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ports in g/Etudes et rapports d'hydrologie 16
of
resources projects
with inadequate data
Proceedings of the Madrid Symposium
,June 1973
Elaboration des projets
d'utilisation des ressources en eau
c dans données suffisantes
Volume I
Unesco - MIMO - 1AHS
Unesco - OMM - AISH
_-
Actes du colloque de Madrid
Juin I973

Studies and reports in hydrology/Études et rapports d’hydrologie 16

TITLES IN THIS SERIES / DANS CETTE COLLECTION
1. The use of analog and digital computers in hydrology: Proceedings of the Tucson Symposium.
June 1966 1 L'utilisation des calculatrices analogiques et des ordinateurs en hydrologie: Actes du
colloque de Tucson, juin 1966. Vol. I & 2. Co-edition IAHS-Unesco / Coédition AISH-Unesco.
2.
Water in the unsaturated zone: Proceedings of the Wageningen Symposium, June 1967 1 L'eau dans
la zone non saturée: Actes du symposium de Wageningen, juin 1967. Edited by 1 edité par P. E.
Rijtema & H. Wassink. Vol. 1 & 2. Co-edition IAHS-Unesco 1 Coédition AISH-Unesco.
3. Floods and their computation: Proceedings of the Leningrad Symposium, August 1967 / Les crues
et leur évaluation: Actes du colloque de Leningrad, août 1967. Vol. 1 & 2. Co-edition IAHS-Unesco-
WMO 1 Coédition AISH-Unesco-OMM.
4. Representative and experimental basins: An international guide for research and practice. Edited
by C. Toebes and V. Ouryvaev. Ptrblished by Unesco.
4. Les bassins représentatifs et expérimentaux: Guide international des pratiques en matikre de recherche.
Publié sous la direction de C. Toebes et V. Ouryvaey. Publié par l'Unesco.
5. 'Discharge of selected rivers of the world 1 Débit de certain cours d'eau du monde. Published by
Unesco 1 Publié par l'Unesco.
Vol. I : General and régime characteristics of stations selected / Caractéristiques générales et
caractéristiques du régime des stations choisies.
Vol. II: Monthly and annual discharges recorded at various selected stations (from start of obser.
vations up to 1964) / Débits mensuels et annuels enregistrés en diverses stations sélectionnées
(de l'origine des observations à l'année 1964).
'Vol. III: Mean monthly and extreme discharges (1965-1969) I Débits mensuels moyens et débits
extrêmes (1965-1969).
6. List of International Hydrological Decade Stations of the world 1 Liste des stations de la Décennie
h'ydrologique internationale existant dans le monde. Published by Unesco 1 Publié par l'Unesco.
7. Ground-water studies: An international guide for practice. Edited by R. Brown, J. Ineson, V. Konoplyantsev
and V. Kovalevski. (Will also appear in French, Russian and Spanish 1 Paraitra
également en espagnol, en français et en russe.)
8. Land subsidence: Proceedings of the To'kyo Symposium, September 1969 1 Affaisement du sol:
Actes du colloque de Tokyo, septembre 1969. 'Vol. 1 & 2. Co-edition IAHS-Unesco / Coédition
AISH-Unesco.
9. Hydrology of deltas: Proceedings of the Bucharest Symposium, May 1969 1 Hydrologie des deltas:
Actes du colloque de Bucarest, mai 1969. Vol. 1 & 2. Co-edifion IAHS-Unesco I Coédition AISH-
Unesco.
10. Status and trends of research in hydrology 1 Bilan et tendances de la recherche en hydrologie.
Published by Unesco 1 Publié par l'Unesco.
11. World water balance: Proceedings of the Reading Symposium, July 1970 1 Bilan hydrique mondial:
Actes du colloque de Reading, juillet 1970. Vol. 1-3. Co-edition IAHS-Unesco-WMO / Coédition
AISH-Unesco-OMM.
12. Results OF research on representative and experimental basins: Proceedings of the Wellington
Symposium, December 1970 1 Résultats de recherches sur les bassins représentatifs et expérimen-
taux: Actes du cowoque de Wellington, décembre 1970. 'Vol. 1 & 2. Coedition IAHS-Unesco 1
Coédition AISH-Unesco.
13. Hydrometry: Proceedings of the Koblenz Symposium, September 1970 1 Hydrométrie: Actes du
colloque de Coblence, septembre 1970. Co-edition IAHS-Unesco-WMO 1 Coédition AISH-Unesco-
OMM.
14. Hydrologic information systems. Co-edition Unesco-WMO.
15. Mathematical models in hydrology: Proceedings of the Warsaw Symposium, July 1971 1 Les mo-
dèles mathématiques en hydrologie: Actes du colloque de Varsovie, juillet 1971. Vol. 1-3. Co-
edition IAHS-Unesco-WMO / Coédition AISH-Unesco-OMM.
16. Design of water resources projects with inadequate data: Proceedings of the Madrid symposium,
June 1973 1 filaboration des projets d'utilisation des ressources en eau sans données suffisantes:
Actes du colloque de Madrid, juin 1973. Vol. 1-3. Co-edition Unesco-WMO-IAHS / Coédition Unesco-
OMM-AISH.

qesign of
water resources projects
with inadequate data :
Proceeúings of the Madrid Symposium
.lune 1973
Elaboration des projets
d’utilisation des ressources en eau
sans données suffisantes
A contribution to the Iniernaiional Hydrological Decade
Une contribution a la Décennie hydrologique internationale
Con resurnenes en espano1
Volume I
Actes du colloque de Moúrid
.luin 1973
Unesco - WMO - LAHS 1974
Unesco - OMM - AISH

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views of the publishers.
The designations employed and the presentation of the material do not imply the
expression of any opinion whatsoever on the part of the publishers concerning the legal
status of any country or territory, or of its authorities, or concerning the frontiers
of any country or territory.
Le choix et la présentation du contenu de cet ouvrage et les opinions qui s’y
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Les dénominations employées et la présentation des divers éléments n’impliquent
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de ses frontières.
ISBN 92-3401137-1
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Printed in Spain

PkEFACE
The International Hydrological Decade (IHD) 1965-74 was launched by
the General Conference of Unesco at its thirteenth session to promote
international co-operation in research and studies and the training of spe-
cialists and technicians in scientific hydrology. Its purpose is to enable
all countries to make a fuller assessment of their water resources and a
more rational use of them as man’s demands for water constantly increase
in face of developments in population, industry and agriculture. In 1974
National Committees for the Decade had been formed in 108 of Unesco’s
131 Member States to carry out national activities within the programme
of the Decade. The implementation of the programme is supervised by a
Co-ordinating Council, composed of 30 Member States selected by the Ge-
neral Conference of Unesco, which studies proposals for. developments
of the programme, recommends projects of interest to all or a large
number of countries, assists in the development of national and regional
projects and co-ordinates international co-operation.
Promotion of collaboration in developing hydrological research techni-
ques, diffusing hydrological data and planning hydrological installations
is a major feature of the programme of the IHD which encompasses all
aspects of hydrological studies and research. Hydrological investigations
are encouraged at the national, regional and international level to streng-
then and to improve the use of natural resources from a local and a global
perspective. The programme provides a means for countries well advanced
in hydrological research to exchange scientific views and for developing
countries to benefit from this exchange of information in elaborating re-
search projects and in implementing recent developments in the planning
of hydrological installations.
As part of Unesco’s contribution to the achievement of the objectives
of the IHD the General Conference authorized the Director-General to
collect, exchange and disseminate information concerning research on
scientific hydrology and to facilitate contacts between research workers
in this field. To this end Unesco initiated two series of publications: Studies
and Reports in Hydrology and Technical Papers in Hydrology.
The Studies and Reports in Hydrology series, in which the present
volume is published, is aimed at recording data collected and the main
results of hydrwlogical studies undertaken within the framework of the
Decade, as well as providing information on research techniques. Also
included in the series are proceedings of symposia. Thus, the series com-
prises the compilation of data, discussions of hydrological research techni-
ques and findings, and guidance material for future scientific investigations.
It is hoped that the volumes wil furnish material of both practical and
theoretical interest to hydrologists and governments participating in the
IHD and respond to the needs of technicians and scientists concerned
with problems of water in all countries.
A number of these volumes have been published jointly with the In-
ternational Association of Hydrological Sciences and the World Meteoro-
logical Organization which have co-operated with Unesco in the imple-
mentation of several important projects of the IHD.

PRBFACE
La Conférence générale de l’Unesco, à sa treizième session, a décidé
de lancer, pour la période s’étendant de 1965 à 1974, la Décennie hydrologique
internationale (DHI), entreprise
e
mondiale visant a faire progresser la connaissance
en matiere ci’ vdrologie scientifique par un développement de
la coopération inyrnati nale et par la formation de spécialistes et de
techniciens. Au moment oìi l’expansion démographique et le développement
industriel et agricole provoquent un accroissement constant des besoins
en eau, la DHI permet à tous les pays de mieux évaluer leurs ressources
hydrauliques et de les exploiter plus rationnellement.
I1 existe actuellement dans 108 des 131 Etats membres de l’Unesco un
comité national qui, pour tout ce qui a tratit au programme de la Décennie,
impulse les activités nationales et assure la participation de son pays
aux entreprises régionales et internationales. L’exécution du programme
de la DHI se fait sous la direction d’un Conseil de coordination composé
de 30 Etats membres désignés par la Conférence générale de l’Unesco; ce
conseil étudie les propositions concernant le programme, recommande
l’adoption de projets intéressant l’ensemble des pays ou un grand nombre
d’entre eux, aide à la mise sur pied de projets nationaux et régionaux, et
coordonne la coopération à l’échelon international.
Le programme de la DHI, qui porte sur tous les aspects des études et
des recherches hydrologiques, vise essentiellement à développer la collaboration
dans la mise au point des techniques de recherches, dans la
diffusion des données hydrologiques, dans l’organisation des installations
hydrologiques. I1 encourage les enquêtes nationales, régionales et internationales
tendant à accroître et à améliorer l‘utilisation des resources naturelles,
dans une perspective locale et générale. Il permet aux pays avancés
en matière de recherches hydrologiques d’échanger des informations; aux
pays en voie de développement, il offre la possibilité de profiter de ces
échanges pour élaborer leurs projets de recherches et pour planifier leurs
installations hydrologiques en tirant parti des acquisitions les plus récentes
de l’hydrologie scientifique.
Pour permettre a l’Unesco de contribuer au succès de la DHI, la Conférence
générale a autorisé le Directeur générale à rassembler, à échanger
et à diffuser des informations sur les recherches d’hydrologie scientifique
et à faciliter les contacts entre les chercheurs dans ce domaine. A cette
fin, l’Unesco fait paraître deux nouvelles collections de publications: «Etudes
et rapports d’hydrologie» et «Notes techniques d’hydrologie,.
La collection «Etudes et rapports d’hydrologie,, dans laquelle est publié
le présent ouvrage, a pour objet de présenter les données recueillies et les
principaux résultats des études effectuées dans le cadre de la Décennie
et de fournir des informations sur les techniques de recherche. On y trouve
aussi les Actes de colloques réunis sur ce sujet. Cette collection publie
donc des données, des techniques et des résultats de recherches ainsi
qu’une documentation pour les travaux scientifiques futurs.
On espère que ces volumes apporteront aux hydrologues et aux gouvernements
qui participent à ,la DHI des matériaux d’un intérêt tant pra-

tique que théorique, et qu’elle répondra aux besoins des techniciens et
des hommes de science de tous pays qui s’occupent des problèmes de l’eau.
Certains de ces ouvrages sont publibs en coopération avec l’Association
internationale des sciences hydrologiques ou l’organisation météorologique
mondiale dans le cadre de projets réalisés conjointement par ces orga-
nisations et l’Unesco.

INTRODUCTION
The Symposium on the Development of Water Resources Projects with
Inadequate Data was held in Madrid from 4 to 8 June 1973 for the purpose
of focusing on the methodology for hydrologic studies for water resources
projects with inadequate data and on current practices for the assessment
of design parameters.
The Symposium was opened at the Palacio de Exposiciones on the
morning of 4 June by Miniester of Public Workes of Spain Addresses were
then given by Dr. Dumitrescu on behalf of the Director General of Unesco,
Professor Nevmec on behalf of the Secretary-General of WMO, Dr. Rodier
as President of IAHS and by Dr. Briones, on behalf of the Spanish Na-
tional Committee for the IHD.
The Symposium was attended by 480 participants from 77 countries.
The technical programme, detaimled in the Table of Contents, included
consideration of 3 major areas:
1. Methodology for hydrological studies with inadequate data,
2. Current practices in different countries,
3. Relation between project economics and hydrological data.
Each area was further sub-divided into topics for each of which the
individually contributed papers were abstracted into a general report, orally
presented by an invited expert, and followed by discussion.
Since the individual papers were not presented at the Symposium orally
by the authors, thery are reproduced here in the orden in which
they were reported in each general report under each topic.

üesip d water reswrcee projects with inadequate dati: Pmeeedin.p d the Madrid 8ympoilUm.
June i973 / Blabontion de# projeu d‘utilhition des ressourcci en eau rans domdes nuttlrantei:
Acui du mlloque de Madrid, juin 1973.
Volume I Contents Table des matieres
Foreword/Avant-propos
TOPIC 1.1 . TRANSFER OF INFORMATION FROM OBSERVED POINTS TO
POINTS OF INTEREST, ESPECIALLY FOR THE ASSESSMENT
OF THE CHARACTERISTICS OF DISCHARGES.
POINT 1.1 - EXTRAPOLATION DES INFORMATIONS RECUEILLIES AUX
POINTS OBSERVES A DES POINTS PRESENTANT UN INTE-
RET PARTICULIER, NOTAMMENT POUR L’EVALUATION
DES DEBITS CARACTERISTIQUES.
SOKOLOV, A.A. (U.S.S.R.) GENERAL REPORT
ALBINET, M., CASTANY, G., DELAROZIERE-BOUILLIN, O., JONAT, R.,
MARGAT, J. (FRANCE)
Evaluation et répartition des ressources en e au d’une grande région par
les paramétres hydroclimatiques et hydrog6ologiques ...............
BALEK, J. (CZECHOSLOVAKIA)
Use of representative and experimental catchments for the assement of
hydrological data of African tropical basins .......................
CORMARY, Y - J.M. MASSON. (FRANCE)
Diverses méthodes convergentes pour l’utihtion de l’information a
I’écheUedgionale .......................................
DUBREUIL, PIERRE. (FRANCE)
Le transfert d’infomtion hydrologique a des bassins versants non obser-
vés ....................................................
GARCIA-AGREDA, R., RASULO, G., VIPARELLI, R. (ITALY)
Pluviometric zones and the criteria to define their boundaries for regions
withscarcedata ............................................
OBERLIN, G.R., GALEA, G.C., TONI, J.T. (FRANCE)
Estimation des étiages de bassins non equipés ....................
TIERCELIN, J.R. (FRANCE)
ParamBtres régionaux relatifs aux ressources en eau. Utilisation. PdciPion
d’estimation ............................................... 125
VAN HYLCKAMA, T.E.A. (U.S.A.)
Estimating evapotranspiration by homoclimates ................ 74
1
15
27
47
61
89
103

VOSKRESENSKI, K.P. (U.S.S.R.)
Prinoiph for the computation of tho nuin Ohinctdutb of river wrtm
reaowcea in the abuncl of oburvrtiona on the bu& of goographid
interpoiation of runoff puameten ..............................
VUGLLNSKI, V.S., SEMENOV, V.A. (U.S.S.R.)
Evaiurtion of water IWOIUWW of mounîdn am11 in cani of rhnce or
inadequacy of datr on runoff ..................................
TOPIC 1.2 - THE IMPROVEMENT OF OVERALL HYDR0UX;IC INFOR-
MATION BY SHORT-TERM ADDITIONAL AND PARTICULAR
OBSERVATIONS AND MEASUREMENTS. INCLUDING THE
PLANNING OF THE ADDITIONAL MEASUREMENT CAM-
PAIGN USING HYDROLOGIC DATA SENSITIVITY ANALYSIS
BASED ON PROJECT ECONOMICS.
POINT 1.2 - AMELIORATION DE L'ENSEMBLE DE L'INFORMATION
HYDROLOGIQUE AU MOYEN DE COURTES CAMPAGNES DE
MESURES COMPLEMENTAIRES ET D'OBSERVATIONS PARTI-
CULIERES, COMPRENANT LA MISE EN OEUVRE DE CAM-
PAGNES DE MESURES ADDITIONNELLES UTILISANT UNE
ANALYSE DE SENSIBILITE DES DONNEES BASEE SUR
L'ECONOMIE DES PROJETS.
RODDA, JOHN (U.K.) GENERAL REPORT
BEARD, LEO R. (U.S.A.)
Hydrological data fiii-in and network design ..................
DELHOMME, J.P., DELFINER, P. (FRANCE)
Application du Krigeage a l'optimisation d'une campagne pluviométrique
enzonearide ..............................................
HALASI-KUN, GEORGE, J. (U.S.A.)
Improvement of runoff records in smaller watersheds based on permeabi-
lity of the geological subsurface ...............................
KOVACS, GEORGE. MOLNAR, GEORGE. (HUNGARY)
Determination of snow water equivalent and snowmelt water by
thickness of snow cover data .................................
MEIJERINK, A.M.J. (NETHERLANDS)
Evaluation of local water resources in semiarid hard rock region by using
photo.hydrological indices ....................................
PANT,P.S.,GUPTA, M.G. (INDIA)
Application of satellite cloud pictures in snow hydrology of the Himalayas
and in the estimation of rainfall over India during southwest
monsoonseason ............................................
137
145
153
161
17
191
205
217
233

I
TOPIC I.3A - THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-
SIGNED FOR DATA-SCARCE AREAS. STATISTICAL ME-
THODS AND DATA OPERATION.
POINT I.3A - UTILISATION DES TECHNIQUES DE SIMULATION SPE-
CIALEMENT EWIBOREE POUR DES REGIONS OU LES
DONNEES SONT RARES. METHODES STATISTIQUES ET
TRAITEMENT DES DONNEES.
JAMES, IVAN CHARLES. (U.S.A.) GENERAL REPORT
CORMARY, Y . GUILBOT, A. (FRANCE)
Etude des relations pluie-débit sur trois bassins versants d’investigation . .
CHARANIA, S.H. (KENYA)
Extension of runoff records for small catchments in semi-arid regions ...
DAVYDOVA,A.I., KALININ, G.P. (U.S.S.R.)
Simulation of hydrological samples by natural water flow characteristics
HAMLIN, M.J., KOTTEGODA, N.T. (U.K.)
The preparation of a data set for hydrologic system analysis ..........
LENTON, ROBERTO L., SCHAAKE JR., JOHN C., RODRIGUEZ-ITURBE, IG-
NACIO. (U.S.A.)
Potential application of Bayesian techniques for parameter estimation
withlimiteddata ...........................................
McMAHON, T.A., MEIN, R.G. (AUSTRALIA)
Storage-yield estimates with inadequate streamflow data .............
MARTIN JADRAQUE, VALENTIN. (SPAIN)
Estimation of Gumbel law parameters in small samples ..............
MOSS, M.E.. DAWDY, D.R. (U.S.A.)
Stochastic simulation for basins with short or no records of streamflow
O’CONNELL, P.E., WALLIS, J.R. (U.S.A.)
Choice of generating mechanism in synthetic hydrology with inadequate
data .....................................................
PORRAS, PEDRO., FLORES, ALFREDO. (VENEZUELA)
Stochastic application in ungauged basins for planning purposes .......
ROCHE, MARCEL. (FRANCE)
Homogdnbisation et interpolation des donndes pour un modèle de simula-
tion .....................................................
SHARMA, H.D., BHATTACHARYA, A.P., JINDAL, S.R. (INDIA)
The use of simulation techniques for sequential generation of short-sized
rainfall data and its application in the estimation of design flood ......
241
265
281
293
305
321
335
349
365
311
355
407
419

VISSER, J.H. (LEBANON)
The we of rtochutlc mod& in hydroJgricultunl dwdopmrnt projoct
Libbanon ................................................
WALLIS, J.R., MATALAS, N.C. (U.S.A.)
Rehtivr importuice of decidon vuirbiem in fiood frequency uulydi ...
WEISS, G. (U.K.)
Shot nohe models for aynthetic generation of multimite M y munflow
data .....................................................
WOOD, ERIC F. (U.S.A.)
Flood control ddgn with Limited data . A comparinon of the chsical
andBayesianapproaches .....................................
TOPIC 1.3B . THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-
SIGNED FOR DATA-SCARCE AREAS. THE USE OF MATHE-
MATICAL MODELS.
POINT I.3B - UTILISATION DES TECHNIQUES DE SIMULATION SPE-
CIALEMENT ELABOFSE POUR DES REGIONS OU LES
DONNEES SONT RARES. UTILISATION DES MODELES MA-
THEMATIQUES.
NASH, J.E. (IRELAND) GENERAL REPORT
BERNIER, J. (FRANCE)
Données inadéquates et modeles mathématiques de la pollution en riviere
COOK, SAMUEL P., MBURU, SAMUEL G. (KENYA)
Regional groundwater recharge estimates via meteorological data ......
DELLEUR, J.W., LEE, M.T. (U.S.A.)
A rainfall-runoff model based on the watershed stream network .......
HANN, C.T. (U.S.A.)
Monthly streamflow estimation from limited data ..................
KOREN, V.I., KUTCHMENT, L.S. (U.S.S.R.)
Obtaining deficient information by solving inverse problems for mathe-
maticalrunoffmodels .......................................
ROFAIL, NABIL. (EGYPT)
The mathematical model of water balance for data-scarce areas ........
VILARO, FRANCISCO., CUSTODIO, EMILIO. (SPAIN)
Data acquisition and methodology for a simulation model of the Llobre-
gat Delta (Barcelona, Spain) ...................................
435
449
457
469
485
513
525
53 1
545
551
569
581

Contents
Table des matieres
Volume I
ForewordIAvant-propos ............................
TOPIC 1.1 - TRANSFER OF INFORMATION FROM OBSERVED POINTS TO
POINTS OF INTEREST, ESPECIALLY FOR THE ASSESSMENT
OF THE CHARACTERISTICS OF DISCHARGES.
POINT I. 1 - EXTRAPOLATION DES INFORMATIONS RECUEILLIES AUX
POINTS OBSERVES A DES POINTS PRESENTANT UN INTE-
RET PARTICULIER, NOTAMMENT POUR L’EVALUATION
DES DEBITS CARACTERISTIQUES.
SOKOLOV, A.A. (U.S.S.R.) GENERAL REPORT
ALBINET, M., CASTANY, G., DELAROZIERE-BOUILLIN, O., JONAT, R.,
MARGAT, J. (FRANCE)
Evaluation et répartition des ressources en eaux d’une grande région par
les paramètres hydroclimatiques et hydrogéologiques ...............
BALEK, J. (CZECHOSLOVAKIA)
Use of representative and experimental catchments for the assessment of
hydrological data of African tropical basins .......................
CORMARY, Y - J.M. MASSON. (FRANCE)
Diverses méthodes convergentes pour l’utilisation de l’information à
l’échelle régionale ...........................................
DUBREUIL, PIERRE. (FRANCE)
Le transfert d’information hydrologique à des bassins versants non obcer-
vés ......................................................
GARCIA-AGREDA, R., RASULO, G., VIPARELLI, R. (ITALY)
Pluviometric zones and the criteria to define their boundaries for regions
with scarce data ............................................
OBERLIN, G.R., GALEA, G.C., TONI, J.T. (FRANCE)
Estimation des étiages de bassins non equipés .....................

II
TIERCELIN, J. R. (FRANCE)
Parametres régionaux relatifs aux ressources en eau. Utilisation. Précision
d’estimation ...............................................
VAN HYLCKAMA, T.E.A. (U.S.A.)
Estimating evapotranspiration by homoclimates ...................
VOSKRESENSKI, K.P. (U.S.S.R.)
Principles for the computation of the main characteristics of river water
resources in the absence of observations on the basis of geographical
interpolation of runoff parameters ..............................
VUGLINSKI, V.S.,SEMENOV, V.A. (U.S.S.R.)
Evaluation of water resources of mountain areas in casi of absence or
inadequacyofdataonrunoff ..................................
TOPIC 1.2 - THE IMPROVEMENT OF OVERALL HYDROLOGIC INFOR-
MATION BY SHORT-TERM ADDITIONAL AND PARTICULAR
OBSERVATIONS AND MEASUREMENTS. INCLUDING THE
PLANNING OF THE ADDITIONAL MEASUREMENT CAM-
PAIGN USING HYDROLOGIC DATA SENSITIVITY ANALYSIS
BASED ON PROJECT ECONOMICS.
POINT 1.2 - AMELIORATION DE L’ENSEMBLE DE L’INFORMATION
HYDROLOGIQUE AU MOYEN DE COURTES CAMPAGNES DE
MESURES COMPLEMENTAIRES ET D’OBSERVATIONS PARTI-
CULIERES, COMPRENANT LA MISE EN OEUVRE DE CAM-
PAGNES DE MESURES ADDITIONNELLES UTILISANT UNE
ANALYSE DE SENSIBILITE DES DONNEES BASEE SUR
L’ECONOMIE DES PROJETS.
RODDA, JOHN (U.K.) GENERAL REPORT
BEARD, LEO R. (U.S.A.)
Hydrological data fiil-in and network design ......................
DELHOMME, J.P., DELFINER, P. (FRANCE)
Application du Krigeage à l’optimisation d’une campagne pluviométrique
enzonearide ..............................................
HALASI-KUN,
GEORGE, J. (U.S.A.)
Improvement of runoff records in smaller watersheds based on permeabi-
lity of the geological subsurface ................................

KOVACS, GEORGE. MOLNAR, GEORGE. (HUNGARY)
Determination of snow water equivalent and snowmelt water by
thickness of snow cover data ..................................
MEIJERINK, A.M.J. (NETHERLANDS)
Evaluation of local water resources in semiarid hard rock region by using
photo-hydrological indices ....................................
PANT, P.S., GUPTA, M.G. (INDIA)
Application of satellite cloud pictures in snow hydrology of the Himalayas
and in the estimation of rainfall over India during southwest
monsoonseason ............................................
TOPIC I.3A - THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-
SIGNED FOR DATA-SCARCE AREAS. STATISTICAL ME-
THODS AND DATA OPERATION.
POINT I.3A - UTILISATION DES TECHNIQUES DE SIMULATION SPE-
CIALEMENT ELABOREE POUR DES REGIONS OU LES
DONNEES SONT RARES. METHODES STATISTIQUES ET
TRAITEMENT DES DONNEES.
JAMES, IVAN CHARLES. (U.S.A.) GENERAL REPORT
CORMARY, Y - GUILBOT, A. (FRANCE)
Etude des relations pluie-débit sur trois bassins versants d'investigation . .
CHARANIA, S.H. (KENYA)
Extension of runoff records for small catchments in semi-arid regions ...
DAVYDOVA,A.I.,KALININ,G.P. (U.S.S.R.)
Simulation of hydrological samples by natural water flow characteristics
HAMLIN, M.J., KOTTEGODA, N.T. (U.K.)
The preparation of a data set for hydrologic system analysis ..........
LENTON, ROBERTO L., SCHAAKE JR., JOHN C., RODRIGUEZ-ITURBE, IG-
NACIO. (U.S.A.)
Potential application of Bayesian techniques for parameter estimation
withlimiteddata ...........................................
McMAHON, T.A.,
MEIN, R.G. (AUSTRALIA)
Storage-yield estimates with inadequate streamflow data .............
III

IV
MARTIN JADRAQUE, VALENTIN. (SPAIN)
Estimation of Gumbel law parameters in small samples ..............
MOSS, M.E., DAWDY, D.R. (U.S.A.)
Stochastic simulation for basins with short or no records of streamflow
O’CONNELL, P.E., WALLIS, J.R. (U.S.A.)
Choice of generating mechanism in synthetic hydrology with inadequate
data .....................................................
PORRAS, PEDRO., FLORES, ALFREDO. (VENEZUELA)
Stochastic application in ungauged basins for planning purposes .......
ROCHE, MARCEL. (FRANCE)
Homogénéisation et interpolation des données pour un modèle de simula-
tion .....................................................
SHARMA, H.D., BHATTACHARYA, A.P., JINDAL, S.R. (INDIA)
The use of simulation techniques for sequential generation of short-sized
rainfall data and its application in the estimation of design flood ......
VISSER, J.H. (LEBANON)
The use of stochastic models in a hydro-agricultural development project
inLebanon ................................................
WALLIS, J.R., MATALAS,N.C. (U.S.A.)
Relative importance of decision variables in flood frequency analysis
WEISS, G. (U.K.)
Shot noise models for synthetic generation of multisite daily streamflow
data .....................................................
WOOD, ERIC F. (U.S.A.)
Flood control design with limited data - A comparison of the classical
andBayesianapproaches .....................................
TOPIC I.3B - THE USE OF SIMULATION TECHNIQUES ESPECIALLY DE-
SIGNED FOR DATA-SCARCE AREAS. THE USE OF MATHE-
MATICAL MODELS.
POINT I.3B - UTILISATION DES TECHNIQUES DE SIMULATION SPE-
CIALEMENT ELABOREE POUR DES REGIONS OU LES
DONNEES SONT RARES. UTILISATION DES MODELES MA-
THEMATIQUES.

NASH, J.E. (IRELAND) GENERAL REPORT
BERNIER, J. (FRANCE)
Données inadéquates et modèles mathématiques de la pollution en riviere
COOK, SAMUEL P., MBURU, SAMUEL G. (KENYA)
Regional groundwater recharge estimates via meteorological data ......
DELLEUR, J.W., LEE, M.T. (U.S.A.)
A rainfall-runoff model based on the watershed stream network .......
HANN, C.T. (U.S.A.)
Monthly streamflow estimation from limited data ..................
KOREN, V.I., KUTCHMENT, L.S. (U.S.S.R.)
Obtaining deficient information by solving inverse problems for mathe-
matical runoff models .......................................
ROFAIL, NABIL. (EGYPT)
The mathematical model of water balance for data-scarce areas ........
VILARO, FRANCISCO., CUSTODIO, EMILIO. (SPAIN)
Data acquisition and methodology for a simulation model of the Llobre-
gat Delta (Barcelona, Spain) ...................................
V

Foreword
While the need for hydrological and meteorological data of many types
for the design of water resources projects is obvious, it is often found,
especially in many developing countries, that such data are either lacking
or inadequate.
Recognizing the existence of this problem, the Co-ordinating Counci*l of
the IHD appointed a group of experts (third session, Paris, June 1967) to
study the problem of design of water resources projects with inadequate
data.
Similarly, the Commission for Hydrology of WMlO (third session, Geneva,
September 1968) established a Working Group on Hydrological Design
Data for Water Resources Projects to prepare guidance material on this
subject for the WMO Guide to Hydrological Practices and to maintain
liaison with the IHD group of experts appointed by the Co-ordinating
Council.
As a means of taking stock of the work carried out by the hydrological
community in coping with project design with scarce data, Unesco and
WMO jointly convened a symposium on this subject. The Symposium was
organized with the co-operation of the IAHS and the Spanish National
Committee for the IHD and was held in Madrid from 4 to 8 June 1973 at
the invitation of the Government of Spain.
The Madrid Symposium concentrated on the methodology of hydro-
logical studies for water resources projects with inadequate data and on
current practices for the assessment of design parameters.
The Minister of Public Works of Spain opened the Symposium at the
Palacio de Exposiciones on the morning of 4 June. Addresses were given
by Dr. Dumitrescu on behalf of the Director-General of Une,sco, Professor
Nemec on benalf of the Secretary-General of WMO, Dr. Rodier as President
of IAHS and by Dr. Briones, on behalf of the Spanish National Committee
for the IHD.
The Symposium was atteneded by 480 participants from 77 countries.
The technical programme, detailed in the Table of Contents, included
consideration of 3 major areas:
1. Methodology for hydrological studies with inadequate data;
2. Current practices in different countries;
3. Relation between project economics and hydrological data.
Each area was further sub-divided into topics for each of which the
individually contributed papers were abstracted into a general report,
orally presented by an invited expert, and followed by discussion.

This volume of proceedings was compiled by the Spanish National Com-
mittee for the IHD; it includes all the general reports and individual
papers presented at the Symposium, as well as the discussions. It is issued
as a joint Unesco/WMO/IAHS publication in the spirit in which the three
Organizations have collaborated during the IHD.
Since the individual authors did not present their papers orally at the
Symposium, the papers are reproduced here in the order in which they
are discussed in the general report for each topic.
Unesco, WMO and IAHS wish to record their thanks to the Spanish
National Committee for the IHD for the many contributions of its members
towards the organization of the Symposium, and for the Committee's as-
sistance in the publication of these proceedings.

AVANT-PROPOS
I1 est évident que, pour élaborer des projets d’utilisation des ressources
en eau il est nécessaire de disposer de données hydrologiques et météoro-
logiques de types très divers; or il apparaît que ces données sont souvent
inexistantes ou insuffisantes, notamment dans beaucoup de pays en voie
de développement.
Conscient de ce problème, le Conseil de coordination de la DHI a créé,
lors de sa troisième session (Paris, juin 1967) un groupe d’experts chargé
d’étudier les moyens d’elaborer des projets d’utilisation des ressources
en eau sans disposer de données suffisantes.
De son côté, la Commission d’hydrologie de l’OMM a constitué à sa
troisième session (Genève, septembre 1968) un groupe de travail sur les
données hydrologiques nécessaires à l’élaboration des projets d’aménage-
ment des ressources hydrauliques; ce groupe de travail a été chargé de
formuler des recommandations destinées à figurer dans le Guide OMM des
pratiques hydrologiques, et d’assurer la liaison avec le groupe d’experts
de la DHI créé par le Conseil de coordination.
Afin de faire le point des travaux accomplis par la communité hydro-
logique en ce qui concerne l’élaboration de projets pour lesquels on ne
dispose pas de données suffisantes, l’Unesco et l’OMM ont décidé de réunir
conjointement un colloque consacré à cette question. Ce colloque, organisé
avec la collaboration de 1’AISH et du Comité national espagnol pour la
DHI, s’est tenu à Madrid en juin 1973, à l’invitation du gouvernement es-
pagnol.
Le colloque de Madrid a traité en particulier de la méthodologie des
études hydrologiques sans données suffisantes et des pratiques courantes
utilisées pour l’évaluation des paramètres de calcul.
Le colloque a été ouvert par le ministre espagnol des travaux publics,
le matin du 4 juin, dans le cadre du Palais des expositions. Des allocutions
furent prononcées par M. Dumitriscu, au nom du Directeur général de
l’Unesco, par le professeur Nemec, au nom du Secrétaire général de l’OMM,
par M. Rodier, président de I’AISH, et par M. Briones, au nom du Comité
national espagnol pour la DHI.
480 participants, venant de 77 pays, participèrent au colloque.
Le programme technique, dont le contenu détaillé figure dans la table
des matières, portait sur trois domaines principaux:
1. Méthodologie des études hydrologiques sans données suffisantes;
2. Les pratiques courantes utilisées dans différents pays;
3. Relation entre les données économiques du projet et les données
hydrologiques.

Chacun de ces domaines était subdivisé en thèmes, et sur chaque thème
un rapport général synthétisant les communications individuellles était pré-
senté par un expert, puis suivi d’une discussion.
Les Actes du colloque, établis par le Comité national espagnol pour
la DHI, comprennent l’ensemble des communications individuelles et des
rapports généraux, ainsi que le compte rendu des débats auxquels ils ont
donné lieu. Ils constituent une publication conjointe de l’Unesco, de l’OMM
et de I’AISH, reflétant l’esprit dans lequel les trois organisations ont col-
laboré pendant la DHI.
Comme les communications individuelles n’ont pas été présentées ora-
lement par leurs auteurs, elles sont reproduites dans l’ordre où elles sont
apparues dans le rapport les concernant.
Unesco, l’OMM et 1’AISH tiennent à remercier le Comité national es-
pagnol pour la DHI du concours qu’il a apporté à l’organisation du colloque
et à la publication de ses Actes.

TRANSFER OF INFORMATION FROM OBSERVATION POINTS TO OTHER
POINTS AND DISSEMINATION OF HYDROLOGICAL INFORMATION TO
UNEXPLORED BAS INS
GENERAL REPORT
Prof. A.A. Sokolov
The netv~ork of !yyi;lr;rolugioaì 3bssrvations efist:Ag at present represents
a discrete field of points ivhioh refleots only approximately the *:onstant
variations of hydrologioai elmeats in space and time.
As quite truly notetl Fierre Dubreuil in his report ("the problemi
af traafer of data from observations on a point to one or emother area
and their dissenclnation on territories and subjeots where no observations
wero onrried out, has always been. and is n q one of the central problems
of hydrologf.
This problem is erspecdally important for developing countries where
the network of hydrologioal stations is still hadequate and the existing
series of observationa too brief(ehort).
But wen in developed countries v:ith a well organieed and sufficiently
dense network of statione and
is, and always will be/ a great number of water bodies (or regiodor! whose
regime not enough light is thrown by obsemstion data, as a nwnbre of middle-
sized, and espeoially small water bodies, considerhg their great qurntiQ,
pnill always be examined only selectively.
posts, having operated Por a long time, there
In the Soviet Unior, for eYaqle,accordhg to data Prom detailed
inventarication(2) are numbered about 150 O00 rivera with a length Of
more than 10 h(and if ne inolude the shortest rivers with a l enw balm
10 h, their total number will aminit to 2 960 O00 ) and about 40 300 lske~
with an area erneeding i square h(?:iùif lakes with an area of less thsa
1 sq.km are boluded, their total number amounts to 2 850 o00 ).The
permanently operathg referenoe network of hpirologioal stations inoludes

2
ebout 6200,and the meteomlogioal
-
network more
-
than 10.000 observation points.
The task of hydrology as a soienoe oonsiate h establishing, on
the basis of a selective stuty of water subjects, natural laws of the
hyilfolcgioal regime and the distribution in spaoe of its ohaxeoteristios,
nllowing with a suffioient reliability and preoiwneea neoessaay for praotioe,
to spread hydrologioal data to subjeots or regions with a soarcity or
the absenoe of hydrologioal data.
Here we would like to refer agaiii to the already mentioned report tnf
Fierre Dubreuil whioh stresses that the most important in the problem of
transfer of hydrological data to unexplored basins is the analysis anä atuày
of the laws of the influenoe of natural and anthropogenous faotors on water
regime aad water balanoe, in the establishment of qualitative and quan.t;it&e
relations "Hydrology - environment". The author of the report notes that the
applioation in the computation of the flcod flow and of other elements of
~drologioal reg- Qf numerous empirio formulae, determined for certain
natural oonditions in other regions with different conditions, often results
in gose errors and misoalculations.
The transfer of hydrological data to unexplored basina(regiœna) is
direotly or indirectly relnted to the methodology of mapping the oharaoterietioe
of the 4drological regime applied in hydrology, since the praotioalwcyra and
man8 of such tranefer are generally baeed on the mapping of oharaoteristioe
of the hydrological regime and ita parr reters.
"mo basic methods of dissemination(transfer) of hydrolngioal daea
on mexplored basins(regions) are used with the aid of mapa:
1) Drwiilg of maps of isolinesi 2) Division of a territory into regions
based on the uniformity of hydrologioal cha-e.oteriat$oa of the regime and
its parameters.
The prinoiple of the method of isolines is the assumption of the
presenoe in ths nature of a smooth, colistant chenve of the oharaoteristiorr
of the hydrological regime in space,froin one point to another. The division
into regions, on the oontrary, proceeds from the assuqtion "IIomogeneity"

of larger or smaller territories end O€ 8 sudden, spcqmodio ohange of the
characteristios of the regime between one region and another..
In the publications on hyckology these h o methods of geographic
generalization are often opposed to one another. The appearanoe OC.& critioal
thaf
attitude ooncerning the method of isolines proceeds from the fact w th the
development of the study of smll basins more and more faotors(data) are in
contradiction with the mothesis on which this method is baaed. The smaller
the river basin, more tho oharacteristics of its hydrologioal regime may differ
from the meaning of the isolizes wì-ich suppose their smooth change throughout
the territoryo
--
To this o m be given a greaf number 3f exnrnpleao In the USSR, or the
territory ssturted on the left bank of the Volea, for instanoc, two small
basins louated side by side (1OC-200 8q.h.) have a mean many-years spring
flow of 27 and 97 mn, while on the map of isolines of the mean depth of the
spring flow, at this place is shown an isoline of 6ûnnn.
"aturally, the question arises: w ht indioate isolineat what is their
sigriificanoe and their meming if the runoff of actual basins deviates so much
from them?
In our piiblications (3,u are examined the reasons of the oontradictory
opinions on the effioienoy of the utilization of the method of isolines and
of the method OP division into regions. They are oauaed by a misunderetandhg
and an oppesition of the zonaliw(moth variations) and the eronali-&(sudden,
looal deviafiole) in nature.
- In our opinion sonai and asonal 1-8, as well as the method8 of mapping
based on them methods of isolines and of division by regione, do not
but mutually oomplete eaoh other. The first (isolines) shows the general,
zonal law# 00 distribution of the characteristios of hydrologioal regime through
the territory of closed basins( ooinoidenoe or a small difference of $be aurfaoe
and the eubsurfaoe nater divide), dieplqred in the murse of &ir averaging
for large areas, for whioh the influence of azonal(looa1) factors of the
environment o m be disregardedzThe seaond. permits to reveel the kiternaï,
disorste by its essenoe, structure Of' these avoraged oharaoteristioa,
3

4
conditioned by the influence of local fwtors - geological struotures, slopes,
vegetation, spi1 and grounds, which constitubo the surface of tho basin, ad.
others a
ûno of the rundamental proTLsions of the theorj of hydrological mqping
and of the applioation of t he method of extrapolation of data on unexplored
basizs by means of maps of io3linesS cor.8ists in tha fact that the data uaed
in this chse, concern basins complying with the condition:
A ~ A ( A
ma^ ( 1)
i
-
whsre A - mean value of optimal areas OZ the catchment in which is telerated
the interpolatcion o: hydrological characteristics by mans of isolineai
- A 6, A max respecti/vely the lower and the upper limit of the catchment
area, whose date are unsuitable for drawing m pa of isolines.
Only relation o those basins wiiich comply with the condition(I),the
goographical Iritqrpolation is parrrlssible and, consequently, the hypoaesis O?
a smoth and omstant rwiation of tho oharnoteristics of the hydrological reLi-ie
on tho terr%toFj is correcto
f
In the oatoi-wnt weas rwging from O to A nLio are found other laws. They
ar+ l.aClected in larger or mallsr deflections of the characteristioe of the
ruioff of small rivers fro,^ zonal(ssa the abova exainple i>f 27,37 arid 6ûmi)inevitably
everrrged and as if liberated from the influonoe of the local factorsíspecificities
oí' the environment, according ti, P. hbreuil). Chinlg to the fa& that in small
basins individual peculiaruios of the conditions of the runoff of snow melt
and rainfall plow are sham most sharply(for example, the g mmd o," one basin ia
made of ~and,oi' another - of olay, or one basin is open, another has a forest cover,
etco) the data on the runaff of these basins generally are not suitable for a
geographioal awmarizing with mags of isolines. With the decreaae of the size of
the ce.%chment increases the probabili% of the defleotions, as well as their
importanoeo
The above oan be illustrated by a scheme of defleotions o? the mean annul
runoff(for m my years) of karat rivers from Its zonal significapoe in relation to
the area of the oatchment(fig.1) .These "fork-shaped" shhemes of deflections can
be disclosed also in study%ng the iliflwnce of othar Pactors(for example, the
dekreb of afforestation) on the runoef a d their relatione to the dimension of

of the o.ztohment.
Th oqiplltation of these dsfleotiona is owried out by mane of gemtic
P
rdlatiora of the charaoteristioa of the hy.itrologioal regime pli* the faotors
LieteriliQ tham by the lntroduotion of oorrection factors in th~ zonal oharaoteris-
Lics sf the hydrologioal regime, obtained for the uriexplcred'bssina tlrougk- the
mq~S of isolines.
Y!ie principles of oomputation of the main oharcoterietios of water
resources in the absence or scarcity of )Ij-1romtrioal drrta, based on the above eox:al
ari e.zo-ml geographioal laws of $he rtmDff, ar9 examhed with mre tietails in the
rop0t.t of Rof K.P.Voekresendqr(5)
In the light of the atom, tht> oonolusion drawn in the report of I.Bdek(e;
euòmitted to th3 present Symposim, beco*:es olear end convincing, nariiely, that
the defiliition of reference hy4rologiaal charaoteristios for mexplored sub jecte
(regions)only on the basia of data from r-jpreaentativa and experimental basins,
cwnot be ~eoomaded, i.e. it ie not possible to transfer direotly data Prom
7bswvations on 8-1 catoinnents tr, unexplored large basine.
The author of the report, analysing the data on experimental and represat-
ativu basins of fropioal Mrioa, where under the THP programe were created m re
than 100 represantativ
+
d experimental basins, brawe the ooaoluiona that a joint
StUdy(8nalya~s)dfdata from experimental and representative basine and of those Prom
the standard network is neoesomy. The differenoe between the charaoterietios of the
runoff, obtained on small experimontai catchumts and aidlar characteristioa o? the
bash of a standard netmork, should be carefully analysed.
Considering the influace of fc-eats on the runoff on the basis of dooumente
from experimental investigationa ia Xqa, ;r.Balak drws the conoluaion that a bamboo
or a high mountain forest reduoes the surface runoff a that the replaobanent of
forests by aEricultural farm inoreases the voliono of the runoff. Transevaporation
from forest vegetation mas three times higher than that from aeac;onl;r subnierged
fields.
With the increase OP tho 'egrje c? boggiilp;, accorahg to data from obser
vations in the basin of the river Ilafou, the annual runoff deoreases. in this oonna-
ctiori, 7.Balek underlines that mra attertio- should bo drawn 'to hy~oloy;y of
tropical mmps, as these wrsnps play an i:pcrtoil-t role in ti-9 foranation of the
river rmoff
5

6
At ths same time he notes that the methods of oomptation of the nuirf8
usad at present in the temperate climate should Se revised taking into aooount
the specific oonditione of tropioal oatohrnents.
In the report of VbS. Vuglinsw and V.A. se?mmv(fl are atudied the
specificities of formation and the methods oî .h.anefer of data from observa-
tions to unexplored basins si' the runoff in mountain are-, including the
conmody utilized method of detedning the standard of the annual runoff,
based on the establishment of regional relations of the npdulua of the
annual runoff to tho height of the uatchment, aooording to data from explored
bas ias
The authors note that,pthe computation of the runoff of small oatch-
-.axts th utilization of the relatlon of the nodulus of ruroff to tho height
of tho catohment obtains not always satisfaatory resdtte, whio-h can be
explained by the kifluence of looal faotors in the mouutains~ In this relation
oatchments with the same altitude 0811 differ ccmsirlerably by the conditions of
their formation, as well as by the volun.3 of the ennual runoff.
In suoh cases the auL,hors recomnend to determine the standards of i ki
annual runoff of mountain catcbments witi sufficient or exoessive misture,
by meau of' a joint solution of tho equation of the sate- and heat-balanoe.
Tile runoff is oaloulated bj the differenoe between preoipitation and m po-
transpiration. The definition of the Etasdard annual preoipitation is oarried
out with the applioation of graphs of the relation of preoipitationa with the
altitude, taking into amount the crogaphio speoifioities of the mear
TT oniputation of the etandexde of aruiual evaporation is made by a
i.mre preoi e equation of & ïbhdyko# nihioh takes into aooouut the turbulent
heat exohange th3 baaio paramatera of mhioh are : radiation bCbhaOe, precipi-
tation and turbulent heat-exohange.
The above sohem of computation is used for catolnnents looated in the
lower and middle mmtein bolts. For the higher iiiomrtaine this method oan
be used also, but ki this o w the number OP terma of the water bcilanoe
equation i8 Fnoreased(it i ta oaïouïata the volume of glacier6
ablation, Qf anow pa& melt and a dietinot oeïoulation of evaporation from

various underlying surfaces of the high muutains~o
In this report are also studied the methoCs of oaloulation of the
coeft'ioient of variability of the annual ruuof'f - Cv, ueed in momtait~
ter ri to i- ies
In the report of the group of authors: ML Albitlet, G. Castany, Mpe
Delaroziere-Boulllin, R. Jonac et G. &-gat(B) is exposed the method of appraisal
of the &mers1 water resoiiroos(equstsd by the authors to the mean m ual r-uiorf)
anci the renmable groundwater resources with inadequate data. used by th93 authors
for the territory of Franca and Venezuela.
The authors recomiiend to determine the general water resourcss(Ptreamf1m)
'51~ means of maps of isolines of preci;>itatkon &Td evapotranspiration, by the
?ifference precipitation minus
t
evepotransporation calculated by ths method
Thornthwalte or Turc. Th3 value f these differancea are dotarmined by conventional
scpares. ìVhm there is a grmi. Wference of the factual evaporation,obtained
>y the differeroe precipitatior? minus runoff in a looked discharge seotion line
mid evaporation, calculated by ti- method of Thornthwaite or Turc, a oorreotion
coeri'icient is introduced C, by means of irhich the map OP the rateu evaporatào.on
13 cor reoted.
yhe authore determina the na%iiral resources of groundwaters far the
smis squares of the map as the volume of tha goneral runoff by moans of
"Geological coefficients" de%i,eci as a portion of the unclergound flair in the
generai river ~ f f i
'fl.is approach to th?: cle.:iriitiov of the volurm of renewed resourods
of groundwaters is also utili.eed in the ':SSt? in the publioations of B.I.Kudolin
and o.v.Fopov(9)
The method o? computation of the total runoff by meam ,>P the difference
3recipi :.ation minue evapotranapii.ution arid ths 3ef inition, ori tYAs basig oí' the
so-. ..allad "Clhtio rUnoff",recomnan

8
+
method onl) in regions xh he differenoe-preoipitation rtlh~us wapotranepiration
is stffioiently important.
In the practioal application of the method, the definition of diïferentiated
rr?min@of the "geologiaal coefficient" for eaoh square of the map Ocin giv$ise
to difficultiee, especially when the territory has been inauffioiently stuclied.
In the reporta e€ van E ylch are examined the methods of oompukation
of evapotranspiration in regione with identioal olhtic oonditione (11)
The author indioates that the existing simplified nudele of oalouiation
of evapotranspiration through a limited number OP parameters, for at€UUple, by the
temperature of the aiqmrathwaite ,me@ aid Criddle,and otherr) lead to errore
in the evaluation of the monthly and annual volumea of evaporation m-hg %O
3~% or mare. Contradictions in t h resulta of oalouiationa mado i>r existing
formulae are shown 3x1 fig.1. The author reoommenda, wh& oaloulating evaporation,
a more detailed method with the utilieation of euch parametera M radiation
balance, precipitations, air misture8 wìcity of wind.
To find an isauo to this sAtuation, namely, that the mentioned initial data
are not everywhere available, tho author proposes to utilize the idea of homolhate.
The main point of his proposal oo#neists in the choice of a well-bonm region
mhere the clinlatir; conditions approaoh to the mx-. the conditions of' the area
in whioh the oamputation of evaporation m&be oarried out, nnd where the hitiril
data are misskiq. According to the aseertion of the author, it is possible by thi&
homolinatio method to oaloulate the monthly and m.nual volrpnes of ewlpotrauepira-
tion, differiiig not more than by i@ from those which were measwed.
The author describes the desip sohhome adopted by him for the homoolimatio
maluationa of evaporation which is based on tho equation developed ky Penmaw
(i9fflDiq56) and leer OD improved by Montet((1963)wd Van Baveiï(1966).
It should be noted that this equatior does not take inb aooount the
temperature stratifioation ar-d th e mietur6 cq-tmt of' +,he soil. Therefore it
iB applioable only for computation of potential svapotranspiration from soctions
of the land with an opthal nmisture(irrigation, oapillary subteraraean feeding,
herbagea oloeed up in the stage o f optimel development). Unfortunately the report
doeo not metnion this.

An important particularity used by the author cf the equation oonsista in
the possibility to oaloulate instentaneOue(urgent) values of the velocity of
?vaToration.me author of the (van 'PlC%)of the opini.on that the Us8 OP
the seasonal, montka and even aeeWy mean values OP the inftfal meteorologiosl
factors for the evaluation of the evaporability(%) gives false resultem We have
tr agree ruith thie.
var. iìylcluraa
In the reference equation Fropoaed by is taken into aooount the
resistance of the outleta(aooord3ng to bntwt). The reoalaulation of tho ptential
evqoration by the equation, in nhioh is taùa into aooount the rssisfauoe of
ouilete, has prmed that this equation obtains the best reaults(8ee lower part
cif figme).
This method shoulü be wed oniy for the computation of ahorMenn(hourly)
data. l'o illuetrate his opinio8fn Hylc-s fig.1 in which it o m be seen that
fi oalculated by the mean hourly initial data cire nearer to fhoee measured,
than the data from oalculation through mean daily values of meteorologiod elements.
TO conolude, t!ie author riotee that on the basis of the available climatio
olassification(maps) it is possible to use the homolimati0 method and obtain
reliable evaluations of the potential Jvapotrsnspirntion for insuffioiontly axplored
regions
The defeot of the proposed method of trm-sfer o? data from one region
to another oonaista in a huffioient preciseness of the definition of the hoim-
climate end the absence of reliable homolimatic maps.
We have to atop shortly on two other reports, although different by their
oontent, but having m q
oomn features. We have iri mind the reports of G.R.
Tiercelin (12) and of the maup of airthora R. OaroirnAgreda, G. Raastd.0 and
Viparelli(l3). Their oomnon feature is the statistical aepe& of the problein of
trauef er of hydrologioal data to unexplored basins(regi0ne)
h3 notes ir his report G.R.Tieroelii: ,disregarding minor d*ih, the
methods of ddinition of parameters of the runoff oould be divided eseentitrlly
hito *O grOU2jS 8
1) The establishment of a regional dependame of the value of the defined
9

10
paraueter(man, oodfioient of variation, coeffioient of oorreìation between
adjaoenf temm of a series, eto.) from basio phyaiographio oharaoteristics
(precipitation, evaporation, dimension of the area of the Oabhmonf, height
above sea ïeveï, forests, etc.).
2) A joint analysis of runoff data by a group of bydrologioal identical
catohments(with a similar condition of formation of the runoff).
In the first oaae, the value of the interested parameter for an unexplored
stream is oaloulated by the dependence obtained through the data of the neighbus
h g streams, and in the seoonä oase this value is oonsidered a8 equal to the
arithmetioal mean value f'rorn the seleoted values of parameters of rfvere studied
jointly.
In the work of G.P.Tieroelin is used BP assooiated analysis of data
aooording to som previously seleoted and hydrologioahy identiod rivers, with
the same periods of observation and having slightly differat seleotive vduss
of statistioal parame'ters.In this m ~ ~ is e r determined the regimal signjd'ioanoe
of parameters of the monthly runoff. Data from 12 stations with 49 years of
observatioiis(Prom 1920 to 1968) are wed and are divided iPt0 ho group80
ñegional aues of ertain parameterskoeffioienti of variatiow&ficiat of
htraouolear correlation, obtained by means of averaging for "idmtioal"
regions are reoonunended by the autbr tQ be trenaferred to irnqlored sl;reama
of a given region.
very importtant in this report is the theoretical part devoted fo the
definition of the mean-square-error of the kraasfer of the regiod value of
the parameter to a oompletely ur-explored or Fnsuffioiently studied wateroourse.
The importanoe of the mean square error depends on the qumtlty and
of
hydroïogioaï data for the region(o0oasionaï deviation), as neil as from the
representativeness of the studied river for a given region(deviat5on oaused by
geographioal faotors). The Formulation of this problem tio a large extent reminds
the works of S.N.Kritcky, ?d.F.b&el and $.G.Blokhinov(m in whioh it is also
proposed to oonsider fho complete dispersion of parrunstere of Joint
f
series a8
the result of a oonoerted aotion of the abové) oauses. The praotioa pplication

of the proposed formulae requires a great oare, as their utilization implies
the sigiif;oance of unknown nctuE1 values of dispersions of paraneters and the
correlation between the selocted paraneters. In the presenoe of short series and
1;lieir smdl nimiber the substitiition of actual values by seleoted values o m in
a nunibre of oases oomiderably distort the value of the mean-square-error of e.
i.ogio2irsS inportanoe.
In the work of G.R.Tiercelil? , th0 choice of a group of catchmante
(ciivisiori by regiom) was cwïied out on the basi8 of orly a "Visual'1
comparison of the selective values of the parameters oaloulated for separate
rivers.
In our opinion, a preliminary analysis of the conditions of formtion
of tho riinoff on catchments outlined for ct joint stuciy with a consequent applic-
ation for the final selection of statistioal oritoria of similarity, would
be m re aoourate. This more rigid apprcjach to the seledion of sMlar regions
is applied in the work of R. Garcia-Agrede., G.Rassulo and R.Viparelli(l3),
in which the authors propose to se1eot"plwiorietric zones'by the oonstruction
of "peridssible" 9% confidenoe intervals of paraiiiaters of distribution, determined
according to data from observations in separate points. This more rigid approaoh
will enable w,in a numbor OP cases, +A avoid tho inclusion by mistake in OW
g-oup catohinontJ with heterogonoous ooI?i?itions of formation of the runoff
The arithntetioal mean should hardly be taken always as D regional value. It would
be m re advisable to weigh the selective values of parametars obtained for
separate rivers. For example, the weight ooeffioiente should be taken in a direat
ratio with the areas of attraotion and the length of the utilized series.
In apite of the great preoiseness of regional parameters noted in the
work o," Mr. Tieroelin , whioh in the opinion of the author can be muoh higher
than for parameters obtained in a short series, the proposed method oannot,of
coupse, replace a oareful analysis of initid data - whioh was already stressed
in the report of P. Dubreuil.
In this oorineotion, we would like to refer to the detailed oritioiem
of the method of hodostations(interc;anneotion of series) submitted in the report
of A.I.ChebotareP. and B.I.Serpik on the Leningrad Symposium on Floods and their
Coniputation(l5), with whose opinion, ooncerning the sffioiency of this method,
i quite agreeo
11

12
Besides, the author himelf repeatedly atreeeee th0 neoeseity of a oareful
approaoh to the interconneotion of eerie8 of obeervationa on rims of the
so-called identioal regione.
To conoluäe, it diould be noted fhat bestigatiom on the a;plication
of the mathematioal apparatus for the Lins of epeoe interpolation of hydrologioal
oharacterietios of the multiple liuear oorrelation are oarried out at preaeurt(16,17)
At the sau~ time onethe basis of a multiple linear regreasion, the oonetruc-
tion of a field of isolines of hyärologioal characteristior, impleppsnted by a
conputor with an ewaluation of the preoiseneas of interpolation in q givan point,
ie eventually projeoted.

I. P.Dubreui1. Transfer of hydrologioal information to uuexploreâ river basins
(presented ta the Symposiimi )
2. A.P.Dodt~, ReG. Dubrovina, A.I. i8aevat"Rivers and Lakes of the USSR"
(reference data) Gidrometeoiedat, 1971.
3e A.A.Sokolov "Zonal m d a~oneï factors of the runofP".Coll.nf public. on %drology
No 2, GidrO~teOisdat, 1961.
4. A.A. Sohlov.The theory of hydrologioal mapping. Bull&ln VW, N0.1~1968.
5. K.P.Voükrûs~~e Principles for the computation of the basi4 oharaoteristioa
of water resouroes of rivers with inadequate observationa on the baais of the
geographical interpolation of th^ paraniators of the' runoff (presented to the
Splposf tall$
6, 3.Balek. Utilieation of representativo and experimental catobments for the
weluation of hydrological datri Sron Aifricm. tropical bas- (presented to the
Syqo s id
7. V.S.Vuglinsky and V.A.Semonov. fialualiion of water rmources of mountain
territories in the absence or soarcity of data of the runoff(presonted to the
symposium)
8. M. Albinef, GICastauy, Mr8r belaroeiercBouillin, R. Jonirc,. J. Margat.
Evaluation and distribution of water resources of large regions on the baais of
hydroclimatio and hydrologic oharaoteristioa (presented to the ~psimn)
9. G.I.Kudelin, OiVmPOpOV. Influeuce of olimate on the natural 1-8 of formation
of the groumator fian. Reports OP the soviet geeïogistrr to the 24th session
of th9 International Congresri on ûeology.%ydrogeology and Engitmerlng bolo&,
"Nauka", baoon, 1Wm
10. M.I.LlvovEtoh. Elemnte of water regime of the rivers OP %e Earth. hblio.
of KtU cenisa) sa) Board OP the Qdrsmiteomlogioal Servioe) Ser.IVlvol.18
Sverdlovsk - Leahgrßddr
13

14
Hylckama
11. T.E.A. V a Computation of evapotrmspiration by region8 with
idontical climatic condWions(presented to the Symposium)
12. 1LR.Tiercelb Regional parainstors concarning water resouroes. üsee.heciseneas
of evaïuation(presented to the Symposium)
13. R. Garoia-Agreda, G. Rasuulc, R. Viparelli. Pluviornetrio zones and oriteria
for evaluation of their limits Por region8 w5th insufficient data from observations
(presuntod to the Symposium)
14. S.?l.Kritzky, M.F.Menke1. T.bthod of a joint aaolyeis of observation8 of the
runoff of identical basina. Public. of the CCI (State Institute of Hydrology)
vol.180, kMmmeteoizdat, lr;7G.
15. A.I.ChsLotmev and B.;.SerpZc. Of the passibility of using the
intorconnected seriea of hyàrologioal ohaxaoteristios for the oomputaticn oi t h
runoff:. Internationo1 Sy-fnposiun on Flscds a d their ComputationbGidrometeoiedat,
1969
16. A.V. lbjdestvendcy. The exporimco of bringing the river runoff to a long-term
period by the method of multiple linear correlation. Coll. of public. on ~drolog
No .10.Gidromsteoi~dat,1970.
17. A.G. bbanova, A.V. HojdestvensQ. Space-correlation funotiona of the river'
raoff of the rivers of the Dnlepr b ash coll. of puklio. on ~dr010gy~1Jo.11
Gi drom tuo izdat , 1973

EVALUATION ET REPARTITION DES RESSOURCES EN EAUX D'UNE GRANDE
ABSTRACT
REGION PAR LES PARAMETRES HYDROCLIMATIQUES ET HYDROGEOLOGIQUES
Par: M. Albinet, G. Castany, Mme O. Delaroziere-bouillin,
R. Jonac et J. Margat.
The evaluation and repartition of total and groundwater resources
or a large unit, country, region or groundwater basin, may be rapidly
made with restricted data, by simple calculation, still obtaining a
satisfactory accuracy.
The total water resources, asimilated to the average annual total
runoff rate of the water courses may be evaluated by the specific
runoff. This is calculed, either directly with hydrometric data, or
in the absence of gauging by extrapolation based on hydrogeological
characteristics collated with the values by the climatological exprez
sions (L.TURC, THORTHWAITE).
The groundwater renouvelable resources are egal to the average
annual groundwater flow rate those evaluation tests on the division,
with the help of an index, of the specific runoff. These indes are
worhed out with the help of geological characteristics an hydrogeo-
logical characteristics punctually obtained b,y field tests.
Thus with resticted hydrogeological and hydrometric data and
sufficient data concerning the precipitations, tempertures and geology,
it is possible to obtain a satisfactory knowledge of water resources
which exploitation and planification. Practical results have been
obtained in France and Venezuela.
RESUME
L'évaluation et la répartition des ressources en eaux, globales
et soutterraines, d'une grande unité, pays, région ou bassin hidro-
géologique, peuvent être effectuées rapidement avec des données res-
treintes, par des calcule simples, tout en obtenant une précision
satisfaisante.
Les ressources en eaux globales, assimilées au debit d'écoulement
global annuel moyen des cours d'eau, peuvent être évaluées par le mo-
dule spécifique d'ecoulement total (i/s.km2). Celui-ci est calculé,
soit directement 2 partir des les données hydrométriques, soit, en
l'absence de jaugeages, par extrapolation basée sur les paramètres
hydrogéologiques et confrontée avec les valeurs calculées par les ex-
pressions climatologiques (L.TURC, THORTHWAITE).
Les ressources en eaux souterraines renouvelables son égales au
débit de l'écoulement souterrain annuel moyen, dont l'évaluation repose
sur le fractionnement, a l'aide d'index, du module spécifique
d'écoulement total. Ces index sont étables l'aide des paramètres
geologiques et des caractéristiques hydrogéologiques obtenues ponctuellement
par des essais sur le terrain.
Ainsi avec des données hydrométriques et hydrogéologiques res-
treintes et des données suffisantes sur les précipitations, les tem-
pératures et la géologie, il est possible d'obtenir une estimation
satisfaisante des ressources potentielles moyennes pour la mise en
valeur et la planification, Une realisation pratique a été obtenue
en France et au Venezuela.

16
1 . INTRODUCTION
1.1. Rappel des notions sur l'écoulement de l'eau dans le sol et le
sous-sol. Répartition de l'eau des précirdtatlons.
Le débit de l'écoulement total QT, mesur8 à la station de
jaugeage d'un cours d'eau, exutoire d'un bassin versant, est la somme
de l'écoulement de surface QR dans le réseau hydrographique et de 1'
écoulement souterrain QW, transité par les aquifères du bassin drainé$
L'écoulement de surface, QR, direct, rapide (quelques heures
quelque6 Bows) correspond à la crue de l'hydrogramme d'écoulement.
L'écoulement souterrain, QW, lent,différ$, de parcours com-
plexe dans les aquifères et de longue durée (quelques années à des
centaines, voire des milliers, de millénaires) est à l'origine du
débit des cours d'eau pérennes en absence de précipitations (étiage).
D'oui l'importance de la mesure des débits d'étiage représentant le
déMt minimal moyen de l'écoulement souterrain.
Le débit de l'écoulement total est alimenté par les préci-
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
réelle, ETR et les précipitations totales, PT ( PE = PT - ER). Eh 1'
absence de variation
des réserves (longue période d'observation) le
déficit d'écoulement moyen interannuel E"T est égal à PT - QT.
Les débits de llécoulement total et de 88s deux composants,
l'écoulement de surface et l'écoulement souterrain, sont régis par six
groupes de facteurs conàîtionnelst
-
caractéristiques dee précipitations: intensi téídurée, nature ;
caractéristiques géologiques du sol: lithologie des terrains,
perméabilité verticale, structures;
- c arac t éri stiqu e 6 mo rp bolo giqu e s : rnorphom 6 t 15 e, pen tes , reli e f ;
- cmactéristiques hydrogéologiques: humidité de la zone non
eaturée, profondeur de la surface piézométrique, paramètres hydrauli-
ques des roches réservoirs et de l'écoulement et de6 structures hydro-
géologiques;

- caractéristiques de la couverture végétale.
Ces facteurs, interférant, peuvent @tre ramenés B trois grands
ensembles: hydroclimatologie-hydrométrie, géomorphologie, géologie.
Les caractéristiques géomorphologiques et géologiques du bas-
sin jouent un r8le primordial dans le fractionnement de l'eau des préci-
pitations, d'o.ii la possibilité d'établir des index, utilisables pour 1'
évaluation du débit de l'écoulement total et de l'écoulement souterraint
De m8me il est possible d'établir des index climatiques.
Les réservoirs aquiferes ont un r8ie régulateur du débit de 1'
écoulement souterrain par la faible vitesse d'écoulement déterminée par
la transmissivité et par la mise en réserve temporaire d'eaux souterrai-
nes, fonction de la diffusivité ( transmissivité/coefficient d'emmagasi-
nement) et des conditions aux limites. Les réserves en eaux souterraines
sont donc a considérer pour l'évaluation des ressources en eau.
1.2. Débit de l'écoulement moyen interannuel et ressources en eaux
renouvelabl es.
L'écoulement moyen interannuel, QT, est assimilé aux ressour-
ces en eaux renouvelables, potentielles, moyennes globales. I1 est
déterminé sur une période de 5 à 10 ans:
- directement par traitement statistique des données hydrométriques;
- -
indirectement 6. l'aide d'expressions climatiques mensuelles ( TURC
et THORNTHWAITE) résolues manuellement ou sur ordinateur.
L'écoulement moyen interannuel, QT, erprimé en laine d'eau
2
moyenne, ou module spécifique d'écoulement total (l/s.km ) permet les
interpolations et extrapolations et l'estimation des ressource8 poten-
tielles moyennes des bassins non jaugés.
L 'estimation des ressources potentfelles moyennes globales par
cette méthode est très acceptable pour les besoins de la planification,
comparée aux &mi.uations basées uniquement sur des mesures hydrométri-
ques relatives à de longues périodes.
17

18
1.3. - Débit et distribution spatiale de l'écoulement souterrain mq- interannuel
Le débit de l'écoulement souterrain moyen
assidlé au débit moyen interannuel des aquifères dans le cours d'eau,
peut être évalué par l'analyse de l'écoulement moyen interannue1,QT.
Une méthode de fractionnement, à l'aiae d'index et étalonnage par des
analyses d'hydrogrammes de bassins représentatifs assez homog&nes, a
áté appliquée. Ces index expriment:
index = écoulement souterrain - - c$w en pour cent
écoulement total QT
2. PRINCIPES DE LA METHODE
Une importance particulière est apportée, dans un souci
de planification et d'aménagement du territoire, à la connaissance,
donc A la cartographie, de la distribution spatiale des ressources en
eau, globales et souterraines. Les données hydrologiques disponibles
sont, dans la plupart des régions, insuffisantes pour permettre
une cartographie. Par ailleurs dans bien des cas il serait inte-
ressant de pouvoir estimer les modules spécifiques d'écoulement
de bassins non jaugés.
C'est dans ces perpectives qu'une méthode simplifiée
d'évaluation des écoulements moyens, total et souterrain, par bassin
versant a été mise au point en vue d'une cartographie à petite échelle
applicable B l'ensemble d'une région ou d'un pays.
Son principe, 6es modalités d'application et les résul-
tats obtenus sont présentés sur un exemple concret.
La méthode d'évaluation et de cartographie de 1'8couleumt
a été établie de facon à pouvoir Itre traitée automatiquement. le, ou
les, bassin6 étudiés étant discrétisés en mailles régulières.
2.1. Données de bases nécessaires à l'application de la méthode.
Ce sont:
- surface du bassin versant

- débit moyen interannuel, QT, de la période p, mesuré à l'exutoire
du bassin versant;
- carte en courbes isohyetes des précipitations moyennes interannuelles
PT, de la période p, sur l'ensemble du bassin versant;
- carte de zonalité de l'évapotranspiration réelle moyenne inter-
annuelle ETR, de la période p, sur l'ensemble du bassin versant. E$
théorie n'importe quelle méthode de calcul d'un indice d' évapotrans-
piration réelle a partir des données climatologiques mesurées ponctuel-
lement peut 8tre utilisé. En général, les valeurs de llévapotranspi-
ration réBlle moyenne interannuelle les plus significatives sont ob-
tenues par calcul sur l'pas de temps11 mensuel, soit à partir de la
hauteur des précipitations et de la température par la méthode de
THORNTHWAITE, soit à partir de la hauteur des précipitations, de la
température et de l'insolation par la méthode de TURC mensuelle. Dans
ces deux cas, les calculs doivent Btre effectués mensuellement pour
chacune des années réelles successives de la période choisie. La moyen-
ne interannuelle doit Btre évaluée exclusivement à partir des valeurs
annuelles de 1' évapotranspiration réelle obtenues. Ces opérations
peuvent Etre réalisées automatiquement à l'aideud'un programme de cal-
cul établi au Bureau de recherches géologiques et minières (B.R.G.M.).
2.3. Calcul de l'écoulement total.
Les données de base étant acquises, les cdculs suivants
sont
-
effectués successivement:
calcul de la lame d'eau prkcipitée moyenne interannuelle (en mm),
m, sur l'ensemble du bassin versant, par moyenne des lames d'eau précipitées
-
sur chaque maille;
calcul du déficit d'écoulement moyen interannuel (en mm), ETT,
sur 1 'ensemble du bassin versant, par moyenne des déficits d'écoulement
relatifs à chaque maille lorsque l'on a admis une hétérogén6ité de
ETR dans le bassin (sinon ETT = PT - QT);
19

20
- comparaison
de la différence, PT - QT (données mesurées) avec
hTT. (données calculées) et calcul d'un coefficient de correction c;
C=(PT-QT)/ETT
Puis calage des "bilans" unitaires de chaque maille ( ETT =
PT - QT) sur le débit d'écoulement total du bassin par application du
coefficient de correction, C, à 1'EITR de chaque maille.
- calcul de l'écoulement total unitaire, maille par maille (en mm),
par différence entre la lame d'eau précipitée et la hauteur d'&rapo-
transpiration réelle corrigée.
2.3. Evaluation de la distribution spatiale de l'écoulement souterrain
La distribution par maille de ltécoulement total pour le
bassin étudie étant connue1.trois procédures sont appliquées en fOnC-
tion des données disponibles pour l'évaluation et la distribution spa-
tiale
-
de l'écoulement souterrain.
premier
-
cas:iaxistence d'une carte des index géologques (page 514
l'écoulement souterrain, QW, de chaque maille est obtenu di-
rectement par application des index à la valeur de l'écoule-
-
ment total de la maille;
le calcul de l'écoulement souterrain total du bassin versant
est effectué par sommation des écoulements souterrains de
chaque maillem C'est cette procédure qui a été utilisée pour
l'étude de la P'ranche-Comté (France) objet du cas concret.
"
- deuxième cas: existence d'une estimation de l'écoulement souterrain
total à l'exutoire du bassin versant (valeur obtenue par analyse des
hydrogrammes, selon une convention appropriée) et d'un bassin litholo-
giquement assez homogène. Cette méthode, concevable en théorie est
rarement applicable en pratique, car les bassins assez grands qu'il
faut considérer ne sont généralement pas homogènes.
- troisiéme cas: existence d'une carte des index et de la valeur
estimée de,l'écoulement souterrain total à l'exutoire du bassin versant

- une première valeur de l'écoulement souterrain de chaque
maille est obtenue par application des index à la valeur de
l'écoulement total de la maille;
- calcul de l'écoulement souterrain total du bassin versant
par sommation des écoulements souterrains de chaque maille;
- comparaison de l'écoulement souterrain total avec l'écoule-
ment total, QT, et calcul d'un coefficient de correction C1:
-
QT QW
- application de ce coefficient de correction C1 aux débits
souterrains de chaque maille.
I1 est important de souligner que le débit souterran
calculé pour chaque maille a la signification de l'alimentation spé-
cifique moyenne probable des nappes souterraines dans la maille, par
infiltration de l'eau des précipitations, indépendamment de tout
apport pouvant provenir d'une autre maille.
2.4. Simplifications admises.
La méthode d'éualuation des écoulements, total et souter-
rain, peut fournir des résultats significatifs al elle est appliquée
H des bassins versants de dimensions assez grandes, à partir des
données hydroclimatologiques moyennes interannuelles, établies
sur une période suffisamment longue pour que le rble des réserves,
superficielles ou souterraines, puisse $tre négligé.
De plus cette méthode s'adresse aux bassins versants pour
lesquels il est possible d'admettre que le débit des nappes souter-
raines est drainé essentiellement par les cours d'eau du bassin. Eh
domaine karstique,par exemple il sera nécessaire de grouper les
bassins versants de telle sorte que les transferts d'eau aux limites
des groupements établis soient négligeables.
terrain,
3. APPLICATION DE LA METHODE A UN CAS CONCRET - POSSIBILITES
D 1 AUTOMATI SATION - PROGRAMME FI,&.
Le calcul et la cartographie des écoulements, total et sou-
ont été réalisés pour les bassins versants du Doubs, de la
21

22
Haute baône et de ltxin, sur une période de référence moyenne de 5
ans. A cet effet, une carte des précipitations moyennes et une carte
de 1' évapotranspiration réelle moyenne interannuelle (méthode de TURC
mensuelle) ont été réalisées.
Trois ensembles de bassins versants ont été utilisés, en
fonction des relevés hydrométriques disponibles, pour llapplication
du programme I.L$C. Dans leur définition, tous les bassins versants
é1éi;lentdres présentant entre eux des échanges souterrains ont été
groupés de telle sorte que pour chaque ensemble les limites topogra-
phiques et hydrogéologiques soient concordantes. Les groupements
suivants ont donc été traités:
- bassins versants de la riaute-Saône et de l'Ognon limités aux
stations de jaugeage de day-sur-&8ne (sur la aaôiie) et de Pesmes
(sur l'Ognon). Superficie totale : 5 782 km2, débit total mogcpn
(période
-
i964-iY68): 3 093, 7. lo6 m3/an;
bassins versants du Doubs et de la Loue limités aux stations
de jaugeage de Rochefort (mir le Doubs) et de Champagne (sur la Loue)t
Superficie totale: 6 350 km'; débit total moyen (période 1964-1968):
5 086,3 .i0 6 m 3 /an;
- bassin versant de l'Ain limité à la station de jaugeage de
r*
Chaaey. Superficie totale: 3 630 km2, débit total moyen (periode
1963-1967): 3 944.i06 J/an.
3.1. MaillaRe des bassins
La planche 1 prhsente le maillage adopté pour le Calcul
des écoulements. Les mailles, généralement carrées (sauf aux limites) I
ont une surface de 25 km
2
pbur les bassins du Doubs et de la Haute-
Saône et de 9 km
2
pour celui de l'Ain. Soit au total pour chaque
bassin versant hydrographique: Haute-Sadne et Ognon, 232 mailles;
Doubs et Loue, 254 niailles; Ain, 405 mailles.
3.2. Cartographie de la distribution probable de l'écoulement total
et de 1 'écoulement souterrain.

Les valeurs par maille de l'écoulement total sont directement
fournies par l'application du programme de calcul FLØC.
La planche 1 présente la carte de la distribution probable de 1'
écoulement total pour le territoire étudié. Elle donne les valeurs en
mm par maille de l'écoulement total moyen interannuel (1964-1968) et
les courbes d' (gal écoulement total''.
3.3. CartograDhie de la distribution probable de l'écoulement souter-
rain.
7
L'application des index aux valeurs calculées de l'écoulement
total permet de dresser une carte de la valeur moyenne du débit des
nappes d'eau souterraine de la région étudiée.
Pour l'ensemble du territoire étudié,l'aptitude du sol et du
sous-sol à permettre l'infiltration a été analysée sur la base des
données de la carte lithologique établie spécialement (fig.2) cl partir
de l'examen des hydrogrammes de quelques cours d'eau. Mais les stationr
de jaugeage dont les données on été utilisées sont généralement rap-
portées A des bassins versants étendus, climatologiquement et litholo-
giquement hétérogènes. L'analyse de leurs hydrogrammes n'a donc pu,
dans la plupart des cas, permettre de définir.
tisfaisante l'importance de l'aptitude du sous-sol A permettre l'in-
filtration. Compte-tenu de ces réserves (et de la possibilitk d'affi-
ner cette analyse lorsque l'on disposera des relevés de nouvelles sta-
tions
-
de jaugeage) deux types de domaines ont été distingués:
les domaines od l'infiltration est nbgligeable, le sous-sol pou-
23
avec une précision sa-
vant Btre conddéré comme imperméable à l'échelle de cette étude) Pour
ces-dbmalnes, A l'échelle du l/ 200 O00 la quad totalité de l'écoule-
ment
-
est de l'écoulement de surface. L'écoulement souterrain est nui;
les domaines à réservoirs aqui feres pour lesquele l'infiltration
est possible. L écoulement souterrain représente alors une proportion
plus ou moins élevée de l'écoulement total. I1 est possible de dis-
tinguer:

24
- les domaines 03. l'écoulement souterrain représente une proportion
moyenne de l'écoulement total (9 %) avec les grès du Permien et du
Trias inférieur, les formations marno-calcaires du Crétacé et les dép8tr
giaci air es et fluvi o- glaci air es ;
- les domaines OC l'ecoulement souterrain représente une forte propor-
tion de l'écoulement total (80 %> avec le Crétacé & dominante calcaire
du bassin de l'Ain et les formations alluviales;
- les domaines OU l'écoulement souterrain représente la totalité de
1 'écoulement: formations calcaires du Nuschelkalk et du Jurassique
supérieur et moyen.
Conclusions - Gomparaisons entre les écoulements
mesurés et estimés- Validité de la méthode.
Différents tests ont été effectués sur plusieurs bassins
afin, connaissant les débits mesurés, d'estimer quelle validité ont les
débits calculés. A titre d'exemple, on peut 'citer le bassin versant du
2
Dessoubre (568 km inclus dans le bassin du Doubs) pour lequel ont été
mesurés 14,2 m3/s (1964-1968) et calculée 16,16 m 3 /s.

SRAE - FRANCHE-COMTE
Carte da k dihbutin rribbk da I'iroulmrnt trtilnomlnterannud
(1964-19681

USE OF REPRESENTATIVE AND EXPERIMENTAL CATCHMENTS FOR THE
LSCESSMENT OF HYDROLOGICAL DATA OF AFRICAN TROPICAL BASINS:?
ABSTRACT
J. Balek
Institute of Hydrodynamics,-Academy of Science,
Prague, Czechoslovakia
Extensive hydrological records in tropical Africa are available
mostly for large basins. Since the beginning of IMD observation on
small representative and experimental areas has been started and
valuable short records are already available. A great number of the
tropical basins still remain unobserved. Although the demand for
hydrological data needed for engineering and agricultural develop-
ment is increasing, in many cases it can be found difficult to
provide reliable estimates of the hydrological characteristics.
Considering the high fluctuation of rainfall patterns and high
non-uniformity of the topographical and vegetational cover on small
tropical catchments it cannot be recommended to establish the cal-
culation of the data for the ungauged areas only on the records
from representative/experimental catchments. All the data from the
catchments should be compared and analysed jointly with the records
of standard network in an attempt to obtain regional characteristics
typical for certain topographical vegetational and rainfall patterns.
Examples of the calculation of the data in the Central Africa region
are presented in the paper.
RESUMEN
Utilización de las cuencas representativas y experimentales pa-
ra la evaluación de los datos hidrológicos en las cuencas inobserva
das del Africa tropical.
Los datos más fidedignos y más antiguos existen para las cuen--
cas grandes del Africa tropical. Durante el DHI empezaron las obsef
vaciones de las cuencas experimentales. Hasta el presente muchas -
cuencas tropicales son inobservadas y los cálculos de las caracte--
rísticas hidrológicas para diversos proyectos técnicos y agrícolas
son difíciles. En las pequeñas cuencas tropicales existen signifi--
cantes variaciones en la distribución de las lluvias, topografia y
vegetación y no es posible calcular las características hidrólogi--
cas solamente por la aplicación de los datos de la cuenca represen-
tativa/experimental más próxima. Hay que utilizar todos los datos -
de las cuencas representativas y experimentales y de la normal red
regional que existen en la región, para determinar las caratteristi
cas hidrológicas de la misma región, representativas para predomi--
nantes tipos de la topografía, vegetación y distribución de las llu
vias. Algunos ejemplos sobre la determinación de los datos para lac
cuencas del Africa Central se presentan en el artículo.
+: The research was sponsored by the National Council for Scientific
Research of Zambia. Some unpublished data were obtained by the
courtesy of W.M.O.

28
INTRODUCTION
During the International Hydrological Decade observations
of several representative and experimental catchments were
started in various parts of the African tropics, Data
obtained from these catchments together with the data from
the catchments established as parts of various special
proj ects represent very valuable material for engineering
and agricultural projects in Africa. As a main problem can
be considered how to make best use of the data when they
are applied outside the catchment boundaries. As can be
seen from the compilation of UNESCO (l), long records for
the African tropics are available in most cases for very
large basins. Obviously such data is of very limited use
because the number of big hydrotechnical schemes is rather
small. More frequently the data are required for small
basins as a basis for numerous rural development projects.
For the interpolation between the data from very large
basins and very small catchments there is no standard
method available. As listed by Toebes and Ouryvaev (2) the
main purpose of the representative catchments is fundamental
research, studies of natural changes, hydrological prediction,
extension of records and in the case of experimental
catchments additional effects of cultural changes. Extension
of the records is one of the most important tasks in the
tropics because increasing the network density can be very
difficult, owing to such circumstances as the river
accessability, staff problems, finances, etc. Thus the idea
of concentrating the effort into small areas well instrumented
and observed according to the requested standards, appears
to be very useful, particularly regarding satisfactory results
as obtained in temperate regions. As an example can be given
Volynka catchment located near to the Czechoslovakian,
Austrian and West German borders,

River
Volynka
Sputka
Peklovka
I I I
Drainage
area
km2.
Rainfall Runoff
mm. mm.
I I I
1
385 709 246
105 7 54 304
80 625 143
I I I
331
463
443
25.9
8.8
16. 3
29
I
--q--E-
Evapotrans
Yield
pirat ion
i/ s/ km2.
4.53
mm. - !
The catchments is situated in the Sumava mountains. In the
mountains are also the headwaters of three rivers (Fig. 1).
Supposing that no direct observation would be available, an
estimate can be done according to the relationship obtained
from the representative catchment (Fig. 2):
River
Drainage
km’.
Otava
Blanice
T.Vltava 347 957 550
However, all three rivers have been observed for a long
period, and actual data calculated:
Drainage
Blanice
T. Vltava 34 7 957 514
1
q-T
- I
Evapotrans
Yield
p irat ion
1 / s / km2.
mm .
17.4
Ev apo tr ans
mm .
Obviously, in a region where very little is known on the
hydrological regime of the rivers, results as obtained
indirectly can be considered as satisfactory.

30
Because factors such as snow melting, çoil freezing and
thawing etc. complicate hydrological regimes of temperate
catchments, one would expect that in tropical catchments
even better results can be achieved. However, owing to a
high variability of evapotranspiration, such an expectation
is far from being correct. There are several factors
contributing to the increased evapotranspiration variability:
Precipitation
Above rather monotonous topography of tropical Africa slow
changes in annual rainfall totals can be expected. The
raingauge network is not dense enough to provide a more
complete picture, however, available records support previous
presumption. From the records of a very dense network
established in small areas, follows that the distribution
of hourly, daily, monthly and even annual rainfall totals
is highly non-uniform. In Fig. 3 the distribution of the
annual rainfall in four Zambian catchments, each of them
less than 2 km , has been plotted. The rainfall distribution
was measured by the network of about 60 gauges. The
variability has been observed for 5 years (3) and it is has
been proved that there is no relationship between the
topographical and rainfall pattern. Jackson (4), studying
the interception of Tanzanian forest, proved similar high
variability within a small area. This of course makes it
difficult to apply some theories, such as, for example unit
hydrograph, because the centres of the storms are rather
randomly distributed above the catchments. Thus, identical
runoff volumes produce different types of hydrographs and
identical rainfall totals produce a great variety of runoff
volumes.

Swamps
Origin, size and location of swamps in tropical basins are
other factors highly influencing tropical hydrological
regimes. The total area of African swamps is about 340.000
km2. They have not yet been classified, according to origin
vegetation, geomorphology, and thus the knowledge of their
hydrological role is also very limited. Several catchments
containing swamps have been under intensive observation in
the tropics. In Uganda the evapotranspiration from swampy
vegetation consisting mainly of the papyrus, has been
studied, because it highly influences the water balance of
the Upper Nile basin. In Zambia heaäwater swamps, so called
dambos, forming a significant part of Central African water
resources, are under intensive study. By a comparison of
the results already available it can be concluded that the
influence of the swamps varies according to their storage
capacity, location within the basin and vegetation. Seasonal
distribution of rainfall above the swamps plays an important
role as well. In Fig. 4 rainfall-runoff relationships as
depending on the percentage of the swamps within the Kafue
basin have been plotted. River Kafue has a low gradient-much
below 0,001. From the graph it can be seen how the increased
size of swamps reduces the annual runoff. Such a type of
relationship is valid for the swamps with unlimited capacity
and located in middle and lower courses of the river. On the
other side the headwater swamps behave differently (5). Owing
to the limited storage capacity and high gradient of the
swamps the carryover from year to year is negligible and in
most cases the swamps are emptied before the next rainy
season starts. The swamps are sorrounded by dense woodland
where no surface runoff can occur and the only surface runoff
produced from the catchment is from the over-storaged
31

32
groundwater aquifer in the swampy areas. As compared with
the previous type of swamps, the time of increased evapotrans
piration is rather limited in the headwaters. However,
representative/experimental catchments are frequently located
in the headwaters and thus any extension of the results
toward the lower reaches is very difficult. In Fig. 5 three
curves characterizing the behaviour of the headwater
catchments with swamps of various slopes are plotted. The
lowest line represents flat areas covered by Brachystegia
woodland in the vicinity of the swamps. These areas do not
release any runoff at all. The middle curve characterizes
the runoff from the catchment slope of 3%, containing 6% of
swamps or catchments slope of 6% containing 5% of swamps.
The upper curve represents an area slope of 10% with 20% of
swamps.
The very first attempts to measure the evapotranspiration
from swamps were made by Hurst (6) who concluded that the
evapotranspiration from the Nile papyrus can exceed the
evaporation from free water surface. By some hydrologists
this has been considered as improbable, however recent
measurements support Hurst's conclusion.
Vegetation
As can be seen from the map in Fig. 6, changes of the
vegetational cover generally follow the changes of the
climate. Thus it might be expected that an intensive
observation of catchments established in each of the
climatical/vegetational belts can provide a full picture of
the role of the tropical vegetation. However, a more detail
ed map of any of the regions indicates a great variety of
vegetational types. It may not be difficult to find a
catchment with uniform cover dominant in the region; the
question is whether such a catchment can supply more
representative data than the catchment with non-uniform

cover. For example, in the catchments of tropical mountains
the vegetation varies accordingly with the temperature and
there a catchment covered by all characteristical mountaineous
types is certainly more representative than a catchment
uniformly covered by one type only.
The influence of the African vegetation on ‘che hydrological
cycle has been studied for a long time, In 1949 Wicht (7) drew
up a set of conclusions on the role of vegetation, founding
that the forest will use more water than grass, the consumption
of water by forest depends essentially on the amount of water
available in the soil and that the removal of vegetation causes
an increased discharge. In Kenya actual evapotranspiration/
potential evaporation ratio Et/Eo from various plantations was
measured (8) and in the experimental areas was found that
either bamboo or tall montane forest is an ideal protection
against overland flow, while the replacement of trees by
plantation increased the runoff.
The evapotranspiration from the grassland and woodland has
been measured in Zambian catchments. It has been found that
the trees consume approximately three times more water than
seasonally flooded grassland. The short grass roots have only
a limited chance to consume soil water, while the woodland
trees will tap the water from the groundwater table during
the dry periods. These results were confirmed by soil moisture
measurements and root density analysis (9). It has been proved
also that the Et/€, rate fluctuates year by year and month by
month depending on the meteorological situation, distribution,
intensity and amount of rainfall and groundwater storage
available during the dry season (3). The following table
indicates the fluctuation of evapotranspiration as obtained
for the swamp grasses and Brachystegia woodland in 1969/70:
33

Month
October
November
December
January
February
March
April
May
July
August
S ep t em b er
Y ear
Rainfall
mm
43.18
80.01
414.27
311.92
237.49
29.97
55.12
. O0
1.02
.o0
11.18
1184.1 5
Evapotranspiration
I
Woodland
mm
Et/Eo Grass Et/Eo
65.35 .4 20.07 .1
98.55 .6 43.82 .3
128.54 .9 95.28 .7
185.71 1.2 97.43 .6
197.91 1.5 71.93 .5
237.05 1.4 77.69 .5
152.66 1.1 39.37 .3
111.54 .9 16.81 .i
74.01 .7 6.02 .1
63.77 .5 5.40
N
60.04 .4 5.28
N
1457.00 .8 407.64 .3
The year 1969/70 was chosen as an example because it followed
after a very wet year and the evapotranspiration from the
woodland exceeded the precipitation, owing to the groundwater
storage accumulated during the wet year. The grass in swamps
evapotranspirated approximately the same amount of water as
during previous years. The ratio Et/Eo indicates when the
actual evapotranspiration exceeded potential evaporation.
The values in the table have been determined as limits for
the locations fully covered by the woodland or by the grass.
Supposing the data were applied outside such an intensively
observed area, actual vegetational composition has to be
taken into account, because owing to it actual evapotranspiration
can be found anywhere between the two extrema1
values. The results as obtained in Zambia are representative
for the vegetation of tropical wet and dry highlands. To
obtain a more complete picture, similar experiments should

e performed at least with two high montane vegetational types,
four types of medium altitude forest, two types of swamp forest,
two types of forest savanna mosaic, four types of wooded
savanna, two types of thicket, two types of swamp vegetation
and various types of tropical cultivated areas.
Topography
From flat areas covered by the Brachystegia woodland neithein
surface nor sub-surface runoff has been observed. An occurrence
of flow was observed only from the parts of the catchments
having some pronounced gradient. Mixed vegetation found there
suggest the idea that the vegetational cover is influenced by
the gradient as well. Actual influence of the catchment slope
can be analysed by a comparison of rainfall-runoff relationships
developed for neighbouring catchments of different gradients.
In Fig. 7 there is a family of graphs developed for the
equatorial highland region. Very likely, for dry-wet tropical
highlands the runoff values will be higheri for the same amount
of rainfall, owing to the increased rainfall rates of separate
rainfalls.
Attention should be paid also to the size of the representative/
experimental areas. According to Toebes and Ouryvaev (2) the
recommended size lies between 1 and 250 km2 and rarely exceeds
1000 km2. Frequently the areas less than 100 km2, so called
small catchments, are recommended for experimental catchments,
this being based on the presumption that a certain uniformity
can be guaranteed. As follows from the previous discussion,
the significant factors in the tropical hydrological cycle are
highly variable even in small areas and no catchment is small
enough from the point of view of the uniformity. On the other
hand the extension of data from very small areas is not an easy
task. It can be perhaps concluded that in tropical regions where
only observational network of low density is available, a
catchment of any size can be considered as representative
35

36
providing that a higher accuracy of basic hydrometeorological
data can be obtained from there than from the standard
network Several catchments established within the main area
can increase the amount of information remarkably. A similar
increase can be achieved by the observation of several
neighbouring catchments. Sometimes occurrence of two or more
factors in some extremal forms can produce rather surprising
results. For instance, at one catchment in Kagera basin near
the Tanzanian-Ugandan borders, steep mountains are drainaged
into an extensive swamp. As a result, the runoff coefficient
reaches almost 30%, which is surprisingly high value for the
tropics. Data from such a catchment cannot be applied directly
to the neighbouring basins, however, since a more dense network
has been established there and the effects resulting from the
combination of two extremal factors can be measured and analysed,
the catchment can serve as a representative area as well.
CONCLU SION S
Misleading results can be obtained from direct application of
the data obtained from the experimental catchments in the
tropics. Therefore, whenever possible, data from experimental
and representative catchments should be compared and combined
with the data as obtained from the standard network. Parti-
cularly basic data, such as annual rainfall, runoff and yield
should be compared before any further analysis is carried out.
Any difference between the data as obtained from the catchments
and from the standard network should be fully explained and the
data developed for any cross section within a basin should fit
with the data €or the headwater catchments and for the lowest
observed point as well. Only equal periods of observation
should be used for the comparison, although in some cases it
mean neglecting the long term records. The long term records
however, are to be used later on, together with long term
precipitation records, for the extension of data within a
reg ion.

In table 1 an example of the basic data for the Kafue river
basin is given based on the observation of the experimental
catchments and standard network as well. Conclusions of the
research on the swamp behaviour served as an additional
source of information. Map indicating rivers and swamps is
in Fig. 7 (Headwater swamps are too small and cannot be
traced in the map, however it has been estimated that they
cover at least 10% of the basin headwaters). Once the data
€or hydrologically significant cross sections such as
confluences, swamp inflows and outflows, observed cross
sections etc. have been estimated, the calculation of the
data €or any point within the main basin is easy and more
reasonable.
According to the UNESCO survey and other sources, more than
one hundred representative/experimental catchments have
been established in various parts of the African tropics.
More information can be obtained from them providing that
the materials is collected and analyzed jointly.
More attention, should be paid to the hydrology of tropical
swamps, because they play an important role in tropical
hydrology.
The results available from severa$ experimental catchments
indicate that various hydrological methods currently used
in temperate regisns need to be reviewed, regarding special
conditions existing in tropical catchments.
37

i.
2.
3.
4.
5.
6.
7.
8.
9.
38
--
REFERENCES
--_____-__________-_--_----__- 1971. Discharge of selected
rivers of the world. UNESCO, Paris.
Toebes, C., Ouryvaev, V., 1970. Representative and
experimental basins. UNESCO, Paris.
Balek, J., Perry, J., 1972. Luano catchments, first phace-
final report. National Council for Scientific Research,
TR 28 Zambia.
Balek, J., Perry, J., 1973. Hydrology of seasonally inundated
African headwater swamps, Journal of Hydrology. In print.
Jackson, I.J., 1971. Problems of throughfall and interception
assessment under tropical forest. Journal of Hydrology 12.
Hurst, H.E. Le Nil. Paris.
Wicht, C.L., 1949. Forestry and water supplies in South
Africa. Dept. Agric. S. Afr. Bul. 33, p. 58.
Pereira, H.C. 1962. Hydrological effects of changes in land
use in some East African catchment areas. East Afr. Agric.
Forestry Journal 27.
Maxwell, D., 1972. Root range investigations. National
Council for Scientific Research, TR 26 Zambia.

Y
ri
Lc
O
39

djONnä 1WlNNVjO WU
O m
ul
O
I- O
O

RNNLJQL RFIINFQLL 1966- 67, LUQNO - CQTCHMENTS .-
v Gauging Statlms
Levei at lûûûmm
Fig.3

O
P 8 8 8 a

Fig ,6

ABSTRACT
REGIONAL VALUATION OF IIYUROLOGlCAL INFORMATION
Y. Cormary - J.M. Masson
This information, inadequate by its very nature, is basically obtained as
temporal series of climatic, hydrometric and water level data, of different
durations.
A diagnosis about data coherency and the subsequent possibility of regional
interpolation, might result of different methods.
1.
2.
3.
RESUME
The mere report on a map of the parameters (of probability for instance)
related to classical hydrologic variables. For the 12 months of monthly
variables this yields however too many values. A solution might be the
fitting to the 12 values of a parameter, of a FOURIER of which only the
2 to 4 first coefficients may be conveniently retained for cartography.
Report on a map of the basic stochastic processes parameters. Probability
distribution of many hydrologic var,iables are derived from those proces-
sed and thus depend on the interpolated parameter values. Furthermore
these processes make a better use of the existing information and a gaghg
point. Analysis of coincidences between different recorded series may also
improve the estimation parameters on the short ones.
Principal componentes analysis (mainly on climatic data) which through an
interpolation of covariance matrix, outline some regional tendancies and
a quantitative interpolation at any point.
4. Analysis of variance of regressions between flows and rainfalls for instance
on n different watersheds, in order to obtain the effect or morphological,
geological, vegetation, soil factors and therefore transform
a basically qualitative information into a quantitative one.
Each of these methods has been actually employed on the rivei- Allier (France).
L'information, insuffisante par nature, est représentée essentiellement par
des series chronoligiques de données climatiques, hydrométriques et piézométriques
plus ou moins longues et nombreuses. '
Un diagnostic d'interpolation sur la cohésion des mesures et les possibilités
d'interpolation géographique peut s'effectuer de plusieurs manières différentes:
1. Cartographie des paramètres des lois de distribution de variables hydrologiques
classiques. Pour des variables mensuelles, ceci aboutit à prendre
en compte un nombre trop important de paramètres. Une solution consiste à
ajuster des séries de FOURIER pour représenter l'évolution des paramètres
au cours de l'année. I1 suffit alors de cartbgraphier 2 à 4 coefficients,
les autres étant des constantes caractéristiques de la région.
2. Cartographie des paramètres des processus stochastiques de base, qui permettent
entre autre de retrouver les lois de distribution des variables
hydrologiques, mais décrivent mieux qu'elles la totalité du phénomène
(pluies ou crues) et utilisent mieux la totalité de l'information disponible.
3. L'analyse de la concomitance des phénomènes entre séries de meme nature ou
non permet de mieux estimer les paramètres de ces processus (modele de renouvellement
double).
3. Analyse des composantes principales (sur les données climatiques essentiellement)
qui permet de dégager les quelques tendances régionales predominantes
et permet une interpolation en tout point.
4. Analyse de variance des coefficients de n régressions ajustées entre variables
hydrologiques (pluies et débits par exemple) sur n bassins versant dif
férents pour mettre en évidence l'influence des caractéristiques physiqueset
morphologiques (géologie, végétation, pente, ... ) et prendre en compte de
manière quantitative une information de type naturaliste essentiellement
qualitative.
Chaque méthode a été utilisée au cours d'une étude du bassin versant de la rivi$re
Allier.
-
Yves CORMARY - Ingénieur Agronome - Laboratoire National d'Hydraulique - Professeur
Associé à l'Université des Sciences et Techniques du Languedoc - MONTPELLIER (Fran-
ce).
Jean-Marie MASSON - Inggnieur Agricole - Maître Assistant à l'Université des Scien-
ces et Techniques du Languedoc - MONTPELLIER (France).

48
I - INTRODUCTION
En hydrologie, l'information est constitu6.e esscntiellement
par des mesures de paramètres hydrométriques, climatiques et piézo-
métriques. Ces mesures sont des variables liges à l'espace (l'endroit
uù ia mesure a été effectuée) et au temps (l'époque où eiie a et6
effectuée).
Sur un emplacement donné, la mesure d'un paramètre est
faite de manière continue ou à intervalles de tmps rh~.,iillc.r. 1.a
suite des mesures au même emplacement constitue line séric rhronologique.
Régionalement, au cours des années, les emplacemerts des
points de mesure changent et on se trouve finalement en présence
de séries chronologiques plus ou moins longues et plus WI moins
nombreuses.
Cependant, quand un problhe relatif à l'eau se ;rose -et
le nombre des problèmes qui se posent ne cesse d'?iipeii'-er .avec
le développement économique de nos régions-. I1 n'mistc p:.iti,uc-
ment jamais à 1' emplacement souhaité les infnrmationn nécessaires
pour résoudre le problème, ou tout au moins ces infarnations sont
insuffisantes. Un des moyens de pallier ce manque de donz6es en
quantité suffisante et au bon endroit, consiste à rnobi1is.c.r toute
l'information disponible sur la région environnante.
En France, l'importance de cette valorisation dc l'infor-
mation régionale n'avait pas échappé au conlté "Actior. Concertée
Eau" qui proposait comme sujet d'htude en lu65 d'abord- les
problemes de ressources de maniere scientifique et ri:.iwalc et
de procéder à une synthèse sur un bassin d'assez vaste fiinen!,ion.
Le souhait du comite se concrétisa par un cf'iltïat d'Etude
passé entre la Dflégation GCnérale .? 12 Xcrkerchc Cr i.'u;i:ifique et
Technique (D.G.R. C.T. ) et le tahoratoire F:.:.7ticriai d"1ydrsuliqrie
(E.D.F.). Le bassin choisi fut celui de 1'ALLT.IR (14 O00 km2j et
parmi les sujets étudiés, on relevait : "ia mise sur pied d'iriie
étude des lois de distribution à l'6chelle régionale, permettant
de créer de nouveaux nodoles dc repr6sentatinn, d'extrnpoler
l'inforination et de déterminer une hiérarchie de l'iii:>:-&t dc
chaque station".

L'étude effectuée par le Groupe Hydrologie du L.N.H. et
la Faculté des Sciences de Montpellier a fait l'objet d'une publication
de synthese sous forme d'un atlas cartonné et illustré intitulé :
"Méthodes d'.études régionales des reesources en eau".
Nous développons ici quelques pages de cette publication,
pages consacrées des méthodes d'analyses régionale3 de l'information.
-
11.- INTERPOLATION REGIOXALE DE L'INFORXATION ET ANALYSE DE LA COHEREK'CE
SPAT1 ALE.
i 1 - 1 - pf~se_elcom~qe_be_lljnooimsq~oo_Ear c arto9laE-~-_régionale -
A/- Au moyen de processus s tzchastiques simples.
Une série chronologique sur une station, par exemple
la série des observatiors journalières des précipita,tions ou
des débits, présente une structure bien particulière qu'on
peut représenter au moyen d'un processus stochastique.
Le type de processus le mieux adapté à beaucoup de
phénomènes est un processus de r~nouvellement.
4
hrru t eu r
des pluies y1
T1
Tt
y3
49
temps
Pour tenir c'ompte des variations saisonnières, on dé-
coupe l'année en périodes OU le processus est à peu près sta-
tionnaire : les Ti successifs, durées entre les événements,
doivent &tre indépendants et suivre la même loi de probabi-
lité de densité f(T). Les Yi successifs doivent être égale-
ment indépendants et suivre la même loi de probabilité de
densité g(y).

50
I1 s'agit donc de trouver ces lois de probabilité des variab!es
fondamentales T et Y et d'estimer leurs paramètres, ce qui nous ramène
aux méthodes classiques de la statistique avec cependant ces différenres
- On mobilise toute l'information. Ainsi pour la pluie, on
tient compte aussi bien des jours secs que des jours pluvieux
- Ces variables fondamentales sont observées en grand riombre,
ce qui rend leur analyse statistique beaucoup plus valable que l'ana-
lyse de variables déduit:., fonction souvent compliquée des variables
fondamentales (les pluies ou les débits maximums par exemple).
. Un modèle a été ainsi construit pour représenter la sixcessiun
des pluies journalières à une station.
I1 suppose que la hauteur d'une pluie journalière est la SOIIIIIIP
d'averses fictives se produisant à des instants aléatoires et rpportant
des quantités de pluie aléatoires. Si le nombre des averses par jour
pluvieux suit une loi de POISSON et que les hauteurs des averses
fictives suivent une loi exponentielle, les hauteurs de pluie journa-
lière doivent suivre une loi des fuites (loi I G ama zéro), ce qui
est bien vérifié siir les stations de l'ALLIER. On peut alors estimer
2 paramètres du processus.
p = hauteur moyenne des averses élémentaires fictives
= nombre moyen d'averses élénientaires fictives par jour de
pluie.
c
Ceci, à partir de la moyenne P 24 et de la variance b224
des pluies de 24 heures non nulles.
p=--- *24
L
P24
P=
-
2 P 242
/I 224
Les deux autres paramètres du modèle sont :
TI
T2
durée moyenne des episodes secs
durée moyenne des épisodes pluvieux.
qui permettent de connaître en terme de probabilité ilétat sec ou
pluvieux d'un jour connaissant l'état du jour précédent ; on a en effet
G ér if i é le caractère markovien de la matrice de transition des états.
On a :

La cartographie des différents paramètres sur l'ensemble
des stations du bassin de l'Allier s'crganise bien et permet l'in-
terpolation des parametres du modèle pour n'importe quel point du
bassin, ainsi que le montrent les graphiques ci-joints concernant
le mois d'octobre.
D'une manière généraìe.on remarque :
- l'influence atlantique (Ti court, T2 long,/U fort, faible).
- l'influence méditerranéenne IT, long, T2 court,,U faible,/=, fort).
Le processus peut s'appliquer aussi aux crues et aux t'empé-
ratures moyennant la fixation d'un seuil.
E/- .Au moyen de processus stochastiques associés.
Exposé simplifié du modèle associant crues et pluies (;enou-
vellement double).
Chaque processus est défini B partir de la. distribution des durées
ectre événements H1 et de celles des grandeurs des événements
(z, 1 -
Par convention, il y a un événement "crue" ou "pluie" quand
la variable dépasse un seuil choisi pour que, en particulier, les
caractéristiques de grandeur (volume total, valeur de pointe) enregistrées
pendant que la variable est au-dessus du seuil, et les caractéristiques
de distributibn dans le temps (durée séparant 2 év6nements
homologues) ne soient pas autocorrélées niais que les 2 6x76nements
pluies-débits soient aussi fréquemment associés que possible.
- pour certains événements (Xi et X2) il y a concomitance entre 1
et II (Pi cas). Sur les grandeurs (zl et Z2) est évaluée la corréletion
pluies crues
I
x2
S
4
x2 s
4
51

52
- pour d'autres événements (X 3, P3 cas et X4, P4 cas) ii n'y a
pas concomitance. Ceci s'explique entre autre chose par le fait
que nous sommes obligés de définir un seuil de dépassement cons-
tant sur chacune des deux variables pluies et débits quelle que
soit la saison ou la saturation du sol.
- Marche h suivre :
Sur une courte période T1 on évalue à la f ois les paramètres
du rencuvellement simple simultané sur chacun des deux processus, on
estime égzlement la corrélation existant entre les grsideurs hydrométriques
-.t les grandeurs pluviométriques (Zl, Z2) soit & . Sur la
série longue (Ti + T,) on estime les paramètres du proce;ci;s pluies
L
seul.
Soit ûT1 un paramètre donné du processus crues, moyenne du
nombre annuel de crucs par pérjrJde, des débits maximums, ou des voiumes
de crue,évalué sur une série courte de T1 et accompagné de
sa variance d'érhantillocnage, soit Var (8 ). La prise en consid&-
T1
ration du procescus pluies (sur T, + Tz) nous permet alors, d'évaluer
de nouvelles estimations des paramètres du processus crues ;
estimations dites améliorées, qui sont accompagndes de nouvelles Trariances
d'échantillonnage plus réduites appliquhcs da.is la théorie
du renouvellement.
D'autre part, ccs améliorations seront d'autant plus effica-
ceti que les proportions :
a =
seront plus fortes.
+ +
P1
et b =
P1
p3 pl p4 pl
Le résultat final rend compte de la superpositioii du proces-
sus X, (Z,) et X (Z ) dans les lois déduites sur les crues
3 3
C/- Autres .néthodes d'analyse de la cohérence temporelle des séries chro-
nologiques.
Dans certains cas, il suffit de mettre seulement en évidence
la simultanéité des événements : l'association statistique des épi-
sodes pluvieux h deux stations ou celle des crues et épisodes plu-
vieux sur un bassin. Dans d'autres, les deux séries de données peu-
vent être considérées comme respectivement les entrées et les SOT-
ties d'un système de transformation dont on cherche à identifier à
la fois la structure et les paramètres (transformation des pluies
en débits sur un bassin restreint, des débits amont en débits aval
sur.un bief, de la pluie en variations d'un écoulement issu de nap-

pes). Suivant la nature des hypothèses (ou des connaissances) le
linéarité, l'invariance dans le temps ou suivant le niveau des
entrées du système, la solution est facile, difficile ou impossi-
ble.
Si le système de transformation n'a pas ou ne prétend pas
avoir de signification physique il suffit d? détzrminer la fonc-
tion (dite boîte noire) de transformation la plus efficace. Les
transformations de Laplace, de Fourier, etc. répondent à ce pro-
blème de même que les études dc type 'diener sur les fonctions
aléatoires (en utilisant les estimations des fonctions de corré-
lation et d'nutocorrélation sur plusieurs couples, entráes-softies.
L'analyse spectrale a aussi conme intérêt d'expliciter mieux que
les procédés classiques la structure des corrélations et l'effica-
cité d'un échantillonnage de mesures un pas de temps déterminé.
Le calcul automatique permet essentieliement les raìciils ma-
triciels, le tirage au hasard (simulation) et la répétition infinie
des tatonnements fastidieux. Les applications en sont :a généralisa-
tion des calciils d'amélioration des stations cuuztes en fonction de
plusieurs stations longues, i'utilisation des techniques de l'analy-
se factorielle pour étudier les liaisons entre variables ou entre
groupe de variables. Ce qui met an évidence les redondances entre
variables ou entre mesures et permet de retenir les plus significa-
tives (analyse discriminante ou canonique.
Ces méthodes élémentaires en dehors des services apprécia-
bles qu'elles rendent par elles-mêmes font parties intégrantes de
méthodes plus élaborées que l'ordinateur permet de maîtriser et sur-
tout d'appliquer à un ensemble important de données hydrornétéorolo-
giques (plusieurs postes, plusieurs variables).
11.2- F.lodele rigional statistique -___-_- _--_ des pluies _-___-_-_____-_-
mensuelles.
I1 s'agit d'expliciter la cohérence spatiale des précipita-
tions mensuelles, afin, par exemple d'optimiser le réseau de mesures.
Sur l'Allier 30 stat'ions avaient fonctionné pendant 40 ans simultané-
ment. Une telle information peut facilement se condenser sous la forme
d'un tableau carré 30 x 30 dont chaque élément m i j représente soit
la variance, soit la covariance, entre la station i et la station j.
Une méthode dite "des composantes principales'' permet par
un opérateur linéaire matriciel A d'élhments a i de passer des 30 va-
riables aléatoires initiales X (pluviométrie des 40 années successives)
à 30 variables aléatoires Y indépendantes, dont la matrice de corréla-
tion est ,cette fois-ci diagonale. C'est-à-dire ne comportant que des zé-
ros pour les covariances. De plus, cette matrice rassemble dans les tous
premiers texnies dc la diagoiinlc toutr la variation coiiteniie dans 1 'in-
formation ini tiaie. En revcnant dans 1 'espace initial chaque variable
53

54
aléatoire X de départ peut s'écrire sous la forme d'une combinaison
linéaire des 3 ou 4 composantes les plus importantes, dites principa-
les.
Les coefficients de ces combinaisons linéaires indiquent
la part prise par chaque station L'la constitution de chacun de ces 3
ou 4 facteurs essentiels et indépendants. D'autre part chaque station
est pour chaque mois caractérisée par un petit nombre de paramètres
que l'on peut cartographier de manière cohérente.
x =
de réa-
lisa-
tioris
Notons m..
Xl
m.
= variance (Xi)
= covariance (X.,X.)
N = 30
= 40
ij 1 J
X 1,'
On peut rassembler ces mii, m. en une i.iatrice de covarian-
1j
ce c'est-à-dire un tableau carré
mil' m12' m13 ............................. m
1N
m2,, mZ2 ....................................
............................. m.. ............
Ji
..............................................
%i .......................................... m
N,N
Cette matrice symétrique (m.. = m..) comporte des éléments
a
trop nombreux pour en dégager les partiidlarii!es, et on va tenter par
une transformation de les réduire.
Puisque les dernières composantes principales Y sont assi-
milables B des constantes, on peut écrire grâce aux propriétés de la
matrice d'éléments a. (matrice unitaire dont l'inverse égale la trans-
pos6e) : ij
3
k
xi = aij
yj + c

où C est en particulier une constante ùe centrage tenant compte en moyen-
ne des composantes négligées, K étant le nombre de composantes principa-
les retenues.
Les Y. n'étant pas corréìés, (3) signifie que ia pluie men-
suelle Xi a la station i est une combinaison linéaire i.e, une somme
pondbrée d'effets Y. non corrélés, ce qui suggère que ces effets sont
ceux des régimes climatiques ind6pendants et dominants sur le bassin.
Pour caractériser fl, on reporte sur-la carte du bassin aux
N stations longues, les valeurs des coefficirnts a. ... On cons-
11
fate que ces valeurs s'organisent, présentent une direction systématique
de variation, perturbée, ce qui Pst normal, par le relief du bassin. On
procdde. de meme pour Y P4,.puisque dans 1,'Allier ces 4 composantes
occupent 80 à 90 $2iey?i variance totale.
Les cartes obtenues suggèrent qu'on assimile Y1, quiprend k
lu: seul 70 % de la variation, aux influences climatiques dominantes ve-
nant du Nord-Ouest. Y,, avec 10 & 15 $ de la variation et un gradient des
courbes orients Sud-XÕrd, est assimilable aii climat méditerranGen-
Enfin Y et Y reprbsenteraient les influences continentales
3 4
assez importantes dans les vb!lées de l'Allier.
Ces transformations permettent :
1.- de reconstituer les pluies aux points sans mesures. Les valeurs des
Coefficients a; . comme nous 1 'avons vu peuvent s'interpoler Yynopti-
quement. I1 estJalors possible de procéàer d'abord pour chaque année
et à l'aide de la pluviométrie des stations longues au calcul de la
réalisation des quatre ou cinq composantes principales pour chaque
mois. Ensuite, les coefficients 8. lus sur les K cartes permettent
14 .
inversement de calculer la pluviom trie en un point quelconque à par-
tir des K (quatre ou cinq) composantes précedenment calculées.
La théorie permet d'expliciter les erreurs résiduelles dues
au fait qu'on se limite aux quatre ou cinq composantes qui expliquent
80 à 90 % de la variance totale.
2.- d'estimer les corrélations entre les pluies mensuelles de deux points
quelconques du bnssin. On calcule d'abord les variances et covariances
pour ces deux points à partir des coefficients aij et de la variance
des composantes principales (valeurs propres) correspondantes- Ce cal-
cul permet celui de corrélation et débouche sur des indications objec-
tives concernant la gestion du reseau (puisqu'on pcut déterminer a la
fois l'information ajoutée par chaque poste & la connaissance de la
lame d'eau et l'étendue ,de la "zone d'influence'' de ce poste).
3.- Le calcul de la loi de probabilité de la lame d'eau sur une surface
quelconque (bassin versant) puisque l'on connaît la pluviométrie dr
toutes les iascs 6I6iiiPntaires quc l'on pcut dbcouper dans le bassin
versant eii mPme teinpc que leur corrélation.
55

56
4.- calciil de la loi de probabilité d'une sécheresse simultanée à plu-
sieurs stat

L'autocorrélation des ddbits (et des pluies) s'apprécie
sur l'ensemble des stations et conduit à ui,e fonction annuelle lissée.
La simulation se fait par un processus de Markov d'ordre 1
compte tenu de la matrice des intercorrélations, les erreurs étant ti-
idpc dans des lois normales centrées. Les moyennes, écarts types pour
chaque station et chaque mois se déduisent de la cartographie des quel-
ques paramètres du lissage précédent.
Lorsqu'il s'agit de simuler en tenant compte des séries
historiques de pluies, il faut pallier la faible autocorrélation des
pluies : un tirage des erreurs ayant une forte corrélation avec les
débits générés au mois précédent est substitué au tirage au hasard.
III.- MODELES KEGIOSAL'X AShLYTIQL"o8.
m.1- A l'échelle - - annuelle.
-___
Sur le bassin versant de l'Allier et à condition de consi.dé-
rer la meme période de temps, les débits annuels D sont linéairement
liés aux précipitations annuelles P (23 bussins étudids). Les cocffi-
cients de r6gression et les ordonnies 9 l'origice ont des val:%urs
qui dépendent des caractéristiques physiqucc des bassins. Les carac-
téristiques qui ont le plus d'influence sont : la pente moyenne et
le pourcentage de terrains incultes.
Ces variables ont un? signification discutable dans la me-
sure OU elles en intkgrent beaucoup d'autres (géologie, altitude,
etc.).
Une analyse de variance - effectuhr en fonctioii de 2 ou 3
modalités des deux caractéristiques prépondéran-tes - nous a per-
mis de choisir statistiquement les coePfi.cients de régression et
l'ordonnt5e à l'origine a retenir suivant les modalitós qui slave-
rent reprisenter des cas différents.
- Résultats de l'analyse de variaLice sur IPS relations
pluie-débit 1'6ciielle aniiuellr.
30
7 30
I

58
Ces liaisons peuvent être utilisges pour allonger des séries
ou mieux pour obtenir les paramètres des lois de distribution des
débits annuels. Sur un bassin supposé ne comporter aucune mesure,
on retrouve ìn. moyenne à 1 6 près mais on sous-estime ia vnriance
de 40 5.
m.2- A l'échelle de la crue.
~
Une crue peut être assimilée à un volume modulé dans le temps.
- Le rrridement de la pluie conditionne le volume R. Sur l'Allier, le
meilleur type de liaison trouvé entre R et la pluie totale P est
U = a.Fb. avec Q débit avant ia crue ; b, c et a sont des
coefficirnts dont la'valeur varie d'un bassin à l'autre et peut
etre reliée aux caractéristiques physiques et morphologiques.
Les valeurs de a et b dépendent surtout de carartéristiques
morphologiques et physiques : s'. de forêts, de labours, géologie,
surface... C'ne analyse de variance faite eri fonction de plusieurs
modalités de 2 de ces caractéristiques, permet de déterminer les
coefficients iì prendre en considération.
- La modulation dans le temps peut être étudiée moyennant une hypo-
thèse de linéaiitl psr la théorie ¿!e l'hydrogramme unitaire.
Sur l'Allier, les hydrogrammes unitaires trouvés sont commo-
dément représentés par l'équation d'une courbe (Pearson III) qui est
définie grâce à deux paramètres o( et K qui sont estimés à partir
des moments H, et M2.
Ces valeurs, différentes d'un bassin à l'autre peuvent être
reliées par analyse de variance à des caractéristiques morphologi-
ques telles que : la longueur et la pente du "rectangle équivalent",
le pourcentage de la surface du bassin occupée par les gneiss, l'hyp-
Sométrie et la surface du bassin versant.
Sur un bassin non jaugé, à partir des caractéristiques physi-
ques il est donc théoriquement possible, pour une pluie donnde, ales-
timer le rendement et la modulation de la crue correspondmte.
Les mêmes modalités d'approche permettent de relier les pa-
ramètres des corrélations des débits minimums annuels de 30 j (Q30
en l/s/km2) avec un facteur climatique (calculé à partir de P et
1IE:T.P.) à la géologie et Èi la surface des bassin.

IV.- CO?;CLUSION.
Ces dernières méthodes en cours de développe'ment, rejoi-
gnent la théorie du contrôle qui est aussi une des bases de l'opti-
misation économique. Ceci contribue à créer un outil et un langage
commun aux économistes et RUX hydrologues. En meme temps, se fait
jour, chez ces derniers en particulier, le souci d'expliciter "la
valeur ajoutée" non seulement de leurs méthodes (ou modèles) mais
aussi de leurs mesures et même de l'organisation de celles-ci (ré-
seaux). Cette valeur ne peut s'expliciter qu'à travers une intégra-
tion au plan des décisions économiques, intégration qui met en jeu
d'autres variables beaucoup plus mal connues que la variable hydro-
logique.
L'exemple actuel le plus préoccupant qui peut concréti-
ser ce problème de l'élaboration de l'information pour SR mobilisa-
tion en vue d'un objectif prxcis, est bien entendu celui de la pol-
liition. Dans ce domaine il est clair que l'emploi d'un modèle quel
qu'il soit suppose dès le départ une méthode d'acquisition des don-
nées conçiies.en fonction du modèle. I1 ne peut être seulement ques-
tion d'utiliser l'information statistique issue d'un paramètre iso-
lé à la significat.ion très fluctuante dans le temps et suivant la
valeur d'zutres paramètres et dérivan$ au cours des années sous l'in-
fluence des progrès de l'industrie. Une exploration préalable, par
simulation sur le modèle' envisagg devrait permettre de définir cet-
te stratégie d'acquisition des données. Celui-ci permet d'explorer
les conséquences en particulier biologiques et écbnomiques des va-
riations de tel ou tel facteur dont l'homne a la maîtrise (soutien
des étiages ou modifications de la charge polluante).
C'est-à-dire l'évolution rapide vers l'intégration et
l'interprétation, dans le domaine de l'eau ,des diverses discipli-
nes axées sur l'étude spécifique soit des ressources superficielles
ou souterraines, soit de la pollution, soit des besoins oil des pro-
blèmes économiques. Ces domaines sont encore assez séparés et il en
résulte un effort d'adaptation permanent.
t *
t
Cette note evoque divers points développés dans une publication de
synthèse du Laboratoire National d'Hydraulique et du Laboratoire
d'Hydrologie de Montpellier, éditée sous l'égide de la Délégation
Générale à la Recherche Scientifique et Technique, ouvrage de 133 pages
intitulé "Méthodes d'gtude régionale des ressources ell eau. Application
au bassin dè l'Allier'', dont les auteurs principaux sont MM. CORMARY,
BERNIER, MASSON, LOBERT, DAUTY, SAUCEROTTE, etc... Cette publication
synthétise un bon nombre d'6tudes méthodologiques dont le but est la
valorisation régionale de l'information.
59

ABSTRACT
LE TRANSFERT D'INFORMATION HYDROLOGIQUE
A DES BASSINS VERSANTS NON OBSERVES
Par
Pierre DUBREUI L*
The lack of sufficient hydrological datas is generally more
important in the basins of small area and located in poorly
developed countries. To estimate the water resources in such
basins, we have to do a transfer of information from "similar
basins" for which we have enough datas. This transfer may be by
analogy when the regional density of hydrological information is
too slight; that's made up by a qualitative analysis of the geo-
morphological factors, which are similar or not, and of their
influence on water resources, between the project basin and the
similar ones. When the regional density of hydrological datas is
higher -old hydrometric network and/or numerous representatives
basins- the transfer will be easier, using stochastic relations
between dependent hydrological variables and explicative variables
fo the physical environment; practically, in this case, we can
establish and utilize regional graphs and norms. Some practical
examples show the possibilities and limits of the two methods of
transver.
--
RE S UME
L'abscence de données hydrologiques suffisantes est d'autant
plus aiguë que les bassins versants sont de faible superficie et
situés dans des contrées peu développées. L'estimation des res-
sources en eau sur de tels bassins exige un transfert d'informa-
tion depuis des bassins de comparaison oh l'on possède des don-
nées. Ce transfert peut être analogique lorsque la densité réeio
nale d'information hydrologique est faible; il consiste en une
analyse qualitative des éléments géomorpholofiques comparables
ou dissemblables et de leurs effets sur les ressources en eau
entre bassin du projet et bassins de comparaison. Lorsque la den
sité régionale d'information hydrologique est élevée -réseau hy-
drométrique ancien et/ou nombreux bassins représentatifs- le
transfert fait appel aux liaisons stochastiques entre variables
hydrologiques dépendantes et variables du milieu physique expli-
catives et se matérialise par des normes ou abaques régionaux.
Des exemples précis et utilisés des deux méthodes de transfert
illustrent leurs possibiiitês et leurs limites respectives.
* Chef du Département de la Recherche Appliquée-au Service Hydro
logique de 1'O.R.S.T.O.M. - France.
-

62
Les pays dans lesquels il y a encore de nos jours absence d'infom-
tion hydrologique sont en quarstité de plus en plus réduite.
L'estinmtion des ressources en eau ne peut y etre faite, a priori,
que d'une manière grossière en procédant par analogie avec d'autres pays dotés
eux d'information hydrologique ; ce transfert analogique d'information est
évidement beaucoup moins stir que celui auquel on peut procéder dans un pays
ou une région non dénué d'infomation hydrologique, la méthodologie restant la.
m& come on le verra plus loin.
Mis 3. part ces exceptions, la p1upal-t des pays disposent d'informations
hydrologiques fournies soit pr les réseaux hydrométriques, soit par les
bassins représentatifs. Ces infornations hydrologiques,quelle que soit la
densité des dispositifs de mesures,ne concernent qu'un certain nombre de bassins
versants. I1 y a toujours des bassins versants non observés, même dans les pays
dotés d'excellents réseaux de mesures. Or, les besoins de connaissance de la
ressource en eau se posent aussi bien pour les bassins des réseaux que pour les
bassins non observés. En effet, les réseaux de mesures ont été généralenient miis
en place peu à peu au cours de l'histoire, l'implantation des stations s'effectuant
en considération des besoins médiats. Dans certains cas, une planification
préalable de l'implantation des stations du réseau a pu ¿?tre réalisée en
tenant compte de certains objectifs à moyen terme de l'utilisation des eaux.
Malgré tout, la croissance des besoips en eau est telle,au cours des années
présentes de la seconde moitié du erne siècle,que les ressources en eau sont
recherchées là OU, il y a vingt ou trente ans, il paraissait ne pas y avoir de
problème et oh, par conséquent, aucune station de mesures ne fut implantée.
On peut donc dire aujourd'hui que l'hydrologue doit partager son
temps entre l'analyse des informations collectées sur les bassins observés et
l'estimation des mems informations sur les bassins non observés.
Si le problème de cette estimation se pose sur un grand cours d'eau
drainant un bassin de superficie importante, il est à peu p&s certain que l'on
trouve en amont et en aval du lieu d'estimation - c'est-à-dire du site d'un
projet d'aménagemnt hydraulique - une station d'observation. Dans ces condi-
tions, le transfert d'analogie est facile puisque les caractéristiques hydrolo-
giques du lieu d'estination sont comprises entre celles des stations d'observa-
tions dont elles diffèrent d'ailleurs assez peu.
La majorité des problèmes d'estjmation se posentpour des bassins non
observés c'est-à-dire pour des bassins versants de superficie faible à modérée
sur lesquelles n'existent aucune station de mesure. La résolution de ces
problemes exige le recours à l'information disponible dans des bassins voisins
de la &me région climatique. Le transfert d'information repose sur le postulat
selon lequel deux bassins auront des caractéristiques hydrologiques identiques
si leur milieu physico-climatique - leur environnemnt - est le &m.

Le problème consiste donc à analyser ce milieu physico-cbtique, en dégager
les paramètres susceptibles d'influencer les caractères hydrologiques afin de
mettre en évidence le r8le de ce milieu sur lesdits caractères.
Si l'on dispose, dans une région climatique homogène, d'une informa-
tion hydrologique abondante et de bonne qualité, ia méthode de transfert
consiste à utiliser un ensemble de liaisons numériques ou graphiques établies
entre variables hydrologiques et paramètres de l'environnement.
Si l'information hydrologique régionale est insuffisante ou si
l'ensemble précédent de liaisons l'hydmlogie-milieult n'a pas été élaboré, la
méthode de transfert est purement analogique et qualitative puisqu'elle ne
peut estkr les caractères hydrologiques du lieu d'estimation que par analogie
avec ceux du ou des bassins observés ayant l'environnement le plus comparable
avec celui du bassin non observé.
On examine successivement ces deux méthodes de transfert de l'informa-
tion hydrologique en s'appuyant sur des exemples concrets.
1. Relations entre variables hydrologiques et paramètres de l'environnement
Sur un plan général, le problème consiste en l'établissement de
relations entre des variables hydrologiques V1, V2.. . définies, a priori, et
certains paramètres Pl, P2". P du milieu physico-climatique, de la forme
n
V1 = f (Pl, P2...P
k ) de telle sorte que l'écart résiduel soit minimal.
Le problème n'est pas nouveau. Déjà au début du neme siècle, l'hydrologie
considérée aujourd'hui com classique avait abordé le problem en
élaborant diverses formules d'écoulement ou explicatives de variables hydrologiques.
La littérature consacrée à ces formules est abondante ; on en trouve
un bon catalogue dans l'ouvrage de G. REMENIERAS rl] y
011 peut citer :
a) les formules donnant le déficit d'6coulement annuel moyen en
fonction des précipitations et de La température annuelles moyennes comme celle,;
Cie COUTACa\IE et TURC ou celle de THOFCNTHWAITE prenant en considération le bilan
mensuel entm pluie et évapotranspiration.
b) les formules donnant h s caractéristiques de l'hydrogram unitaire
de cyue - temps de réponse en fonction de la longueur du bassin, débit de pointa
en fonction de la surface, de la durée de la pluie et de l'état du bassin ...
etc . - come celles de SEDER établies dans la région des Appalaches aim
U.S.A.
63

64
c) les formules donnant le débit ma-1 d'une cme de fréquence
choisie, soit établie de manière rationnelle c om celle de CAQUOT
(Q = KI? Cn A', K fonction de la fréquence, C coefficient de ruissellemnt,
I pente et A surface du bassin), soit établies expérimentalement come celles
des italiens GIEWEUI, KENTURA... etc ... qui reliaient débit et surface
de bassin, temps de concentration de l'écoulenient et surface et pente du bassin.
L'utilisation abusive de ces formules a conduit à de nombreux déboires.
I1 faut, en effet, considérer qu'elles ont été établies à partir de données
expérimentales en quantité limitée et en provenance d'une certaine région et
qu'il était illogique de les appliquer à des bassins situés dans des régions
d'environnemnt différent. En outre, les paramètres explicatifs pris en compte
étaient peu nombreux et pour certaines uniquemnt du domaine climatique ; pzr
conséquent, leur application ne pouvait donner que des résulta-ts d'autant plus
erronés que les particularités de milieu étaient importantes.
On admt aujourd'hui que le domine d'utilisation de ces formles
doit @tre limité à la région de laquelle proviennent les données expérimentales
ayant contribué à leur élaboration OU à des régions d'environnement comparables.
I1 est, en effet, évident que si la liaison proposée est de la forme
V = f (Pl, P2.0ePk)J c'est que les paramètres du milieu P à P ne sont pas
1 kS1 n
influents sur V mais c'est aussi à l'inverse que &.-dite liaison n'est utilisa-
1
ble que dans une région OU les valeurs de P à
k+l
P
n
ne sont pas différentes de
celles de la région d'élaboration de laAite liaison.
Au cours de la seconde moitié du erne siècle, les mesures hydrométriques
ont été intensémnt développées tandis que, parallèlemnt, les utilisateurs
des eau mnifestaient des exigences croissantes quant à la connaissance de la
ressource disponible - précision accrue, diversification des variables -o
Les relations régionales entre variables hydrologiques et paramètres
du milieu doivent de nos jours etre établies à partir de toute l'infomation
disponible critiquée et s'appuyer sur une analyse poussée du milieu.
Rassembler, analyser et critiquer 1' information hydrologique régionale
disponible est aujourd'hui une opération longue et délicate. A l'O.R.S.T.O.M.,
la mise au point d'une monographie de grand bassin hydrographique demnde 4 à 5
ans (SENEGAL, NIGER, CHARI.. .) , la synthèse de quelques 200 bassins représenta-
tifs demnde encore plus de temps. A partir du moment oh lqon exige une bonne
précision des relations hydrologie-milieu, l'analyse critique de consistance
des données est indispensable quelle qu'en soit la durée ou la complexité. C'est
peu pourquoi ces relations régionales, tant attendues par les planificateurs
et les utilisateurs de la ressource en eau, ne voient le jour que très lentemnt,
beaucoup plus lentement que les formules précédemment évoquées.

Ceci est d'autant plus regrettable qu'en l'absence de telles relations, l'utili-
sateur est amené, pour chaque projet, à. solliciter l'avis de l'hydrologue qui
se trouve contraint d'opérer au coup par coup par simple transfert analogique
dont la précision des résultats est moindre. I1 paraft urgent qu'un effort
prioritaire soit décidé en vue de l'établissenient rapide de ces relations
régionales dans tous les pays possédant déjà une information suffisante.
L'0.R.S.T.O.M. a concentr6 une partie de ses activités sur cet objec-
tif au cours des dix dernières années.
Dès 1965, C. AWRAY et J. RODIEX [2] établissaient un ensemble de
graphiques permettant l'esthtion des crues décennales à l'issue de bassins
versants de 2 à 200 km2 en Afrique occidentale intertropicale, à partir de
l'information collectée sur quelque 60 bassins représentatifs exploités de 1
à 5 ans.
Le tableau suivant décrit sommairemnt le contenu de ces graphiques.
: Variable expliquée : Fonction : Paramètre explicatif :
.--------------------:------------------:---------------------.
: decemale : précipitation
: Coefficient de missel-: décroissante : logarithme de la
: lenient : surface
Hauteur de l'averse croissante : Hauteur annuelle de :
: Temps de montée, temps croissante 11
de base et coefficient :
: de forme de l'hydro- :
' gram.
Le miLieu physique était pris en compte par l'intermédiaire de
groupes climatiques (subdésertique à végétation steppique, tropical à végétation
de savane plus ou mohs arbode, équatorial à végétation forestière) à
l'intérieur de chacun desque1 étaient constitués des sous-groupes homogènes
de relief et perméabilité, ces deux paramètres étant d6finis par rangement en
classes arbitraires d'aptitude croissante RI à R6, Pl à P5. Ainsi rien que
pour le coefficient de ruissellement décennal y avait-il près de 25 relations
graphiques pour les seuls groupes de climats subdésertique et tropical.
65

66
Cette synthèse est actuellenient en cours de révision et d'extension
à partir des infomtions collectées sur plus de 200 bassins représentatifs,
en essayant d'expliciter numériquement les liaisons graphiques et en intmdui-
sant tous les pardtres du milieu par le biais de régressions multiples ou de
composantes principales. Le problème du choix des variables et de l'interdépen-
dance des paramètres du milieu a nécessité des études préalables [3] .
On peut également mentionnéfdeux autres exemples de synthèses régionales
élaborées à partir d'informations issues cette fois des réseaux hydrométriques,
après mise en forme de cefis-ci dans des monographies de bassins. Ces
synthèses, ayant conduit B des noms hydrologiques pour aménagements hydrauliques
régionaux, ont été réalisées en collaboration avec J. HERBAUD et G. GIRARD
[4,5] , l'une au CESLRA état du nord-est du BRESIL, l'autre en ALSACE (France).
Elles concernaient pour l'une tous les bassins de 100 à 10.000 h2, pour l'autre
tous ceux de 15 à 3.000 km2.
Une quantification aussi accentuée que possible a été effectuée pour
la prise en compte des paramètres du milieu, ce qui a permis d'établir des
abaques à plusieurs paramètres sans que l'on ait systématiquement numériser les
liaisons.
Le tableau suivant donne une vision globale des liaisons établies,
le paramètre explicatif principal figurant toujours avant les paramètres
secondaires en corrigeant l'effet e
Les domines d'application de ces deux ensembles de liaisons régiona-
les sont évidement très différents. Celui du Jaguaribe concerne un climat
tropical austral semi-aride, à 600-1000 mn de pluie et terrains cristallins ou
gréseux sous savane arbustive plus ou moins défrichée. Celui d'Alsace correspond
au climat tempéré semi-continental B hiver net, avec 800 & 2500 m de pluie
(effet modéré de la neige de 1000 à 1800 m d'altitude) sur terrains cristallins
ou gréseux sous cultures ou foflts à conifères dominants.
On constate cependant certaines similitudes dans les paramètres
explicatifs principaux (surface drainée, hauteur annuelle de pluie) des principales
variables (écoulement annuel et crue décennale) ce qui rejoint et confirm
globalement l'orientation prise par les awburs de formules. Mais les influences
secondaires du milieu sont assez spécifiques : r8le de la pente et de la for&
en Alsace, de la nature géologique du sous-sol au Brésil. Enfin, les coefficients
ùes équations de liaison sont également spécifiques d'un domaine d'application.
Alors que les formules appliquées sans discernement peuvent conduire
L des estimations erronées de 100 et 200 $, l'utilisation des ensembles de
liaisons régionales %ydrologie-enviromeIilenttr assure une précision de 20 à
50 % dans les résultats.

: Variable expliquée : Paramètres explicatifs : Forme de la liaison :
: A - ALSACE
1. Ecoulement moyen
annuel
1.1. Hauteur annuelle de:
précipitations
1.2. Taux de forets :
2. Ecart-type de . ' 2.1. Surface du bassin 1
11 écoulement S
annue 1
4
3. Débit spécifique : 3.1. Surface S
-1 de crue : 3.2. Hauteur annuelle de:
décennale Q precipitation
3.3. Taux de for&
4. Rapport des 4.1. Surface
pointes de crue i
centennale et
décennale
5. Part de l'écoule- : 5.1. Hauteur annuelle de:
ment d'été dans : précipitation
1' écoulement
annue 1
: 5.2. Taux de for€%
67
linéaire croissante :
linéaire croissante
se400 h2 : linéaire i
décroissante :
S>~OC b2 : linéaire
constante :
4,33
Q = 1950. S
linéaire croissante
linéaire décroissante :
en dessous d'un certain:
seuil d'indice de pente:
croissante
linéaire croissante
(liaison diff &ente sur:
terrains cristallins et:
sédimentaires)
linéaire croissante si :
s>75 km2
: B - JAGUARIBE
1. Ecoulement moyen I 1.1. Surface du bassin I L : A S-Oj1'
annue 1 S
i 1.2. Hauteur annuelle dei A = k P" avec n>l
pr6cipitation P :
1.3. Taux de terrains Linéaire décroissante
sédimentaires
(gres>
: 1.4. Degré de défriche- : linéaire croissante :
ment
i

68
2. Variabilité de
l'écoulement
(rapport K entre
une fréquence
donnée et la
moyenne )
3. Débit maxjml
spécifique de
crue décennale Q
4. Rapport de pointe
entre crue
annuelle et
décennale
: 2.1. Surface
: 2.2. Ecoulement moyen
: 3.1. Surface S
:Q=BS
: 3.2. Hauteur annuelle de: B croSt linéairement
précipitation P : avecP
: 3.3. Taux de terrains : linéaire décroissante
sédimentaires
(gres>
: 3.4. Fornie du chevelu : effet croissant si
: radial, décroissant si
: Ilen adte"
: 4.1. Surface
: croissante
croissante
4,484
: croissante
La généralisation de synthèses régionales de ce type pmttra non
seulemnt de mieux répondre à toutes les demandes des utilisateurs de l'eau
mais également d'améliorer les ensembles de liaison eux-mêmes en précisant les
limites de leur champ d'application et de comprendre les causes qui font que
crest tel paramètre plut8t que tel autre qui ici ou là explique mieux les
caractéristiques hydrologique s.
2. Transferi, analogique de l'information
Lorsqu'un bassin versant non observé est situé dans une région dans
laquelle une synthèse de l'information hydrologique disponible a conduit à un
ensemble de liaisons du type de celles qui viennent d'@tre décrites, ou s'il
est situé dans une région d'environnement comparable, l'estimation des princi-
pales caractéristiques hydrologiques de ce bassin est chose aisée. I1 suffit
d'en calculer les paramètres du milieu utilisés dans les liaisons hydrologie-
environnenient et d'appliquer celle s-ci.

