />
A<br />
Ah0<br />
A
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I<br />
L<br />
3<br />
3 -<br />
L<br />
Y<br />
3<br />
L<br />
o<br />
a<br />
281
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0 -<br />
O O<br />
9- i%' x x o N<br />
!li' E l<br />
O 0 I<br />
L J<br />
I I<br />
I I<br />
I<br />
I<br />
I *<br />
I<br />
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I<br />
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I<br />
I<br />
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ae n<br />
oe<br />
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282
283
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IL<br />
IL<br />
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284<br />
I
I l<br />
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285
ABSTRACT<br />
NEW MODELS OF FREQUENCY LAW OF RUNOFF<br />
STARTING FROM PRECIPITATIONS<br />
J.R. TEMEZ<br />
Professor<br />
1.T.O.P.Col<strong>le</strong>ge-Madrid<br />
Two interesting applications of a new hydrometeorological<br />
method are developed with scientific rigour.<br />
From <strong>the</strong> frequency law of annual precipitations, we can<br />
deduce symply <strong>the</strong> law of runoff, and with precission, proved<br />
in all tne esperimental verifications. To do it, we must know<br />
or estimate <strong>the</strong> potential evapotranspiration on <strong>the</strong> basin (ETP)<br />
and <strong>the</strong> minimum effective precipitacion (Po).<br />
An analogous reasoning, yet simplier, allow us to convert<br />
<strong>the</strong> frequency law of maximum precipitations in <strong>the</strong> law of<br />
volumes of superficial runoff in floods. The only necessary<br />
datum is <strong>the</strong> minimum effective rainfall PL, analogous to <strong>the</strong><br />
Po *<br />
We can simplify <strong>the</strong> calculations with special paper of<br />
doub<strong>le</strong> sca<strong>le</strong>. In <strong>the</strong>m, <strong>the</strong> statistic function of runoffs is<br />
<strong>the</strong> same as precipitations if we read each one in <strong>the</strong> correspondent<br />
sca<strong>le</strong>.<br />
The profit of this methodology, evident in bassins without<br />
data of discharge is also important when we know <strong>the</strong> registers<br />
of flow because it makes easy models of adjustment more reasonab<strong>le</strong><br />
than <strong>the</strong> classic one of Galton, Goodrich and so on, and<br />
in this way we avoid nonsensical extrapolations in <strong>the</strong> intervals<br />
of large and small values.<br />
--<br />
RE S U ME N<br />
Dos interesantes aplicaciones de un nuevo método hidrometeo<br />
rológico se desarrollan con rigor científico.<br />
A partir de la <strong>le</strong>y de fre'cuencia de las precipitaciones --<br />
anua<strong>le</strong>s, se deduce la de aportaciones de manera sencilla, y con<br />
precisión, demostrada en todas las comprobaciones experimenta<strong>le</strong>s.<br />
Para elio solamente se necesita conocer o estimar la evapotranspiración<br />
potencial en la cuenca (ETP) y la lluvia mínima eficaz<br />
(Po).<br />
Un razonamiento análogo, aún más simp<strong>le</strong>, permite convertir<br />
la <strong>le</strong>y de frecuencia de máximas precipitaciones en la de volumenes<br />
de escorrentía superficial en avenidas. El Gnico dato necesa<br />
rio es la lluvia mínima eficaz PL, análoga a ia Po.<br />
Los cálculos se simplifican con pape<strong>le</strong>s especia<strong>le</strong>s de dob<strong>le</strong><br />
escala. En ellos, la función estadística de aportaciones es la -<br />
misma de precipitaciones con tal de <strong>le</strong>er cada una en la escala -<br />
correspondiente.<br />
La utilidad de esta metodologia, evidente en cuencas sin da<br />
tos de aforo, es también importante cuando existen registros fo-<br />
ronómicos, pues facilita modelos de ajuste más raciona<strong>le</strong>s que --<br />
los clásicos de Galton, Goodrich, etc., y se evitan así absurdas<br />
extrapolaciones en los intervalos de grandes y pequenos valores.
2 88<br />
1. MOTIVATION<br />
The precipitations, phenomenon of general type, are adjusted<br />
correctly to <strong>the</strong> also general classical frequency laws: Gumbel (maximum<br />
rainfalls), Gauss (annual precipitations*), and so on, altering <strong>the</strong> mean value<br />
and <strong>the</strong> dispersion from some places to o<strong>the</strong>rs.<br />
These precipitations are transformed partially in runoffs, but is<br />
fundamentaly a deterministic process, peculiar of each basin in relation to<br />
<strong>the</strong>ir edafogeological and climatical characteristics. Therefore <strong>the</strong> regime of<br />
runoffs can not be defined with <strong>the</strong> statistic functions, at present in use, which<br />
ignore <strong>the</strong> concrete parameters of each basin, significatives of <strong>the</strong> hydrological<br />
cyc<strong>le</strong>. These functions, by virtue of <strong>the</strong> liberty that <strong>the</strong>ir indetermined coef-<br />
ficients give <strong>the</strong>m, are ab<strong>le</strong> to adjust to <strong>the</strong> experimental points only in <strong>the</strong> -<br />
interval of <strong>the</strong> intermediate values, but <strong>the</strong>y point <strong>the</strong>ir inadequate conception<br />
out to represent <strong>the</strong> hydrological phenomenon in <strong>the</strong> band of small and large<br />
values where <strong>the</strong> disadjustments are significative and many times nonsensical,<br />
such as what happens when <strong>the</strong> runoff of low frequency are negatives and <strong>the</strong><br />
high ones exceed notably <strong>the</strong> cipher of <strong>the</strong> precipitations. Their inadequateness<br />
is manifested more c<strong>le</strong>arly in basins of irregular regime than in <strong>the</strong> regular<br />
ones, since <strong>the</strong>se last are easy to adjust any curve to <strong>the</strong> reduced range of<br />
variation of <strong>the</strong> registers not indicating <strong>the</strong> erroneous extrapolations which are<br />
done about <strong>the</strong> extreme values.<br />
On <strong>the</strong> o<strong>the</strong>r hand, in <strong>the</strong> basins without registers of flow, we<br />
must define correctly from <strong>the</strong> precipitations, not only <strong>the</strong> mean flow, but also<br />
<strong>the</strong> o<strong>the</strong>r parameters which are characteristical of <strong>the</strong>ir hydrological regime<br />
like functions of frequency of runoffs floods and so on.<br />
These considerations have moved <strong>the</strong> author to develop a new hydro<br />
meteorological method of precision and scientifica1 base, though of<br />
simp<strong>le</strong> application.<br />
The artic<strong>le</strong> shows two interesting applications of this metodology,<br />
which will be <strong>the</strong> object of an exhaustive treatment in a later publication.<br />
* The author will propose in a later publication a modified Gauss law.
2. TYPE OF FREQUENCY LAW OF THE ANNUAL<br />
RUNOFFS PROPOSED BY THE AUTHOR<br />
289<br />
When we treat of regulating high percentages of <strong>the</strong> mean flow of<br />
a river, <strong>the</strong> decisive fact is <strong>the</strong> frequency law of <strong>the</strong>ir annual runoffs, lossing<br />
importance in <strong>the</strong> study, <strong>the</strong> precise know<strong>le</strong>dge of <strong>the</strong> monthly variations, more<br />
sensib<strong>le</strong> to <strong>the</strong> geological characteristics of <strong>the</strong> basin, on <strong>the</strong> o<strong>the</strong>r hand deci-<br />
sives when <strong>the</strong> volumes of water to regulate are low.<br />
Therefore it has a big interest to determine <strong>the</strong> said law starting<br />
from <strong>the</strong> precipitations of <strong>the</strong> basins without registers of flow, or by <strong>the</strong><br />
convenient adjustment to <strong>the</strong> experimental points if <strong>the</strong>re are registers of<br />
flows. The method that is exposed later on, solves <strong>the</strong> prob<strong>le</strong>me in both cases.<br />
The balance of water in a period like <strong>the</strong> water year,<br />
permits to establish <strong>the</strong> relation<br />
A = P - E where A = annual total runoff<br />
P = annual precipitation<br />
E = annual and actual evaporation<br />
This equation applicab<strong>le</strong> to <strong>the</strong> individual values of one or several<br />
years, suggest us also about <strong>the</strong> relative configuration of <strong>the</strong> frequency of run-<br />
offs and rainialls laws.<br />
For very big values of P, <strong>the</strong> actual evaporation will be identified<br />
with <strong>the</strong> potential one, practically constant from some years to o<strong>the</strong>rs (in <strong>the</strong><br />
humid years is smal<strong>le</strong>r than in <strong>the</strong> dry ones). On <strong>the</strong> o<strong>the</strong>r hand, in <strong>the</strong> extre-<br />
mely dry years all <strong>the</strong> precipitation will evaporate from its own basin E = P, and<br />
A = O. In intermediate conditions E and A will increase with p.<br />
The qualitative sight of <strong>the</strong> precipitations and runoffs laws is<br />
represented by <strong>the</strong> figure 1. We can calculate <strong>the</strong> frequencies of A from <strong>the</strong><br />
frequencies of P if we know <strong>the</strong> values of 6 (P) that in this way it has <strong>the</strong><br />
significance of a middling evaporation to that precipitation.<br />
That relation 6 = 6 (P) according to <strong>the</strong> considerations done previoly,<br />
will be represented in <strong>the</strong> figure 2 and different from some basins to o<strong>the</strong>rs<br />
in relation to <strong>the</strong> capacity of retention of water on <strong>the</strong>ir soil R and <strong>the</strong> regime<br />
of temporal distribution of <strong>the</strong>ir climatical variab<strong>le</strong>s.<br />
The method proposed consists in establishing <strong>the</strong> family of curves<br />
i=$<br />
(- P<br />
ETP ETP)
290<br />
among which we must se<strong>le</strong>ct <strong>the</strong> most suitab<strong>le</strong> in each concrete case.<br />
We can make that se<strong>le</strong>ction in relation to - R , but <strong>the</strong> curve,<br />
according to what was said before, is also conditioned E by <strong>the</strong> temporal<br />
distribution of <strong>the</strong> precipitation and <strong>the</strong> evapotranspiration, which is variab<strong>le</strong> from<br />
a meteorological zones to ano<strong>the</strong>r. Therefore we think more practical to<br />
represent <strong>the</strong> influence as a <strong>who<strong>le</strong></strong> with all <strong>the</strong>se variab<strong>le</strong>s by <strong>the</strong>ir inmediate<br />
effect Po (figure 2), minimum effective precipitation from which frequency<br />
corresponds a null runoff.<br />
Having present <strong>the</strong> definition of 6 , <strong>the</strong> figure 2 is transformed in<br />
<strong>the</strong> figure 3.<br />
The calculations of <strong>the</strong> frequency laws of precipitation and runoffs<br />
in many basins with registers of flows were made, and in all of <strong>the</strong>m we found<br />
out that <strong>the</strong> values, correspondent to a same frequency, are combined realy<br />
by a relation of <strong>the</strong> type schematized in <strong>the</strong> figure 3 arid has <strong>the</strong> following<br />
expression:<br />
A = O para P < PO y para P > PO<br />
that permits to transforme <strong>the</strong> frequency law of precipitations in <strong>the</strong> frequency<br />
law of runoffs.<br />
The figure 4 contrasts <strong>the</strong> results obtained by this process and<br />
<strong>the</strong> usual ones at <strong>the</strong> basin of <strong>the</strong> Guadalmellato river. In front to <strong>the</strong> good<br />
adjustment of <strong>the</strong> author’s law, Galton gives nonsensical high values, higher<br />
including to <strong>the</strong> precipitations, in <strong>the</strong> interval of high frequencies, whi<strong>le</strong> Good<br />
rich in <strong>the</strong> interval of small values, decisive in <strong>the</strong> studies of regulations of a<br />
river, recomends negatives ciphers including for frequencies higher to O, 10.<br />
The functions of Goodrich, and specially Galton, ignoring <strong>the</strong> physical sense<br />
of <strong>the</strong> hydrological phenomenon, are not capab<strong>le</strong> of simultaneously adapting<br />
to <strong>the</strong> total range of values.<br />
The adjustment of <strong>the</strong> author is repeated in <strong>the</strong> figure 5 with<br />
doub<strong>le</strong> sca<strong>le</strong> of ordinates: <strong>the</strong> normal and <strong>the</strong>ir transformation according to<br />
formula (2). In this way <strong>the</strong> law of frequency of runoffs must be <strong>the</strong> same as<br />
that precipitations provided that to read each one in <strong>the</strong> correspondent sca<strong>le</strong>;<br />
effectivelly we can verify that <strong>the</strong> experimental points as much as <strong>the</strong> preci-<br />
pitations as <strong>the</strong> runoffs are confounded and are distributed in <strong>the</strong> grapht round<br />
about to an only curve.
291<br />
Guadalmellato is an examp<strong>le</strong> of <strong>the</strong> many empirical cornprobations<br />
carried out, which demostrate <strong>the</strong> big precision of <strong>the</strong> method proposed here.<br />
1)<br />
2)<br />
3)<br />
4)<br />
The procedure of calculation consists in <strong>the</strong> following steps:<br />
Calculation of <strong>the</strong> frequency law of precipitations,<br />
Determination of <strong>the</strong> potential evapotranspiration value of <strong>the</strong><br />
basin (ETP) deductib<strong>le</strong> from <strong>the</strong> evaporimetrical measures (or<br />
charts) existing in <strong>the</strong> zone.<br />
Valoration of <strong>the</strong> minimum ei'fective precipitation Po. The data of<br />
flows, if <strong>the</strong>y exist, must orientate <strong>the</strong> computations to choose<br />
<strong>the</strong>parameter Po more convenient. On <strong>the</strong> contrary <strong>the</strong> valoration<br />
of Po must be guided by <strong>the</strong> values obtained in o<strong>the</strong>r basins gauged<br />
of that zone and based in <strong>the</strong> capacity of retention of water in <strong>the</strong>ir<br />
soil. Po is equal to <strong>the</strong> capacity of retention of <strong>the</strong> soil plus a<br />
function of <strong>the</strong> climate which sign can be positive or negative<br />
depending on <strong>the</strong> cases. The author prepares an orientative tab<strong>le</strong><br />
of values of Po, though a hydrologist can estimate sufficiently<br />
exact that cipher with so dear physical sense, based only in <strong>the</strong>ir<br />
experience and in a superficial know<strong>le</strong>dge of <strong>the</strong> characteristics<br />
of <strong>the</strong> basin.<br />
In humid climates where <strong>the</strong> precipitations of low frequency are<br />
much higher than PO, is enough a gross approximation of this<br />
last parameter.<br />
Once that data are known, <strong>the</strong> runoff A with frecuency F is obtain-<br />
ed in relation to <strong>the</strong> precipitation P of that same frequency by <strong>the</strong><br />
formula (1) or <strong>the</strong>ir equiva<strong>le</strong>nt (2).<br />
It must not be forgotten that <strong>the</strong> know<strong>le</strong>dge of <strong>the</strong> frequency law<br />
determines automatically <strong>the</strong> value of <strong>the</strong> mean runoff. Inversely we can<br />
choose <strong>the</strong> parameter Po with <strong>the</strong> condition to proporcionate a mean runoff<br />
equal to <strong>the</strong> previous valuation by o<strong>the</strong>r procedure.<br />
In any case, if <strong>the</strong>re are registers of flow <strong>the</strong> values of ETP and<br />
Po will be needed to get <strong>the</strong> best adjustment with <strong>the</strong> experimental points.<br />
To programme <strong>the</strong> method to <strong>the</strong> use of computers will yet be<br />
easier <strong>the</strong> calculation.
292<br />
3. CORRECTIVE RUNOFF<br />
Let us imagine that after a serie of years of intermediate<br />
characteristics, a year so dry is produced that all <strong>the</strong>ir precipitation is<br />
evaporated and not producing any runoff. In spite of that circunstance, <strong>the</strong><br />
flows of <strong>the</strong> river will be not necessarily null, since <strong>the</strong>y can feed from <strong>the</strong><br />
underground reserve (or superficial) of <strong>the</strong> basin decreasing in agreeab<strong>le</strong> to<br />
<strong>the</strong> curve of exhaustion of <strong>the</strong> base flow.<br />
The corrective runoff, or variation of <strong>the</strong> reserve from <strong>the</strong> end<br />
of a water year to <strong>the</strong> end of <strong>the</strong> following one, will be an analogous function<br />
to <strong>the</strong> represented one in <strong>the</strong> figures 6 and 7: with high values of <strong>the</strong>precipi-<br />
tation, <strong>the</strong> reserve will increase and discharge of <strong>the</strong> river, diminish in this<br />
quantity; on <strong>the</strong> contrary with low values it will decrease, to feed <strong>the</strong> super-<br />
ficial flows; in mean raining years <strong>the</strong> reserve will not change.<br />
For smal<strong>le</strong>r frequency than <strong>the</strong> correspont one to <strong>the</strong> frequency<br />
of minimum effective precipitation F (PO), <strong>the</strong> total runoffs are identified with<br />
<strong>the</strong> corrective ones, responding to terms fundamentally different so far mention<br />
ed. Nei<strong>the</strong>r <strong>the</strong> potential evapotranspiration, ETP, nor <strong>the</strong> Po has now any<br />
incidence in <strong>the</strong> phenomenon, as nei<strong>the</strong>r <strong>the</strong> value of P; <strong>the</strong> law is determined<br />
by <strong>the</strong> curve of exhaustion of <strong>the</strong> reserves of <strong>the</strong> basin, as well as by <strong>the</strong><br />
frecuencies of <strong>the</strong> initial state of said reserves and of PO.<br />
The classical functions of frequency are not ei<strong>the</strong>r capab<strong>le</strong> of<br />
being adapted simultaneously to <strong>the</strong> interval of usual values, prevailing <strong>the</strong><br />
direct runoff of <strong>the</strong> year, and to <strong>the</strong> different interval of small values corres-<br />
pondent to <strong>the</strong> variation of <strong>the</strong> reserve. These functions treat <strong>the</strong>m indiscrimi<br />
nately with an intermediate dull adjustment in which ignoring this real duplicity<br />
to get out of orbit <strong>the</strong> dry runoffs, which are precisely <strong>the</strong> decisive ones in<br />
<strong>the</strong> regulation process of a river.<br />
The artic<strong>le</strong> will not extend in <strong>the</strong> detail of this corrective runoff<br />
which in several cases is necessary to have present, whi<strong>le</strong> in <strong>the</strong> o<strong>the</strong>rs, on<br />
<strong>the</strong> contrary, it has litt<strong>le</strong> importance, like what happens in impermeab<strong>le</strong> basins<br />
with litt<strong>le</strong> variation of <strong>the</strong>ir reserves of a year to <strong>the</strong> next one and of which<br />
mean value and pluviometrical regularity are at <strong>le</strong>ast moderated, in this way<br />
<strong>the</strong> probability of precipitations close to <strong>the</strong> minimum effective one PO is<br />
extremely small not participating on <strong>the</strong> computations. The substractive term,<br />
that according to <strong>the</strong> graph of <strong>the</strong> figure 6, must be applied also to <strong>the</strong> zone of<br />
strong precipitations, represents a percentage very small of <strong>the</strong> total runoff<br />
in <strong>the</strong>se dates which is not worthwhi<strong>le</strong> considering.
4. MAXIMUM ACTUAL EVAPOTRANSPIRATION<br />
ON A DRY CLIMATE<br />
293<br />
As it has been exposed previously, <strong>the</strong> precipitations increase<br />
<strong>the</strong> availabilities of water for <strong>the</strong> evaporation and this one will increase up to<br />
<strong>the</strong> potential evapotranspiration, or more exactely to <strong>the</strong> potential evapotrang<br />
piration of <strong>the</strong> humid years which is about O, 9 times <strong>the</strong>ir mean value. The<br />
previous affirmation is evident to humid climates, but not so to <strong>the</strong> dry ones.<br />
In climates like <strong>the</strong> mediterranean one, <strong>the</strong>re is a season of <strong>the</strong><br />
year (Summer), when <strong>the</strong>ir potential evapotranspirations are maximum<br />
whi<strong>le</strong> <strong>the</strong> precipitations are practically null as much in dry years as in <strong>the</strong><br />
p<strong>le</strong>nty ones. It exists in <strong>the</strong>se dates a permanent deficit of precipitation; <strong>the</strong><br />
actual total evaporation of <strong>the</strong> year does not reach ever to <strong>the</strong> value of <strong>the</strong><br />
potential one and its maximum value will be <strong>the</strong> potential evapotranspiration<br />
of <strong>the</strong> period of precipitations (ETP) p increasing in <strong>the</strong> capacity of retention<br />
of water on <strong>the</strong> soil (R) evaporating in posterior dates.<br />
Is very important that in <strong>the</strong>se cases <strong>the</strong> ETPwhich intervenes in<br />
<strong>the</strong> formulas be replaced by <strong>the</strong> maximum actual evapotranspiration ETP*<br />
where (ETP)* = (ETP)p t R.<br />
It could be said that <strong>the</strong> ETP of <strong>the</strong> formula will in any case be<br />
<strong>the</strong> <strong>le</strong>ast of <strong>the</strong> following values:<br />
1)<br />
2)<br />
potential total evapotranspiration of <strong>the</strong> year<br />
potential evapotranspiration in <strong>the</strong> period of rains increased<br />
in <strong>the</strong> retention of <strong>the</strong> soil<br />
5. FREQUENCY L AW OF VOLUMES OF MAXIMUM FLOODS<br />
The relation between <strong>the</strong> total rainfall P’ and <strong>the</strong> volume of<br />
superficial runoff A’ is of <strong>the</strong> type schematized in <strong>the</strong> figure 3 for P - A, but<br />
now, treating of a phenomenon of short duration and strong concentration of<br />
rainfall, exist <strong>the</strong> following differences:<br />
. The evaporation in so short time and in an atmosphere of big<br />
relative humidity is worth<strong>le</strong>ss and not altering <strong>the</strong> process.
2 94<br />
The essential e<strong>le</strong>ment is <strong>the</strong> quantity of water that can be retain<br />
ed in <strong>the</strong> soil, characterized by a minimum actual rain Pó,<br />
similar in idea to <strong>the</strong> Po of <strong>the</strong> annual runoffs but with ciphers<br />
much smal<strong>le</strong>r.<br />
According to <strong>the</strong> documentation of <strong>the</strong> Soil Conservation Service<br />
of EE. UU. and verified with several studies of <strong>the</strong> author, <strong>the</strong> relation is:<br />
If P’ GPO , A’= O andif P’ >Pó<br />
universal law in relative values to Pó, which is <strong>the</strong>ir only indetermined<br />
parameter (figure 8).<br />
The previous one, suggests <strong>the</strong> creation of a new special paper<br />
(figure 9) with sca<strong>le</strong> of frequency according to Gumbel and doub<strong>le</strong> sca<strong>le</strong> of<br />
ordinates: <strong>the</strong> normal one and <strong>the</strong>ir transformed as:<br />
deducted from <strong>the</strong> equation (4).<br />
x = It (it JTx)<br />
If we draw <strong>the</strong> frequency law of maximum rainfalls on <strong>the</strong> mention<br />
ed paper, that same straight will define <strong>the</strong> frequencies of <strong>the</strong> volumes of flood<br />
A’ reading <strong>the</strong>m in <strong>the</strong> correspondent sca<strong>le</strong>. The method can not be simp<strong>le</strong>r.<br />
The figure 9 shows an examp<strong>le</strong> of application to <strong>the</strong> basin of <strong>the</strong> Cheliff at<br />
Algerie.<br />
The hydrologists defenders of <strong>the</strong> analytical and non-graphical<br />
adjustment, only have to transform <strong>the</strong> law of maximum precipitations accord-<br />
ing to formula (3) or <strong>the</strong>ir equiva<strong>le</strong>nt (4).<br />
There upon <strong>the</strong>se volumes are related only to surface runoff and<br />
to obtain <strong>the</strong> total ones is necessary to increase <strong>the</strong>m in <strong>the</strong> correspondent<br />
groundwater runoff, worth<strong>le</strong>ss however in <strong>the</strong> interval of <strong>the</strong> high values, <strong>the</strong><br />
most interesting to <strong>the</strong> calculations.<br />
In summary, once <strong>the</strong> frequency law of maximum rainfalls is<br />
defined, <strong>the</strong> process of calculation of <strong>the</strong> volumes of flood only need <strong>the</strong><br />
estimation of <strong>the</strong> value of <strong>the</strong> minimum effective rainfall PA,
295<br />
The <strong>book</strong> "Design of Small Dams" of <strong>the</strong> Bureau of Reclamation<br />
take in <strong>the</strong> information of <strong>the</strong> Soil Conservation Service and facilitates a tab<strong>le</strong>s<br />
which suggests values of <strong>the</strong> Pó, principally in relation to <strong>the</strong> nature and thick-<br />
ness of <strong>the</strong> soil, although modified by <strong>the</strong> type of cultivation; this <strong>book</strong> explains<br />
that each concrete case will depend naturally on <strong>the</strong> humidity of <strong>the</strong> soil in <strong>the</strong><br />
initiation of <strong>the</strong> rainfall and <strong>the</strong> values of <strong>the</strong> formula are considered to an<br />
intermediate conditions in <strong>the</strong> dates of presentation of <strong>the</strong> floods. The author<br />
in keeping with <strong>the</strong> <strong>the</strong>ory of <strong>the</strong> Soil Conservation Service but precisely by<br />
that remarkab<strong>le</strong> influence of <strong>the</strong> initial humidity of <strong>the</strong> soil, <strong>the</strong> Pó of <strong>the</strong><br />
formula has to change also in relation to <strong>the</strong> climate and in this manner, o<strong>the</strong>r<br />
things being equal, it will be higher in a dry one than in o<strong>the</strong>r humid one, where<br />
<strong>the</strong>re is a big probability that at <strong>the</strong> beginning of <strong>the</strong> rainfall, <strong>the</strong> soil would be<br />
in proximate conditions to <strong>the</strong> saturation of <strong>the</strong> water.<br />
If data of flows could exist, <strong>the</strong> experimental points of <strong>the</strong> volume<br />
of superficial runoff in <strong>the</strong> maximum flood of each year of register, could<br />
advice <strong>the</strong> Pó to choose to obtain <strong>the</strong> best adjustment.<br />
The conditioning factors of Pó and of Po are basically <strong>the</strong> same<br />
and it must not be forgotten, since any information that w e can orientate in <strong>the</strong><br />
estimation of one (for examp<strong>le</strong> <strong>the</strong> tab<strong>le</strong>s of <strong>the</strong> Soil Conservation, Service) can<br />
be used in <strong>the</strong> determination of <strong>the</strong> o<strong>the</strong>r.<br />
In <strong>the</strong> order of magnitude it can be said that Po is aproximately<br />
fifteen times greater than Pó ; a study directed in establishing with greater<br />
precision this relation would be interesting.<br />
6. CORRELATIONS PRECIPITATIONS-RUNOFFS<br />
The relations:<br />
2<br />
(P - Po) 2 (P'- Pó)<br />
A' =<br />
A = Pt ETP - 2 Po Y P't 4Pó<br />
are rooted in <strong>the</strong> essence of <strong>the</strong> hydrological cyc<strong>le</strong> and have a big physical<br />
signification. Besides obtaining its specific end in <strong>the</strong> transformation Of<br />
frequency law, it also makes c<strong>le</strong>ar <strong>the</strong> types of correlation more adequate<br />
to <strong>the</strong> individual values of <strong>the</strong>se variab<strong>le</strong>s.
296<br />
7. LIMTS OF THE METHOD<br />
This method, as any o<strong>the</strong>r hydrometeorological one, is not<br />
strictly applicab<strong>le</strong> to a singular basin with appreciab<strong>le</strong> captures of water from<br />
o<strong>the</strong>r zones or <strong>le</strong>akages towards <strong>the</strong>m, since <strong>the</strong>ir flows are conditioned also<br />
by precipitations outside <strong>the</strong> said basin.<br />
It is conceived for regimes fundamentally rainy and it has not<br />
been studied for any possib<strong>le</strong> adaptation to <strong>the</strong> snowy ones.
Fig. I . ESQUEMA DE SITUACION RELATIVA DE LAS LEYES DE FRECUENCIA DE "P"Y "A':<br />
RELATIVE SITUATION SCHEME Op FREQUENCY LAWS OF "P" AND "A",<br />
i<br />
4_<br />
ET P<br />
Fig.2 . ESQUEMA DE VARIACION CE 6.<br />
VARIATION SCHEME OF 6.<br />
P = Precipitacidn anual de frecuencia F.<br />
A= Aportacion especifica onud dr lo misma frecunieia<br />
F<br />
5 :Diferencio entre P y A de ia misma frecuencia F.<br />
ETP: Evapotranspiracidn pciencid.<br />
R,= Precipitacidn a cuya frecuencia F ( Po) corresponde<br />
uno aportación nula.<br />
P= Annwl pr.cipitol)an at treginncy F.<br />
297<br />
A= Annwl rprcific totalrunoff Or <strong>the</strong> SQIY fp.p-y<br />
F.<br />
P I<br />
ETP<br />
6 : DiffWlnCe beîueon P and A of iha saw m<br />
ETP x Patentid ewpotranspira tion.<br />
y F.<br />
PO= Precipitation ta which frequrnoy F( Po) Oorrewponh<br />
o null iotalrunoff.
!<br />
-f<br />
I<br />
--- Ley da Goodrich.<br />
Ley d. ßolton<br />
---<br />
4<br />
I<br />
0 1<br />
5<br />
O Puntos experimenta<strong>le</strong>s de atoro.<br />
Puntos rperimnto<strong>le</strong>r de lluvia.<br />
-7-<br />
l !<br />
i i<br />
Fig. 4 r 5 - AJUSTE EN LA CUENCA DEL GUADALMELLATO<br />
ADJUSTEMENT AT GUADALMELLATO BASIN.<br />
-<br />
--- Qoodrich's low<br />
--- Galton's br<br />
o Annual runoff miperiiaentol peints.
AA<br />
P A<br />
299<br />
Fip. 6 y 7 - ESQUEMA DE APORTACION CORRECTIVA DE LA LEY DE FRECUENCIA DEBIDA A LA VARIACION<br />
DE L& RESERVAS.<br />
CORRECTIVE CONTRIBUTICW SCHEME OF FREOUENCY LAW WING TO VARIATION OF RESERVES<br />
P =Procipitacibn anual de frecuencio F , P =Annual precipitation of frequency F<br />
A= Apartacidn especifica M U O I de la misma frocww ' A =AnnuOJ aQocific t#alrUnolf of <strong>the</strong> s- tre-<br />
cia F. qurncy F<br />
ETP = Evapotranipiroci6n potwcial, 1 ETP- Patrntlal rwpotronspiration<br />
PO = Procì~itoci~ 0 cuyo frccunicia F(Po) corres- , PO = Proclpitotion to which frequency F(Pe) carrespande<br />
una apwtación nula. , pondi a null totalrunoff.<br />
AA = Aportación carroctiva. AA = Corroctlve totalrunoff<br />
&,= Máxima valor de la aportaci6n correctivo , AO = Maximum valu# at correctivo totalrunoff
30G<br />
- A'<br />
PA<br />
4<br />
3<br />
2<br />
i<br />
O<br />
a -RELACION ENTRE P' Y A' DE UNA MISMA FRECUENCIA.<br />
RELATION BETWEEN P' AND A' OF THE SAME FREQUENCY.<br />
- P'<br />
?A ~<br />
6 T-<br />
5 1<br />
44<br />
.- . ~- . -- -<br />
Bassin of <strong>the</strong> Cheliff river ( ALGERIE 1<br />
Pk = 35 IlMn.<br />
3 1<br />
I P'with frequency F<br />
I- -- - -- -<br />
ol , i c<br />
. ~~ -<br />
Experimental pointa P'<br />
FIP. 9. GRAFICO ESPECIAL CON DOBLE ESCALA. EN EL UHA MISMA RECTA REPRESENTA LAS LEYES DE<br />
FRECLÆNCIA DE P' Y A'.<br />
SPECIAL GRAPHIC WITH DOUBLE SCALE.ON WHICH THE SAME STRAIGHT LINE REPRESENTS THE<br />
FREQUENCY LAWS OF P' AND A'<br />
P' = Precipitación toto1 de un aguacero de frecuen- P'vTotal precipitation of o rainfall with frequency F.<br />
cia F<br />
A' = Volumen de escorrentia auperficio( de b misma A', surface runoff volume of <strong>the</strong> rame frequency F<br />
frecuencia F en mdrimaa avenidar.<br />
in maximum floods.<br />
Ph= Precipitación de un opuacero do frecuencia - PA= Precipitotlon of o rainfall with frequency F( Po 1 to<br />
F ( PO ) o lo que correrpondo una escorreniio which corresponds o null rurtoce rurloff.<br />
superficiai nulo
ABSTRACT<br />
TRAITEMENT OPERATIONNEL DES DONNES PLUVIOMETRIQUES<br />
ENTACHEES D'ERREURS OU INSUFFISANTES<br />
R. Trendel - Der Megreditchian - Mme Rulliere<br />
The Bureau of Water of <strong>the</strong> National Meteorology, at present<br />
applies a method that allow us, under certain hypo<strong>the</strong>sis, to obtain<br />
<strong>the</strong> equation of lineal multip<strong>le</strong> regression, which permits to<br />
calculate <strong>the</strong> <strong>the</strong>oretical values of <strong>the</strong> monthly rains. The<br />
application of this formula is possib<strong>le</strong> by <strong>the</strong> existance of base<br />
data, corresponding to each season an index actual value/<strong>the</strong>ore-<br />
tical value that is useful to correct.<br />
In this way, we can calculate <strong>the</strong> values of <strong>the</strong>oretical rain,<br />
that allow to correct and comp<strong>le</strong>te <strong>the</strong> series, and also <strong>the</strong> rainy<br />
periods in season without data or with inadequate data. It is<br />
carried out an analysis of correlation to establish <strong>the</strong> degree of<br />
guaranty of this method and to choose <strong>the</strong> parameter to use in <strong>the</strong><br />
different possib<strong>le</strong> hypo<strong>the</strong>sis and, particulary, <strong>the</strong> iterative<br />
method of Van Isacker.<br />
-- RESUME<br />
Le Bureau de l'Eau de La Météorologie Nationa<strong>le</strong> applique<br />
actuel<strong>le</strong>ment une méthode opérationnel<strong>le</strong>, découlant sous certaines<br />
hypothèses de l'équation de regression linéaire multip<strong>le</strong>, permettant<br />
de calcu<strong>le</strong>r des va<strong>le</strong>urs dites "théoriques" des pluies mensuel<strong>le</strong>s.<br />
L'application de cette formu<strong>le</strong> est rendue possib<strong>le</strong> grâce à<br />
l'existence d'un important fichier de norma<strong>le</strong>s. A chaque estation<br />
correspondant un indice (va<strong>le</strong>ur réel<strong>le</strong>/va<strong>le</strong>ur théoriqye), il est<br />
alors aisé de repérer <strong>le</strong>s va<strong>le</strong>urs qui divergent trop a l'intérieur<br />
d'une même zÔne d'homogénéité.<br />
On calcu<strong>le</strong> ainsi <strong>le</strong>s va<strong>le</strong>urs de la pluie "théorique"<br />
permettant de comb<strong>le</strong>r <strong>le</strong>s données manquantes et de pallier <strong>le</strong>s<br />
erreurs <strong>le</strong>s pius grossières.<br />
De même, pour <strong>le</strong>s précipitations, on calcu<strong>le</strong> <strong>le</strong>s va<strong>le</strong>urs<br />
mensuel<strong>le</strong>s et s'il y a lieu, cel<strong>le</strong>s des episodes pluvieux, pour<br />
<strong>le</strong>s poster fermés ou insuffisants.<br />
On effectue une analyse de corrélation pour étayer <strong>le</strong> degré<br />
de validité de la méthode opérationnel<strong>le</strong> et effectuer <strong>le</strong> choix des<br />
paramètres à utiliser. On examine <strong>le</strong>s possibilités offertes dans<br />
ce domaine par la méthode des composantes principa<strong>le</strong>s, en parti-<br />
culier sous la forme itérative de Van Isacker.<br />
Différents critères sont éga<strong>le</strong>ment testés pour déce<strong>le</strong>r <strong>le</strong>s<br />
va<strong>le</strong>urs douteuses, éventuel<strong>le</strong>ment entachées d'erreurs.<br />
Certaines indications sont fournies sur la répartition<br />
rationnel<strong>le</strong> du réseau pluviométrique.
302<br />
I - DETECTION AUTOMATIQUE DES ERREURS :<br />
1 - Pl~~e_<strong>the</strong>oriq~e_'encuel<strong>le</strong><br />
Le Bureau de L'Eau de la Météorologie Nationa<strong>le</strong> a mis au point<br />
une méthode permettant la critique automatique dos données pluvio-<br />
métriques. El<strong>le</strong> est appliquée dans <strong>le</strong> domaine relativement peu<br />
étendu d'un département français; el<strong>le</strong> utilise une formu<strong>le</strong> empi-<br />
rique permettant de calcu<strong>le</strong>r 1a"pluie théorique" mensuel<strong>le</strong> pour<br />
une station donnée, en fonction de la somme pondérée des va<strong>le</strong>urs<br />
réel<strong>le</strong>s de la pluie mensuel<strong>le</strong> aux autres stations du département.<br />
considéré, <strong>le</strong>s coefficients de pondération étant <strong>le</strong> rapport du<br />
seuil de référence de cette station 5 celui des autres stations.<br />
La formu<strong>le</strong> proposée est de la forme :<br />
O0 .~<br />
Pth - p luie théorique mensuel<strong>le</strong> de la station à étudier<br />
m - seuil de référence de cette station<br />
ms - seuil de référence de La station s<br />
Pr(s)-pluie réel<strong>le</strong> mensuel<strong>le</strong> de la station s<br />
n - nombre de stationssans données manquantes utilisées<br />
pour.l'interpolation<br />
La formu<strong>le</strong> précitée décou<strong>le</strong> de la formu<strong>le</strong> utilisée en analyse<br />
objective pour l'interpolation de Gandine.<br />
Les seuils de référence des stations pluviométriques ont ete<br />
obtenus en partant des norma<strong>le</strong>s mensuel<strong>le</strong>s publiées par la Météoro<br />
log i e Franc a i se C.13<br />
______________---_--<br />
Indice d'homogénéité<br />
On appel<strong>le</strong>ra indice d'homogénéité <strong>le</strong> rapport "pluie réel<strong>le</strong><br />
mensuel<strong>le</strong>/pluie théorique mensuel<strong>le</strong>" calculé pour une station<br />
donnée. IL est bien evident que ce rapport sera nul pour une sta-<br />
tion ne possédant aucune donnée pendant <strong>le</strong> mois étudié, et qu'il<br />
sera inférieur 5 la va<strong>le</strong>ur réel<strong>le</strong> si la station possède des don-<br />
nees manquantes.<br />
Lorsque l'on porte <strong>le</strong>s va<strong>le</strong>urs des indices de toutes <strong>le</strong>s sta-<br />
tions du département sur une carte, i l apparait des zÔnes d'homo-<br />
généite bien délimitees 5 l'intérieur desquel<strong>le</strong>s ces va<strong>le</strong>urs sont<br />
très voisines; ce qui veut dire que, dans ces zÔnes, Les pluies<br />
sont fortement corrélées. Les stations ne s'inscrivant pas dans<br />
La répartition spatia<strong>le</strong> des indices sur <strong>le</strong> département sont jugees<br />
douteuses; tel<strong>le</strong> est la base de la critique proposée.<br />
_______ ~<br />
~<br />
2 - Pluie mensuel<strong>le</strong> _______ -_----<br />
estimée<br />
Pour I estimation de Ta pluie mensuel<strong>le</strong> des stations possbUdi~t<br />
une série incomp!&te, nous utilisons la somme pondérée des indjccs<br />
d'homogénéite relatifs aux trois stations complètes <strong>le</strong>c. plus ~r'oches<br />
multiplice par la pluie <strong>the</strong>orique de la station en question.<br />
. . ./
Ob<br />
La formu<strong>le</strong> est de La forme :<br />
'est - pluie mensuel<strong>le</strong> estimée de,la station à étudier<br />
KS - facteur de pondération relatif à la station s<br />
CS - indice d'homogénéité de la station s<br />
303<br />
Pth - pluie théorique mensuel<strong>le</strong> de ta station à étudier<br />
Le facteur de pondération utilisé ici est fonction de l'inverse<br />
des distances entre stations.<br />
3 - ------------ Décalages et --------- anomalies -<br />
Pour rechercher des anomalies ou décalages éventuels, souvent<br />
dus à des erreurs de transcription, nous appliquons <strong>le</strong> principe<br />
suivant :<br />
On considere qu'une va<strong>le</strong>ur journali&re (nous la noterons flk)<br />
est décalée ou anoma<strong>le</strong> si :<br />
a) O, alors que la va<strong>le</strong>ur journalière pour chacune des<br />
trois stations <strong>le</strong>s plus proches est supérieure<br />
ou éga<strong>le</strong> à 1 mm.<br />
b) vk<br />
supérieur à 3 mm. avec une va<strong>le</strong>ur journlière nul<strong>le</strong><br />
aux trois stations <strong>le</strong>s plus proches.<br />
Dans Les deux cas, i l faut que <strong>le</strong>s conditions suivantes soient<br />
vérifiées pour un jour donné;<br />
nombre de stations of = O 1<br />
nombre total de stations<br />
G?T<br />
4 - ------ Cumuls<br />
nombre de stations où ,><br />
nombre total de stations<br />
3 mm. 1<br />
O0 Cj<strong>le</strong>st(j) - p luie journalière estimée au jour j a la station<br />
étudiée<br />
Fil-(s,j> - pluie réel<strong>le</strong> journalière La station s Le jour j<br />
- pluie cumulée de la station étudiée<br />
n,<br />
"1<br />
- <strong>le</strong>r jour des données cumulées<br />
"2 - dernier jour du. cumul ("1 \< j < n2)<br />
Dans la formu<strong>le</strong> (I), nous utilisons un seuil de référence établi<br />
a partir des norma<strong>le</strong>s établies par Angot en 1913 pour la periode<br />
1850-1900. Pour <strong>le</strong>s stations n'existant pas à cette époque, ce<br />
seuil est obtenu par la méthode du tracé des isohyètes. Pour avoir<br />
des va<strong>le</strong>urs aussi précises que possib<strong>le</strong>, nous corrigeons reguli&rement<br />
ce seuil de reference au fur et 3 mesure du développement<br />
du fichier.<br />
Pour cela, nous calculons <strong>le</strong>s moyennes mensuel<strong>le</strong>s des données<br />
du fichier, en tenant compte s'il y a lieu, des observations manquantes.<br />
Ces moyennes considérées comme pluies réel<strong>le</strong>s dans la<br />
formu<strong>le</strong> (1) permettent de calcu<strong>le</strong>r <strong>le</strong>s indices d'homogénéite qui<br />
devraient être voisins de 1. Si <strong>le</strong>s coefficients appartiennent à<br />
l'interval<strong>le</strong> (0,90 ; 1,îO) la norma<strong>le</strong> est acceptée, sinon el<strong>le</strong><br />
est modifiée.<br />
II - REMPLACEMENT DES DONNEES MANQUANTES OU ABERRANTES PAR DES<br />
VALEURS C ALC ULEES.<br />
------ 1Pre méthode - ----- :<br />
En cas de donnees manquantes, la formu<strong>le</strong> (2) permet de calcu<strong>le</strong>r<br />
la pluie estimée. Si la différence "pluie estimée-pluie réel<strong>le</strong>"<br />
est négative, <strong>le</strong>s données manquantes de la station ne sont pas recherchées<br />
car, dans la plupart des cas, el<strong>le</strong>s correspondent a des<br />
traces ou 5 des cumuls oubliés.<br />
Pour calcu<strong>le</strong>r ces données, nous utilisons toujours <strong>le</strong> même<br />
principe de pondération, ce qui conduit A la formu<strong>le</strong> :<br />
Test(j) - pluie estimée du jour j de la station étudiée<br />
V%(s,j) - pluie réel<strong>le</strong> journalière de la station s <strong>le</strong> jourj<br />
R m - pluie réel<strong>le</strong> mensuel<strong>le</strong> de la station à étudier<br />
- nombre de pkriodes distinctes de donnees manquantes<br />
"1 1 - <strong>le</strong>r jour de donnees manquantes de la lierne pbriode<br />
n2(<br />
- dernier jour de cette période(nll
305<br />
--------<br />
REMARQUE<br />
Cette méthode ne donne pas toujours des résultats acceptab<strong>le</strong>s.<br />
a) Lorsque la station dont on veut calcu<strong>le</strong>r la pluie estimee se<br />
trouve à la frontière séparant deux zônes de répartition spatia<strong>le</strong><br />
différentes des indices d'homogénéité, la pondération<br />
utilisée relative aux trois stations <strong>le</strong>s plus proches, traduit<br />
une distribution particuliere des poids qui peut s'éloigner de<br />
la réalité.<br />
b) Alors que <strong>le</strong>s indices d'homogénéité caractérisent <strong>le</strong>s précipi-<br />
tations mensuel<strong>le</strong>s aux stations, ils entrent dans <strong>le</strong> calcul de<br />
la pluie journalière estimée.<br />
Pour toutes ces raisons, i l a été nécessaire de réduire l'échel<strong>le</strong><br />
du temps. La critique automatique des données pluviométriques<br />
est maintenant appliquée aux épisodes pluvieux. L'efficacit6 de ce<br />
procédé a déjà éte vérifié par l'étude de certains mois ne présentant<br />
qu'une seu<strong>le</strong> période pluvieuse.<br />
Pour pallier ces difficultés, nous avons mis au point une deuxième<br />
méthode permettant d'orienter Le choix du météorologiste.<br />
------------<br />
2ème methode :<br />
El<strong>le</strong> est basée sur la recherche d'un rapport de proportionnalité<br />
moyen entre la somme des précipitations correspondant aux périodes<br />
des données manquantes et la différence du total mensuel comp<strong>le</strong>t<br />
avec cette somme.<br />
I Nous utilisons ici <strong>le</strong>s trois stations <strong>le</strong>s plus proches sans<br />
données manquantes.<br />
Appelons Pr(l), Pr(2), Pr(3) <strong>le</strong>s totaux mensuels respectifs de<br />
la première, seconde et tro.isikme stations.<br />
DI, D2, D3 la somme des précipitations tombées respectivement<br />
à ces trois stations durant <strong>le</strong>s périodes considérées.<br />
P/,(s), la différence Pr(s) - Ds (s variant de 1 3 3).<br />
Nous calculons : 3<br />
K=Z K s L (5)<br />
s=' P;(q<br />
OU Ks est un facteur de pondération, fonction de l'inverse de la dis-<br />
tance séparant la station étudiée de la station s.<br />
Connaissant P:, <strong>le</strong> total mensuel incomp<strong>le</strong>t de la station étudiée,<br />
et La va<strong>le</strong>ur de K d'après la formu<strong>le</strong> (5), nous pouvons ecrire :<br />
ob D représente la somme des précipitations correspondantes aux<br />
jours des données manquantes pour la station étudiée.<br />
Pour calcu<strong>le</strong>r <strong>le</strong>s quantités journalibres manquantes, i l suffit<br />
de reprendre la formu<strong>le</strong> (4) utilisée pour la première méthode en<br />
remp1,açant par D.<br />
(6)
0<br />
Y I<br />
. 0<br />
I-<br />
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C 2<br />
i1<br />
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.I- U<br />
c<br />
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- .e W<br />
N N N<br />
n n n<br />
I I<br />
c .<br />
o<br />
4 L<br />
O<br />
e.<br />
I<br />
52<br />
306
1 - 1 1 I<br />
~:ooooooooooooooooooooooooo 00000<br />
n~ooooooooooooooooooo~~oooo 00000<br />
t<br />
Q:000000000000000O0000Q0000 00000<br />
t<br />
yI toe, 00.0 oo oooo 0-0 oo o oo a a o o o oo.oee<br />
.I<br />
*~00O0O00000000000000000000 00000<br />
t<br />
s-~ooooooooooooooooooooooooo ooooa<br />
t<br />
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e<br />
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-i1 O.<br />
W O<br />
-<br />
a<br />
Y<br />
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-a 23<br />
cn<br />
v-<br />
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m<br />
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z<br />
T<br />
a<br />
308<br />
I I I - CALCUL A PARTIR DES STATIONS EXISTANTES DES VALEURS MENSUELLES<br />
ET DES EPISODES PLUVIEUX POUR LES POSTES FERMES OU INSUFFISANTS<br />
Les développements précédents nous permettent de :<br />
1 - calcu<strong>le</strong>r <strong>le</strong> seuil de référence d'une station dont <strong>le</strong>s données<br />
n'existent que pour une période très courte (2 ou 3 ans)<br />
sous réserve cependant qu'el<strong>le</strong>s ne soient pas erronées.<br />
2 - déterminer d'après la formu<strong>le</strong> (1) La pluie théorique d'une<br />
station B l'aide de son seuil de référence et des données<br />
disponib<strong>le</strong>s pour <strong>le</strong>s autres stations du département<br />
3 - calcu<strong>le</strong>r la pluie estimée d'après la formu<strong>le</strong> (21, (Le météo-<br />
rologiste pouvant éventuel<strong>le</strong>ment l'obtenir B partir de la car-<br />
te des indices d'homogénéité) '<br />
4 - rechercher <strong>le</strong>s pluies journalières en utilisant <strong>le</strong>s formu<strong>le</strong>s<br />
(1) (2) et (4) dans Le cadre des épisodes pluvieux.<br />
Les impératifs de l'élaboration d'un fichier valab<strong>le</strong> de pluvio-<br />
métrie avaient déterminé l'adoption dans la pratique opérationnelie<br />
de la méthode simp<strong>le</strong> de critique des données, que nous venons d'ex-<br />
poser.<br />
Parallè<strong>le</strong>ment B cela, Le Bureau de L'Eau a poursuivi des recher-<br />
ches théoriques afin d'elucid.er <strong>le</strong> degré de validité de la méthode<br />
adoptée et <strong>le</strong>s améliorations qu'il convenait de lui apporter.<br />
IV - 'LE PROBLEME DES DONNEES MANQUANTES C8,3,43<br />
Quatre methodes ont été vérifiées sur un fichier donnant <strong>le</strong>s<br />
hauteurs des pluies journalieres en 15 stations des Côtes du Nord<br />
pour <strong>le</strong>s mois de Janvier de 1'1 années consécutives (de 1961 2 1971)<br />
1 - Analyse en composantes principa<strong>le</strong>s<br />
__________-I_---------------------<br />
Le principe de La méthode est <strong>le</strong> suivant :<br />
On passe des données initia1es:r;ii va<strong>le</strong>ur de la pluie <strong>le</strong> jour<br />
i B la station j aux données centrees'réduites =;i 3c.i , où<br />
m<br />
ni % SJ<br />
x.j:A Xij , Sj '2 2 (lu- .<br />
i:4 b-4<br />
<strong>le</strong>s va<strong>le</strong>urs manquantes étant remplacées par la moyenne mensuel<strong>le</strong><br />
de la station concernée. On calcu<strong>le</strong> <strong>le</strong>s va<strong>le</strong>urs propres A; et Les<br />
vecteurs propresci de la matrice de corrélation. Les composantes<br />
principa<strong>le</strong>s sont alors déterminées par la transformation linéaire<br />
des données initia<strong>le</strong>s 3 l'aide de la matrice des vecteurs propres.<br />
On effectue une reconstitution approchée du fichi'er initial en<br />
ne conservant que <strong>le</strong>s premibres composantes principa<strong>le</strong>s. Les va-<br />
<strong>le</strong>urs manquantes sont alors remplacées par <strong>le</strong>s va<strong>le</strong>urs ainsi re-<br />
constitu6es.<br />
L'efficacité de la méthode est mesurée B l'aide du coefficient
30 9<br />
ry<br />
où x est ta vraie va<strong>le</strong>ur, ta va<strong>le</strong>ur reconstituée,gx L'écart<br />
moyen quadratique pour La station concernee, N Le nombre des "trous"<br />
supplémentaires introduits dans <strong>le</strong> fichier de façon aléatoire afin<br />
de tester la methode, La sommation étant etendue à toutes <strong>le</strong>s va-<br />
<strong>le</strong>urs de x correspondant aux trous supplémentaires.<br />
Une étude expérimenta<strong>le</strong> a montre que <strong>le</strong> nombre optimum de com-<br />
posantes principa<strong>le</strong>s retenues pour reconstituer <strong>le</strong> fichier initial<br />
était n = 4.<br />
---_--_-__-<br />
--___- ----------<br />
2 - Analyse des correspondances<br />
La méthode est analogue 3 la première mais, au Lieu de passer<br />
aux variab<strong>le</strong>s centrées réduites =Li -3C.L on utilise La trans-<br />
6J<br />
formation classique en analyse des correspondances<br />
I *<br />
où xi.=- L.r..est la te jour i pour L'ensemn<br />
jz4 Y<br />
1<br />
b<strong>le</strong> du département, x.j=-g x~j est La moyenne mensuel<strong>le</strong> à la<br />
1 m ir4<br />
station j et x..=-P =.j=L 2 ~~.<br />
n j:d<br />
rn ir4<br />
Les va<strong>le</strong>urs propres de la matrice de corrélation des nouvel<strong>le</strong>s<br />
V ariab<strong>le</strong>s tij sont très voisines: 1,24 . 10-2
v -<br />
310<br />
I -4<br />
Pour maximiser {()o on minimise La forme quadratique x V X .<br />
La condition de minimum est ainsi<br />
X'V'1dX = O<br />
si t'on pose XI, =Z,--- ~n <strong>le</strong>s va<strong>le</strong>urs connues et "&+d j - - - -1<br />
7t, <strong>le</strong>s va<strong>le</strong>urs inconnues (manquantes) la condition de minimum<br />
s'-écrira<br />
.-<br />
On obtient n-k équations à n-k inconnues pour determiner <strong>le</strong>s<br />
va<strong>le</strong>urs inconnues x p+4 > ---- =,.<br />
b) On ne connait pas Vxx. On calcu<strong>le</strong> alors <strong>le</strong>s covariances, coup<strong>le</strong><br />
par coup<strong>le</strong>, en effectuant la sommation pour <strong>le</strong>s va<strong>le</strong>urs simultnnement<br />
non-manquantes pour chaque coup<strong>le</strong> donné. On obtient ainsi une<br />
matrice Vxx, qui n'est pas nécessairement definie positive étant<br />
donné que <strong>le</strong>s indices de sommation =ont étendus à des ensemb<strong>le</strong>s<br />
differents. On diagonalise ensuite Vxx, on range Les va<strong>le</strong>urs.propres<br />
par ordrededécroissance et on ne conserve que cel<strong>le</strong>s d'entre<br />
el<strong>le</strong>s qui sont supérieures 5 un seuil positif donne. Un seuil optimal<br />
semb<strong>le</strong> exister qui est d'.autant plus grand que la tail<strong>le</strong> de<br />
l'échantillon est petite.<br />
On a presenté dans <strong>le</strong> tab<strong>le</strong>au 1 <strong>le</strong>s va<strong>le</strong>urs du coefficient 6'<br />
(c-f 7) en fonction du pourcentage de trous dans <strong>le</strong> fichier pour<br />
<strong>le</strong>s trois premikres méthodes retenues.<br />
Le tab<strong>le</strong>au Z permet de comparer, pour la quatrième méthode, Le<br />
gain obtenu en remplaçant la .donnee manquante, non pas par La va<strong>le</strong>ur<br />
moyenne mais, par la va<strong>le</strong>ur reconstituee à l'aide de cette<br />
méthode, <strong>le</strong>s crit&re_c de qualité étant respec.tivement,E(r-;E)',<br />
E(~c-2)~<br />
et -4-t*Cr,~).<br />
RECHERCHE DES ZONES HOMOGENES DE PLUVIOMETRIE PAR UNE METHODE DE<br />
VISUALISATION DE MATRICE D'INTERDISTANCE<br />
Soit N points dans l'espace RP . On construit La matrice symetrique<br />
NxN dont <strong>le</strong>s termes sont <strong>le</strong>s distances euclidiennes entre<br />
points, mesurés dans l'espace R<br />
P ;<br />
On recherche une représentation plane des N points XI, Xz, ...,<br />
XN deRP à l'aide de N points images YI, Yz,-Yp de Rz, de façon<br />
à ce que la distance entre deux points images Yi et Yj soit La<br />
plus proche possib<strong>le</strong> de la distance entre Xi et Xj. ,<br />
Pour cela, on part d'une configuration arbitraire des Yi et on.<br />
deplace ces points de façon a minimiser un critere de type Xa<br />
entre distances reel<strong>le</strong>s et distances images. Cette methode ne neces<br />
site pas de diagonalisation de la matrice de distanceset peut être<br />
aisement mise en oeuvre. La pr8cision de La visualisation diminue<br />
quand <strong>le</strong> nombre de points augmente. I l semb<strong>le</strong> toutefois possib<strong>le</strong><br />
de traiter des matrices 15Ox15C avec une précision de reppesen'<br />
tation satisfaisante.
Pourcentage de trous<br />
dans <strong>le</strong> fichier<br />
Méthode des composantes<br />
principa<strong>le</strong>s<br />
Analyse des<br />
correspondances<br />
Méthode de<br />
regression<br />
15,19 19,6 24,3 29,4 32,9<br />
o,33<br />
31 1<br />
0,34 0,26 0,35 0,51<br />
O ,33 0,36 0,20 O ,33<br />
0,30 O ,30 0,21 0,32 0,46<br />
Tab<strong>le</strong>au I - Va<strong>le</strong>urs des 6% en fonction des pourcentages de<br />
trous dans <strong>le</strong> fichier<br />
tations<br />
1 o1<br />
161<br />
471<br />
1131<br />
1211<br />
1271<br />
1581<br />
1681<br />
1711<br />
2101<br />
21 51<br />
2231<br />
2281<br />
2 62.1<br />
--<br />
E(X -%y<br />
381<br />
335<br />
755<br />
343<br />
797<br />
359<br />
5 79<br />
253<br />
43 4<br />
391<br />
327<br />
41 6<br />
390<br />
761<br />
3621<br />
51 2<br />
T'ablcau 2<br />
JANVIER<br />
E(*- ;y 1 - r2<br />
2795 0,14<br />
1454<br />
2864<br />
1318<br />
1844<br />
649<br />
3453<br />
1632<br />
21 59<br />
1236<br />
1477<br />
2533<br />
2791<br />
2418<br />
1699<br />
O ,23<br />
0,26<br />
0,26<br />
0,43<br />
0,55<br />
0,16<br />
0,15<br />
0,20<br />
0,32<br />
0,23<br />
0,16<br />
0,14<br />
O ,31<br />
0,30<br />
:(x -ry<br />
319<br />
177<br />
179<br />
1 o1<br />
191<br />
97<br />
409<br />
145<br />
220<br />
160<br />
3 85<br />
2 88<br />
186<br />
161<br />
221<br />
JUILLET<br />
E(X-%)=<br />
1542<br />
Y65<br />
992<br />
1126<br />
1027<br />
257<br />
1311<br />
1262<br />
2285<br />
822<br />
1 3 2'ï'<br />
1413<br />
1448<br />
1511<br />
85 1<br />
0,21<br />
0,18<br />
0,18<br />
0,09<br />
0,18<br />
O ,38<br />
0,31<br />
0,11<br />
0,10<br />
0,19<br />
0,29<br />
o ,20<br />
0,13<br />
0,1Î<br />
0,26
312<br />
A partir de 24 stations notées A,B,C,-, X dans la région Sud-<br />
Ouest de la France (Fig 2), on a construit 12 matrices de distances<br />
(une par mois) B partir des hauteurs de pluie journali6res re<strong>le</strong>-<br />
vées dans chaque station. Ces douze matrices ont été visualisées;<br />
on trouvera en exemp<strong>le</strong> (Fig 3)<strong>le</strong>s deux visualisations Décembre et<br />
Mai. Il a eté possib<strong>le</strong> de trouver quatre g'roupes de stations voi-<br />
sines qui se retrouvent sur chaque visualisation.<br />
Ces groupes ont été reportes sur la carte et correspondent A<br />
des zbnes pluviométriques homogènes en ce sens que deux stations<br />
d'un même groupe cont"proches" vis 3 vis de la pluviométrie.<br />
Ces th6mes de recherche ont été développés sous l'egide du<br />
Bureau de L'Eau et rendus opérationnels avec La participation<br />
active de Mrs. B.RAMBALDELL1, J.F. ROYER, J.C. BARESCUT, J.ZIRPHILE.<br />
Notons en conclusion.que ces recherches se poursuivent au<br />
Bureau de L'Eau et que d'autres methodes d'analyse des données<br />
sont éga<strong>le</strong>ment étudiées dans <strong>le</strong> but de parvenir 3 une meil<strong>le</strong>ure<br />
connaissance du champ de pluviométrie.<br />
REFERENCES BIBLIOGRAPHIQUES<br />
CI1 ANGOT A. (1911-1914) Anna<strong>le</strong>s du Bureau Central Meteorologique<br />
de France<br />
C21 BUCK S.F. (1960) A method of estimation of missing values in<br />
multivariate data suitab<strong>le</strong> for use with an e<strong>le</strong>ctronic computer.<br />
Journal of <strong>the</strong> Royal Stat'istical Society, Series 8.22<br />
pp. 302-306<br />
C31 AFIFI A.A., Elashoff R.H. (1966). Missing observations in<br />
multivariate statistics. Journal of <strong>the</strong> American Statistical<br />
Association. 61 pp. 595-604.<br />
C41 KELEJAN H.H. (1969). Missing observations in multivariate<br />
regression : e fficiency of a first order methods. American<br />
Statistical Association Journal. 65 pp 1609-1616.<br />
C51 SAMMON J.U. (1969). A nonlinear mapping for data structure<br />
analysis. IEEE Transactions on computers - Vol C - 18, N' 5<br />
pp. 401-409.
I<br />
Fig. 1 - ndiccs d'homogéneitE<br />
5'<br />
I<br />
ARNE<br />
I<br />
31 3
314<br />
Fig.2 -<br />
12 BIS<br />
Fig.3 - Exemp<strong>le</strong> de visualisation des matrices d’interdis-<br />
tances.
INFLUENCE OF INADEQUACY OF HYDROLOGICAL DATA<br />
ON PROJECT DESIGX AND I'ORMULATION<br />
GENERAL REPORT<br />
by<br />
Leo R. Beard ( 1)<br />
NATURE OF DATA INFLUENCE ON PROJECT DESIGN<br />
In evaluating <strong>the</strong> effect of data inadequacy on water resources project<br />
design, it is important to recognize that a moderate error in project size that<br />
might result is not necessarily accompanied by a proportional over-all loss in<br />
project net benefits. As a matter of fact, <strong>the</strong> difference between benefits<br />
derived from almost any water resources project and <strong>the</strong> costs of that project<br />
changes very litt<strong>le</strong> over a relatively large range of project size in <strong>the</strong> vicinity<br />
of optimum project size. However, it is in this range that added uncertainty<br />
in design reduces project net benefits on <strong>the</strong> average, beca:ice net benefits<br />
decrease in both directions from <strong>the</strong> optimum, and, even thoiigh increased expected<br />
cost due to uncertainty is usually a smaìl fraction of <strong>the</strong> total project cost, it<br />
can be large enough to justify care and extra cost in obtaining data for more<br />
reliab<strong>le</strong> design.<br />
When project design <strong>le</strong>vel is quite different from tconomic optimum<br />
(and this can occur because of financial constraints, political constraints, and<br />
o<strong>the</strong>r factors) , <strong>the</strong>n <strong>the</strong> net project benefits change vei *J rapidly with errors in<br />
design magnitude, but <strong>the</strong>se errors tend largely to carit-el in <strong>the</strong> expectation<br />
computation. Hence, in general but not always, errors in determining over-all<br />
project size have far <strong>le</strong>ss than a proportional effect on project net benefits,<br />
provided that <strong>the</strong> project operation can be modified as necessary to make effective<br />
use of <strong>the</strong> project facilities under conditions different from those anticipated<br />
during design.<br />
On <strong>the</strong> o<strong>the</strong>r hand, ra<strong>the</strong>r minor inadequacies in data can have an unex-<br />
pectedly large effect on <strong>the</strong> over-all project size. In flood control design, for<br />
examp<strong>le</strong>, errors due to data inadequacies can cause differences as great as a<br />
factor of 2 or 3 in estimating extreme flood sizes corresponding to specified<br />
exceedence probabilities. In <strong>the</strong> case of drought regulation (water supply) , a<br />
change in magnitude or duration of a prolonged drought can result in differences as<br />
great as a factor of 2 or 3 in <strong>the</strong> amount of supp<strong>le</strong>mentary supply (usually storage)<br />
(l)Technical Director, Center for Research in Water Resources, The University of<br />
Texas, Austin, Texas, USA.
316<br />
that must be provided for. Here again, though, project design magnitude<br />
does not necessarily respond linearly to changes in flood or drought magnitude,<br />
because cost and benefit considerations have a strong dampening or stabilizing<br />
influence.<br />
INFLUENCE OF DATA INADEQUACY ON PROJECT SIZE<br />
Considering <strong>the</strong>n, that <strong>the</strong>re is no simp<strong>le</strong> relationship between data<br />
inadequacy and project net benefits, it is safe to say that evaluation of <strong>the</strong><br />
effects of data inadequacies on design requires a detai<strong>le</strong>d study of <strong>the</strong><br />
inadequacies and all of <strong>the</strong> interrelated factors that influence project design.<br />
Such detai<strong>le</strong>d studies have been demonstrated in ra<strong>the</strong>r simplified applications<br />
in work cited by Mr. lames and used as a basis of <strong>the</strong> studies described by<br />
Mr. James.<br />
In his paper, “Data Requirements for <strong>the</strong> Optimization of Reservoir<br />
Design and Operating Ru<strong>le</strong> Determination ,” Mr. James develops <strong>the</strong> <strong>the</strong>ory<br />
and some practical demonstrations for determining <strong>the</strong> optimum <strong>le</strong>ngth of<br />
stream gaging stations where <strong>the</strong>ir value for reservoir design and operation<br />
alone is considered. In effect, <strong>the</strong> question to be answered is, how soon<br />
should gaging records be started if a project will be constructed at some<br />
distant time in <strong>the</strong> future. His basic solution is first for a known future<br />
construction time, and <strong>the</strong>n he considers uncertainty in <strong>the</strong> time of construction.<br />
Benefits of gaging records are a function of increased efficiency of design and<br />
operation.<br />
Although much simplification of <strong>the</strong> design and operation prob<strong>le</strong>ms is<br />
assumed, <strong>the</strong> concepts developed by Mr. James are of fundamental importance.<br />
It is interesting to note that optimum record periods are in <strong>the</strong> order of 25 to<br />
50 years, but <strong>the</strong>re is insufficient information in <strong>the</strong> paper to determine whe<strong>the</strong>r<br />
<strong>the</strong> basis of <strong>the</strong>se results is real or largely assumed. Perhaps <strong>the</strong> author could<br />
elaborate on this.<br />
Stream gaging records are of value for many things o<strong>the</strong>r than project<br />
design. It would be helpful if <strong>the</strong> author could express some opinions on whe<strong>the</strong>r<br />
o<strong>the</strong>r benefits exceed <strong>the</strong>se or are ra<strong>the</strong>r minor. It would seem off-hand that our<br />
great heritage of hydrologic data could not have been justified many years ago on<br />
such grounds alone, and yet w e know Chat <strong>the</strong> body of data that now exists is<br />
invaluab<strong>le</strong>.
INFLUENCE OF DATA INADEQUACY ON METHODOLOGY<br />
317<br />
In addition to affecting <strong>the</strong> size of a project, data inadequacies can<br />
greatly influen..e <strong>the</strong> methodology used in planning and designing a project.<br />
Professor Reid o in his paper, "Tiie Design of Water Quality Management<br />
Projects with Inadequate Data, " points out that ma<strong>the</strong>matical models must<br />
be built with availability of data in mind, that <strong>the</strong>re is never as much data as<br />
needed, and that <strong>the</strong> only defense against inadequate data is judgment. He<br />
describes a number of water quality models very briefly in <strong>the</strong> form of ma<strong>the</strong>-<br />
matical equations, but does not attempt to describe <strong>the</strong>ir purpose or application<br />
or to delineate <strong>the</strong> need for data in each case. Perhaps he could elaborate on<br />
this. He expresses some thoughts on <strong>the</strong> cost of waiting for more data before<br />
designing a project, and points out that an important e<strong>le</strong>ment is <strong>the</strong> zost of<br />
postponing <strong>the</strong> stream of net benefits from <strong>the</strong> project.<br />
Dr. Reid suggests 8 quality parameters that are commonly measured in<br />
<strong>the</strong> U.S. with adequate reliability and accuracy and at reasonab<strong>le</strong> cost. It<br />
would be helpful to discuss <strong>the</strong>se in relation to <strong>the</strong> models described, with<br />
particular attention to data gaps that would exist if only <strong>the</strong>se paranieters are<br />
measured.<br />
Dr. Reid also suggests a time sca<strong>le</strong> for a progressive pollution abate-<br />
ment program, showing abatement of lake eutrophication by 1980, reuse by<br />
1990 and recycling by 2000. This is apparently for <strong>the</strong> United States, but<br />
would be of interest to o<strong>the</strong>r countries. It would help if some of <strong>the</strong> abbrevia-<br />
tions used would be explained, if distinction between reuse and recyc<strong>le</strong> is<br />
explained, and if <strong>the</strong> basis for or origin of <strong>the</strong> tab<strong>le</strong> were stated.<br />
METHODS USABLE WITH INADEQUATE DATA<br />
Two o<strong>the</strong>r papers prepared for this session describe specific methodology<br />
that should be used when data inadequacies exist.<br />
In <strong>the</strong> paper, "Designing Projects for <strong>the</strong> Development of Ground Water<br />
Resources in <strong>the</strong> Alluvial Plains of Nor<strong>the</strong>rn India on <strong>the</strong> Basis of inadequate<br />
Data, " Sarherwal describes generalized ground-water yield criteria, developed<br />
for guidance in developing ground-water supplies in <strong>the</strong> Punjab until such time<br />
as systematic data on ground-water reservoirs becomes availab<strong>le</strong>. The develop-<br />
ment of high-yield crops has occasioned a marked increase in ground-water<br />
exploitation as an assured supply for critical irrigation needs. In order to<br />
fur<strong>the</strong>r increase <strong>the</strong> use of ground water effectively, studies based on such<br />
criteria are essential.
318<br />
The criteria described are based on approximating <strong>the</strong> pertinent<br />
components of <strong>the</strong> hydrologic cyc<strong>le</strong>. Of primary concern are those components<br />
associated with rep<strong>le</strong>nishment of <strong>the</strong> ground-water supplies. Formulas are<br />
given for <strong>the</strong> amount of rainfall that contributes to deep percolation, seepage<br />
from lined and unlined canals, recharge from water courses, and return<br />
seepage from irrigated fields. It was found that horizontal movement of ground<br />
water is very small compared to vertical recharge and could <strong>the</strong>refore be<br />
ignored in this set of approximate criteria. Water withdrawal criteria consist<br />
of generalized values for evaporation from water-logged areas and draft from<br />
various types of wells.<br />
Planning of new wells is based on a water balance study using <strong>the</strong>se<br />
generalized criteria and a safety factor dependent on <strong>the</strong> region. An examp<strong>le</strong><br />
of criteria application is given for <strong>the</strong> Bist Doab Tract.<br />
Mr . Sarherwal supports his paper with an abundance of background<br />
material indicating <strong>the</strong> importance of this subject to <strong>the</strong> economy, to <strong>the</strong><br />
ecology and to social conditions in India. It would appear that some elaboration<br />
on <strong>the</strong> ro<strong>le</strong> of surface water development in conjunction with ground water<br />
management would also be very useful in such an outstanding paper.<br />
In <strong>the</strong>ir paper, "Design of Water Resources Projects with Inadequate<br />
Data in India - General and Particular Case Studies," Banerji and La1<br />
describe a variety of methods used in India for <strong>the</strong> estimation of design<br />
floods and for monthly and seasonal runoff quantities.<br />
Rough approximations of seasonal runoff from monsoon rainfall are<br />
obtained with Strange's Tab<strong>le</strong> of runoff ratios for apparently arbitrary<br />
categories of good, average and bad catchments. A <strong>le</strong>ss arbitrary method<br />
of estimating runoff from rainfall uses Khosla's Formuld, which simply<br />
substracts monthly evaporation and transpiration loss from monthly precipi-<br />
tation. The loss is a universal, unique function of average monthly temperature.<br />
A modified formula is given for calculating annual loss from annual temperature<br />
in order to compute annual runoff from annual rainfall.<br />
A third technique for obtaining runoff from rainfall is <strong>the</strong> correlation<br />
of short runoff records with rainfall on an annual or monsoon-season basis<br />
and <strong>the</strong>n estimating runoff for all <strong>the</strong> years of rainfall reconl. The fourth<br />
technique uses <strong>the</strong> standard unit-hydrograph method for relating runoff to<br />
rainfall.
319<br />
Methods of estimating peak flows include empirical formulas<br />
relating maximum observed floods to size of catchment area , envelope<br />
curves of maximum floods and regional flood frequency analysis. Criteria<br />
are given for obtaining probab<strong>le</strong> maximum precipitation and unit-hydrographs<br />
for ungaged catchments. An interesting exponential recession technique<br />
in lieu of <strong>the</strong> unit-hydrograph technique is described.<br />
The authors do not discuss <strong>the</strong> degree of development or of flood<br />
protection that is needed for various types of structures, so it is difficult<br />
to visualize how <strong>the</strong>ir criteria would be applied for a great variety of<br />
structures such as culverts, <strong>le</strong>vees, dams and spillways, where <strong>the</strong>re is<br />
a great range in <strong>the</strong> degree of safety needed. Also, <strong>the</strong>y do not indicate <strong>the</strong><br />
degree of adequacy of <strong>the</strong> methods and whe<strong>the</strong>r fur<strong>the</strong>r development of<br />
methodology or increased amounts of data would substantially improve <strong>the</strong><br />
reliability of project design. It appears that <strong>the</strong>y are in an excel<strong>le</strong>nt<br />
position to render judgment in this matter, and perhaps <strong>the</strong>y would do so<br />
in <strong>the</strong>ir discussion at this session.<br />
MINIMUM DATA REQUIREMENTS<br />
There is always <strong>the</strong> question as to <strong>the</strong> minimum data required for any<br />
design, and, of course, this varies with <strong>the</strong> type of project and <strong>le</strong>vel of<br />
development. In a paper, "Minor water Resource Projects Formulation<br />
on Micro Hydrological Data for Standardization and Quicker Execution<br />
in Developing Areas: Guidelines, " received through written communication,<br />
Mr. Sikka discusses <strong>the</strong> prob<strong>le</strong>ms of data needs in reldtion to development<br />
of projects of moderate size. He supplies a list of miriimum data requirements,<br />
which should be of value to countries outside of India as well as to India.<br />
These include topographic and soil mapping, many types of hydrologic data<br />
and data on irrigation efficiency. Special emphasis is placed on <strong>the</strong> fact<br />
that past drought periods can be exceeded in <strong>the</strong> future and that this should<br />
be taken into account in design.
320<br />
Mr. Sikka discusses environmental impacts and <strong>the</strong> needs for indices<br />
of environmental conditions and for value weights that can be related to<br />
economic efficiency benefits and costs. He lists water and air quality,<br />
wilderness and scientific areas, es<strong>the</strong>tic features and wildlife habitats<br />
as environmental e<strong>le</strong>ments of principal concern. He stresses <strong>the</strong> conservation<br />
of water resources through more efficient application of irrigation water. He<br />
also discusses <strong>the</strong> conjunctive development of surface and ground waters and<br />
related data needs.<br />
Mr. Sikka brings up <strong>the</strong> question of <strong>the</strong> adequacy of basin-wide studies<br />
of surface waters in conjunction with ground-water aquifers that extend beyond<br />
<strong>the</strong> boundaries of river basins. This is a ra<strong>the</strong>r common circumstance, and<br />
it is apparent that <strong>the</strong> scope of surface water studies must be extended where<br />
ground water is a substantial e<strong>le</strong>ment and where horizontal movement of ground<br />
waters across <strong>the</strong> river basin boundaries is significant. This emphasizes <strong>the</strong><br />
importance of obtaining surface and ground-water data extending far beyond<br />
river basin boundaries in some studies.<br />
INFORMATION CONTENT OF DATA<br />
Many of <strong>the</strong> effects of data inadequacies discussed thus far in this<br />
general report are direct influences that are relatively easy to understand.<br />
There are some subt<strong>le</strong> effects that are litt<strong>le</strong> understood and yet can have<br />
major impact on project design.<br />
Weber, Kisiel and Duckstein in <strong>the</strong>ir paper, "Maximum Infonnpon<br />
Obtainab<strong>le</strong> from inadequate Design Data: From Multivariate to Bayesian<br />
Methods," discuss some <strong>the</strong>oretical aspects of a subiect that is critical in<br />
<strong>the</strong> use of inadequate data and has considerab<strong>le</strong> impact even where substantial<br />
data exists in many applications. They examine <strong>the</strong> effect of possib<strong>le</strong><br />
inapplicability of <strong>the</strong>oretical assumptions underlying techniques such as linear<br />
regression, discriminant functions, canonical correlation, principal component<br />
analysis, factor analysis and cluster analysis. In many cases, departure of<br />
data from underlying assumptions such as linearity or normality will cause<br />
erroneous results. More markedly and more generally, confidence estimates<br />
will be in error.<br />
The authors discuss <strong>the</strong> comp<strong>le</strong>xity that is introduced into Bayesian<br />
analysis by uncertainties in <strong>the</strong> basic assumptions and cite some degree of<br />
success in applications to discriminant analysis.
321<br />
Remarks relative to <strong>the</strong> interpretation of <strong>the</strong> results of principal<br />
component analysis are interesting. Attempts to identify <strong>the</strong> physical<br />
significance of <strong>the</strong> components or to use <strong>the</strong> components in subsequent<br />
regression analysis bring up serious questions. The general reporter feels<br />
also that such attempts would constitute a misapplication of <strong>the</strong> technique<br />
(as <strong>the</strong> authors may feel also).<br />
This paper does not attempt to answer <strong>the</strong> prob<strong>le</strong>ms but simply identifies<br />
<strong>the</strong>m. It should be of great value if it occasions attempts by <strong>the</strong> authors or<br />
o<strong>the</strong>rs to find answers to <strong>the</strong>se prob<strong>le</strong>ms. It would be useful if <strong>the</strong> authors would<br />
comment on <strong>the</strong> effects that inapplicability of assumptions may have on <strong>the</strong><br />
stability of maximum-likelihood solutions. The general reporter has witnessed<br />
cases where highly erratic results were obtained through use of maximum-likelihood<br />
parameters that were apparently sensitive to <strong>the</strong> form of a distribution function<br />
and where <strong>the</strong> data were not known for sure to fit <strong>the</strong> assumed distribution.<br />
GEOGRAPHIC CONS IDERATIONS<br />
None of <strong>the</strong> papers in this session discuss <strong>the</strong> differences in <strong>the</strong><br />
various geographic regions that affect <strong>the</strong> adequacy of data. It is known that<br />
many rivers are very stab<strong>le</strong> and that a relatively small amount of data can be<br />
adequate for fairly reliab<strong>le</strong> hydrologic determinations. On <strong>the</strong> o<strong>the</strong>r hand,<br />
<strong>the</strong>re is almost never sufficient data for evaluating <strong>the</strong> runoff potential of<br />
some highly erratic streams where flows some years may be ln0 to 1000 times<br />
as great as flows in o<strong>the</strong>r years.<br />
Also, <strong>the</strong>re is a difference in <strong>the</strong> nafure of data transfer potential in<br />
various geographic regions. In regions where genera! storms or general snow-<br />
melt floods predominate to produce high conelations among hydrologic events<br />
within <strong>the</strong> region, data at a long-record site may be used to effectively extend<br />
data at a short-record site. In this manner, short records can be made to serve<br />
for long records to a large extent. It should be noted, however, that this permits<br />
estimates whose reliability is limited to that obtainab<strong>le</strong> with <strong>the</strong> longest records<br />
of <strong>the</strong> region.<br />
On <strong>the</strong> o<strong>the</strong>r hand, where great hydrologic heterogeniety exists, such<br />
as where small-area storms predominate, information might be transferred if<br />
<strong>the</strong> rainfall-runoff process can be mode<strong>le</strong>d accurately. In this case, <strong>the</strong>re is<br />
a virtually unlimited amount of information in a region that might be assemb<strong>le</strong>d<br />
to yield estimates far more reliab<strong>le</strong> than those obtainab<strong>le</strong> from <strong>the</strong> longest<br />
records. At present, <strong>the</strong> technology does not exist for effectively assembling<br />
such data, but <strong>the</strong> potential certainly is <strong>the</strong>re.
322<br />
SUMMARY<br />
In summary, it is not at all obvious how data inadequacies can affect<br />
design without making a detai<strong>le</strong>d. study and without a thorough understanding<br />
of <strong>the</strong> factors Involved. Errors due to data inadequacies can accidentally<br />
improve a design, but <strong>the</strong> expectation is that better data will produce better<br />
designs, as long as sound policies and technology are employed.<br />
Many important contributions are contained in <strong>the</strong> papers for this<br />
session, and <strong>the</strong> authors are to be congratulated on <strong>the</strong>ir efforts. They have<br />
studied <strong>the</strong> need for and value of basic data and <strong>the</strong> impacts of data deficiencies<br />
on techniques and on design adequacy, and have defined new prob<strong>le</strong>m areas<br />
where special considerations are required in <strong>the</strong> use of small data samp<strong>le</strong>s.
ABS TRACT<br />
DESIGN OF WATER RESOURCES PROJECT WITH INADEQUATE<br />
DATA IN INDIA - GENERAL Q PARTICULAR CASE STUDIES<br />
S.Banerji" E V.B. Lal* - INDIA<br />
India has rich experience in successful construction of water<br />
resources projects with inadequate data. Whi<strong>le</strong> rainfall data of<br />
considerab<strong>le</strong> <strong>le</strong>ngth are availaB<strong>le</strong> in or around <strong>the</strong> catchment, runoff<br />
observations are usually availab<strong>le</strong> for 10 years or <strong>le</strong>ss, Commonly<br />
some gauge site some distance away from <strong>the</strong> dam site may be availab<strong>le</strong><br />
Data on soil moisture, infiltration, and evapotranspiration are<br />
almost non-existent,<br />
The paper, based on a study of several important reports rela-<br />
ting to many projects situated in different climatological, topo-<br />
graphical and geological regimes, describes <strong>the</strong> practices followed<br />
in: (i) transferring rainfall data from a hydrologically similar<br />
region to <strong>the</strong> reservoir catchment by short term correlation, Ciil<br />
establishing correspondence between rainfall and runoff by applying<br />
a regional empirical formula, or by first deriving a regression<br />
equation for rainfall vs, runoff for <strong>the</strong> small period for which simul-<br />
taneous records of both parameters are availab<strong>le</strong> and <strong>the</strong>n applying<br />
it to longer rainfall records for getting <strong>the</strong> discharge series (iii)<br />
tranferring gauge discharge relationship of a distant site to <strong>the</strong><br />
dam site to work out time distribution of inflows, and <strong>the</strong> peak flow.<br />
For <strong>the</strong> latter, a method evolved for estimating maximum flood in <strong>the</strong><br />
Narmada and Mahanadi rivers, principally based on assessing <strong>the</strong><br />
contribution due to different zones of catchment each extending to 1<br />
day's flow time, has been descrìbed for <strong>the</strong> Benefit of monsoon regions.<br />
RESUME<br />
Les Indes ont une grande expérience dans la réalisation d'amlna-<br />
gements des eaux h partir de données insuffisantes, L'observation<br />
directe des débits ne porte en général que sur des périodes de moins<br />
de 10 ans, tandis qu'on dispose souvent de données de précipitations<br />
sur de longues périodes. D'autre part, il est courant de disposer<br />
d'une échel<strong>le</strong> limnimétrique 3 quelque distance du sîte du barrage pro-<br />
jeté, alors que <strong>le</strong>s données sur l'humidité du sol, 1iinfPltration et<br />
ì'êvaporation sont presque ìnexfstantes.<br />
L'étude présentée est basée sur plusieurs rapports importants<br />
relatifs à des projets situés dans des régions qui présentent des<br />
conditions variées en climatologie, topographie et geolog2e. Les<br />
auteurs décrivent <strong>le</strong>s méthodes employdes pour (31 transférer <strong>le</strong>s<br />
données de précipitations d'une région hydrologiquement analogue au<br />
bassin qui alimente <strong>le</strong> r&servol'r, Ci'il &tqblir uqe correspondance<br />
entre <strong>le</strong>s precipitations et l'écou<strong>le</strong>ment par l'application d'une formu<strong>le</strong><br />
empirique régiona<strong>le</strong> ou d'une Iquatioy de r@gress.Con, calcul@es<br />
sur la période d'observation commane des pluzes et des débits et utilisées<br />
avec des données de prgcipitations de longue durée pour obtenir<br />
une extension des debits, (iii) déterminer la relation hauteur-débit<br />
au droit du barrage 1 partir d'une relation établie a une station siT<br />
tuée 3 quelque distance pour estimer la distrìbution des d&bìts et <strong>le</strong>s<br />
pointes de crues, Les auteurs exposent a titre d'exemp<strong>le</strong> une méthode<br />
utilisée pour évaluer la crue maxima<strong>le</strong> des rivibes Narmada et Mahanadi;<br />
cette mlthode, intéressante pour <strong>le</strong>s régimes de mousson, tient<br />
compte de la contribution des différentes parties du bassin, <strong>le</strong><br />
découpage correspondant a un isochronisme journalier.<br />
~<br />
* Scientists, Secretariat of 1.H.D' National Committee<br />
I
324<br />
1.0 introduction<br />
India h a a rich experience in 'successful' construction of water msources<br />
projects with inadequate hydrologioal data. Since 1951, when tha fimt Five Year<br />
Plan commenced, 537 major and mriàium projeots, each having a reservoir etorage of<br />
over 6167 hectare-mtree i.e. 50,000 aumfset, ham been taken up and about 300 have<br />
been comp<strong>le</strong>ted (1). However, since most of <strong>the</strong> gauge ami discharge sites f a<br />
regular observation on Indien riveru have been eet up only after independenoe in<br />
1947, runoff observatioas or even gauge-readings, if at all availab<strong>le</strong> at <strong>the</strong> sito<br />
of a proposed dam were of very short duration, sw, <strong>le</strong>sa than 10 gbm. 'phr redseeing<br />
factor in this situati= has been tknt for most aress of <strong>the</strong> oouutry long records<br />
of rainfall, of 50 years or more ere gen9ra;lI.y availab<strong>le</strong>. Aluso, ooemonlJ aoma gauge<br />
site would be evai<strong>le</strong>b<strong>le</strong> on <strong>the</strong> ccgcerned river som distema away from tkie dai site.<br />
Data on soil moisture, infiltration end evapotranpipiratiai axe praotiedïy non-<br />
existent.<br />
1.1 hia mesent paper is based m e tatu* of neveral intportant reports (vide<br />
appendix 1) mlat- to meny projects of variona siaes rituated in different<br />
ûlbtOlOgiû& topmphioal a d @ûl~iC8l =gimes Of <strong>the</strong> Cmtrj. W pY.æOtiCeS<br />
described relake to <strong>the</strong> following thme oetegoriee of prob<strong>le</strong>m:<br />
(i) 'Jkarisferring rainfall date from an adjacent, lydrologieally nimilar<br />
regiai to tlie reservoir oatchnte<br />
(U) Eatciblisw oormepandeme between rainfall and runoff<br />
(iii)Tramaferring gaw discharga relatioriehip of a distant site to dam site<br />
Mœt of <strong>the</strong>e relate to estimation of runoff vdrmies. Eltimation of p.nk<br />
flore has been discussed separately.<br />
2.0 Indien Pcaotice<br />
It meg be etated hem that 88 <strong>the</strong>re is <strong>le</strong>ge di.arsity not od7 in <strong>the</strong><br />
2.1<br />
size and <strong>the</strong> region of lm&Aon but also in <strong>the</strong> nature of data availab<strong>le</strong> for<br />
differed projeote, lhm are no 8~tau&d8 or mallar8 Wd-8 to -rocmm<br />
prob<strong>le</strong>m of dafa maralty. Then, he8 ban no partioular Pmferenoe for ei<strong>the</strong>r<br />
<strong>the</strong> Mit i@rograph or tb etatietiad Mqoenny distribution In &te- tìm<br />
'dosigm flod8, and, bpending upoe <strong>the</strong> gravity ni wzumxmnnoe of a likely failure<br />
of tb stmûtm, attsmpts hem ben meâe, wimromr m-88- ad poisib<strong>le</strong>v<br />
to arrive at <strong>the</strong> design flood by =me of both thse approdms and a for o<strong>the</strong>rs<br />
of regid applloatian.<br />
1Lo 8 general guideline it h m been pmsuribed that inajar and odium groJects<br />
2.2<br />
should be deeigmd for Probab<strong>le</strong> Msximtri Flood (m) "that wodd reriult from <strong>the</strong>
co&ination of critical msteorslsgicd and hYhO<strong>le</strong>@;iO f oonditians<br />
ca.sidend physi.~c~llg possib<strong>le</strong> in <strong>the</strong> mgid"' Ln oases of SPJQ~ wojects<br />
with mery larga catchm;.nte whem applicaticm Of imit Wai.oS=Qh is bdVhabb<strong>le</strong><br />
a 1OOO-yeer flood esthte ia attempted by fnqwncy onalg.eie from a dkchprgs<br />
aeries at site cmtmted frem &ta of rainfall or o<strong>the</strong>r infomtim that Ippg b~<br />
amilab<strong>le</strong>. In tb CPB~ of germanent ba-8<br />
tu<br />
<strong>le</strong>ss thpn 6167 ha. m <strong>the</strong> design is to be based on <strong>the</strong> Stanbra Pr@d**t<br />
Flood (SPF) that would msdt from I<strong>the</strong> mest severe combination of ~ t e ~ ~ l W i c ~<br />
and hg&ologic omditime, considersd reasonably characteristi0 ef liha =gim<br />
excluding extmmb ram oo?abineticars', er a 1oO-year flmd whia*r IS<br />
For smal<strong>le</strong>r projects design flood ae;y be estimated by approximate end empirioel<br />
methods applicab<strong>le</strong> locally. (2~3)-<br />
3.0 Transferring Rainfall Data from Adjacent Catchment<br />
In Cases where rainfall data for considerab<strong>le</strong> periods are not<br />
availab<strong>le</strong> for <strong>the</strong> catchment upto <strong>the</strong> damsite two tendencies are discerni-<br />
b<strong>le</strong>. If it has been possib<strong>le</strong> to construct a discharge series at <strong>the</strong> dam<br />
site by some technique from discharge data availab<strong>le</strong> elswhere no<br />
outstanding necessity has been felt for precîpitation data for estimating<br />
<strong>the</strong> flood peaks or periodic inflow volumes e.g. Tehri Dam. O<strong>the</strong>rwise<br />
attempts are made to work out precipitation figures for <strong>the</strong> catchment<br />
under consideration. If <strong>the</strong> number of raîngauge of raingauge station<br />
in <strong>the</strong> project catchment is samall, <strong>the</strong> statìons in <strong>the</strong> adjacent region<br />
considered hydrologically similar are utilised for constructing<br />
Thiessen's Polygon for working out weighted average figures of rainfall<br />
in different years within <strong>the</strong> catchment e.g. Hasdeo (Bango) Project. It<br />
is also possib<strong>le</strong> to have a few years'data at some specially set up sta-<br />
tions within <strong>the</strong> catchment and to correlate <strong>the</strong>m with <strong>the</strong> observations<br />
at some stations with longer records, lying outside <strong>the</strong> project catch-<br />
ment but still within a hydrometeorologically similar region. The<br />
short-term correlation thus established is <strong>the</strong>n applied to <strong>the</strong> longer<br />
records of outside stations and <strong>the</strong> series comp<strong>le</strong>ted for <strong>the</strong> project<br />
catchment,<br />
4.0 Establieu C o r m s D o n m y 1 & run& f<br />
Bainfali reoords are @rerally availab<strong>le</strong> for projeet-catohwnte in<br />
<strong>the</strong> form of 24 hour ralnfail amounte obeemd at a fixed hour for moat staticne<br />
and 88 continuous recorde for se<strong>le</strong>cted etations with self-moording raingaugda.<br />
For estimating runoff data <strong>the</strong> follmlng methode am generally follmedt-<br />
a. Regional correlations, like Strange's Tab<strong>le</strong>; b.Khosla's F o d a<br />
C. Regression equations defining correlatia betweem short-term<br />
raipfall runoff data; à. whograph application.<br />
Whi<strong>le</strong> PrrtboC a,b,o, yield estiiiatee of runoff volume, hydrograph application<br />
ia good for eetiaieta of flood volume as well aa flood peaksl.<br />
4-1<br />
l&2&!2w C-OXUl a SWQE'S TABLE<br />
325<br />
It gime permntage of runoff from 10~18ocn-1<br />
rainfalls for different<br />
lnàian catchmanta, which were rathbr subjeotively divided into three oatagories~<br />
'good' 'average' and 'bad'. Thus far a total monsoon rainfall of 1000 e a good<br />
catchment will yield 37.@ runoff, an average oatchnient, 2& ami a bad eatohRIent<br />
18.776. Inspite of <strong>the</strong> faat tìmt <strong>the</strong>ee tab<strong>le</strong>a am ncm very old and o m yield only
326<br />
rough estimates, <strong>the</strong>y ere often applied in <strong>the</strong> projects for assessrnt of runoff<br />
volUries, e.g. Chambal Val<strong>le</strong>y Development Scheme, where such calculations baw also<br />
been checked againat <strong>the</strong> observed data of a few years. €&o (3) has also used Strangego<br />
tab<strong>le</strong> whi<strong>le</strong> working out dischargea for Nagarajuriaegar aiid Srisailm projects.<br />
4.2 sx-&a's Fornu4<br />
Khoela (4) working 'on tb rational concept that runoff is <strong>the</strong> residual of<br />
rainfall after deduction of evaporation and transpiratian loss' aesuiped that<br />
'temperature can be taken to be a comp<strong>le</strong>te maaure of all <strong>the</strong> factore which are<br />
responsib<strong>le</strong> €or <strong>the</strong> loes of rainfall to runoff'. The formula hos no mgional lid-<br />
- ta5cms of applicability.<br />
1<br />
His empirical formula is Ra Pm -Lp wbm Rm, P, Lm am rssp8otiln3ly <strong>the</strong><br />
runoff, rainfall and 'loss' figures for a given month in mu. Lm is taken as equal to<br />
5 'Ern, ilkre Tm is <strong>the</strong> man monthly tempereture in centigrades and is more than<br />
4.5"C. For Tm
327<br />
-<br />
aucounte for ab ut 9% of tlm anatm1 rainfall. Out of <strong>the</strong>e <strong>the</strong>e equstioriey <strong>the</strong><br />
momoon rainfdl-monaoon nuioff equatian gave <strong>the</strong> highemt QOrIdaticeiS coefficient<br />
(0.869) od thia was u. ad to derive <strong>the</strong> amoal runoff from <strong>the</strong> ennuel raiafoll f m s .<br />
4.4 &hr-Dh Applicationr The tuchnique is lairly well haai. Later diseusaions<br />
will ahow how <strong>the</strong> design storm is se<strong>le</strong>cted and its tiiir, distribution obtaineà for<br />
applylag <strong>the</strong> mcipltetion figues to th unit hydrograph.<br />
5.0 -Disc- Be<strong>le</strong>t ioliahip of a Distant site to <strong>the</strong> Barn Site<br />
We piok up tbme o- studies Vix <strong>the</strong> Eirakuà Dam, tts ThiLTi Deia a d <strong>the</strong><br />
NagarJunaaagar to illustrate hou this is being done.<br />
5-1 In <strong>the</strong> Hhalcud Dam project (1947) <strong>the</strong> &am was prapomd to be looeted at a<br />
site near SamboLpur whr8 gauge recad8 existed sinee 1921, but <strong>the</strong>re were no<br />
COrreaPpopdine gauge dieche@ curves. Ebwever, at Earaj, a site som 230 mi<strong>le</strong>s downstream<br />
froaSembalpur, gauge diaßhaqp recards existed sinue 1868. The gaugs madia@<br />
at Sambdpur were corzhlated with <strong>the</strong> gauge readiq at NaraJ, ding due allowance<br />
for <strong>the</strong> tinia. 1- and similar epuea, discbrge c m 8 were prepared for Sdelpur<br />
and checked 8gainst <strong>the</strong> dally discharge obrervatians o M d eince Jm, 1946.<br />
5-2 The Tebri Dam Project (1969j enwbws construction of a dam eor~~s thb river<br />
BhagFratM near Tehrii tha catohnmt arsa upto tb dpn eite ia 7511 8q.h.inaluding<br />
2328 8q.b of constantly snou bound axea. Daily rimr g&ugoa andwe8-U~ âiaaharge<br />
observatiais at tkm damaite .ere availab<strong>le</strong> only fra May, 1964. This Catchment is<br />
a part of tb Ganga cstcharnt in which, at Baire<strong>le</strong>, near Haricbrar about 105 km.<br />
damst- Of 'pehri, deilr aid dia- dot8 Svdlabh €hail l9Olo The<br />
catcent<br />
up%o Raiwala t 23000 8q.b. inoludbg 8450 8q.h- is anow-<br />
bound.The Raiwala data have been utilised to compute runoff at Tehri<br />
in 10-day periods of <strong>the</strong> year. For this purpose <strong>the</strong> runoffs for<br />
different 10-day periods, in <strong>the</strong> period of actual observation of<br />
discharges at Tehri, have been compared with <strong>the</strong> corresponding 10-day<br />
runoffs of Raiwala, assuming a one-day time lag for <strong>the</strong> flow to reach<br />
Raiwala from Tehri. The percentages of Tehri discharges to correspon-<br />
ding Raiwala discharges have been plotted against <strong>the</strong> re<strong>le</strong>vant 10-day<br />
periods for <strong>the</strong> period of observation, 1964-66. These percentages vary<br />
for <strong>the</strong> same 10-day period from year to year due to variation in<br />
precipitation, temperature, humidity, vegetation, soil moisture etc.<br />
and for individual catchments of <strong>the</strong> tributaries of <strong>the</strong> river Ganga<br />
above Raiwala. The required factors have been worked out as below:<br />
AvoroRs of runoff et Tem.<br />
ri<br />
Ave- of runoff at Raiwt<strong>le</strong><br />
~eing r vaime for &ifferet 10- periodi, <strong>the</strong> -off figme at Beirala<br />
have ben canverted to f-a for Tehri for 30 (fra 1936 to 1966).<br />
5.2.1 'Ffiib Rairela Qata haw albo heen wed, ia ocajimotian with <strong>the</strong> ih&-tea<br />
rseard at TekrirL, for estimating <strong>the</strong> flood peds et 'psbri, &a- Baiwda 88 th<br />
etaticus, tfie peroeritege d tias e pcirticular noOb hae been equalla8 or<br />
emeebd <strong>le</strong> plotted agaInet <strong>the</strong> flood 021 a sed-log paper to giw e lozig-tsra data<br />
c m for tb inder itattian. Bor tb short-term for which data ara availab<strong>le</strong> both<br />
for %hi arrd Baida, rimilar o m s u. plot- far both th6 et&ticma. Tbs IOWterm<br />
c m for <strong>the</strong> project 8t&iOn (%-i) L <strong>the</strong>n coiiltriiabd from <strong>the</strong> abow three<br />
oms, and ths flooda of various frequencies &PB obtabd from this OPM. It is,<br />
-
328<br />
hmemr, only apIQ of m w w<br />
adopted in <strong>the</strong> projeot for eetimting tha flood peek.<br />
5.3 Ra0 (5,3) applies a different apprcmh to determim peak flooda at a section,<br />
when discharge data are availab<strong>le</strong> for a diffemnt site al- <strong>the</strong> river. W<br />
princip<strong>le</strong> applied is simp<strong>le</strong>: <strong>the</strong>dischage observed at a dametream aite ieequal to<br />
<strong>the</strong> discharge at an upstream site plue <strong>the</strong> dischar@ contributed by me interniedia*<br />
catcbnt mincis <strong>the</strong> oharial' trough' oapaieitg between <strong>the</strong> two sites. W 'trough'<br />
capacity oan be computed ideally with <strong>the</strong> help of croes eectime of <strong>the</strong> river at<br />
close internals, or o<strong>the</strong>rwise in <strong>the</strong> absence of thie inforretian, by taking <strong>the</strong><br />
average width of <strong>the</strong> river flow at one end, <strong>the</strong> difference in <strong>the</strong> depth of flw on<br />
<strong>the</strong> day of Peak flood and 24 hours before its occurmncc and <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> river<br />
reach into woount. Th? inflow from ths intermediate catchnt q be worked out from<br />
<strong>the</strong> rainfall recorda using strange*s tab<strong>le</strong>. In this way <strong>the</strong> flood aeries at <strong>the</strong><br />
upstream point ia Constructd for a number of years and subjected to frequsnoy<br />
analysis far estimeting tb design flood of a given recurrence interval.<br />
6.0 Esthau= of Ped Floa<br />
Nodly, peak floods are estimated by several mithoda before adopting a<br />
design flood. Such niethods range fra empirical fofiaulae directly giving peak flows<br />
from a oatchniant of given area to <strong>the</strong> elaborate etarm-trailspoeition and aiaximisation<br />
mthode. "hey m y broadly be clamifiad into tuo oategoriest<br />
(a) Non-mteorological =th& (b) Meteorologici1 methods<br />
In <strong>the</strong> non-mteorologieal oategory we may include tb following: Empirioal<br />
formulae, Enveloping Curves, Regional Flood Frequency analysis. In <strong>the</strong> meteorological<br />
category we inelude æ?thoQe that proceed frcm atom analysis. They msy or mqr not we<br />
unit hydrograph.<br />
6.1 .O Non-ktoorolouiad Cateaory<br />
6.1 .1 Enipiriaal F a<br />
(a) Ths noet popul formulae link <strong>the</strong> peak flood with <strong>the</strong> ama of <strong>the</strong> basin,<br />
like Diokm's, Q= CAY4 , for <strong>the</strong> Central and Nor<strong>the</strong>rn India, <strong>the</strong> Byve's,<br />
g CABB, for <strong>the</strong> south India, <strong>the</strong> &lis Q- O0O ."or faa shaped catohmmb ki<br />
<strong>the</strong> Bombay aegica, wherm Q &vea <strong>the</strong> peak ra k- f disohaya in cusecs, A, tis ama<br />
in sq. mi<strong>le</strong>s and C is a coneteat differing fron loeation to looation. &oaueß Of<br />
<strong>the</strong>ir simplicity euch regional formulae<br />
appraaimatiar of <strong>the</strong> likely flood.<br />
still hi wide use for getting a fimt<br />
(b) Quite often, if high flood mark6 ara avrilab<strong>le</strong> with raferrtaae to old trees<br />
or =oient atructplae, or OWE from <strong>the</strong> m ory of th looal inhabitruita, elow-ama<br />
method is employed as (UI aid to gueoo tkm ordar of t b dieoharm. No reliab<strong>le</strong> idea<br />
c m obviously be bad of t b Seetion prevail* at fhs t- of flood fim, end thsm<br />
is diffiaulty in estabbliehin& ttie bed slope, which i8 teken OB equal to tkm SudaCe<br />
Slop establiaha0 from mrka at different points. Kutter's or uamih@' Coeffioient<br />
of rugosity ie eitbr (usumd, or dete-d by t b eubmtitution of ieaemd äata<br />
for a few flood8 In th oonoerned formula.<br />
%Sidea th faot tht eu& formulae are ueeful only for limited regiolipl<br />
applioation, present ri& eoope far eubjectim fagtom in choosing <strong>the</strong> valm<br />
of tïæ constant. Also, it is not paisib<strong>le</strong> to have any idea of tbie probab<strong>le</strong> frequencg
329<br />
of <strong>the</strong> flood so estimated, ao that a partlulm value of C may give a flood which<br />
may be too high for designing a minor work, mey a culvert and too loa for &signing<br />
a spillwey.<br />
6.1.2 Envelope curvesi<br />
Working on tb assumption that basins of similar hgdmlogical characteristics<br />
should produce <strong>the</strong> sana mexlmum floods psr unit of catchment ama, Kanwar Saia and<br />
Karpov plotted data of mm~imum floods in Indian rimiers againet <strong>the</strong> drainage amss<br />
producing thoee floods on a log-log paper, and gave two envelope curves one enveloping<br />
data of South Indian Basins ond <strong>the</strong> o<strong>the</strong>r enveloping data far nor<strong>the</strong>rn and oentral<br />
Indian basina. !be likely maximum flood from a catchment of given area is <strong>the</strong>n<br />
expected to be indicated by <strong>the</strong>se curves. Besides tìm basic inadequaoy that <strong>the</strong>se<br />
curves relate flood potential only with <strong>the</strong> drainage area, <strong>the</strong>y do not provide for <strong>the</strong><br />
occurrence of floods of higher magnitude than those on reoord.<br />
6.1.3 Resi onal Flood Freauew-<br />
Data of all <strong>the</strong> stations (points) in a statietioally homogeneoua region are<br />
combimd to produce a flood-frequenoy oume that is asswd to be valid for <strong>the</strong> entire<br />
region and can thye be applied to determine flood of a retuni period for an imgauged<br />
catchment In <strong>the</strong> region. The simplified procedure recommended by tìm Central Water &<br />
Power Commissian (2) is as follows:-<br />
All stations in <strong>the</strong> regim with flow recorde of 10 years or more are ee<strong>le</strong>cted<br />
and for each etation a frequency curva go- upto a 100-year flood Is constructedby<br />
<strong>the</strong> Gumbel's Ethod, with a confidence-band of 9% reliability. All points am tested<br />
for homogeneity as f ollors:<br />
The ratio of 10-year flood to man annual flood is determined for each point!<br />
this ratio awreged for all points is taiœn to give <strong>the</strong> mean 'lo-year ratio' for <strong>the</strong><br />
ama. The mturn period corresponding to <strong>the</strong> ran 'lO-year ratio' time <strong>the</strong> maen<br />
annual flood Is detemlned from <strong>the</strong> frequency oume of each station and plotted<br />
against <strong>the</strong> number of years of record far that etatim oq a sed-log test graph. If<br />
tiæ pointa far ail tim etations lie between <strong>the</strong> 9% confidence limits, <strong>the</strong>y are<br />
oonsidered homogeneous.<br />
The frequency curves of different stations in a homogsneoua region a m regarded<br />
as different estimates of <strong>the</strong> regimal curve, and tlaey are averaged as follows:<br />
For eaoh statim, flood ratioe (flood of a return period T over <strong>the</strong> niean<br />
mual flood) a m computed for a number of arbitrarily se<strong>le</strong>cted values of T. The rean<br />
of <strong>the</strong> flood ratioe for all stations for a particular period T is taken to represent<br />
<strong>the</strong> flood ratio for <strong>the</strong> r egid curve. The resulting mans for different vaìws of T<br />
are plotted a t b extreiiie value probability paper and <strong>the</strong> best fit line through <strong>the</strong>m<br />
gives tïm mquized regional frequency curve.<br />
The application of thie curve to an wuged catchment rsquims M setimate of<br />
<strong>the</strong> E= BIIILup1 flood for <strong>the</strong> wtchnmnt. This is dom from ano<strong>the</strong>r e m which gives<br />
<strong>the</strong> plot of man umual floods at different statims agai-t tbe corrsepmding<br />
drainage amm. From this C- <strong>the</strong> value Of ttn? likely<br />
flood W d t<br />
<strong>the</strong> are of <strong>the</strong> naxg mgaugad oatobment can be mado<br />
6.200 &teorOlaasop1~
330<br />
records of all <strong>the</strong> precipitation stations in <strong>the</strong> region of, and around, <strong>the</strong> project<br />
catchment, which may ra<strong>the</strong>r subjectively be ree;ardeà 88 hyäromteorologicdly homogeneous,<br />
are studied to sg<strong>le</strong>ct storms of high rainfall covering an area more or<br />
<strong>le</strong>ss equal to or larger than <strong>the</strong> project oatchment. Far this purpose it mu b<br />
neceesary to carry out <strong>the</strong> umtaai Deptharea-Duration (DU> analysis of se<strong>le</strong>cted major<br />
st >rp~s 6nd from <strong>the</strong>re maximum one-day, maximum two-deg, maximum three-day precipitati-<br />
are worked out. These ae<strong>le</strong>oted stom, are <strong>the</strong>n transposed to <strong>the</strong> project<br />
oatcìnrent adjusting <strong>the</strong> precipitation axis dso to an orientation that will give <strong>the</strong><br />
maximun runoff producing effect, if such directional change of storm axis is within<br />
20° from <strong>the</strong> original axis. The storm is <strong>the</strong>n maximised for <strong>the</strong> moisture content by<br />
applying a moisture-adJustiPent factor (maf) defimd as <strong>the</strong> ratio of <strong>the</strong> max. precipitab<strong>le</strong><br />
water over <strong>the</strong> catchmnt, W piex, and <strong>the</strong> precipitab<strong>le</strong> water of <strong>the</strong> storm,<br />
P<br />
W This factor can be worlred out from consideration of <strong>the</strong> repremntatiw dew point<br />
op <strong>the</strong> storm, and <strong>the</strong> mucimm dew point over <strong>the</strong> catchment and <strong>the</strong>n finding out <strong>the</strong><br />
corresponding precipitab<strong>le</strong> waters from <strong>the</strong> 'Pressure Vs Precipitab<strong>le</strong> Water' diagram<br />
between <strong>the</strong> pressure range 1000 mb to 300 mb. Alternatively, in <strong>the</strong> absence of<br />
sufficient data, a multiplying factor lying between 2C$ to 5s ia assumed .<br />
Havhg thue determined <strong>the</strong> design storm, <strong>the</strong> tina-distribution of <strong>the</strong> rainfall<br />
has to be obtained. From DAD analysis maximtua rainfall depths for durations of 6,12,<br />
18,24,36,48 etc. hours are obtained for each of <strong>the</strong> atorpis and expressed a8 percentof<br />
<strong>the</strong> total rainfall, From a study of <strong>the</strong>se pementages suitab<strong>le</strong> distribution for<br />
<strong>the</strong> desiga storm is arrived at. Alternativelyif a limited number of self-moording<br />
rainges are availab<strong>le</strong> <strong>the</strong> ti- distributim cum be obtained from <strong>the</strong> continuous<br />
records. If no self-recerding gauges are availab<strong>le</strong> time distribution based on <strong>the</strong><br />
experience of storma elsewhere in comparab<strong>le</strong> area is adopted. Effective rainfall<br />
for difr'emnt time incremnte is estimated by any of tb usual rays, vis,(a) tha<br />
calculation of infiltration loes by finding <strong>the</strong> total surfme flws from actual<br />
flood-hydrographe and commng <strong>the</strong>m with corresponding rainfall vol~aes e.g.<br />
Tenughat krojeat or (b) by simply assuming a runoff factor and applying it to <strong>the</strong><br />
design stozm values, e.g. Fíasdeo (Bsngo) Project. These effective rainfall values<br />
are <strong>the</strong>n arranged in <strong>the</strong> oritieal sequence which may be a mm or <strong>le</strong>ss sgmnietric<br />
arrangement of valiies with <strong>the</strong> greatest value in <strong>the</strong> midd<strong>le</strong>, or my be determined<br />
by arranging <strong>the</strong> rainfall increaients against <strong>the</strong> ordinates of <strong>the</strong> design d t<br />
hydrograph ln such a way that <strong>the</strong> longest odinate faces tke largest effectiw<br />
rainfall and <strong>the</strong> next largest ordinate faces tb next largeet rainfall increment<br />
and so on, and <strong>the</strong>n reversing this arraageeient to give tb oritical seqrrenœ. It is<br />
<strong>the</strong>n applied to <strong>the</strong> design unit hydrograph, which can ba derivad by eriy of <strong>the</strong><br />
wual nieans,actual obsemtiolls ar syn<strong>the</strong>tic.<br />
6.2.1.1 A recent report (6) suggeete e new Psthod to qatimete t 3 design flood<br />
peak (50-yem recumme) from small oatch~mnta (25 Km to 500 KID 1. It takes into<br />
account se<strong>le</strong>cted besin characteristios (<strong>le</strong>ngth and weighted .Ban BloP of <strong>the</strong> bmid 88 representative of <strong>the</strong> beeh response to tiie storm intaet and <strong>the</strong> atora P-PieterS<br />
like areal to poNt rainfall ratio. The procedure hae been evolved from an -lysis<br />
of short-term diaeharge data (5 to 10 yeare) for 60 drahmege basi- Of different<br />
slopes and sim soattered all over India. It Gen be briefly summed up a8 Pollairs:<br />
The weighiiù mm slope of <strong>the</strong> main stream, defined 88 given belw,<br />
worked out'<br />
L - 2 -<br />
C<br />
= ( Li/+ +L 21 SB 2 + .....<br />
1
331<br />
where Lc is <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> mPin stream ln d<strong>le</strong>s fra th@ maeuremnt site to a<br />
point on <strong>the</strong> main stream near <strong>the</strong> centre of grevity (CG) of <strong>the</strong> catchment area,<br />
and S1,S2 etc. are <strong>the</strong> slopes of <strong>the</strong> stream in <strong>the</strong> remhee of <strong>le</strong>ngths L,,L2 etc.<br />
into which <strong>the</strong> <strong>le</strong>ngth Lc is divided. Lengths axe mesured from <strong>the</strong> topoaheet,ln=l mi<strong>le</strong>,<br />
From <strong>the</strong> value of s, <strong>the</strong> peak rate of flow Qt, in a tc-hour mit graph in cuuecs can<br />
beestimated by th following formiilaer-<br />
-<br />
(i) Qtc I 16000 A%2'3, if s 4 0.0028<br />
(ii) Qtc 320 A 6 , if s > 0.0028<br />
0-9<br />
t, is <strong>the</strong> duration of th rainfall excess given by 255/(Qtc/A)<br />
where A is <strong>the</strong> area of <strong>the</strong> catchment in sq. mi<strong>le</strong>s. For estimeting <strong>the</strong> design rainfall,<br />
a 'design storm by6tograph' tab<strong>le</strong> hae been pmgred giving point-rainfall volume (m)<br />
of 5O-gear return period for durations varying from 15 minutes to 24 hours, and<br />
<strong>the</strong>se are <strong>the</strong>n conmrted to arsal rainfall volume by applying ama1 to point rainfall<br />
ratios that have been worked out earlier by analysing data of 12 àense networks. To this<br />
areal rainfall a uniform loss rate is applied which is determimd from <strong>the</strong> empirical<br />
relatioioohips which have been deriwd for different types of soil. This rill .determine<br />
tb rainfall exoess in t, hours, and <strong>the</strong> Qtc value multiplied by this excess would<br />
give <strong>the</strong> design flood peak.<br />
6.2.2<br />
Bowever, <strong>the</strong>se formulae need to be tested fur<strong>the</strong>r by <strong>the</strong> field events.<br />
Meteoro l wcal cat BPON i without usina t b U.G<br />
Banerji md Mantan (7,û) ham adopted a new approach for eStiIU8ting volume<br />
and peak of runoff from data of atoras. They have studied flood in <strong>the</strong> Namada and<br />
<strong>the</strong> ~hanaäi oatohmants. Studring hydrographe of a number of floods (including ia floods), thy find out <strong>the</strong> ti- base of <strong>the</strong> hydrographs after <strong>the</strong> bese flow has been<br />
eeparated; it was 6 days in each of <strong>the</strong>se oaseu. The basin is <strong>the</strong>n subdivided into<br />
zone8 of 1-day travel time each, by using <strong>the</strong> following equation (9)<br />
1 .i5 0.38<br />
Tc = L /7.700 ii<br />
where Tc is <strong>the</strong> time of ooncentration, calculated for all <strong>the</strong> min tributarie8 to tirs<br />
points of outflow in hours, L is tìm <strong>le</strong>ngth of <strong>the</strong> remoteat point in <strong>the</strong> zone to tb<br />
out<strong>le</strong>t point in feet ani II ia <strong>the</strong> diffemzuu in e<strong>le</strong>vatia betwen <strong>the</strong> waterslmd out<strong>le</strong>t<br />
and <strong>the</strong> moet distant ridge h feet.<br />
It io assumed that <strong>the</strong> contribution to <strong>the</strong> flood-vol- from each ~ OPYI is<br />
depeudent on <strong>the</strong> average ama1 depth in ewh didelm, ths scdl moiet- cmdition<br />
and <strong>the</strong> retenticm orpmity; <strong>the</strong> proentsge contribution of flood vol- from elch<br />
zone is th- made independent of <strong>the</strong> %d catchment characteristioB. It has ben<br />
fur<strong>the</strong>r aaaued that tb infiltration or retention deo<strong>le</strong>ame from upatream to dom-<br />
8tnam zoma so that if K i8 <strong>the</strong> Storage facta (considered a8 <strong>the</strong> fmtiOn Of<br />
ecierage preci itatian &pa a p p a r ~ aa runoff ) in <strong>the</strong> zone nearest to <strong>the</strong> point<br />
of outflcnr, l! is tb starogb faotor fm <strong>the</strong> nth diviaion m y from <strong>the</strong> outflm point.