La plus grande prudence s'impose si l'on n'est pas sûr de l'hornogénéit6 de
l'environnenient du bassin avec celui de la région étudiée et si les paramètres
du bassin ont des valeurs extérieures au champ couvert par ceux-ci dans ia-
dite région : toute extraplation hors du strict domaine d'application est
risquée et ne peut etre effectuée qu'après une reconnaissance géomorphologique
du bassin et de la région de référence.
Beaucoup plus fréquemment le bassin non observé est situé dans une
r6gion pour laquelle on ne possède pas de synthèse de Itinfomation hydrologique,
laldite synthèse nécessitant de longs et délicats travaux d'analyse critique.
Ainsi en France, en dehors de l'Alsace, aucune région n'a fait jusqu'ici l'objet
d'une telle synthèse systématique. Certes, l'analyse de l'information hydrolo-
gique n'est pas restée au point zéro et beaucoup d%ydrologues régionaux sont
A I& intuitivement d'esthr des caractères hydrologiques de bassins non
observés. Ce transfert analogique n'a l'inconvénient que de devoir etre refait
?I chaque demande et d'&re dépendant de la qualité ou de l'intuition de
l'hydrologue,donc d'@tre imprécis et inconsistant.
Malgré ces défauts, il reste la seule méthode d'estimation en
l'absence de liaisons régionales établies.
Le processus opérationnel est le suivant :
a) reconnaTtre le bassin concerné et analyser son environnenent
physico-climatique,
b) rechercher dans la région des bassins observés ayant des environnements
aussi comparables que possible avec celui du bassin concerné,
c) analyser les variables hydrologiques des bassins de comparaison
ainsi sélectionnés,
d) procéder au transfert analogique des variables hydrologiques des
bassins de comparaison au bassin concerné.
Ce transfert est la seule opération originale de ce processus. En
réalité, il s'appuye implicitenient sur l'hypothese que les valeurs des variables
hydrologiques vont évoluer des bassins de comparaison au bassin concerné CO~E
elles évoluent dans les régions connues c'est-à-dire en fonction des parmètres
du milieu. I1 s'agit donc jntuitivewnt de déceler les paradtres explicatifs
principaux de l'hydrologie régionale,puis d'estimer le sens et l'intensité de
leur action pour transférer les variables hydrologiques.
69

70
Si l'on peut réaliser cela sans trop de difficulté pour les paramètres classi-
ques tels que la hauteur annuelle de précipitation et la surface, il n'en est
pas de mhiie des autres facteurs (pente, perméabilité des terrains, cou~rt
végétal ... ) au sujet desquels on peut simplement dire que leur effet sera
croissant ou décroissant sans pouvoir préciser de combien. On limite les risques
d'erreur en choisissant, si possible, des bassins de comparaison dont les fac-
teurs principaux - surface, pluie annuelle - sont proches de ceux du bassin
concerné, sachant que l'effet des facteurs secondaires est de l'ordre de
grandeur de l'imprécision de l'estimation de la variable hydrologique d'après
les facteurs principaux.
Nous avons été anen6 à plusieurs reprises à réaliser des transferts
analogiques de cette sorte pour des problems d'hydraulique agricole en France
l'issue de très petits bassins versants ; par exemple :
- barrage réservoir à l'issue d'un bassin de 300 km
2
pour un Syndicat
intercommunal d'adduction d'eau (région du Centre Ouest de la France)
- barrage en terre pour plan d'eau touristique l'issue d'un bassin
de moins de 20 h2 (versant atlantique des Pyrénées).
N'importe qui aurait pu utiliser à l'occasion une formule classique
d'écoulement j le risque d'erreur aurait certainement &é énorme. Le transfert
analogique évite l'erreur grossière bien qu'il ne perniette pas d'atteindre la
précision d'emploi des Liaisons régionales hydrologie-milieu quand elles existent,mais
à la condition qu'il soit effectué par un hydrologue doté d'un sens
critique aigu, connaissant l'hydrologie régionale et capable de détecter les
effets secondaires de l'environnement (géomorphologie, nature des sols . etc ... ).
3. Conclusion
Les abaques régionaux et le transfert analogique dtinformation permettent
d'estimer bs principales variables hydrologiques d'un bassin non observé
avec une précision qui peut satisfaire le planificateur ou l'utilisateur de
l'eau qui procède
un aménagement simple et modeste. Si l'aménagement est
complexe - réservoir à but multiple, régularisation interannuelle - son coût
s'accroft et la precision requise de lthydrologue également. Les méthodes exps6e:
ici deviennent alors caduques au-delà du stade de l'avant-projet OU des études
préliminaires. I1 est alors indispensable de doter le site d'aménagemnt d'une
station hydrométrique pour affiner les estimations. Cela est pwsque toujours
possible car, entre les études préliminaires et le projet définitif, st6coulent
bien souvent plusieurs années dont l'hydrologue pourra tirer profit s'il a été
avisé en temps utile du problème et de la précision souhaitée.

Ref érence s bibliographiques
1. RAS G. - i960 -
de l'Ingénieur1' Coll. du Lab.
Nat. d'Hydraulique, Eyrolles édit. Paris
2. RODER J.A., AWRAY C. - 1965 - "Premiers essais d'étude générale
du ruissellement sur les bassins expérimntaux et représentatifs
d'Afrique tropicale" A.I.S.H. Symposium de Budapest - Public. no 66,
vol. 1, pp. 12-38
3.
4.
5.
DUBRF;UIL P. - 1970 - '%e rôle des paramètres caractéristiques du
milieu physique dans la synthèse et l'extrapolation des données
hydrologique s recueillies sur bassins représentatif A o I. S. H a
Colloque de Wellington, N. Zél., Public. no 96, vol. I, pp 583-590
DUBREUU, P., GIRARD G., HERBAUD J - i968 - 'Nonographie hydrolo-
gique du bassin du Jaguaribe" Coll. 'Némoires de l'ORSTOM1' no 28,
21 x 27, 385 P.
DUBREUIL P., HERBAUD J. - 1970 - Yontribution à la connaissance
quantitative des modifications du régime hydrologique sous l'effet
du taux de boisement à l'aide de deu exemples : le bass+ alsacien
du Rhin, et le bassin du Jaguaribe (Brésil)" - S.H.F. XIeme
journées de l'Hydraulique - Paris - Tome I, question III, rapport
8.
71

ABSTRACT
ESTIMATING EVAPOTRANSPIRATION BY HOMOCLIMATES
T.E.A. van Hylckama"
Data for planning of water resources projects in arid or
semi-arid climates are generally inadequate. It is here that
the evapotranspiratia term plays an important role in the
hydrologic cycle. Estimating this term by various empirical
formulae using only measured or estimared air tgmperatures
and length of growing season often leads to erroneous results.
It is better to use parameters, such as net radiation, humidity,
wind speeds and rainfall characteristics, obtained from regions
with climates similar to that of the region under study. Such
homoclimatic regions have soils and vegetation of a comparable
nature because both are largely a result of the climate itself.
Examples of the transfer of parameters to determine evapotrans-
piration by the use of homoclimates show that such monthly and
yearly values are, at most, 10 percent larger or smaller than
the measured ones, a significant improvement over empirically
determined values which often are more than 30 percent off.
RESUME
Dans les pays arides ou sub-arides, les données nécessaires
2 l'établissement des projets hydrauliques sont presque toujours
insuffisantes. C'est dans ces pays également que le terme éva-
potranspiration tient un rôle important dans le cycle hydrolo-
gique. Son estimation à l'aide de differentes formules empiri-
ques, qui ne tiennent compte que de la température de l'air et
de la durée de la saison culturale, conduit souvent à des re-
sultats erronés. I1 est préférable d'utiliser les valeurs de
parametres plus efficaces, comme le rayonnement net, l'humidi-
té, la vitesse du vent et le régime pluviométrique, obtenues
dans des régions ayant des climats analogues à celui de la ré-
gion étudiée. On peut penser que des régions de climats voisins
ont des caractéristiques de sols et de végétation voisines, car
ces deux éléments sont en grande partie le résultat du climat
lui-même. L'auteur présente des exemples de transfert de don-
nées pour le calcul de l'évapotranspiration, basé sur ces consi
derations. Les valeurs mensuelles et annuelles obtenues diffé-
rent de moins de 10% des valeurs mesurées, alors que l'emploi
de formules empiriques simplistes fournit des résultats que
différent souvent de plus de 30%.
>* Research Hydrologist, U.S.Geologica1 Survey
Texas Tech University, Lubbock, Texas.

74
NOMENCLATURE
BV
E
Ea’ Eo
H
L
*a
da
k
P
r
r
a
r
U
a
Z
a
z
A
Y
N Y
turbulent transfer coefficient (g cm-2 min-l mb-l) or
(10 kg m-2 min-’ 0.1 kPa‘l).
rate of evaporation (g cm-2 min’l, mm hr-1, or cm day-l).
actual (or measured) rates and computed potential rates.
net radiation (cal cm-2 min-l or 41867.4 j m-2 min-1).
latent heat of vaporization (about 585 cal g-1 or 2.46 x 106 j kg-l).
temperature of the air at height za (meters) (OC).
saturation pressure deficit of air (mb or kPa + 10) = the difference
between saturation and actual vapor pressure.
Von Kármán coefficient taken as 0.41.
ambient pressure, assumed constant for the sites discussed at 983 mb
or 98.3 kPa.
correlation coefficient.
external resistance (sec cm-1).
stomatal or canopy resistance (sec cm-1).
windspeed at elevation z (cm min’l).
a
elevation above surface (m or cm).
roughness parameter (cm).
first derivative of saturation vapor pressure versus T (mb OC1).
psychrometric constant (mb
a dimensionless number dependent upon T
for p = 983 mb Aly = -0.32 + exp 0.045
and p;

I INTRODUCTION
Evapotranspiration and hence the potential evapotranspiration term plays an
important role in the hydrologic cycle, becoming more important as the climate
gets drier. Harrold 111 estimates that 75% of all the precipitation falling on
the conterminous United States goes to evapotranspiration, but in arid lands this
percentage can approach 100. Hence the estimate of potential evapotranspiration
(E,) "is an essential requirement in the assessment of total available water,
regional water balance and irrigation demand" 12 1.
Although the parameters governing potential evapotranspiration are well
known, in areas with inadequate hydrological data, the model required to estimate
Eo quantitatively becomes difficult to construct. Often rather serious simpli-
fications have been assumed at the cost of accuracy in the prediction of the
information needed in water resources projects 131. Hounam 141 presents a few
examples of "approximations and over-sirnpliïications with regards to procedures
or data. For example vapor pressure of the bulk air is sometimes substituted
for surface vapor pressure with considerable loss of reliability, net radiation
may be estimated from sunshine or even cloudiness and air temperature, whilst
the advective term, which can be quite significant in areal evaporation, is
neglected in most methods."
The desire to have a quantitative estimate of Eo regardless of the paucity
of parameters has resulted in a plethora of empirical equations (= models) which
are valid only (if at all) for areas or regions where the empirical correlation
between Eo and one or more parameters was established.
Blaney-Criddle 15 I , who derived a formula for irrigated areas and Thornthwaite,
whose equatiori is based on data from humid climates 16 l.
75
Two examples may suffice:
Yet such equations are often used for estimating potential evapotranspira-
tion when the information needed is inadequate. One has an idea of mean monthly
temperatures and rainfall, either on the area under study itself or in the
neighborhood, and determines Eo by the use of one or the other of these empirical
equations.
A fairly convincing example that such methods lead to unsatisfactory results
is presented in figure 1. This graph, based on data presented and discussed by
Cruff and Thompson 171, illustrates the very poor agreement between six different
models used to estimate potential evapotranspiration. The reason for the discrepancies
is to be found in the characteristics of the evapotranspiration
phenomenon. Plants, and to a certain extent soils, respond to such inputs as
radiation, vapor pressure and winds in a rapid and nonlinear fashion. Taking
seasonal, monthly, or even weekly averages of such parameters and use those to
estimate Eo leads necessarily to erroneous results. This is especially true
when the advective term, combining wind speed and vapor pressure,becomes dominant.

76
It seems that there is a better approach especially if homoclimatic maps,
such as the ones being discussed below, are available. One searches for an
area with a climate as similar as possible to the one of the area under study,
but which has, in addition to the climatic characteristics, also data available
that allow a computation of potential evapotranspiration with a satisfactory
degree of accuracy.
In the following we shall first discuss climate classification and review
the literature on homoclimates, then show that the so-called combination method
enable-s one to obtain very satisfactory estimations of potential evapotranspira-
tion arid finally present an example of the proposed method.
I I HOMOCL INATE S
The earlier climatologists, such as Köppen Island Lang 191, classified
climati,:; mostly by certain relationships between mean annual temperatures and
rain€,'l. Later mean monthly values were taken into account and climates were
Classified by the march of temperature and mean monthly rainfall throughout the
year 1101. Still later aridity indices were used 1111 and in 1948 Thornthwaite
introduced the concept of potential evapotranspiration. Climates now are often
characterized by diagrams, combining graphs of temperature and precipitation, or
evapotranspiration and precipitation [ 12 I . Stations having similar climatic
diagrams are called homoclimes 1131. By extention the term has come to mean a
"region climatically similar to another specified region" I14 I .
Meigs 1151 was probably the first and maybe the only one to use the word
homoclimates in this sense. He used the 1948 Thornthwaite system and his maps
of the homoclimates of arid lads are rather crude and on a scale (about 1 to 3Q
x LO6) too small to be of much practical value.
In Arid Zone Research XXI UNESCO 1161 presents a much more detailed set of
maps which are called bioclimatic maps because "the purpose---is to exhibit for
a particular region a synthesis of the climatic factors of special importance
to living creatures".
subject in itself'' and mention that 26 meteorological elements can affect the
climate, the environment and therefore, a particular animal or plant species 1171.
They fully realize that, at the present time, insufficient information on all
these items is available, but continue: "fortunately however there is one fact
which is firmly established: namely that of all the elements in the environment
those of most importance for living entities, plants in particular, are warmth
and water".
The authors say that "climate is an extremely complex
Unlike most other climate geographers, however, they were not
content to use the ombrothermic diagrams alone, (ombros = rain), but used a
xerothermic index which includes the effects of rainy days, days with mist an6
dew, and allows for atmospheric humidity. This is an attractive compromise
between the 26 elements and the use of only monthly averages of temperature and
precipitation. A day, for instance, with 5 centimeters of rainfall in one hour

has an effect vastly different from a day
12-hour period. A day with an average of
same as one with the same temperature but
humidity of 80 percent.
with a 5 centimeter drizzle during a
15OC under a dry clear sky is not the
under clouds and with relative
Using the diagrams and indices mentioned above, UNESCO I i6 I distinguishes
33 different climates, ranging from true desert (for instance in Libia) to glacier
climates such as in the high mountains of Austria. There are four maps of a
scale of 1 to 10~000,000 of the dry regions in South Africa, South America, the
southwest of North America and the southern parts of Australia. Two others on
a scale of 1 toS,b00,000 cover an area from the AtlaIitic (long ?OoW) to points
west of Karachi (Pakistan) (long 72OE) and from northern Italy (lat 25ON) to
well into the Sahara and also covering the Arabian Peninsula (lat 14ON).
Another source that, at times, could be used to find homoclimatic regions
are the 15 volumes "World Survey of Climatology" 1181. However, due to the preferences
of 11 sub-editors and numerous authors, the classifications are not
consistent, ranging (for example in Volume 8) from the 1918 Köppen system to
classifications according to dynamic concepts from the viewpoint of air-mass
mixing and transformation I19 I .
Terjung's maps of isanomalies 1201 might eventually help to improve homoclimatic
maps. For the present time the author states: 'I--- the rather crude
maps presented here and their cursory examination should not be considered
definitive or qualitatively accurate".
It is perhaps unfortunate that in none of these sources useful data on
ioeasured potential evapotranspiration could be found, and an example of the
applicability of the method of using homoclimates to estimate potential evapo-
transpiration had to be taken from the semiarid southwestern parts of the United
States.
III THE COMBINATION CONCEPT AND SHE CANOPY RESISTANCE
As mentioned earlier, longtime averages of meteorological parameters are
inadequate to estimate potential evapotranspiration. The dynamic characteristics
and the sensitivity of Eo to environmental parameters are most clearly demonstrated
by the correlation method, aïso known as the eddy flux, eddy transfer or covar-
iance method 14, 221.
In this method measurements have to be taken with a
frequency of a few seconds or less. Another method, not as sensitive to be sure,
but quite suitable for our purpose it seems, is a method that combines the energy
budget with a mass-transfer term 123, 24, 251. A complication arises when the
plants, even under conditions of potential evapotranspiration, react to the
environment and seem to control transpiration by means of opening or closing the
stomata 126, 27, 281.

78
It must, first of all, be shown that potential evapotranspiration can be
estimated uite accurately by the use of the combination formul developed by
Penman 1237 and improved by Monteith 129, 301 and van Bavel 125 . The latter,
following the method first used by Penman 1311, derived the fol owing expression
for the instantaneous evaporation rate:
E = '/L
B in this equation is defined as:
V
(A/y) H + L Bv da
A/Y + 1
cai cm-2 min-'
Because expression (2) is based upon standard wind-profile theory, van Bavel
warns that it applies strictly to adiabatic conditions only. But he points out
that the combination model (1) has reduced the criticality of (2).
This model predicts potential evapotranspiration from wet bare soil and from
alfalfa covered soil with great accuracy over hourly periods, as was convincingly
shown by van Bavel I25 1 . However, when used to compute evapotranspiration from a
stand of saltcedar (Tamarix pentandra) , I32 1 there were fai-rly large discrepancies
when computed and measured values were compared.
discrepancies was immediately evident. It is a Weil-known fact that over tall
vegetation, the roughness length (2,) varies with the wind speed, 133, 341 and a
zero displacement length must be incorporated in equation (2). Alternatively,
a modified roughness lengths can be used, and this was done in the present
computations 135 I .
One of the reasons for the
With this modification the discrepancies are smaller but the
results are still not very satisfactory. A typical example is given at the top
of figure 2.
Inspection of the data further showed that the largest deviations between
computed and measured evapotranspiration occured under conditions of high wind
speeds. This indicated that there could possibly be a stomatal or other type of
resistance inside the plants, but most likely a closing of the stomata under
conditions of high evaporativity 1241. It is possible to measure diffusion
resistance directly on most broadleaf plants by the use of one or other type of
porometer 136, 371. These instruments cannot be used on the small scale-like
leaves of saltcedar which are less than 2 millimeters long and 1 millimeter wide.
Xonteith (301, however, has shown how external and stomatal resistant-es can he
estimated from microclimatological data. The combined energy budget and mass
transfer equation then becomes:

in which the external resistance is:
and the stomatal resistance:
r a = (log, (z/zO)l2 / U k2
r = (A/y + 1) (Eo/Ea - 1) x ra (5)
A recomputation of potential evapotranspiration with equation (3) shows that a
muili closer agreement between measured and computed evapotranspiration can be
obtained.
Notice that equation (5) contains the potential as well as the measured
evapotranspiration, but once rs has been computed, it was found that it very
highly correlated with wind speeds, and also (but less) with vapor pressure
deficits. Using the resistances obtained from the equation (5) for one set of
data, potential evapotranspiration could then be computed for other sets of data
and such values are plotted at the bottom of figure 2.
IV AN EXAMPLE
In order to demonstrate how the application of homoclimatic data may help to
estimate Eo, a comparison will be made between evapotrenspiration rates measured
in evapotranspirometers near Buckeye, Arizona (lat 33ON, long 113OW) with those
computed with data available from a homoclimatic area about 50 kilometers to the
east near Tempe, Arizona. Thus we pretend that the needed parameters at Buckeye
were not available and we use those from a homoclimatic region.
Hourly data for 3 days were available from technical reports issued by the
U. S. Water Conservation Laboratory 138, 391. Figure 3 shows a typical example
of hourly values measured at Buckeye, compared with those computed from the Tempe
data. Figure 4 presents a comparison between two sets of computed data. Measured
hourly data for 9 April were not available but, as figure 2 shows, the computed
values are quite valid. Note, incidentally, that on this day in early spring
there were a few hours with dew (negative E's). Not only are the correlation
coefficients very high (0.94 for figure 3 and 0.92 for figure 4) but the regression
equations indicate nearly 1:l relationships. For data of figure 3 we have:
E, = 0,19+0,86 Eo, and for figure 4: Ea = 0,02+0,94 Eo. The t values for both
equations are well above the 1% confidence limits: respectively 6.3 and 4.0.
Student's t value for the 1% limit and 22 degrees of freedom is 2.8, 1401.
That the combination method is valid strictly for short-time data has already
been mentioned. The fact is clearly shown by the data in table 1. In the left
79

80
two columns the sum of 24 hourly values of evapotranspiration rates is given as
mi.llimiters per day. In the right two columns the rates are given as computed
from mean daily averages of the parameters in equation (3). As can be seen the
sum of the hourly values are not only very close to another but also compare
favorably with the measured values.
The data available did not allow to extend the computations to months or
years. However, the agreement on hourly and daily bases makes it very likely
that, on monthly and yearly bases, even better agreement can be obtained.
V CONCLUSIONS
Serra 1411, remarks "climatology and hydrology are two very different
disciplines: if the first is ãescriptive and 'static', the other studies---the
'dynamics' of water and working methods used for the first will have little
chance to fít the second.---The climatologist works with an 'average year'. To
establish his---classification indices he will use the average temperature of
each of the twelve months of the year---. The hydrologist by contrast must
follow from day to day---the living reality of a phenomenon." The phenomenon
Serra refers to is,of course,the evapotranspiration.
If however, the climatic classification is detailed enough and one has in
one part of such an area sufficient information, quantitatively as well as quali-
tatively, on the parameters that drive the evapotranspiration, then it is reason-
able to assume that in other parts of this homoclimate the same data are applica-
ble, at least within acceptable limits. It might of course be necessary (and it
is nearly always possible) to correct for latitude, elevation and exposure.
What we are dealing with seems to be an integration between climate and
meteorology, something that Kisiel 1421 had in mind when he wrote: "The future
of hydrology rests on our ability and willingness to undertake the last integrative
effort on a continuing and adaptable basis. This effort is particularly
urgent if one accepts the thesis that each watershed or basin is a law unto
itself. Transferability of laboratory knowledge to the field and of knowledge
from one watershed to another or from one climate to another rests inexplicably
on our ability to provide a mathematical foundation to the cycle of model building
and its parts.''
The present paper shows that, in principle, the use of homoclimates is
possible and reliable for effectively estimating evapotranspiration rates with
the model presented above. The difficulty lies in the fact that there are so
few homoclimatic maps and those few do not always use the best method of classi-
fying the climates. There obviously is a great need for more and more reliable
homoclimatic maps. These maps should show the locations of stations where com-
plete sets of microclimatological data can be obtained for estimating the poten-
tial evapotranspiration with a desirable degree of accuracy.

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of bare soil surfaces. Tech. Rep. U. S. Army Electronics Conmiand 2-67P-1.
van Bavel, C. H. M., (1967). Surface energy balance of bare soil as
influenced by wetting and drying. Tech. Rept. U. S. Army Electronics Conmiand
2-67P-2.

40. Fisher, R. A., & Yates, F., (1943). Statistical tables for biological,
agricultural and medical research. Oliver and Boyd Ltd., London, table ïïI.
41. Serra, P. L., (1954). Le controle hydrologique d'un bassin versant, in Soc.
Hydrotechnique de France, Pluie, Evaporation, Filtration et Ecoulement,
Compte Rendu des Troisièmes Jourdes de l'Hydraulique, pp. 29-35.
42. Kisiel, C. C., (1969). Mathematical methodology in hydrology, in Chow, V. T.
(Dir.), The progress of hydrology, Proc. of First Internat. Seminar for Hydrol.
Professors, Urbana, Illinois, pp. 362-399.
83

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Fig. 4
Comparison of 24 hourly values of evapotranspiration
computed with equation (5) using Buckeye data (Y> and
Tempe data CX), (9 April '66).
87

88
TABLE 1. Water use by saltcedar in millimeters per day
Sum of 24 Computed from
hourly values Measured mean daily values
Dates Tempe data Buckeye data Buckeye* Tempe data Buckeye data
-
1966
9 April 11.0 11.5 10.4 14.9 14.4
28 April 14.6 15.6 15.4 18.7 18.2
3 May 13.9 12.2 13.5 17.0 15.8
*The lysimeters at Tempe were allowed to dry out so no potential evapotranspira-
tion data were available.

PLUVIOMETRIC ZONES AND THE CRITERIA TO DEFINE THEIR
ABSTRACT
BOUNDARIES FOR REGIONS WITH SCARCE DATA
by
García-Agreda R., Rasulo G., Viparelli R.
A zone is defined "pluviometric zone" if the parameters of
the rainfall distribution function assume the same value in all
of its points, or vary with continuity from one point to another
according to their location.
Consequently, if it is necessary to estimate the rainfall
distribution at a point, only the information derived from
pluviometers of the same pluviometric zone is useful.
By refering particularly to regions in which there is a
scarcity of data, the authors point out that, in order to define
the boundaries of the pluviometric zone pertaining to a given
point, it is necessary preliminarly to formulate a working
hypothesis based on climatic maps in which also the geomorphology,
soils and vegetation are considered.
RESUME
Une zone est défine "zone pluviométrique" si les paramètres
de la loi de probabilité des pluies ont la même valeur dans
toute la région ou ils varient d'une façon continue d'un point
à l'autre.
Par conséquence, s'il faut estimer la répartition statisti-
que des pluies en un point, on peut utiliser seulement les in-
formations tirées des pluviomètres disposés dans la même zone
pluviométrique.
En particulier, en se rapportant aux régions pour lesquel-
les on a peu de données, les auteurs soulignent que, dans le
but de définir les lignes de contour de la zone pluviométrique
qui comprend 1.e point considéré, il fau-t d'abord formuler une
hypothèse de travail qui se base sur des cartes climatologiques
dans lesquelles on considère aussi la géomorphologie, les sols
et la végétation.

90
Symbols
1: Let us indicate by:
- h : the annual rainfall depth at any point;
- y : the log of h;
- O{h} and {y} : the distribution functions of h and y;
- MChl; 0th) and y{h} = m:
aih}
respectively the mean, the
standard deviation and the coefficient of variation of the
probability distribution of h;
- M {y}, oiy} and 02{y): respectively the mean, the stan-
dard deviation and the variance of the probability distribution
of y.
Let us also indicate by:
- hi with 1 c i < n: the n values of h registered during
the observation period;
- yi with 1 < i c n: the n values taken by y = log h;
- h, s{h} and gCh}: respectively the estimates of MIh},
o{h} and yth};
- 7, sty} and s’{y): respectively the estimates of MCy},
aiy} and 02{y};
- 71 and 72, Sf{y} and s;{y): respectively the confidence
limits of and s‘{y] with a tollerance level of 95%;
Assume h is distributed with a good approximation according
to the log-normal law 113 [2].
Consequently, y is distributed according to the normal law
the parameters M{y} and o{y} which characterize its distribution
are connected to M{h} and y{h} by the equation:
and
equations:
By estimating the parameters 7 and s2{h} by means of

and
n
I-
t Yi
n
n
n - 1
the confidence limits of 7 and s2{hl could be expressed by means
of ecuations:
in which t0,025 and ~0,025, to,g75 and ~0,975 are respectively
the percentiles of t and x corresponding to the probability 0,025
and 0,975.
In t roduc ti on
(3)
2: From direct measurements taken at each single point A,
B ... of a region, it is possible to deduce only estimates of
the values that M{h} and y{h) assume at the said points.
Takin into account the fact the said estimates could
deviate from the real value due to sampling errors and that in
technical problems the average rainfall depth distribution on
given surface must be known, it is necessary:
a) to improve the said estimates by decreasing the uncer-
tainty with which they were determined;
b) to estimate M{h) and y{h} and consequently the annual
rainfall depth that occurs with a given probability, even at
points where no pluviometers had been installed.
The two problems become greater in regions where only a
few measuring stations are available and for most of them with
a few years of observation.
91

92
Hydrological Similitude Criteria and Pluviometric Zones
3: The rainfall depth registered, at a generic point A, for
a given event occurs due to the evolution of meteorological
conditions that have their repercussions also on the rainfall
depths that occur in the same event in a more or less extended
zone around A. As it is known, for different environmental
conditions, such as those connected with the morphology of the
zone, the rainfall depth that occurs during the same event in
different points, could be highly different; however, in passing
from one event to another, at least normally, the said environ-
mental conditions excercise a differential action that acts
always in the same direction.
Finally, the rainfall depths h registered at a point A, are
affected both by meteorological factors common to the entire zone
and acting with a variable intensity from one rainfall event to
another; and by the environmental factors that are invariables in
time, but, normally, variable from one point to another,. The
deviations that are observed among the values that h assumes in A,
year after year, depend upon the variability in time of the
meteorological factors; while the deviations that are noticed
among the values that h assumes, with the same probability,
respectively in A and in each of the other points of the zone
around A, depend upon the variability of environmental conditions.
Consequently, if in a zone characterized by common meteoro-
logical factors k pluviometers are installed, in agreement with
what has been said by other authors [l), it is safe to suppose
that in passing from one pluviometer to another, the variation
coefficient y{h} remains constant.
Therefore, ify{h} is constant, it derives, from equation
(11, that even 02{y} remains constant.
At this point, we will say that a greater number of pluviom
eters belong to the same pluviometric zone if the variance assumes
a common value u “Cy}.
As it is known, the definition of a pluviometric zone and
its connected hypothesis are to be considered in a statistical
way.

Precisely, it cannot be excluded that at each single point
the variance 02iy} could differ from the value assumed as
the value to characterize the zone; however, due to the fact that
for each single point only an estirnate s2{y] of 02{y) could be
had it is evident that:
1) the deviation s2{y} - ~''{y), that is observed for single
point between s2{yl and o''{yI, could be caused partly, s*{yI -
- 02{y), by a sampling error (a non-significant part of the
deviation between s2{y} and o"{y) and partly, 02{y] - ot2{y), by
the real difference between 02{y} and or2{y) (the significant
part 1 ;
2) that, however, the deviation significant part is always
modest and such that s'{y] - a"{y} s2{y} - 02{yl t o'íy} -
- C I ~ ~ would I ~ ] range around values that s'{y) - 02{y} would
assume.
4: To determine the pluviometric zo'nes that lie a given
region, the methodology to follow could be divided in theree
phases.
An attemp to formulate a working hypothesis delimiting the
single zones is made during the first phase.
By deducing the best estimate of s'2{y} of the value that
the variance o'2{y) assumes in all the points of the zone during
the second phase, the working hypothesis is formulated.
In doing this, the different significance that the series
of data, obtained in each pluviometer of the zone, have, must be
taken into account depending on the number of a data that appears
in each one of them.
Particularly, if k pluviometers lie in the zone, having
indicated by s:{y}, with r being variable from 1 to k, the
variance estimated for each single pluviometer from the nr data
registered in it, the best estimate of s''{y} could be obtained
by means of equation:
93

94
In the third phase, finally, by assuming that ot2{y}=
= s'2(y) we proceed on to the proof of the working hypothesis
thus formulated, checking by means of equation (6) that the
single estimates differ from the single value with differences
that could be attributed solely to sampling errors.
Naturally, in this process we have supposed that data
collected in each pluviometer of the zone are not correlated
among themselves [3].
5: As an example, let us refer to Morocco
In fig. 1, the assumed working hypothesis
of the region in pluviometric zones is reported
of the division
In fig. 2, shows for some zones a statistical control test
of the validity of the working hypothesis.
As it can be observed, the proof has been carried out by
reporting on a diagram, whose ordinates represent the values of
s2{y} and whose abscissas represent the number n of observation
years :
a) the estimate st2{y} that characterizes the zone;
b) the range of confidence delimited by the two curves
s: (n) and s', (n) corresponding to the said value of s'2{y} or,
briefly, the confidence band of s2{y);
of the zone.
c) the point (n, s2{y}) corresponding to each pluviometer
As it can be observed from the diagrams, as a proof of the
assumed working hypothesis, the points lie within the confidence
bands.

Adaptability of the Climatic Charts for the delimitations of
Pluviometric Zones.
6: In fig. 3 are reported, with different simbols, the
division of Morocco in pluviometric zones, as indicated
previously, and the division in climatic zones as it deducted
from the Meigs Chart [4].
As it can be observed, if the arid zones corresponding to
the Massif of Atlas, labeled by the indez .(1), and the semiarid
zone between Anti Atlas and Hamada du Dra, labeled by the index
(21, are excluded, a noticeable correspondence exist between the
pluviometric zones and the climatic zones reported by Meigs.
On the other hand, the disagreements mentioned previously
could be easily explained if we consider that climatic charts
are deduced by taking also into account geomorphology, the soil
and the vegetation.
In fact, for the zone (11, the lack of vegetation that has
induced Meigs to define it arid, could be attributed not to fewer
precipitations but to the presence of a calcareous massif that
prevents the formation of a vegetative soil.
On the other hand, for the zone (2) constituted by a large
depression delimited by the Chain of the Massif of Atlas on one
side and by Hamada du Dra on the other side, the presence of
vegetation that has induced Meigs to define it as smiarid, could
be attributed not to waters caused by rain that falls directly
on the zone, but to waters that rush there from nearby zones.
Pluviometric zones of Bolivia and Saudi Arabia
I: As it has been said in the previous paragraph 6, to
formulate working hypothesis regarding the delimitations of
pluviometric zones for regions where only few pluviometers are
functioning and, moreover , in most cases, functioning solely for
a short observation period, it could be useful to use climatic
charts.
95

96
Thus, to define the pluviometry of Bolivia and sone zones
of Saudi Arabia, we assumed the working hypothesis that the
pluviometric zones coincide with the climatic zones (figs. 4
and 5).
Particularly, for Bolivia we used the chart published by
UNESCO for the arid and semiarid zones [4], and the Trewartha
and Robinson chart for the humid and sub-humid zones [5]. For
Saudi Arabia only the chart published by UNESCO was considered
*
phase :
As is shown in the above mentioned figures, in the first
Only pluviometers functioning for a long period of obser-
vation have been considered;
the estimates s2{y} have been deduced from the data
collected in each zone;
the estimates have been divided in groups and the location
of each pluviometer has been labeled with a different symbol
according to where s2{y} lies.
From the figures we observe:
1) in passing from one to the other climatic zone, the
values of s2{y} lie in different groups;
2) if in a zone more pluviometers are functioning, the
corresponding values of s'{y] lie, either in the same or conti-
guous groups.
Consequently, by taking into account the definition given
of the zones, it is safe to assume the hypothesis that the
climatic zones coincide with the pluviometric zones even for the
said regions.
Consequently, in the second phase of elaborations, still
taking into account the data relative to pluviometers functioning
for a long period of observation, the working hypothesis for each

single zone has been formulated, by assuming as estimate st2{yl
of the variation ot2{y} that characterizes the zone, the value
deduced by means of equation (7).
Finally, inthe third phase, by using also the data fur-
nished by the pluviometers functioning for a shorter observation
period, to verify the working hypothesis, it was checked that
the deviations between, st2{y} and the value s2{y} deduced for
each pluviometer could be attributed so,lely to sampling errors.
In both cases, from the few data available, the hypothesis
that the pluviometric zones coincide with the climatic zones is
sufficiently ascerIained.
Pluviometric Sub-zones
8: As it has been stated by other authors [l] whenever the
estimates of M{h} deduced for each pluviometer from the data
registered during the observation period, it has been possible
to distinguish, in each zone, one or more sub-zones.
In each of the said sub-zones, when passing from one point
to another, the values of the estimates fi either show that:
a) they scatter around a single value M{h}, or that
b) they scatter around values of M(h3 that vary in function
of either one of the parameters which represent the morphology
of the sub-zone (particularly, in the cases considered, the land
elevation (2)).
In the first case, each single pluviometric sub-zone has
been characterized by indicating the value fit taken by arithmetic
average of fi corresponding to the single pluviometers. In the
second case, each individual sub-zone has been characterized by
specifying the variation law of M{h} as function of z and by
indicating the values 6' and that according to the
(1)
(2)
mentioned variation law correspond to the highest and lowest
elevation of the sub-zone pluviometers.
97

98
paper:
In the fig. 6 are reported on a diagram on logarithmic
a) the points (6, g'{h}) which represent the pluviometric
sub-zones, for the first case;
b) the intervals delimited by the points (hl (1) Y g'(h1)
and (E1(*), g'{h]) for the second case.
As it hast been found by other authors (61, when passing
from one region to another, and for each region from one pluvio~
etric zone to another, the variability increases as the average
annual rainfall decreases.
Instead, as it has been said previously, in each single
pluviometric zone the variability expressed by means of ylh} or
u2{y} is completely independent from an eventual variability of
the average rainfall.

RE FE RCN C ES
[l] VIPARELLI C. : "Idrologia applicata all'ingegneria".
Parte II Fondazione Politecnica del
Mezzogiorno d'Italia, Napoli (1965).
[2] . MARKOVIC R.D. : "Probability Functions of best fit to
Distributions of annual Precipitation
and Runoff".
Hydrology papers, Colorado State Univer-
sity Fort Collins, Colorado (Aug. 1965).
[3] PENTA A., ROSSI F.: "Objective Criteria to declare a
Series of Data sufficient for technical
Purposesff.
Simposio sobre proyectos de recursos hi-
dráulicos con datos insuficientes. Ma-
drid (1973).
[4] CHOW W.T.: "Handbook of Applied Hydrology".
Mc Gram-Hill Book Company. Pg. 24-3 e seg.
(1964).
[5] TREWARTHA G.T.; ROBINSON A.H. , HAMMOND E.H.: "Elements of
Ge o graph y It .
Mc Gram-Hill. Book Company (1967).
[6] HAZEN, ALLEN,: "Variation in annual rainfall".
Eng. News, vol. 75 n. 1 (1916).
The pluviometric data used in this paper have been taken from:
- Institut Scientifique Chérifien du Maroc.
- Servicio Nacional de Meteorología e Hidrología de Bolivia.
- Empresa Nacional de Electricidad de Bolivia.
- Ministery of Agriculture and Water of Saudi Arabian Kingdon.
99

I
l
i
I
l

PLWIO?,ETRIC ZONES AND TIE CRITWIA TO DEYIN2 TRTIR EOiZIDARJ2S ?CI!?
ZEGIONS ::'ITH SCARCE DATA
by
Garcia-Agreda R., Rasulo G., Viparel?.i 9.
Fig. 2
N F
8 8
C3
N O
m
9
h
- x
U
c
x
m

y
Garcia-Agreda R., Rasulo G., Viparelli R.
PLLT'IOXIX'RIC ZOPJ'XS AND TFIE CXITWIA TO DEFINE THEIR BOUNDARIES FOR
EEGIONS WITH SCARCE DAIA
-.
uic. 2
-- __ I
rP
M
l
m
N
O
m
o)
N
I I
m m m
- C U T -
;;:
m *
N N
I
I
l o
o O
O
E
Y

8
12
16
20
24
68 64 60
I
68 a4 60
Fig. 4
FLLWIO?3TTiIC ZONES hi TIE3 CRITERIA TO DEFINE THEIR BOUNDARIES FOR
REGIONS WITH SCARCE DATA
Garcia-Agreda R., Resulo G., Viparelli R.
by
F--..
U 11111 Km aiici
__
24

ì , l
I \
O
h-l
ul
N
O
N
Ln c

jr
Garcia-Agreda R., Rasulo G., Viparelli R.
PLLVIOEETSIC zoms urn THE CRITERIA TO DEFINE THEIR BOUNDARIES BOR
REGIONS WITH SCARCE DATA
O

ABSTRACT
ESTIMATION DES ETIAGES DE BASSINS NON EQUIPES
par G.R. OBERLINB, G.C. GALEA* et J.T. TONI**
The first data collected on the different creeks of the Or-
geval representative watershed (104 km2), showed a great disparity
between low water specific discharges. This disparity subsisted
after adjustement of man influences (such as pumping and throws).
The differences contrasted with the visible simplicity and homo-
geneity of the watershed surface and with the supposed favourable
outline of the ground water catchment. That is to say, even with
these propitious conditions in appraising the hydrological charac-
teristics (good network and problem seemingly easy), the represen-
tativity, i.e. the extrapolation of the results to similar
neeghbouring creeks, was found at fault. To resolve this point
without additive equipment, gauging rounds were undertaken during
low water seasons on a great number of creeks. The comparison of
these groups of instantaneous discharges, measured on the same day
at different places, ser off the behaviour of each watershed. In
some cases, the analyses of these observed differences allowed the
elaboration of general rules and led to pratica1 conclusiones, of-
ten quantitative. As some sections of the measured creeks belonged
to permanent network, and some other conditions having been satis-
fied (especially: a great enough number of measurements during
each low water season), the unknown characteristics of the unex-
plored creeks have been evaluated from these of the permanent
stations.
RESUME
Les premieres mesures effectuées sur les divers sous-bassins
constituant le bassin représentatif de l'0rgeval (104 km'), ont
fait entrevoir de trss importantes différences de débits spécifi-
ques d'étiage. Celles-ci subsistaient après correction des in-
fluences humaines (pompages et rejets). Ces différences contras-
taient avec la simplicité et l'homogénéité apparentes du bassin
en matière de caractéristiques physiques et avec 1 'aspect favora-
ble de son hydrogéologie. Autrement dit, même dans ces conditions
optimales d'estimation de caractéristiques hydrologiques (bon
équipement de mesure et probleme a priori simple), la représenta-
tivité, c'est-à-dire l'extrapolation de résultats aux bassins
voisins et semblables, était mise en échec. Pour résoudre le pro-
bleme sans équipements nouveaux, des campagnes de mesures volan-
tes d'étiage ont été réalisées sur un certain nombre de cours
d'eau. La comparaison de ces ensembles de débits instantanés, me-
surés simultanément en divers lieux, a précisé les différences de
comportement des bassins. Dans certains cas, l'analyse de ces dif-
férences a pu suivre des règles générales et conduire à des con-
clusions dont certaines étaient quantifiables. Comme plusieurs
stations permanentes (équipées) étaient incluses dans ces cam-
pagnes, et que certaines conditionsavaient été satisfaites (en
particulier: un .?ombre suffisant de jaugeages au cours d'une
même saison d'étiage), les caractéristiques inconnues des bassins
non équipés ont alors pu être estimées.
$; C.T.G.R.E.F., Parc de Tourvoie, F - 92160 ANTONY.
*J. ,. ,. Direction Départementale de l'Agriculture. PARAKOU (Dahomey)

104
INTRODUCTION
Dans l'étude des basses eaux, en hydrologie, on se heurte toujours
à une première difficulté concernant la qualité des données de débits de basses
eaux. Cette difficulté résiste remarquablement bien aux dliorations apportées
au fonctionnement des stations hydrométriques, même bien équipées. Sur les
bassins versants d' investigation de l'orgeval, par exemple (surfaces variant
de 7 ?i 104 km2), malgré d'efficaces et importants travaux (11, 18 procédure
habituelle de mesure des hauteurs d'eau, puis de traduction nhauteur - débit"
n'est pas toujours performante en très basses eaux.
Une solution très généralement utilisée consiste alors à effectuer un
grand nombre de mesures instantanées de débit (Jaugeages) et h Interpoler ent-e
ces mesures toutes les fois OU cela est possible, c'est-à-dire lorsque la dé-
crue n'est pas influencée par une crue, si minime solt-elle. Dans cette procédure,
l'équipement de la station hydrométrique (échelle; iimnigraphe. etc.. . )
n'est guère utilisé, sinon de façon qualitative fi]. Par généralisation de la
méthode on peut envisager de réaliser ces campagnes de jaugeages sur des cours
d'eau réellement non équipés et en espérer des résultats autres que ponctuels.
1. RESEAU DE MESURES EPISODIQUES D'ETIAGES
Les premières mesures conventionneiìes effectuées en lgtj2 SUT les 4
stations équipées du bassin de l'orgeval (Minj stère de l'Agriculture, France)
s'étaient évidemment heurtées awc difficultés hydrométriques citées plus
haut. De plus, elles avaient décelé de très grandes différences dans les dé-
bits spécifiques [g, I1 était difficile de savoir si ces différences avaient
une origine hydroaéologique (hétérogénéité dans la répartition des réservoirs
souterrains) ou étaient simplement dues aux aléas des Jaugeages, voire des
influences humalnes (pompages, re Jets, retenues, etc.. . ). A priori, l'excep-
tionnelle homogénéité du bassin en matière de géologie et de formationsde svr-
face (51,Fi~. la) &ait en contradiction avec la premièrc hypothèse. Néan-
moins, les mesures de type extensives mentionnées dens l'introduction ont
été commencées dès 1963.
La liste des stations concernées par ces mesures est donnée dans le
tableau 1 et la Fig. lb présente leur répartition sur la surface du bassin.

2. 8EMERS RESULTATS ISSUS DE SINPLES CORRELATlONS INTER-STATIONS
2.1. Préliminaire : Dans tout ce qui suit, nous appellerons débits d'étiages
tout débit provenant du drainage d'un réservoir souterrain (même proche du
sol), à l'exclusion dé tout ruissellement (de surface, direct, retardé,
105
etc...), et de tout écoulement "hypodermique". Nous faisons l'hypothèse que
nous sommes dans ces conditions lorsque la dernière crue est distante de
plusieurs jours (crues estivales tr&s modestes), ou de Plusieurs semaines
(crues moyennes et fortes), gour des bassins de 10 à 100 Km?: Ne connais-
sant pas la repré;.ntativité dans le temps de ces jaugeages instantanés,
nous corrélons deux h deu lec jaugeages de même date. Pour simplifier, nous
n'avons pasétudié toutes les combinaisons 2 à 2 réalisables avec le groupe
des 9 ou 10 stations. Enfin, nous n'avons pas corrélé les débits spécifiques
mais les débits absolus. Les grandes différences de comportement des divers
bassins montrent en effet que la notion de surface du bassin superficiel
nia sans doute pas grand chose h voir avec les dimensions des réservoirs
souterrains générateurs des débits d'étiage.
2.2. Rksultats : En général, sur tous les graphiques construits (environ 20), on
observe une dispersior. assez forte (Fig. 22). Elle est même très forte sur
toute corrélation concernant Mélarchez. Elle n'est acceptable qu'à l'inté-
rieur du groupe des 3 stations aval du ru des Avenelles (Gouge, Avenelles,
Theil) et de la station du Croupet.
Même en faisant abstraction de la dispersion propre aux erreurs
de mesure (les jaugeages d'étiage sont délicats et peu précis), l'ensemble
des points reste très dispersé.
ia première conclusion à en tirer est ia suivante : les conditions
d'alimentation des différents réservoirs souterrains (à l'origine des débits
d'étiage) ne sont pas homogènes sur l'ensemble du bassin, malgrdla taille
-
réduite (100 Km2) de ceiui-ci.
En regardant de plus près on constate que, pour une année donnée,
la dispersion est moins grande, et parfois même faible. D'ou la seconde
conclusion : pour une période d'étiage continue donnée (un été), les c oa-
tions d'alimentation (l'hiver prbcédent) se révelent stables dans le temps.
Pour la plupart des bassins, cette dernière conclusion doit cepen-
dant être nuancée quand on prend en compte les basses eaux tardives (zutorne).
Ces dernières sont d6jà influencées par les premières pluies d'hiver (les

106
débits remontent, ou bien leur baisse diminue ou s'annule), et les corré-
lations montrent des comportements très différenciés selon les bassins :
les points correspondant à des dates tardives (Octobre &. Décembre) sont
souvent rassemblés d'un Seul côté du nuage de points, pour une année donnée.
Nous dirons que les bassins qui "profitent'' rapidement des premières pluies
d'hiver ont une alimentation plus superficielle que les autres. D'où la troi-
sième conclusion : les bassins amont ont une alimentation ñettement plus super.
ficieìïe que les autres (résultat classique) ; ce caractère supérficieì est
surtout marqué au-dessous de i5 km2 (exutoire situé au-dessus des argiles
vertes) ; surface égale, le ru du Rognon est plus "superficiel" que le ru
des Avenelles : le ru de Bourgogne semble être intermédiaire entre les deux,
mais la probabilité d'une assez forte rétention de surface (forêt) rend cette
conclusion aléatoire.
2.3. Aspect méthodologique : Des observations précédentes nous déduisons un graphiqi
caractéristique (Fig. 23) d'une corrélation entre les étiages instantanés de
deux bassins voisins, mais à système hydrogéologique (d'alimentation d'étiage)
différencié. Il y a une forte dispersion globale, mais l'évolution est cohé-
rente à l'intérieur d'une année donnée (i ou j ou k).
3. CORRELRTION INTER-STATION PAFI WUBLES CUMUL5
3.1. Résultats qualitatifs : Etant donné la forte dispersion des corrélations to-
tales 2à 2, il était difficile d'en tirer des conclusions sur les abondances
relatives des bassins corrélés. Les courbes de doubles cumuls ont donc été
tracées. Elles confirment d'abord les conclusions précédentes : hétérog6néité
non négligeable d'une année à l'autre, bonne homogénéité à l'intérieur d'une
année avec courbure, caractéristique de 1' influence des premières pluies Aiver
Néanmoins, une bonne tendance se dessine sur la plupart des graphiques et per-
met de tirer des conclusions sur l'abondance relative des étiages. D'ou la
quatrième conclusion, en raisonnant en débit spécifique (k surface égale) :
le ru des Avenelles est nettement plus abondant que le ru du Rognon ; le ru de
Bourgogne est legèrement plus abondant que le ru du Rognon : dans le bassin du
Rognon, le ru du Petit Courcy (ferme Plessier) est nettement plus abondant w e
le haut Rcgnon ; dins le bassin des \venelles, c'est le ru de 1'Etang qui
apporte l'essentiel des étiages par rapport un bassin de Mélarchez insignifi;

107
On constate une quasi identité entre les bassins d'étiage faible
et ceux qui profitent rapidement des premières pluies d'hiver (rus "super-
ficiels"). I1 en est be même entre ceux b. étiage pur (coeur de l'été) abon-
- dant et ceux dont les eaux restent basses tardivement.
A noter, enfin, que le caractère de "superficialité" attribué aux
bassins de Mélarchez et de Pierre Levée et déduit d'une-réponse rapide aux
premières pluies d'hiver, se décèle également pour la partie, aval du bassin.
Ce sont les pentes plus fortes dominant l'extrêmité aval des thalwegs (en
particulier au droit de la station des Avenelles et du Champ de Tir) qui
sont probablement à l'origine de cette (faible) croissance relative des
étiages de fin d'été.
3.2. Aspect méthoddogique : La courbe type d'une corrélation par double cumul
entre deux bassins voisins a été représentée sur la Fig. 32. On a bien
entendu fait l'hypothèse de l'existence de différences classiques entre
i- i-
les nappes alimentant les étiages (- abondantes, - superficielles, etc...).
Les caractéristiques d'abondance ont +té mises entre parenthèses
sur la Fig. 32 car la liaison "abondance-profondeur (des nappes)" observée
sur l'ûrgeval, n'est pas générale, même si elle est fréquente.
4. ESSAI DE COMPAXAISON QUAhTITATIVE ENTRE BASSINS
Dans les tableaux qui suivent nous avons essayé de consigner, sous
forme condensée et parfois numérique, les conclusions présentées auparavant.
Chaque tableau se rapporte à une des 4 stations principales du bassin. Les
diverses caractéristiques d'étiage déterminées sur ces 4 bassins principaux
fi] fi] fi] (1/ pourront ainsi être approximativement transformées pour
s'adapter à tel ou tel bassin non observé de manière continue ($9 8 & 9).
Le rapport d'abondance moyenne K noté en colonne (4) est une estimation de la
pente moyenne de la courbe des doubles cumuls (9 3), éventuellement affranchie
des anomalies de cette courbe. Pour souligner les différences de comporte-
ment des bassins, ce rapport est calculé avec les débits spécifiques.
Ce coefficient K est donc (aux rapports des surfaces près) le rap-
port entre les moyennes des mesures d'étiages épisodiques effectuées en 2
stations. ia signification statistique de ces moyennes est a priori tout à
fait particulière ; on verra au § 32 qu'elle peut être rattachée à une carac-
téristique générale,

108
4.1. Etiages instantanés comparés à cem de la station équipée de Mélarchez :
--------_-e============;
- ----- ~ -___
Car adere
Cours d'eau Station
(1)
Avenelles
(Fosse Rognon) Mélarchez
.---_I-----_--_..----------
Etang. ........ Croupet
Petit Couroy.. Bibartault
Rognon.. ...... Pierre k v
ßourgogne.. ... Ch. de Tir
Rognon. .......
Avenelles
Fosse Rognon 1.
(2 1
Bibartault
Gouge
Rognon ........ Ch. de.Tir
Avenelles..... Avenelles
Orgeval.. ..... Theil
Etang. ........ Croupet
Rognon.. ...... Ch. de Tir 43,4
Avenelles..... Avenelles 45,7
Orgeval ...... Theil 104
0,3 à 0,4 í (
0,6 (
superficiel
des nappes
(5 1
-
-------..--_-
assez
prof ondes
II II
superfiddks
assez prof.
It 11
% par rapport au bassin de référenc5 Tableau 41
superficiel
un peu +
superficiel
pius
:orréiation
;res mauvaise
4.2. Etiages instantanés comparés à ceux de la station équipée de la Gouge :
Nous n'avons étudié que les bassins de surface pas trop petite par rapport
à celle de la Gouge, pour ne pas introduire de trop grosses différences.
Cours d'eau I Station I I 'Kf=
1
._______r____
Caractere
superficiel Remarques
des nappes"
(5) (6)
Avenelles
f par rapport au bassin de référence. Tableau 42
1
)corrélation
)ves bonne
1

_______-___ _--_-_-----
_-_____-I__ --I---
-----
Cours d'eau Station
(1) (2 1
Avenelles.. Avenelles
_______-__--..-----------
Rognon ..... Ch. de Tir
Orgeval ... Theil
-____________ ---
Grgeval ..., Theil
_________________-------
. Ch. de Tir
Rognon . ___________________----
__ - _______ -_- --
109
4.3. Etiages comparés à ceux des stations équipées des Avenelles et du Theil :
5. INFLUENCE DES UTILISATIONS HUMAINES DE L'EAU
K"
superficiel
des nappes*
Remarques
(4) (5) (6)
plus
Os4 superficiel
0,75 semblable
Ces observations portent sur 8 années et, pour chaque été, sur des pé-
riodes de plusieurs mois, il nous semble que, sauf exception (cf. ci-dessous),
les influences humaines sur les étiages (pompages, retenues d'eau, etc ...)
ne peuvent pas avoir faussé les conclusions précédentes. De par leur irrégu-
larité (les équipements se modifient, les lieux de pompage se déplacent, les
volumes prélevés ou rejetés sont très irréguliers dans le temps) elles sont
partiellement & l'origine de la forte dispersion mentionnée au début du 5 2,
mais nous avons veillé à ne tirer des conclusions que sur les tendances,
affranchies des irrégularités locales et instantanées. I1 faut noter jci que
cette étude avait déjà été envisagée en 1966 avec les mesures réalisées
cette date. Etant donné la dispersion observke, les résultats avaient
tellement décourageants que ces cûmpacnes de jaugeages épisodiques ont failli
être abandonnées. I1 n'a donc pas fallu moins de 8 ans pour arriver à u'a.f'-
franchjr de ces variabilités locales et percevoir les tendances.
Des estirriations rapides sur les rejets possibles de la commune de DOUE
(alimentée depuis ].'extérieur du bassin) dans le ru de l'Etang, ou sur les
pompages de COUIOhPlIEK3 dans le PU du Rognon (en amont du Champ de Tir), ont
abouti & des influences mriximales de quelques 5, sauf pour le ru clii Ro:--i?n
au Champ de Tir dans le bassin duquel 11 1/s sont captés en quasi-permanence
par la ville de COULOMMIERS. Ces influences ont été ndgligées dans la
été

11 o
6.
présente étude qui fournit simplement des ordres de grandeur pour les
comparaisons entre bassins dans leur stade actuel de fonctionnement.
Une autre influence peut-être non négligeable concerne le captage
d'une partie des eaux de Pierre Levée, en aval immédiat de la station,
par un puits qui ne devrait d'ailleurs servir que pour écrêter les hautes
eaux, c'est-à-dire à partir d'un certain débit, largement supérieur a m
basses eaux concernées dans cette étude. Ceci pourrait cxpliquer la fai-
blesse du ru du Rognon à Bibartault (§ 31). I1 n'est pas posiible actuel-
lement d'estimer les d6bit.s ainsi dérivés et donc de corriger les mesures
faites à Pierre Levée.
Pour le ru du Rognon au Champ de Tir, compte tenu de ce que nous ne
savons pas où iraient les 11. l/s captés s'ils étaient libres de s'écouler
naturellement, nous préférons travailler sur les débits observés.
VARIATIONS DES ETIAGES D'AMONT EN AVAL
Du fait de l'organisation propre d'un réseau hydrographique, lequel
est constitué de divers tronçons réunis en des confluents, les surfaces
contrôlées par un ru d'amont en aval subissent des discontinuités (conflu-
ent) qui rendent délicates les études de variation des caractéristiques
hydrologiques d'amont en aval. Ceci est particulièrement vrai lorsque, à
un confluent, se réunissent deux tronçons de caractéristiques très diffé-
rentes. En toute rigueur, il faudrait faire apparaitre ces discontinuités
dans les résultats.
En première approximation, nous négligeons ces nuances et les Fig. 6a
et Gb qui suivent présentent un ordre de grandeur de la variation des débits
d'étiage d'amont en aval sur les deux principaux rus du bassin : Avenelles
et Rognon. Ces courbes ne donnent qu'une indication de la fourchette obser-
vée sur les 8 ans d'observation. La courbe centrale (point M) n'est pas une
courbe de vraies moyennes statistiques, lesquelles ne peuvent être calcu-
lées étant donné la distribution anarchique des dates de jauseaye sur ces
e ans. D'autre part, pour une date donnée, l'ensemble des débits observés
d'amont en aval ne. forme pas nécessairement une courbe "parallèle" aux
limites ou B la courbe centrale esquissés.
Le caractère plus redressé des courbes du ru du i?ognon s'explique par
le fait que, en allant de l'amont vers l'aval, il rencontre des rus pr-ogres-
sivement plus abondants ($ 4 1). Ln situation est ir.versée porir le
-

u des Avenelles à partir du confluent "ru de 1' Etang - ru de
Fosse Rognon".
111
Les figures 6a et 6b peuvent servir à estimer des moyennes fictives de
débits d'étiages mesur6,s en des stations non observées, ce qui permet d'es-
timer leurs rapports K ($ 4).
7. ASPECTS HYDROGEOIDCIQüES
L'abondance des débits du ru des Avenelles à la Gouge était expliquée,
Jucqu'à présent, par le fait que, situé au-dessous d'un affledrement d'argiles
vertes, ce cours d'eau récupérait l'essentiel des infiltrations stoppées par
cet horizon imperméable des argiles. Or, nous voyons que les débits sont
déjà importants un peu au-dessus de cet affleurement, sur le ru de 1'Etang
à Croupet par exemple.
L'examen de la carte géologique (Fig. la) montrait d'autre part une
forte présence de sables de Fontainebleau dans la partie aval du ru del'Etan@:
(Ouest et Nord de la butte de DOUE).
De même, ce sable (par ailleurs présent çà et là dans tous les limons de
Brie) serait également plus abondant à l'amont du ru du Petit Couroy. Or
dernier est, après le ru de l'Etang, le second "chateau d'eau'' pour les
étiages du bassin. Ta station de Bibartault est, par ailleurs, sitde net-
tement au-dessus de l'aff leurement des argiles vertes.
ia liaison entre sable et étiages paraissait à envisager et nous avan-
cions l'hypoth&se que les étiages du bassin de l'0rgeval étaient moins le
résultat d'un drainage des limons localisé immédiatement au-dessus des argiles
vertes, que le résultat du drainage des nappes éparses qui peuvent être loca-
lisdes dans toute l'épaisseur des limons, mais qui sont simplement plus abon-
dantes dans les zones oÙ le sable est plus fréquent.
Quant aux étiages assez abondants du ru de Bourgogne, il pourrait s'agir
d'une influence bénéfique de la forêt sur le volume des étiages, résultat qui
commence a etre admis un peu partout (pour la zone tempérée), malgré les nom-
breuses controverses toujours en cours sur ce suJet.
Depuis peu, une campagne geophysique de sondages électriques a montré
qu'il n'y avait guère de lentilles de sable, mais des lentilles de calcaire
et meulière de 3rie. La signification géologique change, mais le rgsultat est
quasiment le même pour 1'hyclrol.ogue. Une campagne de mesures épisodiques ue
niveaux de puits a d'ailleurs confirms ces hypotheses.
ce

11 2
Dans tout ceci il faut noter l'extraordinaire différence de comporte-
ment en étiage de bassins qui, en l'absence de ces mesures de débits
partielles et épisodiques, étaient communément considérés comme remarquable-
ment (voire exceptiotuîeììement ) homogènes. Ceci est un avertissement sérieux
qui doit rendre extrêmement circonspect dans toute interpolation ou extrapo-
lation de résultats à une échelle régionale.
8. CARACTEXiISTIQUES D'ETIAGFS DES BASSINS NON EQUIPES
I1 reste à présent à utiliser les résultats ci-dessus pour obtenir
quelques caractéristiques d'étiages sur les bassins secondaires non contrô-
lés en permanence. Ceci suppose d'avoir auparavant élaboré les caractéris-
tiques correspondantes des bassins de référence, équipés.
8.1. Résultats observés sur les bassins équipés :
8.1.1. MéLhgdgs-:
L'étude de la forme des courbes .de tarissement est décevante, ce
qui n'est guère étonnant quand on considère la petite taille des bassins
et l'hétkrogénéité des aquifères d'alimentation : ensemble de nappes
plus ou moins superficielles et locales irrégulièrement distribuées dans
l'espace. I1 n'est donc pas possible d'estimer les volumes emmagasinés
avec une précision acceptable.
L'étude des étiages à l'échelle de temps du mois civil n'est pas non
PlUS très intéressante, étant donné l'inexistance d'une véritable saison
sèche piuviométrique : ï'ûrgevaï est soumis à un climat où les pluies
d'été sont aussi nombreuses et importantes que celles d'hiver. Certes,
la fonction de rendement (coefficient d'écoulement) est très basse en
été mais, s'agissant de petits bassins, l'influence de ces petites crues
d'été est fondamentale sur les débits. I1 en résulte que les périodes
d'étiage peu supérieures à 5 ou 10 jours sont relativement rares ; et
celles de 3 jours consécutifs que l'on peut rencontrer sont toujours 2
cheval sur 2 mois civils.
Les seules caractéristiques intéressantes sont celles qui s'expriment
en fonction des débits journaliers. Les plus connus snnt les débits classés,
notés Dc et les minimums de débits moyens sur N jours notés Vcn
n
(N = 355 - n). Sur l'Orgeva1, nous avons. pris l'habitude fi] d'y ajouter
un troisibme type, not8 QCn et appelé "débit caractéristique de période

continue". BI étiages la définition de ces QCn est la suivante :
113
QC d'une année est, le minimum des débits Journaliers maximums des
n
périodes de N jours consécutifs (N = 365 - n). Cette définition, un peu
complexe, recouvre'en fait une caractéristique de type "seuil", facile
à déterminer sur un graphique (Fig. 811).
8.1.2. Eésultgtz :
Les distributións des 10 valeurs annuelles détehnées, pour chacune
des 4 stations de références, sur la période 19ó2-1g0, e& très irrégu-
lière : non seulement il n'est pas raisonnable d'y ajuster des lois,
mais l'extrapolation de la simple distribution expérimentale F(Q) n'est
même pas envisageable. Dans ces conditions, les seuls résultats synthé-
tiques que l'on puisse avancer sont les valeurs moyennes et extrêmes
observées, en précisant qu'ils sont relatifs k 10 années d'observation,
le tableau 812 ci-dessous récapitule les résultats et la Fig. 812 pré-
sente les valeurs moyennes.
I1 faut noter que les débits caractéristiques QC et VC pour
n n
n = 335 correspondent k des données "mensuelles", mais pour un mois
"mobile", affranchi des limites civiles de début et fin de mois.
O, 493
O, 576
O, 652
1,oe
Note : pour QCn,et VCn, durée de 1.a Période = (365 - n) jours.
le

QCn en 1/s
8.2. Estimation des caractéristiques des bassins non équipés :
En examinarit les r6sultats rgcumks sur la Fig.fi12et en les confrontant
aux termes de comparaisons (K essentiellement) présentés au $ 4, on cons-
tate que :
- la décroissance de chacune des.3 courbes quand n croît se fait à peu
près selon la même pente pour les 4 bassins équipés ;
- les courbes de valeurs moyennes sur 10 ans VC (n) et Dc (n) sont très
n n
proches ; celle de QCn(n) est nettement distincte ;
- la connaissance des 3 points : QC335 , VC335 et Dc permet de dél-imi-
. 365'
ter un triangle représentant quasiment tous les résultats du tableau 812;
- les rapports d'abondance spocifique K définis au $ 4 correspondent à peu
près aux rapports K1 des QC (spécifiques) ; ceci est compréhen-
335
sible car de toutes les ca,ractéristiques d'étiages déterminées au $ 81,
ce sont les QC qui sont le plus éloignés des minimums instantanés
335
et les moins éloignés donc de cette moyenne d'étiages mesurés qui a
servì au calcul du rapport K. ;
- Les rapports K et K entre les deux o.utres caractéristiques définissant
2 3
le "triangle" cité précédernent sont différents de K et Kl, mais leurs
sont approximativement proportionnels et selon un coefficient indépendant
du bassin à l'intérieur d'un même type de bassins (superficiel ou profond)
ces rapports sont cependant variables avec le bassin de référence uti-
lisé ; ils sont présentés dans le tableau 82.
Le tableau 82 peut être complété en utilisant les propriétés notées
ci-dessus et 1es.connaissances qualitatives du bassin (0 $ 2 et 3) : les
rapports K1 à 5 estimés y sont notés entre 2arenthèses. A l'aide de ces
rapports et des résultats des bassins de référence, on a estimé, pour les
6 bassins non équipés, les tro'is débits caractéristiques (QC 335' vc335 et

DC délimitant les triangleybbservés sur la Fig. 812. Certains bassins
365
(Croupet et Rognon au Champ de Tir) non équipés avaient été comparés ?i deux
ou trois bassins de référence et les estimations concordent de maniere satis-
faisante, sauf pour l'estimation du Champ de Tir (Rognon) & partir du Theil
qui est faible,
I1 faut noter que seul un souci d'économie à limité les comparaisons
entre bassins équipés et non équipés ; il y avait en fait des données suffi.-
santes pour comparer chacun des 6 bassins non équipés 5 chacun d:s 4 bassins
équipks de référence.
115
La Fig. 82 récapitule les estimations. Compte tenu du caractère aléatoire
des estimations de K et Y on a également cherché à respecter approximative-
ment la forme des "triangles" qui étaient à peu près égaux sur les données
observées (Fig. 812).
MELARCHEZ
2 5'
Tableau 82 ,
Rapports entre débits caractéristiques d'étiages et
moyennes des étiages instantanés
(les rapports estimés sont entre pareritheses)
' ea11
295 a 3
(1)
3
O,? à O,¡
4 à 5
.
2
4 à 5
fia4
lCUARCHEZ (Y = M) 7 Km2 débits spécifiques
G O U G E (Y = G) 24,7 Km2
I I

11 6
9. DISCUSSION
L'absence de mesures continues sur les bassins non équipés ne permet pas
de tester la précision des estimations faites au $ 82. Néanmoins, la méthode
de comparaison (doubles-cumuls) ayant été appliquée aux 4 bassins observés de
manière continue, on a là un moyen de tester partiellement la méthode : les
estimations sont bonnes, voire excellentes, mais il était nécessaire de dis-
poser d'au moins 2 ou 3 bassins pour connaître les relations entre les
Ki (i = 1 à 3) et K.
Compte tenu de la méthode employée, et de 1'irrégul.arité des distribdions
$$ 81.2), il n'est pas bon de l'appliquer aux valeurs extrêmes observées
l'on ne pourra suffisamment bien mesurer ni la fréquence des rgsultats (repérés.
ou estimés), ni les intervalles de confiance correspondants.
Mises à part les courbes de double-cumuls qui nous semblent etre une
&ape nécessaire et fondamentale (elles permettent de c'affranchir des nom-
breuses irrdgularités locales propres aux étiages et de percevoir la tendance),
la suite de l'analyse pr6scntée ne prétend à aucune originalité et il serait
possible d'utiliser les données autrement, par exemple en étudiant les liai-
sons entre les jaugeages instantanés et les débits mensuels.
Si ces cam-a.gnes de jnugeag'en épisodiques n'avaient -as 4té réa.lis4sJ 1-3
dobits d'étiage auraient été estimés directement & parth- des 4 bassins n'user-
vés, en appliquant la règle habituelle d'égalité de débit sphcifique, la
car

117
connaissance géologique conduisant à diviser le bassin en deux groupes : type
"MELARCHEZ" pour ceux dont l'exutoire est situé au-dessus du niveau impermé-
able des "argiles vertes!, type "GOUGE-AVENELJXS-THEIL" pour ceux dont l'exu-
toire est situ6 au-dessous. A titre d'exemple, le tableau 9 présente les deu
types d'estimations pour le QC 335 *
Bass ins
MELARCHEZ
GOUGE
AVENELLES
THEIL
CROUPET
PIERRE LEWx
BIBARTAULT (P. Courcy)
" (Rognon )
CHAPii &TIR (Bourgogne)
Tableau 9
Comparaison des estimations possibles du QC
335 __---_--_____I
II II II
(Rognon 1
--_-_----_____________
__---_-___
observés
On voit sur le tableau 9 qu'en l'absence de ces jaugenges isolés les
estimations de débit d'étiage auraient été complètement fausses pour les bas-
sins du CROUPET et de BIEARTAULT (Petit Couroy), et très médiocres pour les
2 bassins du CHAMP de TIR, l'écart pour le ru du Rognon au CHAMP de TLH
n'étant que très partiellement réduit par une évmtuelle correction des
débits (au grand maximum + O,25 l/s.W), suite aux captages de COUI13Kt4TERS.
CONCLUSION
Des camnsgnes de ,jaiiFeages 6pisodiqiies en basses eaux pcrrnettent d'c qf iv-r
certaines caractéristiques d'étiages. de cours d'eau non observés en perniônence,
SOUS réserve Ce satisfaire à un certain nombre de conditions. I1 est d'abord
n6cessaire d'effectuer de nombreuses mecurcs (quasi simultanées en tous les
,

11 8
points étudiés) et pendant un assez grand nombre d'années (cycle saisonnier),
de maniere à s'affranchir des incertitudes propres aux mesures d'étiages et
des hétérogénéit6s d'alimentation des. réservoirs souterrains. Ensuite il faut
gén6ralement se 1imit.er h l'estimation de caractéristiques moyennes, les irré-
gularités citées ne permettant guère
Enfin le jaugeage, lors de ces campagnes, de stations observées par ailleurs
en permanence (équipées) est indispensable pour faire dépasser aux résultats
le stade sommaire d'une moyenne de mesures instantanée
que d'observer de$ moyennes à terme.
(de sighification
statistique inconnue) et permettre l'estimation de caractéristiques classiques.
REMERCIEMENTS : Nous remercions ici M. HIAVEC Robert, Chef de la Division Hydro-
loffie du CTOREF et. M. DUEFEUIL P., Inspecteur de Recherche & l'ORSTOM, qui ont
6th les instigateurs de ces campapes de mesures épisodiques. Nol.re reconnals-
sance va aussi à MM. TESSIER, TOL?N%, ROSIQüE (.Ta et

BASSIN DE L'ORGEVAL
Lôgende
@ Statione Hydrometrlques
1 Y6lnrchez
2 Gouge
3 Avenellei
4 Thell
5 Croupe<
6 Plerrelav&
e .
10 .
FIg 2 Réseau des étiagee
119

W
o
120

\
I-
L :-
121

122
dibitr Q
I I

123

ABSTRACT
PARAMETRES REGIONAUX RELATIFS AUX RESSOURCES
EN EAU. UTILISATION. PRECISION D'ESTIMATION
par J.R. TIERCELIN
Di vis i on H y drologie
Centre Technique du Génie Rural, des Eaux et des Forêts
(C.T.G.R.E.F.)
Ministère de 1'Agricultur.e
et du Développement Rural. France
Experience has shown that some parameters relative to monthly
and annual discharges are often similar between the various gauging
stations of a network. Calling "regi'onal value of a parameter" the
arithmetical mean of the values of this parameter in the various
stations, one supposes that this regional value can be used even in
places where no measurement are available. Theory and pratica1
aplication show that some results obtained in this way reach a very
interesting accuracy for people in charge of water management and
designers of water resources projects.
RESUMEN
Las observaciones muestran que ciertos parámetros sobre los
flujos mensuales y anuales varían poco entre las diferentes estacio
nes hidrométricas de una red. Llamando por definición "valor regio-
nal de un parámetro" a la media aritmética de los valores tomados
por este parámetro en las diferentes estaciones de la red, se hace
la hipótesis de que este valor regional conviene, si se utiliza ba-
jo ciertas condiciones, a sitios sobre los que no existen observa-
ciones. La teoria y la aplicación a un caso concreto muestran que
ciertas estimaciones obtenidas de esta manera son, debido a su pre-
cisión, muy interesantes para los responsables de la reordenación
del agua y para los proyectistas encargados de idear los equipos hi
drdulicos.

126
L'utilisation des données d'un réseau en vue d'effectuer des synthkses
régionales pour divers paranktres hycirologiques Ect une méthode pratiquée
depuis longterps tn ce qui concerne les crues, meis égzlenent utilisable
dans l'estimation des apports en eau [i]. En principe les valeurs grises
p.? les lararrètris étudias vprient evec les. conditions physiques et c3.L-
matiquos des aiffé~znts bassins, ce qui conduit à recourir k clas corrxia-
t ions mlt iples .
L'étude rnenoe dans le Sud-Ouest de 12 France montre que certzins ?a?-mktres
ont un? v-lour qui varie très ?eu B'm bôssin versant 5 l'autre,
r21

.2.3. Résultats des estimations :
127
Dans les tableaux qui suivent nous présentons les valeurs obtenues
p3ur certains pararetres régionzux, ainsi que l'estimation de l'erreiir
que l'on com3et en appliqurnt une valeur régionale d'un paramètre à un
point de la r5gion concernée. Pour donner une représentation concrète
de chaque valeur de procision, celle-ci est exprimée par la longueur
d'une série d'cbservations qui fournirait la même variance d'erreur
pour le même paramètre.
En ce qui concerne d'abord les moyennes des logarithmes des débits
mensuels, la prScision sbtenue est tra? faible pour que le résultat ait
de l'intérêt, et ceci en raison de la tro- forte variabilité spatiale
de la pluviodtrie (il pourrait en être autrement dans une raon moins
accidentée).
Pour ce qui est des vdriances drs logarithmes des dobits mensuels,
les résultats sont complétk par la valeur du coefficient de variation
des débits naturels, lié ?i Is varirnce v des logarithmes par l'expression
:
x=dex?(v) - 1 (cf. zar ex. 121).
~n outre, ï'expression L = ex? (t fi- v/2) fournit le
rapport d'un débit de fréquence quelconque au module, en appelant t
la valeur de la variable normale centrée réduite pour cette fréqu, once.
Par ailleurs, dans l'exemple traité, les résultats relatifs aux 12
stations 6tudiées se regroupent nettement en deux ensembles correspondont
respectivement à deux sous-régions : Massif-Central (stations 1, 2, 3, h,
7, 6 du schéma d'ensemble), et Pyrénées (stztions 5, 6, 9, 10, 11, 12).

128
2.4. Conclusion sur les résultats obtenus :
L'utilisation de paramètres régionaux s'avère très fructueuse
pour certains paramètres. Ainsi, dans la région étudiée, et SOUS les
conditions qui seront examinées ci-après, en une station même dépour-
L
cients de variation des dkbits mensuels et des coefficients de corr6-
lation sériels, est la même que si on avait disposé d'une trentaine
d'années d'observations.
En partant de séries courtes, le résultat obtenu est encore plus
intéressant en valeur relative. Ainsi avec 10 ans d'observations
(1959-1968) , les variances d'erreurs correspondent à une dizaine
d'années équivalentes. Pour la variance dans le Massif Central, le
nombre d'années équivalentes est même égal h 13. Ce résultat surpre-
nant est une illustration concrete de la notion de stations-années.

III - CONDITIONS D'APPLICATION
3.1. Utilisation d'un paramètre régional en une station :
129
Un paramètre régional est estimé à partir d'un réseau de
stations dominées par des bassins versants présentant des carac-
téristiques physiques plus oit moins variées.
Pour avoir le droit d'utiliser des paramètres régionaux en
un point de la région étudiée, il'faut qudes caractéristiques
du bassin versant concerné entrent ?i peu près dans la gamme des
caractéristiques physiques des bassins versants dominant
stations du réseau, sinon les variances d'erreurs calculées
n'ont aucune signification.
3.2. Combinaison avec d'autres méthodes d'estimation hydrologique :
On peut faire grief à l'utilisation de valeurs régionales de
ne donner des résultats,intéressants que pour certains paramètres
et de ne pas s'appliquer en particulier à l'estimation de modules
ou de moyennes de logarithmes des débits (du moins dans l'exemple
d'application traité). En fait, il faut observer que ces derniers
paradtres peuvent etre estimés.par diverses autres méthodes, meme
en.des points oh il y a peu ou pas d'observations.
Dans ces conditions , l'utilisation de valeurs régionales
apparaft comme un complément des méthodes existantes pour l'esti-
mation des ressources en eau, en vue de connaftre de façon précise
les paramètres de dispersion et de corrélation sérielle, qui sont
en général estimés avec une précision médiocre lorsqu'il y a pe'
ou pas d'observations.
3.3. Application d'autres régions :
La dthode est théoriquement utilisable à partir de n'importe
quel réseau de stations observée$ simultanément. Néanmoins, pour
qFe la précision soit intéressante, il faut ,utiliser des groupes
de stations suffisamment homogènes, ce .qui peut conduire à diviser
la région comme cela a été fait dans l'exemple d'application précédent.
Moyennant cette précaution, il est vraisemblable que la
dthode est applicable ?i n'importe quelle région du globe, tant
pour les débits que pour les pluies.
--------&o--------
NOUS tenons h remercier, à. l'occasion de cette public ation :
- Mme OBERLIN, du CTCREF, qui a effectué une grande partie du travail de
programmation sur ordinateur,
n M. BERNIW, d'Electricité de France, dont les conseils ont permis de mener
h bonne fin les calculs de variance d'erreur,
- M. HLAVEK, Chef de la Division Hydrologie du CTGFtEF, dont les observations
ont conduit B améliorer la rédaction de la note,
- M. de BEAUREGARD, d'Electricit6 de France, MM. EUICLE et BmIERE des 'Cir-
conscriptions Electriques Sud-Ouest et Centre-Ouest, qui nous ont fourni
les données nécessaires B l'étude.
les

130
I - PRECISION D'ESTIMATION
A N N E X E
1.1. Position du problème :
Considérons n stations étudiées simultan6ment durant m années.
Nous étudions pour un mois pkticulier p de l'année les débits mensuels
sous la forme d'une variable qui doit 6tse comparable entre les différentes
stations : en pratique il s'agira soit du débit moyen mensuel
spécifique, soit du logarithme de cette grandeur. Soit A cette vari-
able pour la station de rang J ; &e donne lieu m réalisations
1. i
m
paJ , paJ, ... paJ, à partir desquelles nous déduisons l'estimation
d d'un paramètre (par exemple moyenne ou variance de A ). Les obser-
PJ P J
vations et les variables aléatoires entrant en Jeu sont figurées dans
le tableau ci-dessous.
j=q
Nous posons par définition c o m paramètre régional la valeur
n
pdj/tl, moyenne des valeurs relatives aux différentes
.P J
... et a sont des estimations
stations. es valeurs iì &, ... P
des valeurs théoriques u 1s ... ... et pu.
1 i m
pa1 ..... pai ..... Pal
:1 :i 'm
paJ ..... paJ ..... paJ
:
:1 :I 'm
pan ..... pan ..... Pan
P J
Valeur
théorique
Le problème qui se pose est le suivant. En un emplacement J'
de la région concernée, différent des emplacements qui ont servi
h l'estimation de la valeur régionale, on décide d'appliquer le
paramètre régional 0. En fait, théoriquement, ce qui nous intéresse
est 1a.vraie vaïeur'inconnue u I prise par le parmètre à l'empia-
cement J', et le problème consJste donc à estimer la variance de

l'erreur - ) commise en attribuant à un bassin quelconque J' la
pUJ '
valeur régionale du paramètre.
L'erreur ( d - ) résulte elle même de la composition de deux
erreurs indépenbteg
131
- l'erreur d'adéquation ( u - u), de nature purement physique, provenant
P J' P
du fait que la valeur régionale standard n'est pas parfaitement adaptée
au bassin j' ;
- l'erreur d'échanti'llonnage ( u - a), de nature purement statistique.
P P
1.2. Résultats généraux :
Pour avoir une estimation de l'erreur d'adéquation, nous posons le
postulat suivant : le bassin J' présente vis à vis du standard régional
une différence du &me ordre de grandeur que les bassins 1.. . J.. . .n
en regard de ce standard (Ce p6lnt délicat e t fondamental pour l'application
de la méthode sera discuté au $ ci-après). Dans ces conditions,
nous poso s que l'erreur d'adéquation, exprimc5e par l'écart quadratique
(u - uJi)', est donnée par l'expression :
(1) = (pu - puJ)2/n
J=
Cet écart quadratique moyen, ajouté B la variance de 1'échantIllOnnage
de d; donne l'expression théorique de la variance d'erreur totale :
(2)
A u = u + var ( Q)
P P
Le problème est maintenant de rattacher cette expression aux observations.
Pour cela nous allons calculer l'expression de l'espérance mathématique
0 )2 en fonction des valeurs théoriques u et puJ.
E ( ~ Q - ~ J P
Décomposons les espérances de carrés et de produits en faisant res-
sortir les variances et covariances :
E ( p *) ~ = u2 + var ( Q)
P P
2
E(b )= u2+var(h)
P J P J P J
E ( 0. Q ) = p~.p~J + cov ( Q,
P P J P P
û
J
)
Comme 0 est la moyenne arithmétique des valeurs
P
la dernière ligne s'écrit : n
19... pQkS s.
Q
P n'

132
Eh reportant ces expressions dans (3), il vient ,:
ce qui permet d'exprimer ( u- u )* en fonction des observations :
P P J
et en effectuant pour toutes lesvaleurs de J la sommation (1) :
L'erreur totale donnée par (2) s'exprime donc par la formule :
et en remplaçant les expressions théoriques par leurs estimations, nous obtenons
en définitive :
Par ailleurs, la poursuite des calculs exigera le recours à la matrice des co-
variances liant les variables aléatoires A et relatives à deux stations
quelconques J, k. L'estimation sans biad de PAkcette covariance est fournie
Nous ne pouvons d'ailleurs écrire cette expression qu'en admettant en
principe que le débit du mois p de l'année i à une station est quasiment indé-
pendant du débit du mois p de. l'année i - 1 ?i la &me station, de faconia dis-
poser pour chaque variable aléatoire A d'une série de réalisations a
indépendantes entre elles. PJ P J
II - VARIANCE D'ECHANTILWNNACE DE LA MOYEN'NE
2.1. Expression de la variance d'échantillonnage :
1
Les
i
paramètres sont donnés ici par l'expression : d -(paj + ...
+ am)/m. PJ P J
paJ '*
P J
m

(6)
ia covariance de a.et û est donnée par ì'expression :
PJ p k
&
m
&
2
i' i")
m CBV =
côv 'PaJ' pak
133
Parmi les termes de la sommation, on peut distinguer ceux pour les-
quels i' = i", et qui correspondent ?i des variables relatives à la même
année, et ceux pour lesquels i' # i", correspondant à des années diffé-
rentes. Pour ces derniers, lorsque j = k, nous avons posé l'approximation
d'indépendance (ci. supra) ; nous poserons a fortiori la dm hypothèse
pour J # k.
k
11 vient donc simplement dans ces conditions : c8v (ptìJ,pûk)=pCJ/m
et en particulier : & ( CI ) = c /m.
P J P J
ce qui d!aprbs (4) permet d'écrire l'expression de la variance d'erreur :
n n
A> = 1) (pO-paJ)2/n -2 pcj/m
j =I J=l
De même N( 0) o servations independantes. hypothétiques donnent pow
variance d'erre& &2. Les variances étant d es le rapport des effectifs
d'observations :
nous obtenons en définitive l'expression cherchée :
j,q P P J

(8)
134
111 - YARIANCE D'EcHANTILLONNAGE DE LA VAAIANCE
3.1. Expression de la variance. d'6chantillonnage :
Considérons les définitions de variabies et de paramètres, ainsi
que les résultats obtenus au 5 1 pour l'ensemble des parametres. Pour
qu'il n'y ait pas d'atnbigulté avec les résultats du 5 2 relatifs aux
moyennes , nous remplaçons ici toutes les lettres "u'' par les lettres
Il 11
v.
Le parametre empirique dont nous étudions la variance est l'esti-
mation sans biais de la variance théorique. A partir des observations,
cette estimation sans biais s'écrit :
D'après [3], nous avons pour covariance de 0 et 0 en supposant
normales les variables aléatoires A etpAk P. J ~ k '
P J
côv ( 0 o ) = 2 ( Ck)2/m
P J ' P ~ P J
et en particulier
J2
(9) v&? (pvJ) = (pcJ) /m
(10)
d'ou, d'après la formle (4) dans laquelLe on remplace les lettres "u"
par des lettres "v" ; la variance d'erreur cherchée :
Si on juge plus commode de se référer aux écarts-types qu'aux variances,
on déduira l'erreur A\ sur l'écart-type à partir de
P
l'erreur &v sur ia variance en posant l'approximation :
P
#2&$/ps , ce qui donne :
i=/
14 # A-v/4pol P
3.2. Nombre d'années équivalentes :
Le raisonnement est analogue à celui du 0 2.2.

Une série de m années Inddpendantes donne par variance d'erreur en
moyenne (loi normale) :
Compte tenu en outre de ia relation : ,&:V/ q v
= m/N (4)
b nombre d'années fictives cherché est donné en définitive par la
relation :
IV - VARIANCE D'ERREUR DU COEFFICIENT DE COW1ELATION SERIELLE
135
Théoriquement il serait concevable d'appliquer ici des raisonnements
analogues à ceux qui ont été faits pour les moyennes et les variances.
En pratique 'le calcul paraft Inextricable et surtout nécessite l'obtention
préalable des corrélations croisées entre les observations de chaque
mois p à chaque station avec les observations du mois p-1 à toutes les
autres stations. k coût du calcul serait en définitive hors de proportion
avec son intérêt.
4.1. -ne supérieure des variances d'erreur :
Dans la recherche de la variance d'erreur affectant un paramètre régional
quelconque 0, on peut obtenir une borne supérieure de cette variance,
utilisable pour n'importe quel parametre.
2
En effet, d'une part, l'expression 2 9 - pû,) /n est un maJorat de
J -1
(pu - puJ)
2
l'erreur d'inadéquation U =
/n, puisqu'elle résulte de la
P
ComtJinaiSon entre cette erreur d'inadéquation et les erreurs d'échantillonnage
sur pQ, et pûl...pûJ...pûn.
D'autre part, nous obtiendrons une borne supérieure-de la variance
d'échantillonnage de par le raisonnement suivant.. Nous avons :
P
n A
var ( û) = var
P [ +. ... + pûJ .... + pûn)/n]
=var (a + .*.
P l
+ .*
2
+ ,fln,/n

136
~ e s stations constituant le rbseau sont relativement homogènes ;
ainsi on peut supposer que var . , varpûj,. . var Q . sont du &me
pn
ordre de grandeur K. Dans ces conditions, les expressions telles que
covar ( ) pour J k auront pour major.ant K. I1 vient dans ces
P J ' P ~
conditions : var < K
Une estimation de l'ordre de grandeur K sera fournie par moyenne des
termes var 0 d'ou :
P J
var (,fi) < i var û / n
j -GI PJ
En définitive, nous obtiendrons l'expression générale suivante,
constituant une borne supérieure de la variance d'erreur d'un parametre
j -1 PJ
1 n --
Références bibliographiques :
[l] BENSON M.A. & MATALAS N.C. (1967) - Synthetic hydrology based on regional
statistical parameters. Water resources research. Vol 3 n" 11,
.
[2] AITCHISON J. & BROWN .J.A.C. - The lognormal distribution. CAMBREGE
University Ress.
[3] ANDERSON - An introduction to multivariate statistical analysis. WIIM.

PRINCIPLES FOR THE COMPUTATION OF THE MAIN CHARACTERISTICS OF
RIVER WATER RESOURCES AT THE ABSENCE OF OBSERVATIONS ON THE
BASIS OF GEOGRAPHICAL INTERPOLATION OF RUNOFF PARAMETERS
ABS T RACT
K .P. Voskresenski
State Hydrological Institute
Leningrad, USSR
At the absence of hydrological data river runoff parameters
may be determined by means of geographical interpolation of
their values computed by observations on other rivers of the
given area. Thus it is possible to obtain principal characteris-
tics of runoff determining the rate of possible development of
rmiver water resources, category and dimensions of the projectei
hydraulic structures, On the basis of the mentioned principles
methods for river runoff computation have been developed in the
USSR for the whole territory of the country.
RESUME
En l'absence de données hydrologiques directes, les paramè-
tres de l'écoulement peuvent être déterminés par interpolation
géographique des valeurs observées sur d'autres rivières de la
même région. On peut obtenir ainsi les principaux paramètres de
l'écoulement qui déterminent les possibilités offertes par l'uti
lisation des ressources en eaux de surface et permetten de fixer
les caractéristiques hydrauliques des aménagements. Sur la base
de ces principes, on met au point en URSS des méthodes de calcul
de l'ecoulement de surface pour l'ensemble du pays.

138
The problem of river runoff computation in connexion with
water resources development and engineering projects with inade-
quabe observational data is very important for many countries
of the world.
It is known, that hydrological observabions are made on
a relatively small number of stations and never cover all the
rivers intended for water resources development. It is often
difficult to predict what rivers will be used for water manage-
ment in future therefore they are ungauged from the hydrological
point of view. Thus, in case of any particular problem on hydrau-
lic engineering difficulties arise because of the absence of
long-term hydrological observations on a particular river.
In case of absence or inadequacy of observations basic
characteristics of river runoff may be determined only by in-
direct methods based on the use of information on water regime
of other rivers in the given region or onlargerterritory.
In the Soviet Union there have been developed and introduced
into practice methods for the computation of river runoff para-
meters essential for water management projects in region in-
sufficiently gauged from the bydrological point of view.
The developed methods provided computation of river runoff
parameters in any region of the USSR with different climatic
conditions from sub-tropics to the arctic zone.
In case of absence or inadequacy of hydrological obse.mations
basic water resources characteristics may be determined by
means of geographical interpolation (in some cases - by extra-
polation) of river runoff parameters computed by a small number
of basic points with long-term observation series usually estab-
lished on main rivers of the country.
This method is physically based on distinct variations of
climakic features of river runoff, i.e. water balance elements
according to geographic zones.
Latitudinal climatic zonation is the basic law of geographic
environment variations. In general it i8 explained by cosmic
reasons determining the amount of solar radiation in different
areas of the world; the latitudinal zonation to a great extent
also depends on the total aiimospheric circula-Lion determining
water cycle on continents and islands, on the location of
continents and on the direction of sea currents.
In accordance with the location of climatic zones in plains
and altitudinal climatic belts in mountains it is possible to
observe latitudinal and altitudinal variations of water balance
elements, i.e. precipitation, evaporation and runoff. This offers
a basis for the plotting of maps of runoff or it5 main parameters
used for the determination of river water resources characteris-
tics in case of data absence. Runoff parameters of ungauged rivers
may be also determined by means of direct interpolation of their
values between the values obtained for basic points with long-
term observation series.

139
Hydrological parameters interpolation is made in accordance
with areal change of climatic factors of runoff with the account
of non-climatic effect of the environment, i.e. topography,
geology, soils and vegetation and permanent morphometric basin
characteristics, i.e. drainage area, slope, etc.
Mean runoff within any region aepending mainly on climatic
features of the region may greatly differ from its actual value
within the limits of individual river basins. In some cases nonclimatic
factors become predominant and the role of climatic
factors becomes subordinate theref ore mean runoff may exceed
the climatic norm or be less than this nom. Local factors
effect is best revealed on small rivers. With .the increase of
river basin the effect of local factors is averaged and in case
of its optimal value, runoff depends only on non-climatic
elements. On the other hand, the increase of basin area above
some definite limit causes great difference in runoff value
in different basin parts and discrepancy between its averaged
value and the climatic norm. This is explained by the fact that
large river basins are usually located within several geographic
zones
Thus interpolation of runoff over territory is possible
only for rivers with basin areas within the limits of
Am7A y A,
me K (1)
where: An+~is mean optimal basin area when runoff interpolation
is possible; Am and AK indicate its upper and lower limits
respectively.
Optimal drainage area is different in various geographic
regions. It depends on a combination of natural conditions determining
river runoff. The optimal drainage area for any region
is established experimentally.
Difference in runoff of individual rivers for any area determined
by the map depends not only on the basin size but on the
peculiarities of methods for runoff maps plotting. It substantially
differs from the methods of other water balance elements
mapping. Unlike maps of precipitation anci evaporation when data
are related to the observation points while plotting the maps,
maps of runoff are prepared by its values related to the basin
centre since water discharge measured at the discharge site is
the averaged value of runoff from the basin upstream this site.
Therefore a discrepancy is possible in runoff values determined
by the map in the basin centre and in its periphery areas. The
difference in runoff values will tend to decrease simultaneously
with the decrease of basin area.
Thus, within definite limits of basin areas gradation runoff
depends on the size of this area. The value of a critical area
in any region above which runoff is subject to no changes,
may be determined by the graph of relations between runoff and

140
basin area. It is evident that runoff values are plotted on
the graph in relative units, i.e. as depth of runoff from the
whole basin (in mm) or as specific discharge (in l/sec per
1 sq.km).
Geographical interpolation method may be used to de termine
basic runoff parameters showing the rate of possible development
of water resowces of the river, as well as the types and catego-
ries of the projected hydraulic structures, i.e. annual runoff,
annual streamflow distribution, maximum discharges, Low (minimum)
flow or periods of no flow in the river.
The dependence of different runoff characteristics on basin
area is different, In its general case it may be expressed by
equation
M.3
where: M is specific runoff from basin area A; Q is parameter
expressing runoff value independent of basin size;
n is the index of runoff reduction with the change of basin
area.
For mean annual runoff within the limits of optimal areas
ha? ; for the modulus of maximum discharge independent of
basin size nLd ; for the mndulus of minimum discharge n>i
Proceeding from the stated character of runoff reduction
maps of mean annual runoff are plotted by the data related to
the rivers with basin areas emtceeding; the lower limit of the
optimal area. These data are reduced to a long-term period on
the basis of correlation with other points having long-term
observation series and located in the given region or even
beyond its boundaries.
The duration of a long-term period is supposed to be
sufficient if standard error of mean runoff does not exceed the
accuracy of measuremenets and annual runoff computation (in the
USSR it is accepted to be equal to !%).
Data on large rivers are used only to control the correctness
of plotting runoff isolines system. For this purpose runoff
determined by the map as mean weighted value is compared with
the actual mean runoff at the outlet obtained by measurements.
Since the value of runoff on small rivers with basin areas
less than the optimal value may be less because of the effect
of prevailing non-climatic factor or it may exceed mean runoff
value in the given Brea, a correction should be introduced to
runoff determined by the map. The value of corrections is deter-
mined by local graphs of relations between runoff and basin
area.
For the USSR area two types of mean runoff reduction from
small basins have been established. In the zones of water
surplus and variable moistening river runoff from basin leas

than the optimal area tends to decrease due to incomplete
drainage of ground water within river basins. On the contrary
in arid zones runoff tends to increase with the decrease of
basin area due to decrease of losses by evaporation.
Appropriate corrections have been determined for rivers
in ùifferent geographic regions.
To determine mean runoff of ungauged mountain rivers
local graphs of relations between runoff and the altitude are
usually usea. Mean basin elevation essential for this purpose
is obtained from topographic maps. As a rule, within the limits
of every geographic region there are several local dependences
of runoff change with the altitude. The number of these graphs
depends not only on the range of altitudes and mountain slopes
exposure, but also on the number of observational points in the
given region. Their increase leads to new local graphs. Thus,
the available graphs are averaged for some territory.
Normal runoff is the main water resources characteristic.
But when planning water resources development it is essential
to obtain data on runoff for wet and dry years with different
frequency of occurrence. In the practice of hydrological computations
in the USSR probable runoff values are obtained by
distribution curve of Pearsan III in its integral expression
i.e. frequency curve. Normal runoff, coefficient of variation
(C ) and coefficient of asymmetry (CS) are frequency curve
pallameters. In case of observational data available the parameters
are computed by mathematical statistics methods. In case
of data inadequacy these parameters are established by geographicalinterpolation
method.
The computation of variation coefficient of annual runoff
is based on the account of effect of climatic factors variability
and factors of natural runoff control. Experimental data
show that runoff variability tends to increase with the debrease
of its value. Therefore maximum variations of runoff are observed
in arid regions, while minimum ones - in the zone of water
surplus. The normal runoff itself may serve as an index of nlimatic
variability.
141
Among the factors of natural runoff control the capacity of
river basin is of the greatest importance; it determines under-
ground water storage. The basin area is an indirect index of
basin capacity .
An empirical formula has been obtained for the whole USSR
territory with the account of the two mentioned factors:
here:bfo is normal runoff (l/sec per 1 sq.km); A is basin area
?sq.km);B is parameter computed by substitution of She values
known for the river-analogue in the given area into equation (3).

142
The meaning; of coefficients of asymmetry in case of the
absence of observations is determined by the ratio of Cv and Cs
established by the rivers-analogues in the given basin. If
no analogues are available in the zones of water surplus or
variable moistening the follbwing ratio is accepted Ce = 2 C
and for arid zones C, = 1.5 + 1.8 C,; for extremely arid re#&ns
cs = 1.5 c,.
When computing maximum discharges in case of no observations
it should be taken into account that the flood character on
rivers in any geographic region is mainly determined by clima-
tic features and therefore data obtained from observations on
some rivers are extended to all the rest of water courses of
the same region.
Different empirical formulae are used for maximum discharge
computation, their parameters are determined by observational
data on some rivers of the region under consideration.
Rational formulae are widely used which are based on the
account of maximum or extreme rainfall intensity during flood
concentration; in general they may be given as follows:
where: K is coefficient of dimensionality; h is maximum rate
of rain or snow melt during lag-time ‘i ; dis coefficient of
runoff during the same interval.
The time of flood concentration is often determined by
empirical relations between this value and river length or
basin area. Runoff coefficient is accepted by the analogy with
flooãs on other rivers proceeding from the general nature of
top cover and topography.
For practical computations it is reasonable to use reduction
foriïiulae of a general type:
where: maf-is maximum specific discharge;
Je - is extreme specific discharge if A-0 and c= i
C -is addition to basin area taking into account the
character of runoff maxima variations in case of
small basin areas;
n. -is the index of maximum runoff reduction.
The parameters in the formula are established on the basis
DQ processing of data on maximum discharges in the given region.
Minimum (low) flow is determined by the rate of underground
water drainage by rivers. The amount of underground water
discharging into rivers depends on the number and capacity of
aquifers cut through by river channel. The depth of erosion cut

143
usually tends to increase with the increase of basin area. Therefore
maximum runoff varies with the change of basin area.
In this case the optimal basin area is supposed to be the
area when rivers cut through all the aquifers of the given
region. It is possible to plot a map of minimum flow for such
rivers to be used for computations.
For small water courses local graphs of relations between
minimum flow and basin area are established.
The rate of wa.ter resources development of some rivers
is determined by the duration of no flow period. Such rivers
occur in arid and permafrost zones. The duration of dry period
is also determined by the basin area size.
The experience of the use of indirect methods for the computation
of main characteristics of water resources of the USSR
rivers shows the eqediency of their use in countries with
different climates and different physiographic features.
1. Voskresenski K.P., Norma i izmenchivost godovogo atoka rek
Sovetskogo Soyuza (Annual runoff norm and vari-
ability for the USSR rivers), Hydrometeorological
Publishing House, Leningrad, 1962, 545 p.
2. Voskresenski K.P. Gidrologicheskie raschety pri proektiro-
vanii sooruzheniy na malykh rekakh, ruchiakh i
vrememykh vodotokakh (Hydrological computations
for engineering projects on small rivers and
temporary mater couraes), Hydrometeorological
Publishing House, Leningrad, 1956, 468 p.

EVALUATION OF WATER RESOURCES OF MOUNTAIN AREAS IN CASE OF
ABSTRACT
ABSENCE OR INADEQUACY OF DATA ON RUNOFF
Vuglinski V.S.
State Hydrological Institute
Leningrad, USSR
V.A. Semenov
Kazakn Research Hydrometeorological Institute
Alma-Ata, USSR
Normal annual runoff, as water resources indicator, may be
determined for mountain areas on the basis of taking into account
the laws of runoff distribQtion over territory and according to
altitudinal zones established by the observational data from the
gauged rivers. These laws are connected with latitudinal and
longtudinal zonalities, with differences in the nature of the
underlying surfaces and slopes exposure relative to moisture
carrying air fluxes. These laws are quant'itatively expressed by
regional dependences of normal runoff upon mean basin elevation
and the rate of its glacierization. Another method, providing
the determination of normal runoff also in case of complete
absence of hydrometric data, is based on a combined solution of
water and heat balance equations with the account of the energy
component of the water cycle. Data from standard meteorological
network are used for computation.
RES UME
Le débit moyen annuel, en tant qu'indice des ressources en
eau, peut être déterminé dans les régions montagneuses en se
basant sur les lois de distribution établies pour l'ensemble du
territoire à partir des données obtenues aux stations de jau-
geages, en tenant compte d'une división par zones d'altitude.
Ces lois sont liées à la situation géographique (longitude et
latitude), qui se traduit par des différences dans la nature du
sous-sol et dans l'orientation des pentes par rapport 3 la di-
rection des masses d'air humide. Elles se traduisent par des
relations régionales quantitatives entre le ddbit moyen d'une
part et l'altitude moyenne du bassin et le pourcentage de gla-
ciers d'autre part. Une autre méthode, permettant d'évaluer le
débit moyen en l'absence totale de données hydrométriques, met
en jeu la résolution de deux équations, relatives l'une au bi-
lan hydrologique, l'autre au bilan thermique tenant compte des
termes énergétiques du cycle de l'eau. Les calculs sont effec-
tués à partir des données fournies par le réseau m6téorologique.

146
A hydrometric network is very scarce in mountain areas since
they are hardly accessible. Methods for the evaluation of surface
water resources in case of inadequacy or complete absence
of observational data are based either on the account of the
laws of space distribution of normal annual runoff @pical of
the gauged regions or on the application of an appropriate
design scheme.
Space distribution of water resources (undisturbed by man's
activities) in mountains and in plains is the result of hydrometeorological
factors interactions (precipitation, air temperatue,
evaporation) with underlying surfaces. But unlike plain areas
where latitude and distance from the sea serve as main factors
of heat and moisture ratio chasacterizing water resources, the
orography becomes the main factor of river runoff formation in
mountains. The effect of topograpb on river runoff results in
its direct influence on the flow velocity down the channels,
depending on the slopes of watersheds and bqsins top cover. But
the most important effect of topography on water resources is developed
by its influence on water balance elements ( recipitation,
evaporation, change of water storage in river basinsl;. This
effect is of a particular importance in mountainous arid zones
of Asia. For example gross precipitation in the mountains of
Middle Asia, Kazakhstan and Mongolis ranges from 150-lOO mm and
less in areas protected from humid air masses (hollows, slopes
of unfavourzble orientation) up to 1500-2000 mm and more on
favourably oriented slopes of periphery mountain ridges relative
to air fluxes. The increase of precipitation according to elevation
and simultaneous losses by evaporation on high elevations
due to low air temperatures stipulate the improvement of conditions
of river feeding characteristic for mountain areas as far
as the basin elevation increases.
In connexion with the stated above, methods based on the
establishment of relations between runoff and orographic peculiarities
of the location have been accepted in the USSR for the
evaluation of water resources in poorly gauged mountain areas.
These orographic peculiarities are as follows: elevation, slope
and orientation of the region relative to the direction of moistw?e
transfer.
For the evaluation of mean annual runoff
Q as an index of
areal water resources the relations between specific discharge
and elevation of the watershed, which in majority of cases is
expressed as mean weighted elevation (H) are widely used.

147
These relations are established for every region nn %hi? bapis
of data obtain9d for gauged watersheds and are uwed ta /?cl;om-ine
normal runoff of ungauged watersheds in the a2progriat;e repien.
Sirice high mountain areas are very poorly gaiged tbs cvaluri-
tion of water resources for such areas Is madß according im
extrapolated portions of the dependences Cj E f (II). Data on
precipitation, ablation and liquid glacia?. runoff are used %o
make extrapolation more reliable .
In case of data on glacierization w&lable they are ursd
both for the extrapolation of dependences Q. = f (HI an0 ?or
direct conput;ation of mean annual runoff iron relat;ively smdl
high mountain areas. According to ths investigations made by
V.L. Schultz /1/ the rise of such empirical relations provi

14 8
and subsoils- m is the exponent of runoff reduction; I is mean
basin slope [ '/oo).
When river basins are composed of karst rocks the effect of
other azonal factors on river runoff may be neglected and only
runoff changes caused by karst may be taken into account. Hence,
for example, an appropriate correction (with negative sign) to
zonal runoff for the Kazakh folded area is computed by empirical
e quat ion :
(2)
where: Q is correction (l/sec per 1 km') due to karst effect.
For the evaluation of zonal normal runoff the maps of isolines
of normal runoff are used; these maps are compiled by
observational data mainly from the basins fully located within
one climatic zone. The method of isolines is usually preferable
in case of natural water resources evaluation for large river
basins and for poorly gauged mountain areas as a whole.
Very few maps of isolines of normal runoff plotted for
particular mountain areas are available in the USSR. Since the
initial &ta are linited, these maps are small-scaled, mainly
of 1 : 2 500 O00 scale not more; these maps are hardly suitable
for the estimation of normal annual rynoff from small and middlesize
watersheâs not exceeding 1000 km . Thus, the method of isolines
provides a sufficiently accurate determination of normal annua3
runoff mainly for large mountain watersheds (more than 1000 km 1.
The method of collective analom based on the graphs of relations
between runoff and mean basin elevation is used bn majoritg of
cases for the computation of runoff from middle-size basins, i.e.
more than 500-600 km2. When computing runoff from small basins
and often larger basins the use of the two mentioned methods
is not always reasonable. It is often explained by inadequacy of
initial information on runoff and by a considerable effect of
azonal factors in mountains; in this connexion even basins with
similar elevation within the same mountain region may differ
greatly in the conditions of runoff formation and its quantitative
characteristics.
In such cases the determination of normal annual runoff
from mountain watersheds located in conditions of sufficient
and excessive moistening is made by a combined solution of
equations of water and heat balances. An indubitable advantage
of this method is in the fact it ensures a relatively accurate
determination of normal annual runoff not only from large
mountai basins but from watersheds with the areas not exceeding
1000 km 9 .
The computation is based on the equation of mean long-term
annual water balance where normal annual runoff is determined

149
by the difference between precipitation P and evaporation E:
Q= P-E (3)
When usin@; this equation it is essential to obtain a
reliable accuracy in determination of noml annual precipitation
and evaporation. The method is applicable for such mountain
areas where the available hydrometeorological network provides an
objective evaluation of precipitation distribution compared with
runoff. In this case it should be kept in mind that evaporation
is less variable over area and altitudinal zones compared with
runoff and it ILK' be computed for mountain watersheds with a
sufficient accuracy .
It should be noted that in equation (3) underground water
exchange with adjacent watersheds is not taken into account1 As
a rule, this component is not big in mountains especially in
permafrost zone. But in cases when its valiles are commensurable
with the other values of equation (3) the account of this
component is essential.
The determination of one of the parameters in equation (3)
i.e. normal annual precipitation, is made with the use of
graphs of precipitation and elevation with the account of local
orographic peculiarities. In this case correction should be introduced
for the initial data which take into account the underestimation
of precipitation by standard precipitation gauges.
Computation of normal annual evaporation is made by equation:
where: W is radiation balance of the moistened surface; Wa is
tubule& heat exchange; L is latent heat of evaporation; e
is base of natural logarithms; th is hyperbolic tangent.
Equation (4) is a precised version of M.I. Budyko's equation
/3/ due to Wa value.
In the right of equation (4) three unknown parameters are
intrmduced: P, W and W .
The way of de$ermina!ion of normal annual precipitation ie given
above .
The value6 of radiation balance of the moistened surface may
be taken from appropriate maps or computed. For many areas within
the USSR territory there exist design formulae for W
determina-
tion according to latitude and elevation of the loca%LQ. In
particular, for Trans-Baikal area /4/ such formula may be
presented as follows:

150
where: '9" is mean watershed latitude; h = (H - SOO ) is the
exceedence of mean watershed latitude over 500 a.s.1. When
computing radiation balance of the moistened surface of
mountain watersheds its variations according to the exposure
and steepness of slopes are taken into account.
The uetermination of Wa as well as Wp is made either according
to appropriate maps or, in case of available data on mean
long-term monthly air temperature, water vapour pressure and
total cloudiness, by a combined solution of heat balance
equation and the equation of Magnus. This method is presented
in detail in some publications /5,6/. When using equation (4)
it should be noted that mean long-term annual values of W
and W are rather stable characteristics slowly changing {ver
territory a d altitudinal zones.
After computing normal annual evaporation it is possible to
estimate mean long-term annual runoff by equation (3).
It should be noted that the presented scheme of computation
may be changed for watersheds located in low and middle-height
mountains. As to watersheds covering high mountain zone, this
method of combined aolution of water and heat balances for
normal annual runoff determination may be applied as well;
the difference is that the number of terms in water balance
equation increases ( it is essential to take into account the
ablation of glaciers, melting of snow fields, separate account
of evasoration from different types of underïying surfaces
in high aountains, i.e. ice, snow, talus and rocks).
The evaluation of long-term variations of surface water
resources of poorly gauged mountain areas is usually made &y an
analytical frequency cume of annual river runoff. The values
of runoff variation coefficient C, essential for its plotting
are evaluated by their regional empirical relations be tween
normal runoff, mean weighted elevation of the watershed or the
glacierization area. These relations are established according
to observational data from the gauged rivers, and the coefficient
of asymmetry C is established by the ratio of this parameter
and coefficient of variations for the gauged rivers of the region.
Ìimpirical dependences of variation coefficient of annual
runo?f and . the determining factors for ungauged mountain areas
are usually
expressed by equations:

where: a, b, c and r are regional parameters; H is mean
weighted elevation of watershed; is exponent of watershed
glacierieation(percentage from the total draina@ area),
The selection of the equation depends on the character of
151
river feeding. The dependence of variation ûoeff icient of annusl
runoff and specific river discharge (equation 6) are used maim for areas with a considerable portion of rainfalls in mountain
river feeding.
When estimating C, for ungauged mountain rivers of the arid
zone where the effect of snow melt water is of particular
importance, the preference is given to the relations of variation
coefficient and mean weighted watershed elevation (equation 7).
For rivers located in basins where glaciers cover more than
IQ% of the drainage area the dependence of C, upon mean weighted
elevation is usually broken and the preference is iven %o empirical
relations between Cv and basin glacierization ? equation 8).
If there are no data available on the amount of glaciers then
instead of glacierization rate for C, determination of ungauged
mountain rivers its indirect indices are sometimes used showing
the relations between the area of altitudinal zone where glaciers
are located and the area of the whole basin.
For the determination of variation coefficient of-annual run-
off of mountain rivers the equation recommended by LP. Voskre-
senski /7/ is used as well :
where: X is regional parameter.
The coefficient of asymmetry of mean annual runoff for ungauged
mountain rivers is usually accepted as C, = 2Cv.
RE F E R E N C E S
1. Schultz V.L. Reki Srednei Asii (Middle Asia rivers). Pt.1 and
2, Hydrometeorol. Publ. House, Leningrad, 1965.
2. Lavrentiev P.F., Semenov V.A., Khitrunova M.S. Uchet sredaei
vysotg vodosborov, ikh orientatsii i azonalnykh faktorov
podstila jushche i poverknosti gri rasc hetakh srednego
godovogo stoka rek Severnogo Kazakhstana (The account of
mean basins elevation, their orientation and azonal facGor’P
of the underlying surface when computing mean annual runoff
of north Eazakhstan rivers), Trana. of Kaz. NIGBdI,
VOI. 41, 1971.
3. Budyko M.I. Teplovoi balans zemnoi poverkhnoeti (Heat baleme
of the Earth’s surface). Hydrometeorol. Publ. House,
Leningrad, 1956.

152
4. Vuglinski V.S. Yetodika rascheta radiatsionnogo balansa
gornoi territorii i ee primenenie na primere
basseina r. Vitini (Methods for the computation of
radiation balance of mountain area and its applica-
tion illustrated by the Vitim river basin). Trans.
of GGI, 1972, ~01. 199.
5. Vuglinski V.S. Raschet normy godovogo stoka neisuchennykh
gornykh rek s primeneniem uravneniy vocino o i teplovo-
go balansov (na priiaere basseina r. Vitimy. (Compu-
tation of normal annual runoff of ungauged mountain
rivers WI th the use of equations of water and heat
balances !strated by the Vitfm river basin).
Trans. of &(GI, 1972, vol. 200.
6. Anàreyanov V.G. Vnutrigodovoe raspredelenie rechnogo
stoka (Annual stream flow distribution), Leningrad,
Hyàrometeorol. Publ. House, 1961.
as Voskresenski K,P. Horma i izmenchivost godovogo stoka rek
Sovetskogo Sojusa (Normal annual runoff and its
variations for the rivers of the Soviet Union).
Leningrad, Hydrometeorol. Publ. House, 1962,

IMPROVEMENT OF HYDROLOGICAL INFORMATION FC?. FX.?:LCT
DESIGN BY SHORT TERM MEASURES -
1. Introduction
GENERAL REPORT
by
Dr. John Rodda
At the present time, when the amount of attention given to all asjxcta of
th:: environnent is growing rapidly, the collect ion of enviromental
information is increasing in importance, especicaìly for use as o m ~f the
bases Of measures for enviromental protection and combatting pollution.
Kere the hydrologist and meteorologist are amongst the more fortam.tc of
environmental scientists: they probably have at their dislwaal a lager
body of information relevant to their needs than is available tc oihcr
scientists in their particular fields. On the other hand it is true to c . ~
that many watcr resources projects are designed with inaclc::imte data, indcd,
sometimes with virtual1.y no date zt all.
likely that wrong decisions will be taken, that wrone critcria will be
selected and that inappropriate and uneconomic designs will be adopted.
The end product can be a water resources system which pstly or entirely
fails to meet the objectives that were foreseen for it, the bexr‘its it
pro&uces,bcaring little relation to the capital invested.
2. zata and Networks
The classic response to this situation is to collect noce and m ro data
for thc national archive. Aniassine; a large quaiitily or^ ìucirologi.:al
information is even seen as an end in itself and virtus.lly mJ’ 5.ncrease
in the total is considered of value.
particularly where national data collection propmr,:es are nct plmned
scieiitifically.
types, generally rainfdl and streamflow records, other LW’C~S heina very
largcly neglected; for example, sediment surveys and soil muistiire records.
St&tions in the data network are often badly distributed, t5m-C arc
differences in the lengths of records and their quality is frcqixntly
suspect. Such networks usually produce information inefficienti.;? and
uneconomically - the oppoeite of the true objective of network &si@.
Scientific design would produce a system whioh would add the moi;:
know1ede;e for the least effort. This sytem would ncit only consïf;t of
station-type time series observations, but also of surveys of various
kinds, including questionnaires 2nd oensusee. It would not be ti riKi8 c;’stcrn,
but one that would be altered and amended in response to needa and as
objectives change.
Lui if th.e purpose of a network can be stated clczrly .then itn ?tasip is
likely’to be fecilitated.
Inadequste duta m ke it more
This is not alwRjr:; the CasQ,(
Usually the bulk of the information is of cno or two
üf ooume networks iisly have various objectivce
F’or project design purposes a network would have

I54
a different form and composition from a network installed for researoh
purposes, although both would be COmPOnCntO of and contributors to the
national network which would itself provide the overall information framework.
in the oontext of this symposium it is important to consider firat
the form of the national network and the attributes that would fit it best
to the needs of project deoign end second the project nbtwoa itself.
3. Data PrODertißS
Langbein (1972) wpsted that water data, And thus the n dork for
acquiring them, have three intrinsio properties:
and continuity. Imoartialitg relates to the aganoy or Bgenoiee that
operate the network and archive the infomatttion from it. In its
Perception of data problems, the agency itself tends to introduce R bias
in the data, a bias towards ita own speaialty. Por example M organieatioii
concerned with water suppl.y wouiû tend to dieregard infomation about
floods and the means of colleoting these data. One solution to this
problem is for basic data oollection to be the remit of en agenay without
opcratiûïìal or cxccative roles, such ôs onc ifi-dve3 in reeearch.
Relevance of the data then becmes important because thie type of data
agency is one stage removed from the problems to which the data arc applied.
On the other hand, this avoids what Langbein calls the "squeeking wheel
principle whereby attention is continually uircctod toward8 mrrently urgcrì:
problema at the expense of the existing balance of the network and its
oapacity to be employed for solving future as yet -own problems,
Continuity follows from the fact that hydrological datu are time-dependent,
hence their collection needs to be oontinuous. Continuity ia at risk
at times of national atresa euch as during tine of war, -turd disaster
or finanoiai stringency.
agencies are required to alter their progremmos and those not directed to
olear and easily recornisable objeotives tend to be curtailed.
4. The Fctwork
impartiality, relevance
Organisationai change ia also a haed;
Ideally the oountrywide hydrologiosl network should preoeed development,
invariably the reverse is true. Most national networks have resultßd
from ad hoc responses to particular problems. Pow networks Beem to have
reached the optimum in termo of distribution of stations, types of data
and form of amhive. Perhaps the difficulty of deoiding what the
optimum is one reason for this, although Dswdy et al (1972) put forward the
idea of the leoel of information being optiwl when decisions involviq
this infomation become insensitive to its m her inorease. !Phis ooucept
has the difficulty that the optimum information level and thus tho optimum
network differs for different objeotivee, so that it m w not be readily

applicable to a malti-purpose countrywide network.
problem of scale and the fact that the component parts of the hydrolo&al
network may have developed separatoly crnd to differing degrees.
155
the evaporation network and the water quality network are unlikely to have
been co-ordinated and this may apply to the other variables.
5. Scale of Networks
The factor of soale ia pu1 important point to consider in project design for
there are differences between information needs on national and local
scales. At the national level tho network would consist of long term,
bench mark primary stations for sampling in the main, variations in time.
The distribution of theso stations would relate to the degree of the
country's development and its hydrological heterogeneity. in other words
a country with uniform climate, geolot;y and relief, a small number of
inhzbitants utilizinl: few of the resources woulù most probtrbly possess
a less developed network than oneaith diVerBe physical features, a
large population and a strong industrial base.
lhe twu couiitrias would present a siuiLr GGZ~IYLS:, bUt both crtrcrkz
should necessarily be capable of utilization, at the vem least, for
accounting for the resource and for the warning of hazards.
1Òcal scale stations would tend to be of a short term, secondary tyae
(Gandin 1967) established to sample variability in space.
network would be a major part of this secondary network.
secondary network would mostly serve current information needs, the baoio
countrywide network would satisfy future demands (Laagbein 1965).
6.
Use of Basic Network for Roject Desim Purposes
Information from the basio network can be employed to provide estimates
of hydrological variables for any Given point within a country and can
thus be applied for project desi-, purposes. The estimâtion may be
undertaken SbjJly bf interpolation between isopleths on B countrywide
map, constructed from the basic network. observations.
data fram the network can be applied in a mapping technique suoh as
the application of the grid system for storage and processing hydrological
information from a large area and its use in relating the hydrolgical
variables to the area's physical characteristics (Solomon et al 1968).
Maps of mean annual precipitation, temperature and waporation were
constructed by this method and then employed with measures of the
topograpliy to develop a map of runoff.
There io also the
importanoe of maps, maps also being important to that regionalisation
type of approaoh. For example, measureß of the pertinent eurface or
For exanple
The information needs of
At the
The project
Whereas the
Alternatively
This approach stresses the

156
subsurface features of an area whioh can be mapped or measiired in the
field are related to a otatistic Of the hydrological vari;:ble in queetion.
Relationships between mean annual rainfall amounts and niemures of -Lhe
topo6.raph.y such ae alevationplope and exposure have been widely de.lcrmined,
likewise relations between the mean annual flood and catchment characteristics
including area,channel slope anù drainé@ density.
The paper 'sIrnprovement of runoff records in smaller watershede, based on
permeability of the geological subsurfacess by Dr Bala&Xun follows this
2
type of appl-oach. Records from basins less than 25OKu1 in area locaied
in li!? USA and Central Europe were used in thio study. P~ak runoffs
(m3 sec -' Km2) of 100 year return period were related to basin size, the
geological character of those basins assessed from permeability
certain storm sires and intensities and also to the slope of the basin.
Dr Halasi-Kun siicgests there is evidence for a significant correlation
betuem permeability and peak runoff but that the geological effect
2
"fades" for basins larger than 235h . This paper also ermines 50 year
2
low flows in the same bapjns (1 sec-' Km ) and again relates ihese flow6
to p3OlOC;iC;rl characteristics. The author concludes that including
permeability improves this type of approach and of course he is correct
in the sense that the inclusion of any further uncorrelated but
quantifiable catchment characteristics is a step forward.
he hacl included as much of his basic data that it was possibla to
publish.
It does not deal with estimation from catchment characteristics but with
reconstruction of records from a shorter period of more complote records
applied to a longer period of more limited information. This is the
paper by Ih. Kovecs and Dr Kolnar: "Determination of snow water
eqyivslent and enow melt water by thickness of snow cover data''
One wishes
Another paper submitted for this section of the programme which (]:!ala
uith information from the basic network is not strictly in the mine cztegorg.
Snow
depth has been observed at about 1000 stations in Hungary for 100 years
and since 1960, water equivalent has also been measured at 60 stations.
From studios of the bulk density of fresh snow ( Y min), mow saturatcd
with capillmy water ( Y k) and melting mow ( 5' ma), snow depth and the
number of layers of enow developed during accumulation (R)o tanned the
critical bulk density, and R.
between y max and R and dao a method for obtaining the duration of
melting from air temperature records during the molting period. These
relations are then applied to the hindcasting of snow water equivalents
from depth measurements and to forecasting duuration of the melt period
and potential volume of melt water.
Then they arrive at a similar relation
The predicted snow water e-ivalent?:
are oomparcd with meamred volumes at one site for part of 1963 and the

match between them seems reasonably good.
provided for the enow melt cdoulations.
15 7
The paper by Ur Beard œHydrological dzta fill in and Network Desi&* is
similar to the previous one in that it deals with the extension of
records from stream gaugingetations with long recoi.de to stations with
only short ~%cordS. A stoolustic model, which can accept montly data, is
based upon multiple linear regressions, using fransformed variables, and
these are derived from each station for eachGalendar month. To illustrate
what happens as a result of chanoe variations in small samples, 10,000
5-year sample8 were drawn from a normal population and their means and
standard deviations oalculated. For each sample items were perated and
their location in the parent population identified. It was found that in
the oase of extremes tco many extreme vdues were generated indicating a
bias in the estimates of extremes.mzde from small samples. To overcome
this bias a transom hinotion was generated.
(equation 2) showing the nucber ~f itefie 3: iiaedid in the shrt-tem
recorà than can improve the accuracy of the short-term mean so that it ia
reliable as the mean obtained l'rom the lon&er reoord Values of I, are
tabulated for various correlations and samplo ßizes against the longer
record length (table 2). Four different %year soquencee were selected
from 40 years of reoord at one station and for each of tho four cases tho
remaining 35 years were filled in.
for the 40 years of reconstructed record ware compared with the actual
40 year mean. The process wzs repeated Sor 3 other stations and a matrix
is presented showing 8 comparison of statistics derived from these recodo.
The author concludes that for correlations above 0.95 short records need
not be continued beyond 5 years but belar 0.8 short records should be
oontinued. Between these values a study of regional variations would
reveal the relative advantage of continuing existing stations or starting
new ones.
A similar cornpanson is not
An equation is given
Then the mean flow for the %years and
Arising from studies like theee is the question of how estimateo compare
with field measurements. Nash and Shaw (1966) in a study of United Kingdom
floods, disoovered that even a single year of discharge records produced
a more reliable guida to the mean annual flood than the methods of
estimation then in current use. lore recently the UK Flood Study Team
have found (Sutcliffe 1973) that estimated mean annual floods are within
2 30$ of the mean of the measured annual maxima for the catchments
studied.
t

158
7.
Short Term Instrumental and’Observational kessurez
There are a number Of constraints to the design of a project network,
time probably being the most important. Usually there are only 2 or 3
years between the time a Project is conoeived and the time when the
design has to be finalised. The risks involved in employing these
2 or 3 years of information is then a maximum but the risks diminish as
the record length inCrcaseS.
more records may be COStly in terms of loss of benefit from the water
resources system. At some Point a balance will be struck‘ between risk
and benefit: this point will depend upon factors such a8 the type of
project and the Proportion cf the resource to be develo2ed. Amongfit the
other constraints are those of finance, the skills available and the
location of the project. Uith adequate funds a well-equipped team can be
brought together and the project placed on a firm footing.
location in terms of OlimatC and topopa& can be a very considerable
handicap even with a well funded project.
However, deferring a project to accumulate
Depending on the nature of the projeot and the informstion it requires
the defiign of the network hinges on, the answers to a nimber of
An unfavourable
questions:
1. How is the information to be obtained?
2. How many sites need it be obtpined from?
3. Where are these sites to be located?
No papers were submitted describing an:! advances in inckwmentation or gouncï
based survey techniques that might be applied to project design.
are new instruments and new methods that could be employed to acquire
information for pro jeot deoign. Batterj operated magnetic tape recording
rain gauges and telemetering gauges proùuce more information more rapidly
than conventional instruments, Automatic weather stations and automatically
operated neutron probes do the same in the fields of evaporation aild soil
moisture measuremcnt and then there are automatic dilution gauging devices
for atream flow measurement to say nothing of the other methods of river
gauging that do not require the conventional stilling well and structure
in the channel.
recent years, but there was only one paper submitted to this Section in
this category.
Remote sensing techniques have dvaiced eiiormously in
Hhat about the use of aerial photography,radar and the
various forms of imagery from satellites?
exception is the paper by Dr leijerink “Svaluation of local water resouses
in a semi-arid hard rock region, by using photo-hydrological indices”.
Yet there
They are not mentioned. The
By interpreting aerial photographs an assessment was made of the local
water resourcës in part of the Cuddapah Basin in south India. Following
field surveys of the area‘s geologyssoils and land use the next stop was

to divide the basin into hydrologically homongeous 1andBczpes. The
hyydrology of these landscapes was deduced from the photographs from
159
features affected by surface flow and from the characteristics of the
superficiel deposits and solid geolow. For example within a particular
landscapo the yield of a well is assmiod to be directly related to the
size of the irrigated area.
for one landficape and also the recharge areas for those wells;
relationship between these factors giving a guide to yield &B a function
of recharce area.
and the results ct the interpretation were checked in the field for the
different relationships.
The question of the number of sites to be sampled is frequently answered
in terms of the funds available for installing and oparating the netuork.
For areas without records of any cort,arriving at a number is particularly
difficillt for the number and location of stations hincos OA the distribution
of the hydrologiczl variable.
according to a predermined grid or according to the distribution of elevation.
Another method would be to delimit areas of homogeneous topograpbjand
~eolocr and to site one station in each area.
exist it is usually far simpler to deterinine where tÒ site additional
gauges.
This is one of the topics discussed in the paper by Kessiers Delhomme and
Delfiner: "Applicaton du Krigeage a' l'optimisation du'une coinpagne
Pluviometrique en zone aride". The subject of this paper is the
2
Kadjemeur Wadi in the east of Chad , a basin 245Km in area containing
33 rain gauges. Here the technique of Kriging is employed to determine
the o ptim weights of the gauges in thc network for the calculation of
the mean basin rainfall.
of the method and then they apply it to the description of a storm on
6 AU~UQ~ 1966. Thc map obtained by ttìa technique of kriging niötohes the
hand drawn isobyetal map very well; in general it produces a broader
smoother interpretation. A comparison of the estimates of the mean basin
rainfalls is given for Krigingand three other methods, the "hiessen,
mithmetic mean and planimetering methods. In general the results are sirnilar
but the concentration of gauges on the western side of the basis distorts
the arithmetic mean results ifi some storms.
the problem of where to locate an extra page.
gauges
the barain or in the centre.
The areas irrigated by wells were determined
the basin, where the gain of information is a maaimum b? construoting
isopeths of gain fiyre 8 shows where these two points axe located - on
the
A similar exercise was under-taken for surfzce wa.ter
One method would be to site stations
Where some stations already
The authors provide a background to the theory
Finally the authorsooncider
From the distribution of
subjectively one would choose a site at the south eastern end of
Kriging ~ 110~s determination ofthe point in

160
the south eastern boundary and in the centre of the basin. There are
various metliods for computing the mean basin .rainfall th8-t have been
advocated recently - various forms Of surfaïri fittinc Sor example.
problcm is that all these methods rely on the accuracy of the point
rainfall measurements which we kno~ as being far from accurato. The
question of' what is the true mean basin rainfall remalns unanswered.
CONCI Ir3IONS
In the verj lengthy title to thio section in the prop-mme for the
symposium, two separate topics were raised, first the improvement of
hydrological information by short term measures and second the value
of such measures, particularly as expressed by project economios. While
one might argue that the first topic is covere? by the five papers
reviewedobviously,the absence of any papers for the second is a
significant pointer to tho need for work on thin topic. The reviewer
proposes that UNESCO and Mi0 should consider strengthening activities
in this field by appointing a rapporteur to prepare a guide to methods
that may be applied to this problem.
References
Langbein W B
Dawdy D R
Gandin L S
Langbein W B
Solomon S I
Nash J E and
Shaw B L
1972 "Water Da-ta Today and in Prospect"
flydroloaical Sciences Bulletin
Vol. 17 110 4 PP 369-385
The
Xoss !: E & Matalas N C 1972 9'Application of Systems Anulysis
to Network Design"
in Casebook on yydrolopical Network Eesia Practice
(Mitor U B Langbein)
wE",O Chapter III - 4.1
i967 "On the PlanninE of Metero1o:cicsl Networks
ifil0 Commission for Clirnatoìoa 4i>p
1965 "Nationzl Networks of Eydro1o;:ical Data"
Denouvillies J P, Chart E J, kloolley J A Cadou C
1968 "The Use of the Grid Square System for Computer Estimation
of Precipitation, Tenperature and Runoff"
Water Resources Reseerch
VOI 4 NO 5 pp 919-926
1966 "Mood Frequency as a Function of Catchment Characteristics'
S.vm.sosium on River Flood Hydrolorn
Institute of civil Engineers, London pp iiFi36
Sutcliff J V 1973 Personal comniunication.
I%ì JORN c RODDA Institute of IIydroloa present Department of the Dnvironment
Wallingford, Berks. Address: 2 Karsham Street, LOiLO€T SW1
May 1973 ~~~LUm>

ABSTRACT
HYDROLOGIC DATA FILL-IN AND NETWORK DESIGN
Leo R. Beard
A study for the Texas Water Development Board in the
USA develops techniques for transferring streamflow data
from locations of long record to locations of short record
and uses such techniques to determine the relative value of
continuing current records or establishing new stations.
Multisite stochastic generation techniques are adapted to
the problem of filling i,n missing data by use of recorded
data at many other locations in the region. Several weaknes-
ses of stochastic data analysis techniques are studied and
new procedures are developed to overcome these weaknesses.
Results of the study are to be used for planning streamflow
measurement programs.
RESUMEN
Un estudio hecho para el Texas Water Development Board
en Los E.E.U.U. desarrolla t6cnicas para transferir datos de
estaciones de largo período a estaciones de corto período y
demuestra el valor relativo para continuar estaciones o esta
blecer nuevas estaciones. La generación de datos probabilís-
ticos para reconstituir el periodo histôrico en varias esta-
ciones en una regibn es demostrado por medio de otras esta-
ciones en la región. Varias deficiencias en el uso del aná-
lisis de datos probabillsticos son estudiadas y nuevos proce
dimientos son desarrollados para sobreponerlas. Los resulta-
dos del estudio serán usados para el planeamiento de progra-
mas en el estudio del cauce en ríos.
(1)
Technical Director, Center for Research lin Water Resources,
University of Texas, Austin, Texas, USA.
(1)

162
THE DATA FILL-IN PROBLEM
In planning the design and operation of water resources projects, it is
necessary to test the plans on the basis of at least 40 or 50 years of stream flow
that can reasonably be expected to occur in the future. Many projects are
influenced by stream flow and other hydrologic quantities that occur at several
locations simultaneously. Accordingly, adequate testing of a design or operation
plan requires 40 or more years of simultaneous hydrologic events at several
locations. Usually it is desired to use for this purpose recorded past flows
adjusted, if necessary, to future conditions. In many regions, even the best
hydrologic records are very short, and in all regions there are very short records
that must be extended for planning purposes. Detailed discussions of the use of
synthetic streamflows in addition to historical streamflows are contained in
references 1 and 2.
Also, in anticipation of future water resources studies, it is necessary to
determine whether to continue records at existing hydrologic stations or to
establish new stations with available resources. It is the purpose of this paper
to describe a study made by The University of Texas for the Texas Water Development
Board in the USA wherein techniques were developed for filling in missing data and
for evaluating short records in relation to long records..
THE DATA FILL-IN MODEL
The computer model used in the study is one developed in the Hydrologic
Engineering Center of the Corps of Engineers and described in reference 3. It
accepts monthly stream flow, rainfall, evaporation or other hydrologic quantities
as variables. The computation procedure consists of:
a. Transforming all variables to logarithms
b. Transforming all logarithms to form normal distributions
c. Deriving, from the data, multiple linear regression equations for
estimating missing quantities from the preceding quantity at the same station and
the current or preceding quantity, depending on availability, at all other stations.
d. Estimating missing quantities using the appropriate regression equation
and a random component, and applying the reverse transform to obtain hydrologic
quantities.
In order to preserve the variance and the correlation matrix relating all
variables, it is necessary to introduce a random component whose standard
deviation is equal to the standard error of estimate of the regression equation.
The model uses a different regression equation for each station and for each
calendar month at that station, and this regression equation can change every year
depending on the availability of data at other stations during the current and

preceding months, Detailed discussion of the data fill-in techniques and
associated mathematical problems is contained in reference 4.
S HORT-RECORD EFFECTS
163
When several hydrologic variables are analyzed simultaneously, it is
usual that some records are very short and that records at some stations do not
coincide in time with records at other stations. Because these short records
and their apparent interrelationships can be very misleading (due to unrepresenta-
tive occurrences within the short time period), it is necessary to provide controls
in the mathematical model so that unreasonable effects will not be generated.
Also, it is necessary to devise estimates of intercorrelation for those pairs of
variables where simultaneous data are not available.
Each element of the correlation matrix is computed using simultaneous
values of each pair of variables after they have been transformed to normal. For
those stations where no simultaneous values exists, correlation coefficients
are estimated by examining the common correlation coefficients that each of these
variables has with each of the other variables in the system. This yields information
by which the maximum and minimum logical correlation coefficient between the
two variables can be established. After this has been done for all other variables,
the correlation between these 2 stations is established as an average of the logical
maximum and minimum values. This is a necessary step for completing the correla-
tion matrix fromwhich regression equations must be computed.
Another short-period effect that can have serious consequences in planning
is the instability of the mean and standard deviations of the logarithms of
hydrologic quantities. It is possible that, when records are as short as 4 or 5
years, unusually extreme values can occur. When this happens, extrapolation to
40 or 50 years can result in unreasonably extreme quantities being generated.
Similarly, in such short records, it is possible that no large or small events would
occur, in which case extrapolation to long periods might not include events that
would normally occur in such periods.
Table 1 illustrates what happens as a result of these small-sample chance
variations. Here, 10,000 5-year samples were drawn from a normal population,
and the unbiased mean and standard deviation were computed for each. Then,
for each sample, 5 items were generated using these sample statistics, and their
location in the true parent population identified. In the fourth line under ratios, it
is shown that far too many extreme values were generated in this manner. Thus,
there is a significant bias in estimates made from small samples. In order to
overcome this, the following empirical transform function to be applied to generated
devia tes was developed:
X I = x - . ~~x~/(N-U~'~
in which I
X
X
N
=
=
=
adjusted deviate (absolute value)
generated deviate (absolute value)
sample size

164
II)
LI
6
H
O
O
O
O
Ln
Ln
al
N
m
al
a
-4
.-4
E
cn
LI
O
w
v>
al
m
>
al
U
-4
n
ry
O
r:
O
-4
rn
LI
al
a
II)
-4
n
nl9mœI-
V, ri19
m
8
-I
-
b-
O
c b

The last line under ratios in Table 1 illustrates that this formula produces a
very nearly normal distribution of values generated from a large number of
small-sample statistics.
STABILITY PROVISIONS
165
The model used in this study for data fill-in includes a number of
features that are necessary in arder to produce stable projections when using
short and intermittent records. When the correlation matrix that was derived
as discussed above is used in constructing a regression equation, it is entirely
possible that the assembled Correlation coefficients will be mutually inconsistent,
since they are not based on simultaneous data. If this occurs, it simply means
that the quantity to be estimated is over-defined and that some of the inconsistent
data must be removed for the purpose of estimating that particular quantity. This
is accomplished automatically in the computer by testing for consistency and,
when the correlation coefficient exceeds unity, eliminating that variable in the
equation which has the lowest direct correlation with the quantity to be estimated.
This elimination process is continued automatically until the correlation
coefficient becomes less than unity.
Even though the correlation'matrix is consistent, it can still be highly
unstable. This occurs usually when 2 of the explanatory variables are highly
interdependent. One indication of this condition is the occurrence of very
high regression coefficients of opposite signs for those 2 variables. The test
for this condition uses the beta coefficient, which is the regression coefficient
that results when each of the variables is adjusted to unit variance (it thus
measures the direct degree of impact of each variable on the regression estimate) .
If any beta coefficient exceeds 1.5 , variables are eliminated from the regression
study until this condition no longer exists. In this manner, a primary cause of
generating unreasonable quantities is eliminated.
REQUIREMENTS FOR MATHEMATICAL ACCURACY
Experience with the use of this model for hydrologic data fill-in has
indicated that mathematical accuracy, integrity, and continuity are absolutely
essential in order to avoid unreasonable estimates. Many attempts have been
made to smooth the statistics and correlation coefficients from month to month
throughout the year in order to stabilize the estimates, but these have usually
resulted in mathematical problems that could not be readily overcome. Attempts
have also been made to adjust coefficients within the correlation matrix in such
a manner as to remove inconsistencies and increase stability, but these also
have resulted in erratic computation. It has become apparent that the regression
equation for estimating a missing value must be used exactly as calculated
from the observed or filled-in data.

166
The transform function used for converting flows to normal has also
been a source of serious mathematical difficulty. If the data being transformed
are highly skewed, transformed values can become highly erratic, particularly
for small samples. In order to stabilize this transform, hydrologic quantities
whose lower limit is zero are first transformed to logarithms (after adding a
small increment). The size of this increment is then adjusted so that the
skew of the logarithms does not differ much from zero. Then the approximate
Pearson type III transform function appears to be completely adequate for
transforming the logarithms to normal. However, when the skew coefficient of
the untransformed values differs from zero by more than a value of about 0.5,
very serious transform problems can occur.
V&UE OF DATA FILL-IN
It can be shown that adjustment of short-record statistics by use of
long-record correlated data can result in improvement of the accuracy of the
mean value in accordance with the following equation:
in which
N1 = number of items in short record
N2 =
R , =
number of items in long record
cross correlation coefficient
N1 = number of items that would be needed in the short record to
obtain an accuracy of the mean that is equivalent to that
obtainable by the adjustment.
Some values obtained with this equation are illustrated in table 2.
In filling in missing values of monthly streamflows by correlation with
long-record stations, correlation coefficients vary from month-to-month, so
there is not a simple relationship that will show how much value is obtained
by extending short records in this manner. However, a group of 4 stations having
40 years of simultaneous data was used to estimate the increase in reliability of
average-flow estimates based on 5 years of data correlated with 40 years of data
at near-by locations. The experiment was conducted as follows:

Table 2
Theoretically Equivalent Sample Size
for Computing Equally Reliable Mean Value
167
Correlation Coefficient
Sample
Size .5 .8 .9 .95 .98
Sample Size of Related Variable = 40
5 6.4 11.4 17.2 23.8 31.3
10 12.3 19.2 25.5 31 .O 35.8
20 22.9 29.4 33.6 36.4 38.5
40 40 .O 40 .O 40 .O 40 .O 40 .O
Sample Size of Related Variable = 100
5 6.6 12.8 21.7 35.1 57.1
10 12.9 23.6 . 36.9 53.3 73.7
20 25 .O 41 .O 56.8 71.9 86.3
40 47.1 64.9 77.8 87.2 94.4
Starting with one station, four different 5-year sequences were selected
from the record. For each of these, data were filled in for the remaining 35 years.
For each of these 5-year sequences, the mean flow for the 5 years and the
mean flow for the 40 years of filled in sequence were compared with the mean
flow for the 40 years of actual record at the station. Standard errors from the
40-year recorded mean were computed.
The ratios of the standard error of the 40-year filled-in data mean to
the standard error of the 5-year data mean are shown in table 3 in the row
designated as obsenred. This process was repeated for each of the 4 stations in
order to obtain the 12 observed values of table 3.
The expected ratios shown in table 3 were computed as the inverse
ratios of the square root of effective record lengths computed from equation 2.
The standard-error ratios thus obtained are somewhat larger than expected,
partly due to the fact that the 40-year record mean is not the true long-term
mean and partly due to the variation of monthly correlation coefficients from the
correlation coefficients of annual flows shown in table 3.

168
Short
Record
Sta tion
1685
Correl coef
Observed
Expected
1675
Correl coef
Observed
Expected
1710
Correl coef
Observed
Expect ed
1730
Correl coef
Observed
Expected
Table 3
Ratios of Standard Error of Fill-in
Mean to Observed Mean for 5-year Records
Correlated with 40-year Records
1685
.97
.75
.42
.82 .79
1 .o9 1 .O3
.65 .68
Long-Record Station
1675 1710
.97 .82
.51 .82
.42 .65
.79
.95
.68
.73 .75 .78
.74 .92 .30
.73 .72 .69
1730
.73
.80
.73
.75
.86
.72
.78
1 .O7
.69
Although the results shown in table 3 are somewhat erratic due to the use
of small samples and a small number of cases, it is apparent that the fill-in
process described herein is generally valid and that table 2 can be used as a
general guide in determining whether to continue short records or to start records
at new locations where data are also needed. The advantage of the monthly fill-
in model over a simple adjustment of mean flows is that realistic variations of
annual streamflow patterns for interrelated stations can be developed for use in
simulation studies.
Unless correlation coefficients between short-record and long-record values
are well above 0.5, there appears to be very little gain in reliability through
correlation. Where there is good correlation, the gain in reliability that can be
expected through maintaining a short record for a longer period (such as continuing
a 5-year record until it is 10 years long) is a function of the length of near-by iong-
record stations.

169
Where correlation coefficients are well above .95, short records need
not be continued much beyond 5 years ,but the near-by long record should
be continued as long as greater reliability is needed. Where Correlation
coefficients are much below .8, short records should be continued. Between
these limits, the relative value of continuing a short record or starting a new
record depends on the unreliability of estimating flows at ungaged locations,
which concerns an area of study beyond the scope of this paper.
CONCLUSIONS
The stochastic data fill-in model described can be used to estimate
monthly values of missing hydrologic data at short-record locations where
longer records exist in the region. The value of the fill-in procedure is a
function of the correlation between the short-record and long-record data and
the relative lengths of record, generally as expressed in equation 2. This
relation, as illustrated in table 2, can be used to determine whether to continue
short records or establish new stations. It appears from table 2 that short
records need not be continued beyond 5 years (unless hydrologic conditions
change) where near-by records are continued that correlate at the .95 level
or better. Where the correlation coefficient is below .8, records should generally
be continued. Between these two values, a study of regional variations would
be needed to determine the relative advantage of continuing existing stations or
starting new ones.
ACKNOWLEDGMENT
The study upon which this paper is based was supported by the Texas
Water Development Board. Computation assistance was furnished by R.V.
Juyal and J.W. Barron. Opinions and conclusions expressed are those of the
author.
1.
2.
3.
4.
REFERENCES
Beard, Leo R. (1965) Hydrologic Simulation Procedures in Water Yield
Analysis, Sixth Congress, International Commission on Irrigation
and Drainage, New Delhi, pp 22.103 - 22.116.
Weiss, Arden O. and Beard, Leo R. (1971) A Multi-Basin Planning
Strategy, Water Resources Bulletin, Journal of the American Water
Resources Association V.7, No.4, pp. 750-764.
Beard, Leo R. (1965) Use of Interrelated Records to Simulate
Streamflow, Journal of the Hydraulics Division, American Society of
Civil Engineers, September 1965, pp. 13-22.
Beard, Leo R., Fredrich, Augustine J. and Hawkins, Edward F. (1970)
Estimating Monthly Streamflows within a Region, National Water Resources
Engineering Meeting, American Society of Civil Engineers, Preprint 1125.

ABSTRACT
APPLICATION DU KRIGEAGE A L'OPTIMISATION
D'UNE CAMPAGNE PLUVIOMETRIQUE EN ZONE ARIDE
-
J.P. DELHOMME, P. DELFXNER
In arid areas, hydraulic planning must often be performed in a
few years: install a rain gauge network, strengthen it if necessary and
determine the major features of the basin, mainly the volume of precipi-
tation and its geographic distribution. It seems impossible to utilize
the usual elaborate statistical methods because they appeal to time COT
rrelations which can hardly be inferred, Indeed, after an initial pro-
gram of precipitation measurements for a basin, data for only a sh-ort
time interval are available, and regional climatological statiwn are
commonly too far removed geograplìically to andd useful ingormqti'on. TO
solve the interpolation problems , only the spatial stxuctgre 08 preci-
pitation on the basin itself can 6e considered. Kriging provides the
best linear estimates based on the experimental data, and this under
very few assumption. In particular, it avoids the traditional assump-
tion of second order stationarity, used in optimal filtering for exam-
ple, and which is not justified in many cases. Moreover, Kriging per-
mits quantification of precision of estimation and provides a solution
to the problem of optimal location of new points of measurement, accor
ding to a criterion of maximum gain of information,
RESUME
Lors d'une étude d'aménagement hydraulique en zone aride, on ne
dispose souvent que de quelques années pour implanter un réseau pluvio-
métrique, le renforcer si besoin est, et cerner les caractéristiques ma
jeures du bassin, principalement le volume d'eau tombé et sa réparti-
tion. Les techniques statistiques élaborées traditionnellement semblent
alors d'un emploi difficile car elles font intervenir des corrélations
temporelles dont l'inférence statistique est quasiment imposible. En
sffet, aprbs une première campagne de relevés pluviométriques sur le b a
ssin, on n'y possède que de tr8s courtes séries chronologiques et les
stations ciimatologiques régionales sont souvent trop élognées géogra-
!hiquement pour apporter une information &ellement valable. Pour trai -
ter les problèmes d'interpolation, on ne peut donc prendre en considéra
tion que la structure spatiale de la pluviométrie. Le Krigeage permet
ie trouver les meilleurs estimateurs linéaires construits sur les va-
leurs expérimentales, et ce, sous des hvpoth8ses tres larges: en parti-
)ulier, l'hypothèse classique de la stationnaritg du second ordre, di-
fficilement admissible dans bien des cas, n'est pas nécessaire. Le Kri-
Ceage permet en outre de quantifier la précision de notre estimation,
?t appoiyte une sol-ution au problème de l'implantation optimale de nou-
reaux points de mesure selon un critère de gain maximal d'information.

17 2
Pour l'hydrogéologue, les précipitations sont non seulement
descriptif du climat, mais aussi, et surtout, l'élément constitutif
du débit des cours d'eau.
A ces deux aspects fondamentaux correspondent deus types
d'approche différents d'une épisode pluvieux. I1 s'agit d'une part
d'estimer en tout point du bassin la hauteur de précipitation pour
avoir une vue d'ensemble de la répartition spatiale de l'averse et
pour en localiser les épicentres, d'autre part, d'intégrer cette
hauteur de précipitation sur toute la surface afin d'evaluer la qua2
tité d'eau tombée sur le bassin durant ce laps de temps.
Dans les deux cas, on ne dispose au départ que des indications ponctuelles
recueillies aux stations pluviométriques. Si 1 'on veut obtenir des évaluations
correctes ã partir de ces données en nombre limité, on doit attacher une
grande importance au choix d'une méthode d'estimation qui soit adaptée aux
buts poursuivis et présente le maximum de fiabilité. Que signifierait une quantité
d'eau calculée avec 100 .d'erreur? Comme dans tout calcul physique, une
valeur numérique n'a de sens F qu'accompagnée d'un intervalle d'incertitude. Si
la précision n'est pas satisfaisante, il conviendra ã 1 'avenir d'installer de
nouveaux pluviomètres. Quelle serait alors leur implantation optiqale? Ces
questions trouvent une réponse satisfaisante dans le cadre de la théorie du
krigeage de G. MATHERON (i), (Z), (3).
I1 n'est pas place ici pour un long exposé théorique que le lecteur
pourra trouver dans les ouvrages de G. MATHERON cités en références.
Aussi a-t-on préféré en montrer une application au cas concret d'une
campagne pluviométrique en zone aride.
PRESENTATION DU CADRE DE L'ETUDE
Les données utilisées ont été empruntées ã une campagne de 1'ORSTOM
dans la région Est du Tchad (4) en 1965-66. Durant la saison dds pluies a lieu
la recharge de nappes souterraines de faible importance qui fournissent 1 'essentiel
des ressources pendant la saison sèche.
Afin d'accroître cette recharge, un projet de construction de barrages
de suralimentation sur certains ouadis a été décidé, les études de reconnaissan-
ce hydrologique devant s 'étendre sur deux années.
On a retenu le cas du bassin de l'ouadi Kadjemeur d'une superficie de
245 km2 et présentant de faibles dénivellées (inférieures ã 100 m.).
Les conditions climatiques sur ce bassin versant sont assez difficiles
?i estimer ã partir des stations climatologiques régionales (Fig.l), du fait de
la rapidité des changements de régime climatique dans la région: en 400 km du
Nord au Sud, on passe du régime sahélien sud d'Abeche au régime saharien de Fada
Les périodes d'observation sont très inégales (Abeche: 31 ans, Guereda: 12 ans,
Iriba: 8'ans, Biltine: 15 ans, Arada: 8 ans, Fada: 32 ans), et les corrélations
d'une station à l'autre ne sont pas satisfaisantes.

173
On ne peut donc prendre en compte que les données recueillies sur le
bassin lui-même, où l'on dispose de 33 points de mesures: 3 pluviographes, 19
pluviomètres association et 11 totalisateurs (Fig.2).
LES BASES CONCEPTUELLES DU KRIGEAGE
Le phénomène étudié est considéré comme une fonction Z associant une
valeur numérique Z(x) à tout point x d'un certain domaine du plan ou de 1 'espace.
On connaît les valeurs prises par Z aux points expérimentaux xl, x2, ...., x
Selon les cas, on cherche ã estimer:
N'
i) la valeur ponctuelle Z(xo) au point xo
2) la valeur moyenne sur un domaine S, soit i Is Z(x)dx
3) la valeur moyenne pondérée de Z, soit:
Z, = j' Z(x)p(x)dx avec p(x)dx = 1
Pour cela, on se'-donne un estimateur Z" de la valeur exacte sous forme
d'une combinaison linéaire des données disponibles:
N
z* =J xi Z(Xi)
1 =1
I1 y a de multiples facons de choisir les coefficients de pondération
xi: tout le problème est de déterminer les meilleurs possibles.
A cet effet, on peut se laisser guider par des considérations physiques.
La qualité de l'estimation doit dépendre de deux facteurs: le nombre et la disposition
spatiale des points de mesure d'une part, la continuité, la régularité
du phénomène étudié, de l'autre.
Pour le premier point, il est clair que l'estimation est d'autant meilleure
qu'il y a plus de données expérimentales. Mais 1 'effectif du réseau de
mesure n'est pas forcément déterminant. Interviennent également la disposition
relative des points expérimentaux entre eux et leur localisation par rapport au
domaine a estimer (point ou surface). Par exemple, pour estimer une quantité
globale sur une région, il est en général préférable d'avoir moins de points mais
disposés de façon uniforme que beaucoup de points agglutinés dans une seule zone.
La conclusion est inverse si 1 'on désire une estimation locale au voisinage précisément
de cette zone la mieux échantillonnée.
Le second point est plus subtil et négligé dans la plupart des métho-
des utilisées actuellement en hydrologie. Une fonction s'interpole d'autant
mieux qu'elle est plus régulière. S'agissant par exemple d'estimer une valeur
ponctuelle Z(xo), il n'y a aucune raison d'utiliser la même formule d'interpola-
tion quand on travaille sur des pluies annuelles ou des pluies journalières.
Dans un cas la valeur au point x diffère peu de celles des points voisins, dans
1 'autre, le phénomène est plus ctaotique et les points lointains apportent une
information non négligeable.

174
Comment tenir compte de la régularité de la variable?
Les méthodes fonctionnelles de 1 'analyse mathématique ordinaire ne sont
guère utilisables pour les fonctions traduisant un phénomène naturel. Celles-ci
ont un comportement spatial bien trop complexe, trop erratique pour se laisser
décrire ii 1 'aide d'expressions analytiques classiques. Pour souligner cette particularité,
G. MATHERON (1) propose de donner a de telles fonctions le nom de
"vari ables régionalisées".
Une façon commode ii la fois sur le plan conceptuel et pratique de traiter
une variable régioiiziisée est de raisonner en termes probabilistes. On
considère la variable régionalisée comme une "réalisation de fonction aléatoire",
c'est ii dire comme le résultat d'un tirage au sort dans un ensemble de fonctions.
Pour préciser cette idée, supposons qu'on range dans un même groupe un ensemble
d'averses analogues, autrement dit, un ensemble de fonctions Zi(X) associant à
chaque point x la hauteur de précipitation en ce point. La fonction aléatoire
Z est telle que pour tout indice i et tout point x du domaine:
z(x,i) = z.(x)
1
Au tirage au sort de l'indice i de l'averse correspond la fonction numérique or-
dinaire Zi(X), c'est ii dire une réalisation de la fonction aléatoire Z. Ainsi
sont fixées du même coup les valeurs prises par la fonction en tous les points
de son domaine de définition, expérimentaux ou non.
Dans le cadre de cette hypothèse, les notions statistiques telles que
moyenne, vari ance , covari ance ou auto-corrél ati on prennent un sens précis.
E le symbole "espérance mathématique", on a:
Soit
E CZ(x)l = m(x) moyenne
E [Z(X)-m(x)]* = D2 [Z(X)]
vari ance
E [Z(x)-m(x)] [Z(y)-m(y)] = K(x,y) covariance
K(X,Y)/=) .JK(y,y) = P(X,Y) auto-corrélation
On voit que p(x,y) se déduit directement de K(x,y), la réciproque étant fausse.
On utilisera donc plutôt K(x,y) qui contient plus d'information.
Pour procéder valablement à 1 'inférence statistique de la moyenne et
de la covariance aux différents points de l'espace, il faut disposer de chroni-
ques suffisantes. Lorsque ce n'est pas le cas, come dans l'exemple de Kadjemeur,
des hypothèses supplémentai res sont nécessaires. Les méthodes optimales du type
de celle du filtrage de WIENER (5), introduite en météorologie par L.S. GANDIN
(6) se placent dans 1 'hypothèse où la variable est "stationnaire d'ordre 2": la
moyenne m(x) est constante et la covariance ne dépend pas séparément des points
d'appui x et y, mais uniquement du vecteur x-y:
E [~(x)] = m
E [Z(x)-m] [Z(Y)-~] = K(x-Y)

175
Ces hypothèses peuvent être trop restrictives. On sait par exemple que
les précipitations sont plus abondantes en altitude qu'en plaine. Par conséquent,
dans le cas général d'une région à relief varié, leur moyenne m(x) présente une
"dérive" et ne peut être considérée comme constante. Par ailleurs, il apparaît
que les calculs d'optimisation n'exigent pas que la variable elle-même, mais
uniquement ses accroissements y possède une covari ance stationnai re.
Ceci étant, les hypothèses du krigeage sont 1 es sui vantes :
1) m(x) n'est pas forcément constante, mais est suffisamment régulière
pour être représentée par une expression de la forme:
k
m(x) = 1 a, f'(x)
1
1 =o
Les fonctions f fxì sont choisies à 1 'avance foolvnomes. fonctions
trigonométriques; etc.. .) ; les al sont des cÒef~cienG inconnus
Une telle formulation englobe le cas le plus simple où la moyenne
est constante. La "dérive" m(x) se réduit alors à:
O
m(x) = ao f (x)= ao
fo(x) étant la fonction identiquemint égale à 1.
O
On supposera toujours que f 1, car cela implique que l'erreur
d'estimation Z-Z* est une combinaison linéaire d'accroissements
de Z(x)
2) Seconde hypothèse: 1 a variance des accroissements Z( x+h) -Z( x)
ne dépend que du vecteur h. On pose:
y(h) = i D2 [Z(x+h)-Z(x)]
~ ( h ) est le vario ramme. Cette fonction du vecteur h renseigne
sur 1 'isotropie +
ou anisotropie de la variable régionalisée.
A direction fixée, elle indique comment varie, en moyenne quadratique
l'écart de valeurs prises en deux points x et x+h
lorsque la distance h augmente. A une variable très régulière
correspond un variogramme très continu, et inversement.
Ces bases définies, il est possible de résoudre tour à tour les différents
problèmes posés.
KRIGEAGE DES ISOHYETES
Soit Z(x) la hauteur de précipitation tombée sur un territoire pour une
période déterminée. Afin d'estimer la valeur ponctuelle Z(x,) , on cherche parmi
les estimateurs linéaires construits sur les données expérimentales z*=xhiZ(xi)
celui qui minimise 1 'erreur quadratique moyenne E[Z*-Z(xo)]2. Or: 1
2
E [Zf-Z(x0)] = D2 [Z*-Z(x0)] + [E[Z'-Z(x0)]]'

176
Le premier terme D2 [Zf-Z(xo)] est la variance de l'erreur.
fonction du variogrannne:
Elle s'explicite en
D2 rhiZ(xi)-Z(x0)] = - 1 1 h-X.y(xi-x.) t 2 1 hiy(xi-xo)
ij 1 J J
i
Le second terme [E[Z*-Z xQ)]I2 est le carré de l'erreur moyenne.
moyenne représente un biais et il faut donc l'annuler.
E [fc-z(x0)] = E
D'après les hypothèses faites sur la dérive:
d'où:
Si l'on pose:
m(xi) = 1 alf 1 (xi) et rn(xo) = 1 alf 1 (x,)
1 1
E [z*-z(xO)l = c al
1
'if 1 (xi) - f'(xO)]
1 1
1 hif (xi) = f (x,)
1
bc 1 = O, 1, ...., k
Cette erreur
him(xi) - m(xo
l'erreur moyenne sera nulle quels que soient les coefficients a qu'il ne sera
1
pas nécessaire de connaître.
Minimisant 1 'erreur quadratique moyenne sous ces k+l conditions, on
obtient le système de krigeage où figurent ktl paramètres de Lagrange ul:
(SI)
1
1 h.y(xi-x.) t 1 ulf (xi) = y(xi-xo)
j J J 1
1 hjf
1
(Xj) = f
1
(x )
O
j
(i=l, ..., N)
(l=O,l,. ..,k)
Ce système est régulier, donc admet une solution unique, pourvu que
les f (xi) soient linéairement indépendants sur 1 'ensemble des points expérimen-
taux (cf.(l) ou (2)).
A l'optimum, la variance d'estimation a pour expression:
D2[Z*-Z(Xo)] = 1 Xjy(x.-x ) t 1 plf 1 (x,)
j 1
On remarque que cette variance ne dépend que du variogramme et des
solutions hi et pl du système de krigeage, c'est à dire uniquement de la struc-
ture du phénomène et de la disposition des points de mesure.

177
L'exemple qui a été retenu est celui de 1 'averse du 6/8/66, la plus
importante de l'année. I1 a été traité sur ordinateur à l'aide du programme
BLUEPACK mis au point à Fontainebleau(79,Le bassin de 1 'ouadi Kadjemeur ne présentant
pas un relief très marqué, la pluviométrie n'y possède pas de dérive systématique.
On a donc pris pour seule fonction de base fo 5 1.
Le variogramme est linéaire avec une discontinuité à 1 'origine.
O pour h = o
ríh) =
en mm2. 20.4 t 11.23 h pour h # O en km
I1 a été d'it plus haut que le variogramme est d'autant plus continu que
la variable est plus régulière.
La discontinuité à l'origine du y(h) traduit une irrégularité à petite
échelle. Ce phénomène a été observé depuis longtemps par les hydrométéorologues
qui 1 'expliquent par les perturbations locales, l'instabilité du mouvement de
l'air au voisinage du sol et l'arrivée de la pluie sur le pluviomètre par rafales
irrégulières. A la limite, si on connaissait parfaitement les hauteurs d'eau en
tout point, il serait probablement impossible d'en tracer la carte, les fluctua-
tions locales interdisant tout tracé continu.
Prise entre la fidélité aux valeurs expérimentales et la nécessité de
dégager des grands traits représentatifs du phénomène, la cartographie manuelle
exige en permanence des choix plus ou moins arbitraires. Ainsi sur 1 'averse du
6 Août (Fig.4), il n'a été tenu aucun compte de la hauteur 27.6 mm mesurée au
pluviometre n"26, alors que les cotes extrêmales 55.5 et 54.5 mm ont été scrupu-
leusement respectées.
Le krigeage, pour sa part, accorde aux valeurs expérimentales une importance
directement 1 iée au degré de structuration du phénomène.
La carte de la Fig.3 a été obtenue après estimation par krigeage aux
noeuds d'une grille régulière.Le pluviomètre n"29 (OU la hauteur mesurée est de
55.5 mm) n'a ainsi contribué que pour environ 63% dans 1 'estimation du point de
grille le plus proche. L'influence, non négligeable, des autres stations a ramené
ce point de grille à une valeur de 48.7 mm.
A cette estimation est attaché un écart-type de l'ordre de 6.25 mm, ce
qui rend cette valeur parfaitement compatible avec la valeur expérimentale voisine.
Sur 1 'ensemble du bassin, les écarts-types d'estimation sont compris entre 5.5
et 14 mn, la zone la plus mal connue étant bien entendu la partie Sud-Est.
ESTIMATION DE LA LAME D'EAU MOYENNE SUR UN BASSIN
A l'heure actuelle, trois méthodes sont utilisées: le planimétrage
des cartes tracées manuellement, la moyenne arithmétique simple, la méthode des
polygones de THIESSEN.

178
La précision de la première méthode est directement liée à la qualité du
tracé de la carte. Mais elle dépend aussi du soin de l'opérateur: de l'attention
qu'il a portée au comptage des carreaux du papier mi1limétré.o~ à éviter les a-
coups dans le maniement du planimètre.
La moyenne arithmétique est le procédé de calcul le plus simple, pour ne
pas dire le plus simpliste. Soit Q la quantité totale d'eau qui s'est abattue
sur le bassin be surface S. Si Z(x) est la hauteur d'eau au point x, la hauteur
d'eau moyenne Z a pour expression:
7 = Q/S soit 2 = i Is Z(x)dx
Faire une moyenne arithmétique simple sur les Z(x):
c'est perdre de vue que seules les-quantités d'eau sont additives, et non les
hauteurs. Pour qu'une telle moyenne ait un sens, il faudrait que tous les pluvio-
mètres soient équivalents, qu'ils représentent en quelque sorte chacun l/Nième du
bassin ,
Les polygones de THIESSEN (8) procèdent d'une analyse physique plus
sérieuse. Ils reposent sur 1 'hypothèse qu'une station est représentative de 1 'ensemble
des points du bassin pour lesquels elle est la station la plus proche -
voir Fig.2. L'idée de base est en fait plus générale et ne repose pas réellement
sur la forme géométrique des polygones. Soit en effet une partition quelconque
du bassin en "zones d'influence'' de surface Si:
s = s, + s, i- .... t SN
Dire que la valeur expérimentale Z(xi) est représentative de la zone Si, c'est
poser:
L'estimation de la quantité d
Q' =
et la lame d'eau moyenne a pour valeur:
eau totale est alors:
Sur le plan formel, 1 'estimateur 2* n'est autre qu'une moyenne pondérée des Z(xi).
Ce qui importe en vérité, ce sont les poids Si/S et non la géométrie des zones
d ' i n f 1 uence .
On est ainsi tout naturellement amené à rechercher les poids Xi qui
opti mi sent 1 'es ti mateur :
t*= 1 Xi Z(Xi)
1

179
En procédant de façon analogue au cas du krigeage ponctuel, on montre que les hi
sont solutions du système:
Comme on choisit toujours pour fo(x) la fonctior constante identiquement
égale 5 1, la première des conditions sur les fonctions f (pour 1=0) s'écrit
si mpl emen t :
I:xj=l
j
La variance d'estimation a pour expression, 5 1 'optimum:
De même que pour le krigeage ponctuel, cette variance ne dépend que de
la structure de la variable et de la configuration des points expérimentaux. Par
conséquent, si le variogramme est connu, on peut calculer la variance d'estimation
sans avoir besoin de la valeur des hauteurs d'eau. C'est cette propriété remarquable
qu'on utilisera pour localiser un nouveau point de mesure par la "méthode
du point fictif".
Sur l'ouadi Kadjemeur, les calculs ont été effectués pour les 13 épisodes
pluvieux de 1966. Vu le faible nombre de points de mesure, il était difficile
de procéder 5 1 'inférence statistique du variogramme averse par averse, d'autant
plus que parfois certaines données étaient manquantes. Prendre brutalement
le variogramme moyen sur l'ensemble des averses eut été faire violence à la nature
car ces averses diffèrent par leur intensité et leur dispersion. Une hypothèse
plus raisonnable a été d'admettre que les variogrammes des épisodes pluvieux
sont proportionnels:
Yk(h) = Wk
oùyk(h) est le variogramme de la kième averse, y(h) le variogramme moyen et Wk
le coefficient de proportionnalité. Cette relation équivaut 5 admettre qu'il y
a conservation des corrélations spatiales sur le bassin. En notant s la variance
expérimentale des hauteurs d'eau de la kiëme averse et 3 la moyenne !es sz, il
en résulte que:
Ok = s;/s;T
-
,

TABLEAU I
xi (en a)
Thiessen (9)
averse
1
2
3
4
5
6
7
8
9
10
11
12
13
effectif
21
28
33
33
33
33
33
33
33
33
33
31
17
moyenne
mm
13.81
34.69
5.34
38.25
1.06
2.33
21.81
4.47
32.29
O .58
O .40
22.35
6.35
1.0 1.0 0.Y 1.2 1.1 1.b 1.1 5.3 3.3 5.U U.6
1.1 1.1 0.8 1.1 1.5 1.5 8.3 3.4 3.8 2.1 1.1
mm2
19.58
64.20
31.90
73.39
1.31
10.96
225.61
82.86
152.21
1.08
1.13
46.46
45.75
o .34
1.10
0.55
1.26
o .o2
0.19
3.88
1.42
2.62
o .o2
o .o2
O .80
O .79

181
On constate une similitude assez nette qui justifierait a posteriori
l'intuition de THIESSEN.
De façon à permettre une comparaison plus complète des différentes méthodes
évoquées, on a porté sur un même graphique (Fig.6) les évaluations des lames
d'eau pour les 13 averses, par moyenne arithmétique, par planimétrage, par
THIESSEN et par krigeage. De part et d'autre de la bissectrice des axes,sur
laquelle figure la valeur krigée, on a indiqué la fourchette à 2 écarts-types.
I1 ressort que trois méthodes donnent des résultats à peu pres équivalents.
Seule la moyenne arithmétique se singularise, en particulier sur les
averses du 9 Août, du 13 au 14 Septembre, du 11 Août et du 23 Juillet, où les
valeurs estimées se situent en dehors de la fourchette pourtant très large de 4
écarts-types. Avec la moyenne arithmétique, tous les pluviomètres ont même importance;
le réseau étant plus fourni à l'Ouest, il y a systématiquement sousestimation
ou sur-estimation de la lame d'eau selon que 1 'épicentre des averses
se situe à l'Est ou à 1 'Ouest.
Que retenir de ces comparaisons?
Une fois écartée la moyenne arithmétique, il semblerait,du moins sur
l'exemple traité, que l'avantage du krigeage ne soit pas très net. Pourtant,
c'est la seule méthode autour de laquelle a pu s'articuler la comparaison, grâce
au calcul d'erreur. En outre, les auteurs ont pu remarquer que la méthode de
THIESSEN, et dans une moindre mesure le planimétrage, s'avèrent en pratique longs
et fastidieux. Le krigeage quant à lui ne nécessite que l'investissement d'un
programme.
OPTIMISATION DU RENFORCEMENT D'UN RESEAU PLUVIOMETRIQUE
Remarquons tout de suite que cette question n'a de sens que si le but
poursui vi a été cl ai rement défi ni .
Dans le cas d'une reconnaissance en vue d'un aménagement hydraulique,
1 'hydrologue a pour objectif l'étude de la relation pluie-débit: il s'intéresse
donc en premier lieu à la quantité d'eau tombée journellement sur le bassin.
La variance d'estimation par krigeage a permis de donner la fourchette d'incertitude
avec laquelle cette quantité peut être calculée. Elle fournit donc tout
naturellement 1 'indicateur de précision nécessaire pour:
1) décider de 1 'opportunité de renforcer le réseau,
2) déterminer 1 'emplacement optimal d'un éventuel pluviomètre
suppl émentai re.
Pour ce faire, on utilisera la méthode du point fictif. A la question
"comment implanter au mieux un nouveau point de mesure" peuvent être attachées
certaines contraintes. Si ce choix ne se pose que parmi un certain nombre de
points présélectionnés selon un autre critère (accès facile, relevé aisé,. . .),
on implantera fictivement un pluviomètre en chacun d'eux et on déterminera le
gain dn précision correspondant.

182
On procedera de même le long d'un cheminement si 1 'on a décidé a priori de .retenir
une ligne caractéristique du terrain (piste d'accès, accident de terrain,. ..).
Quand au contraire, 1 'on ne possêde aucun a priori , on tracera, toujours
de la même manière, une carte d'isogain en précision sur l'estimation de
la lame d'eau. C'est la solution qui a été adoptée pour l'étude du cas de
1 'ouadi Kadjemeur.
LOCALISATION OPTIMALE D'UN PLUVIOMETRE SUPPLEMENTAIRE
Soit U; la variance d'estimation ã partir des 33 pluviomètres existants,
de la hauteur d'eau tombée en moyenne sur le bassin pendant une averse.
Si on implante fictivement un nouveau pluviomètre en un point M, cette
variance d'estimation prend une nouvelle valeur U ~M) < U:. Le gain en précision
peut être défini comme:
G(M) =
uO
Si 1 'on ne tenait aucun compte de 1 'existence de corrélations spatiales, on donnerait
comme gain correspondant à un 34e point de mesure: 1/34 -3%. Quant à la
localisation, elle serait indifférente à 1 'intérieur du bassin.
Considérant un domaines, le krigeage permet de déterminer le point M
où ce gain est maximal.
Pour Kadjemeur, le domaine retenu a été choisi de sorte qu'il englobe
le bassin et ses abords immédiats. Le lecteur est invité à se reporter à la
Fig.7 pour voir où il aurait lui-même implanté une nouvelle mesure.
I1 suffit de comparer les Fig.7 et 8 pour constater que sur de nombreux
points, "1 'intuition" était insuffisante pour appréhender le problême d'une mani-
ère globale.
Le gain maximum est de 13% au 'lieu des 3% donnés par une analyse sommaire.
L'optimum absolu est situé en bordure du bassin, alors qu'au centre, une
grande zone est dépourvue de point de mesure.
Pourtant si l'on revient aux coefficients du krigeage ou ã ceux de
THIESSEN, force est de constater que ces résultats vont dans le sens d'un
"soulagement" des pluviomêtres de poids les plus élevés: on tend vers une égalisation
de la contribution des différentes stations, ce qui satisfait le sens
physique de 1 'hydrologue.
Enfin, l'examen de la carte isogain (Fig.8) sur l'ensemble du domaine
est três instructif par lui-même. I1 apparaît qu'il vaut mieux implanter judicieusement
un pluviomètre à l'extérieur du bassin plutôt que d'une manière
redondante ã 1 'intérieur.

CONCLUSION
183
Dans une zone aride mal reconnue, 1 ' hydrométéorol ogue ne di spose pas
de longues chroniques aux stations régionales; il se voit contraint de n'utiliser
que les données qu'il a pu recueillir pendant une ou deux campagnes
annuelles, pour calculer les lames d'eau sur son bassin versant.
Formalisant et généralisant la méthode des coefficients de THIESSEN, le
krigeage a permis, en fonction d'un objectif précis - estimation locale ou glo-
bale - de trouver les poids optimaux ii affecter aux différents pluviomètres.
Un intervalle de confiance a été associé à chaque estimation.
Aussi le krigeage a-t-il permis de poser en termes de gain de précision
le problème du renforcement d'un réseau: il donne objectivement le meilleur
endroit pour implanter un pluviomètre supplémentaire et la nouvelle précision avec
laquelle pourra être estimée la grandeur étudiée.
La méthode présentée est d'un emploi très souple: le coût d'implantation
peut varier selon 1 'emplacement, le bassin peut également être découpé en
sous-bassins d'importance différente pour 1 'écoulement.

184
BIBLIOGRAPHIE
Matheron, G. (1965). Les variables régionalisées et leur estimation,
Paris, Masson et Cie
Matheron, G. (1969). Le krigeage universel, Fontainebleau, Cahiers du
Centre de Morphologie Mathématique , Fasc .1
Matheron, G. (1970). La théorie des variables régionalisées et ses applications,
Fontainebleau, Cahiers du Centre de Morphologie Mathématique,
Fasc .5
Roche, M.A. (1968). Ecoulement de surface, alimentation de nappe et
transport sol ide des ouadis Fera, Kadjemeur et Sofoya, Fort-Lamy, ORSTOM
Wiener, N. (1966). Extrapolation, interpolation and smoothing of stationnary
time series, Cambridge, Mass., M.I.T. Press
Gandin, L.S. (1963). Objective analysis of meteorological fields, Leningra
Israël program for scientific translation
Delfiner, P., Delhomme, J.P. (1973). Présentation du programme BLUEPACK,
Fontainebleau, Ecole des Mines de Paris, note interne
Thiessen, A.H. (1911). Precipitation averages for large areas, Monthly
Weather Rev. , vol. 39, n07, p. 1082
Del finer, P. (1968). Cartographie et morphologie des précipitations
considérées comme variables régionalisées , Université de Grenoble
(10) Delhomme, J.P. (1970). Présentation d'une méthode objective d'interpolatio
pour la construction de cartes isopiézométriques , Douai , Communication au
Groupe d'Etude de Bassins Versants Souterrains
(11) Delhomme, J .P. (1971). Traitement géostatistique des données piézométrique
le krigeage en hydrogéologie, Fontainebleau, Recyclage en hydrogéologie
mathématique
(12) Delfiner, P. (1973). Analyse du géopotentiel et du vent géostrophique par
krigeage universel, Paris, Note EERM - Météorologie Nationale

Fig.1 - Situation du B.V de l'ouadi Kadjemeur
185

186
Fig.3- Carte obtenue par krigeage

2
mm
4
100
o 1 5 10
c
km
Fig.5 -Variogramme moyen bp~vt~da~n&

188
I
Fig.6- Comparai.mi dei différ-iites méthodes d'estirnaticin globale

189

P
..

IMPROVEMENT OF RUNOFF RECORDS IN SMALLER WATERSHEDS BASED ON
ABSTRACT
PERMEABILITY OF THE GEOLOGICAL SUBSURFACE
Dr. George J. Halasi-Kun
Chairman, Columbia University Seminars
on Pollution and Water Resources, New York, USA
In smaller watersheds with an area less than 250 km2, the recorded
hydrologic data are scarce or non-existent because, Correlating
the extreme runoff data with the characteristic permeability of
the different geological formations can not only improve the va-
rious methods for simulation or interpretation of hydrologic in-
formation from other areas but also provides additional improve-
ment in hydrological data gathering. An accoynt is given about
tentative average values in millidarcys for permeability of diffe-
rent geological formations based on selected bibliography and
previous experience. Further correlation of peak runoffs in Central
Europe and in the Northeastern coastal area of the United States
with their specific geological subsurface is discussed.
Finally, it is pointed out that the geological subsurface as a
characteristic of the peak runoff does not apply at ail for water-
sheds with an area over 300 km2. Similar but less clearly defined
correlation can be found between the lowest runoff and the storage
capacity of the geological formations.
RESUME
Dans les petits bassins, de surface infgrieure à 250 km2, les
données hydrologiques enregistrdes sont rares ou nlexistent %as.
L'étude des corrélations entre les mesures d'écoulement extremes
et la perméabilité caractéristique des différentes formations géo-
logiques peut améliorer les diverses méthodes de simulation ou
d'interprétation des reqseignements hydrologiques provenant
d'autres régions, ainsi que le rassemblement des donnees h.flrologi-
ques. On a essayé de donner des valeurs moyennes de perméabilité,
en l'millidarcyt', pour différentes formations géologiques en se
basant sur une bibliographie sélectionnée et sur l'expérience anté-
rieure. D'autres corr&latio?s, entre les écoulements de pointe en
Europe Centrale et sur la Cote Nord-Est des Etats-Unis, et leurs
sous-sols géologiques, sont aussi discutée:. Enfin on montre que,
pour des bassins d'une surface supérieure a 300 km', la structure
géologique souterraine ne constitue pas du tout une caractgristique
des écoulements de pointe. Des corrélations analogues, mais moi'ns
clairement définies, peuvent se rencontrer entre les écoulements
minimaux et la capacitd de stockage des formati'ons g@ologiques,

192
(1) INTRODUCTION
fi
In smaller watersheds with an area less than 250 lan", the recorded
hydrologic data is scarce. Generally, no data for longer
period is at hand, when the area is planned for development. In
accordance with various studies conducted in moderate climatic
conditions in Central lcurope and in the Northeastern United States
of !\merica, it seems that the peak runoff values of these drainage
basins are highly dependent on the geologic subsurface where the
permeability of the rock formations providesthe ground water storage,
or their impervious surface preconditions the extent of the
lake and swamp areas of the region cl]. Both these characteristicr
directly influence the drainage density in the areas with different
permeability faotor of the geological formations [2], as can
be demonstrated, for instance, by the hydrographioal map of Southweet
Germany from the Upper-Danube region [Figure 1).
Another claeeio example can be the river training and flood
control program of 1840-1950 in the Carpathian Basin in Central
Europe where -- even in a large watershed like the Danube at
Orshova, Romania (576,240 km- area of drainage basin with a yearly
average rainfall of 900 mm) -- the diminishing of lake and swamp
area by extensive drainage and flood control, the maximuin annual
flood increased by 15~6 in the observed 110 year period (Figure 2)
while the lake and swamp area decreased from llo7 to 3% of the
watershed. In the Carpathian Basin (318,030 1ans of the Danube
drainage basin related to the observation station Orshova, Romania)
39,000 km2 agricultural land was reclaimed in the flood and
marshland region. (Figure 3 shows the reclaimed area for the 110
year period including the two lakes of that basin.)
A similar effect on peak runoff was observed in the State of
New Jersey, U.S.A. in a territory of 20,295 h2, This obse ation
is bas d on 67 gaging stations for watershed of from 25.6 2 to
512 Inn 8 with an average yearly rainfall 1125 mm (Figure 4: Adjustment
factor for effect of lakes and swamps on peak runoff in New
Jersey 1897-1972 compared with data developed from the Danubian
basin at gaging station Orshova, Romania 1840-1950).
(2) hhXDdtma SURFACE FLOW
Researohes conducted in the pa t decades concerned maximum
flow in watersheds with area 250 km' or less, where the geological
conditions, topographic characteristics and rock formations permit
an evaluation of peak rates of runoff from smaller watersheds.
Such research revealed a Olear influence of these factors on the
peak runoff. In accordance with the findings abroad (in Central
Europe in an area of 49,008 h a) 13) and in the Northeastern Uni-

193
ted States (in an area of 20,295 km2) [23 the correlation of the
100 year peak flow with the geologic subsurface, the topographic
conditions (slopes) and the size of the watershed can be put in
the following equation:
Q = C.A-e, where
Q = 100 year peak runoff value in m3fsec.km2
A = area of watershed in km2
C = coefficient depending on the geological subsurface with value:
In Central Europe [3] for 100-125 mm/day point rainfall
intensity and 15 km by 50 km recorded storm patternfrom
1 to 10.2 (Figure 51
In Northeastern United States of Amerlca [I] for 2OQ-250
mm/day point rainfall intensity and 30 km by I05 km
recorded storm pattern -
from 7.3 to 147 (Table 1)
e = exponent of the watershed area depending on the topographic
character of the watershed (0.35 -plains; 0.37- slightly
hilly plains; 0.44-0.46 -steeper hills and moderate mountains;
0.50 - Alpine type mountains).
Analyzing the figures of the geological runoff coefficient,,
the result seems to be identical in both areas studied, if we
assume that the point rainfall intensity and the size of recorded
storm pattern have a direct influence on the peak runoff (Figure
5). Furthermore the vegetative cover showed a 2 5% effect on the
peak flood. Similar influence was observed concerning the form of
watershed Ct5% for fan-shaped and -30% elongated form of drainage
basin). Urbanization alters the geological character of the surface
and has a direct relation in increasing the surface peak flows
41 *
(3 1
GROUND WATER STORAGE CAPACITY
It is obvious that the rate of surface runoff must be related
in an inverse way to the permeability of the geological subsurface
of the watershed; and the quality and quantity of ground water
storage is directly dependent on these condltlons. Based on over
70,ûOû well-record files of domestic and industrial wells through-
out the State of New Jersey, U.S.A.
(area 20,295 km22 for the
period 1947-1972, the ground water availability in rock formations
from Precambrian thorough Triassic in age and from unconsolidated
sediments from the Cretaceous to the present, can be estimated.
Comparison of large statlstical samples of well-records i’n the
rock formations to a depth of as much as 550 m has provided a means
of estimating the ground water potential of areas underlain by

194
specific rock types [5, 6, 7, 81. Several of these estimates of
ground water availability have been tested against the experience
in areas of suburban development during times of drought 1961-1966.
There us sufficient consistency in the results to indicate that
underlying rock and sediment types may be determined from well
data where they are otherwise concealed by soil and overburden.
The gathered statistical data on ground water availability
in New Jersey is based al60 on pumping tests and records of the
wells, and their figures can be accepted on the assumption of an
average yearly rainfall 1125 mm -- which can drop for two consecu-
tive years of' drought to 850 mm. Ground water is available in the
northern half (rock country1 of the area studied only to a depth
of 180 m below the surface. Boring tests proved that below that
level, there is a very marked decrease in fractures and fissures
from which water can be obtained. There is also evidence, of
course, that certain fracture zones may give abundant water at
great depth, but these fractures are those such as the Triassic
border fault or others that have been weil known. In the coastal
half (coastal plains) of this examined region the limit, in
accordance with the aquifer layers, is from 200 to 1000 m (from
West to East) below the sealevel 8 .
Evapotranspiration and interception average 450-560 mm yearly,
and the yearly average runoff is up to 550 mm from the annual
precipitation. The ground water availability indicator has a value
from O to 450 mrn yearly, depending on the permeability and storage
capacity of the geological formations 191.
Despite the fact that the estimate of the regional availabi-
lity is complicated by factors such as recharge or transmissability
frow adjacent areas, there is clear evidence of correlation of
permeability of geological formations with surface peak runoff and
ground water availability. The comparison can be based only on
average values of permeability because they are measured under
difficult conditions and in various geological formations which
are similar to those formations used in establishing runoff for-
mula coefficients and ground water availability indicators (Ta-
ble 21. ït must be pointed out that the various formations are
also, in general, already mixed or interwoven even in smaller
drainage areas. The uneven surface weathering, artificial imper-
vious surface due to urbanization, the disintegrated underlying
rock formations at various depths and possible faults add to the
difficulty of establishing a practical average value of permeabi-
lity, ground water availability or surface runoff even for a
small waters he d.

195
The various studies showed that the geological subsurface has
an effect only on smaller watersheds with an area of 250 km2 or
less. The "geologic" character starts to "fade away" when the
size of the watershed is larger than 235 km2; and for a watershed
of over 340 km2 the use of formulas based on geologic conditions
is not recommended because other factors affect the flood flow,
and the influence of geological factors is negligible. Even more
confined in area is the ground water availability estimate where
the practical upper limit may be less than 200 km2, depending on
the surface conditions and on the complexity of the subsurface
rock formations.
(4) LOWEST SURFACE FLOW
The lowest flow in smaller watersheds -- whose importance is
to find out the pollution effect of various polluting sources
depends , similarly, on the permeability of the geological subsur-
face, the length of drought, the frequency and'the amount of rain,
the storage capacity of the aquifer layers, the evapotranspiration,
the temperature of the atmosphere and of the soil, the retardation,
effect of forests and of lakes and the elevation of the watershed
not to mention transfer of available water from one watershed to
the other. From these few factors is also evident that reliable
data about lowest flow are even more scarce for smaller watersheds
than the peak flow. In general, the surface watey records contain
prery li'mited amount of data pertaining to lowest flows.
Despite these circumstances, there is sufficient evidence to
develop also a formula for lowest runoff based on records from
both areas collected especially from the drought periods 1921 and
2947 in Czechoslovakia [IO, 11, 12) and 1961-1966 in New Jersey
15. 6, 7, 81 as it follows:
Q z CSA-~, where
Q lowest runoff (50 years?) value in l/sec.km2
A = area of watershed in km2
C coefficient depending on the geological subsurface (Table 31
In Central Europe from O to 5.58
In New Jersey, U.S.A. from O to 5.75
e 0,065; the exponent indicates an almost even distribution of
the lowest runoff regardless of the size of watershed and the
available data and values were not sufficient evidence to
evaluate the influence of slopes and topographic configuration
of the drainage basins in further details.

196
In Both examined areas in the plain region, where the aquifer
sediments prevail and reach considerable depth as far as 1000 m
below sealevel, the runoff coefficients decrease in value because
the surface runoff is absorBed by these highly permeable layers.
As a contrast to this phenomena in ayeas such as these along the
Atlantic Coast in New Jersey or the valley of the Danube and the
Tisza in Southern Czechoslavakia, the ground water collected, even
from distant area, "spills over" fpon its subsurface storage and
keeps the surface runoff values up to 10.4 lfsec,km2 in periods of
drought. This outcropping of the gpound water can be observed also
in the surface streams of the "Fine Barrens" area of Southern New
Jersey. Unfortunately, the data concerning low runoff is far less
available than that for, peak runoff or well-records, This makes
further evaluation dad calculation extremely difficult, Therefore,
this method of lowest runoff computation may be considered only as
an estimate for planning and developing purposes.

197
Brm I OGRAP IIY
1. T-Ialasi-Kun, G.J. (1972). “Data Collecting on Water Resources
and Computations of MaximUm Flood for Smaller Watersheds, If
Simposio Internacional Yobre la Planifioacion de Recursos
Ridraulicos - Ponenoias, Volume I, Mexico.
2. rrerak, ñ!. , Stringfield, V.T. (1972). Karst, Amterdam-London-
New York, Elsevier.
3. Halasi-Kun, G.J. (1968). Die Ermittlu g von Hbchstabflüssen
für Einzugsgebiete kleiner ala 300 b3 im Bereich der Slowakei,
Braunsohweig, Leichtweiss-Inst itut.
4. nalasi-Kun, G.J. (1969). ”Correlation Between Precipitation,
Flood and Windbreak Phenomena o9 the Mountains, 1’ Proceedings
of University Seminar on Pollution and Water Resources, Vol. I,
~ e w York - Trenton, Columbia University - State of New Jersey,
5. Miller, J. (1973). Geology and Ground Viater Resources of Sussex
County ... , Geol. Bulletin No. 73, Trenton, State of New Jersey.
6. Widmer, K. (1905). Geology of the Ground Water Resources of
Mercer County, Geoï. Report No. 7, Trenton, New Jersey ~eol.
Surve y.
7. Rhodehamel, E.C. (1970). A Hydrologic Analysis of the New Jersey
Pine Barrens Region, Water Resources Circular No. 22, Trenton,
State of New Jersey and U.S.G.S.
8. Barksdale, H.C., Greenman, D.W., Land, S.M., Milton, O.S.,
Outlaw, D.E. (1958). Groundwater Resources in the Tri-State
Region Adjaoent to the Lower Delaware River, Special Report
No. 13, Trenton, State of New Jersey.
9. Halasi-Kun, G.J. (1971). llAspects hydrologiques de la pollution
et des reBsources en eau, dans lee domaines urbains et industriels,lt
Actes du Congres: Scienoes et Techniques An 2000, Paris,
SICP.
10. Dub, O. (1957). Hydrologia, hydrografia, hydrometria, Praha-
Bratislava, SVTL.
11. Halasi-Kun, G. J, (1949) Hydrologia, Kosice.
12. Halasi-Kun, G.J. (1954). Voda v polnohospodárstve(Water in Agriculture),
Bratislava, SPN.
13. Davis, St.N., Dewiest, R.J.M. (1966). Hydrogeology, New Yorlc-
London-Sydney, Je Wiley 81 Sons .
14. Linsleg, R.K.Jr., Kohler, M.A., Paulhus, J.L.H. (1968). Hydrology
for Engineers, New York - Toronto - London, hiCGraW-Hill.
15. Water Resources Data for New Jersey: 1961-1971, Part 1: Surface
Water Records (1962-1972). Treiiton, U.S.G.S.

198
FIGURE I: HYDROGRAPHICAL MAP OF SOUTHWEST GERMANY FROM
THE UPPER-DANUBE REGION.

YEARS
FIGURE 2: MAXIMUM' ANNUAL FLOOD OF THE DANUBE RIVER AT
ORSHOVA, ROMANIA, 1840-1950.
0 SERIES OF PEAK FLOOD DISCHARGES.
@ AVERAGE PEAK FLOOD.
@TREND OF THE AVERAGE PEAK FLOOD.
199

LAKE AND SWAMP AREAIIN PERCENT OF DRAINAGE AREA
201
FOR WATERSHEDS OF 25.6-512km2 (BASED ON 67GAGHG STATIONS IN NEW JERSEY,
USA I 1897 - 1972)
------- FOR IIWTERCHED OF 576.240 km2 (DANUBE AT ORSHOVA,ROMANIA, 1840-1950 WE
TO LAND RECLAMATION AND FLOOD CONTROL)
FIGURE 4: ADJUSTMENT FACTOR FOR EFFECT OF LAKES AND SWAMPS ON PEAK RUNOFF
IN NEW JERSEY, U.S.A. AND DANUBE AT ORSHOVA, ROMANIA

202
Table 1: Peak Runoff Coefficient in Various Hydrogeologic Regions:
Hydrogeologic Regions: Peak Runof
(formations) in Central Euroue:*
(1) Kaolinite, Clay in-
cluding argillaceous
Triassic or Tertiary
Paleogene Flysch
(2) Paleozoic Shales,
Schist and Mesozoic
Mar 1
(3) Igneous Rocks
Tertiary Marl
(5) Weathered Igneous
Rocks, Limestone, Tuff
(6) Mesozoic Triassic
Brunswick Pormat ions .
(7) Mesozoic Cretaceous
Clayey Sands, Tertiary
Eogene Clayey Sands
(8) Tertiary Miocene Sands
and Quaternary Moraines
(9) Tertiary Neogene
! (for peak runoff
17 5-18.2
14
10
7
6
7
7
1 9-2 5
1.9-2 5
1
Coeff icient
* Ratio of combined effeot for point rainfall i tensity and size
of storm pattern in the two observed regions 113:
1 to 8.2 = Eastern Czechoslovakia to New Jersey, U.S.A.
147
100
81
70
69
37
30
26
12
7.3

Table 2: Ground Water Availability in Various Hydrogeologic
Regions and Their Average Permeability:
(format ions )
(i) Kaolinite, Clay in
eluding argillaceous
Triassic or Tertiary
Paleogene Flysch
(2) Paleozoic Shales,
Schist and Mesozoic
hfarl
(3) Igneous Rooks
(except Basalt,Diabase
Sandst ones y Meeozoio
't'riassic Stockton Form
(4) Dolomite, Besalt a
Tertiary Marl
(5) Weathered lgneoae
Rocks, Limestone a Tuff
(6) MeeOZQiO Triassio
Brunmiak Formations
(7) Mesozoio Cretacteou
Clayey Sands Tertiary
Eogene Clayey Sands
(8) Tertiary Miocene 3 nde
and Quaternary Moraine 'I
(10) Quaternary Beach
Sands (cape May Form.)
Woundwater Availability
in New Jersey, U.S.A.
(in mm/year [5,6,7,8)):
Y
d
17 - 25 7
less than 47
63
87 - 125
150
200-225
2 50
300
3 50
455
203
verage Permea-
ility in milli-
arcys 113,144 ) :
1
2
1-1.9
2
2.5
42
3
62
7
102-lP2
102-142
18.22
Values are based on over 70,000 well-reoord files of domestic and
industrial wells of the State of New Jersey from the period of
1947-1972. Further information especially for regions (2),(3)y
(6) and (71, is in references ba6,TY8,13,143. The form of data
on average permeability makes easy comparison with Figure 5.

204
Table 3: Lowest Runoff Coefficient in Various Hydrogeol. Regions:
ifydrogeologic Regions:
Lowest Runoff Coefficient
( f orrnat ions)
tn Central Europe: I in New
I
Jersey, U.S.A.;
(for lowest runoff values in ï/seo.km2)
(i) Tiaoïinite, Clay inc
luding argillaceous
Triassic or Tertiary
Paleogene Flysch
O *3-O 6
0-0 26
(2) Paleozoic Shales,
Schist and h4e s oz oio
Marl
3) Igneous Rocks
I except Basalt,Diabase)
0-1 70
Sands t ones , Mes oz oio
Triassio Stockton Form.
(4) Dolomite, Basalt an
Tertiary Marl
*
1-2
O 17-0 79
(5) 'Neathered Igneous
Rocks, Limestone, Tuff
(6) Mesozoio Triassi0
Brunswick Formations
(7) Mesozoic Cretaoeous
Clayey Sands, Tertiary
Eogene Clayey Sands
(8) Tertiary Miooene Sands
and Quaternary Moraines
(9) Tertiary Neogene
Sands, Mesozoic Cretaoeous
Magothy-Raritan Formations,
aternary River Drift
O) Quaternary Beach
ande (cape May Form.) -
**
4-5 68
bless than 0.3"
O. 62-0 91
2 71-5 75

-- ABS TRACT
-
DETERMINATION OF SNOW WATER EQUIVALENT AND SNOWMELT WATER
BY THICKNESS OF SNOW COVER DATAS
George Kovács - George Molnár
Res-earch Institute for Water Resources Develapment ,
Budapest, Hungary'
In studies on the accumulation- and melting process of snow bulk
densities have been determined for fresh snow (Ymin), for snow saturated
by capillarv water (y,> and for melting snow (ymax). Correlation
studies have shown the magnitudes of y and Ymax to depend
greatly on the number (RI Qf snow lagreps, Tke equations insolying the
bulk densities listed above form the Basi's of tEe computation charts
prepared by the authors. These can be applied to two tñus far unsolved
problems: I. the reconstruction of past time serpes of water equi
valent values for observing statìons w?tk data on tñe thi'ckness of the
snow cover only, and 2. forecasting the duration of the melting period
and of the volume of snow-melt water form data on the thickness of cover
and the air temperature predicted for the melting period.
Au cours de l'analyse du phénomene d'accumulation et de fonte de
e les auteurs ont déterminé le poids volumétrique (y min,
) initial de la neige fraiche, le poids volumétrique (y ,
) de la neige a capìllaire-saturatton, le poids volum&tbique
(y max, [g/cm3] 1 de la neige en fonte, Les analpee correllati'onalles
ont démontré que les valeurs de y et y max dépendent d'une mesure con
k
sidérable du nombre de couches de\la neige (7). Ce sont les equationsexprimant
les poids volum~tri'qne prê=nt&ea pl.pe- EaPt qui servent de
base au diagrama des auteurs, ce dernier permettant la solution de
deux problemes jusqu'alors irrésolus: 1. Réalisation rétrospective des
séries de données, dépendantes du temps, pour l'équivalent neige-eau
dans le cas ou l'on n'a procédé qu'a l'observation de l'épaisseur de
la neige, 2. Pronostics concernant la durse de la fonte et de la quantité
d'eau qui s'y produit, a la Lase d'une épaisseur mesurée et d'une
température prévue pour la période de fonte.

206
ïNTPnODUCTIOIJ
In water budget calculations, on the income side the snow or
the meltage of snow is of great importance. The forecasting of
spring floods and undrained runoff waters, the planning of reser-
voir operation during the melting period and other problems re-
quire more accurate information on snow accumulation and melting,
the continuous observation of the water content stored in the snow
cover, $he recovery of past data and the reconstruction of snow
water reoords with the help of observation data available. Above
all the thickness of mow cover should be considered which has
been continuously observed at more than 1000 stations for nearly
100 years in Hungary. The network of water equivalent measuring
stations, however, works only from 1960, comprising presently 60
s tat i onse
The paper summarizes the investigations'aiming at the dis-
covery of the relation between snow cover thickness and snow-water
equivalent on the basis of the numerous (about 200 O00 per eeason)
data for the 12 years between 1960 and 1971.
1. THE PROCESS OF TIB DEVELOPWT, ACCüMüUTION
AND MELTING OF SNOW
1.1 The development and accumulation of snow
Snow crystals are formed by the hexagonal ice prisms deposit-
ed around the concentration cores (soot, dust, etc.) overcooled to
-15 - -25OC in the high regions of the atrnosphere.Lom temperatures
(-10 - -15OC) and high moisture contents are conducive to the for-
mation of crystals containing large, ramifying pore apace and to
the deposition of these crystals. Under reversed circumstances
(temperature around O°C, low moisture content) so called cylindric
crystals and needles of ice will develop and deposit densely pack-

ed, with high bulk density on the soil surface Il, 4, 103.
The distribution examination performed using numerous data to
determine the bulk density rg/cm31 of fresh snow having an
intact crystal structure and containing no capillary water at all,
yielded the following result:
*min
P 0.118 i 0.028
3
Ig/cm 3
From the beginning of accumulation the snow cover becomes
continuously more compact and its bulk density increases accord-
ingly. This is the consequence on the one hand of the closer and
closer agglomeration of snow crystale and, on the other hand, of
the increase of capillary water content stored in the pore spaces
between the crystals. The increase in bulk density is caused by
the combined effect of external (temperature, sunshine, wind,etc.)
and internal (the weight of snow, etc.) factors to which the snow
cover is exposed C3, 5, 6, 7, 8, 91.
Obviously, the discharge of snow-melt water can only start
when the capillary pores between the crystals have already been
saturated with water.
The bulk density of mow saturated with water is called the
critical bulk density (a,)
1.2 The process of mow melting
According to the correlation examinations performed to de-
termine the critical bulk density (rk), the development of rk con-
siderably dependa on the number CR) of snow layers developed dur-
ing accumulation. During a cold spell following a temporary melt-
ing period with a duration of a few days, a lager of ice will de-
velop on the surface which layer separates the old and the newly
fallen fresh mow. In the case of repeated recurrence of this
phenomenon the snow layer is dissected into clearly distinguish-
able layers, the number(R) of which is a good indicator of the
207

208
periodicity of accumulation and melting. The variations - i.e. the
periodic fluctuation - in the snow thickness time series is a good
basis for estimating the number of layers. The reliability of es-
timation remarkably increases, if, besides the snow thickness re-
cord also the air temperature record is available.
The relation between the numerous data of the critical bulk
density (ak) and of the number of snow layers - both types of data
obtained from the 12 years long period between 1960 and 1971 - is
described by the following regression equation:
rk = 0.153 +
3
0.050 R 2 0.025 Edcm 3
Performing the correlation examination separately with the
data obtained from the mountaina and the lowlands, the following
equations were obtained:
'a
km, t ain
* 'lowland
(2)
= 0.160 + 0,042R 2 0.032 Ig/cm33 (3)
t 0.146 + 0.052R 2 0,028 Cg/cm33
It is interesting to note that the slope of the curve is some
20 % flatter in the mountains than in the lowlands. This is prob-
ably due to the fact that on the slopes in the mountain areas, the
water of a short melting period can immediately flow down, so that
here the ice layers frozen subsequently will be relatively thinner
than in the lowlands.
Melting starts at the instant of capillary saturation,i.e. at
the development of the critical bulk density. In the course of
melting, the ice crystals are merged and destroyed progressively
and become eventually completely liquid. During this process the
bulk density of snow grows continuously until its maximum value is
reached. The maximum bulk density (rma) is calculated using the
regression equation:
max = 0.213 + 0.054R $; 0.035 Cg/cm33 c 5)

209
The duration (mo) of meltinq - as verified by the examina-
tions performed - depends primarily on the temperature conditions
prevailing during the melting period and on the amount (A h) of
snow-melt water.
In the course of studies aimed at the discovery of the rela-
tion between the different values of temperature and the amount of
meltage numerous potential relations were tested, of which
Ah = f(K) K æ tmax + tmin
3
( 6)
has been selected as the beat, in which tmax and tDin are the
maximum and minimum temperatures, respectively, prevailing during
the individual days of the melting period.
The daily temperature value can be computed with Eq.(6) is
called rneltinR heat standard (K, ['CI).
To calculate the duration of melting the relation
can be used where is the daily average melting heat standard of
the week after melting has started.
It will readily be seen that the knowledge of the expected
average maximum and minimum temperatures forecast in Hungary for a
week by the National Meteorological Service is essential for pre-
dicting the duration of the melting period.
2. NEW CALCULATION METHODS
Using the research results demonstrated above two as yet un-
solved problems can be approached:
- The creation and reconstruction of mow-water equivalent time
series - essential in hydrological practice - for snow measuring

21 o
stations where only snow thickness observations are or were per-
formed. In this way the water equivalent time series can be ex-
tended in time for the duration of snow thickness observations.
- Forecasting the volume of snow-melt water and the length of the
meltirg period on the basis of the snow thickness measured and
of the air temperature forecast for the melting period.
To solve these two problems the chart shown in Fia.1 has been
constructed, the use of which and the course of calculation are
demonstrated using the data obtained in 1963 at one of the snow-
-water equivalent measuring stations - Dombori puszta - in the
lowlands. Since snow thickness and the water equivalent were ob-
served simultaneously, the checking of the methods is also pos-
sible.
2.1 Producing the snow-water eauivalent time series of the Deri0.d
examined from of snow thickness data
In the period examined - from the 11th of January to the 9th
of March - snow was stored continuously. Within the period, = as
revealed by the snow thickness time series in the upper part of
Fig. 2 - three greater intermediate and from the 2nd of March a
final melting occured.
a] To produce the snow-water equivalent time series the snow
bulk density time series must be reproduced first from the snow
thickness data available. in the calculation the bulk density of
the freeh snow is supposed to be
in all cases.
rmin = 0.118
3
idcm 3
During accumulation, when snow thickness is increasing, the
average bulk density of the whole snow cover is calculated suppos-
ing that the increment has also a bulk density of 0.118 g/cm3.

211
E.g.: On the 19th of February the snow cover is 25.2 cm thick and
has a bulk density of 0.320 g/cm 3 . Next day an additional
amount of 1.6 cm snow cover fell. Thus the composition of
the mow layer is calculated with the layers having
and
25.2 cm (94 %) thickness and 0.320 g/cm3 bulk density
1.6 cm (6 %) thickness and 0.118 g/cm3 bulk density
i.e. total 26,8 cm thickness and 0.308 g/cm3 bulk density.
On the first day (nk) of the particular temporary and of the
final meltings the critical bulk density ( k) of the snow cover is
taken into account with the corresponding number of layers.
In the example, using part A) of the chart:
rI.i5.
ìr 11.12.
r 11.21.
r 111.2.
= 0.202 g/cm3 because R = 1
E 0.252 g/cm3 because R = 2
t- 0,302 g/cm3 because A = 3
E 0,351 g/cm3 because R = 4
(These values are the solutions of Eq.(2) substituting
R = 1, 2, 3 and 4, respectively.)
On the last day of the particular temporary and of the final
meltings the maximum bulk density of the snow cover is taken into
account with the corresponding number of layers (see part A./ of
the chart). Thus
bx.is.
r11.19.
ìf 11.23.
r111.9.
= 0.267 g/03 because R 5 i
= 0.320 g/cm3 because R = 2
o 0.374 g/cm3 because R = 3
= 0.428 g/cm3 because R = 4
(These values are the solutions of Eq.(3) substituting
B e 1, 2, 3 and 4, respectively.)
Accordingly, the skeleton of the bulk density time series is
formed by the values rk and Y, chosen as the function of the
value min and of the number of layers. Intermediate values can

21 2
only be estimated. In the period of accumulation the method al-
ready demonstrated is used. in the melting period i.e. when the
thickness of snow cover decreases,a linear interpolation consider-
ing the change of thicknees is made to choose values betweenthe
critical and maximum bulk density.
With these considerations the whole time series of bulk den-
sity can be produced.
b) Once the lime series of snow thickness (v) measured and of
bulk density(J-) calculated are available, the time series of wa-
ter equivalent (h) can be computed by the following equation:
h Cmml = Cg/cm31 . 10 v Ccml (8)
In Fig. 2, the data series of bulk density and of water equi-
valent computed with the method demonstrated above are compared
with the time aeries of data measured.
2.2 Forecast of snow meltinq
To demonstrate the method of forecasting the knowledge of
only the snow cover thickness and of air temperature is suppoeed.
a] First the initial water equivalent (ho) of the snow at the
start of melting is to be determined. This can be read directly
from diagram B of the chart.
In the example, at the start of final melting on the 2nd of
March the thickness of mow cover containing 4 layers was 21.0 cm.
A reading in the direction of the fat line on the chart yields an
initial water equivalent of
ho = 74.0 2 6 Cmm],
i.e. melting is expected to produce
74.0~6 rn of snow-melt water
(the water equivalent actually measured was 71.6 mm, i.e. only 4 %

less than computed).
b) With the water equivalent obtained as described above, and
using the data on minimum and maxim air temperature forecast for
the melting period, the duration (mo) of the melting period is
estimated by means of part C of the chart.
The meltina heat standard (K) needed for using the chart is
calculated from Eq46).
In the present example, for the week following the start of
melting = 5.6OC was obtained.
Consequently, by reading in the direction of the fat line, a
melting period of
rn =3 8 2 1 days
O
is predicted for the snow cover having a water equivalent of
74.0 mm. (The actual melting period measured was 7 days).
Note that the forecaet described should be repeat.ed daily
with the latest data to make continuous allowance for changes in
the weather.
s x a t
From the results of error analyses performed for checking the
two methods and from the first experiences gained with their ap-
plication, the methods described for calculating the water equi-
valent and forecasting snowmelt appear to be of practical interest.
21 3

21 4
REZERENCES
Cil Karo1.B.P.: Snow cover (in Russian)
Gidrometeorológitsheskoe izdátelstwo
Leningrad, 1949.
L23 Kovács.Gy.: The altitude system of snow conditions in the
winter 1968-69. (in Hungarian)
Annual report on the work of VITTJKI 1972, Buda-
pest
L31 Kovács G The development, accumulation and melting of
snow, the measurement and calculation of these
features ( in Hungarian)
Bogdhffy, be Pályázat . 1973, Budapest
141 Kuzmin.P.P.: Physics of the snow cover (in Russian)
Gidrometeoizdat, 1957.
151 Péczels,Gy.: The consideration of the accumulation and melt-
ing of snow in the analysis of the precipitation
system of catchments (in Hungarian)
Idójárás 1969/1, Budapest
161 Sa1amin.P.: The examination of snow melting in the Bükk
mountains in Hungarian)
Id6járás 1 4 56/5, Budapest
173 Sa1amin.P.: The problems of the examination of snow melting
(in Hungarian)
Discussion. Department of Agricultural Sciences,
Hungarian Academy of Sciences, 1-3. IX. 1956.
183 Sa1amin.P.: The influence of the relief on the accumulation
and melting of snow (in Hungarian)
Hidrológiai Köalöny 1960, Budapest
191 To1lan.A.: Determination of Areal Values of the Water Equi-
valent of Snow in a Representativ Basin
(in English)
Mordisk Hidrologisk Konferenc, Stockholm 1970.
Li01 Yosida.2.: Physical Studies on Deposited Snow I-IV.
(in Enalish)
Gechanical Properties.
Contributions from the Institute os Snow Tem-
perature Suence, Sapporo, 1956.

RECONSTRUCTION OF THE WATER EQUIVALENT
TIME SERIE8 OF JANUARY TO MARCH, 1963 AT
DOHBORI PUSZTA.
0,600
0,500
w 0,100
- C0b:PU TED
-_ OBSER VF D
21 5

o
216

EVALUATION OF LOCAL WATER RESOURCES IN AN SEMI-ARID,
HARD ROCK REGION BY USING PHOTO-HYDROLOGICAL INDICES
ABSTRACT
By: A.M.J. MEIJERINK
The local water resources of a semi-arid, hard rock
area have been evaluated in a rapid, but approximate way,
by using information derived from aerial photographs and
from field observations.
The sizes of irrigated areas of open, wide diameter
wells, and of small reservoirs, have been taken as indices
for the storage of groundwater and for the runoff generating
capacity of small watersheds up -to a size of 40 kms2.
Geological and geomorphological influences on the
groundwater storage and the recharge of the wells, could be
established.
By means of a judgment of the catchment characteris-
tics, the relative runoff could be estimated.
RESUMEN
The limitations of the use of the indices are discussed.
Los recursos hfdricos locales de una zona semi-árida,
de rocas de baja permeabilidad, han sido evaluados de una
forma rápida pero aproximada, mediante el uso de información
obtenida de fotografías aéreas y de observaciones de campo.
Los tamalios de las áreas irrigadas por pozos abiertos
de amplio diámetro y por pequeños estanques, han sido toma-
dos como indices de la reserva de agua subterránea y de la
capacidad generadora de la escorrentia de pequefias cuencas,
hasta de 40 km2. de extensión.
Se pudo establecer las influencias geológicas y geo-
morfológicas en las reservas y en la recarga de los pozos.
Evaluando las características de las cuencas se pudo
estimar la escorrentla relativa.
Se discuten asimismo las limitaciones 'del uso de los
indices.

218
I. Introduction.
The aim of this paper is to show the use of photo-interpretation for the
assessment of the local water resources in a semi-arid region, underlain by hard
rocks with little storages.
The region studied is a part of the Cuddapah Basin in south India (see figure 1 ),
where the agriculture depend on the meager local water resources.
By means of photo-interpretation the distribution and the relative quantities
of the ground- and surface water resources could be studied.
The study consisted of a simple differentiation of the area in more or less
hydrologically homogeneous units and of an analysis of the irrigated areas within
the various terrain unit s.
The sizes of the irrigated areas are in fact a control of the interpretation
procedures and also an indicator of the hydrological situations.
Approach.
The following approach for the interpretation procedures has been adopted:
____________________------_-_
Differentiation of the region.
The region has to be divided in certain 'landscapes'. Each landscape has
its own complex of gross hydrological processes and gross water resources
which differ from those in adjoining landscapes.
Within each landscape there are a number of smaller land components which,
on a reconnaissance scale, may be considered to be hydrologically homogeneous.
The land components may be of erosional or denudational nature such as a
dissected alluvial fan y an inselberg complex with the surrounding
embayments y etc.
The land components may be sub-differentiated, if necessary, in land elements.
The land elements are described here as terrain units which are closely
associated with simple hydrological processes.
An example of a land element in this study is: an area with well terraced
agricultural fields, which are capable of storing appreciable quantities of
surface runoff.
Practical, rather than theoretical considerations are at the base of this scheme
of land differentiation.
However, the 'landscape' which has been described above, may be compared with
Verstappen's 'Main geomorphological Unit (1 96fl), with the 'Land System' of the
Oxford Working Group (Brink et al. 1966) and with the 'Mesnosti' of the Eussian
Authors (Vinogradow 1968).
In the area of study, the boundaries of the landscapes follow closely the
boundaries of the main lithological units.
The close association between the landscape and the geology in the Cuddapah Basin
is caused by two facts:
1)
2)
The lithologies are markedly different from each other and large outcrop
areas are formed because of structural conditions.
The denudational development of the area under predominantly semi-arid
conditions has resulted in the adjustment of the geomorphology to the
pronounced geological fact ors.

- B. Hydrological evaluation.
219
Various landscapes occuring in the Basin, have been delineated and within the
landscapes a sub-differentiation was made of the land components and occasionally
also of the land elements.
The groundtruth of the interpretation, was in this particular case already
available, because the author had carried out previously field checking of
interpreted geology and geomorphology of the basin during four field seasons.
Moreover, a soil survey of representative strips in each landscape had been
made by a third party.
The possible hydrological significance has been estimated of the interpreted and
mapped land components and land elements. The estimation is subjective.
Details of the features, caused by overland flow and by concentrated sheet flow,
as observed on the aerial photographs, have been used, as well as some other
photographic characteristics. However, far more reliance was placed on the
possible hydrological behaviour of the soils, weathered zones, superficial
deposits, particulars of the lithology, etc.
The photo-interpretation is thus mainly useful as a means of rapid inventarization
and for the study of the inter-relationships of the interpreted features.
The results of the mapping and the hydrological evaluation have then be compared
with the outcome of the analysis of the index 'irrigated area'.
In this text the emphasis is placed on the illustration of the use of the index
in various terrain conditions, rather than on a description of the mapping
procedures.
The results obtained in three landscapes are discussed; first the groundwater
occurrences, then the surface water resources.
Nature of the index.
This index is actually a composite index, as the irrigated area is not only
influenced by the available groundwater in the zone near the surface
(the rocks are impermeable in unweathered conditions), but also by the irrigation
practices. In the area, there are thousands of open wells, out of which irrigation
water is lifted by animal traction. The open, wide diameter wells pumped for a
number of hours are then left to recover for more than 24 hours.
For the sake of briefness, a few examples are discussed.
Because the agriculture is still carried out in a traditional manner, it has been
assumed that the yield of the well is directly related to the size of the irrigated
area.
How accurate this relationship is, within a given landscape, has not been
investigated.
As is discussed further on, it is not possible to compare the acreage per well
of one landscape with another, because of differing soil conditions, crop
rotation and water application.
However, within a landscape the irrigation practices seem to be uniform.
Therefore, the following discussion pertains only to the use of the index within
a landscape.

220
8-
&ample showing the effects of geomorphology on the occurence of
groundwater.
-
Well clusters and the irrigated areas have been investigated on the Vempalli
calcareous shales. A part of the area is shown in figure 2.
It may be noted that the width of the recharge zone on the pediments, is
related to the width of the irrigated areas. This relationship once established
can be used to indicate potential irrigable areas by means of mapping the
pediments in the area.
Field observations have shown that the depth of weathering and of sheetwash
deposits, increases on the pediments in downstream direction.
On the upper parts, the unweathered bedrock is close to the surface and little
infiltration can take place. However, firther downstream till the central
drainage line is reached, the sheet flow may infiltrate partly, and recharge the
- limited - quantities of groundwater.
The relationship between width and recharge zone and width of irrigated area has
been found useful for locating 'under irrigated' areas in this particular
landscape.
- G.
=ample showing the use of statistical test for the evaluation of factors
which influence the occurrence of groundwater.
In a pediments landscape, the well indices have been used to compare the influences
of the rock types on the well yields.
Four sample areas have been selected at the downstream parts of the pediments, in
a rather narrow zone near ephemeral rivers, in order to minimize the influence of
the morphological position.
The four sample areas are underlain by slates, by sericitic schists, by biotite
schists and by gneisses.
An analysis of variance shows that the differences in the sample means are not
significant at the 5% level. The sample sizes varied from n = 12 to n = 24.
Hence it is concluded that the influence of the lithology in this area on the
well yield is not significant. It should be noted however, that the rock types
are rather impermeable anyhow. The similarity of the well yields is attributed to
the effects of weathering, type of soils,calcareous and siliceous crusts and to the
deposition of sheet wash deposits.
A little south of the sample areas, approximately 200 la2, fossil aeolian sands
lare covering the pediment surfaces. The thickness of the sand cover varies from 1
to over 20 meters.
It can be expected that the well yields are higher than in the area without sands,
because of the good infiltration possibilities in the sands and the higher specific
yields of the sandy medium.
This expectation is confirmed by the indeces. However, no significant differences
in the well yields in this area could be detected, after the indices had been
sampled in four areas. The sample areas have been selected on bxoad drainage
divides and near ephemeral channels.

- D.
Ekample showing the possibility of predicting approximate well yields
f l o m _ s _ l l m p i e _ _ - ~ ~ - ~ ~ _______________
~ - ~ ~ ~ ~ ~ - ~ ~ ~ ~
In the three discussed examples, the nature of the recharge area was of
interest. In the granite landscape along the western margin of the Cuddapah
Basin, it was possible to delineate the 'catchment' areas of individual wells,
and thus compare the size of the 'catchment' or recharge area with the size
of the area irrigated by the wells.
The landscape in which three sample areas have been selected, consists of
convex interfluves and concave to flat valley bottoms. On the interfluves
the depth of weathering varies from O to 5 meters. The soils are red coloured,
loamy sands to sandy loams with stone lines. The soils in the valley bottom
are grey coloured gritty clay loams to sandy clays.
Althou& large outcrops may occur in or next to the valley bottom, the average
depth of weathering seems there to be higher. Inselbergs are found scattered over
the area.
2 21
The recharge area of individual wells have been sampled wherever the local
relief was sufficiently high to delineate the drainage divides on the interfluves
and where the irrigated areas could be differentiated from the surrounding
non-irrigated fields.
The three sample areas are shown in figure 1, from which it may be noted that
the mean annual rainfall of the sample areas is about the same.
The three areas seem to be similar in geological and geomorphological respects.
The areas irrigated by wells are shown in figure 3a, and have been plotted as
a function of the recharge area in the graph of figure 3b.
This graph shows the combined results of the three sample areas.
In all the three sample areas the correlation coefficients are significant at
the 5% level (Spearman rank correlation statistic), while no significant
differences have been found between the three correlations (kskall and Wallis
test of variance). For practical purposes, the line of best fit, shown in the
graph may be used as a guideline for the estimated yield of open wells as a
function of the recharge area in the sampled landscape.
The scatter of the plotted points indicate approximately the degree of accuracy
of the estimate.
Remarks.
These few examples show how the index irrigated area may serve as a check on the
expectations of the water occurrences and the approximate quantities, within
well defined landscapes.
It is not possible to compare the indices of one landscape with another, because
the index is very sensitive to variations in irrigation practice, type of irrigated
soils and the type of crops.
If the yields in the various landscape have to be compared, the index has to be
transposed in actual well yields.
However, by employing this index as a control on the expectations, the value of
the photo-interpretation is increased and the amount of field works is greatly
reduced.

222
1x1. Surface water.
- ____________________________I___________-------------------------
A. The use of the index 'area irrigated by water stored in small reservoirs'.
While scanning the aerial photographs, an apparent relationship was noted
between the size of the catchment and the size of the irrigated areas below
reservoirs. The reservoirs are usually constructed across the main drainage
line, and are thus in a position to store the full runoff, provided of course,
that the capacity of the tanks is sufficiently high.
The reservoir capacities cannot be estimated on the aerial photographs
accurately, because they are very shallow, usually less than 2 meters deep.
However, it may be supposed that the irrigated areas are closely adjusted to
the average available quantities of water in the reservoirs.
Thus, the irrigated areas are used as an index, or measure, for the runoff
from the catchments. The selected catchments are smaller than 40 h2.
Comparisien of the indices are only possible in a well defined landscape.
Differences in irrigation practices, type of irrigated soil, etc., prohibit
the comparision of the indices from two or more different landscapes with
each other.
Therefore, the index is mainly used to investigate whether variations of the
catchment characteristics, within a landscape, are associated with variations
in the index.
Before embarking on the discussion of the analysis, it may be useful to
illustrate briefly the rainfall factors, the magnitude of the evaporation and
the type of runoff.
- -------I------_-_-------------
B. Rainfall, runoff and evaporation.
Inspection of the daily rainfall records of a few stations in the area shows
that most of the rainfall occurs during the summer monsoon (SW monsoon).
The maximum daily rainfall in the winter monsoon (NE monsoon) is much lower,
usually less than 20 mms. per day. Mean monthly rainfall varies from 150 mms.
during the SW monsoon to 5 ms. during the dry months.
The frequency of the maximum daily rainfall, based on the partial seriesis
shown in figure 4. The days with more than 30 mms rainfall (arbitrary standard)
have been used for the compilation.
Field observations show that isolated showers tend to produce flashy runoff in
this semi-arid area, where the catchments have little storage possibilities.
However, according to the local population, the tanks get filled up mainly by
prolonged rainfall. It is not uncommon that in the SW monsoon, during three
consecutive days with rainfall, more than 100 mms are recorded.
Such occasions cause overflow of the reservoirs.
On the other hand, during dry years the tanks may not get filled up, or may not
be replenished for the irrigation of the rfpening crops.
Analysis of the irrigdted areas in some catchments, as measured on the aerial
photographs, indicated that the tank capacities are not capable of storing the
most important runoff events, This was found by comparing the irrigated areas,
expressed per unit of catchment area, of two or more tanks in single catchments.
The downstream irrigated area was larger in 10 out of 11 cases.

Although the runoff may be prolonged when successive rainy days with high
rainfall amounts occur, the runoff decreases rapidly after the cessation of the
rainfall. Most of the runoff seems to occur in the form of direct runoff with
very little interflow (throughflow) and base flow.
2
The ephemeral rivers of the small catchments (up to 40 lan ) have dried up
practically within a few days.
The evaporation of the area is high. The highest mean monthly evapotranspiration
in the area is 200 to 220 ms, and the minimum mean monthly value is 11 cms
(at the time of the winterrains).
bhenthemeanyearly evapotranspiration figures, which are based on the Modified
Penman fomla, are compared with the rainfall figures, the water shortages in the
area become obvious.
The average depth of ihe tanks is usually very small (< 1.5 meters), but the
size of the tanks is comparatively very large (5 - 50 hectares).
Monthly evaporation rates of more than 10 cms reduce therefore the effective
storage of the tanks appreciably.
Factors that influence the size of the irrigated area.
From the above discussion, it is obvious that the index 'size of irrigated
area' is an inaccurate measure for the runoff of the catchments.
It is difficult to say for example, to what duration and frequency of the
discharges, the index is related.
For the evaluation of the use of the index the following argument has been
used:
If the index is a perfect measure for the runoff production of the watersheds,
a perfect correlation between the index and the size of the catchments can be
expected. Variations in the relationship should be attributable to variations
of the hydrological effects of the landcomporients.
Actually, it is the variation caused by the land components within a landscape,
that is of interest in this study.
However, the imperfectness of the index may be demonstrated by pointing out
the following sources of error:
1. Original differences in the capacities of the reservoirs, because of
topographical differences, construction of the overflow, etc.
2. Reduction of the original reservoir capacities by sedimentation.
Age and history (breakages, desilting operations) of the tanks may differ
within a landscape, also the sedimentation rates per unit of catchment area.
3. Minor differences in the irrigation practices, water management.
4. Interpretation and measuring errors on the aerial photographs.
The variation caused by these 4 factors cannot be evaluated without detailed
measurements in the field.
Despite the fact that the mentioned factors are an important source of error,
in all the four sample areas, a significant correlation has been found between
the parameters 'sise of catchment area' and 'sise of irrigated area'.
223

224
The influence of the land components on the index is discussed here for one
large landscape, the 'Cumbum Landscape'.
In aiiother landscape, the 'landscape on the eastern basement complex', rather
similar results have been obtained, and need therefore little elaboration.
However, on the third landscape, the one on the granites in the west, where
the land components seem to be equally distributed in the landscape, significant
differences have been found in two sample areas.
In the other landscapes of the Cuddapah Basin, no sufficiently reliable
measurements could be made for a proper analysis.
--______________________________________----_--------------------
The Cumbum landscape, an example of the runoff variation within a landscape.
Descript ion.
The landscape on the shales, siltstones and phyllites with occasional
limestone beds of the Cumbums, forms a separate landscape in the Cuddapah Basi
although the geomorphology of the landscape is not uniform.
In some parts of the area, weathered remnants of large but thin alluvial
fans are found, supporting a dense vegetation of grassland and dense shrub.
Other parts may consist of eroded terrain of varying relief and skeletical
soils. South of this area extensive remnants of an old weathered planation
level are found.
Estimation of the relative runoff from the aerial photographs.
The runoff producing and runoff-storing land components in the watersheds
have been interpreted and mapped.
Land use features and geomorphological elements have been mapped separately,
but have been plotted on a single map.
The joint effects of the land use and geomorphology on the runoff has been
evaluated by means of arbitrary standards:
Land use features such as fields surrounded by earthern walls, behind small
retention structures, etc. are capable of storing runoff and the areas with
many of such features have been classified as 'areas with low runoff'.
Overgrazed, poorly cultivated fields on sloping land, fall in the class
'high runoff'. The other areas have simply been classified as medium runoff.
Similarly, geomorphological elements, such as old weathered fans, thick
slope deposits, buried pediments and elements like heavily eroded soil
bare outcrops, true pediment slopes etc, fall in two opposing classes; high
and low runoff. The remaining elements with no evident extreme hydrological
behaviour fall in the medium class.
The percentages of the areas falling in the three land use and in the
three geomorpho12gical classes are then determined, and a final judgement
puts the catchment in one of the three categories.
The hydrological evaluation of the land components cannot be done without
sufficient field howledge. The procedure is arbitrary, and the results will
therefore vary from one observer to another.
Analysis.
The graph of figure 5a shows the index 'irrigated area' as a function of the
catchment area. The line of best fit has been established by graphical
correlation (Linsley, Kohler, Paulhus 1949).
For all the watersheds, shown on the graph, the runoff class has been
estimated.

2 25
The runoff class is now compared with the deviation of the plotted position
of the points on the graph of figure 5a with the line of best fit.
Points which are below the line of best fit are called negative deviations,
those above the line, positive deviations.
Theoretically, possitive deviations should be associated with high or medium
runoff classes, negative deviations with medium or low runoff classes.
The results of the comparision are shown in figure 5b.
Ideally, in the case of high runoff, all points should fall in the positive
range and the cases of low runoff should fall in the negative range.
The cases of medium runoff should have a symetrical distribution and the
magnitude of the deviation should be limited.
As can be judged from the graph of figure 5b, no ideal result has been
obtained, although the mediam values (see the graph) of the three runoff
classes are at three different levels ( + 36, - 38 and - 96 ).
It should be remembered that the procedure is arbitrary and that there are
many causes of variation. The procedure followed, i.e. the author's
judgment, leads to an underestimation of the storages and losses in the
catchments. However, the method is not ment for an estimation of the
absolute runoff production, but for an estimation of the relative differences
of the runoff in the various catchments within a landscape.
For this purpose, we feel that the results of the comparision of the runoff
estimates with the deviation of the index, are satisfactory, so that the
runoff classification may be applied in this landscape with some confidence.
The landscape on the granites and example of unexplained differences in the
index.
Catchments and irrigated areas have been measured in two samples areas on
the granites, along the western margin of the Cuddapah Basin.
!Che areas, denoted here with the northern and the southern area, are 250 lans
apart. The mean annual rainfall in the two areas is to same, between 600 and
700 mms. Only small differences, if any are expected in the frequencies of the
partial series.
As has been discussed earlier, in the section on groundwater, both areas seem
to be highly similar in geological and geomorphological aspects.
The relationships between the catchment area and the size of the irrigated areas,
is shown in figure 6.
The Kendall rank correlation coefficient for the northern sample area is 0.78,
for the southern area 0.90. Both the correlations are significant at the 5% level.
However, the two sample areas show a marked difference between the indices
'irrigated area' as a function of catchment size.
The Muskall and Wallis test showed that the difference of the two relationships
is significant at the 5% level.
The explanation of the difference is difficult.
Its has been tried to explain the difference by some additional photo-measurements
of factors, that might influence the size of the irrigated area.
Within the catchments, the percentage of outcrop areas in the two sample areas
have been compared, but no significant difference has been found.
It was reasoned that the percentage of outcrop area would influence the runoff
(and thus the index), because of the very low storage possibilities on the
outcrops. The number of wells in the irrigated area have also been compared with
the irrigated area. It was thought that the wells, which re-use the water from
the reservoirs, could be of influenoe. However, no significant correlation has bt.,:.?
found.

226
The explanation of the difference in the relationships may be hidden in
factors, which are not measurable on the aerial photographs.
In this particular area, it is suggested that perhaps, the age of the reservoirs
is an important causative factor.
In the southern area the reservoirs may be older than those in the northern
area, so that the smaller irrigated areas in the southern area, may be explained
by a reduction of the reservoir capacity by sedimentation.
Discussion of the results.
In the area of study, the index 'area irrigated by open wells and by small
reservoirs' provides a useful means of control of the hydrological significance
of the photographic interpretation procedures.
However, the use of the index requires a good deal of local knowledge of the
terrain characteristics, irrigation practices, etc.
The index should be used in a careful way and the purely empirical character of
the index restricts its use to well defined landscapes.
However, a number of practical applications of some tested relationships have
been found for some landscapes in the Cuddapah Basin.
The experience and confidence gained by the study of the landscapes and by the
analysis of the index has been used for the hydrological evaluation of those
landscapes, for which no sufficient indices could be sampled.
It is believed by the author, that for similar semi-arid,areas on hard rocks
an approach along the same lines could give useful results, particularly when
no appropriate hydrological data are existing.
The indices are rather typical for the area investigated, but have been used
for other areas in india as well. In regions where such indices are not
existing, or are not very meaninghl, the evaluation of the landcomponents has
to rely on other field observations. The field observations and measurements
may consist of oral information of water level fluctuations in wells,
determination of approximate yields, measurement of the base flow discharges,
perhaps the estimation of the bankful discharges, and so on.
The approach consists, in short, of:
1.
2.
3.
4.
5.
Interpretation of geology, geomorphology and of aspects of soils,
land use and vegetation.
Differentiation of the regior. in 'homogeneous hydrological landscapes'.
Provisional hydrological evaluation of the land components.
Field checking of the interpretations and the collection of approximate
hydrological data. The field work and the collection of data should have
been planned on the basis of the interpretation results.
Comparision of the results and final, approximate evaluation of the
influences of the land components for the development of the local water
resources.
-0-o-o-

Ref e re n ce s
2 21
Brink A.B., Mabutt J.A., Webster R. and Beckett P.H.T. (1966)
Report of the working group on land classification
and data storage. Milit. Engng. Exp. Establ.
Christchurch, England. Engng. Report no. 940.
Verstappen H.Th. and Zuidam R.A. (1969) I.T.C. system of
geomorphological surveys. I.T.C. textbook of P.I.
VII, 2. 49 pp.
Vinogradov B.V. (1968) Airphoto methods in geographical research
in the U.S.S.R. Photogrammetria 23 pp 17-94
Linsley R.K., Kohler M.A. and Paulhus J.L.H. (1949) Applied
Hydrology Mc. Graw Hill, New York

228
I = GRANITE LANDSCAPE , II = CUMBUM LANDSCAPE ,
III = LANDSCAPE ON EASTERN BASEMENT COMPLEX -
A = ANANTAPUR , K = KURNOOL , C = CUDDAPAH
700 = ISOHYET
Figure 1 - Map showing location of landscapes and the outline of the
Cuddapah Basin.
Fiyre 2 - Irrigated area in relation to the width of the recharge area.
( dots = open wells, white = pediments, hatches = hills ).

Figure 3a - Sample area on the landscape on the granites, showing drainage
divides, thalwegs, wells (open circles), sampled wells (dots)
and the corresponding recharge and irrigated areas.
The schematical section indicates the depth of weathering.
R E C H A R G E A R E A
Figure 3b - Irrigated area as a function of recharge area, for three sample
areas in the landscape on the granites.
229

;"O1
1.2
1 .o
4
w
m
4.8
FI
W
U
H
œ
œ .4
U
.2
2 3 o.
,? -
+
(scale for III )
I I I i ,
0.4 1 .o 2 4 6
I I I I I L I , , I
0.4 0.6 0.8 1.0 2 4 6 ô 10 years
B E C U R R E N C E I N T E R V A L
Figure 4 - Frequency curves of high daily rainfall (partial series), for
three stations:
I = Cuddapah, II = Kurnool, III = Anantapur.
4 a 10 16 20 24 28 ~m2.
D R A I N A G E A R E A
Figure 5a - Area irrigated by reservoirs as a function of the catchent
area for the Cumbum landscape.

Km2,
2.8
2.4
2.0
w
m
4
1 .6.
!a
W
I 3
-4
o 1.2
H
a
e:
H .a
.4
0
rd
5
+
200' -
2 120
3
$ 40-
3
w o-
M 40 -
.rl C
E! 120
.rl
*
rl
200
-
-
-
- IIII.I...IIL
no. of catchments Y
ii
MEDIUA
LOW
Figure 5b - Results of comparison (see figure 5a and text).
S A M P
I. N.E.
II. R A
L E A R E A S :
of K U R N O O L
Y A C H O T I
i/,-/ . ..
/ +
12 16 20 24 28
D R A I N A G E A R E A
231
Figure 6 - Areas irrigated by reservoirs as a function of the catchment
area. Two sample areas are shown of the landscape on the granites.

ABSTRACT
APPLICATION OF SATELLTTE CLOUD PICTURES
IN SNOW HYDROLOGY OF TKE AIMALAYAS AND
IN THE ESTIMATION OP RAINFALL OVER INDIA
DURING SOUTKWEST MONSOON SEASON.
P.S. PANT and N.G. GUPTA
Many of the rivers in the Indi'an suB-continent bave their ori-
gins in the Himalayas. An important souTce of water supply for these
rivers is, therefore, from the melting of the snow in the upper cat-
chment o€ these rivers. As much of this region is inaccessible, the
conventional observations are not availaBle, Examination of the sate-
llite Television Pictures has revealed tñe possibility of estimating
the snow coverage over the river basi'na in tke Hihalayas and thereby
to estimate tlìe contriBution of snowmelt to tñe flow in tEese rivers.
An attempt has also been made to estìmate the 24 hour rainfall amounts
over the plains of India duTing tñe soutñwest monsoon seanson, using
cloud imageries taken By wather satelli'tes. Tiìese estimates are found
to be in reasona6le agreement witñ tñe values obtaihed by isohyetal
analysis of actual Tai'nfall 8ata. Tfie iesnlts'oBtaihed look promising.
If confirmed from an analysis of several rainstorms, these will find
wide application in estimation of 24 hour average areal rainfall over
data sparse river catcñments.
RESUME
Plus de rivieres en le SuB-continent de l'Inde ont leur origi-
ne dans les Himalayas. Une importante source de l'approvisionment
d'eau pour ces rivieres est, donc, la fusion de neige dans la captation
supérieure de ces rhi'eaes. Comme Seaancoup ae cette .pêg?on eat Pnacce-
ssible, les observations de convention ne sont pas disponibles. Un ex5
men de l'images têlévision du satellite a rdv'?lE! la possibilité pour
estimer la enneigement sur les garages de rivières dans les Himalayas
et ainsi pour évaluer le contribution pap la fusion de neilge an coule
ment de ces rivilres. Un effort a &!té aussi Sai't pon? estihex la préci
pitation pendant 24 heures sur les plaines ae l'Inde pendant la sud
-ouest mousson par 'nauge photograpñ2es de satelliye du temps, Ces eya-
luations sont d'accord raisonagle avec les valers obtenues par les ana
lyses isohyetales de actuels prScipitation donnees. Les resultants se
présentent bien, S'il est confirmé d'une analyse de plus de tempete de
pluie, ces trouverent large application pour entilmer la précipitation
asrienne pendant 24 heures sur la captation de rivières oh les données
sont rares.

1. Ii@RDDUCTIDW
1.1 &UIY rivers in India originate in the IIimalayas. The important
some of water supply for these rivers, is the snow over the upper catchment
amm. Wrefore mapping the sncw cover and its variation mer the Himslayas
is vital to forecast the stream flow in theae rivers, which in turn is of groat
inportance for generation OP power and irrigation through these rivers.
1.2 Large p&s of the upper catchent of these rivera are inace8sibi.e
mas and monitoring tho snow cover ard precipitation in these areas by comentional
methods is difficult: With the advent of Polar Orbitting Satellites the
possibïìity of mdtoring the abme-mentioned hylromerteorological parameters by
reaiute sensing techniques has arisen. In the cloud imagery obtained through
the satellites anow aver the Himalaya can be clearly recognbed.
R is also
reiativalg easy to identify individual river valleys on these satellite picturm
d e r cloud f'roe conditions.
1.3 Ih case of rivers whose stream flow ia mainly dependeat on precipitation
it is inportant to evaluate a~ acourately as p0~sibI.e the distribution
of precipitation with area and duration k? order to obtain run-off from
precipitation. 51 case of @lood forecasting the adàitional problem involved is
that the 24 hour raFtiEaU data should be available eqmdi.tiousiy at the forecasting
centre,( Since owll established conmsUnicatian links are nut available
for ail the river catchment areas, it WU be advantageous if atleast a rough
t38thIl&e of aerial precipitatiun (average) can be obtained frm in8tan'tBneoUS
cloud imagery for calculating run-off.
1.4 Before one can make an attempt to derive the 24. hour precipitation
m the basis of m instantaneous satellite cloud picture, one bas to knm
the characteristics of rainfall ono ia trying to estimate. Over large parts
of Pidia more than 75 per cent of the annual rainfall is received during the
southwest mansoon season (June to September). During this seasong the rdnfall
is not corrtinuou~ but ~ C W S in spells laethg for about 5 to 7 days. This Is
uauolly associated with the occurrence of depressiona and their movement roughly
dong the mowoon trough of lw pressure. These depressions which irsueliy
fonn near the he4 of the Bay of Bengal cause locally heavy faUs varyiiig fra
7 to 20 cm in 24. hour, The rainfall associated with them extends mer areas
aa ïarge as 100,ûûû to 200,ooO sq. hs; In hilly areas rainfalls as high as
25 to 35 cm in a day are recorded in assmiation with these depressions.
Mansoon rabif& in general sbowe two diurnal peaks one in the afternoon d
another in the early morning hours. It ia also noticed that the monsoon clod
pattern, p&iCixlarly tbme comected with situations af monsoon depressions,
do not show much äiffereme between the afternoon and the morning. It is therefore
felt that the single satdite cloud pictures from the orbitting weather
satellites can provide a fairly reasonable estimate of the +hour rainfall,

235
1.5 in ader to attain reasombh auccees in our attempt to relate
these satellite cloud bagery with aerial distribirticn of precipitation, we &ve
choeen only cccaaim Of raSn storms durhg the p1~oon season. Mher ue have
&o restricted our attention to the plejm thua avo%dhg the complications that
VU set in hilly areas, With these restrictions it is hopd that a reasonable
degree of aucc888 can be achiwed,
2. SNOW KyDRDi&GY OF HIULAYAS
2.1 Monitoring of the snow cover over Hbhyaa far hy&dogical piurpases
ha gained hportance in Tecent yoara in connection with cmstruction and
operation of the dams cmatructed mer the rivers originating from the Himelayas.
!i'he techniques wed in the application of satellite data for snaw mapping are
basically those of Simple photo-htarpretaticm i.e., detailed vieu inapetion
of individual photogmpha to identify the river comes and large valleys to
detenuine the aerial extent of 8now cover and to make an estimate of the height
of snowline.
2.2 A duly au the aasessment of water flow ki River Sutlej by
Satellite picturea was conducted by Gupta and abbi (1 ) . The aver Sutlej
originate from Lakes B$Las d Mansarovar in Tibet and foUons a aourse of about
@û hm towards west-north-wemt through mountainm terrain before ananating in
the plains of Punjab.4 The monthly werage discharge data of the Bempur stream
gauge site, located near the Hhalagaa, ahow that the water discharge which ia
2000 - 3ûûO c1i8808 during the vinter mcmtha @acember-Febninry) reaches the
peak velue during July when It b more than ten tkes the winter -off.
w u be seen frcm the mdhly average discharge data for the mar 19669 given
in T&ie ï, that while tbe monthly average of water dischare;@ valu@ during
wMer months do not vary much from year to year, there are large variati- in
t b e for the anow melting period.
Table 1
MûNIHLY A m WER DBCXAWE SN CUSiES DIFERE3CE; 33 THE VfiUEs
MONPH
196F3
1 %9
OF WATER nmcmìm
1968u1969
JanUary
31 88
2359
829
February
3068
2665 43
Mmh
394
3658
3 O5
d g d
5944-
5063
881
Hay
11 6a3
13868 -2260
June
31 893
4r5892
-1 2999
July
32/62 47081 -14319
Bug;&
2153 2
N63 2
-131
-
00
September 1m
17869
-5592
October 5w
7357
1450
Naoeniber
3760
45@
-788
Decrember
2800
3500
-700

236
-tion of the satellite pictures for the above two years showed that BLICUdation
of snow aver the western ELimalaya~ OCCW during the maths of
Novembe-F'ebruary. The snow-malt b- spring i.e., from the m&ha of
m h when the valleys and river courses start becorning viaible distinctly.
The minimum snow cover ocam in the post-eionsoon period after the nœrth of
Septeaiber. Figure 1 shows sane of the river courses in the aateïïite picture
of 9 June 1969 cover- northern and western Himalayas. Examination of day to
day satellite Pictures revealed that well sustained precipitation activity mer
the Sutlej basin during the spring and pre-monsoon aeason of 1969, cdributed
to higher values of water discharges from the month of May aL1WBPds. Th basin
was also affected by the recurdng monsoon depressionsin the month of September
I969 whereas the year 1968 was marked by the rar3y withdrawal of mmsoon from
the region.'
2.3 Snow cover stdy of northern snd western Himalayas conducted by
Srinivasan and Raman (2) with the help of selected NnIIBLE3 ïV and ESA4
Pictures of 1969-70 bas confirmed that accmulatian of snow starte from
the month of November and reaches the reixhum during Janu;iry-February wilh
snow line generally at 213 l0n.b The snow melt begins in 4ril ad the mlnimimi
snow cover was found to occur ki this stdy in Augmt-October with anow line
rising upto about 5 km. The tearporai. sequence of aIIMBu9 III Image Diesector
Camera mea (IDCS) pictures of Northern ñhiìayas for the period of April
196Waaueuy 1970 given in a report of NASA an NïMEiiJS (3) dematrates t b
minimum of the snow cover aver the &dus river basin in the Himalaya dipring
Augwt-September. The B~QV accmùlaticm mer the area cormences thereafter
till the month of Bpril when mdting of snow begins.
2.4 maph for the river Brahmputra, rivers originating fmn the
Centrai and W e r n Himalayas do nut have lmg courses along the mountab.
These are therefore not 80 distinctly vieible in satellite *dea a9 the
rivera in northern end western Himalayas. Since the winter precipitatiar over
these regions is much less canpared to northern anl western Hiinalayaa t b
magnitude of snw cover difference during the winter and summep semons ie not
significant, especially so over central Hhaiayas. Examinatiosi of snow cover
over the valleys in these regions, therefore shows that the height of the
snowline is generally between 3.5 -5.0 Kms. The contribution of anw-mdt to
the water discharge of rivers in these regions e m therefore be expected to
be less than that in the western Himalafraa.'
3. EsTmIDN OF RAINFALL DURmG THE SOUl!ME3T MNSOON SEASON
3.1 In recent pars the satellite iimgeries bave been increasi~gly
used to derive the relatianship Wween the cloud pattelas and the WcipittLtion
distribution over the data sparse areas of tropics. In India an early
attempt was made by Hulshrestha and Gupta (4) to st* the rainfall. and oled

patten associated with the mansoon depression. The nephandpis prepared by
U.S. Weather Bureau were &tilb& by B-tt (5) to estimate the maithly rainfa
in the Australian Region. The radiation data of TïBIS III waa utilised
by &inbird (6) to derive the relatimmkip betuetm the height of clorid topa and
precipitation depths. Similar studies for rainfall estimates have elso been
conducted in &her parks of the globe.
3.2 Tn order to establish a reasonably valid relation between the
instantaneous eateììite cloud picture and the averxe areal precipitation over
a particular area, it ha~ to be ensured that the particular clouds together with
their pattern will have an influence an the precipitation which we are trying to
estimate and there will be no other significant development or changes which will
&e our inference invaìid. B is also necessary to avoid atleast at the first
instance local peculiarities like the existence of marked features of orography,
so that the important factor that influences precipitation over the area is
mostly the clouds and their patterns. As already explaincd at paras 1.4 and
1.5 ahove, we have therefore chosen occasions of rainstom only during the
monsoon seaon.
3.3 The =A-9 cloud pictures (taken around 0900 Carr) covering our
regimi for the 1969 and 1770 mansoon seasons were examined in conjmction with
the hrs rainfall record at 0300 GMP of next day. This has lad to develop
ment of the foilowing relations between cloud characteristics and mera
average precipitation range.
Table 2
S.luo. Cloud cover Organisation and1 Aerial distribu- Probable range of
or appearance tion of rainfa 2L+ hr rainfall in cms.
7. Overcast
2. -ao-
3. Broken to
overcast
4. ao-
5. Scattered
NO organisation
smooth stratiform
appearance.
widespread
ûrganised spiralling
bands, convect ive
appearance.
&+
Convect ive Fairly widebands
spread along
the bands
Mainly stratiform
with embedded bright
convective patches
Fairly widespread
a) convective
appearance
Scattered
b) stratiform scattered
appearance.
1-3 cma
237
7-12 C ~ S
with scattered
falls 712 C ~ S
7-12 C ~ S
1-3 cme with
scattered falls
of 6 6 cas.
4-6 c m
1-3 ciü~

238
The probability range of 26 hour rainfall given in the abme table are in accordance
with the criteria followed k? the Mia Meteorological Department to
define the rainfall oharacteristics as moderate, rather heavy, heavy wd very
heavy with precipitation ~IIIOUWLS k? the range of 1-3, &6, 7-12 and more than
12 cms respectively. I3 will be seen from the taDie that while the organised
convective clod bands can cause the rainstonas, the stratiform clou3s give
widespread moderate rains .
3.4 M a r a Aypm et al (7) Sttdied two rainStroma which 00curr.d
during the south-west mcmaoon season of 1970. One of these waa over the
eastern ottar Pradesh and was of two day8 duration i.e. 14-15 September. Tt
caused floods in the River Ganges and its tribuLaries. The other rainstorm
atxurred an 5-7 September 190 and cawed severe flocrds in the Nmda basin.
Abbi et ai also studied (8) the I&- storm. These rainstom were aastxfated
with well marked monsoon depressions, fn the fomer case the depression w m
more or less dation- wer the area during these two dap before recurving ki
a northerly direction. The later depression followed a track which was aïmg
the river basin, The ESSA-9 Satellite pictures correspondhg to rainstmms
recorded on the above dates am shown at f- 2 to 6.
3.5 The fht step in the process of estimation of 24. hour rainfell
mer any partbular area ia an &hatian of the relative wupied by
differed typa of clomis, mhly brlgkt commative and i3ttratifonn. For thie
p ~ o san e overlay consisting of a fine mesh grid was prepared. By placing
this mer the clod pictures and couabhg the nmber of squi3.e~ occupied by the
&ove types of CloudEl ia the overall area under consideration, the relative
areas occupied by different typa of clouds were obtained. These relative
areas were muïtiplled by a factor of 10 in the caere of bright convective and
by 0 in the case of stratiform. Tbe total number thus Mved for the area
der consideration represents the average 2&hr rainfall over the area.
Estimates of 24. hr. rainfail derived from satellite cloud pictures
and corresponding values obtained from isohyetal anaiysis of actual rainfail
recorded are given bdw at Tables 3 and 4.
Date of
satellite
picture
13-6-70
1&%70
Table 3
Bright Stratiform
cìouä type cio&
coverage coverage
Dtiniated -rial Precipitation
average 24. hour in cms an tim
rainfa in cms basis of isofor
1,8O,ooO sq.hs hyetal maw
for 1,4D,ooO eq.
a$
8%
rsra
75%
43 5mo& = 50.9
100
-0
L.0
7 05

Table 4
239
Date of Total Bright Stratiform &sthated aerial Precipitaticm in
satellite cloid cioud type COVB- average 24. hoar cmie on the besi8
Pictun, aoverage ccmer.!ì%e rage, rainfall in cm of isokyetal mapa
for 2,30,0ClOaq.Kmc1 for 2,00,000
sq. Elms,
4-Cp70 33% 2s %
5-9-70 93%
6-9-70 65%
75% 15%
Z3QMZZ= 100 2.9 483
389[10*27112 100 4.3
3s Bi can be seen from the above table that the average aerid rain-
fall estimiatea from the satallite cloud pf.ctms apee fairly well with the
precipitatia depth8 b@ed 011 isohyetal anSlyeiaJ The value6 esthter3 in the
case of rainstrom over Utta Pradesh are hadever found on the higher e2de where-
as estimates in cae of Narmeda basin ~zle lai=,' Thia will sugg8st that the
multiplloation factors appïieä for esthatuig the & hr rainfall. will differ
from catchment to catchment,' Thia ie understandable because local factors
pïey an import& roli in the amount of precipita'cion that c m actuall;g be
redised from a particular type of cloud.
While the resdts indicate that thi8 approach is prOimiahg we have yet
to establish by applying the above criteria to many more s fma th& the fac-
tors are valid,'
4. CûNCLUS1DMS
This preliminary study has shown that Satellite cloud pictures can
be utilised for estimating the 2+hr rainfall which can be utilised for
arriving at atleast preliminary estimate of river discharge for initial
decision making in flood foreoasting.
cloud pictures @trimI reasonable estimates of smw cover and snow line for
utiilisation in t h esti.mates of snow-melt contribution to river
dia charge ,
8.0
337
It is also encouraging that satellite

240
1, -ta, M.G, and Abbi, S.D.S(l9"l). ABsessmmt of water flow in
river Sutlej by Satellite Pictures, Vayu kdal, V0l.l No.3,
1 13-1 17.
2.1 Srhivasan, U and Fiaman,S. (1 972). Satellite Pictures in the
etudy of sntm hydrology mer Western Himalayas, Indian ~ J.bt.
&phSeiCs., vo1.23 No.3, Pp.335-3.44e
3, The beet of Nimbus (1 VI) %P. Prepared for W4, Goddard
Space Flight Center, lularyland, contract No. NAS 5-10343
u Kubhreetha, S.M. and Gupta, M.G. (1 964). SataUite &My of
an in&&wnsoon depression, Indian J .Met .Geoph'rs. , Vol.15,No0.2,
pp. 175-182.
5. Barrett, &C(1%0). The estimation of monthïy rainfail from
satellite data, Mon.üeath.Rev., Vol.%, No.4, pp.322-327,
6. Rainbird, A.F. (1 969). Some poterrtial amlicatiane of meteorological
satellites in flood forecasting, Hydrological Forecasting,
W.M,O. Tech. Note No.92, pp.73-80.'
7;Harihara mar, P.S., Abbi, S.D.S. and Hem ñaj (1971)
Rainfall and floods during 1970 southwest monsoon period,
mdim J,M&.GeO&yS., Vol228 k.1, m.141-1@
8. Abbi, S.D.S et al (1972). RainfaU study of the unprecidented
fïoods of September 1970 in the Marmaäa basin, Mekeomlcytical
Monograph, Hydrolowfio,-2/1 972,

FIGURE-1
ESSA-9 PICTURE OF JUNE9,1969 SHOWS THE 5NOW
COVERED MOUNTAIN RANGES OF NORTHERN AND
WESTERN HIMALAYAS. DUE TO THE MELTING 8F SNOW
FROM THE LOWER VALCEYSj MANY RIVER COURSES
ARE VISIBLE IN THE PICTURE.
241

24 2
F IGURE-2
ESSA-9 PICTURE OF SEPTEMBER 4,1969. AREA
CONSIDERED FOR ESTIMATION OF AERIAL 24-
HOUR RAINFALL IS SHOWN BY BLACK LINE.

FIGURE -3
ESSA-9 PICTURE OF SEPTEMBER 5,1969. AREA
CONSIDERED FOR ESTIMATION OF AERIAL 24-
HOUR RAINFALL IS SHOWN BY BLACK LINE.
243

244
FIGURE - 4
ESSA-9 PICTURE OF SEPTEMBER 6,1969. AREA
CONSIDERED FOR ESTIMATION OF AERIAL 24-
HOUR RAINFALL IS SHOWN BY BLACK LINE.

FIGURE- 5
ESSA-9 PICTURE OF SEPTEMBER 13, 1969. AREA
CONSIDERED FOR ESTIMATION OF AERIAL 24-
HOUR RAINFALL IS SHOWN BY BLACK LINE.
245

24 6
FIGURE - 6
ESSA-9 PICTURE OF SEPTEMBER 14, 1969.
AREA CONSIDERED FOR ESTIMATION OF
AERIAL 24-HOUR RAINFALL 15 SHOWN BY
BLACK LINE.

THE USE OF SIMULATION TECHNIQUES, ESPECIALLY DESIGNED FOR
DATA-SCARCE AREAS STATISTICAL METHODS AND DATA OPERATIONS
Introduction
General Report
by
Ivan C. James, II
U.S. Geological Survey
Simulation is not new to the field of water resources
system design. The mass-curve analysis devised by W. Rippl
ninety years ago continues in use as a graphical-simulation
methodology for reservoir sizing. Allen Hazen made a lasting
contribution to reservoir design techniques sixty years ago by
introducing the concept of a probability distribution of annual
within-year storage requirements. Following this, there were
few substantial changes in water resources design techniques
until the potential of the synthesis of operations research,
and the then newly developing digital computers were recognizea
in water resources planning and design. Within this last twenty
years following this synthesis there has been an explosion in
the size and number of directions of water resources research.
Simulation has continued to be a widely used planning and
design tool. The advent of high level programming languages
and the continuing increases in processing rates with each new
computer'generation has made it feasible to simulate systems of
an incredible complexity. Simulations have been performed to
test the responses of large scale river basin developments,
salinity control projects, aquifers, estuaries, and stream-
aquifer systems to changes in design and operating variables,
just to name a few applications. Current efforts to simulate
world-wide weather systems will dwarf these aforementioned
simulation studies in terms of computations and data require-
ments.
Indeed, maybe we should stop to question this growth in
comp1exit.y of simulation models. Have the requirements of our
models outstripped the growth of our data base? Has the ability
to build complexity and "realism" into our model$ exceeded our
ability to interpret the results and make useful decisions from
them? The answers to these questions depend upon one's objec-
tive framework. I would argue that from the viewpoint Of economic
efficiency, the first question presents a well posed, though not
necessarily mathematically trivial problem. Some of the papers
of this very symposium are providing encouraging, though somewhat
limited, results on the question of optimal amounts of informa-
tion for decision problems. The second question has much less
of an analytical foundation. Marginal benefits from increasing
the complexity of a model cannot be estimated if it is not known
that the increase in complexity is converging to the "true
nature" of the process being modeled. Perspective on this point
might be increased by recalling the title of Tocher's book,
The Art of Simulation. Unless the field of general systems theory
develops some applied branches, the construction and evaluation
of simulation mdoels will remain an art.

248
Large scale rivex bssin simulation models require hydrologic
input traces at many points. Additionally, there may also be
requirements for other hydro-metrological input traces such as
temperature, salinity, precipitation, solar insolation, and wind
speed. In order for the response of the simulation model to be
similar to that of the real system, generated input traces must
maintain statistical relationships among themselves as are found
in the natural data.
Long complete natural records would be ideal, but are not
often available. In the more typical case there is a mixture
of record lengths and record quality, and not unusually the
entire absence of a needed record. Even where all records cover
a concurrent base period, the realization of the process during
that period may exhibit such a pathologically singular behavior
that a deterministic design using those data would be unwise.
The problem, then, is to go from short records of varying
lengths to long records. In doing so, one must establish a
criterion for comparison among alternative techniques for infill-
ing and generation of records. Philosophically we might use as
a criteria the requirement that the decisions that are based on
the simulation be the same as if long ,natural records were avail-
able. This criterion is not measurable and hence the usually
accepted proxy ha y/b5?n3jhe maintenance of low order moments
and correlations.- - - More recently it has been suggested
that other statistics might be pertinent to some design situa-
tions.41 ?/ 61 :/ The nurst coefficient is one of these which
may have importance for the design of long term storage carry-
overs .g/
There are a large number of uncertainties to be considered
in the planning and design process. Uncertainties of the future,
such as population, demand, technology, personal preferences,
political choice, and hydrologic outcome plague us. As hydrolo-
gists, we have tended to concentrate upon this latter source of
uncertainty without a good perspective of our limited input
into the total decision making process. Even in dealing within
our domain of hydrologic uncertainty, we can further subdivide
this into the inherent stochastic uncertainty of the future
events and our misspecification error in modeling the process.
Making optimal decisions in the face of the inherent
stochastic nature of the process is the justification for our
detailed analysis and study of these processes; however, there
are numerous opportunities for the introduction of the misspeci-
fication error in this process. Let us list a few:

1.
2.
3.
4.
Failure of the simulation (design) model to capture
the relevant characteristics of the real-world system.
Failure of the decision process to optimize the objec-
tive.
Selection of an inappropriate or incorrect model for
generating the input to the simulation.
Sampling errors for the parameters of the flow
generating models.
The papers of this session must be evaluated primarily with
respect to these last two sources of error. The other sources
of uncertainty should still be kept in mind.
Review and Summary of Papers
S. H. Charania Extension of Runoff Records for Small Catchments
in Semi-arid Regions.
249
The Thomas-Fiering model is used for generation of synthetic
monthly streamflow traces for two small catchments, one the
Wakefield River in Australia, and the other the Kongoni River
in Kenya. Transforms are applied to the streamflow data until
the resulting values are approximately normally distributed.
For the Wakefield River, the transform is the log of the square
root of the flow.
The generation of normally distributed random number6 is
accomplished by a rather unusual technique. The area under the
normal distribution is divided into 100 equal sub-areas by ver-
tical lines. The average distances to each of these two bound-
aries on each sub-area are tabulated for selection by use of the
computer generated uniformly distributed random number. More
commonly used methods include averaging a number of uniformly
distributed random numbers to approximate normalcy, or normaliz-
ing transforms such as the sine-cosine and Haddamard matrix
transformations.
Statistics of generated flows are checked. On the two
streams tested, 23 of the 24 monthly means and 19 of the 24
monthly standard deviations of the generated data fall within
the 95% confidence intervals. Skewness and kurtosis are appar-
ently less weìl preserved.

250
M. J. Ilamlin and N. T. Kotteyoda The Preparation OZ a Data Set
for Hydrologic System Analysis
Development of the water resources of the Wye and Severn
River basins required a large scale simulation model. Genera-
tion of input data for the simulation model was difficult due
to widely varying record lengths and the necessity of adjusting
records from the gaging site to the sites of potential interest.
Additionally, it was felt necessary to generate daily flows.
This was accomplished by first generating five day average flows
and then disaggregating this into the five daily flows which
would approximately maintain the relevant statistics.
Development of records for the base period required:
1.
2.
3.
4.
5.
6.
adjustments to natural conditions of regulated
records.
adjustments of short records to a base period.
adjustments to another point on a stream based on
drainage area and effective rainfall ratios.
combinations of the above methods.
construction of entire records at ungaged sites based
on drainage and effective rainfall ratios applied to
nearby streams. (Note: No stochastic component
was added.)
construction of records by summing lagged upstream
records and making the usual ratio adjustments.
(Note: No attenuation was used.)
These adjusted base period records consisted of a long sequence
of pentad (5-day average) data at all points of interest.
All extensions of records to the base period are based
upon a bivariate synthesis using one of the two long term sta-
tions as the independent variable. The model is designed to
maintain the seasonal means and standard deviations and the
serial and cross correlation coefficients. It might be noted
that this method does not maintain cross correlations between
sets of extended records. For a simple example, note what
happend when two stations X and Y are extended based on a
station 2. Without loss of generality let all means be zero
and all variances unity. Then:

Y = p 2 4. (1 - P&) l/Z 6
Y=
where E, 6, are NIID(0,l). The cross correlation between X and
Y is then:
The cross correlations generated by the authors is represented
by the first term on the right hand side, while the physically
possible values are defined by the equation for all -1 p 6 5 ~ 1.
I doubt that the last word is in on the data infilling question,
but the method of Crosby and Maddock?/ looks promising.
251
A nbmber of other ad-hoc procedures were used to maintain
certain characteristics. The higher correlation of lowflows
was approximated by selecting a threshold value below which the
correlation was increased. Crossing properties were maintained
by adjustment of the skew coefficient to higher values than
found in the historical data. Daily data was obtained by
interpolation and noise addition on the pentad data.
The authors propose generating synthetic records by first
generating records for the two major long term stations and
then infilling the other records using the base period statis-
tics.
Roberto L. Lenton and John C. Schaake, Jr. Potential Application
of Bayesian Techniques for Parameter Estimation with Limited Data
The authors review the use of Bayesian techniques for parameter
estimation. Bayes theorem is a formalism for incorporating a prior
probability distribution with sample information to achieve a posterior
probability distribution which gives appropriate weights to
both the prior and sample information. Prior distributions aan üe
constructed from subjective judgments, information transfer, or
a combination of these.
Bayesian decision making requirea the selection of an action
such that the expected loss of utility is minimized. Thus, loss
functions must be constructed for the parameters with probability
distributions.
An example of reservoir sizing using u first-order autore-
gressive model is given. A beta distribution was fitted to serial
correlations ùerived from 140 rivera of the world. Diffuse prior

252
probability distributions were assumed for the two parameters whici
contained information on the first two moments of flow. The Bayes
estimator is compared to maximun likelihood estimators for several
sample record lengths under an assumed quadratic loss function.
As Bayesian techniques come into more use in hydraulic design
there seem to be some remaining questions of the method of their
use. Selecting a data base prior as the authors did should con-
sider more of the physical makeup of the basin because invariably
the size, shape, and geology should tell one that there is more
to be known about the basin correlation structure than that given
by the worldwide distribution. If working in a smaller region,
one must also consider that his sample and the data upon which
the prior was based suffer from similar time sampling biases due
to interstation correlation.
M. E. Moss and D. R. Dawdy Stochastic Simulation for Basins
with Short or no Records of Streamflow
The authors show the application of a first-order auto-
regressive-moving-average (ARMA) model to the generation of
streamflow record for reservoir design. The method is partic-
ularly applicable where no records exist, but regioiial rela-
tionships can define the mean5 and variances of monthly flow,
and the means and variances of monthly effective basin precip-
itation. The mean design size as determined by the use of the
sequent-peak algorithm on fifty synthetic records of 58 years
length is found to be essentially the same as that determined
from the historical record of the same length.
The paper also demonstrates an example of a seemingly
growing area of research in hydrologic model building. This
area is characterized by a synthesis of ideas from determinis-
tic model builders on how the components of a basin's hydrol-
ogy should operate with stochastic modeling techniques. Note
how the assumption that the basin releases base flow as a
linear reservoir allows for the model parameters t.> be estimated
as functions of precipitation parameters.
One difficulty in using the model comes from its requirement
for means and variances of effective monthly basin precipitation.
These data are not among the commonly available weather records.
Mean and variance of total monthly point precipitation are, or
could be, mapped for many reqions; however, the reduction of
these values to the model input parameters would require adjust-
ments for basin size and probably also basin shape and orienta-
tion with predominant st.orm paths. This obstacle could be
economically surmounted if the model was to be used extensively
in one region.

T. A. McMahen and R. G. Xein Storage Yield Estimated with
Inadequate Streamflow Data
A seventeen year streamflow record is extended using a
modified Boughton rainfall-runoff model and an 84 year daily
rainfall record. Gould's stochastic model is applied to the
extended record to determine storage requirements.
Boughton's model is similar to several other rainfall-
runoff models, being of the conceptual-component type. Infil-
tration, evapotranspiration, surface runoff and groundwater
components are computed as functions of storage in the three
conceptual zones of interception storage, uppersoil storage and
lower soil storage. The lower soil storage zone is subdivided
into two subzones, each with baseflow discharges to obtain a
base flow with a double recession constant. The nine model
parameters are estimated by a standard function minimization
procedure using the split sample technique such that one-half
of the rscord is used for calibration and the other one-half
for estimation of the fitting error. The criterion to be
minimized in the calibration procedure is not stated.
The relative information content is checked to show that
there is a gain of information about the mean due to the exten-
sion. It would seem that storage requirements are also sensi-
tive to the variance and serial correlation. The information
content of these statistics were apparently not checked.
253
Reservoir capacity for 50% and 90% drafts with 5% chance
of failure were made with Gould'c stochastic storage model.
A comparison of these results with those obtained by behavioral
analysis (reservoir routing) of the synthesized flow record
gives similar results at the 50% level of development without
correction for the effect of serial correlation, but a much
smaller storage requirement for the Gould model (30% of behav-
ioral value) at the 90% level of development. Correcting the
Gould model for serial correlation increases the storage require-
ment to 86% of the behavioral value.
The authors attribute the remaining discrepancy to being
beyond the range of the serial correlation correction procedure.
At such a high level of development, some hydrologists might
argue for models with higher persistence than the lag-one
Markov model.

254
Pedro Porras G. and Alfredo Flores E. Stochastic Application in
Ungaged Basins for Planning Purposes
Water resources planning is described as being dynamic.
Feedback from each iteration can be used to define requirements
for more detailed information. The first version of the National
Plan of Development of Water Resources required an inventory of
surface runoff. The approach for the second version of the plan
was stymied due to inadequate data, hence the application of
stochastic methods were tried.
In designing these methods, several generalizations from
the data were helpful. Some of the physiographic factors affect-
ing precipitation were more influential on rainfall quantity;
others were more influential in affecting the distribution of
precipitation throughout the year. Ratios of monthly to annual
precipitation were often similar in different zones. Data
deficiencies made the construction of monthly isohyetal maps
a difficult task.
All data were reduced to percent of the 10-year average at
that station and grouped into four sets each with a 500 mm/yr.
range in average precipitation. It was found in low rainfall
areas that the variance increased with the mean rainfall, while
there was no relationship at high rainfalls. In either case
the Gumbel distribution was found to fit the data best.
When the data were rescaled to common minimum and maximum
values for the 10-year historical record, it was found that the
cumulative marginal distributions were essentially the same.
Thus monthly rainfall at ungaged sites was generated by use of
a transition process and rescaling with the 12 sets of maps of
monthly maximum and minimum values and map of average rainfall.
For a given precipitation, the marginal distribution of evapo-
transpiration was found to be normal. This provided a mechanism
for generating irrigation requirements from the generated rain-
fall. A two parameter model was used to compute monthly runoff
from monthly rainfall. A computer program was written to carry
out these computations for 2 minute-of-angle grid points on
the maps.
One point to which the authors may want to address some of
their comments is the use to which their results will be applied.
The procedure they have described generates results indepen-
dently at neighboring grid points. Thus some possibly important
properties of the generated irrigation requirements such as the
covariance among the grid points in a region are lost. This
information may be of considerable importance when estimating
the distribution of regional water demands.

Marcel Roche Standardization and Interpolation of Data for a
Simulation Model
255
The author finds that it is necessary to review and
recompute all records when constructing a base period set of
data for input to a sìmulation model. This involves obtain-
ing the original gage-heights and applying the rechecked shift
and datum corrections. Inspection of discharge rating curves
and hydrograph cumparison with nearby stations provide addi-
tional subjective checks on the quality of the records. Examples
of substantial errors have been found using these methods.
Water quality computations also require checking, although
of a different nature. Often one must combine records derived
from conductivity measurements as well as partial and complete
chemical analyses. Precipitation records should be checked by
double mass curve analysis for any systematic errors. Original
records should be checked for random transcription errors.
These operations are necessary to obtain monthly values of
these parameters for the period of record.
The extension of these records to cover the base period may
be accomplished hy using regression analysis. To preserve the
variance, a random component must be added back onto the regres-
sion estimate. This process, however, can produce negative
flows and other problems. The author suggests instead estab-
lishing a line of relation through the origin of the form y = Ax
such that the variance is preserved.
For some basins it is possible to improve the prediction
of the missing data by including an index of local rainfall as
a factor in the multiple regression. An intuitively reasonable
rainfall index is a weighted sum of previous rainfall amounts
where the weights decrease in some geometric progression with
time since the event.
Variances of estimated salinities are preserved by empir-
ically estimating the marginal distribution of salinities for a
number of flow classes, and generating from the marginal distri-
bution appropriate to the flow class.
Finally, the problems of adjusting records from points of
collection to point of need, such as Hamlin and Kottegoda faced,
must be solved. This is complicated by the necessity of main-
taining an additional continuity relationship for the total
dissolved load.

256
Several of the pgints that the author brings up deserve
some discussion. In the U.S., the annual computations of surface
water records are rechecked and compiled after a five year accum-
ulation. Thereafter it would be unusual to recover any remain-
ing errors. Statistical interpretation of historical water-
quality records in the U.S. has sometimes been difficult because
the chemical analyses were done on composited samples. The exist-
ence of several methods of compositing added to this difficulty.
Have the hydrological services of other countries had this
problem? The autiior admits to the inelegance of his practical
techniques for maintaining variance, but one might also wonder
if other unmaintained parameters such as the covariance prop-
erties might be important to the decisions resulting from the
simulation model.
H. D. Charma, A. P. Bhattacharya, and S. R. Jindal The use of
Simulation Techniques for Sequential Generation of Short-Sized
Rainfall Data and-its Application in the Estimation of Design
Flood
The authors attack the problem of, synthetic generation of
the 6 one-hourly rainfall values for the maximum annual storms.
These were then used for computing flood peaks which would
presumably include worse conditions in the catchment than those
experienced in the typically 10-20 years of record available.
The historical data used were the 6-hour annual storms recorded
at New Delhi in the 1956-1965 period.
The rainfalls of an annual storm are assumed to result from
an autoregressive process of the form:
Xt = r
+ t,t-1 Xt-l
This model was used on the 10 years of data shown in table I to
obtain the statistics shown in table II. Unfortunately, an
error, possibly in programming, seems to occur in the generating
model such that the process takes the form:
X = E (E generated from a uniform distribution)
1 1
and
- X = X t + r l < t L 6
t,t-1 Xt-l
hence, the only variation of the hourly rainfall increments is
introduced by way of El and the variance of any hourly increment
is then:
Et

where r
1,Q
E l
and the variance of the total storm rainfall is
This gives a variance of the totals of annual storms of about
37 compared to the sample variance in the historical data of
about 430. This would seem a sufficient reason for the dis-
crepancies between the historical and generated data shown in
figures 2 and 3.
Perhaps the authors could respond to the questions:
1. Why was a uniform distribution selection for E ?
1
2. Why was no random component added on to each hourly
value, independent of the other hourly values?
J. H. Visser The Use of Stochastic Models in a Hydro-Agricul-
tura1 Development Project in Lebanon
257
Consistent monthly temperature, rainfall, and streamfiow
data were needed for a model used to simulate the operation of
an irrigation project. The purpose of the simulation model was
to provide an economic evaluation of the project and a design
sizing of the reservoir.
The historical data consisted of several 30-year rainfall
records, some 15 year temperature series, two 14-year and 13
three-to-five-year streamflow series.
The data generating mechanism has several features peculiar
to the length of record and type of data being generated. Square
root of precipitation and log of discharge were the transforma-
tions chosen to approximately normalize the distribution of
these data. Strong annual but weak monthly correlations between
precipitation and streamflow led the authors to the following
method of monthly streamflow generation. Annual streamflows
were first generated based on a regression with annual precipi-
tation. An autocorrelated series of monthly flows is then

258
generated and adjusted so that its sum is eque1 to Lhe previously
generated annual flow. Temperatures are generated to maintain
their serial correlation and a cross correlation with precipita-
tion.
For the short streamflow records, not enough data were
available for estimation of the mean, variance, serial anã cross
cross correlations for each calendar month. The monthly means
were removed and this series extended on the basis of oqe of
the long term flow records which had been similarly transformed.
J. R. Wallis and N. C. Matalas Relative Importance of Decision
Variables in Flood Frequency Analysis
The authors present interim results of a Monte Carlo simu-
lation of the process of fitting flood frequency curves to data
generated from known distributions. The ultimate objective,^ of
the study is the development of strategies for optimal selection
of flood frequency analysis techniques given the loss function,
length of record, sample flood statistics, and a prior distri-
bution over possible frequency distributions for floods.
The results presented by the authors are the probabilities
of best fit of either the normal, log-normal, or Gumbel dis-
tribution to data generated in every point in the experimental
hyperspace:
distribution: normal, Gumbel;/S;gTmal with
skew = 1/4, 1/2, 1, 1.14, 2, 2
record length : 10, 30, SO, 70, 90 years
plotting position: Weibull, Hazen
fitting criteria: minimum sum of squares, minimum sum of
absolute deviations
It should be noted here that probability of best fit is a measure
of the flexibility of a distribution in fitting a set of data and
gives neither a connotation of better fit to the distribution
that generated the data nor any measure of how well the fitted
distribution estimates the T-year flood.
A quick glance at the results allows for some possibly
interesting interpretations. The maximum probability of select-
ing the correct distribution where the real world is normal
comes from the use of the Weibull (W) distribution and the mini-
mum sum of absolute deviations (MSAD) fitting criterion. Simi-
larly if the real world is Gumble then selection of Hazen and

259
MSAD for short records and Woibull-MSS (minimum sum of squares)
for longer records gives the maximum probability of the under-
lying distribution being of best fit. For all of the log-normal
distributions, the MSS criteria with Weibull for short and Hazen
for long records maximized this probability.
What is apparent is that there is no dominant strategy for
selection of plotting position and criteria. The selection of
these two factors then has an effect on the analysis to deter-
mine the "best-fitting'' distribution. Perhaps the U.S. Water
Resources Council should wonder how the acceptance of the
Weibull plotting position and the MSS criteria influenced
their decision to use the log-Pearson type III distribution
in flood frequency analysis.
Discussions of the theoretical issues involved in the
selection of a plotting position formula can be found in
LangbeinlO/, Benson=/, and Appel=/.
G.Weiss Shot Noise Models for Synthetic Generation of Multi-
site Daily Streamflow Data
This paper is another example of a synthetic gcnerating
mechanism which has a physical interpretation. The shot noise
process is a particular linear filtered Poisson process. For
those familiar with unit hydrograph theory, the psocess may be
described as the convolution of a negative exponential shaped
hydrograph with a time series of rainfall events that have a
Poisson occurrence and an exponential distribution of magnitude.
This generating mechanism was selected to give a first-order
autoregressive process which would reproduce recessions.
Analytical resolutions of problems in parameter estimation
and conversion from a continuous to a discrete-averaged time
series are obtained. A generalization to two site generation
is presented which maintains a cross-correlation. The general-
ization to multiple sites is not given but could possibly be
derived.
Some shortcomings in the generated data required adjustments
in the process. The skews were found to be too high, and the
monthly variances too low. The suspected reason for these
results was because the model did not consider the base flow
component. A double shot noise process was developed which was
the sum of two independent shot noise processes with different
sets of parameters. One might imagine that this physically

260
represents a surface runoff model superimposed on a base flow
runoff model. The break in this line of physical interpretation
comes because each process has a separate time series of pulses
or rainfall. A more intuitive physical model might be one in
which a fraction of the rainfall went into the surface runoff
mechanism and its complement into the baseflow mechanism.
I realize that this may complicate the parameter estimation
problem. The author is invited to give his assessment of the
problems and benefits from extending the model in this manner.
Eric F. Wood Flood Control Design with Limited Data - A Compar-
ison of the Classical and Bayesian Approaches
Classical and Bayesian techniques are compared in the design
of a flood control structure. The author makes two reasonable
assumptions about the distribution of floods: (1) Floods above
a base level can be assumed to occur as a Poisson process; and
(2) The upper tail of many right-side unbounded frequency dis-
tributions is approximately exponential. From these assumptions
is derived an approximate cumulative probability function for
the floods above the base level:
where
z = flood magnitude above the base level
v = arrival rate of floods above the base level
a = reciprocal of mean of floods above the base level
t = time horizon
This model is the basis for estimation by both the classical and
Bayesian techniques.
ln the classical technique, the parameters V and a are
estimated by maximum likelihood techniques. This uses only
the site record and no other information.
In the Bayesian technique, the parameters are estimated
by first forming prior probability distributions on the param-
eters based on regional studies and subjective judgement. Bayes
equation is used to incorporate the sample information into the
prior distribution to obtain a posterior distribution on these
Parameters. Iii the example results of a regression analysis

261
are used for estimating the parameters of the gamma-l prior
distribution of (Y. Since large flood events are correlated,
this method may underestimate the variance of (Y. Subjective
judgement based on personal experience is assumed to provide
the information for the parameters of the gamma-1 prior distri-
bution of v.
Caution should be exercised when interpreting the economics
of the design application example. Note that these costs assume
the particular model correct, and are not measures of efficiency.
For example, at the optimum level of protection there is an
equal marginal trade-off between protection costs and damage
costs; therefore, the evaluation of the design based on the
classical model using the Bayesian model to estimate flood
damages would give expected flood damages of less than the
$7 x lo5 value resulting from the less expensive protection
work designed on the basis of the Bayesian model.
Bayesian decision theory as demonstrated in this example
may have much merit as a tool for incorporating information
from regional studies and small samples for decision making in
data scarce areas.
Summary
Synthetic data generation for infilling and extension of
records is a difficult task when the analyst hac a mixture of
types, lengths, and quality of available historical data.
The approaches developed by these authors attest to this
variety of available data and to the various particular require-
ments for input data of their simulation models and planning
procedures. The literature is replete with examples of tech-
niques developed for special problem applications.=/ =/ E/
It was previously noted that in their data infilling and
extension procedures several authors used methods which main-
tained only one of the relevane cross-correlations. Multisite
synthetic data generation also has problems. Fierings/ dis-
cusses some of the earlier attempts at overcoming the problem
of inconsistent correlation matrices. More recent investiga-
tions of this problem?/ =/ have led to serious questions
about the feasibility of consistent parameter estimation for
the more complicated flow generating models=/. F@r practical
reasons, one must achieve a compromise between elegance and
feasibility in these extension procedures.
Difficulties remain in the problem of how much and of
what type of data are really needed for models used in decision

262
processes. Decision theory tools have only provided answers
for simple and often analytic models. The extension of these
tools into the pre-posterior 'analysis of data requirements for
simulation models may be computationally prohibitive An example
of an approach is given by Young, Tseng, and Taylore/-
Moss and Dawdy, and Weiss have proposed essentially new
statistic models which use some physical interpretation from
the basin in parameter estimation. Is this to be a new emphasis
in model research? Wallis and Matalas, McMahon and Meir, and
Wood are interested in the sensitivity of model and analytic
selection on design results. Does this question have any poten-
tial for being answered? These and other questions deserve some
discussion.
In this short time, I have attempted to cover a few of the
main points which the authors of the 12 papers have documented.
These short synopses cannot do justice to the research and
intellectual effort that was necessary in approaching these
very pressing and practical problems. I urge each of you to
read the papers. Perhaps this discussion can provide some
insights that will be helpful in that task.
of
1.
2.
3.
4.
5.
To the authors I offer my apology for any mistakes or errors
either emphasis or interpretation.
References
Matalas, N, C., 1967, Mathematical assessment of synthetic
hydrology, Water Resources Research, v. 3, no. 4, pp. 937-945.
Fiering, M. B., 1965, Streamflow Synthesis, Harvard University
Press, Cambridge, Mass. 139 p.
Beard, Leo R., 1965, Use of interrelated records to simulate
streamflow, J. Hydraul. Div., Amer. Soc. Civil Eng., 91,
pp 13-22.
O'Connell, P. E., 1971, A simple stochastic modelling of
Hurst's law: Proceedings of the International Symposium on
Mathematical Models in Hydrology, Warsaw.
Rodriguez-Iturbe, Ignacio, Jose M. Mejia, and David R. Dawdy,
1972, Streamflow simulation 1. A new look at Markovian
models, fractional Gaussian noise, and crossing theory:
Water Resources Research, v. 8, no. 4, pp. 921-930.

263
6. Mejia, Jose M., Ignacio Rodriguez-Iturbe, and David R. Dawdy,
1972, Streamflow simulation 2. The broken line process as
a potential model for hydrologic simulation: Water Resources
Research, v. 8, no. 4, pp. 931-941.
7. Carlson, R. F., A. J. A. MacCormick, and D. G. Watts, 1970,
Application of linear random models to four annual stream-
flow series, Water Resources Research, V. 6, no. 4, pp.
1070-1078.
8. Wallis, J. R. and N. C. Matalas, 1972, Sensitivity of res-
ervoir design to the generating mechanism of inflows:
Water Resources Research, V. 8, no. 3, pp. 634-641.
9. Crosby, D. S., and Thomas Maddock, III, 1970, Estimating
coefficients of a flow generator for monotone samples of
data: Water Resources Research, v. 6, no. 4, pp. 1079-1086.
10. Langbein, W. B., 1960, Plotting positions in frequency
analysis in Dalrymple, Tate, Flood-frequency analyses:
U.S. Geological Survey Water Supply Paper 1543-A.
11. Benson, Manuel A., 1967, Average probability of extreme
events: Water Resources Research, v. 3, no. 1, 225 p.
12. Appel, Charles A., 1968, A note on the average probability
of extreme events: Water Resources Research, v. 4, no. 6,
1359 p.
13. Moreau, David H., and Edwin E. Pyatt, 1970, Weekly and
monthly flows in synthetic hydrology: Water Resources
Research, v. 6, no. 1, pp. 53-61.
14. Pentland, R. L., and D. R. Cuthbert, 1971, Operational
hydrology for ungaged streams by the grid square techniqumr
Water Resources Research, v. 7, no. 2, pp. 283-291.
15. Benson, M. A., and N. C. Matalas, 1967, Synthetic hydrology
based on regional statistical parameters: Water Resources
Research, v. 3, no. 4, pp. 931-935.
16. Fiering, M. B., 1968, Schemes for handling inconsistent
matrices: Water Resources Research, v. 4, no. 2, pp. 291-297.
17. Matalas, N. C., and J. R. Wallis, 1971, Correlation con-
straints for generating processes: Proceedings of the Inter-
national Symposium on Mathematical Models in Hydrology,
Warsaw.

264
18. Slack, J. R., 1972, Bias, illusion, and denial as data
uncertainties: Proceedings of the International Symposium
on Uncertainties in Hydrologic and Water Resources Systems,
Tucson, Arizona.
19. Young, G. K., M. T. Tseng, and R. S. Taylor, 1972, Data
selection for environmental simulations - A water tempera-
ture example: Water Resources Research, v. 8, no. 5,
pp. 1226-1233.

ABSTRACT
ETUDE DES RELATIONS PLUIE-DEBIT
SUR TROIS BASSINS VERSANTS D'INVESTIGATION.
Y. C OWRY - A. GUI LBOT
Research basins are useful way to study hydrologic cycle
On three of them,with areas between 109 and 250 km2,the
authors have stablished a model relating rain and runoff
This model is able to simulate mesured runoff series
from rain series and provide a better understanding of hydrologic
mechanisms at the scale of this basins.
ïhis paper suggests a méthodology which,applied to many
ba.sins,should permit the identification of the relations between
the physical caracteristics of a basin and the model parameters
identification which is necessary in order to apply this model
to ungaged basins.
RES UME
Les bassins versants d'invectigatioh constituent en ecx
mêmes un outil de recherche privilégié en ce qui concerne les
mécanismes mis en jeu par le cycle hydrologique naturel.
Sur trois d'entre eux.de superficie comprise entre
100 et 250 km2,les auteurs ont établi un modèle de liaison pluie
débits permettant la reconstitution des séries de débits observés
à partir des séries concomitantes de pluie et autorisant une
mellleure connaissance des mécanismes hydrologiques considérés
A l'échelle de ces bassins
L'approche du cycle hydrologique à nécessité diverses
opérations telles que:
-choix du schéma hydrologique et mise au point du modele
-réglage du modele et mise au point d'an processus de
determination numérique des parametres ,
-vérification de la validité du modèle par comparaison
aux séries obserdes tau niveau des caractéristiques statistiques
des principales grandeurs hydrologiques'
-étude de la convergence des méthodes d'optimisation en
présence d'erreurs aléatoires sur les données d'entrées
-analyse spatiale et temporelle des séries entrée-sortie
(choix du pas de temps des entrées et determination du décalage
pluie-débit par analyses spectrales)
Cette étude définit une méthodologie générale d'utilisa-
tion qui devrait permettre,à long terme,l'identification des
relations liant les caractéristiques physiques d'un bassin et les
parametres du modele,identification nécpssaire dans le ras d'ap-
plication du modèle à des bassins non contrôlés
COWRY Yves - Ingénieur
Agronome - Laboratoire National
d'Hydraulique E.D.F. - Professeur Aesocié à l'Université des
Sciences et Techniques du Languedoc - Montpellier (France)
GUILBOT Alain - Ingénieur - Laboratoire d'Hydrologie -
Université des Sciences et Techniques du Languedoc - Montpellier
(France)

2 66 I. GENERALITES :
Dans l'étude de la liaison pluie - débit,il s'agit
d'élaborer généralement un modele, type "boite noire" qui:
considérant la séries des pluies comme"entrée",permet d'obtenir
une"s0rtie"concordant sensiblement avec la série chronologique
concomitante des débits observés
On peut alors envisager plusieurs types de modeleS.tels
que les modeles linéaires classiques obtenus par corrélation,
analyse multivariables,analyse factorielle..,,les modeles à
élément central linéaire (basé sur l'hypothèse de l'hydrogramme
unitaire) ou les modeles conceptuels qui,en quelque sorte,font
la synthèse générale.
La stucture d'un modèle conceptuel est fondée sur la
connaissance ou la pseudo-connaissance des phénomenes en jeu
dans le cycle hydrologique. .
On suppose,par exemple,que les taux et les vitesses de
transfert de l'eau de pluie par telle ou telle partie de cycle
hydrologique sont asservis à l'état de remplissage de la zone
correspondante par des fonctions à un ou deux parametres.
La sortie résultante,en l'occurence la dérie des débits
calculés,est comparée à la sortie observée daps le systeme réel,
c'est à dire la série des débits observés à l'exutoire du bassin.
Si la concordance ne semble pas satisfaisante,on modifia
les psrametres des foictions des divers sous-systemes,jusqu'à
obtenir une corredpondance satisfaisante entre les séries observées
et calculées
Ceci ne devrait etre fait,non pas dans le but d'un calage
spécifique permettant d'obtenir l'hydrogramme d'un bassin
particulier,mais dans l'optique d'une recherche de liens entre
les valeurs des parametres du modele et les caractéristiques du
bass in.
I1 est donc nécessaire d'une part d'appliquer le même
modèle à de nombreux bassins,d'autre part que tout modele conceptuel
soit,au départ,aussi simple que possible et que des modifications
ne lui soient apportées que sf la nécessité absolue apparaisse.(raisons
physiques ou amélioration évidente de la reproduction)
Un systeme simple,parceque dans un schéma élaboré,il
y aura de fortes chances que le modele comporte deux sous-systeme
tout à fait équivalents et il sera extremement délicat de lever
l'indétermination sur l'attribution de la valeur des parametres à
l'un ou l'autre de ces sous-systèmes,ensuite parceque seul un
modele simple permettra l'identification parometres-caractéris-
tiques du bassin et donc son utilisation sur des bassins non
jaugés.