The total daily runoff *Fit at <strong>the</strong> out<strong>le</strong>t can <strong>the</strong>n be sriPiPed up as<br />
Shew An i8 t h area and Pn-, ia t b average precipitation recorded in <strong>the</strong> nth<br />
division. The value of K is se<strong>le</strong>oted, by trial and error, from past records of<br />
discharge for which simultaneous precipitation data are a<strong>le</strong>o availab<strong>le</strong>. A graph<br />
is <strong>the</strong>n Plotted between values of K obtained for different perioda and corns-<br />
ponding antecedent catchnt rainfall.<br />
hydrograph, but utilises all <strong>the</strong> sante <strong>the</strong> es8ential underlying princip<strong>le</strong> that <strong>the</strong><br />
ordinates of two hydrographs for <strong>the</strong> 8amB basin and similar tine bese are proportional<br />
to <strong>the</strong>ir respective volumes of runoff. The correspondence may be effected between<br />
tkie biggest storm on record of which only rainfall data are availab<strong>le</strong> and any or<br />
all availab<strong>le</strong> hydrographs if <strong>the</strong> oharacteristios of stom are meteorologically<br />
similar to tke outflow point.<br />
Heferences<br />
For working out peak runoff rates (7) <strong>the</strong> method does not need a unit<br />
1. India, Irrigation and Power Projecte (Five Year Plans) 1970, Govwrzuœnt of India,<br />
&inistry of Irrigation and Power.<br />
2. Eetimation of Design nood, bcoimPsndeà Procedures 1972, Govt. Gf India,<br />
Central Water 4 Power Commission.<br />
3. bo, G.ii~. 1569 Modern Trenda in Hydrologic Computations, New Celhi<br />
Central Water and Pmer Commission.<br />
4. Khoela, Ad. 1949 Analysis and Appraisal of Data for t h Appraisal of water<br />
Resources. Central Board of Irrigatia Jour. pp 410-422.<br />
5. Ha0 G.A.H. 1967. Computation technique for Probab<strong>le</strong> Maximum Flood Discharge<br />
at place in <strong>the</strong> river whi<strong>le</strong> gauge dischare data is availab<strong>le</strong> for anotber<br />
pïaoe with special referena to dam on Krishna river, India. Proc. Int. Sgmp.<br />
Floods and <strong>the</strong>ir Computation, Aug. 1967, Leningrad, u-s-sj.~*<br />
6. 1973 Flood Estimation Directorate, Central Water & Power Commission, New Delhi,<br />
&sign Office Report No. 1/1973.<br />
7. Bansrji, Sdhton, D.C. (1967). On estimating peak discharges correeponding<br />
to heaviest redorded atom in a oatchment. Ind. Jour. Wt. and Ceoph. V01.17<br />
Spl. N0.M 297-306.<br />
6. Banerji, S.Manton, D.C. (1967) Determination of <strong>the</strong> distribution<br />
of rainfall floods in large catchments using hydrometeorological<br />
data. Unesco Int. Symp. on Floods and <strong>the</strong>ir Computation, Lenin-<br />
grad.
d<br />
z n<br />
Q)<br />
8<br />
- - V A v<br />
333
n n<br />
4 4<br />
- n<br />
4 4<br />
rl<br />
I
ABSTRACT<br />
DATA REQUIREMENTS FOR THE OPTIMIZATION OF<br />
RESERVOIR DESIGN AND OPERATING RULE DETERMINATION<br />
James, Ivan C., Ir<br />
U.S. Geological Survey, Washington, D.C., USA<br />
Approaches to <strong>the</strong> design of multipurpose reservoirs have usually<br />
assumed a given set of operating ru<strong>le</strong>s. Conversely, studies of oper-<br />
ating ru<strong>le</strong>s have often taken reservoir size as fixed. In only <strong>the</strong><br />
former case have estimates of <strong>the</strong> optimal data requirements been made.<br />
This paper gives <strong>the</strong> estimates of and compares <strong>the</strong> optimal <strong>le</strong>ngth of<br />
data sequences for reservoir design where operating ru<strong>le</strong>s are fixed,<br />
for operating ru<strong>le</strong> determination where reservoir design is fixed, and<br />
for <strong>the</strong> combined determination of operating ru<strong>le</strong>s and reservoir size<br />
for a multipurpose reservoir where <strong>the</strong> benefit is a piecewise-linear<br />
function of storage and re<strong>le</strong>ase. A strategy is developed for <strong>the</strong><br />
economically efficient design of <strong>the</strong> combined program of additional<br />
data col<strong>le</strong>ction and project deferment. The shape of <strong>the</strong> benefits<br />
foregone versus time function is such that project deferment is<br />
usually optimal only where very short hydrologic records exist, and<br />
<strong>the</strong> effect of an uncertain project inception date is to increase <strong>the</strong><br />
optimal-<strong>le</strong>ngth of <strong>the</strong> data sequence.<br />
RESUMEN<br />
Métodos para el diseño de un embalse multipropósito usualmente<br />
han asumido un juego fijado de reglas de operación. Reciprocamente,<br />
los estudios sobre las reglas de operación frecuentemente han ini-<br />
ciado con un tamaño fijado de embalse. Estimaciones de los reque-<br />
rimientos Óptimos de datos se han hecho solamente en el caso ante-<br />
rior. Este artículo presenta las estimaciones del largo Óptimo de<br />
series de datos para el diseño de embalses con reglas fijadas de<br />
operación, para la determinacìón de reglas de operación cuando el<br />
embalse se fija, y para la determinación junta de reglas de operación<br />
y tamano para un embalse multipropósito en que los beneficios son una<br />
función contìnua por arcos de abastecimiento y descarga. Una estra-<br />
tegia se desarrolla para el diseño eficiente economicamente de un<br />
programa junto de aplazamiento del proyecto y recopilación de datos<br />
adiciona<strong>le</strong>s. La forma de la función de beneficios renunciados contra<br />
tiempo es tal que el aplazamiento del proyecto usualmente sea Óptimo<br />
cuando existen solamente registros hidrológicos muy cortos. El efec-<br />
to de una fecha incierta del comienzo del proyecto es crecer el largo<br />
Óptimo de la serie de datos.
336<br />
Introduction<br />
Fundamental to any development process is an information base<br />
for use in making planning, design, and operational decisions.<br />
This input has measurab<strong>le</strong> costs and benefits as do <strong>the</strong> o<strong>the</strong>r inputs<br />
such as planning resources, capital, and site values for alternate<br />
uses. Economic efficiency requires that <strong>the</strong> balance between <strong>the</strong><br />
inputs of a development process be such that <strong>the</strong> marginal returns<br />
on all inputs are equal. These marginal returns should be equal<br />
to <strong>the</strong>ir marginal costs where budget constraints are not active,<br />
or equal to <strong>the</strong>ir shadow costs when <strong>the</strong>y are active. Viewed as<br />
ano<strong>the</strong>r input, information may be conceptually hand<strong>le</strong>d as any<br />
o<strong>the</strong>r input. As Weiner [i] succinctly states:<br />
"Information is only one of many development inputs;<br />
development, in turn, is but a transformation process<br />
adopted in order to reach certain objectives. Infor-<br />
mation is, thus, purely an instrumental objective and<br />
not a final purpose in itself, a basic fact we some-<br />
times tend to forget. "<br />
Information requirements for project development include such<br />
diverse factors as hydrology, future prices and extent of markets,<br />
climatology, topography, soil classification, demand <strong>le</strong>vel, geol-<br />
ogy, demography, and political trends. In reviewing this list of<br />
information requirements, it can be seen that hydrologic data,<br />
particularly those of a stochastic nature such as streamflows<br />
have distinguishing characteristics that require a different treat-<br />
ment than o<strong>the</strong>r information inputs such as economic, demographic,<br />
political, and physiographic data. Climatic and hydrologic phe-<br />
nomena require ra<strong>the</strong>r long data sequences to develop suitab<strong>le</strong><br />
representations of <strong>the</strong>ir generating mechanisms. In contrast,<br />
programs for <strong>the</strong> col<strong>le</strong>ction of physiographic information such as<br />
topographic, soil, and geologic data can be deferred until shortly<br />
before <strong>the</strong>se inputs are needed in <strong>the</strong> planning process. Models<br />
for projecting economic, demographic, and political trends into<br />
<strong>the</strong> project life horizon heavily weight <strong>the</strong> latest inputs, thus<br />
<strong>the</strong>se data are usually col<strong>le</strong>cted only shortly before <strong>the</strong>ir use.<br />
Planning <strong>the</strong> hydrologic data col<strong>le</strong>ction program requires <strong>the</strong><br />
longest <strong>le</strong>ad time and is usually accomplished when a high degree<br />
of uncertainty exists about o<strong>the</strong>r project information inputs.<br />
Efforts devoted to hydrologic network design cannot rely on highly<br />
formal tools in <strong>the</strong> absence of <strong>the</strong>se o<strong>the</strong>r inputs. An alternative<br />
is <strong>the</strong> development of heuristic ru<strong>le</strong>s based on generalizations<br />
from results of pilot studies of optimal data record <strong>le</strong>ngth for<br />
specific planning and design situations.<br />
The relationship between <strong>the</strong> timing of investments in data<br />
col<strong>le</strong>ction and <strong>the</strong> time stream of benefits gained through <strong>the</strong> use
337<br />
of <strong>the</strong>se data in <strong>the</strong> design and o'peration of water developmecii<br />
projects is an important consideration when <strong>the</strong>se time streams of<br />
costs and benefits are discounted to a common point in time.<br />
Discounted marginal costs of <strong>the</strong> first years of stream gaging are<br />
much higher than <strong>the</strong> costs of gaging just before construction.<br />
Design and construction produce large sunk costs for which <strong>the</strong>re<br />
is litt<strong>le</strong> recovery from incorrect decisions un<strong>le</strong>ss <strong>the</strong> designs<br />
have incorporated a high degree of option f<strong>le</strong>xibility such as<br />
through staged construction or o<strong>the</strong>r often costly methods of main-<br />
taining decision liquidity.<br />
The prob<strong>le</strong>m for <strong>the</strong> designer of a water development project<br />
is to maximize net project benefits subject to exogenously<br />
supplied constraints. The decision variab<strong>le</strong>s pertaining to <strong>the</strong><br />
water supply design of a reservoir are usually <strong>the</strong> size of storage,<br />
re<strong>le</strong>ase target, and operating ru<strong>le</strong>s for determining specific<br />
re<strong>le</strong>ases. Herein, <strong>the</strong> decision variab<strong>le</strong>s are divided into those<br />
physically immutab<strong>le</strong> design values'such as sizing, and those<br />
operating ru<strong>le</strong> variab<strong>le</strong>s which could conceivably be changed to<br />
ref<strong>le</strong>ct <strong>the</strong> results of new information.<br />
Whereas design sizing requires historical data, <strong>the</strong> use of<br />
information for defining operating ru<strong>le</strong>s may be ano<strong>the</strong>r matter.<br />
Additional data col<strong>le</strong>cted as normal requirements of project oper-<br />
ation may be used to update operating policies. On a <strong>the</strong>oretical<br />
basis, sequential decision <strong>the</strong>ory provides a methodology for a<br />
f<strong>le</strong>xib<strong>le</strong> and continually updated operating policy. In practice,<br />
however, political and institutional constraints make changes in<br />
operating policy a difficult and expensive process.<br />
The determination of trade-offs between capital and operating<br />
expenditures is a straightforward process when marginal benefits<br />
are known. Planning decisions, by <strong>the</strong>ir very nature are fur<strong>the</strong>r<br />
removed from <strong>the</strong> time stream of benefits than are design decisions.<br />
Litt<strong>le</strong> is known about <strong>the</strong> mix of planning process input resources<br />
which achieve an optimal design from <strong>the</strong> viewpoint of econonic<br />
efficiency. A large effort cannot be expended to determine <strong>the</strong><br />
optimal <strong>le</strong>ngth of record at every possib<strong>le</strong> site: ra<strong>the</strong>r, simp<strong>le</strong><br />
guidelines are needed that answer such questions as:<br />
a) How much streamflow data is optimal at a site €or <strong>the</strong><br />
expected design decisions and economic parameters?<br />
b) What is <strong>the</strong> most efficient operation of a gaging<br />
station when <strong>the</strong>re are uncertainties in decisions and<br />
parameters?<br />
c) I€ <strong>the</strong> gaging station has already.been operated longer<br />
than <strong>the</strong> optimal <strong>le</strong>ngth, what factors would justify<br />
à3scontinuing or retaining <strong>the</strong> station?
338<br />
An approach to <strong>the</strong> determination of optimal <strong>le</strong>ngths of gaging<br />
for <strong>the</strong> design sizing and operating ru<strong>le</strong> determination of irriga-<br />
tion reservoirs is presented herein. This system was chosen for<br />
several reasons. In <strong>the</strong> western United States irrigation reser-<br />
voirs receive <strong>the</strong> majority of <strong>the</strong>ir inflow in <strong>the</strong> spring months<br />
from rainfall and snow-melt runoff. Usually only small quantities<br />
of natural flow are availab<strong>le</strong> during <strong>the</strong> peak demand months of<br />
July and August. The essence of such a system can be captured by<br />
a model in which inflows must be stored for satisfying demands in<br />
later seasons or future years. This permits <strong>the</strong> use of annual<br />
flows and operating ru<strong>le</strong>s which are nonseasonal in nature. Bene-<br />
fits from irrigation projects are also usually more easily measured<br />
than for o<strong>the</strong>r types of water supply.<br />
The value of data in a particular situation is often limited<br />
by constraints in <strong>the</strong> decision space. For examp<strong>le</strong>, if arab<strong>le</strong> land<br />
were limited because of availab<strong>le</strong> streamflow, reservoir design<br />
would become sensitive to <strong>the</strong> irrigation requirements, not mea-<br />
sures of <strong>the</strong> availab<strong>le</strong> streamflow. Systems which have a high<br />
degree of operational f<strong>le</strong>xibility may have low sensitivity to <strong>the</strong><br />
availability of data since operating policies can be changed to<br />
consider streamflow data col<strong>le</strong>cted after project construction.<br />
The Value of Data for Reservoir Design and Operation<br />
The seminal work on <strong>the</strong> determination of optimal data <strong>le</strong>ngths<br />
for project design is that of Dawdy, Kubik, and Close [2]. This<br />
work has been extended by considering <strong>the</strong> effect of discounting<br />
and benefits foregone by Moss [31, Herfindahl 141, and Tschannerl<br />
151.<br />
For <strong>the</strong> most part, <strong>the</strong>se studies have dealt with <strong>the</strong> situation<br />
where operating ru<strong>le</strong>s were given and design siting was <strong>the</strong> primary<br />
decision variab<strong>le</strong>. Hydrologic data also have value for <strong>the</strong> deter-<br />
mination of optimal operating ru<strong>le</strong>s.<br />
Operating Ru<strong>le</strong> Determination<br />
The transform of state variab<strong>le</strong>s such as storage and inflow<br />
into re<strong>le</strong>ase values is accomplished through operating ru<strong>le</strong>s.<br />
Perhaps <strong>the</strong> simp<strong>le</strong>st and most often used operating ru<strong>le</strong> for analyt-<br />
ical purposes is <strong>the</strong> Z ru<strong>le</strong>, where <strong>the</strong> re<strong>le</strong>ase R is defined by:<br />
Minimum { S + I, T 1 for S + I<br />
Vm<br />
Maximum { T, S + I - Vm for S + I > Vm<br />
where S is <strong>the</strong> carry-over storage, I is <strong>the</strong> inflow, T is <strong>the</strong><br />
target, and Vm <strong>the</strong> maximum conservation storage. This ru<strong>le</strong> would<br />
Se optimal if losses were linear with <strong>the</strong> deficits below <strong>the</strong> target<br />
value and no benefits or losses accrued from re<strong>le</strong>ases greater
339<br />
than <strong>the</strong> target value. Though <strong>the</strong>se may be ra<strong>the</strong>r restrictive<br />
conditions, <strong>the</strong> Z. ru<strong>le</strong> has proved to be a useful conceptual tool.<br />
In recognition of nonlinear loss functions and seasonal differ-<br />
ences in expected inflows, Maass et. al. -161 describe modifica-<br />
tions to <strong>the</strong> basic re<strong>le</strong>ase ru<strong>le</strong> cal<strong>le</strong>d <strong>the</strong> pack ru<strong>le</strong> and <strong>the</strong><br />
hedging ru<strong>le</strong>. Young 171 and Hall and Howells [8] used regression<br />
analysis on optimal deterministic re<strong>le</strong>ases derived from dynamic<br />
programming to define re<strong>le</strong>ase ru<strong>le</strong>s. Russell 191 uses dynamic<br />
programming to determine <strong>the</strong> form of an optimal operating policy<br />
following a development similar to that of Gessford and Karlin 1101.<br />
Both investigators assumed serial independence in inflow sequen-<br />
ces. Gablinger and Loucks [ill, Loucks and Falkson 1121, and<br />
Loucks (131 consider design and operation of storage facilities<br />
where flows can be described by Markov transition processes.<br />
The Z ru<strong>le</strong> for re<strong>le</strong>ase determination is not optimal when<br />
marginal losses increase with <strong>the</strong> amount of <strong>the</strong> re<strong>le</strong>ase deficit<br />
or where significant benefits accrue from competing reservoir<br />
purpo'ces such as recreation or water power. One method for devel-<br />
oping operating ru<strong>le</strong>s is to se<strong>le</strong>ct from a set of operating ru<strong>le</strong>s<br />
which are defined as a function of state variab<strong>le</strong>s such as storage<br />
and target draft. Parameters values of <strong>the</strong>se re<strong>le</strong>ase ru<strong>le</strong>s can<br />
<strong>the</strong>n be optimized in conjunction with reservoir design parameters.<br />
The se<strong>le</strong>ction of a particular functional form of <strong>the</strong> re<strong>le</strong>ase ru<strong>le</strong><br />
will depend upon <strong>the</strong> objective function, and <strong>the</strong> nature of <strong>the</strong><br />
assumed mechanism which generates <strong>the</strong> inflows. For o<strong>the</strong>r than<br />
extremely simp<strong>le</strong> prob<strong>le</strong>ms, this determination is non-analytic.<br />
where<br />
The operating ru<strong>le</strong> chosen for this study was:<br />
R = Maximum'{O.O, Minimum (aA - ßB+T, Si+J)}<br />
A = Maximum (0.0, S+I-T'Vol<br />
B = Maximum-{O.O, T+V -S -I}<br />
O i<br />
si e: Minimum {V ; s +I)<br />
rn i-1<br />
and CL and ß are re<strong>le</strong>ase ru<strong>le</strong> parameters, R is <strong>the</strong> re<strong>le</strong>ase to bene-<br />
ficial uses, and ifo is a minimum target conservation storage. In<br />
o<strong>the</strong>r words, <strong>the</strong> re<strong>le</strong>a'se is <strong>the</strong> target modified by a fraction a<br />
of <strong>the</strong> end-of-season contents above minimum pool <strong>le</strong>vel Vo or by a<br />
fraction 6 of <strong>the</strong> end-of-season contents below minimum pool <strong>le</strong>vel.<br />
This re<strong>le</strong>ase ru<strong>le</strong> is more sensitive to project economics than <strong>the</strong><br />
2 ru<strong>le</strong>, yet adds only <strong>the</strong> two parameters a and $ to <strong>the</strong> optirniza-<br />
tion prob<strong>le</strong>m.<br />
The previously defined parameters a, ß, T, and V , <strong>the</strong> active<br />
storage volume above Vo, were determined by <strong>the</strong> methoa of general
340<br />
function minimization described by Berman 1141, using a reservoir<br />
simulation model which returned <strong>the</strong> negative of discounted net<br />
project benefits for any set of <strong>the</strong> four parameters from <strong>the</strong> opti-<br />
mization routine.<br />
The se<strong>le</strong>ction of a project benefit function is not an easy<br />
task. The marginal economic response of crops to marginal water<br />
applications vary widely. For this analysis a linear benefit<br />
function was used with <strong>the</strong> following coefficients.<br />
C1 .O016 $/m3/ short term loss for end-of-season<br />
storage below V<br />
0<br />
C2 .O004 $/m3/ short term benefit for end-of-season<br />
c3 .o12 $/m3/<br />
storage above V<br />
O<br />
short term loss for re<strong>le</strong>ases below<br />
target<br />
C4 .O04<br />
3<br />
$/m<br />
short term benefit for re<strong>le</strong>ases above<br />
target<br />
C6 .O057 $/m3/yr long term benefit for target re<strong>le</strong>ases<br />
Assumed marginal capital cost3 for providing reservoir storage C<br />
5<br />
ranged from .O017 to .O04 $/m /yr. All annual benefits and<br />
costs were discounted at 6% to <strong>the</strong> starting point of project<br />
benefits.<br />
Analysis of Data Requirements for Irrigation Projects<br />
Five gaging stations were se<strong>le</strong>cted which had been used in<br />
<strong>the</strong> design of existing irrigation project reservoirs. The his-<br />
torical data were extended with generated operational hydrology<br />
to a <strong>le</strong>ngth of 200 years. The reservoirs were sized and operat-<br />
ing ru<strong>le</strong>s determined for all 100, 50, 33, 20, 15, and 10-year<br />
sequences of <strong>the</strong> 200-year sequence, and on 8 and 5-year segments.<br />
for some of <strong>the</strong> projects. For each set of determined parameters,<br />
<strong>the</strong> reservoir was simulated for <strong>the</strong> 200-year period and discounted<br />
values of <strong>the</strong> objective computed. These values were <strong>the</strong>n averaged<br />
across all equal-<strong>le</strong>ngth segments of record. Average objective<br />
function values were <strong>the</strong>n plotted against segment <strong>le</strong>ngth and a<br />
curve smoo<strong>the</strong>d in as shown for an examp<strong>le</strong> in figure 1. To <strong>the</strong><br />
values of this curve are added <strong>the</strong> present value of <strong>the</strong> cost of<br />
gaging, and a total cost curve results. The optimal <strong>le</strong>ngth of<br />
record occurs at <strong>the</strong> minimum of <strong>the</strong> total cost curve.<br />
Operating ru<strong>le</strong>s were held constant at <strong>the</strong>ir optimal values,<br />
and <strong>the</strong> analysis repeated for reservoir sizing alone. Also,<br />
reservoir size was fixed and <strong>the</strong> analysis repeated for operating<br />
ru<strong>le</strong>s alone. For clarity, only <strong>the</strong> objective function values
and not <strong>the</strong> total cost curves are shown in <strong>the</strong> published figure<br />
for <strong>the</strong>se analyses.<br />
These latter analyses were done to attain comparability with<br />
previous studies on <strong>the</strong> worth of data where operating ru<strong>le</strong>s were<br />
fixed and only reservoir sizìng allowed to vary.<br />
Discussion of Results<br />
341<br />
The present value of <strong>the</strong> cost of a gaging station record<br />
which is assumed to cost $2000 per year of operation is given in<br />
tab<strong>le</strong> 1 assuming an interest rate of 6%. Thus, in a 50-year<br />
gaging record, <strong>the</strong> first 10 years has a marginal cost of $580,000-<br />
$310,000 = $270,000 as compared to <strong>the</strong> original $20,000 investment<br />
for that record. The effect of discounting <strong>the</strong> gaging costs is<br />
to make <strong>the</strong> marginal cost of gaging much higher than might ordi-<br />
narily be perceived.<br />
The results from <strong>the</strong>se studies are tabulated in tab<strong>le</strong> 2.<br />
For <strong>the</strong> economic parameters used, several results are apparent.<br />
The optimal data <strong>le</strong>ngth for both design sizing and operating-<br />
ru<strong>le</strong> determination is considerably longer than <strong>the</strong> optimal data<br />
<strong>le</strong>ngth for design sizing alone, but only slightly longer than<br />
<strong>the</strong> optimal data <strong>le</strong>ngth for operating ru<strong>le</strong> determination. There<br />
is a strong correlation between optimal data <strong>le</strong>ngth and size of<br />
stream, as is seen in figure 2.<br />
If a gaging station is already in existence and has already<br />
achieved its planned "optimal" <strong>le</strong>ngth of record but <strong>the</strong> project<br />
has not been yet built, <strong>the</strong>n <strong>the</strong> decision prob<strong>le</strong>m must be viewed<br />
from a different economic viewpoint. For this, <strong>the</strong> present value<br />
of <strong>the</strong> marginal benefits of an additional year of record must<br />
equal or exceed <strong>the</strong> cost of that marginal year of gaging. The<br />
right-hand column of tab<strong>le</strong> 2 is <strong>the</strong> point on <strong>the</strong> marginal benefit<br />
curve that equals <strong>the</strong> marginal cost of gaging. For examp<strong>le</strong>, if<br />
<strong>the</strong> decision maker designing a reservoir on <strong>the</strong> Smoky Hill River<br />
near Arnold, Kansas was one year away from construction, <strong>the</strong>n he<br />
should continue to gage if <strong>the</strong> total availab<strong>le</strong> record is <strong>le</strong>ss<br />
than 55 years, even though it may exceed <strong>the</strong> apriori optimal<br />
<strong>le</strong>ngth of 30 years.<br />
In previous work by James, Bower, and Matalas í151, it has<br />
been suggested that <strong>the</strong> total variability of meeting a water<br />
quality target was more strongly influenced by economic and<br />
political uncertainties than hydrologic uncertainty. This was<br />
based upon a multivariate sensitivity analysis of output measures.<br />
Such an analysis is similar to measurement of <strong>the</strong> type A error,<br />
which is an apparent error in design caused by incorrect economic<br />
parameters evaluated at those incorrect parameters. The type B
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343<br />
error, or efficiency loss, is <strong>the</strong> pertinent measure to compare<br />
against efficiency losses from inadequate gaging records. The<br />
type B error is <strong>the</strong> loss measured with <strong>the</strong> true economic param-<br />
eters of a system whose desfgn was optimìzed with <strong>the</strong> incorrect<br />
parameters o<br />
The type A and type B errors resulting from 40% errors in<br />
each of <strong>the</strong> cost parameters used are shown in tab<strong>le</strong> 3.<br />
Tab<strong>le</strong> 3. Type A and type B errors resulting from 40% error<br />
in cost coefficients for a reservoir on <strong>the</strong> Smoky<br />
Hill River near Arnold, Kansas<br />
Cost Coefficient Type A error Type B error<br />
(% of net project benefits)<br />
c1 recreation 3.1 < .1<br />
c2 recreation < .1 < .1<br />
c3 short run deficit .6 < .1<br />
c4 short run re<strong>le</strong>ase 22 2.5<br />
c5 res e rvoi r<br />
construction<br />
10 < .1<br />
C6 long run re<strong>le</strong>ase 60 11<br />
For comparison, on this same project <strong>the</strong> efficiency loSS for<br />
incorrect record <strong>le</strong>ngths of 5, 10, and 15 years is 0.4, 1.4, and<br />
3.7% resPectivelY. Project feasibility is usually quite sensitive<br />
to type A errors and to <strong>the</strong> interest rate used for discounting<br />
project costs and benefits to a common time name. Type A-and B<br />
errors were not estimated for errors in interest rate, however,<br />
on this project a change in interest rate from 6% to 5% would<br />
change <strong>the</strong> optimal record <strong>le</strong>ngth from 30 to about 35 years.<br />
This analysis has taken <strong>the</strong> manager's viewpoint of a project<br />
gaging network, whose objectives are overall economic efficiency.<br />
Budget constraints are not active. The network manager seldom has<br />
precise information<br />
is <strong>the</strong>n to minimize<br />
years in <strong>the</strong> future<br />
m<br />
C P(t)L(t--r) is a<br />
ta0<br />
<strong>the</strong> project will be<br />
on <strong>the</strong> date of project inception. His prob<strong>le</strong>m<br />
<strong>the</strong> losses by starting gaging at a point<br />
such that <strong>the</strong> expected loss<br />
minimum, where P(t) is <strong>the</strong> probability that<br />
started t years in <strong>the</strong> future, and L(t-T) is<br />
<strong>the</strong> total loss function as sho& in <strong>the</strong> examp<strong>le</strong> in figure 1.<br />
Water resources development projects also must consider o<strong>the</strong>r<br />
criteria than economic efficiency. Considerations of regional<br />
income distribution and <strong>the</strong> degree of risk aversion in <strong>the</strong> parties<br />
receiving <strong>the</strong> economic benefits will temper gaging network<br />
decisions.
344<br />
If <strong>the</strong> analysis presented herein is to be used to determine<br />
ghe optimal sìarting time for a gaging station at a previously<br />
yngaged site, some estimates of <strong>the</strong> flow characteristics will<br />
bave to be made. Estimates of flow characteristics prior to gag-<br />
ing can be made for sites in <strong>the</strong> Unfted States using regional<br />
gelationships presented in Thomas and Benson n61. Additional<br />
gaging information can <strong>the</strong>n be incorporated into <strong>the</strong> prior ecti-<br />
wgtes using Bayesian analysis.<br />
cgnclusions<br />
Optimal record <strong>le</strong>ngths for determining reservoir sizing and<br />
ekerating ru<strong>le</strong> parameters can be determined by computing expected<br />
project benefits from designs resulting from varying <strong>le</strong>ngths of<br />
design data. Optimal record <strong>le</strong>ngths for <strong>the</strong> combined process of<br />
reservoir sizing and determining operating ru<strong>le</strong> parameters are<br />
significantly longer than for reservoir sizing alone under a<br />
fixed optimal operating ru<strong>le</strong>.<br />
The optimal <strong>le</strong>ngth of record increases with size of stream,<br />
ganging from 7 to 47 years for <strong>the</strong> five str ams used in this study,<br />
9<br />
which range in discharge from .25 to 27.1 m /sec.<br />
Econcmic efficiency requires that <strong>the</strong> marginal worth of col-<br />
ìecting additional streamflow data be greater than <strong>the</strong> marginal<br />
costs when <strong>the</strong>se two values are discounted to a common point in<br />
%ime. Hence <strong>the</strong> decision prob<strong>le</strong>m of discountinuing an existing<br />
gage is different than <strong>the</strong> decision prob<strong>le</strong>m for <strong>the</strong> optimal<br />
gtarting time for <strong>the</strong> gage because <strong>the</strong> discount factors applied<br />
gp <strong>the</strong> initial year are larger than those applied to <strong>the</strong> final<br />
year.<br />
References<br />
Wiener, Aaron, (1972). The ro<strong>le</strong> of water in development,<br />
New York, McGraw-Hill, Inc.<br />
Dawdy, D.R., Kubik, H.E., and Close, E.R. (1970). The value<br />
of streamflow data for project design - a pilot study,<br />
Water Resources Research, v. 6, no. 4, pp. 1045-1050.<br />
MOSS, M.E. (1970). Optimum operating procedure for a river<br />
gaging station established to provide data for design of a<br />
water supply project. Water Resources Research, v. 6, no. 4,<br />
pp. 1051-1061.<br />
Herfindahl, Orris C., (1969). Natural resources information<br />
for economic development, Baltimore!, Johns Hopkins Press.<br />
Tschannerl, G., (1971). Designing reservoirs with short<br />
streamflow recoräs, Water Resources Research, v. 7, no. 4,<br />
pp. 827-833.
6<br />
7.<br />
8.<br />
9<br />
lu.<br />
1 .<br />
IL<br />
13<br />
14<br />
15.<br />
16.<br />
Maass, Arthur, et. al. (1962). The design of water resource<br />
sy~tems, Cambridge, Harvard University Press.<br />
345<br />
aung, G.K. , (1967). Findiiig reservoir 01 crating ru<strong>le</strong>r. Jour.<br />
f <strong>the</strong> Hydrau1ic.s Div., Am Soc. of Cìvi I Enqr.. v. 93. no.<br />
HY6, pp. 297-<br />
Hall, Warren A., and Howell, David T., (1963 . The uptimiza-<br />
tion of sing<strong>le</strong> purpose reservoir design with <strong>the</strong> appii ition<br />
of dynamic programming to syn<strong>the</strong>tic hydrology samp<strong>le</strong>s, Jour.<br />
of Hydrology, v. 1, pp. 355-363.<br />
Russell, C. Brad<strong>le</strong>y, (19121. An optimal policy for operating<br />
d multipurpose reservoir, Operations Research, v :O, no. 6,<br />
pp. 1181-1189.<br />
Gessford, John, and Karlin, Samuel, Optimal policy for hydro-<br />
el-ectric operdiicns, pp. 179-200 in Arrow, Kenneth, J.. Karlin,<br />
Samuel, and Scarf, Herbrrt (1958). Studies in <strong>the</strong> ma<strong>the</strong>matical<br />
<strong>the</strong>ory of inventory an..; productions, Stanford, Stanford Univer-<br />
sity Press.<br />
bablinger, Moshe, and Loucks, Daniel , (1970). Markov models<br />
for flow regulation, JOUI. of <strong>the</strong> Hydraulics Div., Am. SOC.<br />
Civil Engrs., v. 96, no. HY1, pp. 165-181,<br />
Loucks, D.P., and Falkson, L.M., (1970). A comparison of Some<br />
dynamic, linear and policy iteration methods for reservoir<br />
operation, Water Resources Bul<strong>le</strong>tin, v. 6, no. 3, pp..385-399.<br />
Loucks, D.P., (1970). Some comments on linear decision ru<strong>le</strong>s<br />
and chance constraints, Water Resources Research, v. 6, no. 3,<br />
pp. 668-671.<br />
Berman, Gerald, (1969). Lattice approximations to <strong>the</strong> minima<br />
Jf functions of several variab<strong>le</strong>s, Jour. of <strong>the</strong> Assn. for<br />
omputing Machinery, v. 16, n,,. 2, pp. 286-294.<br />
James, I.C., 'II, Bower, B.B., and Matalas, N.C., (1969 ,<br />
Relative importance of variab<strong>le</strong>s in water resources planning,<br />
Water Resources Research, v. 5, no. 4, pp. 1165-1173.<br />
Thomas, D.C., and Benson M.A., (1970). Generalization Of<br />
streamflow characteristics from drainage basin characteristicsr<br />
1v.S. Geol. Survey Water Supply Paper 1975, 55 P.
U<br />
346
O 10 20 30 40 50<br />
OPTIMAL LENGTH OF RECORD<br />
347
ABSTRACT<br />
THE DESIGN OF WATER QUALITY MANAGEMENT PROJECTS<br />
WITH INADEQUATE DATA<br />
George W. Reid<br />
Regents Professor<br />
University of Oklahoma<br />
One of <strong>the</strong> increasingly important e<strong>le</strong>ments in <strong>the</strong> design of<br />
water resource projects is, of course , <strong>the</strong> management of quality<br />
and a technology that was almost purely hydrological and hydraulic<br />
is now being expanded to include what might be classed as <strong>the</strong><br />
environmental and edological impact areas and systems, So, it is no<br />
longer sufficient to understand <strong>the</strong> interrelationships, flows and<br />
transports, but to this must be added <strong>the</strong> impacts on ttbe lisJrng and<br />
nonliving water, and peripherral environments; with a need to<br />
develop ecological models or more specifically, water quality models,<br />
Unfortunately, <strong>the</strong>re is rarely adequate data to properly describe<br />
<strong>the</strong>se interrelationships, The methodology used for hydrological<br />
studies involving inadequate data such as <strong>the</strong> transfer of okserved<br />
points to points of interest; short term interise studies; &,Y use of<br />
simulation techniques, can and are being used in quality management<br />
modeling, Perhaps more basic is an understanding of data requirements,<br />
using <strong>the</strong> system approach, <strong>the</strong> sequence of events ar- il) prob<strong>le</strong>m<br />
formulation, (2) symbol1 modeling, (3) data col<strong>le</strong>ctlon, (4) analysis<br />
and (5) design. (See Figure 1) Frequently, <strong>the</strong> order is ,hanged,<br />
particularly <strong>the</strong> entire process will start with availab<strong>le</strong> data.<br />
The comp<strong>le</strong>xities, of course, arise due to <strong>the</strong> fa.t that <strong>the</strong><br />
I rocesses associated with water quality management: hydraulic,<br />
hydrologi,al, chemical, biological and ecological -- are extremely<br />
and imperfectly understood. So, that is a comp<strong>le</strong>x reality, with a<br />
great many variab<strong>le</strong>s on which <strong>the</strong>re is availab<strong>le</strong> veri poor measures<br />
and which <strong>the</strong>mselves interrelate in ways very inadequately<br />
understood -- must be measured and appropriately related to be useful,<br />
Certainly, one recognizes <strong>the</strong> superiority of an expli-it quantifiab<strong>le</strong><br />
data and models over intuitive models and hurlChes. The<br />
alternatives to such a model, based on partial know<strong>le</strong>dge, is a mental<br />
model, based on <strong>the</strong> mixture of incomp<strong>le</strong>te information and intuition<br />
similar to those controlling most political decisions. A ma<strong>the</strong>matical<br />
mcdel deals with <strong>the</strong> same incomp<strong>le</strong>te information availab<strong>le</strong> to an<br />
1 1 *uitive model, but through organization of information from many<br />
iifferent sources into a closed loop at last analyses is permitted<br />
and data needs studied,
350<br />
RESUMEN<br />
Uno de los e<strong>le</strong>mentos altamente importantes en el diseño de pro<br />
yectos de recursos de agua es, desde luego, el manejo de la calidad<br />
y tecnologia que fue casi puramente hidrolögica e hidráulica y está<br />
siendo ahora expandida para incluir lo que debe de ser clasificado -<br />
como áreas y sistemas de impacto ambienta<strong>le</strong>s y ecológicos. Asi que -<br />
ya no será suficiente entender las interrelaciones, flujos y trans--<br />
portes, pues a éstos deben de ser agregados los impactos en las ---<br />
aguas con y sin presencia de formas de vida y los ambientes perifi--<br />
cos; con la necesidad de desarrollar modelos ecológicos o más especl<br />
ficamente modelos de calidad de agua. Desafortunadamente, rara vez -<br />
existen datos adecuados para describir propiamente estas interrela--<br />
ciones. La metodología usada para estudios hidrológicos incluye in--<br />
formación inadecuada, ta<strong>le</strong>s como el cambio de puntos observados a -<br />
puntos de interés; estudios intensos de corto plazo; o uso de técni-<br />
cas de simulación, pueden y han sido usadas en modelos de manejo de<br />
calidad. En el modelado existe siempre una cierta incompatibilidad -<br />
entre puntos de sustancia y generalidad; requerimientos de informa--<br />
ciÓn y la representatibilidad del mundo real. El objetivo desde lue-<br />
go, es proveer por medio de una abstracción idealizada un comporta--<br />
miento aproximado el cual es siempre un compromiso entre simplicidad<br />
y realidad. En años recientes una gran cantidad de modelos han sido<br />
desarrollados, pero, desafortunadamente parece haber un alto grado -<br />
de polarización. En un extremo, hay un e<strong>le</strong>gantisimo y sofisticado mo<br />
delo basado en técnicas econométricas requiriendo un alto grado de -<br />
especificación de información, que en la realidad no existe, Por --<br />
otro lado del espectro, los senarios dependen casi muy poco de info;<br />
mación, más sobre conceptos. La necesidad básica es para modelos en<br />
aìgfin lugar entre los dos extremos que están construidos usando in--<br />
formación existente y que puedan ser responsab<strong>le</strong>s a las necesidades<br />
de las agencias de acción, Es en esta realidad en la cual el autor -<br />
ha desarrollado una serie de modelos de calidad de agua. Los proyec-<br />
tos siendo modelados son generalmente de una natura<strong>le</strong>za tal que la -<br />
realización final ocurrirá bastante después de la partida de los di-<br />
señadores y por tal los procedimientos de evaluación directa son im-<br />
posib<strong>le</strong>s, necesitándose de alguna forma de evaluación o integridad -<br />
interna. El prob<strong>le</strong>ma es que usando cuanta información esté disponi--<br />
b<strong>le</strong>, para 50 a 100 años a la fecha y haciéndolo de manera que no sea<br />
tan e<strong>le</strong>gante que se convierta en un modelo dogmático. El autor ha -<br />
desarrollado una serie de modelos respondiendo al desafio. La esen--<br />
cia de la metodologia es reconocer la comp<strong>le</strong>jidad de un prob<strong>le</strong>ma y -<br />
trazar una combinación de técnicas de investigación de operaciones,<br />
técnicas deterministicas, asi como m’etodos empîricos, fenomológicos<br />
y analiticos. Modelos para sistemas de rios responden a la polución<br />
organizada de cuatro maneras: bioquimica, biodegradab<strong>le</strong>, sedimentos<br />
nutriciona<strong>le</strong>s, incluyendo modelos adiciona<strong>le</strong>s para flujos urbanos y<br />
poluciiin dispersa, así como flujos rura<strong>le</strong>s. Todos los modelos usaron<br />
información existente y ésto los sitG‘a para modelado pronosticab<strong>le</strong> -<br />
de nivei de las cuencas siendo computarizados y sistematizados y es-<br />
tán siendo usados en prob<strong>le</strong>mas específicos en el Suroeste de los Es-<br />
tados Unidos.
Prob<strong>le</strong>m Formulation: To arrive at a water resource project design, <strong>the</strong> number<br />
of variab<strong>le</strong>s is enormous, and <strong>the</strong>y are mostly nonlinear. The structure of<br />
<strong>the</strong> system is more hierarchical than functional, and many of <strong>the</strong> parameters<br />
and variab<strong>le</strong>s are unquantified at present, certainly those associated with<br />
ecology. None<strong>the</strong><strong>le</strong>ss, to some degree, a merging of disciplines and <strong>the</strong><br />
increased use of <strong>the</strong> system approach has been taking place in <strong>the</strong> study o€<br />
urban systems, and it is not just a matter of col<strong>le</strong>cting data and figuring<br />
out what one has.<br />
Lf one looks at <strong>the</strong> type of models being postulated €or <strong>the</strong> design of<br />
water quality systems today, it will be seen (Figure 2) that <strong>the</strong>y fall<br />
within a spectrum ranging from erudite ma<strong>the</strong>matical models at one end of<br />
<strong>the</strong> spectrum to scenarios at <strong>the</strong> o<strong>the</strong>r. In <strong>the</strong> first case, <strong>the</strong> ma<strong>the</strong>-<br />
matical models may be rigorously developed in a ma<strong>the</strong>matical sense, but<br />
are all too often of litt<strong>le</strong> use in describing a real comp<strong>le</strong>x system in<br />
inadequate data. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> scenario model - litt<strong>le</strong> data,<br />
numerous ideas --may accurately depict <strong>the</strong> significant e<strong>le</strong>ments of <strong>the</strong><br />
real system, but it is of litt<strong>le</strong> use to <strong>the</strong> engineer-planner because he<br />
cannot manipulate it. or quantify it.<br />
The target one should try to hit is a reasonab<strong>le</strong> and useab<strong>le</strong> balance<br />
hetween <strong>the</strong> po<strong>le</strong>s of intuition and se<strong>le</strong>cting hard data. One would like<br />
to be ab<strong>le</strong> to use <strong>the</strong> ma<strong>the</strong>matical rigor of <strong>the</strong> physical scientist and.<br />
at <strong>the</strong> same time, give equal weight to <strong>the</strong> heuristic insight of <strong>the</strong> social<br />
scientist.<br />
The result would be a useab<strong>le</strong> model for a system design. So,<br />
perhaps, or certainly, for planning purposes, one is dealing with <strong>the</strong> lowest<br />
<strong>le</strong>vel of quantification that allows good estimates and <strong>the</strong> lowest <strong>le</strong>vel of<br />
comp<strong>le</strong>xity which gives a reasonab<strong>le</strong> picture of <strong>the</strong> real world system with<br />
<strong>the</strong> lope of expounding in both directions.<br />
The dpplication of ma<strong>the</strong>matical modeling techniques to water quality<br />
management can significantly aid <strong>the</strong> decisionaakers to arrive at better<br />
decisions. Thus, modeling provides re<strong>le</strong>vant facts and alternatives, <strong>the</strong><br />
decision-maker chooses <strong>the</strong> strategy. Operational mode<strong>le</strong> are still prim-<br />
itive, primarily becaiise of <strong>the</strong> probilistic or random nature of <strong>the</strong><br />
physical processes involved in waste diffusion. One is sometimes inclined<br />
to be skeptical of <strong>the</strong> value of increasing model sophistication which<br />
often seems to have progressed much fur<strong>the</strong>r than our understanding of <strong>the</strong><br />
rmrnp<strong>le</strong>x real world situation; all models currently proposed in <strong>the</strong> literature<br />
have enormous data requirements which far exceed those data usually availab<strong>le</strong>,<br />
and which, for <strong>the</strong> most part, must be derived from actual measurement.<br />
Many parameters in <strong>the</strong> more sophisticated models are simply not known in<br />
actual situations.<br />
The water quality management design prob<strong>le</strong>m require:<br />
1. The cause and effect relationship between pollution from any<br />
source and <strong>the</strong> present deteriorated quality of water in <strong>the</strong> estuary.<br />
2. Forecasting variation of water quality due to <strong>the</strong> natural and<br />
man-made causes.<br />
3. Methods of optimal management, including treatment and flow<br />
regulation to control <strong>the</strong> quality in <strong>the</strong> estuary for muaicipal, industrial,<br />
agricultural, fisheries, recreation and wild life propagation.<br />
4. Chemical, biological, hydrological, hydraulic, at <strong>the</strong> same <strong>the</strong>,<br />
same place, and same accuracy.<br />
3 51
352<br />
Models In modeling <strong>the</strong>re is always a certain incompatibility and representativeness<br />
of <strong>the</strong> real world. The aim, of course, is to provide through<br />
an idealized abstraction an approximate behavior of <strong>the</strong> system which always<br />
is a compromise between simplicity and reality. Water quality models can be<br />
used to simulate, describe and predict, and programming <strong>le</strong>ading to optimization<br />
of design. Programing which <strong>le</strong>ads to policy requires an explicit<br />
set of objectives, or an objective function to maximize benefits or minimize<br />
costs. Simulation does not require explicit results. So, simulations are<br />
misunderstood, if one expects to use <strong>the</strong> numerical projections and values.<br />
Using numbers is wrong if it <strong>le</strong>aves <strong>the</strong> impression that design projections<br />
are in any way predictions of <strong>the</strong> future. It is helpful, E as a prediction<br />
but to get one to realize how short-sighted -- how present-oriented - images<br />
of <strong>the</strong> future ordinarily are, but extrapolltion of present trends is a time-<br />
honored way of looking into <strong>the</strong> future.<br />
Most peop<strong>le</strong> intuitively and<br />
correctly reject extrapolations -- <strong>the</strong> point is that it provides indications<br />
of <strong>the</strong> system's behavioral tendencies and as an analysis of current trends,<br />
of <strong>the</strong>ir influence on-each o<strong>the</strong>r, and of <strong>the</strong>ir possib<strong>le</strong> outcomes.<br />
Models may be classified usefully by areal extent into national, regional<br />
and local . At <strong>the</strong> highest, or national <strong>le</strong>vel, data is necessary for broad<br />
planning purposes, such as to determine an overall <strong>le</strong>vel of water pollution,<br />
to determine <strong>the</strong> total investment necessary for pollution abatement, to<br />
determine national policies and to project <strong>the</strong> prob<strong>le</strong>ms into <strong>the</strong> future.<br />
At <strong>the</strong> second highest <strong>le</strong>vel, <strong>the</strong> regional <strong>le</strong>vel, all of <strong>the</strong> above information<br />
is necessary, plus <strong>the</strong> particular information needs for <strong>the</strong> region. The<br />
third, local <strong>le</strong>vel, consists usually of checking <strong>the</strong> operation of waste<br />
treatment plants to insure compliance with regulations and statutes.<br />
Thus,<br />
due to <strong>the</strong> different requirements and objectives, a data program which may<br />
be optimal at one <strong>le</strong>vel, is usually far from optimal at some o<strong>the</strong>r <strong>le</strong>vel.<br />
Un<strong>le</strong>ss a c<strong>le</strong>ar objective has been set, <strong>the</strong>re is no guarantee that all<br />
critical bits and bytes of information are col<strong>le</strong>cted, and that <strong>the</strong> ga<strong>the</strong>ring<br />
of use<strong>le</strong>ss data is minimized. Similar calssification classification can be<br />
made with relation to time.<br />
Hypo<strong>the</strong>tical attempts to describe <strong>the</strong> intricate relationships between<br />
nutrients, phytoplankton, zooplankton, fish, detritus, bacteria and maninduced<br />
waste loads. There has resulted a great variety of models. One of<br />
<strong>the</strong> first developed, classical Streeter-Phelps equation, describes adequately<br />
<strong>the</strong> deoxygenation and reoxygenation in <strong>the</strong> river. The familiar form of <strong>the</strong><br />
oxygen sag equation is:<br />
-<br />
Do -<br />
-<br />
where: D oxygen dificit at time t<br />
-<br />
oxygen deficit at time zero<br />
BOD at time zero<br />
Lo<br />
t = time (distance) in days<br />
deoxygenation coefficient<br />
kl =<br />
k2 = reoxygenation coefficient<br />
This equation has been expanded to provide for evection and diffusion; algae<br />
growth, beuthal deposits, etc., into, inreality, impossib<strong>le</strong> data requirements.<br />
The basic need is for models somewhere between two po<strong>le</strong>s that are built using<br />
existing data and as such can be responsive to <strong>the</strong> needs of <strong>the</strong> action agencies.<br />
Tt is in this realm in which <strong>the</strong> author has developed a series of water quality<br />
models. The projects being mode<strong>le</strong>d generally are of such a nature that <strong>the</strong><br />
ultimate realization will occur long after <strong>the</strong> departure of <strong>the</strong> designers, and<br />
as such direct validation procedures are impossib<strong>le</strong>, necessitating some form
of internal validation or internal integrity. The prob<strong>le</strong>m is one of using<br />
what information is availab<strong>le</strong> for a 50-100 year future, and doing it in<br />
such a fashion that it is not so e<strong>le</strong>gant that it becomes a classroom make-<br />
believe world, The essential thread in <strong>the</strong> author's methodology is that of<br />
recognizing <strong>the</strong> comp<strong>le</strong>xity of a prob<strong>le</strong>m and drawing on a combination of OR<br />
techniques, deterministic techniques, as well as imperical, phenomological,<br />
and analytical methods. River system models respond to organized pollution<br />
L,I modes.<br />
There are suggested six categories of stream responses: biodegradab<strong>le</strong>,<br />
nutritional, bacterial, solids, persistant of slowly degradab<strong>le</strong> chemicals<br />
and <strong>the</strong>rmai. The response of a given stream to <strong>the</strong>se categories can be<br />
formulated; or <strong>the</strong> reverse. given an instream criteria (RQS), allowab<strong>le</strong><br />
effluent quality can be calculated. The specific criteria now can be<br />
grouped under response headings; for nutritional, one might se<strong>le</strong>ct N, P,<br />
NIP, or AGP, etc. If primary treatment is established as a lower con-<br />
straint on <strong>the</strong> effluent, <strong>the</strong> solids criteria can be de<strong>le</strong>ted; and fur<strong>the</strong>r,<br />
if a public health constraint on toxic and bacterial <strong>le</strong>vels can be exercised,<br />
fosir ra<strong>the</strong>r than six responses can now be used <strong>le</strong>aving a four-by-four matrix<br />
'o be examined.<br />
TABLE I<br />
Municipal Industrial Agricultural Recreational<br />
Biodegradab<strong>le</strong><br />
Nutritional<br />
Control<strong>le</strong>d by D. O. <strong>le</strong>vels<br />
Control<strong>le</strong>d by N and P <strong>le</strong>vels<br />
Thermal Control<strong>le</strong>d by Temperature increases<br />
Persistent<br />
Chemical Control<strong>le</strong>d by Salt, CCE's or ABS, etc.<br />
353<br />
So, a response/use matrix, changing with time will set goals; based on a<br />
matrix such as <strong>the</strong> one in Tab<strong>le</strong> I "d alternative socio-operated projections.<br />
A linking technical basin model can be built and operated to provide <strong>the</strong> optimal<br />
use of water resources, and of necessary treatments; or in pianning for<br />
ruture population increases and <strong>the</strong> concomitant increased use of water, it<br />
is possib<strong>le</strong> to build ma<strong>the</strong>matical models depicting <strong>the</strong> optimum treatments and<br />
stream flows necessary to meet ths RQS. The one-to-one input-output relationships<br />
f-r he four c-tegorius vf waste discharges follows with <strong>the</strong> Low Flow<br />
Augmentation FA), associated with each treatment <strong>le</strong>vel (mi) , will be QL,<br />
QN, Qp and Q,. This is a terminal flow in MO. T'Li is a fraction where i<br />
refers to BOD, N and P.<br />
BIODEGRADABLE MODEL (L)<br />
Y PE or P A (P)<br />
Q, y+ (i-Y) CS - RQsDO<br />
(i) where:<br />
Y = Fraction of total population in SMA's<br />
E = Efficiency term, Point LoadIUniform Load<br />
PE = Population Equiva<strong>le</strong>nt Ln millions<br />
P = Percentage discharge to river, expressed
- as a fraction, Decision<br />
Variab<strong>le</strong> (1-TL)<br />
Cs = DO saturation <strong>le</strong>vel 6 given temperature<br />
A = 942,900 relates to stream characterk2<br />
4 * istics<br />
where n is essentially <strong>the</strong> number of reoxygenized volumes, Ir <strong>the</strong> reaeration<br />
2<br />
constant, L <strong>the</strong> reach, V <strong>the</strong> velocity -- <strong>the</strong>se valués will change as <strong>the</strong><br />
stream Ltself is subject to management.<br />
ACCELERATED RITROPHICATION MODEL<br />
Z'P (1-%-1.44 (1-5) (TLL3250) (3)<br />
Qp = 2-P (1-TLp) - .27 (1-5) (TLL 1080)<br />
F, ROS<br />
THERMAI. MODEL (T)<br />
Qr =<br />
ATw - C<br />
AT +C<br />
Q<br />
AQ<br />
= Thermal Dilution Required, MGD<br />
where:<br />
Qp or QN = Nutritional Dilution Required, MGD<br />
(4)<br />
Z = Relative portion impounded and<br />
effected by RQS, <strong>le</strong>vel<br />
- P = Population, millions<br />
T$ or %<br />
-<br />
Phosphorus or Nitrogen removal <strong>le</strong>vel<br />
expressed as a decimal<br />
F or Fp BOD/, Ratio divided by optimum<br />
N<br />
combining ratio<br />
RN = BOD removal <strong>le</strong>vel expressed as<br />
a decimal<br />
RQS, or RQS, = Acceptab<strong>le</strong> <strong>le</strong>vel, RQS determined<br />
by RQSAGp<br />
A Lw - Allowab<strong>le</strong> temperature difference between added flow and RQSt (t-RQSt)<br />
A TQ =<br />
-<br />
Allowab<strong>le</strong> temperature change (RQST - To)<br />
( Ratio of K/Vx when K Geometric mean for Bowmen's ratio and V =
subsidance velocity<br />
AQ = Waste Flow, MGD<br />
CONSERVED OR PERSISTENT SHEMICAL MODEL (C)<br />
These models, though’cast in terms of dilution requirements, can be<br />
altered, given a diluted <strong>le</strong>vel to provide permissib<strong>le</strong> loadings. The<br />
models (2) thru (6) are based on organized (sewered) pollution. Models<br />
for storm drainings or dispersed pollution have also developed such as:<br />
DISPERSED POLLUTION MODEI. (D)<br />
Y2 = 4.8 + 0.0827X2 + 0.489X8 (7)<br />
where Y is BOD<br />
3<br />
Y = 2.36 - 0.188 1nX + .310 Inxl0<br />
5<br />
where Y is ON and Y6 is PO, in<br />
5<br />
Y6 = 2.90 + .OOOOSX1 - .OOOlX, - .0137X8 - .741Xll<br />
and Xi = population<br />
X2 =. population density<br />
X = number of households<br />
3<br />
X8 = comercial establishments<br />
Xl0 = streets<br />
Xll = environmental index<br />
Models (2-9) can be used to relate waste inputs to stream responses under<br />
varying municipal stream characteristics and against varying goals (RQS).<br />
Many technical models are availab<strong>le</strong> to project flows (Q), and o<strong>the</strong>r stream<br />
characteristics it2, L, V, et.. but a final model is needed for evaluation of<br />
<strong>the</strong> effects of <strong>the</strong> rural upstream watershed programa on downstream runoff to<br />
comp<strong>le</strong>te <strong>the</strong> set. Such a model was developed for <strong>the</strong> Congress in 1969. ls2<br />
For details of model. development see, THE OUTLOOK FOR WATER, Wollman and<br />
Bonem, The John Hopkins Press, Baltimore & London, 1971, Appendix C., p. 203.<br />
This was a special consultative report to <strong>the</strong> Secretary of <strong>the</strong> Interior,<br />
October, 1967.<br />
355
356<br />
UPSTREAM USE MODEL (U)<br />
Y=-16+XX<br />
1 3 - u7x2<br />
Where:<br />
Y =i percentage of nonna1 runoff<br />
X1 -i percentage of normal precipitation<br />
X2 * percentage of watersheds control<strong>le</strong>d by hydraulic structures<br />
Xj = annual above one inch precipitation<br />
In <strong>the</strong>se equation, <strong>the</strong> simp<strong>le</strong> Phelps equation (1) has been reduced to:<br />
n<br />
Expanding this to<br />
dL = % dO = f dO<br />
ao E 2 E a20<br />
-iP<br />
- kdL - kn L" + ka (Cs -C) -<br />
three dimension, (x, y, z,) would require:<br />
at - xa: ++<br />
EZa2 o<br />
i- -<br />
- kic , etc.<br />
ax ax<br />
(10)<br />
(11)<br />
That is to say, <strong>the</strong> load equals <strong>the</strong> capacity. Distribution factors are<br />
added, load is put in terms of peop<strong>le</strong>, PE's, etc. This is useab<strong>le</strong>. On <strong>the</strong><br />
o<strong>the</strong>r hand and by way of contrast, O'Connor uses a one dimensional, differentia1<br />
equation, first involving:<br />
(13)<br />
Also <strong>the</strong> evaluation of E's, U, Ki, etc. in terms of velocity, solar energy,<br />
depth, turbidity, etc. 3<br />
The effectivehess of models is, of course, acceptance. Actually, very few<br />
models have been used. Limitations of applying <strong>the</strong>m to "real" systems are<br />
rooted in many factors, most related to data inadequacies; <strong>the</strong> acquisition<br />
of proper data, adjustment of non-homogenity, or inconsistency, to MIU~ a<br />
few.<br />
SYSTEMS ANALYSIS AND WATER QUALITY, Thoman, Environmental Science Service,<br />
New York, 1972.
Every model, or system, is always embedded in a larger system in space<br />
or time, so one is limited to se<strong>le</strong>ction of a free body cut and exogenously<br />
determined parameters. Finally, serious factors, mostlv associated with<br />
social values cannot, at present, be quantified.<br />
An efficient use of models thus, argues for different models to answer<br />
different question. For examp<strong>le</strong>, one for sediments, one for social costs,<br />
etc. The systems process is iterative and continues whi<strong>le</strong> <strong>the</strong> models are<br />
refined and until satisfactory results are obtained.<br />
The flow of information for all <strong>the</strong> mested models eventually <strong>le</strong>ads to <strong>the</strong><br />
decision process. Forward and feedback information flows take place between<br />
models until <strong>the</strong> alternative se<strong>le</strong>ction and information developed is accepted<br />
for decision-making.<br />
As illustrated, <strong>the</strong>re is no attempt to "hang" all<br />
models toge<strong>the</strong>r. More important, different <strong>le</strong>vels of data, can be used in<br />
each mode, providing homogenity in each model.<br />
DATA<br />
The data must support <strong>the</strong> models. Some of <strong>the</strong> questions for which answers<br />
are needed are, goals, include,:<br />
1. What significant parameters of water quality should be measured,<br />
for an a<strong>le</strong>rt system, for treatment plant control, for a quality forecasting<br />
system, for a river management system?<br />
2. What should be <strong>the</strong> periodicity or time interval in col<strong>le</strong>cting<br />
specific data?<br />
3. What are <strong>the</strong> cross correlations of <strong>the</strong>se parameters?<br />
4. Are <strong>the</strong>re any synergisLic relationships between <strong>the</strong> parameters?<br />
5. What is being accomplished to develop instrumentation that can<br />
gage quantitatively those essential parameters, such as BOD, that are not<br />
being measured automatically at <strong>the</strong> present time?<br />
So, <strong>the</strong>re are all sorts of data, much of it redundant. One needs a model<br />
to discover needs, costs, etc. The process is shown graphically in<br />
Figure 3.<br />
Data has a cost, col<strong>le</strong>ction and deferral of decisions.<br />
The quantity of information col<strong>le</strong>cted should be increased so long as <strong>the</strong><br />
present value of <strong>the</strong> investment opportunity (or cost savings if this is <strong>the</strong><br />
use to which <strong>the</strong> information is put) is increased by more than <strong>the</strong> cost of<br />
<strong>the</strong> information.<br />
The expected value of a decision will be low with litt<strong>le</strong> data availab<strong>le</strong>, but<br />
will rise with more data availab<strong>le</strong>. With litt<strong>le</strong> data availab<strong>le</strong>. <strong>the</strong> solution<br />
often would be overstated (resulting in unused capacity) or understated<br />
(resulting in lost opportunity), thus reducing <strong>the</strong> expected present value of<br />
tht opportunity. For small enough quantities of data, <strong>the</strong> expected value<br />
will be negative.<br />
The conclusion that<strong>the</strong> decision take place when <strong>the</strong> cost of getting one more vear<br />
of information is equal to <strong>the</strong> resulting increase in expected present value.<br />
The cost of getting one more year of data is made up of two e<strong>le</strong>ments, <strong>the</strong>
outlay during <strong>the</strong> during <strong>the</strong> year to get <strong>the</strong> data, k, and interest on <strong>the</strong><br />
expected present value of <strong>the</strong> opportunity one would experience if a year of<br />
waiting is not included. That is, if V(t) is <strong>the</strong> basic function, one should<br />
not wait until its rate of increase, V’(t), is equal to [rV(t) + LI, where<br />
r is <strong>the</strong> rate of discount ( <strong>the</strong> rate of return on investment).<br />
Several conclusions are evident. First, it never will pay to wait for<br />
”comp<strong>le</strong>te” information. Second, an extremely important e<strong>le</strong>ment of <strong>the</strong><br />
prob<strong>le</strong>m is <strong>the</strong> cost coming from postponement of <strong>the</strong> stream of net revenues<br />
from <strong>the</strong> decision. This factor means it does not pay to accumulate data<br />
until <strong>the</strong> increment in expected value is equal to <strong>the</strong> annual cost of <strong>the</strong><br />
data.<br />
Experience in <strong>the</strong> United States has resulted in <strong>the</strong> common utilization of<br />
only eight water quality parameters that are thought to satisfy <strong>the</strong> re-<br />
quirements of reliability, accuracy, and low maintenance. These parameters<br />
are dissolved oxygen, pH, turbidity, conductivity, temperature, OñP, solar<br />
radiation intensity and chlorides. Time sequence is important. Parameters<br />
needed today may not be <strong>the</strong> correct ones torrorrow.<br />
TABLE II<br />
---<br />
TIME SCHEDULE FOR WATER POLLUTION ABATEMENT<br />
Secondaw BOD N&P TDS Thermal<br />
- Time<br />
Treatment Eff Eff Eff - Ef f<br />
1960<br />
1970<br />
X<br />
X X<br />
1980<br />
1990<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X X<br />
2000 X X X X X<br />
Criteria Fish KJlls Eutrophi- Reuse Recyc<strong>le</strong><br />
Water Treatueur. cation<br />
Prob<strong>le</strong>ms<br />
Figure 4 suggest a water pollution abatement time sca<strong>le</strong>; that is, <strong>the</strong><br />
standard will be upgraded with time, and <strong>the</strong> resource must be used<br />
within <strong>the</strong>se constraints.<br />
One is still concerned with <strong>the</strong> frequency with which data should be<br />
col<strong>le</strong>cted, <strong>the</strong> optimum locations of col<strong>le</strong>ction, <strong>the</strong> provisions for data<br />
storage and <strong>the</strong> resources for analysis of <strong>the</strong> data. The use of a shortterm<br />
survey approach or establishment of a minimal number of permanent<br />
stateions. An analysis of historical data will yield insight into those<br />
parameters which require continuous analysis because of significant fluctuations<br />
and help to identify those locations which best identify changing<br />
conditions in <strong>the</strong> receiving water.<br />
In contrast to <strong>the</strong> monitoring of a simg<strong>le</strong> point over a long period, studies<br />
can be concenrrated over shorter times but more intensive. There is a<br />
questi01 ,f manual col<strong>le</strong>ction versus continuous, automatic recording. All<br />
parameters of interest can be determined on a continuous basis and <strong>the</strong> results
transmitted to a central storage facility, whi<strong>le</strong> water quality parametere<br />
that can be economically and dependently measured in <strong>the</strong> field are still<br />
somewhat limited.<br />
CONCLUSIONS<br />
Briefly, models to illucidate design parameters should be built with<br />
availab<strong>le</strong> data in mind. By a process of separating and nesting, submodels<br />
can overcome inconsistencies. If goals are precisely stated as<br />
to function, various parameters can be represented by what ia availab<strong>le</strong>.<br />
The author has developed a series of models using very general data,<br />
<strong>le</strong>aving a latitude of alternative data.items to define a parameter. Data<br />
has a cost, col<strong>le</strong>ction and opportunity or decision errors also cost. If<br />
inadequacies continue, short-term intensive studies are justified, ei<strong>the</strong>r<br />
now or backward, for examp<strong>le</strong>, point reviews can be used. Manual systems<br />
can be replaced by automatic monitors; all eight suggested parameters<br />
hand<strong>le</strong>d by e<strong>le</strong>ctrodes. Generally speaking, however, automatic monitors<br />
tend to provide more data than are needed, because noone dares to turn<br />
<strong>the</strong>se expensive machines off or set <strong>the</strong> sampling interval to such a time<br />
interval that meaningful deviations can be recorded.<br />
One never has adequate data, nor can one afford to wait for it. So, models<br />
must be made using every device availab<strong>le</strong>, recognizing that <strong>the</strong> final<br />
result will still involve uncertainty and risks, and require judgement -<br />
<strong>the</strong> only defense against inadequate data.<br />
359
360<br />
SELECTED REFERENCES<br />
1. Biswas, Asit K. PROCEEDINGS, INTERNATIONAL SYMPOSIUM ON MODELLING<br />
TECHNIQUES IN WATER RESOURCES SYSTEMS. Volumes 1 and 2. Ottawa,<br />
Canada: Environment Canada, 1972.<br />
2. Herfindahl, Oris C. NATURAL RESOURCE INFORMATION FOR ECONOMIC<br />
DEVELOPMENT. Baltimore: John Hopkins Press, 1969.<br />
3. Krenkel, Peter A. (ed.). PROCEEDINGS OF THE SPECLAZTY CONFERENCE<br />
ON AUTOMATIC WATER QUALITY MANAÇEMENT IN EUROPE, No. 28.<br />
University, 1971.<br />
Vanderbilt<br />
4. Mancy, Khalil H. (ed.). INSTRUMENTAL ANALYSIS FOR WATER POLLUTION<br />
CONTROL. Ann Arbor, Mich.: Ann Arbor Science Publishers, Inc., 1971.<br />
5. Public Health Service. U. S. Department of Health, Education and<br />
Welfare. SYMPOSIUM ON ENVIRONMENTAL MEASUREMENTS, VALID DATA AND<br />
LOGICAL INTERPRETATION. Cincinnati, Ohio: Public Health Service, 1964.<br />
6. Public Health Service. U. S. Department of Health, Education and<br />
Welfare. SYPPOSILJM ON STREAMFLOW REGULATION FOR QUALITY CONTROL.<br />
Cincinnati, Ohio: Public Health Service, 1965.<br />
7. Thomas, William A. (ed.). INDICATORS OF ENVIRONMENTAL QUALITY.<br />
New York: P<strong>le</strong>num Press, 1972.
#<br />
PROBLEM<br />
FORMULATION<br />
DESIRED DATA<br />
V<br />
DATA<br />
COLLECTION<br />
ADEQUATE DATA<br />
I<br />
ANALYSIS<br />
1-SIMUUTION<br />
2-PROGRAMING<br />
- L<br />
V C<br />
r<br />
L<br />
DESIGN<br />
CRITERIA -<br />
Figure 1.<br />
c<br />
t<br />
361
362
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C<br />
.- N:<br />
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.- 8<br />
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o)<br />
UI<br />
.-<br />
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3<br />
0"<br />
*<br />
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2<br />
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o<br />
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0)<br />
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363
ABSTRACT<br />
DESIGNING PROJECTS FOR THE DEVELOPMENT OF GROUND WATER<br />
RESOURCES IN THE ALLUVIAL PLAINS OF NORTHERN INDIA ON<br />
THE BASIS OF INADEQUATE DATA,<br />
BY<br />
B. K, SABHERWAL<br />
Utilization of ground water potential to develop irrigated<br />
agriculture in <strong>the</strong> alluvial plains of Nor<strong>the</strong>rn India through<br />
"Push button" water wells has played a vital ro<strong>le</strong> to bring about<br />
<strong>the</strong> Green Revolution for meeting country's food deficit. But <strong>the</strong><br />
positive development on <strong>the</strong> food front is only a phase. Continuing<br />
population growth and <strong>the</strong> resultant increase in demand for food,<br />
fibre and o<strong>the</strong>r services obtaining from water use are adding to <strong>the</strong><br />
water requirements <strong>the</strong>reby underlining <strong>the</strong> urgency to hasten<br />
execution of projects capab<strong>le</strong> of delivering assured water supply to<br />
meet <strong>the</strong> demands of high yielding varieties (-HYV] crops, This can<br />
be achieved by installing more water wells in <strong>the</strong> alluvial plains<br />
of India rich in ground water potential. Ground water resource<br />
though it gets rep<strong>le</strong>nished annually, is not an inexhaustib<strong>le</strong> resource,<br />
Ecological responsibility makes it incumbent on <strong>the</strong> planners of<br />
ground water development projects that this precious resource, IS<br />
not exhausted due to over exploitation, Surface waters are tangib<strong>le</strong><br />
and <strong>the</strong>ir potential can be predicted upto reasonab<strong>le</strong> certainity on<br />
<strong>the</strong> basis of long term observations of flow in channels. Assessment<br />
of ground water potential on <strong>the</strong> o<strong>the</strong>r hand is quite complicated.<br />
The difficulty arises on account of <strong>the</strong> fact that ground water<br />
relates to that invisib<strong>le</strong> part of hydrologic cyc<strong>le</strong> which occurs<br />
beneath <strong>the</strong> land surface. Evaluation of ground water resource to a<br />
high degree of accuracy is a multi discipline study involving,<br />
col<strong>le</strong>ction, analysis and syn<strong>the</strong>sis of hydrological, geological,<br />
meteorological, geophysical, hydrochemical data, computing quantums<br />
of recharge, discharge and balance of ground water in a basin or a<br />
sub-basin and correlating <strong>the</strong> results with <strong>the</strong> changes in ground<br />
water <strong>le</strong>vels and its regime, A comprehensive study of this type is<br />
time consuming and costly, In view of <strong>the</strong> latest developments in<br />
ground water hydrology <strong>the</strong> availab<strong>le</strong> hydrological and geological<br />
data is not adequate enough for a comprehensive and precise<br />
assessment of ground water potential though exploitation of ground<br />
water in India commenced quite some time back. On <strong>the</strong> o<strong>the</strong>r hand<br />
preparation and execution of plans and schemes for <strong>the</strong> exploitation<br />
of ground water cannot be held over till <strong>the</strong> comp<strong>le</strong>tion of such a<br />
study which may take four to five years, ît has <strong>the</strong>refore become<br />
necessary to adopt some reasonably accurate methodology to evaluate<br />
<strong>the</strong> ground water potential with <strong>the</strong> help of <strong>the</strong> availab<strong>le</strong> data and<br />
plan ground water exploitation projects on its basis though at <strong>the</strong><br />
same time keeping margin for subsequent adjustments when better<br />
data becomes availab<strong>le</strong>, Appraisal techniques and adopted criteria<br />
for an approximate evaluation of ground water balance in water tab<strong>le</strong><br />
aquifers are described with particular reference to <strong>the</strong> Bist Doab<br />
Tract of <strong>the</strong> State of Punjab-India which has an area of 9000 sq.<br />
kilometers and where 80% of annual rainfall occurs in <strong>the</strong> months of<br />
July to September. Significant part of <strong>the</strong> assessment study is <strong>the</strong><br />
recharge to ground water from <strong>the</strong> annual flow of about 1.25 M.A.F.<br />
of; surface water thorough a net work of unlined and lined irrigation<br />
canals and its ultimate spillage in <strong>the</strong> cropped fields. On <strong>the</strong>
366<br />
discharge side is <strong>the</strong> drawal by approximately, 0.1 million existing<br />
shallow and deep water water wells which are ei<strong>the</strong>r e<strong>le</strong>ctrically or<br />
diesel driven. The e<strong>le</strong>ctrically driven wells have unmetered e<strong>le</strong>ctric<br />
supply, <strong>the</strong> tarrif being on <strong>the</strong> basis of horse power of <strong>the</strong> e<strong>le</strong>ctric<br />
motor. For both <strong>the</strong> tupes of water wells log <strong>book</strong>s recording <strong>the</strong><br />
number of hours a tubewell operates are not being maintained by <strong>the</strong><br />
private owners, This aspect fur<strong>the</strong>r adds to <strong>the</strong> prob<strong>le</strong>m of working<br />
out accurate drawals from and return seepage to ground water in<br />
tubewell irrigated fields, In <strong>the</strong> absence of adequate data to<br />
correctly evaluate ground water potential and pressing necessity to<br />
exploit <strong>the</strong> potential for food production statistìcal or empirical<br />
methods have been adopted to work out ground water balance and <strong>the</strong>n<br />
apply a reasonab<strong>le</strong> safety factor to take care of short comings in <strong>the</strong><br />
approach. In <strong>the</strong> project areas water tab<strong>le</strong> fluctuations are also<br />
being observed more frequently to closely watch <strong>the</strong> effect of<br />
additional draft,<br />
RESUME<br />
L'utilization du potentiel des eaux souterraines pour <strong>le</strong><br />
développement de 1"agriculture irriguées dans <strong>le</strong>s plaines alluvia<strong>le</strong>s<br />
de l'Inde du Nord au moyen des puits d'eau du button-préssoir a<br />
joué un rô<strong>le</strong> vital pour accomp<strong>le</strong>r la "Revolution Verte'' afin de<br />
satisfaire <strong>le</strong>s besoins deficitaires des aliments du pays. Mais <strong>le</strong><br />
développement positif sur <strong>le</strong> front de nourriture n'est qu'une phase.<br />
La continuation de la croissance de la population et l'augmentation<br />
resultante du besoin de nourriture, tissus, et des autres services<br />
utilizant l'eau necessitent <strong>le</strong>s besoins de l'eau supp<strong>le</strong>mentaires,<br />
ainsi soulignant l'urgence de l'ëxecution des projets capab<strong>le</strong>s de<br />
l'alimentation fourniture assuT+e d'eau pour subvenir la demande des<br />
récolte de haute z-eqdement. On peut satisfaire cette demande en<br />
installant plus de pults d'eau dans <strong>le</strong>s plaines alluvia<strong>le</strong>s de l'Inde<br />
du Nord, riches en potentiel des eaux souterraines. Des ressources<br />
des eaux souterraines quoiqu'el<strong>le</strong>s se remplissent chaque année, n'est<br />
pas une ressource ingpuissab<strong>le</strong>. La responsabilité écologique <strong>le</strong> rend<br />
obligatoire aux planificateurs des projets des eaux souterraines de<br />
voir que cette ressource prlcieuse ne seppuisse pas, en raison de<br />
sur-éxplóitation. Des eaux de surface sont tangib<strong>le</strong>s et on peut prédire<br />
<strong>le</strong>ur potentiel jusqu'une certitude raisonnab<strong>le</strong> sur la base des obser-<br />
vations à long terme de l'écoulament des eaux dans <strong>le</strong>s canaux. L'esti-<br />
mation du potentiel des eaux souterraines par contre est bien compli-<br />
quee. La difficulté s'8<strong>le</strong>'ve en raison du fait que l'eau souterraine<br />
se rapporte à cette partie invisib<strong>le</strong> du cyc<strong>le</strong> hydrologique qui se<br />
fait au-dessous de la surface de la terre. La nature héterogene des<br />
formations géologiques à travers <strong>le</strong>squel<strong>le</strong>s l'eau souterrasne circu<strong>le</strong><br />
rajoute à la complzxitê du problame. YaloTisation des ressources des<br />
eaux souterraines a une haute degr6 d'exactitude est une étude de dis-<br />
ciplines multip<strong>le</strong>s comprenant recuîl, analyse et syn<strong>the</strong>se des données<br />
hydrologiques, géologiques, méteorologiques, géophysiques et hydro-<br />
-chemiques, calculant <strong>le</strong>s quanta de récharge, d@scñarge et <strong>le</strong> bi'lan<br />
l'eau souterraine dans un bassin ou sous-bassin et mettant en corréla-<br />
tion <strong>le</strong>s résultats avec des changements dans <strong>le</strong>s niveaux d'eau soute-<br />
rraine et son régime. Une étude detaillée de cette type demandes plus<br />
de temps et est coûteause. En vue des plus derniers dêveloppments dans<br />
l'hydrologie de l'eau souterraine la données hydrologiques et géologi-<br />
ques disponib<strong>le</strong>s ne sont pas assez pour une estimation complJte et
367<br />
exacte du potentiel des eaux souterraines, bien que l'exploitation<br />
des eaux souterraines commence il y a quelque temps dans <strong>le</strong> passé.<br />
D'un autre cote, la préparation et l'dxecution des plans ou schemes<br />
pour l'exploitation des eaux souterraines ne peut pas &tre arrêtées<br />
jucqu'a la complétion d'une tel<strong>le</strong> étude quì puisse prendre, 4 ou 5<br />
ans.Donc, i1 est devenu nécessaire d'adopter une méthodologie<br />
raisonnab<strong>le</strong> exacte pour estimer <strong>le</strong> potentiel des eaux souterraines<br />
avec l'aide des donnêes dìsponsib<strong>le</strong> et planifier des projets<br />
d'exploitation des eaux souterraines, au même temps en retenant une<br />
marge pour <strong>le</strong>s modificatìons subséquentes quand p+us de données<br />
seront disponlb<strong>le</strong>s. Les technìques d'estìmatïon, pour une valorisa-<br />
tion approximative de balance d'eau souterraine decrite avec une<br />
réference particulière à BIST DOAB tracte Etat de Punjab en Inde qui<br />
a un terrain de 9000 kilometres-carres et ou 80% de pluie annuel<strong>le</strong><br />
arrive aux mois de Juil<strong>le</strong>t.Septembre. La partie signìficative d'6tude<br />
estimative concerne la récharge 2 l'eau souterraine de l'écou<strong>le</strong>ment<br />
annual d'environ 1.25 M.A.F. (million acre pieds) d'eau de surface<br />
par un réseau de canaux d'irrigation alignes et non alignes, et son<br />
utilization ultîme dans <strong>le</strong>s champs cultivés, A cbtk de déchargement<br />
1.0 million des existants puits d'eaux qui sont opérées soit par<br />
e<strong>le</strong>ctricit6 soit par essence. Des puits mechanizes par e<strong>le</strong>ctricit6<br />
assurent une alimentation d'eau sans compteur d'e<strong>le</strong>ctrictricite, <strong>le</strong><br />
tarrif étant basé sur <strong>le</strong> C.V. des moteurs eléctrlques, Pour <strong>le</strong>s deux<br />
types de puits d'eaux, des carnets à rég<strong>le</strong> concernant <strong>le</strong> nombre des<br />
heures qu'un puit S opére, ne sont pas tenus par <strong>le</strong>s propriétaires<br />
prives. Cet aspect ajoute encore au problème de calculs des puise-<br />
ments exacts de l'eau souterraine dans <strong>le</strong>s champs irrigués au moyen<br />
des puits à moteurs é<strong>le</strong>ctriques, Dans l'absence de données de valo-<br />
riser correctement <strong>le</strong> potentiel d'eau souterraine & la nécessité<br />
pressante d'exploiter <strong>le</strong> potentiel pour la production de nourriture<br />
<strong>le</strong>s methodes empiriques et de statistiques ont Btd adoptées pour<br />
retrouver la balance d'eau-souterraine et d'appliquer un facteur<br />
raisonab<strong>le</strong> de sÛréte de bein rendre compte des fautes dans la mainère<br />
d'aborder, Dans <strong>le</strong>s regions sous observation on étudie aussi tres<br />
souvent <strong>le</strong> niveau de variabilité d'eau pour remarquer de pres<br />
l'effect d'eau puisée en supp<strong>le</strong>ment,
368<br />
1. UTROWCTICN<br />
1.1 India i6 <strong>the</strong> seventh largeet country in tbe<br />
World. ït's area is 328 million hectarest 3-28 million bqtlue<br />
iiiïorn9ters) with a population of 547 million (1971)<br />
Agricultural out put accounts for half of thr country's Gross<br />
23: tioneï droductí GPW.<br />
1.2 In <strong>the</strong> year 1947 wheu <strong>the</strong> country was divided,<br />
<strong>the</strong> major irL-i;jation syskais únd %cod piav~ucLi? ?reas were<br />
lost to rtkistm resuïtiag ILI a deficit of 4 dillions tmms of food grains. India had tbrefcm to iwort ,ILL t;.c's.us<br />
from <strong>the</strong> major wheat producing countries of <strong>the</strong> world till<br />
<strong>the</strong> advent of Green devolution recently brought about by<br />
<strong>the</strong>'incrrased utilization of country(s surface water resources<br />
for irriyqtion from 93745 priïlion ~1 m (76 million acre ft.)<br />
in IL361 to 222000 milîion CU m ( 186 miliion acYe a.) at<br />
Grosent ?nd tof ground water lli000 Pilllion CU m (I30 million<br />
ecre ft.) üse í~f high yielding variety (W) seePS of<br />
cercnls hke wheat ,rice,wiize,Jawar and Bjra hes Ris0<br />
hastened to a great extent <strong>the</strong> tremendous increase III<br />
'food cut put. Cevelopmgnt Of rtwarf varieties of wheat made<br />
Possib<strong>le</strong> following <strong>the</strong> introduction of valuab<strong>le</strong> genetic<br />
material from bxtco 141 1962 has alone increased <strong>the</strong> production<br />
of thb important cereal from neerly 12 to 23 million tonnes<br />
within a period of about five yeam.<br />
1.3 Eilt <strong>the</strong> maximm production per unit of any<br />
Particular variety of m d seed is <strong>the</strong> result of a set of<br />
cultivation practices proper doses o9 ioputs prophylactic<br />
and curative measures to check <strong>the</strong> atta& OP insects, pests<br />
and disewes end above all adequate irrigation at proper time.<br />
2. INDIA-PH!EICAL AND OTHE=TI FJ3AlWRES.<br />
2.1 Physio raphially Indkais main land can be<br />
divided into six divisfons comprishg of i-<br />
i) <strong>the</strong> Himslayan mountains<br />
ii) <strong>the</strong> indo-Gangetio Plains<br />
iii) <strong>the</strong> Central Hiagi Unda<br />
<strong>the</strong> Decm Plateau<br />
<strong>the</strong> Eastern Coastal Belt<br />
vi) <strong>the</strong> Western Coastal Belt<br />
2.2 The Himla moimtains are of comparatively<br />
recent origin. The Deccan Eteau end <strong>the</strong> CentraR Hi@ Lande<br />
are composed of ancient rocks. The Plains are hilt up<br />
of layers of sends, clays of molo loally very recent &te.<br />
The metern anditestem Coastal befts comprise of deltaic<br />
and sedimentary marine deposits.<br />
2.3<br />
About 7of of <strong>the</strong> country's ama is under lain by<br />
hard rock with a thin soil cotrer at top derived fra l%o<br />
weal<strong>the</strong>rinn of rocks. IO 1i, mainly <strong>the</strong> Indo-Gan etic Phkis and<br />
<strong>the</strong> two deltaic Eastern and Westem Coastal dts which are<br />
made up of alluvial solls and sedfmentwy deposits varying in<br />
thickness from a few hundred feet ln <strong>the</strong> coastal belts to<br />
thousands of feet in <strong>the</strong> Plains.<br />
2.4 Ailuviai soils are suitab<strong>le</strong> for agricultum<br />
and respond well to artificial irrigation. Being generally<br />
permeab<strong>le</strong> in character and having laysrs of coarser deposits<br />
also provide under ground storage for seepage water. NO<br />
wonder <strong>the</strong> Indo-GangetLe Plains, 8nd <strong>the</strong> tu0 coestal belts<br />
though accomt for on1 l./3 of <strong>the</strong> oauitry'e laad rnam ?ut<br />
suwort atmut of tL caintrps pornlation.<br />
2.6<br />
The maor snow fed riveris of tu country naaiely<br />
<strong>the</strong> triaitaries of <strong>the</strong> U&s, <strong>the</strong> cianges and <strong>the</strong> Bwhaii Putra<br />
flow rlugyishly through <strong>the</strong> indo-Gangetic Blain. The main rivers
369<br />
flowing to <strong>the</strong> coastal belts are <strong>the</strong> Xarkda and <strong>the</strong> Tapti on<br />
<strong>the</strong> western side Cmd <strong>the</strong> Maha Nadi, <strong>the</strong> Godavari, <strong>the</strong> Krishana<br />
rind <strong>the</strong> Cavery on <strong>the</strong> eastern sir%. All <strong>the</strong>se rivers outfall<br />
into sea. The rfvers also provide irririation supklies to <strong>the</strong><br />
vast net work of m al systems part of which was constructed<br />
about a century back. host of <strong>the</strong> old canals are designed as<br />
Il mn of <strong>the</strong> riveril schemes and are unlined. The unlined cana.ls<br />
act es additicnal souY'ce of recharge to ground water besides<br />
seepage from rivers, streams and rainfall.<br />
2.6 In dia s clima. te ranges from con t in en ta. 1 to<br />
oceanic, from extrems of heat to extrerns of cold, Prom high<br />
a.ridity a.nd negligib<strong>le</strong> rainfall to excessive humidity and<br />
torrential rainfall. Sauth destemi monsoons in summer accounts<br />
for mre than 85% of <strong>the</strong> precipitation and that too in a<br />
short span of about 4 months. The great diversity in wea<strong>the</strong>r<br />
conditions and uncertainity of rainfall results in <strong>the</strong> preva<strong>le</strong>nce<br />
of draught condition in about one third of <strong>the</strong> country.<br />
3. GROW D :,! AT-g 2 17s<br />
3.1 In <strong>the</strong> face of variability and tirircliability of<br />
rsinfall and also lack of' adequate storage support for some Of<br />
<strong>the</strong> major canal irrigation schemes, tappin:? of zround water<br />
resource through iiells and tubewells for intensive agriciiltu re<br />
has pla ed a yitzl ro<strong>le</strong> in ushering <strong>the</strong> (Green Revolution<br />
particu Y a.rly in <strong>the</strong>se parts of <strong>the</strong> country where low or badly<br />
distributed rainfall is quickly lost thrm$l evaporation Ixit<br />
where g-ound watnr potential is availab<strong>le</strong> stored in alluvial<br />
deposits<br />
3. 3 Cultive.tion of high yielding varieties and<br />
intmsive cropping dernad water at <strong>the</strong> right time and of <strong>the</strong><br />
rewired quantity. These pre-requisits have made <strong>the</strong><br />
cultivators in areas with copious ground watcr supplies take<br />
to <strong>the</strong> instcllatïon of <strong>the</strong>ir own', push aittontt irrigation<br />
systems. The water scarcity during <strong>the</strong> yesr 1365-67 which<br />
created draught conditions almost all over <strong>the</strong> country acted<br />
as catalyist to boost up exploitation of ground water poteiitfal<br />
thmu& diesel or e<strong>le</strong>ctrically operated tubwells 100 feet<br />
to 200fbet dealfor <strong>the</strong> protection of Crops.<br />
3.3 The Govemrcent also rose to <strong>the</strong> occassion and<br />
undertook to provide large saa<strong>le</strong> loan finance to <strong>the</strong> cultivstors.<br />
for <strong>the</strong> installation of tubewells on <strong>the</strong>ir farms in areas where<br />
<strong>the</strong> ground watcr potentialities were promising. The result is<br />
that at present an investment of about Rs.2000 crores hac Filreacbr<br />
bean Lade in <strong>the</strong> field of ground water exploitation in <strong>the</strong><br />
country. Most of this investment has taken place in private<br />
sector.<br />
3.4 The following tab<strong>le</strong>, indicates <strong>the</strong> progress Of<br />
insxellation of tubwells in <strong>the</strong> a3untry:-<br />
(In thousands I<br />
250 - 1965 A969 - 1971 -.- (anticipated)<br />
Mo.of private tubewells 3 100 279 470<br />
rio.of diesel pumps 66 471 837 1150<br />
No.of e<strong>le</strong>ctric pump sets 19 513 1080 1620<br />
Total 88 1084 2196 3240
370<br />
3.5 The spectacular development of ground water<br />
utilization in <strong>the</strong> country has been influenced by a npmber of<br />
factors namely <strong>the</strong> rzcognition by farmers of <strong>the</strong> importmt ro<strong>le</strong><br />
played by ground water in sustaining modern agricultural<br />
techniques, incrceaed availability of institutional credit for<br />
financing <strong>the</strong> ground water exploitation programme, rapid<br />
e<strong>le</strong>ctrification Of rural areas, local avai<strong>le</strong>bility of technical<br />
how how to drill well8 with machines, and indiyenously<br />
manufactured pumps, motors and o<strong>the</strong>r equipmat for <strong>the</strong><br />
construction of wells and above all large sca<strong>le</strong> village road<br />
deve lopmen t p rogr amme .<br />
3.6<br />
Heavy investments in gound water exploitati.<br />
schemes and <strong>the</strong> involvement of <strong>the</strong> Government back institufional<br />
credit far <strong>the</strong> purpose has made it incunibent to plan and execute<br />
this programme of utmost natio'lal imoortance duly sumorted by<br />
proper assessment of ground water potential.<br />
4. HYDROLOCEIC CYCB.<br />
4.1 All <strong>the</strong> waters in existance en be located by<br />
what is ïmìwn as 1) hydrologic cyc<strong>le</strong>n or 11 Nater Cyc<strong>le</strong>". This<br />
C c<strong>le</strong> involves total earth system comprising of <strong>the</strong> atmosphere,<br />
d e hydro-sphere Snd <strong>the</strong> lithosphere. The activities of <strong>the</strong><br />
n ilater Cyc<strong>le</strong>" are vast extending from an average depth of about<br />
half a mi<strong>le</strong> in <strong>the</strong> lithosphere to abcut 10 mi<strong>le</strong>s in <strong>the</strong><br />
a tmsphere .<br />
4.2 Hydrologic cyc<strong>le</strong> is greatly influenced by <strong>the</strong><br />
geologic history of a particular area. If <strong>the</strong> geology consists<br />
of alluvial fmnatlons, water will occur in <strong>the</strong> openings<br />
between granular XcFosits; EUT; if <strong>the</strong> area fOr~tiOnS are rocky,<br />
<strong>the</strong> ground water Will be found in decomposed parts of ro&B,<br />
freotures or in tabular openings in soluab<strong>le</strong> rocks or opening<br />
in lava formed by flow or gas expansion during solidification.<br />
Guide lines to evaluate ground water potential in alluvial<br />
formatias have only been discussed in this p3Per.<br />
4.3 Ground water originates from surface water and<br />
gets renewed or recharged with <strong>the</strong> down vard percolation Of<br />
precipitation, flow in stream, cana<strong>le</strong>, return flow fra irri,ated fields etc. Propm assessment of this valuab<strong>le</strong><br />
resource fomd in Permeab<strong>le</strong> geQlOgac formations and in motim<br />
through <strong>the</strong> voids or pore spaces in an area requires working<br />
out its total storage and quantities whir& are annually pumped<br />
out or rep<strong>le</strong>nished into <strong>the</strong> ground water reservoir. bality Of<br />
grOUnd water .<strong>le</strong>o requires to be known. Comprehensive studies<br />
and explorati-ns -re necessary to evaluate <strong>the</strong> potential to a<br />
hfgh degree of accuracy.<br />
5. APPRbACH TO WORK OUT GRWdD WATER BULLANCE;<br />
ON TI% &.SIS OF INADECrJATE DATA.<br />
5.1 hthodology for <strong>the</strong> precise eva2natioB of ground<br />
vater potential is quite complicated. The difficulty arises 691<br />
account of <strong>the</strong> fact that ground water relates to <strong>the</strong>.t invisib<strong>le</strong>
371<br />
pert of hydrologic cyc<strong>le</strong> which occurs 'beneath <strong>the</strong> land surface.<br />
9etero~~eneoUs nature of <strong>the</strong> ,(.f?ological format ions through which<br />
ground water moves e-dds to <strong>the</strong> coarp<strong>le</strong>xity of <strong>the</strong> prob<strong>le</strong>m.<br />
5.2 It ha.s been observed by pump tests that in Punjab<br />
which is <strong>the</strong> Nor<strong>the</strong>rn-Western part of <strong>the</strong> Indo-Gangetic Plain<br />
alluvial materials constitute an extensive hetrogeneous and<br />
a.nisotropic unconfined aquifers . Discharge from tubwells as<br />
deep a$ 300' results in- draw-down of water tsó<strong>le</strong> over larye<br />
prea. ?nd is sustained by dawrltering of surface watr,r recharne,<br />
such condit iow jrevail through out <strong>the</strong> top aauffers ~f slluaiurn.<br />
5.3 Planning and designing of ground vmte? development<br />
through small and medium sized tubewells (1.10 to 200 feet<br />
deep) in <strong>the</strong> ground water &sins and sub-basins o0 <strong>the</strong> indo-<br />
Gmjetic plain ' cím <strong>the</strong>refore be ,compared to re:.err)ir prObLem.<br />
This approach cal-1s for drawing iipm <strong>the</strong> fresh water tab<strong>le</strong><br />
a,quifem upto <strong>the</strong> Safe Yield which should not. exceed ^che long<br />
term mean -annual supply or recherge involving wet and dry years.<br />
In view of lack of comp<strong>le</strong>te data <strong>the</strong> genersl. i"om CJf bhe squrtie.cn<br />
of hydrologic equilibrium in thcs project areas has been simplified<br />
cmd suitably adjusted to arrive at worh.b<strong>le</strong> ,g:rourid ;iater hiance.<br />
In areas having ground water quality prob<strong>le</strong>m Safe Yield cannot<br />
be equated to mean annual recharger<br />
5.4 Installation af tubewci-11s upto 300 feet for<br />
irriz8tion is being practised in India since 19.34-36. In<br />
edditìon, tubwells vere a.lso install.ed Por municlpal,rai%ays<br />
Tnd indust, ia1 use. Geological Survey of India, state Zrri%ticn<br />
&partmats and Central Gróund -.:a.ter Board have bom iminte.in1ng<br />
oeological, hydrological, geochemical and o<strong>the</strong>r ground water<br />
data of a rudimentary character. Irrigation Departnlsnts have<br />
also meintahed record of water tab<strong>le</strong> fluctuations keduced to<br />
mean sea <strong>le</strong>vel ( 1%.Lr) from a net work of observa.tion wells.<br />
kmccipitatScn, racord is kept by Indian bieteorological Department.<br />
Ifit no are?wlse systematic investi.zations and exploration to<br />
a-ssess ground water potential were conàucted. In <strong>the</strong> absence<br />
of adequate ùata to evaluate ground water potlontkal on <strong>the</strong> basis<br />
of lat$st deVelOPI~ent6 in grmd water hydrology and pressing<br />
necessity to exploit ground water potential statistica.1,<br />
analytical and empiriel wtbods were resorted to arrive at<br />
preliminary quantitative evaluatîon of ground water balances in<br />
<strong>the</strong> pro,je ct areas<br />
5.5 Ground water balance te Plan schemes was<br />
computed on <strong>the</strong> collp,ction and malysis of <strong>the</strong> following basic<br />
data in project areast-<br />
1. Village -wise iocatim5 and o<strong>the</strong>r details of existing<br />
tu heiael<strong>le</strong><br />
2. Col<strong>le</strong>etion of reliab<strong>le</strong> litholo- of tubewer-ils.<br />
3. Iso-pstch featums E@ revea<strong>le</strong>d by litho-lons and<br />
geologid correlation of strata upto <strong>the</strong> availab<strong>le</strong><br />
depths to broadly understand t.he geometry of aquifers.
372<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
il.<br />
10.<br />
11 0<br />
12.<br />
13.<br />
,Samp<strong>le</strong> observatiais of pumping rates o? existing tubewells<br />
by ushg simp<strong>le</strong> devices(0rifice or ri,qht Ang<strong>le</strong>d '6-notch) .<br />
dimp<strong>le</strong> surveys to assess pumping hours for surn'lier and<br />
winter crops to work out present ground water draft<br />
in <strong>the</strong> Gi'oj ect .-;.rea.<br />
Locatim of raingaue stations m d annual ra.infel1 datn.<br />
for <strong>the</strong> last 20 to 30 years. '<br />
::eighted mean average annual rainfall fill;ures for<br />
different blocks of <strong>the</strong> area by Theisson method.<br />
Locations of existing pugln?JdischPrp sites on stresms,<br />
drains and CO! <strong>le</strong>ctfon of run off data monthwise.<br />
Availab<strong>le</strong> ground water qiality data to demrcate fresh<br />
and inferior ground water zones.<br />
Location of existing cmel irriyation s,Ftem m d data<br />
about <strong>the</strong>ir <strong>le</strong>n$.hs ,. sectims,( lined/unlined) desi<strong>le</strong>d<br />
dischzcrges, actual flow time and areawise.<br />
Yeriod-wise flow in enals Rt <strong>the</strong> point OP entry into<br />
and exit from <strong>the</strong> project area.<br />
Locations of existkg water tab<strong>le</strong> observation wells<br />
and past data of tiatcr tab<strong>le</strong> fluctustbas for pre<br />
and post monsoon periods<br />
hater tab<strong>le</strong> depth data to delineate high water tab<strong>le</strong><br />
areas . Cropping pet tern, cropkning cslander and water<br />
14<br />
requirements for summer and winter crops.<br />
15 . :,orking out zvera- value of specific yield of <strong>the</strong><br />
formations, ei<strong>the</strong>r by pump tests or empirically.<br />
6. TJXH1vICk.L CRITl3RI.A FOF GROTD ?dUKE PLZ./L".JJCI:<br />
C0PîJT;RTIDN.<br />
Technical criteria adopted to work out ground<br />
water balance is given as under:-<br />
6.1 Rechars oroni R&infall<br />
Unconfined %quifers get recharwd from local<br />
rainfall. Based on <strong>the</strong> sQiil<strong>le</strong>s ccnducted in <strong>the</strong> Ganges &sin<br />
for period 1937-78 to 19Sû-51 a relationship vas evolved to<br />
WO& out net penetmtim of rain water to water teb<strong>le</strong> in<br />
alluvial areas<br />
2/5<br />
Rp t 2*O(R-15)'<br />
'Ihere R average annual rainfall Zn inches.<br />
Rp = annial rainfall penetration to water<br />
tab<strong>le</strong> in inches.<br />
ïhis relationship applies to areas having<br />
annual rainfall in excess of 15".<br />
6.2 beemge from Canals<br />
seepage loss values from unlined canals based on
373<br />
experimental data are given below. These figures are inclusive<br />
of evaporation losses which form only a small proportion:-<br />
i) In <strong>the</strong> :tete of Uttar Brzdech-ïndia it is about 8 Cs.<br />
(GU ft/:iec.jfor ordínary clay loam to about 16 cs.for<br />
sandy loam Per million 5;q.feet of wcttad parameter.<br />
i,verage king 10 CS ./k.qt .<br />
li> In <strong>the</strong> Wz1iarashtr.a State-India <strong>the</strong>se sire calculated at<br />
15 Cs./M sat. for discharse upto 250 CS. anil lO.&s!./M%ft<br />
for hiisher discliarzes.<br />
iïi) In FunJab and Haryana state -India =lue, of ..eepa
374<br />
alopes of <strong>the</strong> $round water tab<strong>le</strong> in <strong>the</strong> $lains <strong>the</strong> quantity of<br />
subterrainem flow g ~tthg $.h and out of project are- is<br />
negligib<strong>le</strong> 2s conipared to vertical recharge from rainfall,<br />
cmal PB~PU~L, return flow from irrigated fields etc. Henthis<br />
item has been omitted from computatiuns on both sides<br />
of hydraulic equation amialy due to' <strong>the</strong> non-availability of<br />
adequate data.<br />
6 r? Ground wqtrr loss due to non bene3tçkl<br />
etrapo-tranepiration in water logged areas.<br />
EvagotreEspiration losses ars related to depth<br />
of water tab<strong>le</strong> froin th@ ground surface 81.14 vegetatkq cover.<br />
Airing July to October period when recharge to grow-d water<br />
reservoir is maximum Qe to rainfall and high supplies in rivers<br />
and canals etc. water tab<strong>le</strong> rises towards ground surface. In<br />
<strong>the</strong> riverain md water Jogged tracts, <strong>the</strong> depth of Found water<br />
veries between zero to 6 feet below lanu surface. in 1963,<br />
'J.S. B.RI conducted experiments/ obervctiuns for salvaging<br />
ground water t6i?g evawqsteü from ground LI' transpired non<br />
beqeficially by vegetation In <strong>the</strong> central part of san-his Val<strong>le</strong>y,<br />
Ceil'tral Coloredo ( U.S+) t Graphs were plotted, correlating E.T.<br />
loss to ground water depth E.T.loss th negligib<strong>le</strong> if water tab<strong>le</strong><br />
is lowered to 12.5k<br />
Persistance of higher water tab<strong>le</strong> 5r water<br />
logged amas indicates that recharge to ground waster is equiva<strong>le</strong>nt<br />
to <strong>the</strong> E.T.loss or may be $ven morg. Pending detai<strong>le</strong>d studies,<br />
it would be reasonably pod planning to draft grouad water<br />
within <strong>the</strong> limits of water est9rnated to be lost through evapotranspirat<br />
ion.<br />
6 08 Draft froor <strong>the</strong> exhthg state (deep) and<br />
Arivate (shallow ) tubewells.<br />
The State tubewells ( 1.5 to 2r0 CS. cawcity)<br />
ere planaad to operate at 22 hours a day for 240 days in a y6arr<br />
Henœ dmal draft from state Tubewells varies frm 660 acre ft.<br />
to 880 acre ft. Shallow tubeiwslls (0.2 to 0.8 cs.cepacity) for<br />
abut 800 to i000 hours per year. Draft fra <strong>the</strong>se wells cm b<br />
taken as 15 acre ft, to acre e, per well per year. por wells<br />
driven by animal power dralpt Is taken W 5 acre ft. whi<strong>le</strong> for<br />
drinking sugply wells in villages <strong>the</strong> draft 1 acre ft. has been<br />
adopted per mimam.<br />
6s 9 SpaqIpg q$ shallo~ tubewells, '<br />
Closdl spaaed tubewells muse mutual hydlaulic<br />
interfpence due to ovQr fapphg 09 <strong>the</strong>ir ocrnes of depression.<br />
In ~ thi&y populated &we&9 of <strong>the</strong> IndolGangetic Plph@ Qr<br />
abry Basin ?mm ho&&hgs are very small < abcut 10 to 15 acres<br />
per head ) . The tuWells aro of 0.2 to 0.3 cubic feet/ $eco<br />
dfaeharge and 100'-190' deep. These wells do not run more than<br />
10 to 15% of <strong>the</strong> time in a year. After tests results minimm<br />
epacinp of such wells has been kept at abut 600 feet 6
375<br />
6.10 Grotmd WatEr Ealmce<br />
Computing <strong>the</strong> item of ,annual recharge and<br />
discharge as por criteria discussed above if a balance is<br />
struck 5 first estimation of <strong>the</strong> balance of T2cbyze potcntntlal<br />
in a Project crea beccmes laown for planning fur<strong>the</strong>r explcit-ticq.<br />
A reasotinb<strong>le</strong> factor of safety can be adopted to p<strong>le</strong>n exploitetim<br />
of <strong>the</strong> comnuted ground water ImiBncs which tcakes care of <strong>the</strong><br />
saps in <strong>the</strong> availab<strong>le</strong> data or <strong>the</strong> appraisal approech. The fa.ctor<br />
will depend on area conditicns.<br />
7. GHOTjND b~dkii BLLA.?JiCG IN B145T DCAB 'I'R.'ICT<br />
7.1 Bict Doab is a triangu1a.r part of <strong>the</strong> Punjab<br />
StF.te ( India ) enclosed <strong>the</strong> rivers Sut<strong>le</strong>j and <strong>the</strong> Beas on<br />
t?:c ciäes and aivalu hi1 9 s (lover iïj.ml8yan ranyes) on thc<br />
third side. Three districts of <strong>the</strong> state ncmely Jullundur ,<br />
doshiarmr and Kapurthala are located in th;c tract .<br />
7.2 The area of <strong>the</strong> trect is 9900 Fq. Kilometers<br />
mostly mugrising of alluvial plpin except <strong>the</strong> 8 mi<strong>le</strong>s wide<br />
belt of Shivalik Hills on <strong>the</strong> XGrth-eestorri ,?id&. depth of<br />
alluvium in <strong>the</strong> plain as revea<strong>le</strong>d by seismic surveys is thousends<br />
feet.<br />
7.3 Gmeral water tab<strong>le</strong> is abut 20 to c% feet in <strong>the</strong><br />
plains. Avcrzge ?.nnual rainfall in hilly region 1s 1200 mms<br />
tJhils in <strong>the</strong> plains it varies between 914 mm tc 635 mm. 80% of<br />
<strong>the</strong> rainfpll occurs during <strong>the</strong> monsoon period.<br />
7 04 The soils are ferti<strong>le</strong> and <strong>the</strong> ??SB 112s Copious<br />
gromnd water supplies. There are abcut 0.2 million irriqatiw<br />
tubwells clnà dugwells in <strong>the</strong> area. Quality OP grmnd water is<br />
qood for cultivation.<br />
7.5 The tract is also irri,gated tl-iraigh Bist mab<br />
eanal which draws its supplies fYom <strong>the</strong> barrage on tho river<br />
Lutkj at Rupar. Abut 1.25 M.A.F. of water is used annually<br />
for cultiva.i;ion. The net work of canal system Measures 7,54.28<br />
Lilcirieters out of which 34.5i) Km. is lined. mcthcr feature<br />
of ti<strong>le</strong>; area is ;iutiierous hilly utrea=( clioec) which descend<br />
from <strong>the</strong> :,hivz~lik hills and fiord only &i.ing monsoon period<br />
w ith flashy uischw .;es<br />
7.6 The recharge and discharge computntions ta<br />
work out <strong>the</strong> ground Kater bzlance in <strong>the</strong> m ea dis+;rictwise are<br />
talxilated 88 per stateuats i & 11. .A safety factcr 0.60 has<br />
been cùopted to <strong>the</strong> computed figures to arrive a.t <strong>the</strong> exploitab<strong>le</strong><br />
grourd water stential. The water ttib<strong>le</strong> f1uctuatir:ns end <strong>the</strong><br />
rainfall hi <strong>the</strong> area 8i-e being closely observed to watch <strong>the</strong><br />
strezses m d strains covered by ground water exb>lcitat*on on<br />
<strong>the</strong> shallow unconfined aquifers under water tab<strong>le</strong> conCAtions<br />
8. ca4crusCáu<br />
8.1, Demographic trends indicete that <strong>the</strong> Lndia's<br />
populatun is likely to inn,ease to 700 millions by <strong>the</strong> end of<br />
<strong>the</strong> present dea&. On <strong>the</strong> kcis of <strong>the</strong> piiojecte8 growth rate
376<br />
cf 1.45 per tbausand per anliuam during 1981-85 <strong>the</strong> population<br />
wmld rise to 300.millions in <strong>the</strong> year ZOO0 A.D. i.e. about<br />
S5$ increase over <strong>the</strong> 1371 population. Keeping in view <strong>the</strong><br />
expected improvement in <strong>the</strong> standard of living of <strong>the</strong> peop<strong>le</strong><br />
during <strong>the</strong> intervening period, <strong>the</strong> food and fibre requirements<br />
will increase by abwt 100% of 1371 production. Suck an<br />
enormous increase in <strong>the</strong> production is possib<strong>le</strong> through intensive<br />
agriculture and bringing additional areas under irrigation by<br />
optima utilization of <strong>the</strong> water resources (Surface and qround )<br />
8.2 Tnis will eventually result in intensified drawels<br />
of ground waters from shallow as well as deep aquifers. Ground<br />
water resource though it gets rep<strong>le</strong>nished annually, is not an<br />
inexhaustib<strong>le</strong> resource. Ecological respansi bility mkes it<br />
incumbent on <strong>the</strong> planners of ground water development projects<br />
that this precious resource: is not exhausted due to over<br />
exploitatbn arid is so utilized that it also remakis ava.ilab<strong>le</strong><br />
ïor <strong>the</strong> yencrstion to come. Therefore aremise potential of<br />
ground wa.ter anà its safe yield both from shallower and deep<br />
aquifers neads to te assessed as accurately as possib<strong>le</strong> to<br />
prepa1.e realistic exploitat ion plms and schemes. This aspect<br />
has ben duly recognized and separate state <strong>le</strong>vel orgpnizations<br />
cmpï-is ing of hydrologists, hydrometeorologist, geologist,<br />
agronomist, geoph scist and drilling engineers have been set up<br />
to carry out deta s <strong>le</strong>d investigationa. These detai<strong>le</strong>d studies<br />
will however take time.<br />
However to maIntaln <strong>the</strong> continuity of<br />
N grow more food 1) compaign exploitation of ground water<br />
recharge &lance may be planned on <strong>the</strong> %sis of Safe Yield worked<br />
out with <strong>the</strong> approximations and applicatmn of safety factors<br />
suited to each project area.<br />
1.<br />
mmmv CES<br />
Report of <strong>the</strong> Irrigation Commissian, 1972, Volume-I,<br />
Ministry of Irrigation and irower, New Delhi.<br />
2.<br />
3.<br />
Krishnm, M.S.,m Geology of India and Emma<br />
Tolm, C.J.,l) GroundWater " .<br />
4, Ehattacharya, R.P. 1) Ground Water supplice, dep<strong>le</strong>tion of<br />
water tab<strong>le</strong> and penetration of rain water to ground water<br />
tab<strong>le</strong> in Western Uttar Pradesh ( India )IIo<br />
5. U.S.G.S. Water-supply paper, 1608-G Anplycis of Aquîfcr<br />
Tests in <strong>the</strong> Punjab Region of West Pakistan<br />
6. tJ.s.';.S. :uater supkly papc'r, 1608-G '1 Ground Water<br />
Hydrciogy of <strong>the</strong> Punjab, West Pakistan 1~1th '=mphasiS<br />
of ?rcb<strong>le</strong>ms caused by Canal Irriggtion II .
7.<br />
8.<br />
9.<br />
10 o<br />
11 b"<br />
12<br />
33.<br />
14<br />
15 o.<br />
377
37 8<br />
ITEMS<br />
-<br />
ISWlhRpI<br />
-<br />
1.410<br />
.7ss<br />
b.30<br />
-<br />
D.310<br />
0.oy<br />
0.27<br />
0.06<br />
&IO<<br />
o .ai<br />
ODs'<br />
o .41<br />
o .Ia!<br />
STATEMENT I
A DRAFT<br />
CROSS DRAF T(1+2+:<br />
1 41<br />
II 9<br />
6SC<br />
7 S2<br />
96<br />
2s<br />
. Il<br />
379<br />
0.309<br />
o. 132
380<br />
S E\SM\C L \ NE’S<br />
RE F LE CTI ON<br />
REFRACTI ON<br />
h<br />
8-<br />
TEST WELL LOCATION 8<br />
CONTOUR INTERVAL 02KM<br />
DATUM M.s L<br />
hLLUV\kL PU\H5 O<br />
?LRYLbRY (SM\\uhL\KS)
MAP 15<br />
381
ABSTRACT<br />
IMPROVED TECHNIQUES FOR WATER RESOURCE SYSTEMS DESIGN<br />
J R SEXTON<br />
D G JAMIESON<br />
WATER RESOURCES BOARD, READING, ENGLAND<br />
Flow data inadequacy can take different forms. One extreme is<br />
<strong>the</strong> comp<strong>le</strong>te lack of any information but <strong>the</strong> more usual case is<br />
insufficient <strong>le</strong>ngth of record since very long sequences of flow<br />
data are required to evaluate <strong>the</strong> yield and reliability of water-<br />
-resource systems with confidence. Using traditional concepts of<br />
failure and reliability, all water-resource systems are being<br />
designed on inadequate data with only <strong>the</strong> degree of inadequay<br />
varying between schemes, The use of simulation as a design technique<br />
has necessitated a more rigorous definition of reliability which<br />
accepts <strong>the</strong> lack of data yet maintains a means of comparing <strong>the</strong><br />
reliability of different schemes both in terms of frequency and<br />
magnitude of failure, A new definition of reservoir reliability<br />
has been used for <strong>the</strong> hydrological design of <strong>the</strong> Wash Estuary<br />
Storage, a proposed series of pumped-storage reservoirs in south-east<br />
England.<br />
RESUMEN<br />
La insuficiencia de datos de flujo puede tomar formas distin-<br />
tas. Ocurre el caso extremo de la falta total de información, pero<br />
lo más usual es la duración insuficiente de registro puesto que se<br />
necesitan cantidades inordenadas de datos de flujo para que se eva-<br />
IÚen confianza la eficacia de sistemas de recursos hidráulicos. Em-<br />
p<strong>le</strong>ando conceptos tradiciona<strong>le</strong>s del fracaso y de la eficacia, todas<br />
las instalaciones de recursos de agua se han concebido con datos de<br />
flujo inadecuados, con grado de insuficiencia como sola variación<br />
entre ellas. El uso de simulacibn como modo de diseñar sistemas com-<br />
p<strong>le</strong>jos de recursos de agua exige definición más riguroso de eficacia<br />
que mientras acepta la falta de datos de flujo mantiene sin embargo<br />
un medio de comparar la eficacia de un proyeqto con otro y en térmi-<br />
nos de su frecuencia de ella y en grado de su fracaso. Un concepto<br />
de esos -la frecuencia de poTcentaje cumulativo- se ha emp<strong>le</strong>ado en<br />
el disefio hidrológico Ifel depósito del estuario del Washff, serie de<br />
depÖs*itos de reserva a bomba en el sudeste de Inglaterra,
384<br />
INTRODUCTION<br />
The analysis and study of water resource systems can be conveniently<br />
subdivided into three stages, planning, design and operational.<br />
Each stage has its own specific flow data requirements and what maJr be<br />
adequate for one stage could well be inadequate for ano<strong>the</strong>r.<br />
planning stage, a large number of possib<strong>le</strong> combin&t&ons of sources are<br />
evaluated but not in detail: <strong>the</strong> requirement for hydrological data is<br />
minbal, since <strong>the</strong> yields of individual sources need only be determined<br />
approximately. The most promising combinations of sources are subsequently<br />
examined in considerably more detail at <strong>the</strong> design stage.<br />
This stage is concerned with aspects such as frequency, probability and<br />
reliability all of which make considerab<strong>le</strong> demands in terms of data<br />
quantity and quality. The requirement is for long period of flow<br />
records which may have a time increment of a day or more.<br />
At <strong>the</strong><br />
In <strong>the</strong> oper-<br />
ational staze, <strong>the</strong> data requirement emphasis changes from long-term<br />
flow records to shorter but more detai<strong>le</strong>d flow records perhaps even on<br />
an hourly basis.<br />
This paper is concerned with <strong>the</strong> relationship between <strong>the</strong> assessment<br />
of reliability, <strong>the</strong> definition of failure and flow data inadequacy<br />
at <strong>the</strong> design stage. Flow data can be inadequate in many ways: it may<br />
be that <strong>the</strong>re is no data or Rot enough data, or <strong>the</strong> wrong data has been<br />
col<strong>le</strong>cted. Data can be of inadequate quality or have too coarse a time<br />
increment between successive values. To sunmiarise, inadequate data is<br />
an occupational hazard to all those involved in <strong>the</strong> hydrological design<br />
of water-resource systems. However, with traditional concepts of<br />
reliability and what constitutes a failure, <strong>the</strong> prob<strong>le</strong>m of flow data<br />
inadequacy will remain for a very long time.<br />
In <strong>the</strong> planning of water resources for England and Wa<strong>le</strong>s, many<br />
diverse types of sources such as pumped-storage reservoirs, multipurpose<br />
reservoirs, rivers, aquifers and estuarial storage are being<br />
considered. &ch proposed source is n? longer considered in isolation<br />
hut as part of a much larger water-resource system.<br />
stances <strong>the</strong> individual yield of <strong>the</strong> proposed source loses importance<br />
since it is <strong>the</strong> yield of <strong>the</strong> system as a <strong>who<strong>le</strong></strong> that requires evaluation.<br />
The increase in <strong>the</strong> sca<strong>le</strong> of <strong>the</strong> prob<strong>le</strong>m caused by consideration<br />
of a water-resource system as a <strong>who<strong>le</strong></strong> has outdated many of <strong>the</strong> traditional<br />
techniques for analysing <strong>the</strong> performance of a resernoir: some of<br />
<strong>the</strong> implicit assumptions have been made invalie by <strong>the</strong> comp<strong>le</strong>xity of .<br />
modern water-resource systems, o<strong>the</strong>r assumptions have never been valid.<br />
mHOD OF ANALYSIS<br />
In <strong>the</strong>se circum-<br />
owing to <strong>the</strong> comp<strong>le</strong>xity of <strong>the</strong> water-resource systems currently<br />
envisaged and <strong>the</strong> lack of <strong>the</strong>oretical techniques cspab<strong>le</strong> of analysing<br />
such systems, simulation is considered to be <strong>the</strong> only viab<strong>le</strong> method of<br />
analysis. A simulation model of a proposed water-resource system can be<br />
constructed by joining appropriate component models of particular types
385<br />
of reservoirs iri an ofder corresponding to <strong>the</strong> physical system.<br />
Examp<strong>le</strong>s are given in Fi$ures 1 and 2 of component models for a pumpedstorqe<br />
reservoir and a pumped aquifer. It should be appreciated that<br />
not all <strong>the</strong> links indicated in <strong>the</strong>se models need be included since in<br />
<strong>the</strong> specific application some can be set to zero.<br />
The simulation is structured in a general form with physical constraints<br />
such as <strong>the</strong> capacity of <strong>the</strong> reservoir, maximum pumping capacity,<br />
minimum residual flows in rivers etc treated as input vaxiab<strong>le</strong>s. The<br />
model can <strong>the</strong>n be used to find <strong>the</strong> frequency with which <strong>the</strong> system fails<br />
to meet <strong>the</strong> specified demands and <strong>the</strong> sensitivity to changes in any of<br />
<strong>the</strong>se or o<strong>the</strong>r input variab<strong>le</strong>s in terms of frequency of failing to meet<br />
specified iieiads. The relative importarice of each data ita! xc thus<br />
be determined and <strong>the</strong> effect of data inadequxy can be qusntirird i?i<br />
terms of confidence limits on <strong>the</strong> resulting reliab<strong>le</strong> yield. LÅoFeovcr,<br />
since <strong>the</strong> w a ~ in which a water-resource system is managed will dfect<br />
<strong>the</strong> reliability of <strong>the</strong> system, different operating ru<strong>le</strong>s can be compared<br />
and evaluated.<br />
Since <strong>the</strong> design of <strong>the</strong> system is concerned with rare events,<br />
large amounts of historic or syn<strong>the</strong>tic flow data have to be routed<br />
through <strong>the</strong> models. Consequently <strong>the</strong> component models have to be relatively<br />
simp<strong>le</strong> to keep camputing costs down and <strong>the</strong>refore <strong>the</strong>y are essentially<br />
accounting procedures with lags and attenuation built in.<br />
Given adequate data it is possib<strong>le</strong> to include both conservative and<br />
degradab<strong>le</strong> water quality parameters in <strong>the</strong> model. The build up of<br />
pollutants in various parts of <strong>the</strong> system can be monitored in <strong>the</strong> seme<br />
w a ~ as <strong>the</strong> quantity of water and <strong>the</strong> performance of <strong>the</strong> system can be<br />
depicted as histograms of both quantity and quality of water (Figure 3).<br />
In this WEIJ <strong>the</strong> interactions between water quality and quantity can be<br />
investigated.<br />
BPPLICBTION<br />
The techniques described [i) are being used in <strong>the</strong> hydrological<br />
evaluation of <strong>the</strong> Wash Storage, a pumped-reservoir scheme in <strong>the</strong> estusry<br />
of <strong>the</strong> Great Ouse, a river in south-east England (Figure 4). The preliminary<br />
estimate for <strong>the</strong> total capital cost of <strong>the</strong> scñeme is<br />
2140 O00 O00 at 1971 prices. The work outlined here forms a small part<br />
of <strong>the</strong> e2 900 O00 feasibility study though much of it will have application<br />
even if estuary storage is rejected. A schematic diagram of <strong>the</strong><br />
<strong>who<strong>le</strong></strong> system is given in Figue 5 with symbols defined in Tab<strong>le</strong> 1. The<br />
comp<strong>le</strong>xity of <strong>the</strong> comp<strong>le</strong>te system has necessitated <strong>the</strong> division into<br />
three interlinked subsystem namely, <strong>the</strong> Welland and Nene, <strong>the</strong> Great<br />
Ouse and <strong>the</strong> Wash Storage. The first two subsystems define <strong>the</strong> potential<br />
input to <strong>the</strong> third.
3 86<br />
The Welland and Nene subsystem which comprise8 <strong>the</strong> right hand portion<br />
of Figure 5 is a model of a pumped-storage reservoir, %pingham (now under<br />
construction), in conjunction with a confined aquifer, <strong>the</strong> Lincolnshire<br />
Limestone.<br />
Water will be pumped into lbpingham from both <strong>the</strong> River<br />
Welland and River Bene when <strong>the</strong> flows axe in excess of specified minimum<br />
values. Rnpingham can be used for a variety of purposes including meeting<br />
direct-supply requirements as well as regulating <strong>the</strong> lower Welland to<br />
enab<strong>le</strong> it to support downstream abstraction. Some of <strong>the</strong> water from<br />
hpinghm will be returned to <strong>the</strong> Nene aa effluent, upstream of <strong>the</strong><br />
intake pumps for Ehpingham.<br />
Water from Rnpinghem will also be used to<br />
maintain <strong>the</strong> flow in <strong>the</strong> River G<strong>le</strong>n. The Lincolnshire Limestone is used<br />
mainly for direct-supply in conjunction with abstractions from <strong>the</strong> Welland<br />
but any spillage from <strong>the</strong> aquifer helps to maintain <strong>the</strong> flow in <strong>the</strong> G<strong>le</strong>n.<br />
The possibility of artificially recharging <strong>the</strong> aquifer from <strong>the</strong> lower<br />
Velland has been included.<br />
The Great Ouse subsystem comprises <strong>the</strong> <strong>le</strong>ft hand and upper centre<br />
portions of Figure 5. The model is a simulation of an existing pumpedstorage<br />
reservoir, Grafham Water, in association with an unconfined<br />
aquifer, <strong>the</strong> Great Ouse Chalk. Grafham Water is rep<strong>le</strong>nished by pumping<br />
water from two points on <strong>the</strong> Bedford Ouse, a tributary of <strong>the</strong> Great Ouse.<br />
Agairi, <strong>the</strong>re is an e<strong>le</strong>ment of recirculation since some of <strong>the</strong> water<br />
supplied direct to a demand centre is returned as effluent upstream of<br />
<strong>the</strong> reservoir's intake pumps. The Great Ouse Chalk aquifer has been<br />
model<strong>le</strong>d as six interlinked unconfined aquifers. In a scheme shortly to<br />
be promoted all <strong>the</strong> sub-aquifers are to be used for direct-supply and<br />
river regulation. Obviously pumping water from an unconfined aquifer<br />
will affect <strong>the</strong> natural outflow from <strong>the</strong> aquifer to <strong>the</strong> tributary.<br />
Poreover, if <strong>the</strong> aquifer is drawn down, <strong>the</strong> possibility of seepage<br />
through <strong>the</strong> bed of <strong>the</strong> tributary exists. Both <strong>the</strong>se effects have been<br />
incorporated in <strong>the</strong> model.<br />
The lower centre portion of Figure 5 is a schematic representation<br />
of <strong>the</strong> proposed first two stages of <strong>the</strong> Wash Storage which comprises <strong>the</strong><br />
third subsystem. Water could be pumped from both <strong>the</strong> Great Ouse and <strong>the</strong><br />
lower Nene.<br />
The possibility of having sea-water recirculation schemes<br />
on both <strong>the</strong> Great Ouse and lower Nene has been included. This enab<strong>le</strong>s <strong>the</strong><br />
low-flow constraint at <strong>the</strong> tidal limit of each river to be zero.<br />
Sgn<strong>the</strong>tic flow data generation techniques [27 have been used for<br />
this invastigation. Currently <strong>the</strong> historic flow record on <strong>the</strong> River<br />
Nene has been used as <strong>the</strong> master series and all o<strong>the</strong>r subsidiary flow<br />
sequences have been obtained by regreseion on <strong>the</strong> logarithmic values of<br />
flow. hproved multisite daily data generation techniques are being<br />
developed under contract o] and will be used when availab<strong>le</strong>. Prior to<br />
being used as <strong>the</strong> master series, <strong>the</strong> Nene record was corrected for all<br />
upstream abstractions and effluent returns to obtain <strong>the</strong> 'natural' flow<br />
series.
INADEQUACY OF FLOiV DATA<br />
387<br />
A simulation model such as that used in <strong>the</strong> hydro1oglc.d design<br />
of <strong>the</strong> T.3h Stor2.p rcyui.:>e., a coneiderab<strong>le</strong> amount of information as<br />
input data. It is inevitab<strong>le</strong> that some of this data will be inade-<br />
quate in one form or ano<strong>the</strong>r. The usual case is where some inîorma-<br />
tion is availab<strong>le</strong> but in insufficient quantity to estimate input<br />
paxmeters reliably, m d for some parts of <strong>the</strong> system <strong>the</strong>re is a<br />
comp<strong>le</strong>te absence o€ data. To amplify <strong>the</strong>se prob<strong>le</strong>ms specific ex-<br />
amp<strong>le</strong>s which have been encountered in <strong>the</strong> bdrological desi,m of <strong>the</strong><br />
Wash Storage axe given toge<strong>the</strong>r with <strong>the</strong> way in which <strong>the</strong>y have been<br />
partly overcone.<br />
INADESUACY DUE TO HU DATA<br />
in modelling an unconfined aquifer such as <strong>the</strong> Great Ouse Chalk<br />
it is evident that when punping <strong>the</strong> aquifer for ei<strong>the</strong>r water-supply<br />
or river regulation, <strong>the</strong> natural outflow from <strong>the</strong> aquifer to <strong>the</strong><br />
river will decrease. However, pumping <strong>the</strong> aquifer will have no<br />
effect on <strong>the</strong> run-off from <strong>the</strong> non-aquifer portion of <strong>the</strong> catchment.<br />
It is <strong>the</strong> combination of <strong>the</strong>se two flow components that is measured<br />
by <strong>the</strong> downstream gauging station. In short, if <strong>the</strong> aquifer is to<br />
be developed by pumping, it is neaessaxy to have two inputs, <strong>the</strong><br />
recharge to <strong>the</strong> aquifer and <strong>the</strong> run-off from <strong>the</strong> remaining portion<br />
of <strong>the</strong> catchment when only one measurement of <strong>the</strong> combined effect is<br />
availab<strong>le</strong>. No details on <strong>the</strong> natural recharge of <strong>the</strong> aquifer were<br />
known.<br />
The aasumption was made that <strong>the</strong> downstream flow comprised two<br />
flow recimes, a slow response from <strong>the</strong> aquifer itself and a fast<br />
response from <strong>the</strong> remainder of <strong>the</strong> catchment. Having separated oyt<br />
<strong>the</strong> base flow component, <strong>the</strong> overall 'proportion of base flow to<br />
surface flow for <strong>the</strong> period of historic record was ascertained. The<br />
surface flow component alone was cross-correilated with <strong>the</strong> corresponding<br />
historic flow data for <strong>the</strong> master station on <strong>the</strong> River Nene. The<br />
cross-correlation was performed on <strong>the</strong> logarithmic flow values which<br />
gives weighting to <strong>the</strong> low flows and avoids <strong>the</strong> difficulty caused by<br />
zero flows. This'relationship was <strong>the</strong>n used to generate <strong>the</strong> surface<br />
flow component direct. The base flow component could not be treated<br />
in a similar manner since this was a measure of <strong>the</strong> output from <strong>the</strong><br />
aquifer ra<strong>the</strong>r than <strong>the</strong> input.<br />
It was assumed that <strong>the</strong> temporal distribution of <strong>the</strong> surface<br />
flow component was indicative of <strong>the</strong> periods when natural recharge<br />
occurred. Therefore <strong>the</strong> surface flow component was sca<strong>le</strong>d by <strong>the</strong><br />
overall ratio of base flow to surface flor and used as input to <strong>the</strong>
388<br />
recharge process. This data stream was attenuated by an exponential<br />
delay function to simulate porous-media flow prior to adding <strong>the</strong><br />
percolate to <strong>the</strong> water already in storage. The delay induced by this<br />
process was made equal to <strong>the</strong> observed mean delay between rainfall<br />
ind <strong>the</strong> resulting maximum well <strong>le</strong>vels.<br />
The aquifer above <strong>the</strong> threshold constraint defining when channel<br />
loss occurred, was model<strong>le</strong>d as a sing<strong>le</strong> linear storage. Consequently<br />
<strong>the</strong> natural outflow from <strong>the</strong> aquifer to <strong>the</strong> river is proportional to<br />
<strong>the</strong> mount of water in storage, <strong>the</strong> storage coefficient being derived<br />
from <strong>the</strong> base flow recession. In this W¿QJ <strong>the</strong> effect of pumping <strong>the</strong><br />
aquifer was to reduce <strong>the</strong> amount of water in storage <strong>the</strong>reby reducing<br />
<strong>the</strong> natural outflow from <strong>the</strong> aquifer without interfering with <strong>the</strong><br />
surf ace flow component.<br />
INADEQUACY DUE TO INJCOIJPLEZ'E MTA<br />
Although a historic flow record was availab<strong>le</strong> close to <strong>the</strong> proposed<br />
abstraction point on <strong>the</strong> Ely Ouse, it would have been of litt<strong>le</strong><br />
use for <strong>the</strong> hydrological design of <strong>the</strong> Wash Storage even if it had<br />
'been an accurate flow record. The river acts as a source of supply to<br />
both industrial and agricultural consumers as well ag a disposal<br />
system for treated effluents. No detai<strong>le</strong>d records have been kept of<br />
abstractions or returns and consequently <strong>the</strong> record can not be adjusted<br />
to obtain natural flows. Ideally it would have been fax simp<strong>le</strong>r to<br />
have used <strong>the</strong> natural flow record at this station and account for <strong>the</strong><br />
net changes as time progressed ra<strong>the</strong>r than to have to construct a<br />
simulation model of <strong>the</strong> entire river basin. In this specific case,<br />
however, development of <strong>the</strong> chalk aquifer necessitated a simulation of<br />
<strong>the</strong> entire basin. Fortunately better quality flow records existed on<br />
ail of <strong>the</strong> important tributaries which were all upstream of <strong>the</strong>.major<br />
industrial and agricultural demands.<br />
INADXQUACY DUX TO INSUFFICIENT DATA<br />
Traditionally <strong>the</strong> criterion for assessing <strong>the</strong> reliability of a<br />
reservoir system has been <strong>the</strong> mean recurrence interval between failures.<br />
This concept of return period has generally been defined quantitatively<br />
in one of two ways, namely, a once in T year event where T is typically<br />
50 or 100 yeam or in terms of probability where it is said that <strong>the</strong>re<br />
is a 100 per cent chance of failure occurring in any one year. Assum-<br />
T<br />
ing that reservoir failures axe rare events and that <strong>the</strong> time between<br />
failures has an exponential distribution, <strong>the</strong>se two definitions are
equiva<strong>le</strong>nt and <strong>the</strong> probability of m failures within n years is given<br />
by :<br />
-0<br />
phn> = e- Mrn<br />
n!<br />
consequently <strong>the</strong>re is a 37 per cent chance of <strong>the</strong>re not being a once<br />
in T year event in any T year period of record.<br />
Even in <strong>the</strong> recent past attempts have been made to isolate low<br />
flow events with return periods of 50 or 100 years from a short<br />
period of historic flow data. The usual <strong>le</strong>ngths of <strong>the</strong>se records<br />
typically range from 20 to 50 years. These <strong>le</strong>ngths of record axe<br />
totally inadequate for isolating such rare events and consequently<br />
very litt<strong>le</strong> codidence can be placed in <strong>the</strong> results obtained. For<br />
exmp<strong>le</strong>, to be 9% certain that an estimate of return period is<br />
within 2 10 years of a 50 yeas return period would require 2000<br />
years of data and to be 9@ certain that <strong>the</strong> estimate was within<br />
f 5 years would require no <strong>le</strong>ss than 11,000 years of data. bioreover,<br />
even to be 9% certain that <strong>the</strong> return period was in <strong>the</strong><br />
r,mge of 50 years to 100 years would require 1600 years of data.<br />
These data requirements show <strong>the</strong> absurdity of tho present reliability<br />
criterion. It infers that all water-resource systems are<br />
designed on inadequnte data with only <strong>the</strong> degree of inadequacy<br />
varying between schemes.<br />
Even if a once in T yeas low flow sequence could be isolated,<br />
<strong>the</strong>re is no guarantee that this would produce a once in T year<br />
failure rate in a reservoir system designed to withstand such an<br />
event. Shortkves in water supply are not independent events due<br />
to <strong>the</strong> effect of storage. If a reservoir has fai<strong>le</strong>d one year and<br />
has not recovered it is more likely to fail in <strong>the</strong> follovnng ye:=<br />
than if it had been full at <strong>the</strong> start of <strong>the</strong> year. Consequentlg<br />
reservoir failures come in groups ra<strong>the</strong>r than comp<strong>le</strong>tely random<br />
sequences and an event <strong>le</strong>ss severe than a once in T year flow<br />
sequence closely following on a similar loa-flow sequence could<br />
ceuse <strong>the</strong> system to fail. The occurrence pattern of <strong>the</strong>se extreme<br />
low-flow events is <strong>the</strong>refore as important as <strong>the</strong>ir severity and<br />
individual events should not be taken from <strong>the</strong> historic record and<br />
used in isolation when designing a reservoir system. Unfortunate-<br />
ly <strong>the</strong> historic flow record provides just one realisation of <strong>the</strong><br />
occurrence pattern at a given point and <strong>the</strong> probability of <strong>the</strong><br />
historic sequence being repeated in <strong>the</strong> future i3 infinitesimal.<br />
Consequently even if <strong>the</strong> <strong>who<strong>le</strong></strong> historic record were used and even<br />
if it contained what were considered to be extreme events <strong>the</strong>re is<br />
no guarantee that this would enab<strong>le</strong> a realistic prediction of <strong>the</strong><br />
reservoir's reliability to be made.<br />
389
390<br />
The difficulty of determining 'rare1 events from 'short' data<br />
cannot be overcome. Recently <strong>the</strong> use of syn<strong>the</strong>tic data generation has<br />
al<strong>le</strong>viated some of <strong>the</strong> prob<strong>le</strong>ms. The historic flow &ta is used to<br />
estimate <strong>the</strong> parent population by modelling statistical chaxacteristics<br />
and many syn<strong>the</strong>tic samp<strong>le</strong>s can be generated each of which is<br />
equally as likely to occur in <strong>the</strong> future as <strong>the</strong> historic record was to<br />
have occurred in <strong>the</strong> past. In this way vaxious occurrence patterns<br />
may be obtained and long perio&of syn<strong>the</strong>tic data can be ?%gzìxdd as<br />
producing a larger samp<strong>le</strong> from <strong>the</strong> infinite population of possib<strong>le</strong><br />
flows than <strong>the</strong> historic record affords. With <strong>the</strong> larger samp<strong>le</strong> <strong>the</strong>re<br />
is a. correspondingly increased chance of <strong>the</strong> record containing (rare1<br />
flow events providing a 'true' model has been used. However, <strong>the</strong><br />
syn<strong>the</strong>tic data can only be as representative of <strong>the</strong> parent population<br />
as <strong>the</strong> historic data. If an untypical historic record has been used<br />
or <strong>the</strong>re is insufficient data for <strong>the</strong> reliab<strong>le</strong> estimation of node1<br />
parameters <strong>the</strong>n litt<strong>le</strong> confidence can be placed on <strong>the</strong> generated<br />
sequences and in particulm on inferences about extremes within <strong>the</strong><br />
data.<br />
in looking for a suitab<strong>le</strong> design criterion we must accept <strong>the</strong><br />
lack of data and use a criterion that can be estimated with more<br />
confidence from <strong>the</strong> same &ta. Ra<strong>the</strong>r than defining failure as a<br />
reservoir or aquifer becoming empty, an event which would understandably<br />
be accepted only raxely, <strong>the</strong> introduction OP rationing of<br />
water supplies can be used as <strong>the</strong> definition of failure. This would<br />
occw when only a certain amount of water remained in store and would<br />
obviously be to<strong>le</strong>rated more frequently. In practice a reservoir<br />
would not be used at normal demand until it was empty. Instead a<br />
<strong>le</strong>vel of storage would be reached below,which <strong>the</strong> supply would be<br />
rationed. If rationing could be accepted, say, every twenty year3<br />
<strong>the</strong>n this would be a more frequent event and one has a correspondingly<br />
increased confidence in <strong>the</strong> design.<br />
Ano<strong>the</strong>r shortcoming of <strong>the</strong> return period criterion is that it<br />
gives no indication of <strong>the</strong> magnitude of <strong>the</strong> shortage. For examp<strong>le</strong><br />
in figure 6 <strong>the</strong> reservoir fai<strong>le</strong>d only once in <strong>the</strong> first case whereas<br />
in <strong>the</strong> second case it fai<strong>le</strong>d twice. Therefore although <strong>the</strong> first case<br />
is c<strong>le</strong>arly <strong>the</strong> more severe condition <strong>the</strong> concept of return period<br />
indicates <strong>the</strong> second is worse as it has twice as many shortages.<br />
This is only.to illustrate a point but in practice reservoir failures<br />
do group toge<strong>the</strong>r which poses <strong>the</strong> prob<strong>le</strong>m of deciding whe<strong>the</strong>r such<br />
a series should be consideyed as a sing<strong>le</strong> failure or a number of<br />
individual failures. Therefore return period is not ideally suited<br />
to describe <strong>the</strong> pattern in which reservoir shortages occur. An alternative<br />
criterion i~ required which must be a measure of both <strong>the</strong><br />
frequency and magnitude of failures. It must be f<strong>le</strong>xib<strong>le</strong> enough to<br />
allow for a variab<strong>le</strong> definition of failure (as <strong>the</strong> introduction of<br />
rationing is somewhat subjective) and it must be simp<strong>le</strong> to calculate.
3 91<br />
The concept of cumulative percentage frequency (CPF) of a<br />
.specified failure <strong>le</strong>vel being reached meets <strong>the</strong>se requirements.<br />
CPF measures <strong>the</strong> percentage of time that <strong>the</strong> reservoir is at or<br />
below a spec.ified storage. It will not however differentiate<br />
between sw one &y of failure every year or a one hundred day<br />
failure every hundred years. Fibwe 6 shoirs that <strong>the</strong> first case<br />
would have a CPF of and <strong>the</strong> second 100 (x + y1 which<br />
V V<br />
'correctly assigns <strong>the</strong> <strong>le</strong>ss severe shortage to <strong>the</strong> latter.<br />
The CPF value for any reservoir state can easily be obteined<br />
from <strong>the</strong> storage histopans already referred to (Figure 3). By<br />
rerunning <strong>the</strong> model with different demands a graph showing <strong>the</strong><br />
CPF of vmious storage <strong>le</strong>vels for different demands can be con-<br />
structed (Fibwe 7). Given a reservoir <strong>le</strong>vel at which rationing<br />
would be introduced <strong>the</strong> relationship between quantity of water<br />
and reliability can be obtained. In this way <strong>the</strong> effect of dif-<br />
ferent policies on reliability can be easily determined in toms<br />
of yield and <strong>the</strong> definition of failure does not need to be pre-<br />
judged.<br />
CONCLUSION<br />
Hydrological design criteria are based on rare events and<br />
<strong>the</strong>re will always be some degree of inadequacy in flow data.<br />
Syn<strong>the</strong>tic flow àata is only a partial solution because <strong>the</strong><br />
techniques are dependent upon <strong>the</strong> assumption that <strong>the</strong> historic<br />
samp<strong>le</strong> is representative of <strong>the</strong> infinite popultxtion of flows.<br />
Even <strong>the</strong>n, <strong>the</strong> historic data will only contain limited infoma-<br />
tion on long-term periodicities and persistencics which are<br />
important when examining rare events.'<br />
Where no flow information is availab<strong>le</strong> <strong>the</strong>re seems to be<br />
litt<strong>le</strong> alternative to improvisation. This inay take <strong>the</strong> form of<br />
transposition of data, scaling flow data or estiinating data<br />
indirectly as in <strong>the</strong> case illustrated. Any improvisation should<br />
always be treated with suspicion and attempts made to verify it<br />
if possib<strong>le</strong>, Failing this, simulation can be used, at a cost,<br />
to ascertain <strong>the</strong> sensitivity of <strong>the</strong> system's performance to this<br />
input. If <strong>the</strong> outcome is insensitive to that specific input <strong>the</strong>re<br />
is litt<strong>le</strong> cause for concern. If on <strong>the</strong> o<strong>the</strong>r hand <strong>the</strong> outcome is<br />
sensitive to that input,at <strong>le</strong>ast it shows where <strong>the</strong> àata col<strong>le</strong>ction<br />
effort should be concentrated.<br />
Ano<strong>the</strong>r way of improving <strong>the</strong> confidence in <strong>the</strong> prediction of<br />
reliability, given a limited amount of flow data, is to choose a<br />
better design criterion by changing <strong>the</strong> definition of failure.
392<br />
Hence <strong>the</strong> proposal is made that <strong>the</strong> introduction of rationing should<br />
be used as <strong>the</strong> definition of failure since this would be to<strong>le</strong>rated<br />
more frequently than <strong>the</strong> comp<strong>le</strong>te emptyiw of <strong>the</strong> reservoir. Nore-<br />
over, by changing <strong>the</strong> concept of reliability to one which is both a<br />
measuce of frequency and magnitude of failure ra<strong>the</strong>r than just <strong>the</strong><br />
frequency of failure enab<strong>le</strong>s two schemes to be compared objectively<br />
even :?hen based upon a small amount of flow data. Thus <strong>the</strong> cam-<br />
bination of syn<strong>the</strong>tic flow data generation, introduction Qf ration-<br />
ing au <strong>the</strong> definition of failure and cumulativa percentage frequency<br />
as a masure of reliability helps to overcorce <strong>the</strong> prob<strong>le</strong>m of<br />
inadequate flow data.<br />
ACiO-f?LEIlc~<br />
The authors thank <strong>the</strong>ir Director, Sir Norman Bowtree, for<br />
permission to publish this paper in which <strong>the</strong> views expressed are<br />
those of <strong>the</strong> authors and not necessarily those of <strong>the</strong> Hater Resources<br />
Board.<br />
1. Jamieson, D.G., Radford, P.J. and Sexton, J.R. (1973).<br />
The Hydrological design of water-resource systems.<br />
Water Resources Boosd. (To be published)<br />
2. Bloomer, R.J.G.B. and Sexton, J.R. (1972). The<br />
generation of syn<strong>the</strong>tic river flow data. Water Resouroes<br />
Boad publication No. 15.<br />
3. Weiss, G. (1973). Shot noise models for syn<strong>the</strong>tic<br />
generation of multisite àaily streamflow data. Symposium<br />
on Desi,v of Water Resouxces Project with Inadequate Data,<br />
Uadrid.
TâBLE 1<br />
LIST Q SYNBOLS ASSOCIATZD WITH FIGURE 5<br />
D hand centre<br />
E Effluent retumi<br />
R Naturd recharge<br />
AR Artificial recharge<br />
S Seepage or spill-<br />
I Natural inflow<br />
L "ranslational delay<br />
P Precipitation<br />
V Evaporation<br />
t b P<br />
tc- AtiPinimum-flow constraint<br />
3 93
394<br />
a<br />
II minimum flow constraint<br />
FLOW<br />
II minimum flow constraint<br />
A<br />
SECOND RIVER FLOW - -<br />
FIGURE 1 Component Model of a Pumped-Storage Reservoir<br />
FIGURE 2 Component Model of a Pumped Aquifer
3 95<br />
NUMBER OF DAYS RESERVOIR AT THAT STORAGE<br />
i<br />
W<br />
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397
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398
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3 99
MAXIMUM INFORMATION OBTAINABLE FROM INADEQUATE DESIGN DATA:<br />
FROM MULTïVARIATE TO BAYESIAN METHODS<br />
ABSTRACT<br />
Jean Weber1, Chester C.Kisie12 and Lucien Duckstein2<br />
An overniew is given of some <strong>the</strong>oretical and empirical issues<br />
involved in designing water pesource projects in <strong>the</strong> face of inade-<br />
quate data. The primary focus is on multivariate analysis of samp<strong>le</strong>s<br />
whose properties are not consistent with <strong>the</strong> assumptions of <strong>the</strong><br />
analysis, The multivariate models discussed include multip<strong>le</strong> linear<br />
regression, discriminant functions, canonical correlation, principal<br />
components, and factor and cluster analysis. Each of <strong>the</strong>se models<br />
is discussed in terms of its assumptions, data requirements and<br />
applications in hydrologic research. The Bayesian approach to para-<br />
meter estimation and decision making is introduced for <strong>the</strong> purpose<br />
of considering both <strong>the</strong> uncertainty due to inadequate data and<br />
economic losses.<br />
RESUME<br />
Les auteurs exposent queJques ccFsidérations généra<strong>le</strong>s, théo-<br />
riques et empiriques, sur 1'élaboratio.i des projets d'am€nagement<br />
des eaux quand on se trouve en prdsence de donnges insuffisantes.<br />
Ils mettent l'accent sur <strong>le</strong>s problèmes que pose l'analyse multiva-<br />
ride lorsque <strong>le</strong>s échantillons qui lui sont soumis ne répondent pas<br />
aux hypo<strong>the</strong>ses de base de cette analyse. Les modè<strong>le</strong>s multivariés<br />
dont il est question comprennent: <strong>le</strong>s régressio?s linéaires, l'ana-<br />
lyse discriminatoire (variab<strong>le</strong> dependante discrete), la corrélation<br />
canonique, <strong>le</strong>s composantes principa<strong>le</strong>s, l'analyse factoriel<strong>le</strong> et<br />
l'analyse groypde. Chacun de ces mode<strong>le</strong>s est examiné sous l'ang<strong>le</strong><br />
de ses hypo<strong>the</strong>ses de base, des données qu'exige sa mise en oeuvre<br />
et de ses applications en recherche hydrologique. L'approche bayé-<br />
sienne de liestirnation das paramètres et de la décìsion, permet<br />
d'introduire a la fois l'incertitude due à l'insuffisance des données<br />
et ses conséquences économiques.<br />
lprofessor, Department of Management, University of Arizona, Tucson,<br />
Arizona 85721.<br />
2Professors, Department of Systems and Industrial Engineering and<br />
Department of Hydrology and Water Resources, University of Arizona,<br />
Tucson, Arizona 85721.
402<br />
1 .O Introduction-<br />
This DaDer considers L.e prob<strong>le</strong>m o .-recastinq o hvdroloaic variab<strong>le</strong>s for<br />
water resoke projects when <strong>the</strong> data-are inadequati, thät is, when <strong>the</strong>re is a<br />
mismatch between data and model. This mismatch is considered in terms of multivariate<br />
methods of data analysis. Mismatch implies a discrepancy between model<br />
structure and structure suggested by <strong>the</strong> data and/or data inadequacy in relation<br />
to model requirements. Several types of data inadequacies are considered in <strong>the</strong><br />
context of models frequently used in hydrologic research. The discussion is from<br />
two related points of view; it considers limitations of a model in terms of <strong>the</strong><br />
assumptions on which it is based and sensitivity of <strong>the</strong> predictions of a model to<br />
data inadequacies of various types. These considerations are inextricably related<br />
since <strong>the</strong> more restrictive <strong>the</strong> assumptions of a model are, <strong>the</strong> more likely<br />
it is that data obtained are inadequate for estimating <strong>the</strong> parameters of <strong>the</strong> model.<br />
Uncertain input information for <strong>the</strong> design of water resource systems is <strong>the</strong><br />
result of <strong>the</strong> inability of hydrologists to model large basins in substantial detail<br />
as projected by Freeze (1972) and a result of <strong>the</strong> "curse" of small samp<strong>le</strong>s in<br />
developing space-time series models and probability density models of flow, precipitation,<br />
temperature and evapotranspiration. Prob<strong>le</strong>ms of extending data at a<br />
design site and to ungaged sites are of long standing concern.<br />
<strong>the</strong> implications of assumptions in mu1 tivariate statistical methods applied to<br />
<strong>the</strong>se prob<strong>le</strong>ms is important to subsequent steps of coping with <strong>the</strong> consequent<br />
assumptions and offering alternatives and decision strategies.<br />
1.1 Model Building and Its Assumptions<br />
An awareness-of<br />
When data such as streamflow are obtained, it is almost always for <strong>the</strong> ulti-<br />
mate purpose of designing or operating a structure (bridge opening, dam, drainage<br />
structure); one intermediate step consists of predicting or forecasting future<br />
events (floods or droughts) using a model. The sequence of events in accumulation<br />
of know<strong>le</strong>dge for predictions can be characterized as follows: some know<strong>le</strong>dge is<br />
obtained by observations, a preliminary <strong>the</strong>ory or hypo<strong>the</strong>sis (for examp<strong>le</strong>, log<br />
normal probability density function (pdf) of flow) is formulated on <strong>the</strong> basis of<br />
<strong>the</strong>se observations, additional data are obtained perhaps more systematically, <strong>the</strong><br />
<strong>the</strong>ory or hy o<strong>the</strong>sis is revised and/or refined (for examp<strong>le</strong>, log Pearson type III<br />
pdf of f<strong>le</strong>ws!, additional data are obtained, and so forth. As this interaction<br />
between <strong>the</strong>ory and data proceeds, <strong>the</strong> <strong>the</strong>ory becomes more reproducib<strong>le</strong> and per-<br />
haps <strong>le</strong>ss general and <strong>the</strong> data required for its verification or modification also<br />
become increasingly accurate, so that <strong>the</strong> design process may be started without<br />
having to use large safety factors to compensate for uncertainty.<br />
At some point, after accumulation of sufficient supporting data, a <strong>the</strong>ory or<br />
hypo<strong>the</strong>sis is generally accepted and, un<strong>le</strong>ss subsequent <strong>the</strong>ory and/or obser-<br />
vations strongly indicate o<strong>the</strong>rwise, <strong>the</strong> <strong>the</strong>ory is used for prediction of a design<br />
quantity such as <strong>the</strong> 50-year flood Q(50). By this time <strong>the</strong> <strong>the</strong>ory is frequently<br />
referred to as a model. As a <strong>the</strong>ory becomes generally accepted, even tentatively,<br />
<strong>the</strong> purpose of obtaining data gradually shifts; data are used <strong>le</strong>ss as a basis for<br />
reformulating <strong>the</strong>ory and more as a basis for estimating <strong>the</strong> parameters of a model<br />
whose form has been determined, at <strong>le</strong>ast in most respects. Unfortunately. it is<br />
frequently tempting to accept a <strong>the</strong>ory and corresponding model prematuwly,
especially if <strong>the</strong> urgency of making predictions or forecasts is compelling (for<br />
examp<strong>le</strong>, in a decision to be made at once on <strong>the</strong> building of flood control works,<br />
a water supply reservoir, or hydroe<strong>le</strong>ctric power dam).<br />
Premature acceptance of a model can have very serious consequences, particularly<br />
since <strong>the</strong> model is likely to be idealized to <strong>the</strong> point of being unrealistic<br />
or to hold only under very restricted conditions, such as time invariance of a<br />
watershed (Foge1 et al., 1971). Any model is an oversimplification of reality.<br />
This is inevitab<strong>le</strong>, because <strong>the</strong> purpose of <strong>the</strong>ory is to simplify reality, which<br />
is enormously complicated, by abstracting from it those e<strong>le</strong>ments that explain a<br />
large proportion of <strong>the</strong> observed phenomena. ' Watershed models certainly are in<br />
this cateogry. Acceptance of a model thus involves a compromise between realism<br />
(e.g., a distributed model involving all details of <strong>the</strong> water cyc<strong>le</strong>) and simplicity,<br />
represented by a lumped model. In economics and behavioral science, <strong>the</strong><br />
predictions of a <strong>the</strong>ory or model are said to be appropriate, ceteris paribus,<br />
that is, o<strong>the</strong>r things being equal (not varying). Under control<strong>le</strong>d laboratory<br />
conditions extraneous variation can be minimized; in <strong>the</strong> real world, it generally<br />
cannot. Thus , in using a rainfall-runoff model for forecasting streamflow,<br />
for examp<strong>le</strong>, it is important to know how robust <strong>the</strong> model is to its assumptions,<br />
including <strong>the</strong> ceteris paribus assumption which, for examp<strong>le</strong>, precludes urbani- .<br />
zation. That is, it is important to know whe<strong>the</strong>r minor perturbations in <strong>the</strong> conditions<br />
under which a model is applied have relatively small or relatively large<br />
effects on its predictions. C<strong>le</strong>arly, if <strong>the</strong> predictions of a watershed model are<br />
sensitive to changes in a variab<strong>le</strong>, this variab<strong>le</strong> should be included in <strong>the</strong> model.<br />
Unfortunately, a model developed for one set of conditions is frequently used<br />
under quite different conditions without consideration of <strong>the</strong> inadequacies of <strong>the</strong><br />
data obtained under those conditions. A linear rainfall-runoff model that has<br />
been shown to yield a correct design of a culvert draining 50 km2 cannot be extrapolated<br />
to a 500 km2 watershed. Here <strong>the</strong> predictions of <strong>the</strong> model may be quite<br />
erroneous.<br />
One of <strong>the</strong> major difficulties in choosing and using models for decisionmaking<br />
in hydrology or o<strong>the</strong>r engineering design is <strong>the</strong> mismatch between availab<strong>le</strong><br />
models and availab<strong>le</strong> data. The implications of <strong>the</strong> mismatch are not c<strong>le</strong>arly<br />
understood. Most of <strong>the</strong> analytic derivations of <strong>the</strong> properties of deterministic<br />
and statistical models (for examp<strong>le</strong>, sampling distributions of estimators , tests<br />
of significance, standard errors of estimates and predictions, and so forth) are<br />
based on assumptions that are almost always violated in applications. These<br />
assumptions include linearity in <strong>the</strong> parameters of <strong>the</strong> system and of estimation<br />
equations, normality of population distributions, independent random sampling,<br />
large samp<strong>le</strong>s (for applicability of asymptotic results) and a variety of assumptions<br />
concerning <strong>the</strong> covariance or correlation structure of multivariate observations<br />
and/or <strong>the</strong>ir errors. In most applications at <strong>le</strong>ast one of <strong>the</strong> assumptions<br />
of <strong>the</strong> model is violated; <strong>the</strong> re<strong>le</strong>vant concern should thus not be with<br />
<strong>the</strong> properties of <strong>the</strong> model when its assumptions are satisfied, but should be<br />
with <strong>the</strong> properties of <strong>the</strong> model when its assumptions are violated in various ways<br />
and to various extents. The research in this area is not nearly sufficient to<br />
provide practical guidelines (Dhrymes et al., 1972).<br />
Specific assumptions and <strong>the</strong>ir violations are discussed in <strong>the</strong> next sections<br />
for several models. In addition to inadequacy with respect to <strong>the</strong>se assumptions<br />
data may be inadequate in several more general respects including samp<strong>le</strong> size,<br />
missing observations, measurement errors , secondary variab<strong>le</strong>s.<br />
403
404<br />
These types of data inadequacy occur, for examp<strong>le</strong>, in <strong>the</strong> method of regionalization<br />
used by <strong>the</strong> U.S. Geological Survey (USGS). Assume that a stream flw characteristic, say <strong>the</strong> 50-year flood Q(50), is needed at an ungaged site for<br />
design purposes. Regionalization may be used to calculate Q(50) in those cases<br />
for which a data col<strong>le</strong>ction network for <strong>the</strong> region has been in operation for a<br />
certain time. One method of regionalization, used by <strong>the</strong> USGS, relies on re-<br />
gression analysis (Thomas and Benson, 1970).<br />
characteristic Q(50) is regressed upon basin characteristics, such as basin area,<br />
precipitation, channel slope, e<strong>le</strong>vation, forest cover and soil index. Then,<br />
given <strong>the</strong> basin characteristics of <strong>the</strong> ungaged design site, <strong>the</strong> regression equat-<br />
ion is used to predict Q(50) for that site.<br />
For most regional data col<strong>le</strong>ction networks, many sites have record <strong>le</strong>ngth of<br />
<strong>the</strong> order of 20 years or <strong>le</strong>ss; small samp<strong>le</strong> bias thus is quite substantial, so<br />
that distributions o<strong>the</strong>r than <strong>the</strong> normal distributions should be used to compute<br />
<strong>the</strong> logarithm of <strong>the</strong> flow Q(50); see Met<strong>le</strong>r (1972). The basic reason for using a<br />
regional regression is that data are missing in space at <strong>the</strong> design location and<br />
large sca<strong>le</strong> physically based models of combinations of river basins are non-exis-<br />
tent to help in augmenting <strong>the</strong> data. Although calibration curves of flow veysus<br />
gage height are periodically recalculated, <strong>the</strong>re are difficulties associated with<br />
sediment flows and with recording of flow data. All <strong>the</strong> variab<strong>le</strong>s entering <strong>the</strong><br />
regression equation constitute by definition secondary data (primary data is <strong>the</strong><br />
flow itself).<br />
2.0 Multivariate Models and Data Inadequacies<br />
The follwing sections concern multivariate models frequently used in<br />
hydrologic research and focus on model limi tations arising from inadequate data.<br />
The multivariate models discussed include multip<strong>le</strong> regression, discriminant<br />
functions, canonical correlation, principal components ,:and' factor and cluster<br />
analyses. Each of <strong>the</strong>se multivariate models, with its assumptions, data require-<br />
ments and applications, is discussed in <strong>the</strong> following sections. Primary refere-<br />
nces on <strong>the</strong>se models are Anderson (1958),'Christ, C1966), Dem ster (1969) Dhr mes<br />
(1970 and 1972), Harmon (1967), Johnston (19631, benta (1971!, Morrison (19671,<br />
Press (1972), Tryon (1970), Zellner (1971).<br />
2.1 Multivariate Linear Regression<br />
First, <strong>the</strong> desired streamflow<br />
The purpose of multivariate linear regression analysis is to obtain an<br />
equation -%-for predicting <strong>the</strong> value of a dependent variab<strong>le</strong> 1 as a linear<br />
function of a vector of k independent variab<strong>le</strong>s &=[Xi ,. . , Xk]. The criterion<br />
for obtaining <strong>the</strong> vector k=[bi, . , &] is that <strong>the</strong> sum of squared errors<br />
(y-@) - '(Y-Xi) be minimized.<br />
Linëarregression and analysis of variance, which can be viewed as a special<br />
case of linear regression, are <strong>the</strong> only multivariate models for which <strong>the</strong>re has<br />
been considerab<strong>le</strong> investigation of <strong>the</strong> effects of violation of <strong>the</strong> assumptions.<br />
The assumptions. of <strong>the</strong> multivariate linear regression model ~=XB+E can be stated<br />
as follows: E sN(0,~) where c=$I is <strong>the</strong> covariance matrix of<strong>the</strong> multinomial<br />
(N) population, & Ts non-stocFastTc, and has rank k
405<br />
The number of observations is n, so 1 is n x 1, X is n x k, is k x 1 and E is<br />
n x 1. Thus <strong>the</strong> ässumptions are linearity, normality of errors, serial indëpen-<br />
dence of errors, nonstochastic independent variab<strong>le</strong>s, and an observation matrix<br />
of full rank with <strong>the</strong> nunber of independent variab<strong>le</strong>s <strong>le</strong>ss than <strong>the</strong> number of<br />
observations.<br />
The assumptions required for estimating a model are generally much weaker<br />
than <strong>the</strong> assumptions required for inferences concerning <strong>the</strong> estimates. In<br />
particular, normality assumptions are usually required for inference but not<br />
for estimation (say of <strong>the</strong> 6's). This is <strong>the</strong> case for multivariate regression.<br />
Since estimates are of litt<strong>le</strong> use without know<strong>le</strong>dge of <strong>the</strong>ir distributional<br />
properties, <strong>the</strong> assumptions stated for mu1 tivariate linear regression and for<br />
models discussed subsequently are those required for standard tests of significance<br />
and confidence intervals.<br />
The effects of violating <strong>the</strong> various assumptions of multivariate linear regression<br />
are summarized in Tab<strong>le</strong> 1; also included in <strong>the</strong> tab<strong>le</strong> are procedures<br />
for detecting violations of assumptions and proposed al ternatives for remedying<br />
detected violations.<br />
2.2 Canoni cal Corre1 ati on<br />
The purpose of canonical correlation analysis is to study linear relationships<br />
between two sets of variab<strong>le</strong>s l'=[Yi ,. . ,Y,] and L'=[XI ,. . ,Xq]. Peck<br />
(1972) uses <strong>the</strong> analysis to determine whe<strong>the</strong>r 12 meteorological parameters (like<br />
vorticity, vertical velocity, wind speed at different e<strong>le</strong>vations and from various<br />
directions, temperature differences, etc.) are sufficient to predict variations<br />
in orographic winter precipitation patterns without <strong>the</strong> need for storm typing.<br />
Nimnannit (1969) uses <strong>the</strong> technique to relate spring runoff at a set of stations<br />
in <strong>the</strong> target region (where clouds are seeded) to runoff at a set of stations in<br />
<strong>the</strong> control region (no seeding); <strong>the</strong> urpose is to assess <strong>the</strong> effectiveness of<br />
wea<strong>the</strong>r modification. Torranin (19725 investigates <strong>the</strong> potential of <strong>the</strong> method<br />
for (1) forecast of monthly precipitation of three large areas of <strong>the</strong> U.S. west<br />
coast and (2) forecast of seasonal snowmelt ruiroff for three gaging stations in<br />
<strong>the</strong> Fla<strong>the</strong>ad River Basin in Montana. The applications in hydrology have been<br />
very few in nunber.<br />
The analysis obtains vectors and such that <strong>the</strong> correlation between &'Yand<br />
b'& given by<br />
a'YX'b<br />
. ---<br />
r =<br />
Ja'Y 'Y a b 'X 'Xb<br />
------<br />
is maximized, subject to <strong>the</strong> normalizing conditions C'L'Y~ = 1 = d'&'Xb.<br />
Subsequent vectors are obtained such that for each successive vector %e canonical<br />
correlation is maximized subject to normalizing conditions and <strong>the</strong> condition of<br />
independence with respect to previous vectors.<br />
The number of canonical correlations between r=[Yl,. . .,Y,] and X'=[Xi,. . ,Xq]<br />
is min !p.q), although in practice usually only <strong>the</strong> first few canonic3 correlates<br />
are of interest. For <strong>the</strong> special case when ei<strong>the</strong>r or is scalar, that is, consists<br />
of only one e<strong>le</strong>ment, canonical correlation is equiva<strong>le</strong>nt to multip<strong>le</strong> correlation.<br />
Except in this special case, canonical eorretatim analysis is not useful
406<br />
for prediction but is of value only to aid in formulating a modez; this is in<br />
contrast to hydrologic uses mentioned above.<br />
Under <strong>the</strong> assumption that Y' and X' are jointly normally distributed, <strong>the</strong><br />
joint significance of sets of cänonicar correlations can be tested using a likelihood<br />
ratio statistic. Unfortunately, <strong>the</strong> exact distribution of this statistic<br />
is complicated. An approximate large samp<strong>le</strong> distribution has been obtained, but<br />
its convergence properties have not been studied. Thus , inferences concerning<br />
canonical correlations can appropriately be made only on <strong>the</strong> basis of large<br />
samp<strong>le</strong>s from a multivariate normal population. No information is availab<strong>le</strong> concerning<br />
<strong>the</strong> nature and extent of <strong>the</strong> effects of violations of <strong>the</strong> assumption of<br />
normality on <strong>the</strong> distribution of canonical dorrelations. Canonical correlation<br />
analysis thus appears of limited use for building models for eventual use in<br />
design with limited or inadequate data, in contrast with <strong>the</strong> hydroiogic uses<br />
mentioned above.<br />
2.3 - Discriminant Analysis<br />
The purpose of discriminant analysis differs from <strong>the</strong> purpose of multivariate<br />
linear regression analysis only with respect to <strong>the</strong> type of prediction<br />
required for <strong>the</strong> dependent variab<strong>le</strong>; in regression analysis <strong>the</strong> dependent variab<strong>le</strong><br />
is continuous and its value is to be predicted, whi<strong>le</strong> in discriminant analysis<br />
<strong>the</strong> dependent variab<strong>le</strong> is discrete and its classification is to be predicted, for<br />
examp<strong>le</strong>, classification of watersheds.<br />
For <strong>the</strong> case of a dichotomous dependent variab<strong>le</strong>, discriminant analysis can<br />
be computed as a special case of multip<strong>le</strong> regression analysis by using a dumy<br />
variab<strong>le</strong> having values zero and one for <strong>the</strong> dependent variab<strong>le</strong> and point biserial<br />
or biserial correlations between <strong>the</strong> dependent (dummy) and independent variab<strong>le</strong>s.<br />
The regression coefficients obtained by this type of analysis are proportional<br />
to <strong>the</strong> coefficients obtained by discriminant analysis.<br />
The follwing discussion concerns discriminant function analysis for a<br />
dichotomous dependent variab<strong>le</strong>; <strong>the</strong> discussion can readily be extended to a<br />
dependent variab<strong>le</strong> having more than two categories.<br />
Suppose that <strong>the</strong> independent variab<strong>le</strong>s &=[Xi ,. . ,Xk] are jointly normally<br />
distributed in each of two populations with mean vectors and g and connnon<br />
covariance matrix c of full rank k. If <strong>the</strong> prior probabilities of each population<br />
(pop) are equal ana <strong>the</strong> costs of misclassification are equal, <strong>the</strong>n <strong>the</strong> probability<br />
of misclassification is minimized by using <strong>the</strong> following ru<strong>le</strong> for classification<br />
of an observation g<br />
classify in pop 1 if ~'~+(~1+g)'~<br />
classify in pop 2 if K'L
are extremely complicated and its convergence properties have not been investi-<br />
gated. The affects of nonnormality of & are not known. Thus discrinimant function<br />
analysis is appropriate only when x is normally distributed for each population<br />
and, in addition, its application to small samp<strong>le</strong>s is appropriate only if <strong>the</strong> popu-<br />
lation mean vectors p~ and u and <strong>the</strong> comnon population covariance matrix are<br />
known.<br />
2.4 Principal Components<br />
The purpose of principal component analysis is to reduce <strong>the</strong> dimensionality<br />
of K=[Xl ,. . ,Xk] on <strong>the</strong> basis of dependence among <strong>the</strong> variab<strong>le</strong>s. For examp<strong>le</strong>.<br />
Fiering (1964), in his work on extending <strong>the</strong> sing<strong>le</strong>-site streamflow syn<strong>the</strong>sis<br />
model to <strong>the</strong> multi-site case, applied <strong>the</strong> technique to a river basin with p gaging<br />
sites each site having an n-year record of annual flaws. Craddock (1965)<br />
applied <strong>the</strong> principal components method to monthly temperature series from 1680<br />
to 1963 for Central England. O<strong>the</strong>r applications include increases in sediment<br />
discharge from 31 watersheds after two major floods in nor<strong>the</strong>rn California<br />
(Anderson, 1970), sediment network design in California to insure accuracy of<br />
predicted sediment yield (Wallis and Anderson, 1965), establishment of <strong>the</strong> uniformity<br />
of a hydrological region in Northland, New Zealand (Blake et al., 19-70),<br />
derivation of a water yield model from monthly runoff data (Snyder, 19631,<br />
identification of watershed factors from annual precipitation and runoff data of<br />
watersheds in Coshocton , Ohio and Riesel, Texas (Diaz et al., 1968) , and shortrange<br />
forecasts of river stage or discharge on <strong>the</strong> river Kolyma, U.S.S.R.<br />
(Nechaeva and Mukhin, 1968).<br />
Ma<strong>the</strong>matically, principal components analysis transforms <strong>the</strong> X's to a set<br />
of variab<strong>le</strong>s which are pairwise uncorrelated and of which <strong>the</strong> first has maximum<br />
possib<strong>le</strong> variance, <strong>the</strong> second has maximum possib<strong>le</strong> variance subject to <strong>the</strong> condition<br />
of being uncorrelated with <strong>the</strong> first, and so.forth. Principal components<br />
are estimated on <strong>the</strong> basis of a random samp<strong>le</strong> of n observations as follows. The<br />
first principal component of & is denoted by Li=X a~ and g, is obtained such that<br />
Z'1L1=g'lX1h1 is maximized subject to <strong>the</strong> normaTizing constraint g'1&1=1.<br />
n e secona principal component Q=X- a is <strong>the</strong>n obtained by determining 7uch<br />
that g&= am2X-'X9 is maximiaed sdject to <strong>the</strong> normalizing constraint ti23=1<br />
and <strong>the</strong> independence constraint &'I 3 = O. This procedure is repeated until <strong>the</strong><br />
k principal components have been obtained.<br />
Large samp<strong>le</strong> distributional prooerties of principal components have been<br />
obtained assuming that has a multivariate normal distribution with a covariance<br />
structure such that <strong>the</strong> covariance matrix c has k distinct characteristic roots.<br />
The effects of nonnormality and <strong>the</strong> covergence properties of <strong>the</strong> large samp<strong>le</strong><br />
distributions have not been investigated. Small samp<strong>le</strong> distributional properties<br />
of principal components are not known; this again limits <strong>the</strong> use of this technique<br />
for <strong>the</strong> prob<strong>le</strong>ms considered here.<br />
In many cases determination of <strong>the</strong> number of principal components needed to<br />
account for a reasonably large proportion of <strong>the</strong> variance in X is a matter of<br />
judgment on <strong>the</strong> part of <strong>the</strong> investigator. Even if <strong>the</strong> investTgator is willing to<br />
make this decision on judgmental ra<strong>the</strong>r than statistical grounds and he concludes<br />
that a relatively small number of principal components seem to account for a<br />
reasonably large proportion of <strong>the</strong> variance in X, <strong>the</strong>re is still <strong>the</strong> prob<strong>le</strong>m of<br />
interpreting <strong>the</strong> principal components in terms of <strong>the</strong> original variab<strong>le</strong>s.
40 8<br />
hfortunately, pi.incipal components are not ahap interpretab<strong>le</strong> and this hae<br />
been a deterrent to <strong>the</strong> extensive use of principal components in developing<br />
models.<br />
The use of principal components as independent variab<strong>le</strong>s in regression<br />
analysis has been suggested for <strong>the</strong> purpose of reducing <strong>the</strong> dimensionality of 5<br />
and thus avoiding prob<strong>le</strong>ms with degrees of freedom and for <strong>the</strong> purpose of<br />
circumventing <strong>the</strong> prob<strong>le</strong>ms resulting from multicollinearity in X.<br />
applications see earlier references in this section as well as Singh's (1970a)<br />
application for predicting infiltration in an aspen-grassland watershed in<br />
southwestern Alberta, Canada. Al though principal components have been used as<br />
independent variab<strong>le</strong>s in regression analysis by numerous investigators, <strong>the</strong>re is<br />
no generally accepted procedure for determining <strong>the</strong> number of principal com-<br />
ponents to be included in such analyses, nor is <strong>the</strong>re agreement concerning whe<strong>the</strong>r<br />
it is acceptab<strong>le</strong> to include one of more of <strong>the</strong> original x variab<strong>le</strong>s in addition<br />
to principal components.<br />
In spite of <strong>the</strong>se shortcomings and limitations, a principal component<br />
analysis could possibly <strong>le</strong>ad to a better use of insufficient or correlated data<br />
for hydrologic prediction. For examp<strong>le</strong>, it offers an alternative to <strong>the</strong> method *<br />
of regionalization described elsewhere in this paper.<br />
2.5 Factor Analysis<br />
The purpose of factor analysis is to account for <strong>the</strong> covariance structure<br />
of a set of observab<strong>le</strong> random variab<strong>le</strong>s in terms of a minimal number of unobservab<strong>le</strong><br />
or latent random variab<strong>le</strong>s referred to as factors. Among hydrologic<br />
applications have been those that sought decision ru<strong>le</strong>s that resulted in reduced<br />
inventory and survey costs for specific areas and prob<strong>le</strong>ms, as in <strong>the</strong> study of<br />
<strong>the</strong> chemistry of groundwater quality (Dawdy and Feth, 1967), in <strong>the</strong> design of a<br />
hydrologic condition survey in <strong>the</strong> TVA system (TVA, 1965), in parameter screening<br />
for watershed analysis (Shelton and Sewell, 1969), in predicting reservoir<br />
losses in cavernous terrain (Knisel, 1970) and in reducing a set of edaphic<br />
variab<strong>le</strong>s for a soil (Singh, 1970b).<br />
Factor analysis estimates <strong>the</strong> coefficients to be us& in expressing each<br />
response 'variab<strong>le</strong> as a linear combination of a small number of unobservab<strong>le</strong><br />
common-factor variab<strong>le</strong>s and a (latent) specific variab<strong>le</strong>. The common factors<br />
generate <strong>the</strong> covariances among <strong>the</strong> observab<strong>le</strong> variab<strong>le</strong>s (responses) and each<br />
specific term contributes only to <strong>the</strong> variance of <strong>the</strong> particular associated<br />
response variab<strong>le</strong>. The coefficients of <strong>the</strong> common factors, estimated by factor<br />
analysis, are not required to be orthogonal and <strong>the</strong>ir matrix is unique only up<br />
to multiplication by an orthogonal matrix. The observations are assumed to be<br />
a random samp<strong>le</strong> from a multivariate normal population of full rank and <strong>the</strong> nunher<br />
of common factors is assumed to be known; both of <strong>the</strong>se requirements limit <strong>the</strong><br />
use of <strong>the</strong> technique in hydrology. The factor analysis model can be written as<br />
- X=nl+c where X is pxl, is pxm, 1 is mxl and is pxl, There are thus p<br />
response variames L'=[Xi, ..., X,], m common factor variab<strong>le</strong>s r=[Y, ..., Y,] and p<br />
specific-factor variab<strong>le</strong>s E'=[E~ ,. ..,E 1. The matrix<br />
For hydrologic<br />
gives <strong>the</strong> factor loadings<br />
where aij is <strong>the</strong> loading of <strong>the</strong> ith regponse variab<strong>le</strong> on <strong>the</strong> je common factor<br />
variab<strong>le</strong>. The conunon-factor variab<strong>le</strong>s l'=[Y1 ,. . .,Y,] are independently<br />
distributed N(0, 1). The specific-factor variab<strong>le</strong>s E'=[E~ ,. are independently
distributed N(0,q~~). Factor analysis estimates <strong>the</strong> e<strong>le</strong>ments of <strong>the</strong> loading<br />
matrix A. Maximuil: likelihood estimates can be obtained assuming that is<br />
multivariate normal with covariance matrix L=@&' of full rank p.<br />
Note that principal components can be viewed as a particular solution of<br />
<strong>the</strong> prob<strong>le</strong>m of factoring <strong>the</strong> covariance matrix. The- principal components<br />
solution ignores variance associated with a specific response variab<strong>le</strong> and requires<br />
<strong>the</strong> factors (components) to be orthogonal and of decreasing importance in<br />
accounting for (common) variance in <strong>the</strong> response variab<strong>le</strong>s.<br />
Assuming normality, <strong>the</strong> adequacy of <strong>the</strong> m-factor model can be tested for<br />
large samp<strong>le</strong>s using a likelihood ratio test of <strong>the</strong> null hypo<strong>the</strong>sis E=&+$'<br />
against <strong>the</strong> alternative hypo<strong>the</strong>sis that is any symmetric positive definite<br />
matrix. In most applications <strong>the</strong> number of common factors is not known and<br />
successively larger numbers of factors are extracted until <strong>the</strong> goodness of fit<br />
hypo<strong>the</strong>sis is accepted or <strong>the</strong> computing routine fails to converge. Successive<br />
tests used in this procedure c<strong>le</strong>arly are not independent and <strong>the</strong> statistical<br />
properties of <strong>the</strong> result are unknown.<br />
Factor analysis has been used since <strong>the</strong> beginning of <strong>the</strong> twentieth century<br />
to study <strong>the</strong> covariance structure of multivari te observations. Many variations<br />
of <strong>the</strong> model discussed above have been propose 3 and many estimation procedures<br />
have been developed. Unfortunately, factor analysis, in any of its forms, may<br />
be very difficult to interpret in practice. Part of <strong>the</strong> difficulty arises from<br />
<strong>the</strong> fact that, regard<strong>le</strong>ss of <strong>the</strong> method of extimation used, <strong>the</strong> factor solution<br />
is unique only up to a rotation of <strong>the</strong> axes. Various criteria, notably those<br />
involving simp<strong>le</strong> structure, have been suggested for obtaining <strong>the</strong> rotation most<br />
readily interpreted; in practice, considerab<strong>le</strong> subjectivity may be involved in<br />
applying <strong>the</strong>se criteria, even if <strong>the</strong>ir appropriateness is not in question.<br />
Ano<strong>the</strong>r difficulty in factor analysis arises from <strong>the</strong> fact that evaluation of<br />
factor scores for use in subsequent analyses is not uniquely defined; several<br />
intuitively appealing approaches have been suggested, but <strong>the</strong>re are no apparent<br />
criteria for choosing among <strong>the</strong>m. Thus <strong>the</strong>re are serious prob<strong>le</strong>ms involved in<br />
interpreting <strong>the</strong> results of factor analysis and using <strong>the</strong>m in subsequent analyses.<br />
In addition, relatively litt<strong>le</strong> is known about <strong>the</strong> sampling properties of<br />
<strong>the</strong> estimates obtained in factor analysis (see Matalas and Rieher (1967) for<br />
hydrologic discussions of this issue). The test for appropriateness of structure<br />
assumes normality and large samp<strong>le</strong>s; unfortunately, <strong>the</strong> alternative hypo<strong>the</strong>sis<br />
for this test may not be <strong>the</strong> most interesting alternative in many applications.<br />
There is considerab<strong>le</strong> evidence that factor analysis can give meaning<strong>le</strong>ss results<br />
if its assumptions are ignored.<br />
As an examp<strong>le</strong> of <strong>the</strong> last point, consider Rice's (1970) use of variab<strong>le</strong>s<br />
describing <strong>the</strong> physiography of experimental basins on <strong>the</strong> San Dimas Experimental<br />
Forest in sou<strong>the</strong>rn California. His goal was to identify variab<strong>le</strong>s which would<br />
be useful in flood prediction. He notes "that <strong>the</strong> hydrologist might be better<br />
rewarded if he turns his efforts toward developing physiographic variab<strong>le</strong>s<br />
which better portray hydrologic processes ra<strong>the</strong>r than relying on a ma<strong>the</strong>matical<br />
artifact such Bs factor analysis to appraise <strong>the</strong> utility of various expressions<br />
of basin physiography." This point is borne out in a non-hydrologic study by<br />
Armstrong (1967); he finds that, whi<strong>le</strong> factor analysis "explains" a large proportion<br />
of <strong>the</strong> variances, it fails to identify <strong>the</strong> known factors in <strong>the</strong> model!<br />
409
41 O<br />
2.6 Cluster Analysis<br />
The purpose of cluster analysis is to group multivariate observations according<br />
to various cri teria based on <strong>the</strong>ir degrees of homogeneity and .heterogeneity.<br />
In hydrology, cluster analysis can be used to classify watersheds, flaw regimes,<br />
and climates. Bogardi et al. (1972) have used it to group statistical properties<br />
of monthly water <strong>le</strong>vels in Lake Balaton (Hungary). Hydrologic appli-<br />
cations are very few.<br />
cl us i on in mu1 ti vari ate regression.<br />
The o<strong>the</strong>r multivariate methods discussed above assume that <strong>the</strong> variab<strong>le</strong>s<br />
belong to particular populations and that <strong>the</strong>se populations have specific<br />
(usually normal) distributions.<br />
Cluster analysis can help to identify variab<strong>le</strong>s for in-<br />
In cluster analysis <strong>the</strong> variab<strong>le</strong>s are not assumed<br />
to have even <strong>the</strong> minimal structure of belonging to particular populations and<br />
<strong>the</strong> purpose is to establish appropriate populations as a basis for structuring<br />
<strong>the</strong> variab<strong>le</strong>s.<br />
Techniques of cluster analysis have been developed, almost exclusively, not<br />
only for computer application but also on <strong>the</strong> basis of computer analysis. Al-<br />
though ma<strong>the</strong>matical rigor is minimal and statistical inference is almost non-<br />
existent for cluster analysis, very useful results have been obtained in appli-<br />
cations. Because of its (lack of) assumptions concerning population structures<br />
and distributions, cluster analysis is applicab<strong>le</strong> to a wide variety of hydrologic<br />
and o<strong>the</strong>r prob<strong>le</strong>ms;. its results can be useful if <strong>the</strong>y are recognized as tentative<br />
and if even tentative conclusions are based only on results from large samp<strong>le</strong>s.<br />
There are several questions or decisions that must be considered in any<br />
cluster analysis: <strong>the</strong> number of clusters must be determined, <strong>the</strong> cluster<br />
boundaries must be established, <strong>the</strong> method for handling correlated variab<strong>le</strong>s must<br />
be specified, <strong>the</strong> technique for examining similarities must be chosen, and so<br />
forth. Several approaches have been proposed for each of <strong>the</strong>se aspects of cluster<br />
analysis. Which criteria or ru<strong>le</strong>s of thunh are most appropriate depends on <strong>the</strong><br />
prob<strong>le</strong>m. Regard<strong>le</strong>ss of <strong>the</strong> techniques and criteria chosen for cluster analysis,<br />
<strong>the</strong> investigator usually examines successive computer printouts and uses his<br />
judgment to al ter apparently poorly se<strong>le</strong>cted cri teria and techniques. Compared<br />
with <strong>the</strong> o<strong>the</strong>r multivariate analyses discussed, cluster analysis is more of an<br />
art and <strong>le</strong>ss of a science, but so is engineering design under uncertainty assoc-<br />
iated with insufficient data.<br />
2.7 Bayesian Inference<br />
The preceding discussion of multivariate models is entirely from <strong>the</strong> point<br />
of view of classical sampling <strong>the</strong>ory. Several of <strong>the</strong>se models have been analyz-<br />
ed from <strong>the</strong> Bayesian point of view and <strong>the</strong>se results are summarized in <strong>the</strong> follo-<br />
wing discussion. Bayesian inference incorporates, with samp<strong>le</strong> infomation, <strong>the</strong><br />
investigator's prior information concerning <strong>the</strong> sampling distributions of <strong>the</strong><br />
parameters to obtain point or interval estimates. More general, however, is<br />
Bayesian decision <strong>the</strong>ory that incorporates both prior information and a loss<br />
function with samp<strong>le</strong> information in order to obtain parameter estimates or to<br />
determine <strong>the</strong> optimal decision. Bayesian analysis is intuitively appealing; in<br />
many applications <strong>the</strong> investigator has considerab<strong>le</strong> prior data or experience as<br />
a basis. for prior parameter distributions and in most applications he has at
<strong>le</strong>ast a general idea of <strong>the</strong>?(economic) loss function associated with inaccurate<br />
estimation. Unfortunately, <strong>the</strong> results for many multivariate Bayesian methods<br />
are complicated and at best are applicab<strong>le</strong> only for large samp<strong>le</strong>s. However,<br />
since many classical results also have this limitation, Bayesian methods may be<br />
p:i ferab<strong>le</strong> because of <strong>the</strong>ir f<strong>le</strong>xibility in incorporating prior distributions and<br />
loss functions in <strong>the</strong> estimation of parameters or in determining optimal decisions.<br />
Also, human beings are better at estimating prior distributions than at estiniatlng<br />
posterior distributions (Ferre11 , 1972).<br />
There has been considerab<strong>le</strong> application of Bayesian methods in mu1 tivariate<br />
linear regression analysis. Bayesian point estimates of <strong>the</strong> regression coefficients<br />
can be obtained with or without incorporating loss functions and Bayesian<br />
interval estimators (credibility intervals) can be formulated.<br />
As discussed in section 2.5, <strong>the</strong> maximum likelihood factor analysis solution<br />
is unique only up to a rotation of <strong>the</strong> axes; <strong>the</strong> use of subjective information<br />
in a Bayesian analysis is an intuitively appealing basis for eliminating this<br />
ambiguity. Unfortunately, <strong>the</strong> technical difficulties involved in obtaining<br />
numerical solutions have thus far precluded use of this approach.<br />
The Bayesian approach has also been considered for canonical correlation<br />
analysis; unfortunately, even for <strong>the</strong> simp<strong>le</strong>st assumptions with respect to both<br />
<strong>the</strong> prior distributions of <strong>the</strong> parameters and <strong>the</strong> sampling distributions of <strong>the</strong><br />
data, <strong>the</strong> Bayesian results for canonical correlation analysis are so complicated<br />
that <strong>the</strong>ir applicability is extremely limited.<br />
The most notab<strong>le</strong> success of Bayesian methods in multivariate analysis thus<br />
far has been for discriminant functions. As summarized above, a number of methods<br />
based on <strong>the</strong> sampling <strong>the</strong>ory viewpoint have been proposed for discriminant<br />
analysis, but <strong>the</strong>se results are unsatisfactory for use with small samp<strong>le</strong>s. The<br />
Bayesian approach provides a useful and simp<strong>le</strong> al ternative.<br />
Consider <strong>the</strong> case of classification into one of two mutually exclusive populations.<br />
Denote <strong>the</strong> populations by Pi and P , <strong>the</strong> vector of observations by<br />
x'=[xl, ..., xk]. <strong>the</strong> density functions by fl(Kf and fp(&) and <strong>the</strong> prior probabi-<br />
Tities by p1 and p2 where p +p2=1. The costs of misclassification are C(211) if<br />
an observation from P1 is classified in P2 and C(112) if an observation from P2<br />
is classified in Pl. The problAem is to determine a classification ru<strong>le</strong> of <strong>the</strong><br />
following form: partition K into regions R1 and R2 such that if ZERI, <strong>the</strong> observation<br />
is classified in Pi and if x~R2, <strong>the</strong> observation is classified in P2.<br />
The expected cost of misclassTfication is given by<br />
and <strong>the</strong> corresponding ,classification ru<strong>le</strong> is<br />
f+X) C(l MP2 f+x) C(112)PZ<br />
R2:<br />
R1:q-g 'copl<br />
< cop1<br />
411<br />
This classification ru<strong>le</strong> involves <strong>the</strong> densities f1(&) and f2(&) which may not be<br />
known. Assume that fi(&) and f2(&) are multivariate normal with mean vectors PJ<br />
and and common covariance matrix c. Then <strong>the</strong> above ru<strong>le</strong> can be written<br />
c(1 PIP, c(1 12)P2<br />
R, :L'i.-+(IL,+q) 'L>log, R2:~'6-+(~1+4) '&
41 2<br />
1<br />
where 6 = E- (ply) and x'6 is <strong>the</strong> discriminant function. If p1=p2=% and<br />
C(112)3(2ll),'-this reducësto <strong>the</strong> ru<strong>le</strong>, given in <strong>the</strong> discussion of <strong>the</strong> sampling<br />
<strong>the</strong>ory appLoack to discriminant analysis. As noted in that discussion, samp<strong>le</strong><br />
estimates x x and 2 may be used to obtain from <strong>the</strong> samp<strong>le</strong> data without<br />
know<strong>le</strong>dge d'i' -2 and c.<br />
Bayesian aiscriminant analysis can easily be extended to o<strong>the</strong>r cases; for<br />
examp<strong>le</strong>, fl(x) and f2(&) may have some form o<strong>the</strong>r than <strong>the</strong> normal distribution<br />
or <strong>the</strong>re may be more than two populations into which an observation may be<br />
classified. Finally, for <strong>the</strong> sake of comp<strong>le</strong>teness, we should mention <strong>the</strong> use of<br />
Bayesian decision <strong>the</strong>ory in design to imbed uncertainty in parameters resulting<br />
from inadequate samp<strong>le</strong>s into a loss function (Davis et al., 1972; Davis et al.,<br />
1973).<br />
3.0 Sumnary and Conclusions<br />
In this overview, we have critiqued <strong>the</strong> current status of multivariate<br />
methods of data analysis because of <strong>the</strong>ir central position in making estimates<br />
and predictions of both hydrologic and econometric (e.g., cost) inputs to design<br />
of water resource systems. The design implications of many of <strong>the</strong> assumptions '<br />
in <strong>the</strong>se methods remain to be evaluated - a task of importance to many profes-<br />
sional disciplines.<br />
models without considering <strong>the</strong> assumptions involved, we believe that use of<br />
Bayesian decision analysis, whi<strong>le</strong> not <strong>the</strong> final answer, may be a viab<strong>le</strong> alter-<br />
native for anticipatinq poor design. Bayesian analysis offers f<strong>le</strong>xibility in<br />
incorporating prior (subjective) know<strong>le</strong>dge about probabil i ty distributions on<br />
design parameters and it encourages <strong>the</strong> design engineer to invoke his general<br />
ideas of loss functions associated with inaccurate estimation in many specific<br />
design prob<strong>le</strong>ms. The focus is on <strong>the</strong> consequences for a specific use and not on<br />
a precise design estimate. The latter has a subt<strong>le</strong> linkage of probability and<br />
utility, depending on one's value structure, but Bayesian decision analysis<br />
encourages specific consideration of each in an open manner. With <strong>the</strong> continuing<br />
emphasis on environmental impact evaluation, such an approach seems timely and<br />
necessary in <strong>the</strong> face of small samp<strong>le</strong>s of hydrologic and o<strong>the</strong>r environmental<br />
data.<br />
4 .O References<br />
In contrast to <strong>the</strong> current tactic of using multivariate<br />
Anderson, H. W. 1970. Principal components analysis of watershed variab<strong>le</strong>s<br />
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for Hydrologic Sciences. Publ. 96, pp. 404-416.<br />
Anderson, T. W. 1958. An Introduction to Multivariate Statistical Analysis,<br />
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Armstrong, J. S. 1967. Derivation of <strong>the</strong>ory by means of factor analysis or<br />
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41 3<br />
Blake, G. J., A. D. Cook and D. H. Greenall. 1970. The use of.principa1 component<br />
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Bogardi, I., L. Duckstein, and C. C. Kisiel. 1972.. Distribution of dynamic<br />
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Christ, C. F. 1966. Econometric Models and Methods. New York: John Wi<strong>le</strong>y<br />
and Sons, Inc.<br />
Craddock, J. M. 1965. A meteorological application of principal component<br />
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Davis, D., C. Kisiel and L. Duckstein. 1972. Bayesian decision <strong>the</strong>ory applied<br />
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Davis, D. R., L. Duckstein, C. Kisiel, and M. Fogel. 1973. A decision<strong>the</strong>oretic<br />
approach to uncertainty in <strong>the</strong> return period of maximum flaw<br />
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Dawdy, D. R. and J. H. Feth. 1967. Application of factor analysis in <strong>the</strong> study<br />
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Dhrymes, P. J. et al. 1972. Criteria for evaluation of econometric models.<br />
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Dhrymes, P. 1970. Econometrics: Statistical Foundations and Applications,<br />
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Johns ton, J. 1963. Econometri c Methods. New York: McGraw-Hi 11 , Inc.<br />
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PRATIQUES COURANTES POUR L'EVALUATION DES CRUES ET<br />
DES DEBITS D'ETIAGES PRIS EN COMPTE DANS LES PROJETS<br />
Rapport General<br />
P ar<br />
Marcel ROCHE<br />
Pour &borer <strong>le</strong>s données hyärologiqueis néceseaires à la mise au<br />
point diun projet d'aménagamcmt des eaux, on dispose fina<strong>le</strong>ment de trois<br />
graisda types d'approche :<br />
- u'utilieer que <strong>le</strong>e observatione concernaut <strong>le</strong>s débite et re-<br />
cueillie6 au site m be de l'amhgement ou & p roat6 ;<br />
- utili- 6ge<strong>le</strong>ment <strong>le</strong>s données climatiquee ãieponib<strong>le</strong>s BUF <strong>le</strong><br />
baadn, notemnent <strong>le</strong>s re<strong>le</strong>vée de précipitatione i<br />
. 88 mvlr de formu<strong>le</strong>s r6giona<strong>le</strong>e1 ou de corr6lationa. Pour a-<br />
trapo<strong>le</strong>r des reeultate recueillis d dee statione hgdrom&riqUeE<br />
et/ou plu~iométrigues tdtu&s BiUeure äaus ia regiou, à 1li.n-<br />
ttieur OU d l'extbieur du basain fluvial.<br />
Le premier typa groupe <strong>le</strong>e dthodee dites Wirscteetl, tandis qUe<br />
<strong>le</strong>e d ew autree pmoèdemt de 1'Cvaluation indirecte. La m&hode "la plu8<br />
aireate" consiste en lkdyse statietique d'un Bchentillon, par exemp<strong>le</strong>
420<br />
de débits maximaux annuels t el<strong>le</strong> nécessite, pour que l'évaluation ait un<br />
sens, une infoimation riche, des m emes précises de débite portant sur de<br />
longues périodes.<br />
Le second type de méthodes ccmaiste en fait à l96tendrett des échan-<br />
tillons existante de debits, portant BUT dee périodes courtee, en utilisant<br />
<strong>le</strong>s relations qu'on peut dkager entre <strong>le</strong>a dábits et <strong>le</strong>s donnbe climatiques<br />
disponib<strong>le</strong>s sur une dur& beaucoup plus longue. C'est @ll'exteneion des don-<br />
néedi, qui fait largement appel aux méthodes de r¿greesion et à l'anaïyse<br />
multivariate, mais aussi dans certaine cas aux modè<strong>le</strong>s dits conceptuels. On<br />
peut rattacher à ce type <strong>le</strong>e m&hodes qui consistent à effectuer <strong>le</strong>s trans-<br />
formations ou à appliquer <strong>le</strong>s régressions & u11 évènement climatique de pé-<br />
riode de retour connue, ou considéré comme un maximum possib<strong>le</strong>.<br />
Le tmi&me type, enfin, rassemb<strong>le</strong> <strong>le</strong>s méthodes d'interpolation<br />
ou d'extrapolation géographique. Ces méthodes vont de la simp<strong>le</strong> analogie &o-<br />
morphologique et climatique, aux raffinements de l'analyse factoriel<strong>le</strong>, en<br />
passant par <strong>le</strong>s tab<strong>le</strong>e et abaquee régionaux.<br />
U paraft se dégager de cette &um&ration une id& simp<strong>le</strong> de l'uti-<br />
lisation dee m6thodee en face de l'information disponib<strong>le</strong> ; il semb<strong>le</strong> évident<br />
a priori que <strong>le</strong>s m6thodes de type I n'ont de sens qu'en cae d'information très<br />
abondante, que od<strong>le</strong>s du type II correspondent A une information hydrologique<br />
réduite, mais à une informetion cïimatoïogique consistante, et que <strong>le</strong> type<br />
III groupe <strong>le</strong>s mgthodes utilie8ss lorsqu'on ne dispose de pratiquement rien.<br />
On pourrait en conclure que la m&hodologie de type I devrait être exclue<br />
d'un symposium tel que celui-ci.<br />
En fait, <strong>le</strong> jugement doit être plue nuancé, car on commence à ten-<br />
ter des régionalisatione BUF <strong>le</strong>e lois de distribution, donc à rechercher par<br />
là des estimations indirectes, a melanger dane <strong>le</strong>e études statistiques des<br />
dondes de conaietances t Ae différentes. C'est pourquoi nous rQerverom,<br />
ciam notre expod, une piace à l * m & e statistique.<br />
Chaque m&hode d'heluation fait appel à des outils de calcul qui<br />
ne lui sont pae forchent propre. C'est aind qu'un modè<strong>le</strong> 6 structure détor-<br />
ministe, ì'hydmgranmie unitaire par examp<strong>le</strong>, peut etre utilisé pour une ex-<br />
tension des donnhe, pour u11 celcul de transformation d'una, averse de fr&<br />
quence donnét, ou pour une extrapolation &graphique.
ûn pourrait envisager, dans ce rapport, une présentation par<br />
"outil de ~ elcul~~ ; il nous a paru plus cmfonne au eujet du eymposiuc<br />
d'aùmter un ordre d'exposé qui se rattache, autant qu'il mit possib<strong>le</strong>,<br />
à la quantité d'information disponib<strong>le</strong>.<br />
421<br />
Les sujets proposés dans la question qui fait l'objet de ce rap-<br />
port général se rappsrtent aux débits extr&nes, c'est-à-dire aux étiages<br />
et aux crues. I1 faut bien reconnaître que <strong>le</strong> premier sujet nia guère tenté<br />
<strong>le</strong>s ~ãpécialistes, puisque un 6eUl rapport, eur <strong>le</strong>s quatorze que nous avona<br />
examinés, traite des basses eau, Fina<strong>le</strong>ment, cela n'est pas tel<strong>le</strong>ment sur-<br />
premt ; <strong>le</strong>s basses eaux sont liées de très pr&3 aux problèmes d'hydrogh-<br />
logie ; or ce n'est que dans quelques cas trèer particuliers qu'il est possi-<br />
b<strong>le</strong> de dégager des paraadtree morphologiques et climatiques simp<strong>le</strong>s qui<br />
soient en relation directe avec ces pmblèmeo et qui permettent une tramp-<br />
sition &,panhique suffisamment précise. Par contre, u11 auteur aurait pu<br />
être tent6 par l'aspect "extedon des donnée^'^, qui dierpoee d'une méthodo-<br />
logie courante sinon riche, du moina 88888 efficace ; cela ne s'est pa6<br />
produit.<br />
I1 était précisé enfin que <strong>le</strong>s expos6s devaient se rapporter à<br />
des llpratiques courantes", ce qui semblait exclure <strong>le</strong>s sujete de recherche<br />
et certaim procédk de calcul non tota<strong>le</strong>ment dégagés de <strong>le</strong>ur phase exp&i-<br />
mentaie. Aussi n'Fneieterona noua par3 tmp LNF certaine aapecte qui no-<br />
ont été soumie ; ceci ne veut pas dire que noua <strong>le</strong>e trouviane peu dignes<br />
d'inter&<br />
Dana notre prbentation des rapporta, nous par<strong>le</strong>rone d'abord des<br />
étiages, puis des crues.<br />
Le eeul rapport traitant de ce pmblème est celui de<br />
MM. Vladimirov et Chebotarev [I3]- Lee autemrs définiasent avec m in <strong>le</strong>s
422<br />
en<br />
variab<strong>le</strong>s par <strong>le</strong>squel<strong>le</strong>s on caractÓriae <strong>le</strong>e basses eaux/G.R.S.S. :<br />
- d&it journalier minimal de l'annw,<br />
- d6bbit moyen meneuel minimal de l*annh.<br />
Chacune de ce6 oarlab<strong>le</strong>e est définie pour choune dee deux saisons<br />
de basses eaux : cel<strong>le</strong> d'hiver et cel<strong>le</strong> d'&&-automne.<br />
Les auteurs exposent ensuite ï'infïuence, sur ces débits de bM6e6<br />
eaux, des caractéristiques climatiques, en insistant Bur <strong>le</strong> A<strong>le</strong> du gel, et<br />
des carúct&istiques phyeiographiquuee, en notant <strong>le</strong> r81e particulièrement<br />
important des lacs, des rnaraic, du sol, du sow-sol, du karst. 110 définie<br />
sent enmite <strong>le</strong> cadre régionel dans <strong>le</strong>quel va s'exercer la méthadologie :<br />
<strong>le</strong>s bassins sont classés en<br />
- petits bassins : jusqu'a 1 o00 1 500 h2en zone de plaine hu-<br />
mide, jusqu'à 2 000-2 500 h2 dan6 <strong>le</strong>s zones de montagnes oc <strong>le</strong>s<br />
zones de plaines peu humides, jutSqu'8 5 OOO-10 o00 km' dans <strong>le</strong>s<br />
régiow arides ou pour <strong>le</strong>s rivières soumises à un gel intense ;<br />
- bassins inoyene : jusqu'8 75 O00 km2.<br />
Pour <strong>le</strong>e petite bassins, on applique, dana un cadre strictement<br />
régionali66 6 partir des caractériatiques phydographiquee et climatiques,<br />
une formulo de la fome Q = a (A ;I. fIn oÙ Q eat <strong>le</strong> dBit meneuel minimal<br />
moyen (dit l~omuiì~l daps <strong>le</strong> texte) en d/s, A ia eurface B.V. tan d, f un<br />
correctif tenant compte d, la non coincidence hentuei<strong>le</strong> du bassin topogra-<br />
phique et du bassin souterrain ; a ot n srnt de6 paramdtres régionaux. Le<br />
passage du débit % od1* 6 dee débits de fr8quences donnbs er'effectue par<br />
deux méthode8 diffbentes.<br />
?our <strong>le</strong>e baemns moyenne, on utilise des cartes d'isopl&<strong>the</strong>s (iflolines)<br />
pour la détexmination du débit d'étiage meneuel ; ces cartes sont établies<br />
pour <strong>le</strong>s va<strong>le</strong>ur8 moyennes (norme<strong>le</strong>s) et pour la frbence 80 % de dépasscirent.<br />
coefficient de p-.<br />
Les étiages joumdiers sont dikìuits de6 étiages menaue<strong>le</strong> par un<br />
Il s'agit donc d'un proc6dd8 de formu<strong>le</strong> @kd8giona<strong>le</strong>1@ dont <strong>le</strong>s parani&<br />
tres sont d&erminb gar un catalogue, et non l ib à dee caractéristiques
423<br />
ghorphologiques et climatologiques meeurab<strong>le</strong>s. C'est pratiquement <strong>le</strong> seul<br />
moyen de e'en eortir en matière de basaes eaux, pour los raieons que noua<br />
avo- deja indiqubs. I1 est dommage que l'auteur n'ait pea indiqué quel-<br />
ques va<strong>le</strong>ure de cosfîiciente r6gioll~yx, ni montré sur un exemp<strong>le</strong> la concor-<br />
dance des réniltate du calcul avec des dombs observks.<br />
Lorequ'on dit qu'on détermine dee crue8 en l'absence de données,<br />
c'est faux. Sane donnéeel on ne caicüie ria du tout. Simp<strong>le</strong>ment, on cherche ici<br />
.4 trveer des do~8es existantee ou à établir des relations entre <strong>le</strong> phé-<br />
nomène qu'on &tudie et des données d'une autre nature. Toute L1infonnation,<br />
et par suite +out <strong>le</strong> sérieux de6 edhations qui en dbdent, tiennent dam<br />
Ce8 d O~h6.<br />
C'est pourquoi nous regrettons un peu qu'il ne se mit pas trouvé<br />
d'auteur pour exposer <strong>le</strong>a méthodee, pourtant de pratique bien courante, per-<br />
mettant de reconetituer UTI certain nombre d'6v&nementa marquanedu passé.<br />
Noue verrou tout 6 l'heure qu'on commence a se prkoccuper d'insérer dane<br />
des échantillons r4guliere de dbbite d maw de crue5 annuel<strong>le</strong>s, dee obser-<br />
vations discontinues, parfoie t mnquh et souvent entachées d'erreur6 beau-<br />
coup plus grandee que cel<strong>le</strong>ede l'&chantillon régulier. Lorsque <strong>le</strong>e donnéee<br />
sur <strong>le</strong>s ornes sont rame et surtout portent SUT de5 périodes très courtee,<br />
il devient extrhment important de dispaser du plus grand nombre de tel<strong>le</strong>s<br />
obaervatiom.<br />
chi peut aauvent trouver dana <strong>le</strong>s archivea, dane la preme, ~ ur<br />
de6 télégrammes adminietratifel dee indications parfoiPs trèB préci898<br />
oertaines 5mdee crues. On peut éga<strong>le</strong>ment mener des enquatee 6ur place,<br />
auprès dee riverains et em a'intbrsemant aux d&hhs& ou autres marques<br />
laiss6es par ces orue~). 11 y a 1a toute une m6thodologie dont nom no pou-<br />
viam peu ne pas tout BU moina signa<strong>le</strong>r l'existence.
424<br />
des crues réel<strong>le</strong>ment observées e d traditionnel<strong>le</strong>ment un outil pour informa-<br />
tion riche. Des études relativement récentes ont pourtant 6tk men688 afin<br />
de voir s'il n'Était pas possib<strong>le</strong>, en cas de données rares,<br />
- soit de compléter l'échantillon de données rkgulierement recueillies<br />
par des observations sporadiques et/ou tmnqubs,<br />
- soit de faire de la transposition régiona<strong>le</strong> sur <strong>le</strong>s lois stutisti<br />
ques el<strong>le</strong>s-mêmes<br />
Bien entendu, <strong>le</strong>e deux ne smt pae incompatib<strong>le</strong>s.<br />
La première question a été traitée, au moins partiel<strong>le</strong>mmt, dens la<br />
cornunication de M. Morven N. keee 183. L'auteur considère deux aspects de<br />
la question. Dans <strong>le</strong> premier, il suppose qu'on dispose d'une série n o d e<br />
d'observations (enregistrements par exemp<strong>le</strong>) au cours de laquel<strong>le</strong> un certain<br />
nombre de maximums annuels n'ont pas &é observés, par exemp<strong>le</strong> par suite d'uno<br />
déficience d'appareillage : on cornaft seuìernent , pour <strong>le</strong>s crues concernhs,<br />
<strong>le</strong> seuil inf&ieur du débit, seuil supposé constant eur la période. C'est <strong>le</strong><br />
cas pratique du sty<strong>le</strong>t d'enregietreur qui sart des limites du tambour ou de<br />
la tab<strong>le</strong> dbulante.<br />
Le second aespect est celui de la station pour laquel<strong>le</strong> on dispose<br />
d'une skie d'observations régULiBres de N années, courte mais complète,<br />
des maximums muela de cruea. On connaît par ail<strong>le</strong>urs, sur une p aode de<br />
M autres annha, <strong>le</strong>s n d8bits maximaux de crues ayant d&ad un seuil 5 ;<br />
et on est certain que p e a t <strong>le</strong>e N-n anahs reetantes de cette période, aucun<br />
débit n'a atteint ou d6paee6 xh. Ce eont là des hypo<strong>the</strong>ses assez restrictives<br />
qui correspondent au cas pratique des échel<strong>le</strong>s de hautes eaux Wloitées Pon-<br />
dant u11 certein nombre d'années avant que <strong>le</strong>s services se préoccupent des<br />
basses et moyennes eaux ; ce peut (tre aussi <strong>le</strong> cas de certaines marques<br />
relatives a des cmes historiques.<br />
Lee deux prob<strong>le</strong>mes sont trait& par la méthode du maximum de vrai-<br />
semblance, suivant la technique indicph pm Kendall aux paragraphes 32.15<br />
et suivants de son ouvrage. L'auteur donne un exemp<strong>le</strong> d'application a la ri-<br />
vière Avon a la dation de Bath (U.K.). Le but de l'étude<br />
eat de déter-<br />
miner <strong>le</strong> gain d'information apport& par <strong>le</strong>s observations tronquées ou <strong>le</strong>s don-<br />
nks que l'auteur qualifie d'historiques ; ce gain d'information est estimé
425<br />
ici par la r&ction apport& à 1'eITeur etandard d'estimation pour différent;<br />
quant i <strong>le</strong>s.<br />
Cette 6tude est fort int&resaante, bien qu'on puisse ne pas être<br />
p<strong>le</strong>inement d'accord avec touteei <strong>le</strong>s hypathèees introduites par l'auteur. El<strong>le</strong><br />
est susceptib<strong>le</strong> d'importants prolongements vers d'autres formes d'information<br />
tronquée ou sporadique, mais <strong>le</strong> traitement rieque alors de ne pas être aussi<br />
simp<strong>le</strong>.<br />
En ce qui concerne la aeconde question, la traueposition des lois<br />
statistiques a été mainteefois tenth, souvent avec succ8s. Nous nous permet-<br />
tona de rappe<strong>le</strong>r <strong>le</strong>s 6tudes de U, Oktay hanoglu sur <strong>le</strong>5 mo~u<strong>le</strong>s pluviom&<br />
trique8 de IlAfrique de l'Ouest (Cahiers de l*O.R.S.T.O.M., hérie @droiogie>.<br />
MM. Herbst, Van Biljon, Olivier et Hai1 noue présentent une c odcation sur<br />
la régionalisation des paramètres das lois de distribution des crues 151.<br />
Les auteurs prennent comme m>dè<strong>le</strong> statistique la loi log-gaiinna<br />
incomplète (log-Pearson III), tout en hoquant la possibilité d'effectuer<br />
<strong>le</strong>s mi3mee opératiom en Re basant mr une lo' de Qumbel. La méthode consiste<br />
à<br />
- calcu<strong>le</strong>r. '.es paP810dtime8 des lois pour toutes <strong>le</strong>s longues séries<br />
diaponib<strong>le</strong>B,<br />
- apposer que chacun de ces paramatres d6pend deun certain nombre<br />
de facteur@ gbmorpholagiques et climatiques (il cite la super-<br />
ficie du bm&n, la pluie annuel<strong>le</strong> moyenne, la pente moyenne,<br />
la longueur de la rivière, la pluie menquel<strong>le</strong> maxima<strong>le</strong> médiane,<br />
un facteur de fome mais n'utilise dans la suite du texte<br />
que La surface du bassin A et la pluie annuel<strong>le</strong> moyenne R),<br />
- appliquer pour chaque parametre de la loi une régression multip<strong>le</strong><br />
avec <strong>le</strong>a facteurs retanus ; <strong>le</strong>s coefficients de la r6gression<br />
sont calculés par <strong>le</strong>a moindres carrés et l'opération constitue<br />
en fait un "lieeage gbgraphiquelt des va<strong>le</strong>urs de ces paramètres.<br />
En r&äiité, <strong>le</strong>e auteum neiitilisent pas <strong>le</strong>s param¿tres fLgurant<br />
dana l'expresdon mathhatique de la Loi, &is la moyenne (des logarithmes<br />
dee débite), ll&cart-type et 10 coefficient d1msiymétrie, ce dernier étant<br />
du reste trait6 de fapon trèe diffhnte. Le pint <strong>le</strong> plus important au calcul
426<br />
se rapporte à la variance d'estimation de te<strong>le</strong> paramètres et dee quanti-<br />
qui en dkoulmt, toujours parr l'intermediaire de la loi log-ggnnaa.<br />
Les auteurs donnant quelques résultats obtenue en Afrique du Sud.<br />
Lo cadcation de MM. Davis, Ducketein, Kisiel et Fogel /2]<br />
traite du problhe de la distribution de la pkiode de retour correspondant<br />
au déparssement d'un maxipnrm ou diun volume de crue donni, en partant des don-<br />
nées sur <strong>le</strong>s précipitations. La méthode se rapporte plut8t au type II,<br />
d e <strong>le</strong>s techniques de calcul expos6ee relhnt enti8rement de l'analyse<br />
atatidique. La formu<strong>le</strong> de transformation pluies-débits adoptée pour <strong>le</strong>s<br />
volumes de cnie e& de la forme Q = C (R - A). Dana l'analyse de &bili-<br />
té conduite par dmulation, <strong>le</strong>s auteurs ne se préoccupent que de C et trou-<br />
vent, come il fallait s'y attendre, une énorme influence de la variance<br />
d'estimation de ce paramètre sur la variance d'estimation de la période de<br />
retour. I1 n'est pas pmuvé, come semb<strong>le</strong>nt <strong>le</strong> supposer <strong>le</strong>s auteurs, que<br />
la variance de A n'ait qu'une influence négligeab<strong>le</strong>.<br />
2.2. - kJB-hodes_de rn-1; = *&e-d-og gee ~ o ~ . g<br />
Dane Bon sena <strong>le</strong> plus littéral, l'extension Eee d o ~ de h d6bits<br />
consiste en l'opération suivante.<br />
- Une variate Y &tant définie mivant <strong>le</strong> phhomene i@rologique<br />
qui intbesse et <strong>le</strong> probl8me qu'm a à résoudre (par emmpie<br />
Y a d&it maximal instantané de l'année) ;<br />
- on dispose d'un échantillon de n va<strong>le</strong>urs de Y obtenues par l'Obsemation<br />
directe des débits ;<br />
- on dispoee d'un échantiiìon de N > n<br />
va<strong>le</strong>urs de une 01 ph-<br />
sieurs oaractgristiqueim cìimatoiogiques XI, U .. . Xk(pm 8xezap<strong>le</strong> :<br />
averse dmaìe annuel<strong>le</strong>, indice de pluies antk6dentee mtte<br />
aveme, etock de neige sur <strong>le</strong> baasin au d h t de la fonte) N<br />
contenant n i<br />
(StJ - on cherehe à établir une r8grulsolon multip<strong>le</strong> (au simp<strong>le</strong>) entre<br />
11 et XI, X2 ... xk (ou seu<strong>le</strong>ment XI) :
427<br />
- on appïiqtm la régreseion trouvée aux Io-n va<strong>le</strong>ur8 de XI ... non<br />
contenues dane la période commune de n années ;<br />
- on a ainsi une nouvel<strong>le</strong> ebie de N va<strong>le</strong>urs de Y, plus longue que<br />
la &rie origina<strong>le</strong> de n va<strong>le</strong>urs, laquel<strong>le</strong> on peut appliquer<br />
11analyse statistique.<br />
OU BDN - on &ablit un modè<strong>le</strong> déterministe de transformation pluies-débits<br />
(par exemp<strong>le</strong> hydrogramme unitaire + fonction de ruissel<strong>le</strong>ment) ;<br />
- on applique ce mode<strong>le</strong> aux donn&s climatologiques XI ..., et on<br />
procède comme pour <strong>le</strong>e données reconstitubs par régression.<br />
On doit noter que, ni par une méthode ni par l'autre, on ne :ail;<br />
de transposition ou interpolation gbgraphique. Les mode<strong>le</strong>is, qu'ils soient<br />
régressionsll ou de structure déterministe, sont appliquh aux bassi116<br />
mêmes pour <strong>le</strong>squels ils ont et6 établis.<br />
Le problème du gain d'information se pose dans <strong>le</strong>s deux cas. I1<br />
e~t bien évident qLe <strong>le</strong> nouvel Qchantilion de n va<strong>le</strong>urs de X observées<br />
+ N-n va<strong>le</strong>urs de X calculés n'est pas équiva<strong>le</strong>nt, du point de vue quantité<br />
d'information, à un échantillon de N va<strong>le</strong>urc de X observées, mais '1 un &chan-<br />
tillon de NI va<strong>le</strong>urs, avec n < Nu < N. Si on a procédé par régressions et<br />
qu'on se soit m angé pour que ces régressions répondent à peu près aux con-<br />
ditions suivantes :<br />
- homsc8aasticit6,<br />
- linéarité,<br />
- distributions margina<strong>le</strong>s n odes, en opérant bei changements<br />
de vexiab<strong>le</strong>s ou dea anamorphoeesp <strong>le</strong> gain d'information, c'est-d-dire la wan-<br />
tit6 NI-n, peut être faci<strong>le</strong>ment &du61 en comparant <strong>le</strong>s variance8 des estima-<br />
tiom.<br />
Si on a utili& un modè<strong>le</strong> d&erministe, cette &elmtian n'est pas<br />
imgdiate. Ii est nécetseaire de rechercher empiriquement la loi des écarts<br />
rkiduels, ou la corrélation entre va<strong>le</strong>urs obeervbe et ve<strong>le</strong>~~W calculées
42 8<br />
en se baeent ~ u1 <strong>le</strong>e n ann&s d'obmrvatioiiei conmnme0. il faut dire que bien<br />
souvent on ne fait pas cette remherche et on se psisee de 1*6valuation des in-<br />
terval<strong>le</strong>e de confiance.<br />
L'avantage du moda<strong>le</strong> déterministe, (notamment de ì'hydrolp.Emime uni-<br />
taira, est de fournir la totalité de l'hydrogramme de amel donc simultané-<br />
ment <strong>le</strong> débit maximal, <strong>le</strong> volume ruisselé et la fome. Taiidie que ces Biéments<br />
(au moins débits maximaux et volumes) doivent &re étudiés séparbment par une<br />
méthode strictement 1% régressionet'.<br />
On peut élargir la notion d'extension des données en coneidérant non<br />
plus In totalité de l@échantillon des Xi, mais des ensemb<strong>le</strong>s de va<strong>le</strong>urs de ces<br />
données sé<strong>le</strong>ctionnée par des critères statistiques (averse de fr6quence donnée,<br />
par exemp<strong>le</strong>), ou bien cornidbée comme représentant dee situations partidi&-<br />
rement défavorab<strong>le</strong>s eu égard au but poursuivi (précipitation maXima<strong>le</strong> probn-<br />
b<strong>le</strong>, par exemp<strong>le</strong>).<br />
Cae&<br />
une attitude très répandue, qui correspond bien à la con-<br />
ception moderne des crues de projet. Mais el<strong>le</strong> ne - J~B pas s- poser quelques<br />
problèmes, ourtout quand on procède par hydrogramme unitaire. h effet, sur<br />
lfenser;ib<strong>le</strong> des v:trLat?s Xi, une seu<strong>le</strong> peut &tre introduite avec sa fréquence.<br />
Supposons qu'on désigne par XI l'averse décenna<strong>le</strong> et qu'on prenne pour X2 ...<br />
X& des va<strong>le</strong>urs moyennes, ou médianes ; quel<strong>le</strong> probabilité peut-on attribuer<br />
à la crue obtenue par application du modè<strong>le</strong> à l'ensemb<strong>le</strong> des Xi ? On convient<br />
souvent que la probabilité de la crue est la même que cel<strong>le</strong> de l'averse.<br />
C'est certainanent faux, mais dans quel<strong>le</strong> mesure ?<br />
M. Beran, ciana la c odcation qu'il noua soumet IlIltente de ré-<br />
pondre a cette question. La e.ynthèse qu'il propose pour 1'Wdrogmme e& tout<br />
à fait classique. Le choix de l'averse est fait à partir de ia reìation<br />
hauteur-dicrée-fréquence. La distribution des crues obtenues à partir de cette<br />
averse est étudiée par une simulation effectub pour toutes <strong>le</strong>s combinaisons<br />
possib<strong>le</strong>s d'un choix de<br />
- 12 va<strong>le</strong>urs de 1~ durée de l@averBe,<br />
- 3G 8Ch&~ de hyétogmrmee,<br />
- 12 va<strong>le</strong>urs d'un indice d'humidité du bassin.
La conclusion de l'autemr eat que, ai l'on utildm dee va<strong>le</strong>urs<br />
m8dianes pour la durée de Le pluie, la r-ition de cel<strong>le</strong>-ci au esin de<br />
l'averse, <strong>le</strong> taux d'infiltration, la crue obtenue a une fr6quence voiaine<br />
de cel<strong>le</strong> qui a 6tB choidc pour la hauteur de precipitation. La forme àu<br />
hyéto,p-amme ne semb<strong>le</strong> pas jouer un grand A<strong>le</strong>.<br />
429<br />
MM. Kitmaita et Haehimoto 171 exposW?i'au Japon on part de l'ana-<br />
lyse statistique des prboipitations de deux jours pour <strong>le</strong>e petite bassins,<br />
ou de trois JOWS pour <strong>le</strong>s grands. Lee auteurs attachent une très grande<br />
importance à la distribution de la ?hie à l'intérieur de ces interval<strong>le</strong>s ;<br />
l'étudc de cette ustribution sat rai'.? AU RB de temps horaire et ibdécri-<br />
vent une méthode d'élaboration du hy6tograrame de projet qui met en jeu un<br />
"facteur d'ag-an disse ment^^ et un hyétogramme dit lfreprésentatifl1 qui n'est<br />
autre qu'un hyétogrme naturel observé lors d'une averse récente. Autrement<br />
dit, on sé<strong>le</strong>ctionne une "formea* qu'on applique à la hauteur de pluie déter-<br />
minée par l'analyse statistique.<br />
Pour cette détermination, la période de retour choisie dbend en<br />
fait du type c'e projet et des conditions économiques, socia<strong>le</strong>s et politiques<br />
dans <strong>le</strong>squel<strong>le</strong>s il est envisagé. La durée de cette période est de cinq à sent<br />
ans pour un projet d'égout, de vi.?& ans pour un petit bassin urbdn, de<br />
cent rms pow un projet sur une gl
430<br />
se aont tous cantonnb daOs deux aspects particuliers du problème : la trans-<br />
position de l'hydmgramme unitaire et l'utilisation de formu<strong>le</strong>s régiona<strong>le</strong>s<br />
dbivées de la méthode rationnel<strong>le</strong>.<br />
Noue rappe<strong>le</strong>rone que, parmi d'autres, on peut considher la méthode<br />
des courbes enveloppes comme une methode de tramposition géographique, quand<br />
el<strong>le</strong> est assortie d'une "formu<strong>le</strong> de r6gionalisation", come c'eet <strong>le</strong> cas<br />
pour <strong>le</strong>s abaques de I'ranmu-Rodier (Cahier6 O.R.S.T.O.I., ewe hydrologie).<br />
D'autres méthodee pourront 8tre conatmites H partir du catalogue des crues<br />
exceptionnel<strong>le</strong>s de l'U.N.E.S.C.O., lorsque celui-ci pourra enfin voir <strong>le</strong> jour.<br />
De tels catalogues, lorsqu'i<strong>le</strong> comportent dee descriptions sufff-<br />
ates dee cara& Bristiquee climatiques et gbmrphologiquee du baaseiin (sans<br />
toutefoia trop compliquer <strong>le</strong>e C~OSOE)~ constitueraient par <strong>le</strong>ur sede exis-<br />
tence un outil de tout prender choix pour l'bvaluation dee crues en l'absence<br />
de &m&s insuffisantee. Cseat m8me à vrai dire la seu<strong>le</strong> choae qui actuel<strong>le</strong>-<br />
ment fasse vrdnent ahfaut.<br />
Rappelons enfin que la transposition des loi8 de distributions des<br />
crues pourrait atre traitée BOU cette rubrique. Nous avons pr8î8i.h en par<strong>le</strong>r<br />
6 propos de l'analyse etatietique,davantage pour uno queetion de m&hodologie<br />
que dans un souci de préeentation logique.<br />
L'hydr-ogranme unitaire paraît $tre encore, malgré ses dé-<br />
tracteurs, un inetment de choix pour l'evaluation des crues mar <strong>le</strong>e petits<br />
baseins. Noue n'ailone pas ici emtemer une foie de plue une äi~cuseion am<br />
la d6finition de ce derni- terme. Noue amne d6jà parlé de e m utilisation<br />
à un mame batwin, il s'agit maintenant ae voir comment on peut trawposer<br />
<strong>le</strong>e résuitate.<br />
Cette traaepdition est eseentiel<strong>le</strong> dans la m6thoùologie<br />
ConcermELllt <strong>le</strong>e cruce dee petit8 bamuins. ûn 88 bute bien qu'il n'est pas<br />
poesib<strong>le</strong> d'entretenir des rbaux de longue dude sur la totalité des petits<br />
bassiris d'un pays. I1 n'est m8me pas toujours possib<strong>le</strong>, pour chaque petit<br />
projet, de mettre en oeuvre sur <strong>le</strong> bassin correspondant des observations<br />
d'une densité suffisante, pendant une dur& suffieante pur l'application
de l'bydrogrme unitaire au bassin lul-m&ue.<br />
431<br />
Pour <strong>le</strong>e petits bassins, l'inauffisance, et mbe l'absen-<br />
ce tota<strong>le</strong> de donnbs au lieu d'utilisation, est donc la >e. La préparation<br />
des données hydrologiques pour <strong>le</strong>s projets consiste donc à échantillonner<br />
un certain nombre de basains, dits reprbentatifs, correepondaat B un nombre<br />
suffisant de conditione climatiques et morphologiques. Ce erant <strong>le</strong>e résultats<br />
recueillis sur ces baasins qui permettent la mise en oeuvre des différentes<br />
méthodes de trampsition.<br />
H. ~~äier, sa communication [Il] , présente ia mho-<br />
dologie mise en oeuvre par 1'O.R.S.T.O.M. pour <strong>le</strong>s paye tropicaux. Cette mé-<br />
thodologie est bask sur <strong>le</strong>s résultats obtenus par l'exploitation, pendant<br />
des durées éga<strong>le</strong>s ou supérieures A trois ans, de plus de lo0 ensemb<strong>le</strong>s de<br />
bassins représentatifs. El<strong>le</strong> ee rapporte surtout aux zones mahéliemes et<br />
tropica<strong>le</strong>s, mais des résultats sont ¿ga<strong>le</strong>ment disponib<strong>le</strong>8 pour <strong>le</strong>a zones dé-<br />
sertiques et pour <strong>le</strong>e zonee équatoria<strong>le</strong>s.<br />
Les parametree sé<strong>le</strong>ctionnés pour représenter la forme de<br />
l%ydrograme eant :<br />
- <strong>le</strong> temps dû bue (Tb) OU durée du d,esûllment,<br />
- <strong>le</strong> temps de montée (GI,<br />
- <strong>le</strong> rapport K du d6bit de pointe au d.bit moyen de lib-<br />
drogranune de ruissel<strong>le</strong>ment.<br />
Le volume ruisselé est évalué à partir de la hauteur tota<strong>le</strong><br />
de l'averee par l'intermédiaire d'un coefficient de ruissel<strong>le</strong>ment i$.<br />
L'analyse des rewiltats disponib<strong>le</strong>s a permis de lier <strong>le</strong>s<br />
paramètres %, l& et i à certaines caractéristiques gbamorphologiques du<br />
bassin, soit :<br />
- la Burface du bassin,<br />
- une clame de relief (R) d8terminb A partir d'un indice<br />
-<br />
de pente,<br />
une clame de pennbbilit6 6valub 4 l<strong>le</strong>etime.
Lee relations sont prkentbee SOUS fome d'abaques. Suivant<br />
l'&umbation ci-deesus, ces abaques ne tiennent pas compte explicitement du<br />
rd<strong>le</strong> pourtant importaut de la couverture végéta<strong>le</strong>. C'est que, dam <strong>le</strong>s régions<br />
étudiée^, cette couverture v¿&a<strong>le</strong> abend eaeentiel<strong>le</strong>iacsnt de la zone clima-<br />
tique dane laquel<strong>le</strong> se t mwe <strong>le</strong> bmsin. Come <strong>le</strong>s jeux d'abaque eont établis<br />
par Bones climatiques, loensemb<strong>le</strong> tient compte simplicitemciait dea conditions<br />
de végétation. Dane <strong>le</strong>e cas particuliers, il cmvient au epéciaìiste d'gvaluer<br />
ie ltcoup de pouce" à donner pour tenir compte d'une anomalie de ce8 cmditione.<br />
Le coefficient de fome K est &du6 mivant <strong>le</strong>s zones<br />
climatiques, la eurface du baesin, mu8 forme de tab<strong>le</strong>au.<br />
Les abaques fournissent des va<strong>le</strong>ur6 moyennes de6 parmètres<br />
c,iLL ALI^^..: L. 1x1 ir; q 3vc- 'iverse äécr-~qal-.<br />
ans une communication Is] consacrée e o u t à ilinfìuence<br />
du degré d'urbanisation eur <strong>le</strong>s cme~ des petits bassine, M. Hall propose<br />
une méthode de r6gionalisation des hyärogrmmes unitaires. I1 part d'un hydro-<br />
grme unitaire dimension en utilisant comme paramètre d'khel<strong>le</strong> des<br />
-- *<br />
<strong>le</strong> temps 'r retctrd l i (la?). TL est dors exprimé en fonction du<br />
llrapport de bessin" Z O L /p, où 2 e& en km, L est la longueur du cour6<br />
d'eau principal, en km, et S la pente moyenne du cours principal en %.<br />
W. Eelliwell et Chen, dans <strong>le</strong>ur c odcation [4] présen-<br />
tent 6gaìsment une m6thode de traneposition r8giona<strong>le</strong> b a h ar un bydrogrme<br />
BBPB dimmion. Leur pmblhe e& absolument typique du cbantp dgapplication de<br />
la m6thode de l'bydmgramme unitaire. Les rivières de la colonie de Hong Kong<br />
sont très nombreueee pour un ai pctff territah (1 o00 kd de terres) et la<br />
tail<strong>le</strong> de <strong>le</strong>ure baeeina est hidemuent td6 reduite. Il n'est pas concevab<strong>le</strong><br />
d'utiliser dane ces conäitions la fermu<strong>le</strong> classique du réseau hydrologique.<br />
Lea hydrologues de Hong Kong ont donc s6<strong>le</strong>ctionub quelques<br />
baseins reprbentatife dont ilpl ont 6tudi6 en détail <strong>le</strong> comportement hydrolo-<br />
gique. Le8 auteurs dbivent <strong>le</strong>e mathodee d'analyse utiliekm qui mnt d'ail-<br />
<strong>le</strong>iir8 trh claesiques, sauf que <strong>le</strong> pa^ de tamps tde caurt nécesmire (15 am<br />
ou mine) a créé quelques difncuitb par euite de la ree&¿ de6 enregistre-<br />
ments pluviographiquem exploitab<strong>le</strong>s. L'hydmgramme aan~ dimension eet obtenu,<br />
comme chez Hail, en multipliant <strong>le</strong>s ordannies par <strong>le</strong> Lag, en divisant <strong>le</strong>s
abscisses par <strong>le</strong> h g et Bll r ~ e <strong>le</strong> ~ tout t un volume unité.<br />
433<br />
L'analyse a conduit, pour chaque baeain btudi6, à un hydro-<br />
graimne unitaire moy'en, dont <strong>le</strong> tempe de retard (Lag) a 6th mie en relation<br />
avec l'indice L L, /r8, 0ii E est la longueur de la rivière principa<strong>le</strong>, Lc<br />
la distance <strong>le</strong> long de cette rivière entre l'exutoire et <strong>le</strong> point <strong>le</strong> plus<br />
près du centre du bsrasin, 9 la pente moyenne du cour8 principal. Le Ia[:<br />
a été mio auSei en relation avec la siarface du baeain, et cette régression<br />
donne du rede um meil<strong>le</strong>ure corrélation (0,92 contre O,%).<br />
de LI méthode mt;.onnel<strong>le</strong>.<br />
Touteo <strong>le</strong>s formu<strong>le</strong>e prbentkei par <strong>le</strong>s auteurs sont d&riv&s<br />
EIM. Jarmwathma et Pinkayan dkrivent dana <strong>le</strong>ur rapport [6]<br />
<strong>le</strong>s abthode6 de d cul dee crues utiliebea en Thai'lande pour <strong>le</strong>s petits bas-<br />
si-. Apre6 avoir rappelé la pauvreté dea donnke ditiponib<strong>le</strong>e dane ce pays,<br />
ils adreseent quelques crítiquea à ia formuìe rationnel<strong>le</strong> cl as rip^^<br />
Q P C i A et lui préfèrent la fozmu<strong>le</strong> de Mc Math Q p: A C i @/A) "' qui in-<br />
trodult la pente du tasdn.<br />
La codcation de M. Pereira 9 damie entre autres qual-<br />
que8 indíaatione mur i'utilieation de la methode rationnel<strong>le</strong> au Erbil, notam-<br />
ment den va<strong>le</strong>urs du coefficient de miesel<strong>le</strong>ment. L'auteur y donne &a<strong>le</strong>ment<br />
dea rsnseignsme&e Bur <strong>le</strong>@ tsmpe de retour dOpt68 dane ca, pp d-raE <strong>le</strong><br />
type de l*amhgement et l'erivimmemnt, a r l'intensité dee pluies au &&fi,<br />
BUF <strong>le</strong>a P.W.P.,sur l*eatimation des volumes N imelb à partir deer prkipita-<br />
tio-<br />
en U.R.S.S.,<br />
(fonmi<strong>le</strong> du Soi1 Conservation Service).<br />
d e<br />
M. Sokalov 12 hoque l'emploi de l'hydrogrmme unitaire<br />
fait une place plue large & la méthode rationnel<strong>le</strong>, ainsi<br />
qu'à des foruniLee empiriques de la forme
434<br />
où sax est <strong>le</strong> débit maximal spécifique en m 3/s.km 2 , q un paramètre qui exprime<br />
<strong>le</strong> débit spécifique extrême lorsque la gurface A du bassin tend vers zéro.<br />
C est en fait égal à 1 ; n varie de 0,15 - 0,30 pour une crue de fonte de neige<br />
à O,5 - O,7 pour <strong>le</strong>s crues dues à de vio<strong>le</strong>ntes averses loca<strong>le</strong>s.<br />
M. Won [ 141 expose <strong>le</strong>s méthodes utilisées en République de<br />
Corée. I1 propose une formu<strong>le</strong> qui procède .?i la fois de l'hydrogramme global<br />
(sinon utilitaire) et de la méthode rationnel<strong>le</strong> :<br />
- = CY A R/T<br />
qo<br />
dans laquel<strong>le</strong> g, est <strong>le</strong> débit maximal, qo <strong>le</strong> débit avant la crue, C un coeffi-<br />
cient de forme de l'hydrogramme, 9 <strong>le</strong> coefficient de ruissel<strong>le</strong>ment moyen, A la<br />
surface du bassin, R la pluie tota<strong>le</strong>, T la durée de la crue. I1 propose d'autres<br />
formu<strong>le</strong>s concernant <strong>le</strong> temps de concentration, la courbe intensité-durée, la<br />
durée critique de la pluie (t =
435<br />
M. Rendon Herrero a choisi, pour sa codcation bo] un eujet<br />
bien particulier. Il s'agit du transport de sédiments en suspension €tudi6<br />
à l'échel<strong>le</strong> de l'averse. L'auteur met d'abord l'accent sur l'importance des<br />
apports latéraux de sediments (Washload), mitit par l'brosion en nappe (sheet),<br />
soit par <strong>le</strong> ravinement (gully) par rapport aux dát&iaux du lit mis en jeu<br />
durant <strong>le</strong> transport. Lee r6sultatcs sont interpr8t6e par des techniques ana-<br />
logues & cel<strong>le</strong>s de l'hydrogranmie unitaire (s6dimentopame unitaire). Une<br />
application est faite au bassin de Bix<strong>le</strong>r Run (U.S.A.)<br />
CONCLUSION<br />
I1 eet certes Intbeesant de mettre au point des méthodes d'analyse<br />
de plus en plw bborées pour essayer d'amocher <strong>le</strong> moina mal possib<strong>le</strong> <strong>le</strong>s<br />
caract6riatiques des crues et des basses eaux, lorsqu'on dispoae de donnbs<br />
rarea ou peu précisoa. Xais il ne faut pas trop se faire d'illueion BUT la<br />
portée réel<strong>le</strong> de cette tentative, ni oublier que toute la confiance qu'on<br />
peut attribuer d une eetimation rénide dana la quantité d'information , c'est-<br />
&-dire fina<strong>le</strong>ment dane la masee et la qualité des donnéee diqmnib<strong>le</strong>e. La com-<br />
titution de cette infomation n'est pai3 te&&, il est faia de dire qu'el<strong>le</strong><br />
ne pose plus de probl8mes.<br />
Ea matière de cruet3 par exsmp<strong>le</strong>, ce qui fait <strong>le</strong> plus défaut danri la<br />
plupart des paye, e out lorsque <strong>le</strong>s rivihs y epnt diffici<strong>le</strong>s, torrentiel<strong>le</strong>s<br />
et inetab<strong>le</strong>s, c'est une bonne connei~aance des débits dee plus grandes crues<br />
connues. L'organisation d'un service hydrologique efficace n'est pae une petite<br />
affaire : el<strong>le</strong> demande une grande compbtence, un soin de tous <strong>le</strong>s instants et<br />
une certaine aportivité. El<strong>le</strong> demande awai de l'argent et c'eet L4 que rbide<br />
souvent la plus gnrnde difficult).<br />
<strong>le</strong><br />
Notse conolwion aera donc que/meiUeur moyen de supp<strong>le</strong>er d la ca-<br />
rence dee donube hyàrologiques est encore de s'attacher à la Buppremion, ou<br />
tout au moins d la diminution de cette carence.