267
II.LES BASSINS ET LES DONNEES:
L'étude porte sur trois bassins expérimentaux présentant
des caracteres morphologiques,géologiques et pédologiques
bien différenciés.
-le bassin de ia DIEGE,affluent de la DORDOGNE,
d'une superficie de 225 km2.Géré par EDF depuis 1960 puis par
le Laboratoire d'Hydrologie de l'université des Sciences et
Techniques du Languedoc,c'est un bassin montagneux.,cristsllin,
bien boisé et soumis à des influences océaniques et méditerranéennes.
-le bassin de l'ORGEVAL,affluent du GRAND MORIN
d'une superficie de 104 km2 Géré par le C.T.G.R.E.F(Ministere de
l'Agriculture) depuis 1962,c'est un vaste plateau limoneux,coJver
dans sa majeure partie de culture et soumis à des influences
océaniques et continentales.
-le bassin de l'HALLUE,affluent de la SOW-,
d'une superficie de 219 km2.Géré depuis 1966 par le B.R.G.F,
c'est un bassin de relief modér6,formé de craie recouverte de
limon et principalement mis en culture.11 est soumis essantiel-
lement à des influences océaniques.
Ces trois bassins étant des bassins expérimentaur,les
données étaient caractérisées d'uns part par un volume important
d'informations disponibles,d'autre part par une qualité de l'enregistrement
et du dépouillement (à de rares exceptions pres)
Le choix d'une pluviométrie représentative fut fait,
soit en fonction de nos propres connaissances du bassin(B.V de
la DIEGE),soit en fonction des conseils de l'organisme de gestion
(BV de i'OXGEV?,L),soit apres une analyse spatiale de ia pïuviom6triecB.V
de 1'HALLUE).
Le choix de l'indice d'ETP a,par contre,été mené de m2-
niere quelque peu arbitraire et de façon indépendante pour les
tois bassins ce qui semble une erreur,compte tenu de l'importance
effective de sa variance interannuelle et de son niveau moyen
Remarqueune méthode systématique de dépouillement a
été mise au point et utilisée dans le cadre de cette étude.
I1 s'agit de.traduire l'enregistrement pluviométrique
ou limnimétrique dans un systeme (X,Y) sur machine D.MAC puis
de transformer ces données"digita1isées en donées de pas de
temps voulu (2h,10 mn..)
III.LES MODELES:
Dans le cadre d'une précédente étude,plusieurs mcdeles
avaient été élaborés et testés par le Laboratoire.
Trois d'entre eux ont été'retenus et rendus opérationnel
Ce sont les modeles DIEGE.MER0 et CREC

268
---- IV.LES METHODES EMPLOYEES:
4.1.1:Méthodes des composantes principales appliquée à
la détermination de la représentativité de l'.information pluviométrique
:
La méthode d'analyse en composantes principales permet
de substituer k vecteurs X de n composantes corrélées entres elles
à k vecteurs Y de p composantes indépendantes avec p

269
4 . 2 s ~ :
Les parametres des fonctions des divers sous-systemes
des modeles fproduction,transfert) sont rarement déterminés a
priori de façon précise.Nous avons accompli un effort tout particulier
pour mettre au point une technique de détermination numérique
de ces parametres,technique devant etre assez générale
pour etre appliquée systématiquement 3 n'importe quel bassin en
assurant une convergence réelle et rapide vers un optimum objectif.
La méthodologie que nous proposons,testées initialement
sur séries fictives,donc currespondsnt à une structure de modele
et un jeu de parametres définis,semble particulièrement intéressante:
1.Définition de la zone de variation de chacun
des parametres (en fonction de la nature du bassin et des result
ats des analyses préalables des séries d'entrées)
2.Tirage au hasard,d'abord dans une loi uniforme
puis dans une loi normale avec diminution de la variance
de ia loi en cas de succes(ceci afin d'éviter Loe recherche systématique
à partir d'un faux minimum)
3.Recherche "direcre",avec rotation des axes
de co~rdonnéos("nûCENaRû~~:.~.~c~erche séquentieiie effectuée successivefiient
sur chacun des axes de coordonnées(correspondant
chacun à un parametre) suivant un pas d'exploration modifié selon
les échécs et les succes rencontrés.Si,dans toutes les directions
ont a enregistré au moins un succes suivi d'un échec,on défini
alors la nouvelle direction du premier axe comme étant celle
joignant !e point initial et le point final.La direction des
autres axes est obtenue par la méthode d'ortogonalisation de
SCHMIDT.
4.Recherche fine par ta méthode du gradient
conjuguée (Powell) lorsque la précédente méthode ne converge
que tres 1entement.La méthode de POWELL utilise la méthode des
directions conjuguée mais modifie le procédé de base afin d'accélérer
la vitesse de convergence vers l'optimum tout en définissant
un processus de recherc!:e le long d'un axe.
Ce dtverses méthodes appliquées en cascade permettent
la réduction d'un critere d'écart cho.si afin d'assurer une
reconstitution satisfaisante et homogene sur une période déterminée,(le
critere choisi est de la forme
1 IQobs-
F = -5
Qca$ IOobs - OmovPd
9
N Qobs moyen
N étant le nombre de mois de la période,>de calage et Qmoyen le
module de la période de calage
Ce choix a été fait dans le hut de rendre préférentiels
les écarts SUU les valeurs extrêmes.En effet,dans cette expressio
plus on s'écarte duidébit moyen,plus l'écart relatif est pondéré
par une valeur importante,et cela,aussi bien pour les faibles
débits.que pour les crues.

270
Exemple d'application de la méthode d'optimisation proposée
En appliquant un jeu de paramètres à une série de données pluviométriques
journalières, nous avons généré une série de débits fictifs journaliers
à l'aide du modèle CREC.
Nous nous sommes ensuite proposé de reconstituer cette série de
débits en utilisant la méthode précédemment décrite.
La réelle convergence de la méthode, tant au niveau de la fonction
critère qu'au niveau des paramètres, semble montrer son efficacit6 dans le
cas de séries parfaitement adéquates, sans erreúrs de mesure et avec un
modèle vrai.
x1
x2
x3
x4
x5
X6
x7
Résultats de la recherche des paramètres
du modèle (;I:EC par la méthode préconisée
Erreur
Valeur Etape 1 Etape 2 Etape 3 relative %
vraie
O. 069 O. 0597 0.0765 O. 0693 O. 4
o.. a43 O. 7322 o. 5871 o. a435 O. 06
o. 0212 0.3922 0.0361 9.0221 4
O. 0344 o. ooao O. 0272 O. 0343 O. 3
3.902 6.5784 4.5951 3. a290 1.9
7.992 6. a303 5.5790 a. 020 O. 4
15.146 27.7554 4.6670 13. aaao 8.3
O 636 58 1.5
[Interprétation des résultat4
L'adéquation du modèle CREC à l'étude de la liaison pluie-débit
sur les trois bassins étudiés peut s'accompagner d'une tentative de jus-,
tification.
Le schéma proposé par ce modèle présente au niveau du transfert
une zone que l'on peut qualifier d'hypodermique et une zone souterraine.
I1 apparaît que, pour le bassin de l'HALLUE, l'écoulement calculé
provient pour une part essentielle de la zone souterraine, ce qui est en
accord avec l'influence prépondérante des variati.ons de la nappe phréatique
sur les débits observés sur ce bassin.
De même pour le bassin de l'ORGEVAL, drainé artificiellement
(drainage agricole) et ne présentant pas de réserves souterraines importantes,
la majeure partie de l'écoulement calculé provient de la zone
définie comme "hypodermique'' (l'alimentation de la zone souterraine
semblant être une constante du bassin).
Enfin, sur le bassin de la DIEGE, l'écoulement hypodermique est
là aussi essentiel. De plus, deux remarques sont à faire : d'une part ce
bassin peut présenter dans le cas d'une saturation importante du sol
accompagnée de pluies intenses, du ruissellement "superficiel" (crue historique
de 19601, ce que l'on retrouve au niveau du schéma du modèle CREC,
d'autre part, il semblerait que l'alimentation des réserves souterraines

(faibles dans cette région) ne se
seuil de teneur en eau de la zone
I1 Y a donc une cohérence
produisent qu'à partir d'un certain
hypodermique.
certaine entre la nature des divers
271
bassins et le comportement hydrologique du modèle proposé.
Le manque de politique homogène au niveau du choix de l'indice
d'ETP ne peut malheureusement pas permettre la comparaison de la fonction
de production sur les trois bassins et un effort reste à faire quant à
ce choix.
Conclusion
Si, sur le plan opérationnel, les modèles utilisés se moritrent
,différents au niveau de l'application (performance, sensibilité,....), ils
restent tous discutables sur le plan conceptuel, puisqu'ils fixent a
priori, en l'absence de foute veritable information intermédiaire entre
la pluie et le débit, le schéma du cycle hydrologique.
Néammoins, cette approche a permis de mettre en évidence l'ad+"-
tion de certains schémas du cycle hydrologique pour représenter plusieurs
bassins, en autorisant une extrapolation temporelle (35 ans sur la DIEGE).
Dans un esprit d'application de ces méthodes 2. des projet; d'aménage-
ment des ressources en eau sans données suffisantes, il resterait à :
- dégager des groupes de bassins justiciables de chaque schéma
- définir des critères d'adéquation a priori d'un bassin à un
schéma déterminé
- caractériser chaque bassin par des index mesurablesmou analy-
sables en absence de longues séries de données, et dont la détermination
déboucherait sur l'appréciation quantitative des paramètres d'un modèle
global,
Ceci permettrait le choix d'un modèle (schéma et valeurs des
parametres) capable de i-zprésenter le comportement d'un bassin non jaugé,
dont on pourrait, à partir des séries climatologiques dispoqibles, simuler
1 'écoulement.

272
R E F E R E N C E S
(1) F. AUBIN - A. GUILBOT (note HYD 13/72 et 14/72)
- Application de l'analyse spectrale - bassin de la DIEGE
- Influence d'erreurs aléatoires sur la convergence d'une méthode
d'opthisation. Tentatives de filtrage des séries chronologiques
(2) Y. CORIíARY - A. GUILBOT (HYD 6/71)
Etude générale de quelques modèles détermises de relations
pluie-débit
(3) Y. CORMARY - A. GUILBOT (HY3 44/70 - SHF - NOV. 1970)
Méthodes d'optimisation des paramètres des modèles déterdnistes
(4) Y. CORIlARY - A. GUILBOT (HYD 16/71)
Processus d'optimisation en quatre étapes applicable B la recherche
des paramètrss des modèles déterministes
(5) Y. CORMARY - S. RAMBAL (HYD 32/71)
Relations pluie-débiG, bassin versant de 1'HALLUE B l'échelle
journalisre et à l'échelle bi-horairE
(6) Y. CORMARY - G. GALEA (HYD 27/71)
Relations pluie-débit, bassin versant de l'ORGEVAL, à l'échelle
j ourna 1 i gr e
(7) Y. CORMARY - M. ANGLES (HYD 7/71)
Relations pluie-débit sur le bassin de la DIEGE à l'échelle
journalii2te et à l'échelle bi-horaire
(8) Y. CORMARY - M. LARINIER (HYD 34/71)
Etudes théoriques des processus d'infiltration, d'évaporation
et de drain8ge. Bibiiovraphie. Schemas d'approche du cycle
hydrologique
(9) Y. CORMARY - M. LARINIER ( HYD 33/71) .
Utilisation du catalogue des sols pour la prédétermination des
parametres dans les modèles HûLTAN et HANON
(10) G, GALEA (thèse de 3ème cycle - 1972)
Etude des relations pluie-débit sur le bassin de 1'ORT;EVAL
(11) M. ANGLES (thèse de.3Eme cycle,- 1972)
Etude des relacions pluie-déhit sur le bassin de la DïEGE
000

Eva po ro. t ion
Inférieures aux
po
&
s s ib i 1 i t 6 s à I nbu:, rp 1 ion
i(épartition de ia
pluia absorbée dnns'
los rdsorvoira S ut 1.1
7 PARAMETRES
MODELE CWEC 273
RESERVOIR
Percolation
x, ct x2 tarissrment dos r6servoirs II et G
x oL x4
3
x" et x4
3
x7
rél~artition do In pluio entre les réservoirs S et H
Supérieures aux
po s s ibil i t é s d I ab sorption
'I
rui s s e 11 ein en t
suparficiol
aliaiontntion du r6servoir G (A partir du résorvoir H)
rEductj on do 1lIc.T. P.
2 po~ciinbt.rab détoriiiinaiit 1 'atsoi'ption DETA at CAM>IA.
7
1
I
EX UT01 RE

274
U
a
10 PARAMETRES
MODELE MERO
-
Débordement 1 rcolation dan
. (Ministore de 1 lAgriculture ferahlion
xl, x2, ‘x3 , x4 répartition des surplus antre G, Q et L flu&dBt$c)
x58 “61 x7 , x8
décrue des réservoirs G et Q
X 9 1 xl0 rcmplissage maxirnum de U et L

MODELE DIEGE
STOCK MOYEN
des IO JOURS
PRECEDENTS
i
4 I
I
7 PARAMETRES 8'
. . XI ,x~,xj parambtroa de 'ddcruo
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S?OCI< JOURNALIER
275
I ,
),:" &&&cg
%?brX5,X7 parambtroo de rolotion stock-ddbit r & '3.
x6 .' ~arambtro de rdduotion do l(ETP
' ' ' _".

276
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MODELE CAEC
CRITIOUE DCS DEBITS GENIRES
imk1
4 3 9
B""Y0I journo1ior
MODELE rmRo xviii
critiquo dos rcsultats
1
ANNUEL JOURNALIER
I
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ANNUEL JOURNALIER I i
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MODELE CR@C CHIIIOUE i suits)
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M3DCLE PSCRO
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BASSIN DE L'HALLUE
critique (suite) XIX
MDDCLE ANNUEL
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MODELE CREC METHODE DE POWELL
MODULES ANNUELS
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DEBITS MOYENS MINIMUM$
. SUR IO JOURS CONSECUTIFS
BASSIN DE L'ORGEVAL
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279
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m'l 1 Periode 63-69
16 -
14 - ob:erves
- CO caicules
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Période de calage : 1964 - 1966
MODELE CREC
METHODE DE POWELL
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AMEß

EXTENSION OF RUNOFF RECORDS FOR SMALL CATCHMENTS IN SEMI-ARID REGIONS
ABS TRACT
S. H. Charania
A mathematical model developed by Thomas and Fiering is used
for the purpose. This model is based on statistical principles to
produce an unlimited record which has the same statistical properties
as the original record. The statistical characteristics of data used
are the means, standard deviations, volumes, skewness, variances and
kurtosis of flows. Briefly, the main steps for the synthesls of data
available or their transformations are as follows:
(al determina'ion of statistical properties;
(b) determination of frequency distributions;
(c> regression and correlation analysis of data (or their
transformati0ns)with a detailed study of the confidence
limits of lines of regression and statistical parameters;
(dl generation of random numbers;
(el synthesis of flows using the above model.
Tn order to speed up the work and to get more accurate results,
computers are used for almost all the calculatlons; various computer
programmes (Fortran IV), required for the development of synthesized
data, are developed. The data tested on the model are from small
catchments in semi-arid regions of Kenya and Australia.
RESUME
Un modèle mathématique a été mis au point par Thomas et Fier-
nig. I1 est conçu pour produire une série illimitée ayant les mêmes
propriétés statistiques que la série d'observations disponible. Les
caractéristiques statistiques prises en compte sont: les moyennes,
les dcarts types, les variances, les coefficients d'assymétrie et
d'aplatissement des distributions de dlbits. Les principales étapes
pour la synthese ou la transformation des données disponibles sont
résumées ci-dessous:
(a) Détermination des propriétés statistiques.
(b) Détermination des distributions de fréquence.
(c) Etude des regressions et corrélations concernant les données
ou leurs transformées; étude détaillée des limites de confiance
des courbes de régression et des paramètres statistiques.
Cdl Production de nombres au hasard.
(e) Synthèse de séries de débits ä partir du modkïe élaboré.
Presque toutes les opkrations sont faites sur ordinateur afin
d'obtenir des rgsultats rapides et précis. Plusieurs programmes écrits
en fortran IY ont dtd mis au point pour l'obtention des donnees syn-
thétiques. Le modèle a 6te appliqué 3 des données recueillies SUT de
petits bassins des rbgions semi-arides du Kenya et dfAustralie.

282
INTRODUCTION
Hew and untapped souroes of water need to be developed and
controlled to meet the inoreasing demand for water. For this we
require long records of flows with exact sequence of hydrological
events. Such long records are, unfortunately, not available, es-
pecially in the semi-arid regions. In semi-arid regions of Africa
and Australia usually only 20 to 30 years of records are available.
Short reoords can be extended to any required period of time
by simulation techniques whioh involve certain types of mathematical
or logical models that describe the behaviour of the system over ex-
tended periods of real time.
There are two variants of simulation, operational gaming
and the Monte Carlo techniques. The Monte Carlo technique is the
one used in this paper.
PRINCIPLES OF THE MODEL BY TñOMAS AND FIERING
Data used
The streamflows considered should be independent values
and also of a long period. Therefore annual, monthly and daily
streamflows can be oonsidered; annual flows are definitely in-
dependent flows, but they are often not used when the period of
record is too short. Daily flows are not used if they are not
independent flow values. Monthly flow values generally satisfy
the requirements of independence and adequate periods of record and
are used.
The model rewiremsnts
The data used should be serially correlated and normally
distributed. If the historical flows are not normally distributed
appropriate transformations are or should be used to make them so.
In such cases, the synthesized flows should be converted to their
original form.
The model
The method assume8 that the streamflows are made up of two
components, the deterministic and the random. Therefore the model
(a reoursive equation) for unit time interval (one month) to generate
flows is:

I _- QJ =mean flows of consecutiva months.
Qí~+i) AND
Q; AND Q (i+,) =flows during the months i and ;+f.
BJ =regression coefficient.
ri =norma2 random variate.
aj+- =standard deviation of flaws.
In computations j runs cyclically from 1 to 12 and the
index i from O to 12 times the number of record years to be
generated minus 1.
Determination of Ti
283
The area under the normal distribution curve is divided
into 100 equal areas, each bounded by a vertical line parallel to
y-axis. The values of X at the boundary lines repreeent the bound-
ary values of Ti for each area with equal probability. A typical
vglue of Ti for each area is determined by calculating the mean of
the boundary values. Out of 100 typical values of Ti one is selected
for use in the model using the random numbers.
CRITERIA FOR SELECTION OF THE CATCHEIEñTS
Two important phrases in the title need interpretation as
follows r
(a)
(b)
'for a small reservoir' - The size of the catchment is restricted
to between i30 to 1300 square kilometres.
'a eemi-arid region' - The rainfall in the catchment is
limited to less than 508 milimetres per annum.
The oatchments selected for the paper are Wakefield River
catchment in Australia and Kongoni River catchment in Kenya.
NOTES ON THE SELECTED CATCHMENTS
Wakefield river catchment
The catchment with an area of 420 square kilometres lies
in south Australia. The general relief of the catchment consists
of undulating plains with ridges ranging from 150 to 305 metres.
The highest point in the catchment is 600 metres.

284
The stream, 40 kilometres long, drops Prom 500 to 220
metres and has two major tributaries, the Pine creek and Skillolgale
creek. It is considered perennial but flashy.
500 mm.
Average rainfall on the catchment is between 400 to
Temperatures are high in summer with normal annual range
from 40 to 50 degrees farenheit (4.5OC to IOOC).
Relative humidity is also high in summer with a peak of
80 per cent.
Soils in the catchment are normal malee soils of a pinkish
brown colour and light sandy texture. These are rich in lime but
poor in humus and phosphate. Rocks in the area are mainly lime-
stone.
Monthly flows and their totals from 1953 to 1968 are
available as are the monthly rainfalls for the five stations in
and around the catchment.
Eongoni river catchment
A catchment with an area of approximately i5 square kilo-
metres on the north west slope of Mount Kenya. The source of the
river is at an altitude of 2600 metres and the station at 2000
metres. The stream passes from the forest area into grasslands.
The stream, with no major tributaries, is about 13 kilometres
long and perennial.
Average rainfall in the catchment is between 500 to 600
milimetres, with heavy falls in April and November.
It is hot in the catohment throughout the year with mean
temperatures around 60 degrees farenheit (i5.5OC).
Daily and monthly flovs with their totals from i932 to i969
excluding 1954, 1955, and 1956 are available for the catchment and
also the monthly rainfalls of three stations in and around the area.
ANALYSIS
Statistics of flows
These assist finally in establishing the success of the
method used. The following properties are required for each
month of the year in recordt

(a) total volumes of flows;
(a)
(c)
(d)
(e)
(f)
(g)
means of the flows and/or their transformed values;
confidence limits of the means;
variance and standard deviations of flows and their
confidence limits;
skewness of the flows - these show the degree of departure
of the distribution of flows from symmetry. There is a
dimensionlees measure but that i8 not used. The absolute
and relative skewness are caloulated;
kurtosis of flows - this measures the degree of spread of
the data and is calculated using moments;
285
trends of flows - the calculation is based on the principle
of least squares.
Frequency distribution
This is a very important aspect of the procedure os there
is a condition that the data used should be normally distributed.
There are various methods to check whether the data is normally
distributed or not. Por the paper the following procedure was
followed:
(a>
(b)
(o)
check the skewness and kurtosis of the flow data;
if the skewness is not equal to zero and kurtosis is not equal
to 3.0 the data is transformed and the statistics rechecked.
The txansformations are carried until the values of skewness
and kurtosis are 0.0 and 3.0 respectively;
the data or the transformed values are plotted on the
probability paper using plotting position ia/a. If the
data give a straight line it is aesumed that the values
are normally distributed.
Bemession and correlation analysis
A statistical relationship between the flows in different
months is determined. A linear regression equation Y - A + Bx is
developed for each month; the constants A and B are calculated as
follows t

286
Confidence limits of the regression constant B and the
regression line are also determined.
Coefficient of correlation (or covarianoe), an expression
for the degree of scatter in regression is calculated using the
following formula:
Generation of random numbers
The random numbers are used to obtain the random component
of the model. There are alternative methods to generate a sequence
of random numbers. The latest technique is the generation of psuedorandom
numbers by digital computer methods. This produoes numbers
that are
(a) uniformly distributedl
(b) statistically independent;
(c) reproduaible;
(a)
non-repeating for any desired length.
The validity of random numbers is tested by different
methods but frequency test method is the most convenient.
For the model we require random numbers that are normally
distributed. The uniformly distributed random numbers are there-
fore transformed to be normally distributed. These are then ad-
justed for the mean and standard deviation.
Calculations for the generation of streamflows for the
two catchments are illustrated in Figures 1, 2 and 3.

DESCRIPTION OF RESULTS
287
Wakefield river catchment: Monthly flow values are used
as historical records available for the simulation of records.
These values are not normally distributed so they have to be trans-
formed. It is found that the logs of square roots of flows are
normally distributed.
The total flows show that in i4 years of records available,
the maximum annual runoff is 10.25 a lo3 cum/D. The years 1954,
1957, 1959, 1962, 1965 and 1966 have very low runoffs compared to
other years. The lowest runoff observed is 2.30 x lo3 cum/D.
August has the highest total monthly flow of 88.5 x lo3 cum/D,
the lowest flow is in February of 2.98 x 103 cum/^.
The means, variances and the standard deviations follow a
pattern which is similar to that for the volumes; on the other
hand, skewness and kurtosis have completely different patterns:
this could be due to accumulated errors and hence these values
are only taken as a guide. The confidence limits of the statistiical
parameters are very useful in the final check.
It is found that thereis very little scatter of values
in regression and correlation analysis; the correlations are
poor for a couple of months but for the rest they are very high
indeed.
Konsïoni river catchmentr In this oase also, the monthly
flows are used. in the records of 30 years, the highest flow is
recorded in Aprilr 41.5 x lo6 cum/D; the lowest is in February:
zero flow.
All statistical parameters, except for skewness and kurtosis,
follow the same pattern as that for volumes; again, the skewness and
kurtosis seem to have accumulated errors and cannot be relied upon.
COBCLUSIOIPS
The correlation of flows appear to be extremely good.
The usefulness of the stochastic model for low flows, by
Thomas and Fiering, is well demonstrated. 39 thie paper the model
was used to simulate flows from the small experimental catchments
of Kongoni river in Kenya and Wakefield river in Australia, and for
both catchments, the parameters of synthetic flows lie within the
95 per cent confidence limits of the historical flows.

288
The results therefore indicate that the model can be
successfully applied to the flows from small catchments in
semi-arid regions. Howevex, when applying the above theory, the
following points should be consideredt
(a>
the initial flow values should be reliable to avoid any
accumulation errors in the final results#
(b) the frequency distribution of th+ flaüdg if the initial
flows are not normally distributed, appropriate transform-
ations have to be used to normalise them;
(c 1
the selection of a value of 't'r this is an important
part of the model; random numbers are used to achieve the
purpose as explained in the paper;
(a) negative flows: on synthesis, negative flows are obtained.
If the transformed values used initially are in the log form,
then no changes have to be made; if the initial transformed
values are not in the log forra, then the negative values have
to be removed by replacing them with zero values.
References
Fiering, M.B. (1961). Queing theory and simulation in reservoir
design. Journal of Jïyd. Div. B.S.C.E., Vol. 87, pp. 36-59.
Thomas, H.A. and Fiering, M.B. (1962). Mathematical synthesis
of streamflow sequences for the analysis of river basin by
simulation. Chapter 12 in 'Design of water reaourceE systems'
by Maas et al, Harvard.
Beard, L.R. (1967). Hydrologic simulation in water analysis.
Journal of Irrigation and Drainage Div. A.S.C.E.
Smty, T.L. (1961). Elements of Queing theory. McGraw Hill book
company Inc. New York.

Figure i - Wakefield river catchment
289

290
,-
Figure 2 - Analysis of Kongoni river flows.
I J A S O N D -
Fm-3
no
.O.,”.

I,
f =(SKU(NGl)+ SKUiNG)
12) (-10)
I READ STREAXí1.J) I=it d.14 I
1 READ ARE4 U:.3)E? hORHPL CURVE 1
__-
I
DETERVINE VPLXS U-: STAri?AkD FiORtJAL LARIATE FOR
AREA UNGE P.ORMAL CUR\.€ FRCM 0.01 TO O5
I STCAAil) = VkRAAil)+r0.5 I
'
..
ss- xxxx ciz,rr,
CSLV(l2,l) =xxxxi1z,141
Ir11
P= 1 A
CALL THE RCNWM iiü:A;ER GNERATCG
..
S = 16.70
NG2=100-NG+l
/21 (-10) NO=NG2- 1
T =CU(UíNG2) t SKL'ih'GJ!)
/2a
Figure 3 - Flow ohart for the synthesis of flows
291
-
flG2=lGC - D
T ;(i353 t SSL!;N13!:
/z O)

ABS TRACT
SIMULATION OF HYDROLOGICAL SAMPLES BY NATURAL WATER
FLOW CHARACTER1 S TI CT ICs
A.I.Davydova, G.P.Kaliniii
The paper is concerned with long hidrological series which have
a specified distribution and are characterized by basin annual flows
rather than by random number sensors. The theoretical basis for cons
truction of a numerical sequence is a combined analysis of mean an-
nual flow probabilities for groups of basins (incompletely homoge-
neous) selected to suit certain correlational estimates. By using va
rious techniques the basic statistical characteristics of initial t i
me distributions are taken into account, The length of such series
depends on the amount of data on flows of rivers under study which
are grouped by certain criteria.
Simulation of hydrological series by natural water flow characteristics
does not require any method to allow for the effect of a
preceding value on the law whereby a subsequent one is distributed.
Error is not accumulated in simulated series as they grow,
On a examiné dans le rapport en question la technique de cons-,
truction de longues séries hydrologiques 2 diktrìbùtion calculée par
caractéristiques de l'écoulement annuel de bassins isolés, et non
pas au moyen de capteurs de nombres occasionnels. C'est bien l'ana-
lyse réunie de probabilités de valeurs annuelles moyennes de l'écou-
lement par groupes de bassins (pas tout 2 fait homogJnes1, choisis
suivant les estimations corrélatives determinées, qui sert de base à
la construction de succession numériques. Les différents procédés
mis en oeuvre permettent de tenir compte des élements statistiques
principaux des distributions de départ temporeles. La longueur de t e
lles séries est fonction du volume de l'écoulement des fleuves du
monde, attirés au calcul et choisis selon les critères définis.
En simulant les réalisations hydrologiques par caractéristiques
naturelles de l'écoulement fluvial, point n'est besoin de recourir à
une telle ou telle méthode de prise en considération de probabilités
de la valeur précédente sur la loi de distrifiti'on de prohEY2litds
de la valeu: suivante, Aucune atcumulation de l'erreur nia 12eu dans
les séries a simuler au fur et a mesure du prolongement de celles-ci'.

2 94
Variou8 ways to extend the initial hydrological data are
used h water flow calculation and forecasting. Short time
serie8 of hydrological observations of ten fail to give adequate
c harac teristic 8 of river flow and ensure reliable calculations.
Hydrological data are exbended by probabilistic techni-
ques, among sbich the Monte-Carlo method ia the most widely
U8ed. The esseqfe of the latter is that artificial curves of
probabilistic processes can be obtained by generation of random
numbers distributed by a certûin law. Note that a model of the
flow process thus obtained should have hundreds or thousands
of terms to include all basic features of the process probabili-
ty distribution f mtions. The applichtion of this technique
for hydrological calculations was thoroughly developed by
G.G.Svanidee [9J . The range of application of the Monte-Carlo
insthod was considerably extended by other soviet researchers[2,6].
This paper deals with construction of long hydrological
series with discrete time by using annual flow dharacteristics
rather than rarrlom number aensors.
Water flow Q at a certain cross-section of a river is
regarded as a function of time t. Eet be the number of ar-
bitrary time instants t, .. m , t, for an arbitrary number oî
basins x alia n values of flow. Any specified value can be
expressed in terms of the probabiliby that it d1L not be ex-
ceeded. Probability distribution func tions for non-excees of
annual flow values, mathematical expectation and other statis-
tical characteristics of each series x are given as
B,(t,* P, , ... t, Pn)* dere P i8 the probability distribu-
tion density associated with the fumtion Fx.
The approach consists in consecutive combination of river
flow probability distributions for individual basins. in dohg
so various techniques are employed to inClde basic statistical
characteristics (such as mathematical expectation, coefficients
variation, asymmetry aid correlation) of the initial time dist-
ribUtiOM of the flow.
Conditional probability distxibutions of a combined space
am time sequerice are effectively used in *at is known in
hydrology as the year-point- method in which effbiercy criteria
for combination of time series have been developed and
used in plotting a faired empirical distribution curve.
Combined analysis m thods for incompletely homogeneous
hydrolo&ical characteristics used in calculation of aiaximum
flow hawe been developed br S.V.fulitsky and ?uí.B.bnkel' 181.
In this case the theoretical scheme for the construction
of a numerical sequence is a combined analysis of probabilities
of mean annual flows that do not substantially vary over the
period covered for groups of basins (incompletely homogeneous)

selected by certain correlation estimates.
295
Time series Oi river flow characteristics are grouped by
values of the coefficients of correlation (r) betaen annual
flows ob~erved and calculated for pabs of successive years.
A certain relation between flows of Mfvidual years is established.
The differences in correlation coefficients of river
flows indthin a basin depend on the physical nnn gec,rapnical
conditions &er which the flow vas furmå. When rivers are
grouped by intra-series c osrelation indices, the physical and
etatistical homogeneity of the flow series selected is to
some extent alloued for.
A group may izlude basins differing in the water content.
Modular coefficients are employed to make flow indices of large
and small basins conmeasurable.
Three groups covering the r range from O through 0.45
have been selected (Table I) with intra-series correlation as
a criterion, Calculations for other values are equally possible,
!Bible 1
Rivers Grouped,
by Averaging Intervals of Bhst Self-Correlation Coeff ic lents
Averaged-
values, r
+0.10 +o. 25 +O.W
The first group CO rises rivers whose flow intra-series
correlation is Or r ~'3.220 and iarludes 40 basins, chbfly
in Europe and North America, with coeff icients of variation
F ranging from 0.20 to 0.60,
The second group (+0.21 C r 5 +0,35) includes 45 qivers,
chiefly in Ada and &8t Europe , with variation coefficients
ranging from 0.10 to 0.40, i.e. below the range for the first
group.
The third group is characterized by coefficients of eor-
relation between annual flows of successive years ranghg
from 0.36 through 0.45 and ircludes 28 flow series with coefw
ficients of variation from 0.20 to 0.40. The flow series in
tius interval and duration, The latter varies from 40 to 150
years.
Over recent years the statistical analysis of correlation
coefficients between neighboring terms of a series as a fu- tion of tinis intervals has revealed that %he de4pendeDC8 does
exist in most cases 5,6] . Furthermore, studies [ 21 of inherent

296
and random errors in calculating this coefficient by standard
formulae show that the error8 may erneed 0.07. Values of first
self-correlation coefficieubs r in grouping flow seriee are
selected in a certain variation range , Table I.
To prove or disprove interdependence of these series in
each group, interseries-correlation coefficients R were calculated
for 1921-1955. For some series, the comelation was
found to be as low as iO.30. In order to meet the criteriw
of indepeideme of samples, a hydro&ogicAl LIiqueme generated
should consist of flow series from a certain grouping, the
correlation coefficients of which are close to zero. This is
one of the conditions for lack of simltaneity in flow variationa
of rivers analysed in groups, which results in an illcrease
of the overall data contained in combined hydrological series.
It should be noted, however, that the application of nor-
mal correlation techniques to flow variation etudies may pro-
ve an improper practice because of possible nonlinear rela-
tions. The depiidemes between values observed in initial hyd-
rological series can be curvilinear. Therefore for some hyärolo-
@cal series Cl 1 the initial characteristics ofQthad to be
normalized.
krmalized series thus obtained vm-e used to calculate
correlation coefficients R. These -re compared with the coef-
f icients obtainsd from actual flow characteristics. Formulae
of normal correlation were used to find the proximity betwgen
the associated correlation coefficients. This compromise can
be justified by the lack of more refined techniques of estimati
ing relations betaieen gamma-distributed random values. The am-
lysis has shown that correlation coefficients obtained directly
from series of observations and normalised series are close;
for this reason first values of R were used in the calculatfoas.
Flow series with inter-series correlation coefficients
below 0.3, were tabulated h each group. These data lead to
the assumption that relations between flow characteristics of
these basins are immaterial. !he averahed coefficients B and
the standard values of the totali- of series for each group
analysed are shown in Table 2.
Table 2
---_--------
--_-- I
Average Values of B and dR for the Groups of Rivers
---
-------- - ___- .-_- - ~
>-- __-
--.-
-
fiange of self-correlation coefficient variation
--- I"_ -II _-_-_ - ---- --__ --------- --------
O 5 r L +0.20 +0.21 5 r 5 +0.35 0.36LrC+O.45
""_ . ----_-
k2 0.129 0.134 0.157
¿fi 0.062 O. 081 O. 076
----__--------------_________II__ -
_* .- . . ---- --- --_ - -- ----

297
This table proves the absence of any substantial relation-
ship betwen the flow series analysed, Now,in each group the
flow series are combined in simulated sequ~11ce8. Thus from the
first group (O L, r 6 +0.20) a sequerce of 841 terms m s formed,
the second group 40.21 5 r5t0.35) gave a sequeme of 620 terms,
and the third group (+0.365 r-L+0.45) yielded a sequerice of
530 terms. These flow characteristics are transformed using
the coordinates of the selected type of distributions into the
curves of the event probability security P. The ordinates of
securie cumes are computed wieh an allowme for coefficients
of variation and asymmetry of each flow series. In this case
the structure of the sequence of segueities comguted for the
entire set of series is indeperdent of the flow variation and
normal flow in individual basins. If flow series included in
one sequeme are regarded as S&@pl0S of indepenient random va-
lues, then the corresponding values of flow security are also
i ndep e nde nt rand om value s.
For each series x security curves were computed for taro
types of distributions
1. Values of securities in a tuee-pararnter Kritsky-
&&e1 gamma-distribution P . This distribution was obtained
b replm ing the variable x'%# the gama-distribution equation
2'71. %e va iable is related to the initial value by the equa-
lity Z E a r6 , where a and b are parameters to be deter-
mined on the basis of experimrnial eviüeme (corresponding to
C and C ). The equation of the distribution cume for y is
ix this tase :
y(x) = -- aa a8 b
b ,
f (4
where a s and r(a) is the symbol of gamma-fulirtion.
-k
The ordinates of the security curve expressed by this
equation are always positive when y = O and P = 10%. The shape
behaviour of this distribution permits aqy relations betraeen
a and b, i,e.between the variation coefficient Cv and the asymetry
coefficient Cs
cE3 (--- = I, 1.5, 2.0, 2.5, ... 6 )
Three-parameter gamm-distribution curves fit I@ 11 the
flow series with high values of Cv.
2. P, obtained from generalized curves proposed by
Kaliain pahose studies substantiated the generality of the probabilitptheoretical
schematics dereby various samples of
flow are formed1 41 . Using the formulae K = f (P,C,) ,
%- and the flow data for many rivers, the depedemies
K=%v
I( S(Cv) , K5% = f (Cv> , etc. for annual and mximum flows were
1%obtained
separately for each value of secul$.ty (P=l%; P=5%, etc.

298
The tables of ordfnates of generalized curves for distribution
of annual flow esess probabilities were then compiled.
3. Fapirical values of mcurities for the entire sequerce
were obtained by the saression
Pem = m
-e-. qoos ,
&tI
where m is the point in a sequence of n numbers.
Thus, me have two theoretical aqd one empifica1 distributions
&ich make up the long hydrological series constructed.
Bor lack ~î npace and large sizes of the tables, the values
of simulated samplings cannot be shown this paper.
Let us now proceed to comparison of simulated empifica1
and theore tical distributions of securiQ probabilities.
For a series of 841 ternis, the representativi of the
security probability distributions obtairied mas es 3 mated for
Pea (empirical), PLM (Kritsky-bnkel) and Paen (generalized)
because a choice of the distribution curve type may considerably
affect the distribution of flow values varyi- in the
..
probability of excess.
The following versions of estimates -re consideredt
1) Uniform quantile distribution of Pe,, PK-M, Pge, .
2) Alternation of series of increased anfi decreased water
contents in simulated series.
3) Convergence of the distributions obtaiiigd with respect
to standard deviation.
4) Comparison of sampled spectra by the distribution types.
The first estimace was to reveal the homogeneity of strutture
ad uniformity of security distribution of simulated flow
series. !The number of hits of security curves in distributions
Pen, PK-BiI, Pgep ws compared in terms of arbitrary quantile
security probability distributions with respect to a fixed
quantile do not coim:i.de for the three types. In most cases,
kïowver, the deviation from the average velue does not exceed
10%. The greatest variations are associated dCii quantiles of
15-2O% security.
The statistical mthod of serial tests 103 was used in
the analgcsls of the probability of an event t periods of higher
or lower wter content as against the normal one) by the types
of distribution. In this method, for several. samples of 100
terms each in this case, the number of years (elements) m, with
flow values above the average om ard % with values below theaverage
ani! calculated. Elemsnts of the same kind bounded on
both sides by elements of another kind form series of U.

299
Lf the values of U and the mathematical expectation E differ
substantially, then the no-radom nature of the event is pro-
ved. In this case the number of series should be greater or
smaller than E by a value exceeding 33, (dispersions) of the
sample.
With the probability of higher and lower mter content
studied in this way, ma obtained the relations of series in-
dices, expectation and dispersion which are given in Table 3.
The simulated series were analysed by both the technique dec-
cribed in th& pa er and the Monte+Darlo method applied to
the Krìtsky-Menke? distribution curve.
Table 3
Serial Test Method Parameters for Different Types of
Becurity Distribution of a Simulated Sequeme of 841 Terms
9
41 8 405 w5 4û6
?i! 423 43 6 43 6 435
U 3w 370 3 67 371
E 421 421 42 1 421
D 14.5 44.4
CI------------------------------..
14.4 14.4
. -- -L _. . ---_-u--------
The numerical values of U ani E are seen to differ for ail
the three typs of diskibution by more than 3D an&%imilar
ratios. The qualitative indices obtaimd may prove, firstly,
that the distribution of elements above or below the norm tn
series analysed is not random anã, secondly, that the parameter
values I%, , 9, U, E, D obtained for samples of 841 terms
indicate uniform conditions for the event probability in all
the distribution types analysed; in other words, we have essentially
several representations of the sanie process.
In estimating the representativity of the hydro10 'cal
series obtained we analyse the convergence of eqlrica8Land
theoretical securities for the entire series. This estimate
was obtained from the r.m.s. deviation between the theoretical
and empirical distributions of securities, in per cent, for
various quankile intervals aad for the entire series. In all
the intervals the r.m.6. deviation does not exceed 1.5.
The security of securities curves of the simulated sequernes
(Pig.1) obtaimd for distributions Pgen have a similar
trend with oniy minor differences.

300
On the whole, comparison of empirical and theoretical cur-
ves for tne entire sequsnce proves the representativeness of
the hydrological sequence obtained.
Spectral analysis proved useful in studying the structure
of the simulated sequemes. For comparison of quantirbative
indices, the initial iflormation was furnished by the same
sample of 841 terms obtained for Various modifications of
annual flow distsibution. Mote that in calculation of spectral
density sequences of securiw probability in per cent, were
used. Computations were made for nonfeired aIid faired estimates
of a spectrum using the eqression [3] I
values of the self-correlation fumtion, HI - maximum
where shift, sé ordinal number of the shift, K = 1,2, ... , m.
'phis equation makes it possible to obtain a dispersion
spectrum for each frequency bad as percentage of tho total
dispersion of the time series under etudy. A fairea estimate
of dispersion spectral denSity was obtained by using the Hanning
fairing weight fu= tionr -
i= o, I, 2, ... , m
D1 = O
- Big.2 represents plots of non-faired , S@), and faired,
S(p), spectra computed for the 841 terms aad frequencies
f = O 0.1, ... , 0.2 &. If the delay of m includes less
than i- of the samp e (80 terms), the shift of estimates of
S(p), S(p) being small.
Because the simulated series contain compositional space
and time information on the flow, the spectrum of these series
follow9 the pattern of white noise and is scattered among all
f requemies.
Fluctuations of S(p) and s(p) in individual samples can be
found by computing the nean value of the spectrum, the dispersion
ad the r.m.8. error when frequencies are changed. The
latter charac teristic is computed from thsore tical spectrum
for which is taken a sampled spectrum computed with the use
of Table 4.
Table 4
Sampled Spec tra Estimates
--c-_---_---------------------- - -L---._*_-_-_---------
Non-f aired Baked
IO. 48 10.00 10.45 10.80 10.00
!%$ersion 5.81
6.02 4.18 5.01 4.72
%ba* 0.660 0.782 O 0.660 0.782 0

These estimates are evideme of homogeneous structures
of the simulated sequences for various groupings by the m e s
oî distributions.
301
The analysis leads to the coralusion that hydrological
series constructed from natural flow characteristics contain
a vast body information on combination and duration of periods
differing in water content and may prove useful in estimating
possible ranges of flow variations for certain rivers* The
proposed assessmnt of laws governing hydrological variations
may prove helpful in tackling various *ter industry problems.

302
REFERENCES
1. Alekseev, G.A. Ob'ektivq'e metoäy vyraklnivaniya i normali-
zatdi c orrelyataionnykh myazey. Gidrometeoixdat ,
Leningrad, 1971.
2. Vodnoenergethheskiye raachyoty metodom Monte-Carlo.
Ed. by Resnikovsky A.Sh. "Energiya" , Moscow,
19690
otts 7l
3. Jenkins, %---
G. Spectral Analysis and its Application.
4. Halinin, G.P. Problemy glogalnoy gidrologii. Gidrometeo-
iedat, Leningrad, 7968.
5. Kalinin, G.P., Davydova, A. I. Pro~tramtvenno-meiaeMoy
analiz tsiklichmsti atoka rek. Vestnik MGU, ser.
Geographiya, No.4, 1967.
6. Kilasoniya, A.I. IC voprosu vybsra nachala gidrologicheskogo
goda pri vodokhozyastven4ykh i vodnoenergetioheskikh
raschyotakh. Trudy GrmMIZ, XVIII, 1969.
7. Eritsky, S.N., &&el, LF. Vybor krivykh raspredeleniya
veroyatnostey dl raschyotov rechnogo stoka.
IZV. AN SSSR, OTg No. 6, 1948.
8. Kritsky, S.N., Unkel, M.F. O mtodike sovmestnogo analiza
nabluàenig ea stokom gidrologicheskikh skhodnykh
basseynov. Trudy GGI, Issue 180, Gldrometeoizdat,
Leningrad, 1970.
9. Svanidze, G.G. Osnovy raschyota regulirovaniya rechnogo
stoka metodom &nl;e-Carlo. Metsniereba, Tbilisi,
1964.
10. Yanko, Y. Bdstematiko-statisticheakiye Tablitsi,
Gosstroyizdat, Mosc OW, 1961

303

304
200 - I
1
I
I . - 7- :*u
0.1 0.t 0.8 0.4 0.6
I 1 I I I I ' m
10 N) 30 40 bD 60 Po 80
7 I I -
Y i .
Fig.2. Simulated flow sample spectra for a generalized cume:
(a) non-faired spectrum S(p)i (b> faired spectrum i@),

"THE PREPARATION OF A DATA SET FOR HYDROLOGIC SYSTEM ANALYSIS"
ABSTRACT
M.J. Hamlin, B.Sc. D.I.C. M.ASCE. M.I.W.E.
N.T. Kottegoda, B.Sc. Ph.D. M.I.C.E. M.I.W.E.
The potential of a complex water resource system can often be
determined only by the construction of a mathematical model which is
then used to simulate the operation of the system. Where flow data
is inadequate the input data for the model may present the most
challenging aspect of the design. The use of data generation techniques
to supplement historic records to obtain a complete data set, pseudo-
historic in character, provides a possible solution. Totally synthetic
data sets, based only on the statistics of existing flow data, can be
produced to provide an alternative approach. The operation of the
computed model must be based on a relevant time unit. In most schemes
studied in the United Kingdom decisions need to be taken daily and it
is therefore appropiate for the data to be in daily form. In spite of
considerable research effort methods for the generation of sequences
of daily records are still inadequate and pentads have been commonly
used. These represent the aggregated flows over a five day period and
are used for generation purposes. The five daily totals are subsequently
sub-divided to give daily flow values.
RES UME
On peut souvent déterminer le potentiel d'un système complexe
de ressources en eau en se contentant de construire et d'utiliser un
modèle mathématique de simulation. Quand les données sur les apports
sont insuffisantes, l'établissement des données d'entrées peut repré-
senter l'aspect le plus ardu du calcul. L'emploi des techniques de
génération de données pour suppléer aux observations historiques et
obtenir une série de données complète, méthode de caractère pseudo-
historique, fournit une solution possible. Des series totalement syn-
thétiques établies uniquement à partir de la statistique des données
existantes, peuvent être elaborées et fournir une autre solution. Le
fonctionnement du modèle exige le choix d'une unité de temps appropriée.
Dans la plupart des cas étudiés dans le Royaume Uni, des décisions
doivent être prises journellement; il convient donc que les données
soient journalières. Malgré un effort de recherche considérable, les
méthodes destinées à créer des séries d'observations journalières
restent insuffisantes et on doit fréquemment utiliser des données
pentadaires. On génère ainsi des écoulements pentadaires que l'on
subdivise ensuite pour obtenir des valeurs journalières.

306
Introduction
For the development of the potential resources of the Wye and
Severn river basins it is necessary to investigate both the design of
individual components of the system and the operation of these individual
componenils in an integrated system. The operational study requires the
construction of a complex simulation model.
Thig paper describes only one
aspect of the problem namely the provision of a set of compatible data as
input to the simulation model. The work was undertaken for the Water
Resources Board who were! responsible for the main operational study.
Theoretically a multisite generation model represents the most
attractive solution. In practice for daily or even five daily flows this
presents problems for which there are no immediate solutions and alternative
possibilities had to be sought. The first of these involves the use of
historic and pseudo historic flow records to build up a set of data for each
of the nodal points of the system and this was the course which was adopted.
A second possibility is the generation of wholly synthetic data based on the
statistics of existing gauging stations where a structure is created of
primary and secondary sites which are linked using a standard bi-variate model.
The choice of time unit needs considerable care and units of five
days were agreed as being appropriate. However these tend to over estimate
the resources and for a detailed consideration of critical periods it is
necessary to sub-divide these into five daily values preserving, in as far as
is possible,,all the relevant statistics of the daily time series.
The Physical System
A diagramatic sketch of the two river basins is shown in figure 1.
Both-rivers rise in mid Wales and flow South into the Bristol Channel. They
are both used for water supply purposes but there are substantial untapped
resources and the possibilities of inter-basin transfers both between the
Wye and Severn but more importantly from the Severn towards South East
England are of considerable national interest. In the first instance the
mathematical model was to be operated using data for the 38 years from
1932-1969 inclusive. These contain a number of well known low flow sequences
and in particular the periods 193314 and 1949. For this purpose it was
necessary to produce compatible sets of data for all the nodal points in the
system. Some of this data was available from historical records for the full
period. At other points only partial records were available and there were
a number of points devoid of any flow records. The sketch identifies a number
of typical points within the system although these do not represent the total
number for which data was obtained. The points are classified from A to F
as follows.
A)
Records at these stations had been collected for a number of years and
existed for the full period 1932-1969.

B) Gauged records exist for the full period 1932-1969 but required
adjustment to allow for the effect of reservoirs in upland sub-
catchments.
C)
Records exist for only part of the period and need infilling to
complete the 1932-1969 sequence.
307
D) Gauged records exist for only part of the period and require both
infilling and adjustment to allow for reservoirs in the upland subcatchments.
E) No records are available for these catchments but they can be deduced
from neighbouring catchments using the relationship
where % and A are the areas of catchments E and A
A
respectively I
and RIE and RIA are the effective rainfalls of
catchments E and A respectively
This relationship is purely deterministic and suffers from lack of a
stochastic component. However since the stochastic component cannot
be evaluated there are no means for including it.
F) No record is available for this catchment and flow values can only be
deduced from upstream catchments using the following relationship
is the total catchment area down to point F
Al, A2, A are the areas down to points, 1,
2 and 3 respectively
RIF is the effective rainfall on Area AF
RI1, RI2, RI3 are the effective rainfalls on Areas
Al, A2 and A respectively
3
Q,, Q2, Q, are the flows at points, 1, 2 and 3 respectively
“19 n are the times of travel from points 1, 2 and 3
2’ “3 to point F respectively.
The adjustments necessary to allow for reservoirs in the upland
sub-catchments were calculated using a simple accounting technique which made

308
allowance for the flow times from reservoir to the nodal point. The choice
of generation model and method of infilling data required considerably more
attention.
The bivariate model
As all records extend up to December 1969, bivariate synthesis was
used to infill the earlier parts of records where this was necessary1. For
this purpose pentad data at a satellite station, S, where the flow record is
short is linked to the data at a key station, Ky which has a long and
reliable record,through a bivariate model. If such a model is to be
acceptable statistically it should maintain the coefficient of cross
correlation, rks, between the stations, the lag one serial correlation
coefficients, rk and rs of the two stations and the five day seasonal means
Mkj+l and Msj+l, and seasonal standard deviations, Skj+l and SSj+ly at the two
stationS.in season j+i, 1 i j 5 73, corresponding to time t+i. A bivariate
model can be expressed by
In this particular application as synthesized data was required at
station S only, Kt+l represents the historical flow at station K at time t+l
in pentad units and St+l represents the concurrent synthetic flow at station
S with St as its antecedent value. The variab)e Xt+l at time t+l is given by
1
xt+l = (B/Ssj) (St - Ms.) + i1 (1 - B2) ........................... .(2)
3 t+l
where qt+l is a series of non-autocorrelated numbers with zero mean and unit
variance and a distribution which is estimated from the distribution of the
historical data at the satellite station, S. As shown by Fiering2, the three
correlation coefficients, rk, rs, rks, should be incorporated in the constant
B as follows:-
2 -1
B = (1 - rks) (rs - rk Xs2) ...................................... .(3)
In the first instance the Clywedog data at Llanidloes (station 1 in
Table 1) was extended using the bivariate model and the data from the Elan
Valley Key station. The cross correlation coefficient is 0.89 and it was
found that a three parametric gamma distribution (Pearson type 3) fits the
Clywedog data. These parameters were estimated and read into the main program.
When evaluating reservoir storages from synthetic data at the
satellite station certain defects in the model were found. A visual comparison
of concurrent historic flows at the two stations in dry years such as 1959
showed similarities in the low flows which were not reproduced by the
bivariate model in its original form. This discrepancy was observed in the

pattern of low flows in the synthetic data prior to 1959,e.&,in the critical
dry period of 1933 and 1934. In Fig 2 a comparison is made between 6 years of
historical data at a Key Station, Kt, and concurrent data at a satellite
station, St, which is partly historical (years 4, 5 and U) and partly
synthesised (years 1, 2 and 3) without adjustment. The patterns of concurrent
sets of high flows are random in both parts but the long runs of low flows
in historical dry years were not maintained inathe synthesised data. In a
separate analysis seasonal values of serial correlation were computed but no
significant differences were found. This may be attributed to the fact that
in this climatological zone the times of commencement of the dry and wet
seasons and the lengths of seasons are highly stochastic variables.
However, it nas found that when the flows are below a certain
threshold value?defined with respect to the data at the key station, which is
TT1 in Fig 2, the standardized values of flows at the two stations are
highly correlated so that these are nearly equal. The level TT1 was found by
trial and error and the data generation was repeated with the new criterion
that when the flow in the key station is below the threshold value, the
standardized flows at both stations are equal or alternatively they are
different by a very small random component.
A further refinement was included to establish a recession curve on
runs of low flows. This was estimated empirically and an average value was
read into the program so that the droughts just resemble the historical
droughts. An examination of the adjusted synthesised data showed that
differences in concurrent low flows such as those illustrated in Fig 2 were
eliminated.
Choice of probability distribution
309
In studies dealing with extreme flows and the persistence of high or
low flows the probability distribution of the data is fundamental and except in
the case of certain annual series the distributions of historical data are
significantly different from the Gaussian or normal type. This is because the
distribution of river flows in a historical sample is bounded by zero or a
positive value at its left extremity and has a long tail on the side of
increasing flows.
For this reason, the distributions are said to be positively
skewed. Furthermore, the coefficient of skewness tends to increase inversely
with the time unit on which the series of data is based, which means that the
skewness in pentad data is more than,, in, say, monthly data.
Kottegoda3 has shown that the incorporation of a gamma distribution
in a monthly data generation model could yield realistic flow sequences of
synthetic data. In particular, reservoir storage requirements evaluated from
some of these sequences surpass that from the historical records. In the case
of pentad data, a wider range of distributions to include Pearson's Type I,
-111 and VI functions are necessary in order to model the empirical
distributions4.

310
The underlying generating process in the bivariate model used in
this study is autoregressive of the type
k 1
xt = -1 a.X t-j + nt (1 - R2) ...................................... (4)
J -1
in which Xt, an autocorrelated cycle-free seriespand nt, a random series,
have zero mean, unit variance and non-identical distributions, aj are the
autoregressive parameters, k is the order of the process, R is the coefficient
of multiple correlation and t is a point on the time scale.
In an unpublished study,a comparison is made between crossing and
other properties in historical pentad data and data synthesised using an
autoregressive model and other models of recent origin in all of which the
skewness in the random component, rit, is varied. The crossing properties of
particular interest in hydrology are shown in Fig 3. A sequence of 5 day
river flows, R, which varies with time t is intersected at two levels, viz.,
RU, an upcrossing level above which the flow is higher than the mean flow
and RD, a downcrossing level below which the flow is lower than the mean.
Because with increasing time R rises above the level Ru at 4 points, there
are 4 upcrossings with respect to Ru. Similarly there are 2 downcrossings
with respect to RD. The mean length of the horizontal bases of the 4 shaded
areas above the Ru line is called the mean surplus run length. The total
area of the shaded parts or sums of ordinates within them is the total
surplus run sum.
In the same way the mean length of the intercepts at the
RD level is the mean deficit run length and the total area below the RD level
is the total deficit run sum.
The crossing properties of historical pentad data of the river Wye
at Rhyader and synthesised data based on an autoregressive model are shown
in Fig 4. For this analysis the 33 year historical record is divided into
3 equal samples of 11 years so that sampling variations, that are commonly
found in data of this type could be seen.
The properties of Synthesised data
are based on the means of results from ten 11 year non-historical sequences.
If a Gaussian distribution is used in the model the numbers of downcrossings
are far in excess of the numbers expected from the historical data. The
ratio of skewness applied to the nt series to that estimated from the
historical data ought to be between 1.0 and 2.0 if a realistic number of
downcrossings is to be obtained. A further increase in skewness results in an
undesirable reduction in downcrossings. The necessity of providing f r
greater skewness in the nt series than in the historical data was shown by
Thomas and Fiering5 , who investigated the storage-yield relationship.
approximation to the optimum skewness as obtained by analysing the independent
residuals, Zt, where
k
Zt = Xt -j=l I: Xt-j .................................................... (5)
3,4
has been shown in other studies
An

Another point in favour of using the appropriate value of skewness
is the large number of negative values generated when the skewness is too low
as is the case if the normal distribution is used. On the other :-:A Li che
skewness applied is excessive, not only ari. Fcg-tive values totally eliminated
but the lowest flows are too high as can be seen in the top left hand diagram
in Fig 4. In addition the numbers of downcrossings at various levels are
much fewer than those in the historical records.
31 1
The deficit run lengths and run sums are also shown to be dependent
on the skewness but when compared to the numbers of downcrossings, the change
with respect to skewness is in the opposite manner. With regard to upcrossings
at levels above 50 mms., an increase in skewness tends to 'correct' the
synthesizeddata but more skewness is required than for the low flows. This
may be achieved by incorporating two distributions in the model, one for high
flows and the other for low flows6.
There seems to be no basic difference in the results if.the underlying
model is changed from the autoregressive type. Skewness is the more
important criterion. This is generally true of all pentad flow series from
this climatological zone. Full results will be published in the near future.
Infilling of data
'
The ten stations at which pentad data was extended are listed in
Table 1. The number of years of synthesised data range from 5 to 29 years.
The two key stations used in the synthesis are given. In certain cases the
choice of key station for the synthesis is obvious because of close proximity,
e.g., the Wye at ñhyader and Elan at Caban Coch. In other cases, the key
station was selected on the basis of the best cross correlation coefficient.
The type of distribution adopted was ascertained from a preliminary programme
and it is seen from the table that Pearson's Type 3 or 1 functions provide a
good fit to the data. The best fitting distribution is determined by the
Kolmogorov-Smirnov two sample tests between synthesised and historical data
at the satellite stations. The cross correlation coefficien.ts between
concurrent records at the key stations and satellite stations range from 0.79
to 0.96 for the historical periods at the satellite stations. Comparative
values were obtained in respect of the synthesised data at the satellite
stations and concurrent historical data at the key stations. The table also
indicates that the lag one serial correlation coefficient is naintained by
the model.
The flow diagram in Fig 5 shows the basic structure of the programe
"Two station 5 Daily" which was used for the infilling of the pentad data at
the ten stations. The programme contains twelve subroutines. Subroutine Fxy
ascertains the threshold value, below which it is desirable to incorporate
higher cross correlation in the standardised data in order to maintain
realistic low flow sequences in the synthesised data.

312
Subdivision to daily data
Any system which is subject to daily changesin the operating
strategy must have daily inputs. Initial studies can be undertaken using
sets of data having a pentad or monthly time unit but ultimately this time
unit has to be reduced.
The method used was developed by Green7, a Ph.D student at the
University of Birmingham, who was interested in a river pollution model for
which daily river flow data was essential. Pentad data is broken down into
daily flows by interpolation and to the interpolated values is added an error
term whose purpose is to maintain the statistical characteristics of the
actual daily flow data. The long term characteristics including floods and
drought sequences are preserved in the five daily model.
The synthesis is carried out in two stages. Actual daily data is
accumulated into five-day averages which are then subdivided into synthetic
daily values. The synthetic values are compared with the actual daily values
so that the success of the parameters used in the process and of the process
itself can be measured. Once the composition of the error term can be
adequately described the synthetic five-day averages are broken down to yield
synthetic daily flows.
Alternative data sets
The use of an historic sequence of data enables the system to be
operated so that a whole range of possible planning decisions can be
investigated. Each plan is compared against alternative plans based on the
same set of input data. When an optimum plan has been identified it is
desirable that the operational decisions and their consequences should be
tested agai.nst other possible sequences of input data. The historic data can
only be used to investigate what would have happened in the past. Since the
flows will never be repeated in an identical sequence there is no possibility
of the same set of decisions occurring in the future.
For this purpose it is proposed that a number of synthetic flow
records should be produced. These will consist of compatible sets of data
for the two major stations in the system namely Bewdley and Elan Valley.
When these have been prepared, data at the other existing nodal points can be
obtained using the basic statistics given in Table 1.
Conclusion
The procedure outlined in this paper shows how water resource systems
may be designed in spite of the inadequacy of historic flow records. An
extension of the method allows for wholly synthetic sets of data to be
prepared so that the future effect of possible flow sequences can be studied.
It is an essential feature of these synthetic sets that whilst on the one hand

they reproduce the statistics of the historic data they also, on the other
band, contain sequences of rare events which are not disclosed in the
original record.
Acknowledgements
The authors wish to acknowledge the,financial assistance and
encouragement of the Water Resources Board and also of their colleague
Dr. Kelway and Mrs. Ross who yere:responsible for the major task of data
handling.
References
1.
2.
3.
4.
5.
6.
7.
Hamlin, M.J. and Kottegoda, N.T., (1971) "Extending the record of the
Teme", Jour. Hydrology 12, pp 100-116.
Fiering, M.B., (1964) "Multivariate technique for synthetic hydrology"
Jour. ASCE, 90, No HY5, pp 43-60.
Kottegoda, N.T., (1970) "Statistical methods of river flow synthesis
for water resources assessment". Proc. Inst. Civ. Engrs., Supplement
(xviii) , Paper 7339s.
Kottegoda, N.T., (1972) "Stochastic five daily stream 'flow model",
Jour. ASCE, 98, HY5, pp 1469-1485.
313
Thomas, H.A. dr. and Fiering, M.B., (1963) "The nature of the storage-
yield relationship", Operations Research in Water Quality Management,
Chapter 1 of Report of the Harvard Resources Group to the U.S. Pub. Health
Service, Cambridge, Mass.
Kottegoda, N.T., (1972) "Flood evaluation - can stochastic models provide
an answer?" Int. Symp. on Uncertainties in Hydrologic and Water Resource
Systems, Tucson, Arizona.
Green, N.M., (1973) "A synthetic model for daily streamflow", Jour. of
Hydrol, (in press).

314

Lake
D
FIG. 1
catchment 1
Not to scale
315
Avon

3-
31 6

L a
t
4
c
J
317

13A31 3NISSûä3 NhM 'I3A31 3NISSûä3 NMûCi
t"
O
r:
t -
?3Am 3NISSOä3 dn
'I3Aa-I 3NISSOXI dn
.?
"
.? "
.? "
I
'4 "
I
M
I I
I I
i 1,
a"
I
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I I
M M
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31 8

[EVALUATE HARMONIC FITTED MEANS AND STD. DEVS. Subroutine Frier P'
t
IREAD DATA AT SATELLITE STATION 1
t
319
~- __.
TATION I
t
[COMPUTE SERIAL CORRELOGRAM OF RAW DATA Subroutine Correl
+ 4
ICOMPUTE SKEWNESS COEFFICIENTS OF FIVE
[COMPUTE CROSS CORRELATION COEFFICIENTS. Subroutine Sat I
t
[FIND RUNS BELOW AND ABOVE MEAN. Subroutine Runs I
t
[OPTIONAL - STALL ANALYSIS ON HISTORICAL DATA. Subroutine Stall 1
[OPTIONAL - DURATION ANALYSIS. HISTORICAL. Subroutine Dur
GENERATE & TRANSFORM RANDOM NUMBERS TO PEARSON TYPE OR LOGNORMAL
CHI SQUARE TEST Subroutine Rannor
IDURATION ANALYSIS ON HISTORICAL AND SYNTHESIZED DATA. Subroutine Dur. 1
1
t
t '\
t
J
GENERATE DATA USING CONCURRENT DATA AT KEY STN. SET THRESHOLD VALUE
FOR REALISTIC LOWFLOWS.. Subroutine f XY
I
COMPUTE MEANS, STD. DEVS., SKEW COEFFICIENTS OF SYNTHESIZED DATA.
FIND HIGHEST AND LOWEST VALUES. Subroutine Percen.
t
IPUNCH CARD AND LINEPRINTED OUTPUT OF SYNTHESIZED DATA I
b
[ CO>íF'üTE CROSS CORRELATION COEFF. OF SYbTHECIZED AND M W DATA. Subroutine Sat11
[OPTIONAL - STALL ANALYSIS ON SYNTHESIZED DATA. Subroutine Stall
ISERIAL CORRELOGRAM OF SYKTHESIZED DATA. Subroutine Correl
t
[COMPARE RUNS IN SYNTHESIZED AND HISTORICAL DATA. Subroutine Runs
I
1
IKOLMOGOROV-SMIRNOV TWO SAMPLE TEST OF SYNTHESIZED AND HISTORICAL DATA.
I Subroutine Smir
+
I
'I
I
1
*r

POTENTIAL APPLICATION OF BAYESIAN TECHNIQUES FOR PARAMETER
ESTIMATION WITH LIMITED DATA
ABSTRACT
Roberto L. Lenton, John C. Schaake Jr.
and Ignacio Rodriguez-Iturbe
Department of Civil Engineering
Massachusetts Institdte of Technology
The use of Bayesian Techniques for parameter estimation can
potentially improve the available limited hydrologic data by taking
into account not only the information contained in the historical
sample, but also all the information coming from other sources, both
objective and subjective. At the same time, project economics can be
considered by the use of a loss function which specifies the serious
ness of choosing an estimate which is not the true one. For example,
these techniques can be applied to the estimation of the parameters
of a first order autoregressive model. Moreover, if the hydrologist
is willing to make certain simplifying assumptions and limit his pro
blem to the estimation of only the autocorrelation coefficient, then
comparatively simple estimators result. The comparison between Bayes
and Classical estimators for p on the basis of the risk function and
the expected risk shows that the Bayes estimator is considerably mo-
re advantageous, especially when the sample is of a limited duration.
RESUMEN
La utilización de técnicas Bayesianas para la estimación de
parámetros puede mejorar la informacibn limStada existente, aï tener
en cuenta no sólo la información contenida en la muestra histórica
sino también toda la información proveniente de otras fuentes tanto
objetivas como subjetivas, A la vez, se pueden considerar los aspec-
tos económicos mediante la utilización de una función de pérdidas
que especifica la seriedad de escojer un esti’mado que no es el verda
dero. Por ejemplo, se pueden utilizar estas têcnicas para la estima-
ci6n de los parámetros de un modelo autoregresivo de primer orden.
Mas afin, si el hidrólogo está dispuesto a .realizar ciertos supuestos
simplificadores, y a limitar su problema a la estimación del coefi-
ciente de autocorrelación solamente, entonces se pueden obtener esti
madores relativamente simples. La comparación entre los estimadores
Bayesianos y clbsicos para el parametro p, en Base a la función de
riesgo y al valor esperado del riesgo, demuestra que el estimador Ba
yesiano presenta considerables ventajas, especialmente cuando la mues
tra es de una duración limitada.

322
INTRODUCTION
Various models have been proposed in the past for modelling the
stochastic nature of the hydrologic processes; the purpose of using these
models has been to aid in making better investment and management decisions
regarding Water Resource projects.
Two factors are decisive in the choice of a model: the available in-
formation and the problem to be solved. Once the model has been chosen, how-
ever, these two factors must continue to be considered. The only control the
hydrologist has over his model is in the estimation of its parameters. Hence
the estimation technique should both use the%vailable information in the
most efficient way, and in some manner take into account the problem at hand,
Unfortunately, the classical methods of estimation do neither, and in this con-
text have two important defects:
1. They can only take into account the information contained in the
historical sample. Evidently the hydrologist is limiting himself by not in-
troducing information from other sources which could reduce the uncertainties
of estimation.
2. They produce values that are independent of the economic consequences
of erroneous estimates. It appears evident that it would be more rational
to assess the opportunity losses to be undergone by estimating a parameter
erroneously, and then use as a criterion for estimation the minimization of
those expected opportunity losses.
Bayes Theorem seems to provide a framework for approaching the problems
that have been indicated. The first point is taken into account by providing
an "a priori" distribution on the parameter of interest. This prior
distribution encompasses all infomation that is not in the historical sample,
providing an assessment of both the most likely values and the degree of uncertainty
of the parameter in question. The prior information enters the
estimation procedure via Bayes Theorem, which expresses simply that
where
P (Om a p (O) p (Y/@) (1)
Y = Vector of sample observations
O = Parameter
p(O/Y) = "Posterior" pdf of O I given Y
p(O) = Prior pdf of O
p(Y/O) = Likelihood function for the parameter O .
The posterior pdf now replaces the likelihood function as the means
for making inferences about the parameter, and as a means for taking into
account the second problem that was noted - i.e., the economic consequences
of erroneous estimates.
There are iïitroduced by means of a loss function R (O,@), which specifies
the opportunity loss which is undergqne when O, the true value of the
parameter, is erroneously estimated as O . Hence the Bayesian criterion
A

consists of choosing the value
losses :
or
o =
A h
h
0 that minimizes the expected opportunity
min [a(@,@)] (2)
o
6 = min a(;,@) p (@/Y) do
u n
where fi is the region of the parameter 0 .
(3)
323
Since it is evident that the Bayes approach depends rather heavily
on two factors, the prior distribution and the loss function, these two points
will be discussed below in greater detail.
THE PRIOR INFORMATION
If the prior pdf is to adequately represent all information other
than that contained in the sample, it must be assessed with great care. The
first question that must be answered is the source of ,this non-sample infor-
mation.
One reasonable source of information could be a collection of past
records from other river basins on the value of the parameter of interest.
The analysis can be performed on the frequency histogram of observed values,
by fitting a known distributional form to it. This, of course, is only valid
for non-dimensional parameters (such as the coefficient of variation or the
first order autocorrelation coefficient), the basic assumption being that
there are physical reasons which tend to make some values of the parameter
more likely than others, as reflected in the collection of records. As a
first approximation, the hydrologist could analyze world-wide data; if he
is not satisfied, he could regionalize this information or classify it, taking
into account only rivers of similar characteristics to the one he is studying.
Another source of information could be the analysis of physical cha-
racteristics of river basins which are related to the parameter of interest.
By regression on these characteristics, a measure of the mean and variance
of the parameter can be obtained, and a probability distribution fitted to
it. This is the approach used by Wood (1973), in another paper presented at
this conference,to derive prior information on exceedance flows. It also
might be possible to derive a prior distribution on the basis of theoretical
considerations if the relationship between the parameter and the physical
characteristics of the basin can be modelled. The model would give a measure
of the mean value to assign to the prior distribution, whilst the variance
must be obtained by assessing the reliability of the hypothesized model.
Finally, the hydrologist's judgement and experience must necessarily
enter the picture. When a "data-based'' prior is used, the exact form of the

324
prior pdf is tempered by the hydrologistvs subjective assessment; when
no data is available, the hydrologist can approximate a prior distribution
on the parameter from "introspection, casual observation or theoretical ob-
servations" (Zellner, 1971); he must be extremely careful, however, in
determining that the dispersion in his prior pdf properly represents his
true state of knowledge or ignorance,
THE LOSS FUNCTION
The loss function k(6,o) has been defined as the function that
specifieg the opportunity loss that obtains when the hydrologist "acts" as
though O were the real parameter value, when in fact O is. These
opportunity losses represent the difference between the benefits actually
to be obtained from a given Water-Resources project and the greater value
that would have been realized had the true parameter value been known, It
is seen from this definition that the economic losses are a consequence of
the decisions or ('actions'' that the hydrologist recommends on the basis of
his estimate; for example, these decisions could consist of constructing a
reservoir of a certain storage capacity or, in a more complex system, of
constructing a series of reservoirs, irrigation sites, power stations and
diversions of a certain size or capacity,
Formally, the loss function can be obtained in the following manner.
(Pratt, Raiffa and Schlaifer, 1965). LetAthere be a set A of acts a
(which are a consequence of the estimate 8 ), a set fi of parameter
values O , and a value function (e.g. National Income Net Benefits) Vt
with values vt(a,O). For every parameter point, the greatest of these
values is mpx vt (ayo). Therefore the opportunity loss of any particular
act a given that the parameter is O is
!L(a,O) max vt(a,O) - vt(a,O)
a
where a, is the optimal act a for O .
Finally, if
be expressed
as is the optimal act a for 6 , then (5) can
These ideas are illustrated in Figure 1.
Thus, to determine the structure of his loss function, the hydro-.
logist must first evaluate the value functions associates with his problem.
Application to a Reservoir Sizing Problem
A simple, though common problem in Water Resources Engineering is the
determination of the optimal storage capacity of a reservoir to provide re-

325
gulated flow to an irrigation area of a given size with a given target demand.
The action to be taken in this case consists of constructing the reservoir of
a certain capacity S; the value functions could consist of the net benefits
derived from the irrigation system.
These can be computed on the basis of
the long-term benefits derived from the operation of the irrigation site, the
cost of the reservoir and of the irrigation system, and the short term losses
which occur when the water supplied by the reservoir is insufficient to meet
the irrigation target.
As th2 reservoir size is increased, the short term losses decrease at
the expense of reservoir Costs, and therefore the problem essentially consists
of a trade-off between these two fac'tors.
A discrete set of value functions can be easily determined in this
case through simulation for a discrete number of parameter values
(oi, i = l,Z,...,n) and for a discrete number of design storage capacities,
or actions, (aj, j = 1,2,...,n). Using the assumed flow mod21 with parameter
oi, and setting the reservoir capacity at aj, the system net benefits
Vt(Oiy aj) can be determined after simulating for an appropriate period of
years.
This technique was applied to determine the loss function for a hypothetical
problem of determining the optimal storage capacity of a reservoir
to supply an irrigation system in the Rio Colorado Basin in Southern Argentina
(see Lenton, Rodriguez-Iturbe, and Schaake, 1973) ; the assumed model was the
first-order normal autoregressive model with parameters p ,a2 and p . The
loss function for p , assuming and u' equal to the sample values, is
shown in Figure 2.
It should be pointed out that the loss function on all 3 parameters
of the model could be determined using exactly the same technique, although
a 6-dimensional matrix would be required.
EXAMPLE APPLICATION: THE FIRST ORDER AUTOREGRESSIVE MODEL
The Bayesian methodology for parameter estimation has been applied
quite successfully to the case of the first-order normal autoregressive model
by Lenton, Rodriguez-Iturbe and Schaake, (1973), the basic results of which
are summarized below.
The first order normal autoregressive process may be expressed as
where y, = annual flow at year t
w = independent normally distributed random-variable with zero
mean and unit variance

326
p = mean of the process
u2 = variance of the process
p = first-order autocorrelation coefficient of the process
The Bayesian analysis begins with the assessment of the prior in-
formation. In this case, in order to make the posterior analysis mathema-
tically tractable, the model was reformulated in terms of the parameters
v', U , and p, where v' is the inverse.pf the coefficient of variation.
This approach has the considerable advantage of permitting the assumption
of independence between the parameters at an "a priori" level, and hence
the prior pdf could be expressed
The prior pdf on the parameter p , p(p), was derived from a collection
of past records from 140 rivers of the world, gathered by Yevjevich (1964).
A Beta distribution of the form
was fitted to the histogram of values of p derived from that collection,
resulting in the following values of kl and k2
kl = 9.888
k2 = 14.499
Assuming prior ignorance about the values of v' and U , (8)
was expressed as
It should be pointed out that it is possible to incorporate informa-
tion on the parameter v' as well, utilizing the same collection of records
from which the prior information on p was derived, and fitting a Beta pdf
to the frequency histogram. However, this procedure has the disadvantage of
not permitting a marginal analysis on the parameter p , which can be con-
siderably useful, as will be seen further on.
The likelihood function for the parameters was derived in the usual
manner, and multiplying the prior pdf by the likelihood function for the 3
parameters of the process, the posterior pdf was shown to be

where
Equation (11) is therefore the key equation for the optimum estima-
tion of the parameters of the first order autoregressive model. To do this
in a practical design problem the following two steps must be undertaken:
327
1.) Derive the loss function &(6,0), where O and 8 are now
3x1 column vectors, by application of Equation (6). The value functions can
be obtained through simulation, as shown in the Rio Colorado example.
2.) Obtain the optimum parameter estimates 0 by solving Equation
(3), by substitution of (11). In practical applications, this minimization
procedure must be undertaken numerically. Note that the determination of the
optimal estimates immediately gives the optimal action a6 through the
value function Vt (a,O), as indicated in Figure 1.
Some Simplifying Approaches
The procedure outlined above may be considerab1:r simplified if the
hydrologist is willing to limit his problem to the optimum estimation of only
the parameter P ,the other parameters being estimated by classical proce-
dures. This approach may be justified by noting that the variance of these
estimates is usually quite small.
The marginal posterior pdf for P can be obtained in this case by
integration of Equation (11). However, a much simpler expression, and one
that produces almost identical estimates, can be derived by making the trans-
f o mat ion
The model can now be expressed as
Xt = y, - lJ (12)

328
The marginal posterior pdf for P for this process can be shorn
to be (Thornber, 1967; Rodriguez-Iturbe et al., 1972)
where
T
a = C x 2
t
t=o
T
a 1 = - 2 C xt x t-1
t=l
a2 = I xL
t-1
t=2
Further fundamental simplificatiom can be made if the hydrologist
is willing to fit a simple functional form to his loss function. In this
case the minimization of Equation (3) can be determined analytically. Some
distribution-free results are available in the literature (Raiffa and
Schlaifer, 1960), for example, if the loss function can be assumed quadratic,
then the optimum estimate is the mean of the posterior
function can be assumed linear, i.e.
pdf. If the loss
then the optimum estimate is obtained from the value that satisfies
kU
P(P/X) = -
%+ ko
where P(p/X) is the posterior cdf for p .
Lenton, Rodriguez-Iturbe and Schaake (1973) extensively studied
the properties of the Bayes estimators under the quadratic and linear loss
functions, with various degrees of asymmetry. Basically, interest centered
around comparing the performance of Bayes estimators with that of some clas-
sical estimators. The criteria for comparison were the risk functions

R(p) and the expected risk B.
where
By definition,
R = Region of the sample vector Y
Y
Hence the expected risk is
The prior pdf P(P) used was that given by Equation (9) with the
Yevjevich data parameters.
Some selected results are reprinted in Tables 1 and 2. They
show the Bayes estimator to be considerably superior to the Maximum Liks-
lihood (ML) estimator in all cases, especially in the presence of limited
data.
Table 1
Comparison of Estimators under the Quadratic Loss Function
Sample Length
329
The sensitivity of the Bayes estimator to the form of the loss function
was also studied. The Bayes estimator was found to be remarkably robust;
for example, for sample lengths of 10 years, if the coefficient
ko was
erroneously determined as ko = 4 instead of ko = 0.25, the Bayes estimator
still performed almost twice as well as the ML estimator,under the expected

330
risk criterion.
Table 2
Comparison of Estimators under the Linear Loss Function
Furthermore, practically no difference in the expected risk was
found when the symmetric linear loss function was used instead of the
quadratic. In general, it was concluded that errors in the form of the
loss function are less important than errors in the degree of asymmetry
of the loss function. This observation tends to justify the simplification
of fitting the loss function to a given functional form, provided that the
correct degree of asymmetry is preserved.
CONCLUSIONS
The Bayesian framework for parameter estimation permits the hydro-
logist to correct the defects of classical methods of estimation through
the consideration of fi sources of information and through the considera-
tion of the economic consequences of erroneous estimates.
Infonation obtained from sources other than the historical sample
is incorporated in the prior pdf; however, the source of information must
be analyzed very carefully. The sample information enters the estimation
procedure via the likelihood function. The economic consequences of erroneous
estimates are taken into account by means of a loss function; a general tech-
nique for obtaining this function for a hydrologic design problem is presen-
ted.
The general Bayesian approach to parameter estimation can be applied
to the first order autoregressive model. In general, this procedure permits

the optimum estimation of the 3 parameters of the model. However, if the
hydrologist is willing to make some simplifying assumptions and limit
his problem to the estimation of P , relatively simple estimators can
be derived. These estimators present considerable advantages over the
classical estimators.
ACKNOWLEDGEMENTS
The work was supported by the Office of Water Resources Research,
Office of the Interior, United States Govprnment, under Grant No.
14-31-0001-9021.
1.
2.
3.
4.
5.
6.
7.
8.
REFERENCES
Wood, Eric F., (1973). Flood Control Design with Limited Data -
,A Comparison of the Classical and Bayesian Approaches, Presented
at the Symposium on the Design of Water Resource Projects with
Inadequate Data, Madrid, June 1973.
Zellner, A. (1971). An Introduction to Bayesian Inference in Eco-
nometrics, J. Wiley &Sons.
Pratt, J.W., H. Raiffa, and R. Schlaifer (1965). Introduction to
Statistical Decision Theory, McGraw-Hill.
331
Lenton, Roberto L., I. Rodriguez-Iturbe and John C. Schaake, (1973).
A Bayesian Approach to Autocorrelation Estimation in Hydrologic
Autoregressive Models, Ralph M. Parsons Laboratory, Report No. 163,
M.I.T.
Yevjevich, V. (1964). Fluctuations of Wet and Dry Years, Part II,
Analysis by Serial Correlation, Colorado State University Hydrology
Paper No. 4, Fort Collins, Colorado.
Thornber, H. (1967). Finite Sample Monte Carlo Studies: An Auto-
regressive Illustration, J. Am. Statist. ASSOC., September 1967.
Rodriguez-Iturbe, I., J. Valdes, R. Lenton and D. Valencia, (1972).
Bayesian Hydrological Model Building, Proceedings of the ïnter-
national Symposium on Uncertainties in Hydrologic and Water Resource
Systems, Volume II, University of Arizona, Tucson, Arizona.
Raiffa, H, and R. Schlaifer, (1961). Applied Statistical Decision
Theory, M.I.T. Press.

332
Figure 1 . The value functions and the determination
of the loss function.
\
a

_c,
-p" -0 6 -0 2 02
R

STORAGE-Y1,ELD ESTTMATEC WITH INADEQUATE STREAMFLOW DATA
T.A. McMahon and R.G. Mein
Department of Civil Engineering, Monash University, Australia.
ABSTRACT
Inadequate streamflow data may result from short records.
Various techniques can be used to deal with this paper the design
problem of estimating the storage-yield relationship for a large
reservoir on a stream having only seventeen years of data is conside-
red, There are two parts to this problem; the extension of the
streamflow record and the estimation of storage capacity.
For the example studied, the streamflow record was extended
from daily rainfall using a simple rainfall-runoff procedure
(Boughton's digital computer model modified with a groundwater
component). The model was fitted to half of the available record and
validated against the remaini'ng half. The agreement between estimated
and historical data was better than that resulting from a month by
month regression analysis between the historical flows and the flows
at an adjacent site (with much longer streamflow records).
In the part Gould's stochastic model is used to determine
storage capacity. The mothod is independent of the initial conditions
and takes into account seasonality and monthly serial correlation.
Results are compared with those obtained using a behaviour analysis.
RESUME
L'insuffisance des données portant sur le débit peut résulter
de la trop courte durée des relevés. L'on peut adopter différentes
techniques pour y remédier. Dans le présent article, nous étudierons
plus particulierement le probleme de l'élaboration d'une méthode
permettant d'estimer les relations débit-accumulation dans le cas
d'un grand réservoir situé sur un cours d'eau pour lequel lienregis-
trement des données ne remonte qu'a dix-sept ans. Le problème peut
être divisé en deux parties: ia prolongation des données concernant
le débit d'eau et l'estimation de la capacité d'emmagasinage.
Dans le cas précis, la série des données concernant le ddbit
d'eau a été prolongé au moyen de données pluviométriques grace 3
l'utilisation d'une simple procédure pluviométrie-écoulement (modèle
numdrique de Boughton modifié par l'addition d'un autre facteur,lleau
souterraine). Le modèle a St6 ajusté en utilisant la moiti8 des
données disponibles, pour être ensuite contrôle par comparaison ayec
l'autre moitié. L'accord entre les données estimées et historiques
s'est révelé très supérieur à la corrélation, établie grâce à une
analyse par régressions mensuelles entre les débits observés et ceux
qui ont été obtenus dans un site voisin pour lequel les relevés por-
tent sur une période beaucoup plus longue.
Dans la deuxième partie le modèle stochastique de Gould est
utilisé pour déterminer la capacité d'emmagasinage. Cette méthode est
indépendante des conditions initiales et elle ti'ent compte des fac-
teurs saisonniers aussi bien que de la correlation sérielle mensuelle.
Une comparaison est faite entre nos résultats et ceux obtenus par la
méthode d'analyse dite "de comportement'' (behaviour analysis 1.

336
INTRODUCTION
Inadequate streamflow data may result from measurement errors and
shortness of record. In this paper we are concerned with the latter problem.
Specifically, we take a seventeen year streamflow record, considered
inadequate for storage estimation, and show how this can be extended using
a relatively simple deterministic rainfall-runoff model.
In the second part a stochastic storage model is used to estimate the
capacity of a single reservoir €or various regulating conditions and
probabilities in failure.
The methodology is illustrated using the ïñomson River catchment at
'Ihe Narrows (ref. no. 225210, iat 37O 53'S, long 146O 24'E) in Victoria,
Australia. This is a 518 km2 forested catchment (Fig.1) with elevations
varying between 400m and 1560m.
In the south west, granodioritecountry gives
rise to deep loam soils whereas the remaining area is sedimentary rocks with
shallow soils. Mean annual catchment rainfall is 1540mm which varies from
below 1OOOmm to more than 2500m over the catchment.
Three standard daily read rain gauges (Fig.1) are available with
records extending from 1886 to 1970. A fourth gauge with data from 1942
onwards is also available. Stream heights at The Narrows have been recorded
automatically since 1954 and masured gaugings have been made up to 531n~s-l.
However, discharges up to 150m3s-l have been estimated. No long term
evaporation measurements are available within or near the catchment. The
closest station is at Elelbourne, 130 km to the west.
STREAMFLOW DATA ESTIMATION
Basically there are two methods available for extending streamflow
data at a gauging station. The first method consists in correlating the
flows at that station with those at a nearby station with long records.
From the additional records and the regression equation/s relating the two
stations, flows at the station with the shorter record period are estimated.
Searcy (1960) explains clearly the use of simple analytical and graphical
regression procedures to do this. Multiple regression methods are sometimes
used. Examples of these are given by Brown (1961) who illustrates the
procedures with monthly data for the Snowy Mountains region of Australia.
The second technique is based on deterministic rainfall-runoff models.
Of the many digital computer models now available, only one - Boughton's
model - has been used extensively in Australia. Because of this and because
it utilizes daily inputs, we adopted the Boughton model for extending the
seventeen years of streamflow at The Narrows, in an attempt to improve upon
the results obtained using the standard regression method.
In relation to data estimation, one question which arises is whether
data should or should not be extended. In circumstances where consistency
between record lengths is required, data extension is mandatory. However,
a check on whether statistically one is better off extending data can be made

y computing the relative information content using the procedure given by
Fiering (1962). Checks made for this study showed extension of the data is
beneficial.
337
In this paper the terms data extension and data estimation are used
synonymously to des cribe procedures in which equivalent historical estimates
are made. Herein, we calculate historical monthly flws for the period 1886
to 1353. On the other hand, data generation is an analytical tool in which
a model, which represents the stochastic streamflow process, produces flow
sequences that statistically are no different to the historical sequence,
but cannot be adopted to represent the observed flow record.
Boughton's Rainfall-Runoff Model (Boughton 1966, 1968)
?he Boughton model simulates for a catchment daily surface runoff from
daily rainfall inputs and is operated in three distinct cycles - wetting,
drying and drainage. The wetting cycle is only considered on rainfall days,
but the drying and drainage cycle operate every day.
a) Model structure.
The model consists of four storages representing interception, upper-
soil, drainage and the lower-soil zones (Fig.2).
The interception store represents water stored on vegetation during
rain periods. It fills during the wetting cycle and evaporates (at the
Potential rate) during the drying cycle.
When the interception store is full, excess rainfall is admitted to
the upper soil store which represents the moisture holding capacity of the top
soil. Water is lost from this store during the drying cycle by
evapotranspiration.
The drainage store fills during the wetting cycle only after the upper
soil store is full. This is intended to represent water in the upper soil
which can later drain under gravity to the lower soil zone. If the drainage
store is filled (i.e. the soil is saturated), surface runoff occurs. During
the drainage cycle the drainage store is depleted by water transferring to
the lower soil store. No evapotranspiration occurs from the drainage store.
The lower soil store represents water held in the sub-soil zone.
Drainage from the drainage store adds to the volume in storage, whilst
evapotranspiration and deep percolation deplete it.
üp-dating of the moisture status of the stores occurs daily.
b) Infiltration.
The model utilizes a relation similar to Horton's infiltration
equation:
-k . SS
f = fc + (fo - fc) e . (1)

338
where f = daily loss rate,
fo = loss rate when soil is at wilting point,
fc = limiting value which the loss rate
approaches at high soil-misture levels,
k = an exponent, and
SS = lower-soil moisture level.
Thus infiltration is a function only of the lower soil moisture status. When
this is low, the rate of infiltration from the drainage to the lower soil
store is high, and vice versa.
c) Evap o t rans pi r a t i on.
As well as evaporation from the interception store, evapotranspiration
takes place from the upper and lower soil stores. Evaporation need is first
met from the interception store, and if that need is not filled, evapotranspiratic
then takes place simultaneously from the upper and lower soil stores. The rate
is a function of both the evaporation potential and the soil moisture status
of each store. This approach follows the work of Denmead and Shaw (1962)
and is shown schematically in Fig. 3.
d) Surface runoff.
In the model no attempt is made to simulate the time sequencing of
surface runoff. Runoff occurs only on days of rain. The algorithm for
estimating daily runoff volume is:
Q = P - f tanh (P/f)
where Q = daily surface runoff,
P = daily rainfall less interception and upper soil
store requirements, and
f = daily loss rate.
e) Groundwater.
As proposed by Boughton, the model yields surface runoff only. Ground-
water loss whiclr is a function of the moisture status of the lower store '
accretes to deep seepage. No base flow occurs.
To overc8me this deficiency in the model, the authors substituted for
the lower soil store shown in Fig. 2, the modification shown in Fig. 4.
That is, the lower soil store is divided into two parts, each contributing
to base flow. nie lower section of the store must be full before the upper
section can hold water.
On the assumption that there is no deep groundwater
loss from the catchment, base flow was represented by the following
equations and added to the surface runoff component.
. . . (2)

Qb = kl S1 if SS E Slmax
Qb = kl S1 + k2 S2
Qb = Sp + kl S1 + k2 S2 if SS > Slmm + SZmax
339
. . . (3)
. . . (4)
. . . (5)
where Qb = daily base flow,
S1,S2 = soil moisture levels of each section of the
lower soil store,
S = maximum capacity of lower soil store sections,
simax> 2max
SS = lower soil moisture status,
Sp = spill from lower soil store if the total capacity is exceeded,
kl,k2 = base flow recession constants for the lower soil store
sections determined from the streamflow data.
This particular algorithm was chosen because it represents on a daily basis
the double sloped hydrograph recession limbs observed for The Narrows flow
data.
f) Parameter estimates and optimization.
Before the Boughton model can be used to predict runoff, values for
several parameters must be determined. nie usual method for this is to estimate
values, run the model, and compare the predicted and observed values of runoff.
Then changes are made to the parameters to see if the agreement can be
improved. There were nine parameters for which values were required, namely
the capacities of the interception, upper soil, drainage and lower soil
stores, S2, k2, fo, fc, and k. (kl was determined from base-flow recessions
in the data). Systematic variation of the values of the parameters (optimization)
is made to determine the values.
The most common optimization procedure is the steepest descent method.
Another procedure is the simplex method. Boughton (1968) and Nedler and Mead
(1965) discuss these techniques. Model parameters are determined normally
using half the available conmon rainfall and streamflw record period, the
remaining half being used to test the model parameters by comparing the
computed flows with the observed ones.
Further details of the model can be found in Boughton's papers (1966,
1968a,b) and in Pattison and McMahon (1973).
Results.
The model was applied on a daily basis to the Thomson catchment. For fhe
period 1886-1971, daily catchment rainfalls were estimated from the daily ram gauge readings using Theissen weightings. AS daily streamflow data at The
Narrows were available from 1954 to 1970, the period 1954-1962 was used to
define model parameters, the remaining eight years was used as an independent
test of their adequacy.

340
Daily catchment evaporation estimates for the study period were estimated
by applying monthly sunken tank pan coefficients taken from Wiesner (1970,
Table 18) to kïbourne pan data. In addition to Melbourne being 130 km from
the catchment, data was measured initially using a sunken tank but after 1967
by American class 'Al pan. No suitable class 'A' pan coeffecients are
available for the site. This results in the open surface evaporation estimates
from 1967 to 1970 being uncertain.
From Table I it is seen that the computed monthly flows compare
favourably with the observed values. In making this assessment, the simplicity
of the model, the difficulties in estimating catdiment evaporation and the
normal accuracy of rainfall and streamflow data were all considered. The year
1954 is difficult to simulate because of the effect of the choice of initial
conditions, while the uncertainty of the correct evaporation values for 1967
onwards is certainly a factor in the estimates for that period. A further
comparison is shown in Table II for the period 1954-1970 between historical
flows and those calculated using the model and those calculated from monthly
regression analysis between The Narrows data and its most reliable set of
adjacent flows. The model results are better.
RESERVOIR CAPACITY ESTIMATION
Joy and McMahon (1972) have reviewed a large number of procedures for
computing the capacity of a single reservoir and conclude that Gould's
method is a satisfactory design tool.
Gould's Stochastic Storage Model (Gould, 1961)
Gould's technique is classi£ied as a stochastic approach and is based
on the pioneering work of Moran (1959). Using discrete time units, Moran set
up a simple mass balance of water in storage as follows:
at+l = Pt + Xt - Yt . . . (6)
where Xt,Pt+l = reservoir contents at the beginning and end of
tth discrete time period,
= inflow during tth time period, and
Xt
Yt = release during tth time period.
By neglecting seasonality and annual serial correlation and dividing the
reservoir and streamflow into a number of equally sized zones, Moran was able
to obtain a system of equations (the coefficients of which are equivalent to
the transition matrix of stored contents) describing the cumulative probability
of stored contents. The solution of these equations is the steady state
condition of stored water.
In Gould's technique, the reservoir is also divided into a number of
zones. The transition matrix - the relation of the volume of water in storage
at time t to the volume stored at time (t+l) - is obtained by routing each

year of the historical flow record through a storage of specified size, a month
at a time, beginning each year in each zone. (Twenty zones were used in this
study). Thus seasonality and monthly serial correlation are automatically taken
into account. Releases from the reservoir can be varied seasonally or in any
other specified manner. However, annual flows are assumed independent. By
recording the starting zone, finishing zone and the number of failures, the
transition matrix of stored contents and the conditional probabilities of
failure within the year subject to the reservoir contents at the start of
the year are built up. From the transition matrix, the steady state content
is obtained.
341
As applied bj, Gould, the conditional probabilities of failure were
based on annual failures determined from monthly flows. This results in an
over-estimation of the required storage size. In the procedure used here,
the method is modified so that the conditional probabilities of failure are
determined by monthly failures from monthly flows (see Joy and McMahon, 1972).
Space precludes an adequate description of Gould's procedure. It is
set down clearly in example form in Appendix I of the original paper (Gould,
1961).
In the procedure annual flows are assumed independent. However, Gould
provides an equation to correct for this if the annual serial correlation is
significant. A limitation of the technique is that it requires computer
facilities for solution. On the other hand, an advantage is that the
probability of failure is independent of the initial starting conditions.
Moreover, because of the assumption of annual independence, records with
missing years of data can be utilized without recourse to data extension
techniques.
Results.
Gould's procedure, modified as noted above, was applied to the 85 years
of estimated streamflow data for conditions of 5% probability of failure and
constant draft rates of 50% and 90% of the mean monthly flow. In this context,
5% probability of failure implies that 5% of the time the reservoir is unable
to maintain a specified constant draft (other equivalent terms are yield,
release, regulation) which is defined as a percentage of the mean flow.
Storage estimates are given in Table III and are compared with
estimates based on a behaviour analysis. In the behaviour analysis, changes
in the volume of water stored were examined on a monthly basis by adding inflows
to, and subtracting releases from the water stored in a reservoir of finite
capacity. Probability of failure was defined as the proportion of time units
that the storage is empty to the number of units of historical flow run
through the storage. In the behaviour analysis, it is assumed that the
reservoir is initially full.
At 50% draft, the storage estimates are similar. On the other hand, at
the higher draft the Gould estimate is about 30% of the behaviour value. At
low drafts such as 50% of mean flow, annual serial correlation is unimportant

34 2
(Table III). Nevertheless, at high drafts with long draw .down periods extending
over years, annual serial correlation must be taken into account. On adjusting
the 90% value for the annual serial correlation of 0.34 using Gould's correction
(Gould, 1961), the value is increased to 86% of the behaviour estimate. It
should be noted, however, that 0.34* is beyond the range of correlations
(O - O. 25) used by Gould in deriving the correction procedures.
Consequently
the correction factor is based on an extrapolation and this probably accounts
for the difference in storage estimates at the 90% probability level. It has
been shown elsewhere (Joy and McMahon, 1972) that Gould's method is a
satisfactory procedure for estimating storage capacity.
Because of the long length of record (historical plus estimated) and the
low variability of Thomson River flows (for example the annual coefficient of
variation is 0.42 compared with the more variable Australian rivers with values
over l.O), the behaviour storage estimate is considered to provide a reasonable
check on the Gould results. However, in normal situations where either the
available record is shorter or the stream more variable, the behaviour procedure
is not necessarily a satisfactow analytical tool. Some comments on this
aspect may be found in McMahon, Codner and Joy (1972).
CONCLUSIONS
In this paper we have endeavoured to illustrate the use of a
relatively simple deterministic computer model by Boughton to extend an
inadequate data sequence to one more acceptable in length. A modification
which allowed the model to account for base flow was described.
Reservoir capacity estimates were computed using Gould' s procedure
again with slight modifications. Results were compared with those found
using a behaviour analysis.
ACKNOWLEDGEMENT
nie authors wish to thank the Melbourne Metropolitan Board of Works
for providing the rainfall and streamflow data for this project.
Mr, G. Codner, Department of Civil Engineering, Monash University, provided
the authors with the regression equations used to obtain the comparative
results in Table II.
*The value of 0.34 annual serial correlation is much higher than usual for
Australian streams. It is not possible to determine whether the Boughton
model itself contributed to this high value.

REFERENCES.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Boughton, W.C. (1966). A Mathematical Model for Relating Runoff to
Rainfall with Daily Data, I.E. Aust., Civil Engg. Trans., CE8(1),
pp. 83-93.
343
Boughton, W.C. (1968a). Evaluating the Variables in a Mathematical
Catchment Model, I.E. Aust., Civil Engg. Trans., CElO(1) pp. 31-39.
Boughton, W. C. (1968b). A Mathematical Catchment Model for Estimating
Runoff, Jour. Hydrology (N.Z.), 7(2), pp. 75-100.
Brown, J.A.E. (1961). Streamflow Correlation in the Snowy Mountains
Area, Jour I.E. Aust., 33(3), pp. 85-95.
Denmead, O.T. & Shaw, R.H. (1962). Availability of Soil Water to
Plants as Affected by Soil Moisture Content and Meteorological
Conditions,Agron. Jour. 54(5), pp. 385-389.
Fiering, M.B. (1962). On the Use of Correlation to Augment Data,
Jour. Amer. Stat. ASSOC., 57, pp. 20-52.
Gould, B.W. (1961). Statistical Methods for Estimating the Design
Capacity of Dams, Jour. I.E. Aust., 33(12), pp. 405-416.
Joy, C.S. 6 McMahon, T.A. (1972) Reservoir Yield Estimation Procedures,
I.E. Aust., Civil Engg. Trans., CE14(1), pp. 28-36.
McMahon, T.A., Codner, G.P. 6 Joy, C.S. (1972). Reservoir-Storage Yield
Estimates Based on Historical and Generated Streamflows. I.E. Aust.,
Civil Engg. Trans., CE14(2) (in press).
Moran, P.A.P. (1959). The Theory of Storage, Methuen, London.
Nedler, J.A. & Mead, R. (1965). A Simplex Method for Function
Minimization, Comp. Jour., 7, pp. 308-313.
Pattison, A. & McMahon, T.A. (1973). Rainfall-Runoff Models Using
Digital Computers, I.E. Aust., Civil Engg. Trans., CE15 (in press).
Searcy, J.K. (1960). Graphical Correlation of Gauging Station Records,
Geological Survey Water -Supply Paper 1541-C, Washington.

344
TABLE I : COMPARISON OF BOUCHTON MODEL RESULTS WITH HISTORICAL VALUES
FOR OPTIMIZING PERIOD AND INDEPENDENT TEST PERIOD.
Parame ter
Standard deviation
(w/month)
Corre lati on
coe f fi cient
between historical
and predicted
Optimizing Period (1953-62) Test period
Historical Boughton His torical
Value Mode 1 Value
1 4 8 I 49
1 38
40 42 32
- .94
(1963-1970)
Bought on
Model
TABLE II : OBSERVED STREAMFLOW PARAMETERS (FOR PERIOD 1954-1970) COMPARED
WIM "HOSE ESTIMATED USING REGRESSION ANALYSIS AND BOUGHTON MODEL
Parameter His torical
Value
Me an
(mm/mon th)
Standard deviation
(m/mon th)
Corre lation coefficient
between observed and
predicted I
- .92
TABLE III : STORAGE ESTIMATES FOR 5% PROBABILIW OF FAILURE
(Millions of cubic metres)
I Draft I Gould Model I Behaviour Analysis I
1
* Values in brackets include Gould's correction for
annual serial correlation.
86
40
37
43
41
.89

o km
Legend
Daily read rain
A Gauging station
U
~~
5
Granodiorite
Clay, silt stone, silty sandstone
FIG. 1. 'IHOSON RIVER CATCHMENT
34 5

346
Lower Soil Store
-
FIG. 2. SCHEMATIC DIAGRAM OF BOUGITON MODEL
-

Actual
ET
Rate
Ip potential
ET
Rate
t
eld
pacity
FIG. 3. COMPUTATION OF ACTUAL ET RATE FROM POTENTIAL ET RATE
IN BOUGHTON MODEL. (Figure shows graphically the
method of calculation for moisture level S
for potential rate p)
Lower Sub-store 2
Soi 1
Store ---
Sub-store 1
Infiltration Evapotranspiration
Baseflow 2 (kzSz)
Baseflow 1 (klSi)
-
FIG. 4. MODIFICATION OF 'IHE LOWER SOIL STORE TO OBTAIN A BASE
FLOW WITH DOUBLE RECESSION CONSTANT.
347

ABS TRACT
ESTIMATION OF GUMBEL LAW PARAMETERS IN SMALL SAMPLES
Valentin Martfn Jadraque
Civil Engineer
A complete study of the distribution law of Gumbel for ex-
treme values is realized and the methodology of estimation of the
characteristical parameters, mean and typical desviation in the
case of samples of few extension, which serve to determine the
typical parameters u and u of this law,
RES UM EN
Se realiza un estudio completo de la ley de distribución de
Gumbel para valores extremos y la metodología de estimación de los
parametros caracterfsticos, media y desviación típica en el caso
de muestras de poca extensión, los cuales sirven para determinar
los pardmetros u y u de esta ley.

350
Iii hydrological studies, and especially when studying
the maximum annual flood of a river, this aleatory variable
is considered as distributed according to the Gumbel law.
However, the Gumbel law has more general applications, and
its use is considered satisfactory as distribution of aleatory
variables which are extremes (maximum or minimums) of a certain
phenomenum produced in time.
Tile study we are making is partly a reminder of the main
properties of this variable, such as its distribution function,
density function, moment generating function, and estimation of
the moments regarding the origin, estimating the mean, variance
and typical deviation, and partly a development of the study on
estimation of the characteristic mean and typical deviation
parameters in the case of samples of small extension, which in
turn help to find the typical iy and u paramcters of this law.
The aleatory variable 5 with Gumbel distribution is one
whose distribution function F (x) is:
F (x) = Prob (5‘”) = e
-e-% (x-u)
(1)
wliereMand u are parameters to be determined in each case, and
whose estimation is analysed later on.
The distribution function (i), as all distribution functions,
fulfils the properties :
F (x,) 5 F (x2 1 if x1 x2
F (-”) = O
F (+p.) = 1
The density function f (x) will be:
The distribution method is made by making:
f’ (x) = FI’ (x) = 0

Taking Neper logarithms in (i) and deriving, one obtains:
F" (x) = @.e-=' IF' (x) -oc.F(x) ] whereby;
Thus :
FI' (x) = O implies F' (x) = O(, F(x)
e = 1 and therefore, X mode
The poment generating function
(t) = E (e 5 * t, =
Making
-I?¿. (x-u)
-00
-- 1
Y = - ?
dy = -U. x. dz
ex = eU.y *
= u
Thus :
u. t t o
ys(t) = eu*t* (5 ! = e .r (i - (3)
'p5(O) = 1
351
- "(x. u)
. d. x
[
To calculate the moments in respect of origin &k , we recall
tiiat :
% = 'f k (0) = -$'f5(t4
5
t=O
and therefore:
y&(t> = 1 - o(1 . t + -. t2+ ... +- o(k* tk+ ...;
Taking Neper logarithms in (3) , we get:
d2
l ! 2! k!
LnV5 (t) = u. t -+ Ln r (i - t ) (41
On the other harid, it is known that:
r(i+x)=x! =lim nx
n-+n
(i+&-)
J
j = l

352
Where
C = 0,5772156 ... P Euler constant .
If we call
O0
We will get:
and, the re for e :
O0
K1=r+, c
Ks = - s2
a2 '
..............
...........
Sr.(r - i)!
xr =
ar
.............
ïim ( i .L i i 1 .L ... L I . L
n-+Eo 2 3 n "

Where hr is the cumulant r-esimo.
Since iíl = dl = ,cam
The typ ical devi at i on
tnus :
R
D(5) = + = ka^
Having obtained a sample of values xl, x2 .. . , xn, and
A
estimating the meanPx from this sample and the typical
A
deviation o-x , the estimation of the parametersGy u is
made in accordance with the above formulas:
=fc. - 0,45QOS. a;C
With this theore tical reminder , we shall now analyse the
estimators study centred on the meany, and the typical
deviation cxL
353
The sample mean xn is, as we know, an estimator/ X centered,
of the population meanPx . In fact:
-
x1 I xp 1 ... 1 x n
With x =
n n

354
x1 1 x2 1 ... .! x
E(^ ) - E(Xn) = E( "1 = 2 .[E(xl) 1 E(x2) .! ... 5
TK - n n
In the case of the sample typical deviatioii Sxn, this
A
estimatorcx, is not centered in the typical population deviation
vX and we will tkrefore try and find
&, , which
depending on n, make - 3
5Lxn- 1 .
En En En -- n n
be an estimator centered on x.
Accordingly, it must be verified that:
Where 5 is the Gumbel generical aleatory variable with
dis tribut ion function
-d.(X - u)
-e
F(x) = e
(d7 O) (-QiX,C .!a)
Let us consider the variable 7=a.('5- u) In other words:y=o

with' Sxn =
n
- n
Considering Yn as estimator of /"y , we get:
E($,) =-O(.E(X ) - d. u = .(PX - 1.1) = c
n
To calculate E (S ) , samples (y1, y2 .. . y,) are
Yn
formed, with extension n of the aleatory variable 0 , whose
distribution function is +(y) = Proh( ,r y) = - e-Y
e 7
(reduced Gumbel distribution o(= 1 u = O).
Thus, for the simulation procedure , an aleatory number
zi of the rectangular distribution (0,l) is formed, and
making
-Yi
z.= e-e or in other words:
1
Yi = - I+LZi) one obtains a value yi of the aleatory
variable 12 .
Having fixed the value of n (extension of the sample) and
obtained k samples of extension n, with sufficiently large k,
we will get:
355

Number of
sample
356
Number of
extension n
Sample mean
1 -
€(y 1 = c = 0'5772 -. .% Yni - Y,* 1 ... Y,!;
n li
?'lie value of En will be:
n
-
= Y,
and a centered estimation ofpx and TX will be:
ci ,XT 1 x2 .L ... L
'n - -
Px = Il - xn
Typical sample desviation

Let us see another procedure to estimate the typical
population deviation G X.
x,).
357
Let us consider the extension sample n : (xi x2 ... xi ...
Let us take a smaller to larger scheduling of the form:
kl< "6 . . . ..(Xi& * * sk,
Cons i der ing the "quas i - ranges" :
ox =? n -xl=x rnáx - Xmin = range
rix = 5
n-1 - '2
rZx = - >i3
and in general:
I r
hx = %-h
If we have obtained k samples of extension n, by
simulation in the Gumbel reduced Taw, and in each of
them the "quasi-range" r , we will get:
11Y

358
Let ? nh be the coefficient - function n - such that:
Therefore:E(rhy--)
o( .?nh
Arid thus
Pnh =
In this case a centered estimation of pr and G, would be:
L As 1 ... -L x -
* = x
n n
Concluding, we can say that, given a sample of extension n
(Xi x2 ... Xn ):
1) A centered estimation ofrx is:
I x2
-
1 ... 1 xn -
= x
n
n
2) Centered estimations of 6 y are:
Precisely as both estimators ox, and b, are centered
inKx , in each case it would be convenient to take the one
whose variance is less, in other words, where:
h

we get:
or in other words: :
the variation coefficients of S and r respectively,
Yn hY
the above expressions stands as follows:
And therefore,
Within the "quasi-ranges" rhx, as all the quotients &
Prix
are centered estimators of cx and since r = H.rhx,
hY
we get:
359

360
For two particular values hl and h2 of h, we get:
and therefore:
One concludes, in all cases, that the centered estimator
of Q to be used between SX, I
D Or, ____ 'hlx
En qnh 1
r
or, e will be the one for which the corresponding
variation coefficient V(S ) or V(rhly) or V (r h2Y ) is less.
Yn
With these grounds and criteria, we have obtained the following
results for k = 20.000 samples of extension n, by statistical
simulation of samples of Gumbel reduced law values:

-,- i
I
.
. . . .
....... -1..
. __
i ¡ , . -4
!
. I /
. .- .~ - __ .
.
361

U o
o
U
VI
l.3
U O
m
E
.rl
U
VI
2
a
o
VI
O
c) s ci
E
ci x
> o
TI
,-i
d
u
.FI
.d d
bX
- .d O
U
u
VI
O
3
rl
o
a
r(
5
I

According to the above, and bearing inmind that:
+ 3 I?

fi
F)
C
C
a
364

ABSTRACT
STOCHASTIC SIMULATION FOR BASINS WITH SORT
OR NO RECORDS OF STREAMFLOW
by M. E. Moss and D. R. Dawdy
U.S. Geological Survey, Washington, D.C., USA
Stochastic modeling of streamflows is a powerful tool
in water resources systems desing. Statistics for simulation
which are based on short records may be highly uncertain. A
method is presented for the development of a stochastic model
for an ungaged site. The means and variances of the monthly
streamflows can be based on regional estimates or on physical
characteristics of the basin. The autocorrelation structure
is based on rainfall records and physical characteristics of
the basin alone. Application of the method to the design of a
reservoir is presented. A comparison is made with a reservoir
design based on the recorded flows at the site.
RESUMEN
Los modelos estocásticos son una buena herramienta para
el estudio de los recursos hidráulicos pero los métodos de SL
mulación basados en series de pequeña extensión son muy peli-
grosos se presenta un método de modelo estocástico en el cual
la media y la varianza de los valores mensuales de caudal se
obtienen por comparación a escala regional de las caracteris-
ticas físicas de las cuencas.
La estructura de autocorrelación se basa en los valores
de precipitación y características físicas de una sola cuenca
se aplica este método al dimensionamiento de un embalse y se
compara con los valores que se obtienen a partir de los cauda
les medidos en el emplazamiento de la presa.

366
Introduction
In many parts of the world there are little or no streamflow da
ta. Even in areas where there is a relatively good set of streamflow
data, projects for water resources development are desired for sites
where the data do not exist. Oftentimes regional relations are developed
to interpolate or, more rarely, to extrapolate streamflow characteristics
to ungaged sites. The less data there are, the less
accurate are the regional relations based on those data, However, re
gional relations often are the only basis for design. Thus, the mean
flow, variance of flow, and other statistical characteristics may be
related to drainage area, mean rainfall, or other physical basin mea
sures. For instance, in the United States of America, multiple regre
ssion relations are developed i .e,, c1) fram which can be computed
mean flows and variances of flows for each month in the year, as
well as for the total year.
Stochastic simulation of streamflow is coming into widespread
use for project design (2). The statistics required for stochastic
simulation usually are based upon streamflow records collected at or
near the project site. The accuracy of the statistics estimated for
a stochastic model are deteymined by the length of the streamflow re
cords, Therefore, the regional relations mentioned earlier are a tool
for supplementing the data base for stochastic simulation, In fact,
a reional relation may be superior to records collected at the site
for estimation of some statistical parameters (3). For data scarce
areas regional relations may be the primary source for such estima-
tes.
Stochastic simulation models require a knowledge of and
estimation of the persistence of streamflow. By persistence is
meant the degree to which streamflow today affects streamflows
in the future. This is sometimes called a carry-over effect.
For the first order autoregressive models often used for stream-
flow simulation, the first order autoregressive coefficient is
sufficient to describe persistence. For more complex models,
other measures of persistence may be needed. However, experi-
ence to date indicates that regionalization methods currently
used are inadequate for the statistical estimation of persistence
characteristics 141. This is partly because they are strongly
dependent upon subsurface geology, for which no simple regionali-
zation techniques have been developed. Therefore, a methodology
is needed which can be used to estimate the correlation structure
of streamflow sequences. This correlation structure could then
be combined with the regional relations to develop the parameters
for streamflow simulation models for use for project design in
data-scarce areas.
-

Method of Approach
367
Model choice and parameter estimation are the keys to
effective use of stochastic streamflow sequences in hydrologic
designs such as that of sizing a reservoir. The mixed-autore-
gressive-moving-average (ARMA) model described by Moss [5]
provides a scheme by which parameter estimation may be performed
with a minimum of hydrologic data. It is a model that preserves
many of the statistical characteristics that are commonly associ-
ated with streamflow sequences. ARMA models can be developed
that are covariance stationary and that preserve the memory of
the streamflow process for longer periods than do the more com-
monly used autoregressive models 161. A scheme with such
attributes would seem to lend itself to the development of
data for design decisions in those cases where actual hydrologic
data are too few to provide adequate solutions.
A first order ARMA model for streamflow may be defined as
M n = a M + b Pn-l + c Pn
n- 1
where M and P are the streamflow and effective precipitation,
n
respectively, for the nth time interval and a, b, and c are
coefficients that are related to the basin characteristics.
Moss 151 has shown that, if the baseflow from a basin behaves
as a linear reservoir, that is
-kt
Qt = e QO
where Q is discharge at time, t, and k is a constant, related to
the geoiogy and measuring the streamf low recession rate , the ARMA
parameters can be evaluated as
-k
a = e (3) I
k (Tn-l)
c i l - r e
n
and where r is the ratio of infiltration to effective precipi-
tation and qn is a measure of the time distribution of effective
precipitation during the nth time interval. T is in reality a
random variable, so that the ARMA model does n8t strictly des-
cribe the streamflow process. However because T is usually
restricted in its variations, the ARMA approximaeion may still
be useful.
(5) ,

368
In many instances meteorologic information either in the
form of raw-data or regional relations, such as maps, is avail-
able where hydrologic data are not. Under such circumstances
the parameters b and c can be defined as the expected values of
equations 4 and 5, and the meteorologic information can be used
to fit parameters to the ARMA model in order to generate syn-
thetic 5treamflow sequences. Parameters b and c were defined
in this study by assuming the T for each month was a uniform
random variable with a range from zero to one. These sequences
can be used in design procedures, such as the sequent-peak
algorithm of Thomas [71, in the same manner as actual observed
discharge records. The model is not a perfect transfer mechanism,
however, and the resulting designs will contain modeling errors
in addition to the time-sampling errors that are inherent in the
meteorologic data. Similar time-sampling errors would be inclu,ded
in actual streamflow records were they available. Judgement of
the model as a design tool should be relative to the best avail-
able alternative because of exact design methodology does not
exist.
If the ARMA model is used in the manner described above as
a monthly streamflow generator, the statistical moments of
streamflow that will be preserved in the long run can be extracted
from equation 1. The resulting correlation structure has been
described by Moss [51. For the mean monthly streamflows a series
of twelve linear equations is required:
EIMnI = aEIMn-ll + bEIPn-lJ + cEIPnl, n = 2,12;
12 12
E[Mn] = c E[Pn]
n= 1 n= 1
where E[-] is the expected value or average of the variable con-
tained within the brackets. Similarly, for the variance and
covariances of monthly streamflow
Var [M ] = a2 Var IEln,ll + (b2 + 2abc) [Var Pn-ll
n
where Var 1.1 is the variance of the variable contained within
the brackets, and

= 1,12 i y = k-2-ke-2k- 2e -2k + 4e-k ] / Sk2
369
The coefficient 0.13 that appears In equation 8 was defined
empirically by Monte Carlo methods by MOSS i51. The solutions of
these three sets of equations, although not necessary for the
implementation of the model, yield estimates of the streamflow
characteristics that can be examined for reasonableness prior
to the design step.
Data requirements for testing the model
In order to test the model in a realistic design procedure,
an existing 58-year streamflow record for the Toccoa River ne'ar
Dial, Georgia, USA, was routed through a sequent peak algorithm
to determine the reservoir capacity that would be required to
meet the monthly water demands shown in figure 1. The demands
described in figure 1 are hypothetical, but they vary seasonally
in a realistic manner. The average demand is about fifty percent
of the average streamflow, estimated from the existing record,
for the Toccoa-Dial site.
The ARMA model was subsequently used to generate 50 equally
likely sequences of 58 years of monthly streamflow. For each
synthetic sequence a reservoir capacity, which could be compared
with that defined by the actual record, was determined. Precipi-
tation records from an existing station, Blue Ridge Dam, that is
approximately 5 miles downstream from the streamgaging station
were used in conjunction with Thornthwaite 181 estimates of
evapotranspiration to estimate the mean monthly effective precipi-
tation for each month. The standard deviations of monthly pre-
cipitation for each month were assumed to be equal to those of
the measured precipitation at the Blue Ridge Dam site. The use
of variance of point precipitation as a measure of variance of
precipitation over the basin tends to ovesestimate this parameter;
however, because effective precipitation, which is the difference
between precipitation and evapotranspiration, probably has a
higher variance than precipitation, the assumption of variance
of point precipitation as a surrogate for variance of basin-wide
effective precipitation should not be unreasonable. The twelve
estimates of mean effective precipitation and the twelve esti-
mates of standard deviation of effective precipitation in
conjunction with the assumption of log normality of effective
precipitation were used to synthesize the 58-year records of
effective precipitation, which are converted to synthetic stream-
flow records by the use of equation 1.
Fitting of Parameters
For the application of equations 6-8 to convert an effective
rainfall to a runoff record, four inputs are necessary. First,
of course, is the record of rainfall itself. Second, rainfall
must be converted to rainfall excess by abstracting losses.

370
This was done for the Toccoa basin through use of the Thornthwaite
equation to estimate evapotranspiration. The Thornthwaite
equation was chosen for reasons of simplicity. If adequate data
were available a more accurate estimation might be made, such as
by the Penman formula. Third, the separation of effective precipitation
into direct runoff and infiltration must be performed
by selecting r . Studies by the Tennessee Valley Authority
n
(Eklund, C. D., oral communication) indicate an average value
of approximately 0.7 for r. This value was used for each month.
Stochastic Simulation Results
The utility of the methodology for deriving parameters for
stochastic generation of streamflow was tested next. The parameters
shown in figures 2-4 were used with an ARMA model to generate 50
synthetic sequences of 58 years in length, the same length as
the historical streamflow record. The sequent peak algorithm
was then used with the demand curve of figure 1 to generate
design reservoirs for each of the 50 sequences. The 50 design
reservoirs for the synthetic sequences were then arrayed into
a probability distriaition, as shown on figure 5. The distri-
bution of design sizes is approyirnately normal. The mean value
of the design size is fortuitously close to that size which was
based upon the recorded flows.
The apparent reason for the excellent agreement between
average simulated and recorded design sizes probably results
somewhat from compensating errors. The design storage is
equivalent to about one-seventh of the mean annual flow. The
excesses of demand over supply occur mainly during August to
October. During that critical period the model overestimates
both mean flow (figure 2), which tends to decrease storage
requirement, and variability of flow, which tends to increase
storage requirement (figure 3). The covariance structure is
closely reproduced.
Figure 5 indicates that the stochastic simulation is rela-
tively realistic with respect to reservoir design size, which
is related to the variance and the correlation structure of
flows. The fact that the actual record and the average of the
synthetic records yield about the same design size indicates
that the structure of the runoff series is maintained adequately.
Therefore, in data-scarce areas, this approach may be a tool for
use in project design.

1.
2.
3.
4.
5.
6.
7.
8.
References
371
Carter, R.F., (1970). Evaluation of the surface water data
program in Georgia, U.S. Geol. Survey open-file report.
Fiering, M.B., and Jackson, B.B., (1971). Synthetic
Streamfiows, Water Resources Monograph No. 1, American
Geophysical Union, 98 p.
Ber:son, M.A. , and Matalas, N.C., (1967). Synthetic Hydrology
based on regional statistical paramerers, Water Resources
Research, 3(4), pp. 931-935.
Thomas, D.M., and Benson, M.A., (1970). Generalization of
streamflow characteristics from drainage-basin character-
istics, U.S. Geol. Survey Water Supply Paper 1975, 55 p.
MOSS, M.E., (1972). Serial-Correlation Structure of Discre-
tized Streamflow, U.S. Geol. Survey open-file report.
O'Connell, P.E., (1971). A simple stochastic modeling of
Hurst's law, Symposium on Mathematical Models in Hydrology,
IASH/UNESCO, Warsaw, Poland.
Fiering, M.B., (1967). Streamflow synthesis, Cambridge,
Harvard Univ. Press, p. 69-73.
Veihmeyer, F.J., (1964). Evapotranspiration, in Handbook of
applied hydrology (edited by V. T. Chow), New York, McGraw-
Hill CO., p. 11-26.

372
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Figure 1. -- Monthly water demand.

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P,

ABSTRACT
CHCICE OF GENERATING MECHANISM IN SYNTHETIC
HYDROLOGY WITH INADEQUATE DATA
by
P.E. O'CONNELL
Department of Civil Engineering, Imperial College,
University of London
and
J.R. WALLIS
IBM T.J. Watson Research Center, Yorktown Heights,
New York 10598
A formidable problem in synthetic hydrology is the choice of a
model which will best represent the generating mechanism of
streamflow, which is unknown. Heretofore, such a choice has primarily
been based on a statistical matching between historic record parameters
and the parameters of the generating mechanism. However, for
short historic records, an equally good match may be obtained for a
number of models and statistical tests are not powerful enough to
determine the appropriate model. Alternative mechanisms, however, may
yield quite different design results, resulting in either overdesign
or underdesign with economic regrets in either case. It is suggested
that an alternative approach to model choice would be a decision
theoretic approach, where the choice of model is based on an economic
regret function, and where the model which gives the minimum overall
regrets would be the appropriate choice. An example of this approach
is given where flows are generated by a lag-one Markoj and an ARI A
(1, O, 11 process and the secuent peak algorithm is u ilised in a
deterministic sense for reservoir design together with simple economic
regret functions.
RESUME
Un problème trzs difficile de lrhydrologie synthetique crest
ltadoption d'un modele lequel reprgsente le mieux le mecanisme gênétatrice
de l'écoulement, lequel est inconnu. Jusqu'ici ce choix été
fondé principalement sur un assortiment des parametres des données
historiques avec des parametres du mécanisme génératrice, Toutefois,
en cas des relevés historiques courts on obtient peutêtre un assortiment
aussi bien en employant plusieurs modeles et les épreuves statistiques
n'onts pas assez fortes pour détermin'e la modèle propre.
Toutefois, les autres mécanismes donnent peutêtre les touts autres
résultats pour dessein, le rêsultat est le sur-dessein ou le sous-
-dessein accompanid en tout cas pap des regrets economiques., On prop'ose
que l'approche alternatif au choix de la modèle est llqpproche de
la théorie des decisions par lequel le choix de la modèle est fond6
sur une fonction des regrets economiques et par lequel la modêle qui
fourni les moins regrets totals sera le choix propre. Un exemple de
cette approche est fourni quand les écoulements sont produi't par un
procéds Markovien de deu1 retard et un proc6dé ASEVA 0, O, 12 et
l'algorithme des pics successifs est utilisé au sens déterministique
pour le dessein du réservoir avec des fonctions de regrets economiques
egales.
Y

378
Introduction
The ootid desig, of a water resource EF:E~~Z reqzres 'slowledge of
future flows within t h elstem, which, i- tu--r., ITlies tkat :Be gcze-xting
process of the tows is kofi. Eowever, the gtrerazirg Srocess of srreamflow
is gererallg &om, 2.~5 lilirelv projectiors 05 %hre zlous, callec -T-thetic
strezzflows, may be gererated using approxic-:iozs to tbe UrCsrlyirg gereratirg
process.
(a)
(b)
The apFros3ation procedure involves. -
the postulation of t'ne underlybg generatirg Frocess ar6 its specification
through a set of przeters,
the estimtion of the parameter values from a historic sequence, or
through some alterrative strategy.
Some generating processes cvreztly availatle, proFrties of historic
secperces, and techniques of paraneter estinztio: inU now be considered
briefly.
Gener2tir.g Trocosses
The postulation of a generatirg process kas 'reretofore been b e d on its
ability to generste sjrthetic stremÎlows resez5lirg historic streznflows in
terms of pärameters whic'n are thougit to inflilerce the äesie OÏ the water
resorce system, [I), which necessitates that tte 3rocess re3reserts a redistic
model of streamflow. The lag-oze ?%rkov process 'caä 5ee3 rather widely
accepted as beizg ca-pble of fulfillizg this l2:ter role mtll discrete time
fractional Gaussiul noise (dfGn) was aävocateä as s more resiistic moäel of
strezÏlow [2]. The spent for dfG3 as a ereriting process oÏ streãmflow
finds its roots in the work of Hurst [:I, [4f u50 foud that, for some 800
geopkrsical tine series, including streanflow
- - 1 1
S
h
where Iz/S is temed the rescaled ra^ge a d n is th-e record ler-gth. The
expone'it h in equation (1) was found to have a a7e:age value OÏ 0.73 uith a
stank-d deviati02 of O.Ca. For the lag-oae 3k-207 process =fi other processes
lying withiri the BroEis Comain of attractioz, h equals 0.5, uhile for dfûn,
h q v assume ayy value in the rGge C < h < 1, vit2 the excepriori of h = 0.5,
and b C.5 of particular interesi. Values o? h > C.5 are
s~-no;nous with long te,- srcistexe, vith the distat psr; exertizg mall
but azable effects on present behaviour.
The gelieration of a Lszmple of d-Gn req-aires infixite Ember of operatios,
azd, con~equeztk~, a~~roxinaiio?s are reTïred ir, oräer to Io-date
dfG2 =s an operational gsreratkg process. Tses? assro-xkatioïs w e quite
:oces of szem3ou dictirct fron the agroe=z:ioz or' t're gezerz;irn - _
refezred to Tr3rLouslr ; -&e a~~z-o.sirsziors 10 è3z äm -2$rss17es sx5sequeitly
used to approxicsite the gererating srocess of sirsmflou.

To date, a =.aber of z_nproximtions to d3n ha7e been >-o?ored. Y%zdelbrot
i53 kzs proposed B fast d% approximatiori, wLich requires less ore~atiozs ;CI
elierzte t?mn the tyge I ard II fractiord noise ap?roxinntio-s Frooosed ;-;tially
LJ , Cut he did rot extend the docuieentäïion to the level iecesszyy for syr-tetic
i~y.gdrolo~. &,ser5 02 the tne II ayproxibtio: Tropsed initidly by Yzlldelkot
aLd Ydis [o], Y'tzhs =à Y&is [7] L57e pro-osed a filtered ,'ractionzl
noise asproximatioil, and have formulated the gezeratizg process for the puTses of sgzthetic hydrology in terns of the ppiation rea, vzriarce, lag-or-e exiocorrelation
u-d h, the Hust coefficienz. The KXP! (l,O,l) process [S) IFS
beeo found by O'CozeU [g] to offer a sirple approfixtion to dSs. The a ~roxicatioi
is sufficiect in the serse, that, for a certais rage of pûzameter Taues
adecute agreemerc uith Burst's law (e-tion (1)) is obtaired uithin sequezces
which are sufficiel;tlg lorg for the purooses of syctììetic hyckolog. More
recently, the broke2 line orocess has been progosed by Kejia et zl [IGJ as e
model of the gezorati-g process of strezflow ; houever, ì-kdelbrot [Il] hzs
shorn that the broke2 line process may be regzzded nìerely as zz ¿?.;prOxiuEtiGZ
to fractional Gaussian noise.
Historic Seauences
Frequently, a historic seauence of a-~uai streamflow has been relied upr
to determine the existence 9. aon-existence of persistence thereir. For this
pu-?pose, tests of significbnce for inde-zdence zre available bâsed on the
theory of runs [12], LI31 ar-d the distrikation of the 12 -one serial correlation
coefficient [1q. However, Wallis a d Patalas [75! have fomd that for
a variety of such tests, the probability of type II error (i.e. the probability
of accepting the rull hypothesis of independeme when it is false) is extrerely
high for the sequexe lengths usually avzilable in hydrology.
of persistence, estimates oi the lag-one autocorrelation tend to be biased
towards zero, with the bias increasing with the intensity of the persistence,
and to this latter Îact may be ascribed o=e of the reasons for the low power
of the Acderson test. Where storage design is to be considered, the econorcic
cor:sequences of ty-e II errors may welì be costly. Assuming that evidence of
persistexe has Leer established, the hisioric sequence h s generdly been
relied upon to &-Ovide reliable evidence as to uhich ge2eratiig Eechanism is
the appropriate ore for the flows. A model of the mderlyicg geserating
necliacism is fitted EO the historic sequence, and goodness of fit testS.De
then enployed to àetermine the adeauacy 05 the model. A set of procedures for
model €itti% ami vdidatios &ve been set out by Box axd Je-&ir?s [SI; however,
while s ~ch procedees may provide reliable resdts for the locger recorded
sequerces U S ~ Y
In the presezce
available in industry E d economics, they are liable to
pyovide misleadkg results for 'short' mual strearflow sequences. A question
mises as to what length of record pay be considered 'short' ; uithout digressing
too much, i; suflices to say th-t as 1or;g-terrn persisterce increases,
the hfornztion coztent of a record [IÓ] decre- =ses.
While the kg-oze autocorrelation has bee? used in the _past as a measure
of Fereistence, it essentially measures ody &ort-te,ri -ersistexce, ad, to
quzzitify long-term Frsistezce, other messures must be used. While h, the
Emst coeîficierit, is a measure of long term -xrsistezce, estimates from sms-ll
sansles gererated a lag oie Karkov process c d arrolaoatiox zo dftrri havc
beer sko~?: to be hi- biased and extrezrly vz-iable, 1V++ a d therefore II-relisbible
for chooaì=& betmees short memoLT processes for ubi@ h = 0.5 and 10%
379

380
memory processes, for which h > 0.5. Xore receLtly, Wallis ard O'Cozell k81
have attempted to separate sequences genented by a lolig nezory LW! (l,O,l)
process from seq-Jences generated by a short meao,ry lag ore Yzrirov process
using the distribution ol US, the rescaled rmgo, deriveä t ~ough Xocte Carlo
simulrtions. For the sequelice le-gths w d l y anilable ic. hydrology, they
foud that reliable sepration was not possible.
Historic sequences have bee:: suggested by Slack 1191 as sometimes providir4
little more tha? ari illugoil of what the wderlyi-4 gezeratizg process is, 2c
a sizgle realization of H stochastic process will rarely 'have Ezaple estimates
of parameters e q d to their populatiozi counter-ts. Irdeed, Clack has also
noted that a gereratirg crocess mg 6ezy itself ir the seXe that the process
may generate firite sänylrs to wnich the gezeratirg process itself W o t be
fitted. Slack 123J has illustrated this point more fully for a miltivaiate
lag-ore Markov process, for which certain constraicts exist OP the raqe of
serial and cross correlations that the process ca3 acceyt.
Coxequectly, the
fact that a historic sequence yields prameter estimates macceptable to a
model cannot be readily taken as evidence that the model is an inappropriate
one.
For the univariate lag-one Markov process, the question of 'self denial'
does rot arise as estiusates of pl, the lag one aitocorrelation, will always
lie in the range -1 < e, < 1, for uhich values of the process is stationary,
and flous will be real-vdued. However, for dfGn, waich for zero mean and unit
variace, is characterized by the covariance structure
where s is the lag and h is the Hurst coefficient, ssmple statistics be
in conflict with the covariance structure of the nodel. Approximations to dfGn
serve to approximate the covariance structure C(s,h), where C(l,h) is uniquely
specified by h. As the pocess is specified bhre unit varizce, then C(l,h)
is the lag one serial correlztion, Q,, which for li > 0.5 must be positive, as
must C(s,h) for s > 1.
Eowever, finite sequences from a gezerating proce?
with a covariance structure approGnating C(s,k! rzq yield h > 0.5 unile QI < O,
or, alternatively, h < 0.5 while el > O, which are incopatible with the
structure of the model. However, the dari model isI\employed primily to
preserve 102% ruII effects 2.s evidenced by values of h > 9.5, a d the evidence
supplied bx QI, which is a measure of short run effects, nag be ignored.
Provided QI > C and h > û.5, the filtered type II ayproxioratioil proposed by
Matalss and Wallis [7] zlloïs the sinultarieous :reservation oÎ estimates of
QI azd h through the ixorporation of an extrs filterizg yameter into the
generating process. As a result, the form of the covariace Îwction for dfGn
not be closely followeä for small s- while for large s, the differerce
shoulà be negligible.
The possibility of 'self-denial' eests for the ARIEiA (1 ,O,l) process.
In or6er to ensure that the process is statiore? ar6 invertible, t)ïe Lwple
space for the paneters of the pzocess, azd a, skown i2 figure (la) is
defired as -1 < # < +I, -1 < 8 < +I. The corres-rciirg Lssrifle -ce for QI
and e2 is sho.ir.11 iri figre (13). Hoïever, fizite -les froa the process
may well yield estimtes oÎ pl u.d e2 lyhg ouiside the zcceptabìe rage.

3 81
As a remit, a historic sequence m o t be relied u-pn to -est the
correct generatkg process for the flous, 2nd ET well lezd to 2 moàel beicg
selected uhich is not reyesentative of the gereratixg rec'mim of streamflou,
Houever, parameter estication t hrow statistics zes-mred is. .zi historic
sequerce is lugely relied upon for matching a gezeraticg process n th a his-
toric streamflom sequence.
Parezeter Estimtion
hs already ooted, a geuerating process is geaerally specified by a set
of population Faeters, denoted as {a] = (aq,u2, ... aE), estimtes of uhich
must be obtained from a historic sequezce before tie generating process may be
rendered operatioral. For the lag-ore Ykrkov process, the set { a 1 usu2i.l~
comprises the me=, vziriaxe, skeuness a d lag-ore autocorrelation 2t each
site, and in the ttiuìtisite case, lag-zero cross-correlatioss betveeri sites.
For qproximatiors to dL%, sn additiorial parmeter, the Hurst coefficieat,
is izcluded. For estimztioa purposes, the mettod of moments is gezErallg
emploFed [I ] with the L.mall sample moment estimate of a parameter, a$, obtained
from a historic sequence or' length n bei% equated to its correspondxg pop dation parameter in the generating process, ai. Such a procedure assumes
that
E [",3 = a
n
i.e. that the estimate ai is statistically unbiased. However, in the presence
of persistence, recent sicdies have shown that this assumption is not justified
with respect to estimates of the variance [21], leg-one autocorrelation [I51
and the Hurst coefficient [I?].
The bias affects the resemblance uhich is
obtei2ed between historic axxi synthetic sequences, and is likely to adversely
influence system desigri Mess some small sample bias corrections are applied
to the estimates. Io order to illustrate this liztter point, the small sample
properties of generatiw processes rnust be considered.
Small Sample Prouerties o0 Ge-eleratirE Processes
!The lag-one biarkov generating process m y be specified as
where p 6 and p are the pophtion mean, variance and lag-one autocorrelation
coefficient, and et is an edependectly distributed random nomal variable
with zero mean ard unit vâriance. Estimation of the paraneters e, 6 and p
will cou be considered.
In equation (2), e may be estimated as the lag-one serial correlation
coefficient uhence,

382
ñouever, avaïL&le estkators of pl yield biased estimates of e [UJ. If
the estimator oÏ QI suggested by Box a d Je_nkins is used, then e approximately
satisfies
For n = 25 and e= 0.3, aen E[@] = 0.21. If equation (3) is rearranged then
E@] + l/n
e =
(4)
1 - 4/n
A
If e is obtahed :rom a historic sequence of size n using the Box and Jenkins
algorithm, and E{Q] is replaced by 6 in equation (4) then the ensuirg
Ii
estizate, e *, hill be Enroximately urtbissed. If Q is used in equation (21,
then estimates of the lzg-o?e autocorrelation measured in synthetic sequences
of size u, e , will satisfy
while, if the length of a synthetic sequence approaches infinity, then
i.e.
with
where
the proper resemblance ia maintained between historic and sythetic sequences
respect to the lag oce autocorrelation.
then while
2
If the small sample estimate of the variance, 6 , is defined as
'> n
s2 = (Xt -1) 2
n- 1
t=l
2
If g = O, then E {s 1 = 62 uhile if Q > O then fh, Q> is positive, uhereby
s2 terds to underestimate 8, with th2 bias iilcreasing as e ixreases afd
n decreases. For LL = 25 c d = 0.5, f(n,Q> = C.963, SO tkt the bias Kill
ger-erdly not be too severe for m-ual streamflou sequences. 3 order to correct
for the bias in E? mezs-zed in a historic sequence, a unbiased estimate oÎ the
populztion variszce 2 %,y be defined as
a2 = s2/f(n,p (8)
A
(5)

vhereupon
E{$] = E{s2J/f(n,,) = 6 2
"2
so ttat 6 wiii be unbised. Hoïever, the foregoing correction- procedure pre-
m e s that e is Lcco~",, &de, k practice, od? ar estfate, 8 , rill be
avzïilzble. .In tkis sitti-tiori =y be corrected for tis ii6 &ea* outlked
and ar: estimate of the vzrkce del'bed as
^62 = s2/f(n, e*> (9)
and that
h
may be evaluated at the expected &lue of Q' which yields
then
Hvïever, neither of the above two assu+ions are liable to hold, c d a2 a
result, difficulty is eilcoutered in definirg irnbiaseci estimate of 6 .
Nevertheless, equation (9) is likely to yield FS estimzte of & which is more
appoxinstely wbissed than the straightformd estimate yielded by eqwtiozi
(51 9
383
A''
Using p, 6 as defired in eauation (9) a d e* as defined in equation (41,
a synthetic seculice OZ size n nay be generated using eywitiol? (2). If g2
denotes aa esthte of the variaxe mewed tkerein usi% equation (5) then :-
2
E($) a s
while if the leqth of a synthetic sequexe aPyoaches kfinity, then aporoxii_
matelg :
E@> -6
2

3 84
A2
where IS is defired via equation (9). Hence tFe correction procedure allows
the growr resenMance betueen historic and synthetic sequences to be ubzirtzhed
aporoemtely.
A further quentie uKch may be of interest in rece--voir desigil studies
is the variance of the sanple mea, 8, uhich for equatiori (2) is given as [2$] :-
nhkh is the variace of the sample mean for an indeperdont rmdom procefis.
However, foz Q > O, the term in braces is positive and greater than unity,
vhereupon 6 will be larger t h for a randon time series. Hence if the iaformation
contert of a sequezce is defized as the reciproczi of the vdance of
the -?le mean [16], thea, as persisteme increases, tke information coatent
relative to the mean decreases. For e= 0.3 aLd n = 25
6m
= 0.0724 cr2
Nevertheless, it should be noted that the lag-one Y!kov process is
esseritially a short memory process for which h = 0.5. Ir. the preseme of locg
term persistence, when 0.5 < h < 1, smdì sam?le biases in estimates of the
variarce ar,d lag-one autocorrelation become more severe. and the vaziame of
the sample mean-tends to i-crease. For dan, the variarce of the szmple mean
is given as
2
2
-- 6
(12)
- 2-2h
n
2
where IS is the variance
result for white noise.
2
of the process. For b = 0.5, 6 reduces
For h = 0.7, for which el = 0.30, and n
3.6246 2
- - = 0.145 6 2
- 25
to the
= 25
Comprison of equations (11) and (12) illustrstes that for dsz, and, consecuentlg,
for agroximitio-s thereto, estimtes of -the -?le me= are auch more varizble
and meliable, than for the lag-one I.'!kov &ort memoq process.
Unfortunately, little is b o m of the s-3u sample u-operties of the aoproximatiocs
to dan proposed bp Mandelbrot 151, ?!!'das ar0 Kdis [7] and Xejia
et al; [IO], and equatiog (12) will only be apFoximatelg true. An -tic
derivation of such proprties would apear to Se a extreselg &iÎficult tz&
in the face of the comaex mthemtical prooerties of tle approdmations. €?QU-
ever; tlie LRDiA (I,O,I) p-ocess is mathemticdly more tractable and some scull,
e l e results s v be derived.