436<br />
111 - M.A. B<strong>le</strong>uw (kglanä)<br />
E.timation of dedp floods and <strong>the</strong> prob<strong>le</strong>ma of equating <strong>the</strong><br />
probability of rainfall and runoff -<br />
R. DAVIS,bCKSTFZN, C. KIISIEL, N. Fo(zEL (U.S.A.)<br />
A decision - <strong>the</strong>oretic approach to uncertainty in <strong>the</strong> return<br />
period of maximum now volumee using rainfall data -<br />
131 M.J. HALL (U.K.)<br />
Syn<strong>the</strong>tic dthydrograph technique for <strong>the</strong> design of flood<br />
alïepiatlon works in urban areas -<br />
[4] P.R. EEUWIEL, T.H. CEW (~ong-~ozy)<br />
A dimeriPiionlEss unitgraph for Hong-Kong -<br />
[5] P.E. HERBBT, S. VAN BiLúûN, J.P.J OLMER, J.H. HAIL (South Africa) -<br />
Flood estimation bp determination of regional parameters from<br />
limited data -<br />
r<br />
[GI D. JWATHANA, S. PINKAYAN (Thailand)<br />
Practice8 of design flood frequency for epiall Watereheds in<br />
!l%arland -<br />
171 T. K0IWSITA, T. HAsHuIoTo (Japan)<br />
-<br />
Design diecharge derived from design rainXall -<br />
[8] W.N. LEEBE (U.K.)<br />
The um of asmoreci data in estimating T - y ~ar floode<br />
-<br />
-<br />
L91 P.P. Pm!uzl?A (Brasil)<br />
Amesment of deeign noode in bradl -<br />
Eo] o. RIBIDON mmmo (U.S.A.)<br />
-<br />
A method for<br />
-<br />
<strong>the</strong> prediction of W o a d in certain d l<br />
wateraheäm<br />
FI] J.A. ROD= (fime)<br />
Méthodes utilidee gour l'kaiuation dee dbits<br />
-<br />
de crue des<br />
petits cour6 d'eau en r8gione tmpica<strong>le</strong>e
[la I A.A. SOICOIDI7 (U.S.S.R.)<br />
Methods for <strong>the</strong> estimationa of meximum dischargea of snowmelt<br />
and rainfall water vith inaàequate observational &ta -<br />
Ilq A.M. VLADMIROV, A.1. CHEBOTARGv (U.S.S.R.)<br />
Computation of pmbabilietic values of lot flow for ungaugeü<br />
rivers -<br />
T.B. WON (Korea)<br />
A study on maximum flood discharge fondee -<br />
437
ESTIMATION OF FLOODS BY MEANS OF THEIR SILT LOADS<br />
ABSTRACT<br />
Modesto Batl<strong>le</strong> Girona<br />
Dr. Civil Engineer<br />
An empirical and experimental formula of very simp<strong>le</strong><br />
structure is studied, to obtain €he flows of maximum floods in<br />
relation to <strong>the</strong> sediment loads that <strong>the</strong> floods produce, depen-<br />
ding only of <strong>the</strong> maximum size of aridities of <strong>the</strong> channel.<br />
This formula can be useful to study also <strong>the</strong> behaviour of <strong>the</strong><br />
river bed, alluvial volume, and so on.<br />
Se estudia una fórmula empírica y experimental de es<br />
tructura muy simp<strong>le</strong>, para obtener los cauda<strong>le</strong>s de máximas cre-<br />
cidas en función de los arrastres que éstas producen, depen-<br />
diendo Únicamente del tamaño máximo de ’aridos del cauce. Esta<br />
fórmula puede ser Útil también para estudiar el comportamiento<br />
del <strong>le</strong>cho de los rios, volumen de acarreos, etc.
440<br />
ESTIMATION OF FLOODS BY MEANS OF TIIliIR SILT LOADS<br />
Based on <strong>the</strong> physical fact that every flood deposits<br />
a mass of arids whose maximum cliametres are proportional to<br />
<strong>the</strong> magnitude of <strong>the</strong> flood, by means of a reciprocal process<br />
an attempt was made to find a way of estimating tne discharge,<br />
obscrving <strong>the</strong> silt loads produced,<br />
Accordingly a very simp<strong>le</strong> network formula has been<br />
obtained. In a series of 15 tests, <strong>the</strong> prevision of <strong>the</strong><br />
maximum floods that have occurred could be made (correspond-<br />
ing to a return period between 100 and 500 years) WITH Ah'<br />
LRKOK bELOW 13%. To do so, one merely has to know <strong>the</strong> maximum<br />
size of <strong>the</strong> river-bed arids.<br />
Besides being a new instrument to calculate floods,<br />
this formula may, as indicated iii <strong>the</strong> "Summary" , open up an<br />
interesting field of investigation regarding mobility of <strong>the</strong><br />
river beds, volume of bed-loads, etc,<br />
1. - FORMULA f'llOPOSEI3<br />
1.1. WORK SCHEME, -<br />
On <strong>the</strong> one hand, <strong>the</strong> maximum silt load diametre<br />
is function of <strong>the</strong> flood Jischargc.<br />
On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> arids are moved by <strong>the</strong><br />
force of <strong>the</strong> silt load, wnicii is proportional to <strong>the</strong> gradient<br />
arid to tho draft.<br />
Considering <strong>the</strong> above two factors, a formula was<br />
sought which related <strong>the</strong> diametre of <strong>the</strong> deposited arid, with<br />
<strong>the</strong> draft and gradient. The ma<strong>the</strong>matical deduction of this<br />
relation is however inaccessib<strong>le</strong> and a semi-empirical formula<br />
was sought, verifying it and deducting <strong>the</strong> unknown values of<br />
same, by neans of experimentation.<br />
Once a relation was obtained between <strong>the</strong> diametre<br />
of <strong>the</strong> arids, tlie.gradiarit aiid <strong>the</strong> draft, this could be<br />
deducted from <strong>the</strong> previous ones, thus defining <strong>the</strong> maximum<br />
<strong>le</strong>vel obtained by <strong>the</strong> flood waters. Since <strong>the</strong> bed-section<br />
is also known, <strong>the</strong> discharge of <strong>the</strong> flood which has borne<br />
along <strong>the</strong> arid through this section, depositing it immediately<br />
downstream, can moreover be obtained.
441<br />
Once <strong>the</strong> purpose of <strong>the</strong> study was specified,<br />
a formula liad to be proposed which would relate maximum<br />
diametre-draft-gradient. The probing was systematized,<br />
and <strong>the</strong> proposed formula was verified and as already<br />
mentioned, <strong>the</strong> unknown coefficients of same wcre verified<br />
with a series of 15 samp<strong>le</strong>s or tests. The margin of error<br />
obtained was found and compared with o<strong>the</strong>r existing methods,<br />
The return period of <strong>the</strong> flood-waters calculated with <strong>the</strong><br />
formula, was sought, defining an inferior limit, The possib<strong>le</strong><br />
limitations of <strong>the</strong> formula due to <strong>the</strong> petrography of <strong>the</strong> arids,<br />
<strong>the</strong> morphology of <strong>the</strong> basins used or <strong>the</strong> non-existence of<br />
certain sizes of arid, were studicd. Finally, <strong>the</strong> conclusions<br />
drawn are summarized, All <strong>the</strong> documentation involvcd in <strong>the</strong><br />
tests, regarding diametres of arids and drafts observed, was<br />
col<strong>le</strong>cted photograpliical ly .<br />
1.2. - MATilEMATIC OBTENTION OF TiíE PROPOSGL) FOIMULA<br />
Une tried to reach a formula, deducing it ma<strong>the</strong>-<br />
matically from <strong>the</strong> silt load force equàtions, but <strong>the</strong><br />
influence on <strong>the</strong> larger arids cannot be defined quantitatively,<br />
nor can <strong>the</strong> protector inter-action which <strong>the</strong> silt loads offer<br />
between <strong>the</strong>mselves, in <strong>the</strong> face of <strong>the</strong> dynamic thrust of <strong>the</strong><br />
current.<br />
Various hypo<strong>the</strong>sis were used, but <strong>the</strong> subsequent<br />
elaboration did not crystallize into any practical formula.<br />
Ori <strong>the</strong> o<strong>the</strong>r hand, adopting one or ano<strong>the</strong>r hypo<strong>the</strong>sis as base,<br />
produced inadmissib<strong>le</strong> differences of above 100%.<br />
In view of <strong>the</strong> above, it was decided to employ a<br />
semi-empirical formula, Its structure was obtained matliematically<br />
but it has been verified and defined from <strong>the</strong> experiments made.<br />
1.3. - UL.1)UCING A SEMI-EMPIRICAL FORMULA<br />
To obtain <strong>the</strong> formula in question, various methods<br />
were applied: a) .- considering <strong>the</strong> dynamic thrust on <strong>the</strong> arid;<br />
b).- balancing <strong>the</strong> silt load forces, The liermanek aiid<br />
Manning formulae were likewise used in one case or <strong>the</strong> o<strong>the</strong>r<br />
to determine <strong>the</strong> mean velocity,<br />
When using <strong>the</strong> Manning formula, <strong>the</strong> possibility was<br />
considered of <strong>the</strong> rugosity of <strong>the</strong> bed 'In" b e in g prop or t i ona 1<br />
to one sixth <strong>the</strong> power of <strong>the</strong> arid diametre. ïliis is correct<br />
in canals, but ir1 natural beds, <strong>the</strong> most accepted formulae of<br />
river hydraulics (I<strong>le</strong>rmanek, Christen, Wiiikel, etc.) do without<br />
<strong>the</strong> rugosity or adopt a constant value of same.<br />
Equations were reached through different channels,<br />
with identical structure:<br />
0b<br />
Ila = -.<br />
u. 1
442<br />
but in which <strong>the</strong> exponents a and b varied in terms of<br />
<strong>the</strong> velocity distribution law, adopted. The degree of<br />
parabolic speed distribution is normally one seventh,<br />
and it was with this value that a arid b were calculated.<br />
The values of "a" and "b" were also deduced in <strong>the</strong> hypo-<br />
<strong>the</strong>sis of supposing a one ninth degree distribution.<br />
ïiie results were:<br />
Degree of <strong>the</strong> speed<br />
distribution parabola 1/9 1/7<br />
E X P O N E N T a b a b<br />
Hermane k 1,28 O,78 1,21 0,71<br />
Manning: n = Cte<br />
Manning: n = Cte 0<br />
10<br />
- Manning: 11 = Lte<br />
Manning: 11 Cte 0 1/6<br />
1,11 0,78 1,OS<br />
1~11 1#11<br />
1,13 1,OO 1,O8<br />
1,13 1,33 1,08<br />
0,72<br />
1#O5<br />
1,OO<br />
1,33<br />
Never<strong>the</strong><strong>le</strong>ss, <strong>the</strong> empirical<br />
-<br />
formula proposed and<br />
verified with tests, was, as will be seen later on:<br />
00,s<br />
. 11<br />
u.i<br />
The formulae obtained in <strong>the</strong> previous tab<strong>le</strong> (where<br />
<strong>the</strong> later experimental definition of B) is required,)<br />
iiave no faithful tradition in <strong>the</strong> reality of <strong>the</strong> river<br />
beds, as <strong>the</strong>y give different values to those of <strong>the</strong><br />
said formula where "a" =<br />
The most approximate<br />
In this, m = 7.<br />
1.4, - DLFINITION OF THE PROPOSED FORMULA<br />
The formula we are trying to verify, wkeby on<br />
experimentam obtaining <strong>the</strong> value of B, this value<br />
should be practically constant, was:<br />
1/2<br />
ii = am<br />
13. i
II = Draft expressed in centimetres,<br />
flm = blaximum diametre of <strong>the</strong> arid in centimetres,<br />
i = Gradient<br />
ki = Coefficient to be defined.<br />
44 3<br />
In each test, knowing <strong>the</strong> draft 11, obtained directly<br />
or from <strong>the</strong> discharge, in a sector, wliicli was termed "control",<br />
tlie 0 of tlie arids deposited in <strong>the</strong> area, if possib<strong>le</strong><br />
immedyately downstream of same, was defined, calculating:<br />
i/ 2<br />
B = am<br />
li. i<br />
The series of 15 tests undertaken, permitted a verification<br />
that coefficient B is almost constant; its value could be<br />
calculated, and at <strong>the</strong> same time tlie exponent 1/2 of 0 was<br />
found to be most suitab<strong>le</strong>.<br />
m<br />
2.- RESULTS OBTAINED, -<br />
2.1. - MET1iOL)OLOGY<br />
First of all, a "control section" must be defined. Knowing<br />
<strong>the</strong> maximum historic flood in a prudential period, <strong>the</strong> draft<br />
corresporiding to H was calculated in it. This iì of <strong>the</strong> control<br />
section, was in certain cases measured directly after some<br />
important flood, through <strong>the</strong> traces of undergrowth and residues<br />
that <strong>the</strong> current <strong>le</strong>ft in tlie river bed shrubbery 0. The gradient<br />
i of <strong>the</strong> span corresponding to <strong>the</strong> control section was sought,<br />
and <strong>the</strong> maximum diametre 0 of <strong>the</strong> arids deposited downstream<br />
in this section was meacurgd. With this information, <strong>the</strong><br />
following was calculated: 1/2<br />
B = k'm tl, i<br />
Control Section:<br />
The control section should be sited in stretches with as<br />
uniform system as possib<strong>le</strong>. Consider <strong>the</strong> influence of bridges,<br />
etc. The arids should c<strong>le</strong>arly define 6,.<br />
Gradients:<br />
The "i" adopted, is that of <strong>the</strong> stretch from 1 to 1,s kms.,<br />
immediately upstream from <strong>the</strong> control section, It is obtained<br />
from plan 1:50.000 of <strong>the</strong> Geographic and Cadastral Institute,<br />
rounding off, if iiecessary, <strong>the</strong> excessive twists in <strong>the</strong><br />
longitudinal section.<br />
- Draft:<br />
The "H" draft is measured from <strong>the</strong> lowest reading of tlie<br />
control section,
444<br />
izlaximiim diametre:<br />
The arids probe area for defining Øm will be chosen<br />
downstream <strong>the</strong> "cnntrol section" and as ncar to this as<br />
possib<strong>le</strong>, so that <strong>the</strong> sizes observed are effectively <strong>the</strong><br />
largest that have passed through this section,<br />
'To define <strong>the</strong> maximum diainetre, only those rounded<br />
or paral<strong>le</strong>l-epipedical shape arids will be suitab<strong>le</strong> , whose<br />
smal<strong>le</strong>st dimension is at <strong>le</strong>ast 2/3 (two thirds) <strong>the</strong> largest<br />
orie, The 0 value will be <strong>the</strong> mean of <strong>the</strong> largest two<br />
dimensions of <strong>the</strong> arid.<br />
The suitab<strong>le</strong> arids for defining 0 should be found<br />
with a minimum density of 1 per every pwo square metres.<br />
Uy density, we understand <strong>the</strong> number of units in sight,<br />
per river bed surface, In some cases, this may even drop<br />
to 1 per 4 square metres.<br />
The choice of arids on which <strong>the</strong> 0 is going to be<br />
measured, demands a careful, critical judgement on same,<br />
considering <strong>the</strong> possibility of it coming from <strong>the</strong> erosioned<br />
sides and not upstream, that <strong>the</strong>y may pertain to demolished<br />
works, or under construction, etc, Certain geological know-<br />
<strong>le</strong>gge of <strong>the</strong> area will always prove most useful. Any kind<br />
of rock excepting slate is suitab<strong>le</strong>.<br />
It should be emphasized that in minimum densities ,<br />
relation 2/3 of dimensions, etc., a qualitative common sense<br />
should always preside over ali rigorist criterion.<br />
It is important to take photographs with sca<strong>le</strong>s which<br />
will act as referenee so as to compare <strong>the</strong> field observations<br />
at <strong>the</strong> office.<br />
One must remember that <strong>the</strong> prob<strong>le</strong>m consists in a trans-<br />
port through <strong>the</strong> control section of <strong>the</strong> arids, which will be<br />
deposited immediately afterwards, and that <strong>the</strong>y may even be<br />
covered by o<strong>the</strong>r finer ones, which have sett<strong>le</strong>d when <strong>the</strong> flood<br />
waters dropped.<br />
Qu a 1 i t y :<br />
To have a p;rudeiit judgement of <strong>the</strong> suitability of <strong>the</strong><br />
tests made, <strong>the</strong>y have been classified into GOOD (G), MEDIUM (M)<br />
and FAIR (F) , in accordance with <strong>the</strong> guarantee deserved by<br />
<strong>the</strong> definition of <strong>the</strong> data obtained li, am and i.<br />
pemarks :<br />
1.- In low river-bed spans, with very slight gradient, it<br />
is often difficult to define this on plans 1:50.000<br />
and ori a smal<strong>le</strong>r sca<strong>le</strong>, <strong>the</strong> bottom oscillations are<br />
excessive. In <strong>the</strong>se cases, <strong>the</strong> river is also usually
very wide, and <strong>the</strong> transversal section presents siiarp<br />
relative off-<strong>le</strong>vels. In this case, <strong>the</strong> definition of 11<br />
must be closely examined, to avoid falling into errors.<br />
2.- In some cases, owing to <strong>the</strong> type of alluvial terrace<br />
in wliicli <strong>the</strong> bed is fouiid, <strong>the</strong> prehistoric and mil<strong>le</strong>nary<br />
arids cannot be differentiated from those corresponding<br />
ti1 <strong>the</strong> latest historic floods.<br />
2.2. KLSULTS OF TIIE TESTS.-<br />
Below, <strong>the</strong> data i, 1i and id obtained in each test, anci<br />
<strong>the</strong> resulting value of B are given:<br />
Test Nomenclature and Site i 11 0 yual Coeff<br />
;.i O ím) (CH) ity icient b<br />
445<br />
1 K. Llémana at crossroads 0,00822 2,81 18 M 1,84<br />
Main road S,Martfn de<br />
Llémaria<br />
2 R. Llémana in Sta, Afra 0,00590 3,45 19 li 2,13<br />
3 ‘<strong>le</strong>r in S.Julián de Kamis 0,00725 3,73 29 F 2,OO<br />
4 Ter in S.Julián de Kamis 0,00725 5,63 60 I: 1,90<br />
5<br />
6<br />
Ter in out<strong>le</strong>t of <strong>the</strong> Dar6<br />
Uñar at crossroads N-II<br />
Madri d - F r an c e li i g hw ay<br />
0,00100<br />
0,00300<br />
7,93<br />
5,42<br />
25<br />
10<br />
F<br />
G<br />
2,OO<br />
1,94<br />
7 R.Uerneda at bridge Riudellots,maximum<br />
flood 0,00237 5,lO 8 G 2,33<br />
8 K.berneda at bridge Riudellots,<br />
flood 11-X-70 0,00237 4,OO 5 G 2,05<br />
9 Tordera in Sai1 Celoni 0,00835 2,52 16 G 1,90<br />
10 Tordera in Tordera 0,00320 2,85 4 G 2,20<br />
11 R. Rifer in San Celoni 0,00120 1,05 7.5 G 2 , 16<br />
12 Corigost in La Carriga 0,00880 2,90 28 M 2,07<br />
13 Cardoner in Manresa 0,00435 4,65 18 G 2,09<br />
14 Llobregat in S.Vicente<br />
de Castel<strong>le</strong>t 0,00345 5,85 22 G 2,32<br />
15 Llobregat in Martorell 0,00209 7,65 10 G 1,98
446<br />
2.3.- VERIFYING THE EXPONENT OF QIm.-<br />
In princip<strong>le</strong>, <strong>the</strong> value 1 was adopteù as exponent of OmD<br />
proposing tiie formula: z<br />
-<br />
0,s<br />
am<br />
li<br />
r<br />
ìiowever, <strong>the</strong> value 0,s was later esteemed and its worth<br />
confirmed as <strong>the</strong> tests were made, owing to <strong>the</strong> scanty dispersion<br />
presented by <strong>the</strong> values of B obtained.<br />
The possibility of <strong>the</strong> dispersion of <strong>the</strong> resulting 13 being<br />
<strong>le</strong>ss with ano<strong>the</strong>r exponent , is consequently likely.<br />
dased on <strong>the</strong> values i, li and 0,,, obtained in <strong>the</strong> tests, <strong>the</strong><br />
b coefficient has been calculated for <strong>the</strong> following exponents<br />
of 0,,,: 0,4ü - 0,45 - 0,50 - 0,55 and 0,67.<br />
,A The typical<br />
E<br />
e<br />
0.55<br />
o<br />
u 0.50<br />
derivations obtained are:<br />
for B = 0, 0140/i H - T = 0,195<br />
for U = 0, i 11 -G = 0,175<br />
for B = 0m H - G = 0,143<br />
for U = 0, li - TT = 0,189<br />
8 0.45<br />
for B = 0, OaU7/i H - o = 0,484<br />
J<br />
0.40..<br />
- Desviación típico<br />
. 0.2 0.3 0.4 0.5<br />
0.35<br />
1 7<br />
0.1<br />
In <strong>the</strong> figure, <strong>the</strong> aforementioned results are represented,<br />
and it can be seen how <strong>the</strong> minimum typical deviation corresponds<br />
to <strong>the</strong> 0,s thus verifying that this value is <strong>the</strong> most suitab<strong>le</strong><br />
as exponent of 0,.<br />
2.4 - CALCULATING THE COEFFICIENT U. -<br />
Having obtained <strong>the</strong> value of U for each of <strong>the</strong> 15 tests madc,<br />
what is proposed for <strong>the</strong> famula in question must be defined.<br />
In order to consider <strong>the</strong> quality of each test in <strong>the</strong> determination<br />
of B, by means of a prodent mean, <strong>the</strong> ones qualified as good<br />
(E) are assigned weight 3, <strong>the</strong> "mediumtv (M) , are given weight 2,<br />
and <strong>the</strong> "fair" (F) , weight 1. Tnus we get:<br />
B = 2,08lY2,08
-<br />
The value of <strong>the</strong> coefficient L4 adopted is:<br />
i3 2,08<br />
Whereby <strong>the</strong> formula proposed will be:<br />
2,08 . i<br />
li = maximum draft in centimetres (see 2.1)<br />
Bm= maximum diametre in centimetres (see 2.1)<br />
i = gradient (see 2.1)<br />
2.5 SIGNIFICATION wrE OF TIE TEST SERIES.-<br />
447<br />
The "signification rate" of tlie test series made, or in o<strong>the</strong>r<br />
words tlie quality of <strong>the</strong> <strong>who<strong>le</strong></strong> ensemb<strong>le</strong> of same and <strong>the</strong> mean i =<br />
2,08 was obtained. Accordingly, tiic likiliood of ano<strong>the</strong>r value<br />
of B, obtained as a result of a new series of tests, being<br />
within specific limits, was calculated.<br />
- (mean) aiid ';r (typical deviation) be <strong>the</strong> monthly<br />
characbgigtics of <strong>the</strong> series of Values of B obtained; tlie number<br />
of representative tests is n = 15.<br />
In tiie interval of possib<strong>le</strong> values of B included between:<br />
cr<br />
and Y + tp .-<br />
if we choose a percentage of probability P (ex. p = 1%) where<br />
tiie value of <strong>the</strong> medn U of a new series of tests is outside <strong>the</strong>se<br />
limits, in <strong>the</strong> tab<strong>le</strong> of function of Student for this 1% and n-1<br />
= 14 degrees of freedom, a value of tp is obtained with which<br />
<strong>the</strong> above mentioned "confidence interval" is defined. The<br />
probability p is <strong>the</strong> "signification <strong>le</strong>vel".<br />
-<br />
x = 2,OG; q= 0,143<br />
,* -<br />
- = 0,03823<br />
m<br />
According to tlie Student t8t" function tab<strong>le</strong>, we get:
448<br />
sigriification Value of tp<br />
Confidence limits<br />
G<br />
<strong>le</strong>vel p for 14 degrees t p . m Be 1 ow Above<br />
of freedom<br />
i¿-tp.m G - X*tp.fl=$y o-<br />
~~~~~ ~<br />
__ -~ ~~~~~<br />
0,l % 4,140 0,16 1,90 2,22<br />
1,o % 2,977 1,11 1,95 2,17<br />
2,o % 2,624 0,lO 1,96 2,16<br />
5,o % 2,145 0,08 1,98 2,13<br />
Thus, <strong>the</strong> probability that <strong>the</strong> mean of a new test series<br />
is between 1,95 and 2,17 is 99%.<br />
Let us set this conclusion out in terms more befitting<br />
our prob<strong>le</strong>m.<br />
The confidence interval between 1,95 and 2,17 admits a<br />
possibility below 1 %, that <strong>the</strong> 13 obtained in a new series is<br />
1,95; this represents a discrepancy of 0,11 in respect of <strong>the</strong><br />
mean of 2,OG of <strong>the</strong> experimented series.<br />
This difference of 0,11 means an error of<br />
o 11 I<br />
2,06 *'Oo' = 5B3 %<br />
in <strong>the</strong> appreciation of <strong>the</strong> drafts. Let us see how much this is<br />
?il<strong>le</strong>n translated into flood water discharges:<br />
and if <strong>the</strong> section is approximately rectangular T I1 , Whereby:<br />
Ob7' i1l2,ki.b = 0,34 b.i 1/2 , 1i1,75<br />
Qi = 0,34 li ,<br />
Thus an erro,,i2 draft of 5,3% multiplies <strong>the</strong> discharge<br />
obtained by 1,053 = 1,095 which means an error of 9,5%.<br />
Therefore, <strong>the</strong> quality of <strong>the</strong> series of tests made and<br />
consequently that of <strong>the</strong> B value adopted, can be defined as<br />
follows:<br />
The probability<br />
-<br />
that in a new series of tests, <strong>the</strong> value<br />
of B defined for <strong>the</strong> formula proposed means an error in <strong>the</strong><br />
determination of discharges, below 9,5%, in respect of those<br />
obtained with B 2,08, is 99%.
2.6. - MAXIMUM ERROR AND COMPARISON WITH OTIIER METIfODS. -<br />
In practice, to determine <strong>the</strong> maximum historic flood,<br />
applying <strong>the</strong> proposed formula, a certain number of tests<br />
should be made in certain o<strong>the</strong>r control sections of <strong>the</strong> bed,<br />
and finally obtain a mean, In this way, <strong>the</strong> inevitab<strong>le</strong> errors<br />
and discrepances will be compensated for, and which will take<br />
place on defining <strong>the</strong> gradient (i) and in particular <strong>the</strong><br />
maximum diametre (e,) with which to enter into <strong>the</strong> formula.<br />
The number of verifications will depend on <strong>the</strong> exactness<br />
to be obtained, and on common sense, in face of <strong>the</strong> data<br />
defined in each "control section". On an average, four tests<br />
may prove sufficient.<br />
Supposing that all <strong>the</strong> tests made correspond to a same<br />
bed, we can find <strong>the</strong> maximum possib<strong>le</strong> error by mixing<br />
toge<strong>the</strong>r <strong>the</strong> results of <strong>the</strong> <strong>who<strong>le</strong></strong> series.<br />
The five tests with highest B values are 2,33<br />
2,20 -2,16 - - 2,32 -<br />
2,13; <strong>the</strong> mean is B 02~23.<br />
Similarly, <strong>the</strong> 5 tests with lowest B values are 1,90 -<br />
1,34 - 1,98 - 2,OS - 2,09; <strong>the</strong> mean is B = 1,99.<br />
In both series, those tests classed as "medium" (M) and<br />
"fair" (F) have been omitted,<br />
The difference between <strong>the</strong>se means and <strong>the</strong> value B = 2,08<br />
of <strong>the</strong> formula is:<br />
2,23 - 2,08 a 0,15<br />
2,08 - 1,99 = 0,09<br />
Considering <strong>the</strong> most unfavourab<strong>le</strong> case where <strong>the</strong> tests<br />
have given <strong>the</strong> 5 highest values of 8, whereby <strong>the</strong>ir mean would<br />
be B = 2,23 instead of U = 2,08, this means an error in draft<br />
appreciation of ;.tW O 15 .lo0 = 7.2%.<br />
-<br />
As seen in 2.5, this indicates that <strong>the</strong> real discharge<br />
has been multiplied by:<br />
1 , 0 7 2 ~ ~ 1,129 ~ ~ 1,13<br />
which means a 13% error,<br />
To conclude: THE MAXIMUM ERROR OBTAINED ~ViiliN ESTIMATING<br />
'ï1ii.i IIISTORIC FLOOD DISCHARGE, ACCORDING TO VERIFICATIONS MADG<br />
WITH THE PROPOSED FORMULA, IS 13%.<br />
O<strong>the</strong>r calculation methods:<br />
Wit11 t h statistical method, <strong>the</strong> errors for return periods<br />
449
450<br />
above 50 years, may be around 20 to 3090.<br />
Specifically in <strong>the</strong> floods study of <strong>the</strong> Congost river<br />
made at <strong>the</strong> liydrographic Confederation of tlie East Pyrenees ,<br />
by <strong>the</strong> Measurerncnts Service, using <strong>the</strong> historia method, and<br />
for a 50 year return period, a discharge of 160 m3./sec. is<br />
obtained with <strong>the</strong> Gumbel method and 135 rn3./sec. with ano<strong>the</strong>r<br />
law of distribution, which means a difference of 20% between<br />
iaoth methods. Applying <strong>the</strong> tational method to <strong>the</strong> same study,<br />
biie gets a discharge of 305 m3./sec. for <strong>the</strong> same return period.<br />
2.7. - RETURN PERIOD<br />
The return period of <strong>the</strong> floods of <strong>the</strong> various tests<br />
was also studied, to try and relate it with <strong>the</strong> discharges<br />
foreseen with <strong>the</strong> formula, and at <strong>le</strong>ast obtain a lower limit<br />
of samc.<br />
The return period of <strong>the</strong> floods for which <strong>the</strong> formula<br />
has been verified is:<br />
Test 1 -----__----_-_- 400 years<br />
Test 2 -------_-_---_- 400 years<br />
Test 3 -_----_----_--- 95 years<br />
Test 4 --------------- 95 years<br />
Test 5 --_------_---_- 95 years<br />
Test 7 -_------------- >70 years<br />
Test 8 -__------_----- >70 years<br />
Test 9 ---------_----- 90 years<br />
Test Il--------------- 1000 years<br />
Test 12--------------- 160 years<br />
Test 13--------------- 180 years<br />
Test 14--------------- 180 years<br />
Test IS--------------- 180 years<br />
In face of <strong>the</strong>se figures, it would appear that <strong>the</strong> lower<br />
limit of tlie return period is 100 years,<br />
ilowever, it must be remembered that this conclusion is<br />
based on a series of i3 tests.<br />
As indicated in 2.8, <strong>the</strong> influence of wear throughout time<br />
is very scarce and does not change <strong>the</strong> return period of <strong>the</strong><br />
floodwaters,<br />
It can <strong>the</strong>refore be said that <strong>the</strong> fbod obtained generally<br />
has a return period between 100 and 500 years.
2.3. - STUDY OF TIIE POSSIBLE LIMITATIONS<br />
Due to <strong>the</strong> petrography of <strong>the</strong> arids observed and <strong>the</strong>ir<br />
possib<strong>le</strong> erosion:<br />
There is a possibility that <strong>the</strong> determination of <strong>the</strong><br />
maximum diametre d has only been made with certain rock<br />
types, as <strong>the</strong> o<strong>the</strong>fs were excessively worn by <strong>the</strong> erosion,<br />
If this has occurred, in o<strong>the</strong>r areas where <strong>the</strong>re are no<br />
arids of <strong>the</strong> most resistent types, false results of 0,<br />
could be obtained,<br />
To approach this prob<strong>le</strong>m, <strong>the</strong> petrographic classification<br />
of <strong>the</strong> arids was made, based on <strong>the</strong> photos obtained<br />
in <strong>the</strong> determination of <strong>the</strong> 0, of cach test, and which<br />
defined 0, with <strong>the</strong> following symbolics:<br />
A - Sandstone Co - Conglomerates Gn - gnes<br />
B - Basalts G - Granite and P - Slate<br />
C - Limestone<br />
eruptive rock Q - Quarzite<br />
The types of arid used in cach determination of <strong>the</strong> Ibrn<br />
were:<br />
451<br />
‘rest 6 ------------ 2A + 2G + 3Q<br />
Test 7 ------------- A + B + 2Co+ G + P + Q<br />
Test 8 ------------ 3B + 2G<br />
Test lo------------ (2A + B + CO + 2G * 34<br />
(A + B + G + 4Gn + 2A.<br />
Test Il------------ íG + 34<br />
(3A + 44<br />
Test 12- - - - - - - - - - - - 2A + 2G + Cn + 2q<br />
Test IS------------ SA + 2C + 2G + 3Gn + Q<br />
Test 14------------ 2A + 4C<br />
Test 15------------ SA + 14C + 24<br />
Test lb------------ (2A + 2C + Gn<br />
(2C + P<br />
Making a tctal used of: 26A + 13U + 32C + 3Co + 16G +<br />
+ 9Gn + 22Q<br />
In view of <strong>the</strong> above results! evidently <strong>the</strong> rock type<br />
does not influence <strong>the</strong> determination of <strong>the</strong> diametre, since<br />
all <strong>the</strong> classes appear as maximum arids (8 ) in considerab<strong>le</strong><br />
number (except <strong>the</strong> slates which will be dipcussed below) and<br />
it can <strong>the</strong>refore be supposed that <strong>the</strong> erosion, for <strong>the</strong> effects<br />
of <strong>the</strong> method adopted whep determining <strong>the</strong> 0, iii a way affects<br />
any type of mtk.
452<br />
This is because <strong>the</strong>se arids are not from <strong>the</strong> bed downstream<br />
and <strong>the</strong>y consequently only suffer <strong>the</strong> effects of <strong>the</strong> river<br />
erosion with high waters or normal flooding, which take place<br />
intermittently arid not very frequently. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong><br />
large adjustment obtained in <strong>the</strong> value of B makes one suppose<br />
<strong>the</strong> influence of <strong>the</strong> erosion in tlie various arids could not be<br />
important.<br />
It must however be emphasized tiiat <strong>the</strong> slate, as definers<br />
of tlie 0 only appear twice, as a logical consequence of <strong>the</strong>ir<br />
greater tensitivity to <strong>the</strong> environment, as is deduced from <strong>the</strong><br />
above tab<strong>le</strong>. For this reason, what lias been said in <strong>the</strong> above<br />
two paragraphs cannot be applied to <strong>the</strong> slate arids.<br />
We can consequently say that <strong>the</strong> definition of Om, is<br />
independent of <strong>the</strong> arid petrography except in <strong>the</strong> case of slates,<br />
which should not be used for <strong>the</strong> determination.<br />
ihe to <strong>the</strong> region studied:<br />
All <strong>the</strong> tests made have been in <strong>the</strong> provinces of Barcelona<br />
and Gerona. Therefore extrapolation to ano<strong>the</strong>r type of basin<br />
could present doubts,<br />
In <strong>the</strong> enclosed tab<strong>le</strong> , <strong>the</strong> chief geographic characteristics<br />
of <strong>the</strong> tested basins are indicated.<br />
However, it should be stressed that <strong>the</strong> only thing that<br />
can change <strong>the</strong> validness of <strong>the</strong> proposed formula, is <strong>the</strong> arid<br />
which defines 0, and according to <strong>the</strong> above paragraph, it has<br />
been considered that <strong>the</strong> formula is valid for any type of<br />
rock (except slate).<br />
On <strong>the</strong> o<strong>the</strong>r hand, although not definite, it is most<br />
significant that in a later test made in <strong>the</strong> Guaro river of<br />
tlie basin in Sou<strong>the</strong>rn Spain, near Vé<strong>le</strong>z-Malaga, <strong>the</strong> value of<br />
<strong>the</strong> U coefficient obtained is:<br />
-<br />
B = 2,14<br />
whereas that adopted is B 2,08 which reprecents a 3% error.<br />
Although not conclusive, this result opeiis up a hopeful field,<br />
awaiting an extension of <strong>the</strong> tests to o<strong>the</strong>r regions.<br />
üue to <strong>the</strong> lack of arids:<br />
Thecase may arise of <strong>the</strong>re being no arids of a diametre<br />
superior to a certain size, as <strong>the</strong>y do not exist in <strong>the</strong> bed or<br />
because <strong>the</strong>y are retained by some dam or weir. The lack of such<br />
arids does not produce any change since <strong>the</strong> floods correct <strong>the</strong><br />
gradient of <strong>the</strong> river according to <strong>the</strong> existing sizes. In short<br />
stretches where <strong>the</strong> local effect of a weir modifies tlie gradient,<br />
this should be taken into account,<br />
b
The formula should be applied to river-beds whose<br />
possib<strong>le</strong> mobi<strong>le</strong> bottom during <strong>the</strong> flood later permits <strong>the</strong> maximum<br />
arids deposited by <strong>the</strong> flood peak to be discovered.<br />
If a mobi<strong>le</strong> bottom of considerab<strong>le</strong> thickness is produced,<br />
by means of burrows, <strong>the</strong> thickest deposits made by <strong>the</strong> flood<br />
peak should be reached, aiid if <strong>the</strong> sediment thickness is very<br />
important , <strong>the</strong> gradient adopted should be corrected. However ,<br />
none of this kind have been experienced in <strong>the</strong> tests.<br />
453
O<br />
4<br />
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VI<br />
P,<br />
E-<br />
t.4<br />
o><br />
><br />
.FI<br />
d<br />
x<br />
+J<br />
.rl<br />
rl<br />
O<br />
U<br />
a<br />
a<br />
fl<br />
3<br />
d<br />
0<br />
3-<br />
V<br />
+<br />
45
3.- SUMMARY<br />
To obtain <strong>the</strong> draft (H) that has been produced in <strong>the</strong><br />
maximum historic floods, knowing <strong>the</strong> size of <strong>the</strong> arids (Om)<br />
which may have been hau<strong>le</strong>d along by this flood-water,<br />
across a section termed "control", and <strong>the</strong> bed gradient (li)<br />
<strong>the</strong> following formula is proposed:<br />
o, 5<br />
which should be experimentally checked when <strong>the</strong> coefficient<br />
U is defined.<br />
455<br />
The above formula could not be obtained ma<strong>the</strong>matically;<br />
The most reached expressions of type:<br />
- <strong>the</strong> one nearest <strong>the</strong> proposed one is that where a 1<br />
0,67.<br />
and b =<br />
To define U, a series of 15 tests was made, obtaining<br />
B = 2,08.<br />
There was a possibility of <strong>the</strong> proposed formula not being<br />
correct, which would occur if <strong>the</strong> value of B obtained in <strong>the</strong><br />
tests was variab<strong>le</strong>. However <strong>the</strong>se values ali varied around 2,33<br />
and 1,84. The typical deviation of <strong>the</strong> series was = 0,143,<br />
The series also helped to contrast <strong>the</strong> favourability of <strong>the</strong><br />
exponent of 0, since <strong>the</strong> 0,s produces <strong>the</strong> minimum typical<br />
.devi at i on.<br />
The "signification <strong>le</strong>vel" of <strong>the</strong> mean of <strong>the</strong> series of<br />
tests made, corresponding to <strong>the</strong> "confidence interval" between<br />
B = 1,95 and B = 2,17, is 99%. Expressed in o<strong>the</strong>r terms, it<br />
means that <strong>the</strong> l ikgmd of ano<strong>the</strong>r value of B, defined by a<br />
new series of tests, having ari error in obtaining discharges,<br />
<strong>le</strong>ss than 9,5% (regarding those obtained with B = 2,08) is 99%.<br />
THE MAXIMUM ERROR IN DETERMINING DISCHARGES, ACCORDING TO<br />
TIIE SERIES OF 15 TESTS, IS 13%.<br />
The formula proposed is <strong>the</strong>refore:<br />
O m a ~ 5<br />
li =<br />
2,08.i<br />
li - Draft of <strong>the</strong> maximum historic flood in crns. defined<br />
according to 2.1.<br />
0,- Maximum diametre of <strong>the</strong> arid in crns. defined according to 2.1.
456<br />
i - Gradient of <strong>the</strong> bed, defined according to 2.1.<br />
2,08 - Coefficient with dimensions L<br />
-l/Z<br />
,<br />
Tlie methodology to define <strong>the</strong>se figures is indicated<br />
in greater detail in 2.1.<br />
The return periods of <strong>the</strong> floodwaters estimated with<br />
<strong>the</strong> formula are generally between 100 and 500 years,<br />
The control section is that in which i, II and <strong>the</strong><br />
silt loads Ibm which have crossed it, are defined.<br />
In <strong>the</strong> river bed, whose discharge one wishes to define,<br />
a series of tests will be made in accordance with <strong>the</strong><br />
precise ùegree of exactitude, and <strong>the</strong> guarantee that <strong>the</strong> data Bm<br />
and i offer. An acceptab<strong>le</strong> number may be four.<br />
The formula is valid for any type of arid, except <strong>the</strong><br />
slates which should not be used to define firn.<br />
Tlie formula was applied to beds whose mobi<strong>le</strong> bottom<br />
during <strong>the</strong> flood was sufficiently scarce to permit <strong>the</strong><br />
maximum arids deposited by <strong>the</strong> flood peak to be later dis-<br />
covered. If this mobi<strong>le</strong> bottom <strong>le</strong>aves <strong>the</strong> arids correspond-<br />
ing to <strong>the</strong> maximum discharge hidden, by <strong>the</strong> smal<strong>le</strong>r arids,<br />
work may be done as indicated in 2.8, but in this study it<br />
was not necessary to experiment with buried arids.<br />
The 15 tests were made on <strong>the</strong> Catalan slope, The formula<br />
also appears acceptab<strong>le</strong> in o<strong>the</strong>r regions, but it has only<br />
been verified with a test in Malaga.<br />
This study does not pretend to have exhausted <strong>the</strong><br />
subject, but merely initiatesa new field of operations.<br />
The series of tests can be expanded, The value of B can<br />
be adjusted more in accordance with <strong>the</strong> variations of am<br />
and i. The case of bed with far thicker mobi<strong>le</strong> bottoms<br />
may be studied , during <strong>the</strong> flood as indicated in 2.8, analysing<br />
<strong>the</strong> buried arids. The observation of <strong>the</strong> arids need not<br />
be restricted to <strong>the</strong> surface, By means of burrows <strong>the</strong><br />
diametres of <strong>the</strong> arids of <strong>the</strong> lower layers can be obtained,<br />
thus extending <strong>the</strong> period studied. It may be used to measure<br />
floods, previously photographing a panorama of <strong>the</strong> river bed<br />
arids, with sufficient detail and after <strong>the</strong> flood water to<br />
be studied, with ano<strong>the</strong>r new photograph, establish <strong>the</strong> size<br />
of <strong>the</strong> silt loads contributed by it. The dimensioning of <strong>the</strong><br />
protection rockfills of <strong>the</strong> river bed is defined with <strong>the</strong><br />
proposed formula arid for <strong>the</strong> slopes, <strong>the</strong> pertinent corrections<br />
need merely be made, In terms of <strong>the</strong> draft of each flood,<br />
<strong>the</strong> silt laden arids cari be foreseen and with <strong>the</strong> granulometries<br />
of <strong>the</strong> bed, <strong>the</strong> sedimentation volume can be defined.<br />
The longitudinal section can be studied in terms of <strong>the</strong>
ADDITIONAL NOTE: Afìer <strong>the</strong> present doctoral <strong>the</strong>sis was<br />
approved, <strong>the</strong> author continued making a series of tests<br />
in various points in Spain, obtaining <strong>the</strong> following results:<br />
1. - Guadalquivir basin (Jacsi)<br />
-Kiver Guadalbullón in Mengibar; B -<br />
2.- Ebro basin (Calatayud)<br />
-River Jalón in Cetina;<br />
3.- Ebro basin (Calatayud)<br />
-River Jalón in Ateca;<br />
2,lO<br />
u<br />
-<br />
2,lO<br />
B 2,lS<br />
As can be seen, <strong>the</strong>se values, added to <strong>the</strong> one obtained<br />
in <strong>the</strong> basin in <strong>the</strong> South of Spain (Malaga) in <strong>the</strong> river<br />
Ve<strong>le</strong>z Guaro, with B = 2,14, make solidly based hopes arise<br />
that <strong>the</strong> formula is applicab<strong>le</strong> for all types of basins.<br />
At <strong>the</strong> same time, it brings <strong>the</strong> number of tests made<br />
up to 19, verifying <strong>the</strong> proposed formula,<br />
457
458
ESTIMATION OF DESIGN FLOODS AND THE PROBLEM OF EQUATING THE PROBA-<br />
BILITY OF RAINFALL AND RUNOFF<br />
M.A. Beran<br />
Floods Stuies Team, Institute of Hydrology, Wallingford, Berkshire,<br />
England.<br />
ABSTRACT<br />
Where data on river discharge are scarce it is a common engi-<br />
neering design practise to concoct a design flood with <strong>the</strong> aid of<br />
rainfall depth-duration-frequency information and a catchment res-<br />
ponse model. Two major waknesses of this approach are (.a) <strong>the</strong> pro-<br />
b<strong>le</strong>m of <strong>the</strong> sensitivity of <strong>the</strong> design to <strong>le</strong>gitimate changes in <strong>the</strong><br />
design assumptions and (b) <strong>the</strong> uncertainty of preserving <strong>the</strong> nomi-<br />
nal rainfall return period in <strong>the</strong> design flood. A solution to <strong>the</strong>-<br />
se prob<strong>le</strong>ms is proposed which makes use of a computer simulation<br />
investigating <strong>the</strong> sensitivity of flood magnitude to variations in<br />
return period, storm duration, temporal rainfall intensity pattern,<br />
infiltration loss rate, base flow and unit hydrograph shape. An es<br />
tension to <strong>the</strong> sensitivity analysis allows an estimate to be made<br />
of any quanti<strong>le</strong> of <strong>the</strong> distribution of flood magnitude based on<br />
sampling across all causative rainfall and antecedent conditions.<br />
RESUME<br />
El est courant, lorsque <strong>le</strong>s données sur <strong>le</strong>s débits sont insuf<br />
fisantes, que l'ingénieuc élabore la crue de projet 'a partir de<br />
l'information qu'il possede sur la distribution des pluies, en uti<br />
lisant un modè<strong>le</strong> de transformation pluies-débits. Les deux inconvk<br />
nients majeurs de ce procédé concernent (a) la sensibilité de l'am$<br />
nagement a la variation des paramètres du projet, (b) la conserva-<br />
tion de la période de Tetour (ou de la probabilité) lorsqu'on passe<br />
de la ptuie de projet a la crue de projet. L'auteur propose une so<br />
lution a ces problèmes, en utilisant une simulation pour recher-<br />
cher la sensibilité de la grandeur de la crue aux variations de la<br />
période de retour, de la durée de l'averse, de la configuration du<br />
hyétogramme, de la capacité d'infiltration, du débit de base, de<br />
la forme de l'hydrogramme unitaire. Une extension de cette analyse<br />
de la sensibilité permet d'estimer n'importe quel<strong>le</strong> quantité de la<br />
distribution des crues, en se basant sur un échantillonnage des<br />
pluies et des conditions antécédentes.
460<br />
3 . INTRODUCTION.<br />
Modern engineercg pract
2. THE SAMPLING PROCEDURE.<br />
461<br />
The procedure follows closely <strong>the</strong> steps used to estimate <strong>the</strong> design flood.<br />
(a)<br />
(b)<br />
(c)<br />
Determine a nominal return period<br />
Choose a storm duration and calculate <strong>the</strong> total depth of<br />
rainfall from <strong>the</strong> depth-durat ion-frequency relationship.<br />
Distribute <strong>the</strong> total rainfall within <strong>the</strong> duration to form<br />
<strong>the</strong> gross rainfall hyetograph.<br />
(d) Subtract from this an infiltration loss to form <strong>the</strong> net<br />
rainfall hyet ograph.<br />
(e)<br />
(f)<br />
Convolute <strong>the</strong> net rainfall hyetograph with <strong>the</strong> unit hydro-<br />
graph to form <strong>the</strong> design inflow hydrograph.<br />
Process <strong>the</strong> inflow hydrograph and extract <strong>the</strong> particular<br />
flood magnitude measure of interest.<br />
In practical engineering application an arbitrary sing<strong>le</strong> choice is made at each<br />
step (a) to (f); in <strong>the</strong> procedure described in this paper, however, <strong>the</strong> choice<br />
is made from a se<strong>le</strong>ction of possib<strong>le</strong> values, each one with a frequency proport-<br />
ional to its probability of occurrence. Figure 1 illustrates <strong>the</strong> procedure as a<br />
tree diagram on which <strong>the</strong> "sing<strong>le</strong> choice" method would be represented by a sing<strong>le</strong><br />
pat h.<br />
As implied in figure 1 <strong>the</strong> continuous distributions of variab<strong>le</strong>s such as<br />
rainfall duration are "discretized" so that each variab<strong>le</strong> is made to assume only<br />
one of a finite number of possib<strong>le</strong> values to each of which a probability weight<br />
is attached. Twelve values of duration, 36 temporal intensity patterns and 12<br />
values of catchment wetness index (Cm .- and index of antecedent conditions<br />
governing infiltration loss and base flow) are used. In a separate study to<br />
provide rainfall information (Appendix 1) no dependences were noted between <strong>the</strong><br />
rainfall variab<strong>le</strong>s and this assumption was made throughout <strong>the</strong> simulation. This<br />
means that <strong>the</strong> weights associated with each samp<strong>le</strong>d variab<strong>le</strong> value was itself<br />
invariab<strong>le</strong>; for examp<strong>le</strong> <strong>the</strong> weights associated with each of <strong>the</strong> 12 CWI values is<br />
<strong>the</strong> same for 3 hour as for 48 hour duration storms. This particular consequence<br />
might represent some departure from actuality as, in <strong>the</strong> United Kingdom, both<br />
are seasonab<strong>le</strong> variab<strong>le</strong>s.<br />
However assuming independence and discretizing allowed considerab<strong>le</strong> simpli-<br />
fication in <strong>the</strong> programming and allowed <strong>the</strong> associated weights of each of <strong>the</strong><br />
12 x 12 x 36 combinations to be calculated from <strong>the</strong> product of <strong>the</strong> weights of<br />
each of <strong>the</strong> contributing variab<strong>le</strong>s. This product weight is associated with <strong>the</strong><br />
flood magnitude in calculating statistics or assembling data into histograms.<br />
To summarise, <strong>le</strong>t pi be <strong>the</strong> weigM(or probability) of <strong>the</strong> ith duration,<br />
Di ; <strong>le</strong>t qj be <strong>the</strong> weight (or probability) of <strong>the</strong> jth hyetograph distribution,
462<br />
Hj; <strong>le</strong>t rK be <strong>the</strong> weight (or probability) of <strong>the</strong> kth CWI, CK; and <strong>le</strong>t QijK<br />
be <strong>the</strong> flood magnitude resulting from <strong>the</strong> combination of Di , Hj and Cu. Then<br />
under <strong>the</strong> assumption of independence <strong>the</strong> weight or probability to be associated<br />
with QijK is Wijk = pi qj rn and <strong>the</strong> expected flood magnitude is calculated from<br />
B E pi qjrn Qi~n , whi<strong>le</strong> <strong>the</strong> mean flood magnitude following al1 storms of say<br />
Fh'e fourth duration is calculated from E C W Qijh (Figures 2A and 2B).<br />
j K 41~<br />
Tab<strong>le</strong> 1 shows <strong>the</strong> results of <strong>the</strong> simulation for <strong>the</strong> IO -year return-period<br />
at Burbage and Grendon. The contingent distributions show <strong>the</strong> effect of different<br />
assumed values on <strong>the</strong> peak discharge. One noticeab<strong>le</strong> result is that changes to<br />
<strong>the</strong> rainfall variab<strong>le</strong>s have small effect on <strong>the</strong> average peak discharge showing<br />
that <strong>the</strong> design flood would be insensitive to variations in hyetograph pattern<br />
or storm duration. This is not to say that floods resulting from storms following<br />
particular combinations of duration and hyetograph pattern cannot be found that<br />
depart from <strong>the</strong> average, but as can be seen from <strong>the</strong> low standard deviations of<br />
peaks contingent on chosen CWI values centrally chosen rainfall variab<strong>le</strong>s will<br />
introduce litt<strong>le</strong> bias into <strong>the</strong> design flood. It has been found that this same<br />
effect is even more marked when <strong>the</strong> measure of flooding being investigated<br />
involves seme e<strong>le</strong>ment of storage.<br />
On <strong>the</strong> o<strong>the</strong>r hand, small changes in <strong>the</strong> CWI have a marked effect on <strong>the</strong><br />
resulting flood. It happens that a CWI value chosen to be near <strong>the</strong> median of <strong>the</strong><br />
distribution of CWI would have yielded a peak discharge only 5% in excess of <strong>the</strong><br />
expected flood.<br />
Figure 3 shows some of <strong>the</strong> histograms of flood peaks following <strong>the</strong> 100-year<br />
storm. These are noticeably negatively skewed and <strong>the</strong> modal value is typically<br />
20% to 30% in excess of <strong>the</strong> mean. The inference from this is that a sing<strong>le</strong> choice<br />
of each of <strong>the</strong> variab<strong>le</strong>s is likely to yield a flood that exceeds <strong>the</strong> average flood.<br />
The sharpness of <strong>the</strong> histograms contingent upon CWI and <strong>the</strong> discrete sampling is<br />
responsib<strong>le</strong> for <strong>the</strong> spikey nature of <strong>the</strong> o<strong>the</strong>r histograms.<br />
3. RAINFALL AND DISCHARGE DISTRIBUTIONS.<br />
It had been noted in Section 2 and Figure 3 that <strong>the</strong> probability distribu-<br />
tion of floods following rainfalls of fixed return period is negatively skewed.<br />
One might anticipate from this that T-year return-period storms tend on average<br />
to give rise to more floods with return period <strong>le</strong>ss than T-years than floods of<br />
return period greater than T-years.<br />
To test this and to derive <strong>the</strong> flood distribution <strong>the</strong> simulation was gener-<br />
alised to samp<strong>le</strong> <strong>the</strong> distribution of storm depths. Instead of sampling only<br />
storms of depth and duration such as lie on a line of equal return period <strong>the</strong><br />
sampling is now conducted across all combinations of storm depth and duration.<br />
The depth-duration-frequency is again used in order to calculate <strong>the</strong> probability<br />
of occurrence of any combination (Figure 2C).<br />
Figure 4 shows a comparison between <strong>the</strong> flood frequency relation as derived
463<br />
from <strong>the</strong> two simulations and from recorded flood peaks. In <strong>the</strong> case of Grendon<br />
Underwood <strong>the</strong>re is an apparent tendency for <strong>the</strong> simulated relation to underestimate<br />
<strong>the</strong> flood discharge based on <strong>the</strong> recorded peaks. although independent<br />
evidence from regional analyses has suggested that <strong>the</strong> distribution as estimated<br />
from <strong>the</strong> six only annual maxima would overestimate floods quite severely. However<br />
<strong>the</strong> agreement with Burbage Brook, a small upland catchment in <strong>the</strong> Derbyshire<br />
pennines with 43 years of data, is ra<strong>the</strong>r better. At small return periods <strong>the</strong><br />
generalised simulation produced lower flood values than <strong>the</strong> expected flood<br />
following storms of that same return period.<br />
4. CONCLUSIONS.<br />
A technique has been described whereby <strong>the</strong> solution of several prob<strong>le</strong>ms<br />
pertinent to hydrological design in regions of inadequate data may be approached.<br />
In particular, <strong>the</strong> sensitivity of <strong>the</strong> design flood to design assumptions can be<br />
assessed. Experience with <strong>the</strong> technique suggests that <strong>the</strong> size of <strong>the</strong> flood is<br />
determined more by <strong>the</strong> total depth of <strong>the</strong> rainfall than by its temporal distribu-<br />
tion through <strong>the</strong> storm's duration. Correct choice of loss rate is in consequence<br />
most important.<br />
It appears that median values of duration, temporal distribution and loss<br />
rate yield a design flood not far removed from <strong>the</strong> overall average flood follow-<br />
ing <strong>the</strong> T-year storm. Because of <strong>the</strong> skewed nature of <strong>the</strong> flood distribution a<br />
random choice of duration etc. would be more likely to yield a design flood<br />
ra<strong>the</strong>r larger than <strong>the</strong> overall average.<br />
The ability of <strong>the</strong> technique to reproduce to<strong>le</strong>rably well <strong>the</strong> flood magni-<br />
tude frequency relation could be of very great value at a site where flow data<br />
are scarce, whilst even at a well-endowed location <strong>the</strong> simulation result may be<br />
used with profit to augment <strong>the</strong> flow record.<br />
Whi<strong>le</strong> attention has been concentrated on peak discharge as <strong>the</strong> measure of<br />
flooding it should be emphasized that <strong>the</strong> technique is suited to more comp<strong>le</strong>x<br />
design criteria. The hydrograph may be treated as an inflow and routed through<br />
<strong>the</strong> scheme and so <strong>the</strong> actual design criteria of interest mqr be calculated.<br />
Examp<strong>le</strong>s are:-<br />
(a) volume between inflow and outflow hydrographs for reservoir<br />
freeboard design; (b) time to peak for a flood warning scheme; (c) volume over<br />
a threshold <strong>le</strong>vel for a <strong>le</strong>vee design.<br />
The technique may also be adopted to use an entirely different catchment<br />
response model such as that inherent in <strong>the</strong> rational formula, a multip<strong>le</strong><br />
regression equation or conceptual model although it can be expected that <strong>the</strong><br />
data requirements will be ra<strong>the</strong>r different from those of this investigation.<br />
5. FUTURF RESEARCH.<br />
The simulation appears promising as a tool for assessing <strong>the</strong> sensitivity of<br />
design floods to variations in <strong>the</strong>ir causative factors and in estimating <strong>the</strong><br />
magnitude-frequency relationship for small return periods. However <strong>the</strong> technique<br />
has not succeeded in reproducing <strong>the</strong> observed rapid growth in flood discharge<br />
with increasing return period and it is here that fur<strong>the</strong>r research is being
464<br />
direct ed.<br />
It is felt that <strong>the</strong> disparity between <strong>the</strong> definition of storms used to<br />
determine <strong>the</strong> distribution of depth and duration (Appendix A - Introduction)<br />
could be responsib<strong>le</strong> for <strong>the</strong> "slow" growth and so long term autographic rainfall<br />
records are to be. analysed to provide information on <strong>the</strong> distribution of <strong>the</strong> type<br />
of storm used for <strong>the</strong> duration statistics.<br />
Fur<strong>the</strong>r investigation into dependencies between <strong>the</strong> variab<strong>le</strong>s could produce<br />
results wkich would affect <strong>the</strong> aiscñarge distribution. For examp<strong>le</strong> seasonal simulation<br />
would reduce <strong>the</strong> coincidence of winter storm types with low summer CWI's<br />
and vice versa.<br />
The dependence of CWI on losses and base flow is essentially statistical<br />
and this source of variability could be preserved fn <strong>the</strong> simulation by <strong>the</strong> addition<br />
of a random quantity to <strong>the</strong> values predicted from <strong>the</strong> best fit lines.<br />
6. ACKNOWLEDGEMENTS.<br />
Although few references have been cited <strong>the</strong> labours and opinions of o<strong>the</strong>rs<br />
have played no small part in <strong>the</strong> development of <strong>the</strong> procedure. Col<strong>le</strong>agues and<br />
consultants of <strong>the</strong> United Kingam Floods Studies Team Dr. J.Y. Sutcliffe,<br />
Professor J.E. Nash, Mr. M.J. Lowing, Mr. C. Cunnane, Mr. R.T. Clarke and<br />
Mr. A.F. Jenkinson have all provided advice and encouragement. Mrs. J. Haworth<br />
was responsib<strong>le</strong> for <strong>the</strong> FORTFAN computer program and <strong>the</strong> numerical experiments<br />
were run on <strong>the</strong> ICL 1906A of <strong>the</strong> Science Research Council's computing laboratory.<br />
7. REFERENCES.<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
NASH,J.E. ; "Frequency of discharges from ungauged catchments".<br />
Trans.A.G.U. , Vol. 37, No. 6, December 1956.<br />
CHOW,V.T. and RAMASESHAN,S. ; "Sequential generation of rainfall and<br />
runoff data". Proc. A.S.C.E., Journ. Hyd. Div. , Vol. 9, HY4, July 1965.<br />
EVANS,T. ; "River Eden flood relief studies". Feasibility report by<br />
Sir M. Macdonald and Partners for Kent River Authority. Chapter 4,<br />
September 1971.<br />
DYCK,S. and KLUGE,C. ; "Investigations on <strong>the</strong> structure of frequency<br />
distributions of floods". I.A.S.H. Warsaw, Vol. 3 , July 1971.<br />
EAGLESON,P.S. ; "The dynamics of flood frequency". Trans. A.G.U.,<br />
Water Resour. Res..Vol. 8, No. 4, November 1972.<br />
LECLERC , G. and SCHAAKE , J. C. ; "Derivat ion of hydrologic frequency<br />
curves from rainfall". Water Resour. Res. (in print).
TABLE 3<br />
FLOOD DISCHARGE FOLLOUNG 3 O-YEAR RETURN PERIOD STORMS.<br />
GFiENDON UNDERWOOD<br />
MEAN STANDARD<br />
DEY<br />
3<br />
m3/s m Is<br />
Overall 5.9 2.0<br />
Constant duration storms.<br />
1 hour<br />
3 hour<br />
6 hour<br />
9 hour<br />
12 hour<br />
15 hour<br />
18 hour<br />
21 hour<br />
24 hour<br />
30 hour<br />
36 hour<br />
48 hour<br />
4.5<br />
5.7<br />
6.2<br />
6.2<br />
6.2<br />
6. o<br />
5.9<br />
5.7 .<br />
5.0<br />
4.4<br />
4.0<br />
4.1<br />
I .6<br />
2.0<br />
2.1<br />
2.0<br />
2.0<br />
1.9<br />
1.8<br />
1.7<br />
1.5<br />
1.3<br />
1.2<br />
1.2<br />
Constant Quarti<strong>le</strong> Type<br />
I 5.8 2.0<br />
II 6. o 2.0<br />
III 5.9 2.0<br />
IV 5. a 2. o<br />
Constant CWI<br />
i5<br />
35<br />
50"<br />
60<br />
1.6<br />
2.8<br />
55 70 3.9<br />
70 ao 4.7<br />
80 go 5.2<br />
90 100 5.6<br />
100 i10 6.1<br />
i10 120 6.6<br />
120 130 7.1<br />
130 140 7.7<br />
i40 150 8.3<br />
150 165 9.4<br />
0.2<br />
0.3<br />
O. 4<br />
0.5<br />
0.6<br />
0.7<br />
O. 7<br />
o. 8<br />
o. 8<br />
0.9<br />
o. 9<br />
1.0<br />
Recorded data<br />
Graphical '-<br />
fit 12.0 I<br />
MaX.<br />
Likelihood1 3.1 3. O<br />
MAXIMUM MINIMUM<br />
DISCHARGE<br />
m3/ s *3/s<br />
11.3<br />
7.7<br />
10.0<br />
33.3<br />
11.3<br />
11.1<br />
11.0<br />
10.9<br />
9.8<br />
9.1<br />
8.4<br />
8.9<br />
13.3<br />
10. 5<br />
10.5<br />
11.1<br />
1.9<br />
3.4<br />
4.8<br />
5.7<br />
6.4<br />
6.9<br />
7.5<br />
8.0<br />
8.6<br />
9.3<br />
10. 1<br />
11.3<br />
1.0<br />
3.0<br />
3.2<br />
1.4<br />
3.6<br />
1.6<br />
3.5<br />
1.6<br />
3.6<br />
1.4<br />
1.2<br />
3.1<br />
1.2<br />
1.0<br />
1 .O<br />
1.0<br />
1 .o<br />
1.0<br />
I. 8<br />
2.1<br />
2.3<br />
2-5<br />
2-7<br />
2.8<br />
3. O<br />
3.2<br />
3.4<br />
3.8<br />
4.6<br />
MEAN<br />
3<br />
m /s<br />
6.7<br />
4.8<br />
6.4<br />
7.3<br />
7.3<br />
7.0<br />
6.8<br />
6.5<br />
6.3<br />
5.8<br />
5.2<br />
4.7<br />
5.3<br />
6.2<br />
6.7<br />
6.8<br />
6.9<br />
1.3<br />
2.0<br />
2.9<br />
3.7<br />
4.4<br />
5.1<br />
5.8<br />
6.5<br />
7.3<br />
8.1<br />
9.0<br />
11.0<br />
BURBAGE BROOK<br />
STANDARD<br />
DEY<br />
3<br />
m /s<br />
3.9<br />
3.4<br />
3.8<br />
2.0<br />
3.9<br />
1.8<br />
1.7<br />
1.7<br />
1.6<br />
1.5<br />
3.4<br />
Y .2<br />
3.4<br />
1.8<br />
1 .8<br />
1.9<br />
2.0<br />
0.2<br />
0.3<br />
0.4<br />
0.5<br />
o. 6<br />
o. 6<br />
0-7<br />
0.8<br />
0.9<br />
1.0<br />
3.1<br />
1.2<br />
--<br />
8.6<br />
8.2 1.6<br />
* First figure refers to assumed CWI at Grendon, second to Burbage Brook.<br />
46 5<br />
MAXIMUM MINIMUM<br />
DISCHARGE<br />
3<br />
m /s<br />
14.0<br />
8.8<br />
13.5<br />
13.4<br />
14.0<br />
13.9<br />
13.7<br />
13.5<br />
13.4<br />
12.5<br />
33.7<br />
31.0<br />
12.3<br />
12. i<br />
12.7<br />
12.4<br />
14.0<br />
1 .6<br />
2.7<br />
3.7<br />
4.7<br />
5.6<br />
6.6<br />
7.5<br />
8.4<br />
9.4<br />
10.4<br />
13 -6<br />
14.0<br />
m3/5<br />
0.6<br />
0.6<br />
0.9<br />
1.3<br />
3.1<br />
1.2<br />
1.1<br />
1.1<br />
1 .O<br />
0.9<br />
0.8<br />
0.8<br />
0.9<br />
o. 6<br />
0.6<br />
0.6<br />
0.6<br />
0.6<br />
1.2<br />
1.7<br />
2.2<br />
2.4<br />
2-7<br />
2.9<br />
3.2<br />
3.5<br />
3.9<br />
4.4<br />
5.9
466<br />
APPENDIX A - DATA REQUIREMENTS.<br />
INTRODUCTION.<br />
Statistical distributions were required for <strong>the</strong> three modes of rainfall<br />
variability: depth, duration and temporal variability- for each catchment invest-<br />
igated. In order not to predetermine any of <strong>the</strong> variability modes it was necess-<br />
ary to define a storm in a manner unlike that of <strong>the</strong> customary rainfall depth-<br />
duration-frequency diagram. The definition was expressed k terms of <strong>the</strong> condi-<br />
tions for starting and ending a storm: a storm was considered to begin at <strong>the</strong><br />
onset of rain and to end when in <strong>the</strong> preceeding Y hours not more than X mms of<br />
rain occurred. X and Y were chosen to represent <strong>the</strong> conditions under which a<br />
flood hydrograph would return to near base flow and allowed sbrt spells of zero<br />
rainfall to occur within a storm event.<br />
Hourly analysis of catchment average rainfall was availab<strong>le</strong> from three<br />
catchments; Grendon Underwood, Coalburn and Plynlimon (Wye). Sufficient records<br />
were availab<strong>le</strong> to permit an investigation hto statistical distribution of storm<br />
durations and temporal patterns but not to conduct an investigation into storm<br />
depth. For this e<strong>le</strong>ment of <strong>the</strong> simulation, results of a depth-duration-frequency<br />
analysis of <strong>the</strong> entire country were availab<strong>le</strong> from A.F. Jenkinson (Ref. Al).<br />
The catchment response model is one currently under investigation by <strong>the</strong><br />
Floods Study Team. A relation between catchment wetness index (CWI) and total<br />
storm losses, and CWI and base flow was used. A unit hydrograph based on recorded<br />
unit hydrographs from <strong>the</strong> catchments were convoluted with <strong>the</strong> gross rainfall <strong>le</strong>ss<br />
losses.<br />
DETAILS OF THE SIMULATION DATA.<br />
(a) Rainfall depth: The basic equation used to relate <strong>the</strong> T-year return period<br />
rainfall of any duration (MT) to that of <strong>the</strong> five-year return period rainfall<br />
(~5) is<br />
MIM5 = (T/5)'<br />
where c is <strong>the</strong> "growth factor" and is related uniquely to M5 which is mapped<br />
for <strong>the</strong> entire United Kingdom. O<strong>the</strong>r necessary information required by <strong>the</strong><br />
simulation and provided in Ref. A1 concerns areal reduction factors to convert<br />
point to areaì rainfall.<br />
(b) Rainfall duration: This distribution is dependent upon <strong>the</strong> storm definition<br />
and for <strong>the</strong> values X = 2 m s, Y = 5 hours used for both Grendon Underwood and<br />
Burbage Brook simulation is given below<br />
STORM DURATION 1 3 6 g 12 15 18 21 24 30 38 48<br />
(HOURS )<br />
RELATIVE FREQUENCY 5 12 26 20 13 30 8 1 1 2 1 1<br />
(PER CENT)
It was found that <strong>the</strong> distribution was very similar for hoth upland and lobiland<br />
rainfall stations and varied slowly with changes to X and Y, longer storms<br />
becoming commoner as <strong>the</strong> conditions for ending a storm were relaxed.<br />
467<br />
(c) Temporal distribution of storm rainfall: Several alternative schemes for<br />
describing <strong>the</strong> hydrograph shape were investigated. The one chosen was due to<br />
F. Huff (Ref. A2) in which four quarti<strong>le</strong> types are recognised depending upon in<br />
which of <strong>the</strong> four quarters of <strong>the</strong> storm duration tiïe largest rainfall fell. The<br />
fine detail of <strong>the</strong> hyetograph is samp<strong>le</strong>d by plotting all curves of <strong>the</strong> same<br />
quarti<strong>le</strong> type on a graph showing accumulating fraction of storm depth against<br />
fraction of total storm duration. Composite storms can <strong>the</strong>n be constructed by<br />
connecting points which are exceeded by lo%, 202, 30% etc. of all storms. Sampl-<br />
ing from <strong>the</strong>se composite storms is analogous<br />
from a distribution function in order to samp<strong>le</strong> a variab<strong>le</strong> in proportion to its<br />
frequency of occurence,and were used by <strong>the</strong> sfmulation. The shapes of <strong>the</strong> campos-<br />
ite storms were found to be insensitive to changes to <strong>the</strong> storm definition and<br />
were nearly indistinguishab<strong>le</strong> between upland and lowland catchments. The percent-<br />
age frequency of <strong>the</strong> four quarti<strong>le</strong> types were 12% type I, 32% type II, 35%<br />
type III, 21% type IV.<br />
(d) CWI distribution: CWI is calculated in nuns. from <strong>the</strong> soil moisture deficit<br />
(SMD) as computed by <strong>the</strong> Meteorological Office (Ref. A3) and a five day anteced-<br />
ent precipitation index (API5) using a daily decay constant of 0.5. The formuïa<br />
usedwasCWI = 125-SMD+API5. It had been observed in a recent study (Ref. Ab)<br />
that wet day rainfall and SMD were statistically independent and SO <strong>the</strong> end of<br />
month values were adopted as representative of all cw? values. Oxford data nr&S<br />
used to provide <strong>the</strong> distribution for Grendon Underwood and Buxton for Burbage<br />
Brook. In <strong>the</strong> simulation a linear relation with CWI was used to calculate total<br />
storm losses and <strong>the</strong> reciprocal of <strong>the</strong> temporal variation of CWI as <strong>the</strong> storm<br />
progresses (assuming no evaporation to increase SMD) determined <strong>the</strong> loss rate<br />
curve. An exponential relationship with CWI determined <strong>the</strong> base flow.<br />
REFERENCES.<br />
Al<br />
A2<br />
A3<br />
A4<br />
to sampling at regular intervals<br />
JENKINSON,A.F. ; "Meteorological office progress report. January 1972".<br />
Report prepared for Floods Study Steering Committee.<br />
HUFF,F.A. ; "Time distribution of rainfall in heavy storms". Water Resour.Res.<br />
Vol. 3, No. 4, fourth quarter 1967.<br />
GRINDLEY,J. ; "Estimation and mapping of evaporation". 1970 I.A.S.H.<br />
symposium, Reading, I.A.S.H.<br />
BEM,M.A. and SUTCLIFFE,J.V. ; "An index of flood-producing rainfall based<br />
on rainfall and soil moisture deficit". Journ. of Hydrology, Vo1.17, 1972<br />
pp 229-236.
46b<br />
FIGURE 1
DuraîK<br />
M 1 etc.<br />
j<br />
~<br />
~<br />
FIGURE 2A FIGURE 2B<br />
WïES<br />
1 1 Consider case where two variab<strong>le</strong>s only aFFect discharge Q, for examp<strong>le</strong> storm duration<br />
and CUI (Figure 24).<br />
For each Combination of duration and CWI a value OF Q md B probability of occurrence CU<br />
be calculated. For exsmp<strong>le</strong> combining <strong>the</strong> duration in <strong>the</strong> Fourth interval, Dy, with <strong>the</strong><br />
CWI in <strong>the</strong> second interval. C2, a discharge q(H) and a probability p(H) - p(D,,)xp(ci) arc<br />
ïoud<br />
'-1 Summing ail <strong>the</strong> probabilities in each discharge interval a discharge distribution [Fimi<br />
20) may be COOStmCted.<br />
) This concept can be generalised to samp<strong>le</strong> From Fur<strong>the</strong>r variab<strong>le</strong>s.<br />
I<br />
9<br />
Discharge den<<br />
/ / / /<br />
wm0. Durat ion<br />
a) Depth uld duration are plotted on <strong>the</strong> base plane ( Piwe 2c)<br />
b) Each coibtiatim ia assoeiited nith a probability of occurmce u) givm by thr depthduration-frequency<br />
diagni.<br />
c) Contingent on each depth duration ccmbination B diatribution of diachugai like Pipure<br />
OB CM be visudiaed on <strong>the</strong> vertical discharge arii.<br />
d) Integrating such densities above all points on <strong>the</strong> bue m e 011 locu. or =qual retur,,<br />
period yields <strong>the</strong> results of Section 2.<br />
0) Integrating over <strong>the</strong> entire base plane Yields <strong>the</strong> dihributim of diichuge of Section 3.<br />
FIGURE 2C<br />
469
470<br />
>-<br />
u<br />
æ<br />
W<br />
x<br />
E<br />
s<br />
i=<br />
0,<br />
W<br />
œ<br />
20-<br />
18.<br />
+<br />
l<br />
i 4<br />
I<br />
!<br />
I<br />
4<br />
I<br />
l<br />
l I<br />
PEAK DISCHARGË- M"/S<br />
Burbage Brook-Floods following 100-year Rainfalls<br />
Distribution of all floods<br />
Floods from storms of given duration<br />
Floods from storms of given CWI ---<br />
FIGURE 3
Q/C<br />
2.2<br />
2.0<br />
1.5<br />
1.0.<br />
O. 5<br />
O<br />
Return period-years<br />
I I I I I I<br />
2 33 5 IO 20 50 100<br />
Mean annual<br />
flood<br />
Most likely peak following storms of given return period<br />
\<br />
/'Simulated flood peaks<br />
following storms of given<br />
Peaks have been standardised by <strong>the</strong> arithmetic mean of<br />
<strong>the</strong> recorded annual maxima 5.39 mYs.<br />
Plotting position corresponds to expected value of order statistic.<br />
Graphical fit to plotted points so'recorded'line misses (1,l).<br />
1 1 I I I I<br />
O 1 2 s 4 5<br />
Reduced variate- y<br />
FIGURE 4<br />
471
ABSTRACT<br />
A DECISION - THEORETIC APPROACH TO UNCERTAINTY<br />
IN THE RETURN PERIOD OF MAXIMUM FLOW VOLUMES<br />
USING RAINFALL DATA<br />
Donald R. Davis('), Lucien Duckstein(t:) Chester C. Kisiel('),<br />
and Martin M. Fogel<br />
The maximum seaaonal rUno#f YolUw Q #or an ungaged atream site is<br />
derived using (1) an event-based rainfall mode1 for thunderstorma, and<br />
(2) a linear rainfall-runoff model. Major emphasis is placed on effect<br />
of uncertainty in parameters of rainfall inputs on <strong>the</strong> return period of<br />
maximum runoff volumes in a season. The event-based rainfall model, derived<br />
previously by <strong>the</strong> coauthors and o<strong>the</strong>rs, has <strong>the</strong> following features:<br />
(1) <strong>the</strong> distribution of <strong>the</strong> number of events per season N is Poisson<br />
with mean m; (2) <strong>the</strong> d' 1s t ribution of point rainfall amount R per<br />
event is exponential with mean llu; (3) N and R are independent. More<br />
explicitly, we obtain a correct distribution function for <strong>the</strong> return pe<br />
riod T (x) under <strong>the</strong> uncertainty in m and u, and demonstrate <strong>the</strong> necessity<br />
0 P following this approach for a decision-<strong>the</strong>oretic analysis of a<br />
water resource design prob<strong>le</strong>m. The approach enab<strong>le</strong>s us to design structures,<br />
relying only on rainfall data, on watersheds with ungaged<br />
streams by taking into account uncertainty of design site parameters.<br />
Also, we cari tailor <strong>the</strong> design to a specific prob<strong>le</strong>m ra<strong>the</strong>r than use a<br />
pre-specified design flood, such as <strong>the</strong> magical lOO-year flood.<br />
RESUME<br />
Le volume d'écou<strong>le</strong>ment maximum est calculé 2 un site non instrumenté,<br />
en utilisant: (1) un mode<strong>le</strong> de pluie d'orage construit par événe<br />
ment; (2) un modè<strong>le</strong> pl,uie-débit linéaire. La maniere dont l'incertitudë<br />
sur <strong>le</strong>s paramètres du modè<strong>le</strong> de pluie affecte la période de récurrence<br />
TQ(x) du volume $'écou<strong>le</strong>ment maximum Q est analysée d'une manière quantitative,<br />
Le mode<strong>le</strong> de pluie d'orage a <strong>le</strong>s caractéristiques suivantes:<br />
(1) se nombre d'événements par saison N suit une distribution de Poisson<br />
a moyenne m; (2) la quantité de pluie ponctuel<strong>le</strong> R par événement<br />
suit une distribution exponentiel<strong>le</strong> de moyenne l/u; (3) N et R sont des<br />
variab<strong>le</strong>s aléatoires indépendantes, Nous obtenons la fonction dti distri-<br />
bution de TQ(x) tenant compte<br />
de l'incertitude sur m et u et montrons<br />
l'utilité de cette méthode pour une application correcte de la théorie<br />
de la d6cision à un problème de planification de ressources en eau.<br />
Nous pouvons ainsi de conceyoir des ouvrages sur des bassing déversants<br />
sans données d'écou<strong>le</strong>ment, a l'aide de données pluvincgtriques, tout en<br />
tenant compte de l'incertitude sur <strong>le</strong>s paramètres. Par ail<strong>le</strong>urs<br />
pouvons spécialiser la conception & chaque cas d'e;tpèce au lieu'd:ItTli<br />
ser une crue standard, tel<strong>le</strong> la magique crue de pêriode de retour centë<br />
naire.<br />
1 Respectively, Assistant Professor and Professors, on joint appointment,<br />
Departments of Hydrology and Water Reaources and Sistems and fndustrial<br />
Engineering, University of Arizona, Tucson, Arizona 85321,<br />
2 Professor, Department of Watershed Management, Same address as in.(l),
474<br />
1.0 Introduction<br />
F<strong>le</strong>,ods or stream discharges are properly described by <strong>the</strong>ir durations and<br />
volumes above a certain flow <strong>le</strong>vel and <strong>the</strong>ir instantaneous peak flows. Of<br />
~I1cs.e three properties, this paper is concerned with <strong>the</strong> uncertainty in <strong>the</strong><br />
return period of maximum flow volumes which is a design parameter for flood pro-<br />
tection and o<strong>the</strong>r structures. In particular, we consider <strong>the</strong> uncertainty due tc<br />
inadequate data on small watersheds (up to 500 lon2).<br />
Jt is well known that <strong>the</strong>re is a good chsnce that a flow event Q.with a<br />
large return period TR may be exceeded at <strong>le</strong>ast once in an R-year design period.<br />
Typically, however, calculated risk diagrams (Gilman, 1964) do not consider <strong>the</strong><br />
uncertainty in <strong>the</strong> return periods of rainfall and flow events. TO a design<br />
engineer, <strong>the</strong> uncertainty of inadequate rainfall or flow data cari result in ei<strong>the</strong>r<br />
overinvestment (overdesign) or underinvestment (economic losses) in <strong>the</strong> design of<br />
flood retarding or retention structures or of water storage facilities (farm<br />
ponds or water supply reservoirs for small towns or industries). The Bayesian<br />
framework presented in this paper allows<br />
logic uncertainty as noted above and for<br />
for an explicit<br />
a methodology<br />
consideration of hydroto<br />
evaluate potential<br />
losses associated with that uncertainty.<br />
Approaches takea to arrive at estimates of <strong>the</strong> return period of hydrologie<br />
f'low properties include:<br />
(a) tipirical fitting of probability density functions to historical data;<br />
in particular, <strong>the</strong> Soi1 Conservation Service (1965) fitted Pearson<br />
Type III distributions ta flow volumes for various time periods in<br />
Arizona. This approach disregards any availab<strong>le</strong> information in precipitation<br />
records or any know<strong>le</strong>dge about <strong>the</strong> rainfall-runoff prccess.<br />
(h) Use of phenomenological relations such as a linear trensformation of<br />
rainfall volume to flow volume as a basis for obtaining probability<br />
density functions (pdf) of flow. The pdf of rainfall volume may be<br />
denrribed empiri~ally (with its consequent uncertainty) or from a<br />
procesc viewpoint .,herein individual rainfall events are mode<strong>le</strong>d as<br />
(c)<br />
a stochastic process along <strong>the</strong> time axis<br />
Use of detai<strong>le</strong>r! dynemical flow equations<br />
(Duckstein<br />
to relate<br />
et al. 1972).<br />
pdf of rainfall<br />
psoperties to pdf of flow properties (Rag<strong>le</strong>son, 1972).<br />
In this paper we use <strong>the</strong> second approach. Herein we build on previaus work<br />
(Davis et al. 1972) where we evaluated <strong>the</strong> ucertainty in <strong>the</strong> return period of<br />
point rainfall amounts from summer thunderstorms. We define an event-based<br />
process in this case as a sequence of thunderstorms in tine. The return period<br />
TR(k) of maximum point rainfall e (with k <strong>the</strong> rainfall smount or value of <strong>the</strong><br />
random variab<strong>le</strong> 5) is derived by considering <strong>the</strong> following e<strong>le</strong>ments of <strong>the</strong><br />
event-based nrocess:<br />
(a) l?hë number 1; of events per season is Poisson distributed with met-a m<br />
(of number of events per season):<br />
b)<br />
(cl<br />
(d)<br />
Rainfall events 53, R,,..., are independent identically distributed<br />
random variab<strong>le</strong>s.<br />
The amo\‘Jit 5 of point rainfall per t\sent is exponentially distributed<br />
with parsmeter u (equal to reciprocal of mean emount rainfall per event):<br />
fR(klu) = ueBuk<br />
N and 5 are indepenaent.
Then, <strong>the</strong> return period of k units of rain in a season, given <strong>the</strong> event-based<br />
parameters m and u, is<br />
Because m and u are uncertain due to small samp<strong>le</strong> size, T is uncertain.<br />
475<br />
To encode <strong>the</strong> uncertainty, <strong>the</strong> posterior distribution of m and u represents<br />
<strong>the</strong> likelihood of <strong>the</strong> values of m and u which produced <strong>the</strong> data. This posterior<br />
is given by <strong>the</strong> conjugate distributions for <strong>the</strong> exponential and Poisson distributions<br />
(de Groot, 1970, Chapt. 9). The distribution that is conjugate to both<br />
of <strong>the</strong>se is <strong>the</strong> gamma:<br />
a a-1 -bx<br />
b x e<br />
gX(xla,b)<br />
-. = r(a) (4)<br />
For <strong>the</strong> Poisson distributution,<br />
x = m , <strong>the</strong> parameter of <strong>the</strong> Poisson and estimated as m.<br />
b = n , <strong>the</strong> number of seasons.<br />
a = &-I , <strong>the</strong> total number of rainfall events in n seasons.<br />
For <strong>the</strong> exponential distribution,<br />
x = u , <strong>the</strong> parameter of <strong>the</strong> exponential and estimated as Û.<br />
a = &-I , <strong>the</strong> total number of rainfall events in n seasons.<br />
b = h/û, <strong>the</strong> total amount of rainfall for <strong>the</strong> mn events.<br />
The resulting F (x)s in each case are posterior distributions and represent<br />
z<br />
<strong>the</strong> likelihood that various values of m and u axe <strong>the</strong> values describing <strong>the</strong> rainfall<br />
process that we are observing, after getting <strong>the</strong> data. These posterior<br />
distributions are used in a computer simulation to develop <strong>the</strong> posterior distribution<br />
of TR(k). The mean of this distribution is <strong>the</strong> expected return period<br />
E[T (k)Tl fo; a k-inch rainfall.<br />
R<br />
Computer results' given by Davis, et al. (1972)<br />
indicate that <strong>the</strong> return period of point rainfall is subject to considerab<strong>le</strong><br />
uncertainty even with 20 years of data. The design and operational implications<br />
are obvious for flood control, dry farming with irrigation, and water supply.<br />
Next, we extend <strong>the</strong> procedure to uncertainty in return periods of seasonal flow<br />
volumes on small watersheds.<br />
2.0 Extension to Seasonal Flow Volumes<br />
If m is <strong>the</strong> total number of runoff producing rainfall events in a summer<br />
season, <strong>the</strong>n <strong>the</strong> exact expected return period T (y) of <strong>the</strong> maximum seasonal<br />
runoff volume Q is, under our previous hypo<strong>the</strong>ses,<br />
T& (ylm,u) = [i-exp I-m + m F (ylu)~~-l (5)<br />
9<br />
where F (ylu) is <strong>the</strong> distribution function of runoff per event CJ which we will<br />
9<br />
9 Q<br />
write F (y) for simplicity. Our approach is to obtain F (y) from <strong>the</strong> distribution<br />
function F (x) of rainfall 5 per event, using <strong>the</strong> linear rainfall-runoff relation-<br />
R<br />
where A are <strong>the</strong> initial abstractions depending on <strong>the</strong> watershed and c is a coefficient<br />
depending on <strong>the</strong> rainfall characteristics for a given watershed, in<br />
particular, a time factor such as <strong>the</strong> maximum 15-minute intensity (Duckstein<br />
et al. 1972).<br />
--<br />
Q<br />
R
476<br />
If we <strong>le</strong>t<br />
p = !-A for R > A I<br />
= o for 5 < A<br />
<strong>the</strong>n Equation (6) becomes = CP -- or y = cx; <strong>the</strong> distribution function of P - is<br />
Fp(x) = 1 -exp (-u(x+A)) for x > O (7)<br />
and thit of €j (Fel<strong>le</strong>r, 1967, Chapt. 2) is<br />
m<br />
FQ(y) = I,"p($) fC(c) dc<br />
because c is a random variab<strong>le</strong> as noted in previous work by <strong>the</strong> coauthors<br />
(Duckstein et al. 1972). Since, physically, we cannot obtain more runoff than<br />
rainfall, <strong>the</strong>n O 5 c 51, and a beta distribution for c seems to be most appro-<br />
Driate:<br />
The uncertainty on a,b will not be considered in <strong>the</strong> present study.<br />
Equations (7), (8) and (9) may be combined to obtain<br />
(9)<br />
To sum up,<br />
Equations (10) and (11) are now substituted into Equation (5) to obtain an explicit<br />
expression of Tg (ylm,u). Because we have <strong>the</strong> sufficient statistics,<br />
fi and a, our know<strong>le</strong>dge of m and u can be expressed as a pdf !Tiao and Box, 1973).<br />
Hence, this encoded uncertainty results in a pdf on T (ylm,u).<br />
3. O Met hodolopy<br />
To obtain <strong>the</strong> pdf of <strong>the</strong> return period on hand, T (ylm,u,n) is a prob<strong>le</strong>m of<br />
transformation of random variab<strong>le</strong>s, where a closed form is beyond reach.<br />
Thus, a simulation approach is used as follows: (a) consider a fixed<br />
yearly maximum flow volume Q = yo, and (b) samp<strong>le</strong> values m,u are drawn from <strong>the</strong><br />
conjugate pars.<br />
-<br />
%(mlfi,n) and gu(uli,n), respectively, as noted in our discussion<br />
of Equation (b), (c) <strong>the</strong>se samp<strong>le</strong> values are substituted into T (y,Im,u) to<br />
obtain one value of <strong>the</strong> return period T<br />
8<br />
and (a) <strong>the</strong> process is repeated to<br />
0'<br />
obtain pdf of T for a fixed y (for examp<strong>le</strong> yo = Q = 0.7 inch in Tab<strong>le</strong> 1).<br />
9<br />
A similar procedure is <strong>the</strong>n used to calculate <strong>the</strong> pdf of (T )-', which is<br />
9<br />
<strong>the</strong> probability of exceedance of y . The design parameter of interest may be<br />
O<br />
ei<strong>the</strong>r T (for sizing a small dam) or (T (estimating long-range replace-<br />
B 9<br />
ment costs of structures.)<br />
Finally, to be considered in a later study is <strong>the</strong> pdf of maximum seasonal<br />
flow Q that corresponds to a fixed return period. Such a pdf may be of interest<br />
for flood plain insurance purposes and can be calculated by <strong>the</strong> same simulation<br />
procedure as above.<br />
Q<br />
9
477<br />
4.0 Results<br />
The results of <strong>the</strong> computer simulation are summarized in Tab<strong>le</strong>s 1 and 2 and<br />
Figures 1 and 2. In <strong>the</strong>se we consider <strong>the</strong> variance of c, representative of conditions<br />
on <strong>the</strong> watershed, and <strong>the</strong> variance in our know<strong>le</strong>dge about rainfall<br />
parameters m and u.<br />
Tab<strong>le</strong> 1 shows that u, <strong>the</strong> average rain per event, is much more important<br />
than m, <strong>the</strong> average number of storms per season, as judged by <strong>the</strong> variance of<br />
T ,(Var !? ), for different values of Var c. We also note <strong>the</strong> following<br />
9 9<br />
(a As Var c increases, EL?,? and Var ? decrease, Thus by not randomizing<br />
9<br />
C <strong>the</strong> estimated return period of Q = 0.7 is much higher. By varying C<br />
<strong>the</strong> variab<strong>le</strong> effects of rainfall intensity and watershed behavior on<br />
<strong>the</strong> return period are anticipated;<br />
(b) Var T increases dramatically when Var $ = O for joint uncertainty in<br />
9<br />
m and u;<br />
(c) The mean reciprocal return period (= exceedance probability = p) and<br />
,. -1<br />
Var T increase rapidly as Var c increases. This result is shown<br />
9<br />
because p is commonly used as <strong>the</strong> design parameter in hydrologic risk<br />
analysis.<br />
These patterns hold for all values of runoff volume used in <strong>the</strong> sensitivity analysis<br />
(Q = 0.5, 0.7 and 0.9 inches of runoff) as shown in Tab<strong>le</strong> 2.<br />
As expected, <strong>the</strong> Var T decreases with doubling of availab<strong>le</strong> data (10<br />
9<br />
to 20 years used in <strong>the</strong> simulation) as summarized in Tab<strong>le</strong> 2. The E[id is only<br />
slightly changed. A more general manifestation of <strong>the</strong> simulated process is evident<br />
in Figure 2 where <strong>the</strong> posterior pdf (of return periods for 0.7-inch runoff) based<br />
on 20 years of data has a much sharpy modal value than <strong>the</strong> posterior pdf based<br />
on 10 years of data; note that mear, T is just to <strong>the</strong> right of <strong>the</strong> mode. Whi<strong>le</strong><br />
9<br />
not shown, <strong>the</strong> posterior pdf's become more peaked as Var c increases.<br />
The effect of increasing runoff Qolume is to increase E[Td, Var T and<br />
9<br />
coefficient of variation CV(T ) as shown in Tab<strong>le</strong> 2. The latter result about<br />
9 I<br />
CV(TQ) also implies that a Var T increases more rapidly than E[Td. It is<br />
9<br />
intriguing to note <strong>the</strong> dramatic effect that <strong>the</strong> introduction of Var has on <strong>the</strong><br />
parameters.<br />
The results in Tab<strong>le</strong> 2 for n = 10 years are shown in Figure 1, a plot on<br />
Gumbel extreme value paper. As previously noted, as Var c increases <strong>the</strong> smal<strong>le</strong>r<br />
"Liil. From <strong>the</strong> tabulated results we note that so-cal<strong>le</strong>d confidence limits for<br />
ea& line would get wider as T increases because Var T increases with runoff<br />
61 9<br />
volume. These confidence limits are narrower for n = 20 years of data as is<br />
evident from Tab<strong>le</strong> 2.<br />
Of interest is <strong>the</strong> modest computer time (maximum of 25 seconds for 20 years<br />
of data) per simulation run on <strong>the</strong> CDC-6400. Given <strong>the</strong> number of uncertain<br />
parweters in this prob<strong>le</strong>m, it does not appear feasib<strong>le</strong> to prepare charts and<br />
graphs for routine design use un<strong>le</strong>ss more exhaustive computer studies are performed.<br />
4.1 Comments on Results A<br />
In contrast to <strong>the</strong> classical empirical frequency approach in deriving T 9,
478<br />
<strong>the</strong> event-based approach outlined here results in evaluation of uncertainty in<br />
.<br />
T from physically meaningful parameters like m and u. This is a much more<br />
Q<br />
efficient use of <strong>the</strong> availab<strong>le</strong> data on rainfall and runoff.<br />
We have seen how <strong>the</strong> design would depend on <strong>the</strong> uncertainty in m and u and<br />
on <strong>the</strong> interaction between*uncertainty in m and u and Var C. The end result, a<br />
posterior distribution on T is of value to inference on hydrologic stochastic<br />
9’<br />
processes as discerned from limited data of value to <strong>the</strong> next important step<br />
of invoking Bayesian decision <strong>the</strong>ory for evaluating design decisions and for<br />
judging if better designs are possib<strong>le</strong> by waiting for-additional cata.<br />
It would be desirab<strong>le</strong> to express <strong>the</strong> moments (ErTJ and Var T ) of <strong>the</strong><br />
Q<br />
posterior pdf in terms of m, u, Q and C, but this is intractab<strong>le</strong>. -The next<br />
approach for thinking about our results in simp<strong>le</strong>r terms is to consider <strong>the</strong><br />
mean and variance of 5 = cp:<br />
ECQI = EC~J CIFI<br />
Var = E2[C] Var P + E2[P]<br />
- Var C + (Var C) - (Var P) -<br />
as given by Benjamin and Cornel1 (1970, p. 169).<br />
equations become E[g = CE[?] and Var 9 = C2 Var p.<br />
When C is not random, <strong>the</strong>se<br />
The variance of Q (and<br />
thus its frequency of exceedance and its return period) is dramatically affected<br />
by randomization of C. It is common in hydrologic design to choose a “frequency<br />
factor” z (or standardized variate) in <strong>the</strong> relation 9 = Ere] + z (Var Q) 1/2 .<br />
.<br />
To contrast properly this classical approach to finding a design flow Q with <strong>the</strong><br />
method outlined in this paper wou1.d require a full-f<strong>le</strong>dged decision <strong>the</strong>oretic<br />
analysis for a specific design prob<strong>le</strong>m. The evaluation would have to be repeated<br />
for each design use of <strong>the</strong> posterior pdf. Much work remains to be done in this<br />
direction.<br />
4.2 Relationship of results to Bayesian decision <strong>the</strong>ory<br />
Let <strong>the</strong> loss function for <strong>the</strong> design of a flood protection structure, say<br />
a dike, be L(h,T) where h is <strong>the</strong> height of <strong>the</strong> dike and T is a design return<br />
period such as T or an exceedance probability (T )-l. The result of our<br />
a 9<br />
investigation was to determine <strong>the</strong> posterior pdf f (t) as given in Figure 2.<br />
Thus, we are now ab<strong>le</strong> to calculate Bayes risk, which corresponds to <strong>the</strong> optimum<br />
design h*<br />
+m<br />
BR(h*) = min L(h,t) fT(t)dt (14)<br />
.<br />
h o<br />
We can also calculate <strong>the</strong> worth of samp<strong>le</strong> information to sharpen <strong>the</strong> estimate<br />
of T (Davis et al. 1972) for each intended use of <strong>the</strong> data. Such studies are<br />
9<br />
<strong>le</strong>ft for <strong>the</strong> sequel. It is very important to emphasize that <strong>the</strong> worth of data<br />
discerned by this methodology is based on <strong>the</strong> economic loss function associated<br />
with a particular design use of <strong>the</strong> data; <strong>the</strong> results are not in terms of <strong>the</strong><br />
variance of <strong>the</strong> return estimate (return period in this case).<br />
5.0 Conclusions<br />
It is important to keep in mind when judging <strong>the</strong> results of <strong>the</strong> research<br />
reported here that we are dealing with maximum flow volumes generated by a sequence<br />
of thunderstorms during a season. Additional work is necessary to extend <strong>the</strong><br />
T
479<br />
approach to o<strong>the</strong>r runoff-producing precipitation events (including snow) during<br />
<strong>the</strong> year. The use of <strong>the</strong> Gumbel distribution in this paper goes beyond its<br />
classical use for <strong>the</strong> instantaneous rainfall and flood maxima during <strong>the</strong> year.<br />
We thus have found <strong>the</strong> following points in our <strong>the</strong>oretical and simulation<br />
arialy,is :<br />
The approach enab<strong>le</strong>s us to design structur'es, relying only on rainfall<br />
data, on watersheds with ungaged streams by taking into account<br />
uncertainty of <strong>the</strong> site parameters,<br />
Using this approach we can tailor <strong>the</strong> design to a specific prob<strong>le</strong>m<br />
ra<strong>the</strong>r than use a pre-specified design flood, such as <strong>the</strong> magical<br />
100-year flood.<br />
Simulation is an appropriate method for evaluating uncertainty in<br />
estimates of physically-meaningful parameters arising in <strong>the</strong> eventbased<br />
approach.<br />
Return period varies with record <strong>le</strong>ngth rainfall, and watershed events,<br />
etc. We have given an event-based approach to evaluate this variation.<br />
The sensitivity analysis demonstrates <strong>the</strong> dramatic importance of uncertainty<br />
in <strong>the</strong> average amount of rainfall per event and <strong>the</strong> importance<br />
of considering variability in <strong>the</strong> rainfall and watershed parameter<br />
cal<strong>le</strong>d C in this paper.<br />
The resats, if encoded in <strong>the</strong> posterior pdf of <strong>the</strong> return period<br />
TI, allow <strong>the</strong> user to exercise inference or to find sensitivity of <strong>the</strong><br />
analysis to design decisions in <strong>the</strong> face of inadequate data. Bayesian<br />
decision <strong>the</strong>ory is <strong>the</strong> framework suggested for undertaking <strong>the</strong> decision<br />
analysis.<br />
The results have implications for design of a variety of hydraulic structures<br />
in both urban and rural watersheds, in temperate and arid climates, and in<br />
regions of <strong>the</strong> world confronted with inadequate hydrologic data. In <strong>the</strong> face<br />
of changing watershed conditions, as reviewed by Fogel, et al. (1972), <strong>the</strong><br />
approach offered in this paper permits exercise of judgment on <strong>the</strong> effects of<br />
lack of know<strong>le</strong>dge and of nonstationary meteorologic and hydrologic parameters<br />
such as m, u and C. In o w jument, classical empirical frequency methods do<br />
not provide such a c<strong>le</strong>ar basis for evaluation. Extension to nonlinear water-<br />
shed models are possib<strong>le</strong> as noted by Duckstein, et al. (1972) and Fogel, et al.<br />
(1972).<br />
6. O Acknow<strong>le</strong>dgments<br />
The work was supported in part by U.S. National Science Foundation Grant<br />
GK-35791 and by a matching grant (Decision Analysis of Watershed Management<br />
Alternatives) from <strong>the</strong> U.S. Office of Water Resources Research. !Che computer<br />
programming skills demonstrated by Joel Friedman have contributed substantially<br />
to <strong>the</strong> realization of <strong>the</strong> results.<br />
7.0 References<br />
BenJanin, J.R. and C.A. Cornell. Probability, Statistics and Decision for Civil<br />
Engineers, McGraw-Hill Book Co., New York, 1970.<br />
Davis, D.R., L. Duckstein, C.C. iíisiel, and M. Fogel. Uncertainty in <strong>the</strong> return<br />
period of maximum events: A Bayesian a.pproach. Proceedings, International<br />
Symposium on Uncertainties in Hydrologic and Water Resource Systems,<br />
University of Arizona, Tucson, Arizona; 1972, pp. 853-862.
480<br />
Davis, D.R., C.C. kïsiel and L. Duckstein. Bayesian decision <strong>the</strong>ory applied<br />
to design in hydrology, Water Resources Research, Vol. 8, No. 1,<br />
February 1972, pp. 33-41.<br />
de Groot, M.H. Optimal Statistical Decisions. McGraw-Hill Book Co., New York,<br />
1967.<br />
Duckstein, L., M.M. Fogel and C.C. Kisiel. A stochastic model of runoffproducing<br />
rainfall for summer type storms, Water Resources Research,<br />
Vol. 8, No. 2, April 1972, pp. 410-421.<br />
Eag<strong>le</strong>son, P.S. Dynamics of flood frequency, Water Resources Research, Vol. 8,<br />
NO. 4, August 1972, pp. 878-898.<br />
Fel<strong>le</strong>r, W. An Introduction to Probability Theory and its Applications, Vol. 2.<br />
John Wi<strong>le</strong>y, New York, 1967.<br />
Fogel, M.M., L. Duckstein and C.C. Kisiel. Choosing hydrologic models for<br />
management of changing watersheds, Proceedings, National Symposium on<br />
Watersheds in Transition (American Water Resources Association) , Fort<br />
Collins, Colorado, June 1972, pp. 118-123.<br />
Gilman, C.S. Rainfall (Section 9). In "Hand<strong>book</strong> of Applied Hydrology,"<br />
Edited by V.T. Chow, McGraw-Hill Book Co., New York, 1964, pp. 9-59.<br />
Soil Conservation Service, Runoff volume-duration-probability analyses for<br />
se<strong>le</strong>cted watersheds in Arizona. Central Technical Unit, Hydrology<br />
Branch, SCS, U.S. Dept. of Agriculture, April 1965.<br />
Tiao, G.C. and G.E.P. Box. Some comments on "Bayes" estimators, The American<br />
Statistician, Vol. 27 (i), February 1973, pp. 12-14.
Tab<strong>le</strong> 1: Sensitivity anaiysikon return period moments as<br />
function of uncertain parameters for rural water-<br />
shed with only 10 years of data.<br />
Jncertain<br />
)arameters Var C<br />
l&U O<br />
.O005<br />
.O05<br />
* O5<br />
)nly m .O005<br />
.O05<br />
O5<br />
nly u .O005<br />
.O05<br />
* O5<br />
f<strong>le</strong>an T (years<br />
?<br />
~<br />
A<br />
Moments of return period T<br />
cv( TQ)**<br />
.,<br />
41.82<br />
39.00<br />
26.33<br />
6.30<br />
35.36<br />
23.41<br />
6.20<br />
37.61<br />
24. i4<br />
6.52<br />
Var I<br />
Q -<br />
538.<br />
442.<br />
153.<br />
2.35<br />
8.30<br />
3.89<br />
t 19<br />
383.<br />
110.<br />
1.96<br />
.555<br />
.539<br />
.470<br />
.243<br />
.o81<br />
.O84<br />
.O70<br />
.521<br />
.435<br />
.215<br />
~~ ~<br />
Reciprocal<br />
return period -<br />
nean variance<br />
.O00299<br />
.o00318<br />
.o00328<br />
.o01809<br />
* Conditions for <strong>the</strong> analysis: A = 0.4 inches, mean C = 0.3 for beta<br />
distribution, Q = 0.7 inches on <strong>the</strong> average; rainfall is<br />
distributed on basis of an exponential distribution for<br />
amounts above 0.3 inches with an average of 14.0 storms/<br />
season and an average of 0.39 inches<strong>le</strong>vent.<br />
*+ Coefficient of variation of T s-<br />
481
Average<br />
runoff<br />
volume Q<br />
0.5<br />
Tab<strong>le</strong> 2: Sensitivity analysis on return period moments<br />
asa fun&ion of rainfall P, <strong>le</strong>ngth of record n<br />
ànd variance of C; both m and u are uncertain;<br />
watershed is rural; conditions are as noted in<br />
Tab<strong>le</strong> 1.<br />
n<br />
(years of<br />
data)<br />
Var <<br />
I<br />
Moments of return period 'f<br />
9<br />
&<strong>le</strong>an (years<br />
I<br />
~ ~-<br />
10 .o01 11.70<br />
58.07 .650<br />
.O05<br />
O5<br />
6.97<br />
2.94<br />
14.90<br />
- 27<br />
.553<br />
* 177<br />
20 .O05 6.25 4.38 ,335<br />
10 O<br />
.O005<br />
.O05<br />
O5<br />
41.82<br />
39.00<br />
26.33<br />
6.30<br />
20 37.44<br />
10 O<br />
.O05<br />
O5<br />
24.36<br />
271<br />
103<br />
6.41<br />
14.29<br />
~~~<br />
538<br />
441<br />
153<br />
245<br />
68<br />
48,085<br />
3 , 602<br />
20 .O05 95.95 1,721 I<br />
2.35<br />
1.09<br />
21.25<br />
.432<br />
.555<br />
.539<br />
.470<br />
,243<br />
.418<br />
.339<br />
.163<br />
.809<br />
.582<br />
.323
I .o<br />
O. 8<br />
v,<br />
w<br />
I<br />
o<br />
z 0.6<br />
-<br />
-<br />
w<br />
z<br />
3<br />
-I<br />
$ 0.4<br />
LL<br />
IL<br />
O<br />
Z<br />
3 0.2<br />
O<br />
2 5 IO 20 50<br />
RETURN PERIOD, YEARS<br />
100 200<br />
Figure 1: The effect of <strong>the</strong> varlance of C on tha Teturn<br />
period of punoff volume.<br />
483
w<br />
3<br />
o<br />
w<br />
CE<br />
LL<br />
484<br />
.25 -<br />
.20 -<br />
.I5 -<br />
.IO -<br />
.O5 -<br />
RETURN PERIOD, YEARS<br />
Figura 2: Posterior prooabilitr densiTy function of return periods<br />
f o 0.7-i’nch ~<br />
runoff of record <strong>le</strong>ngth.<br />
O
ABS TRACT<br />
SYNTHETIC UNIT HYDROGRAPH TECHIQUE FOR THE<br />
DESIGN OF FLOOD ALLEVIATION WORKS IN URBAN AREAS<br />
by<br />
M.J. Hall<br />
Lecturer in Civil Engineering, Imperial Col<strong>le</strong>ge<br />
of Science and Technology, University of London<br />
The development of rural land for urban, suburban or industrial<br />
purposes can radically alter <strong>the</strong> flow regime of <strong>the</strong> catchment area WL<br />
thin which such changes take place. The volume of surface runoff tends<br />
to increase, <strong>the</strong> lag time of <strong>the</strong> flood hydrograph to decrease and <strong>the</strong><br />
peak rate of flow to increase. These ch-anges should be anti’cipated in<br />
<strong>the</strong> design of flood al<strong>le</strong>viation works for catchment areas undergoing<br />
urbanisation, but in general, litt<strong>le</strong> quantitative information is avai<br />
lab<strong>le</strong> on <strong>the</strong> magnitude of <strong>the</strong> effect at different stages of urban de-<br />
velopment. If flow records are availab<strong>le</strong> from several catchment areas,<br />
each of which has reached a different stage of urban development, <strong>the</strong><br />
finiteperiod unit hydrographs derived from <strong>the</strong>se data can be used as<br />
an index to <strong>the</strong> influence of urbanisation. The application of a syn-<br />
<strong>the</strong>tic unit hydrograph technique to flow records from both urban and<br />
rural catchment areas within <strong>the</strong> headwaters of <strong>the</strong> River Mo<strong>le</strong> near<br />
Craw<strong>le</strong>y, United Kingdom, has confirmed <strong>the</strong> feasibility of <strong>the</strong><br />
approach but has shown that more thought is necessary in choosing cai<br />
chment characteristics which ref<strong>le</strong>ct <strong>the</strong> character of <strong>the</strong> urban deve-<br />
lopment.<br />
RESUME<br />
L’utilisation des espaces ruraux pour <strong>le</strong> développement urbain et<br />
industriel peut changer radica<strong>le</strong>ment <strong>le</strong> régime hydrol2gique des bas-<br />
sins concernés. Le volume du ruissel<strong>le</strong>ment tend a croitre, <strong>le</strong> temps<br />
de réponse du bassin à décoitre et <strong>le</strong>s pintes de crues s’amplifient.<br />
Lors de l’élaboration des projets, ces modifications devraint être<br />
prévues et on devrait chercher à atténuer l’effet des crues par des<br />
travaux appropigs, mais on ne dispose en général que d’une informa-<br />
tion très succincte sur l’importance de cet effet aux différents sta-<br />
des du développement urbain. Si on dispose de re<strong>le</strong>vés de débits sur<br />
plusieurs bassins atteints à des degrés différents par <strong>le</strong> développe-<br />
ment urbain, on peut utiliser <strong>le</strong>s hydrogrammes unitaires tirés de ces<br />
données pour constituer des indices concernant l’influence de l’appli-<br />
cation d‘u?e technique d’hydrogramme unitaire synthétique aux débits<br />
observés, a l’issue de bassins urbains et ruraux, dans <strong>le</strong> bassin sup5<br />
rieur de la Mo<strong>le</strong>, pres de Craw<strong>le</strong>y (Royaume Uni), a confirmé <strong>le</strong>s possi<br />
bilités de cette méthode; el<strong>le</strong> a montré aussi que <strong>le</strong> choix des caracy<br />
téristiques du bassin reflétant l‘influence du d6velo;pement urbain<br />
demandait une sérieuse réf<strong>le</strong>xion.
486<br />
I. INTRODUCTION<br />
According to Toynbee [I], almost half <strong>the</strong> World's population had become<br />
urban by 1969. This increase in <strong>the</strong> urban population has been accompanied<br />
by an even more marked expansion in <strong>the</strong> area occupied by streets and buildings.<br />
The development of rural land for urban, suburban and industrial purposes is<br />
characterised by two important physical changes, both of which may have a profound<br />
effect on <strong>the</strong> hydrological cyc<strong>le</strong> of <strong>the</strong> area within which such urbanisation<br />
takes place.<br />
Firstly, <strong>the</strong> area covered by relatively impervious surfaces increases,<br />
<strong>the</strong>reby increasing <strong>the</strong> proportion of storm rainfall which becomes surface run-<br />
off. Owing to <strong>the</strong> concomitant decrease in soil moisture recharge, dry wea<strong>the</strong>r<br />
flows are reduced.<br />
Secondly, <strong>the</strong> natural surface water drainage system of <strong>the</strong> area is invariably<br />
subjected to a variety of changes, ranging from realignment of channels to <strong>the</strong><br />
installation of stormwater sewerage. Since <strong>the</strong> flow velocities in <strong>the</strong> modified<br />
drainage network are generally higher than those observed in <strong>the</strong> orlginal<br />
natural channel system, both <strong>the</strong> time-to-peak and <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> recession<br />
of storm hydrographs tend to decrease as a catchment is urbanised.<br />
The increased<br />
volume of runoff, and <strong>the</strong> shorter time within which that volume is discharged,<br />
inevitably produce peak rates of runoff that are markedly higher than <strong>the</strong> flow<br />
records from a catchment in its previous rural state would tend to indicate.<br />
Although <strong>the</strong> effects of urbanisation on <strong>the</strong> flow regipie of a catchment<br />
area have been appreciated qualitatively for over a decade [2], relatively<br />
litt<strong>le</strong> information has been availab<strong>le</strong> on <strong>the</strong> magnitude of <strong>the</strong> changes brought<br />
about by different forms of urban development. Of particular importance to<br />
<strong>the</strong> engineer concerned with <strong>the</strong> design of flood al<strong>le</strong>viation works for urban<br />
areas are<br />
i)<br />
ii)<br />
<strong>the</strong> frequency distribution of peak rates of flow ; and<br />
<strong>the</strong> shape of <strong>the</strong> flood hydrograph.<br />
The changes in <strong>the</strong> magnitude of <strong>the</strong> parameters of <strong>the</strong> frequency distribution<br />
of annual floods caused by urbanisati)onhave been studi,ed by Carter [3],<br />
Martens [4] and Anderson 151, each of whom approached <strong>the</strong> prob<strong>le</strong>m by means of<br />
regional analysis. Much of <strong>the</strong> work on changes in <strong>the</strong> shape of flood hydrographs<br />
has employed a similar treatment, with <strong>the</strong> finite-period unit hydrograph<br />
(TUH) being used as an index to catchment response. Flow records for<br />
catchment areas in different stages of urban development within <strong>the</strong> same<br />
hydrologically homogeneous region have been used to derive WH's of a predetermined<br />
duratiqn. Se<strong>le</strong>cted parameters of <strong>the</strong>se !CUH's have <strong>the</strong>n been expressed<br />
in terms of pertinent catchment Characteristics using multip<strong>le</strong> linear regression<br />
analysis. The relationships so obtained may <strong>the</strong>n be employed to derive TUHts<br />
for both ungauged catchments and gauged catchments in a more advanced state of<br />
development. For examp<strong>le</strong>, Espey et al [6] used 5 hydrograph parameters : peak<br />
rate of flow ; time-of-rise and base <strong>le</strong>ngth of <strong>the</strong> hydrograph ; and hydrograph<br />
widths at 50 and 75 per cent of <strong>the</strong> peak discharge. Since <strong>the</strong> catchment<br />
characteristics se<strong>le</strong>cted by those Authors did not ipclude any parameter ref<strong>le</strong>cting<br />
changes in <strong>the</strong> surface water drainage system, an empirical coefficient (
L<br />
ref. river<br />
no.<br />
1 Mo<strong>le</strong><br />
- Mo<strong>le</strong><br />
2 Gatwick Stream<br />
3 Ifield Brook<br />
4 Crawters Brook<br />
5 Crawters Brook<br />
487<br />
that simp<strong>le</strong> one and two-parameter linear conceptual models can be used to<br />
advantage in characterising <strong>the</strong> changes iq catchment response caused by urbanisation.<br />
However, a prerequisite to ei<strong>the</strong>r approach is <strong>the</strong> availability of<br />
hydrometric data for a sufficiently large number of urban and rural catchment<br />
areas to effect a regional analysis. Where <strong>the</strong> number of flow records is<br />
limited, techniques which employ as few hydrograph parameters as possib<strong>le</strong> are<br />
an obvious advantage.<br />
In <strong>the</strong> following paper, a dimension<strong>le</strong>ss unit hydrograph technique which<br />
involves <strong>the</strong> use of only one parameter is outlined. The method of approach<br />
is illustrated by means of data from an area in <strong>the</strong> south-east of England.<br />
The paper begins in Section (2) with a brief description of <strong>the</strong> area and <strong>the</strong><br />
availab<strong>le</strong> hydrometric data, and continues in Section (3) with an outline of<br />
<strong>the</strong> method by which TüH's were derived. The regionalisation of <strong>the</strong>se TüH's<br />
is discussed in Section (4). The paper concludes in Section (5) with a brief<br />
diqcussion of <strong>the</strong> existing data inadequacies in Urban Hydrology.<br />
2. DATA PRXPARATION<br />
2.1 Data inventory<br />
Between 1949 and 1969, <strong>the</strong> population of Craw<strong>le</strong>y, a town situated some<br />
30 mi<strong>le</strong>s to <strong>the</strong> south of London, increased from 5,000 to 68,000. The<br />
developed area i? drained by <strong>the</strong> headwaters of <strong>the</strong> River Mo<strong>le</strong>, a south-bank<br />
tributary of <strong>the</strong> River Thames. !Che western side of <strong>the</strong> town drains to Ifield<br />
Brook, whereas <strong>the</strong> centre and eastern sides are served by Crawters Brook and<br />
Gatwick Stream respectively (see Figure 1). A major part of <strong>the</strong> urban area<br />
lies on Weald Clay overlying Tunbridge Wells Sand, <strong>the</strong> latter outcropping to<br />
<strong>the</strong> south of <strong>the</strong> area. The average annual rainfall in <strong>the</strong> Craw<strong>le</strong>y region<br />
(1916-1950) ranges from 750-850 rnrn.<br />
There are 6 gauging stations within <strong>the</strong> area of interest, <strong>the</strong> details<br />
of which are summarised in Tab<strong>le</strong> 1. Both <strong>the</strong> gauging statipns on <strong>the</strong> River<br />
Mo<strong>le</strong> and that on Gatwi.ck Stream are operated by <strong>the</strong>.Thames Conservancy ;<br />
Craw<strong>le</strong>y Urban District Council maintain <strong>the</strong> records at <strong>the</strong> remaining 3 sites.<br />
For <strong>the</strong> purposes of <strong>the</strong> present study, data were availab<strong>le</strong> for all sites apart<br />
from <strong>the</strong> River Mo<strong>le</strong> at Gatwick Airport.<br />
TABU 1<br />
: Details of gauging stations within <strong>the</strong> Craw<strong>le</strong>y region.<br />
Hor<strong>le</strong>y Weir<br />
stat ion cat chment records<br />
from<br />
Gatwick Airport<br />
Tins<strong>le</strong>y Sewage Works<br />
Ifield Mill<br />
Hazelwick Roundabout<br />
Woolborough Road<br />
89.8<br />
31 -8<br />
31 .o<br />
12.3<br />
4.7<br />
2.2<br />
Nov., I961<br />
Nov., 1967<br />
Jul., 1952<br />
Dec., I958<br />
May, 1954<br />
Sew.. 1952
488<br />
The positions of <strong>the</strong> 3 principal autographic raingauges located within<br />
<strong>the</strong> headwaters of <strong>the</strong> River Mo<strong>le</strong> are indicated on Figure 1 along with <strong>the</strong><br />
gauging stations. 2 of <strong>the</strong> 3 raingauges have been in operation since before<br />
<strong>the</strong> first regular streamflQw measurements were taken at Tins<strong>le</strong>y Sewage Works<br />
and Woolborough Road, and records from all 3 raingauges are availab<strong>le</strong> from<br />
before 1961 when <strong>the</strong> two gauging stations on <strong>the</strong> River Mo<strong>le</strong> were brought into<br />
use.<br />
2.2 Se<strong>le</strong>ction of storm events<br />
The first stage in <strong>the</strong> analysis of <strong>the</strong> availab<strong>le</strong> data involved <strong>the</strong><br />
preparation of a short-list of suitab<strong>le</strong> storm events for each of <strong>the</strong> £ive<br />
catchment areas. The criteria used in choosing <strong>the</strong>se events were somewhat<br />
arbitrary, but in genera1,an attempt was made to confine <strong>the</strong> analysis to<br />
hydrographs with well-defined peaks having both a smooth rising limb and a<br />
smooth recession. Rainfall data for each of <strong>the</strong> se<strong>le</strong>cted storm events were<br />
<strong>the</strong>n abstracted.<br />
The raingauge at Broadfield was taken to be representatjve<br />
of <strong>the</strong> rainfall patterns over <strong>the</strong> catchment areas draining to gauging statipns<br />
3, 4 and 5 (see Tab<strong>le</strong> I>, and <strong>the</strong> arithmetic mean of <strong>the</strong> catches at Broadfield<br />
and Gatwick Airport was taken for <strong>the</strong> areas commanded by gaugipg stations 1<br />
and 2. The records from Tins<strong>le</strong>y Sewage Works were only used when no information<br />
was availab1.e at ei<strong>the</strong>r of <strong>the</strong> o<strong>the</strong>r gauges.<br />
The above se<strong>le</strong>ction procedure produced 8 storm events at gauging statipn<br />
3, 11 at gauging station 1, 12 at gauging station 4 and 16 each at gauging<br />
stations 2 and 5, <strong>the</strong> majority of whi,ch were associated with rainfall totals<br />
exceeding 12 mm.<br />
2.3 Baseflow separation<br />
The second stage in <strong>the</strong> analysis consisted of <strong>the</strong> separation of <strong>the</strong> baseflow<br />
component from each of <strong>the</strong> recorded streamflow hydrographs, The procedure<br />
adopted involved <strong>the</strong> plotting of <strong>the</strong> recession limb of each hydrograph on<br />
semi-logarithmic graph paper with discharge on <strong>the</strong> logarirchmjc sca<strong>le</strong>. A<br />
straight line was <strong>the</strong>n fitted by eye to <strong>the</strong> lower portion of <strong>the</strong> curve, <strong>the</strong><br />
point at which <strong>the</strong> recession departed from this straight line being taken to<br />
mark <strong>the</strong> time at which surface runoff effectively ceased. The variatipn of<br />
baseflow with time during <strong>the</strong> storm was <strong>the</strong>n represented by a straight line<br />
joining this point on <strong>the</strong> recession limb to <strong>the</strong> beginning of <strong>the</strong> rising limb<br />
of <strong>the</strong> hydrograph.<br />
The above method of baseflow separation, which is both straightforward<br />
in use and <strong>le</strong>ss subjective than <strong>the</strong> majority of <strong>the</strong> alternatgve procedures<br />
was applied to each of <strong>the</strong> 63 recorded hydrographs se<strong>le</strong>cted for analysis.<br />
The ordinates of <strong>the</strong> resultant surface runoff hydrographs were <strong>the</strong>n abstracted<br />
at I-h intervals for all events at gauging stations 1-4 and at 30-min intervals<br />
at gauging station 5.<br />
These data were subsequently transferred on to 80-<br />
column punched cards along with <strong>the</strong> total recorded rainfalls witpin <strong>the</strong> same<br />
time incrementso
3. DERIVATION OF UNIT H!¿DROGRAPHS<br />
489<br />
There are two distinct methods of approach to determining <strong>the</strong> instantaneous<br />
unit hydrograph (IW) or finite-period unit hydrograph (TUH) of a<br />
catchment area from rainfall and streamflow data [g]. The first of <strong>the</strong>se<br />
methods of approach involves <strong>the</strong> fitting of a linear conceptua1,model to <strong>the</strong><br />
records of rainfall excess and surface runoff.<br />
The 4mpulse response functi.on<br />
of <strong>the</strong> fitted mode1,is <strong>the</strong>n taken to approximate <strong>the</strong> IUH of <strong>the</strong> catchment.<br />
This indirect syn<strong>the</strong>sis approach may be contrasted with <strong>the</strong> more direct methods<br />
of analysis which operate on <strong>the</strong> rainfall excess and surface runoff data to<br />
yield an IUH or TLTH wi,thout <strong>the</strong> need to postulate a model. The harmoni? method<br />
for defining <strong>the</strong> TUH of a catchment [9], which was adopted for <strong>the</strong> purposes<br />
of <strong>the</strong> present study, falls into <strong>the</strong> latter category.<br />
3.1 The harmonic method of unit hydrograph derivation<br />
In order to apply <strong>the</strong> harmonic method, <strong>the</strong> volumes of raiyfall excess<br />
and <strong>the</strong> ordinates of both <strong>the</strong> surface runoff hydrograph and <strong>the</strong> TIM are defined<br />
in terms of harmonic series. For examp<strong>le</strong>, if <strong>the</strong> equally-spaced ordinates of<br />
<strong>the</strong> surface runoff hydrograph are given by yi, i = 1, 2, ...., n,<br />
+ = + C[A~ P cos j - 2xi sin j &]<br />
Yi.<br />
n j n<br />
j=l<br />
If n is an odd number, p = (n-1)/2 and<br />
B = 2 c y<br />
sink -;i<br />
n k<br />
j k= 1<br />
The volumes of rainfall excess, xi, i = 1, 2, ..., m, within <strong>the</strong> same<br />
equal time increments may also be expressed as a harmonic series with <strong>the</strong> same<br />
fundamental period and number of terms if (n-m) zeros are added to represent<br />
<strong>the</strong> terms xi, i = m+l, m+2, ...., n.<br />
This series will be identical in form<br />
to equation (I), but with n harmonic coefficients a, b whose values can be<br />
obtained by substituting rainfall excess volumes for surface runoff ordinates<br />
in <strong>the</strong> equations (2). If <strong>the</strong> TITH is also assumed to have n equally-spaced<br />
ordinates, Le. <strong>the</strong> same fundamental period as that of <strong>the</strong> rainfall excess<br />
and surface runoff data, O'Donnell 191 has shown that <strong>the</strong> harmonic coefficients<br />
a, ß, of <strong>the</strong> harmonic series which defines <strong>the</strong> ordinates of <strong>the</strong> TUH can be<br />
calculated directly from <strong>the</strong> harmonic coefficients A, B, a, b usipg <strong>the</strong> linkage<br />
equations<br />
a.A. + b.B - -<br />
a = - but cio<br />
n<br />
-<br />
j a. +b<br />
~j<br />
- 2 w<br />
Pj - n 2<br />
aj2+bj<br />
1 %<br />
n a<br />
O<br />
eq. (3)
49 O<br />
Substitution of <strong>the</strong> aj, ßj in a series expansion of <strong>the</strong> form of equation<br />
(1) <strong>the</strong>n gives <strong>the</strong> successive ordinates of <strong>the</strong> TUH, ui, i = 1, 2, ...., n<br />
dir e c t ly .<br />
The application of any of <strong>the</strong> established methods of analysis, such as<br />
<strong>the</strong> harmonic method, is liab<strong>le</strong> to produce TUH's which are distorted by highfrequency<br />
oscillations of varying amplitude. PhiiliFpee and Wiggert [IO] who<br />
applied <strong>the</strong> harmonic method to data from 38 storms on 4 drainage basins in<br />
thr vicinity of Detroit, encountered this prob<strong>le</strong>m but offered no explanation<br />
as to its cause. More recent studies by Blank et al [Il], who used <strong>the</strong> Fourier<br />
transform approach, which bears some relatipnship to <strong>the</strong> harmonic method, have<br />
indicated that such oscillations can result from errors in <strong>the</strong> data and are<br />
not necessarily caused by <strong>the</strong> inherent non-linearity of <strong>the</strong> rainfall-runoff<br />
relationship. Blank et al [Il] also show that oscillatory TUH's can be avoided<br />
by applying a low-pass digital filter to <strong>the</strong> rainfall excess and surface runoff<br />
data prior to <strong>the</strong> derivation of <strong>the</strong> TUH.<br />
One of <strong>the</strong> principal advantages of <strong>the</strong> harmonic method is its f<strong>le</strong>xibility<br />
in dealing with storm events which produce such oscillatory "UH's without <strong>the</strong><br />
need to use digital filters. This property of <strong>the</strong> method stems from <strong>the</strong> form<br />
of <strong>the</strong> linkage equations (3) by which <strong>the</strong> aj, ßj of <strong>the</strong> harmonic series representation<br />
of <strong>the</strong> TKH depend only on <strong>the</strong> harmonic coefficients of <strong>the</strong> rainfall<br />
excess and surface runoff data for <strong>the</strong> same frequency. Individual harmonics<br />
may <strong>the</strong>refore be omitted from <strong>the</strong> series representation of <strong>the</strong> TUH without<br />
affecting <strong>the</strong> calculation of o<strong>the</strong>r üj, ßj.<br />
In particular, if <strong>the</strong> ordinates<br />
of <strong>the</strong> TUH obtained by using all <strong>the</strong> aj, ßj exhibit high-frequency oscillations,<br />
truncation of <strong>the</strong> series representation may help to eliminate <strong>the</strong>se<br />
oscillations. However, <strong>the</strong> amount of truncation applied should not be suffic-<br />
ient to cause <strong>the</strong> hydrograph obtained by convolving <strong>the</strong> smoo<strong>the</strong>d Tw with <strong>the</strong><br />
distribution of rainfall excess to depart significantly from <strong>the</strong> original<br />
surface runoff hydrograph.<br />
3.2<br />
Appliiation to Craw<strong>le</strong>y area data<br />
A computer program was written to derive TUH's using <strong>the</strong> harmonic method<br />
described ip Section (3.1) above. The computation began with <strong>the</strong> determination<br />
of <strong>the</strong> distribution of rainfall excess using <strong>the</strong> @-index method. The total<br />
volumes of both rainfall and runoff were calculated and <strong>the</strong>ir difference<br />
averaged over <strong>the</strong> number of time intervals with non-zero rainfall. This<br />
average t'losstt was <strong>the</strong>n subtracted from <strong>the</strong> recorded volumes of rainfall w5thin<br />
each time interval, any negative differences being set to zero. The <strong>who<strong>le</strong></strong><br />
procedure was repeated until <strong>the</strong> difference between <strong>the</strong> total volumes of<br />
rainfall and runoff was <strong>le</strong>ss than 0.25 mm. Having obtained <strong>the</strong> distribution<br />
of rainfall excess, <strong>the</strong> derivation of <strong>the</strong> TUH was carried out according to <strong>the</strong><br />
method outlined in Section (3.1) above. The surface runoff hydrograph was<br />
<strong>the</strong>n reconstituted by convolving <strong>the</strong> derived TUH with <strong>the</strong> distribution of<br />
rainfall excess.<br />
The data from all 63 storm events were processed using <strong>the</strong> full number<br />
of harmonic coefficients in determining <strong>the</strong> ordinates of <strong>the</strong> TUH. The results<br />
obtained were <strong>the</strong>n plotted and compared. The majority of <strong>the</strong> derived TUH's<br />
were found to exhibit high frequency oscillations of varying amplitude. The<br />
storm events which gave rise to such behaviour were <strong>the</strong>refore re-processed<br />
using fewer harmonic coefficients in determining <strong>the</strong> TUH ordinates.
The choice of <strong>the</strong> most appropriate number of harmonic coefficients to<br />
use for any given storm event is largely subjective. Truncating <strong>the</strong> harmonic<br />
series representation of <strong>the</strong> TLTH may remove <strong>the</strong> high-frequency oscillations,<br />
but <strong>the</strong> hydrograph obtained by convolving that TUK with <strong>the</strong> distribution of<br />
rainfall excess should not depart markedly from <strong>the</strong> original surface runoff<br />
hydrograph, particularly in regard to <strong>the</strong> magnitude and timing of <strong>the</strong> peak<br />
flows. The amount of computer time that would have been involved in<br />
systematically reducing <strong>the</strong> number of harmonic coefficients in <strong>the</strong> series<br />
representation of <strong>the</strong> !ì'üñ until <strong>the</strong> fit provided by <strong>the</strong> reconvolved surface<br />
runoff hydrograph was no longer acceptab<strong>le</strong> would have been excessive. A pilot<br />
study using a restricted number of truncated,,series,each having a predetermined<br />
proportion of <strong>the</strong> full number of harmonics, was <strong>the</strong>refore carried out. For <strong>the</strong><br />
majority of <strong>the</strong> storm events, halving <strong>the</strong> number of harmonics successfully dampened<br />
<strong>the</strong> high-frequency oscillations, and gave rise to a reconvolved hydrograph<br />
whose maximum ordinate was generally within 5 per cent of <strong>the</strong> peak of <strong>the</strong><br />
original surface runoff hydrograph.<br />
491<br />
As a result of <strong>the</strong> above analysis, 8 TiJH's were obtained for gauging<br />
station 2, 6 each for gauging stations 4 and 5, 4 for gauging station 1 and<br />
3 for gauging station 3. The changes in flow regime which had occurred at<br />
gauging stations 2, 4 and 5 during <strong>the</strong> period of record were immediately<br />
obvious, and TiJH's for each of <strong>the</strong>se sites were <strong>the</strong>refore grouped according<br />
to <strong>the</strong> dates of occurrence of <strong>the</strong> storm events from which <strong>the</strong>y were derived.<br />
For convenience, <strong>the</strong>se different groupings will be referred to by <strong>the</strong> <strong>le</strong>tters<br />
IfAt1 (for <strong>the</strong> earlier storms) and "BI1 (for <strong>the</strong> later storms). Of <strong>the</strong> 8<br />
separate sets of TUH's, none consisted of <strong>le</strong>ss than 3 hydrographs. The TUH's<br />
within each set were <strong>the</strong>n plotted toge<strong>the</strong>r using a common starting time, and<br />
an "average" TiiH obtained by drawing in a smooth curve through <strong>the</strong> plotted<br />
points, care being taken to ensure that <strong>the</strong> area under <strong>the</strong> curve was equiva<strong>le</strong>nt<br />
to 25 mm over <strong>the</strong> catchment area.<br />
Figure 2 shows <strong>the</strong> smoo<strong>the</strong>d TITH'S obtained for Crawters Brook at Woolborough<br />
Road, and is indicative of <strong>the</strong> change in flow regime which has taken place as<br />
<strong>the</strong> town centre of Craw<strong>le</strong>y has developed over a period of some 15-20 years.<br />
4. REGIONALISATION OF UNIT KYDROGRAPHS<br />
One of <strong>the</strong> simp<strong>le</strong>st assumptions that can be made in regiqnaliTi9g a group<br />
of unit hydrographs is that all TLTH'S of a common duration are reducib<strong>le</strong> to<br />
<strong>the</strong> same dimension<strong>le</strong>ss shape. The scaling parameters that are required to<br />
describe <strong>the</strong> dipension<strong>le</strong>ss hydrograph (of which <strong>the</strong>re are generally two) are<br />
expressed in terms of catchment characteristics by means of a multip<strong>le</strong> linear<br />
regression analyses. The appljcation of this approach,$ <strong>the</strong> present study<br />
is complicated by <strong>the</strong> necessity to include independent variab<strong>le</strong>s which ref<strong>le</strong>ct<br />
thg man-made changes within <strong>the</strong> catchment areas affected by <strong>the</strong> development of<br />
Craw<strong>le</strong>y.<br />
Previous authors who have applied a dimension<strong>le</strong>ss unit hydrograph approach<br />
have differed widely ip <strong>the</strong>ir choice of scaling parameters, For examp<strong>le</strong>,<br />
Commons u23 developed a l'basic hydrograph" with a time base of 100 arbitrary<br />
units, a height of 60 arbitrary discharge units and an area of 1196.5 square<br />
units. The scaling parameters required were peak rate of runoff and total<br />
volume of runoff. In common with many similar pairings, <strong>the</strong>se parameters
49 2<br />
are not entirely independent, and as Diskin [I31 has recently pointed out, a<br />
choice of parameters which satisfy <strong>the</strong> constraint of unit area under <strong>the</strong> TLTH<br />
is to be preferred.<br />
A review of previously-published work on <strong>the</strong> hydrological consequences<br />
of urbanisation showed that, of several possib<strong>le</strong> time scaling parameters, <strong>the</strong><br />
lag time TL, defined as <strong>the</strong> time interval between <strong>the</strong> centroid of rainfall<br />
excess and <strong>the</strong> centroid of surface runoff, has been found to show a consistent<br />
variation with <strong>the</strong> <strong>le</strong>ngth and slope of <strong>the</strong> main channel for both rural and<br />
urban catchment areas 13-53. If, however, <strong>the</strong> constraint of unit area under<br />
<strong>the</strong> TUH is observed, making <strong>the</strong> time sca<strong>le</strong> of each TüH dimension<strong>le</strong>ss by expressipg<br />
<strong>the</strong> timing of all ordinates as a proportion of TL also determines <strong>the</strong> ordinate<br />
sca<strong>le</strong>. Hence, <strong>the</strong> dimension<strong>le</strong>ss unit hydrograph can be specified in terms of<br />
only one parameter, <strong>the</strong> functional form of <strong>the</strong> curve being<br />
ut.TL = f (t/TL) eqe (4)<br />
where ut is <strong>the</strong> ordinate of <strong>the</strong> actual TUH at time t.<br />
The above method of producipg a dimension<strong>le</strong>ss unit hydrograph was applied<br />
to <strong>the</strong> data obtained from <strong>the</strong> 5 catchments in <strong>the</strong> Craw<strong>le</strong>y area. The lag time<br />
of <strong>the</strong> I-h TUH for each catchment was obtained by computing <strong>the</strong> tjme interval<br />
between <strong>the</strong> origin of <strong>the</strong> TüH and its centroid and subtracting O.5h. Following<br />
Carter [3) and Anderson [5], a doub<strong>le</strong>-logari4hmic plot of lag time against<br />
basin ratio was prepared (see Figure 3). Basin ratio is defined by <strong>the</strong> quotient<br />
L@, where L is <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> main channel between <strong>the</strong> gauging station<br />
and <strong>the</strong> watershed (km) and S <strong>the</strong> main channel slope.<br />
S is defined by <strong>the</strong> alti-<br />
tude difference between points located 10 and 85 per cent of <strong>the</strong> main channel<br />
<strong>le</strong>ngth upstream from <strong>the</strong> gauging station divided by <strong>the</strong>ir distance apart [14].<br />
Values of L and S were obtained from 1 : 25000 sca<strong>le</strong> maps for all catchment<br />
areas apart from that of gauging station 5 for which 1 : 500 longitudinal<br />
sections were availab<strong>le</strong>.<br />
In preparing Figure 3, <strong>the</strong> number of data was increased by <strong>the</strong> inclusion<br />
of TUH's for 2 gauging stations on <strong>the</strong> River Wand<strong>le</strong> in <strong>the</strong> sou<strong>the</strong>rn suburbs<br />
of London to <strong>the</strong> north of Craw<strong>le</strong>y (see Tab<strong>le</strong> 2). These hydrographs, which<br />
were obtained by Nash [l5], relate to conditions before and after <strong>the</strong> execution<br />
of channel improvement works. The data tabulated by Nash (loc. cit. Tab<strong>le</strong> 3,<br />
p.323) were assumed to relate to a duration of 30 min. The reduction in lag<br />
time shown by <strong>the</strong> Itpost-works" hydrographs was estimated by Nash from <strong>the</strong><br />
records for 3 adjacent catchment areas whose channels were considered to be<br />
in an equiva<strong>le</strong>nt state to <strong>the</strong> improved conditions on <strong>the</strong> River Wand<strong>le</strong>. The<br />
"post-works" hydrographs were <strong>the</strong>refore syn<strong>the</strong>tic and not derived directly from<br />
recorded data. Never<strong>the</strong><strong>le</strong>ss, <strong>the</strong> data were considered to be useful in providing<br />
an independent measure of <strong>the</strong> influence of channel improvement works without<br />
a simultaneous growth in <strong>the</strong> impervious area within a catchment.<br />
TABLE 2 : Details of gauging stations within <strong>the</strong> River Wand<strong>le</strong> catchment area<br />
(from Nash [IS] 1<br />
river station catchment
493<br />
Also plotted in Figure 3 are <strong>the</strong> lag timebasin ratio relationships<br />
obtained by Anderson [5] for catchment areas in three different stages of<br />
development. Drainage class N refers to natural (rural) areas. Drainage class<br />
B includes areas in which <strong>the</strong> impervious cover ranges from 20 to 30 per cent,<br />
<strong>the</strong> tributary streams are sewered but <strong>the</strong> main channels are retained in <strong>the</strong>ir<br />
natural state. Drainage class U refers to fully developed urban areas having<br />
more than 30 per cent impervious cover and all stream channels comp<strong>le</strong>tely<br />
sewered or improved and realigned. The majority of <strong>the</strong> gauging stations used<br />
by Anderson in deriving <strong>the</strong>se relationships were situated within <strong>the</strong> Washington<br />
D.C. metropolitan area.<br />
Examination of Figure 3 shows that data for gauging stations 2 and 3<br />
exhibit markedly longer lag times than would be predicted from Anderson's class<br />
N relationship. Since gauging station 3 is situated at <strong>the</strong> outfall of a mill<br />
pond covering an area of some 8-10 ha, such behaviour is to be expected. The<br />
presence of a lake of similar size within <strong>the</strong> headwaters of Gatwick Stream<br />
catchment has a similar if not as pronounced an effect on lag time. In contact,<br />
<strong>the</strong> data for gauging stations 1, 4A, 5A, 6A and 7A all show reasonab<strong>le</strong> agreement<br />
with <strong>the</strong> Anderson class N relationship, although bearing in mind <strong>the</strong><br />
channel improvements above gauging station 5 and <strong>the</strong> existing urban development<br />
above gauging station 6, <strong>the</strong> broken line drawn above and paral<strong>le</strong>l to Anderson's<br />
equation is perhaps a better approximation to natural catchment conditions<br />
within <strong>the</strong> Craw<strong>le</strong>y area.<br />
Figure 3 shows that at gauging station 5, an increase in <strong>the</strong> proportion<br />
of impervious cover (as estimated from data supplied by Craw<strong>le</strong>y Urban District<br />
Council) from 5 to 26 per cent is associated with a reduction in lag time of<br />
23 per cent, whereas at gauging station 4 an increase in impervious area from<br />
18 to 27 per cent apparently causes a reduction in lag time of 72 per cent.<br />
The latter anomaly is thought to result from extensive renewal of sewerage<br />
within <strong>the</strong> catchment which took place concurrently with <strong>the</strong> increase in paved<br />
area. The data from gauging stations 6 and 7 indicate that channel improvement<br />
(not including installation of sewerage) can cause a 30-40 per cent reduction<br />
in lag time. The results obtained at gauging stations 4 and 5 are <strong>the</strong>refore<br />
not as inconsistent as <strong>the</strong>y might at first appear.<br />
Figure 3 also shows that <strong>the</strong> lag times for Anderson's developed (class B)<br />
and fully developed (class U) catchments are markedly shorter than those<br />
observed in <strong>the</strong> Craw<strong>le</strong>y area. The lower broken line, drawn paral<strong>le</strong>l to and<br />
immediately above Anderson's class B relationship, is probably <strong>the</strong> best approximation<br />
to <strong>the</strong> behaviour of developed catchments with approximately 30 per cent<br />
impervious cover and improved channel systems within <strong>the</strong> Craw<strong>le</strong>y area that <strong>the</strong><br />
availab<strong>le</strong> data will allow.<br />
Having obtained <strong>the</strong> relationship between <strong>the</strong> chosen scaling parameter and<br />
two readily-computed catchment characteristics, only <strong>the</strong> form of <strong>the</strong> dimension-<br />
<strong>le</strong>ss curve is required to construct <strong>the</strong> I-h TUH for an ungauged catchment<br />
within <strong>the</strong> Craw<strong>le</strong>y area. Accordingly, <strong>the</strong> 8 observed TITH'S whose derivation<br />
was described in Section (3) above were reduced to <strong>the</strong> form of equation (4)<br />
using <strong>the</strong> appropriate observed values of lag time (see Figure 4). A sing<strong>le</strong><br />
dimension<strong>le</strong>ss hydrograph was <strong>the</strong>n fitted by eye to <strong>the</strong> plotted points, care<br />
being taken to ensure that <strong>the</strong> area under <strong>the</strong> curve was unity.
494<br />
In practice, application of <strong>the</strong> method to produce a I-h TUH for an ungauged<br />
catchment may be summarised as follows :<br />
i)<br />
ii)<br />
iii)<br />
measure <strong>the</strong> <strong>le</strong>ngth and slope of <strong>the</strong> main channel of <strong>the</strong> catchment area<br />
from a 1 : 25000 Ordnance Survey map, and compute <strong>the</strong> basin ratio ;<br />
use Figure 3 to estimate <strong>the</strong> lag time of <strong>the</strong> catchment for a particular<br />
stage of urbanisation ; and<br />
given <strong>the</strong> lag time, use <strong>the</strong> dimension<strong>le</strong>ss unit hydrograph of Figure 4<br />
to construct <strong>the</strong> I-h TLTH of <strong>the</strong> catchment.<br />
5. CONCLUDING REMARKS<br />
Since <strong>the</strong> procedure outlined above uses <strong>the</strong> lag time as <strong>the</strong> only scaling<br />
parameter, <strong>the</strong>re is an obvious analogy with <strong>the</strong> sing<strong>le</strong> linear reservoir model<br />
whose storage constant is equiva<strong>le</strong>nt to <strong>the</strong> lag time as defined in <strong>the</strong> present<br />
study. The major difference between <strong>the</strong> two approached lies in describing<br />
<strong>the</strong> TUH by means of a series of plotted points, ra<strong>the</strong>r than an equation.<br />
peak of any TUH constructed from Figure 4 is <strong>the</strong>refore constrained to occur<br />
at a specific proportion of <strong>the</strong> lag time ra<strong>the</strong>r than at a time equiva<strong>le</strong>nt to<br />
<strong>the</strong> duration of <strong>the</strong> unit hydrograph.<br />
According to Rao et al 181, <strong>the</strong> sing<strong>le</strong> linear reservoir model provides<br />
an adequate description of <strong>the</strong> behaviour of both urban and rural catchments<br />
<strong>le</strong>ss than approximately 13 km2 in area. Those Authors obtained an expression<br />
for lag time in terms of volume and duration of rainfall excess and proportion<br />
of impervious cover. The results of <strong>the</strong> present study (in particular, Figure 3)<br />
tend to indicate that, for <strong>the</strong> area under study, proportion of impervious cover<br />
provides a <strong>le</strong>ss than adequate description of <strong>the</strong> man-made changes within a<br />
drainage basin. In <strong>the</strong> absence of additional parameters relating to changes<br />
in <strong>the</strong> channel system, and perhaps distribution of impervious cover with respect<br />
to <strong>the</strong> outfall of <strong>the</strong> catchment, <strong>the</strong> engineer concerned with <strong>the</strong> design of<br />
flood al<strong>le</strong>viation works must rely on diagrams such as Figure 3 whose construction<br />
is unfortunately largely subjective and highly dependent on local know<strong>le</strong>dge<br />
of <strong>the</strong> area.<br />
The urbanisation of a catchment area provides one of <strong>the</strong> most dramatic<br />
examp<strong>le</strong>s of man's interference with <strong>the</strong> hydrological cyc<strong>le</strong>. Whereas <strong>the</strong><br />
expansion of any conurbation creates an increasing water demand for domestic,<br />
industrial and recreational purposes, <strong>the</strong> very presence of <strong>the</strong> urban area<br />
acce<strong>le</strong>rates <strong>the</strong> processes by which locally stored ana precipitated water is<br />
returned to <strong>the</strong> sea. Despite <strong>the</strong> major changes in <strong>the</strong> flow regime of a catchment<br />
area which urbanisation can bring about, relatively litt<strong>le</strong> attention has<br />
been given to <strong>the</strong> quantification of such changes when compared with o<strong>the</strong>r land<br />
use changes,such as that of forest to grassland. Bearing in mind <strong>the</strong> large<br />
sums which have been and are being devoted to flood protection schemes for<br />
urban areas, <strong>the</strong> availab<strong>le</strong> information can justifiably be label<strong>le</strong>d as inadequate.<br />
The
ACKNOWLEDGEMENTS<br />
The study described above was carried out on behalf of <strong>the</strong> Resources<br />
Group for West Sussex County Council. The author wishes to thank Dr. T.M.<br />
Prus-Chacinski, partner, C.H. Dobbie and Partners, for his encouragement<br />
to prepare and permission to publish this paper. The assistance received<br />
from <strong>the</strong> Chief Engineer, Thames Conservancy, Mr. E.J. Brettell, and <strong>the</strong><br />
Engineer and Surveyor, Craw<strong>le</strong>y Urban District Council, Mr. H.J. Lum<strong>le</strong>y, in<br />
providing hydrometric data was also greatly appreciated.<br />
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495<br />
Espey, W.H., Morgan, C.W., Masch, F.D. (1965). A study of some effects<br />
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Espey, W.H., Winslow, D.E., Morgan, C.W. (1969). Urban effects on <strong>the</strong><br />
unit hydrograph, in Moore, W.L., Morgan, C.W. (eds.), Effects of watershed<br />
changes on streamflow, Proc. Wat. Resour. Symp. no. 2, Centre for Res.<br />
in Wat. Resour., Univ. of Texas, Univ. of Texas Press, pp. 215-228.<br />
Rao, R.A., Del<strong>le</strong>ur, J.W., Sarma, B.S.P. (1972). Conceptual hydrologic<br />
models for urbanising basins, Proc. Am. Soc. Civ. Engrs., J. Hydraul.<br />
Div., 98 (KY71, pp. 1205-1220.<br />
O'Donnell, T. (1966). Methods of computation in hydrograph analysis and<br />
syn<strong>the</strong>sis, Recent trends in hydrograph syn<strong>the</strong>sis, Proc. Tech. Meeting<br />
no. 21, T.N.O., The Hague, pp. 65-102.<br />
Philippee, J.T., Wiggert, J.M. (1969). Instantaneous unit hydrograph<br />
response by harmonic analysis, Wat. Resour. Res. Centre, Virginia<br />
Polytechnic Institute, Bull. 15, 36 pp.<br />
Blank, D., Del<strong>le</strong>ur, J.W., Giorgini, A. (1971). Oscillatory kernel<br />
functions in linear hydrologic models, Wat. Resour. Res., 7, pp. 1102-1117.<br />
Commons,, G,G. (1942). Flood hydrographs, Civ. Engrg. (New York), 12,<br />
pp. 571-5720
496<br />
13. Diskin, M.H. (1972). The ro<strong>le</strong> of lag in a quasi-linear analysis of <strong>the</strong><br />
surface runoff system, paper presented at <strong>the</strong> 2nd Internat. Hydrol. Symp.,<br />
Fort Collins, Colorado.<br />
14. Benson, M.A. (1959) . Channel-slope factor in flood-frequency analysis,<br />
Proc. Am. Soc. Civ. Engrs., J. Hydraul. Div., 85 (Kyk), pp. 1-9.<br />
15. Nash, J.E. (1959). The effect of flood-elimination works on <strong>the</strong> flood<br />
frequency of <strong>the</strong> River Wand<strong>le</strong>, Proc. Instn. Civ. Engrs., 13, pp. 317-338.
W<br />
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BASIN RATIO, Z, KM<br />
PLOT OF LAG TIME AGAINST BASIN RATIO FOR THE CRAW!.EY AWEA.
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500
A DIMENSIONLESS UNITGRAPH FOR HONG KONG<br />
P. R: HELLIWELL<br />
Department of Civil Engineering, University of Southampton.<br />
ABSTRACT<br />
T.Y. CHEN<br />
Royal Observatory, Hong Kong<br />
The large number of individual catchments in Hong Kong makes it<br />
impracticab<strong>le</strong> to measure stream flows on all but a small proportion<br />
of streams. Rainfall characteristics and topography are similar over<br />
much of <strong>the</strong> area.<br />
Using data for several storms at ea-h of <strong>the</strong> seven stream gau-<br />
ging stations, a mean dimension<strong>le</strong>ss unitgraph was derived. Basin lag<br />
was used in <strong>the</strong> conversion of both time and discharge sca<strong>le</strong>s. For un-<br />
gauged catchments basin lag can be estimated ei<strong>the</strong>r as a simp<strong>le</strong> func-<br />
tion of catchment size, shape and slope.<br />
This work was based on records col<strong>le</strong>rted in 1964 and 1965, and<br />
was one of <strong>the</strong> first studies made possib<strong>le</strong> by <strong>the</strong> installation of a<br />
network of hydrometric stations in Hong Kong.<br />
RESUME<br />
Le grand nombre de bassins fluviaux du territoire de Hong Kong<br />
fait qu'il n'est possib<strong>le</strong> d'effectuer des mesures de débit que sur un<br />
faib<strong>le</strong> pour centage d'entre eux. La pluviométrie et la topographie<br />
présentent des caractéristiques semblab<strong>le</strong>s sur la plus grande partie<br />
du territoire.<br />
En s'appuyant sur <strong>le</strong>s données recueillies à .ept stations de jaz<br />
geage au cours d'un certain nombre d'averses, on a mis au point un h l<br />
drogramme unitaire moyen sans dimension. Le temps de réponse du bas-<br />
sin intervient dans <strong>le</strong>s conversions à la fois pour l'échel<strong>le</strong> des temps<br />
et pour cel<strong>le</strong> des débits. Pour <strong>le</strong>s bassins qui ne tont pas l'objet de<br />
m sures des débits, <strong>le</strong> temps de réponse peut être estimé soit simp<strong>le</strong>-<br />
ment en fonction de la surface du bassin, soit en fonction de sa tai-<br />
l<strong>le</strong>, de sa forme et de sa pente.<br />
La présente étude est basée sur des observations recueillies en<br />
1964 et 1965; ce fut une des premières qui aient été [,endues possib<strong>le</strong>s<br />
par l'installation d'un réseau hydrométrique dans <strong>le</strong> territoire de<br />
Hong Kong.
502<br />
Introduction<br />
The British Crown Colony of Hong Kong is located on <strong>the</strong> coastline<br />
of China, just inside <strong>the</strong> Tropic of Cancer at latitude 220N and longitude<br />
114O. The land area of <strong>the</strong> Colony is approximately 1000km2, comprising<br />
a section of <strong>the</strong> mainland, <strong>the</strong> islands of Hong Kong and Lantau, and a<br />
large number of very small islands. The total area, including sea, is<br />
approximately 2 500km2.<br />
It is an area of high relief, <strong>the</strong> highest point being over 1OOûm<br />
above sea <strong>le</strong>vel. Drainage lines are short, usually <strong>le</strong>ss than lokm, giving<br />
a large number of small steep catchment areas draining to <strong>the</strong> very long<br />
coastline. In some areas, mast notably in <strong>the</strong> northwest, <strong>the</strong>re is an area<br />
of almost flat land between <strong>the</strong> hills and <strong>the</strong> sea shore, formed by silting<br />
up .If shallow bays, and subsequent uplift of <strong>the</strong> land relative to sea <strong>le</strong>vel.<br />
These areas are intensively cultivated, with vegetab<strong>le</strong> crops replacing <strong>the</strong><br />
traditional rice cultivation where <strong>le</strong>vels are high enough to be c<strong>le</strong>ar of<br />
sea water intrusion, and fish ponds starting to replace brackish water rice<br />
cultivation near sea <strong>le</strong>vel.<br />
Much of this flatter land, particularly near<br />
<strong>the</strong> larger stream channels, is natural floodland, which makes stream<br />
gauging at high flows very difficult.<br />
The vegetation of <strong>the</strong> upland areas is of coarse grasses or mixed scrubland.<br />
Ga<strong>the</strong>ring wood for firewood and traditional seasonal burning of hillside<br />
vegetation tend to degrade <strong>the</strong> cover. Soils on <strong>the</strong> hills are coarse,<br />
thin, and poor. Gullying, sometimes severe, occurs mainly in <strong>the</strong> west of<br />
<strong>the</strong> Colony. In <strong>the</strong> East, <strong>the</strong> grass cover is comp<strong>le</strong>te, and sediment loads<br />
are very low. The geology is predominantly granitic. Soil moisture and<br />
groundwater storage are small.<br />
Climate is seasonal. Winters are cool and dry, although periods af<br />
li-ght rain do occur. Summers are warm (daily maximum temperature up eo 35W,<br />
fvith very litt<strong>le</strong> diurnal variation) and wet. The mean summer half year rainfall<br />
at <strong>the</strong> Royal Observatory is 1850mm and <strong>the</strong> mean winter half year rainfall<br />
is 350mm. Observation-day rainfalls in excess of 250mm occur in most<br />
years. Frosts can occur at <strong>le</strong>vels above 600m, but snow does not fall.<br />
Annual rainfall elsewhere in <strong>the</strong> Colony varies between 1250mm and 3000m.<br />
Water supply has always been a major prob<strong>le</strong>m in Hong Kong. The small<br />
size of catchment areas, <strong>the</strong> seasonal nature of rainfall, and occasional<br />
severe droughts have presented a major chal<strong>le</strong>nge to <strong>the</strong> water engineers(l1.<br />
Of necessity, reservoirs with small direct catchments have been built, and<br />
water has been broughtin from much larger areas by systems of catchwater<br />
channels or tunnels, intercepting many small streams which would o<strong>the</strong>rwise<br />
discharge to <strong>the</strong> sea. More recently, arms of <strong>the</strong> sea have been converted to<br />
freshwater storage, at Plover Cove and at High Island.
Measurement of Streamflow<br />
Measurement of streamflow in all catchments in Hong Kong is obviously<br />
impossib<strong>le</strong>, The approach adopted has been to make measurements in a<br />
relatively small number of basins spread through <strong>the</strong> Colony, and to transfer<br />
data from <strong>the</strong>se to o<strong>the</strong>r basins. This paper describes <strong>the</strong> method used to<br />
generalise flood hydrograph data, and presents <strong>the</strong> resulting dimension<strong>le</strong>ss<br />
unitgraph.<br />
Streamflow is measured by fixed structures. Sharp-edged and crump<br />
weirs of various compound profi<strong>le</strong>s, triangular and trapezoidal flumes,<br />
Parshall flumes and broad-crested diversion weirs are all used. Data are<br />
published annually(2).<br />
Se<strong>le</strong>ction of Records for Study<br />
Examination of records from streamflow stations and of autographic<br />
rainfall records for sites in or near to gauged catchments showed that three<br />
or more storms suitab<strong>le</strong> for analysis were availab<strong>le</strong> at seven stations. There<br />
were seven more catchments which had been or were being gauged, but no suitab<strong>le</strong><br />
records for this analysis were found among <strong>the</strong>m, <strong>the</strong> commonest prob<strong>le</strong>m<br />
being submergence of <strong>the</strong> measuring structure at high flow.<br />
notes on all streamflow stations, and Figure 1 is a map showing stations used<br />
in this study. From <strong>the</strong> tab<strong>le</strong> it will be seen that all catchments are small.<br />
Method of Analysis<br />
Tab<strong>le</strong> 1 gives<br />
The method used was that described in USBR Design of Small and<br />
by Lins<strong>le</strong>y, Koh<strong>le</strong>r and Pa~lhus(~).<br />
Studies of base flow had shown that dep<strong>le</strong>tion could be assumed to be<br />
of <strong>the</strong> type qt = qokt.<br />
arithmetic graph paper.<br />
503<br />
Base flow separation was achieved by plotting on log-<br />
No attempt was made to separate interflow.<br />
Duration of effective storm rainfall was found by applying <strong>the</strong>$-index<br />
technique to <strong>the</strong> hyetograph, having found <strong>the</strong> total storm runoff by<br />
integration of <strong>the</strong> storm runoff hydrograph. Storms with up to five unit<br />
periods of excess precipitation were used.<br />
Successive approximation procedures were used to find <strong>the</strong> unitgraph<br />
ordinates.<br />
In order that <strong>the</strong> period should be <strong>le</strong>ss than one third of <strong>the</strong> rise time<br />
of <strong>the</strong> unitgraph (to avoid instability in <strong>the</strong> computations), it was necessary<br />
to use a unit period of 15 minutes, except in <strong>the</strong> case of <strong>the</strong> smal<strong>le</strong>st catch-<br />
ment where 74 minutes was used.
504<br />
With such short periods, <strong>the</strong> accuracy of timing of <strong>the</strong> chart records<br />
of streamflow and rainfall is critical. In <strong>the</strong> cases of Hok Tau and<br />
C’iung Mei, unshielded, tilting bucket rain gauges were sited on <strong>the</strong> stream<br />
recorder house roof in order to ensure correct relative timing. Unfortunately,<br />
<strong>the</strong> mechanism transferring <strong>the</strong> tipping bucket record to <strong>the</strong> chart proved<br />
u.iieliab<strong>le</strong>, and some records were lost. With <strong>the</strong> o<strong>the</strong>r stations, it was<br />
hoped that timing errors would average out, but <strong>the</strong>re is no evidence on this.<br />
From experience, errors in long-period chart records using chart drives of<br />
ZVknlday can be kept to <strong>le</strong>ss than five minutes by making corrections based<br />
on check observations.<br />
On <strong>the</strong> o<strong>the</strong>r hand, standard daily autographic rainfall<br />
recorders in <strong>the</strong> hands of all but <strong>the</strong> most careful observers can often show<br />
fluctuations from correct time of ten to 15 minutes.<br />
The measure of agreement between various unitgraphs for a given catchment<br />
varied. Two examp<strong>le</strong>s, one showing consistent behaviour, and ano<strong>the</strong>r showing<br />
ra<strong>the</strong>r poor agreement, are shown in Figure 2. Average unitgraphs for each<br />
catchment are shown in Figure 3. These were formed in <strong>the</strong> usual way by<br />
averaging time to peak, magnitude of peak and total duration, sketching in<br />
a mean shape, and adjusting <strong>the</strong> area under <strong>the</strong> curve.<br />
The variation of <strong>the</strong> mean unitgraphs can be seen in Figure 3. A means<br />
of unifying <strong>the</strong>se was required. They were made dimension<strong>le</strong>ss in terms of <strong>the</strong><br />
time to <strong>the</strong> centroid of <strong>the</strong> unitgraph and <strong>the</strong> volume of unit rainfall excess.<br />
Time-axis values were divided by time to centroid of unitgraph and<br />
discharge values were multiplied by time to centroid of unitgraph and divided<br />
by <strong>the</strong> volume of unit depth of runoff over <strong>the</strong> catchment area.<br />
The seven dimension<strong>le</strong>ss unitgraphs and <strong>the</strong> mean dimension<strong>le</strong>ss unitgraph<br />
found from <strong>the</strong>m are shown in Figure 4. The ordinates of <strong>the</strong> dimension<strong>le</strong>ss<br />
unitgraph are listed in Tab<strong>le</strong> 2.<br />
Application to ungauged catchments<br />
The size range of individual catchments included in <strong>the</strong> analysis<br />
adequately covered <strong>the</strong> sizes of catchments found in Hong Kong. Similarly,<br />
geographical distribution was quite good, only <strong>the</strong> eroded area in <strong>the</strong> west<br />
being excluded.<br />
Catchment and stream slope variability was not so well<br />
covered. The Tai Po Tau catchment included some lowland area, as did <strong>the</strong><br />
Kam Tin catchment. The o<strong>the</strong>r five were upland in type.<br />
The time to <strong>the</strong> centroid of <strong>the</strong> unitgraph for <strong>the</strong> seven catchments is<br />
<strong>the</strong> basin lag (i.e. time from centre of area of excess rain to centre of<br />
area of hydrograph of excess runoff) plus half <strong>the</strong> unit period, (lag + 9).
505<br />
The basin lag for <strong>the</strong> mean unitgraph of each catchment was plotted<br />
against catchment area and against-where L is <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> main<br />
s<br />
stream projecked back to <strong>the</strong> catchment divide, as measured on 1:25000sca<strong>le</strong><br />
maps, Lc is <strong>the</strong> distance along <strong>the</strong> stream from <strong>the</strong> gauging station<br />
to a point on <strong>the</strong> main stream nearest to <strong>the</strong> catchment centre of area,<br />
and S is <strong>the</strong> stream slope as estimated by <strong>the</strong> difference in e<strong>le</strong>vation of<br />
<strong>the</strong> main stream at <strong>the</strong> catchment divide and <strong>the</strong> gauging station divided<br />
by L). The correlation coefficients were 0.92 and 0.86 respectively. It<br />
was significant that <strong>the</strong> Kan Tin value fell close to <strong>the</strong> regression line<br />
when slope was included, and off <strong>the</strong> line where area alone was used.<br />
However, in all work using <strong>the</strong> hydrograph, catchment area alone has been<br />
used. Figure 5 6 show <strong>the</strong> relationship. Figure 6 also shows data from<br />
Lins<strong>le</strong>y et ad4j and Design of Small Dams(3).<br />
The equations for estimation of catchment lag in ungauged basins are:<br />
lag = 0.47 areaoss4<br />
with lag in hours,area in km2<br />
lag = 0.36 (3)0.40 with lag in hours and <strong>le</strong>ngths in km<br />
si<br />
To apply <strong>the</strong> unitgraph to any particular storm it is necessary to<br />
estimate a @-index value, or loss rate. Studies of this for Hong Kong<br />
conditions showed wide fluctuations between storms, ranging from 2.5 to<br />
80m/h, with values commonly between 10 and 40mm/h. Judgement must be<br />
used in se<strong>le</strong>cting a suitab<strong>le</strong> value. When reservoir spillway studies are<br />
in hand, a very low value is appropriate. For drainage design, a value<br />
nearer <strong>the</strong> mean would be used.<br />
This dimension<strong>le</strong>ss unitgraph has been used in conjunction with studies<br />
of probab<strong>le</strong> maximum precipitation over Hong Kong(5~6) carried out by <strong>the</strong><br />
staff of <strong>the</strong> Royal Observatory, to check capacity of existing reservoir<br />
spillways and in <strong>the</strong> design of new dams in Hong Kong.<br />
Flood frequency<br />
analysis has also been used but in <strong>the</strong> absence of long records this is<br />
thought to be <strong>le</strong>ss reliab<strong>le</strong>f7).<br />
Conclusions<br />
The dimension<strong>le</strong>ss unitgraph derived by procedures developed in <strong>the</strong><br />
U.S.A. is useful as a design tool. Hydrographs from <strong>the</strong> seven catchment<br />
areas, covering <strong>the</strong> range of sizes and types found in Hong Kong were unified<br />
to an acceptab<strong>le</strong> degree of accuracy using time to centroid of unitgraph in<br />
converting sca<strong>le</strong>s into dimension<strong>le</strong>ss values.
506<br />
Whe<strong>the</strong>r area alone or a more comp<strong>le</strong>x parameter should be used for<br />
predicting basin lag is uncertain. Area alone appeared to be adequate<br />
except in <strong>the</strong> case of catchments with extensive lowland area.<br />
The volume of data availab<strong>le</strong> for this study was small both in terms<br />
of <strong>le</strong>ngth of records used and number of stations.<br />
Acknow<strong>le</strong>dgements<br />
Thanks are due to <strong>the</strong> Director of Public Works, Hong Kong Government,<br />
for permission to publish this paper; to Mr. J. Forth, who later extended<br />
<strong>the</strong> work on floods to o<strong>the</strong>r methods of approach; and to Mr. Wong Shiu Ming,<br />
present holder of <strong>the</strong> post of EngineerIHydrologist, for his valued ascist-<br />
ance in checking <strong>the</strong> data in this paper and providing information.<br />
References<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
Robertson, A.S. and La Touche, M.C.D., Assessing <strong>the</strong> Yield of Hong Kong's<br />
Reservoirs, J. Institution Water Engineers, 23, (1969), 8, 507-519.<br />
Hong Kong Rainfall and Runoff (Annually from 1965), Hong Kong, Water<br />
Authority, Public Works Department.<br />
United States Bureau of Reclamation. Design of Small Dams, (1960),<br />
Washington, U.S. Govt. Printing Office.<br />
Lins<strong>le</strong>y, R.K. , Koh<strong>le</strong>r and Paulhus, Hydrology for Engineers , (1958) ,<br />
New York, McGraw-Hill.<br />
Bell, G.J. and Chin, The Probab<strong>le</strong> Maximum Rainfall in Hong Kong.<br />
R.O. Tech. Mem. 10, (1968). Government Printer, Hong Kong.<br />
Cheng, S. and Kwok, (1966) A Statistical Study of Heavy Rainfall in<br />
Hong Kong. Tech. Note 24, Hong Kong, Royal Observatory.<br />
Design Flood for Hong Kong, HS7, (1968), Water Authority, Public Works<br />
Department, Hong Kong.
-<br />
Altitude Catchment<br />
Station of Crest Area<br />
Name - m -@<br />
I_<br />
Tai Lam Chung 15 16.2<br />
60ft. weir<br />
Sham Tseng<br />
Tai Lam Chung<br />
'A'<br />
Tai Lam Chung<br />
'B'<br />
30<br />
75<br />
63<br />
2.0<br />
0.8<br />
1.2<br />
Contro 1<br />
Compound weir with ogee crest,<br />
loft. low flow section.<br />
30ft. compound weir with ogee crest,<br />
3ft. low flow section.<br />
Compound V and rectang<strong>le</strong> sharp-<br />
crested suppressed weir.<br />
Compound V and rectang<strong>le</strong> sharp-<br />
crested suppressed weir.<br />
Instrument<br />
Staff gauge.<br />
Locally-made float <strong>le</strong>vel recorder.<br />
Staff gauge.<br />
George Kent float <strong>le</strong>vel recorder.<br />
Sloping brass staff gauge.<br />
Munro vertical drum<br />
Sloping staff gauges.<br />
Streamflow Stations in Hong Kong, to 1966<br />
Ob s erving Programe<br />
Frequent staff gauge readings<br />
before Jan. 1950, <strong>the</strong>reafter<br />
continuous recording with<br />
daily observations.<br />
Continuous recording with<br />
daily observations.<br />
Frequent staff gauge readings<br />
before June 1963, <strong>the</strong>reafter<br />
continuous reading with daily first<br />
and <strong>the</strong>n weekly observations.<br />
Frequent staff gauge readings<br />
before June 1959, <strong>the</strong>reafter daily<br />
observations.<br />
Readings Record<br />
Commenced Quality Remarks<br />
Apr. 1948<br />
Fair<br />
Discont inued<br />
May 1955<br />
Jul. 1952 Fair Discontinued<br />
June 1956<br />
Jun. 1958<br />
Good<br />
Jun. 1958 Poor<br />
Shek Pi Tau L<br />
41.6<br />
102ft. long with 4ft. wide broadcrested<br />
weir.<br />
Sloping staff gauge.<br />
Munro vertical drum float <strong>le</strong>vel<br />
recorder.<br />
Daily observations before June 1964,<br />
<strong>the</strong>reafter continuous recording<br />
with bi-daily observations.<br />
May 1960 Poor<br />
Ho Sheung Heung 5<br />
16.9<br />
40ft. long broad-crested weir.<br />
Sloping staff gauge.<br />
Munro vertical drum float <strong>le</strong>vel<br />
recorder.<br />
Daily observations before June 1964,<br />
<strong>the</strong>reafter continuous recording with<br />
di-daily observations.<br />
May 1960 Poor<br />
Tai Po Tau<br />
9<br />
15.2<br />
Broad-cres ted weir.<br />
Sloping staff gauge.<br />
Munro horizontal drum float<br />
recorder.<br />
eve1<br />
Continuous recording from July 1961<br />
to April 1963, daily observations<br />
at o<strong>the</strong>r times.<br />
Sha Tin<br />
100<br />
1.2<br />
Compound sharp-crested rectangular<br />
weir without separating walls. 90°<br />
V notch upstream for low flows.<br />
Staff gauge.<br />
Munro horizontal drum float eve 1<br />
recorder.<br />
Continuous recording from Jan. 1961<br />
to Jan. 1963, daily observations at<br />
o<strong>the</strong>r times.<br />
Nov. 1960<br />
Hok Tau 85<br />
6.0<br />
Compound sharp-cres ted rectangular<br />
weir, without separating walls.<br />
Sloping brass staff gauge.<br />
Munro horizontal drum float <strong>le</strong>vel<br />
recorder before May 1964, <strong>the</strong>reafter<br />
Leupold & Stevens A-35 recorder.<br />
Daily observations before June 1961,<br />
<strong>the</strong>reafter continuous recording with<br />
daily first and <strong>the</strong>n weekly observations.<br />
Dec. 1960 Good<br />
Chung Mei 13 9.1 Compound crump weir with -90° V notches<br />
upstream for low flows.<br />
Sloping brass staff gauge.<br />
Munro horizontal drum float <strong>le</strong>vel<br />
recorders before April 1964, <strong>the</strong>reafter<br />
Leupold & Stevens 2A-35 recorder.<br />
Continuous recording with daily first<br />
and <strong>the</strong>n weekly observations.<br />
May 1962 Good<br />
Siu Lek Yuen 74 2.1 Compound sharp-edged rectangular<br />
weir without separating walls.<br />
'Vertical brass staff gauge.<br />
Munro horizontal drum float <strong>le</strong>vei<br />
recorders before June 1964, <strong>the</strong>reafter<br />
Leupold & Stevens A-35 recorder.<br />
Continuous recording with weekly<br />
observations.<br />
May 1964 Good<br />
Tsak Yue Bu 41<br />
1.6<br />
Compound V and rectang<strong>le</strong> sharpcrested<br />
suppressed weir.<br />
Sloping brass staff gauge.<br />
Leupold & Stevens A-35 recorder.<br />
Continuous recording with weekly<br />
obscrvations.<br />
Jul. 1964 Good<br />
Lo Shue Ling 3<br />
10.8<br />
Parshall flume, 15ft. throat.<br />
Staff gauge. Leupold & Stevens<br />
:A-35 recorder.<br />
Continuous recording with weekly<br />
observations.<br />
Jul. 1964 Poor<br />
Kam Tin 3<br />
11.7<br />
Parshall flume, 25ft. throat.<br />
Staff gauge. Leupold & Stevens<br />
2A-35 recorder.<br />
Continuous recording with weekly,<br />
observations.<br />
Jul. 1964 Fair<br />
Oct. 1960 Fair Dis continued<br />
Aug. 1963<br />
Tab<strong>le</strong> 1<br />
Good Discontinued<br />
March 1963
TABLE 2<br />
Ordinates of <strong>the</strong> 15-Minute Dimension<strong>le</strong>ss Unitgraph<br />
Time + (Lag+:)<br />
o. 20<br />
0.30<br />
0.40<br />
0.45<br />
0.50<br />
0.55<br />
0.65<br />
O. 70<br />
O. 80<br />
0.95<br />
1.00<br />
1 .O5<br />
1.30<br />
1.50<br />
2.00<br />
2.20<br />
2.75<br />
3.40<br />
3.90<br />
5.13<br />
U<br />
lag+2<br />
Discharge x - V<br />
0.05<br />
o. 10<br />
0.19<br />
0.32<br />
0.57<br />
O. 71<br />
1.00<br />
1.02<br />
1.01<br />
O. 73<br />
O .64<br />
0.56<br />
0.38<br />
0.30<br />
O. 18<br />
O. 15<br />
o .o9<br />
0.05<br />
O .O3<br />
0.00<br />
509
Fig. 1<br />
Q<br />
51 O
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/<br />
I-<br />
.-<br />
E<br />
I<br />
.-<br />
t<br />
3<br />
-e<br />
oa N<br />
L<br />
.-<br />
B
O<br />
- - t 2<br />
E .-<br />
I-<br />
+ .-<br />
t<br />
3<br />
I<br />
cv<br />
O
cc ;v<br />
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-+
ABSTRACT<br />
STUDY ,OF MAXIMUM FLOODS 1.N SMALL BASïNS OF TORRENTIAL TYPE<br />
Rafael HERAS<br />
Dr. Civil Engineer<br />
Angel LARA<br />
Civil Engineer<br />
The methodology of study is summarized for small b=<br />
sins of torrential character and it is applied to one of <strong>the</strong> gu-<br />
llies of <strong>the</strong> Gran Canaria island, considering <strong>the</strong> geological and<br />
geomorphological conditions of <strong>the</strong> basin and also <strong>the</strong> principal<br />
physical characteristics of <strong>the</strong> same one. In relation to all <strong>the</strong>se<br />
physical characteristics and of a statistical comp<strong>le</strong>te study of in-<br />
tensities, <strong>the</strong> hydrogram is established for different hypo<strong>the</strong>sis<br />
and <strong>the</strong> type of hydrogram is studied more unfavorab<strong>le</strong> in relation<br />
in relation to <strong>the</strong> duration-intensity-frequency curves of maximum<br />
precipitations in 24 hours.<br />
RESUMEN<br />
S:e resume la metodologia de estudio para pequeñas<br />
cuencas de carácter torrencial y se aplica a uno de los barrancos<br />
de la isla de Sran Canaria, teniendo en cuenta las condiciones geo-<br />
lógicas y geomorfológicas de la cuenca y también las principa<strong>le</strong>s c z<br />
racteristicas físicas de la misma. En función de todas estas carac-<br />
terísticas y de un estudio estadístico comp<strong>le</strong>to de intensidades, se<br />
estab<strong>le</strong>cen los hidrogramas para distintas hipótesis y se estudia el<br />
hidrograma tipo más desfavorab<strong>le</strong> en función de las curvas duración-<br />
intensidad-frecuencia de precipitaciones máximas en 24 horas.
518<br />
1. Generalities<br />
This method has been applied to <strong>the</strong> Tirajana gully, which is<br />
one of <strong>the</strong> most important in <strong>the</strong> south zone of <strong>the</strong> Gran Canaria island. The<br />
high part of <strong>the</strong>ir channel is formed by a big number of gullies which have its<br />
origin to an altitude of about 1,700 m., following <strong>the</strong> receiver basin a direction<br />
sensibly north-west-sou<strong>the</strong>ast. The maximum longitude of <strong>the</strong> channel is 27 km.<br />
and <strong>the</strong> total area of <strong>the</strong> basin is 71,4 km2. Its location in <strong>the</strong> island is ref<strong>le</strong>cted<br />
in <strong>the</strong> graph number 1.<br />
2. Geology of <strong>the</strong> basin of <strong>the</strong> Tirajana gully<br />
The region where <strong>the</strong> Tirajana gully is located is <strong>the</strong> sou<strong>the</strong>ast<br />
of <strong>the</strong> Gran Canaria island.<br />
For its location, it participates of <strong>the</strong> geological characteristic<br />
of <strong>the</strong> half south of this island, appearing on <strong>the</strong> surface <strong>the</strong> most ancient<br />
comp<strong>le</strong>x which have taken part in its formation, such as are <strong>the</strong> Ancient Basalt<br />
of <strong>the</strong> Serie I, of basaltic alkaline-olivinical composition and formed by sub-<br />
paral<strong>le</strong>l running out with pyroclasts intercalated, <strong>the</strong> Trachysienite comp<strong>le</strong>x<br />
with ignimbrites associated, of rhyolithical, pan<strong>the</strong>lithical and trachyphenol-<br />
ithical compositions, and <strong>the</strong> Phonolithical serie, composed in this zone by<br />
running out, pius, end ignimbrites, frequently with laminar parting paral<strong>le</strong>l<br />
to <strong>the</strong> direction of <strong>the</strong> flow.<br />
All <strong>the</strong>se series are located principally in <strong>the</strong> midd<strong>le</strong> zone of<br />
<strong>the</strong> gully, existing also in form of litt<strong>le</strong> cropping out, principally phonolithic<br />
and trachysienite, in <strong>the</strong> high part of <strong>the</strong> basin. On <strong>the</strong> o<strong>the</strong>r hand, all <strong>the</strong><br />
mentioned series are practically impermeab<strong>le</strong> in <strong>the</strong> process of infiltration<br />
from <strong>the</strong> surface and, particularly, <strong>the</strong> series of Basalts I and Trachysienite<br />
Comp<strong>le</strong>x, forming <strong>the</strong> majority of <strong>the</strong> substratum on which are seated <strong>the</strong> most<br />
recent superficial formations.
519<br />
The series Pre-Roque Nublo and Roque Nublo, which have its<br />
maximum power in <strong>the</strong> interior of <strong>the</strong> island appearing largely disseminated<br />
in <strong>the</strong> Tirajana gully, specially in its midd<strong>le</strong> and high zones.<br />
The said series are composed lithologically by angular<br />
fragments constituting xenolithical agglomerate with intercalations of<br />
tephrithical lavas, basaltical running out and sediments. In <strong>the</strong>se series, of<br />
moderate permeability, a quick fall in <strong>the</strong> <strong>le</strong>vel of water is produced, being<br />
<strong>the</strong>refore, its pondage coefficient very law.<br />
The most modern basaltical serie that appear on <strong>the</strong> surface,<br />
is <strong>the</strong> correspondent to <strong>the</strong> Basalts II, of basaltical olivinical composition and<br />
constituted by aa and pahoehoe lavas, more permeab<strong>le</strong> than <strong>the</strong> previous<br />
formations. These lavas cover principally <strong>the</strong> northcast zone of <strong>the</strong> fully<br />
disseminating also in smal<strong>le</strong>r proportion in its midd<strong>le</strong> zone.<br />
Finally, this basin present a genuine characteristic which is<br />
distinguished from <strong>the</strong> contiguous ones, since that a big part of its surface<br />
occupied by sedimental formations, of which, <strong>the</strong> avalanches of various ages<br />
constitute <strong>the</strong> principal cropping out of <strong>the</strong> zone of heading, whi<strong>le</strong> <strong>the</strong> low zone<br />
of <strong>the</strong> basin is covered by deposits of recent alluviums, with bigger porosity<br />
and higher permeability.<br />
3. Physical Data<br />
In order to know <strong>the</strong> characteristics of <strong>the</strong> basin to use <strong>the</strong>m<br />
fundamentally in <strong>the</strong> estimation of its velocity of propagation of maximum<br />
flood, it has been calculated for <strong>the</strong> same one, <strong>the</strong> following characteristics:<br />
, longitudinal section<br />
. surface<br />
. perimeter<br />
. equiva<strong>le</strong>nt rectang<strong>le</strong><br />
, hypsometrical curve<br />
. index of compactness<br />
. index of slope
520<br />
surface perimeter<br />
17.4 km2 57.5 k m<br />
4. Maximum floods<br />
4. 1. General planning<br />
The values obtained have been <strong>the</strong> following:<br />
compac tnes s equiva<strong>le</strong> nt<br />
index rectang<strong>le</strong><br />
1.90 L = 26. 10<br />
1 = 2.74<br />
slope<br />
index<br />
O. 263<br />
The principal prob<strong>le</strong>me presented is <strong>the</strong> absolute lack of<br />
direct data of gauging with sufficient extension and guarantee, as much in<br />
<strong>the</strong> studied basin as in <strong>the</strong> rest of <strong>the</strong> island, <strong>the</strong>refore it is not possib<strong>le</strong> to<br />
study <strong>the</strong> flood from <strong>the</strong> direct data of maximum flows nei<strong>the</strong>r by comparison<br />
with o<strong>the</strong>rs basins, affinitive basins hydrologically. Therefore, using <strong>the</strong><br />
maximum availab<strong>le</strong> data, it has been performed <strong>the</strong> comp<strong>le</strong>te study of floods<br />
by empirical and hydrometrical methods, constrasting each one of <strong>the</strong><br />
estimated parameters with data obtained by direct procedures in <strong>the</strong> Gran<br />
Canaria gully.<br />
4. 2. Empirical methods<br />
In <strong>the</strong> formation of maximum floods intervenes multip<strong>le</strong><br />
causes, whose possibility of coincidence characterizes <strong>the</strong> risk. The surface<br />
of <strong>the</strong> receiver basin is one of <strong>the</strong> causes among <strong>the</strong> principal ones, since<br />
<strong>the</strong>re exists a good correlation between <strong>the</strong> basin area and <strong>the</strong> maximum flood.<br />
Using formulas that could tie directly <strong>the</strong> flows of floods with<br />
<strong>the</strong> surface of <strong>the</strong> basin and o<strong>the</strong>rs in which intervene o<strong>the</strong>rs hydrological<br />
parameters.<br />
Among <strong>the</strong> existing formulas it has been used those which are<br />
in <strong>the</strong> joined chart; <strong>the</strong>se formulas has been se<strong>le</strong>cted in relation to <strong>the</strong> hydro-<br />
logical characteristics of <strong>the</strong> basin in <strong>the</strong> present study. In <strong>the</strong> mentioned
chart it has been given, in <strong>the</strong> same way, <strong>the</strong> values which are <strong>the</strong> result of<br />
its application.<br />
SANTI<br />
GREAGER<br />
FORTI<br />
ZAPATA<br />
423 (Tr = 500 años) KUICKLING 255 (Tr = 100 anos)<br />
520 (Tr = 500 años) TURAZZA 820 (Tr = 500 anos)<br />
626 (Tr = 500 años) HERAS 780 (Tr = 500 anos)<br />
272 (Tr.= 100 años) G. QUIJANO 292 (Tr = 100 anos)<br />
The big dispersion of <strong>the</strong> results obtained of <strong>the</strong> same ones<br />
can be observed.<br />
4. 5. Hydrometrical method<br />
521<br />
This method consists in trying to reproduce <strong>the</strong> meteorological<br />
phenomenon and, in this case, we will use <strong>the</strong> method of <strong>the</strong> isochronal curves,<br />
to which it is necessary to discompose <strong>the</strong> surface of <strong>the</strong> basin in some zones<br />
(si, s2, . . . sn) limited by lines (isochrones) in which <strong>the</strong> water fal<strong>le</strong>n in one<br />
of <strong>the</strong>se ones delays in arriving to <strong>the</strong> point in wich we estimate <strong>the</strong> flood,<br />
sucesive times of value t, 2t, . . . , being our case t half hour.<br />
The velocity of <strong>the</strong> water if fixed by experimental and<br />
empirical methods, in relation to <strong>the</strong> physical data, fundamentally of <strong>the</strong><br />
longitudinal section and index of slope, and o<strong>the</strong>r characteristics peculiar of<br />
<strong>the</strong> basin (vegetation, geology and so on). In our case, we have fixed as<br />
velócity 6 km/hour in <strong>the</strong> low zone of <strong>the</strong> basin, up to an altitude of 600 m.,<br />
above sea <strong>le</strong>vel, and 7 km/hour in <strong>the</strong> high part. Once fixed this one, <strong>the</strong> pointe<br />
are obtained from which delays in arriving <strong>the</strong> water to <strong>the</strong> place studied a<br />
same time and with which, as contour line of a topographical e<strong>le</strong>vation, we<br />
can draw <strong>the</strong> isochronical lines obtaining simultaneously <strong>the</strong> concentration<br />
time, that in our case is of 4.3 hours.
522<br />
If we contrast this time with <strong>the</strong> one given by any of <strong>the</strong><br />
empii ical formulas existing (for examp<strong>le</strong>, Giandotti), we obtain a difference,<br />
by an excess of about 1 hour. This appreciab<strong>le</strong> difference is justified by <strong>the</strong><br />
quantity of sediment load which carry <strong>the</strong> floods in this type of gullies, and<br />
produce a disminution in <strong>the</strong> mean velocity of propagation. The incidence of<br />
<strong>the</strong> considered velocity in <strong>the</strong> flood peak is small, and so is only influenced<br />
by <strong>the</strong> concentration time.<br />
The isochrones once obtained, multiplying <strong>the</strong> area encirc<strong>le</strong>d<br />
among <strong>the</strong> same ones by <strong>the</strong> intensity of precipitation and <strong>the</strong> supposed runoff<br />
coefficient, <strong>the</strong> flow is obtained in <strong>the</strong> studied point due to <strong>the</strong> precipitation in<br />
each one of <strong>the</strong> zones.<br />
They are, <strong>the</strong>refore, necessary <strong>the</strong> data of maximum<br />
intensities of precipitations in <strong>the</strong> basin for a determined period of recurrence.<br />
To realize <strong>the</strong> statistical study of <strong>the</strong> intensity we will use from among <strong>the</strong><br />
several laws of distribution of frequencies which are applied in hydrological<br />
prob<strong>le</strong>ms, Gumbel’s law, which is used principally for distributions of<br />
maximum values. This law has been applied to <strong>the</strong> usab<strong>le</strong> series of <strong>the</strong> interior<br />
stations of <strong>the</strong> basin and to a series of stations of lap. All of <strong>the</strong>m can be seen<br />
in <strong>the</strong> graph number 1. The maximum annual values of precipitation in 24 hours<br />
for several periods of recurrence are ref<strong>le</strong>cted in <strong>the</strong> charts numbers 1, for<br />
<strong>the</strong> stations of <strong>the</strong> interior of <strong>the</strong> basin, and number 2, for <strong>the</strong> exterior ones.<br />
In order to adjust <strong>the</strong> distribution of <strong>the</strong> values of maximum<br />
precipitation in 24 hours and considering <strong>the</strong> probability of coincidence of said<br />
values, <strong>the</strong> Gumbel’s law has been applied to <strong>the</strong> monthly data in all <strong>the</strong><br />
stations, for October, november, december, january, february and march,<br />
resulting to be <strong>the</strong> months of October and november <strong>the</strong> most unfavourab<strong>le</strong> in<br />
relation to <strong>the</strong> floods, as it is deducted of <strong>the</strong> observation of <strong>the</strong> chart number 3.<br />
Also, to contrast <strong>the</strong> distribution of maximum values in <strong>the</strong><br />
basin, <strong>the</strong> isomaximum curves has been designed with <strong>the</strong> values of maximum<br />
precipitation in 24 hours to times of recurrence of 50, 100 and 500 years for<br />
<strong>the</strong> maximum maximorum annual values and for <strong>the</strong> maximum values of<br />
October (which is <strong>the</strong> month of maximum intensity). These isomaximum curves<br />
can be seen in <strong>the</strong> graphs numbers 5 up to 10 and have served like contrast of<br />
<strong>the</strong> values obtained by Gumbel and also to adjust <strong>the</strong> mean intensity of<br />
precipitation and its variation with <strong>the</strong> time.<br />
With regard to <strong>the</strong> runoff coefficient, <strong>the</strong>re is hardly no data<br />
for maximum maximorum flows, <strong>the</strong>refore considering <strong>the</strong> impermeability of<br />
<strong>the</strong> midd<strong>le</strong> and high zone and <strong>the</strong> greater permeability of <strong>the</strong> low zone, we<br />
estimate some runoff coefficients of O. 85, O. 80 and O. 50, respectively, for<br />
each one of <strong>the</strong> three considered zones. To estimate <strong>the</strong>se coefficients, which<br />
could be reached in strong floods which would be produced after several days
523<br />
of considerab<strong>le</strong> precipitation, it has been realize studies with all usab<strong>le</strong> data<br />
and considering <strong>the</strong> physical, geological and geomorphological characteristics<br />
of <strong>the</strong> basin, detached in high, midd<strong>le</strong> and low zones, it has been obtained<br />
mean runoff coefficient of O. 78 that seems to be reasonably adjusted to <strong>the</strong><br />
characteristics of this basin.<br />
The duration of <strong>the</strong> storm is an important factor in <strong>the</strong><br />
determination of <strong>the</strong> maximum flood, <strong>the</strong> maximum value of <strong>the</strong> peak flow is<br />
used to obtain with durations of storm about <strong>the</strong> concentration time. In our<br />
case, we have supposed durations of storm of 1, 2, 3, 4, 5, 6 and 8 hours.<br />
The precipitation for <strong>the</strong> several hypo<strong>the</strong>sis has been estimated in relation to<br />
<strong>the</strong> distribution of <strong>the</strong> maximum precipitation in 24 hours for smal<strong>le</strong>r periods,<br />
obtained from <strong>the</strong> short availab<strong>le</strong> data, which have been contrasted with direct<br />
measures, obtaining <strong>the</strong> following values:<br />
Duration of <strong>the</strong> storm (hours) 1 2 3 4 5 6 8<br />
Precipitation in percentage of<br />
<strong>the</strong> precipitation in 24 hours 35 43 57 69 75 ao 86<br />
Although it is considered litt<strong>le</strong> representative <strong>the</strong> compi<strong>le</strong>d<br />
data of maximum intensities in several stations of <strong>the</strong> basin in study, <strong>the</strong> said<br />
values are kept, putting us in security side. At <strong>the</strong> same time and in order to<br />
procure greater aproximation to <strong>the</strong> actual phenomenon, we can consider three<br />
stretch of different mean intensity coincident with <strong>the</strong> high, midd<strong>le</strong> and low<br />
zones, previously mentioned in <strong>the</strong> estimation of <strong>the</strong> runoff coefficients.<br />
For <strong>the</strong> different hypo<strong>the</strong>sis of duration of <strong>the</strong> flood, <strong>the</strong><br />
intensity, toge<strong>the</strong>r, is distributed in <strong>the</strong> time in such a manner that in <strong>the</strong><br />
hydrograms of duration 1 and 2 hours it is considered all <strong>the</strong> unitary intensity<br />
estimated and for 3 or more hours it has been supposed uniform intensity<br />
during <strong>the</strong> two first hours and decreasing in a 20% each hour more of duration<br />
until reaching a minimum of a 20% in <strong>the</strong> storm of 6 or more hours.<br />
The isochrone curves used in <strong>the</strong> calculation of <strong>the</strong> hydrograms<br />
as well as <strong>the</strong> different zones considered can be seen in <strong>the</strong> graph number 11,<br />
and <strong>the</strong> hypo<strong>the</strong>sis that <strong>the</strong> storm i s produced simultaneously in all <strong>the</strong> basin<br />
has been made since that <strong>the</strong> hypo<strong>the</strong>sis that began in <strong>the</strong> head waters and goes<br />
displacing in <strong>the</strong> direction of <strong>the</strong> gully appears excessively unfavourab<strong>le</strong> for<br />
<strong>the</strong> climatological conditions of <strong>the</strong> basin. Once <strong>the</strong> runoff values, intensity<br />
of precipitation and duration of <strong>the</strong> storm, are fixed, we can obtain <strong>the</strong> flows<br />
due to each zone and <strong>the</strong> accumumulated of <strong>the</strong>se ones give <strong>the</strong> flows that would<br />
reach <strong>the</strong> sea in each moment, supposing an infinite time of rain. Displacing<br />
horizontally this curve in <strong>the</strong> time of duration of rain and calculating <strong>the</strong> curve<br />
difference of <strong>the</strong> two, we obtain <strong>the</strong> actual flows that reach in each moment.
In relation to <strong>the</strong> study realized it has been considered <strong>the</strong><br />
hypo<strong>the</strong>sis (i), applying in all of <strong>the</strong>m some runoff coefficients of O. 80, O. 85<br />
and O. 50 for each one of <strong>the</strong> different zones considered and <strong>the</strong> distribution of<br />
intensities already cited for durations higher than two hours.<br />
As summary of <strong>the</strong> hydrograms obtained, in <strong>the</strong> graphs<br />
numers 12 up to 15 figure <strong>the</strong> correspondent to a duration of storm of 4 hours<br />
and periods of recurrence of 100 and 500 years, <strong>the</strong> same for <strong>the</strong> maximum-<br />
maximorum values of precipitation, as for <strong>the</strong> maximum of October. In <strong>the</strong><br />
chart number 4 appears <strong>the</strong> distribution of intensities in space and time for<br />
<strong>the</strong>se hypo<strong>the</strong>sis.<br />
CONCLUSIONS<br />
As a result of <strong>the</strong> calculations realized by <strong>the</strong> different methods<br />
and considering that <strong>the</strong> hydrograms obtained must be affected by a reducent<br />
coefficient in relation to <strong>the</strong> hypo<strong>the</strong>sis of calculation, in which it has been<br />
considered some maximum values of <strong>the</strong> runoff coefficient and some maximum<br />
intensities which must be reduced due to <strong>the</strong> non-coincidence of <strong>the</strong> distribution<br />
in <strong>the</strong> space and time of maximum values in all <strong>the</strong> stations, we obtain <strong>the</strong><br />
results that can be seen in <strong>the</strong> annexed chart.<br />
(1)<br />
The hydrogram type estimated figure in <strong>the</strong> graph number 16.<br />
The statistical study of maximum precipitation in 24 hours has been<br />
realized in <strong>the</strong> period of 21 years, 1949-50 - 1969-70 and <strong>the</strong> data of<br />
<strong>the</strong> usab<strong>le</strong> stations has been contrasted and, generally, it appears to<br />
have enough guarantee, but by <strong>the</strong> extension of <strong>the</strong> period used, resulted<br />
as a risk to extrapolate for times of recurrence higher than 100 years.<br />
The results obtained are conditioned by <strong>the</strong> empirical-<strong>the</strong>orical<br />
methods used, due to <strong>the</strong> absolute lack of series of maximum flows,<br />
although we estimate that <strong>the</strong> maximum difference with <strong>the</strong> actual<br />
values will not exceed 15%.<br />
The study is only related to maximum values of flood and in it has not<br />
been considered <strong>the</strong> effect of <strong>the</strong> solid flows.<br />
of hydrograms with durations of storm of 1, 2, 3, 4, 5, 6 and 8 hours.
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FLOOD ESTIMATION BY DETERMINATION OF REGIONAL PARAMETERS FROM LIMITED DATA<br />
ABSTRACT<br />
P.H. EERBST, S. VAN BILJON, J.P.J. OLIVIER AND J.M. HALL<br />
A regionalized study of maximum annual flows of different short dura-<br />
tions (including peaks) has been carried out. In view of <strong>the</strong> limited<br />
<strong>le</strong>ngth of record availab<strong>le</strong> at most of <strong>the</strong> gauging stations in <strong>the</strong> region,<br />
an attempt has been made to develop a technique to streng<strong>the</strong>n <strong>the</strong> data<br />
availab<strong>le</strong> at any particular point of interest, by using all availab<strong>le</strong> per<br />
tinent flow data in <strong>the</strong> region. Having chosen <strong>the</strong> extrema1 dislribution<br />
best suited to <strong>the</strong> region, <strong>the</strong> moments of <strong>the</strong> samp<strong>le</strong> (after adjustment)<br />
are correlated with various catchment characteristics. This allows estima<br />
tion of flood magnitude frequency curves at any site of interest within<br />
<strong>the</strong> region, with associated confidence bands. Such frequency curves are<br />
determined for various suitab<strong>le</strong> time intervals which <strong>the</strong>n allows <strong>the</strong> syn-<br />
<strong>the</strong>sis of characteristic flow hydrographs, with a specific probability of<br />
occurrence attached to each, along Mith associated enyelopes correspon-<br />
ding to specific confidence limits, Comparison with hydrographs derived<br />
from rainfall input depths with specified probabplities, subtracting los?<br />
ses, and <strong>the</strong>n using unitgraph methods, <strong>le</strong>ads to <strong>the</strong> conclusion that a bet<br />
ter relation between probability of occurrence of a specific hydrograph,<br />
and its magnitude, can usually be obtained by direct statistical methods,<br />
than by more indirect deterministic techniques.<br />
RESUMEN<br />
Fia sido ejecutado un estudio regionalizado de gastos máximos anua<strong>le</strong>s<br />
de duraciones cortas y diferentes (.incluyendo valores máximosr. En vista<br />
de la limitación de información disponib<strong>le</strong> para la mayoria de las estacio<br />
nec de aforo de la región, se ha intentado desarrollar un método que permita<br />
reforzar dicha información para cualquier punto de interés usando to<br />
dos los registros existentes de la región. Habiendo e<strong>le</strong>gido la distribución<br />
extrema que mejor acomoda a la región, se han correlacionado los mementos<br />
estadísticos de muestre0 (ajustando valores) con ias características<br />
de diferentes hoyas. Esto permite la estimación de curvas “magnitudfrecuencia”<br />
de riadas para cualquier punto de inter’es dentro de la región,<br />
asociadas con bandas de confiabilidad. Ta<strong>le</strong>s cur~as de frecuencia<br />
se han determinado para convenientes intervalos de tiempo las cua<strong>le</strong>s permiten<br />
la sfntesis de hidrógrafos de flujo caracteristicos, relacionados<br />
con probabilidades específicas de ocurrencia, junto con envolventes que<br />
corresponden a limites específicos de confiabilidad. Comparación con hidrógrafos<br />
derivados de precipitaciones de ocurrencia especifica, substrayendo<br />
pdrdidas y usando luego métodos de gráfico unitario, l<strong>le</strong>va a la<br />
conclusión que una mejor relación entre probabilidad de ocurrencia de un<br />
hidrógrafo determinado y su magnitud, puede obtenerse generalmente medi’an<br />
te métodos estadísticos directos en vez de técnicas deterministicas indirectas.
542<br />
INTRDDUCTIOIY<br />
There is no need to stress <strong>the</strong> importance of reliab<strong>le</strong> flood magnitude frequency estimates in mter resource development.<br />
Whilst this prob<strong>le</strong>m is especially highlighted in developing regions, even so cal<strong>le</strong>d developed<br />
countries frequently suffer from a limitation of data on which to base reliab<strong>le</strong> flood flow estimates.<br />
An estimate of a flood with a specified recurrence interval (determined by specified design consideration)<br />
should be accompanied by information concerning <strong>the</strong> reliability of such an estimate, but this has not<br />
usually been <strong>the</strong> case in <strong>the</strong> past.<br />
This paper outlines a methodology by means of which all or most of <strong>the</strong> availab<strong>le</strong> flood flow information<br />
in a region can be rationally analysed and assemb<strong>le</strong>d, by taking into account quantifiab<strong>le</strong> parameters of<br />
characteristics of <strong>the</strong> various catchments in <strong>the</strong> region and relating <strong>the</strong>se to <strong>the</strong> moments of <strong>the</strong> frequency<br />
distr, but ion assumed.<br />
Methods are described by means of which a flood hydrograph with a specified recurrence interval can be<br />
estimaied and a quantitative statement on <strong>the</strong> reliability of such an estimate can be made.<br />
lhe method was<br />
drvtloped partly due to uncertainty about <strong>the</strong> validity of use of <strong>the</strong> method whereby rainfall intensity - dura:<br />
::on curves are applied to a transformation function (e.g. a I - hour Unitgraph) due to <strong>the</strong> many assunptions<br />
necessary in <strong>the</strong> latter approach.<br />
It was felt that where some limited flow information does exist, an approach as outlined woulo provide better<br />
estimates of flood frequencies and flood hydrographs, including also information on <strong>the</strong> reliability of such<br />
estimates. Avoidance of any mention of <strong>the</strong> degree of uncertainty in any such estimate does not remove <strong>the</strong><br />
uncertainty, it only serves to diminish consideration of <strong>the</strong> fact that such uncertainty not only exists but<br />
may be considerab<strong>le</strong>.<br />
FREOUtNCV DISIRIBUTIONS<br />
In developing a methodology for flood frequency estimates (for annual extreme flows of various -hart dura=<br />
tions) on a regional basis, it is essential to decide which frequency distribution should be assumEd to apply<br />
throughout <strong>the</strong> rsgion. This is so, firstly for <strong>the</strong> reason that if a reasonab<strong>le</strong> correlation between moments of<br />
<strong>the</strong> oistribution assumed (whe<strong>the</strong>r <strong>the</strong> variab<strong>le</strong>s be transformed or not) and characteristics of <strong>the</strong> catchent can<br />
be found, <strong>the</strong> same distribution must by force also be used for estimation purposes at some new site of interest<br />
in <strong>the</strong> region.<br />
Secondly it was considered that if a distribution is used which has a third parameter, this would provide<br />
<strong>the</strong> necessary f<strong>le</strong>xibility (adaptability) for <strong>the</strong> distribution to be ‘Iraiaxefi so as to fit that particular<br />
region; <strong>the</strong> third parameter thus being a constant throughout <strong>the</strong> region (for every duration).<br />
Fur<strong>the</strong>rmore, <strong>the</strong> possibility exists that such a third parameter may show some sensib<strong>le</strong> variation if adjacent<br />
regions are analysed in turn, thus promising <strong>the</strong> possibility of a “smoothing” <strong>the</strong>reof, providing <strong>the</strong>re are no<br />
gross geographic discontinuities. The coastal zone, consisting of rivers draining to <strong>the</strong> south eastern sea=<br />
board of <strong>the</strong> Republic is considered suitab<strong>le</strong> for such fur<strong>the</strong>r analysis, a similar but <strong>le</strong>ss comprehensive study<br />
having been carried out for those rivers mainly draining via <strong>the</strong> Orange, Limpopo and Komati river systems [i] .<br />
The region chosen for use as a pilot study which this paper srnarizes, consisted of <strong>the</strong> north eastern part<br />
of <strong>the</strong> zone mentioned and is shown on <strong>the</strong> locality map narked figure 1.<br />
Data from wme of <strong>the</strong> gauging stations with a reasonab<strong>le</strong> <strong>le</strong>ngth of record in this region were used to com-<br />
pare <strong>the</strong> log Gumbel and log Pearson Type III distributions. In <strong>the</strong> latter case <strong>the</strong> data were plotted on<br />
specially made graph paper on which a distribution with a skew equal to that calculated from <strong>the</strong> samp<strong>le</strong> concerned,<br />
plots as a straight line. Camparison of <strong>the</strong> plots <strong>le</strong>d to <strong>the</strong> conclusion that no particular superiority<br />
of <strong>the</strong> one above <strong>the</strong> o<strong>the</strong>r was evident. The log Pearson Type III distribution was <strong>the</strong>refore chosen for <strong>the</strong><br />
reasons mentioned above. It should be stated however, that <strong>the</strong> techniques described in this paper could be<br />
applied equally well to <strong>the</strong> Gunbel or log Gmbel distributions.<br />
Moreover, <strong>the</strong> basic supposition that nature would be so kind as to ensure that <strong>the</strong> distribution of flou<br />
extremes would follow some definite (simp<strong>le</strong>) statistical distribution, should always be remembered for <strong>the</strong><br />
fallacy which it is. Ihis is especially true where two distinctly separate flood producing factors may pertain;<br />
and may operate ei<strong>the</strong>r separately or conjunctively.<br />
The authors feel, along with Harter 121 that <strong>the</strong>re can be IM finality about <strong>the</strong> recommendations made by<br />
<strong>the</strong> U.S. Water Resources buncil 131 concerning <strong>the</strong> log Pearson Type III distribution, but for <strong>the</strong> various
543<br />
reasons stated, and <strong>the</strong> availability of <strong>the</strong> tab<strong>le</strong>s provided by Hartar, <strong>the</strong> exact form of <strong>the</strong> distribution<br />
postulated is of <strong>le</strong>sser importance than is <strong>the</strong> proper utilization and assembly of all <strong>the</strong> availab<strong>le</strong><br />
flow data in <strong>the</strong> region, in a rational manner, so as to ohtain <strong>the</strong> best possib<strong>le</strong> flood flow estimates<br />
and concomitant reliability estimates.<br />
The various durations of extreme flows in <strong>the</strong> region that was investigated in <strong>the</strong> pilot study sunniarized<br />
herein were: peak flow, 1 day, 2 day, 4 day and 6 day average extreme flows. The logarithms (to base 10) of<br />
<strong>the</strong>se extremes were found to have a skew of 0,3 for peak flows, 0,4 for 1 day average extreme flows and 0,5<br />
for 2 day, 4 day and 6 day average extreme flows. It would appear that <strong>the</strong>re may be a relationship between<br />
<strong>the</strong> skew and <strong>the</strong> duration and if this is also found to be <strong>the</strong> case in o<strong>the</strong>r regions this could cnnveivabìy<br />
be used to obtain more stab<strong>le</strong> estimates of <strong>the</strong> skew.<br />
Special graph paper was developed for <strong>the</strong> skew values of 0,l (0,l) 1,O and 1,5. An examp<strong>le</strong> of this is<br />
<strong>the</strong> paper on which <strong>the</strong> graph shown as figure 3 appears. The ordinate has both logarithmic and linear sca<strong>le</strong>s,<br />
and <strong>the</strong> abscissa consists of both <strong>the</strong> emulative probability of exceedence (e.g. of a certain flow magnitude)<br />
and a linear sca<strong>le</strong>, <strong>the</strong> units of which are essentially in <strong>the</strong> number (and decimal fraction) of standard<br />
deviations from <strong>the</strong> population mean p, corresponding to <strong>the</strong> probability of exceedence for <strong>the</strong> particular skew<br />
value in question. This sca<strong>le</strong> is identified as <strong>the</strong> K - sca<strong>le</strong> (K being analogous to Gumbel's reduced variate).<br />
On <strong>the</strong> assumption <strong>the</strong>n, that <strong>the</strong> logarithms of <strong>the</strong> annual extreme flows for <strong>the</strong> various durations are distributed<br />
according to a Pearson Type III distribution with <strong>the</strong> applicab<strong>le</strong> regional skew values, <strong>the</strong> N year<br />
flood can be obtained from <strong>the</strong> expresfion:<br />
X =X . KS . (i)<br />
Here and S are <strong>the</strong> mean and ssandard deviation of <strong>the</strong> logarithms of <strong>the</strong> individual extreme annual flows.<br />
X, is <strong>the</strong> logarithm of <strong>the</strong> N year extreiae flood magnitude, and K is <strong>the</strong> number of standard deviations from <strong>the</strong><br />
population man y that corresponds to <strong>the</strong> exceedence probability for <strong>the</strong> skew value in question (presented<br />
in detail in Harters tab<strong>le</strong>s).<br />
If <strong>the</strong> logarithms of <strong>the</strong> extreme flows are distributed according to <strong>the</strong> Log Pearson III distribution with<br />
a skew ofï= 1 say, <strong>the</strong>n if probability paper designed forö= 1 is used, a straight line draw hereon for<br />
specific values of X and S, will yield a flood magnitude - frequency curve of XIversus K. As K is uniquely<br />
related to <strong>the</strong> probability of exceedence, X, can be read and transformed to yield <strong>the</strong> flou value estimated to<br />
be equal<strong>le</strong>d or exceeded for any specified return period within <strong>the</strong> range.<br />
ESTIMATION Of IHE M@KIIIS OF THE DISTRIBUTION<br />
The prob<strong>le</strong>m <strong>the</strong>refore reduces to estimation of <strong>the</strong> values of i and S for a specific catchent. This is done<br />
by correlation of all availab!e and pertinent measured flow data to catchment characteristics, so as to be ab<strong>le</strong><br />
to obtain best estimates for X and S. The variab<strong>le</strong>s investigated depend upon factors considered ei<strong>the</strong>r as<br />
possibly causal, or as possib<strong>le</strong> contributary factors towards <strong>the</strong> occurrence of extreme flows. Although <strong>the</strong><br />
authors are aware of <strong>the</strong> possib<strong>le</strong> application of factor analysis (or principal cuœponent analysis) here, it has<br />
not been used during this study for various reasons [4'J .<br />
The various independent variab<strong>le</strong>s considered were <strong>the</strong> following: area, mean annual rainfall, average<br />
slope, river <strong>le</strong>ngth, monthly rainfall with a tvo year recurrence interval (log normal distribution assumed) and<br />
a shape factor. The data used in <strong>the</strong> present study are presented in Tab<strong>le</strong>s 1 and 2.<br />
The regression nodels used were-all of <strong>the</strong> general form:<br />
X . a . b log A+ c log R t ................. (2)<br />
It may be noted that, as 1 is <strong>the</strong> mean of <strong>the</strong> logarithms of <strong>the</strong> extreme flous, <strong>the</strong> above formula using only<br />
= 2 Ab RC where og,m,iS <strong>the</strong> geometric mean of <strong>the</strong> extreme flows at<br />
A and R is equiva<strong>le</strong>nt to <strong>the</strong> mdel Q<br />
a specific site. g.m.<br />
In obtaining a fur<strong>the</strong>r estArnate of i (1 )by simp<strong>le</strong> or multip<strong>le</strong> linear regression, a value for <strong>the</strong> variance<br />
of such a fur<strong>the</strong>r estimate of X is always optained. This variance depends not only upon <strong>the</strong> degree of variance<br />
explained by <strong>the</strong> regression model, but also by <strong>the</strong> extent of <strong>the</strong> deviation of any of <strong>the</strong> independent variab<strong>le</strong>s<br />
from its mean. In <strong>the</strong> case of equation 2 above <strong>the</strong> expression for <strong>the</strong> variance will be of <strong>the</strong> form:<br />
VAR (ie) = Residual Variance[l + 7 1 + cZ2 (Log Ae - m)' + cj3 (Log Re - v)'<br />
+ZcZj(Log A e - m > (Log R e - W ] .................. (3)
544<br />
The statistical <strong>the</strong>ory applied here is very c<strong>le</strong>arly set outlin text <strong>book</strong>s on statistics [5,6] .<br />
In short, for every regression equation used for estimation of X or S an accompanying equation is developed<br />
e e<br />
for <strong>the</strong> calculation of VAR (<strong>le</strong>) and VAR (Se).<br />
The variab<strong>le</strong>s mentioned earlier were used in regression models to determine <strong>the</strong> regression equations<br />
that would explaln <strong>the</strong> highest proportion of <strong>the</strong> variance of <strong>the</strong> dependent variab<strong>le</strong>s I and S, for <strong>the</strong> five<br />
durations considered.<br />
from some 150 regression models tested, <strong>the</strong> 10 equations that were se<strong>le</strong>cted as <strong>the</strong> best are presented in<br />
lab<strong>le</strong> 3. Values of ~22, C J ~ and C ~ J are also presented for !se in an equation of <strong>the</strong> type represented by<br />
equations 3 and 4, in order to calculate <strong>the</strong> variance of <strong>the</strong> X *s and <strong>the</strong> S 's. E.g. <strong>the</strong> equation for <strong>the</strong><br />
variance of a fur<strong>the</strong>r estimate of X for peak flow, X by uS"e of equatiog 3 is as follows:<br />
1 P t e 2<br />
VAR ($,e) E Residual Variance [ 1- + 0,1191 (Log Ae-Log A) + 11,6819 (Log R e - W I 2<br />
n<br />
+ 2~0,4545 (Log A e - W ) (Log Re- WR)]<br />
= 0,0454<br />
6 2<br />
(this is for <strong>the</strong> brgenstond Dam site which has a catchment area of 528x10 m and a mean annual rainfall of<br />
900 x 10-3m).<br />
In this analysis it was hoped that <strong>the</strong> monthly extreme rainfall would be a more representative parameter<br />
of <strong>the</strong> flood producing characteristic of rainfall than <strong>the</strong> mean annual rainfall. However, as both of <strong>the</strong>se<br />
parameters explained an approximately aqua1 amount of additional variance, it was considered advisab<strong>le</strong> to<br />
se<strong>le</strong>ct <strong>the</strong> annual rainfall for use in <strong>the</strong> prediction equation. In view of <strong>the</strong> availability on magnetic tape,<br />
on a large sca<strong>le</strong>, of such monthly rainfall data, and <strong>the</strong> understandab<strong>le</strong> hope that an extreme value rainfall<br />
parameter would yield better results, this is a very disappointing result. It is however intended to invati.<br />
gate this aspect fur<strong>the</strong>r.<br />
Hawing estimated <strong>the</strong> value of i and Se, a straight line flood magnitude-frequency curve can be drawn on<br />
<strong>the</strong> graph paper with <strong>the</strong> apprcpriate skew, and X for any value of N within <strong>the</strong> range can <strong>the</strong>n be read from<br />
N<br />
<strong>the</strong> graph. Ihis is done for <strong>the</strong> peak flow and for <strong>the</strong> various durations for which formulae have been developed,<br />
thus allowing for <strong>the</strong> syn<strong>the</strong>tization of a balanced hydrograph. Ihis is a hydrograph constructed symmetrically<br />
around <strong>the</strong> peak. It can <strong>the</strong>n be adjusted along <strong>the</strong> time axis (but with retention of <strong>the</strong> properties derived)<br />
to its proper shape, ei<strong>the</strong>r by means of information an unitgraph shapes [7] or by actual measurement of one<br />
or two reasonably large floods at <strong>the</strong> site in question. Such measurements muld be arranged for at an early<br />
stage of a feasibility study involving a specific site, if no data are availab<strong>le</strong>. It should be noted here that,<br />
according to Nash [8,14 estimation of a unitgraph shape, even from only one good sized flood in a season will<br />
yield more reliab<strong>le</strong> results than that whlch can be obtained syn<strong>the</strong>tically.<br />
RELIABILITY OF ESTIMATES<br />
In constructing <strong>the</strong> flood magnitude-frequency relationship we have:<br />
XN=ie + KSe . . (1)<br />
If and Se are not independent, <strong>the</strong> prob<strong>le</strong>m of estimation of <strong>the</strong> covariance term arises, for which a<br />
formula tuch as for VAR (1 and VAR (Se) has not been developed.<br />
This prob<strong>le</strong>m was solves by use of an artificial population, distributed according to Pearson Type III,<br />
withy = O, Q = 1. Skew, 8 was varied from O to 1,5 i.e. a different artificial population for each skew.<br />
For each skeu an artificial population consisting of 10 O00 terns was prepared, by use of Harters tab<strong>le</strong>s.<br />
Samp<strong>le</strong>s of size W ranging from 2 to 20 were <strong>the</strong>n drawn. Every individual value drawn was, however transfor:<br />
med by addition of unity so that in fact an approximation to a universe with O- = 1 andy = 1 was used.<br />
for each samp<strong>le</strong> size N, an adeguate number of samp<strong>le</strong>s were drawn to define-to a sufficient degree of accuracy,<br />
<strong>the</strong> variance of i, s and cov h,s). for each samp<strong>le</strong> drawn, <strong>the</strong>-values of x and s were calculated and <strong>the</strong>n an<br />
adequate number of such samp<strong>le</strong>s drawn so as to calculate var ( x), var (s) and cov (x,s). for <strong>the</strong> smal<strong>le</strong>r<br />
samp<strong>le</strong> sizes <strong>the</strong> number of sanp<strong>le</strong>s drawn were simply increased indefinitely until it was c<strong>le</strong>ar that <strong>the</strong> result
545<br />
:.J& converying. In this way curves were obtained showing how var (i), var (s) and cov (x,s) varies with saw=<br />
p<strong>le</strong>s size N, ranging from 2 to 20.<br />
The results are presented in figure 2. Not all <strong>the</strong> curves developed are shown, but all <strong>the</strong> data obtained<br />
was used in order to achieve an integrated matching set of curves.<br />
lhe use of <strong>the</strong>se curves, in order to solve <strong>the</strong> prob<strong>le</strong>m encnuntered with <strong>the</strong> existence of <strong>the</strong> covariance<br />
term in equation 4 is eqlained as follows:<br />
From <strong>the</strong> regression equations, values are obtained for Xe and VAR (ie) and also for Se and VAR (S 1.<br />
(See Tab<strong>le</strong> 3).<br />
Therefore<br />
PutX = k ; = k andS =ks-k<br />
e _ i i e 2-2<br />
(but x = s = 1)<br />
2<br />
VAR (x,)=VA!? (k X)=kl var (S )= VAR fk2s)=k
546<br />
Iherefore<br />
and<br />
COY (EL,S:) = k3k4 cov (;,SI<br />
<strong>le</strong>t i' and S* represent <strong>the</strong> best poo<strong>le</strong>d estimates of <strong>the</strong>se parameters,<br />
<strong>the</strong>n we have:<br />
Nie + N'i<strong>le</strong><br />
i* = ............... (6)<br />
N +Nt<br />
s =<br />
(N - l)Se + (N' - 1) Se<br />
.v N+fi'-Z<br />
!if unbiased estimators are used)<br />
Then we have:<br />
xi .<br />
A T<br />
1' .<br />
........... (7)<br />
KSI . (8)<br />
and<br />
VAR (xi) = VAR (2.1 + Kz VAR (S')<br />
. 2K CDV (i*,S*) . . (9)<br />
As before<br />
Therefore<br />
j* =kiandS':ks<br />
5 6<br />
VAR (i*) = k2 var (i)<br />
5<br />
2<br />
VAR(S*) = k6 var (5)<br />
and<br />
COV (i', S') = k k COY (g,s)<br />
56<br />
As i = s = 1, k and k can be calculated. By putting N* = N + W1 and entering figure 2 with this calculated<br />
value for N*,<br />
6<br />
"hues for var (i), var(s) and w v (x,s) can be rea$<br />
These latter three values can <strong>the</strong>n be used to calculate VAR (X,) for any specified K value thus making<br />
possib<strong>le</strong> <strong>the</strong> calculation of <strong>the</strong> confidence limits.<br />
An examp<strong>le</strong> of a flood magnitude - frequency curve (for peaks) is shown as Figure 3 (brynstond site),<br />
along with <strong>the</strong> one Standard deviation confidence bands.<br />
BALANCED HYDROGRAPHS AND DESIGN HYDROGRAPHS<br />
Using <strong>the</strong> approach outlined above, an estimate of peak flow with a specified probability of exceedance and<br />
concomitant upper and lower confidence limits corresponding to one standard deviation (or for any o<strong>the</strong>r confi.<br />
dence <strong>le</strong>vel required) [Io3 may be calculated. lhe same can be done for each of <strong>the</strong> five durations, resulting<br />
in a llbalanced hydrographll (i.e. symmetric about <strong>the</strong> peak) toge<strong>the</strong>r with upper and louer envelopes wrrespon=<br />
ding to <strong>the</strong> desired confidence <strong>le</strong>vel. In o<strong>the</strong>r words, for a confidence <strong>le</strong>vel corresponding to one standard de=<br />
viation this implies that <strong>the</strong>re is a 1 in 6 (or 16%) chance that <strong>the</strong> true hydrograph could be as big, or bigger<br />
than <strong>the</strong> upper envelope.<br />
lhe shape of this hydrograph can <strong>the</strong>n be suitably adjusted along <strong>the</strong> tiw, axis, but preserving its derived<br />
characteristics, so as to obtain an estimate of <strong>the</strong> hydrograph with <strong>the</strong> required probability of occurrence,<br />
but incorporating <strong>the</strong> shape which is unique to <strong>the</strong> particular catchment.<br />
CONCL US IOW<br />
Results of work similar to that described here have in <strong>the</strong> past been used in <strong>the</strong> Department of Water Affairs<br />
in a somewhat different manner [i] first for estimation of peak flou only and lately [il] also for ertime<br />
average flows over durations of somewhat longer periods. It is intended to extend <strong>the</strong> study to <strong>the</strong> <strong>who<strong>le</strong></strong> of<br />
<strong>the</strong> eastern and south eastern coastal zone of <strong>the</strong> Republic of South Africa.<br />
The possib<strong>le</strong> existence of medium term (e.g. from 3 to 7 years) non-stationarity of <strong>the</strong> various flood<br />
magnitude-frequency curves due to such medium term variations in sea temperatures, will also be investigated
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13-<br />
547<br />
,: series of recent flood disasters in <strong>the</strong> coastal zone mentioned makes such a study virtually imperative,<br />
<strong>the</strong> controlling conditions may still not have reverted to normal).<br />
grouping approach will be followed to attempt to improve reliability.<br />
for smal<strong>le</strong>r catchments a stratified<br />
More work is also intended to attmpt to determine <strong>the</strong> particular rainfall characteristic most closely<br />
related to <strong>the</strong> flood producing attribute <strong>the</strong>reof.<br />
lhe possibility of using an analogous approach to that described herein for estimation of o<strong>the</strong>r hydroloqical<br />
parameters is not overlooked, e.g. low flow sequences with specified probabilities, mean annual runoff, etc.<br />
This method also holds promise in rational hydrologic network plannirig [i31 or adaptation <strong>the</strong>reof.<br />
Preliminary comparison of this method with older methods used by <strong>the</strong> Department, in some of which <strong>the</strong><br />
probability of <strong>the</strong> causative rainfall is put equal to <strong>the</strong> probability of <strong>the</strong> resulting runoff hydrograph,<br />
s e m to inuicate that this method is preferab<strong>le</strong>, not only from <strong>the</strong> point of view of accuracy of estimation but<br />
also due to <strong>the</strong> frank admission and quantification of <strong>the</strong> reliability of estimation, and <strong>the</strong> extent of <strong>the</strong><br />
probab<strong>le</strong> errors.<br />
ACKNOWLEDGEME NT<br />
The permission granted by <strong>the</strong> Secretary for Water Affairs to publish this paper is acknow<strong>le</strong>dged.<br />
The assistance provided by J. de Beer of <strong>the</strong> Computer Centre and by A.J. Mul<strong>le</strong>r, J. Botha, S. Fitchet and<br />
L. Eskell in <strong>the</strong> preparation of <strong>the</strong> paper is greatly appreciated.<br />
lhe guidance given by W.J.R.<br />
<strong>le</strong>dged.<br />
A<strong>le</strong>xander during <strong>the</strong> course of preparation of <strong>the</strong> paper is gratefully acknoum<br />
RE FERE NCES<br />
1. Herbst, P.H. (1968). flood estimation for ungauged catchments, Technical Report No. 46, Department of<br />
Water Affairs, Republic of South Africa.<br />
2. Harter, H.L. (1969. A new tab<strong>le</strong> of percentage points of <strong>the</strong> Pearson Type III distribution, Technometrics,<br />
II(I), pp.177-186.<br />
3.<br />
U.S. Water Resources Council, (1967). A uniform technique for determining flood flow frequencies, Bull.<br />
No. 15, Water Resources Council, Washington, D.C.<br />
Haan, C.T. and Al<strong>le</strong>n, D.M. (1972). Comparison of multip<strong>le</strong> regression and principal component regression<br />
for predicting uater yields in Kentucky, Water Resour. Res., 8(6), pp. 1593 - 1596.<br />
Ost<strong>le</strong>, B. (1963). Statistics in Research, Second Edition, Iowa State University Press, Ames, Chapter 8.<br />
Ezekiel, M and Fox, K.A. (1959). Methods of correlation and regrassion#analysis, John Wi<strong>le</strong>y & Sons, Inc.,<br />
New York, pp.320-321.<br />
Midg<strong>le</strong>y, D.C., Pul<strong>le</strong>n, R.A. and Pitman, W.V. (1969). Design flood determination in South Africa,<br />
Report No. 4/69, Hydrological Research Unit, University of <strong>the</strong> Witwatersrand, Johannesburg.<br />
Nash, J.E. and Shaw, B.L. (1%6). Flood frequency as a function of catchment characteristics,<br />
Symposium on River Flood Hydrology, Inst. of Civil Engineers, London, Session C 6, pp. 115 - 136.<br />
Beard, L.R. (1962). Statistical methods in hydrology, U.S. Army Engineer District, Corps of Engineers,<br />
Sacramento, California.<br />
Mode, E.B. (1961). E<strong>le</strong>ments of statistics, Prentice - Hall, Inc., Nw Jersey.<br />
van Blljon, S. (1972). flood volume frequency analysis - Vaal Dam. Internal Report, Department of<br />
Water Affairs, Republic of South Africa.<br />
Thorne, R.B. (1966). River Engineering and Water Conservation Uorks, Buttervorths, London.<br />
Herbst, P.H. and Shaw, E.M. (1969). Determining rain gauge densities in England from Limited data to<br />
give a required precision for monthly areal rainfall estimates, Journal of <strong>the</strong> I.W.E., 23(4),<br />
pp. 218 - 230.
548
549<br />
GAUGE<br />
SEOUENCE<br />
NUMBER ON<br />
ûFFICIAL RAINFALL<br />
GAUGE AVERAGE MONTHLY<br />
NLMBER ANNUAL WITH<br />
AREA<br />
Of<br />
CATCH-<br />
SLOPE<br />
Of MAIN<br />
WATER<br />
LENGIH<br />
OF<br />
MAIN<br />
NUMBER<br />
Of<br />
YEARS<br />
MAP OVER 2-YEAR MENI COURSE WATER OF<br />
CAT CHME NT RE CUR R E NCE<br />
INIERVAL<br />
COURSE RECORD<br />
10+m 10-3n i06$ m/mxiû3 10% YEARS<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
11<br />
17<br />
13<br />
14<br />
15<br />
16<br />
17<br />
18<br />
19<br />
20<br />
21<br />
22<br />
23<br />
24<br />
25<br />
26<br />
27<br />
XlMOl<br />
x2Mo1<br />
XZMOZ<br />
XZM08<br />
Xx109<br />
X2M10<br />
X2Mll<br />
X2M12<br />
X2M13<br />
Xx114<br />
X2M15<br />
XYOI<br />
XW6<br />
w4Mo2<br />
W4M3<br />
m4<br />
wsMo5<br />
W<br />
W5M7<br />
WW8<br />
w5Mo9<br />
WW12<br />
w6MOI<br />
XlMo6<br />
m3<br />
WY10<br />
W13<br />
888<br />
1142<br />
1074<br />
1163<br />
1070<br />
1187<br />
849<br />
833<br />
907<br />
1237<br />
936<br />
1492<br />
985<br />
889<br />
924<br />
929<br />
920<br />
947<br />
868<br />
862<br />
899<br />
905<br />
1003<br />
1233<br />
921<br />
953<br />
1099<br />
204 5444 5.50<br />
254<br />
104 l3;49<br />
231<br />
176 9,47<br />
289<br />
181 35J1<br />
264<br />
280 18,40<br />
289<br />
127 19,42<br />
183<br />
401 16,37<br />
192<br />
88 10,03<br />
208<br />
1502 12,45<br />
302<br />
251 19,19<br />
213 1538 1256<br />
368<br />
174 33,03<br />
226<br />
76 1 17,37<br />
200 7122 5,05<br />
204 5843 7,41<br />
210<br />
448 3,91<br />
221<br />
751 3,65<br />
213<br />
176 14,95<br />
191<br />
536 2,68<br />
203<br />
119 4,79<br />
206 2805 12,29<br />
209 12769 8,02<br />
238<br />
694 12,15<br />
279<br />
585 18,Ol<br />
205<br />
218 4,73<br />
213 2201 5,81<br />
263 1155 10.93<br />
-<br />
TABLE 2<br />
144.0<br />
26,6<br />
32,2<br />
25,7<br />
28,2<br />
15,8<br />
26,6<br />
13,7<br />
86,9<br />
x),6<br />
76,4<br />
17,4<br />
53,6<br />
232-6<br />
162,5<br />
24-9<br />
5690<br />
1593<br />
46,7<br />
27,4<br />
9197<br />
184,3<br />
85,3<br />
49,l<br />
25,7<br />
120,7<br />
88.5<br />
61<br />
19<br />
19<br />
23<br />
12<br />
22<br />
15<br />
14<br />
10<br />
11<br />
12<br />
23<br />
12<br />
18<br />
21<br />
10<br />
21<br />
21<br />
16<br />
18<br />
10<br />
12<br />
12<br />
12<br />
13<br />
13<br />
10<br />
Average of Log of Standard Standard Multip<strong>le</strong><br />
Prediction Equation<br />
(Log A=Y ; log R.2)<br />
10 10<br />
= 0,802Y+l,1372-3,880<br />
P<br />
Area Mean An.<br />
in Rainfall<br />
d m 2 10-3,<br />
2,790 2,991<br />
Error of<br />
Estimate<br />
0,207<br />
Deviation:<br />
Dependent<br />
Variab<strong>le</strong><br />
0,511<br />
‘22 ‘33 ‘23<br />
0,119 11,681 0,454<br />
Correlation<br />
Coefficient<br />
0,92<br />
F<br />
Value<br />
67,5<br />
S D =-O,119Y-O,9692+3,611<br />
il= 0,798’f+0,7262-2,921 2,775<br />
0,115 0,133<br />
0,517 0,134 14,105 0,608<br />
O,%<br />
o,%<br />
5,5<br />
1%,8<br />
S1=-0,092Y-0,4302+1,855<br />
il2= 0,832Y+0,947Z-3,7%<br />
S2=-0,08OY-O,4502+1,865<br />
14’ 0,867V+1,1172-4,479<br />
+O, 070Y -O, %32+1, 559<br />
j6= 0,891V+1,2522-5,026<br />
Sk=-0,081 V-O,3922+1,674<br />
2,775<br />
2,775<br />
2,715<br />
0,073<br />
0,530<br />
0,063<br />
0,545<br />
0,059<br />
0,556<br />
0,075<br />
0,134 14,105 0,608<br />
0,134 14,105 0,608<br />
0,134 14,105 0,608<br />
0,73<br />
0,97<br />
0,74<br />
0,98<br />
0,68<br />
0,98<br />
0.63<br />
11,6<br />
187,8<br />
11,9<br />
248,2<br />
8,8<br />
296,3<br />
6.5
550<br />
KM 20 K) O 20 40 60 80 100 KM<br />
SCALE I I I SCALE<br />
MAP SHOWING POSITIONS OF GAUGING STATIONS<br />
FIGURE 1
551
œ<br />
O<br />
LL<br />
u)<br />
3<br />
5-<br />
LI<br />
E<br />
P<br />
a
PRACTICES OF DESIGN FLOOD FREQUENCY FOR SMALL WATERSHEDS IN THAILLAND*<br />
ABSTRACT<br />
Damrong Jaraswathana<br />
Director of Hydrology Division<br />
Royal Irrigation Department, Thailand<br />
and<br />
Subin Pinkayan<br />
Associate Professor<br />
Asian Institute of Technology<br />
Bangkok Thailand<br />
Based on <strong>the</strong> fact that adequate hydrologic data do not exist and<br />
development of water resources projects cannot be kept waiting until<br />
data are made availab<strong>le</strong>. Thailand shares this fact with <strong>the</strong> o<strong>the</strong>r de-<br />
veloping countries. The hydrologic data conditions in Thailand can be<br />
categorized as follows. These are: (1) none of any kind of data avai-<br />
lab<strong>le</strong> in <strong>the</strong> catchment area; (2) some data availab<strong>le</strong> within neighbou-<br />
ring areas; (3) some data with short period of record; and c4) consl-<br />
derab<strong>le</strong> data availab<strong>le</strong> with low reliability and accuracy.<br />
The purpose of this paper is to present <strong>the</strong> general practices of<br />
hydrologic analyses in Thailand particularly on design flood frequen-<br />
cy in small watersheds. The method which was <strong>the</strong> common practice for<br />
assessing design floods was based on <strong>the</strong> concept of rational formula,<br />
<strong>the</strong> unit distribution graph and <strong>the</strong> design storm obtained by <strong>the</strong> con-<br />
ventional procedures of frequency analysis.<br />
RESUME<br />
Les données hydrologiques sont insuffisantes, mais l'aménagement<br />
des eaux ne peut attendre. C'est une situation que la Thaïlande parta<br />
ge avec d'autres pays en voie de développement. En Thaïlande, on peut<br />
classer comme suit la nature des données hydrologiques: (1) il n'y a<br />
rien; (2) on dispose 4e quelque chose dans des bassins voisins; (3)<br />
on dispose de données sur une courte période; (4) on a une grande<br />
quantité de données qui n'inspirent pas confiance et sont peu préci-<br />
ses.<br />
Le but de cette communication est de présenter <strong>le</strong>s méthodes d'ana<br />
lyse htdrologique habituel<strong>le</strong>ment utilisées en Thailande, notamment<br />
pour l'évaluation des crues de projet sur <strong>le</strong>s petits bassins. Les modes<br />
de calcul <strong>le</strong>s plus fréquents sont basés sur la méthode rationnel<strong>le</strong>,<br />
l'hydrogramme unitaire et la recherche de l'averse de projet par<br />
<strong>le</strong>s procédés classiques de l'analyse fréquentiel<strong>le</strong>.<br />
* Submitted for presentation at <strong>the</strong> International Sympsium on <strong>the</strong> De-.<br />
sign of Water Resources Project with Inadequate Data, June 4-9,<br />
1973, Madrid, Spain.