The ABMB (l,O,l) generaticg process is &fined as
where p and 6 are the nean and sLa&d deviatioa, $ zrd Q are the parameters
of t'ne process, E d qt is u irdeperclent rzdom varille. ne varhce o? /7
must be defined 2s
var 3, = +%E&- I+&- L
Estirates of the parameters jZf 2nd O ~7 be dertved from 2 historic, sequence
throqh estinatirg el z d p2, the hg-0r.e =à lag-two zutocorrelation coefî-
icierts, or tho;@ mir: the more efficient rethod of -&m likelihood. [8]
Alterzztively, e!, and f my be used to defire estimtes of j? and 8. For
aporoximatiors to dr%, estimates of h Lid e 1 Lime bee= &om to be bizsed,
As h is not
vith tke SiEs kcreaskg kith ircresLzg h 2zd el I [17, 151.
385
theoretically de5red ai diff3rer.t from 0.5 for the AXPA (l,O,l) process,
which nevertheless yieldsAE fh] in the raxge C.5 to 1 for moderateJO large
values of n, the bias iil h is difficult to Ruztify. E e bias in el could
possibly be defired ardflim3.ly.k aAsinily fâshion to tkat for the lag-ore
Markov process. Even if the b is in h zid could be defired, the appropiate
bias correctiors m y not be copatiblz [19]. xri alterzative aggroach is to
approximately. rztch observed Qq a d h values iri a =>le of size n with the
aprro3riate Z {el] ard E 17 values for the ARRIT2 (l,O,l) process pre-
defiEed through exteEsive Monte Carlo simulatioris [25]. This approach
obviates the necessity lor bias corrections, but the estiptor of h adopted
for the simulatios, giyen ifi [4),
n
h = K = {hg. (R/S)] / [Log "/2 1 (14)
does not allow sufficient variability in E{ ^h] between afferent ,sets of para-
meter values to effect a reliable match.
If the snail sample estimate of the variace is defined as io equation (51,
then O'Connell 12.1 !ES &om that
2
E(s,) = 6
where pl is the lag-oze autocorrelation defined as
= (1 - $QI($ - 6)
(1 + g2 - a)
(15)
(16)
0'CoI;ileI.l [g] 0-r noted tbt vdues of $ in the rmge 2.50 < j? < 0.95 are of
interest i2 modelling lox te-ai._rersislezìce ; for suc3 values of $ m d for
values of el us-:âïïy ercourtered for amua1 szreaflo*', the bias in s2 is
gererqy h-ge. for L = 25, Q
= 0.3 and #.= 0.85, ?(E, e:, $1 = 0.877,
while, if $ = C.95, f(r, p?,$.> = 0.789. For fixed QI, sz ixrease in j?!
reFGesezts ax +crease ia the iitersity of 10%-term persisterce, as eviderceà

386
by higiler observed values of h, the Hurst coefficient [ 91.
As for the kg-orie Yarkov orocess, zn =biased estinate of ci
2
,
42
6 , may be
defined, a s d g that $ acd 8, or, equivdently, J?f azd QI, are boni
However, in practice, o- estimates of e ard Ø will be available, and a
sinilzr problem to that encountered for the la- one Ymkov process is met in
attemptirg to defize an unbiased estimate of I?:
-
The variance of the sample mean, X for the ARIFA (l,O,l) process i$ given
1253
For el = !¿f, for u ~ c h the ARPIA (l,O,l) process reduces to the lag-one Markov
process, equatiori (18) reduces to equation (11). Values of corresponding
to large values oÏ j! reflect the low frequencies inherent in an approxiFtion
For $ = c.85, PI = 0.3 and n = 25, Um2 = 0.15862 which compares
to dfûn.
trith equation (12) for dfGn with h = 0.7.
Impact of Choice of Generatiig Process on System Design
In the abseme of sufficiently lo= streamflow sequences to determine
whether or.not lozg term persistence exists for a particular stream, and,
lacking =y sound -&ysical basis for the ctoice of a generating process, consideratior
must be even to the influence of choice of generating process on
water resource system design.
Few studies to date have investigated the sersi-
tivity of system äesip to choice of gereratirg process for zmud streanflow.
Wallis and Matalas [: 211 have studied the effects of long tern persistence on
reservoir desigri through assuming prior knowledge of p and 6 2nd assessing the
impact of h on the äesign, W n g accomc of mall sample biases in estimates
of p1 a d 18 in the' Lr analysis. The reservoir desigr: Y ~ S evolved using the
sequent Fe& algorith, which was used to determine the minimum reservoir size
necessary to meet 2 specified level of demd a expressed as a proportioli o€
the observed averzge ITOU over the desi= period. For t'ûe desip considereä,
the required reservo-ir capcity was fourd to depend on the mwitudes of h azd
For equd e-cted vaues of the variance, E id), and the lag-one
el.
autocorreletion, E [e?), in desigr sequezces of length I?, and €or a > 0.80,
approxicitions to 12-m with h > 0.5 yieloed reservoir sizes coxiderably in
excess of those yielded by the lapone E%-kov process, thus e nmising the
relative impact or^ long-term acd short-term persistence on the design.
By consideri--6 the water resource -stem design process, a basis emerges
for the ckoice of a ge3eratizg process. A ge-eratirg process cas be postulated
as beiri; tbt of the real worlä,.ad an oFtimd desigz evolved on this basis.
Assuri$iors may thei be made concerzing tìe ideitity of the real world, acd
the ecected reFets accrui3g from each assmotion my be evaluated. The procedure
w be repeated for each postulated gexeratiF4 process for the real

world, with the ssuaei? generztirg process yielding the ~~ici~~m overall regrets
representkg the 251~ro~riite ckoice. A simil- strategr has Seen employed by
P?t&s acd YaXis [26J for the selection of a frequency distribution for the
evalmtion of a design flood magnitude.
3 07
A critical fictor in the choice of a gere-rating process concerrs the
ercistexe azd the intersity of long-term persistence to be mocelìed in synthetic
sequerces. Co-zequentLv, a set of si&tioz eqerimeris were evolved 25
folious to dete,-mir;e strategies n th respect to long-term persistence, z ~ d
to cetermine wke-her or rot, in the presexe of long-te,- persistence, bias
corrections reed to be ap-lied to estimtes of the varizce azd lag-one auto-
correlatisn measured in historic sequerces in o-der to obtaic realistic àesis
results.
Tuo generatkg processes uere adogted for the simulstion experiments,
the lâg-oxe Markov process represexting short-term persistence and the ARPA
(l,O,l) process represeztiq lorg-term persistelice. For the Latter process,
the izterisity of long-ten persisterce may be cottrolled by the parameter 6,
consequent-, a value OÏ Ø = 0.85 YES M e n as represeEzing a medium intensity
of lo-g-term persistence uhile a value of ff = C.95 was selected to model a
strozg intensity of 10%-te-m persistecce. Heme, the real uorid was assumed
to be lag-one hrkov or ARIKA (l,O,l) with $ = 0.85 or f = 0.95 yielding 3
possible choices for the real world, identified by indices r = l,2,3.
An o ptid design is required for each world, and a proceckire had to be
evolved for evaluating the desip, which was defined as the minimum reservoir
size necessu'y to meet 2 set of tzrget demands over the i?esig period rid,
which was taken as 100 gears. Rather than defire the tzrget demands relztive
to the samole me= of the design sequence, the demar,ds uere defined BS percect-
ages of the population meax of the r ed world so 2s to dlow the design to
reflect more fully the varizbility of the =?ïe mean mong different uorlds.
To permit a com-ison between optimal äesigr-s €or diÎferent worlds, each
world was defined to hâve population parameters p and 6 such that
and
SES2] ,* = 9
&e appro-priate values of 6 to be used i?l the gene=tiag *processes %v be
defired fron eoiutions (9) iLld (17) for the lG-one Nzkov azà ARDA (l,O,l)
processes, respeciively. The sequent ~3s.k algorithm u s use6 to evalute the
reserroir size, xxch YS defired 2s the mir?iia size slick tbt the reservoir
rum C-p at host oxe orer the Cesis, period 'cui; suc3 tkt =e target de-~ds are dwqs met [F]. %e de-d levels iàentiiied by izcices k = 1,2,3 ïere
defeea 2s 75%, 85$ ana 955 of the populatioz cean = 70, which, in przctice,

388
may redt in overdevelopmeat relative to the sample mean for a design sequexe.
ñowever, the sequent algorithm 2s originzllg formillatea by Thomas
and Errrdez 1281 czot hardle levels of develo~ent greater t h unity. In
this sitiztion, the reservoir size was Bc-i-ic defined as for the case of urder-
develo-ert usiLg 2 computer zlgorithm, with the initiâl reservoir storage
neceesr-yy to zvoFd deficiercies beilig assiuned zvailable. For each world, o otid
desi,--s sere defT2ed as the expected reoervoir size for design, sequesces of
lezgth "C-3 for kg-one autocorrelatiors e1 = 0.1, 0.3, 0.5 idectified by iräices
j = l,í!,3 and de-ds 7.5, 8.5 ard 9.5 s m e d uniform over the ciesign period.
The &!orite Carlo eweriments to be perfoned =y now be defined 2s follows.
For each choice of real world the eqected reservoir size is denoted as
E { (r, j,k)) for real world r with lag-one zutocorrelations j arid for level
of äe~elopent k, {r,j,k = l,2,3] , and is defized through repetitive smplizg
of desip sequences of lergth 100 for which equations (19) and (2û) hold for
r,j = 1,2,3.
In order to assess the effects of bias correctious on the regrets, a6 uelì
as essmptions about long-term persistence, the following Monte Czrlo samplirg
proceduzes were defked.
(1) Generate a 'historic' sequence identified by index i and length n =
50, 100 for world r with kg-one autocorrelation 2, decoted as {X I i,n,r,
(2) hssirme the hi,storic sequence is gezerated by an assumed world with index
1 = l,2,3, where the values of the index 1 refer to the same worlds as identical
values of the index r. Estimates of the meari D, variance aí!, ard lag-one zutocorrelation
PI are obtaired, at zhich juncture bias corrections may or may not
be applied to #(i,D,r,j,l) and e (i,n,r,j,l). For 1 = 2,3, knowledge of the
assumed world includes krowledge o$ the paranezer Ø(1).
A 4 fi
(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)
a desim sequence oÎ length nd = 100 is generated, whereupon, for k = '1,2,3, a
design reservoir size is evaluated, denoted as h (i,E,r,j,k,l), which is the
reservoir size for a level of develoyezt k for desiga sequence i gecerated
by a? asmed world 1. The pzraneters of the assumeä world are estimated from
a historic sequecce of length n from a real world r with lag-one autocorrelation
index j. An overciesigii or uaderdesign relative to the opti,d design for the
real uorld may the2 be defked as
Ah(i,n,r,j,k,l) = h(i,n,r,j,k,l) - E{>\ (r,j,k)] (21)
which represents a simple lines loss fìzxtion. Different scales BI and B
may be äo-ied to wsitive zzd neetive losses if necessary. A quadratic goss
function nay be del'iried through scusrirg Ah.
+ReFeat (I), (21, (3) a sufficiently lzrge number of times to enable
A (i,n,r, j ,k,l) ] , the expected positive loss, axd E {A-(i,n,r, j ,k,1)3
the absolute vdue of the eqected negaative loss, to be defined. The expected
overall loss may tiien be defined as
uhere (i,s,r,j,k,l) is abreviated to (*).

3 89
ordzr to pro9erL;;- assess *&e effects 00 an =&?tion about long term
persistence, desigs &oula be evo1;red nskg Cesign seyiecces for ïdch
E f.2) :.g for all sained uorlc?~. Hocever, even i' the bias correctioï
procedures orscussed eelier ere z3flied to +Le small -?le ectinzt-es of pl
arid 8, it uLL1 gellerally rot be ooasible to roet the rosuLrezst E [s2} yy: = 9
within desigz seGuences, if oper&50%s (1) - (3) are fozoveo. However, the
series of operztLo.oris outlir?ed .we pr-y designed to illust,rate the effect
of as-lying bis correctiors, prticularly i- &e preserre of long term persiti
terce, as wen as detercrg the impzct of Frsiatence itself on the resets.
KO woblem is encountered k geEerati=g spthetic sequences of leogth n
for a fixed value of el in
fixed í! a d QI ix the a e
process, the value of bS

390
satisfy
However, the estiaate of the variance satisfies
E {s2(i,n2))
vary with i and must therefore be considered as a random
vci25le iikich satisfies
"2 A
which if 6 (i,nl) end f(n2, e(ilnl) are assumed indepecdent,
= E [$2(i,n,)) E [ f (n2, G(i,nlJ
However, in genera,
A
E {I(%, ê(i,nl)) # f(n2, E[@ (i,n,)I I
as f(n2, @i,n ) is non-l+ezr. Howeveft, if Eff(n2..e (i,nlu is evaïEted
at the expected vdue of e(i,nq), E {e (ilni) , which, if e (i,nl) is an
unbiased estimate, equals E> , equation (2'7) reduces to
E{E [s2(i,n2d] = 6 2 f(n,f')
A
=i E(S~],~ = 9 (28)
which is what is required. However, neither of the two assumptions necessary
to arrive at equation (28) are liable to hold ; nevertheless the departure from
the required result may not be very serious. If, however @(i,nl) is biases,
a larger discreparcy will occur which should accordingly manifest itself in
the overall regets.
The set of oserations (I) - (3) outlined above may be modified to study
the effects of aosLying bias correctiocs separztely to estimtes of the varizce
and the lag-one autocorrelatio2. For emple, a simplified set of experimezts
may be cìefined where the variance for the desis seque-ces, E [s2] 1co is
assumed *-OX?,. in this situation, for each es2imate of the lag-one autocorrelation
@(i,nl), the estimate of the variace, c2(i,nq) used to drive the gererator
is defined as
4
b(i,n,) = E Is2] ,oo/f(n2, p(i,n,)) (29) .
r\
where E {szj is nou a constant but f(n2, @i,nl) is a random variable then,
A
2
E ( G2(i,?)] = E [$2(i,nl) f(n2,p(i,nl)j = E [s Loo= 9
from (29)

Cozsequently, the reqired vcriúzce can be miztahed -2 the desiga sequemes,
while t'le effects of bizs correctionzs ??flied io %-ore autocorrelztio~s q v
more esiïy be zssessed; as caa the hpact of tde time de-perxielit structure of
the floäs.
Cozclusions
The problems associated with choosizg a gecerati=& ?rocees for generatixg
synthetic streafo-is &-re been outlked, aid some of zLe problems miskg a
parmeter estdtion discussed. k the preserce of lorg-tem oersistence,
the sadl Foperrties of gererati-4 processes trey ~ffer markedy from
corres?ordirg poplation guzrititi-es, a d a set of simuï.ztio.ori experime-ts &.ve
bees desir-ed to illustrate the izfluerce of 55s correctiox on the ete er
reso'xce system Cesign -roceSS, while also allowing the coase:xexes of the
ixorrect moiellirg of persisteme to be assessed. €?ro?lerns are enco-tered
in tzaixtaici3g a cozstzrt expecteà varicce ir desigri se-ences between differezt
worlds, uhe- the vzzce is estiozted fyom hisroric sequexes, a=d oriy
approxirate bis correctiozs m2y be ap?lied. Zowever, %te results of the sim-
lâtio2 experineats whez 2vWlable &odd illustrate the effects of qplyirg
bizs coxectiors to the variace ard lag-one zxocorrelfcion, ead provide a
guide es to what choice of gezerati-g mecha5m 29pears to -se the o verd
regrets accsilizg from 2 partich choice of ge'reraticg mec-sm. The o otid
choice of gererzting mechanism wo-dd be conditiod on a particular design
process.
3 91

392
-1 +I
Q
Figure la
-1 +I
QI
Pipe Ib
+I
+I
-1
d
e2

Reterir-ces
I.
2.
3.
4.
5.
6.
7.
8.
9.
10.
II.
12.
13.
14.
15.
16.
17.
&tBLas, W.C. (1967), Hathenaticel assesmerd of sy-thetic hyär~logy,
&ter Resow. Res. f(4), 931-935.
P!delbrot, %B., Udlis, J.R. (19681, No&,
kydrology, Kzter Beso*c. Res. 4(5) , 909-918.
Joseph, a d operatio-di.
Pxst, H.E. (1951), Lozg-term storege capacity of reservoiI’S, Th.us. Am.
SOC. Civ. Engrs., 116, 770-8080
393
”st, B.$. (19561, ?!ethods of using long term storzge h reservoirs,
Proc. Inst. Civ. Fagrs., 1, 519-543.
.
Yadelbrot, 3.B. (1971 ), A fast fractiod G2ïssim noise generator,
Kater Resoil-. Bes. 7(3), 543-5530
Fmdelbrot, B.B., WiLlis, J.R. (1969 1, Computer experiments with fractional
Gmssian noises. Part I - Averages m-d variaces, Viter Resoure Res. 5(l),
228-241.
Yatalas, Nec., Wallis, J.R. (191 1, Statistical properties of multivariate
fractional Eoise processes, Water Resow. Res. 7(61 , 1460-1468.
Box, G.E.P., Jenkins, 6.13. (‘JPO), Time series analysis : Forecasting and
control, S a Francisco, Holden-Day Inc., pp.553=
O’Connelì, P.E. (1971), A simple stochastic modellirg of Burst’s law,
Proc. Interrztionaì Syqmsium on Mzthematjcd Models in Bydrology, Varszw,
voi. I(I), 327-358, ïnt. Ass. Sci. ñydrol.
Kejia, J.M., Rodriguez-Iturbe, I., Dawdy, D.R. (19721, Streamflow simulation.
2 - !Che broke2 lice yocess as a potential mociel for hydrologic simulation,
Water Resow. Res. 8(4), 931-941.
Fadelbrot, B.B. (1972), Broken line process derived as ai- approximation
to fractional noise, Yater Resour. Res. 86). 1354-1356.
Kerdall, M.G. (1946), The advanced theory of statistics, v.2, London,
Charles GriIfin a d Co. Ltd,, ?p. :24-125.
Fisz, M. (7?63), Probability theory *and mathematical statistics, New York,
John Wiley ard Som I~C., pp. 421-423.
kderson, R.L. (1942), Distribution of the serial correlation coefficient,
An. Math. Stat., 13, 1-13.
Vdis, J.R. , Natds, N.C. (1971). Correlopm =absis revisited,
Vzter Resow. Res. 7(6), 1w-1459.
Y!talas, N.C. ‘Langbeir,? Y.B. (19621, Info-tion content of the mean,
Join. Geofir. Res. 6 7(9), S I - H .
Vdis, J.R., Natalas, S.C. (Ig”O>, Sm11 ==?le pro-perties of II a d K -
Estimators oi the Ewst coefficient h, Uster Resour. Bes. 6(6), 1931594.

394
18. Vais, Jog., O'CoTlieU., P.E. (1973), Fi-- reserroir yield - hou reli251e
are historic ?;7drolop~c records? 1-ternatiod Sjmoosium on the ñydrolog7
of Mes, Helsizki, Firland.
19. Slack, J.R. (:972), Bi=, illusion arid derial as data mcerteicties,
Interatiod Spoosium 01: Uzcert-ties fi Hydrologic er6 Vater Resource
System, Tucsx, ArizoE.
20.
21.
24.
25
26.
a.
28.
Slack, J.R. (373), I uould if I could (Self-denid by conditiorial models),
Vater Resour. 2es. 9(1), 247-249.
Wallis, J.B., Ystdas, N.C. (19721, Secsitivitr of reservoir design to
the generat- nechzcìisin of inflows, Yater Resou. Res. 8(3), 634-641.
22. Uallis, J.R., O'Connell, P.E. (19721, Small samole estimation of Q
Yater Resour. Res. 8(3), 707-712.
23- Hatalas, N.C. (19671, Some aspects of time series dgsis in hydrologic
studies, Proc. ñydzology Sym~osiuip ?io. 5 - 'Statistical Kethodc in Hydrology' -
held at McGilì Univi. Kontreal, Feb. 1966, National Research Ccmcil of
Cmda, ppm 41-99.
Brooks, C.E.P., Carruther6, N.C. (1953),
in meteorologi, London, EPSO, pp. 412.
Handbook of statistical methoàs
O'Corsell, P.E. (19'731, The use of ARIMA models in the stochastic modellkng
of long-term persistence, R.D. thesis (in preparation).
Katalzs, N.C., Wallis, J.R. (19721, An approach to formulntina strategies
for flood frequency -sis, Interratiorid Symposium on Uncertainties
in ñydrologic a d Water Resource Systems, Tucson, hizona.
Fierizg, M.B. (1967), Streanflow synthesis, London, NcMiUm, pp. 139.
Thomss, H.A., &den, R.P. (19631, Statistical aadysis of the reservoir
yield relatioo, report, chap. 1, pp. 1-21, Harvard Water Resour. Group,
Cambridge, F!s.

AB ST RACT
STOCHASTIC APPLICATION IN UNGAGED BASINS
FOR PLANNING PURPOSES
By Pedro Porras G. and Alfredo Flores E.
Water resources development planning, being iterative and dy-
namic, requires increasingly detailed basic information as each pla-
nning level is surmounted. Successful planning is closely linked to
the quality and quantity of the basic data. But when short periods
are involved, the field of information is increasingly limited, as
when dealing with monthly values instead of annual values. Moreover,
methods and techniques are not as readily available, and besides,
they are more laborious, as compared with those used in connexion
with long periods. On the other hand, the use of sophisticated te-
chniques, such as hydrologic simulation, is adequate at the project
level but not at the planning stage, since it involves a more care-
ful preparation of incoming data and its attendant remarkable effect
on the cost structure. Besides, it is timeconsuming to the extent of
likely jeopardizing the requirement of keeping planning up-to-date,
The method herein expounded deals with the attainment of average mon_
thly values, covering a standard period of years, for precipitation,
evaporation, net irrigation demands, and runoff, in the various
stretches of the different rivers in a region, using transition pro-
babilities (stochastic techniques], beginning wrth observed annual
precipitation values, The method has been designed for computer solu-
tion.
El proceso de planificación hidráulica, por su carácter itera-
tive y dinámico, requiere de una información básica cada vez mas de-
tallada a medida que se van superando distintos niveles. El êxito de
la planificación está íntimamente ligado a la calidad y cantidad de
datos básicos; pero la consecución de esta información se hace mas
limitada cuando se imponen consideraciones de períodos cada vez más
cortos, como ocurre cuando se trata de valores mensuales en lugar
de los anuales; a esta limitación habría que añadir la menor disponl
bilidad de mêtodos y técnicas que a su vez son m’as laboriosas que
las usadas en periodos largos. Por otra parte, el empleo de técnicas
sofisticadas, como las simulaciones hidrolagicas, son adecuadas a ni
vel de proyecto y no de planificación e implican una preparacian más
meticulosa de los datos de entrada incidiendo notablemente en costos
y consumiendo un tiempo que podria poner en peligro la actualizaci’h
que requiere la planificación. El método que aqui se expone trata de
la consecuciön de valores medios mensuales, pa??a un perPodo tipico
de años, de precipitación, evaporación, demandas netas de riego y es-
currimiento, en los diversos tramos de los di‘stintos ribs ae una re-
gión, haciendo uso de las probabilidades de transl’ción (96cnicas e ~ -
tocásticas) partiendo de los valores anuales observados de precipita
ción. El método ha sido diseñado para resolución por computadora.

396
JUSTIFICATION
For th elaboration f the first version of the National Plan of
Development of Water Resources it was necessary to make a national inventory
of the surface runoff. The characteristics of this first version were sufficient
to know the mean annual volumes which were estimated through runoff
isolines given by Hydric Balance method. One of the factors which determined
the simplicity of application of this method was the number of years of the
chosen period- which allowed to accept the hypothesis that the lateral transmissibility
is negligible.
For the second version of the plan it was necessary to make an inventory
of the surface runoff n mean monthly periods trying to reach a desirable
level of detail but it was not possible to make it directly because
of insufficient information ex sting. This situation compelled to make an
investigation but unsuccessfully. Then it was necessary to change and follow
other methodologies. From this emerged the idea of' making investigations
through the application of stochastic methods to attain better results.
CONSIDERATIONS FOR THE MODEL
The proper characteristics of a region determine its own pluvial
cycle specified by the quantity and distribution of the rainfall. Among these
characteristics must be considered the latitude, longitude, proximity to the
sea or lakes, the topography, land form and so forth. Some of them have
greater influence on the quantity and others on the distribution in the year;
determining dry and wet periods. In the annual rainfall distribution may be
appreciated two essential particularities : one, the annual total percentages
corresponding to each month (they represent similar figures for the different
zones though it does not mean that the quantities of rainfall must be the same)
the other, the sequence or the order of their presentation. The observations
made have proved that in relatively small zones the variations of the rainfall
may be important in quantity but not in its distribution.
Generally the characteristics of the available data offer the opportunity
that a lot of factors allow the elaboration of mean annual isohyetal
maps with better reliability than the monthly ones (which in some cases can
not even be elaborated). Among these factors it is important to include the
following: the zonal variations better determined from the precipitationelevation
relation (topographic considerations); the complete annual totals
series; the totalizer records;
the comparison with mean annual runoff in
gaged watershed; the simplicity in the estimation of lacking data through
technical procedures like double mass curve, etc.

The most frequent difficulties in the elaboration of isohyetal monthly
maps are: (a) the cluster of several months in succession since in many cases
even having the annual totals of the complete measurement series makes difficult
the assessment of monthly averages; (b) it does not exist clear relations
between the altitude of the station and the monthly rainfalls; (c) not always
exist definite relations betwe.en the monthly rainfalls measuredat near stations;
(d) the zones with scattered stations where the mentioned considerations hamper
drawing the monthly isohyetals. Here is then the need to generate the monthly
data through adequate methods.
ANALYSIS OF DATA
397
For the stochastic generation of monthly values of rainfall, Region 1
was dected(Maracaibo Lake Watershed) where 141 stations with 10 years period
recorded were estimated: 1961-70.
In a practical manner, even if the monthly average could not be deter-
mined directly from the records, due to clustering of monthly values, it was
possible to determine more or less easily the monthly maximum and minimum that
led to generate series with standard deviation and mean similar to registered
series according to the checking made at stations with complete recording.
The selected value for the analysis was the percentage of each month
in connexion with the annual average for those 10 years period. A frequency
analysis was made of the group of data gathered each month and by mean annual
ranges of precipitation.
The ranges of precipitation selected were the following: (a) from.
50 mm to 1000 mm; (b) from 1000 mm to 1500 mm; (c) from 1500 mm to 2000 mm
and (d) greater than 2000 mm. Then the following particularities appeared:
(a)
(b)
(c)
(d)
In months of light precipitation ‘the interval of the percentages
variation with respect to the annual average varied with the annual
average.
In months of heavy precipitation no relation was noticed.
In all cases the Gumbel distribution function was the one which fits
better on polygon frequency.
The interval of the percentages variation, each month, is characteristic
of each location. For instance: the interval in January in
Paraguaipoa is permanent and is different to Machiques in the same
month.

398
DESCRIPTION OF THE MODEL
It was determined to construct a stochastic model to generate (from
the average values of annual rainfall) monthly series of 10 values which re-
present a typical cycle of 10 years. This matter has the following purposes:
(a)
(b)
(c)
To use the averages of those series in the elaboration of the mean
monthly isohyetic maps.
To compute (from generated rainfall values) probable values of
evaporation which accompany each rainfall in order to obtain balances
and determine the water requirements for irrigation.
To determine mean monthly rainfall over the watersheds area in order
to estimate mean monthly runoff and obtain average values.
For the generation of rainfall values it was necessary to estimate
the possibility to elaborate a matrix of transition probability to obtain
monthly rainfall values for consecutive years. This was not possible because
the monthly series (when are completed) only consist of 10 terms and just like
it was mentioned previously the number of them is very reduced - this being
one of the reasons why the model was prepared.
The matrix of probability of transition was elaborated then considerin!
previous states equally likely that is to say, without distinction in all
months. It was found that among the polygons of "accumulated relative fre-
quencies" corresponding to the same previous state, but of different matrix
pertaining to nearby regions, there exists proportionality. That is, if we
denominate F(x) the function that describes the best fit to the polygon corres.
ponding to a previous state xi of a matrix [A and F, (x) that corresponding
to the same previous state xi in the matrix zû] (Fig. 1) and if the functions
take the same value for values x equal to s and t respectively that is to say
if
F (s)=F( ( t ) (1)
and calling m and M the lower and upper limits of the interval variation of
the first function and m, and M,
is verified:
to the second one; an important relation
-
-
s - m M - m
t - m, M,- m l

This last expression gives a simple form to estimate the value of the
rariable corresponding to a place when is knownitslimits of variation and a
wilt matrix is valid for the region since the real value of x will be:
x =
s - m
M - m
(Mi - mi) + ml (2 1
TO generate a cycle of 10 years of rainfall values, any value (XO) is
:hosen among the interval of variation of the percentages observed for the
nonth of December so as to begin the process. Boundaries for each month are
the maximum and minimum limits where the percentages of each month change and
the value of mean year rainfall in that place is determined. A random number
is selected from O to 100; with this random number and with the initial value
(xo) we come into the matrix of the transition probabilites and we read the
dalue of x which is changed through the expression 2.
This value of x obtained this way and multiplied by the
,f precipitation and divided by 100, generate the first monthly
dumerically is as follows:
I
xu P
LL\ = J-
-
dhere P: mean year rainfall
100
x;: percentage corresponding to ( 'month) state
i l
With the value x: obtained and a new random number, the proced.ure is
repeated so as to generate x2 and continually untili = 120 (10 years). The
first 12 values correspond to each month of the first year generated; the next
L2 correspond to the months of the second year and continually until the tenth
rear.
lbtainment of the Monthly Rainfall Over Area in a Watershed
399
mean year value
rainfall value.
In a watershed, as soon as the isohyets have been drawn with the mean
mnual values of the available stations, the mean annual rainfall over area
)btained with the average of all the point values inferred will be very close
to the vaiue of the mean annual rainfall which has been obtained through the
danimeter. Based on this it is possible to get the mean annual rainfall on a
datershed, using the known mean annual totals or the generated mean annual
lalues. Nevertheless, applying this procedure to the monthly value generated,
leads to poor results, average excepted, since in two adjacent points it is
impassible to guarantee beginning both series with the representative values
if the same year.
1 typical cycle of 10 years, not only regarding the magnitude of the terms but
regarding the order too.
(3)
In spite of this the values of the series obtained represent

400
In order to attain the series of the monthly values corresponding to
the rainfall over the area already mentioned, the same procedure has to be re-
peated (as it has been explained) for getting the monthly values at a point,
starting from an initial value which is the average value of the initio1
volumes at each point extended uniformly from the mean annual rainfall obtained
according to the precedent explanation and the monthly maximum and minimum
limits also obtained just like averages at each point.
Generation of Monthly Values of Evaporation
The precipitation of the zone was divided in 10 mm intervals and in eoc
one of them where the stations existed, the evaporation was studied and the
polygon of frequence which correspond to each rainfall interval was determine
The distribution of frequency of the evaporation was anolyhed in interv
class of 10 mm. Without making a strict analysis it was observed that the distribution
of evaporation for each rainfall interval is normal. The range of
the variation of the evaporation decrease while the mean value of the class of
rainfall increase. Thus, to generate anevaporation value, as soon as the rainfall
value has been generated, a random number is chosen as well as the interva
to which the rainfall generated pertains; the random number determines the
corresponding evaporation value.
With the pair of series obtained at each point, the accounting balance
is made in order to determine the water requirements for irrigation.
RESULTS
With the aim of getting some indicator of the goodness of the results,
the monthly rainfall values pertaining to places with measurements were generat
and the series were compared through means and standard deviations.
The linear correlation was calculated for each place among the twelve
means generated (one for each month) and the measurements; the same also was
made with the standard deviations.
For illustration here are some results:
!- Coefficient of cor;ela:ion between
The Means Standard Deviations
I
O ,98
o ,?2
San José Bolivar
Boroto
Mochiques Gia.
0,90
O ,96
O ,?3
0,91
0,81
O ,96
0,94
0,89

JNOFF
401
The runoff is generated by the following equations explained in the
irk "Analisis sobre las Relaciones Escurrimiento-Precipitation en Periodos
! 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 )
Pedro Porras G.
ere :
Ei = Si-' -
Y
o
a
4- (Pi-si- 1 -)B.
I-a I-a I
si = s. - 1
a
i (Pi - - s. 1 -) (i-+)
I I 1-a
: Number of the period of time (month)
: Runoff in the period
: Storage to the period
i
,f Coefficients depending on the characteristic of the watershed.
This method has been developed in order to be applied by means of the
e of digital computers and to obtain ranges of mean monthly values which do
t differ more than 10 per cent in general terms.
For the application of this method it is convenient to use maps at
100.000 scales with topography; on these maps previously squared (an adequate
dth among the lines must be no more than 2 minutes) we draw the mean annual
ohyets for a period of 10 years and we build the twelve pairs of maps
rresponding to the isopercentsofmaximum and minimum for each month.
In the region all watersheds have to be shown uptothe places of interest;
e data must be prepared for each watershed.
At the intersection point of the grid previously identified, we must
ad the mean annual precipitation value as well as the maximum and minimum
lues corresponding. This information plus the area of the watershed, initial
lues of precipitation in percentage, the regional matrix of precipitation
d evaporation and the values of alp, S establish the input data to the pro-
O
ss.
(4)
(5)

402
The values of a,/, S must be obtained from nearby watersheds with
O
measurements and similar characteristics. In the procedure to be applied to
get this result and in the selection of these values, we must estimate the
very important physiographic characteristics of the watershed.
RESOLUTION BY COMPUTER
The application of the model requires a lot of computing which is
very time consuming by hand; therefore it was necessary to elaborate a progrc
for digital computer.
The program has been elaborated in the PL/1 Language and it consists
of a main program and five subprograms whose functions are the following:
MA1N PROGRAM
This program consists of six main phases:
It reads the data of the transition matrix of the precipitation and
the evaporation, as well as the characteristic data of the points
of the grid in which the watershed has been subdivided (mean annual
precipitation, initial monthly precipitation, monthly maximum limit
and monthly minimum limit).
It generates stochastically for each point of the grid the precipita.
tion month to month for the selected period through the transition
matrix of the precipitation.
From the monthly precipitation of the point and through the transitic
matrix of the evaporation, it calculates the monthly evaporations at
the point.
If the water requirements for irrigation are required at the point,
the program through a control apply the method "Accounting Balance
of Thornthwaite" using the precipitations and evaporation calculated
in the phases 2 and 3.
It generates stochastically the monthly precipitation for the water-
shed from the transition matrix of the precipitation and taking as
mean annual value and initial value, the average of them already
calculated for the points of the grid of the watershed.
It calculates the runoff and monthly volumes for the watershed
according to the work "Analysis of Runoff-Precipitation Relations
by Monthly Periods" by Pedro Porras, Engineer.

SUBPROGRAMS
Subprogram LENGTH:
It calculates the length of the interval of the probability curves
accumulated by the transition matrix of the precipitation.
Subprogram INTER:
It interpoles values in the transition matrix
Subprogram ESTADI:
It computes the mean and the standard deviation
Subprogram DENERI:
403
It determines the water requirements for irrigation starting from the
method of "Accounting Balance of Thornthwaite".
Subprogram RANDU:
It generates random numbers between O and 1 used in the stochastic
process.
In the Fig. No. 2 attached is presented the Flow Chart of the program.
VERIFICATION OF THE MODEL
For the verification of the model the Socuy River watershed was
selected. The basin is located in the north-west region of the Maracaibo
Lake in Venzuela.

404
Fin) Former state: xi
MATRIX [AI
Fiix) Former state:Xi
MATRIX [d
F,(t)rF(i) ---------
Comparison of distribution functions for equal prior
states of the various matrices
PIOURE A

0
COMIENZO
MATRICES DE
PRECIPITACION Y
EVAPORA CI ON
NUMERO DE
CUENCAS
LECTURA DE
DATOS DE LA
CUENCA
PUNTO
LECTURA OE
DATOS DE LO5
PUNTOS DE LA
CUENCA
I A
::::;-I
PR EC I P ITACI ON
EVAPORACION
I
FIGURA 2
- 9
FIN PUNTO
DEMANDAS
FIN
1 CALCULA P R F
PITACION MEDIA
ANUALY MENSUAL
PARA LACUENCA
IMPRIME
PRECIPI TACION
LECTURA DE
CALCULA
VOLUMENEC
MENSUALES
0
FINAL
~~
DIAGRAMA DE FLUJO

406
6
1
8
IC
5
11.11
41.63
154.31
114.65
it.31
41-61
?.II
145.IC
Il.5C
C.15
itit<
13iGi
I(C.51
2t5.lt
5f.13
45.15
13s.11
?li.lf
13.f<
?(.i1
?.'!
42.21
!liil
5f .*i
31.11
41.25
ii.12
I.i< 1-61
15.15
12.15
1e2.27
il.0
145.41
SC.?<
*;.i?
Il5.1f
(1.11
1.51
111.51
lt.53
ili.13
5ct.14
li?.?;
I

ABSTRACT
HOMOGENEISATION ET INTERPOLATION DES DONNES POUR
UN MODELE DE SIMULATION
par Marcel ROCHE
When the topological schema of a project has been defined, it
is necessary to establish data sets (in quantity and possibly in
quality) wich are to be used for operating the simulation. This is
all the more difficult as basic data are more seldom and of less good
quality. It must 5e begun to gathei as completely as possible data
directly observed at available stations and to severely criticize
those data. The second operation consists ia choosing the period of
reference to be used Cas long as possible) and in establishing an
homogeneous series, for this period, from the basic data Ccorrelations).
Finally, from this homogeneous series at available statïons, an
homogeneous series at the various input points of the model has to
be computed (interpolation in space). The author taRes as an exemple
monthly yields and their mean salinity.
RESUME
Lorsque le schema topologique d‘un aménqgement a étd arrêté,
il convient d‘établir los séquences hj-drologTqnes Cqyantit& et
éventuellement qualité] oui devront être utilisées pour proceder a
la simulatlon. L‘opératinn est d’autant plus df3licate que les donnees
de base sont plus payes et de moïns bonne qualite. On doi’t commencer
par faire un bilan aussi complet quo possible des données directement
observses aux stations disponibles et scumettrc ces donnges à une
analyse critique ~6v’ere. La seconde opérntion consiste 3. choisir las
période de référoncn qu‘on utilisera Cia plus 1Qngue possible), et a
établir pour cette ?-riode une série homogène a partir des données de
base (corréiatioris). Enfin, on doit calculer, à partir de cette série
homogène aux stati~ris disponibles, une s&rie homogene aux différents
points d’entrée di1 nodèlo (interpolation spatiale). L’auteur prend
comme exemple les apports mensuels et leur salinité moyenne.

408
Un modèle mathématique de simulation pour un aménagement intégré est
construit 5 partir d'un plan topologique tel que celui de la figvre 1. Sur ce
plan, les débits à injecter pour faire fonctionner le modèle sont représentés
par les symboles An et ACn, suivant qu'ils sont produits en tête de bassin ou
dans un bassin intermédiaire. On peut, s'il est besoin, leur associer les salures
moyennes correspondantes SAn et SACn.
I1 est donc nécessaire de fournir les valeurs de ces apports et éventuellement
de ces salures sur la plus longue période possible. Les débits sont
mesurés à des stations de jaugeage qui ne coïncident pas toujours et avec toutes
les limites des unités hydrauliques. Par ailleurs, les périodes sur lesquelles
portent le6 observations ne sont jamais les mêmes aux différentes
stations. I1 en est de même pour les observations concernant la qualité des
eaux, avec encore moins de stations, davantage de lacunes et des périodes
plus courtes.
Les opérations destinées à préparer l'échantillon des An et des ACn, et
éventuellement celui des SAn et des SACn seront donc les suivantes (on supposera
dans tout ce qui suit qu'on travaille à un pas de temps mensuel) :
- mise au point des débits moyens mensuels observés aux stations,
- mise au point des salures moyennes mensuelles observées aux stations,
- choix d'une période de travail, dite historique, et homogénéisation
des débits moyens mensuels sur cette période,
- homogénéisation, sur la période historique, des données concernant
la qualité,
- calcul, sur la période historique, des An, ACn, SAn et SACn, par
interpolation géographique, et quelquefois par analogie ; équilibrage
des volumes et des poids de sei.
1.- Débits et salures observds aux stations
Les données provenant des réseaux sont traitdes par les services
qui les contr&lent, de plus en plus par les procddds de l'informatique.
Quand on commence une étude hydrologique pour un amdnagement,
on doit, dans toute la mesure du possible, repartir des
données originales non dlabordes,
les critiquer et les traiter à no:
veau, Pour les hauteurs limnimêtriques, on reprend tous les originaux
des observateurs (lecteurs d'échelles) et les limnigrammes s'ils
existent, On vérifie les calages des échelles en s'appuyant sur les
comptes rendus, les contrôles de zéro, sur tout document disponible,
on se livre au besoin 3 des enquetes sur le terrain; on essaye d'@va
luer la qualitd des relev'es d'apr&s la forme des limnigrammes, la
tenue des feuilles d'observations, en faisant des comparahons avec
d'autres stations etc,
Si on dispose des minutes des jaugeages, il n'est pas sans
intérbt de controler quelques dépouillements; si on releve un purcep
tage important d'erreurs il ne faut pas hésiter 3 les reprendre en
totalité, I1 ne s'agit pas toujours 12 d'un polissage raffiné; nous
pourrions citer un cas dans lequel un contrôle a montrd que 25% des
jaugeages présentaient des erreurs de dépouillement supgrieures 3
20%. I1 faut ensuite refaire la courbe d'étalonnage, ou les courbes
si l'étalonnage a varié au cours de la période d'observation, ce qui
est presque toujours le cas pour les basses eaux,
On reprend alors le calcul des ddbits à l'ordinateur et on
en sort la série chronologique des débits moyens mensuels observés
chacune des stations dans le ou les bassins intéressant le projet,

Si l'on doit tenir compte de la qualité des eaux, notamment de la
salinité, il faut faire sur les observations correspondantes une
opération analogue 8 ,la précédente, mais avec une méthodologie de
contrôle tr&s différente,
409
Les mesures de salinités, par exemple, portent généralement SUT des périodes
beaucoup 21us courtes que celles des observations hydrométriques. Rles sont
souvent disparates dans leurs méthodes dféchantillonnage (techniques de prélèvement)
aussi bien que dans les méthodes d'analyses. Pour ces dernières, on procède
soit par analyse chimique complète, en différenciant les sels dissous, soit par
analyse sommaire : teneur globale en sels dissous mesurée le plus souvent par conductivimétrie,
I1 faut homogénéiser tous ces résultats, après une étude critique
aussi poussée que possible portant notamment sur la confiance qu'on peut attri-
buer aux méthodes clsanalyse pratiquées
et aux conditions dans lesquelles elles
ont été appliquées .e lorsqu'on les connaît. On produit ainsi, pour les .besoins
du modèle, un échantillon de salures moyennes mensuelles à un certain nombre de
stations (l).,
Si on a l'intention d'utiliser la pluviométrie disponible pour étendre la
période d'observation des débits, il faudra procéder également à l'indispensable
étude critique des précipitations. On s'attachera notamment à détecter et à cor-
riger les erreurs systématiques, causes d'hétérogénéité dans les séries, en appli-
quant la méthode des doubles cumuls (2). La encore, il ne s'agit nullement d'un
débat académique ; les erreurs systématiques dans ce genre de relevés ne sont pas
occasionnelles, elles constituent la règle générale. Pour éviter les erreurs de
transcription qui risquent d'affecter les publications officielles, on recomman-
de 15 aussi, dans toute la mesure du possible, de partir des relevés originaux.
2.- Homogénéisation et extension des données I1débitsff
Les opérations précédentes ont permis de constituer un échantillon de de-
bits mensuels portant pour chaque station SUI' une période inférieure ou égale a
n années. L'homogénéisation va consister à choisir une période de référence au
plus égale à n années et, p y l'utilisation des regressions, à étendre les rele-
vés de toutes les stations a ces n années, Les corrélations sont estimées mois
par mois pour chaque couple de stations, afin d'éviter l'influence de l'effet
saisonnier.
I1 est très important, lors de ces estimations, de ne pas fausser les va-
riances des échantillons calculés en utilisant sans autre précaution les vérita-
bles équations de régression. Considérons pax exemple les stations i et j pour
lesquelles on dispose, au mois m, d'une série d'observations communes portant sur
p années, soit, pow: une année k donnée, q.m (k) et qj,m (k). On sait que pour
utiliser aisément les corrélations, il faut que la régression de qj,* en qi,,,,
par exemple, soit linéaire et, dans toute la mesure du possible, homoscédastique,
-C-------C----------___I___L_LC_________-
(7) On lira avec profit, à propos du traitement des mesures de salme, m article
de J. CLAUDE, intitulé "une chaîne de programmes pour le traitement des données
sur la salinitéf1, et publié dans les CAHIERS ORSTOM, série HYDROLOGIE, Vol.
IX, no 2, 1972 -
(2) Voir l'article de Y. BRUNET-MOm intitulé ffEtude de l'homogénéité des séries
chronologiques de précipitations annuelles par la méthode des doubles masses",
et publié dans les CAHIERS ORSTOM, série KYDROLOGIE, Vol. VïII, no 4, 1971 -

410
I1 importe donc au départ de faire en sorte que, par anamorphose ou changement
de variable, ces conditions soient réalisées ; soit x la transformée correspondant
à ,(k)
is
et yk la transformée de q (k). On sait que la régression de y
9 .i9m -.
en x s'exprime par la relation :
YX = Yp + r
avec les notations habituelles.
P Y
p 7
(x-Z)
P
Mais yx ainsi calculée correspond à la moyenne conditionnelle des valeurs
possibles de y pour x donné et non pas à une valeur isolée. Une telle valeur se-
rait donnée par la relation y = yx + € dans laquelle & est une variable aléatoire
qui est souvent considérée comme étant normale de moyenne nulle ; elle est indé-
pendante de x si la condition d'homoscédasticité est réalisée. Négliger E dans le
calcul des débits non observés de la station conduit & diminuer artificiellement
la variance de l'échantillon qu'on aura constitué, d'autant plus que le coeffi-
cient de corrélation est plus faible.
Pour atre correct, si on veut utiliser l'équation de régression à ces fins,
il faudrait d'abord déterminer la distribution de E, puis, au moment de la reconstitution,
calculer yx et lui ajouter une valeur € tirée au hasard dans la loi de
distribution ainsi établie. Cela pose en fait un certain nombre de problèmes pratiques
(apparition de débits négatifs) provenant du fait que les hypothèses de
base ne sont pas vraiment respectées et que l'estimation de l'écart-type de E est
peu précise par suite de la petite taille de l'échantillon qui sert à l'établir.,
POW toutes ces raisons, il est finalement préférable de procéder d'une manière
beaucoup plus simple, certes peu conforme à l'esthétique mathématique, mais qui
respecte assez bien la variance initiale : prendre une droite passant par l'origine
: y = Ax,
I1 est parfois possible d'améliorer la corrélation en tenant compte de la
pluviométrie locale par l'application d'une régression multiple ('IIo Supposons,
pour fixer les idEtes, que la variable dépendante (celle qu'on veut estimer) soit
(k). Le bassin de surface S. qui fournit ce débit peut
'j,m J
-&re inclus dans le bassin de surface Si qui fournit %,m(k), on a
alors S. (si,
J
-Inclure si, on a aïors S. ) si,
J
-n'avoir pas de point commun avec S..
Dans' le premier cas, les pluies tombant sur S. alimentent totalement S. et
3 1
on a peu de chance d'améliorer la régression en les prenant en compte. Par contre,
il n'est pas impossible que la pluie tombant sur le bassin intermédiaire S exi-j
plique une partie non négligeable de la variance de q (k). Dans le second cas,
j,m
les pluies sur Si expliquent au moins partiellement q.m(k) et ne peuvent expli-
quer qj,m(k) que par l'intermédiaire de q
i,m
(k) : il est donc à priori inutile de
--------------------_______c____________--
(1) Pour le détail de l'application des régressions multiples à 1-'hydrologie, on
peut se reporter par exemple à l'article de P. TOUCKEBEiJF DE LUSSIQ'E "Régressions
et corrélations multiples en hydrologie", publié dans les CAHIERS ORSTOM, série
HYDROLOGIE, Vol. VïII, ne 4, 1971 -

411
les introduire. Par contre, qj est la somme de qi et de qj-i, variable expliquée
au moins partiellement par les pluies qui tombent sur le bassin intermédiaire j-i,
l'introduction de ces pluies dans la régression peut donc améliorer l'estimation.
Dans le dernier cas il est évident que, si on dispose de pluies sur S il peut
&tre utile de les introduire.
j'
I1 ne sera donc intéressant d'utiliser des régressions multiples portant
sur les pluies que si les données s'appliquent à un bassin contrblé par i ou par
j, mais pas par les deux à la fois. Il faut toutefois noter que, les bassins et
sous-bassins étant voisins, les pluies, surtout à l'échelle du mois, ont des
chances d'être assez fortement liées et on risque de vouloir faire expliquer à la
variable pluviométrique choisie une partie de la variance déjà expliquée par sio
Dans le temps, l'influence de la pluie tombée le mois m sera ßans doute
prépondérante, mais les pluies des mois antérieurs peuvent avoir une influence
non négligeable. On introduira donc, suivant les circonstances, soit les pluies
du mois (pluie mensuelle à un pluviomètre ou moyenne des pluies mensuelles à plusieurs
pluviomètres), soit un indice pluviométrique défini come une somme
Pm + a P + a2 Pm-2 + ooo décroît quand i augmente, par exem-
1
OU a
m-1
+ a. P
i m-i i
ple en progression géométrique de raison i/2. Eh fait la plupart du temps on se
limitera à la pluie du mois.
Les relations ainsi établies, employées avec les précautions indiquées en
ce qui concerne le respect de la variance, servent à établir une chronique de dé-
bits mensuels sur une période de n années pour toutes les stations* On peut alors
chercher à augmenter la durée de cette période avec le seul secours des données
pluviométriques. Cette opération est préparée lors d.e 11 étude précédente d'homogé-
néisation, mais l'absence de variables explicatives fld6bitsfl peut modifier assez
considérablement l'influence relative des autres variables explicativeso
On peut commencer à rechercher, pour chaque bassin, la relation entre le
débit moyen annuel Qi (k) et la pluie moyenne annuelle Pi (k) estimée par la mé-
thode de miessen si on a plusieurs pluviomètres. La relation Q (P) comporte un
seuil physiquement explicable qui correspond en gros à la précipitation minimale
annuelle nécessaire à l'apparition de llécoulement ; on la désignera par Poo A
ce seuil se superpose une constante qui traduit la diminution de variance due à
l'application de la régression. Comme P n'est pas connu à priori, on nia plus la
O
ressource de faire passer la droite de régression Q (P-PO) par l'origine (on suppose
en effet que la régression est linéaire ;si elle ne l'est pas, il convient
de faire les transformations convenables) o On peut appliquer l'équation de régression
vraie Q = A (P-Po), rechercher la loi de distribution des résidus, et Procéder
au calcul de l'échantillon étendu comme on l'a indiqué pour l'homogénéisation,
avec les mgmes avantages et les mêmes inconvénients. ûn préfère souvent
utiliser un expédient dénué, il faut le dire, de base statistique solide, Au lieu
d'appliquer les moindres carrés aux résidus Q
on les applique
calculé - Qobservé'
aux distances des points représentatifs des couples (&k,
à la droite w(p-pO)
qui ne sera plus alors une vraie droite de régression. On dit qu'on utilise une
''pseudo-r égressionl' .
brsqulon a ainsi mis au point un échantillon étendu de débits moyens annuels,
on reprend la m&me opération à Iféchelle mensuelle, en tenant compte au
besoin de l'influence des pluies des mois antérieurs, ainsi qu'on l'a indiqué

41 2
pour l'opération d'homogénéisation. Ces nouvelles régressions sont surtout des-
tinées à fournir la forme de la répartition des débits dans l'année, car les dé-
bits annuels déduits des débits mensuels ainsi reconstitub sont souvent moins
valables que ceux qui sont obtenus par une regression à l'échelle de l'année ;
il convient cependant de s'en assurer.,
I1 reste à vérifier que les données mensuelles retenues pour les diffé-
rentes stations sont compatibles entre elles, c'est-à-dire qu'en général un débit
d'une station aval doit &tre supérieur ou au moins égal à celui de toute station
amont, que si une station aval AV est placée SUT un cours principal alimenté par
deux bras dont les débits sont contr81és par deux stations AM1 et AM2, les débits
de AV doivent &tre au moins égaux aux sommes des débits de AM1 et AM2. Autrement
dit, on ne doit pas admettre de débit négatif dans un bassin versant intermg-
diaire, sauf éventuellement dans deux cas :
- il y a des pertes physiquement reconnues, soit par infiltration,
soit par évaporation (marais o.oetcooo),
- il y a des stockages naturels importants (lacs .etc.).
Ces cas particuliers mis .$ part, si on constate des débits négatifs ou ri-
diculement faibles, et cela arrive malheureusement assez souvent, c'est que, mal-
gré l'étude critique et la mise en ordre initiale des données, il y a des erreurs
dans l'étalonnage et/ou des erreurs systématiques dans les relevés d'échelle et/
ou une mauvaise répartition de ces relevés dans le temps (observations trop es-
pacées compte tenu du régime), I1 faut revenir sur 1'8tude critique et essayer
de déterminer quelles sont les stations auxquelles on peut faire le plus confian-
ce ; il est nécessaire d'aboutir à un choix, meme si ,celui-ci est un peu arbi-
traire. On considérera comme bons les débits des stations sélectionnées, qui
doivent bien entendu &re compatibles entre elles, et on retouchera les débits
incriminés des autres stations jusqu'à remplir les conditions de compatibilité.
3. - Homogénéisat ion et extension des données Ilsalinit ésf1
Les observations directes sur la salve des eaux sont presque toujours
plus rares, dans le temps et dans l'espace, que pour les débits. On sera donc
appelé à combler plus de lacunes que pour les débits et à procéder à une exten-
sion plus importante des périodes.,
Si les rivieres sont restées en l'état naturel, ou tout au moins au rn&me
degré de rejets susceptibles de modifier la salure, les données recueillies récemment
sont susceptibles d'&tre transposées dans le passé. Sinon la transposition
n'est pas Ithistoriquement1l possible, mais c'est d'importance secondaire. Eh effet,
on ne doit pas, pour la simulation, employer des échantillons lJévolutifslt, car
les résultats qu'on en tirerait n'auraient pas de sens, Au contraire, si, par des
tests quelconques, on s'apercevait que les conditions gbéralss de salure ont
changé, on ne devrait conserver que les résultats les plus récents, meme si la
taille de l'échantillon devait passablement s'en ressentir.
h situation se présente de la façon suivante. Pour tout mois de la période
irhomogènelt et éventuellement llétendue't de l'échantillon historique, on
peut disposer :
- d'une série continue de relevés de salure qui, passée dans une chaPne
de traitements de salinité permet d'établir une série complète
de valeurs des saliires journalières et mensuelles,

413
- d'une sbrie incomplète mais pamqttant le caLcul d'un certain nombre
de salures moyennes jourzi.alj.èrm9
- de relevés sporadiques,
- d'au.cm relevé,
Soit une station pour laquelle on s., &mart toute la pér:ode homogène, la
distribution d'observations journalières mivantes (ûuivant les mois de l'arde
d'exploitation numérot8s Î à -121,
pur les salinités moyennes journdi&rcs :
nsjl nsj2 nsjj nsjg nsj5 nsj6 nsj7 nsj8 nnjq nsjI0 nsj II ns%2 y
poi1.r les débits m,oyer,s journ.diers :
pour les dé5its moyen..^ mensuels :
Chaque terme nqmi est égal au nombre d.fann6es que comporte la serie homo-
gène, mais u2 terme nqji p.fast pas forcément égd à nqmi .multiplié par le nombre
de jours du mois i, puisqu'icn certain nonbre do débits moyens mensuels Ont pu
8tre reccnstitués lars de l'opération dfhomogénéisation, sans qu'on possède au-
cun relevé joiirmlier ,?our les mois corres2ondants. On cherchera dans une pre-
mière étape à reconstiti;er, pour chaque qji à.isponible, le sji correspondant qui
n.'auiait pas &té observ8. Dans une seconde Qtape, on fera la meme opération Szn"
les qmi et les mio
Pour un r6gtne hydr2roiogiqu.e donné, la concentration en sel dissous depend
EU premier chef de la nEtu.re minéralogique du bassin concerné, de l'importance
des nappes soutermines et de 13 vitesse GU transit de Ifeau dans ces nappes, Vi-
.cesse qui intervient sux la durée du contact de cette eau avec la roche- On Peut
aSouter comme paramètre l'agressivité des pr6cipitations mesurbe par leur teneur
en CO2 libre et 1ev.ï degr6 de pureté. Le ph&niimène est donc complexe et il ne
faut pas s'attendre 5 p'iwoir le représenter par des relations simples.
I1 est toutefois logique ds penser que les eaux souterraines sont normale-
men;: plus chargées f3n. sols dissous que les eaux de surface, par suite de leur
contact prolone;& 2.vei les roches. I1 faut donc s'attendre 5 ce que les basses
eaux soient plus chrnqkà cge les débits importants at il est logique qu'il exis-
te m e relation, certes m n tcjnctiomei?-e, entre ia salurs et ïe débit, L'expé-
riencs montre qu'il en est bien ainsi ; elle met. de pl.us en evidence une in-
f?u.eiic? saisomiere sur cntte rels.tion.
?oc? 1'6tabiir OB -pxèd.e mois par mois, ou tout au m.oins trimestre par
trimestre. Pour chaque mais :
- on raycrte tocs les sji obserTr8s en regard des qji qui lem cor-
rnsporifimt ?
- on oowtzte m e grande difipxioy, rnwie avec tendance tres nette
5 i:.-ax aroissmc~ des sj.; avec les ?%$iq

La dispersion est souvent telle que l'utilisation sans précaution d'une ré-
gression poserait des problèmes importants de réduction de variance, davantage
que pour les débits. Pour éviter ces inconvénients, nous avons mis au point la
technique suivante qui a au moins l'avantage de respecter intégralement les pro-
priétés statistiques de l'échantillon.
On détermine un certain nombre de classes de débits,
Classe 1 O à ,qj
Classe 2 lqj à ,qj
------------------
Classe k k-,qj à ,qj
Classe k+l > ,qj ,
de telle façon qu'à l'intérieur de chacune on puisse considérer que l'influence
de la variation du débit sur la salinité est négligeable. Eh associant, à chaque
qj de l~échantillon, la valeur s. correspondante, on constitue autant de 'Iréservoirstt
de salures qu'il y a de classes de débits. Avec les précautions prises,
dans chaque réservoir s est indépendant de qo Les différentes salinités contenues
dans un, réservoir sont identifiées par un numéro.
-
A l'ORSTOM, l'opération est effectuée au moyen du programe 703 pour un
découpage mensuel (comme ici) et par le programme 703 bis si le découpage est trimestriel.
Le résultat est une matrice des salures à trois dimensions dont les
indices représentent
- le numéro d'ordre de la salure dans le réservoir,
- le mois,
- la classe de débit à laquelle appartient le débit associé à la
salinit é.
Cette matrice permet de reconstituer les salues correspondant à tous les
débits moyens journaliers observés pour lesquels il n'y a pas eu de mesure de salinité.
Le programme 704, qui fait cette opération pour un découpage mensuel,
procède de la façon suivante :
a - Wegistrement de la matrice des salures.
- Lecture des débits limites de classeso
- Lecture de la matrice des salures (aanS L'ordre : classe, mois,
numéro de série de la salure) : ECHASA (NOCL, MOIS, K).
b - Lecture des débits journaliers.
- On lit les débits journaliers pour un mois et on les met dans
un veateur à 31 positions DEB (JIo
- Au fur et à mesure de la lecture par carte de quinzaine, on
reperfore les données pour constituer un jeu définitif débits
et salures.
C - Lecture des salures moyennes journalières.
- ~n lit les saïures moyennes pour un mois (ie méme que celui
des débits qu'on vient de traiter) et on les range dans un vecteur
SAL (J).
d - Détermination des salures journalières manquantes.
- Dans une boucle J = l,3l, on teste d'abord DEB (JIo S'il est
négatif, c'est qu'il n'y a pas de débit observé pour le jour

415
J ; il n'est donc pas possible de complèter la salinité et on
passe. S'il est positif, on teste SAI; (J) ; si elle est positive,
c'est qu'il y a observation de sdinité ; on passe,, S'il
est négatif, on complète.
- Pour compléter : on cherche dans quelle classe se trouve
X æ DEE3 (j), soit NOCZ ; on tire au hasard un nombre inférieur
ou égal à NC, nombre de salures classées dans NOCL, soit K,
et on associe à DEB (J) une salure SAL (J) égale 2 ECHASA
(NOCL, MOIS, K).
e - Perforation des salinités sous la m&me forme que les débits observés.
On revient alors à b- pour lire les débits du mois suivant, et
on continue airisi jusqu'à épuisement des données.
L'opération a permis de constituer un échmtillon pour lequel à
chaque débit moyen journalier correspond une salme moyenne journalière. Pour les
mois complets en débits observés, on peut alors calculer les salures moyennes mensuelles,
Restent les mois pour lesquels on a pu reconstituer les débits moyenc
mensuels, sans posséder les débits journaliers (homogénéisation) Pour leur attribuer
une salme moyenne, on procede d'une façon analoge à ce qui précède,
4,- Calcul de l'échantillon historique pour le modèle
Lors du découpage géographique, on s'arrange pou que les stations du réseau
tombent autant que possible à des limites d'unités hydrauliques, Mais cela
n'est pas toujours possible dfune part, et d'autre part les unités hydrauliques
sont toujours plus nombreuses que les stations de mesure, I1 est donc nécessaire
de procéder à une interpolation géographique, et m&ne parfois d'utiliser l'analogie
et la transposition pour calculer tous les An, SAn, ACn et SACn,
Pour le calcul des An et ACn (apports), on comnence par dresser un tableau
donnant, pour chaque unité n, sa superficie et la nat-.ne des apports, Lorsque
l'unité est encadrée en amont et en aval, on fait simplement une répartition au
prorata des superficies, au moins dans un premier stade (interpolation géographique).
Lorsque l'unité est en dehors du réseau des stations, on cherche à lui
attribuer un débit spécifique par comparaison avec d'autres parties mieux connues
du bassin ou avec d'autres bassins que l'on suppose avoir le mbe régime (extrapolation
ou transposition) Cette dernière opération provoque nécessairement une
légère erreur systématique par défaut sur la variance de l'échantillon global
constitué pour le modèle, mais cette influence est presque toujomnégligeable.
Si l'unité hydraulique est confondue avec le bassin versant d'une station
de base, on identifie les apports As à la station a u apports An sur l'unité.
si la station de base est unique et son bassin versant différent de l'uni-
té, les apports An sur l'unité sont obtenus à partir de ceux de la station de
base As par calcul au prorata des superficies des bassins : An = As * Sn/Sso
Si 2 stations de base encadrent la limite de l'mité hydraulique, les ap-
ports An sur l'unité sont calculés par interpolation linéaire entre les apports
As7 et As2 aux stations 7 et 2 : A ds1 -t (As2 - Asl) * (Sn - Ssl)/(Ss2 - Ss7)-
Pour calculer des apports intermédiaires ACn avec 2 stations de base encadrant
l'unité, on applique la relation : ACn = (As2 - AsII * Sn/(Ss2 - SSI)~
Lors des calculs relatifs au 4ème cas, on rencontre parfois quelques difficultés
: au pas de temps mensuel, la différence entre les apports observés à la
station aval et ceux de la station amont peut &tre négative bien que le bilan

416
annuel soit normalement positif. Ceci se produit en particulier lorsque les deux
stations de base sont assez éloignées ou séparées par un bassin intermédiaire
de grande surface comportant des affluents dont le régime hydrologique diffère,
de celui du cours d'eau SUT lequel est située la station la plus amont. On peut
alors procéder comme suit, pour différentes unités hydrauliques situées entre
deux stations de base.
- On calcule les apports intermédiaires mensuels et annuels entre les
2 stations.
- On détermine la distribution temporelle moyenne de cet écoulement
intermédiaire pour la totalit 6 de la période disponible (période d'observations
communes entre les deux stations). On obtient ainsi des coefficients mensuels de
distribution exprimés en $ du module interannuel.
- Pour chaque année de la période de reconstitution, on utilise ces
coefficients pour le calcul de l'apport intermédiaire mensuel à partir de l'ap-
port annuel observé.
- On répartit cet apport intermédiaire mensuel sur chaque unité au
prorata de sa superficie et de celle du bassin versant intermédiaire.
Cette difficulté ne devrait du reste pas se présenter si les apports aux
stations de base ont été soigneusement préparés.
I1 y a de nombreuses façons de calculer les SAn et SACn. On pourrait par
exemple passer par l'intermédiaire des poids de sel transités mois par mois aux
stati'ons de base, et opérer de façon analogue à ce qui a été fait pour les ap-
ports. Cela supposerait une certaine homogénéité dans la production de la salure
pour l'ensemble du bassin, ou tout au moins pour des parties importantes du
bassin facilement délitnitables. Cette condition n'est pas toujours r8alisée et
l'origine de la salure des eaux est souvent localisée,
L'interpolation géographique peut &tre sérieusement améliorée si on dispose,
pour chaque bassin d'alimentation d'une unité, des surfaces des formations
salines, et si on peut établir une relation entre cette surface et l'apport de
sel, Ceci revient à définir pour un mois m donné une relation de la forme :
Sape = fs (Qspe, pS) OU Supe est l'apport ,spécifique en sel, Q l'apport spéspe
cifique en eau et pS le pourcentage de formation saline.
Avec une bonne carte lithographique, on peut assez facilement déterminer
pS pour tous les bassins fournisseurs des unités et pour les bassins contrbiés
par des stations de réseau. Le problème serait alors résolu s'il était possible
de déteminer fs avec une approximation convenable. Cette détermination ne peut
se faire qu'en traçant un faisceau de courbes expérimentales à partir des résul-
tats des stations du réseau. La précision dépend du nombre de mesures disponi-
bles à chaque station, du nombre de stations et de la variabilité de pS, ce der-
nier facteur étant particulièrement important.
Si l'information disponible est insuffisante, ce qui est presque toujours
le cas, il est préférable de procéder par analogie. Nous indiquerons la méthode
utilisée par H. DOSSEUR (0,R.S.T.O.M.). On part de séries d'apports et de salures
déjà constituées pour les stations de base.
Si le bassin de l'unité est confondu avec celui d'une station, on iden-
tifie les concentrations. Sinon, on affecte 5 chaque unité une station choisie
de telle façon que son bassin soit le plus représentatif de celui de l'unité
considérée, compte tenu de sa situation géographique et de sa nature galogique.
Ce choix peut &tre précisé à partir de renseignements concernant la salinité
dans un secteur déterminé (mesures ponctuelles, indications d'ordre qualitatif...).

41 7
On associe à l'unité considérée les réservoirs de d ures élaborés pour la sta-
tion @ lui a été affectée, Pour chaque débit ACn, on détermine une concentra-
tion moyenne SACn par tirage au hasard dans ces réservoirs, suivant la méthode
indiquée antérieurement, avec toutefois une transformation préalable des clas-
ses de débits en classes de débits spécifiques pour tenir compte du rapport des
superficies entre l'unité et le bassin de la station associ&,
Du fait m&me de la méthode utilisée, l1échantillon des SACn présentera
2 peu près sûrement des incompatibilités analogues à celles qui ont été signalées
pour les débits liquides, I1 faudra donc contr8ler, au moyen d'un programme
annexe, que les poids de sel P% = AG * SAC, obtenus pour chaque mois de chaque
année de la période historique sont tels que le PS d'un point quelconque
du réseau hydropaphique est au moins égal au PS de tout point situé à son
amont, et que, si un point i limite à ï'avaï une unité limitée à ll~ont par
des points 1, m r, Psi doit &tre au moins égal à Psi I- PS, I- oooooop PS, ,
Pour tous les mois Ou ces conditions ne sont pas réalisées, il est indispensable
de retoucher la répartition des salinités dans le bassin pour rétablir
la compatibilit 6.
* *
*

41 8
Fig:l - SCHEMA TOPOLOGIQUE

ABSTRACT
THE USE OF SIMULATION TECHNIQUES FOR SEQUENTAL
GENERATION OF SHORT-SIZED RAINFALL DATA AND ITS
APPLICATION IN THE ESTIMATION OF DESIGN FLOOD
H.D.Sharma*, Dr.A.P.Bhattacharya** and S.R.Jindal;t**
The studies based on rainfall runoff data are considerably vi-
tiated in the event of inadequate data, as the reliability o€ the
probabilities of occurrence is reduced, It is, however, possible to
get over the lacuna of inadequacy of data by creating bigger-sized
artificial series of rainfall. The use of such a series gives grea-
ter precision in the estimations or projections based on expected m a
ximum rainfall with specified levels of occurrence and also provides
better insight into possible patterns of behaviour. This is done by
the procedure of sequential generation fo data by the use of simula-
tion techniques. Making use of these, the technique has been applied
for generating rainfall series of 100 nombers on the basis of recor-
ded rainfall data for a period of ten years, The generated rainfall
series was compared with the historical data which showed strong co-
rrelation.
These results have been used for the estimation of design
flood for Yamuna river at Okhla (Delhi).
Les procédés qui consistent à déduire les écoulements des pr5
cipitations voient leur efficacité considérablement diminuée lorsque
les observations concecnant celles-ci sont insuffisantes, par suite
de l'imprécision qui regne alors sur l'estimation des probabilités
de ces précipitations. On peut essayer de tourner la difficulté en
créant artificiellement de longues séries d'observations pluviométri
ques. L'utilisation de telles séries conduit a une meilleure préci-
sion des estimations ou des prédéterminations basées sur la pluie ma
ximale attendue avec une probabilité donnée; elle permet aussi une
meilleure vue suc les schémas possibles du comportement des précipi-
tations. On procede par génération séquentielle des données, en uti-
lisant les techniques de simulation, On donne comme exemple la cons-
titution d'une série de 100 ans à partir d'une période de 10 ans
d'observations. La séries engendrée , comparée avec la sérìe histori
que, met en 'evidence une forte corrélation.
Ces résultats ont étd utilisés pour l'estimation d'une crue
de projet à Okhla, sur le fleuve Yamuna [Delhi).
* Director , Irrigation Research Institute , Roorkee, U .P,
$:* Research Officer , Basic Research Division, Irrigation Research
Institute, Roorkee, U.P.
;'

420
i. INTRODUCTION
1.1 In all iqr-gothetical investigations, particularly in the estimation
of design flood of river basins, it 1s essential to have an idea of the
distribution of rainfall as also the relationship between rainfall a d
runoff. This is, however, not always possible in case of small sized
data, extending over 8ay 10 to 20 years as is usually met vit
oractice, as these may not be representative of the vorst possible
conditions prevaillng in the catchment. On account of such shortcomings,
it is likely that the findings based thereon may not be realistic. This
difficulty may be overcome by resorting to the technique of sequential
generation with the aid of which it is possible to artificially create
larger sized data series.
2. CONCWT OF W?UENTIAL GEWERI'EION
2.1 sequential generation is a statistical process usiag Monte Carlo
methods to produce a random sequence of hydrologic or any other data
on the basis of a stochastic model for the hydrologic process. Monte
Carlo method is an experimental or merical probability method used
for the statistical sampling of random variables. The sequence so
generated makes possible detailed study of the performance of various
hydrologic events, thus helping the development of well balanced hydo
rologic designs.
2.2 Unless the record is too meagre to be considered as a represento-
tive sample, the statistical parameters derived from It should enable
the hydrologist to construct a suitable model that wlll generate
hydrologic information for as long a period of time as desired. Bnce
the statistical parameters of the population of the generated data
are necessarily the same as those estimated from the bistorical date,
the new information is limited
that are inherent in the observed record.
3
errors of measurement and sampling
n

2.3 The procedure of sampling by shuffling; cards which waa among the
srllest techniques can be simplified by the use of random number tables.
naugh random number tables are available as punched cards, with Increase
3g use of digital cornputor, mathematical methods for generating pseu-
Fndom numbers within the computing machine have been developed'in order
2.4
eliminate the need for extensive input of random numbers.
3 the basis of required statistical levels of errors and confidence,
Lthough the optimal size may be determined more realistically by compar-
the cost of the Increased sample size with the benefits of the corres-
4 21
The size of the hydrological data to be generated may be estimated
mding increase in accuracy, provided that the benefit and cost data
:e available.
, ANàLYSIS OF RAINFALL QATA
3.1
The rainfall data analyse8 herein pertain to 6 hour annual storms
?corded at New Delhi for a period of 10 years from 1956 to 1965. They
ive been arranged in such e manner that the storm starts with the first
burly rainfall and ends at the 6th hourly rainfall, although in reality
Le arrangement may be vitiated in some cases by the occurrence of a
Bizzle before the recording of the main _portion of the storm or by
beaks within the duration of the storm. The recorded data may be seen
I Table I.
FORMULBTION OF THE MATHEMûTICAL MODEL
:.1 To develop a suitable model to represent the time degendent
ndom process of the hourly rginfalls, the following non-stationary
rkov-chain niodel(l) was found to be consistently satisfactory.
.) Ven Te Chow, Handbook of Applied Hydrology, pp. 8-93,
McGraw Hill Book Co.

422
......... (1)
where xt x the hourly rainggll of any one of B annual
storms at the t hour,
xt-1 z the hgurly rainfall at the preceding or the
(t-i) h hour,
t = time in hour ranging from 1 to m,
r = Markov Chain Coefficient,
6~ = random component due to hourly rainfall xt ,
For the first hour when t = 1, the trend component r Xt,l become
zero and X1 may be taken to be equal to €1 . The Markov Chah Coeffi-
cient r and the random component €G may be determined from the give1
rainfall data by the method of least squares by fitting a straight
line between Xt and Xt-1.
4.2 For the rainfall data recorded at New Delhi Station, the storm
duration m =6 hours and number of annual storms, Ns10. The distribi
tion parameters, mean and standard deviation of the historical rain.
fall data were determined for each hour and are given in column 2 ai
3 of Table II. The values of the random component et and the Markov
Chain Coefficient r were worked out by the method of least squares
and are shown in columns 4 end 5 in Table II.
4.3 In the present analysis based on sequential generation, the
oractice followed has been to generate 100 pseudo-random numbers fo:
uniform distribution of the first hourly rainfall by I.B.M. Compute:
1401, whose programme is given in igpendix I. These 100 generated
random mmbers of a uniform distribution have been taken as first
hourly rainfalls of 100 storms and have been utilized for computing
100 second hourly rainfalls by the Markov-chain model given in
equation (1).

4.4 The rainfall data have been generated for each successive hour on
the basis of the rainfall in tlx? previous hour according to the Markov
chain model formulated. Knowing the Markov chain coefficient r and
random component 6,for the second hour derived from the historical data
(vide Table II) a random series of 100 second hourly rainfalls can be
comouted by means of equation (i). These 100 generated second hourly
rainfall were then utilized to compute 100 third hourly rainfalls with
the help of Markov chain coefficient r and random component (vide
3
Table II) by using equation (i). This procedure has been repeated for
successive hourly rainfalls until serles of 100 hourly rainfalls for
all the six hours were generated. The involved operations were carried
out on IBM computer, 1620 as per programe given in Appendix II. The
sequentially generated data has been shown in Table III.
4.5 The cumulative probability function P(x) of the variate X may be
obtained by the following equation;
where ,.ho 5 Y & ,h~
.o (2)
fiois the lower limit of the variate X which may be assumed to be zero
an8 is the upper limit of variaue X.
4.5.1
In the present analysis, the total hourly rainfall of annual
storms have been worked out by adding all the six hourly rainfalls for
each storm of historical data as well as generated data as per column 8
of Tables I and III respectively. The cumulative probability per cents
have been evaluated by the use of equation (a for ten storms of the
historical data as ?er column (9) of Table I as also for 100 storms of
the generated data as per column (9) of Table III.
423

424
5. EsTIYVìTION OF DESIGN FLOOD WITH THE AID OF GENERATED RAINFALL S ~ I
5.1
It is possible to derive a series of runoffs from the generated
rainfall series provided that the relationship between rainfall and
off for a particular basin is known. In the present Case, in which
sequential generation techniques have been applied for only on rainfall
station in the Yamuna catchment, vie. New Delhi and for wNch 110 rain-
fall-runoff relationship was available, an assum3tion has been made tha
surface runoff from rain storm is 80 per cent of rainfall during the
period of high floods when most of the catchment is saturated and in-
filtration losses are of low order. Based on this preamble, a series
of runoffs may be assumed to be generated. The abovezentioned series
can be utilised to compute the peak floods with the help of unit
hydrograph developed at the gauge site and other methods.
5.2 The series of 100 peak floods comguted for the river Yamuna at
Okhla (catchment area = 6811 sq. Kms.) shown in column 10 of Table
III has been used to derive the following stochastic model on the
Dasis of princinles of stochastic hydrology reported earlier for
the estimation of design
wnere yo is the design flood and Tk is the recurrence interval.
5.3 From Mg. 4 based on above, the design flood with a recurrence
interval of 500 years works out to 7794.5 cumec for the Yamuna river
at Okhla (Delhi). It may however be 2ointed out that this should be
talen to be more as an illustration of the application of the techn-
ique of sequential generation for the estimation of the design flood
in view of the limitations of the rainfall data for the entire catch-
ent and - ilitv of a rainfall ru ela t i o =hi D
(2) ,,ttZharya>A8P., Jindal, S.R. and RamJ%ff :Estimation of Design--
Flood of the Ganga Fiver by processes of Stochastic hydrology",
U. 2. Annual Besearch rieport, 1967 (Technical Memorandum No. 37) .
1

5. DISCUSSION OF RLiSULTS
6.1 Figure 1 gives a comparison between the worst possible raiaall
;tarm of the historical data and the generated series an the basis of
I gra3hical plot between time in hours and hourly rainfall. It is
.ndicated that there is close Conformity for the entire storm dura-
,ion comorising six hours.
6.2
425
Gra^hical comparison has been made bbtwecn historical and generated
Iata with respect to cumulative probability distributian of rainfall at
he third hour, at which the peak rainfall was rècorded in the observa-
ional as well as seqtientially generated data as per Figure 2. Close
ionformity is indicated between the two distributions.
6.3 similar comlarison has also been made for the two series for total
ix-hourly rainfall for the annual storms as shown in Figure 3. Close
ionformitg is observed in this case as well, both for ehird hourly rain-
'all and total six-hourly rainfall, which provides added evidence regard-
ng the representativeness of the sequentially generated series.
6.4 mom the generated rainfall series, it has been ,possible to derive
cm?
runoff serles which has been utilised toda series of 100 peak dischar-
es. The latter orovide the background for the derivation of a stochastic
ode1 wherefrom a hypothetical 500-year design flood for the Yamuna
iver at okhla (Delhi) may be estimated.
7.1
COI;CLU~IOMS
he size of the historical data, particularly in such investigations
herein this may be a limiting factor for analytical studies.
7.2
The technique of sequentiaï generation may be adopted for increasing
storm rainfall is a time dependent raridom series and may be treated
y El finite duration discrete non-stationary process that is ameneble to
athematical formlation and analysis. For rainfall at New Delhi, the
istorical data of hourly rainfall in the annual storm has been regresented
y nan-stationary Markov-chain model, the data consisting of ten .six-

426
hourly storms.
7.3 A compari n of the historical and generat LI 100 y ar data, both
for third hourly rainfall and total six hourly rainfall, shows that the
sequentially generated series is fairly representative of the charactei
istics of the historical data.
7.4 The generated hydrologic series of rainf'all has been utilised to
estimate the design flood of the Yamuna river at Okhla(De1hi) with a
recurrence, interval of 500 years.
The authors wish to acknowledge the useful help extended by
Messrs Ramjeet and D.C.Mltta1 in the analysis and computational work.
APPENDIX I
Fortran program for the generation of PseudÕrandom
numbers in Uniform Mctribution. 4
SE Q spm FORTUN STATENE2iT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C GEN-RATION OF 100 ?SEUDORANDUM Nuz.[BERS IN
UNIFORM DI SJXtBUTION
10 IALFA- 10**17 -C 3
IRN1 = 10*(10**19-1) -b 7
8% 0.0
N=l
91 READ 95,B
95 FC-WAT (F4.2)
Do 2 I = 1,100
IRN IRNl*IALFA
RSN= IRN
RSJ!N= RUN* 10.0**(-20)
SN = (B-A)* RSTN .) A
R=N+1
IRNI= Im
2RIhT loo, SW
100 FORMAT (2E 16 e 8)
2 COICTIhUE
IF (SENSE ShkTCS O) 92,3
3 GO TO 91
92 STOP 555
END

10
100
APPENDIX II
Fortran program for the generation of i00 slx hourly
rainfall storms for New Delhi Station by Markov-chain
Model.
DIMEESIONS X(iOO),A(lOO) ,B(100) ,C(iOO) ,D(iOO) ,G(100),
DIEIEKSI ONS Y ( 100 ,V ( 100 )
READ 100, (X(I), I = 1,100)
FORMAT (~oF7.4)
cupi = 0.0
SUMA = 0.0
SUMB = 0.0
SUlC = 0.0
m!D = 0.0
SUMG = 0.0
smfl = 0.0
DO 200 I = 1,100
b(1)' 0.973 -k 1*551*X(I)
B(1) 2 15.023 46.694*A(I)
C(I) = 12.297- 0*036*B(I)
D(1) z L.871 + O. 106*C(I)
G(1) = 0.138 4 0.400*D(I)
Y(1) = X(1) + A(1) t BU) t C(l) -k D(I)S G(1)
V (I ) = 664.9 *Y (I )
X(1)
suMx= s w+
SUMA = SUMA t A(1)
SüMl3 = SUMB f B(I)
swc SUMC -t C(I)
SUMD = SUMD + D(1)
SUMG = SUIVIG + G(1)
SUMV = smn +V(I)
PUNCH300, X(I ,A (I 1 , B(I 1 , C (I ) , D( I , G (I ,Y (1
PUNCH350 ,V (I )
350 FORMi1T (FS0.4)
300 FORMAT (7F10.4)
200 CGEJTI NUE
PUNCH 400, SUMX, S W
400 FORMAT (6F12.4)
,"UI\+CH 500,SUMV
500 FORMAT (F 25.4)
STO?
ENI)
, SUME , SUMC, SUMD , SUMG
42 7

428
TABLE I
Historical hourly rainfall data for annual storms for New Delhi Station
1
2
3
4
5
6
7
a
9
10
20; 7.56
13.9.57
29-90 58
6.9.59
5.10.69
24.9.61
20.9062
8.8.63
14.7.64
2.9.65
0.25
1. 80
2.00
‘O. 10
O. #
o. 10
0.30
1.50
O. 40
1.90
O. 50
2.10
3.30
4.60
O. 80
0.40
O. 50
le 80
1.50
7.80
19.30
22. so
42.00
54.20
19.10
8.50
21.50
new
30.00
61.20
14.75
8.10
13. so
3.50
9000
5.50
9.10
22.00
17.20
9.30
4.06 1.02
5030 3060
11.90 5.60
0.50 0.20
2.00 0.40
2.40 0.08
0.10 0.10
1.80 1.50
1.80 0.90
0.70 0.20
39.88
43.40
78.30
63.10
31.70
16.98
31.60
56. So
51.80
81.10
49.2
53.5
96.5
770 8
39.1
20.9
39.0
69.7
63.9
100 o
TABLE II
fa
Parameters of the Markov-Chain Model(hour1y rainfall of annual storms
of New mihi ‘Sation.
Time í t 1 Mean (mm/hour) Stendard Random Markov-Chai n
in hours devi at i on component coefficient
(mmhour 1 et r
1
1 2 3 4
-
b
o. 875 O 8122 .I
2 2.330 2.3522 0.973 lo 551
3 30.620 16o7600 15.023 6.694
4
5
11.195 5.8190 12.297 -0 036
3.056 3.4928 1.871 0.106
6 l b 360 1.8311 0. 138 0.400
II

429

Co
I
431

432
FIG 1 - DISTRIBUTION OF WORST RAINFALL STORM
FOR NEW DELHI
10 30 50 80 90 95 99 99.8 999!
CU M U L A T IV E PR OB AB ILtTY PERCE NT
F IG.2 - CUMLILATIVE PROBABILITY DISTRIBUTION OF
THIFiC 40URLY RAINFALL IN ANNUAL STORMS

E
O
I
-I
4
I-
O
k- 20
433
40 60 80 90 95 98 99 99.8 99.99
CU MU LAT IVE PROBABIL I TY PERCE NT
RECURRENCE INTERVAL (Tk) IN YEARS
FIG.4- STOCHASTIC MODEL FOR ESTIMATION OF
DESIGN FLOOD (Yo1 FROM RECURRENCE
INTERVAL ( TK)

ABSTRACT
THE USE OF STOCHASTIC MODELS IN A HYDRO-AGRICULTURAL
DEVELOPMENT PROJECT IN LEBANON"
by
J.H. Visser
Stochastic modelling techniques were employed in order to
provide long sequences of monthly streamflow and water demand needed
for irrigation scheme design (the historic flow records being too
short to serve this purpose).
The water requirement calculations (with the Blaney Criddle
formula) were based on generated series of monthly rainfall and
monthly mean temperature. The generation "in phase" of these variables
with streamflow, ensured that a dry year was characterised by high
demand and low flow, This strategy of Ilin phase" generation was
preferred to the more usual treatment of assuming a fixed annual
cycle of demands and allowed for a better assessment of the design
parameters and a better economic evaluation.
The Ilin phaset1 series required the generation of
- monthly rainfall (simple model due to absence of
persistance)
- monthly mean temperature (mixed model with auto regression
and linear regression on monthly rainfall)
- annual streamflow (with linear regression on annual
rainfall )
- monthly streamflow, related to the annual flow, using auto
regression (for stations having a historic record of more
than 10 years)
- monthly streamflow, related to the annual flow, using auto
regression of a deseasonalized variable and and linear
regression on the same variable of another (better) station.
(for stations having less than 10 years of record).
Undertaken jointly by the Government of Lebanon and the Food and
Agriculture Organ2sation of the United Nations.

436
RESUME
Des méthodes stochastiques ont été utilisées pour pouvoir
disposer de séries longues dtapport et de demande mensuels, nscessaires
pour Irétude dfun projet d'hrigation. CLes séries historiques
d'apport étant trop courtes pour être utilisées).
Les besoins en eau (calculés avec la formule de Blaney-
-Griddle) ont été basés sur des séries générées de la pluie et de la
température mensuelles. La génération "en phase" de ces variables
avec les apports a fait que l'année sèche se caractérise par une
demande élevée et des apports faibles.
Cette stratégie de génération Ifen phase" a été préférge par
rapport à la méthode plus habituelle dlun cycle annuel fixe de la
demande, et a permis une meilleure &valuation de la gestion diun
projet d'irrigation aïnsi quiune meilleure évaluation économique.
' -
Les séries "en phase" ont nécessité la génération de
- pluie mensuelle (modèle simple vu l'absence de persistance)
- température mensuelle (modèle mixte de régression sérielle
et de régression linéaire par rapport ?i la pluie mensuelle
- apport annuel (régression simpie par rapport à ia pluie
annuelle)
apport mensuel, par rapport à ifapport annuel, avec
régression sérielle (pour les stations ayant au moins 10
années d'observations)
- apport mensuel, par rapport à ifapport annuel, avec
régression sérielle diune variable désaisonnalisée et de
régression linéaire par rapport à la même yariable drune
autre (meilleure] station (pour les stations ayant moins de
10 annges d'observations).

1 - INTRODUCTION
43 7
1.1. One of the objectives of the UNDP/FAO Project LEBANON 13,
concerning the hydro-agricultural development of North Lebanon, was to study
an irrigation scheme of about 7 O00 ha in the KOURA-ZGHARTA region. For the
water supply of this scheme a dam has to be constructed on the Aasfour river.
The reservoir inflow can be provided by the Aasfour discharges together with
part of the streamflow of an adjacent river.
To assess the performance of the design reservoir long series of
streamflow are needed to be routed through such a reservoir, Such long series
of historic records were missing and the presence of outliers (very wet and
several consecutive dry years), made it virtually impossible to establish
with any confidence a return period for these outliers.
1.2. The hydrologic information available in the Project area was
based mainly on the following data :
- -
two streamflow series with 14 years of record
-
thirteen streamflow series with 3 to 5 years of record
several rainfall series of about 30 years of record
some temperature series of about 15 years of record.
A good correlation exists between annual rainfall and streamflow
but low values are found of the correlation coefficient between monthly rainfall
and streamflow. This can be explained by the fact that the response of
the catchments to rainfall has a delay factor of one to two months due to the
presence of snow and / or springs. It was thus impossible to apply the conventional
technique of extending the shorter streamflow records by correlating
them with the longer rainfall records, Unfortunately monthly streamflow data
are needed for reservoir analysis studies,
As the conventional techniques were unable to provide these data,
the use of hydrological modelling techniques became necessary.
1.3. Two main categories of mathematical models can be used in
principle for this kind of problems : Deterministic and stochastic. The first
category permits to extend the length of the historic streamflow series to
the same length as the (longer) historic rainfall record. The stochastic mo-
delling however permits to generate synthetic events of any length adequate
for certain design purposes.
A mixt use of a stochastic input into a deterministic model could
be useful in principle but unfortunately no such valuable generating models
for daily rainfall existed.

43 8
1.4. The statistics, such as the expected frequency of failure
of the design system, depend largely on the variation of the streamflow, i.e.
on the values of the variance of monthly and annual flow. The stochastic model-
ling techniques can improve the estimate of the variances of the shorter
records by using the infoption available in the longer series.
It was for all these reasons that the Lebanon 13 Project decided
to apply stochastic modelling techniques.
1.5. In order to apply generated series of streamflow in the
reservoir simulation studies it was necessary to generate also long series of
rainfall and temperature in order to calculate long series of water demand. To
avoid generating series of streamflow, rainfall and temperature that were un-
correlated, a method of "in phase" generation was adopted, This phasing will
ensure, for example, that during a dry year the values of streamflow and rain-
fall are both low, together with high temperature values resulting in high de-
mand for the year.
1.6. For the calculation of crop water needs the Blaney-Criddle
formula was used in view of the insufficiency of data for the Penman method.
However from the point of view of the methodology, there 2s no objection to
replace the Blaney-Criddle formula by that of Penman or another. The methodo-
logy for adjusting water resources and water demand as applied to North Lebanon
is explained schematically in the attached flow chart.
2 - CHOICE OF TYPE OF STOCHASTIC MODELS
2.1. The stochastic models, described hereafter, for the gene-
ration of long time series for different variables which are mutually in phase,
were proposed by Mr. J. Bernier, Chief of the Statistics Group at the Labora-
toire National d'Hydraulique, Chatou (France) and consultant to the North
Lebanon Project for stochastic hydrology. The models chosen were in response to
the availability of data and other local conditions as well as to the objecti-
ves of the study in particularly to provide input data for the simulation
studies, which explains the use of monthly values of the different variables.
-
2.2. The elements on which this choice was based were :
-
a caracteristics of available data :
streamflow : the presence of 2 series of 14 years of record

-
The
439
- temperature : the presence of some series of about 15 years of
record and a significant value for the corre-
lation between monthly rainfall and temperature
during spring and autumn.
- rainfall : several series of 30 years of record and a good
correlation between rainfall and streamflow on
annual basis but a bad one an monthly basis
(due to snowfall and / or karsticity)
- perennial flow of the rivers.
(the temperature and rainfall series together permit the use of
Blaney-Criddle's formula for the calculation of crop water needs)
requirements imposed by the methodology used for adjusting water resour-
ces and water demand :
- the series of streamflow and demand have to be in phase
- the "being in phase" of streamflow and demand requires automati-
cally the' "in phase" generation of the series of rainfall, tem-
perature and streamflow.
2.3. The short streamflow series (3 to 5 years of record) can
only be used after "deseasonalisation" of the variable (3) resulting in series
of 36 to 60 months of record in which the different characteristics of the par-
ticular months have been neutralised,
2.4. The following transformations of the variables were necessa-
ry in order to be able to use the normal distribution :
- the logarithm of the discharges instead of the discharges
- the square root of the rainfall instead of the rainfall (the
temperature is taken without any transformation).
2.5. The particular features described above led to the applica-
tion of the following generating models :
1) A simple model for monthly rainfall due to the absence of
persistance in the monthly rainfall series. This model uses
only mean and variance of the historic record together with
generated random numbers (eq.1).
2) A mixed model for monthly temperature using autoregression
plus regression on another variable (monthly rainfall) and
a random number generator (eq.2).

44 O
3 - DESCRIPTION OF MODELS USED
3) A simple regression model for annual streamflow using re-
gression o,n annual rainfall plus a random number generator
(for stations with at least 10 years of record) (eq.3).
4) An autoregression model for monthly streamflow using the
relation monthly / annual streamflow to give monthly stream-
flow plus a random number generator (eq.4).
5) A mixed autoregression model for monthly streamflow of a sta-
tionary ("deseasonalised") variable with regression on the
same variable of another station plus random number generator
(stations having less than 10 years of record) (eq.7).
3.1. Generation of monthly rainfall
Hypothesis : normal distribution of
(Pt = monthly rainfall).
The statistical analysis of the historic series of monthly rainfall
gives the values of mt = the mean of the fit of the month t (12 values)
and the s2t = the variance of the fit of the month t (also 12 values),
Generating model :
fitemt -+ st . u (eq.1)
( 6 = o for Pt = 0)
where : u is the random normal deviate with zero mean and unit variance
- u = o, s2(u) = 1
The square root of the rainfall depends thus on mt, st and U.
3.2. Generation of mean monthly temperatures
Hypothesis : normal distribution of Tt (Tt = mean monthly tempe-
rature of month t).
The generator model uses a correlation of temperature with

ainfall and is as follows :
Tt,i = mean monthly temperature of month t (t runs from 1 to 12 and i
represents the ith month after the start generation, i = 1,2 .... >
Ut
K i =
mt
U
vt
al,t
= the mean of the mean monthly temperatures of month t, in period
of record (12 values)
square root of the rainfall of month t,i
= the mean of the square root of the rainfall of month t, in period of
record (12 \talues)
= random nonual deviate with 3 = O and s2(u) = 1
= the variance of the residuals of T in the month t, in period of
record (12 values)
441
and d2,t = partial regression coefficients (for method of calculation
see standard works, e.g. Ven Te Chow "Applied Hydrology", page 8:6û)
For the first value of Tt,l, ill the following fonuula is used :
Tt-1 = M t-1 + G u'
where M and v are known and a single value for the random normal deviate u'
is sufficient to define the value of Ttml.
Once the coefficients of the generator model for mean monthly tem-
perature are known for each month, a sequence can be generated which will be in
phase, at the monthly level, with rainfall.
3.3. Generation of monthly streamflows for stations having at
least 10 years of record
The poor degree of correlation found in the study area between
monthly streamflows and monthly rainfall, due to the effect of snow and karsti-
city of the basin, necessitated first a correlation of the annual streamflows
(14 years of record) and the annual rainfall (the same station as used in para
3.1. above). This was achieved through the following equation based on the
hypothesis of a normal distribution of log Q :

log ~a = M +e( q- m> + K u
M = the mean of the logarithm of the annual flow Qa (14 values of Qa)
m =
u = random normal deviate
íeq.3)
the mean of a (Pa = annual rainfall for the rainfall station 32 years)
v = variance of the residuals of log Qa
In order to adjust to the long series of rainfall (32 years)
the following correction to M was prepared. This type of correction is only
necessary when records of the period of M are not typical of the period of m.
In fact, since the rainfall record (32 years) is longer than
the streamflow record (14 years) and the two are correlated, the estimate M
of the population mean of log Q and the estimate v of the variance of the
residuals can be improved, leadtng to revised estimates M;! and v2 as follows :
2 2
s2 (log Qa> = s1 (log Qa) - r
2
where the suffixes 1 and 2 refer to the 14 and 32 year records respectively.
2
The value of 62 (log Qa) so obtained can be used to derive an
improved estimate v from the relation :
a
(eq. 3b)
2 2
v 2 = (1 r s2 (log Q,) (eq.3~)
All the coefficients of equation 3 being known, a long series
of logarithms of annual flow Q, can be generated and will be in phase with
the annual rainfall.
The next step is the generation of the variate Zt,i = log Qt,i - log Qa (eq.4)
(eq.4)

443
where :
- - Zt = mean of (log Q
t,i log Q a
for period of recorii (12 values, t = 1,2,... 12)
r = correlation coefficient of Zt on ZtWl (12 values)
t
s(Zt> = standard deviation of Zt (12 values)
n
vt = variance of the residuals = - . 2 2
s
n-2
2
u = random normal deviate with u = O, s (u) = 1.
-
(2,) . (1 C. rt)
The application of equation (4) then gives a generated series
of Qtai following.the transformation
-
Qt,i - Qa e ('t,i'
íeq.4a)
for each month. The annual totals of these Qtai should be equal to the values
of Qa generated with equation (3) but this will not usually be so, in
which case the following correction must be made for the generated Q
. tai
(eq. 4b)
where :
E 'a
QL,i - ' Qt,i
Q'a
I
Qt,i
Qa
QIa
= revised value of Qtai (monthly generated flows)
= annual flows generated by eq. (3)
= annual sum of monthly flows Q generated by eq. (4)
t,i
The result of these processes is a generated series of monthly
streamflows which are in phase annually with rainfall which is in turn in
phase monthly with the mean monthly temperatures.
3.4. Generation of monthly streamflows for stations having less
than 10 years of record
For short series equations (3) and (4) cannot be used because
a short series does not permit a sufficiently precise calculation, month by
month, of the mean, variance and correlation coefficient. To avoid this diffi-
culty the observed series of the monthly streamflow Qt i can be "deseasonalized"
to produce a new series Y giving, in the case of 4 yeah of observations, 48
values of Y. To "deseasonalize" one of two transforms was used :

444
Y = log Q - a cos (-
t,i
12
(eq.5)
where (eq.5) : the origin was taken as the month of November (t = 1)
the phase determined in such a way as to place the mean
e = maximum flow in the required month, which also determines
the month of minimum flow which will be six months later.
a
= the amplitude of the variate (the logarithm of the monthly
flows) chosen in such a way as to give the best fit to
the observed maximum and minimum monthly values.
The choice of the transform depends on the shapc of the hydro-
graph. In the Lebanon project equation (5) was found to give less good re-
sults and therefore equation (6) was used.
Each of these equations gives a series of Y which combines all
months. From this series the mean (i) can be calculated and also the variance
(

-
X
=I mean of X (during period of record)
u = randa normal deviate
v = variance of the residuals of Y
44 5
When all the coefficients of equation (7) are known and data have
been generated for the "long" record station as described in para. 3.3, the model
can be used to generate the long series of Yi and so, using the transforms in
equations (5) and (6) above, long series of monthly streamflows Q
t,i'
The resulting values of Qt,+ are related to the monthly generated
flows of a station having a long record which are themselves in phase annually
with the annual rainfall. The annual rainfalls are the sum of the monthly rainfalls
which themselves are in phase with the mean monthly temperatures.
- 4 CONCLUSIONS
4.1. The checks used on the generated time series are the sta-
tistical moments (mean, variance and coefficient of variation) and periodicity
(winter - summer). It was found that the generated time series were in general
of good quality but that the value for the coefficient of variation (i.c. the
variance) was too high. This did not matter in the Lebanon case because the
reservoir simulation studies done with these time series kept us on the safe
side, but for the sake of completeness some kind of correction should be intro-
duced in future.
4.2. All the programmes have been written in Fortran IV for the
IñM 1130 computer in such a way that they can be used separately or in series.
In this latter case the calculation time is about one hour per run. To obtain
as output in one run the results of system simulation (reservoir size, irrigable
area, failures and effects on other water users) the following input data are
needed :
(a) a historic record of rainfall (monthly)
(b) a historic record of mean temperature (monthly)
(c) a "long" (12-14 years in North Lebanon) historic record of streamflow
(monthly
(d)
a "short" (3 to 5 years) historic record of streamflow (monthly)

446
the duration of sunshine as a percentage p of the maximum possible
the monthly crop coefficient K (by crop)
the maximum usable soil moisture storage (by ero;)
the coefficient of growth of the plant (by crop)
the phasing of irrigation development of the whole area, and the
subdivision of the area by crop (in %)
the rate of changedver from the present olive groves to the new crops
coefficient of irrigation efficiency
geometric characteristics of the reservoir
P (e) and (f) are necessary for the application of the Blaneydriddle formula.
e

I II I
Il
m
P 00
I I m
FLOW CHART FOR DATA GENERATION AND SYSTEM ANALYSIS
I 1. II ..... i ace -. 3.7.4 or t a North Ltibonon irriqaion scheme
1-1 i
o' indirti. Bencator
Jaouory 1972 lidopied from Prqe~i Droir~ng AE-2513

RELATIVE IMPORTANCE OF DECISION VARIABLES
IN FLOOD FREQUENCY ANALYSIS
Wallis, J.R.
IBM, Thomas J. Katson Research Center , Worktown Heights , N .Y, , USA
ABSTRACT
Matalas, N. C.
U.S. Geological Survey, Washington, D. C., USA
Monte Carlo simulations were used to assess flood and overde-
sign losses that result from differing choices of assumed frequen-
cy distribution, plotting position, criterion of best fit and
length of record. Probabilities of best fit for an assumed world
distribution, given a real world distribution, are given.
RESUMEN
El método de simulación de Monte Carlo se utiliza para eva-
luar los daños producidos por máximas crecidas en funci’on de las
leyes de distribución de frecuencias, de las estaciones utilizadas
y de la calidad y extensión de las series hidrológicas. De la mues
tra puede obtenerse el valor minimo teórico de los daños estimados.

450
Introduction
In many cases, the design of multipurpose water resource
systems includes flood control as one of the purposes. While the
design process may specify the sizing of flood control structures
as a function of the T-year flood, the design process must cope
with the uncertainty as to the magnitude of the T-year flood.
Given that floods are a random phenomenon, the magnitude of the
T-year flood depends upon the underlying probability distribution
of flood events and the values of the distribution's parameters.
Among the objectives of flood frequency analysis is that of
determining the magnitude of the T-year flood, referred to as the
design flood. While the underlying distribution of floods is
unknown, an estimate of the design flood can be provided. A dis-
tribution may be assumed or chosen in accordance with some criter-
ion of best fit to observed flood sequences. Given an observed
flood sequence of length n, and an assumed or chosen distribution
estimate of the distribution's parameter values, the design flood
can be derived. These estimates are subject to sampling errors,
the magnitudes of which depend upon n, and on uncertainties that
are only partially a function of n. To reduce sampling errors,
longer flood sequences are needed. To acquire longer sequences
through direct observation might necessitate the delays in the
design of the water resource system. Delays would be economically
feasible if over the period of data collection no benefits were
foregone. In those cases where benefits would be foregone effec-
tively longer sequences might be obtained through regional analyses.
However, even with a very large but finite flood sequence,
uncertainty would still exist in the estimate of the T-year flood.
The Uncertainty arises because the assumed or chosen distribution
used to estimate the T-year flood does not necessarily have to
be the correct real world distribution, and in fact if a criter-
ion of best fit is used, the chosen distribution might vary given
another flood sequence of equal length.
By using an estimate of the T-year flood as the design flood,
either of two types of losses is likely to be incurred. The first
type refers to overdesign costs of flood control structures which
would be incurred if the estimated T-year flood exceeded the true
value of the design flood. The second type refers to the down-
stream damages which would be incurred from underdesign if the
true value of the design flood exceeded the estimate of the T-year
flood. In the design process, what is of concern is not how well
.a particular distribution fits an observed flood sequence par se,
but to what extent the two types of design losses are affected by
the choice of a particular distribution. To the designer, the

451
criterion of best fit refers to choosing a distribution to minimize
design losses.
To gain some insight as to the magnitudes and sensitivities
of the design losses to uncertaintfes in the choice of a flood
frequency distribution and estimates of the distribution's param-
eter values, several computer-based experiments, employing Monte
Carlo techniques, are currently being performed. In this paper
the nature of these experiments is briefly discussed, and some
experimental results as to the probabilities of fitting of
observed flood sequences with particular distributions are
presented.
Monte Carlo Experiments
Two sets of distribution functions are considered. The first
set, referred to as the real world set, consists of several dis-
tributions, any one of which may be the underlying distribution
of floods. From the real world set, a distribution function is
chosen where the distribution's parameters values are related to
the mean, u, the standard deviation, o, and the coefficient of
skewness, y. For this distribution, 18,000 flood sequences of
length n are generated.
The second set of distribution functions, referred to as the
imagined or assumed set, contains several distributions, any one
of which may be fitted to observed flood sequences. Each element
of the imagined set is fitted to each of the generated sequences,
and on the basis of various methods for defining plotting positions
and measuring goodness of fit, the particular distribution of best
fit is determined for each generated sequence. From each of these
distributions, the flood having an exceedance probability of 1/T
is determined. These floods are estimates of the real world flood
of exceedance probability 1/T.
Initially, overdesign and underdesign linear loss functions
in terms of the difference between the real world T-year flood
and its estimate are assumed. Nonlinear loss functions will be
considered in subsequent experiments. Given the 18,000 values of
the differences between the real world T-year flood and its
estimates, the probabilities of incurring overdesign and under-
design losses and the expected values of the losses are estimated.
Similarly, these values are estimated for each of the other dis-
tributions belonging to the real world set.
Three flood control design objectives are considered:
1) minimizing the expected overdesign losses, 2) minimizing the
expected underdesign losses, and 3) minimizing a weighted sum
of the expected overdesign and expected underdesign losses. For

452
the third objective, the weights, say CI and ß, where a > O, ß > O,
and o: + ß = 1, are varied. Among the methods for defìning plotting
positions and measuring goodness of fit, the particular method and
measure by which the various design objectives are met were deter-
mined. The sensitivities of the design losses to less than optimal
choices of the plotting position method and measure of goodness
of fit were assessed.
The experiments were carrìèd out for every feasible point in
the following experimental hyperspace:
p = 2600
u = $00
y = O, 1/4, 1/2;*, 1, 1 . 1 4 K 2
n = 10, 30, 50, 70, 90
The results of these experiments are conditional on the distribu-
tions belonging to the real world set. Subsequent experiments
will consider prior information on the real world distribution
function and regional estimates of the distribution's parameter
values.
Probabilities of Best Fit:
Some preliminary results of these experiments are presented
namely, the probabilities of best fit. Both the real world and
imagined world sets consisted of three elements -- the normal
distribution, the log-normal dìstribution, and the Type I extremai
(Gumbel) distribution. For each distribution belonging- to the real
woerfd sat) i8000 sequences were generated f o each ~ feasible
point in the experimental hyperspace. Floods for each sequence
of length n were ranked in order of magnitude from the largest,
having rank m = 1, to smallest, having rank m = n. The flood of
rank m was assigned an exceedance probability, P[m,nl , by both
the "Weibul method," defined as
and by the "Hazen method," defined as
(See Chow: 1964).
P [m,nl = m/(n+l) (1)
W
P,[m,nl = (2m-i) / 2n (2)
For a given element of the real world set and a given element
of the imagined world set, two sets of differences of flood magni-
tudes were formed for each generated sequence. The first set con-
sisted of the differences between the observed, that is generated,

453
floods and the corresponding imagined world floods having exceed-
ance probabilities defined by Pw(m,nJ. Similarly, the second set
of differences was based on exceedance probabilities defined by
PH (m,n).
Two measures of goodness of fit were considered -- the sum of
squares of the differences in flood magnitudes and the sum o€ the
absolute differences in flood magnitudes. For each sequence based
on an element of the real world set and relative to each method of
assigning exceedance probabilities, the element of the imagined
set which provided the best fit to the sequence of floods was
determined, where best fit was defined by each of two criteria --
minimum sum of squares of differences in flood magnitudes and
minimum sum of absolute differences in flood magnitudes.
The probability of the event that a sequence, having a par-
ticular element of the real world set as its underlying distribu-
tion, is best fitted by a particular element of imagined world
set, relative to a particular method of assigning exceedance
probabilities and a particular criterion of best fit, was esti-
mated by N/18,000, where N denotes the number of times the event
occurred and 18,000, the total number of times the event could
have occurred. The probabilities for each of the 36 points in
the event space relative to each feasible point in the experi-
mental hyperspace were determined.
Remarks
For the experimental hyperspace, the estimates of the probabil-
ities of best fit multiplied by 1000 over the event space are given
in Lahles 1 through 9, where MSS denotes minimum sum of squares,
MSAD denotes minimum sum of absolute differences, N denotes the
normal distribution, G denotes the GuInbel distribution, and L denotes
the log-normal distribution. These estimates were based on the follow-
ing additional experimental operating rule -- if the computed value
of the coefficient of skewness, y, for a generated sequence was equal
to or less than 0.007, then the sequence was considered to have been
drawn from a normal distribution. While these probabilities give
some indication as to the power for identifying the real world flood
distribution from an observed sequence, they do not give any indica-
tion of the optimum strategy to use for choosing a design distribu-
tion. Interpretation of these experimental results in terms of
overdesign-underdesign strategies will be the subject of subsequent
papers.
Reference
Chow, Ven T. (1964). Handbook of Applied Hydrology, McGraw-Hill.

454
Table 1. -- Real World is normal
Weibull Hazen
MSS MSAD MSS MSAD
n N L G N L G N L G N L G
~~ ~~
10 527 468 5 602 314 84 523 289 188 556 234 211
30 531 432 36 608 341 51 530 432 38 579 364 58
50 528 455 17 614 360 26 526 465 9 489 387 24
70 526 468 6 613 377 io 526 473 2 595 397 8
90 532 466 2 620 374 5 529 470 1 603 394 4
Table 2. -- Real World is Gumbel with skew = 1.14
Weibuil Hazen
MSS MSAD MSS MSAD
n N L G N L G N L G N L G
10 216 766 i8 287 523 190 212 416 371 245 358 398
30 40 646 314 78 624 299 40 658 303 57 590 352
50 7 612 380 27 642 331 8 717 275 19 629 352
70 2 596 402 10 625 364 2 756 242 7 641 352
90 0 591 409 6 622 372 O 767 233 3 635 361
Table 3 . -- Real World is log-normal with skew = i/4
Weibull Hazen
MSS MSAD MSS MSAD
n N L G N L G N L G N L G
10 441 551 8 521 370 109 436 323 241 474 261 265
30 342 573 85 433 469 99 341 570 89 395 481 124
50 279 659 62 382 550 68 279 683 38 347 583 70
70 234 727 38 339 621 41 232 750 18 313 649 $8
90 205 771 24 313 661 26 204 788 8 290 688 23

Table 4. -- Real World is log-normal with skew = 112
Weibull Hazen
MSS MSAD MSS MSAD
n N L G N L G N L G N L G
455
10 359 630 11 442 421 i37 356 352 292 393 290 317
30 197 643 161 281 554 165 196 641 163 240 556 204
50 116 721 i63 202 655 143 116 780 i04 169 678 153
70 71 789 i40 147 735 117 70 859 71 124 758 118
90 46 837 i17 118 789 93 46 901 53 97 812 91
Table 5. -- Real World is log-normal with skew = v1/2
Weibull Hazen
MSS MSAD MSS MSAD
n N L G N L G N L G N L G
10 302 684 i4 382 461 157 298 375 327 333 315 353
30 i14 662 224 i86 597 218 i14 662 224 i50 582 268
50 48 688 264 106 672 222 48 775 177 83 677 241
70 22 712 266 64 719 217 21 832 i47 49 731 221
90 11 737 253 43 756 201 11 865 125 32 765 203
Table 6. -- Real World is log-normal with skew = 1
Weibull Hazen
MSS MSAD MSS MSAD
n N L G N L G N L G N L G
10 235 750 i5 309 508 183 231 406 362 260 350 390
30 50 658 292 96 631 273 51 663 286 72 594 334
50 13 625 363 39 649 312 13 736 251 27 640 333
70 5 607 388 20 651 330 4 767 229 i4 653 334
90 1 592 406 io 657 333 1 787 212 6 667 327

n
Table 7 .-- Real World is log-normal with skew = 1.14
Weibull Hazen
MSS MSAD MSS MSAD
N L G N L G N L G N L G
10 209 775 17 281 528 192 205 418 376 234 364 403
30 34 651 314 70 638 292 34 660 306 52 594 354
50 7 606 386 24 638 338 7 721 272 i7 633 351
70 2 586 412 9 635 356 2 755 243 6 641 353
90 O 582 418 4 632 364 0 777 223 3 647 351
n
Table 8. -- Real World is log-normal with skew =
Weibull Hazen
MSS MSAD MSS MSAD
N L G N L G N L G N L G
10 169 814 16 231 565 204 166 438 396 193 387 421
30 17 655 328 38 658 304 16 667 317 27 615 358
50 2 620 378 11 657 332 2 727 271 6 648 346
70 0 618 381 3 658 339 O 772 228 2 671 327
90 O 629 371 1 664 335 o 804 196 1 685 314
Table 9. -- Red World is log-normal with skew = 2
Weibull Hazen
MSS MSAD MSS MSAD
n N L G N L G N L G N L G
10 113 865 21 163 614 223 112 473 415 135 427 438
30 4 696 300 11 710 279 4 710 286 8 679 313
50 0 723 277 1 751 247 O 798 202 1 754 245
70 O 770 230 0 795 205 0 860 i40 O 807 193
90 o 810 190 O 826 174 O 903 97 0 842 158

AB STRA CT
SHOT NOISE MODELS FOR SYNTHETIC GENERATION
OF MULTISITE DAILY STREAMFLOK DATA
bY
G, WEISS
Department of Mathematics Imperi'al College
University of London
While multisite models for generating synthetic streamflow da-
ta on a monthly basis have been successfully used, adequate daily
models are lacking, In particular, existing models based on Gau-
ssian processes are unsuitable in reproducing the recessions which
are clearly observable in daily data. The models currently being de
veloped at Imperial College, London, under contract for the Water
Resources Board, England, are based on "Shot Noise" or filtered Poi
sson processes. These processes consist of a series of Poisson
events, each of which generates a pulse of random height and some
fixed recession shape, In the simplest of these models the pulses
consist of jumps which are exponentially distributed in magnitude,
and which decay exponentially with a fixed decay rate. This is a
continuous time first order autoregressive (Markovian) process, and
its instantaneous values have a Gamma distribution. This model can
be fitted to streamflow data so as to preserve the observed means,
standard deviations, serial and cross correlations of daily data.
Using a more complicated model consisting of two shot noise proce-
sses, monthly statistics can be preserved in addition to the daily
statistics. This model gave satisfactory results with data from SO-
me East Anglia sites,
-- RESUME
Alors qu'on a réussi a construire des modgles capables de €OU:
nir artific2ellement des sérri'es ae dé:its moyens mensuels en plusieurs
sites, :n manque encore de modeles satisfaisants pour les.va
leurs journalieres. I1 faut souligner en particulier que les modeles
stochastiques actuels, basés sur des processus gaussiens, ne sont
pas capables de reproduire les décrues qui sont faciles 2 mettre en
évidence dans les relev6s.journaliers. Les modèles qu'on est en
train de mettre au point a l'Imperia1 College (Londres), pour le
compte du Water Resources Board (Angleterre) sont basés sur le
"shot noise" ou processys de Poisson filtr'es. Ces processus se composent
d'une série d'évenements obéissant a une loi de Poisson, dont
chacun produit une impulsion d'amplitude aléatoire dêcroissant suivant
une forme déterminée. Dans les plus simples de ces modèles,
les impulsions consistente en des sauts dont les amplitudes aléatoA
res sont distribuées de facon exponentielle et sont affectées, une
fois produites, d'une décroissa?ce exponentielle dans le temps, la
constante de temps étant fixée a l'avance. Ceci constitue un proce5
sus (Markovien) autorégressif de premier ordre continu dans le temps,
dont les valeurs insta2tannées sont distribuées suivant une loi
Gamma. Le modèle peut etre ajusté aux données disponibles concernant
l'écoulement de facon a respecter les moyennes, les écarts-types et
les corrélations croisées des débits journaliers. En utilisant un
modèle plus compliqué, formé de deux processus ltshot noi~e'~, il est
possible2 en plus des caractéristiques statistiques des valeurs
journalieres , de conserver celles des valeurs mensuelles. Ce-modèle
a donné des résultats satisfaisants pour un ensemble de rivieres
dans l'Est de l'Angleterre.

Introduction
The present work is aimed at supplying multisite daily synthetic stream-
flow data for the British Water Resources Board. The Water Resources Board
is currently developing regional simulation programs to help in the planning
and operation of water supply, and sensitivity tests have shown that at the
level of detail used in these programs, monthly data are insufficient and
daily data are indeed required.
Generation of synthetic monthly data in Hydrology was apparently first
attempted in the Harvard Water Program [I], and has been developed as a useful
tool since ([2], [ 3J,[ 41). Basically, in the methods previously used, the
data or a transformed series obtained from the data are assumed to follow a
multivariate Gaussian 1st order autoregressive process. Some sophisticated
transformations and some higher order regression models have also been used
([5], [6])*
Autoregressive models based on the Gaussian distribution have also been
applied to daily data [7J, [SJ. Such models, however, are inadequate in the
following sense : one of the prominent features of daily streamflows is the
presence of peaks and recessions clearly observed in the data. Yet no model
based on the Gaussian distribution can reproduce these recessions, no matter
what transformation or what order of autoregression is used.
In this paper a class of models which reproduce recessions is introduced.
A simple model from this class, of the same degree of complexity as the
Gaussian 1st order autoregressive process is developed, and its implementation
for data generation described. The theoretical aspects of reproducing recessions
are discussed. Finally, the application of the model to some British
streamflows is illustrated.
Filtered Poisson Processes
Let N(t) be a P&isson process, let Y be a random variable, and let
w(t,y) be some function. Let the sequence ..., 7-1, TO, 71, .O be the times
of events of the process N(t), and let ..., y-1, yo, YI, ... be mutually
independent random values having the same distribution as Y, and all of which
are independefit of N(t). A filtered Poipson process X(t) is defined by :
For further details of these processes refer to CS].
A physical interpretation in hydrological terms can be given to the
filtered Poisson process. The events at random times T ~, given by the Poisson
process can be thought of as beginnings of rain storms. The random value ym
associated with T; could correspond to the amount of water in the rainstorm.
Finally, T and y , will produce a response in the flow given by w(t-.cm,ym)
and thus w?t,y) represents the system transfer function.

459
The foregoing interpretation is only approximate, since rainstorms are
not independent as the Ym'S are required to be in the definition. One can
however imagine that an independent series of climatic events exists initially,
and that w(t,y) is the transfer function which transforms such climatic events
into streamflow.
The Shot Noise Process
A particular linear filtered Poisson process was adopted for modelling
streamflow, which is referred to as the shot noise process. In a filtered
Poisson process, let N(t) be a Poisson process with rate 3 . Let Y be a
random variable with an exponential distribution and with mean 8, and let
w(t,y) = ye-bt, for t > O. The shot noise process is defined as :
The process has three parameters : 9 - the event rate, 8 - the average
jump height, and b - the decay rate. !he process has the following properties
(found by applying theorems in [9J) :
X(t) has a Gamma (Pearson type 2) distribution with parameters (Q, q/b),
and thus X(t) is nonnegative and positively skewed, and has probability density
function :
The moments of X(t) are given by :
From eq. (2) the process at time t+s, X
(s > o)
t+s), can be written as :
The two terms in (5) are independent. The first represents the effect
of events previous to t, and is equal to e-bs X(t). The second includes the
events in (t, C+s) and is the innovation term.
~~(t+s) one has
Denoting the innovation by

460
Thus, the shot noise process is in fact a 1st order autoregressive process
in continuous time. However, it differs from the Gaussian 1st order autoregressive
process in that ES(t+s), instead of being Gaussian, has a skewed distribution
with a positive probability of being exactly zero. This arises when
no events occur in (t, t+s).
Some Aspects of Modelling Recessions
When modelling a stochastic process in hydrology one often makes the in-
exact but not unrealistic assumption of a linear system. This amounts to
assuming that the process X(t) is of the form :
where dY(t) is a completely uncorrelated and independent process, which
describes all the randomness in X(t), and h(t) is the system transfer function.
A usual choice for dY(t) is a Gaussian white noise. The characteristic
feature of the shot noise process is that dY(t) is chosen as zero almost every-
where, except for a series of spikes. These spikes occur at random time
instants determined by a Poisson process and each spike has some random mass.
(Note that in eqns. (1) and (2) summation over the spikes replaces the integral
in (7)).
The choice of the transfer function h(t) determines the autocorrelation
of X(t). In particular h(t) = e-bt gives a 1st order autoregressive process,
and corresponds to a single linear reservoir. In addition h(t) must determine
the shape of recessions in X(t).
However, if dY(t) is chosen as Gaussian white noise, no recessions will
appear in X(t). The absence of recessions may be explained intuitively by the
fact that Gaussian white noise is changing by minute quantities very quickly ;
hence the recession shape of h(t) appears in minute form and is immediately
swamped by the next change in dY (t) . Thus linear Guassian processes cannot
reproduce recessions, irrespective of the form of h(t), and the same will be
true even if a non-linear transformation is used pointwise on X(t).
The ability of the shot noise process to reproduce recessions prompted
its use in the model1ing;of daily streamflows.
Averaged Sampling of the Shot Noise Process
Natural streamflow and the stochastic shot noise process are continuous
time processes. Recorded daily streamflows and the synthetic data to be
produced are on the other hand discrete time processes. The usual approach
in the modelling of monthly data is to consider the data as a discrete sample
of the process, i.e. the values of the continuous process at discrete time
points. However, discrete sampling is inaccurate, since the data are actually
obtained by averaging the flows over the period between the discrete sampling

time points.
461
The difference between the two approaches, of discrete sampling
or of average sampling, is negligible for serial correlations of up to P = 0.5
which are typical for monthly data.
which is typical for daily data.
It is however substantial for e= 0-8
It is assumed here that streamflow follows a continuous time shot noise
process X(t), and that the observed data (and the generated data), are averages
of this process over a period of T = 1 day. The data X,, X2, ... are thus
defined as :
The moments of X. are slightly different from those of X(t), and are given
by : J
SQ
E(X.1 =
J
Var (x.) = - 9 Q2 2 [b-(l-e-b))
3 b (9)
b2
(s 1)
In addition, the averaging changes the shape of the recessions. Whereas
in the process X(t) recessions start from a vertical rising limb, for the
averaged values X the transfer function is of the shape,
j
1 -bt)
f; (1-e
that is the rise is gradual over O \< t < 1.
Fitting the Shot Noise Model to Daily Streamflow
(o < t < 1)
In fitting the shot noise model an approach similar to that employed by
Matalas [3J is used. Values of 9 , 8, b are calculated which preserve the
values of 1.1, o2 and p(l) obsezved in the data. Thus the sample mean, variance
and first serial correlation, p, 82 and g(1) are calculated from the hietorica:
data. These are substituted in equations (9), which are solved for 4,
8, b. The estimated decay rate 6 is solved for from the thigd equation by
numerical methods, and the other two equations yield 8 and 9.
An alternative approach would be to estimate b directly from observed
recessions or from the unit hydrograph of the basin, and to estimate 3 by

46 2
observing times of peaks in the data. This latter approach was attempted for
the British streamflow data at our disposal. However, preservation of observed
p, o2 and (1) using this method did not ensue. A similar approach may
however prove useful for different data, for instance, streams in semi-arid
regions, where data may consist of short records of frequent observations.
Synthetic Generation of Shot Noise Data
Let 3, 8, b be the parameters estimated from historical data. The algo-
rithm for generating synthetic data is as follows :
Denoting by Xt, t = 1, 2, ..., the averaged shot noise to be generated,
and by X(t), t = O, 1, 2, ..., the values of the continuous process, one
obtains from (5) and (IO) :
where the first term in eqns. 11 and 12 is the contribution from events
preceding t , and the second is the contribution of events in (t, t+l).
Starting with an initial value for X(O), XI and X(1) are generated.
X(1) is then used to generate X and X(2) and so on. Assuming XI, ..., Xt
and X(t) have been generated, tge following steps lead to Xt+l, X(t+l).
1) The first terms of (11,121 are calculated, from X(t). X(t) can then be
discarded.
2) Time of last event preceding (t,t+l) need not be remembered.
are initiated by putting m = O, .cm = O.
Event times
3) The next event .cm+l is generated as zm+l = T~+I, where I is a random
number generated from an exponential distribution with mean (1/3 1.
4) If T ~ > + 1 ~ all events in (t,t+l) have been exhausted and so generation
of Xt+l and X(t+l) is complete.
5) For T ~+I < 1, ya+-, is generated as a random number from an exponential
distribution with mean 8.
6) The contribution of y to (11, 12) is calculated as :
m+ 1
1 (l-e-b(l-Tm+l 1 ) and e -b(l-.cm+l), and added to the values
;
of Xt+l and X(t+l) respectively.
7) m is set to m+l and steps 3 to 7 are repeated.
Thus the generation requires only random numbers from exponential distri-
butions, which are easy to create.

Multisite Shot Noise Processes
463
The shot noise process already defined can be easily extended to a
multisite process. Let Xl(t), ..., XM(t), be continuous shot noise processes
at M sites, with parameters Sk, Qk, bk, k = 1, ..., M.
A multisite process incorporating all of the parameters will be defined
by assuming that some of the events occur simultaneously at several sites,
and give rise to correlated jumps yk at these sites.
For two of the sites, k and 1, let the events which occur simultaneously
be at rate Jkl(with 3k1 < 3 k, 3 kl < 3 i), and let the jumps associated
with a simultaneous event, y , yl, have a correlation coefficient ckl. Then
the correlation between \(ty and X,(t) is :
c + I
kl
.& . 2
In this expression the first term shows the effect of the different decay
rates on the cross correlation, (i.e. the effect of the two different recession
shapes), and the other two terms arise from the correlation between the two
series of events and jumps.
By (13) Ski and Ckl can be chosen for each pair k,l, to preserve the
observed cross correlation in the multisite data.
The Double Shot Noise Process
Some difficulties arose in fitting the shot noise process to average
daily flows from some English streams. While the model did preserve the mean,
standard deviation and lag one serial correlation coefficient of the daily
data, when the synthetic data was averaged over months, the synthetic monthly
data had much smaller standard deviations and lag one serial correlation
coefficients than those observed in the historic data. Moreover, in the synthetic
data the recessions decayed too fast towards zero, and too many rises
and recessions were generated.
Inspection of the historic daily ctreamflow hydrographs showed that the
streams modelled have a pronounced base flow component which is not reproduced
by the shot noise process.
Therefore a more sophisticated model was proposed, which assumes X(t)
to be the sum of two independent shot noise processes, Xq(t) with parameters
+A, QI, bl and X2(t) with parameters $2, 82 and b2 (cf equation 2). In
these two process $1, QI, bl are assumed to be larger than $2, 82, b2, so
that Xl(t) has more recessions, higher jumps and a faster decay rate than
X2(t). In physical terms, Xl(t) may be thought of as representing a surface
runoff mechanism and X2(t) as representing a baseflow mechanism.

464
In fitting the model, the six parameters can be calculated so as to
preserve the observed mean, standard deviation and lag one serial correlation
of the observed daily flows, and the standard deviation and lag one serial
correlation of the observed averaged monthly flows.
An Application of the Double Shot Noise Model
Data from the river Nene in East Anglia and some of its tributaries, and
of one tributary of the neighbouring Great Ouse was used to generate synthetic
data. The Nene flows through East Anglia, and discharges into the Wash. It
has a drainage area of 1630 km2, it receives an average annual rainfall of
623 mm, and has an annual runoff of 157 mm.
The historic data consists of 11 years of average daily streamflows
concurrent at 8 sites. !Che double shot noise model was fitted to the data so
as to preserve at each site the overall mean, the standard deviation of the
daily and of the monthly series, and the lag one serial correlation coefficient
of the daily and the monthly series, and so as to preserve the daily cross
correlations between the sites. Seasonality was accounted for through estima-
ting the parameters separately for each calendar month.
Twelve series of synthetic data, each of them equal in length to the
historic record, were generated. Table 1 summarises some of the results from
the historic and generated data. The table includes quantities calculated for
the River Nene at Orton, close to the outflow point, and for the calendar month
of January. Flows are listed in m3/s.
The table gives a comparison between properties of the historic data,
properties of the theoretical model calculated analytically, and properties
of the synthetic data. Column 1 refers to the historic data. Columns 2-4
refer to the theoretical model. Column 2 contains quantities calculated for
the double shot noise model, and the decomposition of these quantities into
the process modelling the surface runoff mechanism (fast process) and the
process modelling the baseflow mechanism (slow process) are given in columns
4 and 3 respectively. Columns 5-7 refer to the synthetic data. Column 5
contains values which are averages of all the twelve synthetic series while
columns 6 and 7 list the lowest and highest values obtained for each quantity
out of the twelve series.
The different rows of the table list the values of several quantities
of interest. The quantities which are starred, are those used in fitting
the model. For some of those quantities the model preserves the historical
value exactly, while others were only preserved approximately, due to numerical
difficulties. e12 is the cross correlation between Nene at Orton and Great
Ouse at Thornborough Mill.
The rest of the quantities listed were not used in fitting the model,
and success in preserving them can serve as a measure of the adequacy of the
model. Of these quantities which were not fitted, the lag two and lag three
daily serial correlation coefficients are extremely well preserved. On the
other hand the skewness of the data was overestimated by the model. It is
very encouraging that the model seems to yield reasonable values of high and
low flows.

465
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466
Conclusions
The shot noise model has been developed as a physically more realistic
model of daily streamflow data than has heretofore been proposed, and in
particular models recession effects which are a prominent feature of daily
streamflow data.
In its basic form, the shot noise process in its conception, statistical
properties and with the associated method of fitting and method of data gener-
ation is as simple and easy to handle as the Gaussian 1st order autoregressive
model.
The use of the double shot noise model for some English streans gave
satisfactory results, and illustrates the adaptibility of this class of models.
It is felt that these mdels, with their emphasis on events and recessions,
could with further research provide a link between deterministic and stochastic
hydrology. Thus studies by deterministic methods of the instantaneous unit
hydrograph and of the mechanism of base flow etc. could provide some of the
parameters needed for a stochastic model based on shot noise processes.
Acknowledgements
This work is financed by the Water Resources Board of England and Wales,
and is being carried out at Imperial College of Science and Technology in
London under the supervision of Professor D.R. Cox of the Department of
Mathematics and Mr. T. OIDonnell and Mr. P.E. O'Connel1 of the Hydrology
Section, Department of Civil Engineering, to whom I am deeply indebted for
ideas and help.

References
1.
2.
3.
4.
5.
6.
7.
8.
9.
467
Thomas, H.A. and Fiering, M.B. (19621, Mathematical synthesis of stream-
flow sequences for the analysis of river basins by simulation, Ch. 12 in
'Design of Water Resources Systems', by Maass, A., et al, London, Macmillan,
pp. 459-493.
Fiering, M.B. (19641, Multivariate techniques for synthetic hydrology,
Proc. Am. Soc. Civ. Engrs., J. Hydraul. Div., Vol. 90, HY5, pp. 43-60.
Matalas, N.C. (19671, Mathematical assessment of synthetic hydrology,
Water Resources Research, Vol. 3, pp. 937-945.
Young, G.K. and Pisano, W.C. (19681, Operational hydrology using residuals,
Proc. Am. Soc. Civ. Engrs., J. Hydraul. Div. Vol. 94, HY4, ppe 909-923.
Beard, L.R. (1965), Use of interrelated records to simulate streamflows,
Proc. Am. Soc. Cive Engrs., J. Hydraul. Div., Vol. 91, Hy5, pp. 13-22.
Moreau, D.H. and Fyatt, E.E. (19701, Weekly and monthly flows in synthetic
hydrology, Water Resources Research, Vol. 6, pp. 53-61.
Quimpo, R.G. (19681, Stochastic analysis of daily river flows, Proc. Am.
Soc. Civ. Engrs., J. Hydraul. Div., Vol. 94, HYI, pp. 43-58.
Payne, K., Newman, W.R. and Kerri, K.D. (19691, Daily streamflow simulation,
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Parzen, E. (19641, Stochastic processes, San Francisco, Holden-Day,
pp- 144-159.

ABSTRACT
FLOOD CONTROL DESIGN WITH LIMITED DATA - A COMPARISON
OF THE CLASSICAL AND BAYESIAN APPROACHES
Eric F. Wood
Department of Civil Engineering
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Water Resource planners usually design flood control structures
by choosing an extreme value model. The model's parameters are esti-
mated from the available streamflow data and design decisions are ma
de by finding the discharge related to a particular level of risk.
In most design problems the data on extreme events is severely limi-
ted, making parameter estimation difficult. Two different parameter
estimation approaches are investigated - classical and Bayesian -
which are applied to a flood design problem for a nortkeastern U.S.
river. The classical approach uses the maximum likelihood cri'terion
for parameter estimation. The Bayesian approach is performed for 05-
jective prior information based upon observations from other rivers
and for subjective prior information derived by considering the
effect of river basin development upon flood discharges. The results
indicate that the classical and Bayesian approaches lead to diffe-
rent design discharges for the same level of risk.
Normalmente los Ingenieros Hidráulicos diseñan las estructuras
para el control de crecidas eligiendo un modelo de valores extremos;
los parámetros del mismo se estiman mediante los datos de aforo y
se diseña mediante la determinación de la crecida asociada con un
cierto nivel de riesgo. En la mayoria de los casos, la información
disponible sobre los valores extremos es muy escasa, haciendo muy
difícil la estimación de los parámetros. Se investigan dos métodos
de estimación de pardmetros - el clásico y el Bayesiano - que se
aplican a un problema de control de crecidas para un rio del nordes
te de los Estados Unidos. El método cl'lsico utiliza el criterio de
verosimilitud máxima para la estimación de parámetros. El enfoque
Bayesiano se desarrolla con informaci8n priori" objetiva basada
en observaciones realizadas en otros ríos, y con información "a
priori" subjetiva obtenida al considerar el efecto del desarrollo
de la cuenca sobre los caudales de crecida. Los resultados indican
que los métodos clásico y Bayesiano llevan a diferentes valores de
caudales de crecida para un mismo nivel de riesgo,

470
INTRODUCTION
In water resources planning the hydrologist's main function is analysis
that lead to engineering decisions. The decision variable is not a hydrologic
variable but a general engineering variable like the height of a dike or the
size of a spillway. The separation of hydrologic analysis and economic
analysis can not occur if efficient designs are to be obtained. In the
decision process the design variables are related to the estimation of hydrologic
variables through a loss function which reflects the different economic
implications of the project.
If one accepts this role for the hydrologist within the planning process
then a number of important qualitative implications follow.* It is useful to
describe complex phenomena such as rainfall or runoff processes by the application
of probability theory only from the point of view that it produces a more
economical design. If streamflows can be treated as random variables thep it
is consistent to treat the unknown parameters of the distributions of streamflows
as random variables. To treat the parameters of distribution of random
variables as random variables is not "permitted" within the framework of
classical statistics.
The extension of this argument is that it is useful and professionally
sound to treat any uncertain factor as a random variable if it leads to better
decisions. This includes variables such as the quality of workmanship in construction,
cost and benefit adjustments due to inflation as well as the more
traditional hydrologic variables.
The designer should consider three types of uncertainty in his analysis -
uncertainty of a probabilistic nature (i.e. frequency of occurrence), statistical
uncertainty due to the limited number of observations from which parameters
are to be estimated, and professional uncertainty arising from incomplete information
concerning the underlying process and its probabilistic representation
(Cornell, 1972).
The methodology of the decision process must be able to con-
sider these forms of uncertainty as well as be able to utilize professional
judgement obtained from related experience with similar projects.
Bayesian analysis within the framework of statistical decision theory
(ñaiffa, 1968) prescribes a methodology for making decisions under uncertainty.
Decision theory allows the decision maker to consider together both the uncertainty
of the modelled process, the quantifying of the decision outcomes and
the preferences for these outcomes. Bayesian analysis is a probabilistic framework
by which the uncertainty in any design variable and the knowledge about
that variable can be considered. This paper is concerned with the latter aspect
of the proposed methodology. The application of Bayesian analysis in water
resources planning is gaining acceptance as decision makers recognize the
* Parallel arguments have been used previously in support of a statistical
decision approach to structural reliability analysis by Cornell (1972).

inherent advantages that combining information sources and treating uncertain
parameters as random variables leads to better designs. In recent years many
researchers have made significant contributions to this area. These include
the work of Bernier (1967), Shane and Gaver (1970), Davis et al (1972a)
Bogardi and Szidarovszky (1972) amongst others.*
Probabilistic Model Formulation
471
One area of particular concern to water resource planners is the analysis
of extreme events, mainly floods. This problem is especially applicable to the
issues raised earlier since data is often scarce, with the consequences due to
an inadequate design often severe.**
The issues we wish to focus upon in this paper is a comparison of the
classical approach and the Bayesian approach to flood analysis and design when
direct observation of extreme events are either scarce, non-existant, or non-
stationary. Substantial urbanization of a river basin introduces non-
stationarity effects into the direct observations, thus decreasing their
information content.
The first step in any analysis, classical or Bayesian, is the construction
(or assumption) of an underlying probabilistic model which represents the
physical process. Consider the hypothetical streamflow trace presented in
Figure 1. The flows of interest are those flows greater than Qo and it is
assumed that flows larger than Qo can be described by a Poisson process (the
time between events are exponentially distributed) with an average annual
arrival rate and the probability distribution of the flows of interest (flows
greater than Qo) can be represented by the exponential distribution
where
This is a fairly general form since the upper tails of many distributions
may be represented as exponential. The proposed model has been used for
extreme flows by Shane and Lynn (1964) and Todorovic and Zelenhasic (1970) and
for rainfall events by (Davis et al (1972b)) and Grayman and Eagleson (1971).
*The numerous papers at the International Symposium on Uncertainties in
Hydrologic and Water Resource Systems, December 11-14, 1972, Tucson, Arizona,
U.S.A. is proof of the growing interest in this field.
**The decision makers may also consider besides economic consequences social
and professional consequences due to failure of a flood control structure.
These may be loss of life, disruption of community services and the loss of
professional prestige. On the other hand, over design commits resources that
could be used on other projects.