554<br />
INTRODUCTION<br />
As far as <strong>the</strong> existing hydrological networks and its operation in Thailand<br />
are concerned, it can be considered that <strong>the</strong> design of water resources pnojects<br />
in Thailand are based on inadequate data. Whi<strong>le</strong> <strong>the</strong> water resources developments<br />
cannot be kept waiting, it is <strong>the</strong> main function of hydrologist to modify<br />
<strong>the</strong> conventional approaches to <strong>the</strong> hydrological assessment in planning of<br />
water resources projects. Application and modification of conventional<br />
approaches may have certain degrees of complication depending upon <strong>the</strong> availability<br />
and limitation of <strong>the</strong> information. Inadequacy of hydrological data<br />
within <strong>the</strong> country may be catagorized as follows: (i) none of any kind of<br />
data availab<strong>le</strong> in <strong>the</strong> catchment area; (2) some data availab<strong>le</strong> within neighbouring<br />
areas; (3) some data with short and/or broken period of record; and<br />
(4) considerab<strong>le</strong> data availab<strong>le</strong> with low reliability and accuracy. Hydrologist<br />
has to make a great attempt and utilized his own experiences in <strong>the</strong> country in<br />
drawing up <strong>the</strong> basic fact, assumption and related know<strong>le</strong>dge to justify <strong>the</strong><br />
hydrological assessment of each case under study. With such attempts <strong>the</strong><br />
justification of a project could be made with only a fair degree of accuracy.<br />
Most of <strong>the</strong> small storage reservoirs in <strong>the</strong> country were planned by applying<br />
<strong>the</strong> modified Conventional approaches to <strong>the</strong> hydrological assessment, in which<br />
<strong>the</strong> only availab<strong>le</strong> information is <strong>the</strong> rainfall data in <strong>the</strong> neighbouring areas.<br />
HYDROLOGICAL DATA PROCUREMENT<br />
It can be stated that hydrological observation has been initiated in<br />
Thailand since 1831, when a staff gage had been instal<strong>le</strong>d at Ayuthya to observe<br />
annual flood inundation within <strong>the</strong> Central Plain areas. The nature of flood<br />
inundation had been a major factor ref<strong>le</strong>cting <strong>the</strong> crop yield during <strong>the</strong> last<br />
century.<br />
Such maximum water <strong>le</strong>vels were indications of water condition in term<br />
of good wateryear, too high flood or drought conditions which, in turn, would<br />
help predicting <strong>the</strong> annual rice harvest.<br />
Later in 1905 <strong>the</strong> streamflow measurement by surface float was introduced<br />
to measure <strong>the</strong> discharge of <strong>the</strong> Chao Phraya river for <strong>the</strong> purpose of planning<br />
of water conservation, diversion and irrigation control works. Development<br />
at that time comprised of diversion schemes in <strong>the</strong> river val<strong>le</strong>ys and tidal<br />
irrigation in <strong>the</strong> delta area. Connected channels between rivers in <strong>the</strong> estuary<br />
were also excavated to conserve water for irrigation and navigation from and<br />
to Bangkok. Scientific approaches applied to planning and imp<strong>le</strong>mentation of<br />
irrigation and drainage schemes were carried out by <strong>the</strong> Royal Irrigation<br />
Department since 1915. Development of water resources was gradually extending<br />
towards headwater and slowly progressed.<br />
Until 1952, when <strong>the</strong> storage work began, <strong>the</strong> modern methods and scientific<br />
standards to be introduced in <strong>the</strong> hydrological investigation were recognized.<br />
A network of comprehensive streamflow gaging stations was set up in major<br />
tributaries where damsites for possib<strong>le</strong> large reservoirs were found. Meanwhi<strong>le</strong>,<br />
numbers of stations were added consecutively in <strong>the</strong> accessib<strong>le</strong> remote areas to<br />
examine <strong>the</strong> runoff and flood yield from watersheds. At present <strong>the</strong>re are over<br />
230 rating stations operated by various government agencies. Among <strong>the</strong>m <strong>the</strong>re<br />
are small number of stations that <strong>the</strong> drainage area is <strong>le</strong>ss than 100 square<br />
kilometers. Many common prob<strong>le</strong>ms of hydrological investigation and data procurement<br />
still exist in <strong>the</strong> country hence <strong>the</strong>y limit <strong>the</strong> expansion of <strong>the</strong> net-
work particularly into <strong>the</strong> smal<strong>le</strong>r watersheds. Among those common prob<strong>le</strong>ms,<br />
<strong>the</strong> limitation of financial support and lack of well-trained personnel are considered<br />
to be <strong>the</strong> main factors. Lack of popularity of work is ano<strong>the</strong>r important<br />
factor <strong>le</strong>ading to have <strong>le</strong>ss fund allocated for hydrological investigation.<br />
5 55<br />
Besides <strong>the</strong> large multip<strong>le</strong>-purposes storage reservoir projects, <strong>the</strong><br />
surface reservoir of comparatively small storage volume cal<strong>le</strong>d "tank" irrigation<br />
projacis were commenced in 1951 in <strong>the</strong> Nor<strong>the</strong>astern Region of Thailand.<br />
Several small watercourses in <strong>the</strong> undulated topography were formed and appeared<br />
to be good sites for storage tanks. The reservoirs range in capacity from<br />
around 40,000 cubic meters up to 18 million cubic meters. Of course, <strong>the</strong> hydrological<br />
investigation of such small basins has never been practiced in <strong>the</strong><br />
region as well as in <strong>the</strong> o<strong>the</strong>r parts of <strong>the</strong> country.<br />
PRACTICES OF DESIGN FLOOD FOR SMALL WATERSHED<br />
To present <strong>the</strong> general practices of design flood for small ungaged water-<br />
shed, a case design by Royal Irrigation Department (1963) of <strong>the</strong> Sattaheep<br />
Tank Project, Thailand, is described below.<br />
It was <strong>the</strong> requirement of <strong>the</strong> Sattaheep Naval Station in 1963, to construct<br />
a small reservoir of 2 million cubic meters capacity for domestic supply. The<br />
proposed damsite has a drainage area of 10.9 square kilometers. None of any<br />
kind of data is availab<strong>le</strong> except <strong>the</strong> rainfall data at Sattaheep, located about<br />
8 kilometers west of <strong>the</strong> basin. Hence, <strong>the</strong> rainfall data at this station were<br />
used in assessment of <strong>the</strong> design flood. Such adoption was based on <strong>the</strong> generally<br />
practices that it was applicab<strong>le</strong> where rainfall characteristics were similar.<br />
Trials had been made by applying <strong>the</strong> empirical formula to determine <strong>the</strong><br />
maximum discharge. The rational formula, Q = C i A, was found to be <strong>le</strong>ss<br />
applicab<strong>le</strong> as its coefficient. C, could not be determined correctly. The McMath<br />
formula, Q ACi(S/A)1/5, was <strong>the</strong>n introduced because it seems to be<br />
more applicab<strong>le</strong> as <strong>the</strong> formula involves <strong>the</strong> basin slope which is one of <strong>the</strong><br />
major factors governing <strong>the</strong> peak rate.<br />
The frequency of design flood cannot directly be determined. In this case<br />
it was assumed to be similar to that of one-day rainfall. From 24-year period<br />
of daily rainfall record at Sattahepp, <strong>the</strong> maximum one-day rainfall amount<br />
of 302.7 mm was observed on 6 October 1957. The computed frequency of occurrence<br />
of this one-day storm rainfall is once in 40 years. From <strong>the</strong> conventional<br />
frequency analysis, <strong>the</strong> 50-year frequency one-day rainfall amount of 320 mm<br />
was adopted in <strong>the</strong> assessment of <strong>the</strong> inflow design flood. Such frequency was<br />
assumed to be that of <strong>the</strong> design flood.<br />
Careful inspection of <strong>the</strong> catchment had been carried out in order to<br />
examine <strong>the</strong> basin characteristics and to estimate <strong>the</strong> concentration time. It<br />
is apparent that <strong>the</strong> time of concentration of such small basin is very short<br />
and usually is much <strong>le</strong>ss than one-day.<br />
The percentage of rainfall as a fraction<br />
of one day was obtained from <strong>the</strong> graph of rainfall recorder. In this case<br />
several storm events were examined and <strong>the</strong> envelope curve was used. The<br />
rainfall amount falling within <strong>the</strong> time of concentration was calculated and<br />
converted into rainfall intensity which is to be used in <strong>the</strong> McMath formula.<br />
The basin coefficient, C, was estimated based on basin characteristics as
556<br />
inspected. The basin slope, S, was determined from <strong>the</strong> availab<strong>le</strong> topographic<br />
map of <strong>the</strong> basin. The design peak discharge obtained in this case study was<br />
43 cubic meters per second.<br />
O<strong>the</strong>r means of assessments were also made for comparison. The unit hybograph<br />
procedure was applied. The assumption of <strong>the</strong> base time of unit hydrograph<br />
is important as it will result in varying peak rate. The storm runoff coefficient<br />
was carefully assumed and flood volume was computed. Peak flow rate was,<br />
<strong>the</strong>refore, obtained by applying triangular distribution hydrograph to <strong>the</strong> flood<br />
volume. The second comparison was made with <strong>the</strong> specific yields of flood<br />
flows obtained from <strong>the</strong> actual streamflow measurements observed in larger watersheds<br />
by <strong>the</strong> Royal Irrigation Department (1965). The flood yield per unit area<br />
computed from those stations were plotted against <strong>the</strong>ir respective drainage<br />
areas. The possib<strong>le</strong> maximum flood yield from smal<strong>le</strong>r watersheds, in term of<br />
cubic meter per second per square kilometeramay be read from <strong>the</strong> logarithmic<br />
extrapolation of <strong>the</strong> envelope curve of specific yield. Such technique will be<br />
one of <strong>the</strong> most reliab<strong>le</strong> indirect approaches if <strong>the</strong> flood yields of small<br />
streams are availab<strong>le</strong> with longer period of record. After several trials were<br />
made, <strong>the</strong> design flood of 43 cubic meters per second were adopted in this study.<br />
The assigned frequency was 50-year. The specific yield of flood flow was around<br />
4 cubic meters per second per square kilometers, which is believed to be<br />
adoptab<strong>le</strong> in <strong>the</strong> area easily affected by tropical depression storms.<br />
CONCLUSIONS<br />
Several modifications of conventional approaches were used in planning and<br />
design of water resources projects in small watersheds in Thailand. The results<br />
obtained by such methods would be satisfied up to a certain degree. New concepts<br />
and statistical techniques which give more reliability are needed to<br />
design of small water resources projects in Thailand.<br />
REFERENCES<br />
1. Royal Irrigation Department (1963). Sattaheep Tank Project, Assessment of<br />
Water for Storage, Hydrology No.137/63, Royal Irrigation Department, Bangkok,<br />
Thailand.<br />
2. Royal Irrigation Department (1965). Mean Annual Discharge vs. Drainage Area,<br />
Envelope Curves of Maximum Recorded Peak Discharge, Specific Yield of Flood<br />
Flow for Rivers in Thailand and Malaya, Hydrology No.186/65, Royal Irrigation<br />
Department, Bangkok, Thailand.
ABSTRACT<br />
DESIGN DISCHARGE DERIVED FROM DESIGN RAINFALL<br />
Takeo KINOSITA<br />
Takeshi HASHIMOTO<br />
A design discharge for flood control in Japan is in general<br />
derived from a design rainfall since discharge data are not suffL<br />
cient for designing. The procedure of derivation and its merits<br />
and demerits will be explained in this report according to fallo-<br />
wing four steps. (i) A design rainfall in a certain return period<br />
is determined by a probability process. (2) Design rainfall dis-<br />
tribution are obtained by enlargement of rainfall distributions<br />
of recent representative storms. (3) A simulation model for runoff<br />
is decided by rainfalls and runoffs of recent representative<br />
storms. (4) A design discharge is determined by <strong>the</strong> simulation mo<br />
del with enlarged rainfall distributions.<br />
RESUME<br />
Au Japon, <strong>le</strong>s données concernant <strong>le</strong>s débits ne sont pas SUL<br />
fisantes pour évaluer <strong>le</strong>s crues de projet; on procède donc généra<br />
<strong>le</strong>ment par l'intermédiaire de l'averse de projet. Les auteurs ex-<br />
posent <strong>le</strong> procédé utilisé, ses mérites et ses inconvénients; il<br />
se décompose en quatre étapes. (1) On détermin: par analyse fré-<br />
quentiel<strong>le</strong> une averse de projet correspondant a une certaine pê-<br />
riode de retour. (-2) Cette averse est distribuée dans <strong>le</strong> temps en<br />
s'appuyant sur des hyétogrammes d'averses récentes considérées<br />
comme représentatives. (3) On choisit un modè<strong>le</strong> de transfoTmation<br />
pluies-débits élaboré à partir d'observations de pluies et de dé-<br />
bits effectuées récemment au cours d'averses représentatives. (4)<br />
On applique ce mode<strong>le</strong> au hyétogramme de projet élaboré en (2).
558<br />
I. Introduction<br />
Japan is located in <strong>the</strong> temperate and humid zone. A river in this country<br />
is comparatively small and its gradient is steep. Floods have occurred<br />
very often since <strong>the</strong> prehistric age and been serious constraints against development<br />
of <strong>the</strong> nation for a long time. Flood control is one of <strong>the</strong> major items<br />
of water resource development works.<br />
It is necessary to col<strong>le</strong>ct and analyze discharge data for design of flood<br />
control projects. Authorized discharge gauging stations are 330 in 120 rivers<br />
in Japan. There are many o<strong>the</strong>r non-authorized discharge gauging stations.<br />
However, land developments and river improvement works have remarkably succeeded<br />
and hydrological eituations of river badins are rapidly changing. This fact<br />
induces that <strong>the</strong> discharge cannot be used for design purpose directly and used<br />
only for verification of a runoff simulation model, and <strong>the</strong> rainfall which is<br />
not affected by human activity is used for design purpose.<br />
Mot only <strong>the</strong> peak discharge but also <strong>the</strong> flood hydrograph are necessary<br />
for channel improvement, design of multipurpose reservoirs and soon. The procedure<br />
to obtain <strong>the</strong> design hydrograph will be discussed in this report.<br />
2. Probability Analysis<br />
Since discharge is originally derived from rainfall, <strong>the</strong> design discharge<br />
for water resource system is determined by <strong>the</strong> design rainfall through <strong>the</strong> runoff<br />
simulati on model.<br />
At <strong>the</strong> first step of this procedure, <strong>the</strong> total amount of <strong>the</strong> design rainfall<br />
within a certain period should be computed by means of <strong>the</strong> probability<br />
analysis. The important assumption of this section is that <strong>the</strong> time series of<br />
rainfall are produced by some stationary stochastic process.<br />
The procedure of this analysis is divided into two.<br />
(9 Sampling from observed rainfall data.<br />
cn) Frequency analysis.<br />
The latter has been discussed by some hydrologists, 80 <strong>the</strong> authors intend<br />
to focus <strong>the</strong>ir attention on <strong>the</strong> practical phase of <strong>the</strong> former. The series of<br />
annual extreme values of rainfall within a certain dulation is se<strong>le</strong>cted from<br />
<strong>the</strong> historical data. The duration in this paper is a period which is significant<br />
to <strong>the</strong> design for <strong>the</strong> water resource system in <strong>the</strong> definite basin, and<br />
cannot be so freely chosen. The rainfall within an adequate duration has <strong>the</strong><br />
closest relation to <strong>the</strong> magnitude of <strong>the</strong> flood discharge, and <strong>the</strong> rainfall with-<br />
in a comparatively short or long duration has <strong>le</strong>ss relation to it.<br />
<strong>the</strong> design duration must be appropriately Chosen according to <strong>the</strong> basin characteristics,<br />
for instance <strong>the</strong> drainage area, <strong>the</strong> channel <strong>le</strong>ngth, <strong>the</strong> slope and so<br />
on.<br />
The net work of daily rainfall observation covers all over Japan, and<br />
daily rainfall data have been recorded for more than thirty years, at some stations<br />
a hundred years. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> network of hourly rainfall observation<br />
is sparser than that of <strong>the</strong> daily rainfall. The hourly rainfall<br />
data have been recorded for twenty years on an average.<br />
Therefore<br />
The credibility of<br />
statistical estimations is dependent on <strong>the</strong> samp<strong>le</strong> size, that is to say <strong>the</strong><br />
<strong>le</strong>ngth of <strong>the</strong> series of observed data. Therefore <strong>the</strong> statistical analysis is<br />
hardly applied to hourly rainfall data. Daily rainfall data are used for de-<br />
termining <strong>the</strong> amount of design rainfall by means of <strong>the</strong> statistical analysis.
5 59<br />
Then, <strong>the</strong> design duration must be an integer multip<strong>le</strong> of a day. As noted<br />
above, <strong>the</strong> design duration muet be se<strong>le</strong>cted ae a time in which <strong>the</strong> rainfall<br />
has a close relation to <strong>the</strong> peak discharge. Since <strong>the</strong> time sca<strong>le</strong> corresponds<br />
to <strong>the</strong> space sca<strong>le</strong> in natural phenomena, <strong>the</strong> duration for a smal<strong>le</strong>r basin must<br />
be a day, and that for a bigger basin must be three days in this country.<br />
A daily rainfall in Japan is defined as a rainfall observed from nine<br />
a.m. to nine a.m. <strong>the</strong> next morning. If a storm stretches over this boundary<br />
of observation, a daily rainfall cannot represent a storm rainfall. 'Two<br />
days'' seems a minimum design duration even for a small basin. The fact that<br />
a big storm in Japan tends to continue more than a day requires this limitation<br />
of <strong>the</strong> minimum design duration. The design duration for <strong>the</strong> statistical analysis<br />
is two days for a small basin and a medium basin, and three days for a big<br />
basin.<br />
The return period for design purpose is not determined by a ma<strong>the</strong>matical<br />
way, but by consideration of economical, political and social situations on <strong>the</strong><br />
basin. The sewerage system design claims for five to seven years as a return<br />
period. For an urban basin, some period above twenty years is se<strong>le</strong>cted as a<br />
return period. A big river basin in Japan requires almost a hundred years'<br />
return period. For spillway design of a dam, about two hundred years' return<br />
period is commonly used.<br />
3. Enlargement of Observed Hyetographs<br />
The amount of <strong>the</strong> probability rainfall for design purpose is determined<br />
as shown in <strong>the</strong> above section. A careful attention should be paid to <strong>the</strong><br />
procedure for distributing <strong>the</strong> amount of <strong>the</strong> probability rainfall to <strong>the</strong> time<br />
axis, because an hourly distribution of flood runoff, a hydrograph, is indispensab<strong>le</strong><br />
for a flood control project, and a hydrograph is derived from an<br />
houry distribution of rainfall, a hyetograph, through a runoff simulation<br />
model. A hyetograph for design is deduced from that of a recent representative<br />
storm. A part of <strong>the</strong> hyetograph observed during <strong>the</strong> representative<br />
storm is se<strong>le</strong>cted aiming at <strong>the</strong> time of occurence of <strong>the</strong> maximum amount of<br />
rainfall, where <strong>the</strong> time is taken equally to <strong>the</strong> design duration.<br />
Suppose N is <strong>the</strong> number of hours within <strong>the</strong> duration, Rpis <strong>the</strong> amount<br />
of probability rainfall and Roi ( for i=1,2, ... ,N is <strong>the</strong> observed rainfall<br />
depth in <strong>the</strong> i-th hour. The enlargement factor R? is defined by <strong>the</strong> following<br />
equation.<br />
Then <strong>the</strong> enlargement factor is multiplied to each R<br />
of design hyetograph.<br />
to get <strong>the</strong> time series<br />
EX'-%, EF'sR, a. ,EF.Rou<br />
This procedure is cal<strong>le</strong>d enlargement of observed hyetograph. Several hyetographs<br />
are derived from several representative observed hyetographs in this<br />
way.<br />
If <strong>the</strong> amount of <strong>the</strong> representative rainfall is almost same as that of<br />
<strong>the</strong> probability rainfall, This procedure is very successful. If not, <strong>the</strong><br />
enlarged hyetograph sometimes shows an unexpected pattern. In order to<br />
avoid such an unexpected pattern, <strong>the</strong>re must be some limitation for enlarge-
560<br />
ment. Several proposals were given for this limitation, but <strong>the</strong>re's no<br />
praiseful one. For this limitation is to be deduced not <strong>the</strong>oretically but<br />
merely empirically. One of <strong>the</strong> proposals is presented in <strong>the</strong> following<br />
paragraph.<br />
A certain domain is set up including <strong>the</strong> basin concerned, From all<br />
<strong>the</strong> rainfall gauging stations in <strong>the</strong> domain, maximum point rainfall values<br />
are se<strong>le</strong>cted about various periods shorter than <strong>the</strong> duration. The enlarged<br />
hyetograph is compared with <strong>the</strong>se values. If <strong>the</strong> enlarged amount during<br />
some period exceeds <strong>the</strong> mimum point rainfall value during <strong>the</strong> same period,<br />
<strong>the</strong> enlarged hyetograph must be abandoned because of <strong>the</strong> rareness of occurrence.<br />
But this proposal raises ano<strong>the</strong>r question. What region is appropriate<br />
as <strong>the</strong> domain? For instance, if we replace Japan with <strong>the</strong> world, <strong>the</strong><br />
se<strong>le</strong>cted value of maximum poin rainfall becomes greater at any period. In<br />
spite of this question, this proposal seems reasonab<strong>le</strong>. Because <strong>the</strong>re must<br />
exist a realistic upper bound on <strong>the</strong> amount of rainfall that can occur on <strong>the</strong><br />
basin within a certain period. An examp<strong>le</strong> is adduced. On <strong>the</strong> upper Kiso<br />
River basin from 1951 to 1971, <strong>the</strong>re were 29 representative storm in which <strong>the</strong><br />
maximum rainfall amount durig48 hours was greater than 100 mm. The maximum<br />
point rainfall values are made into Tab<strong>le</strong> 2. As a result of <strong>the</strong> comparison,<br />
<strong>the</strong> exceedance is noted by symbol 'E* in Tab<strong>le</strong> I. If <strong>the</strong> exceedance has occurred<br />
at some domein, it also occurs at any narrower domain. And yet, in this<br />
case, <strong>the</strong> exceedance is apt to occur in a shorter period than a longer period.<br />
This fact suggests us that <strong>the</strong> hyetograph of a heavy storm is uniformer in its<br />
time distribution than that of a common storm.<br />
4. Runoff Simulation Model and Effective Rainfall Analyeis<br />
In this section, a runoff simulation model is determined, and simultaneously<br />
an empirical ru<strong>le</strong> is derived-on <strong>the</strong> separation of rainfall excess from observed<br />
rainfall.<br />
Among <strong>the</strong> great number of rainfall-runoff convertion schemes, 'Storage Func-<br />
This method is expressed by <strong>the</strong> follow-<br />
tion Method' is commonly used in Japan.<br />
ing two equations.<br />
where sr is <strong>the</strong> storage in <strong>the</strong> basin, qL is <strong>the</strong> outflow from <strong>the</strong> basin, r excemsive<br />
rainfall, K and p are empirical constants dependent on <strong>the</strong> basin, and suffix<br />
denotes delayed variab<strong>le</strong> by a lag time Ta. These constants are neceseary<br />
for <strong>the</strong> runoff simulation of <strong>the</strong> storage function, so <strong>the</strong>y are previously<br />
determined by sets of rainfall and runoff data of <strong>the</strong> recent representative<br />
floods. As is seen in Fq. (2) this method contains a nonlinear process.<br />
'Tank Model Method, aïso aseumes a nonlinear process, and is sometimes<br />
used as a runoff simulation model. Unit hydrograph method has been improved<br />
in this country, and is put to practical use today. But <strong>the</strong> use of unit hydrograph<br />
method ia restricted to <strong>the</strong> basine where <strong>the</strong> assumption of linearity is to<br />
a certain degree appreciab<strong>le</strong>.
The separation of excessive rainfall from <strong>the</strong> observed one is an important<br />
but an awfully suffering work. In Japan, <strong>the</strong> soil moisture of <strong>the</strong> basin general-<br />
ly shows a vio<strong>le</strong>nt variation from dry to wet according to <strong>the</strong> wea<strong>the</strong>r condition.<br />
The soil moisture antecedent to <strong>the</strong> storm strongly governs <strong>the</strong> rising part of <strong>the</strong><br />
runoff hydrograph, sometimes even <strong>the</strong> crest. So <strong>the</strong> effective rainfall analysis<br />
must be carried out carefully for <strong>the</strong> identification of model parameters. How-<br />
ever, at <strong>the</strong> simulation for a design flood, a common value of <strong>the</strong> parameter repre-<br />
senting <strong>the</strong> soil moisture in <strong>the</strong> basin is used.<br />
5. Design Discharge<br />
Finally, <strong>the</strong> design discharge is determined in this section. The enlarged<br />
hyetographs of representative storms are used for <strong>the</strong> runoff simulation. A hydrograph<br />
corresponding to each design hyetograph is computed by <strong>the</strong> runoff simulation<br />
model.<br />
Tab<strong>le</strong> 1: Comparison of Enlarged Hyetographs with Maximum Point Rainfall Values<br />
tom<br />
NO.<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
11<br />
12<br />
13<br />
14<br />
15<br />
16<br />
17<br />
18<br />
19<br />
20<br />
21<br />
22<br />
23<br />
24<br />
25<br />
26<br />
27<br />
28<br />
29<br />
Ra<br />
mm 1<br />
-<br />
137.1<br />
137.6<br />
125.1<br />
117.4<br />
1 IO. 1<br />
114.9<br />
147.7<br />
125.2<br />
106.3<br />
195.9<br />
117.2<br />
107.6<br />
143.6<br />
118.7<br />
173.6<br />
227.5<br />
119.1<br />
148.0<br />
114.3<br />
112.3<br />
144.8<br />
123- 2<br />
169.5<br />
156.3<br />
118.1<br />
136.6<br />
148.1<br />
E3<br />
-<br />
2.19<br />
?.i8<br />
2.40<br />
2.56<br />
2.72<br />
2.61<br />
2.03<br />
2.40<br />
2.82<br />
1.53<br />
2.56<br />
2.79<br />
2.09<br />
2-53<br />
1.73<br />
1.32<br />
2.52<br />
Upper<br />
Liso River<br />
1 3 6<br />
E<br />
E E E<br />
E E E<br />
E<br />
Who<strong>le</strong><br />
Liso River<br />
E<br />
E E<br />
E<br />
E<br />
Chubu<br />
District<br />
1 3 6<br />
E E<br />
Japan<br />
1 3 6<br />
E E<br />
561<br />
The<br />
World<br />
1 3 6<br />
2.03<br />
2.62 E<br />
2.67<br />
2.07<br />
2.44<br />
1.77 E<br />
E<br />
E<br />
1.92 E E E E<br />
E<br />
E<br />
2.54<br />
2.20<br />
2.03<br />
263.9 1.14<br />
- - 225.6 - 1.33<br />
Symbols<br />
EF : Enlagement Factor ( Rf/Ro<br />
Rp : Amount of Two Days' Probability Rainfall at 200 Years, Return Period<br />
RD : Maximum 48 Hours' Rainfall Amount<br />
E : ERLarged Amount exceeds Maximum Point Rainfall Value in this Domain<br />
1*3 and 6 are periods in houds). The estimated value of Rp is 300 mm.
I The<br />
562<br />
Tab<strong>le</strong> 2: Maximum Point ñainfall Values ( mm )<br />
World 350 600 1488900<br />
Although areas of <strong>the</strong>se hydrographs are almost same, peak discharges are<br />
different each o<strong>the</strong>r. The reasom are (i) variety of hyetograph, (n)nonlinearity<br />
of runoff model, and (ai) loss factor Among <strong>the</strong>se, (i) is <strong>the</strong><br />
most predominant. Owing to this fact and to M e it possib<strong>le</strong> to obtain various<br />
hydrographe, enlargement was applied to various representative storm hyetographs.<br />
From <strong>the</strong> above hydrographs, one is se<strong>le</strong>cted as a design discharge. The<br />
e<strong>le</strong>ction of a design discharge itself brings up a brand new prob<strong>le</strong>m, but <strong>the</strong><br />
authors shall <strong>le</strong>ave <strong>the</strong> discussion to ano<strong>the</strong>r occasion.<br />
6. Conclusion<br />
The derivation of a design discharge is explained in thie re art. It is<br />
composed of (1) design rainfall in a certain return period, (27 enlargement,<br />
(3) a simulation model and (4) design discharge derivation. The procedure is<br />
not fixed today. It will be improved everyday with development of hydrology.<br />
Takeo KINOSITA, Takeshi HASHIMOTO : Public Works Research Institute,<br />
Ministry of Construction,<br />
Government of Japan.
AB ST RACT<br />
THE USE OF CENSORED DATA IN ESTIMATING T-YEAR FLOODS<br />
Morven N. Leese<br />
Institute of Hydrology, Wallingford, Berks, U.K.<br />
Types of incomp<strong>le</strong>te data to be found in connection with<br />
flood series are described, and it is shown how samp<strong>le</strong>s contai-<br />
ning such data may be used to estimate <strong>the</strong> parameters of a dis-<br />
tribution describing annual maximum flows. Formulae for <strong>the</strong><br />
standard errors of <strong>the</strong> resulting estimates are also given.<br />
Examp<strong>le</strong>s are taken from a river for which censored data exist.<br />
Preparatory data standardization is described, and <strong>the</strong> parame-<br />
ters estimated using this data are compared uith <strong>the</strong>se estima-<br />
ted using <strong>the</strong> comp<strong>le</strong>te samp<strong>le</strong> only. The marginal value of using<br />
censored data in this context is assessed by means of <strong>the</strong> subse-<br />
quent reduction in <strong>the</strong> standard errors of <strong>the</strong> estimates of<br />
T-year floods for various values of T, and this is related to<br />
<strong>the</strong> effort required to col<strong>le</strong>ct and standardize <strong>the</strong> data.<br />
RESUME<br />
L'auteur traite d'une catégorie de données incomplètes<br />
qu'on peut rencontrer dans l'étude d'une série chronclqgique COZ<br />
cernant <strong>le</strong>s crues. I1 montre comment des échantillons contenant<br />
de tel<strong>le</strong>s données peuvent être utilisés pour estimer <strong>le</strong>s param;-<br />
tres d'une loi de distribution des maximums annuels. I1 donne<br />
éga<strong>le</strong>ment des formu<strong>le</strong>s pour calcu<strong>le</strong>r <strong>le</strong>s erreurs types des esti-<br />
mations qui en résultent. I1 prend comme exemp<strong>le</strong> un f<strong>le</strong>uve pour<br />
<strong>le</strong>quel existent de tel<strong>le</strong>s données tronquées c'est-à-dire défi-<br />
nies par un suil auquel el<strong>le</strong>s sont éga<strong>le</strong>s ou supérieures. I1 in-<br />
dique comment on peut parvenir à une normalisation de ces bonnées<br />
et compare <strong>le</strong>s va<strong>le</strong>urs ainsi estimées pour <strong>le</strong>s parametres a ce-<br />
l<strong>le</strong>s qu'on puet obtenir à partir du seul échantillon des donnees<br />
régulières. L'appréciation du gain marginal d'information dÛ a la<br />
prise en compte des données tronquées revient à évaluer la réduc-<br />
tion de l'écart type qui en résulte pour <strong>le</strong>s estimations des crues<br />
de période de retour T, pour différentes va<strong>le</strong>urs de T. Ce gain<br />
d'information est comparé 'a l'effort nécessaire pour col<strong>le</strong>cter et<br />
normaliser de tel<strong>le</strong>s données,
5 64<br />
1;JTFODUCTION<br />
The design of hydraulic structures for a water resources project depends<br />
in part on estimates of <strong>the</strong> floods which <strong>the</strong> structures may be required<br />
to withstand during <strong>the</strong> project's economic life. The feasibility of <strong>the</strong><br />
project may be examined by comparing <strong>the</strong> cost of building each structure<br />
to <strong>the</strong> requirements of its design flood with its anticipated benefits.<br />
The latter may be realized in terms of reduced damage to <strong>the</strong> structure<br />
itself as well as to surrounding property, and in more effective flood<br />
plain use.<br />
Precision in <strong>the</strong> estimation of a design flood conveys a monetary benefit<br />
by mitigating <strong>the</strong> costs which arise from over or underdesign.<br />
Never<strong>the</strong><strong>le</strong>ss, <strong>the</strong> use of additional data to increase precision will have<br />
a marginal cost which may be greater than its marginal benefit.<br />
circumstances it is necessary to quantify <strong>the</strong> increase in precision, and<br />
if possib<strong>le</strong> to express this increase in financial terms.<br />
hydraulic structures thus involves both hydrologic and economic<br />
considerations which require for <strong>the</strong>ir formulation: estimates of floods<br />
with given return periods; values to be placed on <strong>the</strong> precision of <strong>the</strong><br />
estimates; cost and benefit curves for <strong>the</strong> structure.<br />
In <strong>the</strong>se<br />
The design of<br />
The standard form of data for <strong>the</strong> estimation of floods consists of a series<br />
of annual maxima derived from a continuous flow record. It is proposed to<br />
show how data which is not of <strong>the</strong> standard form may still be used for this<br />
purpose, and that <strong>the</strong> use of additional data of non-standard form'hcreases<br />
<strong>the</strong> precision of estimation.<br />
It is not proposed to discuss in detail<br />
<strong>the</strong> economic implications of <strong>the</strong> increase, but <strong>the</strong> order<br />
of magnitude of<br />
<strong>the</strong> resulting cost-reduction is indicated by means of a simp<strong>le</strong> examp<strong>le</strong>.
ESTIMATION OF T-YEAR FLOOIX - STANDARD DATA<br />
565<br />
Jn order to estimate <strong>the</strong> flood with return period T, sey, (or 'T-year flood'),<br />
it is first necessary to choose a probability distribution representative<br />
of <strong>the</strong> annual maxima. The parameters of <strong>the</strong> distribution may <strong>the</strong>n be<br />
estimated from past records by one of many estimation procedures. The<br />
Gumbel , or doub<strong>le</strong>-exponential, distribution is often used to represent<br />
annual maxima because of a supposed validity on <strong>the</strong>oretical grounds and<br />
although <strong>the</strong>se grounds have been questioned (2), its extensive use justifies<br />
- (=)<br />
its fur<strong>the</strong>r study in this context. It has <strong>the</strong> following form:-<br />
F(r) = exp [-e -J , - o
566<br />
-1/& E N<br />
i = N<br />
+ L<br />
i = 1<br />
P o<br />
where y <strong>the</strong> so-cal<strong>le</strong>d ‘reduced variate’ a is given by:-<br />
i’<br />
The flood of return period T, %, is <strong>the</strong>n given by:-<br />
so that<br />
Pr (x 5 %) = 1 - 1/T,<br />
- %-u = 1<br />
-e-<br />
- 1/T.<br />
e a<br />
m y T = %-u.<br />
-9<br />
a<br />
an estimate oí’ 3 is <strong>the</strong>n<br />
A<br />
A 3 = Q+ a YTa<br />
A A<br />
where
<strong>the</strong> values of whoee e<strong>le</strong>ments are substituted into <strong>the</strong> following:-<br />
h<br />
V mw be approximated by replacing a by a. A pull derivation of <strong>the</strong><br />
above quantities may- be fomd in Gumbel ( 1) and Kimball (4).<br />
NON-STANDARD DATA<br />
Those concerned with maximum flood estimation will be familiar with at<br />
567<br />
<strong>le</strong>ast two types of non-standard data found in connection with flood series:<br />
missing peaks in continuous chart records and historic flood marks.<br />
Hissing peaks occur when <strong>the</strong> flow is so high that <strong>the</strong> recording pen runs<br />
off <strong>the</strong> eàge of <strong>the</strong> chart; whilst it should be possib<strong>le</strong> to estimate a<br />
missing peak discharge from h know<strong>le</strong>dge of <strong>the</strong> <strong>le</strong>ngth of time <strong>the</strong> chart<br />
limit is exceeded, <strong>the</strong> properties of such a method require fur<strong>the</strong>r<br />
investigation.<br />
The approach used here is to assume no more than that a<br />
flood which has exceeded a chart limit has a peak discharge greater than<br />
<strong>the</strong> flow corresponding to <strong>the</strong> flov at <strong>the</strong> chart limit.<br />
Historic flood marks are usually to be found in waìls,bridges or on<br />
specially coktructed flood stones.<br />
which have risen above a fixed point<br />
certain circumstances, it mey be assumed that all such floods have been<br />
marked, and that floods in <strong>the</strong> intervening years for which no marks exist<br />
have fai<strong>le</strong>d to reach <strong>the</strong> fixed point.<br />
These are <strong>the</strong> two types of data to be considered.<br />
They indicate <strong>the</strong> <strong>le</strong>vels of floods<br />
during some historic period. In<br />
values are only specified if <strong>the</strong>y lie on one side of a given threshold.<br />
Samp<strong>le</strong>s which exhibit this property are known as censored samp<strong>le</strong>s, <strong>the</strong><br />
threshold being cal<strong>le</strong>d <strong>the</strong> censoring point.<br />
(6)<br />
They have this in common;<br />
If <strong>the</strong> threshold is fixed,
568<br />
as it is in <strong>the</strong>se caces, and <strong>the</strong> proportion of censored events is a random<br />
variab<strong>le</strong>, <strong>the</strong> censoring is type I; if <strong>the</strong> threshold is a random variab<strong>le</strong>,<br />
but <strong>the</strong> proportion of censored events is fixed, <strong>the</strong> censoring is type II.<br />
A fui1 discussion of censoring is given by Kendall and Stuart(5),<br />
fur<strong>the</strong>r references are given.<br />
The incorporation of a. 'missing peak' or 'historic record' into a<br />
where<br />
standard samp<strong>le</strong> is c<strong>le</strong>arly a type I censoring prob<strong>le</strong>m. The general form of<br />
<strong>the</strong> likelihood function Lc for a samp<strong>le</strong> of n + k values of which n are below<br />
<strong>the</strong> censoring point x and are specified 8s x 1, x2 ... x and k ari? above<br />
C n'<br />
x and arë unkna~n, b BS follows:-<br />
C<br />
where f (x) is <strong>the</strong> appropriate probability distribution function. A similar<br />
expressiori rr,ay be obtained for censoring above a censoring point, and maximum<br />
likelihood equations m4y be obtained from ei<strong>the</strong>r expression in <strong>the</strong> USUEL manner.<br />
ESTIMATION OF T-YEAR FLOODS - NON-STANDARD DATA<br />
Censoring above a threshold (Missing Peaks).<br />
(assumed<br />
n<br />
independent) and k missing peaks which are knm to be above <strong>the</strong> chart<br />
Suppose a samp<strong>le</strong> consists of n annual maxima xl, 5, . . . x<br />
limit xc.<br />
-<br />
<strong>the</strong>n :<br />
The likelihood hction L and maximum likelihood equations are<br />
C
-e JC<br />
where yi = Xi - Y i Y, = X - u ; W = e .<br />
o a<br />
The variance-covariance matrix Y of ac U <strong>the</strong> pumeters estimated<br />
Ca<br />
from equations (9) is <strong>the</strong>n given by <strong>the</strong> inverse of Rc, whose e<strong>le</strong>ments ri<br />
are as follows:-<br />
where c = eTC, and J and K are integrals which require to be evaluated<br />
numerically. They are given by:<br />
J 3 5<br />
5 70<br />
L<br />
i=N+r i=N+r<br />
- l/a (N+r) - E y. + E YieYi<br />
1<br />
i= 1 i= 1<br />
+ <strong>le</strong>T%d = O;<br />
i=N+r<br />
- i/a r-(N+r) + C eyi +<br />
-<br />
-<br />
i= 1<br />
where yi = x.-u ; yh %-u ; u= e - 1 -<br />
a a<br />
(13)<br />
A<br />
The variance-covariance matrix $ for a and B, <strong>the</strong> parameter estimates<br />
estimated from equations (13) is <strong>the</strong>n given by <strong>the</strong> inverse 'of #,<br />
whose eisments r5 are 88 follows:-<br />
- rI2h = - E[a2LogI l/a2 {(N+M (0.4228) + M u(yh - 1<br />
aaau i<br />
where J and K, are expressicm of <strong>the</strong> form of ( 1 1) with h 5 e*<br />
lower limit of integration.<br />
-q 1,<br />
as <strong>the</strong><br />
Equations (9) and (13) are thus <strong>the</strong> modified equations to be used for <strong>the</strong><br />
estimation of parameters f im <strong>the</strong> na-standard floods data described.<br />
have been derived in <strong>the</strong> context of reliability <strong>the</strong>ory, and are given in<br />
(6). Similar expressions may be obtained for distributions o<strong>the</strong>r than <strong>the</strong><br />
Gumbel distributirm by substituting <strong>the</strong> appropriate p.d.f in (7) or its<br />
equiva<strong>le</strong>nt fon censoring belar a threshold.<br />
They
An iterative technique for solving <strong>the</strong> equations (3) for a standard samp<strong>le</strong><br />
may be found in Jenkinson (71, who gives a worked examp<strong>le</strong>. This technique<br />
my be used with slight adaptation for <strong>the</strong> solution of equations (9) and<br />
(13); satisfactory results are obtained if <strong>the</strong> iteration matrix is <strong>le</strong>ft<br />
unchanged, i.e. given values appropriate to an uncensored samp<strong>le</strong>. However,<br />
care should be taken in <strong>the</strong> choice of initial values if <strong>the</strong> proportion<br />
of censored values is high.<br />
THE AVON AT BATH - AN APPLICATION OF THE EQUATIONS<br />
The most satisfactory applications of <strong>the</strong>se modified equations has been in<br />
<strong>the</strong> extension of records to achieve significantly greater precision in <strong>the</strong><br />
resulting estimates.<br />
Avon at Bath, where a set of historic flood marks had been recorded during<br />
a historic period prior to a fairly long continuous chart record.<br />
parametex estimated from <strong>the</strong> recent record alone were compared with <strong>the</strong>se<br />
estimated frcm a combined samp<strong>le</strong> consisting of <strong>the</strong> historic floods and <strong>the</strong><br />
recent records. The data used is shown in t'ab<strong>le</strong> 1, 1940 being <strong>the</strong> date of<br />
<strong>the</strong> beginning of <strong>the</strong> recent record.<br />
One such application was for data from <strong>the</strong> river<br />
It is necessary to perform a number of checks on historic data before<br />
entering it into equations ( 13). for instance, <strong>the</strong> stage-discharge<br />
The<br />
571<br />
relaticnship derived for <strong>the</strong> recent record may require adjustment before it<br />
is applied to <strong>the</strong> historic flood marks, whose site may be at some distance<br />
from <strong>the</strong> modern gauging station. The assumption that floods are marked if<br />
(and mly if) <strong>the</strong>y have risen above <strong>the</strong> threshold makes it necessary to<br />
investigate <strong>the</strong> circumstances surrounding each flood mark.<br />
time-consuming exercise, but it is one which could to a large extent be<br />
carried out by local library or museum staff who have to hand contemporary<br />
evidence such as old newspaper reports.<br />
This mey be 8
572<br />
TABLE 1. Annual Maximum Flooäa Used in <strong>the</strong> Estimation of<br />
a and u in Gumbel's Extreme Value Distribution<br />
(In Cumecs).<br />
* Water<br />
Year<br />
1865<br />
i866<br />
1874<br />
1875<br />
1879<br />
1882<br />
1888<br />
Flood<br />
206<br />
228<br />
12 1<br />
218<br />
264<br />
362<br />
204<br />
375<br />
154<br />
239<br />
302<br />
i86<br />
255<br />
148<br />
Water<br />
Year<br />
1941<br />
1942<br />
1943<br />
1944<br />
1945<br />
1946<br />
1947<br />
1948<br />
1949<br />
1950<br />
195 1<br />
1952<br />
1953<br />
1954<br />
Flood<br />
84<br />
149<br />
' 73<br />
118<br />
128<br />
282<br />
98<br />
i04<br />
1 q3<br />
229<br />
136<br />
116<br />
96<br />
296<br />
Water<br />
Year<br />
1955<br />
1956<br />
1957<br />
1958<br />
1959<br />
1960<br />
196 i<br />
1962<br />
1963<br />
1964<br />
1965<br />
1966<br />
1967<br />
1968<br />
- !he following values were obtained for <strong>the</strong> data of tab<strong>le</strong> 1:-<br />
N 0 32; M m 58; 1 = 48; r= 10; 5 200,<br />
and when <strong>the</strong>se values, and <strong>the</strong> data of tab<strong>le</strong> 1, were substituted into<br />
equations (13), <strong>the</strong> estimates sham in tab<strong>le</strong> 2 were obtained.<br />
Flood<br />
128<br />
107<br />
138<br />
169<br />
169<br />
352<br />
12 1<br />
103<br />
277<br />
110<br />
178<br />
172<br />
31 1<br />
A<br />
Floods with various return periods were <strong>the</strong>n estimated from <strong>the</strong> values of a,<br />
and ;h obtained from <strong>the</strong> enlarged samp<strong>le</strong>, by substitutitm in equation (41, and<br />
<strong>the</strong>ir large-samg<strong>le</strong> stenâard errore were elso calculated from equation (6).<br />
These are shown in teb<strong>le</strong> 3, where values obtained from <strong>the</strong> original srmg<strong>le</strong> am a<strong>le</strong>0 shown.<br />
125
TAñLE 2. Estimates of Paretem of Gumbel's Extreme Value<br />
(1) Estimated Flood:<br />
Large-s amp<strong>le</strong><br />
standard error:<br />
(2) Estimated ' Flood:<br />
Large-samp<strong>le</strong><br />
standard error:<br />
573<br />
Distribution Using (1) Historic Flood Marks and Recent Data<br />
and (2) Recent Data Alone. (In Cumecs).<br />
Parameter<br />
(1) Estimate:<br />
Large -samp<strong>le</strong><br />
standard error:<br />
(2) Estimate:<br />
Large-samp<strong>le</strong><br />
standard error:<br />
a U Samp<strong>le</strong> size<br />
47 128 29 recent values<br />
+ 13 historic values<br />
2.5 27 + 48 censored values<br />
48 128 29 recent values<br />
27 29<br />
TABLE 3. Estimates of Floods with Various Return Periods Using (1)<br />
Historic Flood Marks and Recent Data and (2) Recent<br />
Data Alone. (In Cumecs).<br />
1 Reluni Period I 2.33 (Mean) 10 25 50 100 1000 I<br />
THE AVON AT BATH - THE VALUE OF ADDITIONAL DATA<br />
155 234 278 311 344 , 453<br />
t7 - +12 +i6 219 223 i33<br />
156 236 281 314 348 458<br />
- +Il - +20 - +26 231 - t36 +51<br />
It will be seen from tab<strong>le</strong> 3 that <strong>the</strong> sampling error in <strong>the</strong> 50year flood<br />
estimate was reduced from 10% to 6% by <strong>the</strong> use of historic flood marks. Was<br />
this reduction worthwhi<strong>le</strong> in view of <strong>the</strong> effort required to standardize <strong>the</strong><br />
data?<br />
type.<br />
A number of approaches mw be taken ì.n answering questions of this<br />
Ultimately <strong>the</strong>y involve <strong>the</strong> formulation of expressions for <strong>the</strong> benefits<br />
and c,osrs which arise from acquiring <strong>the</strong> data, and since <strong>the</strong>se c m never be<br />
fully known, <strong>the</strong> prob<strong>le</strong>m of evaluating <strong>the</strong> worth of stream flow data are far<br />
from etrai@t-ionrard.<br />
I
574<br />
Reasonab<strong>le</strong> attempts have been made, however, by E~RS of simplifying assumptions.<br />
une such attempt has been made by Wilson (81, whose method allows <strong>the</strong><br />
estimation of <strong>the</strong> reduction in cost of a small structure consequent upon an<br />
increase in <strong>the</strong> precision of its desis flood estimate. His approach is now<br />
applied to data from <strong>the</strong> A vm at Bath.<br />
It is assumed that <strong>the</strong> total cost C mey be written as C = $+Z2 where C1 is<br />
associated with construction costs and has <strong>the</strong> form:-<br />
failure and has <strong>the</strong> form:-<br />
S<br />
c2 = K2 p,<br />
c1 = XTm> (15)<br />
and C2 is associated with <strong>the</strong> probab<strong>le</strong> future damage resulting from structural<br />
where i is <strong>the</strong> optimum design flood with return period T, K1 and K2 are<br />
constants, and m and s are indices dependent an <strong>the</strong> particular structure.<br />
For smll structures m and 8 may be hssumed eausl.<br />
i 16)<br />
Wilson's formda depends partly on <strong>the</strong> fact that floods with r etm periods<br />
between 5 and 50 )ears may be represented by a power law of <strong>the</strong> following form:-<br />
5 = A$<br />
( 17)<br />
where A is a constant. p is an index which Wilscm suggests may be estimated<br />
><br />
as <strong>the</strong> ratio of <strong>the</strong> 50- to <strong>the</strong> S-year flood.<br />
following formula gives <strong>the</strong> reduction in cost (Ec) of a structure, given <strong>the</strong><br />
precision of <strong>the</strong> estimate of <strong>the</strong> design flood (Ex):-<br />
Ec = E E'<br />
2 x'<br />
Writing n = lb - 8,<br />
_-<br />
<strong>the</strong><br />
It should be noted that this formula applies only to small structures, with<br />
design floods of moderate return periods (ie. between 5 and 50 years).<br />
For <strong>the</strong> river Avon at Bath, p was found to be 0.2; taking FS= 0.75 88 a<br />
typical value, an increase in precision from 10% to 6% may be seen to <strong>le</strong>ad<br />
to a decrease of 1% in <strong>the</strong> cost of a structure with a 50 years design flood.<br />
Whi<strong>le</strong> not a high percentage, this would represent in absolute terms a sum of<br />
money considerably in excess of <strong>the</strong> cost of obtaining and standardizing
<strong>the</strong> data.<br />
More importantly, a similar cost-reduction, by this analysis,<br />
575<br />
wouïd require a fur<strong>the</strong>r 20 years of streamflow data from continuous records,<br />
which might be impractical and would certainly be expensive.<br />
Since <strong>the</strong> incorporation of <strong>the</strong> o<strong>the</strong>r type of nan-standard data considered,<br />
Le, chart censoring resulting in missing peaks, entails no extra cost, and<br />
bearing in mind that flood estimates are likely to be of interest in a<br />
variety of contexts (not only one as in <strong>the</strong> above examp<strong>le</strong>), it may be<br />
concluded that it is on <strong>the</strong> <strong>who<strong>le</strong></strong> worthwhi<strong>le</strong> to use additional data of <strong>the</strong><br />
types described.<br />
Acknar<strong>le</strong> dgement<br />
The author wishes to thank <strong>the</strong> following for <strong>the</strong>ir assistance:<br />
Robin T. Clarke, who initially suggested this study and provided<br />
valuab<strong>le</strong> guidance during its progress; Con Cunnane, who made availab<strong>le</strong><br />
compu+.er programs,'adaptstions of which were used in this work> and<br />
Dr Malcoiz D.Newson who collated and helped to standardize <strong>the</strong> historic data<br />
used in <strong>the</strong> eyamp<strong>le</strong>. This paper is presented by permission of <strong>the</strong> Director,<br />
Institute of Hydrology, Wallingford, Berkshire, U.K.<br />
1. Gumbel, E.J. (1960) Statistics of Extremes Columbia University Press<br />
Iiew York (1959).<br />
2. Moran, P.A.P. (1959) !he Theory of Storage. Methuen and Co. London (1970).<br />
3. Lowery, M.D. and Nash, J.E. (1970) A comparison of methods of fitting <strong>the</strong><br />
doub<strong>le</strong> exponential distribution. Journal of Hydrology IO, 259-275.<br />
h. Kimball, B.F. (1949) An approximation to <strong>the</strong> sampling variance of an<br />
estimated maximum value of given frequency based on fit of doubly exponential<br />
distribution of m&mum values. Ann. Math. Stat., 110-1 13.<br />
5. Kendall, M.G. and Stuart N. (1961) The Advanced Theory of'statistics<br />
Vol.11. Char<strong>le</strong>s Griffin and Co., Ltd, London.<br />
6. Harper, H.L. and Moore, A.H. (1968) Maximum-likelihood estimation, from<br />
doubly censored samp<strong>le</strong>s, of <strong>the</strong> parameters of <strong>the</strong> first asymptotic distribution<br />
of extreme values. her. Stat. Assoc. Jour. 63. 889-901.<br />
7. Jenkinson, A.F. ( 1969) Estimation. of Maximum Floods. Chapter five of<br />
W Technical Report 98, 193-227.<br />
8. Wilson, K. C. ( 1972) Benefit-accuracy relationship for small structure<br />
design floods. Weter Resources Research 8(2), 508-512.
ABSTRACT<br />
ASSESSMENT OF DESIGN FLOODS IN BRAZIL<br />
Paulo Poggi Pereira<br />
The techniques utilized by <strong>the</strong> Departamento Nacional de Obras de<br />
Saneamiento for computing <strong>the</strong> caracteristics of flood to be used for<br />
designing works against inundations are described. Very seldom trus-<br />
tworthy river flood discharge measurements are obtained. In most ca-<br />
ses design flood discharges are estimated with a basis on topographic<br />
data which can be ga<strong>the</strong>red quickly. Until thirty years ago <strong>the</strong> contri<br />
buting basin area was multiplied by a standard unit discharge in or-<br />
der to get <strong>the</strong> design flood discharge. Later on, <strong>the</strong> rational method<br />
was addopted, mainly for designing small canals. This system was con-<br />
siderably improved by <strong>the</strong> execution of .a statistical study of heavy<br />
rains observed in <strong>the</strong> Country. The choice of <strong>the</strong> heigth of some dikes<br />
was based on <strong>the</strong> high water <strong>le</strong>vels attained during ancient floods ob-<br />
served and still remembered by local peop<strong>le</strong>. It has been found neces-<br />
sary to perform more elaborate and time-consuming hydrological obser-<br />
vations and studies for designing dams. The use of ma<strong>the</strong>matical models<br />
is still Incipient but promising. Design floods of different standard<br />
periods of iecurrence are addopted according to <strong>the</strong> type of <strong>the</strong> work,<br />
<strong>the</strong> size of <strong>the</strong> river and <strong>the</strong> utilization given to <strong>the</strong> area to be pro<br />
tected.<br />
RES UME N<br />
Son descritas las técnicas emp<strong>le</strong>adas por el Departamento Nacional<br />
de Obras de Saneamiento en la determinación de las caracteristicas de<br />
las crecidas a ser consideradas en el proyecto de obras contra inunda<br />
ciones. Raramente se consiguen datos de mediciones fidedigna de las<br />
descargas de crecidas de los cursos de agua. En la mayoria de los ca-<br />
sos estimanse descargas de crecidas para el proyecto, con base en da-<br />
tos topográficos que pueden ser obtenidos rápidamente. Hasta treinta<br />
años atrás, el método utilizado consistia en multiplicar el área de<br />
la cuenca hidrográfica contribuyente por una descarga especifica pa-<br />
dronizada para obtener la descarga de crecida para el proyecto. De<br />
ahí en adelante, fue adoptado el método racional, principalmente para<br />
proyectar pequefios cana<strong>le</strong>s. Este sistema fue considerab<strong>le</strong>mente mejora<br />
do por la ejecución de un estudio estadístico de las lluvias intensas<br />
observadas en el país. La altura de algunos diques fue escogida en ba<br />
se de los nive<strong>le</strong>s de agua alcanzados por antiguas crecidas, cuyos ves<br />
tigios perduran todavia y son indicados por los moradores del lugar.<br />
Para el proyecto de represas ha sido necesario realizar observaciones<br />
y estudios hidrologicos más precisos y demorados. El uso de modelos<br />
matemáticos es aún incipiente, no obstante, promisor. También, adop-<br />
tanse crecidas de proyecto con diferentes periodos de recurrencia coz<br />
forme el tipo de la obra, el caudal del curso de agua y los intereses<br />
en juego de las comunidades vecinas.
578<br />
-1 . INTRODUCTION<br />
The Departamento Nacional de Obras de Saneamento - D.N.0.S.of<br />
<strong>the</strong> Brazilian Ministry of Interior, has been building flood<br />
control works for almost40 years.<br />
Such works include channel improvements, dredging and lining<br />
of canals, building of <strong>le</strong>vees, dams, conduits and tunnels.<br />
The first basic step in <strong>the</strong> design of <strong>the</strong>se works is <strong>the</strong> de-<br />
termination of <strong>the</strong> features of <strong>the</strong> floods to be contro<strong>le</strong>d or taken<br />
into account. The main methods that have been used for this purpo-<br />
se are presented in <strong>the</strong> following subtit<strong>le</strong>s. It should be noted<br />
however that not every method reported is still in use.<br />
There is a generalized lack of good reliab<strong>le</strong> hydrometric<br />
observations and measurements. As a consequence, indirect hydrologic<br />
methods have been used as a ru<strong>le</strong> with very few exceptions.<br />
2. RATIONAL METHOD<br />
The rational method is <strong>the</strong> most widely adopted for designing<br />
canals and condui te.<br />
it gives <strong>the</strong> descharge - Q - through <strong>the</strong> equation Q = CIA ,<br />
<strong>the</strong> e<strong>le</strong>merits of which are determined as follows:<br />
The area of <strong>the</strong> drainage basin - A - is obtained from maps<br />
or aerial photographs. When none is availab<strong>le</strong>, field surveys are<br />
made.<br />
The runoff coefficient - c - depends primarily on land use.<br />
As an e)ramp<strong>le</strong> <strong>the</strong> following tab<strong>le</strong> was copied from (i), a recent<br />
D.N.0.S.- O.A.S. publication:<br />
Downtown areas, densely built, with paved streets and sidewalks<br />
C = 0.70 to 0.90<br />
Neighborhood areas, <strong>le</strong>ss densely built, with paved streets<br />
and sidewalks ,C 0.70<br />
Residential areas densely built, with paved streets C -<br />
0.65<br />
Residential areas averagely inhabited C = 0.55 to 0.65<br />
Suburban residential areas, sparsely built C = 0.35 to 0.55<br />
Residential areas with gardens and unpaved streets C = 0.30<br />
Vegetated areas, parks with gardens, unpaved sport fields<br />
c = 0.20<br />
The value of <strong>the</strong> runoff coefficient for <strong>the</strong> drainage basin<br />
is obtained by adding <strong>the</strong> products of <strong>the</strong> fractions of total drainage<br />
area occupied by each land use, multiplied by <strong>the</strong> corresponding<br />
coefficient.<br />
The determination of <strong>the</strong> rain intensity - I - is made through<br />
<strong>the</strong> following steps:
57 9<br />
a) A recurrence interval is chosen, usually obeyine, <strong>the</strong><br />
following criteria (1 and 2):<br />
Rural area Urban area<br />
Small canal (no <strong>le</strong>vees) 5 years 10 years<br />
Large canal (no <strong>le</strong>vees) 10 years 25 years<br />
Small canal with <strong>le</strong>vees 25 years 50 years<br />
Large canal with <strong>le</strong>vees 50 years 100 years<br />
Small conduits for urban drainage 3 or more years<br />
b) The time of concentration is computed by adding <strong>the</strong> time<br />
n eeded by <strong>the</strong> rainwater fal<strong>le</strong>n on <strong>the</strong> remotest part of <strong>the</strong> watershed<br />
e o reach <strong>the</strong> canal or conduit, to <strong>the</strong> travel time necessary for <strong>the</strong><br />
water to flow to <strong>the</strong> point under study. The travel time is computed<br />
by dividínp <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong> canal or conduit by <strong>the</strong> averape<br />
flow velocity.<br />
c) A total depth of rainfall is determined taking into<br />
account <strong>the</strong> chosen recurrence interval and a duration of rain equal<br />
to <strong>the</strong> time of concentration. (3) is resorted to for this purpose.<br />
The ratio rainfall depthtrain duration gives rainfall intensity I.<br />
3. INTENSE RAINS IN BRAZIL<br />
In 1957 D.N.O.S. edited Otto Pfafstetter's "Intense Rains in<br />
Brazil" prepared mainly for applications of <strong>the</strong> rational method (3).<br />
This <strong>book</strong> presents <strong>the</strong> results of frequency analysis of rain<br />
fall vhlues recorded in 98 stations of <strong>the</strong> Brazilian Departamento<br />
Nacional <strong>le</strong> Meteorologia.<br />
Rainfall corresponding to several duration periods of rain<br />
(5, 15 and 30 minutes, 1, 2, 4, 8, 14, 24 and 48 hours, 1, 2, 3, 4<br />
and 6 observation days) were analysed separately for each station.<br />
Recurrence intervals of <strong>the</strong> precipitations - T - were carac-<br />
terized by <strong>the</strong> equation T = n/m, being n <strong>the</strong> total period of obser-<br />
vation and m <strong>the</strong> number of order occupied by <strong>the</strong> rainfall in a<br />
series where all observed intense precipitations were placed in de-<br />
creasing order of magnitude.<br />
This <strong>book</strong> presents diagrams, tab<strong>le</strong>s and formulas that allow<br />
<strong>the</strong> determination of design rainfall for <strong>the</strong> 98 studied stations up<br />
to 1000 years of recurrence intervals. Values pertaining to <strong>the</strong><br />
station nearest to <strong>the</strong> place for where <strong>the</strong> design is being prepared<br />
are usually utilized. For checking representativeness, rainfall<br />
frequency curves of dayly precipitations of this station are some-<br />
times compared with similar curves prepared with data from a non re<br />
cording raingage insta<strong>le</strong>d at <strong>the</strong> actual place of <strong>the</strong> contemplated<br />
works.<br />
4. STANDARD UNIT DISCHARGES<br />
According to (4) <strong>the</strong> rational method was addopted when D.N.QS.<br />
began its activities many years ago reclaiming swamps in <strong>the</strong> neiFh-
5 80<br />
bn,irhood of Rio de Janeiro.<br />
The reasons for this choice were <strong>the</strong> absence of discharge<br />
~~~.,?i~~ements, <strong>the</strong> frequent inexistence of defined streams in <strong>the</strong><br />
swam-is and because it was feared that <strong>the</strong> drainage canals to becons<br />
tructcd would change so much <strong>the</strong> hydraulic caracteristics of <strong>the</strong><br />
watersheds that <strong>the</strong> measurements would not provide a reliab<strong>le</strong> basis<br />
f o r designs .<br />
On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong>re were no recording rain gage charts<br />
fiom ~t.ich rainfall intensities for different rainfall durations<br />
and recurrence intervals could be deducted.<br />
There were only rainfall measurements performed with non re-<br />
rording rain gages for a relatively short period which showed a ma-<br />
ximum precipitation of 120mm for l day (24 hours).<br />
To get rainfall intensity, this observed depth of precipitation<br />
wds supposed to be uniformly distributed through <strong>the</strong> 24 hours<br />
of observation.<br />
So, <strong>the</strong> rain intensity addopted was always <strong>the</strong> same, regard-<br />
<strong>le</strong>ss of <strong>the</strong> time of concentration of <strong>the</strong> various basins. As a con-<br />
sequence, <strong>the</strong> discharge became directly proportional to <strong>the</strong> drain-<br />
age basin area.<br />
The runoff coeficient addopted €or rural basins was 0.7,<br />
obviously for compensating <strong>the</strong> weak rainfall intensity used. For<br />
<strong>the</strong>se<br />
3<br />
values, <strong>the</strong> rational met9od equation gives a discharge of<br />
100 m /s fLt a basin of 100 km .<br />
As a matter of fact this application of <strong>the</strong> rational method<br />
was only a means of justifying <strong>the</strong> standard unit discharge of 1 m3/<br />
1s km2 that was an addopted ru<strong>le</strong> of thumb. The behaviour of <strong>the</strong> u~<br />
lined rural canals designed accordingly has been good. Some ocasion<br />
al flooding has occured but without excessive resulting damage.<br />
Ano<strong>the</strong>r standard unit discharge is 0.5 m Is km2. It reeult-<br />
ed from a design especification asking for pumping rainwater out of<br />
polders within a few days for avoiding <strong>the</strong> breeding of mosquitoes.<br />
Here again rain intensity was not related to <strong>the</strong> concentration time<br />
of <strong>the</strong> drainage basin.<br />
5. DESIGN FLOODS FOR DAM SPILLWAYS - TRIANGULAR UNITGRAPH<br />
The design OP dam spillways is usually based on flood hydro-<br />
graphs. The triangular unitgraph presented in (5) has been used<br />
very often because It presents <strong>the</strong> advantage of doin without hydro-<br />
metric data.<br />
It is believed that peak discharges obtained by this method<br />
are exagerated but flood volumes are correct. Therefore this method<br />
is considered good for cases where <strong>the</strong> dam reservoir retains much<br />
of <strong>the</strong> flood volumes.<br />
3
The following examp<strong>le</strong> is based on recent design computations<br />
of a dam spillway for Nor<strong>the</strong>astern Brazil.<br />
a) The time of concentration was estimated by <strong>the</strong> equation<br />
of <strong>the</strong> “California Highways and Public Works” adapted for metric<br />
units :<br />
3<br />
5 0.95 x (L /<br />
TC<br />
Tc = time of concentration in hours<br />
I, = <strong>le</strong>ngth of watercourse in km measured from divide to<br />
spillway site.<br />
581<br />
H = difference in e<strong>le</strong>vation in meters between spillway site<br />
and divide.<br />
In our examp<strong>le</strong> L = 17 km, H = 400 m and<br />
Tc = 0.95 (lì3 / 4 0 0 ) ~ ’ = ~ 2.5 ~ ~ hours<br />
b) The time in hours from start to peak rate of unitgraph<br />
(T ) was computed<br />
-<br />
as follows for excess rains of 1 and 6 hours<br />
periods (D)<br />
-<br />
T D/2 + 0.6 Tc<br />
P<br />
For D = 1 hour, T 112 + 0.6 (2.5) = - 2 hours<br />
P<br />
For D = 6 hour6, T = 612 + 0.6(2.5) 4.5 hours<br />
P<br />
c) The time in hours from peak rate to end of unitgraph<br />
triang<strong>le</strong> (T,) was computed as follows:<br />
Tr =<br />
-<br />
1.667 T<br />
P<br />
- For D = 1 hour, Tr 1.667(2) = 3.3 hours<br />
For D = 6 hours, Tr 1.667(4.5) = 7.5 hours<br />
d) Peak rates of unitgraphs for 1 mm exceas rainfall of 1<br />
and 6 hours duration periods were computed as follows:<br />
A I<br />
‘p 1.8(T + Tr)<br />
P<br />
= peak rate in m 3 /e<br />
4P<br />
- A = drainage area in km2. In <strong>the</strong> examp<strong>le</strong> A = 97 km2<br />
97<br />
For D =<br />
-<br />
1 hour, 9<br />
10.2 m3~s<br />
‘p 1.8(2 + 3.3)<br />
For D = 6 hours, œ<br />
97<br />
4.5 m3/s<br />
qp 1.8(4.5 + 7.5)<br />
e) The excess rainfalls and corresponding runoff hydrographs<br />
-unitgraphs - are represented schematicaly in <strong>the</strong> annex figurestoge<strong>the</strong>r<br />
with lists of unitgraph discharges corresponding to <strong>the</strong><br />
midd<strong>le</strong> of consecutive one hour time intervals.
6. PROBABLE MAXIMUM PRECIPITATIONS<br />
The spillway of <strong>the</strong> examp<strong>le</strong> would be located uptream of a<br />
large town and <strong>the</strong> failure of its dam by flood overtoping would<br />
cause great property damage and seriously jeopardize human life in<br />
large numbers. Therefore it vas considered appropriate to utilize<br />
<strong>the</strong> maximum probab<strong>le</strong> precipitation for computing <strong>the</strong> design flood.<br />
There were not enough storm data for estimating directly<br />
<strong>the</strong> values of such precipitation. The indirect approximated method<br />
proposed in (6) was used. It is based on suppos ing that maximum<br />
probab<strong>le</strong> precipitation values are identical to those of a region of<br />
<strong>the</strong> United States where rainfalls of 10 years recurrence interval<br />
are <strong>the</strong> same as those observed in <strong>the</strong> watershed under study.<br />
(3) was used for obtaining 10 years recurrence interval prg<br />
cipitôtions from a station nearby <strong>the</strong> spillway site and (7) permil<br />
ed to locate <strong>the</strong> area in <strong>the</strong> United States with equiva<strong>le</strong>nt rain-<br />
falls and also furnished <strong>the</strong> probab<strong>le</strong> maximum 6-hour precipitation<br />
for a 10-square-mi<strong>le</strong> area: 686 mm.<br />
By using charts from (5) values of probab<strong>le</strong> maximum preci-<br />
pitations were computed for <strong>the</strong> drainage basin under study, which<br />
has an area of 97 km2 = 37.5 square-mi<strong>le</strong>, for <strong>the</strong> following listed<br />
periods of duration.<br />
duraticm period<br />
houss<br />
6<br />
12<br />
1<br />
2<br />
3<br />
4<br />
5<br />
computation<br />
88% x 686<br />
107% x 686<br />
50% x 604<br />
65% x 604<br />
76% x 604<br />
85% x 604<br />
93% x 604<br />
rainfall<br />
rnm<br />
604<br />
7 34<br />
302<br />
39 2<br />
460<br />
513<br />
562<br />
Rainfall increments disposed in descending order of intensity<br />
were calculated as follows:<br />
interval duration<br />
hours<br />
rainfall increments<br />
mm<br />
1 P1 = 302<br />
392 - 302 = 90<br />
P2<br />
460 - 392 68<br />
p3<br />
513 - 460 = 53<br />
P4<br />
P5 = 562 - 513 = 49<br />
P6 604 - 562 42<br />
P12= 734 - 604 1130
583<br />
For obtaining <strong>the</strong> design precipitation, rainfall increments<br />
were tabulated in <strong>the</strong> following order as suggested in (5): P6, Pq;<br />
P3, P1, P2, l 5 nad P (see annex tab<strong>le</strong>).<br />
12<br />
7. RUNOFF FCTIMATION AND COMPUTATION OF THE DESIGN FLOOD HYDRC<br />
GRAPH<br />
The computation of <strong>the</strong> design flood hydrograph of <strong>the</strong> exam-<br />
p<strong>le</strong> is presented in <strong>the</strong> annex tab<strong>le</strong> and was made through <strong>the</strong> follo_w<br />
ing steps:<br />
a) Rainfall increments obtained as of <strong>the</strong> preceding cub-<br />
tit<strong>le</strong> were added in order to obtain accumulative precipitation.<br />
b) Accumulative runoff or excess rainfall was estimated by<br />
means of <strong>the</strong> equation of <strong>the</strong> "Soil Conservat ion Servi ce" presented<br />
in (5):<br />
(P<br />
2<br />
- R =<br />
0.2 S)<br />
P + 0.8 s<br />
R = runoff in mm<br />
P = accumulative precipitation in mm<br />
S = maximum potential difference P - R at time of rain's<br />
begining.<br />
S was estimated as 100 mm.<br />
c) Increments of runoff were computed by subtracting <strong>the</strong><br />
accunulative runoff obtained for <strong>the</strong> preceding interval from <strong>the</strong><br />
accumulative runoff obtained for <strong>the</strong> interval under consideration.<br />
d) Increments of runoff were compared with rainfal.1 incre-<br />
ments. The difference between <strong>the</strong>m should attain at <strong>le</strong>ast lmm for<br />
each interval hour. As this did not happen at <strong>the</strong> last tabulated<br />
time interval <strong>the</strong> increment of runoff for that interval was recal-<br />
culated by subtracting 6 mm from Phe rainfall increment.<br />
e) Increments of runoff for each interval were multiplied<br />
by <strong>the</strong> unitgraph discharges listed in <strong>the</strong> annex figuresand <strong>the</strong> pro<br />
ducts were tabulated in <strong>the</strong> corresponding time intervals.<br />
f) The average discharge of <strong>the</strong> design flood in each time<br />
interval was obtained by adding <strong>the</strong> products resulting from <strong>the</strong><br />
previous step for that tf.me interval.<br />
minal.<br />
g) The base flow was not taken into account for being no-<br />
a. STATISTICAL METHODS<br />
Frequency analysis is applied whenever records that allow<br />
its use are availab<strong>le</strong>, for reasons of better precision and reliability.<br />
Gumbel's and/or Hazen's methods are <strong>the</strong> most favored.<br />
D.N.O.S. fi<strong>le</strong>s keep reports of classical hydrological studies<br />
mainly based on frequency analysis of water <strong>le</strong>vel observations<br />
and discharge measurements.
5 84<br />
One of <strong>the</strong>m is an outstandingly interesting examp<strong>le</strong>: <strong>the</strong> de<br />
termination of <strong>the</strong> heigth of <strong>le</strong>vees for protection of <strong>the</strong> city of<br />
Porto A<strong>le</strong>gre against floodings of <strong>the</strong> Guaiba River.<br />
In that reach <strong>the</strong> Guaiba River forms an estuary and its<br />
water <strong>le</strong>vels are dependent not only on <strong>the</strong> river discharges as well<br />
as on <strong>the</strong> water <strong>le</strong>vel ocurring in <strong>the</strong> lagoon where it flows to ,<br />
which can be strongly influenced by winds.<br />
There were litt<strong>le</strong> know<strong>le</strong>dge of <strong>the</strong> e<strong>le</strong>ments involved and<br />
<strong>the</strong>ir effect.<br />
On <strong>the</strong> o<strong>the</strong>r hand <strong>the</strong> Guaiba River water <strong>le</strong>vels had been sys<br />
tematicaly observed since 1899 by means of a staff gage instal<strong>le</strong>d<br />
near downtown Porto A<strong>le</strong>gre. The data so obtained was frequency an=<br />
lysed and, according to Gumbel's method <strong>the</strong> biggest recorded flood,<br />
which occured in 1941, was found to have a recurrence interval of<br />
about 370 years.<br />
Local peop<strong>le</strong> remembered which places had been flooded and<br />
which <strong>le</strong>vels had been attained by <strong>the</strong> water in different places of<br />
<strong>the</strong> town during <strong>the</strong> 1941 flood. With <strong>the</strong>se informations it was<br />
possib<strong>le</strong> to draw a water-surface profi<strong>le</strong>, which was confirmed later<br />
by a hydraulic model of <strong>the</strong> estuary.<br />
It was decided to set <strong>the</strong> crest of <strong>the</strong> <strong>le</strong>vees 1.20 m above<br />
that water-surface profi<strong>le</strong>. No discharge considerations were taken<br />
into account although discharges were estimated by making use of<br />
<strong>the</strong> above mentioned model.<br />
9. MATHEMnTICAL MODELS<br />
Up to present time almost no use has been made of ma<strong>the</strong>matic<br />
al hydrological models for determination of design flood caracteristics.<br />
Recently, <strong>the</strong> Streamflow Syn<strong>the</strong>sis and Reservoir Regulation<br />
(SSARR) Model began being used for forecasting <strong>the</strong> behaviour (flood<br />
and low water <strong>le</strong>vels as well) of <strong>the</strong> Paraguay River and some tributaries.<br />
This model was develloped by <strong>the</strong> U.S. Army Corps of Engineers<br />
which addapted it for <strong>the</strong> Paraguay River basin as part of <strong>the</strong><br />
activities of <strong>the</strong> "Project of <strong>the</strong> Hydrological Studies of <strong>the</strong> Upper<br />
Paraguay River Basin" - a UNDP/UNESCO technically assisted project<br />
for which D.N.O.S. is <strong>the</strong> responsib<strong>le</strong> Brazilian counterpart agency.<br />
The potentiality of SSARR model for evaluating <strong>the</strong> caracte-<br />
ristics of design floods of large rivers is obvious and it is ex-<br />
pected be much used for this purpose in <strong>the</strong> future.<br />
10. CONCLUSION<br />
D.N.O.S. has always used addapted foreign feekiikques for<br />
assessing design floods. On <strong>the</strong> o<strong>the</strong>r hand, local data has been<br />
used as extensively as possib<strong>le</strong>. Methods that did not allow easy
585<br />
'use of this data have not enjoyed preference. Such is <strong>the</strong> case of<br />
empirical formulas for rainfall intensity and flood discharge which<br />
were used only in a few instances.<br />
Elaborate methods have not been much addopted. The main<br />
reason for this may be <strong>the</strong> ra<strong>the</strong>r vague effect of high accuracy<br />
assessment of design flood caracteristics upon <strong>the</strong> economics of<br />
flood control works in most cases, a fact that does not encourage<br />
too many efforts for refining design flood assessment.<br />
REFERENCES<br />
a<br />
1. D.N.O.S. e Organizaqao dos Estados Americanos (1972). Relató-<br />
rio do Estudo para Contro<strong>le</strong> da Erosao no Noroeste do Estado do<br />
Parana, Rio de Janeiro, DNOS.<br />
2. Poggi Pereira, P. (1967). Contro<strong>le</strong> de cheias: custos e benefi<br />
cios, SANEAMENTO, Rio de Janeiro, DNOS.<br />
3. Pfafstetter, O. (1957). Chuvas Intensas no Brasil, Rio de Ja-<br />
neiro, DNOS.<br />
4. Arauja Goes, H. (1942). A Baixada de Sepetiba, Rio de Janeiro,<br />
DNOS.<br />
5; U.S. Department of <strong>the</strong> Interior, Bureau of Reclamation (1960).<br />
Design of Small Dama, Washington, U.S. Government Printing<br />
Office.<br />
6. Pfafstetter, O. (1967). Floods for Spillway Design, Neuvieme<br />
Congres des Grands Barrages, Comission Internationa<strong>le</strong> des<br />
Grands Barrages.<br />
7. U.S. Wea<strong>the</strong>r Bureau (1963). Rainfall Frequency Atlas of <strong>the</strong><br />
United States for Durations from 30 Minutes to 24 Hours and<br />
Return Periods from 1 to 100 Years, Technical Paper NQ 40 ,<br />
Washington, Wea<strong>the</strong>r Bureau, U.S. Department of Commerce.
586<br />
FIGURES<br />
c.ü 5 1 hour<br />
L-<br />
or runoff<br />
-- r<br />
L-<br />
5 87<br />
M<br />
--l<br />
I<br />
N<br />
N<br />
I<br />
d<br />
d<br />
-<br />
-<br />
-<br />
1<br />
O<br />
-<br />
Io<br />
ci<br />
O<br />
c<br />
G<br />
a<br />
rl<br />
ABSTRACT<br />
A METHOD FOR THE PREDICTION OF<br />
WASHLOAD IN CERTAIN SMALL WATERSHEDS<br />
by<br />
Oswald Rendon-Herrero<br />
Present know<strong>le</strong>dge on <strong>the</strong> prediction of washload reveals that with<br />
<strong>the</strong> exception of <strong>the</strong> universal soil-loss equation, and sediment-rating<br />
techniques, a rational method does not exist that can accomplish this<br />
task. A method is presented thar is analogous to Sherman's unit-hydro-<br />
graph method of hydrograph analysis. The ordinates of a sediment dis-<br />
charge graph are divided by <strong>the</strong> excess runoff that mobilized it, prod:<br />
cing a unit sediment discharge graph. When this is done for many storm<br />
events, unit sediment discharge graphs are generated that vary conside-<br />
rably in peak value and shape. The ordinates of <strong>the</strong> latter graphs are<br />
<strong>the</strong>n plotted logarithmically against <strong>the</strong>ir respective excess runoff,<br />
yielding data points that can be fitted by straight lines. Predictions<br />
pf sediment discharge or <strong>the</strong> generation of a sediment discharge graph<br />
for a given excess runoff can be accomplished using <strong>the</strong> resulting<br />
graph, Bix<strong>le</strong>r Run Watershed, Pennsylvania, having a drainage area of<br />
15 square mi<strong>le</strong>s, was se<strong>le</strong>cted as a data source. Granulometric tests<br />
and otti~r related information disclosed that <strong>the</strong> suspended sediment in<br />
Bix<strong>le</strong>r Ruri is predominantly washload. Prediction of washload utilizing<br />
<strong>the</strong> propose? method yielded errors that were considerably <strong>le</strong>ss than<br />
that reported using availab<strong>le</strong> sediment transport formulae and techni-<br />
ques.<br />
RE C U ME N<br />
Actualmente los conocimientos con respecto a la predicción de<br />
"washload" son bastante limitados. Con las excepciones de la ecuación<br />
universal de pérdida de suelo y técnicas sedimentarias (sediment-ra-<br />
ting) todavía no existe un método racional para resolver esta tarea.<br />
El procedimiento presentado es análogo al método de Sherman (Unit hy-<br />
drograph) o sea un análisis hidrográfico. Las ordenadas de la gráfica<br />
de descarga sedimentaria divididas entre el volumen del derrame excesi<br />
vo producen una gráfica unitaria de descarga sedimentaria. Al comp<strong>le</strong>-<br />
tarse este procedimiento para muchas lluvias, gráficas unitarias de<br />
descargas sedimentarias son obtenidas y se notarán las diferencias de<br />
los cambios de valores máximos. Las ordenadas de estas Últimas gráfi-<br />
cas son trazadas logaritmicamente versus sus respectivos volúmenes de<br />
derrame excesivo rindiendo diferentes puntos de dato, los cua<strong>le</strong>s pue-<br />
den unirse con líneas rectas. Predicciones de descargas sedimentarias<br />
dado cierto derrame excesivo pueden observarse en la gráfica obtenida.<br />
El área se<strong>le</strong>ccionada de 15 millas cuadradas, donde los datos fueron ad<br />
quiridos queda situada en Bix<strong>le</strong>r Run, Penns.ylyania. Pruebas granulomé-<br />
tricas y otras informaciones relacionadas indican que la aescarga de<br />
sedimentos en Bix<strong>le</strong>r Run es casi todo "washload". El método presentado<br />
rindió errores de magnitud mínima en comparación con los errores repor<br />
tados por otras técnicas y fórmulas de transporte sedimentarias.
590<br />
INTRODUCTION<br />
Relationships have been developed whereby <strong>the</strong> sediment transport of materials<br />
which are native to a channel can be computed with varying degrees of<br />
accuracy. When <strong>the</strong> sediment transport is primarily composed of <strong>the</strong> lateral inflow<br />
of particulate matter eroded from <strong>the</strong> land surface (washload) in a basin,<br />
<strong>the</strong> relationships derived are no longer velid(lg2). Heretofore, <strong>the</strong> <strong>le</strong>teral<br />
inflow component of sediment discharge was predicted via <strong>the</strong> universal soil<br />
loss equation(*), and sediment rating techniques.<br />
<strong>the</strong>se methods are subject to large errors. The universal soil loss equation<br />
has <strong>the</strong> disadvantage of providing only annual predictions, The need for quantitative<br />
evaluation of washload is of paramount importance at <strong>the</strong> present time.<br />
A method is presented which is applicab<strong>le</strong> to certain small watersheds and<br />
which can enab<strong>le</strong> <strong>the</strong> prediction of sediment discharge on a storm basis. By<br />
"small" is meant those watersheds where <strong>the</strong> spati ribution of <strong>the</strong> rainfall<br />
is uniform over <strong>the</strong> watershed area. Some authors<br />
Predicted quantities using<br />
$3 ,w<br />
define a small water-<br />
shed as being <strong>le</strong>ss than 161.0 or as much as 3219.0 square kilometers in area.<br />
"Certain" refers to <strong>the</strong> sediment discharge graph's locus (sedimentgraph) dependency<br />
on <strong>the</strong> soil type. For general stream conditions, fine-grained and<br />
colloidal materials transported in suspension will yield. a sedimentgraph that<br />
appreciably paral<strong>le</strong>ls <strong>the</strong> shape of its associated hydrograph; under similar<br />
stream conditions, coarser partic<strong>le</strong>s in transport will not result in paral<strong>le</strong>lshaped<br />
discharge graphs. The applicability of <strong>the</strong> series graph method depends<br />
on <strong>the</strong> para!<strong>le</strong>l nature of <strong>the</strong> sedimentgraph and hydrograph for a given excess<br />
runoff. Use ~f <strong>the</strong> adjective "series" is explained in <strong>the</strong> Analysis of Da<br />
section of this paper. The series graph method is analogous to Sherman's F%><br />
unit hydrograph prc,zedure for <strong>the</strong> analysis of a direct discharge hydrograph.<br />
The series graph method is demonstrated using Bix<strong>le</strong>r Run Watershed, a<br />
monitored drainage basin 38.9 square kilometers in area near Loysvil<strong>le</strong>, Penn-<br />
sylvania (Figure 1). Granulometric measurements made of <strong>the</strong> bed, bank, and<br />
suspended sediment, has established <strong>the</strong> sediment transport in Bix<strong>le</strong>r Run as<br />
being predominantly washload. Sediment sampling in <strong>the</strong> Bixier Run Watershed was<br />
begun on February 1, 1954, using a U.S.D-43, and a DH-48 depth integrating<br />
hand samp<strong>le</strong>r(7).<br />
The series graph method is used where <strong>the</strong> quantitative analysis of wash-<br />
load is necessary for <strong>the</strong> prediction of sediment discharge and/or variation<br />
with time. The prediction of total sediment discharge is required for examp<strong>le</strong><br />
where <strong>the</strong> rate of sedimentation can become prob<strong>le</strong>matic. This consideration is<br />
particularly important in <strong>the</strong> allocation of storage volumes in new reservoirs.<br />
WASHLOAD<br />
Due to a series of rainfall-induced erosive processes, particulate matter<br />
eventually reaches a stream course after being transported through a great<br />
variety of distances in a drainage basin. Depending on such characteristics<br />
as, for examp<strong>le</strong>, land slope and <strong>le</strong>ngth, topography, and availability of trans-<br />
portab<strong>le</strong> surficial soils, various-sized partic<strong>le</strong>s can, given amp<strong>le</strong> time, reach<br />
<strong>the</strong> main waterways in a basin.<br />
Depending upon <strong>the</strong> streamflow character, some of <strong>the</strong> eroded materials that<br />
reach <strong>the</strong> stream course as lateral inflow combine with sediments native to <strong>the</strong><br />
channel proper and continue to be transported downstream by <strong>the</strong> prevailing flow.<br />
The lateral inflow of Sediment is known as washload. Sediment transport in <strong>the</strong>
s t-ct'arn may be accomplished by four generally accepted modes depending primarily<br />
upon partic<strong>le</strong> diameter and stream transport capability. The transport modes are<br />
known as contact, saltation, suspended, and solution load. The saltation load<br />
in combination with <strong>the</strong> contact load is generally assumed to comprise <strong>the</strong> bed<br />
load. The sum of <strong>the</strong> suspended, bed, and solution loads is cal<strong>le</strong>d <strong>the</strong> total<br />
load. Of particular note here is <strong>the</strong> fact that <strong>the</strong>re is no sharp line of demarcation<br />
between m terials tifried as bed load ot as suspended load. Some<br />
authors (e.g., Graf ?I), Shen<br />
<strong>the</strong> washload may comprise from 90 to 95 percent of <strong>the</strong> total sediment load.<br />
The scope of this paper is limited so<strong>le</strong>ly to washload. Bed load, and<br />
suspended sediments mobilized from <strong>the</strong> bed, are not considered within <strong>the</strong> con-<br />
text of this paper.<br />
591<br />
, Chow(3)) have indicated that in many instances<br />
THEORY<br />
Of <strong>the</strong> numerous sediment transport equations that have been presented,<br />
none have been derived which account for <strong>the</strong> lateral inflow of water-soil mixtures<br />
(washload) originating from sheet and gully erosion of land surfaces in<br />
a drainage basin. Shen(*) points out, "Finally, none of <strong>the</strong> equations for predicting<br />
suspended load account for <strong>the</strong> washload of <strong>the</strong> stream." Shen(') also<br />
indicates that application of <strong>the</strong> availab<strong>le</strong> suspended sediment transport<br />
equations to stream give rise to substantial error.<br />
The existing sediment transport equations are based so<strong>le</strong>ly on <strong>the</strong> mobilization<br />
of fine particulate concentrations (sediment suspensions) and coarse<br />
layered masses of <strong>the</strong> bed, which are native to <strong>the</strong> stream channel. Of importance<br />
here Is <strong>the</strong> fact that in most instances, <strong>the</strong> quantity of washload derived<br />
from lateral inflm can be substantially greater than <strong>the</strong> suspended sediment<br />
native to <strong>the</strong> bei. Several authors (e.g., Graf (11, Shen(') y Chow(3)) estimate<br />
that <strong>the</strong> bed load contribution to <strong>the</strong> total sediment load is usually on <strong>the</strong><br />
order of five percent, and may in some cases be 'neg<strong>le</strong>cted from total load calculati<br />
ms.<br />
Given <strong>the</strong> flow condition and composition of materials native to <strong>the</strong> bed,<br />
several relationships have been developed that .provide a general relationship<br />
for <strong>the</strong> rate of sediment transport. It is not <strong>the</strong> intent of this paper to<br />
present a development of <strong>the</strong> availab<strong>le</strong> sediment transport (bed load, suspended<br />
load, or total load) equations, since <strong>the</strong>ir basis of derivation places <strong>the</strong>m<br />
outside of <strong>the</strong> realm of washload phenomena and, <strong>the</strong>refore, <strong>the</strong> scope of this<br />
study.<br />
The reader is referred to Graf (1) y Shen(') , and Nordin and McQuivey(8)<br />
for a general development and assessment of availab<strong>le</strong> sediment transport<br />
formulae.<br />
COMPILATION OF DATA: BIXLER RUN WATERSHED<br />
Storm events were chosen according to accepted hydrograph analysis criteria<br />
and which appreciably satisfied certain analogous sedimentgraph analysis con-<br />
ditions. The storm events were primarily classified according to <strong>the</strong> degree to<br />
which <strong>the</strong> locus of fhe sedimentgraphs were defined by sampling. In many instances<br />
sampling in <strong>the</strong> region of <strong>the</strong> crest of <strong>the</strong> sedimentgraph was not accomplished;<br />
<strong>the</strong> Bix<strong>le</strong>r Run project hydrologist <strong>the</strong>refore estimated <strong>the</strong> peak's shape from<br />
<strong>the</strong> relative positions of <strong>the</strong> rise and recession samp<strong>le</strong> points and from know-<br />
<strong>le</strong>dge of previous sedimentgraphs where <strong>the</strong> peak was known. The latter class-<br />
ifications yielded 63 storm events, which were grouped on <strong>the</strong> basis of runoff<br />
derived during winter (October to March) and Sumner months (April to September).
592<br />
UNIT GRAPH DEVELOPMENT (WATER AND SEDIMENT DISCHARGE)<br />
Processing of <strong>the</strong> stage hydrograph and sedimentgraph for inidividual storm<br />
s!i>ents involved as a first step <strong>the</strong> separation of base flow from <strong>the</strong> total dis-<br />
charge. In <strong>the</strong> case of <strong>the</strong> stage hydrograph, baseflow was assumed to comprise<br />
both groundwater flow and interflow. Base flow for <strong>the</strong> sedimentgraph was<br />
assumed to be <strong>the</strong> sediment flow prior to <strong>the</strong> beginning of <strong>the</strong> rise of a sedi-<br />
mrntgraph for a particular storm event. The base flow separation technique<br />
WJS identical for both <strong>the</strong> stage hydrograph and sedimentgraph (see Figure 2).<br />
Point A (or A') on Figure 2 is defined as <strong>the</strong> point where <strong>the</strong> rise of <strong>the</strong> dis-<br />
charge graph begins and is determined by inspection. Line AB (or A'B') is a<br />
tangential straight line projection, continuous with <strong>the</strong> base flm curve pre-<br />
ceding it, emanating from point A ,(or A') and bisecting a vertical line drawn<br />
through <strong>the</strong> peak. Generally, <strong>the</strong> points B and B' were appreciably in phase<br />
for most of <strong>the</strong> storm events considered in <strong>the</strong> analysis. On <strong>the</strong> average, where<br />
such was not <strong>the</strong> case <strong>the</strong> hydrograph peak lagged <strong>the</strong> sedimentgraph peak by one<br />
hour. The point C (or Cl) is determined by drawing tangents on <strong>the</strong> recession<br />
and base flow portions of <strong>the</strong> curves; <strong>the</strong> bisector of <strong>the</strong> tangents intersects<br />
at a point assumed to be at <strong>the</strong> termination of surface runoff C (or C').<br />
Although <strong>the</strong> separation technique utilized in this analysi bitrary , <strong>the</strong><br />
important feature is <strong>the</strong> consistency of its use throughout $3fay3f <strong>the</strong> data<br />
process ing.<br />
The resulting direct flow discharge graph data was <strong>the</strong>n processed by<br />
computer to derive <strong>the</strong> unit hydrograph, unit sedimentgraph, excess rainfall,<br />
and associated sediment mobilized.<br />
Hyetogrqhs were constructed for <strong>the</strong> se<strong>le</strong>cted storms in order to determine<br />
duration for thc derived unit graphs. This was donefor winter and Sumner<br />
rainfall storms ordy.<br />
GRANULOMETRIC MEASUREMENTS OF SUSPENDED,<br />
CHANNEL-BED, AND CHANNEL-BANK MATERIALS<br />
Grain-size analyses of suspended-load , channel-bed , and channel-bank<br />
materials were conducted by <strong>the</strong> USGS District Office, Surface Water Quality<br />
Branch, Harrisburg, Pennsylvania.<br />
The compi<strong>le</strong>d data serves as a basis for comparison of <strong>the</strong> materials transported<br />
during storm events and as a basis for reiative classification of <strong>the</strong><br />
prevailing transport mode (washload, bed load, etc.).<br />
Results of 115 granulometric tests conducted on <strong>the</strong> bed, bank, and suspend-<br />
ed sediment samp<strong>le</strong>s are plotted in Figure 3. Granulometric distributions obtained<br />
from <strong>the</strong> tests generally plot as three distinct bands of points, with a<br />
minor degree of overlapping. For clarity, only <strong>the</strong> arithmetic mean curves are<br />
presented on Figure 3.<br />
These were determined by sumning <strong>the</strong> percents finer than<br />
by weight at a given partic<strong>le</strong> size diameter and material source (bed, bank, or<br />
suspended), and obtaining an arithmetic average.<br />
Figure 3 corroborates verbal communication between <strong>the</strong> project hydrologists<br />
of <strong>the</strong> USGS, Harrisburg, Pennsylvania, and this worker, to <strong>the</strong> effect<br />
that materials encountered in <strong>the</strong> bed are primarily coarse-grained. In many<br />
instances, bedrock is exposed at <strong>the</strong> surface. The suspended material, <strong>the</strong>refore,<br />
can only have had as its primary source of origin <strong>the</strong> watershed's land<br />
slopes. The distinctness with which <strong>the</strong> individual mean curves plot on Figure<br />
3 is also indicative of significantannoring of <strong>the</strong> bed. Armoring is <strong>the</strong> time-
wise removal of fine particulate matter from <strong>the</strong> bed.<br />
Sed imcntgrriph Analysis<br />
ANALYSIS OF DATA<br />
The original premise proposed in this study was that <strong>the</strong> unit hydrograph - .<br />
concept as applied to a direct runoff hydrograph was directly analogous in <strong>the</strong><br />
iinalyiiis of d sedim@negraph, A fami of a tirlit gedinientgraph idas indeed develop.<br />
ed whose standard unit was 1.0 kilogram for a given duration, distributed over<br />
<strong>the</strong> watershed area, analogous in unit-hydrograph analysis to 1.0 centimeter of<br />
excess (effective) rainfall over <strong>the</strong> same area. The shape of <strong>the</strong> resulting<br />
unit sedimentgraphs varied only slightly for different rainfall events of a<br />
given duration, as is anticipated in unit-hydrograph analysis. In order to<br />
utilize such a unit sedimentgraph in generating a sedimentgraph for a particular<br />
storm event, <strong>the</strong> total amount of sediment mobilized during <strong>the</strong> event would have<br />
to be known or estimated. A relationship has been determined between total<br />
sediment mobilized and excess runoff for sing<strong>le</strong> storm events. This is shown<br />
on Figure 4, for runoff events resulting from winter and summer storms.<br />
The latter approach would, <strong>the</strong>refore, entail <strong>the</strong> estimation of total<br />
sediment mobilized on <strong>the</strong> basis of a known or predicted runoff excess as an<br />
initial step, followed by <strong>the</strong> se<strong>le</strong>ction, based on duration, of an appropriate<br />
unit sedimentgraph. The latter can <strong>the</strong>n yield a sedimentgraph by multiplying<br />
<strong>the</strong> individual unit sedimentgraph ordinates by <strong>the</strong> total sediment mobilized.<br />
A simp<strong>le</strong>r approach, however, was adopted which has <strong>the</strong> advantage that consideration<br />
of duration of runoff excess may be neg<strong>le</strong>cted altoge<strong>the</strong>r; <strong>the</strong> relationship<br />
developed, as will be shown, is independent of duration.<br />
Once <strong>the</strong> observed total discharge hydrographs and sedimentgraphs were<br />
graphically convzrted to direct discharge graphs by deducting <strong>the</strong> base flow,<br />
<strong>the</strong> following calculations were performed:<br />
DDi = DTi - DBi (1)<br />
Where DD. is direct water discharge in cubic meters per second (hereinafter<br />
designated as crns)<br />
DT is total water discharge in crns<br />
i .<br />
DB. is base water discharge in crns.<br />
The subskript "i" refers to <strong>the</strong> time at which water discharge values (e.g.,<br />
DTi) are measured on <strong>the</strong> hydrograph's abscissa. For this analysis <strong>the</strong> hydrograph's<br />
time base was divided into "n" two-hour increments.<br />
Similarly,<br />
SDi = STi - SBi ( 2)<br />
where SD, is direct sediment discharge in parts per million (hereinafter<br />
designated as ppm)<br />
ST. is total sediment discharge in ppm<br />
SB: is base sediment discharge in ppm.<br />
Thelmagnitude of <strong>the</strong> ppm units is equiva<strong>le</strong>nt to mg/A units (milligrams per<br />
liter) as long as <strong>the</strong> sediment concentration does not exceed 15,900 ppm (9).<br />
Concentrations greater than 15,900 ppm have to be multiplied by a factor (9) in<br />
order to convert ppm to mg/Q units. The units of.<strong>the</strong> direct sediment discharge<br />
(SDi) are <strong>the</strong>n converted from ppm units to kilograms per day, thusly,<br />
593
594<br />
Where Si is direct sediment discharge in kilograms per day, 86.56 is a factor<br />
for converting <strong>the</strong> sediment discharge to kilograms per day.<br />
Excess runoff and <strong>the</strong> associated sediment mobilized are determined as<br />
follows:<br />
i-1<br />
i=l<br />
Where ER is excess runoff in centimeters per square kilometer of drainage basin,<br />
A is <strong>the</strong> watershed area in square kilometers,<br />
0.1157 is a factor for converting <strong>the</strong> remaining e<strong>le</strong>ments of Equation 4 to<br />
centimeters per square kilometer,<br />
ES is sediment mobilized in kilograms per square kilometer.<br />
Individual unit sedimentgraph ordinates are determined thusly,<br />
usoi =<br />
'i<br />
Yhere USO. is <strong>the</strong> individual unit sedimentgraph ordinate in units of square<br />
kilometer per day. Multiplying USOi by kilograms per squre kilo-<br />
Eaters , yields kilograms per day.<br />
Equation 6, as was previously pointed out, cannot directly be used in <strong>the</strong><br />
fashion of a unit hydrograph ordinate. The method, <strong>the</strong>refore, requires <strong>the</strong><br />
following operation,<br />
SGO.i =<br />
'i<br />
Where SGOi is an individual "series" graph ordinate in units of kilograms per<br />
day per centimeter of excess.runoff per square kilometer. By<br />
"series" is meant that in contrast to a unit sedimentgraph ordinate<br />
which approximately superpose each o<strong>the</strong>r for a given duration, a<br />
series of graphs are obtained which vary considerably in shape and<br />
peak. For <strong>the</strong> purpose of discussion, Figure 5 will hereinafter be<br />
referred to as a series graph.<br />
The series graph lines were developed by plotting SGOi for a given excess<br />
runoff. This entai<strong>le</strong>d some judgment in <strong>the</strong> se<strong>le</strong>ction of coordinate points. The<br />
latter procedure is analogous to se<strong>le</strong>cting a mean unit hydrograph curve from a<br />
number of curves, which in practice generally do not overlap for a given<br />
duration. The judgment used was partly justified by <strong>the</strong> fact that <strong>the</strong> <strong>le</strong>ast<br />
squares line fit of <strong>the</strong> se<strong>le</strong>cted coordinate points (p, p 2 2, etc.) have a<br />
distinct tendency to plot approximately paral<strong>le</strong>l to each o<strong>the</strong>r. This is indicative<br />
of a prevailing trend.<br />
Series graphs were constructed for winter including rainfall and snowmelt,<br />
and for s mer months.<br />
coordinate points were plotted in time groups referenced to <strong>the</strong> peak discharge<br />
(p). Thus p + 2 for examp<strong>le</strong>, refers to <strong>the</strong> direct discharge ordinate two hours<br />
(7)<br />
These are shown on Figure 5. The SGO. versus "ER"
after <strong>the</strong> peak; in all, <strong>the</strong> time increments considered were p,+ 2, p + 4, and<br />
fir <strong>the</strong> summer events only, p + 6. Generally p + 6 represents a negligib<strong>le</strong><br />
discharge quantity, very frequently zero, and was <strong>the</strong>refore assumed to be<br />
zero for winter rainfall and snowmelt events.<br />
This writer is of <strong>the</strong> opinion that for this particular analysis a great<br />
part of <strong>the</strong> data scatter on <strong>the</strong> series graph and Figure 4, can be explained by<br />
<strong>the</strong> mnnncr in which <strong>the</strong> sedimentgraphs were defined by sampling. The data<br />
points are not shown since some overlapping exists as refers to p 5 n lines.<br />
The sedimentgraphs se<strong>le</strong>cted for analysis did not have continuously defined loci.<br />
As a result, graphical interpolation and judgment by <strong>the</strong> USGS, based on experience<br />
and know<strong>le</strong>dge of sediment behavior, were incorporated in drawing <strong>the</strong> sedimentgraphs<br />
between measured points. The observed sediment concentration points<br />
were used as guides. It may be possib<strong>le</strong>, <strong>the</strong>refore, to considerably reduce <strong>the</strong><br />
scatter of points by adequately defining sedimentgraph loci for a given storm<br />
event by more frequent sampling. Most of <strong>the</strong> sedimentgraphs considered herein<br />
generally had from four to six, and at times as many as 10 observed samp<strong>le</strong><br />
points defining th graphs; in many of <strong>the</strong> cases <strong>the</strong> USGS estimated <strong>the</strong> magnitude<br />
and location of <strong>the</strong> peak in its entirety. Consideration of scatter, at<br />
<strong>le</strong>ast in this study, would suggest, that an attempt at explaining <strong>the</strong> variation<br />
due to watershed soil types, vegetative cover, slope, etc., would be meaning<strong>le</strong>ss.<br />
This worker would, however, opine that <strong>the</strong> loci of well-defined<br />
sedimentgraphs would <strong>le</strong>ad to <strong>the</strong> development of series graphs prossessing<br />
<strong>le</strong>ss scatter.<br />
Exsmp<strong>le</strong>s of sedimentgraphs predicted on <strong>the</strong> basis of season and runoff<br />
excess art. shown on Figure 6. Tab<strong>le</strong> I lists comparisons between predicted and<br />
actual eroded sediment quantities in Bix<strong>le</strong>r Run as shown in Figure 6. To<br />
illustrate <strong>the</strong> ;ange of applicability of <strong>the</strong> series graph method to Bixier Run,<br />
variations in excess runoff for snowmelt or rainfall are included in Tab<strong>le</strong> I.<br />
For <strong>the</strong> four storm events considered in Tab<strong>le</strong> I, <strong>the</strong> average error of estimate<br />
for washload ranges from 16.1 to 16.5 percent as determined by <strong>the</strong> series graph<br />
method and <strong>the</strong> ES versus ER graphical relationships, respectively. This is<br />
based on comparisons with ac'tual conditions observed in <strong>the</strong> field. The errors<br />
of estimate computed are all considerably below that reported for similar suspended<br />
sediment load predictions, which may in some cases by greater than 100<br />
percent.<br />
TABLE I<br />
Comparison of Predicted versus Computed Sedimentgraphs<br />
595<br />
Sediment Mobilized Percent Error<br />
Date Excess<br />
Runoff Source of<br />
(tons/sq. km.)<br />
Actual Predicted<br />
Total Sediment<br />
Bases (%)<br />
centimetersf sq.<br />
h.<br />
Runoff BY BY BY BY<br />
Series ES vs. Series ES vs.<br />
Graph ER Graph ER<br />
Method Curves Method Curves<br />
10/19/68 . O35 Win ter-Ra inf a 11 19.6 27.9 27.7 29.6 29.1<br />
031 10/67 .343 Winter-Snowmelt 1523.0 1293.0 1330.0 17.8 12.6<br />
10/04/62 .572 Winter-Rainfall 2710.0 2694.0 2555.0 0.50 5.7<br />
05f07f 56 .O55 Summer-Rainfall 73.8 61.7 91.0 16.5 18.7
596<br />
CONCLUS IONS<br />
This study discloses two important findings for Bix<strong>le</strong>r Run Watershed.<br />
597
598<br />
1.50<br />
1.25<br />
I 1.00<br />
1.75<br />
; peak<br />
\<br />
HYDROGRAPH<br />
?noon 6pm i2pm Gain 12nocn 6pm l2gin 6am 12noon<br />
I_ -I.-.--I-<br />
TIME IN HOURS<br />
FIGURE 2 : TYPICAL STAGE HYDROGRAPH AJW SEDI?KNTGRAPW<br />
(s-roiwi OF MARCH 13, 1963, BIXLER BUN WATERSHED )<br />
5
599
600<br />
3502<br />
a<br />
w<br />
I-<br />
w<br />
I<br />
O<br />
4<br />
Y<br />
W<br />
a<br />
4<br />
350.<br />
u)<br />
\<br />
cn<br />
z<br />
a<br />
a:<br />
W<br />
O<br />
1<br />
y.<br />
- z<br />
.<br />
cn<br />
hl<br />
e<br />
c3<br />
W<br />
N<br />
4 35.0<br />
m<br />
O<br />
z<br />
I-<br />
z<br />
Id<br />
2<br />
I<br />
a<br />
Ill<br />
o)<br />
3.5<br />
(<br />
BIXLER RUN WATERSHED<br />
w<br />
;o098 0.0098 0.098<br />
EXCESS RUNOFF ( ER ) , IN CENTIRIETERS / SQUARE I(Il.Ol\r;ETER<br />
FIGURE 4: SEGIMENT Mû01LjZEü \E:;) VERSUS EXCESS RIINOFF (ER)
I I I I 1 1 1 1 1 I I I I I I I I I<br />
- EIXLER RUN WATERSHED<br />
- <strong>le</strong>gend:<br />
- p peak value.<br />
II I,<br />
p+n n hours before (-1 or after (el peak.<br />
- --summer roinf al I.<br />
- ______winter rainfall. - - winter snowmelt.<br />
0.72C I I I I 1 1 1 1 1 1 I I I 1 1 1 1 1<br />
I<br />
0:00098 0.0098 0.098<br />
6 O1
602<br />
ò A v a p-6 p-4 P-2 P P+2 pt4 pt6<br />
TIME IN HOURS<br />
FIGURE 6 : PREDICTED GRAPH OF SEDIMENT DISCHARGE VERSUS<br />
TI ME
ABSTRACT<br />
METHODES UTILISEES pour 1'EVALUATION des DEBITS de CRUE<br />
des PETITS COURS d'EAU en REGIONS TROPICALES<br />
par J. A. RODIER<br />
For most of <strong>the</strong> tropical small streams studied by <strong>the</strong> author,<br />
<strong>the</strong> floods result from surface runoff, field of application of unit<br />
graphs. Hydrometric networks are use<strong>le</strong>ss for <strong>the</strong> floods of <strong>the</strong>se basins<br />
(<strong>le</strong>ss than 500 km2). Two methods are described:<br />
For area without cyclonic precipitations: <strong>the</strong> depth of <strong>the</strong><br />
storm of <strong>the</strong> frequency choosed for <strong>the</strong> project is computed assuming <strong>the</strong><br />
o<strong>the</strong>r characteristics equal to <strong>the</strong> more frequent values for <strong>the</strong> big<br />
storms. The transformation of rainfall into discharge is made in two<br />
steps: computation of runoff coefficient and flood volume, computation<br />
of characteristics of <strong>the</strong> hydrograph. This incorrect method gives good<br />
results if used with judgement. Empirical graphs and ru<strong>le</strong>s have been<br />
deduced from systematical researches on representative basins for comp~<br />
tation of <strong>the</strong> e<strong>le</strong>ments of <strong>the</strong> flood from physiographical data. A gene-<br />
ral syn<strong>the</strong>sis will permit a better characterization of <strong>the</strong> basins.<br />
In area with cyclones: <strong>the</strong> precipitation depth are estimated<br />
from <strong>the</strong> observations and high values of runoff coefficient are choosed<br />
in rel&tion with observations or envelope curves are drawn from obser-<br />
ved data 5.n <strong>the</strong> world.<br />
RESUME<br />
Pour la plupart des petits cours d'eau tropicaux étudiés par<br />
l'auteur, <strong>le</strong>s crues résultent du ruissel<strong>le</strong>ment superficiel, domaine<br />
d'application de l'hydrogramme unitaire.<br />
Les réseaux hydrométri ues sont sans utilité pour <strong>le</strong>s crues<br />
de ces bassins (moins de 500 km 9 ), Deux méthodes sont décrites:<br />
Pour <strong>le</strong>s régions non affectées par <strong>le</strong>s cyclones: on détermine<br />
l'averse de fréquence éga<strong>le</strong> à cel<strong>le</strong> de la crue du projet, <strong>le</strong>s autres<br />
caractéristiques étant <strong>le</strong>s plus fréquentes pour <strong>le</strong>s tres fortes aver-<br />
ses. La transformation en débit est faite en deux temps: calcul du coe<br />
fficient de ruissel<strong>le</strong>ment et du volume de crue, calcul des caractéris-<br />
tiques de l'hydrogramme. Cette méthode non rigoureuse fournit de bons<br />
résultats si el<strong>le</strong> est employée avec discernement. Des diagrammes ou<br />
des règ<strong>le</strong>s empiriques sont déduits de recherches systématiqyes sur bac<br />
sins représentatifs, pour calcu<strong>le</strong>r <strong>le</strong>s éléments de la crue a partir<br />
des données physiographiques. Une synthèse généra<strong>le</strong> permettra de mieux<br />
caractériser <strong>le</strong>s bassins.<br />
Dans <strong>le</strong>s régions de cyclones: on détermine <strong>le</strong>s averses d'après<br />
<strong>le</strong>s va<strong>le</strong>urs observées et on suppose des coefficients de ruissel<strong>le</strong>ment<br />
tres é<strong>le</strong>vés en rapport avec ces observations, ou on établit directement<br />
<strong>le</strong>s courbes enveloppes a partir des données observées dans <strong>le</strong> monde.<br />
Chef du Service Hydrologique de l'office de la Recherche Scientifique<br />
et Technique Outre-Mer<br />
Conseil<strong>le</strong>r Scientifique à E<strong>le</strong>ctricité de France (DAFECOI.
604<br />
L'étude des ouvrages utilisant <strong>le</strong>s eaux des petites rivières<br />
tropica<strong>le</strong>s ou méditerranéennes présente de très sérieuses difficultés<br />
dès que loon aborde la dbtermination des conditions hydrologiques de<br />
réalisation et d'exploitation des ouvrages, en particulier cel<strong>le</strong> des<br />
d6bits moyens annuels et surtout cel<strong>le</strong> des ddbits de crues.<br />
Les donnhes sur <strong>le</strong> régime hydrologique sont dans ce cas<br />
inexistantes : la densit& des réseaux hydronktriques est faib<strong>le</strong>, <strong>le</strong><br />
nombre de stations amdnagées est nul ou dérisoire. LEh outre, <strong>le</strong>s varia-<br />
tions temporel<strong>le</strong>s des débits sont si rapides que <strong>le</strong>s donn&es de ces sta-<br />
tions sont souvent diffici<strong>le</strong>s à exploiter. Enfin, contrairement & ce qui<br />
a lieu pour <strong>le</strong>s grandes rivières, <strong>le</strong>s crues de faib<strong>le</strong> frequente ne lais-<br />
sent aucun souvenir dans la mémoire des habitants <strong>le</strong>s plus proches.<br />
I1 est très sou-ntinipsib<strong>le</strong> de procéder 2 une étude hydrologi-<br />
que sérieuse sur <strong>le</strong> terrain pour un seul ouvrage car el<strong>le</strong> durerait long-<br />
temps et son prix atteindrait ou dépasserait même celui de l'ouvrage lui-<br />
même<br />
Tout ce que l'on peut faire c'est organiser une tel<strong>le</strong> étude à<br />
l'occasion de la r8alisation d'une série importante de tels ouvrages,<br />
par exempie pour la construction de tous <strong>le</strong>s ponts d'une longue voie<br />
ferrée (chemin de €er transcamerounais), ou d'un grand axe routier, ou<br />
lorsqu'o, ariiinage i la fois 30 ou 50 petits barrages comme cela a 6t6 <strong>le</strong><br />
cas en I!AUî.Q-VOLTA il y a quelques années.<br />
Autrement, on est conduit & utiliser <strong>le</strong>s résultats de synthèses<br />
& caractère g6ogr;pliique.<br />
Nos hydrologues ont souvent rencontré ce problème en Afrique<br />
Tropica<strong>le</strong>, en Am6rique du Sud, dans <strong>le</strong>s f<strong>le</strong>s du Pacifique et de l'Océan<br />
Indien et ils ont mis au point différentes rnbthodes pour la détermination<br />
des débits moyens annuels et des dLbits de crue, premiers 6léments que <strong>le</strong>:<br />
ingénieurs demandent aux hydrologues.<br />
Dans ce qui suit, nous ne traiterons que <strong>le</strong> prob<strong>le</strong>me de la dé-<br />
termination des d&its de crue dans <strong>le</strong> cas de cours d'eau dont <strong>le</strong> bassin<br />
versant couvre une superficie inférieure 200 km2 et plus souvent infé-<br />
rieure à 50 km2. Au-delà de ces surfaces, <strong>le</strong>s m6thodes ne sont plus <strong>le</strong>s<br />
memes. Ellos correspondent souvent en effet la limite d'emploi de l'hy-<br />
drogramme unitaire et des modè<strong>le</strong>s globaux.<br />
Pour ces petits bassins, on considérera deux cas différents sui-<br />
vant la genbse des crues exceptionnel<strong>le</strong>s. Dans <strong>le</strong> premier cas, el<strong>le</strong>s sont<br />
dues a des orages convectifs avec prhcipitations intenses mais d'assez<br />
courte durbej dans <strong>le</strong> second cas, il s'agit de précipitations cycloni-<br />
ques ?ì iiitensité plus faib<strong>le</strong> mais de plus longue durde, <strong>le</strong>s derniers 616-<br />
ments de l'hpisode pluvieux arrivant sur un sol pratiquement saturé.