472
A A
The probability that, in any single occurrence, a discharge z (z = q -
exceeds the discharge z is Pz, where
-a2
Pz = 1 - FZ(z) = e
the process of these occurrences is Poisson with an average arrival
rate VP and the probability that in time t n exceedances of level z will
Z
occur is
n -UP t
P[N = n 1 = (vP,) e
n !
No exceedances of z in time t is just
probability function of z. Substituting
[ -az
F (z) = fvte z>o
Z
z< o
The probability the z = O is equal to the probability that a peak discharge q
is less than Q,. I If z is such that the probability of exceeding z is small
and the arrival rate of such events is small then FZ (z) can be approximated by
-a2
FZ (z) 1 - vte
(3)
P [nz = O] = F (2); the cumulative
P from (2) into
Z FZ(z) gives
The probabilistic model of the underlying physical process serves both the
classical and Bayesian analyst but in slightly different ways.
Assume that there is no uncertainty in the model itself but only in its
parameters CY and V . The classical analyst then obtains point estimators,
V and a , (usually by the maximum likelihood criterion) from the observed
streamflow record. His probabilistic model is
The
the
Bayesian analyst, meanwhile, obtains probability density distributions on
unknown parameters, v and a , from combining all sources of information.
The Bayesian approach to the use of probabilistic methods recognizes that
the subjective information of the analysis is inseparable from the objective
aspects. Subjective infomation is incorporated into the analysis through a
prior probability distribution which reflects the information content. This
prior information is combined with objective information - direct data observations
- to provide the analyst with a posterior distribution. This reflects
all of his information. If the prior information is vague and the sample information
is very good then the posterior distribution of the information will be
(5)
QO)

4'1 3
negligibly affected by the prior. The opposite also holds. The prior information
may be looked upon as that information an analyst wovld use if he had no
observable data. In the design for floods a number of sources of information,
are available. These include such sources as regression equations based on the
physical characteristics of the basin (Benson, 1962) and analytical derivation
of extreme flow dynamics (Eagleson, 1972) as well as engineering experience and
expertise. To ignore these sources is to throw away potentially significant
information which could lead to better designs. The use of diffuse or noninformation
prior is in most cases wrong since it side steps this important
aspect of Bayesian theory.
The prior information and the direct observations are combined through
Bayes theorem
where
f" (a) = t(aJUamp1e) f'(a) (7)
f" (a) is the posterior probability distribution of parameter a
&(alSampie) is the likelihood function of a given the observed
samples.
f'(a) is the prior probability distribution of parameter a.
The posterior distribution, f'l(a), can be found analytically if the prior distribution
is a natural conjugate. To obtain the posterior distribution from a
prior which is not a natural conjugate usually requires the application of
numerical methods.
jugate for both parameters Y and a.
obtained from:
The gamma-1 probability density function is the natural con-
1 -
FZ (2) = FZ
all Y all a
The Bayesian distribution of z, FZ (21, is
(zIv,a) f"(v) f"(a) dvda (8)
where FZ (zIv,a) = Fz (z) of equation (5)
By assuming the posterior distribution on parameter v to be gamma - 1 with
parameters u" , SI' and parameter a to be gamma - 1 with parameters Y", E"
(these aze obtained with natural conjugate priors) permits analytical evaluation
of 1 - FZ (2).
v a

474
where
L ""+1 J
1 - F (2) = Ut 1 +E
z
cI=vII+1
E''
3 = - u"+l
S "
This is the probabilistic model for the Bayesian analysis. It is interesting
to note that the form is completely different from classical analysis model
(equation 6).
Design Model Formulation
The motivation for developing the probabilistic models of extreme events
is to apply them in making decisions. Suppose we are interested in the damage
associated with the exceedance flow z which is larger than the flood pro-
tection flow level r . The total cost is comprised of a damage cost C,(z)
and a protection cost C (r). For our example, let's assume that the damage
cost can be expressed b!:
C,(z) = C1 (z-r) (10)
while the cost of protection can be expressed by:
C,(r) = K + Co r (11)
If the expected criterion is used to evaluate different protection levels then
the expected cost, E[c], of protecting for a flow r is
m
z=r
For the classical model f(z) is:
from differentiating equation (6). Thus the annual exp,ected damage E[CZ] from
flooding when flood protection r is provided is
This assumes that r
FZ(z) is valid.
a
is large enough that the upper tail approximation for

In the Bayesian framework equation (12) applies but with the Bayesian
density function f(z) which is obtained from equation (9) as:
The annual expected damages due to flooding is
again assuming that
Example Application
the upper tail approximation of Fz (z) is valid.
475
The analytical formulations developed here are applied to a river in the
Northeastern region of the United States. The mean of fhe maximum yearly flood
is about 5800 cfs. Exceedance events were considered to be flows greater than
10,500 cfs which is somewhere around the 10 year recurrence intervals. Only
three flows in the 37 years of record (1929 through 1965) exceeded this base
flow.
Bayesian Parameter Estimation
Estimation for a
Prior information on a , the event magnitude distribution, was obtained
from a regression on 36 other Northeastern United States basins. The regression
related exceedance flows to physical characteristics found within any drainage
basin. The following regression was obtained:
.153 o 2.87 A .81 .74 .54 .65
Qm- St
where*
is mean exceedance flow, in cubic feet per second
% is orographic factor
A is drainage basin area, in square miles
S
T
is main channel slope, in feet per mile
is average January, degrees below freezing, in degrees Fahrenheit
St
is percent of surface storage area plus .5 percent
Since partial duration series and annual flood series are virtually identi-
cal above a frequency of about the 10 year flood (Langbein, 1949) the use of the
annual series for the prior was considered to be adequate for this example.
Research is presently being conducted by the author to study the problems of
appropriate prior information.
*The physical characteristics are from Benson (1962) and the streamflow data
from the U.S. Geological Water Supply Papers. (1301-A, 1721-A, 1901-A).
J

From the regression an estimate of the mean exceedance flood, Q and an
estimate of the variance of the mean flood were found to be:
P9
= 734 cfs
QP
5 2
V[Q ] = 4.9 x 10 (cfs)
P
Due to the assumed distribution of the magnitude of exceedance events, the mean
exceedance flood can be related to the event magnitude distribution parameter
by Q = l/a. If Q is assumed to be distributed as an inverted gamma -1 distribetion
with pargrneters v' and R' then a is distributed gamma -1 with parameters
v', $' (biffa and Cchlaifer. 1961); that is
with
V[Qp] = v'>1
(v')2 (VI-i)
This gave parameters v' = 2, R' = 1468. Thus the prior distribution on O! ,
f' (a) is
-1468a
fgYl(a) = e a2 (i468I3
r (3)
The posterior of , f"(a) can now be evaluated by equation (7). Thus
5 -33668a
f" (a) = Ka e
YI
where
(20)
n
The posterior of a is gamma -1 with parameters E' = + zi ; VI' = V I+ n
Estimation for v .
The estimation of prior information on u, the average arrival rate,
involves, as a first step, the estimation of the first two central moments of
the distribution of the arrival rate of a peak flow that will exceed the base
flow Qo. In our example, this base flow was 10,500 cfs. There are some
probabilistic or statistical methods one may use to approach this problem or
the engineer may have said simply "based upon my experience in the area, my
best estimate of v
minus .O25 of .l'I.
is .1 and there is a 50-50 chance that v could be plus or
The implication of that statement is that the standard

deviation is about ,033. If that is accepted for our example and if a g a m - 1
distribution for the prior
-
of V is assumed then
-0'V U'
f' (v) e (s'v) s'
YI
-
r (u'+i)
with u' 8
s' = 92
(21)
The posterior distribution of , fy1" (u) is just
3 .-37 v8 .-92~
f II (y) = v
Y1
-129~
= v" e
(22)
- yhich is g a m - 1 distributed with parameters u" 11, si' = 129, and mean
v= .O85 events per year.
Substituting these into the Bayesian design model of Equation (9) yields
- 1 - FZ (z) 5 .085t
-
Thus the Bayesian model, 1 - F (z), of equation (23) is shown in
2
Figure 2. The effect of considering diffuse prior information (no observations
in no years of data) is also shown in Figure 2.
Classical Proceedures
Application of the classical estimation proceedures is straight forward.
Estimators for both the average arrival rate, v , and the parameter of the
event magnitude distr&bution, a can be obtained by applying the maximum likelihood
criterion. For v , the estimator for v, the likelihood function is:
and the maximum likelihood criterion; 2 = O yields
Similarly for a ;
a,
j = 0 = 3 = .O81
tr
-
37
-5
a = n = 9.3 x 10
zi
i=l

478
A Kolmogorov-Smirnos test on the models using the derived estimators
passed the .10 significance level with ease.
Thus the classical estimator model for our example is
-9.3 10-~~
1 - FZ(z) .O811 e
which is the probability of observing a peak flow z in the next interval of
time. The classical model, 1 - F
Z
(z) represented by equation (25), is compared
to the Bayesian model in Figure 2.
Design Application
Using cost coefficients for Equations (10) and (11) as being:
4
c1 = $10 Icfs
K = $25 x lo4 equivalent annual cost over a
= $102/cfs proposed 50-year project life.
for the classical design proceedures in Equation (16) utilizing the Bayesian
design model and in Equation (14) for the classical design model.
The expected annual cost of providing protection against the 100 year
flood is presented in Table I.
100 yr flood Expected Flood Equivalent Annual
discharge Damages ($1 Protection Cost-50 Yr Life($)
Bayesian Model 17500 7 105 20 105
Classical Model 22500 10.76 lo5 25 105
Table I - Comparisons of Costs and Damages for Bayesian
and Classical Models
For each model the flood which had the lowest expected total cost also was the
100 year flood.
Discussion
Incorporating Non-Stationarity Effects
The problems of flood analysis when non-stationarity has been introduced
into the streamflow records due to increased development of the drainage basin
have not been completely solved. Recent studies by Bras (1972) have shorn
~~

479
increases in flood peaks of developed catchments of between 30% to 115% depend-
ing upon the particular size and shape of the storm. Basin development tends
to remove natural stream storage areas as well as decrease impervious areas and
holding ability of natural ground cover. The effects of changing these charac-
teristics can best be investigated by a deterministic catchment runoff model
that utilizes a stochastic rainfall generator (Harley, Wood and Schaake, 1973).
The Bayesian analyst has a number of options open to him which include a
rainfall analysis, a runoff analysis from the catchment analysis, and other
approaches. He can either utilize the streamflow data or ignore it, applying
his engineering judgment in many ways.
The classical analysis has few, if any, options open to him. The strict
application of his theory permits him only to consider the historical record
which will not apply to the developed basin. If the amount of development is
small, then the historical record may still contain valuable information but if
extensive modifications have taken place and the classical analyst still uses
his historical record then he must be able to defend it.
Conclusions
The role of analysis is to aid decision making. The two approaches presented
here lead to quite different design decisions.
The classical approach restricts the analyst to the observable hydrologic
data to which other information sources can not be added. Furthermore, it is
not possible to include within the analysis other uncertain parameters which
may affect the design.
Instead some other artifical mechanism is used such as
adding a factor of safety to the design variable, designing for the largest
possible event or using some other method which can not be related to a mean-
ingful economic (or social) preference criterion.
Too often too much weight is given to a few observable data points and too
little weight to other available information. Lhe Bayesian analysis is a methodology
which enables the combination of information sources as well as allows
the explicit evaluation of the effect of all sources of uncertainty upon the
decision variables. The application of the Bayesian approach will lead to
better design than will a classical analysis which is restricted to a few observations
and whose conclusions are difficult to interpret.
Acknowledgments
The work was supported by the Office of Water Resourc Research, Office
of the Interior, United States Government under Grant No. 14-31-0001-9021.
References
1. Benson (1962). "Factors Influencing the Occurrence of Floods in a Humid
Region of Diverse Terrain" U.S. Geological Survey Water Supply Paper 1580-B,
Washington, D.C.

480
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13,
Bernier (1967). "Les Methods Bayesiennes En Hydrologie Statistique"
Proceedinvs of the Int. Hydro Symp., September 1967, Colorado State
University, Fort Collins, Colorado, USA.
Bogardi and Szidarovszky (1972)."The Margin of Safety for Compensating
Losses due to Uncertainties in Hydrological Statistics" Proceedings of
the Int. Symp on Uncertainties in Hydrologic and Water Resource Systems.
Dec. 1972, University of Arizona, Tucson, Arizona, USA.
Bras (1972)."Effects of Urbanization on Runoff Characteristics of Small
Basins in Puerto Rico" Unpublished Bachelor of Science thesis, Department
of Civil Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts, U.S.A.
Cornell (1972). "Bayesian Statistical Decision Theory and Reliability-
Based Design" Structural Safety and Reliability, (A. Freudenthal, ed.),
Pergamon Press, New York.
Davis, Kisiel, and Duckstein (1972a)."Bayesian Decision Theory Applied to
Design in Hydrology" Water Resources Research, Vol. 8 No. 1.
Davis, Duckstein and Kisiel (1972b)."Uncertainty in the Return Period of
Maximum Events : A Bayesian Approach" Proceedings of the Int. Symp. on
Uncertainties in Hydrologic and Water Resource Systems. December 1972,
University of Arizona, Tucson, Arizona, U.S.A.
Eagleson (1972). "Dynamics of Flood Frequency" Water Resource Research
Vol. 8, No. 4.
Grayman and Eagleson (1971). "Evaluation of Radar and Raingage Systems for
Forcasting" Ralph M. Parsons for Water Resources and Hydrodynamics T.R. No.
138, Department of Civil Engineering, M.I.T., Cambridge, Mass. U.S.A.
Harley, Wood, and Schaake (1973). "The Application of Hydrologic Models to
Urban Planning" Presented at 54th Annual Meeting, American Geophysical
Union, Washington, D.C., April, 1973.
Langbein a949). "Annual Floods and the Partial Duration Flood Series"
American Geophysical Union Transaction, V. 30, pp. 879-881.
Raiffa (1968). Decision Analysis, Addison-Wesley, Reading, Mass., U.S.A.
Raiffa and Schlaijer (1961). Applied Statistical Decision Theory, M.I.T.
Press, Cambridge, Mass., U.S.A.
14. Shane and Gaver (1970). "Statistical Decision Theory Techniques for the
Revision of Mean Flow Regression Estimates" Water Resources Research,
Vol. 6, No. 6.

16. Todorovic and Felenhasic (1970). "A Stochastic Model for Flood Analysis"
Water Resources Research, Vol. 6, No. 6.
481
17. United States Department of the Interior. "Surface Water of North Atlantic
Slope Basins, throu$-11950", U.S.G.S.
D.C., 1957.
Water Supply Paper f301, Washington,
18. . "Surface Water of North Atlantic
Slope Basins, 1950-60", U.S.G.S. Water Supply Paper 1721, Washington, D.C.,
1969.
19. . "Surface Water Supply of the U.S.
1961-65Jater Supply Paper 1901, Washington,
D.C., 1969.

482
4
Q
W
TIME
O. DISCHARGE Q I EXCEEDANCE DISCHARGE 2 = Q-Qo
Figure 1: Typical Discharge Record Showing Exceedance Events and
Showing the Probability Density Functions for both Discharges
and Exceedance Events.
Qo

O O
rn
O
2
O
O
O U1
O
O
o
O
i
O
Lrl
ri
O
01
O
O
ln
N
483

Authors and Titles;
axe:
THE USE OF MATHEMATICAL (DETERMINISTIC) MODELS
General Report
bY
J. E. Nash
University College, Galway. Ireland.
At the time of writing four papers have been received. These
(1) "A Rainfall-Runoff Model Based on the Watershed Stream Network" by
J.W. Deleur and N.T. Lee, of the School of Civil Engineering, Purdue
University, and the Department of Agricultural l3conomics of the
university of Illinois, respectively.
(2) "Monthly Streamflow EstLnation from Limited Data" by C.T. Haan,
of the Agricultural Engineering Department, University of Kentucky.
(3) "Obtaining of Deficient Information by Solving Inverse Problems
from Mathematical runoff models", by V.I. Koren and L.S. Kutchent,
of the Hydrometeorological Centre of the U.S.S.R.
(4) "The. Mathematical Model of Water Balance for Data Scarce Areas" by
Nabil Rofail, Water Resources Department, Desert Institute of Cairo.
Introduction: In view of the relatively small number of papers it had been
suggested to me by the OrgGisers that I should include some introductory comment
of my own,on the subject of catchment modelling.
However, while the number of
Papers is indeed small, they are all interesting and two of them are of a

486
mathematical nature and will require time to elucidate.
interesting,in that it presents a practical tool developed and used in the
Soviet Union but, as far as I am aware, not generally known in the West.
paper also happens, understandably, to be very difficult to follow, and on
these two accounts, I propose to devote a somewhat disproportionate part of the
time available to its consideration. I feel sure that the other authors will
not consider this in any way a slight and, as I am sure that this distinguished
audience would prefer me to devote any time available to the consideration
of khis interesting technique, I shall keep my own general comments on the
subject of modelling as brief as possible.
Hydrological Modelling:
One of these is particular3
This
The variety of titles among the papers we are considering
reflects the widely different senses in which the term modelling is understood.
Nevertheless, there is a strong common link between them.
that a natural phenomenon such as the conversion of rainfall into discharge, or
the movement of water in a porous medium, is represented by an hypothesis or
model, expressed as a series of operations which are,performed
This would seem to be,
on one
function of time (the input) to convert it to another function of time (the
output), or as a single mathematical relationship, a partial or ordinary
differential equation which must be solved in terms of boundary CQnditions.
The relationship between the three elements may be represented by
Where the relationship is of a causal nature this may be indicated by
6 f fe.c E-
y ?)

I make this distinction because the mathematical solution of such problems is
often against the direction ,of the arrow, which from the mathematical point of
view may,therefore,be considered irrelevant. I shall use the terms cause and
487
effect for emphasis, only when the direction of the arrow is physically relevant
Either diagram represents a relationship between three quantities one only of
which may be unknown in any realistic problem.
The solution sought may be the output, the input, or a description of,
(or parameters of) the operation itself (e.g. a unit hydrograph or the coefficients
of a differential equation).
Generally speakingrit is true in the hydrological context that the operations,
viewed in the direction from cause to effect are stablelin the sense that bounded
causes produce bounded effects and small variations in the causes produce smaller
variation in the effects.
Precisely because of this fact, the inverse operation
discussed by Koren and Kutchment, of the discovery of the cause of an observed
effect, or the discovery of an operation itself, tends to be unstable and small
variations oierrors in the observed output produce larger variations or errors
in the computed cause,or the computed values of the parameters of the operation.
Por this reason the solution of the inverse problem is usually very much more
difficult than the solution of the direct problem.
The Direct Problem:
which arise usually involve questions of convergence of finite difference
solutions of differential equations.
This has at least a logical simplicity and the difficulties
The paper by Nabill Rofail describes the
solution of one such problem which we shall discuss in some deail later.

488
Among the inverse problems it is useful to distinguish three types
(a) The input is unknown
(b) The values of the parameters of the operation are unknowr
(c) The form and parameter values of the operation are unknowr
In the particular case of a lumped linear system where the input, the
operation and the output may be represented by
L
where x(t) is the input, y(t) is the output and h(t) the impulse response, the
three classes collapse to one. For such systems the form of the operation may
be described uniquely by the impulse response of the system and,theoretically at
least,this may be found without prior specification of its form. Therefore the
second and third classes merge.
Furthermore, because of the symmetry in h and
x in the two equations, a symmetry which becomes more obvious when the relationshi1
ià expressed in terms of Lapace transforms through the Faltung theorm,
Ys) = X(sj HU) (3)
the problems of discovering h and x are mathematically the same, so that all three
distinctions vanish.
in the nonlinear problem, however, or when recognition
of the system implies discovery of the coefficients of a partial differential
equation (a distributed system) the distinctions remain valid.
The Lumped,Linear Model:
For functions which are not simple expressions, it is
usually easiest to deal with these models in discrete form.
Eqs. 1 and 2 are
replaced by €Y] = IhJ i4 (4 1
and ius = PI €h3 c=>

where [x] and
at equal time invervals.
[y] are vectors of the input and output respectively, sampled
ordina tes of the impulse response as
h, O 0:
[hJ is a rectangular matrix formed from the
[h] =
Similarly Lx] in eq. 5 is a rectangular matrix formed from the input
ordinates in the same way.
.
The direct. problem of finding fy3 is trivial. The inverse problems of
finding 1.3 of eq. 4 or {XI from eq. 5 are mathematically the same.
Because of the,great stability of the operation in the direct direction,
solution of the inverse problems tends to be very unstable and therefore
489
difficult. Traditional means usually involve a least squares solution;(Snyder)
or the imposition of constraints on the impulse response e.g. harmonic analysis
with a limited number of terms (O’D~nnell). A new method of obtaining a least
squares solution under a constraint is described in the paper by Koren and
Kutchment.
Distributed Linear Models: If the input is distributed in space in a
constant manner the direct problem is essentially the same as that of the
lumped linear system (Kraijenhoff van de Leur, Venetis). If however the
input is arbitrarily distributed in space,the differential equation must be
solved numerically for the direct problem, usually by reducing the problem
to linear difference equations, which are solved at the node points of an xt
plane. An example is provided in the paper by Nabil Rofail.

490
Threatment of the inverse problem to discover the input or the coefficients
in a known linear system expressed as a partial differential equation,are rare.
An attempt to discover the characteristics of an aquifer is described in ref. 6 of
the paper by Koren and Kutchment, and the paper itself gives two examples of such ai
attempt to determine values of the conveyances and cross sectional areas as
functions of space and time in an open channel.
The General Nonlinear Problem: The direct problems are again relatively
straightforward - the nonlinearity complicates the numerical solution of distribute(
systems (partial differential equations) but .the lumped parameters systems are
scarcely affected.
The Inverse Problem involves, generally, postulation of the form of the
operation (i.e. a conceptual model) and estimation of the parameters
successive approximations. The first approximations are inserted in the model
and the output computed.
This is compared with the observed output and a
single expression of the observed errors (the objective function) is systematically
reduced by subsequent trial and error adjustments of the parameters values.
Examples are provided in the papers by Deleur and Lee and Haan. The major'
difficulties with this method are that the set of solutions obtained may not
be unique and, particularly when two or more parameters represent similar
operations,the optimised values are subject to very .high sampling variance.
It would be interesting to speculate whether such problems could be made amenable
to direct least squares approximations, as so often used in the corresponding
linear case.
Theorétically, this would seem possible, but it may be, as seems
to be generally assumed, that the complexity. of the equation representing the
by

dependence of the objective function On the parameters might render its formulation
difficulty. I feel however that this possibility ought to be explored.
Having thus classified the papers according to the nature of the problem discussed
we corne to a consideration of the papers themselves in some detail. These I
would like to take in the order of their classification above.
491

492
A Mathematical Model of Water Balance for Data Scarce Areas
by Rofail
The title of this paper is somewhat misleading.
fact the numerical solution of the linearised equations of motion of groundwater
in an unconfined aquifer.
distributed linear system. Neglecting any vertical component of velocity, the
horizontal components parallel to the x and y axis in a homogeneous aquifer are
proportional to the gradients of the piezometric head (h+z). The constant of
proportionality (k) is known as the coefficient of permeability, (authors eqs.
1 and 2).
The subject matter is in
It is therefore a case of a direct solution of a
The continuity equation (aut4ors eq.3) expresses the fact that the rate
of rise of the surface of saturation
, at a’point in the aquifer, is
proportional to the rate of percolation down to the aquifer at this point, plus
the net rate of flow towards the point (the negative of the divergence). The
constant of proportionality is known as the specific yield and is given the
symbol/h in authors eq. 3. These two equations are combined in the authors eq.4
by replacing the velocity terms by the corresponding gradients of the piezometri
head, yielding an equation in the head only.
3Yhtz) k ukæl , a h
3% b>c
03 1

This equation contains first and second order derivatives of the depth h and
the elevation of the aquifer bed z, and is non-linear due to the occurrence of
- By further assuming that the gradient of h is small relative to that of z,
terms such as (&become ah small relative to s. 9s and are dropped,thus
linearibing the equation (authors'eq. 5)
9 8 - h $:jL- WZ. ah
11
az- - arc- E
bLh hg?, ah .k-- -
-h--, -
bu
DY ay k
493
N =o (Id
This assumption is attributed to"Boussenzq" and is stated to be that the
powers of derivatives of the first order are of a lesser order of magnitude
than the derivatives themselves. This would, of course, be an acceptable assumption
but it does not seem to be that which is in fact made by the author in obtaining
his eq. 5 from eq. 4. Perhaps the authors would like to comment on this.
To emphasise the linearity of eq. 5 in terms of the partial derivatives of h,
the partial derivatives of z (assumed to be known) are written as 7, and fy
and the second order derivatives (also assumed to be known as r, and fiv
in eq. 10. (Note that in the text the quantities 7,, and Vy are incorrectly
stated to be the partial derivatives of h; this in only a printing errer).
In order to advance the solution from time n'to the n+l the author
replaces the equation of motion with two distinct finite difference approximations.

494
The first is applied to the first half of the time interval and the second to
the second half. For ease of comparison I reproduce here the equivalences as
used in the two successive steps.
j and k refer to the node point location in the x and y directions and n
to the number of the time interval. approximation
1st half 2nd half
derivative step.
step
-
These approximations have a certain symmetry, which when the finite differenc
approximations are added for the two half time steps,make the results consistent
with the original equation up to the second order. They have the additional merit
that the finite difference
written implicity in terms
equation for the first half step in time, can be
n 42
of linear combination of (h,-l ., h, and hJ+l)
with the coefficients all known (author's eq. il). Similarly the equation
applied to the second
rt'
half step yields an implicit equation linear in
h, and hrti ail the coefficients again being known (authors eq. 12).
(LI
These implicit equations can be solved as a linear set when the boundary condition8
are provided to yield h for all node points at a single time.
the solution through time.
Repetition extends

"A rainfall runoff model based on the watershed and stream network"
ßy Delleur and Lee
The problem discussed, i.e. that of recognising a lumpedpon-linear model,
is in the inverse category and the method used is that of postulating the form
of the system and optimisation of the parameters.
The authors begin by pointing out that even with forty years of record the
errors in estimating the parameters of a stochastic model of annual flows
may be quite high.
rainfall-runoff process,likewise,require long term series of both the rainfall
and runoff for their calibration. They conclude, therefore, that for regions
with inadequate data one may have to resort to deterministic models either of
the "black box" or of the "physical" type.
to whether the model form attempts to mirror the physical processes or is merely
a linear regression. The authors point to the obvious deficiency of black
box models,that they cannot be transferred from one location to another because
there is an absense of a one to one relationship between the parameters of the
model and the parameters
They mention also that stochastic linear models of the
This distinction is made according
the watershed. They conclude, therefore, that a
495
physical model requiring only a small number of identifiable parameters or a model
based on data which can be obtained in a relatively short time is required.
I am not sure what the authors mean by 'a stochastic linear model of the
rainfall runoff process" nor am I sure that these several models are
alternatives for the same piirpose.
It seems to me that if one requires a
stochastic model in order to generate a time series having the properties of
the observed sample it can scarcely be rubstituted for by a deterministic model
(whether of a black box nature or otherwise) relating rainfall to discharge.

496
Before such a model could be used to produce a synthetic discharge record the
stochastic properties of the rainfall input would have to be computed and a
synthetic rainfall record fed into the deterministic model.
that the problem had merely been transferred rather than solved by the substitution
of the deterministic model.
Thus it would seem
If of course a much longer ,rainfall record was
available this might be useful. The authors intention is to provide a deterministi
model with a small number of parameters preferably identifiable from the catchment
characteristics. I don't .think there would be any argument about the usefulness
of such a model even if it would not substitute for a stochastic model for a
different purpose.
The authors suggest that the availability of modern techniques of photography
and general remote sensing technology make it possible to observe relevant
catchment characteristics on a large scale and therefore to include these in the
deterministic model.
In the authors'model use,is made of the following catchment
characteristics obtained by aerial photography - the plan form of the stream
network, the topography, and the soil type. The model attempts to relate the
areal mean of the rainfall to the discharge at the gauging site, both as functions
of time.
area which is the area contributing at any given instant to the flow at the
gauging station i.e. the function of time representing that portion of the
catchment from which runoff is currently passing the gauging station at the time
in question.
The structure of the model depends largely on the concept of a contributi
Obviously the contributing area is a functi,on of the catchment
wetness and the model for this quantity, described by the authors'eq. 1 is clearly
dependent upon the antecedent rainfall.

The contribut?.ng area at timeibt is A (fdk) and the total catchment area is A,.
I assume that in eq.11 the second negative sign from the right in the numerator
should in fact be positive, and with this interpretation I understand the
assumption of eq. 11 to be that the contributing area expressed as a proportion
of the total catchment area varies with the Nth power of a wetness index, which
is obtained by the sumation up to the time under consideration of the proportion
of the net rainfall in each previous time interval, weighted in an exponential
manner according to remoteness in time. Later it is stated that this equation
is subject to a constraint of continuity, that is, that the total effective
rainfall is equal to the total discharge. It is not explained how khis condition
is fulfilled but if eq. 11 is taken as a statement of proportionality rather than
of equality, giving, therefore, the relative valuaat ail times of the contributing
areas, the constant of proportionality may be chosen to satisfy the requirement
of continuity.
This is my interpretation of the authors'intention. The quantity
B would seem to be a constant loss rate existing throughout the storm. I think,
perhaps, the authors might like to clarify these few points and explain what
happens if the rainfall intensity is less han B.
The next step is to distribute the total contributing area A(7At) along the
channel. of the catchment. The contributing area per unit length of channel
At> is obtained under the following assumptions.
1. Constant velocity at a given time throughout the catchment.
2. Uniform distribution of drainage density.
3. Uniform distribution throughout the catchment of first order streams.
497

498
Under these assumptions the total contributing area at any given time
is distributed according to the distance (or time of flow) from the gauging site,
in th? same way as tli? number of channels is distributed according to distance
from the gauging site. Thus it is possible,on an examination of the stream
system,to define the function al2 6 k) in space and time.
The input of each reach, in each time element, is obtained by multiplying
the appropriate contributing area by the net rainfall intensity (i.e. the
total rainfall intensity minus the loss rate) and this input is routed through
the channel system by a 1inear.method (Dooge and Harley) to give the output
at the gauging station as a function of time.
Because the routing is linear the output due to the input on a given reach
could be represented by a convolution integral (but the kernel may vary from
reach to reach). The total output as a function of time is obtained as the
spatial integral of this convolution integral. The kernels themselves vary
with stream slope, a reference discharge, and a roughness parameter.
To reduce the complexity it is proposed to replace the actual stream network
by a "folded up" one in which (I believe) all stream elements lying the same
distance from the gauging site would be assumed equal to one another in the
properties of length, roughness, slope and reference discharge. They would
also agree, of course, in the depth of contributing area. It is mentioned later
that the roughness and slope parameters are obtained by actual observation but
it is not clear to me how this can be done in the idealised or "folded up" model.

The parameters of the model are:
A catchment area
D and N numerical parameters in eq. 1.1
B the constant loss rate
CZ the roughness coefficient (one parameter only)
QR the reference discharge (one parameter)
SL the main charnel slope (one parameter)
Subsequently in the work, the parameter B was set to zero and the area
49 9
and slope parameters were obtained by physical measurement (I think the authors
might like to explain this a little further) the remaining parameters D,N, CZ
and QR were obtained by optimisation.
Details of the method are not given,
nor are we told what sampling variance of the optimum values was obtained.
We are told,however,that the model was insensitive to D (when D exceeded 0.5)
and a fixed value of D = 0.8 was chosen. QR was found to vary only slightly
between 1.1 and 1.4 cubic meters per second for rainfall values ranging from
2.5. to 14 milimeters. Thus only N and CZ are left as free parameters.
The model was applied to 13 basins in the eastern haî€ of the United
'States and when.the optimised values had been obtained, relations were sought
between these and the catchment characteristics, so that these relations could
be used to provide estima'tes of the parameters values for use subsequently
on wigauged catchments.
N was found to vary with the ratio of runoff to rainfall volumes for the
storm according to authors eq. 8.
# = esp (0.464- R,)/O.24L

500
This ratio itself was found to vary between storms in accordance with the
daily temperature, an index of soil permeability, rainfall volume, and maximum
in t ens it y e
CZ was significantly related to basin area, stream slope and the base
flow value at the time of occurence of the storm.
Using these relations between the model and the catchment to estimate
the parameters of the former for insertion in the model which was subsequently
fed with'the observed rainfall, good results were obtained, hydrograph peaks
being reproduced with an error of the order of 20% in magnitude, and 10% in
timing .

Monthly streamflow estimation from limited data
by Haan.
501
The purpose of the exercise described in this paper is very similar to that
in the paper by Deleur and Lee.
monthly runoff from daily rainfall and the parameters are related to catchment
characteristics by regression equations. The model is of the physical rather
than the "black box" type according to the distinction of Deleur and Lee
and according to the classification I have suggested, the problem is inverse
non-linear lumped.
A four parameter model is used to compute
The structure of the model is not described but we are told that there
are four parameters.
fmax- maximum infiltration rate (cm-hr)
-- maximum daily seepage loss (cm)
'max
c -- "the water holding capacity of that part of the soil, from
which the evapo-transpiration rate is less than the potential
rate, unless this portion of the soil is saturated".
F -- fraction of seepage that becomes runoff.
s
The input to the model is a series of daily rainfall values and average
monthly values of potential evaporation (evapo-transpirat'ion).
The optimisation is obtained by comparing computed and observed values
of monthly discharges and summing the squares of the errors to obtain the
objective function.
in turn.
The search is carried out along the axis of each parameter

502
The model was.app1ied to 27 catchments in Kentucky and South Carolina and
a four percent average error was found in the prediction of the annual discharge.
This of course is not a very efficient test of the model. Details of the results
obtained are not provided - in particular the estimates of the sampling variances
of the optimum parameter values are not provided.
To provide for ungauged catchments,regressions were sought for the optimum
values of the parameters obtained from 17 catchments on certain characteristics
of these catchments. The independent variables were 12 in number (see table 1
of the paper) leaving, it would seem, only 5 degrees of freedöm, though perhaps
even this is an overestimate as covarience terms appear in the regression equations
It would be interesting to learn how significant the coefficients in these
equations appear to be.
Having obtained the regression equation the model was applied to six
catchments not used in obtaining the regressions. The model parameters were
obtained from the regressions and the runoff simulated.
the total runoff for the whole period varied from 1.8 to 11.8.
do not indicate how the model performed over shorter periods, for example of
one year, one month, peak flows, etc.,etc..
Percentage errors for
These figures
On a single catchment the effects of different methods of parameter
estimation are explored.
regression equations asd a percentage error (in the total flow?) of 8.64%
observed.
increased this figure to 10.13% and when 2 or 3 years of records were so used
figures of 2.19 and 9.38 were found.
one.
Firstly, the parameters are estimated from the
Optimisation of the parameters in the first year of record surprisingly
The last optimisation was a rather curious
The parameters were first obtained through optimisation in the first years

ecord and the remaining 21 years Of Output simulated. Next the worst two of
these years, from the point Of View Of agreement between computed and observed
outputs, were noted and the parameters optimised.again,independently in the
records of these two years.
averages of the two sets weighted according to the sum of deviations of
observed and simulated flows.
The final parameters were taken as weighted
using these final values of the model parameters was made the observed
error in the total discharge was only 0.56%.
503
When the simulation for the full period of record

504
"Obtaining of Deficient Information by solving inverse problems for Mathematics
Runoff Models" by Koren and Kutchment.
The authÒrs'definition of an inverse problem is in agreement with %hat which
1 have been using.
problems and mention the lack of uniqueness of the solutions obtained by
postulating the form of the operation and adjusting the coefficient or parameters
by trial and error.
Tikonev which restores the proper posing of the problem and limits the possible
variation of the solution in accordance with "a priori" information on the
s o lu t ion.
They explain the difficulty of obtaining solutions to such
Instead they propose the application of an algorithm due to
Unfortunately, I am not familiar with the sources quoted and my interpretatio
of the method derives solely fromthe present paper.
I would hope the authors
would forgive me if I misinterpret their intention and I would hope that, if
at all possible, time should be provided to allow them to correct me and explain
sezral points of difficulty which I stil1,only very imperfectly,understand.
The method involves the algebraic minimisation of an objective funtion and
is akin to Lagrange's method of undertermined multipliers.
Consider a function F(h) where h is a vector .hl,h2...and suppose
we wish to minimise F(h) subject to a constraint on h, e.g.T(h) = O.
Lagrange's method states that the conditional minimum of F(h) occurs at the
same h ao the unconditional minimum of G(h,a) where

G(h,oc) a F(h) +- 9th) (is 1
A formal algebraic proof is possible but scarcely necessary.
constraint cp(i-i) = O the functions G(h,*) and F(h)
Along the
are identical and therefore
their (conditional) minima agree. But the unconditional minimum of G(h,a)
obtained by differentiating G(h,ch) with respect to h a ndu and simultaneously
equating the derivatives to zero, implies q(h) = O or the general (unconditional)
and conditional minima of G(h,a) agree.
of C(h,5) gives the value of h which corresponds the conditional minimum of F(h).
An optimum value for a is also found.
to be an adaptation rather than the straightforward use of this method. A
series of values of the vector h which minimise G(h,@) for a series of values
of agradually increasing from zero toward the optimum (SC are found by
opt
differentiating. The first of these vectors h (corresponding to = O) corresponds
505
to the unconditional minimum of F(h). The last (corresponding toca*
opt
corresponds to the conditional minimum (i.e. to the constraint fully implemented)
and the intermediate solutions correspond to the partial implementation of the
constraint.
Consequently the unconditional minimum
The method used by the authors would seem
in this way the investigator is enabled to seek about in the vicinity of the
optimum h for one which provides a reasonable compromise between satisfying the
constraints and minimising the function.
in the three examples quoted in the paper the physical problem is reduced,
in one case after the application of much ingenuity, to the solution of a set of
linear algebraic eqs.
4
Q =AZ (I 6)
Where Q and h are vectors and A a rectangular matrix. Assuming redundancy among
the equations a least squares solution could be found by minimising
F(h) 5 /I Ph- all"
(I 7J

506
As shown by Snyder, the solution of this equation is
A*Ah = A*Q (12)
or h = (A*A)-l A*Q (1 9)
In the inverse problem, in the hydrological context (e.g. h is the impulse
response or the input to a linear system) this equation is often badly conditioned
and the h obtained may be seriously distorted by small errors..in Q or A.
In particular, h may fail to conform to some physical requirements (e.g. unit
area or smoothness of the impulse response). The authors method is to minimise
i.e., to find h from
(A*A+ aE)h = A* Q (where i? is the unit matrix)
with preselected values of e(presumab1y increasing from zero.
Obviously the smaller the value of q the more /lh112 is permitted to increase
and therefore increasing q corresponds to increasing the permissible fluctuation
in h.
Presumably a selection is then made between the several h, bearing in
mind that the nearer oi is to zero the nearer h is to the least squares solution
of the equation.
It is clear that the constraint need not be precisely stated. It is sufficiei
that the coefficient of arepresents some quantity which increases with the
undesirable property of h.
the Lagrangian method would probably be the better.
If the constraint can be precisely stated,e.g.rh = 1,
Of the three examples quoted by the authors, the first involves finding the
effective rainfall input given the impulse response and the discharge.
problem as we have seen is identical (even in the constraint) to that of finding
the unit hydrograph given the input and output. The authors mentioned various
The

constraints includkigrh = 1 which would yield
9 =Ikh-QI\ + aZh
and a smoothness constraint /h/ yielding
=Ih-Q/I + ec )I hl
Straightforward application of Lagrange s method would of course yield h = O
which would be useless. The solutions for lesser values of &would permit h # O
2
while restraining /h// .
The second problem discussed by the authors is that of discovering the
coefficients of the Saint-Venant eqs. for flow-in open channels
- .II
;; = $+$*&(e)
as +Fk
+i&($)
507
(d
= 0 )
This problem arises in, for example, flood routing, where it is impracticable
to measure the conveyance and areal relations K(z,x) and F(z,x) for each
cross section.
from observations made on the discharges and waterlevels as functions of space
and time, Q=Q(x,t) and z=z(x,t), during the passage of a particular flood.
ûnce.these relations have been established they may be used directly in subsequent
routing operations.
Instead, smoothed values of these functions may be obtained
The continuity equation integrated with respect to x, provides
Q(x, t >-Q(o, t 1 = & F (i,, t 1%
where 7) is a dummy variable along the length x.
In ?finite difference form,
this equation applied to a reach of channel from K=O to K=i, becomes
f$ (j+l, k 1 + F (j+l, k+l 1-F (j , k 1-F ( j , k+l)]
I< :
It.
~~Q:q(j+l,o) + Q(j+i,i) - Q(j,il-Ki,od
where j and k refer to time and space, respectively.

508
The authors state that this eq. maybe arranged as
-?+
AF = Q
one such equation existing for each discrete time and the vectors running, as it
were, along the channel.
It would seem that 2); has been omitted, but even allowing for this, I cannc
express eq. 24 in this form. Nor does it seem to me that eq. 25 is redundant.
It would certainly be interesting to have this point cleared up, but I think we
can all accept that the finite difference equation can somehow be reduced to a
set of linear equations between the changes in time in F(x) and in distance iii
Q(>r). Such a set of equations would apply for one instant only and would take
the form of
The authors apply the algorithm already expiained,with the constraint that
Ik-FOIr, where F is an initial estimate of F
0-
0)
is minimised for
With regard
chosena's by solving
Q@,F,) = //AF-Q 11 +a IIP-gI
(A*A+aE )F=A*Q+ a EP
is kept small,
(27
to choosinga the authors mention the "method of discrepancy" whit
I don't quite understand, nor can I see how the smooth variation of P with time
can be insured, as P(x) seems to De found independently for each time step.'
Perhaps in calculating F the values of F obtained .in the previous time step may bc
used in the algorithm for F and thus, by constraining //F-Fo1\2 the change in F
O
is distributed regularly over all x's.
Having found, P(x,t), thus, from the continuity equation, and having z(x,t)
C=

already, F(x,z) can be found.
The dynamic equation is similarly used to find K(x,t) and hence K(x,z).
Details are not given by the authors but the computation would seem to be
quite independent of the Computation of P(x,t,).
in their thikd and final exmple the authors deal with the same equations
but Pith different boundary conditions.
discharges are known as functions of time, only at the beginning and the end
of the channel reach, i.e. Q(o,t) and Q(L,t).are known.
z(x,t) is known.
case.
This time they assume 'that the
509
They assume also that
These conditions are more parsimonious than in the former
The difference between the discharges at the ends of the channel reach
is related to the rate afincrease in storage in the channel by the continuity
The right hand side is a single known quantity for sach t he step and may
thus be expressed as a vector in time.
The left hand side, because of the
integration with respect to x, is also a vector in th(unknown).
Assume that P(x,t) can be expressed as a smooth function of space and
time by a Chebishev polynomial.
Ex;>anded, this would be an ordinary polynominal in x and z and in xz with
constant coefficients depending on Aks.

510
To evaluate the coefficients, P(x,z) could be inserted in eq.29 but as this
is a difference equation Li P, the solution would be underdetermined at least
to the extent of the arbitrary constant. Instead of using F, the authors use
the top width B(x,z) and.expand this in x and z as
I€ there are m by n t e m in the expansion of B(x,z),there will be m by n
unknown coefficients A and,therefore,at least this number of equations in
ks
the form of eq.29 must be found. This can be done by taking sufficient time
intervals in the rising and falling hydrograph and evaluating the right hand
side of eq.29 accordingly to yield xl,x2,X3, .
For every k and s the quantity
is known for every.x and t and the integral with respect to x can therefore be
found.
Hence the coefficients of every A in eq.29 after substitution
k.9
can be written down yielding.
%st 'ks
(3 21
= Xt and this can be arranged in matrix form, if (33)
necessary, and I think it would be necessary, with Akis written in vector form
A,l, Ak2, Ak3, ............. Thus the linear equation in the unknown A would
ks
be obtained as
0 Q =x 9
where $ is the matrix of the coefficients Cks in eq. 33 and the solution for
9
8 the vector of unknown A,s found by application of the algorithm.
(Sf 1
(Q* FBI0 = 9*x @a
In this case, the constraint imposed is 191 small. The choice afa(and
therefore of e) seems to be made at the value ofawhere a further change
inawould produce only a minhum change in 8 expressed by eq.36.

Y
(c = 2 p(QP+,) -
J- I
Once F(x,z) has been determined, and remembering that we have already
z(x,t), K(x,z) can be found from the dynamic equation simplified to
and, thus, all the parameters of the equation are available. I cannot quite
follow the authors explanation of this part of the project.
511
be some subtleties here which I am missing,but I can see no particular difficulty
in solving it along the lines I have indicated.
There may
This is an extremely interesting though difficult paper and I hope the
authors will be available to correct iy very inadequate exposition of it and
perhaps resolve some of the difficulties which I have mentioned and others
which may be tròuhling other colleagues.
References
(1) Snyder, W., Tennessee Valley Authority (1961)
"Matrix operations in hydrograph computations"
(2) O'Donnell T., (1960) 'Instantaneous unit hydrograph derivation by harmonic
analysis'IASH (Helsinki) Pub No 51
(3) Kraijenhof van de Leur, D.A., "A study of non-steady ground water flow with
special reference dto a reservoir coefficient"
De Ingenieur , 70(19 1 (1 93 8 1.
(4) Venetis C. "Estimating infiltration and/or the parameters of unconfined
aquifers from ground water level observations" Jour Hyd. 12(1971)

ABSTRACT
DONNEES INADEQUATES ET MODELES MATHEMATIQUES
DE LA POLLUTION EN RIVIERE
Par J.BERNIER
Laboratoire National d'Hydraulique
CHATOU - France
The value of I'inadequate1l information must be judged
relatively to the mathematical tools used and the practical problem
to be solved, Concerning river pollution where the problem is to
design projects as sewage treatment plants for instance, the
limitation of the usual Information collected in situ is shown,
These limitations do no appear with the standard math.ematjca1 model.
Taking in account of more realistic stochastic model allows us to
measure the value of this information and to design experiment for
collecting acceptable data,
RESUME
La valeur de l'information inadequate doit être jugee en
fonction des problèmes à rbsoudre et des outils mathématiques utili-
sês pour cette résolution, En matière de pollution en rivière où il
s'agit de dêfinir les caractdristiques des moyens de lutte comme
celles des stations dlspuration par exemple, on montre les limita-
tions de l'information usuellement recueillie in situ, Ces limita-
tions n'apparaissent pas avec les modèles mathgrnatiques standard,
La prise en compte de modèles stochastiques plus réalistes permet
de mesurer la valeur de cette information et de definir les condi-
tions de collecte de données acceptables,

51 4
I - INTRODUCTION
En matière d'aménagement des ressources en eau, l'insuffisance
des données est souvent le premier écueil auquel on se heurte. Cependant
pour mieux apprécier la validité et la précision des réponses aux questions
posées à l'hydrologue ou l'ingénieur, il faut noter que cette insuffisance
de données n'est en fait que le reflet de l'inadéquation des inhthodes uti-
lisées pour résoudre les problèmes. Ces méthodes doivent etre adaptées >
la nature de l'information disponible ou 2 recueillir. Certes dans bien
des cas les données disponibles doivent être coiilplétées mais l'organisa-
tion de la collecte des données complémentaires, le choix des conditions
opératoires de mesures ne peuvent valablement être &finis qu'en fonction
des méthodes iiiobilisant l'inforination recueillie. Dans ce contexte, certai-
nes iitéthodes usuelles sont particulièreiiient inadéquates. On peut en trouver
des exemples dans le domaine de la pollution en rivière notamment dans
l'étude du inouverrient et des réactions auxquels sont soumises des matières
polluantes en riviere 2 l'aval d'un point de rejet en vue d'apprécier la
capacité d'autoépuration de la rivière compte tenu de ce rejet. Nous trai-
terons ici du seul problème de la pollution biochiinique'caractbris6e par
le bilan d'oxygène.
II - LA METHODE USUELLE
La méthode classique utilise le modèle de Streeter et Phelps
décrivant le bilan dynamique d'oxygène sous la forme de deux équations
différentielles (voir la liste de notations en fin de note).
J

515
Mises sous forme intégrale et en faisant apparaître l'abscisse
longitudinale x prise le long de la rivière et liée au teiiips d'écouìe-
2
nient t et à la vitesse moyenne du courant u pa; t = - , les équa-
tions donnent :
L(x) = Lo e
X
-K3 u
X
- K -
B (x) =ao e ~ ~ ( 2 u -e e
OÙd (x) est le déficit en oxygène : 3 = Cs - C.
U
'J - Kg U)
Ainsi peut-on mesurer l'incidence d'un rejet (spécifiant les
concentrations initiales d'oxygène Co et de demande biologique en
oxygène L ) sur l'autoépuration à l'aval.
La mise en oeuvre courante de ce iiiodèle demande l'estimation
préalable des coefficienb de reoxygénation K2 et de biodégradation K
1
et K3.
On utilise généraleinent des formules empiriques ou quelques observations
recueillies dans le tronçon de rivière étudié. Les multiples formules
empiriques disponibles établies en laboratoire ou sur des rivières de
caractéristiques biochimiques particulieres présentent des résultats
extre'mement dispersés difficilement extrapolables hors des limites du
doniaine où elles ont été établies. I1 reste l'inforniation recueillie
dans chaque cas d'espèce. A la limite Kg et K2 peuvent ëtre calculés
par l'intermédiaire des formules (2) en fonction d'un seul couple d'ob-
servations amont (Lo, Co) et d'un seul couple aval ( L(x), C(x)). Une
telle procédure, trop souvent utilisée pratiquement, est justifiée dans
le contexte du modèle déterministe strict décrit par (2) mais ce carac-
tère déterministe est extrêmement fallacieux come nous allons le voir.
Par ailleurs il importe de se soucier de la cohérence spatiale des mesures,
le décalage des époques d'observations a l'amont et à l'aval devraient
tenir compte du temps d'écoulement t . En pratique cette condition iinpé-
rative est rarement respectée et la non prise en compte de la menie masse
d'eau 2 l'amont et à l'aval entrarne une dispersion notable des observations
et des estimations erronées de KI, K2 et Ka.

516
III - INSUFFISANCES DU MODELE DETERMINISTE DE STREETER ET PHELPS
Le modèle (i), (2) schéiuatise l'hydraulique de l'écoulement :
celui-ci est supposé permanent, uniforme ; de plus on néglige la disper-
sion de polluants dissous ou en suspension dans l'eau imputable au phéno-
mène de diffusion turbulente et aux fluctuations de vitesse dans chaque
section. Cependant l'effet de cette dispersion est surtout notable en
régime transitoire et devient négligeable en régime de pollution permanent
ou lorsque cette pollution (caractérisée par L et CI présente une évo-
lution lente, d ms le cas de rivière à coefficient de dispersion faible
(cf. [i] et [2] ).
L'inadéquation en nature du modèle de Streeter et Phelps provient
surtout des hypothèses caractérisant les phénomènes biochimiques :
- prise en compte de la seule phase carbonée des réactions de
dégradation des matières organiques (non prise en compte de
la phase azotée) ;
- non prise en compte des effets de la photosynthèse de la respi-
ration des boues de fonds, de la sédimentation des matières
polluantes etc ...
Certaines tentatives [a] ont été faites pour inventorier et
modeliser plus completement les phénomènes mais la multiplicité des para-
. .
mètres et les difficultés pratiques d'estimation de ces paraniètres rendent
illusoire l'apparente précision qui semblerait résulter d'un inventaire
exhaustif des mécanismes biologiques et physico-chimiques.
Par ailleurs si les concentrations en oxygène dissous peuvent
être mesurées in situ avec une précision acceptable, il n'en est pas de
même de la demande biologique en oxygène qui n'est pas estimée directement
mais seulement par 1 l intermédiaire d'un test chiriiique, la DB05, effectué
au laboratoire sur des échantillons prélevés en rivière. Des essais ont
montré que l'imprécision de ce test est notable et que la chaîne complexe
des conditions opératoires, depuis le prélèvement en riviere jusqu'a
1 'analyse chimique en laboratoire introduit des erreurs systématiques et
aléatoires importantes. On ne peut donc considérer cette DBO comme une
5
mesure exacte de la consommation d'oxygène in situ mais comme un index
représentatif de cette consommation en plus ou moins bonne comélation
statistique avec elle.

IV - UN MODELE STOCHASTIQUE
517
I1 est classique en statistique de prendre en compte globalement
l'ensemble des phénomènes négligés dans un modèle déterministe schématique
sous foriiie determes d'erreurs aléatoires. Cette conception permet d'intro-
duire la souplesse nécessaire à une bonne adéquation du modèle aux données
d'observations en nature. Les équations (1) sont remplacées pour un sys-
teme différentiel stochastique :
Les paramètres pl et p2 représentent les irioyennes, c'est-à-dire la
2
part systématique. des erreurs dues aux phénoiiiènes négligési o12 et u2
représentent les variances des termes d'erreurs. E, et .C2 sont alors
les erreurs centrées réduites (de moyenne nulle et de variances égales à 1).
Dans ce contexte aléatoire, les paramètres K1, K2, K3 peuvent
être interpréter coime les coefficients de la régression statistique des
variations de concentrations en fonction des grandeurs
L etd et ils
ont une signification statistique plutôt que physique. On peut alors rem-
placer l'index DB05 par tout autre index qui soit en corrélation avec
la demande en oxygène (par exemple : demande chiiiiique en oxygène, carbone
organique total, etc... indexes qui sont plus aisément mesurables que la
DB05). I1 est possible d'intégrer le système ( 3) (cf. [ 41 1. En adinettant
des conditions initiales fixées et go au point origine x = O, on
LO
peut montrer que L(t) et3 (t) sont des variables aléatoires quasi-
gaussiennes dont les espérances niathémat iques et variances sont :
E (L) =
- P2 + (Lo - -) P2 e
K2 Kg
- K3 t
E(d)=-t- Pi
K1
P2
K2 Kg
pci +- '1
- K3-K2
P2
(Lo - -)I
K2
-Kpt
e
+-í-
K1 P2
- K3t
-Lo) e
K3-K2 K3
(4)
(5)

v -
518
Var (LI =
- 2 K3t
v22 (i - e 1
2 2
2 u2 t ui r,
Var ($1 = ( u1 +
2
(K~-K~) K3-K2
formules dans lesquelles
erreurs Cl et E,.
K3
- 2 K t
(i - e 3 )
Kg
- 2 K,t
(i - e 1
K2
K1
(u u r+-
K3-K2 1 2 K3-K2
(i - e 1
Kg +
K2
(7)
r est le coefficient de corrélation entre les
LE MODELE STOCHASTIQUE AVEC ERREURS DE MESURES SUR L et d
Connie nous l'avons déjà souligné, 1' inadéquation du iiiodèle aux
données in situ est liée également aux erreurs de mesures importantes sur
ces données. Pour analyser plus complèteinent le problèiiie, il importe de
préciser les conditions d'observations. Nous ne traiterons ici que de la
méthode usuelle OU l'on observe deux points amont x et aval distants
de x = - xcI.
écrit :
Compte tenu des foririules (4) à (7) le modèle intégré peut être
en regroupant sous forine des constantes a et b les termes indépendants
des conditions initiales dans les espérances inathématiques et sous forme
des constantes p1, p2, p3 les coefficients de Lo et do. Les variables
aléatoires d'écart EL et I!& ont alors des variances u et u2 données
L
par les formules (6) et (7) et qui sont donc indépendantes de ces conditions
2
=2

initiales. En fait ce que l'on observe n'est pas directement
mis deux grandeurs X et Y telles que :
X=L+ 7)
1
Y =a + 7)
2
519
L ou 3
où Il e; 7)2 sont les erreurs de mesures aléatoires de variances respec-
2
tives VL et Va . L'estiiriation in situ des parainetres du modèle stochas-
tique et notanunent des vitesses de reoxygénation et de biodégradation
K2 '
K et K3 , demande la connaissance préalable des variances d'erreurs de
1
mesures. De façon précise on supposera qu'ont été obtenus n ensembles
de grandeurs observables (Xoy X1, Yo, Y
1
) aux deux points amont et aval
du tronçon de rivière considérée. Les paramètres statistiques de ces cou-
_ - - -
2 2 2 2
ples (moyennes Xo, XI' Yo YI , variances S x0, S x13 S yo, S y1 , et
covariances
S&xl3 Sx0y1, etc ... ) permettent l'estimation des coefficients
du inodele au moyen des relations :
P, = sxlxo
2 2
SX, - VL
- -
a = X1 - p 3 Xo
(10)
(12)

520
VI - LE PROBLEME DES ERREURS D'ECHANTILLONNAGE
Dans la plupartdes cas pratiques l'estimation des paramètres du
modèle d'oxygène ne peut être effectuée que sur un nombre
n de répéti-
tions d'observations assez liiriité. I1 importe alors de iiiesurer la précision
de ces estimations par leurs variances d'échantillonnage. Nous ne pouvons
ici développer l'ensemble des formules, nous renvoyons le lecteur à [2]
pour un aperçu sur le problème. En expriniant les divers paramètres comme
des fonctions des variances et covariances
les formules précédentes peuvent permettre le calcul approché de ces va-
riances d'échantillonnages à partir de celles des variances et covariances
estimées (cf. [ 51 ). En ce qui concerne notaniiiient les paramètres pi qui
déterminent les vitesses des réactions biochiiriiques , on aura des formules
de la foriiie :
VI1 - VALEUR DE L'INFORMATION RECUEILLIE EN NATURE
S2
XO
2
S yl, Sxoxl, etc ...
Pour la suite de la discussion il est conimode d'appeler variances
2
d'erreurs d'adéquation du inodele les paramètres uL , mg2 car leur valeur
est liée à l'importance de l'explication des variations d'oxygène dissous
par les paramètres
C, DB05 ... pris en compte. L'interprétation des
variances d'échantillonnages, perinet de préciser le noiiibre d'observations n
nécessaires à l'obtention d'une précision d'estiniation donnée ; on pourra
observer généralement la loi générale de l'augmentation de n en fonction :
- des valeurs croissantes de l'erreur d'adéquation .
- des valeurs croissantes de l'erreur de mesure.
Quels que soient les paramètres de pollution pris en compte, quelles que
soient les procédures opératoires de mesures, il restera toujours des
erreurs d'adéquation et de mesure irréductibles. Dans de telles circons-
tances, on ne peut utiliser les procédures classiques d'estimation des
modèles d'oxygène supposés déterministes et qui n'utilisent que des infor-
mations trop partielles. réduites trop souvent 2 un unique ensemble des
4 valeurs Xo, Yo, XI, Y1. I1 est absolument indispensable de faire des
mesures répétitives en nombre n suffisant. Le modèle stochastique est

alors un guide précieux pour la planification de la collecte de cette
information et des procédures opératoires de mesures.
521
Placé devant un problème de décision, le gestionnaire de la qua-
lité de l'eau d'une rivière aura donc a sa disposition des observations
cohérentes recueillies in situ ; mais ce n'est pas la seule source d'in-
formations disponible. Le gestionnaire dispose également de données plus
ou moins qualitatives sur les vitesses de réactions, de forinules semiempiriques
diverses [ 61 dont la dispersion des résultats est telle qu'elle
ne peut. donner que des ordres de grandeurs assez grossiers. Une telle
information est cependant précieuse si elle permet de réduire le nornbre
d'observations in situ. La disposition d'un modèle stochastique permet
l'utilisation des méthodes bayésiennes [7] , [a] dont le but est l'incor-
poration des inforiiiations de diverses origines dans un modèle quantifié.
Davis, Kisiel et Duckstein [E] ont montré tout l'intérêt de ces techniques
appliquées ?i l'étude des risques,associées aux décisions en matière de
gestion des ressources en eau et au calcul de la valeur économique de
l'information hydrologique qui tend à réduire ces risques. Ia prise en
compte d'un modèle stochastique est la première étape de l'approche déci-
sionnelle dans les problèmes de qualité de l'eau.

522
c :
N O T A T I O N S
concentration en oxygène dissous,
concentration en oxygène dissous 2 la saturation
Cs - C déficit en oxygène
DB05 : demande biologique en oxygène mesurée au laboratoire sur 5 jours
à la température de 2OoC sur un échantillon supposé représentatif
de la rivière
K1 : coefficient de consoinmation d'oxygène dans le modèle de Streeter -
Phelps
coefficient de réoxygénation
K2 :
K3 : coefficient de dégradation de la DBO restante
L : deniande biologique en oxygène restante
x : abscisse longitudinale de la rivière
u : vitesse moyenne de l'écoulement.
moyenne par section de rivière
L

B I B L I O G R A P H I E
[i ] W.E. DOBBINS : BOD and Oxygen relationships in streams
Froc. ASCE Sanit Div. S A 3 - 1964
[2]
523
J. BERNIER - P. LENCIONI : Utilisation d'un modèle stochastique pour
organiser 13 collecte in situ des données de qualité
de l'eau d'une rivière - 15ème Congrès de 1'A.I.R.H.
Ictainboul 1973.
[3] D. LEFORT : Modèles mathérriatiques de pollution en riviere -
La Houille Blanche - nuiiiéro spécial 8/1971
[4] D.R. COX - H.D. MILLER : The theory of stochastic processes
Methuen - 1965
[ 5 ] T.W. ANDERSON : An introduction to multivariate statistical analysis
Wiley - 1958
[6]
M. NEGULESCU - V. ROJANSKI : Recent Research to determine reaeration
[ 71 J. BERNIER
[ 81
coefficient - Water Research - Vol 3 no 3 - 1969
: Les méthodes bayésiennes en hydrologie statistique.
Froc. Intern. Hydrology Symp. Colorado State Univer-
sity - Fort Collins - 1967
D.R. DAVIS - C.C. KISIEL - L. DUCKSTEIN : Bayesian Decision Theory
applied to design in Hydrology - Water Resources
Research - Vol 8 no 1 - 1972.

"REGIONAL GROUNDWATER RECHARGE ESTIMATES VIA METEOROLOGICAL DATA"
ABSTRACT
SAMUEL P.COOK AND SAMUEL G,MBURU
In arid regions the planning of agricultural development
requires estimates of the availability of groundwater, Adequate
detailed data bases are unlikely to exist in most areas of interest
in developing countries, We have attempted to compute the
groundwater recharge potential for East Africa using primarily the
available meteorological data, The procedure has been to generate
a synthetic year by averaging the meteorological data for each
month at each meteorological site over a period of years, Then from
the monthly precipitation, the estimated evapotranspiration and the
estimated run off is subtracted, This computation proceeds according
to an assumed soil moisture storage and transport model whose
throughput constitutes the potential groundwater recharge, Contour
lines of this cuantity are plotted and these can serve as a guide
to rural planners for optimizing the selection of sites for new
development,
RESUME
Aux regions arides quand on fait un plan du development agri-
cole il faut evaluer l'eau souterraine desponible, I1 est tres peu
probably qu'on trouve les donnees de base assez detaillees dans la
plupart des regions en observation aux pays que son en train de se
developper. On a essayé de computer le potentiel de la recharge des
?aux souterraines pour les pays en Afrique de l'Est en utilisont,
?rincipalement, les données meteorologique disponibles, On a produit
sne ann&e des donnges synthetìques en faìsant la moyenne les
ionnêes meteorologiqueo pour chaque mois a chaque hstallatlon meteo-
nologique pendant une periode des annêes, Puis on a soustrait de la
>recipitation mensuelle, l'evaluation de ltevapotranspirat2on et de
L'ecoulement total, Cette computation continue suivant un modele
;uppose de la capacité de l'eau et du transport de l'eau dans le sol
lui evalue le potentiel de la recharge de l'eau souterraine. Les
:ourbes de niveau de cette quantite sont tracées et les organisateurs
lu developpement rural peuvent s'en regler pour optimiser la
;election des situations pour des projets neufs.

526
At the East African Agriculture and Forestry Research
Organization we are interested in estimating regional groundwater
recharge rates to provide basia information for agricultural
planning. Our approach is based on a water balance calculation. The
gmundwater recharge is the residual which reinains after subtracting
from the precipitation the losses due to evaporation and surface and
subsurface run-off. The total precipitation can be computed reasonably
well from the rainfall records. The remaining terms can be estimated.
T. Woodhead, 1. Dagg and D.A. Rijks (1, 2, 3, 4) have
studied extensively the computation of the Penman potential
evapotranspiration from the data sources in Eaet Africa. The actual
evaporative losses may be estimated with the use of a physioal model
of the soil moisture storage and transport system. Por a regional
computation in a predominently arid region the net surface and
subsurface run-off may be taken a8 aero to yield an upper bound on
the potential groundwater recharge. The accuraoy of the final result
will depend on the ohoice of the soil moisture storage and transport
model. This must represent the average regional response to the
stimuli of rain, wind and sun.
Water Balanoe Oalculation 0.f Groundwater Beoharge
GWFli Potential Groundwater Resharge
Pa Precipitation
Eo: Penman Potential Evapotranspiration
E : Actual Evapotranspiration
Q: Soil Moisture Storage
AQ; Change in Soil Moisture Storage
RO: Rn-off
GWR = P - E -.U) - RO
In order to compute the actual evapotranspiration from the
Penman potential evapotranepiration the dynamics of a soil moisture
model m e invoked. Many such models arc possible and future studies
may improve this phase of the work. For the present computation a
nbucket model" has been chosen. This model allemes that moisture
stored in the soil root sone is freely available for transpiration
up to the Penman potential Eo demand. If the root zone soil moisture
is exhausted no further evapotranspiration takes place regardless of
the Penman Eo demand. Furthermore, lhe root abne has a finite
capacity, Qo, for moisture storage. In the event the monthly
precipitation minus the monthly Penman Eo exceeds Qo, downward
percolation of the excess moisture takes place and constitutes the
potential groundwater recharge.

"Bucket Model" of Soil Moisture Stcrage ana Transport
1. Root zone has a maximum moisture storage capacity, Qo.
2. If QSQo, groundwater recharge i? cil.
3. If monthly moisture input plus storage exceeds Qo, the
excess is potential groundwater rechzrge.
We have made use of meteorological data collected by
T. Woodhead, M. Dagg, and D.A. Rijks (iq 2, 3, 4). This data is
based on 80 stations in Kenya, 50 in Tanzaniz and 17 in Uganda, at
ozly a few stations were oomplete records of precipitation, wind run,
and insclation available. Several methods were devised by Wlcndhead to
fill in the blanks. In order to eompute the Penman potential
evapotranspiration, Eo, according to the method of McCullooh (5) the
following inputs are needed:
2
R: insclation in cakories/cm /day
n/N: =?io 3f observed to maximum possible number of daily
smshine hours.
Ta:
zverage screened ambient temperature 'C.
o
Td: mean deily temperature of den point C.
U: win& run in miles per day at 2 meter elevation.
Eased cn fifteen stations a linear regression between R and
n/N has been derived (6, 7). This regressior, was used to derive one
of thene quantities when the other was available from the meteorological
recnrds. When neither R nor n/N were recorded use was made of a,
rehtionship esteblished (1, 2) between the monthly mean of daily
sucshim duraticn and the tot21 cloud amount. Estimates of total
cloud smount are made at most civil airfields. When records of wind
run were not available use was made of e relation established
betaeen Beaufort scale assessments of wind velocity and wind run (3).
After these procedures had been applied to complete the
recr-ds they were processed to construct a synthetic year. Por each
site sverages over the length of the record were made for each given
mocth of the year. Monthly values of precipitation and Penman
potential evapotranspiration were obtained. At each site an estimate
was also made of the model parameter, Qo, the maximum m il moisture
storage capacity in the root zone.
The soil moisture storage and transport system is conceived
of E? a dynamic system whose output is the downward percolating groundwater
recharge which is determined jointly by the system driving funation
3n3 the system physical parameters. The present output depends on the
whcle past behavior of the input. If the syetem ie 3inesr, the system
oiityut is the convolution of the system unit impulse rnspcnse and
tke system c?ri.ving function. If the system is non-licesr, given the
527

528
input we can compute the output ueing the system parameters.
For our bucket model system we proceeded as follows.
If the Penman Eo exceeded the precipitation for several months we
tentatively assumed that the soil moisture was totally depleted at the
end of the dry season. Starting at that month the bookeeping was
begun arid carried out each month for the duration of the synthetic
year. On the other hand, if the precipitation exceeded the Penman
Eo for most of the year the assumption was made that the soil was
saturated with moisture at the end of the wettest period. Then the
oookeeping was completed for the synthetic year. One of these
assumptions always led to a consiatent sat of bookeeping entries.
The monthly groundwater recharge was summed at each site over all
mrnths of the synthetic year. These totals were then noted on a
map and contour lines of equal groundwater recharge were interpolated.
After obtaining the meteorological data the single model
payameter 20 determines the reault. Therefore, the sensitivity of
the cont Ars to a variation of Qo ia of interest. It is evident that
If the soil moisture is not completely depleted at some time during
7-e gear, a change in Qo alone will not affect the groundwater recharge.
For this situation, only the moisture in permanent storage will be
changed. On the other hand, if the sril moiature is completely
exhausted at one time during the year, a change in Qo will change
the throughput by an equal and opposite amount. Therefore, in arid
regions a low soil moisture storage capacity favors groundwater
recharge. Low storage capacity implies that a short intense rain
will rapidly fill up the soil moisture reservoir and the excess will
quickly percolate downward beyond the root zone to the subsurface
storage aquifer. In arid regions vegetative cover is generally less,
which re8'uces the losses due to tranepiration. Deep rooted
vegetation wnuld negata this advantage.
In order to investigate the eensitivity of the potential
groundwater reoharge contours to a change in the value of QG, the
ac.mputations were carried out for two choices of this parameter at
each site.
Oritiaue of the Method
le hare attempted to obtain same idea of the regional
potential groundwater recharge ratea using the available data, mainly
meteorological records. By the use of a simple aoil moisture
storage and transport model we estimate the actual evapotranspiration
from the Penman potential evapotranspiration. The residue from the
precipitat irn after subtracting the evapotranspiration and the
increment tc soil moisture storage we identify as the deep percelation
or potential groundwater recharge. The model we used for the soil
mt-isture is a one parameter model and we have studied the effect on
tSe throughpiit of the choice of this parameter.

Several improvements are poserible in the treatment. A study
co-ild be carried out to improve the accurary of the model. It is easy
tri devise multi-psrameter models. These could be compared to field
me-rsurements. Another refinement would incorporate sdditional input
data. Neutron soil moisture probes are now fairly widely available.
Fer future work such data should be incorporated in the computation.
The use of this additional input can lighten the burden of the model
in the determination of the actual evapotranspiration.
The neutron moisture probe data could be used in the
following way which modifies the present soil moisture model. Soil
misture profiles could be taken at monthly or weekly intervals to a
depth of five meters. The total soil moisture to a fixed depth will
??e totaled fnr each measurement. During the interval between
measurements the precipitation and the Penman potential
zvspotranspiration will be totaled. If during that interval there
i? soil moist*ire storage exceeding the wilting point in the roet zone
than the amel evapctranspiration will be taken as the Penman Eo. If
wt, the actual evap6transpiration will be taken as zero.
With this accounting procedure we could compute the sum of
the deep percolation and the difference between surface and subsurface
run on and runoff. If these last categories are in approximate
halance, then we have the deep percolation or potential groundwater
recharge as the residual, From measurements at a network of sites the
regional maps msy be constructed.
The authors wish to thank the Director of EBBFRO for
aermission to present this paper at the Symposium on the Design 00
Tater Resources Projects with Inadequate Data, Madrid, June 1973.
REFERENCES
I. . Woodhead, T. (1966). Empirical relations between cloud
amount, insolation and sunshine duration in East Africa:
i, E. Afr. Agric. For. J., 2, pp211.
2.
3.
Woodhead, T. (1967) Empirical relations between cloud
amount, insolation and sunshine duration jr? East Africa:
II, E. Afr. Agric. For. J., 2, pp474.
Woodhead, T. (1970) mapping potential evaporaticn for
tropical East Africa; the accuracy of Penmen estimates irem
indirect assessments of radiation and wind speed, Proc.
Reuding Symposium World WgletgFi,B&lanCe I. A.C.H.
. I._ I -
Dagq, M. and Woodhead, T. and Rijks, D.A., Evapcmtirr
jn Enst Africa, nul. I.A.S.H., XV, 1, pp61.
529

530
c .. .
6.
7.
McCulloch, J .S .O. (1965), Tables for the rapid i:omputE+,i.on
of the Penman estimate of evaporation, E. Aï'?. AgTjc. For.
J., 22, pp.286.
Woodhead, T. (1968), Studies of Potential Evaporaticn in
Kenya, Government nf Kenp, Nairobi.
Woodhead, T. (196R), Studies of Potertial Evapuration i.c
Tarizania, Dar es Saham.

A RAINFALL-RUNOFF MODEL BASED ON THE WATERHED STREAM NETWORK
A BS TRAC T
by J.W. Delleur and M.T. Lee*
School of Civil Engineering
Purdue University '
West Lafayette, Indiana 47907, USA
Physical models of the rainfall-runoff process are better
suited than either stochastic or black box models for areas with
limited data, The model parameters must have a physical signifi-
cance, be convenient to obtain and their number should be small,
The framework of a model meeting these objectives is proposed and
is based primarily on the geomorphologic characteristics of the
stream network obtainable from maps or from aerial photographs.
There is analytical and experimental evidence that hydrographs are
dominated by direct runoff from very short overland flow paths
from precipitation on transient, near channel wetlands. This
wetland area is dynamic in the sense that it varies in terms of the
history of the excess of the precipitation over the "B" horizon
permeability, The distribution of the dynamic contributing area
along the main stream is obtained under the assumptions that the
velocity of flow along the stream network is uniform, that the
drainage density is a constant within a given watershed and that the
first order streams are uniformly distributed in the basin, The
runoff from the dynamic contributing area is then routed through the
synthesized stream network to obtain the direct runoff at the basin
outlet,
RESUME
Les modèles physiques des transferts pluies-débits s'adaptent
mieux que les modèles stochatiques ou que les "boîtes noires" aux
régions où les donndes sont limitées, Les paramêtres de ces modeles
doivent être pourvus d'un sens physique, faciles 2 obtenir et leur
nombre doit être petit, Le cad~e du modèle proposé se conforme 2 ces
objectifs et est basé principalement sur les caractéristiques gêomorphologiques
du rlseaa fluvial que l'on peut obtenir de cartes ou
de photographies aériennes, I1 a dtb démontrê analytiquement et
expérimentalement que les hydrogrammes sont en general dominbs par
le ruisellement sur de petits parcours situês dans les zones mouiliges
près des cours d'eau, Ces zones mouillêes son dynamiques dans
le sens qu'elles varient pendant la pluie et avec la saison, Un modèle
mathêmat2que de ces zones est formule en fonction de Ifhistoire
de la précipitation excédant la permsabilité de l'horizon iiB't. La
distribution de ces zones le long du rdseau fluvial est obtenue en
supposant que la vitesse de l'écoulement est uniforme dans le réseau
fluvial, que la densitd de drainage est constante pour le bassin
donne, et que les cours d'eau du premier ordre sont uniformdment
distrlbuh dans le bassi'n, Les ruissellements des zones dynamiques
sont achemint% au travers d'une synthèse du rdseau fluvial pour obtenir
i'ëcouïement à l'exutoire.
* Current Addres,s: Dept, of Agricultural Economics, Univ. of
Illinois, Urbana, 1llinoi.s y 61801 y USA,

532
Stochastic models for the generation Of river flow sequences require long
historical time series for the appropriate calibration ofthe parameters. In
many parts of the world, actual streamflow records are not sufflciently long to
attempt to deflne an elementary model such as a flrst order Markov process for
annual streamflow series. According to Rodriguez-Xturbe [i] for serles shorter
than 40 years the error in estimating the annual man might run from 2% to 20%.
for the variance from 15% to 60%. and for the rank one serial correlation it
might be as high as 200%. It may be fìatile to attempt to develop generating models
which preserve parameters, the estimation of which carries such an uncertainty.
The formulation of mnthly models requires shorter records, but with a
record of i5 years, the error in the rank one serial correlation coefficient is
still of the order of 40%. Btochastic linear models of the rainfall-runoff process
likewise require long time serles of both the rainfall and runoff for their
calibration. It would, therefore, appear that for regions with inadequate data,
one may have to resort to deterministic models. At this point, the choice may
be between a %lack box" type of model and a physical model. Black box models
ceanot be transferred from one location to another as the meaning of the peu-8meters
in terma of their representation of the components of the hydrologic
cycle is usually undeflned. The proper choice appears to be a physical model
which requires a small number of easily identifiable parameters, or a model
based on data w-hich can be obtained in a relatively short time, perhaps by new
techniques.
These physical models could conceivably be formulate&, by making use of information
that is becoming available through remote sensing from airCrart and
from satellites. By means of these new techniques, large areas CBP be observed
and analyzed in a short time, end require a small amount of observations on the
ground. The recent developments in remote seneing technology thue seem to point
to a new direction for hydrologic investigations in -(LB with inadequate data.
Remote sensing from aircraft or from satellite is best applied to observing
or monitoring fairly large area and thus lende itself to the hydrologic studies
of complete watersheds. Images taken at different times cm show changes In the
watershed, such as variations ia the land we.
The potential of remote sensing
for water resources stubies has been discussed by Kiefer and Scherz [2], but the
principal application of remote sensing to hydrology has been through aerial
photography.
The eye can see light from about .4 to .7 microns, but photographscan sense
from about 0.3 to 1.0 microns, thus extending the range to lower and higher wave
lengths. Color end color infrared photography have been used with great success
in forestry .and in agricultural crop identification. [3] Themai scanning op-
erates in the heat emission part o? the energy spectrum in the wan length riPr
3 to 20 mincrons. Pluhowski [4] shoved that with Infra-red Imagery in the 8 to
14 micron range, it is possible to discern thermal contrasts of 1' or 2'C. Thin
technique enables the hyarolo,5lSt to detect areas of &K>inid water dlacharge mad
to identify circulation patternr in large vater bodies.

533
More advanced techniques include mUitiSpeCtra1 scanning and side looking
radar. Multispectral scanners produce as many as 20 separate images in wave
lengths ranging from the reflected infrared region to the ultraviolet region.
These images may then be analyzed by means of computer data processing programs
which classi* the surface materials. This classification is accomplished by
separating materials in a known area according to their spectral response characteristics
and then applying these criteria to unknown areas. [3] The side
looking airborne radar can operate through dense cloud covers. It has been used
in mapping southeastern Pan- and northwestern Columbia, which could not be
mapped by conventional aerial photography becauee of the cloud cover. As an example,
Weaver [5] cites that the meandering pattern of the hiira river was revealed
by this technique.
Black and white, color and color infrared photography combined can be used
to delineate water bodies, rivers and streams, the drainage structure of watersheds,
to give indications on the underlying geology and on the soil types of
the region. Waltz and Myers [6] have shown that there exists a significant correlation
between the optical density measured from an aerial film and soil water
content measured by neutron probes and also between ground water temperatures as
measured through infrared thermal scanner and the soil water content of fallow
or bare soil. Zachary et.al. [7] has applied multispectral. remote sensing to
soil survey research in Indiana.
These techniques may also be used for enalysie
of water quality and for monitoring water pollution. [a] A general review of
the application of remote sensing in the management of earth resources has been
prepared by Colwell [9].
It appears that at present, black and white, color and infrared aerial
color photography, can be used to obtain the basic information regarding stream
networks, water bodies, main geologic and soil features needed in hydrologic investigations.
It also appears that in the near future, more dependable informa-
tion on soil water will become available through remote sensing.
The remote
sensing techniques thus appear to be of particular interest in areas with inade-
quate data, as a substantial area can be mapped in a relatively short time with
a minimum of ground observation.
MODEL FRAMEWORK
It is the purpose of this paper to explore the feasibility of developing
rainfall-ninofi models based primarily on information that can be obtained from
remote sensing aerial photography and to establish a framework for such models.
The simpler observations obtainable f rm the aerial photography being the plan
form of the stream network. the topography and the soil type. the proposed model
is based on these three types of information and particularly on the stream net-
work s
Geomorphologists have developed parameters which describe the topology, the
structure, the planform and the relief of stream networks. [lo] Some of these
parameters can be used ae indices of the hydrologic behavior of the basins since
scmral characterietics of the hydrograph depend upon the efficiency of the

534
drainage networks. For example, the bifurcation ratio (ratio of number of streem
segments of one order to number of stream segments of next higher order) is an
important control over the peakedness of the runoff hydrograph. Another geomorphologic
parameter which affects the moff pattern is the drainage density (smmation
of stream lengths divided by basin area) which is approximately one halr
of the reciprocal of the overland flow length. A high drainage density indicates
a rapid removal of the surface runoff, a decrease in the lag time and an
increase in the peak of the hydrograph.
A model based on the stream network also lends itself to the application or
the dynamic source area concept rather than the application of classical Horton
infiltration theory for the purpose of estimating the runoff-producing-rainfall.
Freeze [li] has shown theoretically that on concave slopes with lower permeabilities
and on all convex slopes, hydrographs are dominated by direct runoff with
a very short overland flow path from precipitation on transient, near channel
wetlands which form the variable response area.
DuMe and Black [12] reported
that the area contributing to the overland flow ie dynamic in the sense that it
varies seasonally and throughout a storm. Nutter and Hewlett 1131 have depicted
the growth of the source area during a storm from areas adjacent to the lover
order streems and gradual4 expanding to the main stream in one direction and to
efflmeral stream in the other. It seems logical to assume that the response
area will depend on the soil type adjacent to the stream and on the antecedent
rainfall.
In view of the complexity that would result from estimating and routing the
runoff in each tributary, it is proposed to synthesize the stream network by
folding it along the main stream in a manner similar to that mea by Lkwge [i41
with the time-area diagram.
Several routing procedures could be used, the %om-
plete linear routing" method of Dooge and Harley [i51 was used because of its
superior accuracy among other linearand emperical methods. It is in the appli-
cation of the routing procedure that the slope of the main stream plays a major
role. For full details the reader is directed to ref. 17.
iVRMULàTION OF THE MDDEL
The model of the contributing area A(iAt) is expressed by the relationship
i;,
i-1
[R(kAt) - B]At + [R(iAt) - B]At
A(iAt) = Ao I
i
T
[R(kAt) - 1
B]At
k=O
where A(iAt) is the contributing (response) ana at time iAt
R(kAt) is the rainfall intensity at time kAt
B is the "B" horizon permeability
D is the fraction of the antecedent rainfall contributing to the
response area
N is a parameter
T is the total nuniber of sampling points of the runoff -&-ogreph
(1)

k
i
is an index to count the time of antecedent rainfall excess,
k*i
is an index indicating the current time
Equation (1) is subject to the conetraint that the continuity equation must
be satisfied, namely the volume of rainfall excess muet be equal to the volume
of direct runoff:
- T T
1 QO(lAt)At 1 A(iAt) [R(iAt) - B]At
i =o i=o
where Q,(iAt) is the direct runoff at the outlet at time Ata
535
The synthesis of the stream network is based, in part, on the observation
made,by Leopold [i61 that there is no definite tendency îor the flow velocity to
have a great change along the length of the stream for a given retuni period or
fkequency. It may thus be assumed that locations having equal distances measured
along the stream network to the outlet, have the same runoff travel time to
the outlet. If it may be further assumed that the drainage density is approximately
uniform within a watersheä, then the total stream length upstream of a
particular point on a stream ia proportional to the tributary drainage area at
that point. Thus the distribution of the travel times is proportional to the
distribution of the drainage areas along the stream reaches, and only the latter
need to be considered. Fig. 1 shows the method of estimation of the distribution
of the drainage arem s(JAL) along the main stream reaches for a idealized
waters he d.
The vol^ of runoff may be obtained by adding the runoffs from each of the
elementary contributing areas. Calling a(jAL, IAt) the dynamic response area
at stream reach jAL and at time iAt, the continuity equation may be written
T T S
1 Qo(iAt)At 1 1 a(JAL, iAt) [R(iAt) - BIAL At (3)
is0 i=O, j=o
where S is the total nimiber of stream reaches.
Assuming further that the first order atreams or the atream Bources are plaiformly
distributed over the watershed, then, at a given time iAts the ratio 0%
the dynamic response area a(jAL, iAt) at stream reach jAL to the tributaPy
drainage area at the same stream reach is equal to the ratio of the total pesponse
area A(iAt) to the total watershed area Ao- Thus
The continuity equation (3) thua becows

536
The direct runoffs from the individual stream reaches are then routed
through the stream network by means of a linear routing procedure. li- 2
shows schematically the routing procedure for a stream reach.
X(JAL, kAt), in reach j at time
-
The input,
kAt is the direct runoff given by
X(JAL, kAt) ao(jAL) A(kAt) [R(kAt) - BI (5)
AO
and the routed outflow from reach,j, at time iAt is Y(JAL, iAt), given by the
convolution integral shown in pig. 2 which Is approximated by the convolution
sum
i
Y(jAL, iAt) 2: Hu(JAL, (i-k)At) X(jAL, kAt)At (6)
k=O
where H (JAL, (i-k)At) is the kernel function or instantsneous unit hydrograph
of the bear routing procedure used. The runoff at the outlet, Bo, is obtained
by summation over the stream reaches
S i
2 (iAt) = 1 1 H,(jAL, (i-k)At) * Ia0(jAL) A(kAt) [R(kAt)-BI - At '
C
(7)
j=O k=O AO
where A(kAt) is given by equation (1). The kernel fbctions for 10 of the most
common linear routing models have been listed by Toebes and Chang [la]. In this
particuiar study the linear channel routing kernel function used is based on a
linearization of the Saint Venant equations developed by Rwge and Harley [15].
This kernel function has three parameters: the stream slope, a reference discharge,
and a roughness parameter.
IMPLEMENTATION OF THE MODEL
The watersheds selected for testing the model are located in the state of
Indiana, near the center of the esstem half of the United States. Thirteen basins
were used with areas ranging from 8 to 400 square kilometers. The drainage
maps for these watersheds were prepared from aerial photographs at the scale of
1:20,000 by the staff of the Airphoto Interpretation Laboratory, School of Civil
Engineering at -due University.
The maps used were at the scale of 1:63,360
(one inch equals one mile). The longitude and latitude of all stream junctions
and stream sources within the basins were digitized and stored on punched cards
by means of an automatic digitizer. The details of assembly and of the storage
of the hydrologic and geomorphologic data on magnetic tapes have been reported
by Lee, Blank and Delleur [is]. Fig. 3 presents a CALCOMP restitution of the
drainage network of a watershed from the data stored on magnetic tape. Also
ahawn on Fig. 3 are the etream link m d the drainage eue8 distributions along

537
the main stream. The rainfall imposed on the dynamic contributing areas is then
used as the input into the linear routing procedure for each main stream link
and then summed over all the stream links. Fig. 4 (right) shows the outflow hy-
drograph obtained by the complete linear routing method using the parameter val-
ue shown (QI3 = reference discharge in d/Sec, CZ = roughness coefficient in
d2/sec, SL = main stream slope). With the exception of the slope, the parame-
ter velues were obtained by an optimization procedure which minimized the differ-
ence between observed and calculated peak discharges and observed and calculated
timesto the peak discharge. The parameters so obtained were correlated with
climatological and geomorphological characteristics of the watersheds with the
following results.
For the watersheds used in this study it was found that the outflow hydrographs
were not sensitive to the choice of D for D > 0.5. A value of D = 0.8
was used. The value of B was taken as zero as the soils were generally impervious
because of their clayey type and high permanent water table in the contributing
areas adjacent to the streams. The value of N was found to be related t6
the moPf ratio, R (ratio of measured runoff to measured rainfall):
r'
0.464 - Rr
for = D = 0.8, B = O
0.242
(8)
The runoff ratio was in turn related to the storm characteristics, the tempera-
ture and an average soil permeability index of the basin by a regression equa-
tion of the type
a 8 6 ~
Rr = Tmin 'I 'max
where T+n is the minimum daily temperature when the storm occurs, Sf =: soil
permeability index determined by assigning soil permeability VaEues to major
soil types occurring in the basin, and calculating the weighted average for each
basin, Pt is the rainfa-ll. volume and P- is the maximum rainfall intensity. The
independent variables in the right hand side of Eq. (9) are listed from left to
right in order of decrearkig significance. Al1 the exponente were negative and
less than one (a = -0.42, f3 = -0.15, y = -0.18, 6 = -0.25). The multiple correlation
coefficient was 0.91.
The roughness coefficient C, was significantly correlated to the basin area,
the stream slope, and the b me flow per unit area by a regression equation of
the t yp
where Bf is the base flow per unit area when the storm occurs, 4 is the drain-
age area and So is the slope of the main stream. The independent variables are
listed in order of decreasing aignificance. The exponents p and v were positive
(1.0 and 1.4 respectively) but A was negative (-0.21)* The multiple correlation
coefficient was 0.64. The value of the reference discharge varied between nar-
row limits, 1.1to 1.4 cubic meter per second for atom wlumas ranging from 2.5
(9)

538
to i4 mm. Making use o? equations 8 throua 10, the model regenerated well the
shapes of the hydrographs; the peak discharges were in general, reproduced within
20% and the times to peak within 10% of the observed values. It should be
remembered that equations 8, 9 and 10 are unlikely to be vali8 outside of the
geographical arca ?or which they were obtained. They indicate, however, the
type of variables which influence the model parameters and their corresponding
sensitivity.
DISCUSSION AND CoNCurSIONS
The framework has been developed for a model which makes it possible to es-
timate the runoff from rainfall and from data obtainable from aerial photography
and from remote sensing. As presented, aerial photograph is needed for the de-
termination of the stream network, the main stream slope and the watershed a r e a n
In addition infrared color photography and/or ground observations are needed to
obtain the soil permeability index, the soil types for the estimation o? the 'B"
horizon permeability and the base flow.
with the rapid progress of the remote sensing technology, it is expected
that, in the near future, the amount of field work necessary may be greatly re-
duced and lidted to calibration areas to obtain the spectral response charac-
teristics needed for the interpretation of the remote sensing scanning.
The rainfall-runoff process in a watershed was simulated by three basic
components: a dynamic contributing area model, a contributing area distribution
curve which integrates the contributing areas along the stream network, and a
linear routing technique. In the proposed dynamic contribution area model the
exponent N, which quantifies the rate of expansion of the response area, Was
found to be the dominant parameter, and was found to be correlated to the -off
ratio. The "B" horizon infiltration and the weight of the antecedent rainfd1 D
were not the primary parameters. In the linear routing, the roughness Parameter
was found to be correlated to geomorphologic parameters and to the baseflow Wr
unit area. The reference discharge did not change significantly from Storm to
storm or from watershed to watershed.
ACKNOh'LEIKMENT
The Work presented herein was supported by Lhe Office o? Water Resources Re search, U.S. Department of the Interior under grant OWRR-B-008-IliD, by the Purdue
Research Foundation under grant XR 5869 and by PiPrdue University. m-
thors wish to exprese their thanks to the sponsors,
REFERENCES
1. Rodriguez-Iturbe, I. (1969) Estimation of Statistical Parmeters for Annual
River Flows, Water Resources Research, 5, pp. 1418-1421-
2. Keifer, R. W. and J. A. Sherz (1971) Aerial photograph for Water reSomceS
studies, Jour. o? Surveying and Mapping Div. , Am. Soc. civil Enva. vole 97,
No- SU29 PP. 321-333.

3.
4.
5.
6.
7.
539
Laboratory for Agricultural Remote Sensing, Purdue University, Lafayette,
Indiana (i968 and 1971) Remote Multispectral sensing in Agriculture, Vol.
3 (Annual Rept., 1968) Res. Bull. No. 844, Agr. Exp. Station, also Vol. 4
(Annual Rept., 1971) Res. Bull. NO. 873, Agr. Exp. Station.
Pluhowski, E. J. (1972) Hydrologic interpretations based on infrared ima-
gery of Long Island, New York, Geel. Surv. Water Supply Paper 2009, U.S.
Govt. Print. Off.
Weaver, K. F. (1969) Remote sensing: new eyes to see the world, Natl. Geo-
graphic, Vol. 135, NO. 1, pp. 47-73.
Waltz, F. A. and Y. I. mers (1970)
Remote sensing of hydrologic resources
in the Great Plains, Rept. #I, Remote Sensing Inst., Univ. of South Dakota.
(Available from IVTIS, NO. PB 195 451)
Zachary, A. L., J. E. Cipra, R. J. Diderickson, S. J. Kristof, and M. F.
Baumgardner (1972) Application of multispectral remote sensing to soil
survey research in Indiana, Lab. for Application of Remote Sensing, Purdue
Univ., Lafayette, Indiana, Print 110972. '
8. Scherz, J. P. (1971) Monitoring water pollution by remote sensing, Jour. ai'
Survy. and Map. Div., Am. Soc. Civil Engr., Vol. 97, No. SU2, pp. 307-320.
9. Colwell, R. N. (1973) Remote sensing in the management of earth resources,
American Scientist, Vol. 61, NO. 2.
10. Strahler, A. N. (1964) Quantitative geomorphology of drainage b&ns and
channel networks, in Handbook of Applied Hydrology, V. T. Chow, Ed., McGraw-
Hill Book Co., pp. 4-40, pp. 44-74.
11. Freeze, R. A. (1972) Role of subsurface flow in generating surface runoff,
2, Upstream Source Areas, Water Resources Res., 8. pp. 1272-1283.
12. Dunne, T., Bnd Black, R. D. (1970) Partial area contributions to storm runoff
in a s-1 New Englua watershed, Water Resources Res., 6, pp. 1296-1311.
13. Nutter, W. J., and Hewlett (1971) Stream flow production from permeable upland
basin, paper presented to the Third Internatl. Seminar for Hydrology
Professors, Furdue Univ., Lafayette, Ind., USA, July 1971.
14. Dooge, J. C. I. (1959) A general theory of the unit hydrograph, Jour. o?
Geophys. Res., Vol. 64, NO. 2, pp. 241-256.
15 Dooge, J. C. I. and B. M. Harley (1967) Linear routing in uniform chmela,
Proc. inti. wdroïogy Symp., Sept. 1967, Fort Collins, Colorado, USA, 1, pp.
57-63.
16. kopold, L. E. (1953) Downstream change of velocity in rivers, Am. Jour. of
Science. 25. PP. 606-624.
17. Lee, M.-T. t&d-J. W. Delleur (1972) A program for estimating muoff from
Indiana Watersheds, Part III, Analysis of geomorphologic data snd a dyndc contributing area model for runoff estimation, Purdue Univ. Water Resources
Res. Center, Lafayette, Ind. Tech. Rept. No. 24.
18. Toebes, G. H. and T. P. Chang (1972) Simulation model for the Upper Wabash
surface water system, Purdue Univ. Water Resources Res. Center, Lafayette,
Ind. Tech. Rept. NO. 27.
19. Lee, M. T., D. ~lank, J. W. Delleur
(1972) A program for eetimting mori from Indiana watersheds , Part II , Assembly of hydroloaic euid geomorphologic
data for small watershed0 in Indiana, Purdue Unio. Water Resource9 Ra#. Ccnter,
Lafayette, Indiens, Tech. Rept. No. 23.

540
a
U
O
O
STREAM REACH
2 4 6 8
STREAM REACH i
FIGURE I DRAINAGE AREA DISTRIBUTION ALON
THE STREAM REACHES

INPUT 5 q (0,t)
INPUT - 1 SYSTEM1 OUTPUT -
INPUT
DE LTA
FUNCTION
q(0.t) = Xf0,t)
-
t
t
OUTPUT = q (L,t)
OUTPUT
2 PHYSICAL D?AGRAM OF UPSTREAM INFLOW
INSTANTANEOUS UNIT HYDROGRAPH
FOR SINGLE STREAM REACH
t
541

m (D U N
542

8 N
f d-
(u
d
O
8
ï v)
4
O
I-
o
W
rr n
âLL
=IA
o- 0
"w z
z3
I-o-
CJ
O
O.
a
œ
œ
n
r
>
I
Q
543

MONTHLY STREAMFLOW ESTIMATION FROM LIMITED DATA
C.T. Haan
-- ABSTRACT
A four parameter, monthly water yield model has been developed
and tested that makes it possible to estimate monthly streamflow
volumes from daily rainfall information. The four parameters of the
model can be determined from as little as two years of observed
monthly streamflow data. This makes it possible to install
temporary, short term stream gaging stations to collect two or
three years of monthly streamflow data and from this data determine
the four parameters of the water yield model. The model uses a
self-optimizing procedure so that the user is not necessarily involved
in the parameter estimation process. A study of 24 watersheds
has also shown that the four model parameters can be related
to soil, geomorphic, and geologic characteristics of the basin.
In this way the parameters for an ungaged basin can be estimated
without requiring any data from that particular basin. Once the
model parameters are determined, long traces of monthlv streamflow
data cán be simulated us ng only daiiy rainfa 1 as a model input.
RESUME
_ _ mis au point un mode
On a développé et
e à quatre paramètres,
pour l'évaluation du rendement mensuel en eau, qui rend possible
une estimation du volume de l'écoulement mensuel à partir de
l'information que constituent les pluies journalières. Les quatre
nnram5tres - - . - - - - - di1 - - mod31e . - - - - - n~iivent - - . - -- - P+PP - - - - d8tPvrninLs - - - - , , -. . - - > - na-etiv r-.- --- ~tin~nmrna-
- ....I"L...tions
aussi restreintes que les résultats de deux ans d'observation
des débits. Cela permet d'installer des stations de jaugeage des
débits, temporaires, à court terme, afin de réunir des informations
portant sur deux ou trois ans, sur les débits mensuels, et, 2
partir de ces informations deduire les quatre paramètres du modèle
de rendement d'eau. Le modèle fait usage d'un procédé qui évalue
automatiquement les informations et les dispose de la meilleure
facon possible (self-optimizing procedure) de sorte que l'usager
n'a pas nécessairement à se préoccuper de l'estimation des paramètres.
Une étude de 24 bassins (watersheds) a aussi montré que
les quatre paramètres du modèle peuvent être reliés aux caracteristiques
géomorphiques et géologiques, ainsi qu'a celles du sol, pour
un bassin (basin) donné. De cette facon, les paramètres pour un
bassin non jauge peuvent être évalués sans que l'on ait besoin de
faire appel a des renseignements sur ce bassin en question. Une
fois que les paramètres du modèle sont déterminés, on peut produire
artificieilement un tracé important (long traces) des débits
mensuels en se servant seulement, pour alimenter le modèle, des
précipitations j ournaïières .

546
The design of surface water supply systems requires an
estimate of streamflow volume characteristics. Ideally, the
designer would like to have available long streamflow records.
Generally, for the design of water supply reservoirs, a long
record of monthly runoff volumes would be sufficient. Unfortunate-
ly these streamflow records are not available for a vast majority
of the streams draining watersheds of up to 1500 square kilometers.
Thus, methods of estimating monthly runoff volumes for these un-
gaged basins are needed.
Watershed models that simulate continuous streamflow records
of fer promise in this area. Several comprehensive watershed
models are presently available [1,2,3]. These models may be
categorized as parametric models in that they contain several
parameters that must be estimated before they can be applied to a
particular watershed. The parameters must be determined manually
and are based on a comparison of observed streamflows with
simulated streamflows.
Realizing the ability of computers to find parameter sets that
satisfy certain objectives and the differences that occur when
different people determine the values for model parameters, Liou
141 attempted to develop a self-optimizing procedure for the
Stanford Watershed Model [i]. Further work on the Stanford Model
was done by Ross [SI in an attempt to relate the optimum model
parameters to watershed characteristics.
Haan [6] has developed a relatively simple runoff model that
enables one to estimate monthly streamflow from daily precipitation
and estimated average daily potential evapotranspiration. The
model contains four parameters that must be determined for each
basin. These parameters are :
f
Smax
Cmax
= maximum infiltration rate (cm/hr).
= maximum daily seepage loss (cm) .
= water holding capacity of that part of
the soil from which the evapotranspiration
rate is less than the potential rate
unless this portion of the soil is
saturated (cm) .
'Associate Professor, Agricultural Engineering Department ,
University of Kentucky , Lexington , Kentucky, 40 506 , USA.

Fs = fraction of the seepage that becomes
runoff (-1.
The optimum values for these parameters are defined to be
those that minimize the sum of squares between the observed and
simulated monthly runoff volumes. Thus, to get the optimum
parameter values , some observed streamflow data is required.
Work with the model has shown that two years of monthly streamflow
data are usually sufficient to obtain a satisfactory set of
optimum parameter values.
The model has the capability of determining the optimum
parameter values for a particular basin when provided with daily
rainfall, average daily potential evapotranspiration by months,
some observed monthly streamflowc for the basin, and a set of
initial estimates for the parameter values.
The optimization procedure that is
univariate technique. The value of the
n
c (Vo - Vs. ) is computed using the
i=l i 1
parameter value. in this expression, n
flow and Vo is the observed volume and
streamflow $or the ith month. Next the
presently used is a simple
objective function
initial estimates for the
is the number of monaths of
Vs. the simulated volume of
vatue of one of the parameters
is chanued bv a fixed amount and the obiective function is
recomputed. &e vaiue of this parameter contiAues to be changed
as long as the objective function is improving (getting smaller).
The other three parameters are adjusted in the same manner one at
a time. Since these parameters are not independent, the entire
process is then repeated one or two times. The result of this
iterative process is taken as the optimum set of parameters.
This model has been tested on 24 watersheds in Kentucky
ranging in area from 1.74 to 1225 square kilometers and on 3 water-
sheds in South Carolina ranging in area from O .ll to 2.27 square
kilometers. The results of these evaluations are given in 161 ,
[71 and [8]. Defining the average prediction error (%) as 100
times the absolute value of the difference between the observed
and simulated average annual streamf low divided by the observed
average annual streamflow, the average prediction error for these
27 watersheds is 4.0 percent. For these watersheds, the average
annual runoff varied from 18.7 to 48.6 cm.
For streams on which there are no records available, at least
two procedures can be used to estimate the optimum parameter values.

548
If sufficient time is available, a temporary stream gaging station
can be established and operated for two or more years. This
station would only have to provide information on the monthly
flows. The data from this short-term gaging program could then
be used in the optimization scheme described earlier.
Jarboe and Haan [7] have used a second technique for estimat-
ing the four model parameters for ungaged basins. This method uses
streamflow information from gaged basins in the vicinity of the
ungaged basin of interest. The optimum model parameters for the
gaged basins are determined and related to measureable character-
istics of the gaged basins. These relationships are then used
to estimate the model parameters for the ungaged basin.
The basin characteristics used by Jarboe and Haan [7] are
shown in Table 1. The four model parameters were related to these
factors using multiple linear regression. Twenty-three watersheds
were included in the study. Six of the watersheds were selected
at random and treated as ungaged basins.
The remaining 17 basins
were used in developing the following prediction equations for the
model parameters :
fmx = 11.83 - 11.51 Smax - 0.0147 SdSb - 0.030 A H
g
- 0.334 PkFc + 0.692 VrPR
= 0.073 + 0.0031 Wc + 0.00075 Iw L - 0.0021 P H
nax a g
+ 0.00011 FcL - 0.0057 V H
r g
C = 7.69 + 0.739 IwSb + 0.011 S H + 0.0243 FcIw
d 9
Fs = 0.325 + 0.0068 L + 0.444 PkSb + 0.00027 PsSd
- 0.018 WcPk
These equations should not be used on watersheds (1) greater
than 100 square kilometers in area, (2) on urban watersheds, or
(3) on watersheds that differ greatly in their hydrologic char-
acteristics from the watersheds used to derive the equations.
(1)
(2)
(3)
(4)

Table 1. Watershed characteristics used by Jarboe and
Haan [7] to estimate the water yield model
parameters.
Geomorphic Factors
A Basin area (km')
percent of basin under forest cover (%)
Percent of basin in lakes and ponds (%)
Slope of the main stream (%)
Length of the main stream (km)
%
Soil Factors
W Average available soil water capacity (cm)
HC U. S. Departnuint of Agriculture, Soil
Conservation Service hydrologic soil
group converted to a numerical index
from 1 to 4 (-1
Sd Average soil depth (cm)
P Average soil permeability (cm/hr)
p:
Average permeability of upper soil horizon
(cm/hr)
549
Geologic Factors
vr
"Rock" volume = mean basin elevation above
basin outlet times the basin area (krn3)
I,
Water availability index (an index ranging
from 1 to 4 depending on the ability of the
material underlying the basin to yield water
to wells) (-)

5 50
Table 2 presents a s-ry of the results of using the above
4 equations to estimate the parameters of the model on the 6 basins
that were taken as unqaged. The simulated runoff values were
obtained by estimating the model parameters from equations 1
through 4 and then using these estimated parameters in the water
yield model to simulate monthly streamflows. The six watersheds
listed in table 2, although actually gaged, were considered as
ungaged and not used in developing equations 1 through 4. These
results indicate that reasonably good estimates of runoff volumes
can be made on watersheds for which no streamflow records are
avai lab le.
EXAMPLE APPLICATION
The South Fork of the Little Barren River in Kentucky was
used to illustrate the application of this model under various
conditions. Streamflow records have been maintained by the U.S.
Geological Survey for this watershed for the period October of
1948 through September of 1970. A U.S. Weather Bureau rain gage
at Edmonton, Kentucky, about 6 1/2 kilometers from the watershed,
was used to provide the needed precipitation input. Some of the
watershed physical characteristics are given in table 3. This
watershed was not selected because of the ability of the model to
simulate its monthly flow, but because of the long gaging record
for the stream that could be used to check the simulated results.
Table 4 summarizes the various simulations made on the South
Fork of the Little Barren River watershed. Methods a through d
are examples of how the model might be used to simulate streamflow
from a previously ungaqed area. Method a required no streamflow
records in that the parameters were estimated from equation 1
through 4. Methods b, c, and d illustrate how the model can be
used if it is possible to initiate a stream gaging program and
collect 1, 2, or 3 years of data respectively on monthly runoff
volumes. In method e the entire 22 years were used in a procedure
described by Haan 161 to obtain the parameter values. In table 4
the percent error is as previously defined, the correlation
coefficient is the simple correlation between the observed and
simulated monthly flows for the entire 22 year period of record,
and the slope is the slope of a simple regression line relating
the observed and simulated flows.

Table 2. Comparison of observed and simulated average
annual runoff for six Kentucky watersheds when
the mode1 parameters are estimated by equations
(1-4).
Observed
Average
Annual
Watershed Runoff
Helton Branch 43.59 crn
McGills Creek 41.50
Perry Creek 34.16
Stillwater Creek 48.59
Little Plum Creek 46.74
N. F. Nolin River 39.90
Simulated
Average
Annual
Runoff
44.63 cm
46.41
33.55
42.98
48.01
43.46
551
Percent Watershed
Error Area
2
2.4 2.20 km
11.8 5.54
1.8 4.45
11.5 62.16
2.7 13.33
8.9 94.28
Table 3. Physical characteristics of South Fork of the
Little Barren River watershed, Kentucky.
Area
Forest cover
Lakes and ponds
Slope of main stream
Length of main stream
Available soil water capacity
Index of USDA hydrologic soil group
Average soil depth
Average soil permeability
Average permeability of upper soil horizon
II Rock I' volume
Water availability index
47.4 km2
62 %
0.11 %
0.32 %
15.96 km
17.68 cm
2.30
84.84 cm
3.02 cm/hr
3.35 cm/hr
2.25 km3
2.0

552
The mean annual runoff for the South Fork of the Little
Barren River is 50.17 cm. Thus an error of 1 percent represents
an average annual error in the simulated runoff of O .5 cm. When
the model parameters were calculated from equations 1 through 4,
the error in the average annual runoff was 4.3 cm. Figure 1 shows
a portion of the simulated and observed streamflows for the watersheds.
The simulated monthly runoff shown in this figure were
obtained using parameters calculated from equations 1 through 4
in Haan's [6] wateryield model.
Again it is cautioned that these
equations may not produce reliable parameter estimates for regions
hydrologically different than Kentucky. The technique of deriving
parameter prediction equations should, however, be valid else-
where.
Methods b, c, and d of table 4 illustrate how a few years of
streamflow data can be used to estimate model parameters which in
turn can be used to simulate long traces of monthly flows. The
variable nature of streamflow from year to year is apparent in
the runoff records from this watershed. As an example the first
three years of the 22 year record produced the highest, third
highest, and sixth highest annual runoff. The average annual run-
off for the first three years was 75.79 cm as compared to 50.17 cm
for the entire period of record. It was these three wet years
that were used in determining the model parameters indicated in
table 4 under methods b, c, and d. This indicates that even
though the years used in obtaining the model parameters may not be
representative, reasonable estimates of streamflow can still be
obtained.
Table 4 also indicates that the accuracy of the simulation
depends on the years used in determining the model parameters.
The fact that using two years of flow data to obtain the
parameter values produced better simulated results for the entire
22 year period than did the parameters obtained from three years
of data is not unusual; however, in general the more years used
to obtain the parameters, the better will be the simulated
results.
Method e consisted of (1) optimizing the model on the first
year of record, (2) simulating the entire 22. years of flow with
these parameters, (3) reoptimizing the model on the 2 years from
the entire 22 year record that produced the poorest fit, and
(4) finally determining the final parameters as a weighted
average of the resulting two optimum sets of parameters where the
weighting factors are the sum of the deviations of observed flows
from simulated flows. The parameters obtained in this manner

Table 4. Methods used to optimize parameters on S.F.L. Barren
River and summary of simulation results.
Method
~
Des cri D ti on
(a)
íb 1
Parameters calculated from equations 1-4.
Parameters determined by optimization on first year of
data.
(Cl Parameters determined by optimization on first two years
of data.
(dl
(e)
Parameters determined by optimization on first three
years of data.
Parameters optimized by Jarboe 181.
Percent Correlation
C
fmax 'ma,
Method Error Coefficient Slope cm/hr cm/day cm
(a) 8.64 0.91 0.93 3.58 0.21 13.33 0.49
(b) 10.13 0.87 0.96 3.45 0.25 16.38 0.54
(c) 2.19 O .92 0.91 3.58 0.20 . . 12.57 0.54
(d) 9.38 O .92 0.89 5.61 0.22 11.81 0.69
(e) O .56 o .91 0.92 3.30 0.22 12.45 0.58
when used with the watershed model were able to simulate the 22
years of record with an average annual error of only 0.56 percent
or 0.28 cm. Obviously this technique cannot be used on a data
scarce watershed. It is included here only to provide an
indication of the ability of the model to simulate monthly stream-
flows.
This model like most parametric hydrologic models, is
in a constant state of change as improvements are incorporated to
make the model easier to use, to reduce computer processing time
and to increase the accuracy of the simulations.
FS
553

554
SUMMARY
Two procedures for using a four parameter water yield
model for simulating traces of monthly streamflaw from watersheds
with either no or very limited streamflow information are
presented. The two procedures are (1) to relate the model
parameters to watershed physical characteristics using stream-
flow data from watersheds located near the watershed of interest
or (2) to establish a short term gaging program on the stream
draining the watershed and use these streamflow records to
determine the model parameters. Once the model parameters are
determined, long streamflow traces can be generated using either
measured or synthetic daily rainfall. These two procedures were
illustrated on a watershed in Kentucky and demonstrated that
reasonably accurate estimates of monthly streamflow can be
obtained.
Acknowledgements: The work in which this report is based was
supported in part by the Kentucky Division of Water and in part
by the Kentucky Agricultural Experiment Station as a contribution
to Southern Regional Project S-53 "Factors Affecting Water Yields
from Small Watersheds and Shallow Ground Aquifers". The paper
is published with the approval of the Director of the Kentucky
Agricultural Experiment Station.

1.
2.
3.
4.
5.
6.
7.
8.
BIBLIOGRAPHY
555
Crawford, N. H. and R. K. Linsley. (1966). Digital simul-
ation in hydrology: Stanford watershed model IV. Technical
Report 39 , Stanford University , Department of Civil
Engineering, Stanford, California.
Holtan, H. N. and N. C. Lopez. (1971). USDAHL-70 Model of
Watershed Hydrology. Technical Bulletin No. 1435, Agricultural
Research Service, U. S. Department of Agriculture, Washington,
D.C. 84 pp.
Tennessee Valley Authority. (1972). Upper Bear Creek
Experimental Project: A continuous daily streamflow model.
Division of Water Control Planning, Hydraulic Data Branch,
Knoxville, Tennessee. 99 pp.
Liou, E. Y. (1970). OPSET: Program for computerized
selection of watershed parameter values for the Stanford
Watershed Model. University of Kentucky Water Resources
Research Report 34, Lexington, Kentucky.
Ross, G. A. (1970). The Stanford Watershed Model: The
correlation of parameter values selected by a computerized
procedure with measureable physical characteristics of the
watershed. University of Kentucky Water Resources Institute
Research Report 35 , Lexington, Kentucky.
Haan, C. T. (1972). A water yield model for small watersheds.
Water Resources Research 8 (No. 1) , pp 58-69.
Jarboe, J. E. and C. T. Haan. (1972). Calibration of a four-
parameter water yield model to small ungaged watersheds in
Kentucky. Paper No. 73-207 for presentation at the 1973
Annual Meeting of the American Society of Agricultural
Engineers, Lexington, Kentucky, June 17-20, 1973.
Jarboe, J. E. (1972). Calibration of a four-parameter water
yield model for use on small, ungaged watersheds in Kentucky.
Unpublished M. S. Thesis in Civil Engineering Library , University
of Kentucky, Lexington, Kentucky.

556
20[ PART OF
- S.F.L. BARREN R.
- RUNOFF RECORD
- -
OBSERVED RO O
SIMULATED RO -
TIME
Figure 1. Example of monthly streamflaw simulation
results using calculated model parameters.

AB ST RACT
OBTAINING DEFICIENT INFORMATION BY SOLVING
INVERSE PROilLEMS FOR MATHEMATICAL RUNOFF MODELS
V.I. Koren and L.S. Kutchment?:
Possibilities are considered for increase of deficient
information for extending observation series by solving the
"inverse problemt1 for mathematical runoff models. The results
of applying the theory of "improperly posed problems'' are
presented. Examples are given for representing hydrological,
geometrical and hydraulic characteristics of the basin by
lumped and distributed parameter runoff models.
RESUME
Les auteurs examinent les possibilités de la resolution
du problème inverse appliquée aux modèles mathématiques d'eco!
lement, en vue de compléter les lacunes des séries d'observa-
tions et d'étendre la période couverte par ces séries. Ils ex-
posent les résultats qui ont été obtenus par l'application de
la théorie des problèmes posés incorrectement. Ils citent des
exemples de détermination des caractéristiques hydrologiques,
topographiques et hydrauliques a l'aide de modèles d'ecoulement
globaux ou matriciels.
:k Hidrometeorological Centre of the USSR.

558
Mathematical modelling of hydrological processes is increas-
ingly used to provide for missing information and to extend hydrolo-
gical time series. Mathematical models are predominantly used for
the solution of the so-called 'direct problem', consisting of deriv-
ation of unknown hydrological variables by solving respective differ-
ential equations with known coefficients and known initial and bound-
ary conditions, In a large number of cases it is necessary to solve
the 'inverse problem' namely to find the coefficients and establish
the initial and boundary conditions using observed values of the
hydrological variables included in the equations. This approach has
as yet gained relatively rare use due to the fact that the solution
of the 'inverse problem' is more difficult than that of theldirect
problem'. The solution of the 'inverse problem' may be circumvented
by multiple solutions of the 'direct problem' for example by the
methods of trial and error and subsequent optimization. Thia may
lead however to a non-unique or inferior solution. The principal
difficulty in the solution of the inverse problem consists in the
fact that it may be incorrectly posed and thus leads to the non-
existence of some or any initial conditions or leadsto a solution
in which a small change of initial conditions (data) due for example
to observational errors, results in major changes in the results.
This has caused in the past a reluctance toward the use of this
method, since the solution being of very low accuracy and high un-
certainty casts doubt on its physical significance.
A number of studies were made in recent years (particularly
by A.N. Tikhonov and his school) aiming at the correct posing of
the problem by establishing the necessary conditions for it.
A.N. Tikhonov has shown that it is possible to u13e a priori inform-
ation on the solution to ensure a continuous dependance of the
solution of an incorrectly posed problem on its initial conditions
and to derive special algorithm:: which prevent bringing out the solution
outside the limits of its uniqueness and of the existence of its initial
conditions. In particular it made possible to solve with sufficient
stability such classical incorrectly-posed problems as the integral
equation of the first type, algebraic systems with improper initial
conditions, the Cauchy problem c?f the Laplace equation and others.
The theory of the 'inverse problem' has thus stimulated the formu-
lation of algorithms used in many scientific and technical fields.
The method was particularly useful in geophysics, where it permitted
the solving, for example, of problems of determination of rock charac-
teristics not accessible for direct measurement as well as restora-
tion of missing information, to cite only the most important points.
The use of this method in hydrology appears also as most promising.
Examples of such studies, used in hydrological practice, are given
below. They illustrate also the principles and possibilities of the
theory of incorrectly posed problems.
1. Determination of the input functions of the models
with lump parameters
Let us suppose that the process of transforming an input h(t)
in the catohment (effective rainfall or an inflow) into an output
Q(t) can be described by the Duhamel integral:

559
where P(t) is some known function of influence. Then having the observations
on Q(t) and knowing the function P( t) (by historic observations or
from physiographic and hydraulic data) it is possible using (1) to derive
h(t). Thus an improperly posed problem is solved - consisting of an integral
equation of the first type. It is possible to solve this problem
on the basis of A.N. Tikhonov's algorithm. Integral (1) is replaced by
a summation according to the method of rectangles and a smoothed functional
curve is constructed:
-b
3
where Q = a vector, designating the Ordinates of the given hydrograph Q(t);
h = a vector of the unknown ordinates h A = a matrix with elements
5'
P ; d= a positive Constant. Finding the minimum of this functional
mk&'it possible to receive a sequence of stable solutions %, which
converge to the accurate solution providing there are no errors in the
given data. However since there are always errors in these, changing
the parameter &(called parameter of regularization) we select such
solution which corresponds best to the a priori information about the
function h(t). For exam e good results are obtained with the aid of
the condition Th(t)dt=$(t)dt.
o
Other kinds of a priori information, allowing the narrowing
of the interval of unknown solutions, may be a suggestion on the smoothness
of the solution, the non-negativeness of the ordinates, the closeness
to some known function and so on. Naturally, the narrower the interval
of the solution, the higher accuraoy will be obtained.
Results in using
functional (2) to determine the input functions of the runoff models,
described by the Duhamel integral, are presented in greater detail
in (3)' where examples of constnicting effective rainfall, hydropower
station releases and snowmelt intensity are treated. Another approach
to the solution of the inverse problems for models described by the
Duhamel integral (linear models with lump parameters) are indicated in
(6).
2. Determination of geometric and hydraulic charactexistics
of river channels using observations of flow
To describe unstea9flow in a river channel Saint Venant
equations may be used:
(3)

560
where 2 (x,t) = stage at point x at time t, Q(x,t) = discharge, K(x,z)
forces of resistance1 g= acceleration of gravity. Because of great
variability of geometry and roughness of the river channels the
functions F(x,z) and K(x,z) determined by the observations in
separate points are not quite representative for the whole river reach,
even with large frequency of observations. Thus a problem of determining
the averaged relations P(x,z) or B(x,z) = aF/aZ and K(x,z) by observations
of flow (the determination of coefficients of the system (3))
is of great significance for the establishment of the most characteristic
geometry and hydraulic properties of the river channel as well as
for ensuring sufficient accuracy of the calculationa. It can be shown
that this problem is improperly posed and for its solution it is
necessary to derive special calculating algorithms. We shall discuss
below two of the approaches tried by us in solving this problem.
(A) The discharges and Water levels are known in a rather large
number of sites.
Integration of the continuity equation (3) with respect to x,
leads to:
Finite differences are substituted for the derivatives and
instead of an integral it is possible to construct for every time moment j
the following system of equationst
In order to solve this system it is necessary to have Q(x,t) F(x,o) and
F(o,t). As the problem is improperly posed the solution of the
system (5) is unstable. For its regularization the solution of
A.N. Tikhonov's functional is with introducing initial approximation.
As a result for every time suchFarefoanä which correspond to the
minimum of the functional.

561
where 2 is the given initial approximation, d= positive parameters,
thus a golution is found which not only secures the minimum of square
deviation of the right part of the system (5) from the left part, but
at the same time it is least deviated from the initial approximation.
The condition of functional extreme gives:
To select the quantitydmethod of discrepancy has been used.
The idea of this methori COnSiStS in conforming the accuracy of the
problem's solution to the accuracy of observed data.
It is supposed that the error0 of the given information forming
discrepancy of the system (5) are known and an d is found which
secures this discrepancy 8'. It is possible to prove that if the
functional (6) is used the parameterdsecuring the given discrepancy
is unique. The initial pproximation can be made in a rather crude
mannerbarticularly for 9 = O), however giving a good initial approximation
contributes toam-aocurate optimum d. Use of the initial
approximations provides great possibilities for improvement of the
solution by introduction of a priori information. Such a priori
information can be an empirical relationship between geometrical and
hydraulic characteristics, observed in separate sites, and different
theoretical formulas (for example, we have used the equation of the
typical form of river Otrinnel derived from the principle of minimum
dissipation of energy).
The values of F (x,t) found according to equation (4) have
been used for determining the characteristics of the resistant forces.
For this purpose the momentum equation has transcribed:
Derivatives with respect to t have been replaced by forward directed
finite differences and the integrals have been replaced by sums de-
rived by the method of rectangles. The resulting algebraical systems
have been solved for all time intervals with the help of the same
algorithm as the system (5) (without the initial approximation).
Aa for determining F(x,t) and K(x,t) the discrepancy has been
taken equal to 5 per cent of the average module from the left integral
equation's part.
This method has been tested on data obtained by special
observations of unsteady movement in the merca river and it has
given satiafactory results (a comparison ha8 been made between the
relations F(x,z) and K(x,z) which have been derived by different
floods by measurements in separate sites) (see figure 1).

562
(B) The stages are known in a rather great number of sites and
the discharges only in the first and the last site.
Let us integrate the continuity equation with respect to
time (in the interval (Ti, Ti+l)) and to distance (in the interval
(0, LI):
to solve it in the form:
where ‘y- the Chebishev polinomials. Let us put (10) in (9):
No terms with zero polinomial are in the left part of the equation (li),
because in this case the integral would be equal to zero. The equation
(li) is therefore not sufficient for the full determination of the
function (10). However it can be used for determining the function
B(x,z), which can be presented:
To find the coeffiaients we shall construct a system of equations
(their number must not b e h w than Y=(n +l)m , and change the limita
of the integration with respect to time in (dl so as to embrace the
whole amplitude of variation of discharges and of stages on the rising
as well as on the falling, part of the hydrograph. Let us write this
system in the matrix forms
Bere is the matrix of +th order, its elements are equal

563
id- vector of the unknown coefficients $(B. x - right part with elements:
Since the system (13) is unstable, ita solution is possible
with A.N. Tikhonov's functional. AS a result the following system is
found :
where%'* - matrix transformed with relation top, E - the unit matrix.
The parameter of regularization d has been determined from the
conditions of minimum of the function
where A.PP,,), A (dp) - j-th elements for the two successive
values JO(. +or determining (ni + i) coefficients entering in (10)
we shall replace in (9) discharge with the product of a cross section
area and the velocity of the current U(x,t) and shall make the proper
integration with respect to time and to distance. Putting in the
resulting equation the relation (10) we shall find:
Here C - matrix of (nI + I) x N-th order with elements
The rest of symbols are the same. lystem (17) is solved by analogy
with system (13). Having determined 3 and it is possible, asing
relations (10) and (12) to find function B(x,z).
This approach has
been tested on the data of special observations in the Svir river.
In figure 2 functions B(x,z) for some sites, calculated by relation (10)
are shown: furthermore widths were determined aocording to topographic
data. For controlling the results of these calculations the discharges
in-the intermediate sites have been determined with the help of equation:

564
These discharges have been found as very close to those observed.
The coefficients received from the different floods have turned out
to be quite simila % and this fact indicates their sufficient stability.
Let vs see now a scheme for determining the hydraulic'characteristics
of river channels. We use the dynamic St. Venant equation, assuming
that the ineLtial terms are equal to zero
Putting (18) into (19) and integrating with respect to distance in
the interval (0,ï) we gett
às earlier we shall find the solution in the form:
Putting (21) into (20) and using Tikhonov's functional by analogy with
the previous one we construct the system of equations for determining
the coefficients Dks:
-b
where D - vector of unknown coefficients, 3- vector with elements
&=Z(t) - Z(t), - matrix of (n2+1).(m2+l) N-th order with elements
The function B(t, t) hae been calculated according to relation
(12) including the earlier determined coefficients Ilks. The found
functions have been oompared with the functions determined by the method
of optimization. It was found that a strong smoothing is observed.
This can be eliminated by taking logarithms in equation (19).

References
565

566
I50
SO
/Iff
96
a 0
e
E
a *'
0
8
e
O
e
8
b

x r acm
I l
I
I
I
I
I
I
15.0 250 2.50
I I I
200 390 4uu
.x= 3.9cm
I I I
i50 250 350
i
/
567

ABS TRACT
THE MATHEMATICAL MODEL OF
WATER BALANCE FOR DATA-SCARCE AREAS
by
Nabil Rofail
The mathematical model of water balance for data-scarce areas
is designed. The solution of this problem is considered of general
type of boundary conditions.
The equations of motion and mass conservation lead to the lin
ear parabolic partial differential equation. The equation is solved
by implicit scheme and the alternating direction procedure is appli
ed for computation. The numerical procedure has second order accu-
rracy and is unconditionally stable. As it contains no iterative --
routines, it is also exceptionally economical in computing time and
memory requiments. Therefore the procedure is recommended for areas
of inadequate hydrological data.
RESUME
L'auteur décrit un modèle mathématique de bilan hydrologique
destiné aux régions pour lesquelles on dispose de peu de données. -
La solution du probleme est envisagée pour des conditions aux limi-
tes générales.
Les équations du mouvement et de la conservation de masse con
duisent à une équation aux différentielles partielles linéaire para
bolique. Cette équation constitue un système implicite qu'on résoud
par approximations successives. Le mode de calcul numérique est in-
conditionnellement stable et permet une précision du second ordre.
Comme il ne contient pas de procédé itératif, il est aussi exception
nellement économique en temps de calcul e; en dimension de mémoire.
I1 est donc recommande pour les régions ou les données hydrologiques
sont insuffisantes.
~~ ~ ~
* Water Resources Dept. Desert Institute, Mataria, Cairo, Egypt.
-

570
Introduction
The system of equations of motion and equations of mass con-
servation for the ground water flow leads to linear pzrabolic di-
fferential equations. In the present work the impervious bed has
been considered of any configuration and the recharge or dischar-
ge from the aquifer has been introduced. Moreover the effect of
boundnries are considered either of a river or of a continuation
of the aquifer of different parameters. The present aaterial deals
with the solution of the balance equation that can be easily app-
lied for areas of inadequate hydrologicd. data.
Formulation of muation
The aquifer is considered homogenous and the effect of the
impervious bed is &%Ven. According to Dupuit assumption, the equ-
ation of motion cam be written as followsj ( see Fig. i) ,
v = - k 2 i h - t ~ ) = - k a h - k dz (2)
"Y "Y
The equation of mass conservation can be considered as follows;

!he dischuge or the recharge to the aquifer is considered as a
:unction of time to the exposed area i.e. N = f ( x,y,t) equat-
tons (1),(2) and (3) proYide the following equation system,
- k d(h+z) ah
- k h QQ(lZz'
- kh --
9X ax
Q'íh+z) - k a(h+z) ah - N =a (4)
9 Y* ay al/
?or simplifying the solution of equation (4), Boussensq acsump-
:ion ( the powers of derivatives of one order, SLTB of lower ma-
pitude than the derivatives themselves ) is applied, therefore
the system will be,
kherefore equation (5) can be written in the alternate farm,
571

512
\A consistent imdicit difference scheme
The three implicit difference scheme has been applied for equ-
ations (6) and (7), considering the grid spacing of A5 and time
intervai At, ( see fig. 2 >. Thus leads to;
AS
and (g), such that;
Equation (10) is found by Taylor's series expansion, to be term
by term consistent with equation (5). The von Neuinann method is
used for examining the stability of the finite deference scheme.
It has been found out that the amplification factor 5 1, that
1mAS
the original component e will not increase with time.
Therefore the scheme can be considered absolutely consistent to
second order accuracy, i.e. to O ( Asz, A tz), and absolutely

unconditionally stable. Thus it is not unexpected as the applied
scheme is implicit.
Alternating Direction Algorithon
Equations (8) and (9) can be written respectiviely in the
following form,
@here, A,B,C and D are coefficients of known values,i.e.,
573

574
Each of equations of equation5 (11) a d (12) forms a tridiago1
al vector system that may be solved using the aìternating directior
algorithm (e.g. Richtmyer and Mooton,1967 1, by introducing auxili-
ary variable E and P in the x sweep as follows:
Introducing (11) in (131, that provides recurrence relations;
)-I , ,$= Cq- A, $+,)(A,%+,+%)
5
= - CJ ( AJ El,, +B,
Method of Computation:
Thensgion of interest in the x-y plane in which the numurical
solution of equation (5) is carried out, is divided by a mesh of g~
id lines. The distance between grid lines need not be the same, ths
is considered one of the reasons for recommending this metnod of cc
putption for areas of inadequate hydrological data? Equation (8) is
solved for the x sweep for the time level (9) to the time level
( n + j$ 2, and equation (9) for y-sweep for the time level ( n +
to the time level ( n 4 1 ). The recurrence can be initiated from
boundpry conditions of any two - point type; two velocities ( con-
tinuiation of the aquifer ), two depths ( bounded by rivers ), one
velocity and one depth.
If the boundary is a continuiation of the aquifer, this could
be illustrated as follows;

.s the boundary point is situated at JJ A x, the equation (14) can
expressed as follows,
r the, case where the boundary is bounded by a river, therefore;
575
The comput.afbncan be applied for the x-sweep by determining the
zfficients P and P from one bounary to the other bounduy and bet-
3n them for all grid points. Thus the new values of the potentials
the time level ( n + % ) can be determined from the recurrence re-
-
;ion (13). These new values of the time level ( n + M ) are used
? the computation of the coefficients of y-sweep and the values of
;ential at the end of time level ( n + 1 ) have been found out ( see
IW chart diagram ). This method is known as the multi - sweep method.

576
Conclusions
A program in AIGOL - 60 has been executed on ICL - 1900 machine
for the water balance equation. The program was tested for different
boundaries ( e.g. river, dyke, .*) of different parameters.
As the method of solution is based mainly on the three implicit
difference scheme that there is no conditions for choosing the dis-
tance between the grid points and the time interval A t. Use this
procedure contains no iterative routines and it has been found out
that this method is exceptionally ecomomical ln computing time and
memory requifements and the solution is considered of high accuracy
Thus this method of computation is recommended for areas of in-
adequate hydrological data.

BOR n = 1 STEP 1 UNTIL nn
I
[COMPUTATION i)F h AT n + $ 1
1 -[h
.y
= Km h
J.
,COMPUTATION OF SWBE2 IN Y-DIHECTION
FOR J = 1 STEP 1 UNTIL JJ
\'
i
COU'UTATION OF SwIEE;p IN X-DIWTIÙN
c
[COMPUTATIUN FOR E & F FOB kk - 1 to i]
1
ICOMPUTAIION OF h AT n+l
I
FLOW CRART DIAGRAM
577

570
Symbo 1 e
h : heigtit of water table above the impervious bed.
u,v: the flow components. in x and y directions respectiv;aly.
k : Coefficient of permeability.
: Coefficient of specific yield.
ïV t intensi- of infiltration to the ground water table.
AS : the grid spacing.
At : the grid spacing on t-axis.
JJ : the number of grid points on x-axis.
kk : the number of grid pointe on y-axis
nn : the number of grid points on t-sis.
3 : any grid point on the x-axis.
k : any grid point on the y-axis.
n : any grid point on the t-axis.
AOKNOWLEDGWT
This work is sponsored in part by the Water Resources Depart-
ment of the Desert Institute, Cairo, Egypt, to which the author is
gratef uì .

Literature
1. Abbott M.B. ( 1967 ). Difference methods, Lecture note, Inter-
national course of Hydraulic Engineering, Delft, Holland.
2. Mitchell A.R. ( 1969 ). Computational methods in partial diff-
erential equations, John Wiley.
3. Nabil Rofail ( 1972 ). The numerical computation of parabolic
equation using inplicit difference scheme and alternating
direction methods, gth Conference on statistics and computat-
ional science, Cairo, Egypt. pp. 572~- 593.
4. Richtmyer R.D. and Mooton K.P. ( 1967 ). Difference mothods for
ijlitial value problems, Interscience.
5. Uri Shamir, (1967). The use Of computers in ground water hydrology,
hydro dynamics Laboratory, Beport NQS. 105, Yasaachusetts.
579

580
t
cr>
4
t
tn
0
r:
3ig.l Diagramatic representation of unconfined aquifer
Y
a R U
J+1 J J-1
k-1 n+l
k
4 k-1 n
a
Fig.2 The three level Scheme
,
'COS- AS 3

ABS TRACT
DATA ACQUISITION AND METHODOLOGY FOR A SIMULATION MODEL
OF THE LLOBREGAT DELTA (BARCELONA, SPAIN)
Francisco VilarÓ Rigo1 y Emilio Custodio Gimena
The Llobregat Delta (Barcelona) is a 80 sqKm., area supplying up to
150 million cubic meters per year of water for industrial, urban and
agricultural uses, in order of decreasing importance. The construction
of a exploitation simulation model has been necessary in order to study
carefully the new problems concequence of a increasing rate of
abstraction, the conversion of extense irrigation lands in industrial
areas, the dredging of a new harbor and the forcoming river regulation
with dams. Historical data were initialy scarce. In one hand they were
restricted to the rainfall and main river discharge knowledge and in
the other hand to some disperse ground water level data and file of
well drillers logs without interpretation. After the classification of
the existing data, some specific studies were iniciated in order to
complete the knowledge of the system and finally the model was
constructed, followed with an important stage of value adjustment,
specially those related to intermediate aquitard properties. The
ajusted model has been used in several stages to forecast the response
to preestablished possible future situations.
Key words: scarce data, model, adjustment, exploration.
RESUMEN
El delta del Llobregat (Barcelona) constituye una zona de 80 km2,
que llega a proporcionar hasta 150 millones de m3 anuales de agua para
usos industriales, urbanos y agrícolas, por orden decreciente de impor
tancia. Ha sido necesaria la construcci6n de un modelo de simulacibn -
de la explotaci6n a fin de estudiar con detalle los problemas apareci-
dos a causa de la cada vez más intensa explotación, transformación de
áreas agrícolas extensas en industriales, apagado de un nuevo puerto y
próxima regulación del río mediante embalses. tos datos histbricos --
existentes inicialmente eran escasos. Por un lado se reducían al cono-
cimiento de la pluviometrla y del caudal del rio principal y por otro
lado a algunos datos esporadicos de niveles del agua y un archivo de -
perfiles de pozos sin elaborar. Se han realizado estudios detallados -
orientados a complementar el conocimiento del sistema y finalmente se
ha construido el modelo con una importante fase de ajuste de los valo-
res estimados, en especial a los referentes al acuitardo intermedio. -
El modelo ajustado ha sido utilizado en varias fases de previsión de -
respuesta ante determinadas situaciones futuras posibles.
Palabras clave: datos escasos, modelo, ajuste, explotación.
( ) Comisaría de Aguas del Pirineo Oriental y Curso Internacional de -
Hidrología Subterránea. Barcelona.
I

582
1.- LOCATION AND BACKGROUND
The Bajo Llobregat is an area spreading from Barcelona
Eastwards and the Garraf Limestone Massive Westwards and SW
(Fig. 1). It is largly occupied by the valley of the Llobregat
river and its delta, whose alluvial formations occupy around
80 Km2., of which slightly over 50 Km2. correspond to the delta
itself.
The proximity to the important urban nucleus of Barcelona,
the fertility of the land, the easy availability of water and
the existence of a big market for its products, have given rise
to and important agricultural and industrial development. Today,
the industry is tending to take the place of farming and both
are rejected by the expanding urban area of the town of Barce-
lona. On the other hand,the current expansion of the Barcelona
harbour, needs new service areas to be prepared, which the Bajo
Llobregat easily offers.
The problems of important water extractions, of increasing
interest in the sands and gravels of the delta and valley for
construction, the additional communication lines, the prolifer-
ation of discharges and tipping of all classes, etc., create a
harmful and apprehensive climate, and leads to the destruction
of the aquifers by emptying and contamination and it may produce
a deep sea intrusion. Its rational administration requires a
good knowledge of the characteristics and hydraulic operation
of the aquifers in the area.
In 1909 a detailed study was made on the groundwater
hydrology of the delta (61, but the systematic and detailed
studies started in 1964, which is the inicial point of a series
of ‘mportant works and reports which are partly listed in the
references. They have mostly been prepared by personnel of the
General Hydraulic Works Board, through the East Pyrenees Water
Committee and the Delegation in Barcelona of the Public Works
Geological Service.
The complicated factors raised the need to have a simulation
model of the aquifer systems available. The Public Works
Geological Service built a R-C (resistors and capacities) model
in 1970 as a first approximation, and almost simultaneously,
the East Pyrenees Nater Board and the Public Works Geological
Service prepared a digital mathematical model of the explotation,
capable of further details and more flexible use (i). The main
problem when building such models lies in the scanty historical
data available, since the systematic control studies of the area
were initiated sistematicaly after 1966.

2.- AQUIFERS IN THE AREA
583
Figure 2, shows the general features of the aquifers in
the area, by means of three cross-sections. In the Llobregat
valley, there is a single aquifer of coarse gravel which divides
up in the delta entrance, into two superposed ones, separated
by a silt-clayey intercalation, which increases in thickness
towards the sea. Thus an upper aquifer, which is mostly a water
table one,and a deep confined aquifer with a weakly semi-
pervious roof are separated. The silt intercalation narrows
and becomes sandy towards the delta margins, and finally
disappears, thus allowing both aquifers to lie directly above
one another, and in easy hydraulic relation (8) (9) (14).
The aquifer of the valley and the deep aquifer of the delta
present areas of high transmissivity where there are important
pumpings, whereas the upper aquifer of the delta support only
reduced explotation.
Both the delta and the valley ape marginated by materials
which may be considered as impervious.
3.- EXTRACTIONS AND HYDRAULIC OPERATION
In figure I, the main extractions and hydraulic conditions
of the model area were shown. In the delta, the two largest
pumping centres are found in Prat de Llobregat and the Free Port,
and they gravitate on the deep aquifer; in the valley they lie
alongside a lower end (Cornella-Sant Joan D'Espi) and neighbour-
hood of Sant Feliu de Llobregat. Other extraction nuclei are
found along the SU edge of the delta, besides other isolated
points, served from both aquifers. The upper aquifer of the delta
receives an excellent recharge from irrigation return flow and
waste water discharge, and it is drained by the sea, the final
stretch of the river, the drains of the airport and the marginal
pumping areas. The aquifer of the valley receives its main
recharge through river water infiltration and from the irrigation
canals, but the permeability of the beds impedes the maintenance
of a direct hydraulic connection, and a non-saturated mediun
exists between water table and the bottom of the surface water.
The deep aquifer of the delta receives the recharge direct from
the valley or from the upper aquifer in the marginal areas or by
vertical infiltration through the silt lens. These relations
and actions can be seen in the double piezometric surface of
figure 3, and are reflected in some detailed studies based on
balance criteria. (9) (10) (141, hydraulic computations (18) (19)
and geohydrochemical evaluation (2) (3) (5) (8).
4.- MOTIVATION OF THE MODEL
Delta groundwater explotation for industrial uses has been
increasing at a rapid pace during the last ten years, at the

5 84
same time as normal extractions for the Barcelona supply have
been dropping as a result of the direct utilization of the
river water, after a suitable treatment. Total extraction
however has gradually increased and it will rise possibly in
the immediate future when it is necessary to reactivate the
urban supply wells to meet growing âemand. The total capacity
of water stored in the aquifer system and easily mobilizable,
is between 100 and 150 million m3., a small figure compared
with the annual extraction which non exceeds 140 million m3.,
and can reach 200 with the present existing pumping capacity.
This means that in the absence of recharge, in a few months,
certain parts of the aquifer dry up or are left with an insuf-
ficient saturated thickness to maintain vel1 discharges. The
river infiltration does not increase when the extractions rise,
as a result of its disconnection with the water table in the main
recharge area (corresponds to the valley), and there is no other
important recharge source except the sea, this inducing a
steadily advancing sea water intrusion. (5) (20).
The study of the effect of new extractions or of different
natural or artificial hydrological, river situations, as a
result of its dam regulation or water transportation to other
areas and also the conversion of farming areas into industrial
zones, is complex. Por this reason it was decided to built a
simulation model to analyse the explotation of the ground waters,
which would also help to assess the river recharge, the sea
intrusion (by indirect evaluation) and the interferences.
The different objectives and variables to be estimated may
be summed up as follows: (4)
Study of the effects of the explotation in certain places,
with or without disappearance of some of the present pumpings.
Study of the artificial recharge effects by spreading and
by wells, and analysis of their technical, economic and
legal feasibility.
Study of the effects and suitability of a recharge litoral
barrier to reduce sea intrusion, in the upper aquifer, in
the deep one, or in both, im selected areas.
Study of the Llobregat river regulation effects and/or
derivation of larger discharges for supply.
Study of the suppression effects of irrigated areas or the
modulation of the irrigation discharges.
Study of the geotechnical problems derived from abandonment
of the main current pumpings.
Study of the operation of the aquifers as local reservoirs
for the most adequate service of demand.

The study of these possibilities includes:
1) Determination of the external balance,
2) Determination of the internal balance.
585
3) Estimation of the fresh water discharges into the sea and
river.
4) Estimation of the sea water encroachment areas and their
possible evolution.
5) Estimation of the deficits which may turn up in the
different zones.
5.- MODEL NETWORK
The shape of the piezometric surface, the distribution of
the pumpings, the plant of the aquifer system and present
knowledge, advised an assymetric network, digital mathematical
model, similar to the one established by the California Water
Resources Department (17) as being the best suited. In accordance
with the already known estimation principles (12) were made the
necessary adaptations for its programming and handling on the
double memory IBM 1630 computer at the Computation Office of the
Public Works Ministry in Madrid, and a series of special
modifications in the boundary conditions. Its constructive and
network details have been published on several occasions (1) (4)
(15). In the delta, the two aquifers are simulated by means of
a double network of polygons (fig. 4) connected by a vertical
conductor branch. The sea condition is established as a constant
level directly for the upper aquifer and by means of a resistent
element (the aquitard) for the deep aquifer. The condition of the
draining river is imposed as om another constant level, and the
river condition in recharge area is established giving a recharge-
discharge set of figures by polygon in terms of the river discharge.
6.- RESOLUTION OF INSUFFICIENCY OF DATA. ADJUSTMENT.
At the time when the need for the model came about, the
number of available data were small, especially regarding the
length of the observation period.
The number of data figures needed is very varied and com-
prises those referring to the geometric form of the aquifers
and their hydraulic parameters, up to those referring to the
temporary and spacial distribution of the extractions, passing
by the infiltration of the rainwater, the river and the irrigations
(6) (7) and they should have a sufficient precision and represen-
tative nature in accordance with the model network. A set of data
should be available in each node and branch of the projected
model.

586
In this case, the geological structure was well known,
owing to a high number of bore-holes (fig. 3) and wells with
filed lithological log, but not SO the hydraulic characteris-
tics of the different formations. These values were fractionary
and corresponded to some precise data of tests in piezometers
and wells made very ofer under difficult conditions, and some
few prolonged pumping tests, with complicated interpretation
due to the notable piezometric fluctuations that are produced,
wich sometimes exceed a metre throughout the day.
With the available data, a plan of isotransmissivities was
completed and a distribution of the seepage coefficient of the
aquitard (intermediary silts) was estimated, based on a few
granulometric tests and geohydrochemical considerations, which
only gave the approximate magnitude.
Clearly a model built under these circunstances,with a
poorly known connection with the river, and for which there was
only a few partial semi-quantitive figures, mostly obtained by
statistical analysis of the river discharges, supply extractions
and levels in valley (131, is a long way from reproducing the
reality, An adjustment process is necessary, based on comparison
of its response to actions taken from the historic series and
comparison with the effects observed in the aquifer. These
actions are the extractions and recharges and the effects are
the piezometric levels.
The adjustment process requires a sufficiently long and
complete set of historic data, in order to complete, correct and
suit the imprecise data, or the estimated or non-existent data.
This adjustment process allows some data to be corrected if
other can be taken as sufficiently precise. Otherwise, no
sole situation is reached, or there is no satisfactory solution
nor one which responds to the real conditions of the prototype
or real system. The set of historic data should be for each
polygon, and this is difficult even in well known areas, and
with a good systematic of measurements. In the case of the Llo-
bregat delta, it was decided to take as "exact" data, despite
certain uncertainties in their determination:
a)
b)
The extractions by pumping anã the recharges by wells and
drains, using as contrast criterion: for agricultural uses,
the irrigated surface anã calculated water needs; for indus-
trial uses, the type of production, number of workers and
real production in those cases where it was known; and for
supply uses, the urbanistic level and population served.
The infiltrations of the rainwater, the irrigation water
and runoff of the nearby areas obtained from water balances
in the soil, and therefore of theoretic type. Nevertheless,
as this is a mild climate area, flat and with classic
irrigation crops, a small error is expected.

507
c) The water losses of the canals by infiltration based on the
inlet and outlet measurements and the irrigation quantities.
In winter, these irrigation quantities are almost non-existent,
wich permits an acceptable estimation.
d) The piezometric surfaces and hydrograms of available ground
water. Most hydrograms have been obtained with eight water
level recorders, plus daily measurements on another six
piezometers, plus monthly readings on a few more points. The
piezometric surfaces correspond to intense and periodical
measurement campaigns of one or two days duration, but these
may have errors due to variations in the measurement hour,
or introduction of some dynamic data or tridimensional flow
areas; nevertheless they are sufficiently valid.
e) River discharges, obtained with certain guarantee at the
upper valley inlet, in Martorell.
f) Geometric dimensions of the modelled units.
The data to be adjusted are:
1. on the model in itself, based on already mentioned
previous values, and with a pre-established variation
margin, taken from existing knowledge.
- Transmissivities of the surface and deep aquifer, with
reduced variations.
- Vertical permeability of the aquitard (intermediary
silts) for which the previous data could be notably
erroneous.
- Porosity of the water table aquifer, only admitting
slight variations in accordance with the lithology.
- Coefficient of elastic storage of the captive aquifers
within a logical margin according to the existing
structure and figures on the interpretation of pumping
tests and the response to sea tide in some water level
recorders of ground waters.
2. on the actions impossed on the aquifer in the adjustment
period, not directly known.
- River recharge, estimated previously from balances,
simplified analysis of the piezometric surfaces of the
valley and a statistical correlation betwen discharges
of the river-supply extractions and levels in the
valley (13).
- Discharge to the river in the final stretch, estimated
by partial balances and sumplified analysis of the
piezometric surfaces. This is a relatively small value.