1. Cas de crues provoquées par des orages convectifs,<br />
605<br />
La mise au point de méthodes pratiques nous a demandé quinze<br />
ans de recherches fondamenta<strong>le</strong>s. Nous passerons rapidement sur ces<br />
recherches pour insister plus particulièrement sur la m6thodologie<br />
proposGe aux ing6nieurs, cette méthodologie n'étant guère applicab<strong>le</strong><br />
que pour des p6riodes de retour de 10 ou 20 ans. L'idée de base est<br />
IQutilisatian de l'information pluviomhtrique existante et plus parti-<br />
culièrement des sc'ries chronologiques de précipitations journalières<br />
et la transformation des hauteurs de prGcipitations en débits de ruis-<br />
sel<strong>le</strong>melit superficiel. Pour des averses de ce type et d'assez faib<strong>le</strong><br />
frcquerice, en réginns tropica<strong>le</strong>s et méditerranéennes, il se produit<br />
g(!nbra<strong>le</strong>rnent du ruissel<strong>le</strong>ment superficiel, ce qui permet l'emploi de<br />
la m6thode de l'hydrogramme unitaire.<br />
1.1. Recherches fondamenta<strong>le</strong>s entreprises.<br />
Les plus importantes ont ét6 <strong>le</strong>s suivantes :<br />
1.l.l.Etudes g&n&ra<strong>le</strong>s statistiques des pluies journalières. En<br />
Afrique Occidenta<strong>le</strong> OU el<strong>le</strong>s ont été <strong>le</strong> plus poussées, el<strong>le</strong>s ont port6<br />
sur 1 O00 stations environ. L'étude simultanée pour un grand nombre de<br />
statioils a conduit a des va<strong>le</strong>urs assez sûres des paramètres des lois de<br />
distribuLion pour des p6riodes de retour de 10 à 20 ans. Eh particulier,<br />
el<strong>le</strong> u cona.iit ?i abandonner, pour cette région du monde, la distribution<br />
de GALTOW tro2 pessimiste,pour une distribution de PEARSON III. Sur <strong>le</strong><br />
plan pratique, on en a déduit une série acceptab<strong>le</strong> de précipitations<br />
journali&res de période de retour 10 ans ou 20 ans.<br />
1.1.2.Etude de l'abattement (Inverse du rapport pour une méme frdquen-<br />
ce, eiitre la hauteur de précipitations en un point et la hauteur de prcci-<br />
pitations sur une surface donnée entourant ce point). Les études menées à<br />
partir de données recueillies sur bassins représentatifs ont conduit a<br />
des ordres de grandeur acceptab<strong>le</strong>s pour la pratique.<br />
1.1.3.Etudes des courbes intensité-durée : ces études faites surtout<br />
à partir des pluviographes des bassins représentatifs ont permis, pour<br />
l'Afrique Occidenta<strong>le</strong>, de donner des courbes-types.<br />
1.1.4.Etude des relations pluies-débits : cel<strong>le</strong>s-ci ont été etudibes<br />
averse par averse pendant plusieurs années sur une centaine de bassins<br />
reprGsentatifs, qui ont éga<strong>le</strong>ment 6té utilisés pour <strong>le</strong>s recherches vi-<br />
sées aux points 1.1.2 et 1.1.3. La méthode des r6sidus a permis de dé-<br />
teniiiner dans chaque cas la hauteur d'eau bcoulée HR ou <strong>le</strong> rapport KR<br />
entre HH et la hauteur de précipitation P eii fonction de P, des condi-<br />
tions d'humidité du sol avant l'averse et de la durée de l'averse.<br />
1.1.5.Etude de la forme des hydrogrammes. Sur <strong>le</strong>s mgmes bassins re-<br />
prdseiitatifs, on a pu appliquer la mgthode des liydrograrimes unitaires<br />
et dbterminer la forme des hydrogranines-types. On en a retenu trois
606<br />
éléments caractéristiques : <strong>le</strong> temps de montée tm , la durée de ruis-<br />
sel<strong>le</strong>ment tg et <strong>le</strong> rapport k entre <strong>le</strong> débit de pointe de l'hydrogramme<br />
unitaire et <strong>le</strong> débit moyen pendant la durée du ruissemment.<br />
1.2. bisthode de détermination des débits de pointes de crue et de <strong>le</strong>ur<br />
volume.<br />
Le cas des fréquences décenna<strong>le</strong>s ou de fréquences voisines<br />
est assez différent de celui de la crue maxima<strong>le</strong> probab<strong>le</strong>. Dans ce qui<br />
suit nous traiterons <strong>le</strong> cas des crues de périodes de retour de 10 ans<br />
ou 20 ans.<br />
De façon généra<strong>le</strong>, on a cherché à mettre au point des méthodes<br />
simp<strong>le</strong>s qui puissent &tre utilisées sans ordinateur. Ces méthodes sont<br />
probab<strong>le</strong>ment très différentes de cel<strong>le</strong>s qui sont élaborées actuel<strong>le</strong>ment<br />
et qui seront vulgarisées dans quelques années, mais de nombreux pays en<br />
voie de développement ne disposent pas, à l'heure présente, de moyens de<br />
calculs suffisants et n'ont pas assez de personnel bien entraîné pour <strong>le</strong>s<br />
utiliser pour des fins hydrologiques.<br />
C'est pourquoi, dans ce qui suit, on adoptera <strong>le</strong>s principes sui-<br />
vants, dont certains sont discutab<strong>le</strong>s, mais qui permettent aux ingénieurs<br />
d'arriver à des résultats utilisab<strong>le</strong>s avec <strong>le</strong>s moyens dont ils disposent.<br />
1.2.1.Principes du calcul : Le point de départ est la série d'obser-<br />
vations de précipitations journalières au poste <strong>le</strong> plus proche de l'ou-<br />
vrage que l'on a & étudier ou un poste pluviométrique correspondant aux<br />
mdmes conditions pluviométriques si la qualit6 des données du pluviomè-<br />
tre <strong>le</strong> plus proche est insuffisante.<br />
Des bassins de 50 km2 sont généra<strong>le</strong>ment assez homogènes, mais,<br />
dans <strong>le</strong> cas de forte différence d'altitude, <strong>le</strong> poste pluviométrique choi-<br />
si devra se trouver à peu près à l'altitude moyenne du bassin et non pas<br />
au niveau de l'esutoire, ce qui rend <strong>le</strong> choix beaucoup plus diffici<strong>le</strong>.<br />
On étudie la distribution statistique des précipitations journalières ce<br />
qui, en région tropica<strong>le</strong>, correspond à peu près à la distribution des<br />
averses orageuses et on détermine l'averse correspondant ?I la fréquence<br />
de la crue (période de retour 10 ans, 15 ans, 20 ans, etc...). On recher-<br />
che, en Ltudiant <strong>le</strong>e 'enregistrements disponib<strong>le</strong>s, quel est <strong>le</strong> schéma lo<br />
plus courant des répartitions des intensités pour une averse donnée, on<br />
examine éga<strong>le</strong>ment quel<strong>le</strong>s sont <strong>le</strong>s conditions moyennes d'humidité préa-<br />
lab<strong>le</strong>s que rencontrent généra<strong>le</strong>ment <strong>le</strong>s fortes crues. Enfin, on trans-<br />
forme la hauteur de précipitation en un point par la hauteur de préci-<br />
pitation moyenne sur une surface en la multipliant par un coefficient<br />
d'abattement inférieur à 1.<br />
Au moyen du mode<strong>le</strong> de transformation des pluies en débits, on<br />
transforme la pluie décenna<strong>le</strong> en crue décenna<strong>le</strong> en veillant bien à ce<br />
que la distribution des intensités de l'averse, l'index représentant<br />
l'humidité préalab<strong>le</strong>, <strong>le</strong> mois de l'année lorsque celui-ci intervient,<br />
correspondent aux conditions <strong>le</strong>s plus fréquentes pour <strong>le</strong>s fortes préci-<br />
pitations. Sur <strong>le</strong> plan statistique, ceci est très contestab<strong>le</strong> : la
607<br />
v8ritab<strong>le</strong> solution consisterait 5 appliquer <strong>le</strong> modè<strong>le</strong> de transformation<br />
pluie/débit à la totalité des averses observées au poste de référence,<br />
sur 40 ans par exemp<strong>le</strong>, et à étudier la distribution statistique de<br />
l'échantillon de crues reconstituées sur 40 ans. Mais cette méthode<br />
serait peu réaliste pour beaucoup de pays en voie de développement<br />
parce qu'il est beaucoup plus diffici<strong>le</strong> de mettre au point un modè<strong>le</strong><br />
valab<strong>le</strong> pour toutes <strong>le</strong>s averses qu'un modè<strong>le</strong> uniquement valab<strong>le</strong> pour<br />
<strong>le</strong>s fortes averses et parce qu'ensuite la reconstitution des crues de<br />
petits bassins pour 40 ans ne peut se faire qu'avec l'ordinateur.<br />
La transformation pluie/débit se fait en deux temps :<br />
lo - calcul du volume de crue par la détermination du facteur KR (voir<br />
1 .I .4.) ;<br />
2* - à partir de ce volume, détermination du débit de pointe par la forme<br />
de l'hydrogramme (voir 1 .1 .5.).<br />
Autant que possib<strong>le</strong>, on a cherché à ramener ce calcul à des<br />
opérations très simp<strong>le</strong>s dans un certain nombre de pays où <strong>le</strong> nombre de<br />
bassins représentatifs était suffisant.<br />
1.2.2.Pratique du calcul pour l'Afrique Occidenta<strong>le</strong> :<br />
Le5 études systématiques visées en 1.1.1. et 1.1.3. fournis-<br />
sent des éléments pluviométriques permettant de déterminer l'averse de<br />
fréquence cherchée avec son diagramme de distribution temporel<strong>le</strong>, pour<br />
la majeure partie de l'Afrique Occidenta<strong>le</strong>. Des études beaucoup plus<br />
partiel<strong>le</strong>s effectuées dans d'autres r6gions du monde ont fourni <strong>le</strong>s me-<br />
mes données.<br />
On réduit ces va<strong>le</strong>urs ponctuel<strong>le</strong>s à des va<strong>le</strong>urs moyennes sur<br />
une surface donnée en <strong>le</strong>s multipliant par un facteur qui décroît de 1<br />
pour une svface S inférieure à 25 km2, à 0,8 pour une surface comprise<br />
entre 150 et 200 km2. Ces chiffres qui ne sont valab<strong>le</strong>s que pour <strong>le</strong>s<br />
orages convectifs des régions tropica<strong>le</strong>s africaines, sont peut-$tre un<br />
peu forts. Ils seront probab<strong>le</strong>ment diminués à la suite de recherches en<br />
cours.<br />
On dispose donc de la hauteur de précipitation P,.<br />
Pour déterminer la va<strong>le</strong>ur de Ks, on a établi des séries d'aba-<br />
ques pour deux types de couvertures végeta<strong>le</strong>s naturel<strong>le</strong>s (liées au cli-<br />
mat), savane et savane boisée d'une part, steppe et savane à épineux<br />
d'autre part. I1 n'a pas encore été possib<strong>le</strong> d'établir d'abaques conve-<br />
nab<strong>le</strong>s pour la forat tropica<strong>le</strong>.
608<br />
Les autres facteurs pris en considération pour la détermina-<br />
tion de KH sont : la superficie du bassin, la perméabilité globa<strong>le</strong><br />
du sol P et la pente R. A défaut d'index quantitatif pour R et surtout<br />
P, on a établi deux classifications : R correspond à des plaines très<br />
plates, RG à des pentes de montagne (pentes longitudina<strong>le</strong>s supbrieures<br />
à 5 $, pentes transversa<strong>le</strong>s supérieures à 20 $)o<br />
Pl correspond à un sol rigoureusement imperméab<strong>le</strong>, P5 à un<br />
sol très perméab<strong>le</strong> (sab<strong>le</strong> ou carapace latéritique très disloquée. Le<br />
graphique 1 d m e un exemp<strong>le</strong> de ces abaques pour des sols imperméab<strong>le</strong>s<br />
(Pl - P2) et des pentes variab<strong>le</strong>s de R2 à R4. Ces abaques ont été &ta-<br />
blies & partir des données des bassins représentatifs. Les va<strong>le</strong>urs de<br />
KR correspondent des pluies de fréquence décenna<strong>le</strong> (Pm vaciant de<br />
&o à IO5 mm) dans ces régions, tombant dans des conditions d'humidité<br />
du milieu de la saison des pluies.<br />
Bien entendu, au cas OU des facteurs secondaires tels que <strong>le</strong><br />
réseau hydrographique, présenteraient des caractéristiques anorma<strong>le</strong>s,<br />
par exemp<strong>le</strong> lit marécageux, on devrait rectifier <strong>le</strong>s va<strong>le</strong>urs de KR en<br />
conséquence.<br />
Le volume de ruissel<strong>le</strong>ment de la crue :<br />
A ce volume il convient d'ajouter <strong>le</strong> volume correspondant au<br />
d6bit de base do1.t on peut avoir une idée sur <strong>le</strong> terrain, sans &tude<br />
hydroiagique très diffici<strong>le</strong>.<br />
Pour la forme de l'hydrogramme,des abaques ont été éga<strong>le</strong>ment<br />
mis au point;. On en trouvera un exemp<strong>le</strong> au graphique 2 qui donne <strong>le</strong><br />
temps de base ou duróe du ruissel<strong>le</strong>ment en fonction de la surface du bas<br />
sin et de l'index de pente pour <strong>le</strong>s m8mes conditions de végétation que<br />
<strong>le</strong> graphique no 1.<br />
La connaissance du temps de base TB permet de calcu<strong>le</strong>r <strong>le</strong> débit<br />
moyen de ruissel<strong>le</strong>ment :<br />
M<br />
'ruis<br />
e<br />
s eli emen t<br />
TB<br />
M est obtenu en m 3 /s.<br />
K (K P F)<br />
Pour trouver <strong>le</strong> débit de pointe s, on utilise un coefficient<br />
étudibi pour <strong>le</strong>s mêmes rbgions sur bassins représentatifs.
609<br />
Pour la couverture végtitaïe steppe ou savane à épineux avec<br />
des va<strong>le</strong>urs de KR pas trop &<strong>le</strong>vées, on trouve des va<strong>le</strong>urs de K variant<br />
entre 2,5 pour 25 km2 à 3,t pour 100 kmz. Si ces va<strong>le</strong>urs de KR sont<br />
supérieures à 50 - 60 $ K varie entre 3 pour 2 km2 et 4,5 pour 50 km2.<br />
de base.<br />
On obtient QM en multipliant $1 par K et on ajoute <strong>le</strong> débit<br />
Bien entendu, si <strong>le</strong> diagramme de répartition temporel<strong>le</strong> des<br />
intensités et si la superficie du bassin sont tels que la crue n'est pas<br />
unitaire, il existe des abaques complémentaires donnant <strong>le</strong> temps de base.<br />
Dans ce qui précède, c'est volontairement que nous n'avons pas<br />
Gtabli de formu<strong>le</strong>s pour repr6senter <strong>le</strong>s courbes des graphiques 1 et 2<br />
auxquel<strong>le</strong>s nous voulons garder un caractère provisoire.<br />
l.Z.3.Limitations de la méthode :<br />
El<strong>le</strong> ne s'applique bien en Afrique tropica<strong>le</strong> que pour des su-<br />
perficies inférieures & 50 - 100 km2.<br />
Comme nous venons de <strong>le</strong> dire, nos courbes sont provisoires et<br />
on met ali point des modè<strong>le</strong>s plus &labor& pour revoir <strong>le</strong>s bases de nos<br />
abaques quL nécessitent encore un sérieux effort d'homogénéisation des<br />
données et des proc6dés de calculs.<br />
Les problèmes de forêt tropica<strong>le</strong> exigent encore un effort im-<br />
portant de recherches sur <strong>le</strong> terrairi.<br />
%fin, il n'est pas très faci<strong>le</strong> de classifier un bassin en caté-<br />
gorie P2 ou P . Des recherches de physique du sol sont en cours pour arri-<br />
ver ?i des règ3es simp<strong>le</strong>s permettant de <strong>le</strong> faire. C'est certainement 1&<br />
<strong>le</strong> point <strong>le</strong> plus diffici<strong>le</strong>.<br />
Pour définir quantitativement des index R une bonne combinai-<br />
son des facteurs géomorphologiques courants doit donner satisfaction.<br />
Actuel<strong>le</strong>ment, cette méthode est très souvent employée en Afrique,<br />
mais, dans bien des cas delicats, il serait plus prudent que <strong>le</strong>s bassins<br />
soient examinés auparavant par un hydrologue confirmé. El<strong>le</strong> présente l'im-<br />
mense avantage d'éviter toute véritab<strong>le</strong> étude hydrologique sur <strong>le</strong> terrain.<br />
1.3. Crue de période de retour supérieure à 20 ans.<br />
C'est là un problème très diffici<strong>le</strong> car la documentation plu-<br />
viométrique est tout à fait insuffisante. Pour des périodes de retour<br />
de l'ordre de 100 ans, une minutieuse étude critique des re<strong>le</strong>vés de nom-<br />
breux postes pluviométriques permet d'aboutir à un ordre de grandeur.
61 O<br />
En Afrique tropica<strong>le</strong>, <strong>le</strong>s averses journalières centenaires<br />
de caractère convectif sont peut-8tre de l'ordre de 200 ?i 3-400 mm en<br />
24 heures, suivant <strong>le</strong>s régions.<br />
I1 reste ensuite à choisir une va<strong>le</strong>ur de KR qui n'est plus<br />
cel<strong>le</strong> des abaques mais qui doit en tenir compte, car tous <strong>le</strong>s bassins<br />
pour de tel<strong>le</strong>s averses ne parviennent pas à la limite de O,&5 - O,9O.<br />
Enfin, généra<strong>le</strong>ment, l'averse dure au moins 5 ou 6 heures et parfois<br />
20 heures, el<strong>le</strong> n'est donc plus unitaire. On utilise donc <strong>le</strong>s abaques<br />
tels que ceux du graphique 2 pour établir <strong>le</strong>s différents hydrogrammes<br />
é<strong>le</strong>hentaires qu'on ajoute après avoir découpé l'averse centenaire.<br />
Enfin, s'il s'agit de la crue maxima<strong>le</strong> probab<strong>le</strong>, il ne reste<br />
plus qu'8 appliquer la formu<strong>le</strong> de FIERSHFIELD OU l'on ajoute à la va<strong>le</strong>ur<br />
moyenne de la précipitation journalière maxima<strong>le</strong> annuel<strong>le</strong> 15 fois 1°é-<br />
cart-type de la distribution de cette précipitation maxima<strong>le</strong>. Mais il<br />
faut d'abord partir d'une série de précipitations journalières de quali-<br />
té suffisante pour en déduire une va<strong>le</strong>ur correcte de l'écart-type.<br />
D'autre part, si cette formu<strong>le</strong> paraft excel<strong>le</strong>nte pour l'Afrique du Nord,<br />
<strong>le</strong>s régions soumises à des cyclones tropicaux, el<strong>le</strong> semb<strong>le</strong> conduire à<br />
des chiffres trop é<strong>le</strong>vés pour <strong>le</strong>s orages convectifs d'Afrique tropica<strong>le</strong>.<br />
Bans ce cas éga<strong>le</strong>ment on revient à l'application de la méthode de l'hydro-<br />
gramme unitaire pour des averses élémentaires successives, mais <strong>le</strong> choix<br />
de la distribution temporel<strong>le</strong> des intensités est délicat. Cequi arrive<br />
souvent c<strong>le</strong>st que d'un bout à l'autre de l'estimation, on arrive à de<br />
tel<strong>le</strong>s cascadea de marges de sécurité qu'il est faci<strong>le</strong> de fournir des<br />
chiffres trop élovés.<br />
2. Crues dues à des averses cycloniques :<br />
2.1. Crues décenna<strong>le</strong>s :<br />
L'averse décenna<strong>le</strong> est plus diffici<strong>le</strong> & définir que dans <strong>le</strong><br />
cas précédent, la distribution statistique est plus diffici<strong>le</strong> à étudier<br />
et <strong>le</strong>s donnkes de base sont plus mauvaises (en cas de cyclone une bonne<br />
partie des pluviomètres débordent), mais dans beaucoup de pays du monde,<br />
on arrive 8 d6finir une va<strong>le</strong>ur à peu près convenab<strong>le</strong> de l'averse décen-<br />
na<strong>le</strong>, on doit alors découper l'averse en averses élémentaires comme au<br />
point 1.3. et on transforme ces averses en crues par la méthode des hy-<br />
drogrammes unitaires. Très souvent, pour <strong>le</strong>s dernières averses élémen-<br />
taires rC, est voisin de 0,9O si la pente est notab<strong>le</strong>, que la couverture<br />
soit forestière ou non. Après calcul, il est bon de comparer <strong>le</strong> résultat<br />
aux crues maxima<strong>le</strong>s connues dans <strong>le</strong> monde en utilisant des diagrammes<br />
tels que <strong>le</strong> diagramme FWCOU-RODIER.
2.2. Crue maxima<strong>le</strong> probab<strong>le</strong>.<br />
Dans ce cas, on ne peut donner que des indications gén6ra<strong>le</strong>s.<br />
Pour l'averse à prendre en considération, on pourra se référer, dans<br />
<strong>le</strong>s pays à fortes averses, aux va<strong>le</strong>urs maxima<strong>le</strong>s mondia<strong>le</strong>s tel<strong>le</strong>s qu'el-<br />
<strong>le</strong>s sont données dans <strong>le</strong> Guide des Pratiques Hydrométéorologiques de<br />
l'OMM ou aux résultats de la formu<strong>le</strong> de HERSHFIELD, mais il sera encore<br />
plus diffici<strong>le</strong> que plus haut d'aboutir à une va<strong>le</strong>ur convenab<strong>le</strong> de l'écart-<br />
type, ceci nécessitera une sérieuse étude critique des rares données<br />
pluviométriques disponib<strong>le</strong>s, en tenant compte du débordement 8ventuel<br />
des pluviomètres. Le reste est plus faci<strong>le</strong> car <strong>le</strong> coefficient de ruis-<br />
sel<strong>le</strong>ment KR est de l'ordre de 0,gO.<br />
Si la région est COMUB pour avoir des averses exceptionnel<strong>le</strong>-<br />
ment fortes, il est normal de prendre en considération des va<strong>le</strong>urs de<br />
précipitations supérieures aux maximums mondiaux connus car, en pays de<br />
cyclones tropicaux, la connaissance des averses de durée inférieure à<br />
48 heures est très incomplète et <strong>le</strong>s maximums mondiaux connus doivent<br />
&tre considér6s comme piut8t provisoires.<br />
ïb générai, dans <strong>le</strong>s cas graves concernant <strong>le</strong>s cyclones tropi-<br />
caux, l'hydrologue arrive a la conclusion un peu décevante qu'il serait<br />
prbférab<strong>le</strong> que l'ingénieur prévoie son barrage de tel<strong>le</strong> façon qu'il puis-<br />
se être submergé par n'importe quel<strong>le</strong> crue.<br />
Qua1 que soit <strong>le</strong> cas étuùié, un examen du terrain orienté vers<br />
la recherche L;ss traces laissées par de fortes crues est n&cessaire.<br />
611<br />
I1 résulte de tout ce qui précède que <strong>le</strong>s ñydrologues ont<br />
encore de nombreuses recherches à faire pour aider efficacement <strong>le</strong>s<br />
constructeurs dans <strong>le</strong>ur tâche.<br />
Quelques références uti<strong>le</strong>s pour <strong>le</strong>s petits bassins de ces régions :<br />
1. 0.M.E.i. (1965). Guide des Pratiques Hydromét&orologiques, no 168<br />
T.P. 82, Genève.<br />
2. RODIER J., AWRGY C. (1965). Estimation des débits de crues décen-<br />
na<strong>le</strong>s pour <strong>le</strong>s bassins versants de superficie inférieure à 200 km2<br />
en Afrique Occidenta<strong>le</strong>, ORSTOM, Paris.<br />
3. HERSHFIXLD D.M. (1963). Estimating <strong>the</strong> probab<strong>le</strong> maximum precipita-<br />
tion. Am. Soc. of Civil hhgineers Transactions, Vol. 128, Part I,<br />
PP. 534-556.
61 2<br />
4. FRANCOU J., RODBR JO (1967), Essai de classification des crues<br />
maxima<strong>le</strong>s observées dans <strong>le</strong> monde. Cahiers d'Hydrologie OHSTOM<br />
vol. IV, ne 3, pp. 19-46. Paris.<br />
5. BENSON M.A., (1968), Measurement of Peak Discharge by Indirect<br />
Methods, OMM ne 225. TP. 11gP Genève.
613<br />
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Fig: 2<br />
Temps d e base en fonction d e R et de S<br />
REGIMES SAHELIENS - SUBDESERTIQUES<br />
I
METHODS FOR THE ESTIMATION OF MAXIMUM DISCHARGES OF SNOW<br />
MELT AND RAINFALL WATER WITH INADEQUATE OBSERVATIONAL DATA<br />
ABS TRACT<br />
Prof. A.A. Sokolov<br />
State Hydrological Institute<br />
Leningrad, USSR<br />
The prob<strong>le</strong>m of floods computation was accepted by <strong>the</strong> UNESCO<br />
Co-ordinating Council for <strong>the</strong> IHD as one of <strong>the</strong> most important<br />
prob<strong>le</strong>ms. For its solution <strong>the</strong> Working Group on Floods and <strong>the</strong>ir<br />
ccmputation was established; it has realized several projects on<br />
<strong>the</strong> THD programme essential for future research on floods, deve-<br />
lopmen: and improvement of methods for floods computation. The pc<br />
per give;: an evaluation of <strong>the</strong> up-to-date state of this prob<strong>le</strong>m<br />
as means fLr its solution.<br />
RESUME<br />
Le Conseil de Coordination de l'UNESCO pour la DHI a estimé<br />
que <strong>le</strong> problème du calcul des crues était tres important. Pour<br />
l'examiner, il a créé un groupe de travail sur <strong>le</strong>s crues et <strong>le</strong>ur<br />
évaluation. Ce groupe a réalisé, dans <strong>le</strong> cadre du programme de la<br />
DHI, un certain nombre de travaux très importants pour l'avenir<br />
de la recherche sur <strong>le</strong>s crues, pour la mise en oeuvre et l'amélig<br />
ration des méthodes de calcul qui <strong>le</strong> concernent. Dans la présente<br />
communication, l'auteur fait <strong>le</strong> point de la situation actuel<strong>le</strong>,<br />
ainsi que sur <strong>le</strong>s moyens de parvenir la solution du prob<strong>le</strong>me.
61 6<br />
At present triere are two uifferent approac,ies to t.ie complitatlon<br />
of fiood discilarge of ungauged rivers cievelopec to a certain<br />
dcbree quite independently. The first approach is based on <strong>the</strong><br />
s~atistical analysis and generalization of field data on<br />
flood runoff; <strong>the</strong> second approach is based on <strong>the</strong> genetic<br />
analysis and syn<strong>the</strong>sis of flood hy&ograph. As was: ctateu<br />
by J. Nemec and M. Moudry (Czechoslovakia) at; <strong>the</strong> Leningrad<br />
Symposium (1967) <strong>the</strong>se two approaches ase gradually being<br />
brought toge<strong>the</strong>r and at; present <strong>the</strong>y are used simultaneously<br />
supp<strong>le</strong>menting each o<strong>the</strong>r Lis,-/.<br />
Humerous design schemes have been proposed to estimate maximum<br />
flqod runoff of ungauged and poorly gauged rivers.<br />
According to <strong>the</strong> princip<strong>le</strong>s of approach and <strong>the</strong> scope of<br />
flood computation <strong>the</strong>se schemes may be divided into 2 main<br />
groups:<br />
3. Empirical or semi-empirical formulae for flood discharge<br />
computation based on <strong>the</strong> account of some of its most important<br />
factors ( e.g., drainage area or maximum precipitation rate),<br />
providing maximum water cìischarge only.<br />
2. Methods considering flood genesis and providing <strong>the</strong><br />
possibility of plotting <strong>the</strong> <strong>who<strong>le</strong></strong> hydrograph on <strong>the</strong> basis of<br />
time inflow of snow melt and rainfall water and its transformation<br />
into runoff as a result of losses by infiltration,<br />
suriace retention, lag along <strong>the</strong> slopes and channel network.<br />
At k-esent, methods of maximum flood discharge computation<br />
on <strong>the</strong> basis of <strong>the</strong> use of di€fereat design formulae are<br />
widel applied. The most important formulae are as follows:<br />
(a? formulae of extreme intensity, or <strong>the</strong> so-cal<strong>le</strong>d<br />
rational formulae, basea on <strong>the</strong> account of maximum or extreme<br />
rainfall intensity during lag-time or flood flov concentration<br />
in its general form:<br />
Qmax =/Cpa,d,A. (1)<br />
where : gp is coefficient of dimensionality; ar is maximum<br />
intensity of rain or snow melt during lag time ( c ); dris<br />
runoff coe ficient during this time interval; A is drainage<br />
area in km 5 .<br />
Pormula (1) is usually applied for <strong>the</strong> computa-t;ion of<br />
maximum runoff for relatively small basins (<strong>le</strong>ss than 200 km ).<br />
To determine design values of' dc curves of maximum precipitation<br />
increase %, are plotted with <strong>the</strong> increase of time<br />
interval Z' ? given as percentage of daily precipitation of <strong>the</strong><br />
same probability of exceedence ( p )<br />
The maximum mean precipitation rate Jcp for any time<br />
interval is estimated by formula:
617<br />
Mean velocity and lag-time down <strong>the</strong> channel and slope6<br />
are computed by simplified formulae of Chezy-Manning or<br />
C he zy -Baz in.<br />
The principal disadvantage of formula (1) consists in some<br />
uncertainw and inaccuracy of <strong>the</strong> determination of lag time<br />
c-<br />
or flood concentration c, is interpreted by iiiJividiia1<br />
scientists in different ways, <strong>the</strong>refore it causes iii-<br />
accuracy of determination of principal parameters az and d,-<br />
appearing in formula (1). Besides, formulae of type (1) do not<br />
take into account <strong>the</strong> remaining flood e<strong>le</strong>ments (duration rise<br />
and fall duration ratio) and do not provide <strong>the</strong> plotting of<br />
%he viho<strong>le</strong> ïlood hydrograph essential for <strong>the</strong> determination of<br />
maxima transformation in ponds and reservoirs;<br />
(b) empirical or semi-empirical reduction formulae of <strong>the</strong><br />
general type:<br />
is maximum specific discharge, m'/sec per 1 km';<br />
9. is parameter comprising <strong>the</strong> extreme specific discharge<br />
if A-O and C = 1.O;cis addition to <strong>the</strong> drainage area<br />
considering non-lineariky of dependence 4 ~mp,aj@@~<br />
within <strong>the</strong> range of small areas of <strong>the</strong> basin; nis <strong>the</strong><br />
eqonent of reduction of maximum specific discharge3 with %he<br />
incrsase of basin area and varying according to experimental<br />
and bheoretical data from n= 0.15 - 0.30 for runoff maxima of<br />
snow mel? water or caused by prolonged frontal rainfalls, to<br />
n = 0.5 - 3.7 for maxima caused by short heavy local storms.<br />
Parameter +,,may be estimated according to <strong>the</strong> extreme rate of'<br />
snow melt or rainfalls for minimum time interval, e.g. 1 hour,<br />
or for snow melt water according to depth of runoff during a<br />
flood.<br />
In <strong>the</strong> first case when C = 1.0 formula (4) may be presented<br />
as fo1lov;s:<br />
where: Q,,,~<br />
where: % is coefficient of dimensionality; % ds maximum<br />
hourly rato of snow melt or rainfall; d, is overland flow<br />
coefficient .<br />
Since does not exceed 10-15 m/hr for snow melt water<br />
and 300-400 mm/hr for rainfall water ( on <strong>the</strong> basis of computation<br />
of heat balance of snow melt), <strong>the</strong>n <strong>the</strong> extreme value of<br />
qe in fo mula (i+) if do= 1.0 and k$= 0.23, may not exceed<br />
2.8 -4.2 m 3 /sec pe 1 lun2 for snow melt water and up to 84-<br />
112 mj/sec per i bS for raidail water.<br />
On <strong>the</strong> basis of <strong>the</strong>se e<strong>le</strong>mentary considerations it is possib<strong>le</strong><br />
to conclude %hat many empirical formulae of type (4) with<br />
parameter 9. exceeding <strong>the</strong> mentioned limits have no physical<br />
substantiation.<br />
Due to some uncertainty of C value in formula (4) when<br />
A -0, this Lormula is sometimes used as follows:
61 8<br />
w ei'3: Cp.6 is parameter (maxim specific discharge in<br />
ì$/sec if &ainage area B=2ûû by:<br />
a is tiie exponent of reduction dcterrnined uy regional<br />
=fy/A/.<br />
dependences ep<br />
For practical computations on <strong>the</strong> basis of generalization<br />
of empirical data a map of regional boundaries is prepared<br />
with similar exponents of I2 .<br />
The advantage of formula (6) consists of <strong>the</strong> fact that <strong>the</strong><br />
value of parameter c,, slightly depends on and<br />
<strong>the</strong>refore <strong>the</strong> mapping of cp,6<br />
possib<strong>le</strong> e<br />
in <strong>the</strong> form of isolines is<br />
In case of eo determination according to <strong>the</strong> depth of runoff<br />
of snow melt water during flood hrnformula (4) may be present-<br />
ed as follows:<br />
where: & is t,ie coe€ficient considering a number cif otlier factors,<br />
in particular, duration and siiape of flood.<br />
Formula (7) is used as <strong>the</strong> basis for <strong>the</strong> computation of<br />
maxinum snow melt water äischarge for <strong>the</strong> <strong>who<strong>le</strong></strong> USSII territory<br />
L4L<br />
Duriw recent years <strong>the</strong> reduct'on scheme is seldom used as<br />
simp<strong>le</strong> regional dependences =*[JI plotted by dependences<br />
enveloping empirical points usually related to larger basin<br />
areas.<br />
In this case it is essential to take into account flood<br />
runoff probability of oxceedence Its parameters are<br />
differentiated according to climatic zones.<br />
Along with <strong>the</strong> basin area which was previously accepted as<br />
almost <strong>the</strong> only maximum runoff factor, numerous important<br />
clinlatic factors are also taken into account, i.e. depth and<br />
sate OP precipitation, snow melt rate, flood runoff depth;<br />
and morphological factors as well, i.e. basin topography, lakes<br />
and swamps areas, river network density, soils and subsoils<br />
composing <strong>the</strong> basin mant<strong>le</strong>, etc.<br />
7iith <strong>the</strong> increase of hydrological information and reliab<strong>le</strong><br />
runoff data from small basins <strong>the</strong> reduction scheme has greatly<br />
consolidated its positions since ibs basic parameters i.e.<br />
reduction coefficient anCr maximum runoff of e<strong>le</strong>mentary (small)<br />
basins have gainea a reliab<strong>le</strong> substantiation. This particular0<br />
concerns maximum runoff of snow melt water;<br />
(c) <strong>the</strong> so-cal<strong>le</strong>d volumetric formulae considering flood<br />
shape and duration, besides maximum ordinate of flood, may be<br />
given as follows:
61 9<br />
where: H is depth of precipitation or snow melt water;<br />
&is total or volumetric coefficient of runoff during flood;<br />
Ttn ndicate general duration of flood or its rise phase;<br />
4 and$ , are coefficients of flood shape, i.e. ratio of maximum<br />
discharge to mean discharge.<br />
Design formulae as (8) or (9) are preferab<strong>le</strong> compared with<br />
formulae of type (1) or (4) since all flood e<strong>le</strong>ments are co-ordinated<br />
but <strong>the</strong>y ara applied only for simp<strong>le</strong> one-peaked<br />
floods for which general or volumetric coefficient of' runoff<br />
is applicab<strong>le</strong>.<br />
Despite <strong>the</strong> existing numerous formulae <strong>the</strong> computation of<br />
rainfall flood runoff for ungauged rivers is not reliab<strong>le</strong>.<br />
Every design scheme proposed is characterized by certain a6vantages<br />
and disadvantages. There exist no accepted design<br />
schemes until now.<br />
formulae is in <strong>the</strong> cornputation of individual flood e<strong>le</strong>ments<br />
without <strong>the</strong>ir co-ordination and without <strong>the</strong> account of genesis<br />
anrid type of flood; <strong>the</strong> latter is essential for hydraulic engineering<br />
projects to make a correct estimation of <strong>the</strong> flood transformation<br />
rate in ponds and reservoirs and <strong>the</strong> amount of<br />
discharte through spillways. These disadvantages never occur<br />
in genetx computation methods for floods of <strong>the</strong> 2nd group<br />
based on <strong>the</strong> account of time variations of inflow of rain and<br />
snow melt waters and <strong>the</strong>ir transformation into runoff hydrograph<br />
as a result of non-simultaneous water lag from different basin<br />
areas.<br />
These methods are based on <strong>the</strong> plotting of runoff transit<br />
curve showing <strong>the</strong> distribution of areas of simultaneous runoff<br />
over time intervals.<br />
The following methods may be mentioned for <strong>the</strong> computation<br />
of floods:<br />
(a) isochrone method based on <strong>the</strong> determination of ordinates<br />
of <strong>the</strong> curve of unit areas distribution (transit curve) by<br />
means of plotting <strong>the</strong> lines of equal transit (isochrones) on a<br />
topographic map o€ river basin;<br />
(b) unit hydrograph method, based on <strong>the</strong> determination of<br />
transit curve by <strong>the</strong> ordinates of <strong>the</strong> observed unit flood<br />
hydro .raphs caused by individual storms;<br />
(cy method of ma<strong>the</strong>matical floods simulation.<br />
The method of isochrones is mainly applicab<strong>le</strong> for floods<br />
computation on small water courses with surface flow prevailing.<br />
&/<br />
A general disadvantage of <strong>the</strong> majority of empirical design<br />
Nhen <strong>the</strong> method of isochrones is applied it is essential to<br />
use topographic map of <strong>the</strong> basin of sufficiently large sca<strong>le</strong>
620<br />
anu ti<strong>le</strong> data on time variations of rain or snow melt water<br />
Inflow, moreover, for small water courses it is necessary to<br />
have data on such variations within a day and this causes certain<br />
restrictions in <strong>the</strong> sphere of its application.<br />
Unit hydragraph method is based, as it was mentioned, on <strong>the</strong><br />
plotting of curve of unit areas distribution according to <strong>the</strong><br />
ordinates of unit floods observed.<br />
Unit hydrograph method is based on <strong>the</strong> lag <strong>the</strong>ory expressed<br />
by <strong>the</strong> so-cal<strong>le</strong>d genetic formula of runoff:<br />
where: Qdt indicates discharges at <strong>the</strong> out<strong>le</strong>t at <strong>the</strong> moment t ;<br />
ht-r is effe tive precipitation per time unit Af at <strong>the</strong><br />
moment k-r; #c indicates ordinates of <strong>the</strong> curve of unit<br />
areas distribution.<br />
Tne unit ,iydrograpli metiiod is cnaracterized by its visuality<br />
and.pa.ysica1 substantiation, it provides <strong>the</strong> plotting of design<br />
flood nyúrograpii according to preciFitation; tiiis resulted in its<br />
wide application in many countries of ti<strong>le</strong> world despite some<br />
draNoacks.<br />
Its sppircasion is aiII1cuII; mainly ow- ‘GO m e inadequacy<br />
of <strong>the</strong> mekhods used for averaging <strong>the</strong> observed individual floods,<br />
separation cf multi-peaked floods and methods for infiltration<br />
rate and flow coefficient determination, There is also some<br />
uncertainty as to <strong>the</strong> applicabili- of <strong>the</strong> unit hydrograph method<br />
to different drainage areas; also prob<strong>le</strong>matic is <strong>the</strong> relationship<br />
between an individual storm duration and flood rise duration<br />
or <strong>the</strong> time of peak shifting relative to rainfall maximum during<br />
<strong>the</strong> flood.<br />
All <strong>the</strong>se factors limit <strong>the</strong> application of <strong>the</strong> unit hydrograph<br />
method.<br />
Lately <strong>the</strong> method of ma<strong>the</strong>matical floods simulation has been<br />
more and more widely used.<br />
For instance, in one of <strong>the</strong> variants of trie method of ma<strong>the</strong>matical<br />
floods simulation applied in <strong>the</strong> USSR <strong>the</strong> analogy between<br />
equation (10 ), describing flood formation resulting from<br />
water lag and summation of individual discharges from different<br />
parts of river basin, and <strong>the</strong> eq,uation describing <strong>the</strong> change of<br />
current in <strong>the</strong> e<strong>le</strong>ctric circuit, was used,<br />
To apply this method practically, it is essential to develop<br />
investigations connected with determining parameters set for <strong>the</strong><br />
specific e<strong>le</strong>ctric analog computer to estimate flood runoff of<br />
ungauged rivers.<br />
The importance of research, computation and prediction of<br />
floods for many countries of <strong>the</strong> world necessitates <strong>the</strong> international<br />
scientific co-operation on <strong>the</strong> prob<strong>le</strong>m of flood flow<br />
computation.
621<br />
This co-operation is in particular exercised under <strong>the</strong> auspices<br />
of UNESCO and WMO within <strong>the</strong> framework of <strong>the</strong> IIID programme.<br />
For this purpose <strong>the</strong> Co-ordinating Council €or <strong>the</strong> IHD<br />
established <strong>the</strong> Ciorking grou:, on floods and <strong>the</strong>ir compuation.<br />
This Working group has studied and generalized, to some extent,<br />
<strong>the</strong> international experience in <strong>the</strong> field of research and<br />
computation of flood flow.<br />
A great contribution was made by <strong>the</strong> International Symposium<br />
on floods and <strong>the</strong>ir computation held in Leningrad at <strong>the</strong> initiative<br />
of UNESCO and with <strong>the</strong> participation of \YMO and IAkIS.<br />
The proceedings of this Symposium were published, <strong>the</strong>refore<br />
<strong>the</strong>re is no need to cite and consider <strong>the</strong>m here /18, 26/.<br />
The review made by WMO on meteorological aspects in computing<br />
flood flow is ra<strong>the</strong>r useful. ‘ïiiic review made up a technical<br />
note on this prob<strong>le</strong>m /25/.<br />
The results of processing of observation data on floods made<br />
at <strong>the</strong> network of IHD stations also appear to be valuab<strong>le</strong>. Taking<br />
into account that in some large areas covered by <strong>the</strong> network<br />
of IIID stations outstanding floods, may always occur. The<br />
UNESCO IHD Working group on floods and <strong>the</strong>ir computation prepared<br />
and published <strong>the</strong> technical note on col<strong>le</strong>ction and processing<br />
of data on floods /27/.<br />
The Working group also developed <strong>the</strong> programme for <strong>the</strong><br />
World Catalogue of very large floods, according to which a<br />
riiimber of countries were entrusted to col<strong>le</strong>ct, process and<br />
publish <strong>the</strong> data on large floods.<br />
Besides, <strong>the</strong> Working group considered it necessary to study<br />
and generalize <strong>the</strong> international experience in <strong>the</strong> field of<br />
floods covputation and is now preparing for publication <strong>the</strong><br />
Technical Note (Case<strong>book</strong>), wiiich would ref<strong>le</strong>ct a great experience<br />
in computation of flood flow, gained in many countries.<br />
These pub lications toge<strong>the</strong>r with <strong>the</strong> Proceedings of <strong>the</strong> Lenin<br />
grad Symposium on floods will serve as a good basis to improve<br />
methods of flood flow computation, which in its turn will<br />
contribute to a more rational and economical se<strong>le</strong>ction of<br />
parameters for hydraulic structures on rivers, <strong>the</strong>ir durability<br />
and resistance to floods.<br />
R E F E R E N C E S<br />
A<strong>le</strong>xeev G.A. (1 966). Sk<strong>le</strong>ma raschetov maximainyh dozhdevyh<br />
raskhodov vody PO formu<strong>le</strong> predelnoy intensivnosti stoka,<br />
(Computation of maximum rainfall discharge by means of<br />
<strong>the</strong> formula of extreme runoff intensity). Transactions<br />
of <strong>the</strong> GGI, vol. 134.<br />
Befani A.N. (1958) Osnovy teorii protsessov stoka i puti<br />
dalneishykh iss<strong>le</strong>dovaniy. (Theory of runoff processes and<br />
directions of fur<strong>the</strong>r research). Transactions of OGMI,<br />
vol. 15.
622<br />
3. Velikanov M.A. (1931) GidromekhanichesQ analiz poverhnostnobo<br />
stoka. (Hyaromechanical analysis of <strong>the</strong> surface runoff).<br />
Geophysics Nos. 1-2.<br />
4. Voskresenski K.E. (1956) Gidrologicheskie raschety pri proektirovmii<br />
sooruzheniy na malyh rekah, ruchjah i vremennyk<br />
vodotokakh (Method osn. i prakt) (Hydrological computations<br />
for structures on small rivers and temporary water<br />
courses), Leningrad, GIMIZ.<br />
5. Kalinin G.P., Milukov ?.I. (1958) Priblizhennyi raschet neust<br />
anovivshe go sy a dvi zhenia v o w h mass . (Approximate<br />
estimation of <strong>the</strong> unsteady water motion), Transactions of<br />
TsW, vol. 66.<br />
6. Kovzel A.G. (1951) Opyt projektirovania hydrografa vesennego<br />
stoka dlya malogo vodosbora. (The designing of <strong>the</strong> hydrograph<br />
of spring runoff on small watersheds) .Tr.of GGI,v.31(85)<br />
7. Kuzmin P.E. (1961) Protsess tayania snezhnogo pokrova.<br />
(Snow cover melting) Hydrometeorological Publishing House.<br />
8. Lvovich M.I. (L34-ö) Protsessy formirovania pavodkov.(Flood<br />
formation), Transactions of GGI, vol. 10.<br />
9. Moklyak V.I. (1965) Pormirovanie maximalnyh raskhodov ot<br />
talyh vod i ih raskhoày (Formation of maximum snowmelt<br />
discharges), Kiev.<br />
10. l'rotoãyakonov M.M. (1966). Oprede<strong>le</strong>nie maksimalnogo stoka<br />
poverhnostnyh vod s malyh vodosborov. (Determination of<br />
maximum surface runoff on small watersheds) Hydrometeos<br />
raological Publishing House Leningrad.<br />
11. Sokoiov A.A. (1963) Maximalnyi stok talyh vod e<strong>le</strong>mentarnyh<br />
bassainov i priroda ego reduktsii. (Maximum snowmelt<br />
runofi on e<strong>le</strong>mentary basins and <strong>the</strong> nature of its reduction).<br />
Transactions of GGI, vol. 107.<br />
12. Sokolov A.A. (1966) Metodika rascheta maximalnyh raskhodov<br />
talyh vod pri otsutstvii ili nedostatochnosti gidrometricheskikh<br />
dannykh (Computation of maximum discharges of<br />
snowmelt water in case of <strong>the</strong> absence or inadequacy of<br />
hydrometric data) Transactions of GGI, vol. 134.<br />
13. Sokolovsky D.L. (1937) Normy maximalnogo stoka vesennikh<br />
pavodkov rek SSSH i metodika ikh rascheta. (Norms of<br />
maximum spring flood runoff of <strong>the</strong> USSR rivers and <strong>the</strong><br />
technique of <strong>the</strong>ir computation). Hyarometeorological<br />
Publishing House.<br />
14. Sokolovse D.L. (1948) Metodika postroenia hydrografa liv-<br />
nevogo stoka PO osaäkam (Plotting of rainfall runoff hydro-<br />
graph on <strong>the</strong> basis of rainfall aata). Transactions of<br />
GGI, vol. 14.<br />
15. Sokolovsky D.L., Shiklomanov I.A. (1965). Haschety hydrografov<br />
pavoàkov s primeneniem e<strong>le</strong>ktronnyh modeliruyshchih<br />
UStroiStV. (Computation of flood hydrographs by means of<br />
e<strong>le</strong>ctronic modelling devices). Transactions of LGMI,<br />
voi. 23.<br />
16. Sokolovslry D.L. (1968). Hechnoi stok, (River flovi). 3rd<br />
edition, Hydrometeorological Publishing House, Leningrad.
623<br />
17. Stroitelnye normy i pravila. (19661,Chast II, rasàel II,<br />
glava 7. Raschetnye maximalnye raskhody vody pri proektirovanii<br />
gidrotehnichesmh sooruzheniy. (Norms and<br />
instructions for civil engineering. Part II, section<br />
II, chapter 7. Maximum design discharges for hydrotechnical<br />
structures). Normy proektivania (CH i II<br />
II - 4. 7-65). MOSCOW.<br />
18. Mezhdynarodnyi simposium PO pavocikairi i ili raschetam.<br />
(1969). (International symposium on floods ana <strong>the</strong>ir<br />
computation) I and II, Leningrad, 15-22, August, 1967.<br />
Hyàrome teorological Publishing House, Leningrad.<br />
19. Ukasania po oprede<strong>le</strong>niu raschotnyh maximalnyh raskhodov<br />
talyh vod pri otsutstvii ili nedostatochnosti gidrometricheskih<br />
nabludeniy. (1966) (Instructions for<br />
<strong>the</strong> determination of maximum design snow melt discharce<br />
in case of <strong>the</strong> absence or inadequacy of hyarometric<br />
data). CH 356-66. Hydrometeorological Publishing<br />
House, Leningrad.<br />
20. Ukasania po opreae<strong>le</strong>niu raschetnyh hydrologicheskih kharac-<br />
teristik, CH 435-72 (1972). (Instructions for <strong>the</strong> esti-<br />
mation of <strong>the</strong> hydrological design values CH 435-72).<br />
Hydrometeorological Publishing House, Leningrad.<br />
21. Ukasatel literatury PO pavodkam i ih raschetam (1967)<br />
(Bibliography on floods and <strong>the</strong>ir computation)<br />
Hydrometeorological YuDlishing House.<br />
22. Chegodaev N.N. (1953). Haschet poverhnostnogo stoka s<br />
mlyh vodosborov. (Estimation of surface runoff from<br />
su.311 watersheds). Tranzheldorizdat.<br />
23. Shiklomanov I.A. (1964). Kaschet transformatsii pavodkov<br />
vodokhranilishchami i prudami pri pomoshchi e<strong>le</strong>ctron-<br />
nogo modeliruyushchego ustroistva (Computation of<br />
flood transformation by ponds and reservoirs by means<br />
of e<strong>le</strong>ctronic modelling devices). Transactions of LGMI,<br />
vol. 26.<br />
24. A<strong>le</strong>xeev G.A. ana Sokolov A.A. General princip<strong>le</strong>s and<br />
methods for <strong>the</strong> computation of flood discharges applied<br />
in <strong>the</strong> USAR. Atti del convegno internaziona<strong>le</strong> (Roma,<br />
23-30 November 1969). Roma, ANDL! pp. 735-747.<br />
25. Estimation of maximum floods. Technical Note No. 98.<br />
LNO v No. 233, TP. 126, T;MO, Geneva (1969).<br />
26. Floods and <strong>the</strong>ir computation. Proceedings of <strong>the</strong> Leningrad<br />
Symposium. August, vol. 1 and 2. UNESCO/USH (1969).<br />
27. Flood studies: an international scuiae for col<strong>le</strong>ction ~ and ~~-..<br />
processing of data. Technica1"papers in hydrology No.8,<br />
UNUSCO. Paris (1971).<br />
28. Gray D-M. -Syn<strong>the</strong>tic-Unit-Hydrographs for Small Watersheäs.<br />
Proc. Am. Soc. Civ. Eng., vol. 87.<br />
29. Lins<strong>le</strong>y R.X., Koh<strong>le</strong>r M.A. and Paulhus L.N. (1949). Applied<br />
Hydrology McGrow Hill Book Company N.Y.<br />
30. Morgan and Johnson (1962). Analysis of Unit-Graph Method.<br />
Journal or Hydraulic Division, 88, NY-5.<br />
31. Sherman L.R. (1932) Stream-Flow Rainfall by Unit- Graph<br />
Method ¡in . Mews Record.<br />
32. Snyder F.F. f1938) Syn<strong>the</strong>tic Unit-Graphs. Trans. Am.<br />
Geophys. Union, vol. 19.
ABSTRACT<br />
COMPUTATION OF PROBABILISTIC VALUES<br />
OF LOW FLOW FOR UNGAUGED RIVERS<br />
Vladimirov A. M., Chebotarev A. I.<br />
State Hydrological Institute<br />
Leningrad, U.S.S.R.<br />
The main characteristics of low flow (minimum daily, mon-<br />
thly and seasonal flows) are investigated. The computation methods<br />
are Sased on a combined use of geographical interpolation and pro-<br />
bability analysis and considering <strong>the</strong> main factors affecting <strong>the</strong><br />
volume aiid regime of low flow. Principal characteristics of low<br />
flow for mkdium-size rivers are determined by maps of flow isoli-<br />
nes, those for small rivers are determined by regional empirical<br />
correlation. Design flow is established by means of transition<br />
coefficients. The principal computation methods discussed are deve<br />
loped for U.S.S.R. rivers.<br />
Les auteurs analysent <strong>le</strong>s caractéristiques principa<strong>le</strong>s des<br />
débits de basses eaux (journaliers? mensuels, minimum saisonnier).<br />
Les méthodes $e calcul font appel a la fois à l'interpolation géo-<br />
graphique et a l'analyse statistique, compte tenu des facteurs<br />
principaux qui influencent l'abondance et <strong>le</strong> régime des débits de<br />
basses eaux. Les principa<strong>le</strong>s caractéristiques des débits d'étiages<br />
des rivières moyennes font l'objet d'une représentation cartogra-<br />
phique; pour <strong>le</strong>s petites rivières on <strong>le</strong>s traduit par des relations<br />
empiriques régiona<strong>le</strong>s. Les débits correspondant à la fréquence<br />
choisie pour <strong>le</strong> projet sont établis en utilisant des coefficients<br />
de transfert. Les auteurs présentent <strong>le</strong>s principa<strong>le</strong>s méthodes de<br />
calcul utilisées en URSS.
2 6<br />
Low flow is one of <strong>the</strong> principal phases of <strong>the</strong> hydrological river<br />
regime. During <strong>the</strong> dry periods, when precipitation is usually at its<br />
lowest, rivers have ra<strong>the</strong>r stab<strong>le</strong> and relatively small discharges. Their<br />
variations in <strong>the</strong> flow hydrograph tend to approximate a horizontal line.<br />
The lowest flow observed during a certain period is generally cal<strong>le</strong>d <strong>the</strong><br />
minimum flow during that period.<br />
Separation of low-flow period in river flow hydrographs<br />
On rivers with distinctly expressed spring snowmelt floods and<br />
autumn floods <strong>the</strong> period of low flow is observed during winter and summerautumn<br />
seasons. Its beginning in summer is determined by <strong>the</strong> end of<br />
spring high-water period, i.e. when <strong>the</strong> intensive rate of decrease of<br />
discharge tends to become smal<strong>le</strong>r. The summer period of low flow ends with<br />
<strong>the</strong> arrival of <strong>the</strong> autumn floods or <strong>the</strong> appearance of ice in <strong>the</strong> river.<br />
In <strong>the</strong> latter case <strong>the</strong> low-flow period is cal<strong>le</strong>d <strong>the</strong> summer-autumn period.<br />
The winter low-flow period begins at <strong>the</strong> appearance of ice events<br />
in <strong>the</strong> river and continues until spring high-water period begins. In<br />
case of no ice phenomena in <strong>the</strong> river <strong>the</strong> winter low-flow period is<br />
assumed to be a period from <strong>the</strong> average data of air temperature falling<br />
down contii:vously through O°C and below it up to <strong>the</strong> beginning of <strong>the</strong><br />
spring high-wcter period.<br />
The low-flow period includes also floods if <strong>the</strong> volume of each<br />
of <strong>the</strong>m does not exceed 10-15 per cent of <strong>the</strong> flow volume for preceding<br />
and subsequent lol.:-flow periods, without taking into account <strong>the</strong> volumes<br />
of floods already included. If <strong>the</strong> flow hydrograph has <strong>the</strong> form of a<br />
saw-like curve (frequent floods of various magnitudes), <strong>the</strong> period of<br />
low-flow includes floods with maximum discharges that are 3-5 times<br />
greater than preceding daily minimum discharges (depending on <strong>the</strong><br />
volume of <strong>the</strong> flood peak).<br />
These criteria facilitate <strong>the</strong> plotting of<br />
river low-flow periods, although <strong>the</strong>y slightly overestimate <strong>the</strong> volume<br />
of low flow.<br />
In <strong>the</strong> U.S.S.R. low flow may be expressed as minimum daily,<br />
minimum monthly (30-day) or minimum seasonal flow.<br />
Seasonal flow is <strong>the</strong> average value of discharge (specific discharge)<br />
for winter or summer-autumn seasons. The minimum monthly flow<br />
is <strong>the</strong> average during <strong>the</strong> lowest ca<strong>le</strong>ndar month in <strong>the</strong> given season.<br />
On rivers with flood regime during winter or summer-autumn seasons when<br />
<strong>the</strong> low-flow period is of <strong>le</strong>ss than two month duration or is interrupted<br />
by large floods, <strong>the</strong> smal<strong>le</strong>st of <strong>the</strong> average discharges during a ca<strong>le</strong>ndar<br />
month may appear 1,5-2 times bigger than <strong>the</strong> minimum discharge. In such
627<br />
a case it is necessary to introduce a temporary correction taking into<br />
account not a ca<strong>le</strong>ndar month, but a 30-day period with <strong>the</strong> lowest flow.<br />
If frequent and considerab<strong>le</strong> floods make it difficult to find out <strong>the</strong><br />
30-day period of minimum flow, it may be reduced to 25-23 days in order<br />
to exclude <strong>the</strong> influence of floods.<br />
This secures <strong>the</strong> genetic homogeneity<br />
of <strong>the</strong> minimum flow of years with varying water volumes, which is impor-<br />
tant in <strong>the</strong> determination of minimum flows for ungauged rivers.<br />
Physiographic factors of low flow<br />
The duration and volume of low flow depend on physiographic<br />
factors which may be divided into two groups: (1) climatic conditions<br />
and (2) factors of underlying surface.<br />
lhe water resources of a certain basin depend on <strong>the</strong> climatic<br />
conditions prevailing in that basin. Precipitation contributes to <strong>the</strong><br />
increase of ground water supply, whi<strong>le</strong> evaporation decreases its recharge<br />
and supply. In winter low-air temperatures cause a considerab<strong>le</strong> freezing<br />
of soils and subsoils and contribute to <strong>the</strong> decrease of underground<br />
flow into rivers. Climatic factors determine areal low-flow distribution<br />
in accordance with <strong>the</strong> low of geographical zonation.<br />
1.i some cases and for small rivers particularly, low flow is<br />
greatly intluenced by local (azonal) factors of <strong>the</strong> underlying surface,<br />
i.e. <strong>the</strong> surface and underground flow contributions (lakes, swamps, soils<br />
and subsoils, karst, etc.).<br />
The influence of <strong>the</strong>se factors may be so considerab<strong>le</strong> and<br />
exceeds <strong>the</strong> influence of <strong>the</strong> climatic conditions.<br />
The most essential factor is <strong>the</strong> permeability of soils and subsoils.<br />
They serve as underground flow reservoirs, detaining water during<br />
high-water periods and re<strong>le</strong>asing it during low-water periods. The<br />
capacity of underground storage is determined by <strong>the</strong> geological structure<br />
of <strong>the</strong> area and its hydrogeological conditions. Loose and porous or<br />
crevassed deposits (sandstone, limestone, shing<strong>le</strong>, and <strong>the</strong> like) create<br />
favourab<strong>le</strong> Conditions for underground storage of water and far its subsequent<br />
re<strong>le</strong>ase during <strong>the</strong> low-flow periods in rivers. Solid clay or<br />
monolithic crystal rocks (granite, gneiss) near <strong>the</strong> surface decrease <strong>the</strong><br />
regulating capacity of <strong>the</strong> storage and reduce <strong>the</strong> low flow. The influence<br />
of karst-affected rocks on <strong>the</strong> regime and volume of low flow is determined<br />
by <strong>the</strong>ir absorption capacity und <strong>the</strong> rate of water yield - <strong>the</strong> bigger<br />
it is, <strong>the</strong> <strong>le</strong>ss is <strong>the</strong>ir influence an <strong>the</strong> low flow.
628<br />
The contribution of underground water reservoirs to <strong>the</strong> flow in<br />
rivers is a factor of extreme importance in <strong>the</strong> study of low flows. In<br />
this respect due consideration should be given to <strong>the</strong> number water con-<br />
tent and regime of aquifers contributions to river flow and <strong>the</strong> dynamics<br />
of underground flow into rivers. These factors determine <strong>the</strong> contribu-<br />
tion of underground aquifers to <strong>the</strong> flow in rivers.<br />
The study of factors affecting <strong>the</strong> volume and regime of low flow<br />
is a necessary prerequisite for <strong>the</strong> successful development of <strong>the</strong> com-<br />
putation methods.<br />
The analysis of <strong>the</strong> influence of main factors on conditions of<br />
low flow formation necessitates <strong>the</strong> division of rivers into small and<br />
midd<strong>le</strong>-size rivers when developing computation methods since <strong>the</strong> process<br />
of low flow formation is different for <strong>the</strong> two types of rivers. Large<br />
rivers are not considered here.<br />
Differentiation of small and midd<strong>le</strong>-size rivers<br />
The quantitative characteristics of a small river may be assumed<br />
to be <strong>the</strong> value indicating <strong>the</strong> extent of aquifers discharge contribution<br />
to total flow, i.e. <strong>the</strong> erosion depth of river channels. The determination<br />
of its îharacteristic is <strong>the</strong> ratio between <strong>the</strong> erosion channel depth<br />
and <strong>the</strong> aquifeis depth feeding <strong>the</strong> river along its <strong>le</strong>ngth up to <strong>the</strong> out<strong>le</strong>t.<br />
Th.e quantitative estimation of <strong>the</strong> influence of main hydrogeological factors<br />
on low flow is difficult to make, whi<strong>le</strong> developing low flow computation<br />
methods involve additional characteristics: correlations between<br />
<strong>the</strong> capacity of underground storage and drainage densities. In similar<br />
regions <strong>the</strong>re is a definite relationship between <strong>the</strong> volume of underground<br />
storage, river channel erosion depths, watershed boundaries and<br />
drainage areas.<br />
Therefore <strong>the</strong> value of river basin area provides an<br />
integrated indicator of morphological and hydrological conditions of low<br />
flow.<br />
In this case, a criterion for <strong>the</strong> term "small river" may be <strong>the</strong><br />
largest (critical) area of <strong>the</strong> basin responsib<strong>le</strong> for <strong>the</strong> comp<strong>le</strong>te drainage<br />
of aquifers feeding <strong>the</strong> river and with <strong>the</strong> enlargement of which no varia-<br />
tion of low flow modulus is observed. The value of <strong>the</strong> critical area is<br />
established by graphs of relationship of minimum 30-day flow modulus with<br />
river basin area for <strong>the</strong> physiographically similar regions.<br />
For <strong>the</strong> U.S.S.R. rivers, <strong>the</strong> critical area of <strong>the</strong> basin ranges<br />
from 1, O00 to 1,500 km2 in flat wet regions and in all mountain regions.<br />
In semi humid zones it rises to 2, 000-2, 500 km2 due to <strong>the</strong> lower depths<br />
of uquifers drained by rivers. In semi arid areas rivers with 5,,000-<br />
10, O00 km2 basin area are classified as small rivers.
Computation of normal low flow of small rivers<br />
In <strong>the</strong> U.S.S.R. computation practice for determining mean low<br />
flow of small ungauged rivers, <strong>the</strong> following equation relating <strong>the</strong> dis-<br />
charge to <strong>the</strong> river basin area is most widely used:<br />
where Q is discharge (seasonal minimum) in m3/sec; A is river basin<br />
area in km2; f is ei<strong>the</strong>r a regional impermeab<strong>le</strong> mean area or a<br />
permeab<strong>le</strong> contributing area outside <strong>the</strong> drainage basin. In <strong>the</strong> first<br />
case, <strong>the</strong> parameter f has <strong>the</strong> sign minus (-), in <strong>the</strong> second case it has<br />
<strong>the</strong> sign plus (+). Under usual conditions and permanent flow availab<strong>le</strong><br />
f = O. a, n are regional parameters characterizing conditions of low<br />
flow formation.<br />
629<br />
The determination of <strong>the</strong> parameters of design equation (1) is made<br />
for <strong>the</strong> regions se<strong>le</strong>cted on <strong>the</strong> base of a careful study of hydrogeological<br />
conditions of <strong>the</strong> basins under study and on <strong>the</strong> analysis of principal<br />
physiographic conditions. For instance, whi<strong>le</strong> dividing <strong>the</strong> territory of<br />
<strong>the</strong> U.S.S.R. into regions <strong>the</strong> following were used:<br />
water beating formations by rivers, hydrological descriptions of conditions<br />
favouiing <strong>the</strong> formation of underground flows of regions, ground flow<br />
map of <strong>the</strong> intcnsive water exchange zone, map of underground flow in percentage<br />
of <strong>the</strong> total river flow and coefficients of underground flow in<br />
percentage of precipitation.<br />
precipitation for warm and cold seasons, data on evaporation, air temperature<br />
for <strong>the</strong> winter season in ice melt regions, topographic map of <strong>the</strong><br />
U.S.S.R., hydrological regionalization of <strong>the</strong> U.S.S.R., map of physiogra-<br />
phic regionalization of <strong>the</strong> U.S.S.R.<br />
account as much as possib<strong>le</strong> all <strong>the</strong> characteristic features under which<br />
low flow of se<strong>le</strong>cted regions is formed. The boundaries of regions with<br />
similar low-flow conditions during winter and summer-autumn, were plotted<br />
along <strong>the</strong> boundaries of sharp change of hyd2logical conditions. For<br />
instance, when in some river basins <strong>the</strong> change of hydrogeological and<br />
o<strong>the</strong>r conditions take place, <strong>the</strong> change in <strong>the</strong> volume of river flow will<br />
not be observed immediately, but gradually whi<strong>le</strong> <strong>the</strong> most notab<strong>le</strong> change<br />
will take place at <strong>the</strong> confluence of two rivers.<br />
map of drainage of<br />
Also used are maps of annual river flow,<br />
All <strong>the</strong>se allowed to take into<br />
In this case <strong>the</strong><br />
region boundary follows <strong>the</strong> watershed divide between <strong>the</strong>se river catchments<br />
across <strong>the</strong> point of <strong>the</strong>ir confluence.<br />
Formula (1) may be used for <strong>the</strong> computation of flow of flat and<br />
semi-mountainous rivers with <strong>the</strong> average accuracy of 152% (for 1 500<br />
points on <strong>the</strong> U.S.S.R. rivers <strong>the</strong> deviation of computed minimum mean<br />
long-term 30-day discharge was 17-2w of <strong>the</strong> actual flow for <strong>the</strong> summer-<br />
autumn season, and for 750 points in winter it was 15%). Taking into
630<br />
account <strong>the</strong> accuracy of determining <strong>the</strong> actual data, use of formula (1)<br />
may be recommended for <strong>the</strong> computation of low flows of rivers of basin<br />
areas not <strong>le</strong>ss than 20 km2 for humid zones not <strong>le</strong>ss than 50 km2 for<br />
semi-arid zones, where <strong>the</strong> low flow volume is ra<strong>the</strong>r small and <strong>the</strong> in-<br />
fluence of various local factors is most evident.<br />
In regions with very<br />
complicated conditions of low-flow formation <strong>the</strong> area should be not <strong>le</strong>ss<br />
that 100 km2.<br />
A wide use of formula (i) in <strong>the</strong> designing practice (5, 6) paoved<br />
it reliab<strong>le</strong>.<br />
In high mountain areas <strong>the</strong> altitude of <strong>the</strong> catchment may be of<br />
a great significance as <strong>the</strong> factor ref<strong>le</strong>cting <strong>the</strong> influence of vertical<br />
zonation upon <strong>the</strong> conditions of low flow formation. Therefore, <strong>the</strong> low<br />
flow modulus for regions similar in hydrogeology etc., is related to<br />
mean basin altitude.<br />
Determination of low flow for midd<strong>le</strong>-size rivers<br />
The low flow volume of midd<strong>le</strong>-size rivers, i.e. those with area<br />
larger than <strong>the</strong> above stated critical area, but not more than 75 O00 km2,<br />
is formea under principal influence of zonal factors. The flow modulus of<br />
<strong>the</strong>se rivers varies smoothly and in accordance with geographical zonation<br />
(1-atitudinal or vertical) over <strong>the</strong> area. Therefore, low flow of midd<strong>le</strong>size<br />
rivers can be determined by maps of flow isolines, made for a certain<br />
characteristic of low flow. The flow modulus relates to <strong>the</strong> catchment<br />
centre, <strong>the</strong> interval between isolines is given in accordance with <strong>the</strong> map<br />
sca<strong>le</strong> and <strong>the</strong> value of flow variation over <strong>the</strong> area. In mountain regions<br />
<strong>the</strong> average catchment altitude is taken into account; flow isolines may<br />
not be closed, but end on <strong>the</strong> side of <strong>the</strong> mountain ridge without passing<br />
over to <strong>the</strong> o<strong>the</strong>r side (due to a great difference in wetness of slopes).<br />
Maps are plotted both for <strong>the</strong> mean and for flows of various frequencies.<br />
For instance, for <strong>the</strong> U.S.C.R. territory <strong>the</strong>re are plotted maps of <strong>the</strong> mean<br />
and S-frequency of minimum. 30-day winter and summer-autumn flows,<br />
which allows to determine <strong>the</strong> flow with <strong>the</strong> average accuracy of 10-20$.<br />
Computation of low flow of different frequencies<br />
For designing purposes <strong>the</strong> characteristics of low flows of different<br />
frequencies are of <strong>the</strong> highest importance. In <strong>the</strong> U.S.S.R. design<br />
flow of 75-97$ frequency is mainly used. Necessary values may be determined<br />
with <strong>the</strong> help of three parameters: mean flow La), coefficient<br />
variation of !Cv ' and skewness coefficient (CS).
631<br />
The second way is by <strong>the</strong> use of a transition coefficient from one<br />
fixed frequency (e.g. 75 or 8%) to ano<strong>the</strong>r. This method has been lately<br />
more and more widely used in <strong>the</strong> U.S.S.R., especially in its application<br />
to low flow, since it is more accurate and simp<strong>le</strong> than <strong>the</strong> method of three<br />
parameters, and since availab<strong>le</strong> hydrometric data allow to generalize for<br />
almost <strong>the</strong> <strong>who<strong>le</strong></strong> territory of <strong>the</strong> U.S.S.R.<br />
The advantage of <strong>the</strong> transition coefficients method is proved<br />
by <strong>the</strong> mere fact thót in this case <strong>the</strong> total mean square root error will<br />
consist of <strong>the</strong> error of <strong>the</strong> flow of fixed frequency (u 1 and <strong>the</strong> error<br />
'P<br />
of transition coefficient A , i.e.<br />
terms:<br />
i.e.<br />
When using three parameters, <strong>the</strong> same error will consist of three<br />
standard error (Qn), error of Cv (c($ and error of Cs (, Cc,),<br />
Tt is evident that <strong>the</strong> error in <strong>the</strong> second instance will be<br />
greater, and if we take into account unreliab<strong>le</strong> methods for determining<br />
coefficients C,, and Cs for any rivers, <strong>the</strong>n <strong>the</strong> advantages of <strong>the</strong> method<br />
of transition coefficients become quite obvious. It is <strong>the</strong> more so,<br />
as in <strong>the</strong> range of frequencies under consideration (7597%) <strong>the</strong> curves of<br />
low-flow frequencies are ra<strong>the</strong>r stab<strong>le</strong>, gently sloping and quite reliab<strong>le</strong><br />
in most cases.<br />
This stipulates a ra<strong>the</strong>r small (for <strong>the</strong> given frequency)<br />
variability of transition coefficient for <strong>the</strong> area and season and, con-<br />
sequently, its high reliability. Thus, for <strong>the</strong> U.S.S.R. rivers <strong>the</strong> value<br />
of transition coefficient from <strong>the</strong> minimum 30-day discharge of 8% fre-<br />
quency to <strong>the</strong> discharge of 75% frequency varies from 1.03-1.06, and for<br />
transition to <strong>the</strong> discharge of 9056 frequency - from 0.83-0.91, i.e. <strong>the</strong><br />
value of coefficient xchanges only by 5-1056 and may be averaged for <strong>the</strong><br />
given frequency over a large area. Its value varies significantly only<br />
for episodically drying or freezing rivers.<br />
The flow of fixed frequency is established by formula (1) or by<br />
maps of flow isolines, plotted for this frequency. Thus, for <strong>the</strong> U.S.S.R.<br />
territory <strong>the</strong>re are plotted maps of <strong>the</strong> minimum 30-day flow of 8056 frequency<br />
(for winter and summer-autumn periods separately) and <strong>the</strong> maps of<br />
flow of limiting season of 75% frequency.<br />
Also determined are <strong>the</strong><br />
parameters in formula (i) for <strong>the</strong> 30-day discharge of 8w and 7546 fre-<br />
quency.
63 2<br />
To determine flow of o<strong>the</strong>r frequencies, a tab<strong>le</strong> of transition<br />
coefficients ;I has been prepared.<br />
Computation of minimum daily flow is made by relationship with <strong>the</strong><br />
value of minimum 30-day flow (normal or fixed frequency flow for se<strong>le</strong>cted<br />
regions) :<br />
Q = K , Q<br />
P p,30<br />
where Op is minimum daily discharge of design frequency. Qp30 is minimum<br />
30-day discharge of corresponding frequency, determined by maps of isolines<br />
or by formula (1). k is <strong>the</strong> regional transition coefficients for <strong>the</strong><br />
given season.<br />
For <strong>the</strong> U.S.S.R. territory <strong>the</strong> value of coefficient k, when<br />
determining <strong>the</strong> minimum daily discharge of 8C$ frequency, varies from<br />
0.59 to 0.90 in winter and from 0.45 to 0.86 in summer-autumn seasons.<br />
Its value depends on <strong>the</strong> degree of river flow dep<strong>le</strong>tion for <strong>the</strong> period<br />
under study and on <strong>the</strong> volume of runoff during low-flow periods.<br />
Determination of minimum daily flow of design frequency is made<br />
by using ti:s above-mentioned coefficients A , since <strong>the</strong> frequency curve<br />
of daily and 30-day discharges vary practically in <strong>the</strong> same manner.<br />
The stated methods for <strong>the</strong> computation of probability values<br />
of river low flow are given in Gosstroy Standards of <strong>the</strong> U.S.S.R.<br />
/5,6/ and are widely used by designing organizations of <strong>the</strong> Soviet<br />
Union.
REFERENCES<br />
1. Vladimirov, A. M.: Minimalny stok rek SSSR (Minimum flow<br />
of <strong>the</strong> U.S.S.R. rivers) Hydrometeorological Publishing<br />
House, Leningrad, 1970, p. 214.<br />
2. Vladimirov, A. M.: Raschetnye minimalnye raskhody vody<br />
(Design minimum discharges) Trans. of GGI, v. 188,<br />
Hydrometeorological Publishing House, Leningrad,<br />
1972, p. 244-272.<br />
3. Kudelin, B. I. (ed.): Podzemny stok na territorii SSSR<br />
(Underground flow in <strong>the</strong> U.S.S.R. territory), MGU<br />
Publishing House, 1966, p. 303.<br />
4. Popov, O. V.: Podzemnoe pitanie rek (Underground river<br />
recharge) Hydrometeorological Publishing House,<br />
Leningrad, 1968, p. 291.<br />
5.<br />
6.<br />
633<br />
Ukazania PO oprede<strong>le</strong>nia raschetnykh minimalnykh raskhodov<br />
vody rek pri stroitelnom prooktirovanii (Instructions<br />
for determination of design minimum discharges of<br />
rivers in engineering projects). CH 346-66, Hydrometeoxdogical<br />
Publishing House, Leningrad, 1966, p. 17.<br />
Ukazania PO oprede<strong>le</strong>niu raschetnykh gidrologicheskikh<br />
’ kharakteristik (Instructions for determination of<br />
design hydrological characteristics), CH 435-72.<br />
Hydrometeorological Publishing House, Leningrad, 1972,<br />
p. 18.
ABSTRACT<br />
A STUDY ON MAXIMUM FLOOD DISCHARGE FORMULAS<br />
Tae Sang Won, PhD.CE., Dr. En.${<br />
This paper describes a new formula for <strong>the</strong> calculation of<br />
approaching velocity of rain water, and a number of new formulas<br />
for <strong>the</strong> estimation of maximum flood discharge which have been deve-<br />
loped by <strong>the</strong> author.<br />
Many empirical formulas, which have limited application,<br />
exist. However, in devising his formulas, <strong>the</strong> author derived <strong>the</strong>o-<br />
retically <strong>the</strong> form of <strong>the</strong> basic maximum discharge formula for <strong>the</strong><br />
case of rivers with no tributaries, and determined stochastically<br />
<strong>the</strong> value of <strong>the</strong> coefficients in his basic formulas using <strong>the</strong> re-<br />
cords of observed measurements. Then <strong>the</strong> author derived <strong>the</strong>oretic2<br />
lly many differe,it formulas for <strong>the</strong> case of rivers with tributa-<br />
ries to fit in <strong>the</strong> actual localities of <strong>the</strong> site under considera-<br />
tion, besides <strong>the</strong> basic formulas. So <strong>the</strong> author's formulas would<br />
be widely applicab<strong>le</strong> for rivers or sewer nets, and also for any<br />
regions, countries, with different locality. The author could con<br />
firm <strong>the</strong>se facts through <strong>the</strong> numerical examp<strong>le</strong>s. The author's fo:<br />
mulas may be used not only for estimating <strong>the</strong> design flood, but<br />
also in flood routing. The author believes that his formulas would<br />
be very helpful in <strong>the</strong> planning of water resources development pro<br />
jects -specially for those with inadequate data.<br />
RESUME<br />
L'auteur présente une nouvel<strong>le</strong> formu<strong>le</strong> pour la vitesse de con<br />
centration d'un bassin et en suggere d'autres pour <strong>le</strong> calcul du dg bit de la crue maxima<strong>le</strong>.<br />
On trouve de nombreuses formu<strong>le</strong>s empiriques dans de nombreux<br />
manuels, mais ces formu<strong>le</strong>s sont d'une application limitée. L'auteur<br />
parvient cependant à asseoir la forme de sa formu<strong>le</strong> sur des bases<br />
théoriques, lorsqu'il s'agit de cours d'eau sans affluents; il pro<br />
cede à l'évaluation des paramètres qu'el<strong>le</strong> contient par ajustement<br />
statistique aux données d'observation disponib<strong>le</strong>s. I1 generalise<br />
ensuite à différents cas de cours d'eau avec affluents. Les formu-<br />
<strong>le</strong>s proposées de'vraient pouvoir être appliquées n'importe où,<br />
aussi bien pour <strong>le</strong>s cours d'eau naturels que pour <strong>le</strong>s réseaux<br />
d'assainissement; c'es ce que l'auteur peut confirmer par des<br />
applications numériques. Les formu<strong>le</strong>s peuv:nt servir non seu<strong>le</strong>ment<br />
au calcul des crues de projet, mais aussi a celui de la propagation<br />
des crues. L'auteur pense que ses formu<strong>le</strong>s devraient rendre de<br />
grands services dans la planification de l'aménagement des eaux,<br />
spécia<strong>le</strong>ment lorsque <strong>le</strong>s données disponib<strong>le</strong>s sont insuffisantes.<br />
fg Professor of Civil Engineering, Seoul National University, Seoul,<br />
Korea.<br />
1
636<br />
I e XNTR001';TION<br />
a<br />
Char<strong>le</strong>s F. Ruff defin& Bmximm probab<strong>le</strong> floodn as follows.<br />
"The maximum probab<strong>le</strong> flood does not mean <strong>the</strong> largest flood possib<strong>le</strong><br />
but a flood so large that <strong>the</strong> chance of its being exceeded is no<br />
greater than <strong>the</strong> hazards normal to all of man's activities." The<br />
author will use here <strong>the</strong> term of 'maximum flood discharge" with <strong>the</strong><br />
same meaning of "maximum probab<strong>le</strong> flood" as defined by Ruff.<br />
It is very important to calculate maximum flood discharge cor-<br />
rectly, and also it i5 a very difficult prob<strong>le</strong>m <strong>the</strong>oretically and<br />
practically. It m y be impossib<strong>le</strong> to establish a plan for flood con-<br />
trol an8 water resources development or sewer nets projects without<br />
reckoning correctly <strong>the</strong> maximum flood discharge or <strong>the</strong> design flood.<br />
There are many methods for calculation of maximum flood dis-<br />
charge, and we have to adopt <strong>the</strong> most suitab<strong>le</strong> method in accordance<br />
with <strong>the</strong> comp<strong>le</strong>teness of <strong>the</strong> data. However,<strong>the</strong> method of calculation<br />
by <strong>the</strong> maximum flood discharge formulas,especially for <strong>the</strong> case of<br />
those with inadequate data, is easy and simp<strong>le</strong> for practicing engi-<br />
nßers. There are many empirical formulas devised by many authors<br />
such 88 Kuichling,Mead,KresnikeDickens,Metcalf and Eddy,Brix,Lauter-<br />
burg,Possenti,Buerkli-Zieg<strong>le</strong>r,Dr.Hisanaga,Kajiyama,and many o<strong>the</strong>rs.<br />
These old fosmulas have been devised empirioally and have limited<br />
application. It will be c<strong>le</strong>ar that one may be unab<strong>le</strong> to apply <strong>the</strong>m<br />
generallj.. Aliso it will not be strange to obtain results which may<br />
be 10 or liio time8 of <strong>the</strong> correct values, according to se<strong>le</strong>ction OP<br />
<strong>the</strong> coefficieqts in <strong>the</strong>se formulais when <strong>the</strong>se formulas are actually<br />
applied to practical prob<strong>le</strong>ms.<br />
Generally speaking,<strong>the</strong> flood discharge depends upon <strong>the</strong> shape<br />
of catchment,drainage area,amount of rainfall and <strong>the</strong> position at-<br />
taoked by <strong>the</strong> heavy rainfall,pemneability,slope of <strong>the</strong> catchment,<br />
shape of <strong>the</strong> water cours6,status of <strong>the</strong> surface,geological status,<br />
etc. Strictly epeaking,such statua of catchment differs from o<strong>the</strong>rs<br />
from se68on to season,for every floo8,even in <strong>the</strong> same catchment as<br />
well as in different draimge basins. In o<strong>the</strong>r words,flood disoharge<br />
üepends also upon <strong>the</strong> inteneity of Fainfall which causes <strong>the</strong> flood,<br />
duration of <strong>the</strong> rainfall and <strong>the</strong> position of <strong>the</strong> oater of <strong>the</strong> lows,<br />
or statua of <strong>the</strong> ground in case of heavy rainfall,vie.,dry ground or<br />
saturated qround,etc.<br />
As <strong>the</strong> maximum flood discharge depends upon many factors, as<br />
stated above, it may be very difficult to express it in a formula.<br />
However,if we can consider <strong>the</strong>oretioally correct value of approaching<br />
velocity of rain water and intensity of rainfall, we may deduct<br />
<strong>the</strong> maximum probab<strong>le</strong> flood by getting <strong>the</strong> rainfall for a certain districtc<br />
The princip<strong>le</strong> of derivation of <strong>the</strong> author's formulas belongs<br />
to this process, and it may be said that this is an approach differe.it<br />
from many scholars who had derived <strong>the</strong> old formulas.<br />
~<br />
* Ruff,Char<strong>le</strong>s F.;nMaximuni probab<strong>le</strong> floods in Pennsylvania Streamst'<br />
Transactions ,American Society of Civil Engineers .Vol .i06,1g4l ,p . 11 53
637<br />
In <strong>the</strong> first step, <strong>the</strong> author thought out a method to ascertain<br />
correctly <strong>the</strong> approaching velocity of rain water. At <strong>the</strong> same time,<br />
<strong>the</strong> author found that <strong>the</strong> Rizha's (Germany) formula,<strong>the</strong> only comp<strong>le</strong>te<br />
one for this purpose, could not be applicab<strong>le</strong> to solve practical<br />
prob<strong>le</strong>m8 as it gives too small values. In <strong>the</strong> second step, <strong>the</strong> author<br />
studied <strong>the</strong> rainfall intensity curve comprehensively, and found out<br />
<strong>the</strong>oretically when <strong>the</strong> maximum flood discharge may occur. In <strong>the</strong><br />
third step, <strong>the</strong> author has <strong>the</strong>oretically derived <strong>the</strong> maximum dia,<br />
charge formulas for rivers with many tributaries by appljing <strong>the</strong><br />
general ru<strong>le</strong>s which he has determined by <strong>the</strong> first and second step.<br />
In <strong>the</strong> fourth step, <strong>the</strong> author determined <strong>the</strong> discharge coefficient<br />
in his formula from <strong>the</strong> actual records. The author was <strong>the</strong>n ab<strong>le</strong> to<br />
calculate <strong>the</strong> value of <strong>the</strong> discharge coefficient, with a great degree<br />
of acouracy, of <strong>the</strong> rivers in Korea and Manchuria.<br />
II. APPROACHING VELOCITY OF RAIN WATER<br />
The approaohing velocity of rain water (U) is defined as <strong>the</strong><br />
mean velocity of rain, water approaching from <strong>the</strong> far<strong>the</strong>st point F in<br />
a river basin to <strong>the</strong> point O where <strong>the</strong> maximum flood discharge is to<br />
be ascertained,in o<strong>the</strong>r words, <strong>the</strong> mean velocity of flow between F<br />
and O (Fig-1). There is only one formula to find such approaching<br />
velocity so far expressed in equation, given by Rizha,Germany, and a<br />
tab<strong>le</strong> given by Kraven,Germany.<br />
1) RIZYA'S FORMULA<br />
0" 72 So'' (1)<br />
where<br />
a= Approaching velocity of rain water (km/hr)<br />
s = H/L<br />
H = Difference of e<strong>le</strong>vation of height between O and F<br />
L = Distance of OF (Length of water course)<br />
2) KRAVEN'S TABLE<br />
- C above 0.01 0.01 0.005 below 0.005<br />
w (kln/hr) 12.6 10.8 7.56<br />
Kraven had expressed only about approximate limite of Cd , <strong>the</strong><br />
author tried to formulate his tab<strong>le</strong>, to pass through <strong>the</strong> medium<br />
points as follows.<br />
3) THE AUTHOR'S FORMULA<br />
The author succeasfully devised a new method to determine <strong>the</strong><br />
approaching velocity of rain water e3 ,<strong>the</strong>oretically which may be<br />
applicab<strong>le</strong> for rivers where <strong>the</strong> hydrographic surveying was comp<strong>le</strong>ted.<br />
Neg<strong>le</strong>cting <strong>the</strong> princip<strong>le</strong> and <strong>the</strong> process of derivation here, <strong>the</strong> result<br />
of <strong>the</strong> author's formula is illustrated as follows.<br />
(2)
638<br />
The value of for rivers in Manchuria calculated using eq(3) is<br />
shown in Tab<strong>le</strong>-i. As we see Tab<strong>le</strong>-1, <strong>the</strong> value of lies between 133<br />
and í77, and we may be ab<strong>le</strong> to recognize that <strong>the</strong> Rizhass formula wil<br />
not be of practical use because of <strong>the</strong> reason that <strong>the</strong> salue of iCt in<br />
his formula is too smll,apparently,compared with <strong>the</strong> normal salues.<br />
ide can also see from Tab<strong>le</strong>-1 that <strong>the</strong> value of K represents <strong>the</strong><br />
bottom slope OP <strong>the</strong> hydraulic siope of <strong>the</strong> river, and this fact coin-<br />
cides with practice. Also it can be seen that <strong>the</strong> value of &, de-<br />
ureases gradually according aa approaching <strong>the</strong> downstream of a river,<br />
and this fact skok's that <strong>the</strong> bottom slope or <strong>the</strong> hydraulic slope of<br />
a rivep generally decreases gradually as we approach <strong>the</strong> downstream.<br />
III. RAPFALL INTENSITY CURVE<br />
h'e can express <strong>the</strong> rainfall intensity by <strong>the</strong> following equation.<br />
Tab<strong>le</strong>-I. The values of & and o<strong>the</strong>rs fop rivers in Manchuria<br />
Name of Rivers<br />
/c Range of S<br />
(400)<br />
Tumen 8.<br />
277 4.82 - 8,06 33,400 487.6 68.50. .LW<br />
Whancheng R.<br />
207 6.72 - 9.64 4,000 160.7 24.92 .l52<br />
Rohe H.<br />
177 3.54 - 4.22 31,455 444.0 70.84 -159<br />
Seasamorin R.<br />
171 2.81 - 2.97 29,927 412.0 72.64 .i76<br />
Sea<strong>le</strong>og R.<br />
149 8.37 - 3.48 510165 767.0 66.71 .O86<br />
Tong<strong>le</strong>og 3.<br />
207 1.70 - 3.16 10,318 33345 313.13 -093<br />
The upatrean of <strong>the</strong> 150 1.83 - 2.30 178,699 . 1040.5 171.74 .O65<br />
min Lacg R.<br />
The midd<strong>le</strong> of <strong>the</strong> Leog 158 1.62 - 1.77 187,250 1199.0 157.49 0132<br />
GhsnF: R.<br />
159 3.90 - 5.51 4,958 163.0 30 . 42 .i86<br />
Icwaslg R.<br />
137 4-36 - 5.26 2,129 94.0 22.65 231<br />
Van R.<br />
149 7.07 -18,66 1,072 102.5 10.46 .lo2<br />
Pa B.<br />
150 3.68 4.06 2,361 178.0 13.26 e 074<br />
Csnkai R.<br />
134 1.43 - 1.96 515 51.0 10.10 .198<br />
F:catchmtmt area<br />
I = ß/(t + 1 ( 4) L:<strong>le</strong>ngth of main water oourse<br />
wher0<br />
t= Bwatim<br />
I = Average intensity of rainfall during duration t<br />
OC,,^ = Any-constant<br />
Eq(4) represents a kind of hyperbola, and <strong>the</strong> constants a ar3.B can<br />
be found by eq(5) by <strong>the</strong> princip<strong>le</strong>s of <strong>the</strong> method of <strong>the</strong> <strong>le</strong>ast squame<br />
n(12t) - (ï)(ït)<br />
d= LI)' - n(I')<br />
B=<br />
(Il(P2t) - (It)(12)<br />
(I)~ - n(I')<br />
E= nwnber of observations<br />
Next <strong>le</strong>t R be total. amount of rainfall aurin$ <strong>the</strong> buration t,<br />
R = It = p t/(t + QL 1 (4)
639<br />
T~b<strong>le</strong>-2 illustrates <strong>the</strong> values of <strong>the</strong> constants d and fi in eq(4)<br />
for various regions. In this tab<strong>le</strong>, those for <strong>the</strong> regions of Korea<br />
and Manchuria show <strong>the</strong> absolute maximum rainfall intensity curves<br />
during those periods, Those for <strong>the</strong> regions marked with <strong>the</strong> asterisk<br />
(*I were calculated by <strong>the</strong> author himself by <strong>the</strong> records of <strong>the</strong> re-<br />
cording gauges.<br />
IV. THE AUTHOR'S MAXIMUM FLOOD DISCHARGE FORMULAS<br />
1) FUNDAMENTAL FORMULA FOR THE CASE OF A RIVER WITH NON-TRIBUTARY<br />
(a) Retardation of Run-off<br />
Tab<strong>le</strong>-2. The values of cc and P in eq(4)<br />
Region d a<br />
(min) (hour)<br />
P b <strong>the</strong> records (minutes)<br />
Period taken Range of t<br />
Seoul ,K 59 0.938 7,860 131 1905 - 1920 5-60 min<br />
Inchon , K 37.5 0.625 8,640 144 ditto 5-240<br />
PymgYanR , K 41 0.683 6,000 100 1914 - 1920 ditto<br />
Pusan , K 106.1 1.77 14,015 233.6 1914 - 1953 10min-24hr<br />
Wonsan , K 75 1.250 7,740 129. 1914 - 1920 5-240 min<br />
Taegu,K * 40.2 0.67 8,711 145.2 1929 - 1953 10min-24hr<br />
Chonj:i,K * 81.1 1.35 15,160 252.7 1918 - 1954 ditto<br />
Kwangju,K * 90.4 1.51 10,866 181.2 1938 - 1954 ditto<br />
Mokpo,K * 101.8 1.70 11,398 190 1916 - 1953 ditto<br />
ChmgCheng,): * 40.5 0.675 5,929 98.8 1937 - 1943 10min-48hr<br />
Sping,r? * 45.1 0.752 8,487 141.4 i934 - 1944 ditto<br />
Tokyo, J 50 0.833 5,500 91.6 1891 - 1911 5-60min<br />
where a d/60 b P 1'3 /60 KæKorea M=Manchuria JsJapan<br />
Prior to deecribing <strong>the</strong> flood discharge formula, <strong>the</strong> definition<br />
of "retardationR must be understood. Now <strong>le</strong>t O and F be <strong>the</strong> point<br />
under coneiäeration and <strong>the</strong> far<strong>the</strong>st point of a catchment respective-<br />
ly, 1 be <strong>the</strong> <strong>le</strong>ngth of water course between O and F , CL) <strong>the</strong> approaoh-<br />
ing velocity of rain water flowing from F to O, tc <strong>the</strong> time of con-<br />
centration,i.s.,<strong>the</strong> time necessary for reaching O from F, tr <strong>the</strong> du-<br />
ration OP rainfall,i.e.,<strong>the</strong> perlob between <strong>the</strong> beginning and enâing<br />
of a reinfall (see Fig-i), <strong>the</strong>n,<br />
t, = l/Ld (79<br />
T tr + t c * tr + l/W í 8)<br />
where T P The time of <strong>the</strong> period between <strong>the</strong> beginning of a rain-<br />
fall and <strong>the</strong> ending of <strong>the</strong> run-off due to <strong>the</strong> rainfall<br />
at <strong>the</strong> point O<br />
It may be better to use <strong>the</strong> author's formula for determining tc<br />
When it becomes tr
640<br />
falling at F reached O, <strong>the</strong> rainfall would have ceased. In o<strong>the</strong>r<br />
words, <strong>the</strong> rainfall causing <strong>the</strong> maximum flood discharge is <strong>the</strong> rain-<br />
fall which falls in a part of <strong>the</strong> catchment.<br />
(b) Fundamental Formula for <strong>the</strong> case of non-Retardation<br />
The fundamental princip<strong>le</strong>s of <strong>the</strong> author's maximum flood dis-<br />
charge formulas have already been described. The author adopted de-<br />
ductive and inductive <strong>the</strong>ories for <strong>the</strong> derivation.<br />
Since it is unab<strong>le</strong> to derive <strong>the</strong> rational equation of <strong>the</strong> flood<br />
discharge hydrograph, <strong>the</strong> author expressed <strong>the</strong> peak of <strong>the</strong> flood<br />
discharge hydrograph for <strong>the</strong> case of non-tributary and non-retardation<br />
by <strong>the</strong> following equation.<br />
- T = Duration of flood tr+ tc<br />
qm- 9. a CfA R / T (9)<br />
where<br />
qm = Peak discharge in flood time<br />
qo = Discharge of run-off in normal time<br />
tr = Duration of rainfall<br />
tc = Approaching time or time of concentration<br />
R = Total amount of rainfall during duration of tr<br />
A = Catchment area<br />
9 = Average run-off factor<br />
C P A coefficient depending upon <strong>the</strong> shape of <strong>the</strong> flood<br />
discharge hydrograph<br />
Now <strong>le</strong>t Fig-,? show a discharge hydrograph during a flood period.Then<br />
<strong>the</strong>, peak disoharge qm-qO may be represented by eq(9)and <strong>the</strong> product<br />
AR of eq(9) shows <strong>the</strong> total run-off during <strong>the</strong> period of flood<br />
T. As this also represents <strong>the</strong> area of DMED of Fig-2 geometrically,<br />
we may affirm that <strong>the</strong> authorls fundamental formula is reasonab<strong>le</strong><br />
anal tically or graphically. Replacing <strong>the</strong> value of R of eq(6) into<br />
ea(9 9 ,<br />
qm - qo=C 9 AB / T = C 9 A b tr /(tr + t, )(tr+ a (10)<br />
We h ow through eq(1O) that <strong>the</strong> peak discharge is a function of tr,<br />
anä it will take a limiting value to make <strong>the</strong> peak discharge maximum.<br />
So differentiating eq(l0) with respect to tr<br />
and.<br />
tr = (unit in hours) (11)<br />
T=t,+ tG= 6 +t, (12)<br />
Hence we know that <strong>the</strong> maximum flood discharge will occur when <strong>the</strong><br />
duration of rainfall t r satisfies eq(l1). Up-to-datesue have taken tr<br />
generally without definite reason as follows: 5 or 10 minutes for de-<br />
sign of sewers, 3 or 4 hours for small rivers flowing <strong>the</strong> vicinity of<br />
a city, 24 home or more for big rivers. However aceording to <strong>the</strong><br />
author's <strong>the</strong>ory, <strong>the</strong> value of tr must satisfy eq(l1) to cause <strong>the</strong><br />
maximum flood discharge. Substituting eq(l1) into eq(lO),
641<br />
If we express in metric units,i.e. ,A(km2),R(~) ,t (hr) ,q (cms), (13)<br />
becomes,<br />
qm-q,= 0.2778C 9 b a A / (t,+fic)( a+<br />
1<br />
where (14)<br />
a,b = any constants depending upon rainfall (see Tab<strong>le</strong>-2)<br />
The value of tc can be found from eq(3) .(see Tab<strong>le</strong>-1)<br />
(c) The value of <strong>the</strong> coefficient C<br />
As stated above, <strong>the</strong> coefficient C in eq(9) depends upon <strong>the</strong><br />
8hape of <strong>the</strong> discharge hydrograph of <strong>the</strong> region . The relation be-<br />
tween <strong>the</strong> kinds of <strong>the</strong> cuPve consisting <strong>the</strong> discharge hydrograph and<br />
<strong>the</strong> value of C is illustrated deductively as fOllOW6.<br />
Tab<strong>le</strong>-3. The value of C found by deduction<br />
Kind of curve C kind of curve C<br />
parabol a 1.5 cosine curve 2.0<br />
triang<strong>le</strong>(strai ht 2.0 probability 2.394<br />
1 ins? curve<br />
Also <strong>the</strong> value of C can be calculated inductively from a dis-<br />
charge hydrograph by using eq(91,whioh gives,<br />
C 0 (qhi-Qo)T /$ARa(Qm=Q,)T/ V<br />
(15)<br />
nhere<br />
V = The volume of run-off represented by <strong>the</strong> area DMED of Fig-2.<br />
The value of C for <strong>the</strong> rivers in Manchuria found by <strong>the</strong> author using<br />
eq(l5) are given in Tab<strong>le</strong>-4.<br />
Tab<strong>le</strong>-4. The value of C for Manchrian rivers found by induction<br />
Name of river Site of Duration of flood taken Value Value<br />
measurement from <strong>the</strong> records<br />
of' T of .c<br />
( hr , day-hr ,day, month, r<br />
Tong<strong>le</strong>og Ho Tida<strong>the</strong>rgtse 15,3rd-21 ,4th,Aug,l9ii 30 1.664<br />
n<br />
Sankankeu 12,10 .e 5,21 ,Sep,1939 257 1 .697<br />
Whan Ho P eidakeng 15,24. -1 9 , 27, AUg,l94O 76 1.543<br />
Main stream C hengs enkong 19,2nd- 6,6th,Sep,1939 83 2.090<br />
of Leog Ho<br />
ditto<br />
ditto 797th- 7,103 Se~r1939 72 1.966<br />
Taitse Ho Whelongbo 12,31Jul-3,3~,Aug,1940 63 1.754<br />
n<br />
n<br />
9,4th-i7,6th,A~g,l 940 56 1 0975<br />
I<br />
n<br />
16,6th-l9,8th, I<br />
51 1.087<br />
n<br />
n<br />
17,2nd- 8 s 5thSSQp ,1939 63 2.137<br />
n<br />
H<br />
9,jth-l6,9th, " " 103 1.806<br />
n<br />
5,6th-l3,9th, Jul , 80 2 . 204*<br />
* show6 <strong>the</strong> value oalculated by estimation because of non-measurement<br />
at <strong>the</strong> vicinity of <strong>the</strong> peak discharge.<br />
(d) The fundamental formula for <strong>the</strong> case of retardation of flow
642<br />
The basic formula for <strong>the</strong> case Of non-retarclation ,mentioned<br />
above, is applicab<strong>le</strong> for <strong>the</strong> case of retardation of flow,too. But it<br />
is necessary to multiply <strong>the</strong> coefficientp due to retardation, viz.,<br />
q,-qo=yCpbf& A /(tc+GL )(a+ Gc 1 (16)<br />
p = f(tc/tr) (17)<br />
It is c<strong>le</strong>ar that <strong>the</strong> value of <strong>the</strong> Coefficient /3 equals to 1 for <strong>the</strong><br />
case of non-retardation, but it beoomes <strong>le</strong>ss than 1 for <strong>the</strong> case of<br />
retardation. The value of p varies inversely with that of tc / t, .<br />
It is necessary to find out a general form of f(tc/tr) for<br />
practical calculation. So <strong>the</strong> author tried to find out <strong>the</strong> general<br />
form of <strong>the</strong> function f(tL/ty) atoohastically using some data ob tained for rivers in Korea by some o<strong>the</strong>r methods. The author would<br />
like to assume <strong>the</strong> general form of <strong>the</strong> function ofp as follows.<br />
J)= (1 + k 1 / ( tc/t,+ k 1<br />
where k = Any constant<br />
Finding <strong>the</strong> value of k in above equation by <strong>the</strong> method of <strong>the</strong> <strong>le</strong>ast<br />
squares, we get k = 4.802 . Accordingly,<br />
p= 5.802 / (tc /tr+ 4.802) (18)<br />
2) 'THE MAXIYUM FLOOD DISCHARGE FORMULAS FOR THE CASE OF RIVERS<br />
WITH TRIBCTARIES<br />
(a) The maximum flood discharge st <strong>the</strong> confluence of a trlbutary<br />
The author found that existence of tributaries affect greatly<br />
<strong>the</strong> peak discharge of flood flow at <strong>the</strong> proposed site of <strong>the</strong> main<br />
stream. Su <strong>the</strong> author derived many different formulas of maximum dis-<br />
charm for <strong>the</strong> case of rivers with tributaries, besides <strong>the</strong> basic<br />
formula for <strong>the</strong> case of thoee with non-tributary. Therefore it would<br />
be said that this is a great approach different from many scholars<br />
who never considered <strong>the</strong> Influence of tributaries in <strong>the</strong>ir tradition-<br />
al formulas.<br />
KOW assume one of <strong>the</strong> simp<strong>le</strong>st case as Fig-3. The discharge<br />
hydrograph for this case may be illustrated as Fig-&. The value of q,<br />
in ~ig-4 shows <strong>the</strong> peak discharge of <strong>the</strong> triùutaryíI), and <strong>the</strong> value<br />
of q2 shows that of <strong>the</strong> main river (II) alone,excluding that of <strong>the</strong><br />
tributary(1) , also Qm shows that of <strong>the</strong> composed maximum discharge<br />
to be occurred at <strong>the</strong> proposed site. The rational equation of <strong>the</strong><br />
curve ,i.e.,<strong>the</strong> true shape of <strong>the</strong> discharge hydrograph is unknown.<br />
But <strong>the</strong> author would like to discuss about <strong>the</strong> shape of <strong>the</strong> curve in<br />
<strong>the</strong> following. Let us consider two cases, one of <strong>the</strong>m <strong>the</strong> simp<strong>le</strong>st<br />
case,i.e.,<strong>the</strong> case assumed that <strong>the</strong> discharge hydrograph consista of<br />
an isosce<strong>le</strong>s triang<strong>le</strong>, and <strong>the</strong> o<strong>the</strong>r <strong>the</strong> case assumed that it consists<br />
of a parabolio ourve, to seek <strong>the</strong> effect of <strong>the</strong> nature of <strong>the</strong><br />
discharge hydrograph which influences on <strong>the</strong> peak disoharge Qn
(i) 'The case of an isosce<strong>le</strong>s triang<strong>le</strong><br />
In this case, it evident from Fig-5,<br />
TE<br />
Um" qp+q,(2 --1<br />
T,<br />
(ii) The case of a parabola<br />
(19)<br />
643<br />
Since it is evident as <strong>the</strong> nature of <strong>the</strong> parabola,at Fig-6,<br />
q = 4qot/T - 4q,(t/TI2 í a)<br />
we can get <strong>the</strong> following equation for Fig-?,<br />
and by dQ/dt = O Q,/TI + q2 /T2<br />
to= 2 (q,/T,' + q,/T:)<br />
Accordingly, substituting eq( 201 into eq( b) , we get<br />
NUMERlCAL EXAMPLE<br />
(20)<br />
An ilìwtration is given here to compare <strong>the</strong> degree of accuracy<br />
of <strong>the</strong> two cbses mentioned above.<br />
Given T2 = 26 hr, TI = 20 hr, = 5000 cms, q = 3000 cms . Then since<br />
Te/T,= 26/20 = 1.3 from eq81) , <strong>the</strong> case of assuming as parabolic<br />
curve, &TA= (5000 + jOOOx1.3<br />
/( 5000 + 3000x1.3x1.3 1 -<br />
7865 cms<br />
Next from eq(19), <strong>the</strong> straight line formula,<br />
Qm = 5000 + 3000x(2 - 1.3) = 7100 cms<br />
Hence,we h ow that <strong>the</strong>re is not any remarkab<strong>le</strong> difference on <strong>the</strong> re-<br />
sults of calculation of <strong>the</strong> maximum discharge whe<strong>the</strong>r we assume <strong>the</strong><br />
discharge hydrograph as straight lines or a parabolic curve through<br />
this numerical examp<strong>le</strong>. Aliso we can imagine that we shall obtain <strong>the</strong><br />
similar results with this numerical examp<strong>le</strong> even in <strong>the</strong> cases we<br />
adopt Borne o<strong>the</strong>r ourves else than parabola for <strong>the</strong> discharge hydro-<br />
graph,e.g.,cosine or probability curve. But adopting <strong>the</strong> oase as-<br />
sumed as a parabolic curve is safer,easier to hand<strong>le</strong>,& reasonab<strong>le</strong>.<br />
So <strong>the</strong> author would like to suggest those of <strong>the</strong> parabolic curve as<br />
<strong>the</strong> general formula in this paper.<br />
(b) The maximum flood discharge formula at <strong>the</strong> confluence for <strong>the</strong><br />
oaee of a river where n-1 tributaries flow into <strong>the</strong> confluence<br />
(F ig-8 1<br />
If we assme <strong>the</strong> discharge hydrograph consists of a parabolic<br />
curve, by <strong>the</strong> srne pricip<strong>le</strong> with that in <strong>the</strong> previous paragraph, we<br />
(0) The maximum flood discharge at <strong>the</strong> proposed site which is<br />
located <strong>the</strong> downstream of a tributary
644<br />
Now <strong>le</strong>t O is <strong>the</strong> proposed site, O' <strong>the</strong> confluence of <strong>the</strong> tributary<br />
in Fig-10, and t, is <strong>the</strong> necesssry tirne for reaching of rain<br />
water from O' to O. If we assume <strong>the</strong> discharge hydrograph consists<br />
of a parabolic curve.FiR-11 ,<strong>the</strong>n<br />
Q = Q;,+ qiez4q, (t-t, 1 /TI - 4 qi( t - t , l2 / TF+4 qr t/T+ - 4 kt2 /Ti (<br />
(d) The maximum flood discharge at 8 proposed site where n-1 tribu-<br />
taries join to <strong>the</strong> main river at its upstream side.(Fig-12)<br />
if we assume <strong>the</strong> discharge hydrograph consists of 8 parabolic<br />
curve, by <strong>the</strong> same princip<strong>le</strong> with that in <strong>the</strong> previous paragraph, we<br />
(e) The ma>;Smum flood discharge at a proposed site where m tribu-<br />
taies flow into this site and n-1 tributaries join to <strong>the</strong><br />
main river at its upetream side. (Fig-14)<br />
This is <strong>the</strong> most general case. If we assume <strong>the</strong> discharge hydro-<br />
graph consists of a parabolic curve, by <strong>the</strong> same princip<strong>le</strong>s, we get<br />
V. CONCLUSION<br />
The maximum flood discharge generally increase toward âown-<br />
stream, a6 <strong>the</strong> result of increment of <strong>the</strong> drainage area. But as <strong>the</strong><br />
approaching time also increases approaching down stream,in o<strong>the</strong>r<br />
words,ss <strong>the</strong> nearer approaching downstrem,<strong>the</strong> greater effect of re-<br />
tardatlon. Accordingly <strong>the</strong> rate of increment of <strong>the</strong> peak discharge<br />
decreases generally approaching downstream; and sometimes,i.e.,in<br />
such oa8e18 where <strong>the</strong> approaching time remarkably increases compared<br />
with <strong>the</strong> inorement of <strong>the</strong> drainage area, not only <strong>the</strong> rate but also<br />
<strong>the</strong> actual absolute value of <strong>the</strong> peak discharge decreases at <strong>the</strong> dom<br />
stream than those of <strong>the</strong> upstream. These fsots are experienced some-<br />
times in practice, In such cases,it was impossib<strong>le</strong> to expreee this<br />
fa& by <strong>the</strong> old formulas. However by <strong>the</strong> author's formulas, it is<br />
ìGYi
645<br />
easy am2 <strong>the</strong>oretically sound to express this fact. Because as we see<br />
<strong>the</strong> author's basic formulas-eq( 9)-(14) , which represent <strong>the</strong> drainage<br />
area A in <strong>the</strong> numerator and <strong>the</strong> factor of <strong>the</strong> approaching time tc in<br />
<strong>the</strong> denominator. So it may also be said that <strong>the</strong> author's formulas<br />
are very <strong>the</strong>oretical from <strong>the</strong> point of view of this fact.<br />
As mentioned above,<strong>the</strong> author derived <strong>the</strong>oretically,i.e.,ration-<br />
ally or stochastically many formulas of maximum flood discharge- <strong>the</strong><br />
basic formulas for <strong>the</strong> case of a river with non-tributary and many<br />
o<strong>the</strong>r different formulas for <strong>the</strong> case of rivers with tributaries. Be-<br />
cause <strong>the</strong> author found that <strong>the</strong> existence of tributaries affect great-<br />
ly not only <strong>the</strong> peak discharge but also <strong>the</strong> entire shape of <strong>the</strong> dis-<br />
charge hydrograph at <strong>the</strong> proposed site of <strong>the</strong> downstream. Consequent-<br />
ly it may be posaib<strong>le</strong>,by applying <strong>the</strong> author's formulas,to find <strong>the</strong><br />
real shape of <strong>the</strong> discharge hydrograph at <strong>the</strong> point under consider-<br />
ation to be occurreti in some flood time.<br />
Some scholars advocate that <strong>the</strong> actual shape of <strong>the</strong> flood dia-<br />
charge hydrograph resemb<strong>le</strong>s to ~ig-16. On <strong>the</strong> o<strong>the</strong>r hand,some o<strong>the</strong>r<br />
scholars insist that it should be resemb<strong>le</strong>d to Fig-17. But <strong>the</strong> author<br />
should say that <strong>the</strong>se <strong>the</strong>ories both advocated by <strong>the</strong> traditional<br />
scholars are those have not been touched to <strong>the</strong> core of <strong>the</strong> true <strong>the</strong>o-<br />
ries. The real shape of <strong>the</strong> discharge hydrograph depends upon <strong>the</strong> lo-<br />
cality Qf <strong>the</strong> point under consideration,in o<strong>the</strong>r words, it depends on<br />
<strong>the</strong> relat4ve position of <strong>the</strong> proposed point and those of <strong>the</strong> conflu-<br />
ences of th? tributaries on <strong>the</strong> mainstream under consideration. Conse-<br />
quently it is resemb<strong>le</strong>d to ~ig-16 in some cases, and also it takes a<br />
shape resemb<strong>le</strong>el to Fig-17 in some casessin accordance with <strong>the</strong> locali-<br />
ty of <strong>the</strong> point under consideration. As stated above,it would be ab<strong>le</strong><br />
to show <strong>the</strong> real shape of <strong>the</strong> discharge hydrgraph just fitted in <strong>the</strong><br />
locality of <strong>the</strong> proposed site by applying <strong>the</strong> author's formulas.<br />
The author's formulas also would be applicab<strong>le</strong> not only for <strong>the</strong><br />
purpose of reckoning of <strong>the</strong> design flood, but also for that of esti-<br />
mation of <strong>the</strong> flood routing for some floods. In <strong>the</strong> case of flood<br />
routing,it would be possib<strong>le</strong> to obtain more correct results by taking<br />
<strong>the</strong> real value fop tr instead of that calculated from eq(l1) in some<br />
cases,i.e.,<strong>the</strong> real value of tr is greatly different from that calcu-<br />
lated from eq(i1).<br />
The author's formulas would be widely applicab<strong>le</strong> for rivers or<br />
sewer nets,& also for any regions,countries with different locality,<br />
and it would be possib<strong>le</strong> to obtain correct and accurate results by<br />
se<strong>le</strong>cting or assuming <strong>the</strong> values of <strong>the</strong> coefficieuits in his fQrUIUlaS<br />
appropriately. Aceoräingly <strong>the</strong> author should like to suggest that <strong>the</strong><br />
author's formulas shall be applied in practioe in many regions and<br />
also for many purpose8 as far as possib<strong>le</strong>.