588
- Discharge to the sea and sea water encroachment values,
according to the aquifer and the coastline area con-
sidered. Measured very roughly due to estimation dif-
ficulties, excepting the central coastal stretch of
the water table aquifer.
The distribution of the recharge between the upper and deep
aquifer of the delta is a result of the adjustment, and also is
the water discharge circulating through the aquitard.
The chief difficulties regarding the adjustment are derived
from insufficient data on levels and a need to account on the
seasonal variations, owing to the great importance of extractions
in relation with the quickly mobilizable ground storage volume
of water. The first piezometric surface useable is at the begin-
ning of 1966, and although another six complete ones and one
partial one were available, their distribution was neither regular
in time, nor covered each of the quarterly periods into which
the year was to be divided up. It was therefore decided to use
the available piezometric surfaces, with minor corrections to
adopt them to the final moment of each quarterly interval,
forming new interpolated piezometric surfaces, based on the data
of the continuous piezometric measurements in some points, already
discussed, trying to maintain the flow shape character.
To complete the quarter figures, the water balance estimations
were made in the soil, and the extractions were calculated from
the inventory according to the annual rate of use and moment the
wells went into operation in some cases, or based on the demand
curves given by some users.
In figure 5, a sample of the result of the final adjustment
process can be seen, taking the 4 years of figures, distributed
into 16 quarterly terms. This final adjustment need 13 stages
with the definitive network. Some prior trials were made with a
simplified network, to know the magnitude and convergence rates
of the estimation process. This adjustment stage incorporated
the modifications suggested by the previous one, mainly modifying
the hydraulic characteristics of the units and the recharge of the
river. Before making a modification, the results obtained were
carefully studied, taking into account previous results with early
stages, and in order to be in accordance with the physical charac-
teristics of the system.
An important result of the adjustment process is not only the
correction of the imprecise data, but obtaining other necessary
data for the model explotation process, previously unknown. Such
is the relation Qr (river discharge) versus IR, thus allowing
the (river infiltration) computation of IR (not measurable) with
available data on QR. The adjustment obtained shows there is this
relation with a sufficient statistical degree of significance.

589
Figure 6 shows the inicial map of transmissivities of the
deep aquifer and the valley gravels and the one obtained after
the adjustment. The differences are not important and in many
cases, the variations are not merely a correction of an erroneous
value, but the adaptation of a precise value (test in bore hole
or well) or of a regional value (pumping test or analysis of
piezometric oscillations) to the dimensions and forms of each
polygon.
7.- UTILIZATION OF THE MODEL
The model has been built for use under different conditions
as those prevailing during theadjustmen process. This creates
various problems. For example, the validity of the model for
other distributions of the pumping or recharge-disoharge, or
those corresponding to piezometric surfaces notably different.
Also, one must consider the validity of the Qr - 1, (river
discharge - river recharge) relation, under different circumstances
of the river system or after conditioning works in the bed. The
adjustment period is rather short, but sufficient to insure
credible results under conditions similar to the adjustment
ones and in time periods not much greater. If one attempt to
simulate 20 years or under pumping conditions with other centres
of extraction, noticeably different as those existing now, the
results would possibly only be semi-quantitative.
One of the recent processes of using the model arose to study
the possibility of temporarily increasing the groundwater
extractions for supply, in the event of a succession of dominatly
dry year combined with a delay in the first service of the new
surface water regulation works of the Llobregatriver (ll), taking
into account the normal pumping increase forecastsfor other pur-
poses. The injuries and needs of redistribution and conditioning
of the pumpings under various foreseen hypothesis have been
assessed, and the sea water encroachment and the later return to
a "normal" situation after these regulation works have been
finished. Some of the possible extraction situations have not
been made as they produce excessive drops which prevent the
pumping capacity of the wells to be maintained.
The use of the model permits the aquifer system of the Bajo
Llobregat to be handled as a regulating reservoir, analysing
the guarantees of the different ground water demands in different
natural or man-made hydrological situations, and a knowledge of
the rate and location of the progressive salinization process or
the effectiveness of the measures adopted to reduce it. These
eventualities were analysed by seven different hypothesis
following the adjustment process (1) (15), including the analysis
of the possible artificial recharge. The model, in its explotation
phase, works with six monthly intervals instead of the quarterly
intervals of the adjustment.

590
8.- CONCLUSION
The careful1 modelling of an aquifer permits a very useful
work tool to be obtained, even though the initiai data is
incomplete or non-existent in certain aspects, provided another
series of sufficiently precise data, or with known error is
available, and which is such that it permits an adjustment
process with a sufficient number of steps.
9. - REFERENCES
1.
2.
3.
4.
5.
6.
7.
Cuena, J. and Custodio, E. (1971) - Construction and adjustment
of a two layer mathematical model of the Llobregat Delta.-
International Symposium on Mathematical Models in Hydrology.-
International Association of Scientific Hydrology.- Varsovia.
Custodio, E. (1967) - Etudes geohydrochimiques dans le delta
du Llobregat, Barcelone (Espagne) - Geochimie, Precipitations,
Humidité du Sol, Hydrometrie. Assemblée Générale de Berne,
1967. Association International d’Hydrologie Scientifique.-
Gentbrugge - pp. 134/155.
Custodio, E. (1969) - Ground water entries in the Llobregat
river delta.- Hydrological Reasearch Documents No. 6, Barce-
lona Water REsearch, Applications and Studies Centre. Speech
in Pamplona 1967 - pp 205/237.
Custodio, E., Cuena, J. and BayÓ, A. (1971) - Problem,
execution and use of a two layer mathematical model for the
Llobregat delta aquifers (Barcelona).- First Spanish - Por-
tuguese - American Congress on Economic Geology.- Madrid-
Lisbon. sep 1971. Section 3. pp 171/198.
Custodio, E., Bayo, A. and Pelaez, M.D. (1971) - Geochemistry
and water entries for study of the movement of ground water
in the Llobregat river delta (Barcelona) - First Spanish-
Portuguese-American Congress on Economic Geology.- Madrid-
Lisbon. sep. 1971. Section 6 pp 51/80.
Custodio, E. and López-Garcia, L. (1972) - Construction and
utilization process of a model. Chapter 5 Basic Theory on
Analogical and Digital Models of Aquifers. Informations and
Sutides, Bulletin, 37, Geologicql Service of Public Works.
Madrid.
Custodio, E. (1973) - Basic data for building an aquifer
simultation model. Chapter 16.1. on Subterranean Hydrology
Omega Editorial. Barcelona (at press).

591
8. Custodio, E. and others (1973) - Compiling of works made
during the period 1966/1972 in the Bajo Llobregat. Water
Board of the East Pyrenees and Public Works Geological
Service. Barcelona (in preparation).
9. Llamas, M.R. and Molist, J. (1967) - Hydrology of the Besos
and Llobregat River deltas.- Hydrological Investigation
Documents NQ 2 - Water Research, Applications and Studies
Centre. Barcelona. Speech in Barcelona (1966).
10. Llamas, M.R. and VilarÓ, F. (1967) - Die Rolle der Grund-
wasserspeicher bei der Wasservorsorgung von Barcelona.-
Das Gas-und-Wasserfach, Wasser Abwasser, Vol. 34. No. 15,
August 1967. pp. 945/953.
11. Martin-Arnaiz, M. (1972) - Report on the explotation
possibilities of the Llobregat river delta aquifers. General
Board of Hydraulic Works. East Pyrenees Water Board. Barce-
lona (prior report).
12. Mc Neal, R.M. (1958) - An asymetrical finite difference
network - Quarterly of Applied Mathematics. Vol. XI. No. 3
1958.
13. Montalbán, F. (1969) - Factorial analysis of the oscillations
of the deep aquifer of the Llobregat river. Hydrological
Investigation Documents No. 6. Water Research, Applications,
and Studies Centre. Barcelona, Pamplona speech (1967).
14. Ministry of Public Works (1965).- Study of the Total Hydraulic
resourcs of the East Pyrenees - Second Report East Pyrenees
Water Board and Public Works Geological Service. Barcelona.
15. Ministry of Public Works - Report on the construction and
application of a mathematical simulation model of the Llo-
bregat delta aquifers.- Study of the Total Hydraulic Resources
of the East Pyrenees. Central Area. Report CE-111.- East
Pyrenees Water Board and Public Works Geological Service.
Barcelona.
16. Santa Maria, L. and Marin A. (1909) - Hydrological studies
on the Llobregat river basin.- Bulletin of the Commission
of the Geological Map of Spain LX 2nd Series.
17. Tyson, H.N. and Weber, E.M. (1964).- Ground water management
for the nations future computer simulation of ground-water
basins - proceedings of the ASCE, Journal of the Hydraulics
Division. New York. Jyly 1964.
18. VilarÓ, F. (1967) - Balance of the present use of the Bajo
Llobregat. Hydrological Investigation Papers No. 2. Water
Investigations, Applications and Studies Centre. Barcelona
Speech in Barcelona, (1966) - pp 155/169.

592
19. VilarÓ, F. and Martin Arnbiz, M. (1968) - Hydric Balance
of the Bajo Llobregat - Hydric Balance Seminar - F.A.O. -
Geology and Mining Institute of Spain. Madrid.
20. VilarÓ, F. Custodio, E., and Bruington, A.E. (1970) - Sea
Water intrusion and water pollution in the Pirineo Oriental
(Spain) - ASCE National Water Resources Engineering Meeting,
Memphis, Tennence. - Meeting Preprint 1122.

Fi g. 1 .- Plano general de situaci& y de extracciones.
General location and pumping map.

O
O 00 O O
Yi
sariaw - soiiaui S~JI~UI- soiiaw
s '"" r
O'' . I ' U
594

-2-
..2-.-
595
-
Escal o-Scale
O 1 2 3 L SKm.
L ogunas pantanosas natural es
Limite de los zonas permeables
Limite del ocuifero
profundo
Isopieza del acuifero profundo I ml.
Isapieza del acuifero suprficial [m 1.
e Sondeo piezomctrico
-.-.-
Natural marshy lagoons
Boundary of the permeable oreas
Boundary of the deep aquifer
A- Isopiestic line Of the drepoquifer(ml
,-2--- Isopiestic line of the upperoquiferIm)
Observation bore- hole
Fi g. 3 - Superficies PiezornCtricas en Abril de 1.967 (s& Custodio) y
situaci& de los sandeos.
Piezometric surfaces in April 1.967 (after Custodio) and 10Cb
tion of the boreholes.

I
596

in
c
I in .e
I - a
I u
I o 4
I C
/* :
Y --
597

598
Tranrmirividad del aeuifero del
valle y profundo del delta en m2/dia
Transmissivity of valley and delta
deep aquifers in sqml day
-- id
- - - - Dato inicial Preliminary figure
1000 Valor ajustadocon Value mstchcd with the
el modelo model
Fig. 6.- Valores de la tranrrtirividaà del acuftero del valle y prohrado &l dal-
ta del Llobregat.
Values of the valley aad delta upper aquifers OP the Llobregat delta.

and in Etudes et rapports d ‘hydrataptie 16
gn of
r- resou rces
inadeq uate
projects
data
Proceedings of the Madrid Syinposiurn
June 1973
Elaboration des projets
d’utilisation des ressources en eau
sans données suffisantes
Volume 2
Unesco - WMO - IAHS
Unesco - OMM - AISH
Actes du colloque de Madrid
Juin 1973

Studies and reports in hydrology/Etudes et rapports d’hydrologie 16

TITLES IN THIS SERIES / DANS CETTE COLLECTION
1.
2.
3.
4.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
1s.
16.
The use of analog and digital computers in hydrology: Proceedings of the Tucson Symposium.
June 1966 / L'utilisation des calculatrices analogiques et des ordinateurs en hydrologie: Actes du
colloque de Tucson, juin 1966. Vol. 1 & 2. Co-edition IAHS-Unesco / Coédition AISU-Unesco.
Water in the unsaturated zone: Proceedings of the Wageningen Symposium, June I967 / L'eau dans
la zone non saturée: Actes du symposium de Wageningen, juin 1967. Edited by / Edité par P. E.
Rijtema & H. Wassink. Vol. 1 & 2. Co-edition IAHS-Unesco / Coédition AISH-Unesco.
Floods and their computation: Proceedings of the Leningrad Symposium, August 1967 / Les crues
et leur évaluation: Actes du colloque de Leningrad, août 1967. Vol. 1 & 2. Co-edition IARS-Unesco-
WMO / Coédition AISH-Unesco-OMM.
Representative and experimental basins: An international guide for research and practice. Edited
by C. Toebes and Y. Ouryvaev. Published by Unesco.
Les bassins représentatifs et expérimentaux: Guide international des pratiques en matière de re-
cherche. Publié sous la direction de C. Toebes et V. Ou-vaey. Publié par l'Unesco.
'Discharge of selected rivers of the world / Débit de certain cours d'eau du monde. Published by
Unesco / Publié par l'Unesco.
Vol. I : General and régime characteristics of stations selected 1 Caractéristiques générales et
caractéristiques du régime des stations choisies.
Vol. II: Monthly and annual discharges recorded at various selected stations (from start of obser.
vations up to 1964) / Débits mensuels et annuels enregistrés en diverses stations sélectionnées
(de l'origine des observations à l'année 1964).
'Vol. III: Mean monthly and extreme dlscharges (1%5-1969) / Débits mensuels moyens et débits
extrêmes (19651969).
List of International Hydrological Decade Stations of the world / Liste des stations de la Décennie
hydrologique internationale existant dans le mmde. Published by Unesco 1 Publié par l'Unesco.
Ground-water studies: An international guide for practice. Edited by R. Brown, I. Ineson. V. KO-
noplyantsev and V. Kovalevski. (Will also appear in French, Russian gnd Spanish / Paraitrg
également en espagnol, en français et en russe.)
Land subsidence: Proceedings of the Tokyo Symposium, September 1969 / Affaisement du sol:
Actes du colloque de Tokyo, septembre 1969. 'Vol. 1 & 2. Co-edition IAHS-Unesco / Coédition
AISH-Unesco.
Hydrology of deltas: Proceedings of the Bucharest Symposium, May 1969 / Hydrolaße des deltas:
Actes du colloque de Bucarest, mai 1969. Vol. 1 & 2. Co-edition IAHS-Unesco / Coédirion AISH-
Unesco.
Status and trends of research in hydrology / Bilan et tendances de la recherche en hydrologic.
Published by Unesco 1 Publié par l'Unesco.
World water balance: Proceedings of the Reading Symposium, July 1970 / Bilan hydrique mondial:
Actes du colloque de Reading, juillet 1970. Vol. 1-3. Co-edition ZAHS-Unesco-WhfO 1 Coédirion
AISH-Unesco-OMM.
Results of research on representative and experimental basins: Proceedings of the Wel1inp;ton
Symposium, December 1970 / Résultats de recherches sur les bassins représentatifs et ex érimen-
taux: Actes du cowoque de Wellington, décembre 1970. 'Vol. 1 & 2. Coedition IAHS-Jnesco /
Coédition AISH-Unesco.
Hydrometry: Proceedings of the Koblenz Symposium, September 1970 / Hydrométrie: Actes du
colloque de Coblence, septembre 1970. Co-edition ZAHS-Unesco-WMO / Coédition AISH-Unesco-
OMM.
Hydrologic information systems. Co-edition Unesco-WMO.
Mathematical models in hydrology: Proceedings of the Warsaw Symposium, July 1971 / Les mc-
deles mathématiques en hydrologie Actes du colloque de Varsovie, juillet 1971. Vol. 1-3. Co-
edition IAHS-Unesco-WMO / Coédit on AISH-Unesco-OMM.
Design of water resources projects with inadequate data: Proceedings of the Madrid sym ,
June 1973 / Elaboration des projets d'utilisation des ressources en eau sans données sufp:z:
Actes du colloque de Madrid, juin 1973. Vol. 1-3. Co-edition Unesco-WMO-IAHS / Coéditiori Unesco-
OMM-AISH.

Design of
water resources projects
with inadequate data
Proceedings of the Madrid Symposium
June 1973
Elaboration des projets
d’utilisation des ressources en eau
sans données suffisantes
A contribution to the Lnternationat Hydrological Decade
Une contribution a la Ecennie hydrologique internationale
Con reshmenes en csuañol
Volume 2
Actes du colloque de Madrid
Juin 1973
Unesco - WMO - IAHS 1974
Uiiesco - OMM - AISH

Published jointly by
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ISBN 92-3-001137-1
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PREFACE
The International Hydrological Decade (IHD) 1965-74 was launched by
the General Conference of Unesco at its thirteenth session to promote
international co-operation in research and studies and the training of spe-
cialists and technicians in scientific hydrology. Its purpose is to enable
all countries to make a fuller assessment of their water resources and a
more rational use of them as man’s demands for water constantly increase
in face of developments in population, industry and agriculture. In 1974
National Committees for the Decade had been formed in 108 of Unesco’s
131 Member States to carry out national activities within the programme
of the Decade. The implementation of the programme is supervised by a
Co-ordinating Council, composed of 30 Member States selected by thc Ge-
neral Conference of Unesco, which studies proposals for developments
of the programme, recommends projects of interest to all or a large
number of countries, assists in the development of national and regional
projects and co-ordinates international co-operation.
Promotion of collaboration in developing hydrological research techni-
ques, diffusing hydrological data and planning hydrological installations
is a major feature of the programme of the IHD which encompasses all
aspects of hydrological studies and research. Hydrological investigations
are encouraged at the national, regional and international level to streng-
then and to improve the u6e of natural resources from a local and a global
perspective. The programme provides a means for countries well advanced
in hydrological research to exchange scientific views and for developing
countries to benefit from this exchange of information in elaborating re-
search projects and in implementing recent developments in the planning
of hydrological installations.
As part of Unesco’s contribution to the achievement of the objectives
of the IHD the General Conference authorized the Director-General to
collect, exchange and disseminate information concerning research on
scientific hydrology and to facilitate contacts between research workers
in this field. To this end Unesco initiated two series of publications: Studies
and Reports in Hydrology and Technical Papers in Hydrology.
The Studies and Reports in Hydrology series, in which the present
volume is published, is aimed at recording data collected and the main
results of hydrological studies undertaken within the framework of the
Decade, as well as providing information on research techniques. Also
included in the series are proceedings of symposia. Thus, the series com-
prises the compilation of data, discussions of hydrological research techni-
ques and findings, and guidance material for future scientific investigations.
It is hopped that the volumes wil furnish material of both practical and
theoretical interest to hydrologists and governments participating in the
IHD and respond to the needs of technicians and scientists concerned
with problems of water in all countries.
A number of these volumes have been published jointly with the In-
ternational Association of Hydrological Sciences and the World Meteoro-
logical Organization which have co-operated with Unesco in the imple-
mentation of several important projects of the IHD.

PRÉFACE
La Conférence générale de l’Unesco, à sa treizième session, a décidé
de lancer, pour la période s’étendant de 1965 à 1974, la Décennie hydrologique
internationale (DHI), entreprise mondiale visant a faire progresser la con-
naissance en matière d’hydrologie scientifique par un développement de
la coopération internationale et par la formation de spécialistes et de
techniciens. Au moment où l’expansion démographique et le développement
industriel et agricole provoquent un accroissement constant des besoins
en eau, la DHI permet à tous les pays de mieux évaluer leurs ressources
hydrauliques et de les exploiter plus rationnellement.
I1 existe actuellement dans i08 des 131 Etats membres de l’Unesco un
comité national qui, pour tout ce qui a tratit au programme de la Décen-
nie, impulse les activités nationales et assure la participation de son pays
aux entreprises régionales et internationales. L’exécution du programme
de la DHI se fait sous la direction d’un Conseil de coordination composé
de 30 Etats membres désignés par la Conférence générale de l’Unesco; ce
conseil étudie les propositions concernant le programme, recommande
l’adoption de projets intéressant l’ensemble des pays ou un grand nombre
d’entre eux, aide à la mise sur pied de projets nationaux et régionaux, et
coordonne la coopération à l’échelon international.
Le programme de la DHI qui porte sur tous les aspects des études et
des recherches hydrologiques, vise essentiellement à développer la col-
laboration dans la mise au point des techniques de recherches, dans la
diffusion des données hydrologiques, dans l’organisation des installations
hydrologiques. I1 encourage les enquêtes nationales, régionales et interna-
tionales tendant à accroître et à améliorer l’utilisation des resources na-
turelles, dans une perspective locale et générale. I1 permet aux pays avancés
en matière de recherches hydrologiques d’échanger des informations; aux
pays en voie de développement, il offre la possibilité de profiter de ces
échanges pour élaborer leurs projets de recherches et pour planifier leurs
installations hydrologiques en tirant parti des acquisitions les plus récentes
de l’hydrologie scientifique.
Pour permettre a l’Unesco de contribuer au succès de la DHI, la Con-
férence générale a autorisé le Directeur générale à rassembler, à échanger
et à diffuser des informations sur les recherches d’hydrologie scientifique
et à faciliter les contacts entre les chercheurs dans ce domaine. A cette
fin, l’Unesco fait paraître deux nouvelles collections de publications:

tique que théorique, et qu’elle répondra aux besoins des techniciens et
des hommes de science de tous pays qui s’occupent des problèmes de l’eau.
Certains de ces ouvrages sont publiés en coopération avec l’Association
internationale des sciences hydrologiques ou I’Organisatioii mMorologique
mondiale dans le cadre de projets réalisés conjointement par ces orga-
nisations et l’Unesco.

Design d water resources projecis with inadequate data: P-dings d the Madrid aympoaium,
June 1973 / Elaboration des projeta d'utilisation dei resswTas en eau aona d onka auffluntes:
Actea du wlloque de Madrid. juin 1973
Volume II Contents Table des matidres
Foreword/Avant-propos
TOPIC II.1A . METHODS FOR STUDIES LN DATA-SCARCE AREAS AND
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF
DATA FOR PURPOSES OF PLANIFICATION OF WATER
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).
METHODOLOGY FOR ASSESSING HYDROLOGICAL CHA-
RACTERISTICS IN DATASCARCE AREAS.
POINT II.1A . METHODES D'ETUDES UTILISEES DANS LES REGIONS OU
LES DONNEES SûNT INSUFFISANTES ET INFLUENCE SUR
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR
L'ELABORATION DES PROJETS DE L'UTILISATION DES
RESSOURCES EN EAU (A L'EXCLUSION DES CRUES ET
DES DEBITS DE BASSES EAUX).
METHODOLOGIE POUR L'EVALUATION DES CARACTE-
RISTIQUES HYDROLOGIQUES DANS LES REGIONS OU
LES DONNEES SûNT RARES.
BASSO, EDUARDO. (UNDPMIMO) GENERAL REPORT
ABIODUM, ADIGUN ADE. (NIGERIA)
Water resources projects in Nigeria and the hydrological data employed in
their planning and development ................................
BASSO, E., ARRIAGADA, A., NEIRA, H., PEREZ DELGADO, M. (COSTA
RICA)
An example of regional co-operation for improving the hydrological and
meteorological information ...................................
CUBAS GRANADO, FRANCISCO. (SPAIN)
Existing methodology for estimating free water surface evaporation ....
CUSTODIO, EMILIO. (SPAIN)
Geohydrological studies in small areas without systematic data ........
DALINSKY, JOSEPH S. (ISRAEL)
Methods of analysing deficient discharge data in arid and semi-arid zones
for the design of surface water utilization .......................
D'OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)
Mapai river hydrological study (Limpopo's river) ...................
D'OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)
Application of Coutagne's and Turc's formulas to southern Mozambique
rivers ...................................................
HERAS, R. (SPAIN)
Report hydrological programa of the Center for Hydrographic Studies for
the investigation of hydraulic resources with insufficient data .........
1
21
35
59
77
95
141
121
155

KARAUSHEV, A.V., BOGOLIUBOVA, I.V. (U.S.S.R.)
Computation of reservoin wdLnrntition .........................
KLIGUE, R.K., MECHDI EL SACHOB (U.S.S.R.)
CilnilritionofrunoffinIraq ..................................
KUZMIN, P.P., VERSHININ, A.P. (U.S.S.R.)
Determination of evaporation in caw of the abmnce or inadequacy of
data .....................................................
PENTA, A., ROSSI, F. (ITALY)
Objective criteria to daclare a aerier of data sufficient for technical pur-
poses ....................................................
QUINTELA GOIS, CARLOS. (PORTUGAL)
Objective criteria used in hydrology with inadequate data ............
SMITH, ROBERT L. (U.S.A.)
Utilizing climatic data to appraise potentiai water yields .............
STANESCU, SILVIU. (COLOMBIA)
Determination of hydrological characteristics in points without direct
hydrometricdata ...........................................
TEMEZ, J.R. (SPAIN)
New models of frequency law of runoff starting from precipitations ....
TRENDEL, R., DER MEGREDITCHIAN, G., RULLIERE, MARIE CLAIRE.
(FRANCE)
Traitement opérationnel des données pluviométriques entachées d'erreurs
ouinsuffisantes ............................................
TOPIC II.1B . METHODS FOR STUDIES IN DATA SCARCE AREAS AND
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF
DATA FOR PURPOSES OF PLANIFICATION OF WATER
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).
INFLUENCE OF INADEQUACY OF HYDROLOGICAL DATA
ON PROJECT DESIGN AND FORMULATION.
POINT II.1B - METHODES D'ETUDES UTILISEES DANS LES REGIONS OU
LES DONNEES SONT INSUFFISANTES ET INFLUENCE SUR
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR
L'ELABORATION DES PROJETS DE L'UTILISATION DES
RESSOURCES EN EAU (A L'EXCLUSION DES CRUES ET
DES DEBITS DE BASSES EAUX). INFLUENCE DU MANQUE
DE DONNEES HYDROLOGIQUES SUR LE CALCUL DU
PROJET ET SA FORMULATION.
BEARD, L.R. (U.S.A.) GENERAL REPORT
BANERJI, S., LAL, V.B. (INDIA)
Design of water resources projects with inadequate data in India. General
& Particular Case Studies ................................... 323
199
207
217
221
24 1
253
265
287
30 1
315

JAMB, IVAN C. (U.S.A.)
Data requirements for the optimization of reservoir dengn and operating
dedetermination ..........................................
REID, GEORGE W. (U.S.A.)
The design of water quality management projecta with inadequate data
SABHERWAL, R.K. (INDIA)
Designing projects for the development of ground water resources in the
alluvial plains of northern India on the basis of inadequate data .......
SEXTON, J.R., JAMIESON, D.G. (U.K.)
Improved techniques for water resource systems design .........
WEBER, J., KISIEL, CHESTER C., DUCKSTEIN, LUCIEN (U.S.A.)
Maximum information obtainable from inadequate design data: from
multivariate to Bayesian methods ..............................
TOPIC 11.2 - CURRENT PRACTICES FOR ASSESSING DESIGN FLOODS
AND DESIGN LOW FLOWS, INCLUDING THE USE OF
SYNTHETIC UNIT HYDROGRAPH, WITH PARTICULAR
EMPHASIS ON MAXIMALISATION AND MINIMALISATION.
POINT 11.2 - PRATIQUES COURANTES POUR L'EVALUATION DES
CRUES ET DES DEBITS D'ETIAGES PRIS EN COMPTE DANS
LE PROJET, COMPRENANT L'EMPLOI D'HYDROGRAMMES
UNITAIRES DE SYNTHESE, AVEC ETUDE PARTICULIERE
DE LA MAXIMALISATION ET DE LA MINIMALISATION.
ROCHE, MARCEL. (FRANCE) GENERAL REPORT
BATLLE GIRONA, MODESTO. (SPAIN)
Estimation of floods by means of their silt loads .................
BERAN, M.A. (U.K.)
Estimation of design floods and the problem of equating the probability
ofrainfailandrunoff ........................................
DAVIS, DONALD R., DUCKSTEIN, L., KISIEL, CHESTER C., FOGEL, MAR-
TIN M. (U.S.A.)
A decision-theoretic approach to uncertainty in the return period of
maximum flow volumes using rainfall data .......................
HALL, M.J. (U.K.)
Synthetic unit hydrograph technique for the design of flood alleviation
works in urban areas ......................................
HELLIWELL, P.R., CHEN, T.Y. (U.K.)
A dimensionless unitgaph for Hong Kong ........................
HERAS, R., LARA, A. (SPAIN)
Study of maximum floods in small basins of torrential type ..........
335
349
365
383
40 1
419
439
459
473
485
501
517

HERBST, P.H., VAN BIWON, S., OLIVIER, J.P.J., HALL, J.M. (SOUTH
AFRICA)
Flood estimation by determination of regional parameten from limited
data ....................................................
JARASWATHANA, DAMRONG., PINKAYAN, SUBIN. (THAILAND)
Practices of design flood frequency for small watersheds in Thailand ...
KINOSITA, TAKEO., HASHIMOTO, TAKESHI. (JAPAN)
Design discharge derived from design rainfall ..................
LEESE, MORVEN N. (U.K.)
The use of censored data in estimating t-year floods .........
POGGI PEREIRA, PAULO. (BRAZIL)
Assessment of design floods in Brazil ........................
RENDON-HERRERO, OSWALD. (U.S.A.)
A method for the prediction of washload in certain small watersheds ....
RODIER, J.A. (FRANCE)
Méthodes utilisées pour l'évaluation des débita de m e des petits com
d'eau en régions tropicales ....................................
SOKOLOV, A.A. (U.S.S.R.)
Methods for the estimation of maximum dischargea of snow melt and
rainfall water with inadequate observational data ..................
VLADIMIROV,A.M.,CHEBOTAREV, A.I. (U.S.S.R.)
Computation of probabiustic valuea of low flow for ungauged riven .
WON, TAE SANG. (U.S.A.)
A study on maximum flood discharge formulas ....................
TOPIC III - RELATION BETWEEN PROJECT ECONOMICS AND HYDROLO-
GICAL DATA
POINT 111 - RELATION ENTRE LES DONNEES ECONOMIQUES DU PRO-
JET ET LES DONNEES HYDROLOGIQUES
BURAS, NATHAN. (ISRAEL)
The cost-effectiveness of water resources systems considering inadequate
hydrologiddata ...........................................
FILOTTI, A., FRANK, G., PARVULESCU, C. (ROMANIA)
Optimization of water resources development projects in case of inade-
quate hydrologic data ....................................
POBEDIMSKY, A. (ECE)
Relation between project economics and hydrologicai data ...........
54 1
553
551
563
517
5 89
603
615
625
635
649
66 1
683

INTRODUCTION
The Symposium on the Development of Water Resources Projects with
Inadequate Data was held in Madrid from 4 to 8 June 1973 for the purpose
of focusing on the methodology for hydrologic studies for water resources
projects with inadequate data and on current practices for the assessment
of design parameters.
The Symposium was opened at the Palacio de Exposiciones on the
morning of 4 June by Miniester of Public Workes of Spain Addresses were
then given by Dr. Dumitrescu on behalf of the Director General of Unesco,
Professor Nevmec on behalf of the Secretary-General of WMO, Dr. Rodier
as President of IAHS and by Dr. Briones, on behalf of the Spanish Na-
tional Committee for the IHD.
The Symposium was attended by 480 participants from 77 countries.
The technical programme, detalled in the Table of Contents, included
consideration of 3 major areas:
1. Methodology for hydrological studies with inadequate data,
2. Current practices in different countries,
3. Relation between project economics and hydrological data.
Each area was further sub-divided into topics for each of which the
individually contributed papers were abstracted into a general report, orally
presented by an invited expert, and followed by discussion.
Since the individual papers were not presented at the Symposium orally
by the authors, thery are reproduced here in the orden in which
they were reported in each general report under each topic.

Contents
Table des matières
Volume II
Foreword/Avant-propos ................................
TOPIC II.1A - METHODS FOR STUDIES IN DATA-SCARCE AREAS AND
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF
DATA FOR PURPOSES OF PLANIFICATION OF WATER
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).
METHODOLOGY FOR ASSESSING HYDROLOGICAL CHA-
RACTERISTICS IN DATA-SCARCE AREAS.
POINT II.1A - METHODES D’ETUDES UTILISEES DANS LES REGIONS OU
LES DONNEES SONT INSUFFISANTES ET INFLUENCE SUR
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR
L’ELABORATION DES PROJETS DE L’UTILISATION DES
RESSOURCES EN EAU (A L’EXCLUSION DES CRUES ET
DES DEBITS DE BASSES EAUX).
METHODOLOGIE POUR L’EVALUATION DES CARACTE-
RISTIQUES HYDROLOGIQUES DANS LES REGIONS OU
LES DONNEES SONT RARES.
BASSO, EDUARDO. (UNDP/WMO) GENERAL REPORT
ABIODUM, ADIGUN ADE. (NIGERIA)
Water resources projects in Nigeria and the hydrological data employed in
their planning and development ................................
BASSO, E., ARRIAGADA, A., NEIRA, H., PEREZ DELGADO, M. (COSTA
RICA)
An example of regional co-operation for improving the hydrological and
meteorological information ...................................
CUBAS GRANADO, FRANCISCO. (SPAIN)
Existing methodology for estimating free water surface evaporation ....
CUSTODIO, EMILIO. (SPAIN)
Geohydrological studies in small areas without systematic data ........
DALINSKY, JOSEPH S. (ISRAEL)
Methods of analysing deficient discharge data in arid and semi-arid zones
for the design of surface water utilization .......................

II
D’OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)
Mapai river hydrological study (Limpopo’s river) ...................
D’OLIVEIRA, EMILIO EUGENIO. MIMOSO, JOAO JOSE. (PORTUGAL)
Application of Coutagne’s and Turc’s formulas to southern Mozambique
rivers ....................................................
HERAS, R. (SPAIN)
Report hydrological programs of the Center for Hydrographic Studies for
the investigation of hydraulic resources with insufficient data .........
KARAUSHEV,A.V., BOGOLIUBOVA, I.V. (U.S.S.R.)
Computation of reservoirs sedimentation .......................
KLIGUE, R.K., MECHDI EL SACHOB (U.S.S.R.)
Calculation of runoff in Iraq ..................................
KUZMIN, P.P., VERSHININ, A.P. (U.S.S.R.)
Determination of evaporation in case of the absence or inadequacy of
data .....................................................
PENTA, A., ROSSI, F. (ITALY)
Objective criteria to declare a series of data sufficient for technical pur-
poses ....................................................
QUINTELA GOIS, CARLOS. (PORTUGAL)
Objective criteria used in hydrology with inadequate data ............
SMITH, ROBERT L. (U.S.A.)
Utilizing climatic data to appraise potential water yields .............
STANESCU, SILVIU. (COLOMBIA)
Determination of hydrological characteristics in points without direct
hydrometric data ...........................................
TEMEZ, J.R. (SPAIN)
New models of frequency law of runoff starting from precipitations ....
TRENDEL, R., DER MEGREDITCHIAN, G., RULLIERE, MARIE CLAIRE.
(FRANCE)
Traitement opérationnel des données pluviornetriques entachées d’erreurs
ou insuffisantes ............................................

TOPIC II.1B - METHODS FOR STUDIES IN DATA SCARCE AREAS AND
THE INFLUENCE ON DESIGN OF THE INADEQUACY OF
DATA FOR PURPOSES OF PLANIFICATION OF WATER
RESOURCES (EXCLUDING FLOOD AND LOW FLOWS).
INFLUENCE OF INADEQUACY OF HYDROLOGICAL DATA
ON PROJECT DESIGN AND FORMULATION.
POINT II.1B - METHODES D’ETUDES UTILISEES DANS LES REGIONS OU
LES DONNEES SONT INSUFFISANTES ET INFLUENCE SUR
LE CALCUL DU PROJET DU MANQUE DE DONNEES POUR
L’ELABORATION DES PROJETS DE L’UTILISATION DES
RESSOURCES EN EAU (A L’EXCLUSION DES CRUES ET
DES DEBITS DE BASSES EAUX). INFLUENCE DU MANQUE
DE DONNEES HYDROLOGIQUES SUR LE CALCUL DU
PROJET ET SA FORMULATION.
BEARD, L.R. (U.S.A.) GENERAL REPORT
BANERJI, S., LAL, V.B. (INDIA)
Design of water resources projects with inadequate data in India. General
& Particular Case Studies ...................................
JAMES, IVAN C. (U.S.A.)
Data requirements for the optimization of reservoir design and operating
rule determination ..........................................
REID, GEORGE W. (U.S.A.)
The design of water quality management projects with inadequate data .
SABHERWAL, R.K. (INDIA)
Designing projects for the development of ground water resources in the
alluvial plains of northern India on the basis of inadequate data .......
SEXTON, J.R., JAMIESON, D.G. (U.K.)
Improved techniques for water resource systems design ..............
WEBER, J., KISIEL, CHESTER C., DUCKSTEIN, LUCIEN (U.S.A.)
Maximum information obtainable from inadequate design data: from
multivariate to Bayesian methods ..............................
TOPIC 11.2 - CURRENT PRACTICES FOR ASSESSING DESIGN FLOODS
AND DESIGN LOW FLOWS, INCLUDING THE USE OF
SYNTHETIC UNIT HYDROGRAPH, WITH PARTICULAR
EMPHASIS ON MAXIMALISATION AND MINIMALISATION.

IV
POINT 11.2 - PRATIQUES COUFUNI"'I'S POUR L'EVALUATION DES
CRUES ET DES DEBITS D'ETIAGES PRIS EN COMPTE DANS
LE PROJET, COMPRENANT L'EMPLOI D'HYDROGRAMMES
UNITAIRES DE SYNTHESE, AVEC ETUDE PARTICULIERE
DE LA MAXIMALISATION ET DE LA MINIMALISATION.
ROCHE, MARCEL. (FRANCE) GENERAL REPORT
BATLLE GIRONA, MODESTO. (SPAIN)
Estimation of floods by means of their silt loads ..............
BERAN, M.A. (U.K.)
Estimation of design floods and the problem of equating the probability
of rainfall and runoff ........................................
DAVIS, DONALD R., DUCKSTEIN, L., KISIEL, CHESTER C., FOGEL, MAR-
TIN M. (U.S.A.)
A decision-theoretic approach to uncertainty in the return period of
maximum flow volumes using rainfall data .......................
HALL, M.J. (U.K.)
Synthetic unit hydrograph technique for the design of flood alleviation
works in urban areas ........................................
HELLIWELL, P.R.,CHEN, T.Y. (U.K.)
A dimensionless unitgraph for Hong Kong ........................
HERAS, R., LARA, A. (SPAIN)
Study of maximum floods in small basins of torrential type ..........
HERBST, P.H., VAN BILJON, S., OLIVIER, J.P.J., HALL, J.M. (SOUTH
AFRICA)
Flood estimation by determination of regional parameters from limited
data .....................................................
JARASWATHANA, DAMRONG., PINKAYAN, SUBIN. (THAILAND)
Practices of design flood frequency for small watersheds in Thailand ...
KINOSITA, TAKEO., HASHIMOTO, TAKESHI. (JAPAN)
Design discharge derived from design rainfall ......................
LEESE, MORVEN N. (U.K.)
The use of censored data in estimating t-year floods ................

POGGI PEREIRA, PAULO. (BRAZIL)
Assessment of design floods in Brazil .............................
RENDON-HERRERO, OSWALD. (U.S.A.)
A method for the prediction of washload in certain small watersheds ...
RODIER, J.A. (FRANCE)
Méthodes utilisées pour l’évaluation des débits de crue des petits cours
d’eau eri régions tropicales ....................................
SOKOLOV, A.A. (U.S.S.R.)
Methods for the estimation of maximum discharges of snow melt and
rainfall water with inadequate observational data ..................
VLADIMIROV, A.M., CHEBOTAREV, A.I. (U.S.S.R.)
Computation of probabilistic values of low flow for ungauged rivers ....
WON, TAE SANG. (U.S.A.)
A study on maximum flood discharge formulas ....................
TOPIC III - RELATION BETWEEN PROJECT ECONOMICS AND HYDROLO-
GICAL DATA
POINT III - RELATION ENTRE LES DONNEES ECONOMIQUES DU PRO-
JET ET LES DONNEES HYDROLOGIQUES
BURAS, NATHAN. (ISRAEL)
The cost-effectiveness of water resources systems considering inadequate
hydrological data ...........................................
FILOTTI, A., FRANK, G., PARVULESCU, C. (ROMANIA)
Optimization of water resources development projects in case of inade-
quate hydrologic data ....................................
POBEDIMSKY, A. (ECE)
Relation between project economics and hydrological data ...........

Foreword
While the need for hydrological and meteorological data of many types
for the design of water resources projects is obvious, it is often found,
especially in many developing countries, that such data are either lacking
or inadequate.
Recognizing the existence of this problem, the Co-ordinating Counci.1 of
the IHD appointed a group of experts (third session, Paris, June 1967) to
study the problem of design of water resources projects with inadequate
data.
Similarly, the Commission for Hydrology of WMO (third session, Geneva,
September 1968) established a Working Group on Hydrological Design
Data for Water Resources Projects to prepare guidance material on this
subject for the WMO Guide to Hydrological Practices and to maintain
liaison with the IHD group of experts appointed by the Co-ordinating
Council.
As a means of taking stock of the work carried out by the hydrological
community in coping with project design with scarce data, Unesco and
WMO jointly convened a symposium on this subject. The Symposium was
organized with the co-operation of the IAHS and the Spanish National
Committee for the IHD and was held in Madrid from 4 to 8 June 1973 at
the invitation of the Government of Spain.
The Madrid Symposium concentrated on the methodology of hydro-
logical studies for water resources projects with inadequate data and on
current practices for the assessment of design parameters.
The Minister of Public Works of Spain opened the Symposium at the
Palacio de Exposiciones on the morning of 4 June. Addresses were given
by Dr. Dumitrescu on behalf of the Director-General of Unesco, Professor
Nemec on benalf of the Secretary-General of WMO, Dr. Rodier as President
of IAHS and by Dr. Briones, on behalf of the Spanish National Committee
for the IHD.
The Symposium was atteneded by 480 participants from 77 countries.
The technical programme, detailed in the Table of Contents, included
consideration of 3 major areas:
1. Methodology for hydrological studies with inadequate data;
2. Current practices in different countries;
3. Relation between project economics and hydrological data.
Each area was further sub-divided into topics for each of which the
indivi,dually contributed papers were abstracted into a general report,
orally presented by an invited expert, and followed by discussion.

This volume of proceedings was compiled by the Spanish National Com-
mittee for the IHD; it includes all the general reports and individual
papers presented at the Symposium, as well as the discussions. It is issued
as a joint Unesco/WMO/IAHS pub,lication in the spirit in which the three
Organizations have collaborated during the IHD.
Since the individual authors did not present their papers orally at the
Symposium, the papers are reproduced here in the order in which they
are discussed in the general report for each topic.
Unesco, WMO and IAHS wish to record their thanks to the Spanish
National Committee for the IHD for the many contributions of its members
towards the organization of the Symposium, and for the Committee’s as-
sistance in the publication of these proceedings.

AVANT-PROPOS
I1 est évident que, pour élaborer des projets d’utilisation des ressources
en eau il est nécessaire de disposer de données hydrologiques et météoro-
logiques de types très divers; or il apparaît que ces données sont souvent
inexistantes ou insuffisantes, notamment dans beaucoup de pays en voie
de développement.
Conscient de ce problème, le Conseil de coordination de la DHI a créé,
lors de sa troisième session (Paris, juin 1967) un groupe d’experts chargé
d’étudier les moyens d’elaborer des projets d’utilisation des ressources
en eau sans disposer de données suffisantes.
De son côté, la Commission d’hydrologie de l’OMM a constitué à sa
troisième session (Genève, septembre 1968) un groupe de travail sur les
données hydrologiques nécessaires à l’élaboration des projets d’arnénagement
des ressources hydrauliques; ce groupe de travail a été chargé de
formuler des recommandations destinées à figurer dans le Guide OMM des
pratiques hydrologiques, et d’assurer la liaison avec le groupe d’experts
de la DHI créé par le Conseil de coordination.
Afin de faire le point des travaux accomplis par la communité hydrologique
en ce qui concerne l’élaboration de projets pour lesquels on ne
dispose pas de données suffisantes, l’Unesco et l’OMM ont décidé de réunir
conjointement un colloque consacré à cette question. Ce colloque, organisé
avec la collaboration de I’AISH et du Comité national espagnol pour la
DHI, s’est tenu à Madrid en juin 1973, à l’invitation du gouvernement espagnol.
Le colloque de Madrid a traité en particulier de la méthodologie des
études hydrologiques sans données suffisantes et des pratiques courantes
utilisées pour l’évaluation des paramètres de calcul.
Le colloque a été ouvert par le ministre espagnol des travaux publics,
le matin du 4 juin, dans le cadre du Palais des expositions. Des allocutions
furent prononcées par M. Dumitriscu, au nom du Directeur général de
l’Unesco, par le professeur Nemec, au nom du Secrétaire général de l’OMM,
par M. Rodier, président de l’AISH, et par M. Briones, au nom du Comité
national espagnol pour la DHI.
480 participants, venant de 77 pays, participèrent au colloque.
Le programme technique, dont le contenu détaillé figure dans la table
des matières, portait sur trois domaines principaux:
1. Méthodologie des études hydrologiques sans données suffisantes;
2. Les pratiques courantes utilisées dans différents pays;
3. Relation entre les données économiques du projet et les données
hydrologiques.

Chacun de ces domaines était subdivisé en thèmes, et sur chaque thème
un rapport général synthétisant les communications individuellles était pré-
senté par un expert, puis suivi d’une discussion.
Les Actes du colloque, établis par le Comité national espagnol pour
la DHI, comprennent l’ensemble des communications individuelles et des
rapports généraux, ainsi que le compte rendu des débats auxquels ils ont
donné lieu. Ils constituent une publication conjointe de l’Unesco, de l’OMM
et de I’AISH, reflétant l’esprit dans lequel les trois organisations ont col-
laboré pendant la DHI.
Comme les communications individuelles n’ont pas été présentées ora-
lement par leurs auteurs, elles sont reproduites dans l’ordre où elles sont
apparues dans le rapport les concernant.
Unesco, l’OMM et I’AISH tiennent à remercier le Comité national es-
pagnol pour la DHI du concours qu’il a apporté à l’organisation du colloque
et à la publication de ses Actes.

METHODOLOGY FOR ASSESSING HYDROLOGICAL CHARACTERISTICS
IN DATA SCARCE AREAS
General Report
bY
Eduardo Basso*
INTRODUCTION
Three of the eight papers reviewed in this Report describe in general the form
of assessing hydrological characteristics in data-scarce areas (Nigeria, Ango-
la and the Central American Isthmus). The other five papers deal with the ap-
Plication of certain particular methods, covering estimation of runoff, evapo-
ration, sedimentation and other hydrological parameters. Therefore, the re-
vision will be made in this order.
The procedure to be followed in this summarizing report consists of present-
ing summaries of the papers followed by a discussion of the main subjects and
by some general comments on the whole subject.
REVIEW OF THE PAPERS
Okavango Basin in Angola. - The paper by Mr. Quintelag** presents the
studies made in Angola and in particular those of the Okavango Basin, which
is one of the big international rivers of the South of Angola. The basin is shown
in Figure 1 of the paper. Its drainage area is about 150 O00 Km2 and most of
the rainfall occurs from October to April. Altitudes range from 1000 to 1800
meters. For the rainfall studies, 28 stations could provide 20 years of records
after completing some shortages by correlation. With these annual values, an
isohyetical map was drawn taking into account altitudes and some climatical
factors. From there, the mean annual rainfall was computed and analysed by
applying the Foster-Hazen method. The result is shown in Figure 2 of the
paper, from which a mean annual precipitation of 950 mm was debermined.
19 flow measuring stations operate in the basin, but only records for 7 years
were available. As the mean rainfall of these seven years is near theaverage
the author concludes that the mean annual flow can be estimated by averaging
the flows of those seven years for every station.
One station operatdby the
South African Services had longer records (25 years) and for it the Foster-
Hazen method was used (Figure 3 of the paper). Finally, Figure 4 shows a
curve indicating the variation of the specific annual flow within the drainage
area.
* Project Manager, Central American Hydrometeorological Project, (UNDP/
WMO). Managua, Nicaragua.
** See list of references at the end of this Report,

2
Niveria's Case. Abiodun's paperg deals first with Nigeria's water policy and
with the institutional arrangements in relation 90 water resources studies.
It later presents some examples of utilization of hydrological data in existing
projects.
The Kainji multipurpose scheme is located on River Niger (Figure 1 of the
paper). Although construction was started in 1964, no water levels were observed
prior to 1959. The precipitation network was also insufficient until in
1953 when new stations were installed allowing a seven year record (1953-59)
from which the rainfall over the catchment area was calculated for this period.
A relatively long record at Jebba, upstream of the dam, could not be used
because of lack of adequate datum information. A correlation between monthly
rainfall and runnoff at Jebba was obtained using the newly observated discharges
at Jebba for seven years and w e -ed for ccmp&ing the discharge from the basin
between Niamey (upstream of the dam site at Kianzi) and Jebba Cbserved and
computed flows for two years are shown in Figure 2 of the paper.
The &ect of lage up to one month were considered in the correlation (equation
1 of the paper). Finally, the discharge at the damsite was obtained substracting
the eetimated runoff on two areas, estimating runoff coefficients of O. 1 and
O. 2 (equation 2 of the paper).
The paper refers them to the problems produced by the lack
cal data in Midwestern Nigeria.
of hydrogeologi-
in Western Nigeria many long and reliable evaporation and rainfall data are
available, but river discharges are very scarce. According to the author, the
standard practice is to base the water scheme design in a conservative form
using a monthly evaporation of 127 mm and computing runoff with the formula
in which Q, is the catchment annual runoff, A the basin drainage area, Rpsn
the basin rainfall value corresponding to correspondjng the probability of -Urtaace.in
5ûy~are Co;'Coefficient of runoff for the basin, estimated at 4%. Abig
dun indicates that variations of this formula are widely used in western Nigeria.
and the uee of a form of it was not used for computing the flood for the
spillway for Asejire Project because of the advice of a foreign consultant. Ins
tead, a runoff of 490 l/sec was used. basedan similar occurrences in other
West African dtreams.
The paper refers also briefly to the Lake Chad basin studies which count with
cooperation from the United States Geological Survey, FAO and UNESCO. In
this case FAO'e efforts have been directed to the harmonization and evaluation
of the data. infra-red aerial photography has been used in connection with
thee e tas ka.
The paper concludes with an appraisal of the studies used and with a brief des-
cription of the future activities in the field of water resources investigations in
Nigeria. Here, the use of new techniques such as rsmote sensing is recom-
mended.
The Central American Hydrometeorological Proiec
The paper by Basso. Arriagada, Neira and Pérez 13. describes the activities
of the Central American Hydrometeorological Project, a co-operative effort
-

etween the countries of the Central American Istbmis and the United Nations
Development Programme acting as executive agency the World Meteorological
Organization. The main objectives of the Project are: (i) Installation of a basic
network of meteorological and hydrological station; (ii) Collection, processing
and publication of the data; (iii) training of personnel by means of courses,
fellowships or through technical publication and manuals; and (iv) The institutional
strengthening of the meteorological and hydrological services of the
area.
At the beginning of the Project (1966) the conditions in the area varied widely
from country to country. In the average the few river discharge measuring
stations had short and sometimes unreliable data, the meteorological network
was poorly distributed and bore no connection with the hydrological network, a
defect that has been also reported in other of the papers under review; ra-
diation, evaporation and rainfall intensity information was completely insuf-
ficient, and --except for one or two countries-- no sediment or water quality
measurements were made at all. A few capable technicians were available,
but extensive training was a pressing need.
The paper describes in some detail the steps taken by the Project, as result
of which the present situation is quite satisfactory for developing conditions.
Of particular interest for the subject of this meeting is the description of some
methods proposed by the project for assessing hydrological characteristics
with insufficient data.
The use of the sediment rating curve, Figure 7 of the paper. has been used for
computing sediment transportation. The remarks on the variation of the coe&
ficient n of the equation C SA Qn with annual precipitation (G:sediment dis-
charge, Q: Discharge; A, n coefficients) are of interest in aMlyZing scarce se-
diment information. Figure 8 of the paper shows the results of some measure-
ments made by the Project, indicating the effect of rainfall and vegetation cover
in the sediment yield. The effect of the destruction of the vegetable cover by a
volcanic eruption should be noticed as a quite particular case.
Flood and rainfall envelopes (Figure 9 and 10 of the paper) have been used as
a first estimate of maximum discharges and precipitation studies. Studies of
regionalized flood frequency analysis are now under way.
Other achievements of the Project include etudies for determining evapotrans -
piration and water requirements for irrigation, studies on runoff forecasting
groundwater studies using a regional analog computer, etc.
The report refers also to the problem of network implementation in areas with
access problems, and the use of prefabricated elements Éhould be noted
(Figures 3 and 4 of the Report). Figures 5 and 6 shows the change in areal
coverage as result of the action of the project. The successful use of modern
mechanical methods for processing meteorological and hydrological information
should encourage other developing countries in the use of these methods.
The report concludes with a remark on the importance of adequate institu-
tional support for these activities, which< imitia1Ly requi res the creation of
concern of the Governments on the importance of meteorology and hydrology.
3

4
Est i mat i ng Water Yiel ds
Smith's9 paper presents an interesting example of estimating water yields
using only precipitation and temperature measurements.
The basic water balance equation applied to a catchment area may by expressed
as:
P R+E+ AS
P precipitation; R total basin outflow, E evapotranspiration and AS
change in storage. For a long period AS becomes negligible, and making
some transformations in equation (1) it is possible to rewrite it as:
Thus, in the long term, the runoff oefficient C is governed by climatic consider-
ations. ln 1967 Guisti and López$ proposed that the mean stream discharge
could be determined as a function of the mean annual precipitation and the basin
climatic index, BCI, defined as:
where P: average monthly precipitation in centimeters and T: average monthly
temperature in degrees centigrade. A relation between C and BCI based in 250
catchments in the United States and Puerto Rico is shown in Figure 1 of the paper.
The use of regional relations between BCI and P as those shown in Figure 2 of
the paper allows to derive C only from precipitation data. The basic C versus
BCI relationship was tested with satisfactory results as those shown in Figure 3
of the paper.
The basic relationships can also be used to appraise the effect of changes of the
precipitation, If subscript 1 represents natural conditions and 2 represented
augmented conditions (in the case of an increase in rainfall) then the gain in runoff
can be written as:
Where, PM L P2/Pi
Jn table 1 the author compares the results of using this method with the results
of ueing hydrologic simulation as reported by several investigators with good
agreement..
Using a reasonable amount of judgment it is possible to determine flow characteristics
other than the mean. Figure 4 shows a comparison of calculated and
observed annual runoff distributions for the Marias de Cygnes River, Kansas,
USA. However, the limitation of this method, as clearly indicated in the text
of the paper, should be considered before using it.
estimation of monthly yields allocating them in proportion to their contribution
to the BCI (a two month running average should be used due to tag problems).
(1)
This also applies to the

The use of the basic relation can also be extended with the help of certain flow
and miscellaneous field measurements.
The paper closes showing the application of the method for appraising the po-
tential yield characteristics of coastal aquifers in southern Puerto Rico and
presenting one example of the adjustments required when the natural conditions
have been changed by man's activities.
Application of Coutagne's and Turc's Formulas
The paper by D'Oliveira and Mip~so6J applies Coutagne's and Turc formulas
for the southern Mozambique rivers.
Coutagne's general rule states:
D-H-KH2
D: Runoff deficit = H - E; H: Mean rainfall height; K! Coutagne's constant
Also C = KH where C: Runoff Coefficient ; &
H
The most probable value of K is obtained by equating to zero the first derivative
of E(C - KH)2, which results in:
Turc's general rule can be expressed as:
P.
H
/.z-g-
Where L: Turc's constant = A t 25T t O. 05T3,
P: Evaporation plus percolation looses ( runoff deficit), H: Precipitation,
A: Constant; T: Mean temperature (In degrees centigrade)
Turc applied his rule for 254 basins, using A 300, finding that in 53% of the
cases the difference between the real and computed D was lees than 40 mm; in
43% of the cases this difference was less than O. 1 of measured D and in 65% the
difference was less than O. 2 measured D.
The application of both formulas to seven basins was divided nto two groups; the
Limpopo River group (Rainfall 450-650 nun; temperatures 18' C-20' C) and the
Incomati, Sabie, Umbeluzi and Usoto Group (Rainfall 800 mm; temperatures
higher than 20').
Detailed results are presented, which can be sumarized as follows:
5

6
Limpopo area
Elephants River
Beit Bridge
Trigo de Morais
All group
C ontanne' s Turc relation
K A = 300
Per cent of D -Dcalc
greater than O. 1 Dcalc
o. O00055
O. 000031
O. 000047
o. 000050
Incomati, Sabie. Umbeluzi and Usoto area
Incomati River O. 0001 50
Sabie River
O. 000131
Umbeluzi River
O. 000145
Usuto River
O. 000162
All group
O. 000140
-the use of Turc's relation with A 300 produces poor results, the authors
present a nomograph (Figure 2 of the paper) to compute the value of A. Using
these new values of the constants the difference between calculated and measured
D is reduced to acceptables levels.
Estimati on of Lvapotranspiration
The paper by Kuzmin and Vershininu deals with the determination of evapora-
tion in case of the absence or inadequacy of data.
Since methods for direct evaporation measurements are still being developed,
computations are the main source of information. These can be divided into
three groups: (i) methods based in the physical analysis of the process, (ii)
methods combining the physical analysis with semi-empirical constants deter -
mined from actual evaporation in representative regions and (iii) purely statistical
methods.
The first group includes methods using heat balance equation. water balance
equation and turbulent diffusion. In the USSR equation (1) which has been
deduced from the simplified equation of the heat balance of the land surface with
the account of Bowen ratio is widely applied.
where: E: Evapotranspiration, R is the measured value of the radiation balance
of the surface, B is the heat income into the soil, L is the latent heat of evapor-
ation, Cp is the heat capacity under constant pressure, H is the atmospheric
pressure, t and e are respectively the differences in temperature and water
pressure measured at two levels above the ground.
Equation (1) should rather belong to the second group than to the first one, since
it does not represent all physical factors that affect the phenomena.
Full water balance is not applied in the practice but in the case of deep water
table. in this case, the fobwing equation is used in the USSR for estimating-
evapotranspiration from non-irrigated fields;
64%
16%

E = X +(W1 . W2) (2)
X: Precipitation; W1 and W2 are the moieture storage in soil at the beginning
and at the end of the design period. Some conditions for using this relation
are indicated by the author. Another partial solution of water balance equation
is the estimation of mean annual sums of evapotranspiration as the dif -
ference between precipitation and runoff. After indicating the possibilities of
methods based on turbulent diffusion problems in deriving a universal equation
are also stated. Therefore, the convenience of equations using non-specialized
observations is evident. One of these for regions - of natural moistening is due
to Budyko:
-Ro XL
;h Ro (cm year-1 ) (3)
4
equation (3) includes only one observational parameter, X: long term average
precipitation (cm/year) R o is the average annual radiation balan e of the undey
lying surface which can be obtained from the m ap of Referenced. L is the
latent heat of evaporation, A method for distributing the mean annual sums
estimated from equation (3) is explained by the ,authors.
Monthly evapotranspiration from irrigated fields are estimated with the help
of simplified heat balance equations. The standard error is about 15% when
special observations are available or about 30% with standard observations.
The use of em+ical relations similar to that of Blaney and Criddle can be
used only if the empirical coefficients are tested and corrected for each point
of their application.
The most simple equations allowing the estimation of evaporation from water,
snow and ice surfaces by means of standard observational data, are the
following binomial and monomial equations:
and
E (a t ab&) (es - e2)
E = A U2 (es - e2)
being: E; evaporation -/day; U, wind speed at the height z above the surface
in misec; es and e2 are the maximum water vapor pressure estimated
from surface temperature and water pressure at 2 meters in mb; A, a and b
are experimental constants. For estimating evaporation from snow the values
of a -0.18 ab=O. 098 z=lOm should be used in equation (8). For lake evaporation
a=O. 14 b=O. 72 and 9m should be used in equation (8). Other cases are also
discussed in the paper.
RESERVOIR SEDIMENTATION
The paper by Karaushev and Bogeliubevag presents a method for estimating
reservoir sedimentation based on the equation of sediment balance as applied to
the whole reservoir or its parts.
The inflow of sediments is computed by observational data or by indirect
methods. The outflow of sediments is computed based in hydraulic and sediment
characteristics. The determination of sedimentation during one year is
reduced to estimating tha portion of the sediment inflow that is accumulated in
the reservoir.
7

8
Equation (1) of the paper shows the computation of sedimentation for any si ze
fraction in a design interval:
m -6
Paj = Z Pi in j - Qter j A tj 10 ils si terj (1)
is1
P aj is the amount of sediments of all size fractions trapped by the reservoir
during k tj; Pi in j is the inflow of sediment for each i-th fraction; Q ter j is
the mean water outflow (m3/s) and Si ter j is the mean turbidity (concentration)
for the time 4 tj and for the i-th fraction of size. Equation (2) to (9) are used
for computing si ter j and are based in hydrod namic considerations and the
reservoir characteristics such as length and depth. The amount of bed load in
the reservoir is computed by equation (10):
Pa bed j = lom3 (R bed in j - R bed ter j) A tj (1 0)
Pa bed j is the weight of bed load in the reservoir (Tons), R bed in j and R bed
ter ' indicate bed load discharge at the initial and terminal discharge sites
(Kgjsec), A tj is the time interval, Bed load, R bed, is computed with Shamovls
equation (equations 11 to 14 the text).
The annual accumulation of all sediment fractions for the first year of reser-
voir operation is obtained by adding the suspended and bed sediments as indicated
in equation (15) of the paper. The value Pai (tons) so obtained is transformed
into volumetric units Wai:
- Ys is the specific weight of the sediment (T/m3). After the first
duced volume W Wa is used for the computations of next year.
For the computation of the chronological variations of sedimentation the Shamov
method is recommended:
Where Wat is the sediment volume in t years; Wal is the sedimentation volume
during the first year, computed as explained before, W a ext is the extreme
volume of sediments in the reservoir, approximately computed by:
Where W is the initial volume of the reservoir, Ur is the area of river cross
section when discharge is close to maximum and up is the maximum cross
section area of the upper pool near the dam.
Surface Water Utilization in Arid and Semi Arid Zones. - The paper by Dalinskyly
shows *e experience of Tahal-Water Planning for Israel Ltd. in various methods
of analyzing stream flows. For planning of utilization the following information
is required: (a) the average volume of annual flows (P.ave),. representing the
stream water resources potential; the average annual feasible utilizable flows
is a portion of this value; (b) the stream's flow regime including flood frequency;
(c) the stream variability within a season, a year, or from one year to another.

For determining the annual flood return periods the author proposes the use of
the well known T = formula; for longer return periods the estimates of
m
order of magnitude of annual flows for longer return periods can be obtained
by extrapolation on probability paper.
The next section deals with the well known flow-durati on curves,
The concept of Ilhorizontal cut" of the stream hydrograph is useful in the case
of a diversion of a stream, as indicated in sketchs 2 and 3 of the paper. The
"horizontal cut1' can be expressed mathematically as:
Qd = Qi when
Qd (ad) max when
mix
Qi 3 Qd) m ax
Where: Q: atreamflow discharge
Qd diverted discharge
(Qd) max: maximum diverted discharge
... (2)
For a period of n years, a series of n annual diverted volumes can be obtained
and the average diverte&annual flow (va) can be calculated for each value of
(Qd) max. The funtion Ud = f (ad) max has the form indicat& in sketch 4.
Three zones can be distinguished in this curve; in zone I Ud is
'mmax
relatively large and almost constant; in zone II the derivative decrease quickly
as (Qd) max increases; in zone III the derivative trends to eeru, when (Qd)max+ Q
Most of diversions will be economically justified in_zone i, and unfeasible in
Zone IU. Formula (3) can be used for calculating Ud from the flow-duration
curve.
Adjustments for baseflows or minimum diverted discharges can be made easily
changing the origin of coordinates.
When there are limitations to the diversion of baseflow discharges, a "double
cuttt is required as indicated in sketch 5. This case will arise when baseflow
is undesirable due to high salinity or other reasons for diversion. A maximum
desirabledischarge is determined generally by sedimentation conditions. The
value of UA, average diverted flow can be computed as:
where ÜB is established by means of a horizontal cut and E by means of a
vertical cut. Funtions Ug and Uc can be easily calculatfi by computer, An
example is shown in Fig. 2, App A. Using equation (5) UG can be calculated
from the flow-duration curve without use of a computer.
Another uae of the vertical cut is presented for planning of diversiomwith limitations
of maximum discharges due to sedimentation:
Qd m Q
for Q 4 Qdmax Qd = O for Q > IQdmax l
The resulting discharge curve is combined with the sediment concentration
flow discharge curve shown in Figure 4, App. A for computing the sediment
transport as detailed in App. B. of the paper.
9

10
Next section deals with the determination of annual storable flows as a .function
of reservoir capacity. Assuming that losses during the rainy season can be
neglected, the following equation applies:
- UR: is the n years' averageannual amount of water stored in the reservoir
(Net average capacity'RN)
UR)^ is the amount of water stored in the i th year;
UR)i Ui when Ui 4 RN
Iud.
i: (RN)i when Ui P (RN)i
Ui: is the annual streamflow
(Rn)i represents the net reservoir capacity in the ith year
Limitations of these relations arc indicated in the text (Neglecting losses, etc. )
Relations between K a n d the reservoir efficiencyhs function of average
Uave
net reservoir capacity are shown schematically in sketch 6. (The meaning of
the three zones is the same as in sketch 4). This analysis is important for
preliminary estimates and/or feasibility calculations. Recent investigations
reported in the paper prove that these relations can be approximately estimated
on a regional basis using as a parameter the dimensionless standard deviatio-
Formulae for computing RN and RN. based in the decrease of the capacity of
the reservoir due to sedimentation are given in other section of the paper. The
paper concludes recommending hydrological investigations to find, on a regional
basis,parameters allowing to represent the main functions discus sed in the
article.
ASSESSING mROLOCICAL CHARACTERISTICS IN DATA -SCARCE AREAS
Estimating flow regime when no data are available. - The first problem with
which the hydrologist has to deal in data-scarce areas consists of obtaining
the hydrological characteristics of the region. First of all, the average discharge
has to be estimated. For this, a wide variation of methods can be used
depending in the availability of information
The worst case consists in a complete lack of information Here the estimates
should be based in observations in similar gauged zones. The estimations made
for Asejire Project seem to belong to this case. However, even in this extreme
case, the scarce available information should not be neglected. Topography,
altitude, shape, orientation, geology and ve@aHecover of the basin are easily
obtained and should be always used.
The most elementary equation is the surface relation:
A
Q=- Qb
Ab
where: Q : Flow at the site under study; Qb : Flow at a base station; A: Surface
of the basin under study and Ab: Surface of the basin of the base station.
This equation, ii obviaidya wrypoor representation of the +noniena, and sbculd be used

only for gross estimates.
When precipitation data is available,this mdhod can be improved by introducing the
precipitation data for the basin under study. (P) and for the basin of the -base
station (Pb). The relation in this case is:
A P
Q a-
A b pb Q b (b)
If the yield of the basin is defined as K e 2, equation (b) becomes:
PA
Q-KbPA (4
A variation of this method, used sucessfully in Chile and Central America" J
consists 8f analyzing the variation of K with the basin conditions, topography,
elevation, vegetation, geology, orientation, etc.. . Figure 1 shows an example
of this method.
Abiodun uses this method in his paper. No explanations, however, are given
for the criteria in selecting K = O. 04 and for the use of a 1:50 years precipitation.
This seems an exaggerately pesimistic estimation and should result
in underestimation of the water resources. However, since as the paper explains
that efforts are been made for enlarging the scope of the hydrological investigations
in Nigeria, it is hoped that soon it will be possible to revise these computations
with more accurata methods.
When the Economic Comission for Latin America decided to make a preliminary
survey of the water resources in the Central American Isthmus, the UNbP/
@Mo project prepared the maps of curves of runoff deficit shown in Figure 2,
wnich allowed first estimates for ungauged areas. The trace of the curves
should take into account the already mentioned physical factors.
A further improvement consists of the introduction of climatic factors. such as
temperature. Examples of these methods are those of Khosla, Langbein,
Coutagne, Turc and the one propeed by Smith in his paper. Application of
these methods with universal constants produce, sometimes, large errors, so
they should be limited to regional use, previously determining their constants
in gauged areas of similar characteristics.
in D'Olivieria's case, the use of Coutagne's rule with the original constants
would hata introduced very large errors in the estimates. The same occures
when A = 300 Le used in Turc's formula.
As Smith shows in his pa er a good relation between precipitation and tempe-
rature (or Flimatic indexf can be found in a Fegionalized basis. The reporter
has added entra1 American values to Smiths relations for Puerto Rico and
Kansas with good results (Figure 3). However, the Constants used in thesb
rnethds elaould be verified-on a regional basis.
A check has been made to all these methods using them to estimate the mean
annual discharge (in mm) of eleven CentralAmerican streams of quite different
conditions. The following methods have been used: Equations a. b and c,
Coutagne, Turc and Smith, and the results are summarized in table i.
11

12
Table L - Comparison of the use of several methods for estimating mean annual
runoff (mm) of Central American streams.
Country Drainage Estimates of mean annual runoff Obserx
and Basin ed
Station sq Km. Eq.a Eq, b Eq, c Coutagne Turc Smith runoff
Guatemala
Candelaria
Honduras
Re. Pimienta
El Salvador
Bande ras
San Marcos
Nicaragua
Dar $0
Tamarindo
Costa Rica
Cachi
El Humo
Palmar
Panamá
David
Majk
849. 5
883.8
432.8
180. O
91 5
165
904.1
135
486 3
1392
321 8
470
720
89 0
560
160
280
2280
26 50
1590
1970
1370
480 550
720 750
840 800
590 620
160 200
250 480
2000 2160
5700 6000
2300 2350
2150 2200
1520 1550
1550
540
7 50
81 O
50
340
21 O0
(6 500)
2270
2880
880
1830 1190 440
1250 320 740
1500 660 500
1500 1000 590
670 50 110
1040 230 500
2250 1700 2000
(5240) (6100) 6270
2420 1800 1970
2620 2150 2650
1670 800 1560
Average
error ‘já 35 21 15 40 39 60
( ) Extrapolations.
Equation (c) gives the best results, followed by the simple areal relation corrected
for taking into account the change in precipitation (Equation b). However, these
are the results for a particular area, Central America, and there is no assurance
that similar results should apply to other regions of the world. The best advice
could be to try several of these methods and check as soon as possible the results
with measurements at the site under study.
Extending short or incomplete records, - The most commonly used method for
extending short or incomplete records is to correlate the records of the station
with the records of a station with longer records. The correlation can be done
with mean annual, mean monthly, mean daily or instantaneous discharges: the
quality of the correlation decreasing in this order. For daily or instantaneous
dischargestlag effects have to be taken into account. in larger basins --as in
the case reported by Abiodun-- lag effects apply also to monthly discharges.
The quality of the correlation can be determined easily by means of simple
statistical tests. This quality depende on the physical and meteorological
characteristics of the basins being compared. in general, correiations between
two stations nearly located over the same river give good results. The fol1 w-
ing results should be expected when the basins of Figure 4 are comparedld:

Basins compared Quality of correlation
1 and 2
2 and 3
3 and 4
2 and 5
4 and 6
5 and 6
Good : Basins of similar form,
size and orientat i on .
Fair : The orientation of the
valley is different.
Poor: Different altitude and
orientation.
Fair : Same form but different
a It it ude.
Poor : Different characteristics
Poor : Different orientation and
altitude.
When no hydrometric information is available, correlation can be tried with
longer precipitation series.
A long time average can be obtained assuming a constant yield of the basin, or:
In this case, long (n year) precipitation records are available, Pt and Qt are
the rainfall and discharge averages over the t years for which discharge data
are available. This method, after checking that Pn = P was used by Quintela
in his paper. However, an interesting verification in that case would had been
correlating the seven year's records with the South African station
Studies made in Chile and in the Central American Isthmus show that the results
of correlation studies are far more reliable than the methods explained
in the preceeding section. However, extreme caution has to be exercised when
records are too short, carefully avoiding to be too influenced by some statistical
indicators. In this case comparison with other methods is an useful auxiliary
tool. Complete verification of the base information should be the starting point
of any extension of hydrological records.
Estimating evaporation and evapotranspiration. - The estimation of evaporation
and evapotranspiration has several important implications in hydrological
studies, such as computations of reservoir evaporation, water balances and of
requirements for agriculture.
Direct measurements are difficult; the U. S. Weather Bureau type A pan, the
mQst frequently used instrument in developing countri es, is not always correctly
read and the relation from pan to lake evaporation remains in doubt . The
development of a simple formula for computing potential evaporation, is therefore
of great importance.
Kuzmin and Vershinin give an excellent summary of formulas used in the USSR..
For data-scarce areas, however, formulas based in the physical interpretation
of tFe fenomena are quite difficult to apply. Equations (2) and (3) are certainly
promising and it would be interesting to have more details on them Binomial
formulas are widely used. Equation (9) lightly different coefficients has
been used in Chile and in Central America wit&
with unsatisfactory results.
13

14
Equation (9), as reported by Kuemin and Vershinin, ly been compared with
Blaney -Griddle, Penman, Hargreaves -Christiansen and Meyer formulas
for five locations in the Central American I sthmus with the following results:
Table II
Evaporation computed with several formulas
Station Madden San José Chorrera Guija G ua t emala
Panamá Costa El El Guatemala Average
Formula Rica Salvador Salvador
USSR, Binomial
lake evaporation 79 8 488 1293 1133 765 89 5
Blaney-Criddle 2062 1749 2041 1910 1647 1882
Penman 1328 1077 1345 1530 1350 1326
Hargreaves-
Christians en 1400 1 O00 1340 1280 1070 1218
Meyer 1394 1227 2147 1577 1628 1594
Potential
Evaporation
(Measured in
pan x O. 77) 1020 1145 1760 1460 1050 1287
in average, the best agreement ie reached Penman formula. However, as the
Hargreaves-Chrintiansen equation was a plied using only temperature, humidity
and precipitation (wind was estimated7 it provides a simple alternative, Blaney-
Criddle and Meyer (a simple binomial formula) give excessive values. The USSR
binomial formula gives very low values, which is probably due to the obvious
differences in climate with respect to the conditions from which the formula was
de rived.
Sediment studies. - Three of the papers show examples of sediment determinations,
which quite frequently have to be made with insufficient information,
The first problem refers to estimating sediment yields from streams without
sediment measurements. Figure 8 of the paper on the Central American Hydro
meteorological Project shows the wide variation, within a regiun, of sediment
yields. Thus, determining sediment transportation loads without field measur -
menta is quite unreliable. Hydrologic and hydraulic information from the
gauging stations "somehow improves these estimates. However, very simple sq
diment measurements allow a relatively acc rat estimate of suspended sediment
loads. A good correlation has been found18)1 between the concentration of a
sample taken with a bottle by U n d of unskilled obskrvers and the mean concen-
tration obtained with conventional sampling.
The sediment rating curves (two examples presented in the papers) allow to
compute the total suspended load in a quite simple form, The points of the curve
relating the solid and liquid discharges have a large dispersion (due to errors

in measurements, differences in the raising and decreasing stages of a flood,
variations in the availability of sediment supply, etc. ), but its mean trend has
been found to be relatively stable, which allows estimates of suspended loads
with series of observations as short as one year.
Determination of bed load presents just the opposite problem. Direct measurements
are difficult and provide in most of the cases non-meaningful results.
Use of well-known formulas is thus encouraged, in spite of the fact that they
give enormous differences. Therefore, the use of several methods is suggested
including, if possible, methods, such as modified Einstein, which use the available
suspended sediment measurements. It is also quite useful to observe the
'!critical discharge", i. e. discharge at which the bed movement starts, which
in some cases can be determined by detection. of stone noise by the stream
gaugers.
Karaushev and Bogeliuva's paper deals with the important problem of predicting
the chronology of the filling of a dam, and esents a new interesting approach
to the problem also studied by Brownlv. However, the main difficulty
as seen by this reporter, is the estimation of "in situ" specific weight of the
settled suspended sediment.
very fine its settlement is very slow and subject to relatively complicate laws.
For this the formula of Reference 14/ can be used:
yT : -k k( T-l T Log T - 1 )
15
The problem here is that when the sediment is
k is a constant depending on the size and mechanical distribution of the material;
YT the specific weight after T years and y1 the s ecific weight of the sediment
("in situ") after one year of settling. Referencelggives values of k and Y
but to the reporteis knowledge, no check of these values have been made for
1,
most of the world. These "in situ" determinations are difficult, since it is
practically impossible to obtain indisturbed samples of submerged clay or lime.
The use of y Ray diffusion probes can be usefull, but they require careful
laboratory calibrations and expensive equipment.
Water Resources Studies. - Dalinsky's paper present an interesting and simple
method for preliminary studies of water resources projects, and should be con-
sidered a preliminary approach to those exposed in other sections of this Sym-
posium.
CONCLUSIONS
Assessing hydrological characteristics in data-scarce areas is indeed a difficult
problem The difficulties in the studies increase inversely with the amount of
information. available, not because of the intrins ic mathematical and operational
problems, but because extremely good judgement is required. Unfortunately
good hydrological judgement depends on the knowledge of the meteorological,
physical and hydrological characteristics of the region under study.
Several excellent examples have been shown of what can be done with scarce
information, but the possibilities of big mistakes appeared also evident. These
can be avoided either with excellent judgement or with the help of a few, but
adequate data. These data do not need to be long term series or sophisticated
measurements, thus can be collected at a relatively low cost. This cost represents
only a small fraction of the eventual overexpenditures or losses from
poorly designed schemes.

16
The ideal, obviously, would be to undertake in each data-scarce area a com-
prehensive meteorological and hydrological survey, as the U DP/WMO projects
in several parts of the world. Evaluation of these projectslg allows to show
several concrete examples where a small investment in these basic surveys has
resulted in economic benefits several times larger than the expenditures in me-
teorology and hydrology.
REFERENCES
Quintela Gois, C. - Some Criteria Used in Hydrologic Studies with Inadequate
Data. Symposium on the Design of Water Resources Projects
with Inadequate Data. Madrid 1973.
Abiodun, A. A. - Water Resources Projects in Nigeria and the Hydrological
Data Employed in their Planning and Development. Symposium on the
Design of Water Resources Projects with Inadequate Data. Madrid 1973.
Basso, E., Arriagada, A., Neira H. and Pérez Delgado, M. - An Example
of Co-operation for Improving the Hydrological and Meteorological Information.
Symposium on the Design of Water Resources Projects with Inadequate
Data. Madrid 1973.
Smith, R. - Utilizing Climatic Data to appraise Potential Water Yields.
Simposium on the Design of Water Resources Projects with Inadequate
Data. Madrid 1973.
Giusti, E. V. and López M. A. - Climate and streamflow of Puerto Rico,
Caribbean Journal of Science, Vol. 7, pp 87-93, 1967.
D'Oliveira Martens, E. E. and Mimoso Loureira, J. J. - Application of
Coutagne's and Turc formulas to the Southern Mozambique rivers. Sym-
posium on the Design of Water Resources Projects with-Inadequate Data.
Madrid 1973.
Kuzmin, P. P. and Vershinin, A. P. - Determination of Evaporation in
case of the Absence or Inadequacy of Data. Symposium on the Design
of Water Resources Projects-with Inadequate Daia. Madrid 1973.
Materialy Mezhduvedomstvennogo Sovetchchania PO probleme Izuchenia
i Obosnovania Metodov Rasheta Isparenia s vodnoi Poverkhnosti i Suchi.
(Materials of Interagency Meetings on the Problem of Study and Substantiation
of Methods for the Computation of Evaporation from Water and
Land Surfaces). Edited by CGI, Valdai 1966.
Karaushev, A. V. and Bogeliulova L V. - Computation of Reservoir Sedi-
mentation. Symposium on the Design of Water Resources Projects with
Inadequate Data. Madrid 1973.
Dalinsky, J S. - Methods of Analysing Defficient Discharge Data in Arid
and Semi-arid zones for the Design of Surface Water Utilization Symposium
on the Design of Water Resources Projects with Inadequate Data.
Madrid 1973.
Central American Hydrometeorological Pro.iect. - Manual de Instrucciones:
Estudios Hidrológicos (Manual of Instructions: Hydrological -
Studies)
Publicación No, 70, San José, Costa Rica 1972.

Central American Hydrometeorological Proiect. - Medida de la Evaporación
(Measurement of Evaporation) Publicación No. 19, San José, Costa
Rica,. 19 68.
Brown, C. B. - Discussion of "Sedimentation in Reservoirs" by B. J.
Witzig" transactions ASCE Vol. 109, 1944, pp 1080-1086.
Office of Indian Affairs, Bureau of Reclamation, Tennessee VaBey Authority
Corps of Engineers, Geological Survey, Department of Agriculture and
Iowa institute of Hydraulic Research. - A Study of Methods Used in Measurment
and Analysis of Sediment Loads in Streams. Report 9 "Density of
Sediments Deposited in Reservoirs", St. PBul District Sub-Office, Corps
of Engineers, Hydraulic Laboratory University of Iowa, Iowa City, Iowa
194.
Central American Hydrometeorological Project. - Estimación Preliminar
del Balance de Aguas en el Istmo Centroamericano (Preliminary estimat-
ion of the water Balance in the Central American Isthmus) Pubiicación
No. 18, San José, Costa Rica 1968.
World Meteorological Organization. - Twenty Years of WMO Assistance.
WMO-No. 338, Geneva, Switzerland 1972.
Central American Hydrometeorological Project. - Deficiendas de Agua
en Centro América B Panamá (Water Defficiencies in Central America
and Panad) Repodprepared by G. Hargreaves as a consultant to the
Central American Hydrometeorological Project. Publication No. 88,
Managua, Nicaragua, 1973.
Central American Hydrometeorological Project. - Empleo de la Muestra
Puntual para la Determinación del Sedimento en Suspensión (Use of the
Puricbial Sample for the determination of Suspended Sediment) Publica -
ciÓn No. 1, San José, Costa Rica, 1967.
Central American Hydrometeorological Project. - Manuel de Instruccio-
nes: Hidrometría (Manual of Instructions: Hydrometry) Publicación No.
47, Segunda Edición, San José, Costa Rica, 1972.
17

18
Figure 1. -
Method for estimating
hydrologic yield of
u ngauged areas
(From Reference lu)
Figure 4. -
Basins used for
checking results of
correlation
(From Reference ,l#

x
20
300
200
Kansas
L
5u
tra1 America
1 I I
1 O0 200 500
P MEAN ANNUAL PRECIPITATION CENTIMETERS
Figure 3. - Central American Values plotted into Smith's BCI f(P)

ABSTRACT
WATER RESOURCES PROJECTS IN NIGERIA AND
THE HYDROLOGICAL DATA EMPLOYED IN THEIR
PLANNING AND DEVELOPMENT
Adigun Ade Abioduna
The need for adequate water supply to meet the demands of
Nigeria's growing population is well known. However, the technical
adviser is seriously handicapped in his planning efforts by the
lack of sufficient information. As a result, different kinds of
data and different levels of efficiency have been employed by the
various agencies which have planned the existing major water related
projects un Nigeria. This investigation shows and intensity of
rainfalls and the attendant floods, small scale project modelling,
projections based on hydrologic data from other but climatologically
similar places, provision of missing data by statistical correlation,
and intensive surveys over short periods to obtain rapid and exten-
sive information. These schemes have been reviewed and the hydrologic
information employed in designing them has been appraised. This study
also shows that Nigeria must intensify her efforts to provide exten-
sive basic data on her surface and groundwater resources if costly
mistakes are to be avoided in the future. A case is also made for the
use of new techniques such as Remote Sensing for rapid identification
and appraisal of these resources.
RES UME
On sait quels sont les besoins du Nigbria pour un approvision-
nement en eau capable de satisfaire les demandes de sa population
croissante. Or il se trouve que le conseiller technique y est sérieu-
sement handicapé, dans son effort de planification, par l'insuffi-
sance de l'information. Les diverses agences qui sont chargées, au
Nigeria, des grands projets d'aménagement des eaux, doivent utiliser
des données disparates ayant des niveaux d'efficacité différents.
L'analyse des problèmes montre que l'gtude de ces projets doit faire
appel -3 l'information locale sur la fréquence et l'intensité des
pluies, et les crues qui en sont la conséquence (petits aménagements),-
aux évaluations tirées des donnés hydrologiques recueillies dans de
régions climatiques semblables, -à l'utilisation des corrélations
pour boucher les lacunes,- a l'observation intensive sur de courtes
périodes pour étendre rapidement l'observation. Des efforts ont été
faits dans ce sens, mais il reste que le Nigeria doit les intensifier
pour rassembler une masse importante de données de bases sur les
ressources en eaux de surfaces et en eaux souterraines, afin d'éviter
dans l'avenir de coûteuses erreurs. On ne néglige pas non plus
l'utilisation des techniques nouvelles, telles que la détection 'a
distance, pour améliorer lainventaire de ces ressources.
* Lecturer, Dept. of Agric. Engineering, University of Ife, Ile-Ife,
Nigeria.

22
1. IBTRODUCTIOH
The developat of water reaiources miithin the pat deeade, in Nigeria,
has concentrated moetly on the prorieion of adequate pipe-borne water for
domeetic and institutional supplies. The trend, however, ia changiiig, and
it is now realieed that water resource8 developent, a8 a ipa$Or economlo
revolutionary tool, ihould Qiphaeise ita total harnesaing, control and
utilization to provide in addition to watar supply, such other benefits 88
hydro-power, irrigation water, flood control, water transportation into and
from the hinterled, fish and wild-life, recreation and pollution abatarsient.
The awaxenees of these needa has provoked riome dee Unking and has,
in part, precipitated the putting together of the Färat ]81962-68) and the
Second (1970-74) National Developent Plans. The objective of the latter,
according to the National Economic Counail, being
"the achievement and mainteamce of the highest poeaible rate
of increase in the standard of living and the creation of the
neceieary conditions to this end, inoluding public support
and awareness of both the potaiti&le that exist and the sac-
rificee that will be required."
The implementation of the variow schemes coatained in them pl- have
experienced sime hardship especially where technical man-power and information
were needed. In many inatances, the technPlogist is often called upon to
make far reaahing professional deaisiona, and quite often, he ia seriouelg
handicapped in hie $Lanning efforts by the lack of scientific information.
This problem of planning dthout facta waa amply stated by Andu (1) about
bore hole drilling (for water) in Yeatern Nigeria:
"1 have emphasised the handicap due to ecantinesa of hydrolo-
gical data; and eince the gigantia Five Year Developnent
Programme cannot be held up becaune of this, the practice
now is to confine drilling to areas with favourable geolo-
gical formations. Time facbr has made any exploratory
test drilling virtually impoaaible. The location of a bore
hole wen in a geologically favourable area is chancy -
and there ia no sufficient guarantee that water of adequate
quantity ehall be etruck. It ie not uncomon to drill far
deeper than expected where the exhibited geological patterme
indieate otherwise...."

In order to achieve thd goals spelled out in the National Dwelopent
Plane, expertise are often imported to analyae our local data or to use
their "ingenuityn to generate needed scientific information on which our
planning and development programmes could rely.
rical data are either scanty, unreliable or absent, and the synthesized
data can only be 88 reliable as the historical but scanty data available.
For many foreign experts, handling the problwie of the tropics is a new
educational experience and most of these techniml consultants, who are
often from temperate climates can only draw on their bowledge and ewe-
rience of their own temperate environment and adapt them to plan for the
needs of the tropical zones.
23
More often than not, histo-
Most of the existing water resources sehemes have been handled in the
nanner enunciated above, and sane of these techniques can in some caees be
referred to as "educated guesen work by the experts. Hence, this paper
examinea, in closer details, a few of the existing water resources projecte
in Higeria with a view to high-lighting the kinds of hydrologic data, analysis,
and the different levels of efficiency that have characterized their planning
and developeat. Such an evaluation should offes some guide-lines for
systematic planning in the future.
2. HYDROLOGICAL DATA COLLECTION
The hydrological data needed to effeat adequate study of water resou~.ces
inolude data on precipitation, evaporation, stream-flou and groundwater.
In Nigeria, the sole responsibility for collecting rainfall data
reste on the Federal Meteorological Service (IPPS).
over lux) rain gauging stations throughout the country, utilizing the
recorded data Rom these statione for water resources planning would require
further interpretation and analyeis. This is so because these stations
uere not established in relation to river basins.
Although M!CS maintains
Furthermore, those co-
llecting the rainfall data such as the local school teachers and looal post
office personnel owe no allegiance to the WITS since the latter never rewards
them in any way or form for their services. Hence, the accuracy and relia-
bility of data collected under the aforeaentioned condition are often in
grave doubt.
The measurement of evaporation data acroas the nation is also done by
the employing some 68 clans A evaporation pans in an area almost 590,000
square kilometres. In additiop, there are three lysimeter statione in Nigeria
- two at Ibadan and one in Zaria. Although reservoirs are being built on 8
continuing baais, and evaporation acrose the land v miw between 102 to 204
eentimetrea a year, the impact of evaporation on the yields of these reservoirs
is probably still not fully realised.

24
Stream flow data are collected by auch ageucies as the IpLand Waterways
Department (IWD) and the Ministries of Work. The former maintains over 100
gauging stations along the major rivers of Higeria for the expresseu purpose
of recardiiig stage heights which are used to determine navigable waterways.
The Ministries of Work on the other hand are mom interested in potential
areas for the location of highway bridges, hence, most of their hydrological
stations are non-self recording. The unavailability of diacharge measurements
or proper rating curves that could be used to interprete the recorded
stage heights has rendered most of the date available unworkable.
In the
Northern States, where there were ZIO reel hydrological net-rrodf until after
1960, most of the rivers are non-perennial and shifting; the latter situatinn
makes it mandatory to provide more than one ratirig curve per Station per
seaBon thus rendering most of the available record difficult to interprete.
The Geological Survey of Nigeria (GSN) is totally responsible for collecting
data on the groundwater resourceB of the nation. Most of GSl's
efforts have been concentrated in the Northern States where there is abundant
supply of grounawater and very little surface water supply. The GSü
in collaboration with the United States Geological Survey, haa carried out
some investigation in the Chad Basin complex, and estimates have been made
of the life of the aquifers in the basin as a result of groundwater mining.
However, no attempt has been made to quantify the annual natural groundwater
recharge or the contribution of the groundwater to river discharge. Information
is also not available on the groundwater flou conditions.
Although the agencies cited above collect vast quantities of data
annually, the fact ia that until very recently, the data collected have been
piecemeal and the hgdrologicd records were never checked nor analyred. In
many cases, the records have no duplicate8 and hence distribution is often
impomsible. This state of affairs is often due to two major factors - lack
of funds and lack of badly needed technical mawpower. This dearth of
adequate hydrological information has not however precluded the planning and
the actual developent of a -ber of major water schemes in Nigeria euch as
the Kainji Dam and Lake Project on River Niger.

30 HYDBOLOCICAL DATA USED IN EXISTING PROJECTS
Sequential generation of hydrological data has been a tool the hydro-
logist haa often used to create synthetic records, in the absence of very
long hiebrical records, that could be used in hia water reaourcea planning
efforts. Since the generated set of data is only as good as the historiaal
set employed in such a synthesis, the historical set should not be too ahort.
In the absenoe of such a hiatorical information, the planning anã derelopent
of the existing water resources schemes in Nigeria have relied on such tech-
niques of hydrological data derivation as - local information, eduaateä
gueas method, projection based on hydcological data from other but climatolo-
gically similar places, provision of missing data by correlation and intenaive
surveya over short periods. A few significant schemes are examined below.
A. KAINJI W B ND D q
The moat wide-ranging water reeourcee project undertaken to date in
Nigeria is the Kainji Dam and kke Saheie on River Niger (Fig.1.). The scheme
was conceived to provide hydro-eleckic pwer, flood control, regulated water
for navigation, and fishery benefits. Although actìml construction started
at the Kainji site in 1964, water levels were never observed there prior to
1959. The two nearest statitma where ologic data were observed on the
Niger prior to 1959 vere Jebba (Nigeria Y and Niamey (Niger), both of which
sandwich the Kainji site and are 906 anä 1630 kilometres respectively amy
from the Atlantic mouth of Xivex Niger.
The pre-construction density of rainfall net work within the catchment
area of the Keinji project was too low to serve as the baais for any reliable
hydrological interpretation. Conaequently, new rainfall uging stations
were established for the project and a seva year record K955f959) was
obtained by the consulting f im (2).
method, the total amount of rainfall on the catchment area was calculated
for the seven year period.
25
Through the application of the Thieasen
Although records of water levels at Jebba were available for the years
1915-24 and 1947-64, the ahiftirig positions of the gauges during those years
&e it impossible to oorrelate the datum points of all the gauges used.
Consequently, a decieim was made to correlate the rainfall with the nia-off
within the Niamey-Jebba catchment, using the newly observϊ atage discharges
at Jebba for seven years, and to employ this correlation curve with the cal-
culated rainfall data to establish a 1939-59 discharge reaord for Jebba.
Owing to the relative insignificant average value of the inflou between
Jebba and KainJi, the obaei-red and the generated discharge data for Jebba
were aaaumed to be the same for linin31 - which ie upstream of Jebba - and
were analyzed accordingly. In establishing a satisfactory correlation between
the rainfall and runoff data, two steps were taken:

26
Because of the wide variation in both the rainfall and the discharge
data betueen gauging stations, only monthly totale were uaed in the
dYSi8. This approach was found to produce smoother oorrelation
curves than thore obtained from daily or 54ay records.
The Jebba-Niamey catchment area i8 extensive, and the runoff contri-
bution to the Niger flow from the Dahomey catchment area takes a
longer time to reach Jebba than the othr catchent6 downstream.
Hence, a sequence of lag time wa8 introduced into the data analysis
to yield an expression heeeby derived as
where
QA,P+= CU~,.O + CP Re&- Ah),$ + C3Rr.n ( 1)
BA,,.,, = Runoff of the month of August from the Jebba-Biamey
catchment srea;
R4.~ U Bainiail of the month of July on the Dahomey catchment
area;
R($-k),s= Bainfall of second-half of July plus that of firsthalf
of August on the Sokoto basin;
e,,. = Rainfall of the month of August for the rest cf the
Jebba-Niamey catchent area; and
'
C,, C2, C are nuioff coefficients far Dahomey, Sokoto and the
rest of Jebba-Niamey catchment area respectively.
The introduction of the coefficients of runoff in the above equation
became necessary as a result of the wide variation in the geographioal
nature of the catchment area.
Through the step enumerated above, the m 4 f f data from debba-iainey
catchment area were deduced for the period 193959, and th6Se w8re added to
the Niamey observed record. The net result is w.2, the hydrograph of the
Niger discharge at Jebba. This figure was in turn used to develop the maas
inflow curve info bke I[a;Lnji. The most important atreaai between gaiaji and
Jebba is River Oïi with an estimted aiktchment correlation ooefficient of
0.2 The remainder of the drainage basin had an estimated runoff coefficient
of 0.1 These coefficients were used by the oonaultants in the equation
wwe
Q,,,,,
= QJebh - -e,, - O.'F?, (2)
Q E river discharge in eubic metres/ulrit of time
P = rainfall in cubia metres/unit of time
to arrive at the mass inflow curve for Lake Kainji.

The daily discharges used in the hydrologfcal analysis are very
interrelated and the peaks are interdependent. Sime these daily discharges
exhibited a tendency towards persistence in succeasive stream flows, the
Goodrich dietributiona were used in the frequency calculations; the latter
were of the exponential type and were similar to the exponential Gabel
distributions.
B. WATER SUPPLY II MIDWESTERN NIGERIA
Water resources activities in the lid-West are centred mostly on
water supply. The latter is tapped, in general, from the various aquifere
which underlie 9% of the State. The Benin sand aquifer has the greatest
potential - about 333 metres thick extending laterally to an appreciable
distance - but the hydrological studies from which the aquifer chacterietics
could be obtained are etill in the planning stage. m y of the ;aquifers,
such as the Benin sand (3) and the Coastal Plain aquifers oan be described
only in the moat general tarma because of the lack of recorded data. There
i8 also no information on the hundreds of veils tht tap water daily from
these aquifers.
C. UTER SUPPLY II WESTERN NIGWIA
The Western Nigeria Water Corporation is entirely responsible for the
planning and the developinent of Water supply in the State. The Corporation
obtains the necessary evaporation and rainfall data, m y of which are very
long and reliable, from the Federal Baeteerological Service in Lagoa. However,
because of the scantiness of data on river discharges, the standard praotice
in those parts of the West, where surface uater has been developed, is to
base the rater scheme design on the following hydrological assumptions in
addition to a very liberal monthly evaporation of 127 mm:
(i) A conservative runoff coefficient of 4s
(ii) A once-in-50 years recurrence probability in rainfall with
where P = Percentage probability W rainfall being equal to or
lesa than a given talue;
m = rank of the year; and
n = number of years of record
The catchment annual ruaoff, Q, which is based on these assumptiom
can be computed from the expreseion
where A = Basing drainage mea:
27

28
= Baein rainfall value correspnâing to the probability of
orne-in-% yeare oocurenae.
Co = Coefficient of m f f for the basin.
]Equation (3) or a forin of it hae been widely applied on the numerous eurfaae
water apply eohetmee in the West. Eowvwr, bey%$ of the vast arai- area
of 7,500 sq. kilometres that is governed by the,project, a form of the equation
(3) shown above was not employed to predict the maximm probable flood.
Instead, Professor M. Parde of the University of Grenoble in France, a
speciaìiet in the field of flood studies, advised the consultants that a
runoff in the order of 490 litres per second per square kilometre is known
to hava occured within West African strema of similar importance ae the
Oehun river on which the achenie is estsb1ished.b Sapply Ibadan aith water.
Hence thb value was used in caltuleting the project's spillway
deeign flood of 3680 cmbic metres per second.
The developaenf of groundwater resonroes in the West has encountered
a number of difficulties. When the existing wells were been developed, the
areal extent of the bed-rock formation was not fully known, and the lithographic
characterietics of the water bearing formations, in many cases, were
etill to be studied. Absence of perfonaance data on the wells has only
aggravated the situation, and in most cases, local informtion on dug weU, was often obtained from the inhabitanto.
The Geological Survey of Nigeria (WN) in collaboration with the United
States Geological Survey haa undertaken aome imestigativ6 work which had led
them to edict a 30 year yielding life for the Lake Chad Baain middle eone
aquifer Kg.3) at a withdrawal rate of 5000 gph with wells placed at 16 kilometres
apart. Huiidred8 of bore holes have been drilled but co-ordinated dayto-day
performance data on these bore-holes are lacking. Most of the information
that can be readily obtained on the basin's aquifere are available only
in special reports. It is also irapossible to undertake a meaninghil study
-
of the basin's aquifere within Bigeria along since four countries share the
bain area. The FAQarpd' -@are assisting the Lake Chad Basin Commiedon
a regional organization of the countries that have territorial claims over
parts of the basin, to
(i) Compile all the available date in the baain;
(ii) melop an analme compu4er model that would miwilate au. the
activities that affect the quantity of water in the basin; and
(iii) Define the various aquifers in the Chad bydro-geological basin and
erdeavour to arrim at a synthesie or composite picture covering
the correlations between the atmospheric, emrface and groundwater
ae well as between individual aquifera.

From hydrological stand point, the Kainji project has been more intenaively
studied than any other water scheme in Nigeria. A number of houn
standard methods were used to develop some reasonable results such as the
correlation established between runoff and rainfall of the Jebba-Niamey
catchment area. Since the project design flows were in prt derived from the
discharge data obtaineä at Niamey, the accuracy of the rating curve used to
determine the Niamey discharges shonld have been verified. The importance
of this project also warranted a longer hydrological record than the 20 year
reconstituted record used. This could have been sequentially generated.
In order to emure enough water supply, coneervative estimates of rainfall,
runoff and liberal estimater of evaporation have been the 8taRdard
practice in the West. But such educated guesaes have not prevented water
shortages resulting froa both drought and under-design for the needs of the
communities served. It appears that these educated guess teahniques haye
never taken into consideration that moat of these watershetie would be opened
up in the near future ae a result of extensive and medhanised farming practices.
The occurence of %aximam possible rain-stOmR vouìd also yield higher peak
flow than most of the existing achemea, except the Aaejire project, have
been designed to handle. And the eubsequent floafiing resulting from the
faracing practices or the maximum rainstorm would not only wipe aut these
water schemes but would also endangaz lives and property.
Groundwater develogaent based on inadequate or scanty data has been
found to be both unecoaoioicnl and frustrating. Typical examples include
bore hole failures at Agbor and Warri owing to the mollapse of cui-,
and poor yield as a result of drilling for water in granite aone at Ijebu-
Ode. The problame of grodwater developaent in the Hid-Vest are eubatan-
tial, and only a through analysis can fore-stall more problema in the
future. And the efforts of Mo within the chad Baain has aided and aacele-
rated not only the haraonizatim of the existing data, but aiso the evaluation
of data concerning evaporation and temperature meaeurenent by infra-red
aerie1 photography.
5. HPDBOLOOIcbL RESS(XIRCE
Planning without facts ehould na longer plague the water resource8
program of the nation, mre especially, as we shift our emphiusis from
single giirpose water supply to multi-purpose schemes such aa the Kainji and
Kano river schemes. The latter, also a victim of hydrological data scarcity
ia enviaaged to pride barrefita in such areas ae irrigation, hydro-pouer and
flood control.
In order to enmare systematic planning in the future, the
newly created Water Resoaaraoa btitute.8haeld eatablieh a Hydrological
resource^ Centre whoee primary hction uill firet be the aollecting and
compiling of the existing piecemeal data and nates that are scattered all over
29

30
the nation. This responsibility should be a continuow one and the inforaur-
tion so collected ahould be published annually and made available to the
public on eale. The centre should also standardice, nation wide, the inetru-
mentation and hyàrological data collecting and recording procedures.
The information available to the centre can be upgraded both in quality
and in quantity through the application of Remote Sensing Technique (RST).
The latter ia currently available through participation in the United Stateet
urth ~esourcea ~echnology Satellites program (EE~TSP). The data ana imegery
obtaiasd though such a pmg~am can be utilised in sweral ways including
the rapid identification and appraisal of our water resource^. The immediate
hydrological investigation required in Nigeria, uuder the EELTSP, includes:
The overall delienation of the aquifers of the Chad Basin and the
pattern of the groundwater movsment in the basin. Such information
can be integrated into the basin'e existing analogne model;
the monitoring of changes in reservoira' levels resulting from
evaporation and changes in rater courses resulting Rom erosibn and
siltation;
%he identification and quantification of the groundwater resources
of Southern Nigeria including the location of the position and
evaluation of the extent of salbwater intrusion along the coastal
aquifers. Data obtained from ERTSP would generate more awareness
of the problem in surface water developeat and would provide
needed information on which conjunctive ground and surface water
programa oould be baed.
6. CoáIcLusIOB
Within the past decade, a =ber of water resources schemes have been
developed, ard in genSral, these schemes hare been plaaned with very limited
hydrological data tbat were often extended through the applicaticm of
statistical techiqueo to provide rational design parameters. In others,
"educated guess" technique wan subatitnted. The net resuit of auch methode
haa been the failure of many water supply schemes to meet demands eapeoially
aurfng the dry seaeon ana the location of unproductive bore-hoiea which had
to b abaadoned.
The future of io~iiy of these a ches osnnot be accurately preàicted at
this point. bwever, there is the urgent need to upgrade the scanty data,
through a contirmoire qioiritoring proc688, on which them schemee were built.
such a step WOUM provide a sound baeie for ature schemer, and would e mme
the propat execution of any modification on existing sc@nms when warranted.

The scarcity of reliable data is mostly due to the acute shortage of
hydrologists and middle-level techniciana in this diecipline. Herne, it
will be necessary for the Water Resources IMtitUte in collaboration dth
some of the eriating Universities to develop and execute achenes uhereby
a large number of such teohnologiate and techniciana might be trained
locally to meet the urgent needs of the nation.
In order to eneure systematic planning in the futare, the power to
collect, compile and hairnonise all the hydrological data in the country
should be vested in a Hydrological Resource Centre. The hydrological
information available to such a centre could include data and imagery
obtained through the use of Remote Sensing Technique.
The information
obtained will enable the nation to accelerate the pace of ita natural
resources developneat.
31

32
1.
2.
3.
4.
5.
6.
7.
8.
9.
BIBLIOGWHY
Andu, J. A. (1965). Exploitation and Developent of Groundvater in
Western Nigeriay Ministry of Works and Trsmeport, Ibadan, Nigaria.
IJEDECO (1961). Niger Dame Project, Vol. 2, Hydrology and Beilervoir
Operation, Report auhitted to the Fedaral Ooverment of Nigeria,
Lagoa
Tahal (Water Planning) Ltd. (1965). Master Plan for Urban and Rural
Water Supply, Report submitted to the Hid-West Ministry of Works and
l!rsnsport, Benin, Nigeria.
Tahal Consulting Ebgineers Ltd. (1969). Akungba-Shapureka-Ido&
Water Supply Sahane, Plauning Report suinuitte8 to the Western Nigeria
Water Corporation, Ibadan.
Tahal and Motor Columbus Ltd. (1961). Ibadan Water Supply - hejire
Daia, Final Design Report submitted to the Western Nigeria Hinietry of
Works and Tranpport.
Miller, B. E., R. E. Johnaton, J. A. Oloni, and J. A. Umma (1968).
Groundwater ñyärology of the Chad Baein in B om and Dikwa Emiratem,
N.E. Mgeria, with special Eaphasis on the Flow Life of the Artmian
System. USGS Water Supply Paper 1757-1, U.S. Govt. Printing Oîîïce,
Washington, D.C.
üHESC0 (1970). Study of Water Resources in the Chad Basin, Report
on the resulta of the hoject, Conclueionis and Recamendatione, PSriS,
fiFrance.
HEDECO (1970). Feasibility Study-gan0 River Project: Report eubdtted
to the %o State Winistriea of Agriculture sad Iaatursl Besotarcea, and
Worka and Sumeye, b o , Isigeria.
Federal Ministry of Infonuation, Iagos (1970). Second Hationel
Developent Plan, Fed. Govt. Printer, Lagos, Hgeria.

3G1 Ma2 of W Africa showing R Niger and its tributaries
---_ -~~drograph
I
I L -
derived Smm rom
4dL
- N+ogTh
derived From &sei
. ved. w* -.rat5
FIG 2 Hydrogragh of R Niger at bbba (1955-571 (Reference 2)
33
..

ABSTRACT
AN EXAMPLE OF REGIONAL CO-OPERATION FOR IMPROVING
THE HYDROLOGICAL AND METEOROLOGICAL INFORMATION
Eduardo Basso*
Andrés Arriagadann
Hcsrnando Neira**
Manuel PBrez Delgado***
T&e Centra8 A ~s~ican Hydrometeorological Project initiat ed
in September 1961 rapresents & co-operative effort among the countries
of the Central Amsricea Isthmus (Costa Rica. El Salvador, Guatemala,
Honduras, Hiceragua and Panam%) and the United Nations Development
Programme, acting aar executivo agency the World Meteorological
Organizatioa. Its objectives are the following: (i) installation of a
basic netW6Pk of meteorological and hydrological stations, (ii)
collection, preceesing and publication of the data, (iii) training
of personnel by neans of course.%, fellowships or through technical
publications end manuals and (iv) the institutional strenghtening of
the meteorological and hydpalogical services in the area. Important
Project activities have been the Paeting of new equipment used in
developed countries in order to study their application to the
characteristics and tropical climate of the area, and the development
and application of methods for meteorology, hydrology and sediment
studies with limited information. It is concluded that their use in
other areas with similar conditions can be useful and that regional
cooperation can be one effective means for coping with inadequate
data through the pooling of individual countries efforts.
RES UMEW
El Proyecto Hidrometeorológico Centroamericano iniciado en -
Setiembre de 1967 constituye un esfuerzo cooperativo entre los paí-
ses del Istmo Centroamericano (Costa Rica, El Salvador, Guatemala,
Honduras, Nicaragua y Panamá) y el Programa de las Naciones Unidas
para el Desarrollo, actuando como agencia ejecutora la Organización
Meteorolögica Mundial. Sus objetivoe principales los constituyen: -
(i) la instalación de una red básica de estaciones meteorológicas e
hidrolbgicas, (ii) la recolecciön, proceso y publicación de los da-
tos, (iii) el adiestramiento del personal ya sea con becas y cursos
o mediante publicaciones y manuales tgcnicos, y (iv) el robusteci--
miento institucioaal de los servicios meteorolbgiccs e hidrológicos
en el área. Actividades importantes del Proyecto han sido la prueba
de nuevos equipos utilizados an países desarrollados para estudiar
su adaptación a lae características y clima tropical del area, y el
desarrollo y aplicacids de métodos para la ejecución de estudios mo
teorológicos, hidrol6gicas y de sedimentación con información limi-
tada. Se concluye estimando que su uso en otras áreas COQ condicio-
nes similares puede ser de utilidad.
* Project Manager, Central American Hydrometeorological Project
** Hydrologist Expert., Central American Hydrometeorological Project
*** Hydrometeorological Expert, Central American Hydrometsorological
Pro j ect

36
INTRODUCTION
The Central American Hydrometeorological Project is a joint effort betweenthe
Governments of the Central American Isthmus ( Costa Rica, El Salvador, Guate
mala, Honduras, Nicaragua and Panamá) and the United Nations Development
Programme. The World Meteorological Organization acts as Executing Agency
The Project started in September 1967, at a cost of 9. 2 millions dollars (3.3
millions UNDP and 5.9 millions Governments), which makes this Project one
of the largest in this field. in March 1973 a second phase of the Project was
started devoted mostly to the Coordination and Consolidation of the activities in
Meteorology and Hydrology. This second phase has a duration of three years
and the global contribution of UNDP adds to 1.3 millions dollars.
PROJECT OBJECTIVES
The main objectives of the first phase of the Project, already completed were
the following:
Installation of 290 hydrometric stations in the six countries.
Installation of 830 climatological stations (60 main, 240 secondary and 530
pluviomet ric).
c) Institutional strenghtening of the Meteorological and or Hydrological Services
and the collection, preparation and publication of the data in both the new
and old stations.
d) Training of the personnel, by means of fellowships, seminars, courses,
publications and on-the-job training.
At the end of the project the number of stations constructed surpassed by far the
goals; more than 350 hydrological and more than 950 of all kinds
of meteorological stations were completed. The achievements in the activities
of data processing and publication as in personnel training were most remarkable.
In some countries, meaningful results were obtained in the important task of
institutional building. In others the present condition is not yet adequate for the
needs of their development, but it is expected that during the Second Phase it
will be possible to complete the necessary arrangements for this. For Co-ordinating
at a regional level the activities in Meteorology and Water Resources Investigations,
a Regional Committee was created. This Cornmiittee, formed by
the presidents of the National Coordinating Committees of the six countries, has
proved to be an excellent arrangement and can be considered a good example of
regional Co-or dination.
PRE-PROJECT CONDITIONS
The conditions before the beginning of the project varied widely from country to
country. However, in some countries it was practically inexistent. Although
about 180 hydrometric stations were in operation in 1966, only a few provided
reliable data. Even these had a very short period of records, normally less.
than five years. In some countries the stations consisted only of a staff gauge,
without bridge or cable for flood measurements, in other cases limnigraplis were
installed without a device for checking the river levels, Sediment measurements
were made in only one country and water quality determinations were made. How
ever, the main deficiencies arised from the methods of collecting and processing

the data. The O. 6 depth method of velocity measurement was used in some cases
introducing errors in the stream gauging data. The discharge rating curves were,
generally extrapolated graphically, and no checks were made for the consistency
of the resulting information. Even though most of these defects were recognized
the counterpart pcrsonnel lacked the means and the influence for improving the
situation.
The situation in Meteorology was similar. Even considering that some countries
had fairly well organized cervices, the network was absolutely insufficient for
the needs of the region. Several Services had only one main meteorological
station, in the principal airport. The number of secondary stations in good work
ing standard were less than 20. The rainfall observation network comprised only
the.main inhabited areas, and even there, only a few recording instruments were
available. Some countries showed a complete lack of rain recorders and others
of evaporation stations. The processing of the data was quite rudimentary, and
their publication with a few exceptions, inexistent. Only five meteorologist with
university degree were available, and all five worked in one country. Practically,
no co-ordination between meteorological and hydrological services existed.
TRAINING
The activities of personnel training at all levels were considered fundamental
and received a preferential treatment from the project.
Training in the Region. Training in the region was done with courses --in-
cluding cour s e s by correspondence - -, seminar s, on-the - job training, confer ence s
and publications. Without including on the job training, approximately 500 people
received formal or informal courses. This does not include the personnel
trained by other WMO projects, such as the Chair of Meteorology at the University
of Costa Rica or the Mobile Center for Training of Meteorological Personnel.
Practically all the graduates of these courses are engaged in activities connected
with the Project.
Fellowships. 37 fellowships with a total of 324 men-month were made available
to the Project, for the preparation of new personnel or for the improvement
of the training of the existing ones. These fellowships were a fundameotal
completement for the local training which was devoted mainly to a large number
of low level technicians. Most of the fellows completed successfully their
studies, and some obtained higher degrees in well known Universities. The importance
given to the practical training must be noted; the course for preparing
technicians in meteorological instruments--Buenos Aires-- must be specially
remarked. Unfortunately, some of the fellows ieft their jobs with the Government
sometime after the completion of their studies, which means that their
and UNDP's effort was wasted. However, the percentage of fellows in this
case was relatively low, 13%. in addition, the Project co-operated actively
to obtain fellowships from national and multilateral sources. in such a way,
64 more fellowships were obtained, without including the ones used for the
courses already mentioned. As a consequence, the total of trained personnel
has been significantly higher than the quantity that should have resulted only
from the fellowships assigned to the Project. Even so, the shortage of capable
personnel can still be noticed, specially in the Meteorological Services.
31

38
Publications. The Project considered that one the most effective forms of
training in a dispersed regional project was the intensive use of technical
publications. in general, the reaction to these publications were encouraging.
and resulted in a large demand of them, both from the countries of the area
and from outside. Their main merit aves to the fact that bibliography in
Spanish was made a.vailable to all counterpart levels.
pared, about 100 technical publications and 60 reports had been released, .To
make the diffusion of Project activities more available a by-mounthly newsletter
was edited, Over 500 copies of each issue were printed, making possible for
all members of the Committee to know the activities of the others. The editorial
activity of the Project stimulated also the publications of the counterpart, increasing
their technical reports and data publication. By far the most important
publication of the Project is the "Manual of Instructions". which has been plan-
ned in four volumes.
three chapters: (1) Field measurements and installations, (2) Data processing
and (3) Sediments. Standards are set for the installations, field measurement
and methods for processiqg the information. The second volume is devoted to
IIHydrological Studies" comprising: (1) Verification and Correction of Hydrolo -
gical Records, (2) Extension of Hydrological Records, (3) Duration and Variation
Studies, (4) Hydrometeorological Studies, (5) Floods. (6) Draughts, (7) Hydrological
Forecasts (8) Hydrological Studies for Power Developments (9) Agricuitural
Hydrology (10) Economic Aspects in Hydrology (11) Use of Mechanical Data
Processing. The third volume refers to "Meteorological Observations" and has
been edited only in a preliminary form. The last volume "Ground Water Hydro-
logy" will be prepared -h the future.
When this paper was prg
The first one deals with "Hydrometry" and comprises
The Manual is aimed to the medium level
technicians and includes several numerican examples, with information of the
area. Special emphasis has been given to the specific problems arising from
the lack of long and reliable records. (1) (2).
EQUIPMENT
. The equipment component, formed the major part of UNDP's contribution,
adding to a total of about 1,9 million dollars.
Meteorological Equipment. The main meteorological stations (type A) were
in general provided with universal wind recorder, mercury barometer, microbarograph,
psychrometer, maximum and minimum thermometers. thermohydrograph,
set of geothermometers, Robitzch actinograph, Campbell-Stokers heliograph,
Piche and tank evaporimeter, tank level anemometer, water thermometer,
raingauge and rain recorder (Figure 1).
The ordinary climatological stations were provided with psychrometer, maximum
and minimum thermometers, raingauge, rain recorder and Piche evaporimeter.
in most of the station of this type evaporation tank, pan level anemometer, and
thermohydrograph were installed and in several, anemograph, heliograph, actinograph
and or soil thermometers were also included. (Figure 2). Some stations
included also agrometeorological instruments. such as soil-moisture gauges,
dew recorders, lysimeters, extrasoil thermometers. etc. Several pre-Project
stations were reinstalled and or completed with new instruments.
The meteoro
logical equipment included also six standard barometers, which were included
in the principal station of each country, replacement parts for the period of the
project and for some time after its completion and equipment for inspection and
maintenance. Also included were equipment for part of a regional laboratory
for calibration of meteorological equipment. The equipment provided was, in

general, of good quality, and adequate for the needs of the Project. As far as
possible equipment of complicated operation or maintenance was avoided, preferring
simple and sturdy ones suitable for tropical conditions. In some cases
'defects were detected, but they were satisfactorily corrected by the manufacturers,
by introducing several changes in the design of the instruments.
Hydrological Equipment. The stations installed included the total or part
of the following elements provided by UNDP: limnigraph --of the float type or
bubble gauge type (manometric) --damping pipe, housings, connections and pack
ings of the limnigraph, sets of staff gauges, cables and accessories (Cable caq for the cableway instailation plus a reasonable quantity of spare parts.
portant to note the fact that the standardized prefabrication of the construction
elements, especially the cableway towers, which were designed for 3, 6 and 9
meters height, allowed a simplification of the construction of the stations, making
easier its transportation and mounting;a fact that was of fundamental importance
to reach isolated and difficult zones (Figure 3). The equipment was designed in
order to ensure a maximum of safety during construction and operation, providing
the cable cars with safety brakes, the towers with stairways protected with
safety rings, etc.. . In addition the publication of standards of construction and
operation aimed to ensure this objective. As a consequence of this, and reversing
the pre-project conditions, serious accidents happened neither during the
constructionnor during the operation of the stations. (Figure 4) The equipment
included a current-meter calibrating tan!!, which was installed at the Universidad
Centroamericana in Managua. Probably due to the careful supervision of
the design and construction of the building, the installations remained undamaged
by the earthquake of December 1972.
instruments for level recordings, three digital limnigraphs were qperated expe-
rimentally for some years. The results of their operation was, in general, LUISA
tisfactory, because of extreme humidity and lack 'of adequate maintenance. This lias
proved that the selection of mechanical equipment was a wise one, and that the
gradual introduction of digital equipment should wait for more development to
solve the observed defects and to allow training of specialized personnel.
39
It is im-
in order to gain experience with modern
Equipment for flow and Sediment Measurement. The Hydrological services
were provided with flow meters, counterweights, winches and cranes, etc. to
ensure the adequate operation of the hydrometric network. Selecting the type of
current meters was also subject of detailed studies, and it was decided to use
both the axial and the Price current meter. after a consideration of their relative
merits. The manual of Instructions (1) of the Project contains instructions regarding
the criteria to be used in selecting one or other instrument. in addition,
measurements made in Costa Rica and El Salvador proved that the difference
between the measurements made with both kinds of current meters is very small.
The Project started and intensive programme of sediment sampling, for which
the acquisition of standardized D49 and DH-48 samplers has been fundamental
for the successful achievement of this goal. In addition the Project provided
construction,laboratory, navigation and transportation equipment.
Data Processing Equipment. This comprises fundamentally two groups: e-
lectronic calculators and peripherical comnutation equipment. The first group
cornprises conventional and programmable calculators, which have been used
preferentialy in the computation of streamflow measurements, hydrograms and

discharge rating curves. The second group includes mainly card perforators
for imput to conventional electronic computers. This equipment will be used as
a base for the future data processing centers planned in the second stage of the
Project.
DESIGN AND CONSTRUCTION OF THE NETWORK
The design of the climatological network was based upon the following crite
ria:
a. The first priority for the main mateorological stations was given to the irriplementation
of the basic synoptic network, which had been planned before the
Project. The remaining main stations were located in intermediate points,
trying to obtain a relative uniform density and a good representation of the different
climates of the area. Preference was given to installation in the main
airports.
b. When possible, an ordinary station was installed in each mayor agricultural
area. In isolated valleys with characteristic microclimates, an effort was made
to asign a station to each of them.
c. To obtain the necessary interrelation between the hydrological and the me- teorological network, at least an ordinary st,ation was assigned to each mayor basin
or sub-basin with co-ordinated operation of the meteorological and hydrdogical
networks.
d. in scarcely populated areas the main consideration was the availability of
observers.
e. Finally, the availability of air, land or water access was a limiting factor
in some jungle, mountainous or isolated areas. The pluviometric network was
planned following the recommendations of the Guide for Hidrometeorological
Practices of WO, with the limitations imposed iy the lack of observers and
the inaccessibility of some regions. Regarding the design of the hydrological
stations, these were located in the following places:
i Near the mouth of the principal rivers and or their main tributaries. ii. in
each main lake. iii. At the outlet of each main lake. iv. Where dams of major
hydraulic works were planned. v. At the entrance of a river to a mayor
valley.
vi. At the crossing of a major river of an international boundary. in
addition, some stations were located in urban and minor basins, based mainly
on utility criteria or, in some cases, for use as representative basins. it
was planned to make sediment measurements in part of the network mainly at
the stations listed under i and iv. The complete plan was co-ordinated at a
regional level and approved by the Regional Committees (3).
The impact of the
Project in the meteorological network coverage can be appreciated in Figure 5
which shows the situation before and after of the Project. A similar compari-
son has been made for the hydrological network in Figure 6.
Sediment Measurements. (4) One of the subjects of main interest for the
Project was measurement of the sediment loads of the rivers, since --with the
exception of Costa Rica-- practically no information was available at the begin.
ning of the Project.
At present., systematic samplings are made in 136 of the
gauging stations in the area. Samplings are made in accordance with the usual
techniques and are later analized in the laboratories established in the six

countries to derive the sediment load. When access problems limit the number
of measurements, some local observers take a point daily sample it was found
that in most of the cases the concentration of this sample correlate well with the
average of the compusite samples. The use of the sedbent rating curve, relating
the solid discharge (G) with the liquid discharge (Q) has been used for completing
the records. Figure 7 shows one typical sediment rating curve and
Figure 8 summarizes some of the first results obtained by the project.
The bed load is computed using several of the usual formulas, and several examples
have been published in arder ta explain the procedure to the counterpart
technicians (12). An interesting result concerning the sediment rating curve is
that the coefficient of the equation G = A an, varies between 1.4 ad 4. O. The
lower values of n (1.4 to 2. O) are associated with rivers crossing arid areas,
and the .value of n in general increases as the rainfall also increases. A theory
for explaining this has been developed by the project (20) and will be the object
of further publications.
'
Water Quality. Although this objective was not originally though of, the
Project has started a minimum programme of measurements of water quality.
At present systematic samplings are made in only 48 stations of Costa Rica,
El Salvador and Guatemala, but it is expected that in the future this programme
will be expanded.
STUDIES AND APPLIED RESEARCH
Most of the problems in hydrology and meteorology in Central America
arise from the lack of appropriate information, therefore this subject falls
directly in the main theme of this Seminar. Although in the area of the Project
a few, very few, meteorological stations existed with information up to the begin
ning of the century, this fact did not help much in the evaluation of water re-
sources and much less for the feasibility studies.
the Project provides the necessary coverage so in most of the cases the problem
is nowof "insufficient data" and not of complete "lack of Information1I. In Central
America it is now possible to undertake the study of the potential resources of a
basin or for estimating the maximum design flow, even considering that the
stations giving an adequate coverage have only two or three years record. With
this information, a model of the weather responsible for the major floods can be
prepared.
model which CaA%e transposed in time to the most intensive storms, knowing
only data at a few rainfall stations and very iimited hydrological information; the
maximum historical gauge levels par example. The above mentioned method is
now being used for the design flood of a large hydroelectrical dam in southern
Costa Rica, and will be published in a future report of the Project.
wind, present weather, meteorological phenomena, temperature and humidit y
obtained at two possible sites for the new airport for Tegucigalpa for short
periods of observation, have established the need of further information for a
meaninful decision. in this case the lack of information on cloud cover and vi-
sibility made impossible a decision as in the previous case. Therefore, it can
be seen that the problems of evaluation with insufficient data differ substantially
from one case to another, and it is impossible to propose fixed solution methods.
The first case shows how the action of the Central American Hydrometeorological
Project has made possible the evaluation of water resources with insufficient in-
formation by means of a closed and co-ordinated work between the meteorologist
41
The network established by
Based in this weather model it is possible to develop an isohyetical
Data on

42
and the hydrologist. in the Symposium, it would be important to recognize this
fact. Special enphasis has to be placed in the fact that in the area of evaluation
of natural resources with limited information, the problems will be solved best
wtth a close collaboration between hydrologists and meteorologists, since it is
impossible to separate the aerial and terrestial phase of the hydrologic cycle.
The lack of hydrological information can be compensated with meteorological
information and viceversa. "Elastic relations" which allow to extrapolate the
few observed data, based on some knowledge of the mechanics of the phenomena,
should be used as far as possible. The fact that the hydrometric data are
based on pluviometric information must not be forgotten, since it provides the
most effective tool for the evaluation of water resources.
The scope of this
paper makes impossible to detail all the studies of the Project. A list of some
of them of which most were published is the following: - Studies for determining
water requirements for irrigation (5) (6) (7) (8). - Studies on runoff forecasting
(9) (10). (Already being used for forecasting the operation of several reservoirs
in the area). Effect of the eruptions of the Irazú Volcano on the sediment discharge
of the Reventazón River (11) (12). - Sediment computations, specially
bed load, for several projects.
for several projects. -
- Assistance for the computation of design flood
Development of methods for estimating floods in the
area. Figure 9 shows some flood envelopes for all the Central American area.
Figure 1 O shows some rainfall envelopes for the area (1 3) (14). -Groundwater
studies with the analog computer were made for the Project at El Salvador (1 5).
- - Water balance studies (16) (17) (18). Figure 11 shows schematically the results
of a preliminary study for all the Central American area. Effect of the
temperature on the sediment load (19) Figure 12 summarizes the result of this
study.
STUDIES WITH INADEQUATE DATA
The inadequacies of data arise from (i) incorrect data and (ii) short or insuf
ficient records. Although coping with this is one of the tasks of the Second
Phase of the Project, efforts for correcting and extending the available data have
been made up to now. The Manual ofInstructions of the Project (2) details the
techniques suggested for this.
Double mass curves are used for a first check of the quality of the data.
When errors are found in the hydrological records they are generally due to incorrect
extrapolation of the stage-discharge curve. Jn this case several methods
for determining this curve are proposed, some based in hydraulic relations and
other in the hydrological balance of the basis.
The filling or extension of these records is made either using simple or mgl
tiple corelation andfor estimating the runoff based in the meteorological data
and basic characteristics.
Up to now, the checking and extension of meteorological and hydrological
records has been made following specific needs, but it is planned to undertake
thin task in a co-ordinated and comprehensive form for all Central American
Isthmus during the Second Pahse of the Project.
DATA PROCESSING AND PUBLICATION
One of the main Project activities has been to ensure the prompt and adequate

processing of the information. This has been achieved, both in meteorology and
hydrology, by means of modern systems based in the use of electronic computers.
Meteorology. The data collected at the stations are directly written in the
computer entrance forms, except where, due to limitations of the observer, this
has to be done in the central office of the meteorological services. The detail of
the forms and instructions for filling them are indicated in Publication No 84 of
the Project (20). The results of reading the graphs of the recording instruments
are also filed on the form. At this stage, the adjustment of the graphs by cornpa
rison with the direct reading instruments has to be made. Finally, before
punching these data on IBM cards, the consistency of the data is checked. This
system allowed the publication of the first meteorological yearbook (21) using
services of a rented computer. In the future, this system will be changed for
one that.wil1 requiere a minimum of services of commercial firms. Plans for
mechanizing the reading of bands and for preparing some secondary processing
are also being taken into account. Each country will prepare its part of the
yearbook on uniform format, so that the preparation of a regional yearbook wili
consist of joining the national parts only.
Hydrology. The action of the Project has allowed the standarization of data
processing, following the usual recommendations in this kind of work Therefore
it is now possible to ensure the reliability of most of the records that are
published. At the same time the deficiencies of the previous data are now evident.
Therefore, the revision of these old data constitutes a fundamental activity
of the second phase of the Project. The Project has proposed a complete niechanized
processing, as indicated in the instructions (2) (22), but for the lack of
computing facilities this objective could be achieved only partially. In practice,
the computation of stream gauging is made mechanically, either by means of
programmable calculators or by conventional computers. The use of small programmable
calculâtors or mini-computers will be extended to the second phase
of the Project. The translation of the graphs of the limnigraphs has been made
up to now by manually, but the rest of the process from there on is more or less
mechanized up to the tables for publication. Mechanization of all this process
is contemplated in the second phase of the Pr'oject. The rest of the processes,
i. e. : rating curves, sediment computations, duration curves, etc.. . , is made
manually or with the use of the few programmable calculators provided up to
date, but the trend is towards to a complete mechanizations of these computations.
The Project has published four regional yearbooks (23). of which the last three
have been prepared with the help of electronic computers. These publications
have received excellent comments by the users of the information. The yearbooks
contain in addition to streamflow records, lake levels, sediment discharges,
water quality, duration curves and flood envelopes.
OUTLOOKFORTHEFUTURE
The impact of the Project on the meteorological and hydrological activities
in the Central American Isthmus has been impressive not only in the amount of
available information, but in the increase of the public concern with the importance
*
of these. The second phase of the project is aimed maidy to completing the
institutional strenghtening necessary to ensure the continuity of the activities re -
quired for providing the basic information needed for the social and economical
43

44
development of the Central American Isthmus. The Project will have at that
time prepared the local Services for providing all necessary information for pr2
ject design Where this information is insufficient, tools will be available for
mbking a reasonable good estimate which will avoid delaying the implementation
of, the Project. When the information is inexistent, criteria for obtaining a mini
mum set of data will be well known to the local technicians. Finally, the mete2
rological and hydrological services will be in a good position for influencing national
polices on natural resources, ensuring a rational and efficient use of them.
1.
2.
3.
4.
5.
6.
7.
8.
9.
1 o.
11.
12.
13.
14.
-
REFERENCES
PHCA. Manual de Instrucciones; Hidrometría, Publicación No 49
PHCA. Manual de Instrucciones; Estaciones Meteorológicas, Publication
No 70.
-
PHCA. Programa Regional de Instalaciones; Publication No 20
- PHCA. Medida de
Publication N"79.
Sedimento s en Algunos Ríos del Istmo Centroamericano
PHCA. El cálculo de los requerimientos de agua en Costa Rica.
tion No 39.
Publica-
Hargreaves, G. Requerimientos de Irrigación y Balance de Agua; Proyec-
to propuesto Arenal, Costa Rica, Publication No 87 del PHCA.
Hargreaves, G. Necesidades y Requerimientos para Irrigación; Comayagua
y Vecindades, Hondruas. Publication No 86 del PHCA.
Hargreaves, G. Deficiencias de Agua en Centroamérica y Panamá. Publication
No 88 del PHCA.
- PHCA. Previsiones de Escorrentía. Publication N"46.
PHCA. Pronósticos Hidrológicos para la Operación de Plantas Hidroeléctr-
'=&tas del Seminario de Managua) Publication N091.
Basso, E. Sediment measurements in several rivers of the Central Ameri-
can Isthmus. Fall meeting of the American Geophysical Unnion, San Fran-
cisco 1971.
- Se dimento en algunos rfos del istmo Centroamericano,
PHCA. Medidas de
Publication No 79.
Basso, E. Some Methods for Estimation of Floods with Limited Information
in One Tropical Area. Second international Hydrology Symposium Fort
Collins, Colorado 1972.
-
PHCA. Envolvente de Precipitaciones en el Istmo Centroamericano, Publication
No 81.

15. PHCA. Factibilidad del Riego con pozos en el Proyecto Usulután
dor, Publication No 25.
16. PHCA. Estimación Preliminar del Balance de Aguas en el Istmo Centroa-
mericano; Publication No 18.
45
El Salva
17. Alghren, L. ; Basso, E; Jovel R. Preliminary Evaluation of the Water
Balance in the Central American Isthmus. Symposium on the Water Balance
in North America; Banff 1970.
18. PHCA. Estimación preliminar gel Balance de Agdas del Lago de Managua.
Publication No 7 5.
19. PHCA. Efecto de la Temperatura en el Transporte de Sedimentos. Publi-
cation No 6 1.
20. PHCA. Curva de Descarga de Sedimentos. Publication No 8.

46
Figure 1
Main Met e or olog ica 1 Stat i on.

Figure 2
O r dina ry Meteor olog ical Sta tion.
A_....
47

48
Figure 3
Typical Hydrometric Installation.

Figure 4
Prefabricated Cableway Tower.
49

50
Figure 5
Meteorological coverage, before and after
the Project.

Figure 6
Hydrological coverage after the PFoject.
51

52
Figure 7
Sediment rating curve.

AVERAGE ANNUAL PREC/P/TAT/ON MM
Figure 8
Results of the Sediment measurements
53

54
Figure 9
Flood Envelopes.

Figure 10
Maximum Rainfall Envelopes.
55

56
Figure 11
Water Balance in Central America

StZE OF PARTlCLES MM
Figure 12
Effect of Temperature in sediment transportation.
57

ABS TRACT
METHODOLOGY EXISTING FOR ESTIMATING
FREE SURFACE WATER EVAPORATION
by
Francisco Cubas Granado
The purpose of this paper is to recount the metodology
for estimating free surface water evaporation and particulary
in the case of a reservoir when studing the regulation curves
theceof or the regulation-exploitation system, for estatistics
and empirical methods.
RESUMEN
El objetivo de este artículo es recopilar los distintos
métodos para estimar la evaporación en lámina libre y particu-
larmente en el caso de un embalse en función de la regulación
que efectue y del sistema regulación-explotación utilizando m5
todos empíricos y estadlsticos.

60
Al1 water returning to the atmospherd due solely to evaporation
procosoen is an important element in the hydrologic cyole. Moreover,
it io a limitin;? factor for the effecient utilisation o9 free surface
mtor (reservoira, lnken, rivers, etc.).
In vie# of its big influonce in the water cycle, wvapqration has
beon the subject of innumerable surveys which, because of the diversity
of sndc pursued in enmh one thereof, have not given rise to R homopmaour
theory that could be accepted unanirnoualy.
pronent papar io to recount the methoùology exinting for estimating
I'rso rJurfaoo MttCr evaporo.tion.
The sole purpose of this
l~nrticularly in the oose of a reßervoir,
vhen studgin,? the re$Tlation curve8 thereof or the re~lation-exploitation
riyntern, formulan are required which nay enable the mapomtion ocourring
to be cotimnted ox evaluated when it occurs on J. large sc¿ile or must
lie taken into account for r.orkinL; out these calculations.
oithcr empirical or on n phgBica.1 basis, may often mitipte the lack of
"in nitu" data.
1.2. %ctoro affectinp: the atmomhere's evaporating Dower
Suah formulas,
The a.tmoephore's evaporating power in tho avapoxation rate, ex-
prossed in millimetres of water, for tho period determined (mm. of
uistor per day, for example).
The atmoaphere'o evaporating povier faotors are: the hypometrio
deficit, temperaature of the water, temepraturo of the air, insolation,
fiperd and turbulence of the wind, barornotrio pressure, the quality Of
Lhc irntar and u1ti tude.
In fact, moct of these po.ra,metere are corelatecl to each other and
th@ nractica.1 formulae used Tor eva.luntinf: evaporation gfily use8 those
:mmmnotera which ame the moot important or easiest to measure.
1.3. b'actorr, afrectinn wa.-aorcr.tion of Tree viater surfacea.
¡?ret? wa.tc,r r:iirfn.cr rv&porntion, ,Tiven that th'e atmosphere's
~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
in aliallow, t!ir ternpcrn.tnre of thp whole eanilg follovrs the law of
thermal variation as np1)lied to itr, surfo.ce.
Iprne imter nurfri.ce evaporation in all the less in hot seaaona m a
pentrr in cold venther, the bi,:ger in area and in 'depth the water
napne ie.
It can bo considered that, ba.nically, free surface water evaporation
dc:)enda
-
on:
'ih enorgy availabla from uo1n.r rr.diation.
- The perceptible heat traiiomitted throunh the air.
- 'l'ho airari cnpncity to trianoport imter vapour.
Tlir diff~rent methods propozcd for estimating waporation may be
poiipod into tiro ccitr:;orieo:
EL) Empirioal methods giving rise to Yormulae baaed, mostly, on
BEI lton,a law with modi í.'icationo to Che fmtors affeating evaporntion.
b) VathodB with n rational basis of physicaï theories that may ha
aummrd up ant
- Ilcthotln baned on vater evaluation, consisting in performing
EL water input and output LaLance with evaporation being
calculated as an unknown in the balnnoe equation.
- ilethocin harad on the evnluation of energy where the bnlence
mnde io a,n energy enterinc and leaving brzlanoe. The cslculation
of evaporation io aimilar to the foregoing g~oup.
- Methodo bnaad on the rnnßri trnnuaort theory vihere cvaporation
in evaluated from the wind cpeed and the vapour preoouro
61
,ymdient between lhe surface i!ntPr and the supwinounbent
lnyerc of air.
The irater or cner.3 bo.l:i.nce methods are theoretically suitable for u88
in cn1culntin:y evaporation in 1a.k~~ and reservoirs. Nevertheless, it is

62
difficult to a.pnly tham in prnotice because of the error committed
in msaaurin(: some terms in the balance.
Nore rocent renenrch hno drmonstratod that over relatively long
prriodo, >t Icaat one month, the potential ovapo-transp'iration is
conota,nt Pnd only depends on climatic factors. This has led the
researchers to Geek empiric formulae depending on these factors.
In pmrcr!.l, empiric rormnlae Iinvc. been nought after by oo-relating
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
-
The temperature of tho air
Incident SOIR.T ro.dia,tion
- Air humidity
- A combination of tho foresing
Howover, many of these îormulno hnvo to be checked in practice
b(?îore using them on uurfnces or areas which are not thoee where
they were first obtained. Their contrast is obtained by oelibrating them
by actual measurements of evapora.tion on the basis of ihatruments al-
ren4dy exintin,? (rafte, tanice, evmorimeters, ato.). Xn fact, the only
prccodure fox direot meanurement of evaporation lies in solving a water
bn lance.
Let uö romnmber that:
. The hycrornetric deficit or atmosphere saturation deficit,
obtained ao n difference between saturating vapour tenaioh Fe lo the
irater nurfme tompera,turr T and the actual vapour tension pa in the
ri.mbisnt air, ir, the main ;,pa.rameter of the atmoophere avnpomting QOWBF.
- 'The hyqrometric condition or r3eiFee E of thc air
rcTrm5.n:: to tho viater nurfoce temperature T is the quotient betmen the

the tencione 1%. 8.nd Pe ( f.
= Fan/fe) and represents the relative
humidity or -tha air.
- The psiahrnmetric diffarence, obta.ined an a difference
b

64
it in not p017::ii~i.e to expect a,nythin!y rno:t*e Lhm an a.pproririo,tioii froin
this typo OP rrtimnte, a.ü IJC? have u.lreedy mid.
The rooults obta.ir.eci in the evapora.tion tanka must be multiplied
by tho trin!c coefficient, which will be peculiar to ench type, in order
to oritir!ia.-tc! the actual evawration.
ïri addition, the pa.rn.meter va.luec wc h3.w to consider are those
exia Lin:: in the air-water surface intoqhase which
to rdiwniire wit1 iro Iin,vo to observo them at the inost ncceosibla points where
i.1; il: niipponcd that their values Co-rrlíite well with those which would
h:i.vr beon obt:iiiiccl in tho micl intarphase.
2.3.1.-n0s forrnula
are generally impoccible
In 1802, I):i.lton deduced that , just likc! other parameters, evaporation
on a. free mter surface is proportioria.1 to the hygrometric deficit.
hi2 cvo.por:i t ion forrnula:
E = (Pc? - Fa) = o( (Fe - Fa.), depending on the hygrometric
or, what ir: tho ao.rne8
II
deficit
= d :pe (i-€), drprndin,y on tho saturatinC;
vapour t enci i on and hjr$romet ri c
de I;r ea,
IIence,

In thc Pirat expression of Uaiton’s forinuls, II represents the total
:)resnurc! (,pm plus water vapour) above the evaporatine ourface. H’s
inl.‘lirence only intervenes ac a corrective term in evaporation problem0
riml m;ry br tiiccanxlsd in R. rirst approximation.
‘i’hc cocPPioicnt 4 in cha.racteri::tic of the meterological station under
o o iin i rl erat i on.
‘J!hi:i formula civon very vRria.ble evo.poration valuos Prom one plme to
i’i.notJior, uhioh limite iti; une.
in irhiolri:
- i3 i: the water evaporated in o. month of n days.
- Po (in mrn. of r.lg.) ir: tho nvera,p saturz,tin,s water vapour
tension nt temnoraturs T. This in obtained from hygrometric
tnbles.
- Fn. (in min. of JIg) in the actual a.verage monthly water vapour
ttrnoion of‘ tho fiar nt the tima of tho T readings. Thin ia
obtained by multiplying Fe by the hypometrio deGree.
- I3 (in min. o.€ !!go) in the Lx1,roniotric axerage monthly pressure.
- T (in QG) ia the a,vora.ge monthly tomporature.
65

66
vhore:
- E (in mrn.) io tho evn7oration in 24 houm.
- U, the :i.riiìnesn. 'Phi:: is calculated by tha equation D=lO@-humidity
at u ntmocphereo.
- V (in miles/hour) io the :i,vera,ye wincl opeed over 24 hours.
- 'I' (in QB), .the ::.vcrnp temperature over 24 hours.
3) Tiorton's eqiiation:
i - p + r Y - i
W-h
with P i o for n inrater ahcet irith 2. crflall ßurfaoe.
4) Rohwer' a equation:
E=O.7'(1 (1./16s-0.01:36 n1.Y. (Fda)

D = averaee barometric pressure.
C numerioal coefficient
1% = prenent vapour pressure
Be sotimating vapour pressure
h m lblative humidity of tho air
P = fraction of time during which the wind is turbulent.
t = nurnber cf days.
Ta = average tempwature cf the air
Tw = average temporature of surfaoe water.
N P monthly wind speed average
'y wind factor,
&3.6. ûbnarvation
The dií'ïiculty in applying these formulae lies in that the
mnjoiity of the vnrisbles appear as nn average vsluo and it is possible
io2 their valueci not to represent their total VQlUe well.
'1.1. Critime on the mothods
Tho methods based on the enerm balance enter more into the
field of research than in that of praoti-1 usage.
contrant or chock purposes.
or tima which are nufficiently long.
67
Yhey can be usied for
They can give evaporation Values over periods
Tho methodo based on the water balonce or OR ,th0 mass twnspcrt
ihoory 1i.m likawins mur@ suitable for uoinc 81 ohecke than fa2 graatioal
utla~a. ThePie are, however, more recomendable for invsetigating svapomticn
over ohort periorln of timo ( a few hours).
'Pheee nothaaa also always hitve to be chaoked &na contrasted, as
hnppene with tho ampirio mothods, on the basis of direot eveporntion mean-
iiromonto, bccnuee without these contTaStB and the modifications resulting
thorofrom, oounte-active resulto may be given.