(p<br />
646<br />
F ¡y -I<br />
e<br />
c, 3<br />
O<br />
Fig- 7<br />
e<br />
t<br />
-+<br />
O<br />
Fi 9 -2<br />
Fig- 8<br />
Y<br />
ot P<br />
O<br />
Fìg -9<br />
Fij-3<br />
t
647
THE COST-EFFECTIVENESS OF WATER RESOURCES SYSTEMS<br />
CONSIDERING INADEQUATE HYDROLOGICAL DATA<br />
Nathan Buras, Ph.D.<br />
The Lowdermilk Faculty of Agricultural Engineering<br />
Technion - Israel Institute of Technology, Haifa, Israel<br />
Introduction.<br />
The question of how much hydrological information is<br />
necessary for <strong>the</strong> design of water resources systems has not<br />
been answered satisfactorily as yet. Perhaps this question<br />
does not admit of a unique answer, but ra<strong>the</strong>r of a range with-<br />
in which <strong>the</strong> specific solution to a given situation may be<br />
found .<br />
In general, one can state intuitively that <strong>the</strong> cost of<br />
a water resources project decreases with <strong>the</strong> amount of avail-<br />
ab<strong>le</strong> hydrological data. For examp<strong>le</strong>, a longer hydrological<br />
trace at a given reservoir site will yield improved estimates<br />
of mean annual discharges and of extreme flows, so that <strong>the</strong><br />
dimensions of <strong>the</strong> dam and of <strong>the</strong> spillway may be reduced for<br />
a given probability of failure during <strong>the</strong> same period of time.<br />
On <strong>the</strong> o<strong>the</strong>r hand, additional hydrological data irivolve in-<br />
creased cost, not only in terms of more gauging stations and<br />
of <strong>the</strong> attendant manpower, but also in terms qf 'osts incurred<br />
to <strong>the</strong> society by delaying <strong>the</strong> design and <strong>the</strong> covistruction of<br />
<strong>the</strong> project until more data is col<strong>le</strong>cted and processed. Schem-<br />
atically, one can show <strong>the</strong>se two cost functions as two curves<br />
intersecting in <strong>the</strong> data-cost space (Figure 1). However, of<br />
practical importance are not <strong>the</strong> individual cost curves, but<br />
<strong>the</strong> parabola which is <strong>the</strong> sum of <strong>the</strong> two functions. We shall<br />
define, <strong>the</strong>refore, as adequate hydrological data <strong>the</strong> amount<br />
of hydrological information corresponding to tho niinimum<br />
ordinate of <strong>the</strong> total cost curve. This definition impli-s<br />
that hydrological data in excess of this amount are as '.nade-<br />
quate as those which are short of it: indeed, <strong>the</strong> effort put<br />
in obtaining this additional information may increase <strong>the</strong> total<br />
cost of <strong>the</strong> project. For this reason, we recommend <strong>the</strong> use of<br />
<strong>the</strong> terms insufficient data for <strong>the</strong> information <strong>le</strong>ss than ade-<br />
quate, and redundant data for <strong>the</strong> information in excess of <strong>the</strong><br />
point of adequacy.<br />
The prob<strong>le</strong>m of adequate hydrological data is part of <strong>the</strong><br />
broader issue of planning water resource; s@erns. Within this<br />
enlarged context, <strong>the</strong> hydrological data is but one of <strong>the</strong><br />
several planning variab<strong>le</strong>s, <strong>the</strong> o<strong>the</strong>rs being socio-economic<br />
considerations, organizational and i.nstitutiona1 structures,<br />
political constraints, and so on. The ro<strong>le</strong> of <strong>the</strong> hydrological<br />
data in a comp<strong>le</strong>x water resources system was investigated relative<br />
to <strong>the</strong> water quality in <strong>the</strong> Potomac estuary [I]. In this<br />
analysis, four planning variab<strong>le</strong>s were considered: (a) hydrological<br />
inputs; (b) models of <strong>the</strong> dissolved oxygen fluctuations
650<br />
in <strong>the</strong> estuary; (c) economic projections of <strong>the</strong> region serviced<br />
by <strong>the</strong> water resources system; (d) water quality objectives in<br />
<strong>the</strong> estuary. Under <strong>the</strong> specific conditions of <strong>the</strong> Potomac, it<br />
was found that <strong>the</strong> performance of <strong>the</strong> planned water resources<br />
system was most sensitive to <strong>the</strong> economic projections, and<br />
<strong>le</strong>ast sensitive to <strong>the</strong> hydrological planning variab<strong>le</strong> (10-<br />
year and 50-year sequences of hydrological data).<br />
Sufficiency of hydrological data.<br />
It does not seem that <strong>the</strong>re is today a generally accepted<br />
method for <strong>the</strong> evaluation of <strong>the</strong> amount of hydrological data<br />
with respect to <strong>the</strong>ir adequacy for planning water resources<br />
systems. However, <strong>the</strong> prob<strong>le</strong>m was recognized for some time and<br />
several approaches toward its solution were developed. One such<br />
approach, based on <strong>the</strong> concept of information content of <strong>the</strong><br />
observed data [2], is oriented toward <strong>the</strong> determination of an<br />
opt,imal .<strong>le</strong>twork of hydrological stations in a region. A some-<br />
what similar approach is based on minimizing <strong>the</strong> sum of variances<br />
of <strong>the</strong> estimates of <strong>the</strong> mean flows at gaging stations in a hydro-<br />
logical network subject to a budgetary constraint [3]. All <strong>the</strong>se<br />
approaches attempt, in fact, to devise optimal strategies of<br />
hydrc logical sampling.<br />
However, when considering <strong>the</strong> L msequences of inadequate<br />
h,droiogical data on <strong>the</strong> cost and effectiveness of water resources<br />
Ftrur?l,!res anfi FroJects, it secas thnt <strong>the</strong> scope of <strong>the</strong> analysis<br />
Iza> to be broadened. This analysis takes into account not only<br />
ali pcc:;ib<strong>le</strong> samp<strong>le</strong> results, but also computes <strong>the</strong> expected<br />
worth 3r expected opportunity loss) of a strategy which assumes<br />
that <strong>the</strong> best decisions (regarding <strong>the</strong> various components of a<br />
water resouI-ces system - to construrit or not to construct) are<br />
dependent upon <strong>the</strong> information content of <strong>the</strong> observed samp<strong>le</strong>.<br />
This approach is cal<strong>le</strong>d preposterior anal sis [4], because,<br />
&hough carried out before <strong>the</strong> samp<strong>le</strong> + in ormation is obtained,<br />
it attempis + assesserior probabilities derived on a particular<br />
sapl e rutcome.<br />
A simp<strong>le</strong> examp<strong>le</strong> will illustrate <strong>the</strong> preposterior analysis.<br />
Suppose that <strong>the</strong> Development Authority of region A<strong>le</strong>ph is considering<br />
thF construction of a major dam. However, <strong>the</strong> Authority<br />
wants ils~ to appraise <strong>the</strong> advisability of obtaining additional<br />
hydrological data thus delaying <strong>the</strong> planning and imp<strong>le</strong>mentation<br />
schedu<strong>le</strong> by a few years. It is estimated that total costs involved<br />
in obtaining <strong>the</strong> additional data, including costs generated by <strong>the</strong><br />
non-availability of water and water derivatives at <strong>the</strong> dam site
651<br />
6<br />
during <strong>the</strong> additional time period, are 15 x 10 Monetary Units<br />
(in short, 15 MMü). The contemplated structure needs an invest-<br />
ment of 160 MMU, whi<strong>le</strong> <strong>the</strong> present value of <strong>the</strong> stream of net<br />
benefits generated by it would add up to 200 W.<br />
The Authority has two options:<br />
al: build <strong>the</strong> dam<br />
a2:<br />
do not build <strong>the</strong> dam<br />
with <strong>the</strong> possib<strong>le</strong> outcomes<br />
el: <strong>the</strong> project is successful<br />
9,: <strong>the</strong> project is a failure.<br />
On <strong>the</strong> basis of past experience and with <strong>the</strong> help of a<br />
firm of consulting engineers, <strong>the</strong> Authority reaches <strong>the</strong> con-<br />
clusion that <strong>the</strong> prior probabilities of success or failure are<br />
p(el) = 0.25<br />
p(e2) = 0.75.<br />
On <strong>the</strong> basis of <strong>the</strong> existing data <strong>the</strong> prior expected<br />
opportunity losses (EOL) can be computed as follows:<br />
Tab<strong>le</strong> 1.<br />
Calculation of Prior Expected 0pportimj.ty Losses<br />
a,: build <strong>the</strong> dam<br />
Probability Opportunity Loss, Wej-giited Oppor-<br />
Out come p(e; - MMu tunity Loss, MNRT<br />
el: success O<br />
û2: failure 150<br />
O<br />
120<br />
m<br />
EOL (u,) -- 120 MMU<br />
a,: do not build <strong>the</strong> dam<br />
Outcome<br />
Probability<br />
p(e4 - )<br />
Opportunity Loss,<br />
ndMu<br />
Weighted Opportunity<br />
LGSS, MMU<br />
el: success<br />
û2: failure<br />
200<br />
O<br />
50<br />
O<br />
EOL (a,) = 50 MMü<br />
opt EOL = EOL (a,) = 50 MMü<br />
5u
652<br />
Thus, with no additional information, <strong>the</strong> best decision<br />
would be not to build <strong>the</strong> dam. In this way, region A<strong>le</strong>ph would<br />
forfeit only 50 NIMU, <strong>the</strong> expected opportunity loss.<br />
Now <strong>the</strong> Development Authority turns to its Hydrological<br />
Service asking its advice regarding <strong>the</strong> nature and usefulness<br />
of <strong>the</strong> additional information which may be obtained& <strong>the</strong> cost<br />
of 15 MMU. The attitude of <strong>the</strong> Hydrological Service is that. by<br />
and large <strong>the</strong> additional data would yield one of <strong>the</strong> following<br />
three types of indications regarding <strong>the</strong> effectiveness of <strong>the</strong><br />
reservoir (in terms of streamflow regulation, hydropower gener-<br />
ation, flood contral, etc.):<br />
X1: increase in effectiveness<br />
X2: no change<br />
X3: decrease in effectiveness.<br />
These variab<strong>le</strong>s could have been measured only when projects were<br />
constructed, whe<strong>the</strong>r successful or not. Thus, <strong>the</strong> Hydrological<br />
Service had in its records a set of joint probabilities P(X.1)gi)<br />
as follows:<br />
J<br />
out co1iie<br />
Tab<strong>le</strong> 2.<br />
Joint Probabili ties<br />
P( XJW; )<br />
'i x1 x2 x3<br />
0,: project successful 0.20 0.05 0.05<br />
û2: project unsuccessful 0.05 0.10 0.55<br />
- - 7<br />
To taï<br />
To tal 0.25 O. 15 0.60 1 .o0<br />
Of course, <strong>the</strong> column totals represent <strong>the</strong> mar inal proba-<br />
bilities of .<strong>the</strong> usefulness of <strong>the</strong> additional data: PTX,) = 0.25,<br />
P(X,) = 0.15, P(X3) = 0.60.<br />
The expected value of <strong>the</strong> information which may be obtained<br />
by <strong>the</strong> additional hydrological data is reached by means of' a dia-<br />
gram, as shown in Figure 2. The set of probabilities appearing<br />
in <strong>the</strong> last branches of <strong>the</strong> decision tree are conditional probabi-<br />
lities p(eiJxj),
653<br />
The amount of 28.9 MMU appearing at <strong>the</strong> node (a) in <strong>the</strong><br />
decision tree represents <strong>the</strong> expected opportunity loss if it is<br />
decided to obtain additional hydrological information and if<br />
optimal decisions would be made on <strong>the</strong> basis of <strong>the</strong> new data.<br />
Comparingbhis amount with <strong>the</strong> 50 MMU obtained under Itno additional<br />
data" policy (Tab<strong>le</strong> I), it appears that it is worth spending<br />
50.0 -'.2&9 = 21.1 MMU in getting more hydrological information.<br />
The difference between <strong>the</strong> outcomes of <strong>the</strong> two policies is cal<strong>le</strong>d<br />
<strong>the</strong> expected value of samp<strong>le</strong> information. The expected net gain<br />
of samp<strong>le</strong> information is 21.1 - 15 = 6.1 MMU, i.e., <strong>the</strong> expected<br />
value of <strong>the</strong> samp<strong>le</strong> information exceeds <strong>the</strong> costs incurred in<br />
obtaining it. The Development Authority concludes, on <strong>the</strong> basis<br />
of preposterior analysis, that it is worthwhi<strong>le</strong> to get <strong>the</strong> addi-<br />
tional hydrological information.<br />
Cost-effectiveness.<br />
Cost-effectiveness is, in fact, engineering economics<br />
[5]. It is concerned with evaluation of a system worth, before<br />
<strong>the</strong> decision is made to construct <strong>the</strong> system. Thus cost-effectiveness<br />
is future oriented, and because of it its mode of<br />
expression is in terms of probabilities and expectations.<br />
With reference to <strong>the</strong> situation represmted by Figure 1 ,<br />
one can relate cost-effectiveness with <strong>the</strong> reciprocal of Cost,<br />
i.e., l/(cost). In this way, <strong>the</strong> lower <strong>the</strong> cost of a project,<br />
<strong>the</strong> higher would be its cost-effectiveness (which wculd also be<br />
a measure of its worth).<br />
Now, <strong>the</strong> cost-effectiveness of a system (as measured by<br />
its worth) increases with <strong>the</strong> amount of information availab<strong>le</strong><br />
at <strong>the</strong> time when <strong>the</strong> system is designed. In o<strong>the</strong>r words, this<br />
is a re-statement of <strong>the</strong> truism that <strong>the</strong> more we know about <strong>the</strong><br />
universe within which we design a system, <strong>the</strong> better <strong>the</strong> chances<br />
to produce a good design. The increase in <strong>the</strong> cost-effectiveness<br />
can <strong>the</strong>n be observed with respect to two major aspects.<br />
(a) F<strong>le</strong>xibility in plarmin . Because of hydrological,<br />
economic, an-gacing <strong>the</strong> planner of a water<br />
resources system, it is desirab<strong>le</strong> to produce a f<strong>le</strong>xib<strong>le</strong> system.<br />
In this context, byY<strong>le</strong>xibilitytl is understood one or more of<br />
<strong>the</strong> following attributes:
654<br />
(i) The possibility of increasing <strong>the</strong> capacity of <strong>the</strong><br />
system by adding additional components of <strong>the</strong> same kind (e.g.,<br />
pwnp stations in a pipeline network).<br />
(ii) The possibility of altering operating policies so<br />
that <strong>the</strong> system may respond to a broader range of demands.<br />
(iii) The possibility of modifying <strong>the</strong> system when <strong>the</strong><br />
nature of <strong>the</strong> demand changes, e.g., when <strong>the</strong>re is a traasition<br />
from irrigation to domestic and industrial uses of water.<br />
(iv) The possibility of constructing <strong>the</strong> system in<br />
stages, so as to respond to increases in <strong>the</strong> demand for water.<br />
(b) Reversib<strong>le</strong> vs. irreversib<strong>le</strong> decisions. The design<br />
of a system or of a component is a one-stage decision process:<br />
<strong>the</strong> size and dimensions are established. If <strong>the</strong> system is im-<br />
p<strong>le</strong>mented, <strong>the</strong> design decision may have irreversib<strong>le</strong> effects<br />
upon <strong>the</strong> emironment, such as <strong>the</strong> transformation of a canyon<br />
of unique scenic beauty into a man-made lake of doubtful<br />
es<strong>the</strong>tic value. The decision to delay imp<strong>le</strong>mentation is<br />
reversib<strong>le</strong>, since it keeps open <strong>the</strong> alternative to construct <strong>the</strong><br />
system. In addition, until <strong>the</strong> first decision is reversed,<br />
additional information may affect several planning details, ar-d<br />
also technologies may be improved in <strong>the</strong> interim.<br />
As an examp<strong>le</strong> of <strong>the</strong> introduction of <strong>the</strong>se two aspects<br />
in <strong>the</strong> planning process, one can indicate thi. Israel Water<br />
Scheme. The planning process was oriented toward increasing<br />
<strong>the</strong> cost-effectiveness of <strong>the</strong> system, especially with respect<br />
to f<strong>le</strong>xibility in design and to <strong>the</strong> reversibility oî decisions [6].<br />
The development of water resources progressed from local ground<br />
water schemes, to regional groundwater projects, finally to <strong>the</strong><br />
construction of <strong>the</strong> major component related to surface water<br />
resources - <strong>the</strong> National Water Carrier.<br />
Planning and design with inadequate data.<br />
The adequacy of hydrological data as defined by Figne 1<br />
represents one aspect of <strong>the</strong> general prob<strong>le</strong>m of <strong>the</strong> consequences<br />
of inadequate hydrological dataon <strong>the</strong> cost and effectiveness of<br />
water resources structures and projects. Although this aspect<br />
can be quantified, it is still somewhat mechanistic.<br />
Ano<strong>the</strong>r aspect would stress <strong>the</strong> linkage between data<br />
and decisions in <strong>the</strong> planning process. Although this aspect
655<br />
also <strong>le</strong>nds itself to'quantification, at <strong>le</strong>ast as far as <strong>the</strong><br />
data are concerned, it seems that it ref<strong>le</strong>cts also <strong>the</strong> quality<br />
of <strong>the</strong> ensuing design (and operating) decisions.<br />
It would be perhaps beyond <strong>the</strong> scope of this paper to<br />
survey <strong>the</strong> state of <strong>the</strong> art in <strong>the</strong> planning and <strong>the</strong> design of<br />
water resources systems with inadequate data. However, it<br />
would be instructive to mention two of <strong>the</strong> more recent contributions<br />
to this prob<strong>le</strong>m: one dealing primarily with surface<br />
water, and ano<strong>the</strong>r related to groundwater.<br />
Wallis and Matalas [7] consider <strong>the</strong> prob<strong>le</strong>m of deter-<br />
mining <strong>the</strong> capacity of a surface reservoir such that a given <strong>le</strong>vel<br />
of demand be satisfied. Observed hydrological data were used to<br />
generate syn<strong>the</strong>tic flow sequences, using two different sequence-<br />
generating mechanisms: (a) a well-known model based on <strong>the</strong><br />
Markovian process; (b) a model developed recently [8] which<br />
assumes <strong>the</strong> process to have a finite memory <strong>le</strong>ngth M and <strong>the</strong><br />
Hurst coefficient h; this is cal<strong>le</strong>d <strong>the</strong> filtered type 2 process.<br />
It seems that for streamflow regulation of up to 80$ of <strong>the</strong> mean<br />
annual flow, <strong>the</strong> Markovian model may be quite useful for <strong>the</strong><br />
determination of <strong>the</strong> minimum necessary storage; for higher degrees<br />
of streamflow regulation, <strong>the</strong> filtered type 2 model with h 7 2<br />
should be used.<br />
Maddock [g] used mixed integer programming methods for<br />
evolving a planning and management model of a groimd water<br />
development project. The model is oriented toward deterqining<br />
three components of <strong>the</strong> overall system: (a) <strong>le</strong>ast cost operation<br />
of existing wells; (b) <strong>le</strong>ast cost spatial and temporal schedu<strong>le</strong><br />
for installing new weììs; (c) <strong>le</strong>ast cost transport system to.<br />
convey <strong>the</strong> pumped water to a central demand point. The methodo-<br />
logy developed is tested on a hypo<strong>the</strong>tical samp<strong>le</strong> prob<strong>le</strong>m in<br />
which ground water development has to satisfy <strong>the</strong> demands for<br />
water of a town. The concept of expected value of opportunity<br />
loss (similar to <strong>the</strong> expected opportunity loss encountered in<br />
<strong>the</strong> preposterior analysis) is used as a measure of how much<br />
<strong>the</strong> errors inherent in estimating <strong>the</strong> model parameters will<br />
affect <strong>the</strong> cost of <strong>the</strong> project in terms of overdevelopment or<br />
underdevelopment. The results of <strong>the</strong> analysis indicate that <strong>the</strong><br />
reduction of uncertainty for <strong>the</strong> purpose of decreasing <strong>the</strong><br />
expected value of <strong>the</strong> opportunity loss should be a balanced<br />
activity, i.e., beyond F- given point, fur<strong>the</strong>r reduction of <strong>the</strong><br />
hydrological uncertainty will not improve <strong>the</strong> decision-making<br />
process un<strong>le</strong>ss <strong>the</strong> economic uncertainty is alao diminished.
656<br />
Concluding; remarks.<br />
The consequences of inadequate hydrological data on <strong>the</strong><br />
cost and effectivepeas of water resources structures and pro-<br />
jects were assumed to have a parabolic shape in <strong>the</strong> data-cost<br />
space. The abscissa of <strong>the</strong> minimum point of thìs vertical<br />
parabola defines <strong>the</strong> adequacy of data.<br />
There are several methods for <strong>the</strong> evaluation of hydro-<br />
logical data with respect to <strong>the</strong>ir adequacy for planning. One<br />
such method using <strong>the</strong> preposterios analysis is presented in<br />
some detail. This method enab<strong>le</strong>s <strong>the</strong> calculation of expected<br />
opportunity loss generated by a program designed to obtain<br />
additional hydrological data, as well as <strong>the</strong> expected value<br />
of <strong>the</strong> samp<strong>le</strong> information. If <strong>the</strong> expected net gain of samp<strong>le</strong><br />
information is positive, it is an indication that <strong>the</strong> existing<br />
hydrologic& data are insufficient.<br />
Cost-effectiveness of projects is briefly discussed,<br />
with some emphasis on its aspects regarding <strong>the</strong> f<strong>le</strong>xibility in<br />
planning and <strong>the</strong> irreversibilityof some design decisions. As<br />
an examp<strong>le</strong> of <strong>the</strong>se asepcts, <strong>the</strong> Israel Water Scheme illustrates<br />
a planning process oriented toward increasing <strong>the</strong> cost-effectiveness<br />
of <strong>the</strong> system.<br />
Fi,nally, how to plan and design water resources systems<br />
with <strong>le</strong>ss th&? adequate hydrological data was illustrated by<br />
two examp<strong>le</strong>s. In <strong>the</strong> first examp<strong>le</strong>, syn<strong>the</strong>tic hydrology was<br />
used to determine <strong>the</strong> capacity of <strong>the</strong> reservoir, but <strong>the</strong> design<br />
was sensitive to <strong>the</strong> type of model used to generate <strong>the</strong> syn<strong>the</strong>-<br />
tic sequence. The second examp<strong>le</strong> related to a ground water<br />
development project.<br />
Ref erences .,<br />
1. James, II, I.C., Bower, B.T. and latalas, N.C. (1969)<br />
Relative importance of variab<strong>le</strong>s in water rescurces planning,<br />
Water Resources Research, 5( 6), pp. 1165-1173.<br />
2. Matalab, N.C. (1968) Optimum gaging station location,<br />
Proceedings, IBM Scientific Computing Symposium, Water and<br />
Air Resource Management, White Plains, N.Y., pp. 85-94.<br />
3. Fiering, P.B. (1965) An optimization scheme for gaging,<br />
Water Resources Research, 1(4), pp. 463-469.
4. Hamburg, M. (1970) Statistical analysis for decision<br />
making, N.ew York, Harcourt, Brace & World.<br />
657<br />
5. English, J.M. (1968) Cost-effectiveness, New York, Wi<strong>le</strong>y.<br />
6. Buras, N. (1971) Utilization of ground water resources in<br />
Israel, Atti, Convegno Internationa<strong>le</strong> sul<strong>le</strong> Acque Soterranee,<br />
Pa<strong>le</strong>rmo, pp. 674-680.<br />
7. Wallis, J.R. and Matalas, N.C. (1972) Sensitivity of<br />
reservoir design to <strong>the</strong> generating mechanism of inflows,<br />
Water Resources Research, 8( 3), pp. 634-641.<br />
8. Mataïas, N.C. and Wallis, J.R. (1971) Statistical pro-<br />
perties of multivariate fractional noise processes, Water<br />
Resources Research, 7( 6), pp. 1460 - 1468.<br />
9. Maddock, III, T. (1972) A ground-water planning model<br />
basis for a data col<strong>le</strong>ction network, International Symposium<br />
on Uncertainties in Hydrologic and Water Resources Systems,<br />
Tucson, Arizonp, pp. 6.3-1 - 6.3-26.
658<br />
cost,<br />
Monetary<br />
Units<br />
~~ ~ ~~~<br />
Amount of hydrological data<br />
Figure 1. The data-cost space.
\<br />
3<br />
R. A<br />
Figure 2.<br />
o. 20/0<br />
659<br />
Decision diagram for preposterior analysis, MMU.
OPTIMIZATION OF WATER RESOURC ES DEVELOPMENT PROJECTS<br />
ZN CASE OF INARE2UATE HYDROLOGIC VATA.<br />
A. Fi<strong>le</strong>tti' ) , G. Faank ' 1, C. Pahvu<strong>le</strong>b eu' ' I<br />
Bucuhebti, Romania<br />
----<br />
I n t h o d u c t i o n .<br />
The phob<strong>le</strong>mb which have to be bolved by hydhaulic<br />
engineeh4 ahe 06 g hed diveaitq and deeibionb in <strong>the</strong>ih<br />
dieldb o 6 actiuity o@en imply a conbidehab<strong>le</strong> hen ponb abi-<br />
lity, not only in hebpect to <strong>the</strong> economic conbequenceb,<br />
but albo to <strong>the</strong> bocial and ecologic eddectb 06 buch de-<br />
cibionb. Being heeded to <strong>the</strong> mabtehing 06 cehtain natuh-<br />
al phenomena, mobtly hu<strong>le</strong>d by btochabtic lam, <strong>the</strong> con-<br />
ception as well a¿ <strong>the</strong> opehation od wateh hebouhceb de-<br />
velopment dthuctuheb depend on <strong>the</strong> deghee od know<strong>le</strong>dge a-<br />
vailab<strong>le</strong> on natuhal data, ebpecially on thobe helated to<br />
hydhologic euenth. The hydhologic hecohdb necebbahy to<br />
hydaaufic engineehs ahc not condined to data &elated to<br />
liquid blow, though data 06 thib type ahe ebbenaal doh<br />
<strong>the</strong>ih acLluity, but ah0 concehn bed-load phocebbeb ,hiveh<br />
and bank dynamics, wintea phenomena etc. Euidently,due to<br />
<strong>the</strong> a<strong>le</strong>atohq chahactes ob mobt hydhologic occuhenceb,even<br />
long ~recohdb 04 pat hydhologic phenomena cannot oddeh an<br />
abbolute baáety a¿ hegahdb avoiding ehhohb and deviation<br />
@om <strong>the</strong> altehnaZiue which could be phoved ab optimal. lt<br />
,i¿ neueh<strong>the</strong><strong>le</strong>sb unanimoubly accepted that, ab <strong>the</strong> volume<br />
and quality 06 in6ohmation on t<strong>le</strong>cohded hydhologie events<br />
incheas eh, in conditionb o 6 i tlb cohhect intehphetation,<br />
<strong>the</strong> phobability 06 cohhect ebtimation od dutute occuhencc<br />
ob hydirologic phenomena inchease4 and, a2 <strong>the</strong> bame time,<br />
<strong>the</strong> hhkb abbumed in taking deCihion6 decaease.<br />
~<br />
'1 Vocto~-Engineeh,ChCed Engineeh od <strong>the</strong> Rebeahch and Ve-<br />
bign Inbtitute doh Wateir RebOuhCeb Engineehing.<br />
I') Engineea, Team <strong>le</strong>adeh at <strong>the</strong> Inbtitute doh tlydsoetec-<br />
thic SXudied and Vebign.<br />
11') Voctak-Engineeh, Section <strong>le</strong>adeh at <strong>the</strong> Rebeakeh and<br />
Debign Indtitute doh Wateh Reb ouhceb Engineehing .
66 2<br />
It muAt be undeiraned that <strong>the</strong> indluence 06 incom-<br />
p<strong>le</strong>te hydirologic data on <strong>the</strong> pobbibieity 06 coirirectlg de-<br />
teamining <strong>the</strong> technological, duncL¿onal and economic pa-<br />
irameteu 06 wateir heb ouirceb development piroject¿ depend¿<br />
in gheat meabuhe to <strong>the</strong> hydirologic chahacteh 06 <strong>the</strong> iriuez<br />
bain, on <strong>the</strong> natuire 06 wateir Ubeb, on <strong>the</strong> type 06 bthuC-<br />
tuheb etc. Theaedohe,any opL¿mizaL¿on method mubt be con-<br />
bidehed in <strong>the</strong> fight od <strong>the</strong> condition4 in which thib me-<br />
thod ib apptied; <strong>the</strong> bpecidic 4itUafiOnb and <strong>the</strong> tenden-<br />
cieb in bolving <strong>the</strong> phob<strong>le</strong>mb menaoncd in thib papeh must<br />
be looked at only ab typical examp<strong>le</strong>b.<br />
Undeh conditionb o6 incomp<strong>le</strong>te hydhologic indoirmat-<br />
ion, <strong>the</strong> methodb Ubually appfied ahe no longeir clbe~jul and<br />
it i8 necebbahy ei<strong>the</strong>h to adapt <strong>the</strong>be methodb to <strong>the</strong> a-<br />
vailab<strong>le</strong> data oh to adopt bimp<strong>le</strong>h phoceduheb which aire<br />
conbibtent with <strong>the</strong>be data. In be<strong>le</strong>cting buch methodb,<br />
6oÆlowing itemb mubt be taken into account:<br />
- <strong>the</strong> methodb mubt mahe integhal ube 04 <strong>the</strong> auaila-<br />
b<strong>le</strong> volume 06 indohrnaL¿on.At <strong>the</strong> bame L¿me it mubt<br />
be kept in mind ithat no phocebbing id ab<strong>le</strong> to cire-<br />
ate quantitatively new in~ohmation and <strong>the</strong>hedoh it<br />
lb ubetebb to thy genehafing in6ohrnat.ion not con-<br />
tained in <strong>the</strong> basic data;<br />
- bebideh hydhologic data irecoirded in <strong>the</strong> aiueir ba-<br />
din bubject to analybib, it i4 pobbib<strong>le</strong> to take<br />
advantage 06 additional indohmaL¿on irecoirded in<br />
neighbouhing hiveir babinb. Indihect hydirology may<br />
be ubed ebpecially in ohdeir to obtain quafitufive<br />
indohmation hegairding <strong>the</strong> beabona1 oh annual dib-<br />
thibufion 06 dlow, <strong>the</strong> pobbibifity od occuirence 06<br />
cehtain hydhologic phenomena in vaaiou4 pehiodb od<br />
<strong>the</strong> yeah, etc;<br />
- <strong>the</strong> method mut not <strong>le</strong>ad to an ampfi6icaLLon od<br />
ehhou od <strong>the</strong> hydhologic basic data but,ab much ab<br />
pobbib<strong>le</strong> to <strong>the</strong>iir attenuation.<br />
A gheat diveuity 06 bituationb exibtb conceirning a-<br />
vailab<strong>le</strong> hydhologic data, covehing <strong>the</strong> <strong>who<strong>le</strong></strong> dietd dirom
663<br />
total lack to an acceptab<strong>le</strong> volume od in6okmation. Theke-<br />
dohe, tkying to bet up cefitain methodb o6 gcneaat appbic-<br />
ation ib haadly to be hecommended. In paincip<strong>le</strong>, it<br />
would be make cokhect to talk about gkoupb ok typed 06<br />
methodo having a common p~ncipîe. Thebe have to be adap-<br />
ted to concirete conditionb and objectiveb 06 each inveb-<br />
tigated pkoject. The methods which can be appLied in cabe<br />
06 inadequate hydkological data may be ceabbidied ab dol-<br />
LoWb :<br />
- methodb baoed on <strong>the</strong> genekation 06 byn<strong>the</strong>tic hy-<br />
dkologic bequenceb, btakting ('ron1 given btatibtic-<br />
al pazameteu od hydkologic phenomena [ Monte Cak-<br />
to methods);<br />
- methodo based on <strong>the</strong> genekation 06 hydhologic be-<br />
quenceb by comelation with kaindall data;<br />
- method¿ based on <strong>the</strong> genehalization o6 kebultb od<br />
watea he6 oukceo engineeking computation4 ;<br />
- methods based on <strong>the</strong> <strong>the</strong>oky od gameb.<br />
Thib kepoht intend6 to pkebent <strong>the</strong> wayd in which<br />
<strong>the</strong>be method4 can be appaed'in dome key pkob<strong>le</strong>mb 06<br />
watek ire4 ouaced engineeking .<br />
Dimenbioning 06 kivek blow hegulating<br />
wokkh debigned dok rneefing ~atek<br />
demandb.<br />
Une od <strong>the</strong> mobt ubual paobtemb, cohkebponding to an<br />
incipient phue 06 watek keboukceb development, i¿ to ed-<br />
timate <strong>the</strong> capability 06 unhegulated UVC CM to meet u b e ~<br />
watek demand.<br />
The bolution od thib type 06 pkob<strong>le</strong>mb id based <strong>le</strong>bb<br />
on <strong>the</strong> detekmination 06 avekage @ow and dependendb gnea-<br />
tty on tow watehb and on chakactehibtic minimum valued 06<br />
dibcchakgeb; <strong>the</strong>be valued can pkebent a gkeat benditivity<br />
to <strong>the</strong> quantity 06 availab<strong>le</strong> in6okmatiav1, to <strong>the</strong> methodb<br />
06 dikect oh indikect detekmination od dtow valued and to<br />
<strong>the</strong> degkee to which ba~ic data have been extkapolated.<br />
Theke6oke, <strong>the</strong> tack 06 adequate hydkotogic data and
664<br />
pahticulahly 04 batib dactohy hydhomethic hecohdb can be,<br />
in thh cue, <strong>the</strong> bouhce od impohtant ehhoh4. The w e 06<br />
cohhelative methoda d6 hibky, as <strong>the</strong> helative valueb 06<br />
low wate~ depend on <strong>the</strong> individual bupply 06 each wateh<br />
couue. On <strong>the</strong> o<strong>the</strong>h hand, <strong>the</strong> u4e o6 bimulated 6low beqU12MCe4,<br />
wually applied doh genehating mean monthly<br />
dib chahgeb, cannot be batib dactohy, bince daily valueb<br />
od minimum dibchahge can<br />
monthly valueb.<br />
conóidehably did6eh dhOm mean<br />
The dub-unitahy Ratio 06 minimum daily dibchahge to<br />
mean monthly dibchahge i4 wually bmal<strong>le</strong>h id <strong>the</strong> phobability<br />
o6 exceeding <strong>the</strong> dibchahge incheaeb and <strong>the</strong> buhdace<br />
od <strong>the</strong> hiuek basin decheaeb.<br />
The ehhoM which can be made in buch cabe4 can be<br />
one 06 <strong>the</strong> dollowing:<br />
- oueh-evalua-ting availab<strong>le</strong> low dlow,which can <strong>le</strong>ad<br />
to a bmal<strong>le</strong>h phobability od being ab<strong>le</strong> to meet<br />
wateh demand od ue~; thib phobabifity might not<br />
be acceptab<strong>le</strong> becaube 06 <strong>the</strong> excebbive lobbeb due<br />
to dhequency and bevehity od wateh bhohtage. It<br />
i4 Wohth mentionning that in bome counthieb (as in<br />
<strong>the</strong> U.S.S.R. ,Czechoblouahia, Romania and o<strong>the</strong>hs 1<br />
<strong>the</strong> phobabifity 04 being ab<strong>le</strong> to meet demand A<br />
ebtablibhed by btandahdb and A ,<strong>the</strong>hedohe, compulb<br />
ohy;<br />
- undeheualuating availab<strong>le</strong> low dlow,which can <strong>le</strong>ad<br />
to a da&e conclubion, that <strong>the</strong> hiVeh i4 not ab<strong>le</strong><br />
to meet demand without blow hegutation and that<br />
a stohage hCbehU0ih Oh a diuehbion dhom o<strong>the</strong>h excedentahy<br />
hiueu h a to be budX.Thib would imply<br />
u4e<strong>le</strong>b4 expenbeb Oh, in <strong>the</strong> bebt Ca4e,UbelQbb immobifibation<br />
od capital.<br />
Ab wateh demand ghowb in compahibon to availab<strong>le</strong><br />
wakeh hebOUhCQ6, <strong>the</strong> neCCbbahY deghee od blow hegula-tion<br />
incheu eb and mutationb emehge concehning <strong>the</strong> bignidicance<br />
06 vahiou categohieh od hydhologic indohmation. In<br />
thib benbe, <strong>the</strong> data helated to <strong>the</strong> auehage inalow doh
665<br />
longeh L¿me pe&¿odb: monthb, beabonb, yeah4 oh even be-<br />
quenceb ad yeau begin to play an ebbential pmt in de-<br />
tehmining <strong>the</strong> pahametehb 06 btohage hequihed.<br />
The indtuence 04 genehat hydhologic data on <strong>the</strong> va-<br />
tue 06 <strong>the</strong>be puhameteu i4 heuea<strong>le</strong>d by <strong>the</strong> geneaal phac-<br />
-Lice 04 wateh hebouhceb engineehb a4 well a4 by home<br />
Apecial hebeahch phoghamb. Thib inbluenee iA made evid-<br />
ent by invehtigating :<br />
- <strong>the</strong> genehat hetations between <strong>the</strong> chahacteaibtic<br />
blow panameteu and <strong>the</strong> bpecidic deghee 06 deue-<br />
topment 06 wateh hebouhceb;<br />
- <strong>the</strong> ben&¿tiVitLj od hebUltb concehn.¿ng blow hegut-<br />
dion at vahiou4 degheeb 06 apphoximation od <strong>the</strong><br />
inadequate hydhologic data.<br />
Thub, a diut aspect 06 thib anatybib concehnb ihe<br />
cohhelaXion bemeen <strong>the</strong> main puhametehb chahactehib.tiC<br />
doh dtow dL¿thLbution: <strong>the</strong> vahiation coeddicient Cu, 2he<br />
coeddicient 06 bkewnebb Cb and <strong>the</strong> coe6dicien.t od sehial<br />
cohhelation ir on one hand and <strong>the</strong> net volume od necebba-<br />
hy ótohage hebehV0iJr.b and <strong>the</strong>ih opehating policieh on<br />
<strong>the</strong> o<strong>the</strong>h.<br />
Vahiou4 sepohth conceaning genehatized Jab ultb on<br />
<strong>the</strong> connection befween <strong>the</strong>be patameteu and <strong>the</strong> magnit-<br />
ude 06 <strong>the</strong> ove&-annual component 04 <strong>the</strong> nequihed btohage<br />
doh a given bade yield have been pubfibhed 111, 121 .<br />
Rebula 06 buch hebeahch ib beAt made evident by<br />
diagrramb phebenting <strong>the</strong> comelation between <strong>the</strong> coedtjic-<br />
iena a, p and B, 06 which an examp<strong>le</strong> ib shown in diguhe<br />
i. Following bymboeb have been uded :<br />
- a, <strong>the</strong> blow aegulation coeddicient 04 <strong>the</strong> deve-<br />
lopment, deiined ab <strong>the</strong> ha.t.io 06 <strong>the</strong> bade yield<br />
od <strong>the</strong> dtohage hebehvoih to <strong>the</strong> auehage dibchahge<br />
06 <strong>the</strong> wateh COUILA~ ;<br />
- p id <strong>the</strong> phobabieity od meeLing wateh demand, de-<br />
dined ab <strong>the</strong> &mit 06 <strong>the</strong> hatio 06 <strong>the</strong> numbeh 06<br />
yeau in which no wateh bhohtage appeau to <strong>the</strong><br />
total numbeh 06 yeau invebtigated;<br />
- 0 c6 <strong>the</strong> btohage coeddicient, dedined ah <strong>the</strong> hu-
666<br />
tio 06 <strong>the</strong> oveh-annual component 06 Atohage to<br />
<strong>the</strong> avesage yeaaly dibchahge 06 <strong>the</strong> hegulated ti-<br />
Veh.<br />
lt mUbt be kept in mind that <strong>the</strong> bame avehage<br />
dib-<br />
chahge 2 could occuh in vahioub beqUQnceb 06 bingutarr<br />
dibchahgeb 06 <strong>the</strong> hivet; thib can be made evident by an<br />
analybd 06 hecohded beqUenCeb 06 dibchahgeb 06 hivehb<br />
phebenting vahioub Valued 06 <strong>the</strong> pahameteu cv,Cb and 4.<br />
Folîowing conc~ubionb may be dhawn dhom hebeahch deveto-<br />
ped in thib 6ield:<br />
- <strong>the</strong> vatiadon coed{i&ent Cv ha4 a dihect inblu-<br />
ence on <strong>the</strong> volume 06 <strong>the</strong> btohage hebehvoih, <strong>the</strong><br />
Atohage coe6bicient B being <strong>the</strong> gheateh, <strong>the</strong> m o u<br />
<strong>the</strong> vahiation coe6,jicien.t incheabeb ; genehaîly<br />
<strong>the</strong> helaAive ghowthb 06 ß ahe gheateh, sometimes<br />
even benbibly ghedek, than <strong>the</strong> gaowth 06 Cv, i6<br />
<strong>the</strong> valuta 06 a a m high. On <strong>the</strong> con;titahy,6oh low<br />
valued 06 a <strong>the</strong> gaowth 06 <strong>the</strong> necebbahy volume 06<br />
btoaage in <strong>le</strong>bb hapid than <strong>the</strong> ghowth 06 <strong>the</strong> va-<br />
hiaiSon coe66icient [ 6iguhe 2) ;<br />
- <strong>the</strong> coe{,jicient 06 bkewnebb cb ha an inveme in-<br />
ence ce on <strong>the</strong> volume 06 <strong>the</strong> btohage hebehvoih;<br />
Zhu, i6 cb incheabeb <strong>the</strong> volume decheaseb coh-<br />
hebpondingly. Genetally, pehcentual heduciSonb 06<br />
B avre Amal<strong>le</strong>h than <strong>the</strong> pehcentual ghowthb 06 Ch;<br />
- <strong>the</strong> coedbicient 06 betial cohhelation h has a di-<br />
heet in6luence on <strong>the</strong> value 06 <strong>the</strong> btoaage; thub,<br />
<strong>the</strong> incheabing 06 h <strong>le</strong>adb to ghowthb 04 B, peh-<br />
centual ghowthb 06 both parrametem being 06 <strong>the</strong><br />
bame ohdeh 06 magnitude.<br />
Similah conbidehationb can a do be made in connect-<br />
ion to <strong>the</strong> beabonal component lyemly component) 06 <strong>the</strong><br />
heqdhed btohage. Evidently, in thib cabe, <strong>the</strong> coe66ic-<br />
Leni2 06 vatiation, o6 bhewnebb and 06 behiat cohhelat-<br />
ion mubt be based on daily, decadal oh monthly avehage<br />
dib chahgeb .<br />
A beeond inteaesting aspect concehnb <strong>the</strong> inbluenee<br />
06 <strong>the</strong> intehval taken in account 60k debign iday,decade,
667<br />
month etc) oh <strong>the</strong> Lime intehval doh which hydhologic data<br />
ahe ebtimated and wateh balance computations ahe undehtaken<br />
on <strong>the</strong> hequihed btohage volume.<br />
Vehy 06ten,waZeh balance id ebtablibhed on a monthly<br />
bad&, taking into account a bequence 06 mean monthly<br />
dlowb COVehing a pehiod 06 ¿evehat qeah4, doh which hecohded<br />
oh indihectly detehmined hydhologic dda ahe a-<br />
vailab<strong>le</strong>. Thib way 06 dealing with <strong>the</strong> phob<strong>le</strong>m imptieb<br />
<strong>the</strong> assumpLion that <strong>the</strong> dhchahge od <strong>the</strong> Jbiveh and <strong>the</strong><br />
demand o{ <strong>the</strong> ueh ahe baihey conbtant duhing a month.<br />
Thib asbumption ib neveh abbolutely cohhect doh <strong>the</strong> di&chahge<br />
od <strong>the</strong> hive&, nOh bometimeb {oh <strong>the</strong> Watch demand.<br />
ZnvutigaLionb undehtaken in thib 6ieÆd bhowed that<br />
in cehtain cabe6 <strong>the</strong> ,time pehiod taken into account hab<br />
a gheat indluence on <strong>the</strong> hebultb obtained concehning <strong>the</strong><br />
volume 06 hequihed btOhUge.lt has been pobbib<strong>le</strong> to utabtibh<br />
cohheîationb bemeen btohageb cohhebponding to<br />
,time pehiodb od a month oh a day ubed in wateh balance<br />
calculations. An examp<strong>le</strong> 06 buch an intehdependence ib<br />
bhown in biguhe 2.The genehat conclubion od <strong>the</strong> tedeahch<br />
i& tha.t bhoht pehiodb, o6 a day, mubt be ubed 46 badie<br />
ame intehvaÆ only i6 <strong>the</strong> heqLUhed btohage,hebulted dhom<br />
pheaminahy computation4 ubing monthly UVehUge valueb c4<br />
Amall. foh gheateh 4tOhage volume4,<strong>the</strong> indluence 06 bhoht<br />
Lime pedo& id negtigeab<strong>le</strong>.<br />
16 <strong>the</strong> invebtigation 06 lahge-bca<strong>le</strong> phoject.4 i4 undehtaken,<br />
u6e od 6 yn<strong>the</strong>Lic s<strong>the</strong>am- {low bequenceb had<br />
btahted to impobe itbet6 even in cae4 in which <strong>the</strong> volume<br />
06 availabÆe hydhologic indohmation would have been<br />
conbidehed adequate. S<strong>the</strong>am-dlOW genehation methodb,<br />
oh Monte Cahlo techniqueb, btah-t dohm cehtdn comphehenhive<br />
hydhologic PaharneteM buch ab avehage dibchahge,<br />
coeddicient 06 VahiatiOn, coe66icien.t 06 behiat cohhelation<br />
etc, ebtimated on <strong>the</strong> babib od a minimum 06 dihect<br />
hecohdb oh by genchalizing hebultb od hydhologic inveb-<br />
LigatiOnb in bimilah (Meu. Ahtidici& time behieb o6<br />
hundhed and even thouband4 od hydhologic yeau ahe genehated;<br />
<strong>the</strong>be include a multitude 06 pobbib<strong>le</strong> bequenceb
66 8<br />
06 day, wet and aveirage yemb, which condeh a high heliability<br />
on <strong>the</strong> hebultb 06 watch balance<br />
calculationb e<br />
and btohage<br />
Such methodb, babed on byn<strong>the</strong>a2c btneam-6low bequenceb,<br />
a m applied on a lahge ¿ca<strong>le</strong> in <strong>the</strong> U.S.S.R.<br />
l4I,l5/, <strong>the</strong> U.S.A. 161 and o<strong>the</strong>h counthieb. ln Romania,<br />
1101 thib method .d being applied doh <strong>the</strong> btudy o6 <strong>the</strong><br />
development 06 lahge hive& babinh.<br />
A hecent hebeahch 131 ib conceancd with <strong>the</strong> iniluence<br />
06 <strong>the</strong> <strong>le</strong>ngth 06 <strong>the</strong> heal hecohd on <strong>the</strong> hequihed<br />
stotrage,in 06 a seasonal oh yeuly dlow hegulation.<br />
The teseahch phogham covehed a gheat vahiety od bituationh,<br />
including di66ehent typeb 06 dlow dibthibufion<br />
and bevehal VatUeb 06 berrial cohhelaZLon and btoirage<br />
coeddicienfi.<br />
The hebultb o6 thih hebeahch Lead to <strong>the</strong> conclubion<br />
(bee diguhea 3 and 4) that <strong>the</strong> hequihed btohage incheabeb<br />
with <strong>the</strong> <strong>le</strong>ngth 06 <strong>the</strong> bequence. The inchease .&<br />
dabteh doil higheh valued 06 <strong>the</strong> behial<br />
stohage cae66icienh.<br />
cohhelation and<br />
Though thib invehtigation w a ~ concetned only with<br />
yeahly (low rregulation, <strong>the</strong> hebula obtained doh high<br />
valueb o6 <strong>the</strong> btohage coeddicient (maximum 1.0) phove<br />
that <strong>the</strong> <strong>le</strong>ngth od <strong>the</strong> dihect hecohd and <strong>the</strong> bequence<br />
06 wet and day pehiod~ have a bignidicant indluence alho<br />
in <strong>the</strong> cabe 06 oveh-annual atohage.<br />
A bthiking examp<strong>le</strong> in this ben~e was uphedented by<br />
<strong>the</strong> evolution od btohage hequihed 60s <strong>the</strong> wateh bupply<br />
06 a big induthial plant in Romania. Wateh balance calculationb,<br />
6ihbt undextaken in <strong>the</strong> eahly 1960-ieb Wehe<br />
based on dihect dlow hecohdb covehing only a pehiod od<br />
15 yeah6, btmfing @om 1947148. Extending thib bequence<br />
by cohhelation with hecohdb in neighbouhing babinb genehated,<br />
intek alia, an ex<strong>the</strong>mely dhy hydhoîogic yeair<br />
(/942/43). 76 thib yeah wab included in <strong>the</strong> hecohd ued<br />
a~ babib doh htonage calculationb, <strong>the</strong> hequihed btoaage<br />
wab neahly <strong>the</strong> doub<strong>le</strong> (220 million cubic meteu) 06 <strong>the</strong><br />
btoxage which would have deemed necebbahy i6 thib yeah
669<br />
had not been included into <strong>the</strong> hecohd (120 million cub-<br />
ic meteu). Ab a matteh 06 dact, <strong>the</strong> additionat hecohdb<br />
06 <strong>the</strong> 6ottowing ten yeau beem to indihm <strong>the</strong> indihect<br />
data obtained doh <strong>the</strong> yeah 1942143; it wad <strong>the</strong>hedohe de-<br />
cided not to include thib yeah into <strong>the</strong> invebtigated be-<br />
quence when btohage caÆcutaL¿onb wehe again undehtaken<br />
on <strong>the</strong> basi4 06 a .tongeh dcquence 06 dihect hecohdb.<br />
In buch bituationb , behideb <strong>the</strong> ub ual methodb 06 ex-<br />
tending <strong>the</strong> hydhotogic time behieb by bimilahity with<br />
o<strong>the</strong>h hiveh badinb, modehn techniques can be applied doh<br />
detehmining <strong>the</strong> optimum decndionb. A botution might be<br />
dound i6 <strong>the</strong> <strong>the</strong>ohy 06 game4 id applied; <strong>the</strong> hecommended<br />
de&ion <strong>le</strong>ad& in thib cade to minimum heghet, taking<br />
into account on one had <strong>the</strong> cobtb ad <strong>the</strong> btohage hueh-<br />
voi& and on <strong>the</strong> o<strong>the</strong>h hand <strong>the</strong> pobbibte damageb. Thib ib<br />
a cla6bica.î cade 06 a game againbt natuhe. Natuke’b se-<br />
action ib not indluenced by <strong>the</strong> phevioub dechion4 con-<br />
cehning <strong>the</strong> management od wateh hebouhceb and ib ei<strong>the</strong>h<br />
a<strong>le</strong>atohy oh hu<strong>le</strong>d by an asbumed pkobabilibtic law 06<br />
dib t&ib utio n .<br />
06 couue, <strong>the</strong> methodb based on <strong>the</strong> <strong>the</strong>ohy 04 gameb<br />
do not bolve <strong>the</strong> inadequacy 06 hyditotogic data. Ubing<br />
<strong>the</strong>m maheb howeveh <strong>the</strong> minimization 06 adveue e66ec.tb<br />
06 inadequate data po4bib<strong>le</strong>. Theh~,joke, buch methods<br />
bhould not be phebented in oppobifion to thobe based on<br />
<strong>the</strong> genehation 06 new data oh on <strong>the</strong> genehatizaaSon o6<br />
cehfain heb uttb od watek heb ouhceb engineehing calculat-<br />
ionb. Both technique4 can be bimultaneoubly applied.<br />
One 06 <strong>the</strong> advantageb o6 <strong>the</strong> <strong>the</strong>oky 06 gameb ib <strong>the</strong><br />
pobbibifity 06 taking into account any inadequate data,<br />
not only od hydhologic chairacte& (do& examp<strong>le</strong>, data &e-<br />
dcilning to <strong>the</strong> dUtUhQ development o{ watek ue~).<br />
Noticing <strong>the</strong> ub e o 6 cohhelaLionb between hydirologic<br />
pahameteu and O<strong>the</strong>& chahacteJùbtic e<strong>le</strong>menX.4 06 <strong>the</strong> hi-<br />
ve& basin: aveaage haindall, altitude, adtjoheAta,tion,<br />
btope etc. bome engineeu thied to ebtablihh genehatized<br />
helaZionbhipb be&een wateir heb ouhceb engineehing paha-<br />
meteu, buch ah <strong>the</strong> kequiheb btohage, and geo-meteoholo-<br />
gic e<strong>le</strong>mena. Such hebeahch has been cashied out in Po-
670<br />
land. Methodb 06 thib type have, neveh<strong>the</strong><strong>le</strong>bb, a limited<br />
appficability, geneaalizationb being pobbib<strong>le</strong> only at a<br />
®ional bea<strong>le</strong>. They may, evidently, give a view on <strong>the</strong><br />
bize 06 necebbahy Wdeh hebOUhCeib development WOhkb and<br />
can be 06 help doh <strong>the</strong> phefiminahy debign 06 ceht&n<br />
bmall btohage damb. Theih ube, without a camparribon with<br />
o<strong>the</strong>s mom elaboaate methodb id howeveh not to be hecorn-<br />
mended in <strong>the</strong> inwedtigation 06 impohtant btohage. hebeh-<br />
VOihb.<br />
Uimenbioning od hydhoe<strong>le</strong>cthic Noirkb.<br />
The utilization 06 wateh poweh id, evidently, hela-<br />
ted to <strong>the</strong> cohhect know<strong>le</strong>dge 06 <strong>the</strong> natuhal potential<br />
and o6 <strong>the</strong> conditionb 06 developing it. in thib conned-<br />
ion, hydhologic data ahe necebbahy not only in ohdes to<br />
detekmine <strong>the</strong> genehal e66ect and edbiciency 06 poweh<br />
plana2 but albo t o ebtablhh <strong>the</strong> development bcheme, <strong>the</strong><br />
enehgetic parrameteu and <strong>the</strong> charractexibtics 06 <strong>the</strong><br />
bthuctuhes. Fah plana luith no btohage oh having a low<br />
deghee 04 6low heguldtion a6 well c~6 {oh <strong>the</strong> intake4 06<br />
becondahy watch divehhionb it can be parrticulahly impoh-<br />
tant to ebtimate c~hhectly <strong>the</strong> daily, and dometimed even<br />
<strong>the</strong> momentarry blow hégime. Condthuction 06 irégime and<br />
dlow dukation cuhveb , chahacte&ibtlc doh kelatively long<br />
pehiod6,may be necebbahy. Such cuhueb alre uded in ohdeh<br />
to ebtablibh <strong>the</strong> deghee 06 utilization 06 <strong>the</strong> avehage<br />
yeahly dh chahge and to detekmine <strong>the</strong> inbtal<strong>le</strong>d capacit-<br />
ieb<br />
The value4 06 <strong>the</strong> hydhologic pahameteu mentionned<br />
in <strong>the</strong> phevioub chapteh: auehage d.ib chahge, coe66icient<br />
06 vahiation, coe6dicien.t 06 bkewnebb and coe6dicient 06<br />
behial comelation, ahe necebbahy in ohdeh to ebtablibh<br />
<strong>the</strong> inbluenee 06 vadoub Atohage volumed on <strong>the</strong> magnitude<br />
and quality 06 e<strong>le</strong>cttic po~eh phoduction and on <strong>the</strong><br />
enehgetic conditions 06 btohage hydhoe<strong>le</strong>ctaic plan&. On<br />
thib bahib, <strong>the</strong> opfimization o6 <strong>the</strong> volume 06<br />
age hebehv~ikb cb aho podbib<strong>le</strong>.<br />
<strong>the</strong> btoh-
FOR. mote advanced dlow conthol, <strong>the</strong> avehage multian-<br />
nual dibchakge i6 <strong>the</strong> hydaologic etement whobe indluence<br />
on <strong>the</strong> economic eddiciency 06 hydhoe<strong>le</strong>cthic plant6 i4<br />
gheatest. In paht, thi6 ib due al60 to <strong>the</strong> opehation 06<br />
hydhoeÆecthic plana2 within 6taong poweh dydtemb , genehal-<br />
ly with inteknational linkd. In buch 6Ybtem4, <strong>the</strong> e66ect<br />
o 6 individual hydaologic 6ituationd i6 attenuated and <strong>the</strong><br />
<strong>who<strong>le</strong></strong> hydhoe<strong>le</strong>cthic poweh availab<strong>le</strong> can actually be uded<br />
in <strong>the</strong> dybtem, even id only doh <strong>the</strong> dilling o6 <strong>the</strong> hebeh-<br />
voim 06 pumped 6tohage powek plana.<br />
Re6eahch on <strong>the</strong> indluence 06 <strong>the</strong> <strong>le</strong>ngth 06 <strong>the</strong> hecoird<br />
on <strong>the</strong> value 06 <strong>the</strong> aveirage dibchahge hevea<strong>le</strong>d that,<br />
604 m o ~ t eutopean wateh couh6e6, time behie6 covehing a<br />
dequence 06 30 yeam ahe u6ually batihbactohy. Conclubiond<br />
Reached at in inve6tigating data concehning <strong>the</strong> a-<br />
veaage didchaqe doh Recoaded Lime behie6 06 vahiou6<br />
<strong>le</strong>ngth6 at <strong>the</strong> OIL6ova gauge on <strong>the</strong> Danube can be bahen<br />
a6 an examp<strong>le</strong>. On <strong>the</strong> basib od 133 yeah long hecoad6<br />
(1838 - 19701 <strong>the</strong> avehage value 604 di66ehent 64 . ~ e b 06<br />
a given Length, covehing 10 - 40 con6ecuL¿ve yeau wehe<br />
calculated and <strong>the</strong> highebt and Lowe62 value6 o6 <strong>the</strong>he a-<br />
vehage6 wehe examined. The irebutRb alre phedented in <strong>the</strong><br />
dollowing tab<strong>le</strong>:<br />
10 124 6090 4520 12.5 -16.5 0.36<br />
15 119 5910 4760 9.2 -12.0 0.28<br />
20 114 5770 4850 6.7 -10.2 0.22<br />
25 109 5660 5040 4.6 -6.8 0.17<br />
30 104 5670 5200 4.1 -3.9 0.13<br />
40 94 5650 5230 4.4 -3.3 0.11<br />
Thehedohe, i6 <strong>the</strong> hydhotogic data ahe not adequate<br />
doh <strong>the</strong> de6ign 06 hydhoe<strong>le</strong>cthic developemntb, a6 a huÆe,<br />
<strong>the</strong>be data dhoutd be extended to a 25-35 yeah6 long time
672<br />
behie, by analogy with o<strong>the</strong>k watek couhdeb ok with kain-<br />
ball. longeh extenhionb 06 hecohdb by bimulation ate<br />
veky kcvrely applied.<br />
Debign and opehation 06 blood<br />
conakol developmenth .<br />
Flood canx7ca.t developmen;td ake kelated not only to<br />
economic bene6itb but aedo to <strong>the</strong> becuhity 06 bocial<br />
Lide in wide axeab. At <strong>the</strong> same time, ab álood conthol<br />
bthuctuheb ane debigned to {ace hydhologic bituatioptb<br />
exceeding ex<strong>the</strong>me hibtohicaî hecohdb, <strong>the</strong> ube 06 cx<strong>the</strong>-<br />
mely accuhate data ib, in phincip<strong>le</strong>, necebbahy.<br />
A ptob<strong>le</strong>m ahibing most dhequently id helated to <strong>the</strong><br />
debign 06 out<strong>le</strong>& o6 6low hegulating bthuctuheb. O<strong>the</strong>&<br />
impoatant phob<strong>le</strong>mb, buch ab <strong>the</strong> debign ,od bhidgeb oh 06<br />
o<strong>the</strong>s btkuctuheb chobbing hivetb, 06 dykeb and 06 blood<br />
detenfion hehehvoitb axe aedo helated to dlood hydhol-<br />
OgY *<br />
Except vehy hahe cab eb, debign hydhologic conditionb<br />
have nevek been hecohded and have to be ebtablhhed<br />
by bpecLal computationb. In thih hebpect, ,two tendencieb<br />
may be kemairhed:<br />
- <strong>the</strong> extkapolation 06 phobabifity dibtkibufion<br />
cuhvcb 06 maximum aecokded ,$toodb ubing ma<strong>the</strong>matical<br />
btatibticat methodb; thib extkapolation has<br />
to keach imposed debign paobabilitiea . O<strong>the</strong>& methodb,<br />
ubed in o.<strong>the</strong>h hydhologic pkob<strong>le</strong>mb 6011 <strong>the</strong><br />
extension 06 tecohded fime bekieb ,buch ab byn<strong>the</strong>fie<br />
blow genekation,ubing Monte-Cakto techniqueb.<br />
MQ applied at a bah amal<strong>le</strong>h bca<strong>le</strong> bok 6lood conak0l;<br />
- <strong>the</strong> geneha~on 06 valueb 06 maximum dibchakgeb<br />
and 06 ,$laod waveb by hain6altlhun-od6 cohhelation;<br />
thib apphoach thiea to compenbate <strong>the</strong> lack<br />
06 hydhologic data by uning hain{all oh O<strong>the</strong>& me<strong>the</strong>ohologic<br />
magnituded doh which, UA ually, longea<br />
hecohdb ake availab<strong>le</strong>. ln <strong>the</strong>ih dimp<strong>le</strong>st 60hm,
67 3<br />
<strong>the</strong>be methodb hephe4enZ ha.in6attlhun-od6 depen-<br />
dencie6 which have been wed doh a long time in<br />
hydhologic indihect btudie6 .Duhing <strong>the</strong> lut yeah6<br />
<strong>the</strong>be methodb have been extended, 6ohmeh 6impte<br />
dependencieb being developed into phy6ioghaphic<br />
haintjalllhun-066 mode&. Thebe mode& have been<br />
ubed ebpecially in <strong>the</strong> U.S.A. and in Fhance.Theih<br />
eidiciency wab pkoved e6pecially doh <strong>the</strong> htudy 06<br />
dlood conthot phob<strong>le</strong>m6 .<br />
The majoh did6iculty haAed by applying genetic mo-<br />
de& conbi6t.h in a pheviou6 detehmknation 06 <strong>the</strong> pahame-<br />
teu 06 each model. FOh thi6 pukpo6c,aecohded dl00d6 ahe<br />
houted thhough <strong>the</strong> model. Theiredom, ube o6 h&n6all/<br />
hun-066 modeh implie6 <strong>the</strong> exhtence 06 a minimum o6 hy-<br />
dhologic hecohdb; it ib, howeveh, ju6t data on dlood6<br />
which ahe dhequently mi6bing @om hecohd6.<br />
ln <strong>the</strong> abbence 06 adequate hydhologic data, oveh6i-<br />
zing most dtood conthol 6thuctuhe6 i6 to be hecommended,<br />
even ifj it’6 con~equence i6 a ubete66 inve~tment o6 ca-<br />
pital, in ohdeir to avoid <strong>the</strong> hibh 06 ovehtopping and 06<br />
eventual bucceeding 6ailuhe od 6thuctuhQ6 181. Thi6 po-<br />
&cy h juhtidied by <strong>the</strong> exponential ghowth 06 <strong>the</strong> den6-<br />
ity 06 economic and 6ocia.t objecaXve6 placed in <strong>the</strong> ama<br />
in Which dtood COnthOt ~66e~tb 06 <strong>the</strong>be bthUCtUhC6 i6<br />
he6ented. Eventual damage6 due to exceedence 06 de6ign<br />
pahameteh6 will Zhu6 be exponentially incheahed in compahibon<br />
Stood.<br />
to actual damage6 cohhehponding to <strong>the</strong> 6ame<br />
Thub, adteh <strong>the</strong> hydirologic excedentahg peiriod 06<br />
<strong>the</strong> la62 yeah6 and pahticu.tah.ty aóteh <strong>the</strong> 7970 and 7972<br />
iloodh, <strong>the</strong> conclubion wa6 dhawn in Romania that 6lood<br />
detention volume6 hebehved in 6tohage lahel, which, a6 a<br />
hu<strong>le</strong>, Wehe phteViOU6ly 06 <strong>the</strong> ûhdeh 06 b - 12 % 06 <strong>the</strong> a-<br />
vehage annual blow a m knbuddicient; incireahing <strong>the</strong>6e<br />
volume6 in butuhe up to 20 8 06 <strong>the</strong> aveaage annual dlow<br />
i4 hecommended. Except 4 ome i6 olated ea6 e6, <strong>the</strong> execut-<br />
ion 06 hubrneuib<strong>le</strong> dyke6 has been comp<strong>le</strong>tely abandoned<br />
bivice 1960.
674<br />
In <strong>the</strong> debign 06 dyke6, <strong>the</strong> gkowth 06 blood <strong>le</strong>vea<br />
due to eliminating <strong>the</strong> natukal deterifion o6 Blood in<br />
<strong>the</strong> dtood plain and to kiVeh bed dynamic¿, pahticulahlg<br />
<strong>the</strong> haióing od <strong>the</strong> base 04 ehobion, obbehved on many wu-<br />
tek couhbeb 06 <strong>the</strong> plain hegiOMb. At <strong>the</strong> tail watekd 06<br />
cehtain btohage kebehvoihb located in countkied with de-<br />
vem climate, inadequate know<strong>le</strong>dge 06 wintek phenornena<br />
may <strong>le</strong>ad to undehebtimafing <strong>the</strong> backwatek phoduced by<br />
<strong>the</strong> mas4 06 ice and to undokbeen dlooding, .i6 no phe-<br />
cauLion4 ate taken.<br />
Finally, bok <strong>the</strong> cased whehe data on <strong>the</strong> genebis 06<br />
6loodb ahe lacking, it i4 pobbib<strong>le</strong> to apply methodb od<br />
<strong>the</strong> <strong>the</strong>ohy 06 gamed. Such methodb have been applied in<br />
Romania 604 bevekal pkojech 171.<br />
Some aspecÁ2 o 6 multi-puhpob e<br />
wutek ked ouhceb development.<br />
One 06 <strong>the</strong> chahac$ehib.i.ics 06 <strong>the</strong> contempokahy wa-<br />
tek he6 Oukceb engineehing conbibtb in <strong>the</strong> multipuhpob e<br />
and comphehen6ive development 06 hiWh buinb, within<br />
an ebtablibhed development bcheme. VaGoub pko jectb ake<br />
phomoted a¿ bepahate deÜeÆopment btages 06 <strong>the</strong> genehat<br />
bcheme, all phojecÁ2 being based on unitaky methodolog-<br />
ieb and being conceived bo as to meet <strong>the</strong> kequikemenÁ2<br />
06 d l wateh ubeIL4 u well as 04 <strong>the</strong> COnthOl 06 deb-<br />
thucfive eddech 06 wateh. The btep by step development<br />
06 a hiveh basin i4 impohtant to <strong>the</strong> phob<strong>le</strong>m dibcubbed<br />
in thib hepoht because 06 it'd pobitive conbequenceb he-<br />
d<strong>le</strong>cted in <strong>the</strong> pobbibility 06 cokhecfing emou, 06 he-<br />
ducing exaggehated hibkb and even 06 attenuafing bail-<br />
ukeb by <strong>the</strong> way 06 conceiving Bututre phOjUÁ2 developed<br />
in <strong>the</strong> bame hiveh basin. Thió i4 helated to <strong>the</strong> 6act<br />
that <strong>the</strong> unitahg management 06 <strong>the</strong> watek kebouhceb ob<br />
a hiveh basin cheateb a <strong>le</strong>bb 4thLc.t dependence 06 each<br />
phoject on <strong>the</strong> dibffkibufion 06 <strong>the</strong> dlow 06 <strong>the</strong> main hi-<br />
veh among vahiou4 thibutahieb and heduceb <strong>the</strong> benbitiv-<br />
ity to emohb in evaluating hydhaulic hCbOUhCe6 in each
675<br />
~pecibic bite. Thib i.4 impohtant, ab global ebtimation<br />
od <strong>the</strong> hCAOUhCeA 06 a kiveh bain i4 UbUal.& <strong>le</strong>Ab hub-<br />
ject to ehhohb than <strong>the</strong> ebamation od <strong>the</strong> heAouhceb 06<br />
<strong>the</strong> thibutahieb. On <strong>the</strong> o<strong>the</strong>h hand,<strong>the</strong> btep by btep de-<br />
velopment o 6 bthuctuheA doh wateir heb ouhced management<br />
makes changed in <strong>the</strong> pahametea od ultehioh phojectb<br />
pobbib<strong>le</strong>, Ao ab to COhJLect ~46ectb 06 Oveh oh undeh-<br />
hizing <strong>the</strong> 4-thuctuheA built at phevioub development<br />
A tag eb .<br />
In <strong>the</strong> cae 06 pkojecth doh which decibionb ahe<br />
made undeh condifionb od inadequate basic data, inveb-<br />
tigai2on 06 <strong>the</strong> pobbibifitieb 06 butuhe sxtenbion od<br />
bh’iUC~UReb d od gheat crteirebt.Such podbibilitied eliminate<br />
<strong>the</strong> necedbity 06 immobilizing capital doh <strong>the</strong><br />
development 06 initially oveh~izc! d bthuctuheb, deaigned<br />
in thib way in ohdeh not to loo¿e <strong>the</strong> pobbibifitieb 06<br />
an advantageou dite.<br />
EhhohA committed due to inadequate hydhologic<br />
data ahC not fimited to <strong>the</strong> design btage o6 hydhaufic<br />
AthUCtUheb. FOh <strong>the</strong> conception od buch bthUCtuheb<br />
<strong>the</strong> lack 06 dihecz hydhologic data can obten not<br />
be avoided. 76 <strong>the</strong> imp<strong>le</strong>mentation od an adequate hydhometeOhOlOgic<br />
6ohecabting nekwohk, including <strong>the</strong> meaAuhing,<br />
thanhmibbion and data ph0Cchbing equipment d advibab<strong>le</strong><br />
in Ohdeh to Opehate a CQhtLn hebChVOih, .¿A puhely<br />
an economic phob<strong>le</strong>m. In thib benbe, <strong>the</strong> technical<br />
oh economic analyAib 06 opehating conditionb 06 vahiOUA<br />
lahge Acate pkojech and <strong>the</strong> indluence 06 a good doheca6i2ng<br />
aybtem on <strong>the</strong>iir opehation Achedu<strong>le</strong>4 <strong>le</strong>ad invatiably<br />
to <strong>the</strong> conclubion that <strong>the</strong>be Aybtemb<br />
culahly e 4 bicient.<br />
ahc pahti-<br />
Thib L¿ not only <strong>the</strong> cabe whehe hydhaufic Aybtemb<br />
debigned 604 dtood canthot oh doh bade yield ahe addected<br />
by <strong>the</strong> lack od adequate indohmation oh dohecabzb<br />
at Auch exteint, that <strong>the</strong>ih opehation accohding<br />
to b chedu<strong>le</strong> d phactically impohbib<strong>le</strong> without imphovcng<br />
<strong>the</strong> indoirmational bybtem. The advantage id evident
67 6<br />
ado when a betteh know<strong>le</strong>dge 06 <strong>the</strong> phobab<strong>le</strong> pahame-<br />
tehb o6 butuhe hydhologic even22 <strong>le</strong>adb only to opehat-<br />
ional imphovemenX.6. Thib cb, doh inbtance, <strong>the</strong> cabe 06<br />
hydkoe<strong>le</strong>ctk-ic plana.<br />
An illwthative examp<strong>le</strong> 06 an in6OhmatiOnal and<br />
donecabX.ln9 netwohk concehnb khe lhon Gate¿ hydhoe<strong>le</strong>c-<br />
thiC plant on <strong>the</strong> Danube. The wateh <strong>le</strong>vel 06 <strong>the</strong> bfoh-<br />
age hcbehVOih at <strong>the</strong> dam vahiab<strong>le</strong>, opehating bchedu-<br />
&A phoviding doa an ab conbtant ah pobbib<strong>le</strong> wateh <strong>le</strong>-<br />
vel & <strong>the</strong> tail 06 <strong>the</strong> Atohage hebChVOih, Upb<strong>the</strong>ani 06<br />
<strong>the</strong> IhOn Gates gohgeb. Thib policy cb due to <strong>the</strong> con-<br />
cenakation in thib ahea o6 bthuctuheb debigned doh <strong>the</strong><br />
photection 06 hipak-ian land and o<strong>the</strong>h development¿,<br />
bthuctuheb which would be oventopped at higheh <strong>le</strong>ve&.<br />
Conbcquently, <strong>the</strong> hydhoe<strong>le</strong>cthic plant bhould hegul&e<br />
<strong>the</strong> wateh <strong>le</strong>vel & <strong>the</strong> dam bon an expected inblow, bo<br />
UA to obtain <strong>the</strong> maximum u.ti.lizab<strong>le</strong> head, to avoid, U<br />
much ah pobbib<strong>le</strong>, <strong>the</strong> 0vehhpit.ling o6 Waxeh and,& <strong>the</strong><br />
bame time, to avoid <strong>the</strong> exceedence 06 <strong>the</strong> badety <strong>le</strong>vel<br />
in <strong>the</strong> photected aheu. Vue to a good dohecabtin9 netwohb<br />
on <strong>the</strong> Danube and on <strong>the</strong> main thibttahiebp upbtlream<br />
o6 <strong>the</strong> phojecf, it w u pobbib<strong>le</strong>,even duk-ing <strong>the</strong><br />
dihbt WO yeahb od opehaL¿ng, to hegutate <strong>the</strong> daily<br />
powea genetation in buch mannet that <strong>the</strong> deviaLion<br />
6hOm <strong>the</strong> <strong>the</strong>ohea2ca.t optimu did not exceed 1.2 %.<br />
The utility 06 <strong>the</strong> exPendituaeh intended to bet<br />
up and maintain an adequate 6onecasLing bybtem, pahticutaaly<br />
within hiveh bain6 whehe dlood deten.tion heb<br />
eh<br />
oiha ahe located has not any mohe to be demonsthated.<br />
Situation¿ may be met in which an inadequate opehation<br />
od OUtkkX.6 dhom Atohage hebehVOihb clln OvehpObe<br />
nohmally buccebbive blood waved, <strong>le</strong>ading to an aggnavakion<br />
06 <strong>the</strong> bituation which would<br />
unhegulded b&eamb.<br />
have occuhed in
C o n c t u b i o n b .<br />
677<br />
The accukacy o6 <strong>the</strong> dolutionb 06 debign and opekat-<br />
ion 06 hydkauîic wokkb depend not only on hydhologic<br />
in6akmation, bu.2 albo on <strong>the</strong> degtee o6 cokkect ebtimat-<br />
ion 05 a multitude 06 o<strong>the</strong>ir 6actou, 60ir inbtance <strong>the</strong> e-<br />
conomic and conjectukal condixXonb ,<strong>the</strong> wateir demand etc.<br />
Thebe iactok obten aire much m o u uncektain than hydholo-<br />
gic evenh. in thib context, in cae 06 ceirtain ubeb 06<br />
hydkautic bthuctukeb, Auch ab hydkoe<strong>le</strong>cakic powea pko-<br />
ducfion, watea tkanópokt, low dlow kegulaaon boa watek<br />
ueb, <strong>the</strong> hydhological in6oirmatian bhouLd be consideked<br />
in <strong>the</strong> hame way a6 o<strong>the</strong>n uncektain basic data, <strong>the</strong> qua-<br />
îity 06 which h a a gtobaî inbluenee on <strong>the</strong> p04bibiti-<br />
ty 06 op~mizing bOLUtiOn4. in buch cabeb, <strong>the</strong> necebbaky<br />
accukacy 06 hydiroLogical data mubt be Looked at in cok-<br />
helaZion with <strong>the</strong> accukacy o6 o<strong>the</strong>k e<strong>le</strong>menh ke<strong>le</strong>vant to<br />
<strong>the</strong> decibion. Them ake howevea alho o<strong>the</strong>k typed 06 hy-<br />
dkaulic b&uc.twreb,buch a6 thobe debigned 60k dlood con-<br />
thol, 60ir which global analybib 06 accukacy 06 basic da-<br />
ta i~ tebb impoatant and hydkologic data have to be ta-<br />
hen bepaately into account.<br />
R E F E R E N C E<br />
1.- PLESHKOV, 1.F. Reguliirovanija kechnogo btoba -<br />
GWdkometeoizdat, Uobkow, I96 1.<br />
2.- DYCK,S.; SCHRAMM,M. Stochabtibche Methoden 6Ük die<br />
Bemebb ung deb Wasb eupeichekhaumeb.<br />
- Mitteilungen deb in-<br />
AZifUteb {Ük W a b eir~ikt.6 cha@,<br />
Nk.28, BekÆLn, 1968.<br />
3.- STEGARUZU, P. CokectiiLe de debite zilnice in<br />
calcuîeÆe de gobpodairike a apeloir.<br />
- Hidkotehnica, Nk. 111972.<br />
4.- SVANIDZE, G.G. ûbnovy k a chety keguîikovanija<br />
aechnogo btoka metodom i\lonte -<br />
Kaalo.- Tbiîibbi, L964.
67 u<br />
5. L’EZNlKOVSKll, A.A.<br />
Vodnoenehgetichebhie ha6 chety<br />
metodom Monte-Kahlo - Enehgija,<br />
idobbow, 1969.<br />
S<strong>the</strong>amdtow Syn<strong>the</strong>dib . - MacMil-<br />
6. FlERlNG, M.B.<br />
7. - VORVEA, A. ; Fl LOTT i, A.<br />
lan, London-Melbouhne, 1967.<br />
PhOb<strong>le</strong>me de gOdpOdahihe a apeloh<br />
CU aplicatic? la bazinul<br />
Bahlu&. -1nbtitutul penthu Planuhi<br />
de Amenajahe bi Consthue-<br />
8.- FILOTTZ, A.<br />
tii Hidtotehnice ílPACHl.15 ani<br />
de activitate. Bucutebti, 7968,<br />
pp. 85 - 96.<br />
Dib cubbiòn deb happohh concehnant<br />
la photcefion de4 eaux en<br />
9. - TEODORESCU, 1.<br />
ea6 de cûtabthûpheb. -GeWabbehbchutz<br />
im Katasttophen~all.Sympobnum<br />
vom 23 - 26 Ohtobeh in<br />
Flohenz. Födehafion Euhopäib cheh<br />
Gewabbehbchutz, Vol. 15, Ziihich,<br />
1969, pp- 92 - 96.<br />
Gobpodahihea Apeloh. - Ceheb,<br />
FILOTTI ,A.; CHlRl AC V.%ucuhebti, 19 73.<br />
10. -S?MON, A; Vl LAN. A GenehcZhea bihUhil0h hidhalogice<br />
daha aUtOcOhe<ie. - Studii de<br />
Economia Apeloh, Vol. 1. lnbtitutul<br />
de Studii bi Cehceta~<br />
penau Zmbunatatihi Funciahe<br />
bi Gobpodahihea Apeloh, BuCuhebti,<br />
1971, pp.311 - 366.
ß, 1<br />
IO<br />
Fig.1.<br />
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0,s<br />
C"<br />
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20<br />
LO<br />
1.0<br />
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as 0.5<br />
. [r=45; c,=2c,<br />
- 1<br />
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28<br />
10<br />
679<br />
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pigc4. Average required etorage a8 a function<br />
of record,for<br />
-<br />
different degrees of<br />
regulation8<br />
c= 0,5 I r 0.3<br />
681
ItRELATIONS BETWEEN PROJECT ECONOMICS AND HYDROLOGICAL DATA"<br />
by<br />
A. Pobedimsky<br />
Economic Commission for Europe<br />
Introduction<br />
In e::ce;j:;, water ilhich is very often considered as a source o;" wealth, m y<br />
cause considerab<strong>le</strong> damage or äisaster and impose a heavy burden on a country's<br />
economy. sorn&jimez it may affect a goup of borderine countries (as for examp<strong>le</strong>,<br />
those locate6 in <strong>the</strong> Danube and Rhine river basins).<br />
The accelzrating rate of population eowth in <strong>the</strong> EC3 countries and <strong>the</strong><br />
economic proLpcss in Lechnologicrì changes, during recent years, causing <strong>the</strong><br />
dep<strong>le</strong>tion of natural resources have all rapidly increased <strong>the</strong> importance of<br />
water resources development which, curing <strong>the</strong> last few decades, hac, become one<br />
of <strong>the</strong> doninnting factors in <strong>the</strong> national economy of most stoLintries. All this<br />
has obliged countries to improve <strong>the</strong>ir water resources nanagement so as to achieve<br />
proper f<strong>le</strong>xibility a d effectiveness corresponding to modern requirerncnts of national<br />
e conomie s.<br />
Mater Phnagement, deals o-lher things with a verj importent component -<br />
hydrological data which define <strong>the</strong> availab<strong>le</strong> water resources to me& national or<br />
regional demnds.<br />
The mtcliing of <strong>the</strong> bdznce or water resources md. neecls, as vel1 as <strong>the</strong><br />
planning and imp<strong>le</strong>mentation of eppropriatc masures to provide <strong>the</strong> nececjsa.0 liater<br />
cupply for a region, have become &ieslionr of hi& priority,<br />
The belances oi' water resources and needs mentioned above, which serve to<br />
elaborate <strong>the</strong> measures io be taken to avoid ne,r;etive c.onsequences for ths populaLion<br />
and <strong>the</strong> regional economy, are now Peing used as a effective tool in mqv LC3<br />
countries. The first internationa!. l4anue.l for <strong>the</strong> compilation of <strong>the</strong>se balances,<br />
now beiny: comp<strong>le</strong>ted by <strong>the</strong> ZCg Conrmi-kLeé on idater Prob<strong>le</strong>ms g d groups of national<br />
experts, emphasizes that in regions with I.imited va-<strong>le</strong>r re::ouxces a high degree of<br />
accuracy in <strong>the</strong>ir assessment is an essential condition ror e. rational economy.<br />
The E.'mual enphasizes <strong>the</strong> importtince of <strong>the</strong> earliest possib<strong>le</strong> organization of<br />
hydrological studies in a river basin or 8 region where an intensive growth oi<br />
uater needs is in prospect. ?he Phniisl defines in <strong>the</strong> following WP.~ <strong>the</strong> economic<br />
impact of reliab<strong>le</strong> hydrological daCC. on ua-ter resources dcvelopnent in particular<br />
cnd on <strong>the</strong> national evonoq in genernlc ?he more reliab<strong>le</strong> <strong>the</strong> iiatter zupply, thi:<br />
smal<strong>le</strong>r i.iill be <strong>the</strong> damage resultin, from cutr in periods of water shortagetf. By<br />
cornparin: losses anU expenditure, it vili, in prlncip<strong>le</strong>, be poosi'u<strong>le</strong> to determino<br />
<strong>the</strong> economic optimum.<br />
Sufficient and accurate hydrological data promote effective vater management and<br />
<strong>the</strong> prevention or didnuation of damage caused by such hydrological phcnomna as<br />
severe floods , ice jm, rnuàflotis, intemive oedirnents, dangerous va.ter pollution etc.
684<br />
On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> intensification of human activities in river basins<br />
rnl :,Lcii, watersheds including <strong>the</strong> increased anount of untreated effluents<br />
uic .liarged FnLo <strong>the</strong> water couse, during recen.'c decades, uraently calls for <strong>the</strong><br />
re1iai;l.e esse:;smnt of availab<strong>le</strong> wa-tor resources. The ECS Nanual mentioned above<br />
states that <strong>the</strong> consequences of h w n activities make advanced hydrolo$cal<br />
research imperative.<br />
considered important.<br />
A re<strong>le</strong>vant improvement of hydrological methods is<br />
5Je understand that this topic is of considerab<strong>le</strong> interest to hydrological<br />
services which must strike a balance between tho cost of gauging stations and<br />
<strong>the</strong> probab<strong>le</strong> futvre benefit that will result from <strong>the</strong> information to be obtained.<br />
TakTng into account <strong>the</strong> special importance of hydrological data for water<br />
resources management, various aspects of <strong>the</strong> development of hydrological networks<br />
were discussed thoroughly at <strong>the</strong> ESE Seninar on Se<strong>le</strong>cted Water Prob<strong>le</strong>ms in Sou<strong>the</strong>rn<br />
Europe convened in Zagreb, Yugoslavia, in October 1971. Certain conclusions<br />
concerning <strong>the</strong> design of hydrological networks and <strong>the</strong>ir improvement were<br />
ref<strong>le</strong>cted in <strong>the</strong> recommendations adopted by <strong>the</strong> ECE Codttee on Water Prob<strong>le</strong>ms.<br />
Taking all this into account it is generally recognized that hydrological data<br />
and well planned hydrological networks are prerequisites for efficient and sound<br />
water resources planning.<br />
The purpose of this paper is to appraise <strong>the</strong> possib<strong>le</strong> economic effect of<br />
insufficient hydrological data on <strong>the</strong> effectiveness of water planning and <strong>the</strong><br />
design of hydraulic engineering structures and <strong>the</strong>ir operation.<br />
It is suggested to consider <strong>the</strong> following main aspects of <strong>the</strong> subject:<br />
The economic consequences of a deficiency of hydrological data on water<br />
planning, construction and operation,<br />
"he impact of a deficiency of hydrological data on main water users.<br />
The economic consesuences of a deficiency of hydroloaical data on water<br />
P ~ construction ~ P and operation<br />
The following questions could be raised in connexion with this aspect:<br />
what is <strong>the</strong> extent of <strong>the</strong> economic impact of insufficient hydrological<br />
records on a project and its subsequent operation?<br />
in particular, what is <strong>the</strong> possib<strong>le</strong> effect on investment in economic<br />
development if hydrological data are not accurate enough and <strong>the</strong> records<br />
are insufficient?<br />
is <strong>the</strong> predominantly quantitative character of hydrological data sufficient<br />
for modern intensive water resources development?<br />
is it reasonab<strong>le</strong> to postpone <strong>the</strong> initiating of water project planning and<br />
<strong>the</strong> construction of water projects if <strong>the</strong> hydrological observations are<br />
insufficient?<br />
The availab<strong>le</strong> information on <strong>the</strong> experience and research in <strong>the</strong> ECE region<br />
shows <strong>the</strong> following facts which could be emphasized in an attempt to answer <strong>the</strong><br />
above questions.
Economic -acts Qf insufficient hydrological data and difficulties caused at<br />
<strong>the</strong> key stages of water resources development:<br />
Mater planning and desim<br />
Generally speaking, poor hydrological data and forecasts made on this basis<br />
can <strong>le</strong>ad to inappropriate proposals for investment in water engineering works and<br />
<strong>the</strong> economic development of <strong>the</strong> region concerned.<br />
The importance of sufficient<br />
data at <strong>the</strong> following staget of planning and designing can be pointed out.<br />
The elaboration of schemes for niltipurpose development of water resources<br />
is greatly dependent on accurate hydrometrical data.<br />
according to which a scheme is to be designed cannot be derived from incorrect<br />
hydrological data.<br />
The appropriate conclusions<br />
It should be underlined that possibilities of considerab<strong>le</strong> miscalculations<br />
exist in preinvestment studies as well as in fur<strong>the</strong>r stages of planning and design<br />
of engineering structures.<br />
characteristic river discharges, approximake methods and empirical fornulas<br />
are used and <strong>the</strong> <strong>le</strong>ngth of time of hydrological observation is relatively short<br />
(considerably <strong>le</strong>ss than 30-40 years).<br />
They may be especially acute when, to estimate<br />
Taking this into account, approximate<br />
methods are being limited to <strong>the</strong> preliminary estimations for <strong>the</strong> elaboration of<br />
schemes of development, but are not reconmiended for <strong>the</strong> design of water structures.<br />
Appropriate attention to this matter has been given in <strong>the</strong> ECL i.ianual<br />
mentioned above.<br />
Different research during recent years has also analysed <strong>the</strong> importance of<br />
hydrological data for large sca<strong>le</strong>, long-term investment for general economic<br />
development.<br />
The conclusions of some of <strong>the</strong>se studies may be summarised as follows:<br />
It is considered that, if <strong>the</strong> period during which hydrological records have been<br />
kept is not sufficiently long or <strong>the</strong> data are not sufficiently accurate, <strong>the</strong>n <strong>the</strong><br />
investment in <strong>the</strong> economic development will be larger than is necessary, or <strong>the</strong><br />
possib<strong>le</strong> production of <strong>the</strong> plant will be <strong>le</strong>ss, due to its smal<strong>le</strong>r size. In both<br />
cases <strong>the</strong> result will be a loss to <strong>the</strong> overall econon&.<br />
Experience shows that various hydrologic parameters could have economic<br />
importance for different stages and purposes of water resources development.<br />
For examp<strong>le</strong> - flow variability and drought occurence for <strong>the</strong> design of storage<br />
reservoirs; flood occurences for <strong>the</strong> design of spillways and o<strong>the</strong>r control works;<br />
<strong>le</strong>sser importance is attached to <strong>the</strong> mean flow.<br />
one a uthod<br />
However, as can be cited after<br />
r/ D. Johanovic. Abstract. Vhe Ro<strong>le</strong> of I-Qrdrologg and Hydromteorolow in <strong>the</strong><br />
Economic Development of Africat! Ma, No. 301, 1971.<br />
2/<br />
K.C. Wilson Cost-benefit approach to hydrometric network planningtt Water Res.<br />
Research. October 1972.<br />
685
686<br />
Wariation of mean annual flow, depending upon hydrologic data, and <strong>le</strong>ngth of<br />
observation <strong>le</strong>ads to overestimation or underestimation in determination of<br />
sizes of water engineering structures, determination of regimes, capacities<br />
of plants, irrigated areas" etc.<br />
In fact <strong>the</strong> following implication of an underestimation of <strong>the</strong> mean annual<br />
flow in <strong>the</strong> planning of water resources developnient might ba indicated:<br />
(a) Smal<strong>le</strong>r sizes of water storage capacity resulting in limited possibilities<br />
of regulation of flows with subsequent adverse impact on:<br />
(i) possibilities of self purification of water;<br />
(ii) availabilities of water supply for drinking and industrial purnoses;<br />
(Xi) potential of production by hydro-e<strong>le</strong>ctrical plants;<br />
(iv) quantity of water for irrigated areas;<br />
(v) navigationaï capaciw of rivers;<br />
(Vi) inadequacy of water storage size:: requires additional investments for<br />
increasing dans, canals, etc. at a later stase.<br />
On <strong>the</strong> o<strong>the</strong>r hand, over-estimtion of inem annui flow can <strong>le</strong>ad to <strong>the</strong> following<br />
implications :<br />
(i) oversizing of iiriter engineering strictures, low efficiency of <strong>the</strong>ir<br />
operation, larger investments in conparison with normal;<br />
(ii) insufficiency of water for designated irrigation area3<br />
(iii) energy production below planned target;<br />
(iv) <strong>le</strong>sser dilution of effluents discharged and slower processes of self<br />
purification of water.<br />
Studies of <strong>the</strong> value of hydrological data are being carried out in several<br />
countries. Comprehensive studies were conducted jointly by <strong>the</strong> United States Corps<br />
of Engineers and <strong>the</strong> Geological Survey (1970) with <strong>the</strong> purpose of evaluaiing <strong>the</strong><br />
iiorth or" hydrological data for determination of <strong>the</strong> optimum water storage conserva.tion<br />
capacity. Some American researchers have developad a ma<strong>the</strong>matical relation betireen<br />
<strong>the</strong> llworthll of data and <strong>the</strong> <strong>le</strong>ngth of periods during which <strong>the</strong>y were recorded.<br />
Some research has been carried out to estimate <strong>the</strong> possib<strong>le</strong> loss due to<br />
inperfect information. The research, for instance considering optimal reservoir<br />
design, analytically defines <strong>the</strong> opportunity loss as <strong>the</strong> difference between net<br />
benefits associated with different hydrological data depending upon streamflow<br />
record <strong>le</strong>ngth. Reservoir designs are obtained by simulating flovs and se<strong>le</strong>cting<br />
that combination of storage capacity and target yield which gives <strong>the</strong> greatest<br />
net benefits. Graphic functions obtained by this research shou that opportunity<br />
losses decrease rapidly due to increasing <strong>the</strong> <strong>le</strong>ngth of streamflow observation,<br />
achiedng very small magnitude beyond thirty years of observation; this conforms<br />
to many practical observations. Various types of reseaxch and observations show<br />
that <strong>the</strong> cost of obtaining addition& hydrological data i.e. increasing <strong>the</strong> <strong>le</strong>ngth<br />
of observation is insignificant in relation to<strong>the</strong> reduction of <strong>the</strong> esrected
opportunity lox.<br />
687<br />
Some CanaCZan research developing generalized computer programmes to relate <strong>the</strong><br />
costs of operating and intensif'ying a hydrometric network to <strong>the</strong> resulting increases<br />
in <strong>the</strong> accurecp oi: <strong>the</strong> three parameters mentioned above should be noted.<br />
The availab<strong>le</strong> experience of differenct ECL countries confirms to an extent <strong>the</strong><br />
conclusions of <strong>the</strong> research mentioned.<br />
-<br />
-<br />
As ior examp<strong>le</strong> in <strong>the</strong> USSR, it is considered:<br />
<strong>the</strong> economic benefits of <strong>the</strong> hydrometeorological service due to which <strong>the</strong><br />
hydrological and meteorological observations can be obtained are a high as<br />
one billion roub<strong>le</strong>s, i.e. 4-5 tines <strong>the</strong> amount that is spent on maintaining<br />
this semice;<br />
<strong>the</strong> introduction of <strong>the</strong> use of hydrological forecasts in national planning<br />
made it possib<strong>le</strong> to raise by 10-15 per cent <strong>the</strong> efficiency of water<br />
installations and to obtain correspondingly higher profits2/.<br />
In <strong>the</strong> UnLted Kingdom and France, <strong>the</strong> relrvant benefits are estimated to exceed<br />
<strong>the</strong> national hydrometeorological budgets at <strong>le</strong>ast 20 timed.<br />
However, in many countries, especially in <strong>the</strong> developing ones, <strong>the</strong> povement<br />
agencies responsib<strong>le</strong> for h@rological observations do not have sufficient funds<br />
availab<strong>le</strong> for <strong>the</strong> development of adequate nationwide hydrological network$. This<br />
insufficiency of financial resources for <strong>the</strong> col<strong>le</strong>ction of hydrological data was<br />
also emphasized at <strong>the</strong> ECE Seminar on certain uater prob<strong>le</strong>ms, convened in Zagreb<br />
in 1971.<br />
The USSR experiences show that in <strong>the</strong> absence of hydrometeorological observation<br />
data in an area se<strong>le</strong>cted for construction, a special hydrometeorological investigation<br />
should be conducted with an expenditure of 2-3 thousand roub<strong>le</strong>s for each million<br />
roub<strong>le</strong>s invested. This amount is considered to be an econom if, as a result,<br />
sufficient hydrometeorological data proved to be availab<strong>le</strong>. Experience in o<strong>the</strong>r<br />
countries confirms that <strong>the</strong> expense of acquired adãitional hydrological data<br />
is much <strong>le</strong>ss than <strong>the</strong> losses involved in water engineering construction,<strong>the</strong> design<br />
of which is based on inaccurate data. In this connexion as a positive experience,<br />
a considerab<strong>le</strong> extension of national and regional hydrological networks is taking<br />
place. It should also be noted that in some countries <strong>the</strong> application of automatic<br />
monitoring stations to control <strong>the</strong> quality and quantity of a river flow has<br />
considerably extended during recent years. This modernization increases <strong>the</strong><br />
reliability of recorded data indispensab<strong>le</strong> for accurate planning and design.<br />
2/ E.J. Tolstikov, Vhe Benefits of Hydroirieteoroloefc Services in <strong>the</strong> UcsRtt<br />
'dl40 Vhe Economic Benefit of National Meteorologic Services'World Wea<strong>the</strong>r<br />
Watch No. 27 - i968<br />
i/<br />
Richard D.A. Kill IXydrological and Hydrometeorological data as essential<br />
parameters for design or economic development projectsf! - ld4l Eo. 301
68 8<br />
In spite of <strong>the</strong> sound persuasiveness of all <strong>the</strong> research mentioned above,<br />
arm<strong>the</strong>r trend of research should be pointed out. Several studies and research<br />
have been devoted to Solve <strong>the</strong> prob<strong>le</strong>m of whe<strong>the</strong>r it is desirab<strong>le</strong> and profitab<strong>le</strong><br />
to postpone <strong>the</strong> development of water resources or <strong>the</strong> construction of certain water<br />
cngineering projects if <strong>the</strong> hydrological data for <strong>the</strong> river or particular site<br />
investigated are insufficient. It is considered by some of <strong>the</strong> authors that, by<br />
postponing <strong>the</strong> construction, <strong>the</strong> realization of <strong>the</strong> net benefits of that construction<br />
is also being postponed.<br />
The availab<strong>le</strong> results of research conclude that <strong>the</strong> risk of such postponement<br />
must be carefully evaluated. Nore than that, some research c<strong>le</strong>arly rejects<br />
postponement as a protitab<strong>le</strong> course of action. It is considered that only an<br />
exceedingly loid discount rate makes <strong>the</strong> minimum total cost of postponement<br />
e.g. for one year, equal to <strong>the</strong> cost without postponement.<br />
Taking into account <strong>the</strong> definite discrepancy between <strong>the</strong>se conclusions and<br />
those previously mentioned we feel that fur<strong>the</strong>r research should be continued,<br />
emphasizing not only purely economic aspects, but also taking into account social<br />
and technical aspects, including first of all <strong>the</strong> prob<strong>le</strong>m of safety of structures<br />
and subsequently of <strong>the</strong> population. The lack of unaniixi~iy, even in a purely<br />
economic approach, concerning <strong>the</strong> postponement of projects is to be noted.<br />
Some authors concludeb/:<br />
tlSince demand for project outputs is presumably growing over time, <strong>the</strong> more a<br />
project is deferred, <strong>the</strong> more quickly it is likely to be used, and <strong>the</strong> greater <strong>the</strong><br />
benefits generated per time period of project life will be.”<br />
The conclusion seems to he very sensib<strong>le</strong>.<br />
Pldn,? and desiminn of flood protection<br />
This cspect deserves special attention, bearing in mind that economic losses<br />
due to floods in <strong>the</strong> river basins of <strong>the</strong> world continue to be very high. Noreover,<br />
<strong>the</strong> fur<strong>the</strong>r extent of <strong>the</strong> economic development of regions being threatened by high<br />
flows <strong>le</strong>ads to fur<strong>the</strong>r growth of economic damage from floods. Just one examp<strong>le</strong>, fron<br />
<strong>the</strong> experience of <strong>the</strong> United States, shows that <strong>the</strong> frequency of floods, causing<br />
major roperty damage of $50 million or more were increased aimst three times since<br />
19L&d Based on <strong>the</strong> currßnt status of flood control works and project conditions<br />
of flood pldm use and development <strong>the</strong> total annual floodcbage potential for <strong>the</strong><br />
nation is anticipated to increase from $1.7 billion in 1966 to $5.0 billion in 20Zoz/.<br />
It is considered that, for elaboration of efficient national flood proLection<br />
policies, <strong>the</strong> data on distribution of river flow durine a year, asml1 as <strong>the</strong> exact<br />
characteristics of floods including maximum discharge, duration, tlme of flood end<br />
volume of <strong>the</strong> flood flow are of considerab<strong>le</strong> importance.<br />
However, <strong>the</strong> determination of <strong>the</strong>se characteristics becomes verydifficult if<br />
hydrological observations are insufficient, as was mentioned above.<br />
m e s<br />
Geophysical Union, bJashington D.C. 1971<br />
IfThe Nations Water Resources11 W.R.S. United States, 1968, 5-2-2.<br />
1/<br />
\J. IIowe naenefit-Cost Analysis for Water System PlanningIl - American
Experience shows that <strong>the</strong> frequency of peak floods, determined on <strong>the</strong> basis<br />
6 89<br />
of a short ceriod of observations cm be several times lower than <strong>the</strong> adequate<br />
vdue, and that <strong>le</strong>ads to considerab<strong>le</strong> damage and to catastrophiee. The foll-owing<br />
consequences of inaccurate flood forecasting should be pointed out as regards<br />
water resources development and planning and designing of flood protection.<br />
(a) Overestimation of flood discharge - <strong>le</strong>ads to unnecessary expenses in<br />
weter engineering construction;<br />
(b) Underestimation of flood discharge - <strong>le</strong>ads to full dektruction of <strong>the</strong><br />
sl.r.ucture, resulting in damage and possibly in loss of human lives in<br />
<strong>the</strong> region.<br />
Thus, proper estimation of possib<strong>le</strong> flood peak, depending upon <strong>the</strong> quality<br />
and accuracy of hydrological data is very important. At <strong>the</strong> samc tine <strong>the</strong><br />
experience of many countries still shows that: <strong>the</strong> continuing groi.rth of economic<br />
damage to individuals and to <strong>the</strong> national economy in many regionu of <strong>the</strong> world,<br />
caused by river floods, can be explained not oniy by <strong>the</strong> spontaneity of <strong>the</strong> flood<br />
phenomena, but also by <strong>the</strong> lack of adeguate organization, insufficient hydrologic<br />
observation and <strong>the</strong> necessary financial means, which very often are considerab1.y<br />
<strong>le</strong>ss than <strong>the</strong> value of <strong>the</strong> damage caused. Talcing this into account <strong>the</strong> ECZ<br />
Cohtiee on Water Prob<strong>le</strong>ms has recently initiated studies on <strong>the</strong> availab<strong>le</strong><br />
experience in rational methods of flood control planning in river basin<br />
development with <strong>the</strong> purpose of extending this experience to al1 ECd countries.<br />
In spite of <strong>the</strong> concept generally adopted that <strong>the</strong> development of sufficient<br />
hydrological observation is very important for <strong>the</strong> planning of proper flood<br />
protection and <strong>the</strong> organization of adequate operational measures to prevent<br />
considerab<strong>le</strong> damage, never<strong>the</strong><strong>le</strong>ss <strong>the</strong> availab<strong>le</strong> infornation from some countricc<br />
indicates :<br />
- a lack of reliab<strong>le</strong> data regarding rainfall intensities and corresponding<br />
stream flow;<br />
-<br />
-<br />
insufficient studg of floods based on a detai<strong>le</strong>d analysis of recorded flows<br />
so that water management authorities in some river basins or <strong>the</strong>ir parts are<br />
unab<strong>le</strong> to arrive at more realistic estimates of expected floods;<br />
<strong>the</strong> insufficiency of <strong>the</strong> empirical and ra<strong>the</strong>r arbitarg methods used up to<br />
now in some countries for estimating peak floods; inadequecy of such methods<br />
has been proved and should no longer)%cceptab<strong>le</strong>, in dew of <strong>the</strong> magnitude<br />
and importance of <strong>the</strong> projects undertaken.<br />
It is also indicated in some countries that from <strong>the</strong> point of view of<br />
safety and also from <strong>the</strong> economlc ang<strong>le</strong>, <strong>the</strong> necessity for current study and<br />
evaluation of <strong>the</strong> magnitude and frequency of <strong>the</strong> occurence of floods in<br />
connexion with <strong>the</strong> economic development of Tiver basins has become essential.<br />
The existence of dams in <strong>the</strong> vicinity of populated areas necessitates <strong>the</strong> closest<br />
study of <strong>the</strong> anticipated probably floods, in order to provide adequate spillway<br />
capacity for <strong>the</strong> safety of <strong>the</strong> dams, and <strong>the</strong> downstream areas.
690<br />
In some countries flood forecasts are not yet of <strong>the</strong> desired accuracy, thus<br />
increasing Lhe danger of economic damage or reducing <strong>the</strong> efficiency of water storages<br />
down <strong>the</strong> river, whi<strong>le</strong> accurate flood forecasting can increase <strong>the</strong> overall potentialitie:<br />
of a multi-purpose project.<br />
The disoytrous floods which occurred in several regions of <strong>the</strong> world in <strong>the</strong> recent<br />
decade, causing considerab<strong>le</strong> loss of life an9 treinendous economic damage, pointed to<br />
<strong>the</strong> need to extenä flood forecasting and warning syotems in many countries, especially<br />
in <strong>the</strong>ir m.ìt vulnerab<strong>le</strong> river basins. Re<strong>le</strong>vant units established by national kjdro-<br />
meteorological services or by central water and power agencies confirm <strong>the</strong>ir<br />
effectivenes: It is considered that ths expenses for <strong>the</strong> maintenance of <strong>the</strong>se unita<br />
is only of mciest cost compared with <strong>the</strong> peins achieved by timely forece-sting. As<br />
an examp<strong>le</strong>, considerab<strong>le</strong> economic benefits are accrued in mqv countries when flood<br />
forecast? nre usea to enab<strong>le</strong> protective measures to be taken against <strong>the</strong> affects of<br />
floods. In <strong>the</strong>se cases, <strong>the</strong> economic benefits through iorecasts are caisulô.ted as<br />
<strong>the</strong> differenw between profits from proteoted zoner or areas.<br />
The use of flood forecasts in <strong>the</strong> IJSCR reduces <strong>the</strong> cost of damage äue (* floods<br />
by 20-30 per 'cent. merience in <strong>the</strong> Uni'reä -:-cates sonfirms that, <strong>the</strong> redudion of<br />
flooe .iamagt aione would far outweigh <strong>the</strong> tow?. cost of .<strong>the</strong> hydrolo@.cai forecasting<br />
serviqe, including <strong>the</strong> proposed network.:-'.<br />
8,<br />
A:$ participants were informed at <strong>the</strong><br />
Uniteu Nations inter-regional Seminar .,r, '?loo6 DamP-Ze Prevention Measures and<br />
Management, convened in Saptenber 1969 i,. Tbidsi, ITSSR, <strong>the</strong> annual savings due<br />
to <strong>the</strong> exlsting flood warning systems in <strong>the</strong> United States exceed $30 million a year.<br />
Experience in India also confirms that <strong>the</strong> early expenditure on <strong>the</strong> maintenace of<br />
<strong>the</strong> unit in <strong>the</strong> central water and power commission as well as <strong>the</strong> cost of <strong>the</strong> necessary<br />
equipment - is only a very moderate investment compared with <strong>the</strong> benefits which have<br />
been brought by forecasting.<br />
Importance of accurate data to show flow formation regardinp human activity at<br />
watersheds of rivers<br />
Attention should be drawn to <strong>the</strong> importance of streng<strong>the</strong>ning research to analyse<br />
<strong>the</strong> influence of human activity in watersheds on river flow. It is considered in some<br />
countries that <strong>the</strong> euccessful design and operation of water engineering structures<br />
could be possib<strong>le</strong> if sufficiently accurate data to show <strong>the</strong> hydrological characteristic<br />
of river basins under natural conditions of flow formation, with regard to <strong>the</strong> sca<strong>le</strong><br />
and direction of changes caused by humans, were availab<strong>le</strong>. The importance of this<br />
aspect has been emphasized among o<strong>the</strong>r publications by <strong>the</strong> ECE Manual mentioned in<br />
<strong>the</strong> first part of this paper.<br />
8/<br />
s/<br />
M.A. Kohïer (United States National Wea<strong>the</strong>r Service) Wase<strong>book</strong> on Hydrological<br />
Network Design Practicet' - WMO No. 324, 1972<br />
Sh.M. Manshard (Central Water and Power Codssion) tTlood Forecasting and<br />
Flood Warning in India". Seventh Symposium - The Civil and Hydraulic Engineering<br />
Dept. Water Resources. May 1971.
II. The inpact of a deficiency of i1ydrolopicai data on main water users<br />
The impact on <strong>the</strong> following water users is considered:<br />
(a) 1)onestic water supply<br />
(b) mdustrial water supply<br />
(c) Xydro-power generation and <strong>the</strong>rmal power plant water supply<br />
(d) irrigation<br />
(e) 1knigation<br />
(f) Fisheries<br />
(g) in-ter pollution control<br />
Ex'üen3.ve research was carried out in i?ariy AC3 countries devoted to t'ne appreisal<br />
of <strong>the</strong> impact of a deficient river flow aroused, among o<strong>the</strong>r reasons, by <strong>the</strong> lack of<br />
hydrological obsorvations and insufficiently accurate forecasts. One such research<br />
project has been carried out during -<strong>the</strong> recent decade 'by <strong>the</strong> central research institute<br />
on water prob<strong>le</strong>ms in <strong>the</strong> USSR (btinsl~)~. Analysing dieierent nethodx of estimation<br />
for <strong>the</strong> appraisal of <strong>the</strong> impact of doficient water flow, <strong>the</strong> authors indicate that, in<br />
son<strong>le</strong> case:, estbation becomes difficillt by reason of too 5hoi-t a period ol 1iyarologica.l<br />
obsemration:;. It is aïs0 pointed out that special estimation techniqucr to be applied<br />
in cases of short availab<strong>le</strong> periods oi hydrological observations, which could be applied<br />
for economic analysis, have not yet been created.<br />
However availab<strong>le</strong> results of research carried out in <strong>the</strong> same corntry suggest<br />
some methoas for use under conditions of deTiCient hydrological dataJ.<br />
11<br />
They atter-pt<br />
to find a relztion between <strong>the</strong> extent of limitation of <strong>the</strong> water siipply and <strong>the</strong><br />
reduction of economic activities (e.g. <strong>the</strong> industrFal output) of a certain enterprize<br />
in order to asress <strong>the</strong> economic damage caused by water shortages.<br />
Domestic water supplx<br />
It is generally adopted that domestic uater supply does not allow in-Lerruptions<br />
i.e. <strong>the</strong> pyaranteed water supply for <strong>the</strong>se consumers must be doze to 100 per cent.<br />
Industrial water siipply<br />
It is considered that <strong>the</strong> magnitude of economic damage connected. with stoppage<br />
or partial reduction of water supply is considerably influenced by <strong>the</strong> quality of<br />
hydrological prognosis.<br />
Some researchers in countries with a centrally planned econoqv suggest to<br />
subdivide <strong>the</strong> dmages, as followsw:<br />
o/ 1.1. I-fechetov, V.N. Pluzhnikov, L.J. Popov IfBalance of Water Resources and Needs -<br />
<strong>the</strong> method of optimal planning of water ressurces dsvelopmentl~ Multiqurpose water<br />
resources development and conservation. Minsk. 196û.<br />
Ir/ V. Andreyanov IiInternal distribution of river flowl'. Gidrometizdat. USSR.<br />
Centrcrl Research Institute of Water F'rob<strong>le</strong>cis. USSR. Minsk. "Miilti-purpose<br />
development and quality conservation of water resources" - 1968.<br />
6 91
692<br />
(a) direct damages (expressed by direct cost of unproductive forms of enterprises<br />
due to water stoppage or shortace);<br />
(b) indirect - measured by <strong>the</strong> loss of profit during <strong>the</strong> time of stoppage or<br />
reduction of water supply.<br />
The lack or insufficiency of research to define a relation between -Lhe extent<br />
of <strong>the</strong> limitation of <strong>the</strong> water supply and <strong>the</strong> re<strong>le</strong>vant economic damago to clifrerent<br />
tLJhnological processes is being noted in some branches of production.<br />
Generation of e<strong>le</strong>ctric enerpy by hydro-e<strong>le</strong>ctric plants and <strong>the</strong>mai power plants<br />
Proper flow predictions are important +o obtain optimum utilization of *dro-e<strong>le</strong>cti<br />
Timely 8n.d accurate ïlood forecasting in periods of lieais.<br />
and tharml. power plants.<br />
rains increase:. tlie overall potentiality of a multi-purpose project. Accurate<br />
forecasting is especially important bearing in nind tho fact that hyciro-power<br />
resources, depcnd on neteorological m u hydrological conditions vhich
693<br />
Il, is emphasized that not onlx <strong>the</strong> entent of reduction o€ water supply is important<br />
here but also .<strong>the</strong> seasonal period of this reduction which could be -
69b<br />
pollutant to <strong>the</strong> river in a period uith lower flow could create dangerous pollution<br />
concentrzìions in <strong>the</strong> lower reaches of <strong>the</strong> -river. %u hydrological forecasting<br />
has <strong>the</strong>refore aezome more iraportant fÒP watcr quality control in <strong>the</strong> river basin.<br />
The o<strong>the</strong>r nspect - is <strong>the</strong> importahce of better organization and modern<br />
instmntation of water quality contrbl trg hyd.rolo~&cal sedces. The ECE Manual<br />
mentioned above gives particular at4ention to water- qdity/data and research<br />
simultaneously with hyàrological resáárch, reqtliring <strong>the</strong> extehsion and hprovement<br />
of <strong>the</strong> system of observations on water”qua1ity in streams and reservoirs.<br />
Information from <strong>the</strong> ECE countries, posticulariy diseussed at <strong>the</strong> ECE Sendnar<br />
in Zagreb, 1971, points out khat considerab<strong>le</strong> improvement in <strong>the</strong>se functions is<br />
being iqîemented in mmy countries,<br />
As regards <strong>the</strong> assessment of economic danage caused by water quality<br />
deterioration due to flow reductionsin rivers, including reasons caused by<br />
untimely and inaccurate hydrological forecasts, <strong>the</strong>re are as yet no methods<br />
of assessment generally recognized or adopted. In some of <strong>the</strong> ECE countries<br />
it is considered that <strong>the</strong> infringemant of sanitary water conditions in a river<br />
basin, due to low flow, and <strong>the</strong> re<strong>le</strong>vant damage dee economi=. assessment.<br />
Fur<strong>the</strong>r comprehensive studies seem to be necessary in tlis field.<br />
The same is said of <strong>the</strong> economic assessment of <strong>the</strong> impact of water quality<br />
deterioration on water recreation. Correspondingly it 1s considered that it is<br />
difficult to substantiate economically permissib<strong>le</strong> standards of concentration of<br />
polluters in a river basin.<br />
However, <strong>the</strong>re is no unanimity on this point. Research in some o<strong>the</strong>r ECE<br />
countries show Lliat <strong>the</strong> increase in <strong>the</strong> present value of direct quantifiab<strong>le</strong><br />
recreation benefits of inproved water quality, for examp<strong>le</strong> in <strong>the</strong> Delaware River<br />
bacin could be as high as $300-350 niillion for <strong>the</strong> highest water quaïity clase<br />
adoptadw.<br />
Sediment control<br />
In e:cicnding and fur<strong>the</strong>r improving hydrological services and <strong>the</strong>ir forecasting,<br />
sediment control shodd not be neg<strong>le</strong>cted.<br />
Fur<strong>the</strong>r water resources development and growth of human activity on <strong>the</strong> watersheds<br />
m e s quantitative and qualitative sediment data very important as a part of<br />
hydrologic controls for different periods of <strong>the</strong> year. The iniportance of this<br />
prob<strong>le</strong>m could be seen from<strong>the</strong> experience of q ECE countries.<br />
It has been recognized by nany countries, that accumulation of sediments in<br />
niagy cases ohortcns <strong>the</strong> effective economic life of water engineering structures such<br />
as water-storagc ’, causes ii tensive wearing of pumps and turbines, requiring in all<br />
cases new capiti investments. As Is known, <strong>the</strong> prob<strong>le</strong>m can be grcatly mitigated<br />
by cidensive and comp<strong>le</strong>x control measures, including inproved forecasting.<br />
12/ A.B. Iïneese and B.T. Bower. AnaJYSeS conducted by <strong>the</strong> University of Pennsylvenia.<br />
Wanaging water quality’t. John Xopkins Press, Baltimore, 1968.
Experience in <strong>the</strong> United States shows that <strong>the</strong> damage from sediments to water<br />
management and national econow reaches more than $500 million a nnuala.<br />
Data on ice and slush ice conditions<br />
Inaccurate forecasting and warning ice and ice slush conditions cause<br />
considerab<strong>le</strong> damage to hydro-e<strong>le</strong>ctric plants and <strong>the</strong> energy consumers,<br />
causing reduction of industrial production in some of <strong>the</strong> regions of <strong>the</strong> midd<strong>le</strong><br />
climatic and rmuntainous zones.<br />
1.<br />
Conclueions<br />
In conclusion we would Uke to emphasise <strong>the</strong> following:<br />
Availab<strong>le</strong> experience and research being carried out in ECE countries c<strong>le</strong>arly<br />
demonstrate <strong>the</strong> considerab<strong>le</strong> impact of insufficient hydrological data and<br />
forecasting in all phases of planning of water resources development and use,<br />
which in <strong>the</strong>ir turn could have an *act on economic development of certain<br />
regions.<br />
2.<br />
695<br />
assess <strong>the</strong> value of hydrological data already have achieved certain positive effects,<br />
for examp<strong>le</strong> <strong>the</strong> development OP ma<strong>the</strong>matical relations between <strong>the</strong> present value of<br />
<strong>the</strong> value of data and <strong>the</strong> <strong>le</strong>ngth of <strong>the</strong>ir record, as well as development of some<br />
methodology to assess <strong>the</strong> impact on main water users.<br />
3. Availab<strong>le</strong> research c<strong>le</strong>arly shows that <strong>the</strong> cost of additional hydrological data -<br />
i.e. increasing <strong>the</strong> <strong>le</strong>ngth of observation - is imd@ficant in relation to <strong>the</strong><br />
expected losses due to insufficient data.<br />
in many countries, especially developing countries, are very often evaluated as<br />
inadequate for proper development of hydrological networks as was revea<strong>le</strong>d and<br />
emphasized by different conferenoes.<br />
4. In spite of considerab<strong>le</strong> research being carried out in ECE and o<strong>the</strong>r countries,<br />
as veli as various studies by specialized organizations, <strong>the</strong> following main<br />
deficiencies in hydrological observations and research are noted in different<br />
countries, which render a certain negative impact on water resources development<br />
-<br />
Availab<strong>le</strong> attempts in ECE countries to create and improve &.sting methods to<br />
and effective use:<br />
However, insufficient funds availab<strong>le</strong><br />
<strong>the</strong> lack of reliab<strong>le</strong> data on rainfall intensities and corresponding streamflowsj<br />
practical usage of empirical and sometimes ra<strong>the</strong>r arbitary methods for estimating<br />
peak floods; <strong>the</strong> lack of scientific studies on evaluation of <strong>the</strong> magnitudes<br />
and frequency of <strong>the</strong> occurrence of floods; insufficient studies and research to<br />
assess t h impact of human activity in watersheds on waterflow; lack of<br />
special estimation techniques to be applied for economic analysis in cases<br />
of availab<strong>le</strong> short periods of hydrolo,.Lcal observations; deficiencies of<br />
u "Xnvironmental Prob<strong>le</strong>ms". Monograph presented by <strong>the</strong> United States Government<br />
to ECE. January 1970.
696<br />
availab<strong>le</strong> methods of long-tem hydrologi.cal forecasts; insufficiency of<br />
research to define a relationship between <strong>the</strong> extent of <strong>the</strong> limitation of<br />
,he water supply and <strong>the</strong> re<strong>le</strong>vant economic damage to different technological<br />
processes; lack of generally recognized methods of assessment of economic<br />
damage caused by water quality detexioration.<br />
Streng<strong>the</strong>ning of <strong>the</strong> research regarding <strong>the</strong> deficiencies listed above Is<br />
considered as important and urgent.