68
In view oi the faet thRt in the imter or energy balance
rn(>thods it is difficult to measure the terms appearing for evaporation
ctiidy on free inter eurfaccs, the use of methods based on the mass
transport theory is more otxongly recomnendad.
3.2. :mos trmmort methods
In masE tmnsport methods, the value of evaporation is entimated
ïroin the wind opeod aiid the vapour pressure ,?radient between the surface
water cind the laysra of air, bu wing a formula of the typez
R = ri fl (u) f 2 (ao - 1%)
iilioro II ir. the pronortionality constant commonly known as the 'ImaSB
tmmiport coePí'iciontlt, 21 and f2 nre knovm functions of the wind apeed
anti vapour prop nure {yadi snt res? cct ively.
The í'o.re{:oin,y formu1t-c ir: rrritten, in its more uma1 forrnt
iihioh enn.bleii E to be calculnted if i:e know the value of H beforehand.
3.7.1. Cp.lcula.tin,s !I
N'E: value in obtained in two waynr
1) Dy catirnntiny: the evaporation by other methods and dividing
i I; by ths product U.(Fe-Fa)
2) Dy obtaining n linear c.clyreocion arqmtion betwnn the chango
[)I' tlio r:tnto OP the wa.tcrl\lI md. thr product U(l%-%), in the following
i.!ayg
A 11 fl.IT. (F'c?-~) 5 C
iiherr bhc coii:;to,rit C giver. the avcrn,Te loss from filtration in the free
1I:l:LPr ::iii..l'n.cc.

69
Thr ficrr t pro.-,edure for i”ind,inf; the mass transport coefficient
mtriiirec n. qrncise water balance crhich forces exact measuring o€ the input
and output or the surface wa.t,or unùer study to be ca,rried out.
not; bc prn.ctica.l in till CRDBB.
This may
The nocond. procedure in cntisfmtory if the losses not due to
rva.pora,tion do not vary to any ,great extent or are not the most part of
tho mtnr loet from thc surface in question.
It ia interesting to look into the aeroAynarnic formula, here,
that cnnbloo T7 to bc calouliited RB n function of the shape (perimeter) and
six0 (nroa) of the free irater nurface, i+moncFt other variables.
3.%.2. The rnntooroloaianl .station rrcruirod Eor viorking out the
a ero dynnmi c met hod
The fo11owin:y rnateriii.1 iu necesnsry for the typical station:
- A Ch:::: A raft on the edge of the water surface.
- ‘l‘wo CUD type wind Buceo
- %io limnigrnphs
- A water tempersture recordar.
- A hy,irothermograph and a pyrmometro.
- Several plqvioTra.phs cet out all around the perimeter of the
ourfnco in question.
J.?. 3. kiorkinrt out the evanore,tion esuation
Ilvnpomtion ia, with rajytrd to i’aorkinc out the formula mentioned
in 3.2.1., ri. diffunion proceso in which the water vapour is transported
froin the tinter nurface to thn Fi.tmocnherc.
Vrrtichl trn.nni)orta.tlon of tho vo.nour depends on thn effective-
naon of the tur.hiilent mixture in the lor:er layers of air, r.nd the main
inl.‘liisnce tharcin in tho wind npcred and ro1ir;hnaon OP the surfa.ce. A turb
ii1an.t coefficinnt ban be found which varies rrith the vind cpeeù Por each

70
tlotcrininotl niirfnco, ir1 thc caso whcrrby the narodynarnio characteristics of
tlic 1::tter romain conntant.
The tramsport of the vapour takes place under the vapour
~)recsure {:radient cet up between thc vapour saturation pressure' at the
curlno

) It iu accci>tcri thnt the laminar sublnyer ia of a negligible
thicknsnn nntl thnt tho turbulent boundary layer extondB below the
wrfaca water.
c) The riincl cpeed profile is given by the law:
m
u(z) = aZ
tihere t h conntc?.nto 2 and fi depend on the etability nnd roghness of the
: : iirf ace.
Combini'iig the equations given for E ana T, we havez
From $ho wind profile i: found:
irliioh a.lloi~n us to irrite:
m-1
dz. E. z
Integrating for
E = <
u2
Throou,yh a,nalap;y trith the fluid flow through & unifom tube,
it c m be shotrn thnt:
(2m+l) (di)
O,:! Q.2 0.4, 1,8
U
nliich civer: rice to tho followinc expreesion for E:
71

72
where: - -
Tho a.vern.go speed value U is equa.1 to the product
(K2. IJ2)
- K1 and 1 2 aro numerical constants.
- V ic the kinetic viscosity.
- e,
X in riven by X with A and P being the area and the
perimeter of the surface under study.
139 expresein,y the aaecific humidities as a function of the
vapour t en0 iam, the evnpora.tion equation becomes t
if 6 is tho thickncco of the turbulent layer, it is easy to show that:
and thon:
ii,o-rotlyrinmic rnet hod.
Kl=m+l;K2= 1 .
m 4 3
Tho ciluationn (1) ; ml (7’) aro the formuin proposed by thio
In order to facililate the calculation of evaporation in small
. ux’hcrs, lhe Col loirinl: hypoihcocs can Uc rnnde:

3.2.5. Conclusiono
a) Give the wind profilo exponent the valus
1
m=s
b) Take a, value of 6 metren a6 the thickness of the turbulent
boundary layerr
Thon, tho mass tranoport coeffioient ie given by2
-4 p
N = (2.62 x 10 ) (A)
o1 2
Thena fortnulno ara vary useful when, st al1 times, tho area
nnd perimeter of the \ranter surface under survey are known. They are very
important, then, for applying to tho study of the evaporation change that
would OCOUT in n re~ervoir in every situation thereof.
In order to oalculate the value of N, th4 values for U, Fe and Fa
obtained at R meteorological station near the zone under survey can be
UCoa.
The inclusion of the perimeter and area in the formula for
cnlcutnting I compenses the variability OS this mass transport cooffioient.
The application of this formula to very irregular ehapad water
ririrfiloao may lead to an excaesiva ca.lculation of evaporation because of
the effect of tho perimeter in the formula. In such caees, to mitigate
.thio exoesn, we c ~ maker n
u. Thort hw&t WIo3 t zmanOo formula
‘Phis expraeoeo evaporation by
(in inches/hour)
73

74
EJ h ere:
- F1 ana B'2 are the vapour pressures (in inchee of Hg) at
nltitudeo hl and hp.
- U1 and U2 are the winü speeds (in m/h) at the said altitudes.
- iri thc nvertrn,Te temperature (inop) of the air between altitudee
hl n.nd h2
3.4. PItnmnn's formula
'Chio haCj tho expresoion:
E I 0.4 (1 4 0.17 U) (Fe - Fa)
whore E io civen in mm/day and the wind veloaity U, at 2 m. height, in
mile5 per hour.

UT i3LIOGRAPlIY
tl;dodon en uso y DU empleo para cdlculo de la eva,potranspira~idn~~,
by Paustino Lonnno Cnrcfa.- February 1964.- Publication no. 23 of the
C.E.11. OP tiio Xiriistry of Public Blorks.
" L'hytl xo loei e d R 1 ' ingeni eur", by C. ii6rn6ni &$as. -Publishad by Eyro 1 l es.
'llI:indbaok of applied i!:;ciro1.ogyW, by Van Te Chon.-Published by Elei.c-Graw Hill.
"Netodos prdcticoo PRM. el estudio hidrologico completo de una cuenca",
by R. fieran.- Published by the C.F.H. of the Ministry of Publio Works.
75

ABSTRACT
GEOHYDROLOGICAL STUDIES IN
SMALL AREAS WITHOUT SYSTEMATIC DATA
Emilio Custodio Gimenan
Frequently are needed studies to profit ground water resources by
means of wells or galleries in areas with non existing data on river
and spring flows and on recharge, but in which injuries may be imposed
on pre-existing water uses. One begins looking for available data in
several kinds of files and inquiring local people. Moreover, the size
of the existing water concessions and their specific use allows the
appraisal of mean and base discharge. The pluviometry is obtained though
the closest stations, and some corrections on judgement. The key problem
is the effective ground water recharge calculation, beeing solved
through the consideration of three independent points of view:
1) modified hydrometeorological balance
2) ground water flow calculation based on existing or estimated data
3) salt balance, specially chloride, based on water - table chemical
analysis and rain water composition
Generally is possible to get coherent results. As an illustration,
three cases are presented:
a) Montroig Area (Tarragona). It is a coastal plain
b) Riera de Carme Basin (Barcelona). It is a limestone formation
c) Famara Massive (Lanzarote, Canary Islands). It is a basaltic
formation in an arid clima
Key words: scarce data, ground water, chemical balance, perameter
estimation, subterranean flow, case histories.
RESUMEN
Con frecuencia deben realizarse estudios para aprovechamiento de -
aguas subterráneas mediante pozos o galerías en zonas en las que exis-
ten datos sobre caudales de ríos y fuentes, ni sobre la recarga, pero
en las que se esperan afecciones a usos ya establecidos. Se procede a
la búsqueda de los posibles datos en los archivos y al interrogatorio
de los habitantes. Por otro lado la importancia de las concesiones --
existentes y su destino permite apreciar los caudales y los caudales -
de base. La pluviometría se interpola a partir de las estaciones más -
próximas efectuando correciones estimativas. El problema clave es el -
cálculo de la recarga eficaz a los acuíferos y se ataca bajo tres pun-
tos de vista:
1) balance hidrometeorológico modificado
2) cálculo del flujo de agua subterránea a partir de datos disponibles
o estimativos
3) balance en sales, en especial en cloruros, a partir de los análisis
del agua freática y de la composición del agua de lluvia.
En general se obtienen resultados coherentes. A título de ilustración
se comentan tres casos prácticos:
1) área de Montroig (Tarragona). Es un llano costero
b) cuenca de la Riera de Carme (Barcelona). Es un macizo calcareo
c) macizo de Famara (Lanzarote, Islas Canarias). Es un macizo basálti-
CO en clima arido.
Palabras clave: datos escasos, agua subterránea, balance químico, esti
mación de parámetros, flujo su,bterráneo, casos reales.
9; Comisaria de Aguas del Pirineo Oriental y Curso Internacional de Hidrología
Subterránea. Barcelona.

78
1. INTRODUCTION -
Frequently, geohydrological studies are made in small basins
where problems of water use exist or are foreseen. To solve
these problems, data is required which has not usually been
compiled or taken down. Usually, there are only a few
pluviometers in the area, and are of dubious reliability; there
are have no measurements of the water courses as they are small
or ephernerous, and the springs or sources have not been
controlled. On the contrary, the exploitations established may
be of a the same order of magnitude of the total available water
resources.
It is not possible to give general working norms, since
there is a very wide range of climatic, geological structural
conditions etc. After setting out some general rules, three
cases will consequently be discussed, showing notable differ-
ences in conditions, discussing the form of operation and the
guarantee of the estimations made.
The main objectives of the work to be carried out may be
summarized as follows:
a) Knowledge of the groundwater flow pattern, including
recharge, circulation and discharge. The identification
of the main aquifers is one of the stages to be covered.
b) Obtain a reasonable hydraulic balance, if possible
coherent with the results of various independent
estimation processes.
c) Analysis of the existing and projected water up-taking
ernphasing the possible interferences between them and
with the water courses and springs, and also, if possible,
obtaining the foreseen user's extraction programme.
It is important to remember that one is obliged to cany ont
these studies should be during certain months along or at the
most within a year; consequently the only data available to
compute the components of the mean hydrological cycle are
those existing at the time. The hydrological data taken during
the study are not mean values, but depend on the climatic
conditions during the study and past actions, and they should
consequently be corrected to obtain a mean or pre-established
situation.
One needs to solve the problem by various channels as
independently as possible. In principle, they may be included
in any of the following three large groups:
a) Hydrometeorological methods.
b) Geohydrochemical methods.
c) Hydrodynamic methods.

Details of these methods will not be discussed in this
paper as their general lines are well known. For further
details, the reader may consult the two volume text:
"Hidrologia Subterránea" coordinated by M. R. Llamas and E.
Custodio, at present being printed by Ediciones Omega, Barce-
lona.
The investigation and special methods are expressly
excluded, since this is not the right place to discuss them,
but the studies to solve the real problems raised and which
require a prompt answer and an order of magnitude of their
confidence. More delicate works can later be set up to find
or affirm the basic estimations and hypothesis.
2.- DATA COMPILATION AND SYNTHESIS
The data compilation and synthesis work is necessary in
any hydrological study, but in small basins with insufficient
data, it assumes peculiar features, since it is frequently
necessary to test all possibilities in various aspects.
First, it should be defined the sort of is necessary data,
to later define the search places where data can be found and
finally establish the methodology of compilation and elaboration.
When discussing the three factors defined in the introduction,
will be specified what data is necessary, if they already exist.
The places where the data can be found vary from one country to
another, and from one place to another and a list of them would
prove very tedious. The official centres officially in charge of
filing and compiling certain types of data, and their publications
should be permanently in mind. The consultation and help of local
experts may prove essential, and also the water-well companies;
furthermore one should not forget the local people as well,
without whose collaboration many important aspects may pass
unnoticed, and even essential data or also some time the main
profited sources and wells.
Except perhaps for very little developed areas, with abundant
water resources, the local people have a noticeable, often
unconcious, knowledge of the local hydrology, of a qualitative
nature, but which may be quantized and built up with adequate
surveys. This method of obtaining data not only saves a lot of
work and time, but perhaps is the only way of obtaining historic
knowledge and erroneous conclusions, by building a logical
structure on not well foundes basis.
A good knowledge of the local idiosyncrasy and people
customs is needed for these tasks, and they should not be given to
under-qualified people who raise suspicions and are not capable
of handling, screening and correcting the information received.
Generally speaking, the local inhabitants are not very
willing in principle to reveal their knowledge, out of fear
79

80
it may prejudice them. The interviewer should be prepared
to “waste time” in winning over their confidence and present
the survey without them noticing it, making notes discreetly.
One should try to get the information to flow out on its
own, just channelling it and loocking for the interesting
details.
It is generally difficult to pass judgement on the data
obtained in this way and it requires a great critical sense,
a good knowledge of the area and a continuous contrasting.
The collaboration of the local inhabitants is most
important in locating springs, bore-holes, wells, etc., and
to establish the most important characteristics of them. On
the other hand, the local corporations and Town Councils are
usually important sources of information.
3.- - OBTAINING THE OBJECTIVES
To obtain the objectives listed in the introduction, it
is frequently necessary to set up a general water balance in
homogeneous part ia1 areas , bearing in mind the limitations
and inevitable errors they contain. One should not only see
an equation between mean values in the word ”balance”, but
also the possible variations in the different values
intervening and their interconnection (SC). This is specially
important when the ground-water reservoir capacities available
are small in relation with the water volumes to be exploited
annually, giving rise to accentuated seasonal effects.
This raises the problem that in one of these small areas
where data is scarce and not very reliable, an elaboration
and definition is necessary, with a depth not common in the
case of large basins.
One of the greatest unknown factors is usually the
infiltration and recharge to the aquifers , which should be
estimated using the best methods available.
4.- -- HYDROMETEOROLOGICAL METHODS
The hydrometerological methods to establish water balance
and define deep infiltration are the classical ones, except
in arid or semi-arid areas, where a daily computation (10)
must be made to avoid excessive errors in monthly data handling.
(*) The autor is conscious of the limitations of the water
balance but feels it is a very useful tool if the person
handling it is aware of its restrictions and errors, and
the variability of the involved magnitudes.

The pluviometry must be obtained through the usually
scant stations available, which generally do not cover the
mountainous parts where the pluviometry is usually greater
than in the lowlands.
A first measure is to correlate the different stations
and complete the series, trying to obtain a definition of
areas with the same rainfall (quantity, distribution and
intensity), making estimated altimetric and topographic
corrections.
Next, the graphs of accumulated deviations of the
pluviometry should be drawn, and these will be the basis of
the study on the springs and water courses discharge and
water-level in the wells. With these relations hips, the
conditions observed during the study will be changed into
mean conditions or those conditions of particular interest
in order to ascertain extent the pluviometric variations
influence the ground waters.
When estimating the surface runoff, the knowledge of
the local people may provide interesting data helping the
morphological appreciations made. Frequently, local people
can tell the heights and frequences of water in the river
beds under various circumstances, and thus draw an initial
scheme of the system. When there are permanent waters is
frequent the presence of manufacturing or irrigation
installations which use them, and in this case they are usually
dimensioned for the base discharge or some figure slightly
higher. A knowledge of this discharge and the user's remarks
are of great importance, as it permits the characteristics
of the surface hydrology to be reconstructed approximately,
based on one or various river flow measurement campaigns in
selected points. The absence of noticeable surface uses by
means of simple derivations, may be a clear sign of temporary
discharges.
Rarely are there homogeneous and well defined crops in
these basins, and frequently there is forest, brush, bare
rock areas and great slopes, and consequently the classic
evapotranspiration estimations are not applicable. Added to
this is the rare availability of the necessary meteorological
data, excepting some thermometric station. Thorntwaite's
method (9) may give an initial idea of the potential value.
Successive balances based on an estimated field capacity,
enable the real evapotranspiration to be calculated by
difference. In low pluviosity areas, with a high evapotrans-
piration capacity, the errors may be very important and a
value of the infiltration plus surface runoff below 10 or 20
per cent of the annual mean pluviometry, may only give a mere
indicative figure. In this case, other balance methods should
be established.
81

82
5. GEOCHEMICAL METHODS
In studies where there is insufficient data, the chernical
characteristics of the ground water may be extremely useful,
if they are correctly interpreted.0ne of the chief advantages
of the geochemical methods lies in the low variability of the
chemical composition of the ground waters, averaging the
annual and seasonal variations, and in the low cost of an
overall indicative analysis if there are sufficient points for
the sampling. The interpretation however, is a delicate affair
and should be made by an experienced person with sufficient
knowledge of the local hydrogeology and geology.
On the one hand, the geohydrochemical methods may help
to establish the paternof the groundwater .flow, comparing the
analysis of various points of water, using graphs (mainly
those of logarithmic vertical columns or Schoeller's;
triangular with three fields, or Piper's; and Stiff's modified
polygonals) (8) and ionic indexes, helped if necessary by dis-
persion diagrams (correlation between two chemical charac-
teristics (8).
From another point of view, the chemical composition of
the ground water may further information on the recharge. For
this, it should be admitted that the aquifer does not notably
modify the salt contents of the infiltered water. To remove
the possible influence of solution and modifying phenomena
such as ionic exchange, redox reaction aggressiveness to
carbonates, precipitation etc., the chloride ion is taken as
reference, which can only be changed by an addition of a new
chloride ion by the aquifer. In alluvial aquifers, limestone,
dolomite, etc., no important additions are expected if for
from the sea.
In this case, all the chloride of the ground water would
come from rain, and therefore we can state: (2) (4) (5) (8):
(P - E) . Ca I . Cs
Where:
lr
P annual mean pluviometry
E = annual mean surface runoff
I = annual mean deep infiltration
Ca mean concentration in rain water chloride
C, = concentration in ground water chloride
It can be easily deduced that:
I (F - E) cs

Some care is required when applying the method. One is
that the activities on ground surface should not modify the
chloride contribution. The method therefore has a dubious
application in intensive crop areas with irrigation or in
areas with disposal and infiltration of important amounts of
direc.t residual waters or though tippers, etc., and also in
immediate coastal areas with direct sea influence.
The Cs value may easily be obtained from the ground
water analysis provided this is stable. If not, the method
should not be applied. The value of Ca is not normally known,
as it is rare to find systematic analysis of the rainwater;
during the study some analysis of this rainwater may be made,
but before taking a content as the mean value, various deter-
minations must be compared, since this content varies each
season according to the origin of the clouds and even within
a same rainfall. During initial estimation attempts, it may
be considered that in areas several dozen km. away from the
sea, the chloride content is generally less than 10 ppm, and
that in areas some hundred kms from the sea, it is less than
1 ppm (8).
In areas near the coast, the variations and contents may
be higher. Close to populated areas, in particular if these
are industrial, high values may also occur.
The soluble salts brought along by the infiltrated water,
may not only come from the rain, but also from the atmospheric
dust and this is another reason for doubt. In principle, the
rainwater collector units should also collect the atmospheric
dust, but at a sufficient height to avoid the local particle
movement at low level.
Experience shows that the method is good in arid areas,
where the rain concentration due to evaporation is high and
the surface runoff is scarce. The result is more problematic
in the more humid areas where the infiltration is an important
fraction of the pluviometry and where the surface runoff is
notable and errors in estimation greatly influence the P - E
value. However, in these areas it is possible to apply the
salt balance in order to separate the components of the
hydrogram of a gauging station, if a sufficient chemical
analysis series is available during a rainfall and the later
period, but the author has no direct experience in such cases
(12) (15).
6. HYDRODYNAMIC METHODS
The hydrodynamic methods try to determine the infiltration
based on the hydraulic characteristics of the aquifer and the
piezometric surface. The most correct way of making the balance
is using a simulation model, but this is generally a detailed
study phase and requires a notable amount of data (13).
83

The methods given herein refer to simple situations within
the study area with a well defined piezometric surface and
with a pattern and slope which scarcely varies throughout the
year, so that an almost stationary situation can be imagined,
with a well differentiated recharge and drainage area. The
method means that the flow of water per unit of transversal
width is equal to the mean recharge upwards. The application
means having the mean transmissivity of the aquifer in the
analysis area obtained by means of some pumping tests and
bore-holes and that the piezometric surface has been observed
in a sufficient number of points to precisely know the mean
gradients. The estimation is a mere application of Darcy's law.
q T. i.
where q = discharge per unit width
T = transmissivity
i = piezometric gradient
Darcy's law is generally valid in most normal circumstances.
7. EXAMPLES
To illustrate the above, three examples have been chosen,
corresponding to studies in areas of less than 100 Km2, one in
a semi-humid area, another in a semi-dry area and the other in
a sub-desert climate. To better locate the data, the example
has been broken down into multiple paragraphs:
7.1 Montroig Area
Location.- S.W. of Tarragona, in the Baix Camp (fig. 1)
Physiographic characteristics.- Flat coastal strip, 4 km
wide, bordered by the mountain range. The water divide line
is 12 km from the sea (1) (3) (5).
Geological characteristics.- Plain of detritic materials
resting on clay formations. Mountain range materials are of
low permeability (1) (11).
Water exploitation.- Traditional use for irrigation. Near
the coast, pumping of 10 m3/year approximately, for supply of
the Vandellos Nuclear Station. New extractions for Tarragona
are ready to start in a short time (3) (5).
Basic problem.- Find out the resources and sea intrusion
process when new well will start pumping.
Existing data.- Scarce, reduced and partial hydro-
meteorological data. Mean pluviometry 400-500 mm in the plain,
higher in the mountain range (1). Almost non-existant hydrological
data; there are no permanent water courses. The existing ones

are short-lived dry creeks. Contributions are estimated be
means of local surveys. Almost non-existant hydrogeological data
prior to the studies for the Vandellós Nuclear Station; later,
values of the transmissivity of the piezometric gradients in a
reduced area were available. The aquifers drain directly into
the sea (1) (5) (11).
Hydrometeorological balance.- Of dubious value owing to
incertitude of data and low infiltration (2).
Geohydrochemical balance.- Good application conditions in
non-irrigable land areas. There is no direct data on the
chloride content of the rain water, but this can be obtained
by comparison with similar areas with data (2).
Hydrodynamic balance.- Ideal conditions for application,
but the interpretation of the pumping tests becomes complex (5).
Results.- The mean recharge obtained by each of the three
methods was as follows, in thousands of m3 per year, per km
of coastline$: ( 2).
a) Hydrometeorological method ....... 600
b) Geohydrochemical method .......... 900
c) Hydrodynamic method .............. 1.100
Method a) foresees a progressive marine intrusion; method
b) foresees a critical situation and method c) a certain
residual flow to the sea, which would stabilize the salt water-
fresh water interfacies in a new position.
Check-ups.- Two lines of piezometers were installed to
control the sea intrusion and three water table elevation
recorders to control the levels were installed. After over
three years exploitation, the interphase movement is more in
accordance with result c) than with the other two. The
hydrometeorological method is excessively pessimistic. The
geohydrochemical method is acceptable in a first approximation,
and the hydrodynamic method is the closest to reality (5).
7.2. Riera de Carme Basin
Location.- S. of the town of Igualada (Barcelona) (fig. 2).
Physiographic characteristics.- It spreads over 100 km2
betwen 250 and 900 m of altitude. The length of the small river
is 25 km.
Geological characteristics.- The materials are clay, silt
and limestone Eocene formations, resting on clay and chalk of
the Keuper formation. The tectonic alteration is important.
There are important travertine and calctuff formations (6).
* In reality, a maximum and a minimum value was calculated.
85

86
Water exploitation.- The discharge of the Riera de Carme
is notably regular. The main ground regulating reservoir are
the alveoline limestones, discharging relatively important
springs. The main spring discharges in Capellades, outside the
basin. There is an intense industrial use.
Basic problem.- Some wells have been chilled from which
100 l/sec., are going to be pumped continuously. Find out
whether it is possible to obtain this discharge in dry seasons
and what type of injuries will be produced to springs and water
courses. The tests have been made in an extraordinarily humid
season, and it is therefore essential to obtain the probable
situation under other conditions.
Existing data.- Various peripherical pluviometric stations,
and only one interior one, at present out of service. Spacial
distribution of the pluviometry relatively regular, around
600 mm/year.
The hydrological data is very scarce. With only one gauging
station operating since 3 years ago. The runoff characteristics
have been reconstructed, based on a inventory, survey of the
canals, seried gauging and comparison with other,basins. The
normal basic discharge of the river is 400 l/sec., which should
rise to 500 ì/sec., if the ground discharge to a nearby basin
is considered. The hydrogeological data are almost non-existant,
except a prolonged pumping test lasting for two months, and '
various tests on bore-holes (6). Various springs have been
regularly gauged and the data has been apparently satisfactorily
completed, by means of local surveys on the field and in
factories.
Hydrometeorological balance.- Not very reliable since most
of the basin has high slopes, with wood or brush, and with
sometimes very permeable materials.
Geohydrochemical balance.-In the main springs area, ground
water has 15 to 24 ppm in Cl-. The scarce rain water samples
show chloride content betwen 5 and 10 ppm. The accuracy is
very low. Direct surface runoff is not known with an adequate
degree of confidence. Data is only indicative. Possibly the
chemical conditions for hydrogram components separation by
means of salt balance discharges are optimum, but has not been
made as the influence of the industrial discharges is not very
well known. .
Hydrodynamic balance.- The important variability of
transmissivity conditions, of the main fractured aquifer and
its complex arrangement, make estimations difficult. The best
way is by a study of discharge recession curves in selected
points.
Results.- Estimation of the total infiltration in millions
of m3/year, including the groundwater discharge outside the
basin (6):
a) Hydrometeorological method ........ 10
b) Geohydrochemical method ........... 18

c) Hydrodynamic method ........ ?
d) Separation of hydrogram
components ................. 15
Check-ups.- No direct check-ups are made, but they will be
obtained after completion of the study wiht the 2-month pumping
test, and related observations.
7.3. Famara M,assive
Location.- N. of the Island of Lanzarote, Canary Islands
(fig. -3).
Physiographic characteristics.- Massive of over 600 m. in
altitude, which forms a notable cliff over the W. coastline.
Spreads over 80 km2. Very scanty vegetation, almost sub-desert
climat e.
Geological charact.eristics.- Tahular hsalts, of more than
1.000 m. thickness, buried cinder cones, very continuous and very
little permeable subhorizontal clay-like levels (almagre].
Water exploitation.- Reserve area of ground waters for
supplying the capital. Collections by means of galleries which
penetrate deep into the massive, with horizontal drills and at
present also some vertical ones, to increase the drainage. Discharge
obtained 2 to 3 l/sec., at present temporarily increased to 20
l/sec. (4)
Basic problem.- Get to know the reserves of the massive and
determine the exploitable discharges, their rate and recession
curves, Assess the possible resources.
Hydrometeorological data.- Sufficient rainfall network, except
in the highest areas, where most of the low infiltration mus-t be
produced. Compiling of data and detailed elaboration by the
Hydrographic Study Centre In Madrid c"]. Mean rainfall below 200 &year.
Hydrological data.- Absence of surface runoff except in
strong storms. There are no direct available.
Hydrogeological data.- Almost non-existent, except in the
galleries where there is a data record and several deep exploration
bore-holes. There are various small springs and oozes, of very
fine or inappreciable discharge, inventoried by the Public Works
Geological Service, and with some chemical analysis.
Hydrometeorological balance.- Of dubious Worth, due to the
highly arid climate and because many suppositions have had to be
made. A daily balance calculates a mean infiltration of 3 mmLyear.
There is possibly no recharge if the daily rainfall does not exceed
20 to 40 mm., which only occurs a few times in a period of several
years.
By reason of the Scientific Study of Water Resources of the Canary
Islands, made by the General Board of Hydraulic Florks of the
Spanish Government, and UNESCO.
a7

88
Hydrogeochemical balance.- Chloride content of the
infiltration water obtained from the analysis of the small water
oozes on the almagres at high altitude (around 300 to 700 ppm.)
and surface wells in Haría (900 ppm). The water of the galleries
is more salty possibly due to basalt contributions by the high
holding time. There are no direct data on chloride content in the
rainwater, but it can be estimated from the data of the island of
Gran Canaria (4).
Hydrodynamic balance.- Made with precautions from the
freatic surface obtained by a careful study of the data on the
bore-holes and galleries, and the hydraulic characteristics of
the basalts obtained by various procedures (4). The hydraulic
gradients at times exceed 10% per cent in a plane at O elevation.
Results.- The rechar e obtained by each of the three
methods in thousands of m 5 /year, are (4):
- Mean
Min.
Max.
-
-
Famara Heights (19 Km2) .... 225 28 5 190
Famara Lows (28 Km2) .... 140 224 84
Marginal plains (25 Km2) . . , 50 75 25
Total (72 Km2) ,. .. 415 584 299
In this case, the best method would appear to be the
geohydrochemical one. There is no direct verification, but
additional information is obtained by means of isotopes and
ambient radioisotopes, apart from a study of the salinity of the
soil and dust, in elaboration,
Co'plementary,- Since the explotation is mainly of
reserves, it has been computed by hidrodynamic study methods of
the recession curves of the gallery discharges, that the water
yield should vary between 0,03 and 0,05. This figures, jointly
with the other data allow to estimate exploitable reserves in the
gallery area, betwen 20 to 60 million m3, The overall transmissi-
vity is 100 m'/day, with a thickness between 200 and 500 m.
8. CONCLUSIONS
In areas with small infiltration in relation to the
pluviometry, the geohydrochemical method applied with
precautions, is a very useful tool which can improve the
hydrometeorological balance method. In more humid areas, the
results are not so clear. The hydrodynamic balance is the best
method but in some cases it needs appropriate conditions for
application, and in any case, it requires numerous reconnoisance
tests to determine the hidrodynamic characteristics of the aquifer.
9. REFERENCES

1. Custodio, E., Molist J., and Martin Arnaiz, M. (1968).
First Report on the works for supply of the Vandellós
Nuclear Station. Geoteclinics Geokgists Consultants.
Barcelona.
2. Custodio, E. (1969). Report on the present state of
the possibilities of the Montroig collections to- supply
water to the Vandellós Nuclear Station (internal report).
3. Custodio, E., Bayo, A., and Orti, F., (1971). Geological,
Hydrogeological and geochemical characteristics of the
coastal aquifers between Cambrils and L'Ametlla de Mar
(Tarrag ona)i. -
on Economic Geology, Madrid-Lisbon. Section 3. pp 1471170.
4. Custodio, E., and Saénz de Oiza, J., (1972) Geohydrological
study on the Famara Massive, Lanzarote. General Board of
Ilydraulic Works. Las Palmas - Barcelona. 2U4 pp.
5,
6.
7.
8.
9.
10.
11.
12.
13.
89
Fi rs t Span i s h - Portuguese -Amer i can Congr es s
Custodio, E., and others (1973). Final Report on the
works to supply the Vandellós Nuclear Station. Geotechnics
Geologist Consultants, Barcelona (being elaborated).
Custodio, E., and others (1973). Geohydrological study of
the Carme Basin. Barcelona. East Pyrenees Water Board
and Public Works Geological Service. Barcelona.
Custodio, E. (1973). Hydraulics of water collections.
Section 9 of Subterranean liydrology. Vol, 1.- Omega
Editorial. Uarcelona (at printers).
Custodio, E. (1973). Geo1iydrocliemistry.- Section 10 of
Sub terruiieanz liydrology , Vol, 2, - Omega Editorial, Barcelona
(at printers),
Martin Arnaiz, M. (1973). Components of the Hydrological
Cycle, Section 6 of SUBterranean Iiydrology, Vol. 1.- Omega
Editorial. Barcelona (at printers).
Mero F. (1969). Approach to daily hydro-meteorological water
balance computations for surface and groundwater basins.-
Proceedings ITC - UNESCO Seminar on Integrated Surveys for
River Basin Development.- Delft. pp 89/116.
Orti , F. (1970). Notes on the hydrogeological prospecting
made for supply of the Vandellós Nuclear Station (Tarragona)
Geological Research Institute of the Provincial Delegation.
Vol. XXIV, pp 75/88 Barcelona.
Pinder, G.F. and Jones, J.F. (1909) Determination of the
ground water component of peak discharge from the chemistry
of total runoff.- Water Resources Research, Vol 5. No 2
April 1969 pp 438/445.
Public Works Geological Service (1972). Basic Tiieory on
analogical digital models of aquifers. (See especially
chapter 5, Process of construction and use of a model
by E. Custodio and L. Lopez-Garcia), Information and
Studies, Bulletin No. 37 - Public Works Geological Service,
Madrid, October 1972. 178 pp.

90
14. VilarÓ, F., Custodio, E., aiid Bruington, A.E., (1970)
Sea Water intrusion and water pollution in the Pirineo
Oriental (Spain). ASCE National Water Kesources Engineering
Meeting, Memphis, Tennesse. Meeting Preprint 112.
15. Visocky, A.P. (1970). Estimating the groundwater con-
tribution to storm runoff by the electrical Conductance
method.- Ground water, Vo. 8. No. 2, March-April 1970.
pp 5/10.

Fig.1 - Situdción del Area de Montroig
Locat i on map of Fdontroi g Area
91

92
Fig. 2 - Sitiiacibri de la Cuenca del Carme
Locat i on of Carme Has i n

Q Al cgranza
93

METHODS OF ANALYSING DEFICIENT DISCHARGE DATA
IN ARID AND SEMI-ARID ZONES FOR THE DESIGN OF SURFACE WATER UTILIZATION
-____ ABSTRACT
bY
Joseph S. Dalinsky
TAHAL - Water Planning for Israel Ltd., Tel Aviv, Israel
Technion, Israel Institute of Technology, Haifa, Israel
This paper surveys various methods of analysing stream flows:
frequency of annual volumes, discharge-volume relationship with
horizontal, vertical and double hydrograph cutting, and calcula-
tion of the storage volumes available as a function of reservoir
capacity.
Application of these methods, which have been successfully
applied by Tahal-Water Planning for Israel Ltd. over the past '
ten years, can generate data for the design of surface water
utilization schemes when flow records are available for only
a few years.
The understanding and application of the general design
aspects, even if only qualitative, enables the planning engineer
to reduce his basic hydrological requirements to less than 10
years duration.
It is proposed that applied hydrological research be di
rected towards evaluation of a number of important hydrologi-
cal design parameters on a regional basis to enable nondimen-
sional curves to be established.
RESUME
L'auteur examine différentes méthodes pour l'analyse du
débit des riviére: fréquence des volumes annuels-relation entre
ces volumes et les débits dérivés pour l'utilisation par tronca
ture'des hydrogrammes, cette troncature pouvant etre verticale,
horizontale, ou les deux à la fois- calcul des volumes stockés
disponibles en fonction de la capacité du réservoir.
Ces méthodes ont été utilisées avec SUCC~S par Tahal-Water
Planning, pour Israël Ltd, au cours des dix derni2res années.
Leur application permet de fournir des donnérs pour l'aménagement
des eaux, lorsqu'on ne dispose de données di'éculement que
pour un'petit nombre d'années.
Une mise en oeuvre intelligenye des aspects généraux d'un
projet, même sous une forme purement qualitative, permet à l'i2
génieur de planification de se contenter, pour les données hy-
drologiques de base, de moins de 10 ans d'observation.
L'auteur propose que la recherche hydrologique appliquée
soit orientée vers l'estimation des parametres hydrologiques -
important à la réalisation des projets. Une telle étude doit
être menée sur une base régionale et déboucher sur l'établisse-
ment d'abaques adimensionnels.

96
INTRODUCTION
The need for water in the arid and semi-arid zones is in most cases
greater than the water resource potential, since there are generally
large areas of good soils that cannot be cultivated as a result of the
scarcity of water for irrigation.
utilization is directed toward maximum exploitation of the lfmited re-
sources at reasonable cost.
Hence, the planning of surface water
The annual yield of surface water resources varies considerably as
a result of the extremely non-uniform climatic conditions that prevail in
arid and semi-arid zones. In many cases, the available source cannot by
itself provide an adequate supply and other solution& must be found,
Possible solutioqs are as follows:
(1) To recharge surface water to suitable groundwater aquifers which
would serve as long-term reservoirs.
will be the varying annual volumes of surface water, in addition
to the natural replenishment, while the output will be an ap-
proximately constant annual rate.
In this case the input
(2) In the case of supply from surface reservoirs fed from intercep-
tion of stream flows, this source can be integrated with some
other certain or steady but limited source. In this case, in
years of adequate flow from the less reliable or variable
source, the yield of the steady source is retained €or use in
those years in which the yield of the variable source is in-
adequate or non-existent.

In such cases, planning should be based on the average flow which
can be diverted and recharged, or stored in the surface reservoir under
dif fereniconditions.
The techniques proposed in the following are aimed at calculating
these average values as a function of planning parameters such as maximum
diverted flow or maximum net capacity of the surface reservoir.
Acquaintance with streamflow regimes is best acquired by study of
hydrographs.
ledge it provides.
although detailed time dependence of discharges (instantaneous discharges)
are the subject of greatest interest; where daily discharges do not under-
go rapid changes (e.g. in rivers, springs, and baseflows), monthly data
may be sufficient.
The more detailed the information, the more exact the ùnow-
Data of hourly or daily flow rates are important,
Hydrograph study provides valuable information on the flow regime
and can lead to diversified techniques of analysis.
The following information on streamflows is essential for the plan-
ning of utilization:
- The average volume of annual flows (U 1, which represent the
ave
stream water resources potential; the average annual feasible
utilizable flows is a portion of this value.
- The stream's flow regime: Is the stream perennial, intermittent,
or ephemeral?
Does it have a significant base flow or only dis-
continuous floods? What is the duration of flow or floods, the
yearly number of floods, and the interval between successive
floods?
97

98
- Streamf low variability, which comprises variability within a season
or a year and variability from one year to another.
Hydrograph analysis can provide the required information s&h as:
monthly and annual flows and their frequencies; flow-duration curves;
annual average flows in relation to diverted discharges - represented by
horizontal and vertical hydrograph cuts; and annual average flows in rela-
tion to possible reservoir capacities.
In many cases the planning has to be done while insufficient hydro-
logical data are available.
the limited data available to be used efficiently and therefore reduce to
a minimum the period of records needed for planning purposes - very often
to less than 10 years, if the period for which records are available can
be taken as representative of climatic conditions.
A. ANNUAL FLOOD BETURN PERIODS
The techniques presented in this paper enable
For practical purposes of surface water utilization planning, annual
flood return periods can be computed by using the established formula:
n + l
TE- m
where: T is the return period (in years);
n is the number of annual flow data;
... (1)
m is the serial number of annual flow data arranged in descending
order, by size.
By using the above formula, return periods approximately equal to
the period for which data are available can be reasonably evaluated.
estimates of annual flows (order of magnitude) for longer return periode
can be obtained by extrapolation on probability paper by using the points
The

which were calculated according to formula (1) as plotting points; though,
for practical purposes, rare annual flows are of little importance, if any,
since in most cases, a project based on rare flows will not be economic,
B. FLOW-DURATION CURVES
Flow-duration curves express the average duration of occurring die-
charges equal to or greater than given values (Q > Q ); or dischargea
i- 1
equal to or smaller than given values (Qi 5 Q21e Schematic representation
of flow-duration curv
is given in Sketch 1.
Sketch 1: Flow-Duration Curve - Schematic Representation
The duration can be expressed as the average number of days per year
on which the said discharges occur, or as the total number of days in n
years (see illustration of flow-duration curves ‘for the Qishon stream in
Fig. 1 in App. A), or as the relative duration (which is similar to
relative frequency ) .
The computed discharges can be hourly or daily averages, or averages
for any period - in accordance with the aims of the analysis and the
nature of the data, In general, average daily discharges will be used
when the daily discharge fluctuations are not appreciable, or wnen the
representative changes are daily changes. For streams characterized by
99
,

100
a flood flow regime - where the flows are of short duration and there is
no significant baseflow - average hourly discharge or averages for even
shorter periods can be chosen. It is customary to express the relative
duration in percentages (p).
The area delimited by Lhe curve Q = f(p), when /Qdp or Z(Q x Ap), is
equal to the average discharge of the stream.
The relations can be easily and economically established when calcu-
lations are made by computer, in many cases as by-product of the computer
analysis of streamflow data. For planning purposes, the direct use of
flow-duration relations or curves is not convenient and their use is limi-
ted to assisting the computation of data needed for drawing
represent the horizontal and vertical hydrograph cuts (as shown in the
following sections of this paper). It should be stressed that: (1) The
flow-duration relations can represent the streamflow character; (2) These
relations can be achieved with satisfactory accuracy in the zone of the
practical importance (the zone where relatively small or medium size dis-
charges occur) using data of a relatively short period (few years, mostly
less than 10 years).
curves which
C. AVERAGE ANIUAL FLOW IN RELATION TO MAXIMUM DIVERTED DISC-GE -
HORIZONTAL CUT
When stream diversion i8 considered, whether by gravity flow or
pumping, the dependence of annual diverted flows on the maximum diverted
discharge is computed using the historical data. The curve representing
the dependence of the average annual diverted flows on the maximum diver-
ted discharges can be considered as the stream's "visiting card".
meaning of the diversion, from the hydraulic aspect, is that all the
The

discharges which are equal to or smaller than a certain magnitude, (Q,),
are being diverted (see Skbtch 2).
the diffeTence [(Q,) - (Qd)max I will overspill, while the diverted dis-
charge, Q, will be approximately constant, at the magnitude of about
(Qd)maxg
Qd i Diversion
discharge
When the discharge (Q,) exceeds (Q,),
Sketch 2: Schematic Layout of Diverdion Sketch 3: Hydrograph Horiaontal Cut
Q
I
From a hydrological point of view this means a "horizontal cut"'of
the streamflow hydrographs (see Sketch 3).
where:
The "horizontai cut" can be expressed mathematically as:
Qi
Qd
(Qä'rnax
is the streamflow discharge;
is the diverted discharge;
is the maximum diverted discharge.
101

102
For every maximum diverted discharge a certain volume can be diverted
every year; for a period of n years - a series of n annual diverted volmes
can be obtained, out of which the average annual diverted flow (Ud) can be
calculated for each value of (Q )
d max'
Sketch 4.
The function cd = f (Qdlmx has the shape illustrated schematically in
When Qd -+ œ, then * U e where U represents the stream poten-
d ave' ave
tia1 (average annual flows).
The curve representing the dependence of Ü on (Q )
d d max
into three m in zones according to the tangent slopes:
Zone I: AÜd/A(Qd)mx is relatively
large and almost constant. The
diversion will be most justified
economically
-
Zone II: AUd/A(Qd)max quickly de-
creases as (Q ) increases.
d max
This is a transition zone.between
Zones I and III.
-
Zone III: AUd/A(Qd),, diminishes
as (Qd)mx increases, and tends
towards zero when (Q ) * m, In
d max
this zone, diversions are usually
not worthwhile o
A
5
CI
v
-
Sketch 4: Ud -
can be divided
I
-H I
Uave
f [(Qd)max]

- Note:
(i) The above relations can be easily obtained from the computer
at small cost.
They can also be calculated without a computer,
in some cases easily (depending on the data).
(ii) If the flow-duration curve is used, expressed by relative
durations (p), and the average nupiber of flow days per year
(ta) are given, then üd = f [ (Qd),] can be calculated by
the formula:
-
- -
where:
t is the average number of flow days per year;
. . . (3)
p is the relative duration (expressed as a fraction) of
discharges equal to or greater than the appropriate Q.
(iii) Althmgh the historical streamflow data will not be repeated
in the future, the calculated diverted volumes represent
fairi,y satisfactorily the amounts and distribution of the
expected volumes for practical purposes of planning.
In planning streamflow utilization by diversion of flows up to a
certain maximum diversion discharge, the above relations, based on a
simple "horizontal cut" of hydrographs, are widely used (see illustration
of horizontal, vertical and double cuts of hydrographs in App. A, Fig. 2).
The maximum diverted discharge is determined according to direct or in-
direct economic considerations (for instance - the limitation of the
diverted discharges in order to minimize the inflow of sedimentary
materials).
When limitations exist with regard to small discharges,
this method cannot be applied (see Section D).
103

1 o4
When water rignts refer to baseflows and/or discharges up to a minimm
diverted discharge, Qo, (when Q > Q only Q is utilized), the computation
O
can be made by translating the pivot of the axes to a new starting point
- - *
(Q0; U >. In this case, new scales will Óe used: Qd (Qd - Qo) and
O - U: = Üd - -
Üo. If the point (Qo; U ) lies outside of Zone I, or at its
edge - the best flows are already utilized, .even though this does not mean
a priori that the proposed scheme will not De feasible.
O
D. AVELUGE ANHUAI., FLOW IN RELATION TO DIVERTED DISCURGES - DOUBLE CUT
When there are limitations to the diversion of baseflow discharges,
or any definite discharges less than a certain magnitude, the relation of
average annual flows to diverted discharges cannot be computed by means of
a simple "horizontal cut",
and horizontal cut, is required (see Sketch 5).
In these casesp a "double cut", i.e. a vertical
Double Cut Horizontal Cut Vertical Cut
Sketch 5: Schematic Representation of Double Cut
Such a case will arise when baseflow is of undesirable quality for
diversion purposes - generally too saline; conversely, during flood flows
tne water is of good quality and can be diverted up to a certain maximum
value, (Qd)max*
In this case the double cut - shown in Sketch 5-A - ie

105
used, When from a certain discharge onwards tiie sediment concentration is
undesirable for artificial recharge or from the aspect of reservoir capa-
city losses, and it is decided not to divert those discharges - a vertical
cut for Q = (Qd)mx is used (see Sketch 5-C).
Since during planning it is still unknown which Q and which (Q )
d max
will be selected, different combinations have to be examined. Tiiis can be
done by establishing a series of curves which describe the relations be-
tween annual average flows to maximum diverted discharges for different
values of Q
[Qo = constant]. This analysis has the disadvantage of being
related to discontinuous values of Qo and the need for repeating the calcu-
lations for each value of Q
involved, many sets of curves'will be required).
O
(when the storage capacity of a reservoir is
Such work is superfluous
and can be limited to the calculation of only two curves - the horizontal
cut curve and the vertical cut curve - if the following equation is used:
- - -
UA = UB - uc
where:
-
UA = fl
- UB = f2
-
... (4)
Qo; (Qd)mxl, represent the double cut';
(Q,),] o is established by means of a horizontal cut;
U = f [Q ] , is established by means of a vertical cut.
c 3 0
It is possible to calculate Ü
A
for any desired combination of (Q
d
)
max
and Qo by the use of the two curves (see Sketch 5-B and C).
Each of the
functions Ü and Û can easily be calculated by computer. These functions,
B C
as calculated for the Qishon stream in Israel, are illustrated in App.&
'pig. 2.

1 O b
-
Function Ü can easily be calculated from U*, without use of a compu-
C
ter, on the basis of a flow-duration curve, when the duration indicates
the average number of days per year of any given discharge or discharges
exceeding the given value.
be expressed by:
-
where:
In this case the relation between the two will
ta and p as in equation (3)
t* is the average number of days per year of a discharge of Qo or
-
more (see Sketch 3); t* = ta x (P)~,.
E. THE USE OF THE VERTICAZ. CUT
In addition to the contribution of the vertical cut curve for
simplifying the double cut technique and its use for planning of di-
versions with constraints of maximum discharges owing to sedimentation
IQ0 (QdImxi Qd Q for Q 5 (Qd1-i Qd = 0 for Q ’ (Qd)maxla the
curves are used for calculating the average annual sediment concentra-
tion or load.
The average annual sediment load can be calculated using simul-
taneously the vertical cut curve and the curve describing the relation
of the sediment concentrations to flow discharges (mostly log-log rela-
tions). The computation is carried out as demonstrated in Appendix B.
The average annual volume of sediment load, (Us)ave, of the stream
is calculated as:
m
... (6)

Accordingly, the average annual sediment concentration, ave(Cv), is
calculated as:
where :
(AUVIj
-
i- (Uslave
ave('v) 'ave
.. (7)
107
-
(ÜVli - (Uv)i-l, indicates the contribution of the discharges
within the limits of Qi-i to Q, to the average annual flow;
-
(U
v
)
i
indicates the average annual flow from the vertical cut
curve [for Q 2 Qil.
[(Cv)avel, is the mean volumetric sediment concentration of the dis-
charges within the limits of Qi-i to Q,.
j indicates the intervals of the discharges (AQ), chosen for
the calculation.
It should be noted that since the significant reduction in reservoir
storage capacity resulting from sedimentation in arid and semi-arid zones
is mostly due to rare high rate floods, there is a need for data of a
relatively long period. In such cases, it is therefore recommended to
use probability analysis in the evaluation of the frequencies. However,
regional analysis supported by analysis of historical flood water marks
for rough estimates of the "maximum historical floods" enables relatively
short period data to be used for evaluating the expected average annual
sediment load order of magnitude (in this case - channel sections with
a stable bed should be chosen; otherwise large mistakes may occur as a
result of marked changes in the channel bed).
F. ANNUAL STORABLE FLOW AS A FUNCTION OF RESERVOIR CAPACITY
The quantities of reservoir-stored streamflows which can be
utilized depend on flows, net reservoir capacity, reservoir operation,
and seepage and evaporation losses.
When the reservoir is to be emptied every year (e.g. there is
a rainy season in which the flows are stored and a dry season when the
stored water is used), it is possible to estimate the annual stored

108
quantities according to annual flows and net reservoir capacity, as long
as losses, at least during the rainy season, are small. When the expected
losses are large, the possible losses must be known or estimated before
calculations can be made; however,this information is often not available.
When losses during the rainy season can be disregarded, the reservoir
can every year store quantities smaller than or equal to its net capacity:
in which
Here:
iL
UR = - n
i-1
(URIi = ui when Ui 2 RN
(UR)i = (%li when Ui - ' (%)i
Ui
indicates the annual streamflow in the ith year - when the
reservoir is on the channe1,and annual diverted flow - when
the reservoir is off the channel;
(%)i representing net reservoir capacity in the ith year;
(U
R
)
i
is the amount of water stored in the ith year;
-
UR
is the n years' average annual amount of water stored in the
-
reservoir (whose average net capacity is %)
n is the number of annual data (calculated by the use of either
observed, historical, or of reconstructed synthetic data).
The quantity ÜR is always smaller than Uave, when Uave designates
the average possible annual inflows into the reservoir, such as average
diverted streamflow or average streamflow.

Annual net reservoir capacity is defined as freel,,annual capacity
up to maximum operational height (e.g. up to the spillway crest). When
the reservoir is operated in consideration of a planned dead storage, net
operational capacity is constant until the dead storage is replete with
-
sediment; i.e. (%Ii = RN = constant. Storage losses are dependent on
two major factors: the volume of annually deposited sediment, and the
volume of water remaining in the reservoir at the end of each year (the
"remaining volume" is generally constant, owing to the reluctance to pump
mud, except in reservoirs which store water for more than one year; the
case of such reservoirs 'is not dealt with in this article).
The following should be noted:
(a) The computations as described give approximate solutions.
If losses (by seepage and/or evaporation) are relatively
large, at least monthly water balances are required in order
to calculate the stored inflows with reasonable accuracy.
(b) The amount,,of water which can be annually utilized also depends
in each case on the operational regime of the reservoir and on
the losses (there is a difference between the utilized and the
stored amounts, since losses occur while the stored water is
being utilized).
(c) In arid and semi-arid zones - in many cases, due to the limited
potential of the stream and the considerable seepage and evapora-
tion losses - surface water utilization is based on artificial
recharge of aquifers. In such cases the reservoirs are used
for regulating and silting purposes; therefore (UR)i can exceed
the net reservoir storage capacity, as it is a product of a
109

number of floods which entered the reservoir after it was
emptied or partly emptied (after every flood the stored water
is transferred to spreading grounds for artificial recharge).
Principally, the calculations, the character, and the analysis
of the relations between FR and RN are the same as dealt with
in this Section.
The average annual volume of stored water (uR) and the reservoir
efficiency (ÜR/uaVe) as functions of average net reservoir capacity (i$)
are shown schematically in Sketch 6. (The meaning of the different zones
is as explained in Section C for Sketch 4)
d
Zond Zonelzone (UR/Uave>
A Zonalzone I Zone
2-
Rd -
Sketch 6: Schematic Representation of U R = f (s) and (ÜR/UaVe) = F (s)
The curves, which summarize the aforementioned influences, illus-
trate the contribution of average net reservoir capacity (Q).
-
Here too,
as in the analysis of the relations illustrated in Sketch 4, different
zones of the curves can be discerned, characterized by the magnitude of
the slopes of the tangents to the curves (AÜR/A% or A(ÛR/Uave)/A$).
These slopes represent the marginal additions of the average annual
quantity of storable water for the addition of a unit of net reservoir
capacity .

-
It is characteristic that as % increases, AÜR/AQ decreases. For
111
high values of RN the value AÜR/A$ is small, as it represents rare flood
flows; its reliability is therefore limited.
An analysis of this kind is of great importance for preliminary
estimates and/or feasibility calculations, since it makes it possible
to find easily the approximate economic solution.
Recent investigations made by the Surface Water Utilization Depart-
ment of Tahal - Water Planning for Israel Ltd. prove that the relationship
- -
between UR and % can be approximately estimated on a regional basis using
as a parameter the dimensionless standard deviation (the ratio u /U
u ave’
where U is the standard deviation, and U is the average annual flow).
U ave
It was found that the ratio ouIU is, in many cases, of regional
ave
character. (The above-mentioned investigations have not yet been con-
cluded and hence cannot yet be summarized).
The function described above is illustrated in App. A, Fig. 3.
G. THE COMPUTATION OF RN AND RN
where :
Annual net reservoir capacity can be computed from the equation:
(%li = - ... (9)
($>i
is the net reservoir capacity at the end of the
ith year;
(%)i-1 is the net reservoir capacity at the end of the
(i-i) th year;
(Rs)i is the volume of the sediment trapped in the
reservoir during the ith year.

112
The volume of the sediment deposits trapped in the reservoir during
a certain year can be calculated from the equation:
where :
(CS) i
(RsIi = (EVIi x Uix(T.E.Ii = - x Ui x (T.Ea)i ... (10)
YS
is the average concentration of the transported sediment
during the ith year, by volume;
(C ) is the average concentration of the transported sediment
s i
th
during the i year, by weight (e.g. in p.p.m);
YS
ui
is the average specific weight of the trapped sediment
(generally approximately constant, depending on sediment
qualities and reservoir operation);
represents the reservoir inflow in the ith year;
(T.E.Ii is the trap efficiency in the ith year - the portion of the
sediment which remains in the reservoir (if there is any
overspill, part of the sediment leaves the reservoir with
the overspill).
For a design period of n years, especially when the value of n is
high (tens of Years), the total loss in reservoir storage capacity re-
sulting from sedimentation can be calculated as:
will be
Therefore, the average annual loss in reservoir storage capacity
-
Rs = - (Rs)i = Uave x (T.E.1 x [ave(Cv)l ."a (11)
n i=l

where :
ave('v)
is the average concentration, by volume,of the sediment
deposited by the transported water at the reservoir
location;
(T.E.) ,the average trap efficiency.
The average net reservoir capacity for a period of n years will be
estimated as:
where :
Ro
is the initial reservoir capacity.
... (12)
11 3
It should be noted that since it is impossible to predict the future
annual flows, there is no other practical possibility of evaluating the
net reservoir storage capacity. For practical purposes, the use of average
net storage capacity (%) is sufficient.
H. RECOMMENDED HYDROLOGICAL INVESTIGATIONS
Hydrological investigations directed towards finding parameters
which enable non-dimensional curves to be established which represent
the main functions discussed in this article, are recommended, especially
on a regional basis.
The reconstruction of such a regional synthetic curve, even though
not "scientifically accurate", will be of great assistance in planning
surface water utilization schemes, and especially in planning the first
stage of such schemes.

114
BIBLIOGRAPHY
This article is based on the experience gained in Tahal - Water
Planning for Israel Ltd., in the last 20 years, &.on the Technical
Reports published by Tahal in Hebr’ew, as also on the foliowing works.
1. Kuiper, E., Water Resources Development, Buttexworths,
London, 1965
2. Linsley, Ray K. and J. B. Franzini, Water Resources
Engineering, McGraw-Hill Book Co., London, 1964
3. Searcy, J. K., Flow-Duration Curves, Manual of Hydrology,
Geological Survey Water Supply Paper 1542-A, Washington D.C.,
1959

APPENDIX 13: CALCULATION OF AVERAGE ANNUAL SEDIMENT. VOLUME
TRANSPORTED BY LOWER QISHON FLOWS
li 5
1. Streamflow hydrographs were used for preparing a %cirtical cut curve"
representing the average annual values of flow (Uc) contributed by
discharges up to any value of QI as explained in Sections D and E,
and illustrated in Fig. 2 of App. A.
2. Simultaneous data of sediment concentrations and instantaneous dis-
charges, supplied by the Israel Jydrological Service, drawn on a
log-log paper enable the construction of a correlation line between
the average sediment concentration and the instantaneous discharges.
In order to be on the safe side, the line was removed toward the
higher concentrations (of each discharge) - see Fig. 4 of App. A.
3.. The calculations are shown in detail in the following table.
-

11 6.
CALCULATION OF AVERAGE ANNUAL SEDIMENT LOAD
EXAMPLE: LOWER QISHON STREAM (ISRAEL)
-
uc
cs
L CU. mf s ec
-
5
O
1
3
5
1 .20
10
15
--
Total
LEGElID:
-
MCMIY r
3.0
6.0
8. O
10.4
II..
12.0
13. O
Q = diccnarge
3,O
3.0
2. o
2.4
1.0
O. 6
1.0
13.0
PPm
400
800
1 , 200
1,700
2,300
2,700
-
300
600
1 , O00
1 , 450
2,000
2,500
4,000
5 x AÜc
ave
tonlyear
APP.
Sneet 2
900
1 , 800
2,000
3 , 480
2,000
1,500
4,000
15,680
U2 = average annual flow volume related to Q calculated by
vertical cut of hydrographc (from Fig. 2 of App. A)
I -
AUc = the interval of Uc contributed by discharge interval
-
Cs = average sediment concentration, by weight (from Fig. 4
of App. A) , high values
- -
(Cs)ave = average CS for discharge interval
4. The result obtained from the calculations shown in the above table,
is that average annual sediment load transported by the Qishon stream-
ilows amounts to about 16,000 ton. Assuming an average trap efficiency
of 90 percent and sediment deposits specific weight of 1.5 ton per cu.m -
the average annual value of sediment trapped and deposited in the planned
reservoir will be about 10,000 cu.m per year ( 16~000x0'9 9,600
1.5
i0,OOO cu.m per year).

Rainy yerre - average: t* > 40 daye I
---Ueriium reinfall years - averaec:
20 1 t* 40 daye
Dry year# - average: t* < 20 daye
dischargee exceeding Q
t* - Number OP daye with discharges cxceeding
1.5 m3Jeec
.,...u. Average for 1940141 to 1964165
FIG. 1: FLOW-DURATION CURVES FOR QISHON
- STREAM (ISUAEL)
FIG. 2: HORIZûNTAL, VERTICAL AND DOUBLE
CUTS OF HYDROGRAPHS OF QISHON
STREAM (ISRAEL)
117
Appendìx A

118
FLG. 3: THE DEPENDENCE OF THE AVERAGE STORABLE
FLOWS AND THE STORAGE EFFICIENCY ON
THE NET AVERAGE RESERVOIR CAPACITY AT
THE UPPER QISHON (UPSTREAM THE HYDRO-
METRIC STATION TO WHICH THE DATA OF
FIG. 1 AND 2 REFER), ISML
Amendix A

Appendix A
11 9

ABSTRACT
APPLICATION OF COUTAGNE'S AND TURC FORMULAS
TO THE SOUTHERN MOZAMBIQUE RIVERS
Emilio Eugénio D'Oliveira Mertens
Joäo José Mimoso Loureiro
Checking of Coutagne's and Turc formulas, was purposed to
obtain values, though approximated, for the annual mean runoff
of the several rivers at southern Mozambique where few gauging
stations exist. Therefore, measured rainfall and temperature
values were collected from the meteorological and gauging stations,
as well as the runoff values observed in the locations.
We conclude from the results obtained that the application
of these rules has given us, with relative guarantee, the annual
mean runoff values, with deviations inferior to 10% which can be
considered as satisfactory.
RESUME
Les formules de Coutagne et Turc ont été utilisées pour
obtenir des valeurs, même approximatives, de l'écoulement moyen
annuel pour les différents fleuves de la région sud de Mozambique
dans laquelle on ne dispose que d'un nombre très limité de
stations de jaugeage.
Les calculs ont St6 effectués à partir des valeurs des
précipitations et des températures mesurées aux stations météorologiques
et pluviométriques, ainsi que des valeurs des écoulements
observées à différentes stations.
Les résultats obtenus montrent que l'application de ces
deux formules donne, avec une précision relative, des valeurs de
l'écoulement moyen annuel. Les écarts sont inférieurs à lo%, ce
qui peut être considéré comme satisfaisant.

The hidrological phenomenons o- greater interest, relating to the hidro-
logical studies of a catchment area under consideration, are namely:-
Rainfall
Air temperature
Relative humidity
Evaporation
Hidrometical records
Flow discharges of streams and runoff
Sediment discharges
The main purpose of a certain hidrological study, consists on the deter-
mination for each one of the observed actions, of the variability principles
thereof at distinguished intervals, analogy principles of the phenomenon itself
from site to site and of the correlation principles amongst the several pheno-
menons.
One of the basilar elements necessary for the planning of an economical
development program is the knowledge of the value and distribution of its hidrg
logical resources.
In Mozambique, registration of the hydric resources has been facing
great difficulties not only in what refers to the extension of the territory but
also, and essentially, by lack of observations of the hidrological phenomenons,
namely the runoff and flow discharges of rivers and water-sources.
From a report presented by Dr. L. Turc on the 3rd.iiidrological Ehgeneer-
ing Congress organized by the 'Societé iiidrologique de France', which took
place in Argel, in 1954, we were suggested to follow the idea of verifying the
possibility in the application of Coutagne's and Rirc general rules, related to
the Southern Mozambique water-sources.
2 - COUTAGNE'S AND TURC GENERAL RULES
These general rules allow us to estimate, by simple calculation, the
value of a catchment area's runoff deficit, provided that rainfall and tempe-
rature are known.

2.
Runoff deficit - is the difference between mean rainfall height
123
pertinent to a certain site in the water-source and the corresponding height
to the flow discharge estimated at the referred site.
2.1 - COUTAGNE'S GENERAL RULE
Being :
H = Mean rainfall height
general rule is
E = Hficient rainfall height, that is, the height which transform
itself theoretically, in the whole, to runoff.
D = Runoff deficit = H - E
C = Runoff coefficient = 2
H
K = Coutape's constant
2
D = H - KH2 being E = KH
and since C = E and D=H-E
H
Now as:
C,D-H or C = KH
H
2
(C - KH)2 = (c 1 - KH + (c 2 - KH 2)2 + (c 3 - KH 3) + ....**
to minimize this sum, it will do equalizing zero to the first derivative:
wherefore:
(C 1 - KH 1) H 1 + (C 2 - KH 2) H 2 + .......= O
CH
CH-KH2 = O K = -
H2
From the above determination it is given the most probable value for K.

124
/3.
2.2 - TURC'S GENERAL RULE
L = Turc's constant
P = Evaporation plus lost by percolation
T =Mean temperature
A = Constant
being
wherefore
H
'dFtJ2
L2
L = A + 25 T + 0,05 T3
The author still precises that applying his general rule in 254
Catchment areas, considering A = 300, distributed towards every climate in the
world, it has been reckmed that values of 0 observed and calculated from the
referred general rule, came out as to the undermentioned results:-
or
or rather
in 53% the cal. D - me86 D < 40 mni
in 43% the cal. D - meas. D < 0,l meas. D
i,n 65% the cal. D - meas.D < 0,2 meas. D

4.
3- REPORT ûF THE CONSIDERED LOCATIONS
125
Described hereyder are the considered locations at the Limpopo's
(incl. Elephant's River), Incomati, Umbeluzi, Sabie and Usuto Rivers (D. 1)
3.1 - ELEPHANT'S RIVER
Location: Maçuço - Mozambique
2
Catchment area: 66.600 Km
Mean rainfall height: 636 mm
Mean temperature: 18,9OC.
Observation years : 1944/45 to 1970/71.
3.2 - LIMPOPO'S RIVER
Location:Beit bridge - R.A.S.
2
Catchment area: i88.000 Km
Mean rainfall height: 481 mm
Mean temperature : 2OoC.
Observation years : 1955/56 to 1963/64
3.3 - LIMPOPO'S AND ELEPHANTS RIVERS
Location: Vila Trigo de Morais - Mozambique
2
Catchment area: 340.000 Km
Mean rainfall height: 541 mm
Mean temperature: 20,2OC
Observation years: 1951/52 to 1969/70
3.4 - INCOMATI'S RIVER
Location: Ressano Garcia - Mozambique
2
Catchment area: 21.600 Km
Mean rainfall height: 832,2 mm
Mean temperature: 18,8OC
Observation years : 1955/56 to 1969/70
./.

126
3.5 - SABIfl'S RIVER
Location: Machatuine - Mozambique
Catchment area: 6.200 Km2
Mean rainfall height: 766,4 mm
Mean temperature: 20,7OC
Observation years: 1955/56 to 1969/70
3.6 - UMBELUZI 1 s RIVER
Location: Goba - Mozambique
Catchment area: 3.100 Km
2
Mean rainfall height: 820,3 mm
Mean temperature: 21,6OC
Observation years : 1955/56 to 1970/7i
3.7 - MAPUTO'S RIVER
Location: Sipofaneni - Swaziland
2
Catchment area: 12.903 Km
Mean rainfall height: 83i,7 mm.
Mean temperature : 22OC
Observation years : 1958/59 to 1964/65
3.8 - Therefore we get two distinguished groups in regarding to pluviosity
and temperature:-
- Limpopofs River Groue - with mean rainfalls between 450 and 650 mm
and temperatures from 18OC to 2OoC;
- Incomati's, Sabie, Umbeluziaid Usuto Group with mean rainfall
values of 800 m. and temperatures higher than 2OoC.
4 - APPLICATION OF COUTAGW'S GENERAL RULE
We have tried Coutagne's general rule for each one of the above
groups and locations therein.

4.1 - LIMPOPO'S RIVER CATCHMENT AREA
127
2
The 340.000 Km of the Limpopo's River Catchment Area relating to
2
Vila Trigo de Morais' gauging station, include the 66.600 Km of Maçuço's
gauging station at Elephants' River and the 188.000
2
Km pertinent to Beit
Bridge location.
4.1.1 - For the 27 observation years at Elephants' River we have reached to
the following type of Coutagne's rule:-
D = H - 0,000055 H (1)
The most probable values for the measured runoff deficits (612,9 mm)
and calculated ones (613,3 mm) differ in 4 mm to a mean deviation of _+ 8,7 mm
and a mean observation error of 7,4 mm.
Extension of this rule for the available 66 rainfall observation
years would be plainly acceptable in view of the fact that for a period of 34
years of which we own the closest possible runoff estimatives, difference is
kept for the measured and calculated deficit.
4.1.2 - For the 9 observation years in Limpopo's River area at Beit Bridge we
reached to the results hereunder, to Coutagne's rule:
2
D = H - OJ00O031 H
(II)
recording the most probable values of the measured runoff deficits (472,7 mm)
and calculated ones (473,7 mm) being the mean deviation and each observation
error of _+ 6,2 mm and 3,7 mm respectively.
4.1.3 - Finally for the i9 observation years in Vila Trigo de Morais, situated
after the confluence with Limpopo's and Elephants Rivers, Coutagne's rule
presents us the following result:-
2
D = H - 0,000047 H
(III)
Measured and calculated runoff deficits have a similar probablest
value, but the mean deviation is increased in ,+ 10,l mm and mean observation
error amounts to f 7,6 mm.
./.

128
/7.
4.1.4 - It is left to determinate now an available Coutagne's general rule
to the entire group of 55 observations, as the principal elements taken to
its calculation - runoff coefficient (C) and mean rainfall (H) - are not
dependent values on those of the referring catchment areas, same being consi-
dered to the runoff deficit.
Ordering the values of the observed mean rainfall, the undermentioned
rule is calculated (Q1):-
2
D = H - 0,000050 H (IV)
Through the same comparative system, it was obtained to the measured
runoff deficits, the value of 560,3 mm and for the calculated ones through
the same rule (IV) 56O,6 mm being 0,3 mm the difference thereof.
Mean deviation of the measured and calculated values amounts to
+ 9,l mm with a mean observation error for each one of 7,2 mm.
-
Seing that the values of medium rainfalls, relating to the observa-
tion periods, are respectively of 636, 481 and 541 mm with a short difference
from the medium normal rainfall, we may conclude that Coutagne's rule of
which coefficient is equal to 0,000050, can be applied to every catchment
area of which medium rainfall is comprehended between 450 and 650 mm. However
it is necessary to point out that its application, in view of the great ex-
tensions that they comprehend, cannot be considered as absolutely precise,
except for mean values.
Application of this rule every year may lead us to
mistake, since it calculates a regular correlation amongst runoff and rainfall
which is not precised in the practice because of the powerful stream of Lim-
popo's River, namely before the confluence with Elephants' River.
4.2 - CATCHMENT AREA'S GROUP OF INCOMATIIS, SABIE, UMBELUZI AND USUTO RIVERS
2
The 43,803 Km of this group comprehend all the rivers which drain
off in Lourenqo Marques' Bay and are located in an area, mean altitudes of
which excede the 800 m. and mean rainfall is estimated between 750 mm and
850 mm.
All the above catchment areas are neighbouring.
4.2.1 - For the 15 observation years of the Incomati's River at Ressano
Garcia rainfall station, we conclude from Coutagne's general rule the next:
2
D = H - 0,000150 H (V)
./.

129
The mean deviation value amounts to 19,7 mm and the mean observation
error to i5 mm for the measured and calculated runoff deficits of 737,3 and
739,O mm respectively.
4.2.2 - For the Machatuinels rainfall station of Sabie's River and Goba's
rainfall station of Umbeluzi's River, respectively with 15 and 16 observation
years pertinent to an identical period, Coutagne's general rule figures like:
D = H - 0,000131 H
2
2 (VI)
D = H - 0,000145 H
To the first location, measured and calculated runoff deficits are
similar (684,7 mm) and mean deviation amounts to _+ 20,4 mm.
At Umbeluzi's River, mean deviation amounts to the decreasing value of
+ 181 mm and difference between the measured and calculated deficits amounts
-
to 718,s and 719,O.
4.3.3 - For the 7 observation years at the Sipofaneni's rainfall station in
Swaziland, the main confluent of Maputo's River, Coutagne's general rule is
as follows:-
D = H - 0,000162 (VI1 )
Measured values (717,3 mm) and calculated ones (717,O) differ from
O,3 mm and mean deviation amounts to &17,3.
4.3.4 - Similary to what has been done in Limpopo, it was arranged the group
of 53 observation years (Q2) in regard to mean rainfall, which alters from
500 mm to 1.540 mm and the exposed Coukagne's rule is:-
D = H - 0,000140 H2 (VIII)
Difference from the results determined by measuring and obtained from
this ru1.e (VIII) is of 1,2 mm with a mean deviation of f 19,3 mm and _+ 15,8 mm
for the medium error of each observation.
To this catchment area's group mean rainfall wherein exceeding 800 mm
and mean small deviations qualifying same as of minor torrentiality, and,
consequently, higher stream regularity, application of Coutagne's general rule
more than granting us accurate values for the annual runoff deficits yet allow
./.

130
/9.
us its application year after year.
5. - APPLICATION OF TURC'S GENERAL RUlE
Application of this rule, such as formed by Turc, that is, considering
A=300, could not be used but running the risk of forming gross estimate errors
seing that in Mozambique, the catchment areas in study have great extensions,
usually.
5.1 - We have tried to both of the groups the application of the general rule
formed by Turc, and have found the next following results:-
5.1.1 - IJMPOW'S CATCHMENT AREA
Difference calc. D - meas. D L
II II L
II II A
II II L
II 11 <
II II L
II II
II II
1
Ls
20 mm -
40 mm -
40 ~UW -
0,Ol m.D -
0,05 m.D -
0,l m. D -
0,l m. D -
0,2 m. D -
5.1.2 - INCOMATI'S, SABIE, UMBELUZI AND USUTO GROUP-(Q2)
Difference calc. D - meas. D c 20 m -
11 Il < 40mm -
II II
> 40 ìüiìì -
11 II < 0,Ol m.D -
II II < 0,05 m.D -
11 II < 0,l m.D -
II 11 > 0,l m.D -
11 Il > 0,2 m.D -
53%
76%
24%
14%
22%
36%
64%
O
3%
66%
34%
1 5%
50%
84%
16%
5.2 - Percentages differ from the obtained values by Turc for his Group of
254 catchment areas and considering the 64% (Limpopols Group) of events
superior to 0,l of measured D, we are not abled to consider the rule as
applicable.
10%

lo.
131
Therefore it was tried to find a rectifying solution of the constants
in function of the mean rainfall and annual mean temperature for every catch-
ment area.
Thus we have traced the graphic shown in D.2, consequence of
succeeding considerations on the measured values.
5.3 - Upon the above application of Turc's general rule, established the
constant A from the graphic, we reached to the following results:
5.3.1 - LIMPOPO'S CATCHMENT AREA (Q3)
Difference calc. D - meas. D < 20 mm
II 11 < 40 mm
II II
40mm
11 II
d 0,Ol m.D
II II < 0,05 m.D
11
II
11
II
II
11
< 0,l m. D
> o,1 m. D
> 0,2 m. D
88%
95%
5%
33%
io%
5.3.2 - INCOMATI'S, SABIE, UMBELUZI AND USUTO CATCHMENT AREAS (a)
Difference calc. D -
II II
II
II
II
II
Il
11
II
11
II
11
11
II
10%
O
O
meas. D < 20 mm - 43%
< 40 iiìüi - 87%
> 40 ïìüü - 13%
¿ 0,Ol m.D - 25%
< 0,05 m.D - 8%
< 0,l m.D - 96%
> 0,l m.D - 4%
> 0,2 m.D - O
5.4 - From the whole of the 108 compared values, we conclude that in 87% of
the cases the difference among the calculated and observed values does not
exceed 40 mm and in 96% the difference does not exceed 0,l from the measured
runoff deficit. Nevertheless, the most relevant results are that in 2% of the
cases the deviation does not exceed 0,Ol meas.D and 53% does not amount to
O,O5 of the observed runoff deficit.
. /*

132
/li.
6 - CONCLUSIONS
In the water-sources situated at the South of Save's River, there
might be applied the Coutagne's and Turc general rules on the following way:
and 700 nun.
COüTAGNE'S GENERAL RULE:
Catchment areas with annual medium rainfalls coaiprehend between 450
2
D = H - 0,000050 H
for mean rainfall superior to 700 nun.
TURC'S GENERAL RULE
2
D = H - o,000140 H
Determining constant A from the graphic
Utility in the application of these rules becomes evident in view
of the non-existence of gauging stations along the multiple water-sources in
the area under consideration, meteorologic and rainfall stations taking their
place instead.
But, seing that for the inventorying of the hidrological resources
is matter of extreme necessity the knowledge, though approximated, of the
annual mean rainfall, and yet because it has become evident through the appli-
cation of the aforesaid rules that precise values amount to less than i%, we
may say that have succeed with the reaching of our main purposes.

133
D1

850
800
750
700
650
6 O0
55 o
500
450
\
\
\
\
\
\
\ '
\ i
\ v
35 O
400
C5 O
500
ln
U
L
C
m .-
U - W
u
O
P
a
.- E
-I
VI
O
.-
a
A (TURC2
ABACO PARA A DEIERMINAFAO
DA CON5fANTE DE TURC

DETERMINAÇ~O
DO COEFICIENTE DE COUTAGNE
Rios Limpopo e Elefantes

DEIERMIN4ÇÁO DO COEFICIENTE DE COUTPGNE
Rios Limpopo e Elefantes
1

DETERMINPÇÃO DO COEFICIENTE O€ COüTAGNE
Rios Incomati, Sabié, ümùeltízi e Maputo
OEFIC E NI
e E SCGAIAI
C
-__
Q tu
-Qm
AQ6
Q,lQ
All-
Q,05
n ,u-
9,08
4.05
-0,l3
0,16
0,11
0,08
0,lQ
0,07
0,08
Li1
0.10
o, 11
0.11
O ,O8
O, 14
0,13
0.09
0,OI
O, 08
o ,oa
0,13
0,15
Od3
o, 09.
0.11
0,12
O ,O9
0,13
Q,16
0,12
0,11
0,l.l
0,12
O J 5
-
-e*-
-
HZ
259d81 - -.
392.599
3 1 W P - .
605.284 ~
606 .a41 -~
624.00
628 -849
649.636
6 22.400
685 584 .
687.241
714.025 .
734.449
736 16 4
17O.8134
774.40Q
7 84.996
792.100 -
792.100
793.881
195.664
802.816
846.400 -.
81.9_41
.
-
C H
-
93L776
25,99
@+5Q -
XL§Q __
5Z,60
34kUlQO- - .- - 6430
3áL O M .. in,oo -
381924 67.98
38 4.400 .- 49,6Q_
419,881
32.01
412.164
83,46
448.900 107.20
A98 329
462.40Q
74,47
1>4,40
463,761 48 ,LO
467,856
485,809
504.100
. 47.88
55 I 76
78.10
132.9QO
73,OO
547.600 . 81.40
549.061
81.51
599-076 -
61,92
. 108,92
101.27
71110
55,SI
64,4t?
65,60
107,64
124,35
109.85
77,13
94.38
105 ,ih
79 120
115,lß
142.40
106,80
9h,01
96,l'
107,~J
138.00
17c,77
-
:EPICI1 drESCOAUEN1
-
O CALCULA
45 2
502
.I 29
5 13
524
16 i)
547
57 1
610
56 0
562
603
622
6Q7
630
6 40
634
655
66 2
6 56
708
56 7
676
713
737
739
749
721
704
738
780
76 2
777
799
77 1
744
785
792
797
79 1
780
813
-
A
OM- o
- 21
- -6
+-L3
-
2
-
lî*
-
-12
-18 -
441
- 36
269
144
324
i18 . 324
-1 8 324
+4 - IL
i26 -6 36
-24 57t
-46 211C
-10 -100
+6 36
-1 O 101
+19 36 1
+LI 171
-6 36
-1 1
-2 4
-9 bl
i17 289
-2 7 729
-19 36 i
+10 100
+ 32 1074
+2 3 520
*23
-12
529
144
-29 8 41
-8 6 1"
+?5 625
+ 7 49
+ 2 4
+2 7
-6
- 36
+ 5
729
36
1296
2 c>
+12 __ 144
+11 121
+7 49
-22 484
+.4 16
- -
-

38
DETERMIN4C$O DO COEFICIENTE DE COUT4GNE
2

QUADRO COMPARATIVO DOS RESULTADOS OBTIDOS
PELA FORMULA DE TURC E DA CONSTANTE TI,
RADA DOÁBACO
ZONA Bios Limpopo e Elefantes mor
3

140
QUADRO COMPARATIVO DOS RESULTADOS OBTIDOS
PELA FORMULA DE TURC E DACONSTANTE TI-
RADA DO ÁBACO
ZONA Incomati, Sabi6, Umùelilzi e Naputo
4

ABSTRACT
MAPA1 HI DROLOGI CAL STUDY ( LIMPOPO ' S RI VER)
EMILIO EUGENIO D'OLIVEIRA MERTENS
JOÃO JOSE MIMOSO LOUREIRO
Lack of observations in the flow discharges and runoff,
taken at the future location of mapai's dam, have compelled
us to the essaying of diversed methodology viewing its
obtention.
It was selected the method of the specifica1 runoff
technic which has conducted us to most consistant and
significant results in conjunction with those observed and
calculated for other locations at the catchment area.
RESUME
L'abscence des observations relatives aux débits et
écoulements measurables au futur lieu du barrage du mapai,
nous a forcé d'essayer diverse méthodologie pour en obtenir.
I1 a ;te choisie la method: de la technique dés débits
specifiques que nous a conduit a des resultats tres concordants
et significatives en conjunction avec ceux observés
et calculés pour les autres lieu du bassin versant.

142
1 - CATCHMENT AREA
1.1 - Site, area, relief and hydrography:
The hidrographic basin of the Limpopo River has its major part in the
territories of South Africa, Rhodesia and Botswana, its area of 412 O00 km2
being devided in the following manner (Drawing 1):
South African Republic .................... 193 500 km2
Rhodesia .................................. 66 O00 km2
Botswana .................................. 73 O00 km2
Mozambique ................................ 79 500 km2
Rounded off, the catchment area is situated beyween 220 and 260
South and 269 and 350 East, its highest altitude being 2.300 metres near the
city of Lydenburg.
In National territory, situated between parallels 210 and 250 South and
meridians 310 and 359 East, the basin has to the North, that of the River Save,
to the South, that of the River Incomati and to the East, that of the River
Govuro, and a coastal strip where a few closed catchment areas are found from
which the water-sources accumulate in lakes.
In Mozambique there is no noticeable irregularity, this occurring only
in the limiting zone to the south of the Limpopo, in a reduced area with eleva-
tions of 400 metres.
In its total length the average height is of 840 metres, its being 977,
964 and 950 metres, respectively in Beitbridge, Mapai and Trigo de Morais.
The average slopes of the course of the water are:
Upper stream ....................................... 2, 50 dlan
Central stream ..................................... 1,80 m/km
Lower stream ....................................... 0,Og m/km
The Limpopo is one of the most important rivers of South Africa and
Mozambique and, as happens with the Incomati River, it is contained in the
lower part of the great drainage area, which includes more than half of the
Transvaal and a considerable part of South Rhodesia.
./.

143
The Limpopo is a strange river, very changeable and capricious, perhaps
due to the influence of the dissimilarity of its hydrographical basin; its vol5
me of water is extremely variable as in dry weather it is very reduced and during
the rainy season, reaches heights of 7 metres which flood large areas of ground
in the central and lower courses. The Limpopo River, when it enters our territory,
has already a definite bed, where it has three large tributaires: on the
right bank, the Elephants River, and on the left bank, the Nuanetzi and the
Changane; it is to these that it owes its permanent volume of water for the
flow from those joining it, its principal supplier being the Elephants River, a
water-source which crosses a region of high rains, its hydrographic basin having
a somewhat impermeable geological configuration.
It belongs to the hydrographical system of the African Continent and it
is of the torrential rate of permanent volume.
The course of the water which takes the name of Limpopo River, is formed
by the junction of the Marico and Crocodile Rivers which have their sources at
an altitude of 1.500 metres to the west of the city of Pretoria.
The principal tributaries of the right bank, all with their sources in
the Transvaal, from the source to the mouth of the Limpopo River are as follow:
River Matablas , Pongola, Palala, Sand, Pafuri (flowing in close to Pafuri,
already in Portuguese territory) and the Elephants River, the largest and most
important which joins it within Mozambique after some 110 kms. On its left bank,
the Limpopo receives large courses of water all with their sources in Rhodesia,
the principal ones being:
River Notwani, Macloutsie, Tuli, Umtzingwane, Bubye, Nuanetzi (which has
already flown about 50 kms in Mozambique) and the River Changane.
1.2 - Geological Aspect, Soils and Vegetation:
In the Limpopo basin, formations are found which belong to different
systems, such as Karroo, Waterberg, Primitive System.
The basin in South African and Rhodesian territory seems to be constituted
of basaltic lava, Serie Ecca, siliceous detrital rocks (sandstone) of brown
red and purple colours, formations of conglomerates, graphite and gneiss.
In Mozambique, the basin is mainly constituted of sedimentary formations.
In a narrow area near the border, volcanic rocks are found, in the upper
course of the Limpopo River and Elephants River formations of the Cretaceous Era,
in the rest, Quaternary formations with alluvium, sandstone, calcarium and sand
deposits.
./.

144
The vegetation in foreign territory is mainly constituted of bush and
grass, of great density in the highlands, and mixed bush and grass plains. In
Mozambique, the vegetation is of the bushy type and plains with some trees,
level grass plains and large stretches of grassy land.
The predominant soils in our territory are: sandy in the coastal area,
salty in the river vales, soils of mananga in the lower Changane and conglome-
rates.
1.3 - Climate:
In respect to the area situated in Mozambique, it appears that the
average annual temperatures are practically the same in almost all the basin,
being 240 C with the exception of the north eastern side, where it goes as low
as 220 C.
On the coastal and north-eastern areas, the average maximum daily tem-
peratures are 300 and 320. C and in the central area 34Q.C.
The average temperature in the hottest month is 280 C. and the lowest
260.c., the annual variation of these averages being between 60 and 9Q.C.
The average temperature in the coldest month is 20% in the central
area, and 18% in thê rest, while the coastal area has an average minimum in
the coldest month of 12%.
The annual average relative humidity in the central area is 65%, increa2
ing to the north and south to reach the highest rate of 75%.
According to the classification of Koppen, the climate of the basin is
in general the dryness of steppes with a dry season in winter, dryness of the
desert in the area of Pafuri, dryness of the steppes in the south of the basin,
and in the coastal area, tropical raininess of a savanna.
The predominant winds in the months of September to February are those
from the East and, during the other months, almost entirely with predominance
from the West.
In the whole basin, one finds that it is situated between the isothermics
of 240 and 170 with the average temperatures of 200, 2003 and 2002 respectively
for the areas of Beitbridge, Mapai and Trigo de Morais.
1.4 - Hydrological Occupation:
Both the South African Republic and Rhodesia have a network of udometric
and hydrometric stations which, for the African Continent, can be considered
dense: one pluviometer for 200 km2 and one hydrometric station for 4 o00 h2.

145
There are readings from 31 Rhodesian udometric stations and from 90
South African posts, the majority of which with more than 30 years of existence.
The more significant hydrometric stations in Rhodesia and South African
not only for the area they cover and their locality, but for the extension of
their records, are:
Rhodes ia :
South Africa:
- River Tuli - 4 144 km2
- Unzimgwane River - 2 533 km2
- Bubye River - 8 029 lan2
A3 MO7 - Eerste Poor - Groot Marico Rivier
A2 M25 - Hardekool Bulti - Crocodile River
A5 MO2 - Vischgat - Palala River
A5 MO3 - Oxenham Ranch - Limpopo River
A7 MO4 - Beitbridge - Limpopo River
A7 MO3 - Zamenkomst - Sand River
A9 MO1 - Schuinshoogte - Luvuhu River
- Liverpool - Olifants Rivier
- (354) - Olifants Rivier
- Manorvlei - Letaba River
- Letaba Ranch - LetkaRiver
- Driehoek - Blyde River
- 8 588 h2 - 21 i09 lan2
- 2 341 h2 - 97 850 km2
-180 O00 h 2
- 6 900 km2
- 912 lan2
- 42 352 km2
- 27 928 km2
- 668 km2
- 4 716 h2 - 2 199 km2
In Mozambique, there are 33 udometric stations, some with a significant
period of regular readings, and 12 stations for the measurement of water volu-
me, some equipped with linmographs, the most significant of those with regular
readings being those of Maçuço (66 O00 km2) and Tiobine (68 450 km2) on the
Elephants River; Vila Trigo de Morais (340 O00 km2), Pafuri (235 930 km2),
Mapai (246 O00 km2), Mohambe (342 780 h2), João Belo (407 970 km2) on the
Limpopo River, and Chibuto (43 200 km2) on the Changane River.
As regards evaporation, there are 6 U.S. Class A Tina evaporemeters in
Rhodesia, 20 Standard Symons in South Africa, and 5 U.S. Class A Tina evapore-
meters and 7 Piche atmometers in Mozambique.
./.

146
Also in the Portuguese part of the basin are 3 lysimetric stations, 3 cli
matological stations, 2 agronomic-climatological posts and 4 climatological posts.
2 - RAINS
2.1 - Introduction:
From the analysis of the normal isohyetal map, it is noted that the basin
is within the ishoyetal extremes of 400 and 1.500 mm., the monthly distribution
of rainfall being divided in a deficient way throughout the year,with a concentrg
tion of about 85% of the total during the months from October to March inclusive.
To determine the average annual rainfall in the Limpopo basin as far as
Mapai, three methods were used:- of the rainy districts; of the area of influes
ce and the isohyetal figures.
2.2 - The method of the rainy districts
The South African Republic is divided into restricted zones by rains of
average equality, certain rainy districts under the same principle also dividing
the areas of Rhodesia.
In accordance with the udometric records for the period 1914/15 to 1963/
/64 (50 years), the average annual rainfall figures were determined in respect
to the basin, taking into account the fall in each district.
From the period of 50 years, the average annual rainfall arrived at was
579 mm, its having been 582, 566 and 560 mm respectively during the past 10, 15
and 25 years.
2.3 - Method of Area of Influence
In accordance with the records existing for the period 1954/55 to 1963/
/64 (10 years) and using the 71 udometric posts, the average annual rainfall was
determined, the figure obtained being 514 mm.
2.4 - Isohyetal Method
With the normal figures - 30 years - recorded at the udometric posts,.
isohyetal curves were interpolated, the areas between the adjacent isohyetal
figures being thereafter measured.
The average figure for the rainfall in the basin thus obtained was 504m.

2.5 - Correlation of the methods:
147
For the common period of 1954/55 to 1963/64 and 1955/56 to 1963/64 of
which the average annual rainfall figures are available, the correlation between
the two methods was determined, the following equations having been obtained:
1954/55 to 1963/64 - x = 0,94y + 99
1955/56 to 1963/64 - x = 1,02y + 62
which give a lineal relation between them, the rates of the correlation being
very significant (0,98 and 0,96) the figures arrived at showing only small dif-
f erences .
2.6 - Resumé:
ANNUAL RAINFfiLL
Averages :
3 - RUNOFF
Period of 50 years ........... 579 mm
Period of 25 years ........... 560 mm
Period of 15 years ........... 566 mm
Period of 10 years ........... 582 mm
Rainiest year ................ 971 mm (1924/25)
Rainiest year w/lOO year occrence
........................ 1 101 mm
Driest year .................. 355 mm (1963/64)
Driest year w/lOO year OCCUT-
rence ........................ 290 mm
3.1 - Elementary Principles:
In view of there not being any measurements of the volume of the Limpopo
River in Portuguese territory, it was necessary to resort to comparative studies,
taking as a basis the specific flowage in the various hydrometric stations,
existing upstream in the region of Mapai.
./*

148
3.2 - Details of the Study
Barrows, in his book "Water Power Ehgeneering" stated that "it is often
possible to consider, without serious error, that the specific volume of a river
is similar to the successive contours along the same river".
also advises that, whenever possible, comparisons and corrections should be esta
blished, not only in respect to the rainfall, but also to the altitude, slopes,
constitution and the rock formations of the soil.
The same author
Based on the measurements made at the Hydrometric stations of the Repu-
blic of South Africa and Rhodesia which cover 195 841 km2, 79,6% of the respective
basin in the area of the barrage of Mapai, we took into account the specific run-
off year by year and we calculated the specific runoff of the locality under
study.
Since 1963, efforts have been made to estimate the average annual runoff
always using the methods of the specific volume but adopting various criteria.
The average annual figure obtained for the period of 12 years was 3 095
million cubic metres, which corresponds to the specific runoff of 12 580 m3
k2 - 1 (chart attached).
Thus we have:
Study in 1963 ( 6 years) specific runoff 13 700 m
3
(k2)-1
Study in 1965 ( 7 years) specific runoff 13 658 m3 (k2)-1
Present study (12 years) specific runoff 12 583 m 3 (k2)-1
If we observe the sequence of the years in which measurements existed
for the 3 studies realised, it will be noted that the period of 12 years includes
an excessively dry year (1963-1964) and one high runoff (1966-67) while there
were no measurements taken in 1965-66 at the fundamental station of study (Beit-
bridge) due to the hydrometric station having been under water (floods in Februa-
ry, 1966) and as a result of which, the figure now obtained is necessarily defec-
tive.
The 1970 study covering a period of 15 years, places the specific average
3 2
runoff as 13 O00 m (k )-l.
For the 246 O00 km2 of the river basin, the figures obtained for the
average annual runoff in the region of Mapai are respectively as follow:
1963 study .............. 3 370 million m 3
1965 study .............. 3 360 million m 3
1970 study .............. 3 200 million m 3
Present study ........... 3 095 million m
3
./.

149
Any of these figures fall within the admissible limits based on the rate
of the runoff observed at the stations of Beitbridge and Vila Trigo de Morais,
the average of which is respectively 0,015 and 0,026.
Thus for a figure of 3 O00 million m3 and for the average rainfall of
579 nun., runoff coefficient is 0,021, which is within the observed limits.
Using the method of Coutagne only for the average annual figures, for
a rainfall of 579 mm as the average over 50 years, we arrive at a runoff of
2 969 10 6 m 3 and for 582 nun as the average for the past 10 years, the figure
of 2 999,7 10
6
m
3 .
Observing and trying all these ways and means, we shall adopt chart
attached hereto for the annual runoff because, as they arise from direct mea-
surements, they fall within all the estimated figures.
3.3 - Monthly Distribution
The monthly distribution is based on a hydrometric station in the Repu-
blic of South Africa (Beitbridge), which already has years of sufficient read-
ings, its area being much like that of Mapai.
Thus we haïe:
4 - FLOODS
October ................ 0,5%
November ................ 0,6%
December ................ 4,6%
January ................ 25,%
February ................ 33,8%
March ................ i8,i%
April ................ 8,s
May ................ 4,3%
June ................ i,%
July ................ 1,1%
August ................ o, 9%
-
September ................ 0,3% 10%
The River Limpopo is typically torrential and as such, not only dries
during consecutive months as it is susceptible to exceptional floods.
Many records of the volume of floods have been compiled in the Republic
of South Africa since 1915, always based on specific volumes and they obtained
. /.

150
measured details which extended to the area of Mapai, gave us the figures of
16 925 m3/s (1933) and 12 792 m3/s (1966).
Thus, using various formulas, one can estimate the volume of floods for
return periods of 100 and 200 years.
100 years 200 years
Water affairs formula ............ 13 243 14 963 m3/s
Mimoso Loureiro formula ........ 14 304
18 375 II
Fuller formula ................. 19 763 21 284 11
(Period 34 years and specific
volumes measured)
Larivaille formula ............... - 15 375 'I
The estimate is thus very difficult and depends a great deal on the
type of barrage to be adopted.
The hydrograph of a maximum flood was also determined, based on the
formula of Giandotti in the calculation of the times of concentration of the
peak and of the swell of the flood, and on the hydrographs of the floods record-
ed at the border (Pafuri) during the years 1955, 1958, 1959, 1966 and 1967.
It was verified that, in the flood of 1966, there was agreement in the
calculated and observed times, because the calculation placed the figure at
153 hours and from observation, at 150 hours for the time of concentration,
there being, however, a difference in the time of the swell of 600 with 724
hours (sketch attached).
5- Evaporation and Solid flows
Using the details measured in Rhodesia, South Africa and Mozambique,
we can place the resulting evaporation at Mapi as 1 344 mm. As to solid volumes,
the figures are few and vary greatly; as for the rest, definitely, confix
med by the complex composition of the hydrographic basin, as for the same volu -
me one obtain 21,8 kg/s and 97,l kg/s, without any meaning.

0
N
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N
151

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1

Relation of hydrological programs of the Center of Hydrographic Studies
for complete studies of hydraulic resources with insufficient data
Dr. Rafael HERAS
1 LIS TA Lista las ochenta columnas de las fichas.
The programme lists the eighty columns of the cards.
2 LIS 65 Lista las ochenta columnas de las fichas, poniendo en la
cabecera de la pagina las columnas numeradas y saltan-
do página cada 65 fichas.
It lists eighty columns of the card, putting on the heading
of the page the numered colum and skipping a page after
every 65 cards.
3 LIFLA Lista fichas con flags.
It lists carde with flags.

156
4 LIMO1 Lista cinta de papel del limni'grafo.
It lists ribbon of the limnigraphe.
5 LICAN Listado de longitudinales.
Longitudinal listing.
6 LIGUI Listado de transversales.
Transversal listing.
7 LGMET Listado de datos geográficos.
Listing of geographical data.
8 LPMET Lista la cuenca, número de estación, aflo y datos de prg
cipitaciones mensuales.
It lists the basin, number of station, year and monthly
rainfall data.
9 LFRNT Lista cinta de papel perforada en código FERRANTI.
It lists perforated ribbon in FERRANTI code.

10 LI-C-CA Lista cabeceras de los canales de aforos.
It lists the headings of the channels of valuation.
11 L-NI-24H Lista precipitaciones máximas en 24 horas.
It lists the maximum rainfall in 24 hours.
12 LTVP Lista tablas de vertidos probables,
It lists tables of probable downpour.
13 LTAC Lista las tablas de alturas-caudales.
It lists tables of altitudes-flows.
157
14 TAVAL Lista los valores de las curvas alturas-caudales y está
preparado para obtener valores que no figuren en las ta
blas, interpolando linealmente entre los dos puntos más
próximos.
It lists the values of altitudes-flows lines and it is
prepared to obtain values not appearing in the tables,
by linear interpolation between the closest points.

158
15
16
TAB- 1
TAB-2
17 TAB- 3
18 TAB- 4
Este programa lista, de las cinco fichas de que consta
la información de cada pozo, lo siguiente: número de pg
zo, fechas de muestreo, número de muestras (laborato-
rio), latitud, longitud, nivel de agua/l, columna de agua,
horas de bombeo, caudal en Ils., procedencia, número
de laboratorio, fecha del análisis, temperatura del aire,
temperatura del agua e indicador.
This programme lists the five cards which contains the
information of each well, as the following: number of
well, dates of sampling, number of samples (laboratory),
latitude, longitude, level of water/l, column of water,
hours of pumping, flow in Ils., origin, laboratory number,
date of analysis, temperature of the air, temperature of
water and indicator.
Lista: número de pozo, fecha de muestreo, calcio, mag-
nesio, manganeso, sodio, potasio, cloruro, sulfato, fluo
ruro, silice, fosfato y carbonato.
It lists: number of well, date of sampling, calcium,
manganese, sodium, potassium, chloride, sulphate,
fluoride, silica, phosphate and carbonate.
Lista: número de pozo, fecha de muestreo, bicarbonato,
nitrito, nitrato, amoniaco, boro, hierro, pH, resistivi-
dad, gravedad especifica, sólidos disueltos, litio, estrog
cio, ni’quel.
It lists: number of well, date of sampling, bicarbonate,
nitrite, nitrate, boron, ammonia, iron, pH, resistivity,
gravity/m, solid in dissolution, lithium, strontium, -
nickel.
Lista: nimero de pozo, fecha de muestreo, cobalto, iodo,
bromo, molibdeno, zinc, plomo, cromo, cobre, vanadio,
mer curio, ar sé nico.

159
It lists: number of well, date of sampling, cobalt, iodine,
bromine, molibdenum, zinc, plumbum, chromium, cop-
per, vanadium, mercury and arsenic.
19 TAB-5 Lista: número de pozo, fecha de muestreo, pH, resisti-
vidad, gas carbónico libre, oxigeno disuelto, dureza, dg
reza (sin carbonatos), alcalinidad, T. A., HC03.
It lists: number of well, date of sampling, pH, resistivity,
free carbonic gas, oxygen disolved, hardness (without
carbonates), alkalinity, T. A., HC03.
20 TNFAG Tablas de interpolación polinómica de cuarto grado.
Tables of polynomical interpolation of the fourth grade.
21 TANG Tablas de senos, cosenos y tangentes.
Tables of sine, cosine, tangents.
22 TNUA Mete en disco tablas de números aleatorios.
It puts in the disk tables of fortuitous numbers.
23 T-V-P Mete en disco tablas de datos hidrológicos.
It puts in the disk tables of hydrological data.

160
24 T-EVA
25 EN-PRT
26 SAPRI
27 PDPRC
28 PC 128
Mete en disco tablas de evaporaciones.
It puts in the disk tables of evaporation.
Dada una serie de estaciones, almacena en disco dichas
estaciones.
It stores in the disk stations which are previously given.
Obtiene un listado de las estaciones almacenadas en disco.
It obtains a listing of stations stored in the disk.
Perfora datos con anos consecutivos para los diversos
programas de regulaciones.
It perforates data with consecutive years for the different
programs of regulations.
Dada una serie de caudales diarios con formato 12 F 6.2,
los perfora con formato 9 F 8.3, para ser utilizados por
el programa I~TCDAP".
Given a series of daily flows with format 12 F 6. 2, it
perforates them with format 9 F 8. 3, to be used by the
program I'TDFAF~~.

29 PRD-I Ordena un máximo de 1.000 datos, de mayor a menor,
con formato 12 F 6.2.
161
It puts in order a maximum of 1.000 data, in descendent
order, with format 12 F 6. 2.
30 PRD-2 Igual que el anterior, pero con formato 9 F 8.3.
The same as above, but with format 9 F 8.3.
31 PRD-3 Ordena números en coma fija, de mayor a menor.
It puts in order numbers with fixed point, in descendent
order.
32 DUPLP Duplica las ochenta columnas de las fichas.
It duplicates the eighty columns of the cards.
33 DUP-Mp Duplica las ochenta columnas, modificando la columna
que se desee.
It duplicates the eighty columns, modifying the column
that is wished.
34 SUMNU Suma o resta un número a una serie de datos mensuales.
It adds or deducts a number from a series of monthly data.

162
35 SUMRE Dadas dos series de datos mensuales, las suma o las
resta.
it adds or deducts two series of monthly data which are
given.
36 SUMAR Suma series de datos hidrológicos.
37 MULTI
38 MULTA
39 MUL 12
It adds series of hydrological data.
Multiplica series de datos por un número fijo y obtiene
el listado, así como las fichas perforadas, con estos nue
vos valores.
It multiplies series of data by a fixed number and it
obtains the listing, in the same way as the perforated
cards, with this new values.
Multiplica los 12 números de la primera fila por cada
uno de los datos, los pone en orden decreciente y los lis
ta en 12 columnas.
It multiplies the 12 numbers of the first row by each one
of the data, it puts them decreasing order and it lists -
them in 12 columns.
Multiplica los datos mensuales por el número que ocupa
el lugar correspondiente a ese mes en la primera ficha,
lista y perfora.
It multiplies the monthly data by the number that occupies
the corresponding place of that month in the first card, it
lists and Derforates.

40 MULAN
41 MRS 12
42 MA TEN
43 SECUA
44 DIS -KM
163
Multiplica los datos mensuales de cada ano por el &me-
ro que ocupa el lugar correspondiente a ese ano en las
primeras fichas que lee.
It multiplies the monthly data of each year by the number
that occupies the corresponding place to that year in the
first cards that is read.
Multiplica, suma o resta dos o una series de datos sin
limitación.
It multiplies, adds or deducts two or one series of data
without limitation.
Eleva una matriz a la potencia enésima.
It elevates a matrix to the n power.
Resuelve un sistema de ecuaciones (40 como máximo).
It solves a system of equations (40 as maximum).
Dada la situación geográfica de una serie de estaciones
por su latitud y longitud, este programa selecciona los
grupos de estaciones a comparar con el criterio de que
las distancias entre las estaciones sean menores de una
cantidad fija.
Given the geographical position of a series Of stations by
their latitude and longitude, this programme chooses the
group of stations to be compared with the criterium that
the distance among the stations be smaller than a fixed
quantity.

164
45 LISDA Lista series de datos mensuales (sin limitación de exten
sión), imprime cabecera y calcula la suma anual y las -
medias mensuales.
It lists series of monthly data (without limitation in its
scope), prints heading and computes the yearly sum and
the monthly averages.
46 MAXLL Dados los valores de precipitación total mensual, dias
de lluvia y los valores máximos en 24 horas, obtiene los
máximos en 24 horas y los dilas de lluvia a escala anual.
Given the values of total monthly rainfall, days of rain
and the maximum values in 24 hours, it obtains the -
maximum in 24 hours and the days of rain in a yearly
s cale.
47 INT-ES Dados los caudales medios mensuales y anuales, la apoy
tación media de los años precedentes y el caudal máximo
(medios diarios e instantáneo y la fecha), obtiene cauda-
les medios anual y mensual, caudal y aportación mensual.
Deduce la aportación y caudal anual, caudal medio y apor
tación media de la serie anual.
Given the monthly and annual average flows, the average
afford of the preceding years and the maximlim flow (daily
and instantaneous averages and the date), it obtains year
ly and monthly average flows, monthly flow and afford. It
deducts yearly afford and flow, average flow and the aver
age afford of the yearly series.
48 INT-EM Dado el volumen embalsado, aportación de salida y media
precedente, obtiene entradas y salidas, reserva, aporta-
ción del año, aportación de entrada y salida, media de e;
trada y salida (aportación y caudal).

49
50
51
INT -CA
ADfDB- 1
ADfDB-2
165
Given the stored volume, afford of exit and the preceding
average, it obtains entries and exits, reserve, yearly -
afford, afford of entry and exit, average of entry and -
exit (afford and flow).
Realiza la misma función, pero sin el caudal máximo
instantáneo.
It performs the same function, but without the maximum
instantaneous flow.
Dadas las series de datos hidrológicos de un conjunto de
estaciones a escala anual, mensual, etc., forma "esta-
ción tipo" (media aritmética de las estaciones de cada
grupo) y a continuación compara cada una de las estacio
nes con su "estación tipo", acumulando las series y dag
do, además de las sumas acumuladas, la relación entre
las acumulaciones de cada estación con las acumulacio-
nes de la "estación tipo".
Given the series of hydrological data of an assembly of
stations at yearly scale, monthly scale, etc., it forms
''type station!' (arithmetical mean of the stations in each
group) and afterwards compares each stations with its
'hype station", accumulating the series and giving besides
the accumulated sums, the relation among the accumula-
tion of each station with those of "type station".
Este programa es análogo al anterior, pero no utiliza -
estación tipo, haciendo todas las comparaciones posibles
en cada grupo.
This programme is analogous to the preceding one, but
it does not use type station, performing all the possible
comparisons in each group.

166
52 A-AC-96 Acumula datos hidrológicos mensuales (de 1 en 1 hasta
96 en 96 meses). Imprime 96 cuadros.
It accumulates hydrological monthly data (from 1 to 1 up
to 96 in 96 months). It prints 96 charts.
53 AC-96-P Este programa es análogo al anterior, pero perfora los
resultados en ficha.
This programme is analogous to the preceding one, but
it perforates the results in card.
54 AM-AC 1 Acumula aportaciones mensuales y ordena de menor a
mayor .
It accumulates monthly affords putting in ascendent order.
55 AM-C 96 Dada una serie de datos hidrológicos mensuales, acumu-
la de 1 a 96 meses consecutivos, y los ordena de menor
a mayor.
Given a series of hydrological monthly data, it accum;I-
lates from 1 to 96 consecutive months, and putting them
in ascendent order.
56 AC-T~D Acumula todos los valores de una serie.
It accumulates all the values of a series.

57 AD~BP Dibuja en el Plotter los diagramas de las acumulaciones
dobles.
167
It designs in the Plotter the diagrams of the double - -
accumula tion.
58 ALAOA Dadas las aportaciones diarias, las lista, ordena de mg
nor a mayor y las acumula.
Given the daily affords, it lists, puts in ascendent order
and accumulates them.
59 TCDAP Dada una serie de caudales diarios, la transforma en -
aportaciones diarias.
Given a series of daily flow, this programme transforms
them in daily affords.
60 TCAP~ Dada una serie de caudales mensuales, la transforma en
aportaciones mensuales.
Given a series of monthly flows, this series is transform
ed in monthly affords.
61 TAPPC Dada una serie de aportaciones mensuales, la transforma
en caudales mensuales.
Given a series of monthly affords, it transforms them in
monthly flows.

168
62 TANAD Dada una serie de aportaciones naturales diarias, la -
transforma en aportaciones derivables diarias.
Given a serles of daily natural affords, the programme
transforms them into daily derivable affords.
63 C-C-D-ES Dada una serie de caudales diarios, obtiene los caudales
derivados diarios en m3/s., medias y aportaciones men_
suales, caudales clasificados, aportación y caudal total
del año.
Given a series of daily flows, the programme obtains the
daily derivable flows in CU. m/s., averages and monthly
affords, classified flows, afford and total flow of the year.
64 C -C -D-CA Realiza la misma función que el programa anterior, pero
con caudales mensuales.
It performs the same function as the preceding programme,
but with monthly flows.
65 C-C-D-EM Dados los volúmenes y salidas diarias de un embalse, ob
tiene las reservas diarias (Hm3), caudales diarios (sali-
das en m3/s. ), media mensual, salida y entrada mensual,
evaporacion y un resumen anual (caudales medios, salida,
entrada, evaporación).
Given the volumes and the daily exits of a reservoir, the
programme obtains the daily reserves (CU. Hm), daily
flows (exits in CU. m/s), monthly average, monthly entry
and exit, evaporation and an annual summary (averages
flows, entry, exit, evaporation).

169
66 C-C-D-E1 Dados los datos de alturas de escala diaria y los valores
de las curvas alturas-caudales de una estación, calcula
los caudales diarios de dicha estación. Los caudales que
no figuran en las tablas se calculan interpolando lineal-
mente entre los dos más próximos.
Además del caudal diario, este programa obtiene los cag
dales máximos y mínimos, los caudales medios mensua-
les y las aportaciones mensuales.
Given the daily scale heights and the values of the height-
flow charts of a station, the programme computes the --
daily flows of said station. The flows that are not appear-
ing in the tables, are calculated by linear interpolation -
between the closest points.
In addition to the daily flow, this programme obtains the
maximum and minimum flows, the monthly average flows
and the monthly affords.
67 C-C-D-E2 Dados los datos de alturas de escala diaria y los valores
de las curvas alturas-caudales de una estación, calcula
los niveles diarios en metros, los caudales diarios en
m3 /s. , medias mensuales, máxima instantánea, aporta
ci& mensual en Hm3, caudales clasificados y un resu--
men de los datos del año (aportación y caudal total y es-
pecifico y caudales caracteristicos).
Given the daily scale heights data and the values of height-
flow lines of a station, the programme computes the daily
levels in meters, the daily flows in CU. m/s., monthly -
averages, instantaneous maximum, monthly afford in --
CU. Hm, classified flows and an annual summary data --
(afford and total flow, specific flow and caracteristic --
flows).
68 C-C-D-ES Dados los datos de alturas de escala diaria y los valores
de alturas-caudales de una estación, calcula los caudales
diarios y el caudal medio anual.
Los datos son los obtenidos en limnigrafo.

170
69
71
72
CURGA
Given the daily scale heights data and the values of height-
flow lines of a station, the programme computes the daily
flows and the yearly average flow.
Data are obtained by the limnigraphe.
A partir de unos puntos base tabula una tabla de gastos.
Lista y dibuja los diagramas de las curvas de gastos.
From a basic point, it tabulates a tables of expenses.
It lists and designes the diagram of the expense lines.
ME Y ME Dadas las aportaciones mensuales de una serie de esta-
ciones de una cuenca, calcula e imprime las medias mec
suales de cada estación y la media de las medias de todas.
MEDIP
Given the monthly affords of a series of stations of a basin,
it computes and prints the monthly average of each station
and the average of all means.
Dados los datos diarios de años de una estación, los
imprime y calcula las sumas y medias mensuales de ca
da ailo y las medias de las medias (mediorum).
Given the daily data of c years of a station, the programme
prints them and computes the sums and monthly averages
of each year and the average of the n_ means (mediorum).
MET, ME Dada una serie de datos mensuales, calcula ias medias
para cualquier periodo.
It calculates the average for any period of a series of
monthly data, which are given.

73
74
75
76
CIC LJD
MED -AN
A-ESP
INF - 1
171
Calcula las medias acumuladas en periodos de n años y
sus relaciones con la media del periodo total.
It calculates the accumulated averages in periods of E
years and their relations with the average of the total
period.
Dada una serie de datos anuales, calculala media para
cualquier pedodo.
Given a series of yearly data, the programme calculates
the average for any period.
Dadas unas series pluviométricas reales, obtiene una se
rie real, media de las anteriores, a partir de la cual ob-
tiene otra de precipitaciones efectivas, de la que se con-
siguen las aportaciones especificas y los caudales en una
cuenca.
Given an actual pluviometrical series, the programme
obtains an actual series, average of the preceding, from
which it obtains another series of effective rainfalls, -
from which it gets specific affords and the flows in a basin.
Dadas las precipitaciones mensuales de una cueBca y los
coeficientes de capacidad de infiltración mensual en mm.,
de humedad inicial del suelo y de superficie en Has., ob-
tiene las aportaciones especificas, infiltración y evapora
cion.
Given the monthly rainfall of a basin and the coefficients
of monthly infiltration capacity in mm., initial humidity
from the earth and from surface in Has., it gets the --
specific affords, infiltration and evaporation.

172
77 INF - 2 Realiza la misma función que el programa anterior, pe-
ro a nivel diario.
It performs the same function as the preceding one, but
in a daily level.
78 EVAP- 1 Dados los volúmenes, superficies y reserva, halla las
evaporaciones.
The programme computes the evaporations, knowing the
volume, area and stock.
79 EVAPT Dado el número de estaciones, superficie de la cuenca
(kmz), aportación media (media de una serie) y precip'
tación media, obtiene la aportación en Hm3, coeficiente
de escorrentia y déficit de escorrentía.
It obtains the afford in CU. Hm., runoff coefficient and
deficit of runoff, knowing the number of stations, area
of the basin (sq. Km. ), average of the afford (average
of a serie) and the average rainfall.
80 REST-l Lista: número total de muestras, media aritmética, va-
lor máximo, valor minimo, desviación tipica, tempera-
tura del agua.
The programme lists: total number of samples, arithmeg
cal mean, maximum and minimum value, standard desvia
tion. temperatura of the water.
81 REST-2 Lista los mismos parámetros para calcio, magnesio, mall
ganeso, sodio, potasio, cloruro, sulfato, fluoruro, silice,
fosfato y carbonato.

82 REST-3
83 REST-4
84 D-FLSI
85 GE~NE
It lists the same parameters for calcium, magnesium,
manganese, sodium, potassium, chloride, sulphate, -
fluoride, silica, phosphate and carbonate.
173
Lista los mismos parámetros para bicarbonato, nitrito,
nitrato, amoniaco, hierro, resistividad y sólidos disuel
tos.
It lists the same parameters for bicarbonate, nitrite,
nitrate, ammonia, iron, resistivity, solid in dissolution.
Lista los mismos parámetros para pH, resistividad, gas
carbónico libre, oxígeno disuelto, dureza, dureza (sin -
carbonatos), alcalinidad, T. A. , HC03.
It lists the same parameters for pH, resistivity, free
carbonic gas, oxygen dissolved, hardness, hardness -
(without carbonates), alkalinity, T. A., HC03.
Dado el perimetro y la superficie por encima de cada co-
ta de una cuenca, obtiene el rectángulo equivalent e, coe-
ficiente de Gravelius, indice de pendiente, pendiente m e-
dia y altitud media de la cuenca.
Given the perimeter and the area above each elevation of
a basin, the programme obtains the equivalent rectangle,
Gravelius factor, pendant index, average of the pendant
and average altitude of the basin.
A partir de coordenadas de puntos fijos dados, de medi-
ciones, direcciones y distancias para una conexión de pu9
tos aislados o múltiples, el programa obtiene las coorde-
nadas de los nuevos puntos.

86
87
88
DIS-2P
HELME
REPPU
89 ARCES
From coordinates of fixed given points, of measuring,
directions and distances to a connection of isolated or
multiples points, the programme gets the coordinates
of the new points.
Dadas las coordenadas de dos puntos, calcula su dis-
tancia.
Given the coordinates of two points, it calculates their
distance.
Convierte coordenadas instrumentales en terrestres
(Helme rt ).
It converts from instrumentais coordinates to terrestrial
(Helmert).
En función de las coordenadas de los puntos calcula azi-
mutes y distancias para replanteo.
In function of coordinates of the points, it calculates azi-
muths and distances to be reconsidered.
Calcula la superficie en planta que abarca cada curva en
un embalse.
It calculates the area which comprises each line in a
reservoir.

90 EAKIN Dada la superficie del tramo máximo de un embalse, su
longitud y la superficie del perfil de sondeo, obtiene el
volumen de esas superficies.
Given the area of the maximum stretch of a reservoir,
their longitude and the area of the profile of sounding,
it obtains the volumes of those surfaces.
91 RAPEM Cubica un embalse según la fórmula RAPEM.
It cubes a reservoir according to formula RAPEM.
92 C~R-AM Dada una serie mensual de datos hidrológicos, calcula
la correlación ortogonal, los momentos de la serie reg
pecto al origen y el coeficiente de correlación.
175
Given a monthly series of hydrological data, it calculates
the ortogonal correlation, the moments of the series with
regards to the origin and the correlation factor.
93 CPR-LM Calcula la correlación lineal mensual, obteniendo el cog
ficiente de correlación, las medias, varianzas, disper--
si&, coeficiente angular y ordenada en el origen de la
recta de regresión.
The programme calculates the monthly linear correlation,
obtaining the correlation factor, the averages, variances,
dispersion, grade angle coefficient and ordinate at the -
origin of the regression straight.
94 CPR-AN Este programa se diferencia del anterior Únicamente en
que, partiendo de los datos mensuales, utiliza solamente

176
los anuales, efectuando la correlación entre ellos con -
arreglo al esquema ya reseñado.
This programme differs from the previous one only in
that starting from the monthly data, it uses only the -
annual ones and performing the correlation among them,
in accordante with the indicated scheme.
95 C~R-DM Con los datos hidrológicos mensuales de tres estaciones,
z, x e y, realiza la correlación doble de x e y con z, ob-
teniendo la ecuación del plano de correlación.
With the monthly hydrological data of the three stations,
z, c and y, it performs the double correlation of x and y
with z, obtaining the equation of the plane of correlation.
96 CPR-PA El esquema es igual al de los anteriores que realizan cg
rrelaciones, dando éste la ecuación de la parábola de re
gresión de y sobre x.
The scheme is identical to the previous one which performs
correlations, , the equation of the parabola of regression
of y on x is given by the scheme.
97 C ~ R - ~ R Dadas dos series mensuales de datos hidrológicos, calcg
la la correlación ortogonal obteniendo la recta cuya suma
de cuadrados de distancias a los puntos es minima.
Given two monthly series of hydrological data, the pro-
gramme calculates the ortogonal correlation, obtaining
the straight whose sum of squares to the points in min-
imum.

98 CfbR-A-p Dada una serie mensual de datos hidrológicos, realiza
la correlación anual ortogonal.
Given a monthly series of hydrological data, it performs
the yearly ortogonal correlation.
99 CPR-48 Realiza la correlación ortogonal para los meses de estia
je (4) y el resto de los meses (8).
It performs the ortogonal correlation for the summer
months (4) and to the balance of the months (8).
100 CPR-LP Dadas dos series de datos hidrológicos, obtiene la recta
de correlación ortogonal entre sus logaritmos.
Given two series of hydrological data, it gets the straight
of ortogonal correlation among their logarithms.
10 1 COR-O-P El esquema es igual al del propama "CqR-qR" y, ade-
más, dibuja la nube de puntos en el Plotter.
The scheme is identical to "COR-OR" programme, in
addition it draws the clouds of points in the Plotter.
102 cpycfb Completa y corrige una serie de datos hidrológicos dan-
do las ecuaciones de las rectas de regresión (y=ai x - bi)
entre las dos series (se puede hacer con una ecuación o
con dos simultáneamente).
It completes and corrects a series of hydrological data,
giving the equation of the regression straights (x=ai x - bi)
between the two series (it can be done with an equation or
two simultaneously).

178
io3 ~p-DRR Completa los datos hidrológicos según la recta de regre-
sión.
It completes the hydrological data according to the reg-
sion straight.
104 IN-D-M1 Inventa datos mensuales de una estación con datos anua-
les, a partir de los datos mensuales de otras dos esta--
ciones.
It creates monthly data of a station with yearly data, star
ting from the monthly data of other two .stations.
105 IN-D-M2 Inventa datos mensuales de una estación con datos anua-
les, a partir de los datos mensuales de otra según la fÓ'
mula B (I) = [A (I) * (SUMB (I) / SUMA (I) 1.
It creates monthly data of a station with yearly data,
starting from monthly data of other station, according
to formula: B (I) = I A (I) * (SUMB (I) / SUMA (I) 1 .
106 COQUI Calcula 20 correlaciones ortogonales, entre elementos
quhnicos, pintando por Plotter los puntos y la recta de
regresión y dos paralelas a una distancia igual a la dis-
persion.
The programme calculates 20 ortogonal correlations,
among chemical elements, drawing in Plotter the points
and the regression straight and two parallel lines to a
distance identical to the dispersion.

179
107 BERKA Dibuja el diagrama de Berkaloff-Scholler por el Plotter,
uno por cada pozo, en una escala logaritmica. Pinta los
puntos de los siguientes elementos: CA, MG, ALC, CL,
SO4, HCOQ + CO3, NO3. y une con segmentos dichos -
puntos.
It designs the diagram of Berkaloff-Scholler by means of
Plotter, one diagram to each well, in logarithmical scale.
It draws the points of the following elements: CA, MG, -
ALC, CL, SO4, HC03 t COQ, NO3, and it joins the - -
mentioned points with the segments.
108 STIF Dibuja el diagrama de Stif.
The programme draws the Stif diagram.
109 PIPER Dibuja el diagrama de Piper. Consta de dos triángulos
equivalentes; en la base del primero se marca en 70 el va
lor CA, en mgl/l, y en los otros dos lados, MG y NA+K,
y mediante paralelas a 10s lados opuestos obtenemos un
punt o.
De la misma forma, en el segundo triángulo pintan en el
lado base, CL, y en los otros dos CO3 t HCH03 y so4 t
t NOP en 70 y mediante paralelas obtenemos otro punto.
Esta operación se repite para cada pozo y para las ocho
zonas.
It draws the Piper diagram. It is composed of two equiva
lent triangles; in the base of the first one, it marks in 70
the value CA, in mgl/l, and in the other two sides, MG
and NA t K, and by means of parallel lines to the opposite
sides obtaining a point.
In the same way, in the second triangle, it marks in the
sidebase, CL and in the other two sides, Co3 + HCHO
and SO4 + NOP in 70 and by parallel lines obtaining another
point. This execution is repeated for each well and for the
eight bands.

180
i10 G ~ K A los valores de una serie de datos hidrológicos, le ajug
ta una ley de Goodrich y contrasta la bondad del ajuste -
mediante el test de Kolmogoroff.
Given a series of values of hydrological data, it fits with
Goodrich’s law and contrasts the perfection of the fitting
by Kolmogoroff’ test.
111 GPK~L A partir de una serie de datos hidrológicos anuales, se
ajusta una ley de Goodrich y se contrasta la bondad del
ajuste mediante el test de Kolmogoroff.
Los datos de entrada son series mensuales. El programa
obtiene las series anuales mensuales, la media, los mo-
mentos respecto al origen de orden 2 y 3, los momentos
centrales de segundo (varianza) y de tercer orden, as( -
como los parámetros de la ley de distribución de Goodrich.
Starting from a series of hydrological annual data, Good-
rich’s law is adjusted and the perfection is contrasted by
Kolmogoroff’test.
The entry data are monthly series. The programme - -
obtains, annual series, monthly series, the average, the
moments with regard to the origin of order 2 and 3, the
central moments of second (variance) and third order, -
the programme obtains also the parameters of the Good-
rich distribution law.
112 GPKAF Dada una serie de datos hidrológicos, ajusta la ley de dig
tribución de Goodrich.
Given a series of hydrological data, the Goodrich distri-
bution law is adjusted by the programme.

113 GPKPL Realiza la misma función que el programa IIGQKOLII y
dibuja las curvas.
It performs the same function that the "GOKOL"
programme and draws the curves.
114 GPK 70 Dada una serie de datos hidrológicos, ajusta una ley de
Goodrich y contrasta la bondad del ajuste mediante el
test de Kolmogoroff.
181
Given a series of hydrological data, it fits with Goodrich's
law and contrasts the perfection of the fitting by Kolmogo
roff ' s tes t.
115 GUMB 1 Ajusta una ley de Gumbel a una serie de datos hidrológi-
cos. El programa nos da los diversos valores que resul-
tan de la ley de Gumbel, ajustada para tiempos de recu-
rrencia de 5, 10, 25, 50, 100, 500 y 1000 anos ylista --
los datos originales clasificados de menor a mayor, asig
nándoles a cada uno la frecuencia 2n-1 / 2 N, donde n es
el número de orden y N el total de datos.
Datos de entrada (12 F. 6. 2). Anuales.
This programme fits the Gumbel's law to a series of --
hydrological data. The programme gives us several values
according to the Gumbel's adjusted law, for time of --
recurrences 5, 10, 25, 50, 100, 500, 1000 years and it
lists the original data classified in a crecent order assign-
ing to each one a frecuency equal to 2n- 1 / 2 N, where n is
the number of order and N the total number of data.
Entry data (12 F 6. 2). Yearly.
116 GUMB 2 Realiza la misma función que el programa "GUMB l", pg
ro los datos de entrada son (9 F 8.3).
It performs the same function as "GUMB l", but the entry
data are (9 F 8. 3).

182
117 GUMB 3
118 GUMB 4
119 GUMB-P
Realiza la misma función que el programa "GUMB 1".
pero los datos de entrada son mensuales.
It performs the same function as the "GUMB l", but the
entry data are monthly data.
Ajusta una ley de Gumbel a una serie de datos mensua-
les. Obtiene mensuales y máximos anuales.
This programme fits the Gumbel's law to a series of
monthly data. It obtains also monthly and maximum
annual series.
Realiza la misma función que el programa "GUMB 1"
y además dibuja la nube en el Plotter.
It performs the same function as the programme "GUMB 1"
and draws the clouds of points in the Plotter.
120 C-C-D Dados los datos diarios de aportaciones naturales, se ob
tienen datos diarios de aportaciones derivadas para dis-
tintos caudales de derivación, con los que se obtienen ds
tos mensuales de aportaciones naturales y derivadas. Con
estas parejas de datos se ajustan unas curvas, que sirven
para obtener datos mensuales de aportaciones derivadas
cuando sólo se tengan datos mensuales de aportaciones -
naturales.
Given the daily data of natural affords, the programme -
obtains daily data of derived affords for different flows -
of derivation by which are obtained monthly data of natural
and derived affords; with these pairs of data, some lines
are adjusted, which are used to obtain monthly data of
derived affords, when only monthly data of natural affords
are had.

183
1 .21 GAMMA Dada una selección de 10 valores equidistantes de la fun_
ciÓn g amma de X (con 16 cifras significativas) y sus seis
primeras diferencias en el intervalo (1, 2), obtiene por
interpelación cualquier g amma de X.
Given a selection of 10 equidistant values with the func-
tion g amma of X (with 16 significative digits) and their
six first differences in the interval (1. 2), the programme
obtains by interpolation any gamma of X.
122 NUMRE Dada una selección de 10 valores equidistantes de n y
sus seis primeras diferencias en el intervalo (-0. 75,
4.25), obtiene la función inversa de la función de Good-
rich.
Given a selection of 10 equidistant values of n and their
six first differences in the interval (-0. 75, .4. 25), the
programme obtains the inverse function of the Goodrich
function.
123 AM-C 4P Ajusta una ley de frecuencias parabólicas a los diez m e-
nores valores de una serie de aportaciones clasificadas,
sacando el valor de la aportación correspondiente a una
garantía dada.
The programme fits a law of parabolic frecuency to the
ten least values of a series of classified affords, obtain-
ing the value of the afford corresponding to a given - -
guarantee.
124 CD SOO Calcula las aportaciones acumuladas correspondientes a
una garantía dada, obteniendo la curva de seguridad.
It evaluates the accumulated affords corresponding to a
given guarantee, obtaining the safety lines.

184
125 CD SO1
126 CD 502
Calcula las aportaciones acumuladas durante un afio,
obteniendo la curva de seguridad.
It calculates the accumulated affords during a year,
obtaining the safety line.
Calcula las aportaciones acumuladas, después las clasi-
fica considerando los siguientes periodos:
Oct Nov Set
Oct + 1 Nov + 1 Set + 1
_ _ _ _ - - - - - - - - - - -
Ott + k NO~ + k Set + k
y obtiene después las aportaciones acumuladas corres -
pondientes a una garantia determinada. Posteriormente,
calcula las demandas acumuladas para los mismos peri2
dos y seguidamente las superficies evaporantes corres-
pondientes a un volumen cualquiera vi, tomando como sg
perficie evaporante en un mes la media aritmética de las
correspondientes al estado inicial y final, y aplicándolo
a la evaporación unitaria mensual.
A partir de estas pérdidas obtiene, para la curva de ga-
rantia dada, las pérdidas totales para cada periodo.
Finalmente, calcula por iteración la curva de seguridad
por meses, según la ecuación
siendo :
Ei (G) = volumen embalsado al principio del mes (i)
para las curvas de garantia G.
Dik
Pik
= demanda real acumulada desde el principio
del mes (i) durante k meses sucesivos.
= pérdidas del embalse acumuladas desde el
principio del mes (i) durante k meses suce-
sivos.
Aik (G) = aportación del embalse acumulada desde el
principio del mes (i) durante k meses suce-
sivos, que tiene una probabilidad G de ser
superada.

185
It evaluates the accumulated affords, and afterwards it
classifies them considering the following periods:
Oct Nov Set
Oct t 1 Nov + 1 Set + 1
_ _ _ _ _ _ _ - - - - - - - -
Ott t k NO~ t k Set + k
computing afterwards the accumulated affords corresponcj
ing to a determined guarantee. Lately, it evaluates the -
accumulated demands for the same periods and thereafter
the evaporating areas corresponding to a given volume vi,
taking as the evaporating areas in one month, the arithm-
etical mean corresponding to the initial and final state -
and applying this to the unitary monthly evaporation.
Starting from this losses, the programme obtains for the
given line of guarantee, the total losses to each period.
Finally, it evaluates iteratively the safety line by months,
according to the equation
where:
Ei (G) = stored volume at the beginning of the month (i)
for the guarantee line G.
Dik
'ik
= actual accumulated demand since the begin-
ning of the month (i) during k consecutive -
months.
losses in the reservoir accumulated, since
the beginning of the month (i) during k conse-
cutive months.
Aik (G) = accumulated afford in the reservoir since the
beginning of the month (i) during k consecutive
months, that has a G probability of being over
pas sed.

186
127 CDSSE
128 REGCV
Calcula las curvas de seguridad de un embalse, para -
cualquier nivel de garantía de suministro de una deman
da dada en función de la serie histórica de aportaciones
de hasta 60 anos de duración.
It evaluates the safety lines of a reservoir, to any level
of giiarantee of supply of a given demand in function of
the teorica1 series of affords up to 60 years of duration.
A partir de la serie de aportaciones mensuales en un pun
to, calcula las capacidades de embalse estricto mediante
el método de las diferencias acumuladas, por la expresión
c = q - Aki.
El mismo programa distingue dos casos:
a) Regulación a caudal constante
Mediante el programa obtenemos el principio y la
duración del período de vaciado del embalse, del
intervalo de meses sucesivos que da el máximo VE
lor positivo a la suma de las diferencias ni q - Aki,
siendo qi el caudal minimo continuo garantizado -
durante el periodo considerado. Calcula también el
volumen medio regulado en % de la aportación m e-
dia y en Hm3, el caudal regulado y las capacidades
de embalse estrictas para asegurar estos caudales
en tanto por ciento de la aportación media y en Hm3.
b) Regulación con caudal variable
En este caso el caudal para hacer el cálculo de la -
regulación es variable en cada uno de los meses. -
El programa nos da el principio y la duración del -
periodo de vaciado del embalse, volumen medio re
gulado en 70 de A m y en Hm3, capacidades de embF2
se estricto en 7' de A m y en Hm3 y, además, el nu-
mero de Has. regables con los volúmenes medios -
regulables.
Starting from the series of monthly affords in a point, the
programme calculates the capacities of strict reservoir
according to the method of the accumulated differences, -
by the expression C z ni q - Aki.
The same programme distinguishes two cases :

a)
Regulation at a constant flow
187
By means of the programme we obtain the beginning
and the duration of the period of emptying the resec
voir, the interval of following months which gives
the maximum positive value of the sum od differen_
ces ni qi - Aki, being qi the minimum continuos -
flow guaranteed during the period under consider-
ation. It calculates also the average regulated - -
volume in 70 of the average afford and in CU. Hm,
the regulated flow and the strict capacities of --
reservoir to assure these flows in percentage of
the average afford and in CU. Hm.
b) Regulation with variable flow
In this case the flow to perform the calculation of
the regulation is variable in each month. The pro-
gramme gives us the beginning and the duration of
the emptying period of the reservoir, averages -
regulated volume in 70 of A m and in CU. Hm, capa
cities of the strict reservoir in 70 of Am and in CU.
Hm in addition, the number of Has irrigables with
the average regulable volumes.
129 REG25 Realiza la misma función que el programa "REGCV",
pero la entrada de datos está calculada para que regule
estaciones durante 25 horas.
The programme performs the same function as the pro-
gramme "REGVF", but the entry data is evaluated to
regulate stations during 25 hours.
130 REGVA Dada una serie histórica de aportaciones mensuales, unos
consumos, una serie de precipitaciones mensuales sobre
el cultivo, unas capacidades de embalse máximo muerto
y dando distintos porcentajes del consumo, calcula las -
variaciones de volumen embalsado, los dé€icits y verti-
dos, después de abastecer los regadios con unos consu-
mos determinados, a los que descuenta la precipitación
sobre el cultivo. El programa tiene en cuenta la evapora
ciÓn mensual del embalse.

188
Given an historical series of monthly affords, some -
consumptions, a series of monthly rainfall over the crop,
a capacities of maximum dead reservoir and giving dif-
ferent porcentages of the consumption, this programme
calculates the variations of volume of the reservoir, the
deficits and emptying, them to supply the irrigated land
with a determined consumptions, deducting the rainfall
on the cultivation. It has in consideration also the month-
ly evaporation of the reservoir.
131 REG-RA Estudia la regulación para riegos y abastecimientos de
forma análoga al REGVA.
This programme studies the regulation for irrigation
and supply in the same way to the REGVA programme.
132 REG-K2 Estudia la regulación conjunta de un sistema de embal-
ses considerando evaporación, para lo que utiliza los -
siguientes datos:
a)
Las series de aportaciones en uno, dos tres o c u ~
tro embalses de los que se trata de efectuar una -
regulación conjunta.
3
b) Los consumos mensuales en Hm , suponiendo que
se consume anualmente el 100% de la aportación -
media de cada embalse.
c)
d)
La evaporación mensual en cms.
Las caracteristicas de los embalses, ecuaciones
de las curvas alturas -volúmenes, superficies -voli
menes, capacidad total y volumen de embalse - -
muer to.
El programa realiza entre las aportaciones de cada em-
balse sorteos equiprobables de 5 en 5 años, hasta 1000
años, y a las series de 50, 100, 150 - 1000 les aplica el
proceso de regulación conjunta, en hipótesis de consumo
de diversos % de la aportación media, dando como res-
tado el 70 de fallos en cada serie de anos para cada uno -
de los embalses considerados.

133 REG-SU
189
It studies the compound regulation of a system of reser-
voirs considering evaporation, usind the following data:
a)
The series of affords in one, two, three or four
reservoirs of which it treats to realize a compound
regulation.
b) The monthly consumption in CU. Hm, supposing -
that one hundred per cent of the average afford is
used yearly in each reservoir.
c) Monthly evaporation in cms.
d)
The characteristics of the reservoirs, equations
of the height-volume lines, area-volumes, total
capacity and volume of the dead reservoir.
The programme performs among the affords of each
reservoirs equi-probable casting lots every 5 years
until 1000 years, and to the series.50, 100, 150, 1000
the programme uses the compound regulation process,
in the hypothesis of different 70 of consumption of the
average afford, given as a result the percentage of -
failures in each series of years for each reservoir under
consideration.
Este programa estudia la regulación sucesiva de una se-
rie de embalses, sin limitación de número, utilizando las
curvas de regulación del programa anterior y en la hipó-
tesis de que la capacidad de embalse se utiliza para reg5
lar la aportación de la cuenca propia y los caudales no re
gulados aguas arriba, obteniéndose como resultado los -
volúmenes regulados por cuencas parciales y totales.
This programme studies the sequential regulation of a
series of reservoir, without limitation of number, using
the regulation lines of the preceding programme and -
under the hypothesis that the capacity of reservoir is -
used to regulate the afford of its own basin and the non
regulated upstream flows, obtaining as a result the regu
lated volumes by partial and total basins.

190
134 REG-KI
135 REG-K3
EAM
RYPJU
RELLO
RESE
EBBE
136 CMAR
137 EAM
Igual que el REG-KI sin embalse muerto.
The same as REG-KI, but without dead reservoir.
Realiza la misma función que el programa "REG-KI",
calculando directamente varias hipótesis.
It performs the same function as the programme "REG-
K2", evaluating directly several hypothesis.
Dados los valores de abastecimiento, riegos en valor
absoluto y 70, calcula los consumos mensuales.
Given the values of supply, irrigations in absolute value
and percentage, it calculates the monthly consumption.
Estudia la explotación de hasta seis embalses, interco-
nectados entre ellos por una red principal de canales de
conducción, que se simula mediante una malla de 24 nu
dos.
Utiliza una serie de aportaciones generadas por sorteo
aleatorio teniendo en cuenta o no la autocorrelación de
las aportaciones anuales.
Los Órdenes de desembalse se establecen en función de
los vertidos probables de cada uno de los embalses y de
la demanda a satisfacer. Se tiene en cuenta las pérdidas
por evaporación en los embalses y la capacidad de las -
conducciones.

191
EAM It studies the development of up to six reservoirs, inter
connected among them by a principal net of channels of
conduction, which are simulated by a mesh of 24 knots.
The programme uses a series of generated affords by
fortuitous casting lots having present or no the self-cor
relation of yearly affords.
The orders of emptying are established in relation to the
probable emptying of each reservoir and of the demand
to satisfy. Having present the losses by evaporation in
the reservoir and the capacity of the conductions.
138 RYPJU Este modelo simula la explotación y la producción ener-
gética de un conjunto de aprovechamientos.
Se aplica a un sistema de aprovechamientos (embalses y
saltos hidroeléctricos) situados sobre dos rios en forma
de Y, al cual se pueden añadir caudales regulados en --
otras cuencas o detraer caudales regulados por el sist:
ma.
A partir de una serie de aportaciones generada por sor-
teo aleatorio y teniendo en cuenta la autocorrelación, se
establecen los Órdenes de desembalse en función de las
demandas y de los vertidos probables en cada embalse.
El modelo calcula las producciones en todos los saltos
y la garantia de suministro de la demanda prevista.
This model pretends the exploitation and energetic pro-
duction of an assembly of utilizations.
It is applied to a system of utilizations (reservoirs and
hydroelectric waterfall) located on two rivers in the form
of Y, to which it could be added regulated flows in other
basins or take away flows regulated by a system.
Starting from a series of affords, generated by fortuitous
casting lots and having present the self-correlation, the
order of emptying in relation to the demand and the pro-
bable emptying in each reservoir is established.
The model calculates the productions in all waterfalls
and the guarantee of supply of the calculated request.

192
139 RELLO
140 RESE
Este modelo simula la explotación coordinada de los re-
cursos superficiales y subterráneos.
Supone la existencia de un embalse subterráneo del que
se puede extraer un caudal uniforme prefijado, en fun-
ción de los estados de los embalses del sistema.
Utiliza una serie de aportaciones generadas por sorteo
aleatorio, teniendo en cuenta la autocorrelación de las
aportaciones anuales y las curvas de seguridad de un -
embalse equivalente a la suma de los embalses del sis-
tema, determinadas mediante el programa CDSSE.
La explotación se simula teniendo en cuenta las pérdidas
por evaporación en los embalses y obtiene la garanda de
suministro de la demanda de abastecimiento junto con -
los valores de la extracción anual media del acuifero y
del periodo de máxima duración de la extracción máxi-
ma prevista.
This model pretends the coordinated exploitation of su-
perficial and underground resources.
It assumes the existence of an underground reservoir
from which a uniform flow can be extracted fixed in ad-
vance, depending on the state of the system of the resec
voir.
It uses a series of generated affords by fortuitous casting
lots, having present the self-correlation of the yearly
affords and the safety lines of a reservoir equivalent to
the sum of the system, which is determined by the pro-
gramme SAFLI.
The exploitation is simulated, having present the losses
by evaporation in the reservoir, the guarantee of supply
of the demand, together with the values of the annual -
average extraction and of the period of maximum calcu-
lated extraction.
Este modelo simula un sistema de explotación con varios
embalses situados sobre una misma corriente, uno de -
los cuales puede ser el origen de un aprovechamiento -
hidroeléctrico.
La explotación se establece a partir de una serie de apor
taciones generadas por sorteo aleatorio y teniendo en -

141 EBBE
193
cuenta la autocorrelación de las aportaciones anuales,
en función de las curvas de seguridad de un embalse tg
tal equivalente para atender a una demanda de usos cog
suntivos. Al mismo tiempo que calcula la garantía de su
ministro de la demanda prevista obtiene las produccio-
nes históricas en todos los saltos, distinguiendo la enec
gi’a de puntas y la energía producible en horas llenas en
el periodo critico (nov. a feb. ) de máxima demanda ener
gética. Ordena los valores de la energía optenidos y as{
puede suministrar los valores de la energia de distinta
calidad garantizada en el per

194
of the calculated reservoirs. It calculates the values of
the possible overflows at the same time that it calculates
the guarantee of supply of the calculated demands and the
energetic productions and consumptions in the hydroelec
trical utilizations of the basin.
142 LAMI 1 Estudia la laminación de un embalse, supuesto un nivel
inicial determinado y pudiendo utilizar uno o varios si-
temas de desagües, en función de las caracteristicas -
del embalse y de la crecida.
It studies the lamination of a reservoir, supposing an
initial level determined in advance and being possible
the use of one or several drainage systems, in function
of the characteristics of the reservoir and the flood.
143 HIDR 1 Calcula el hidrograma para diversas hipótesis de inten-
sidad horaria de precipitación, coeficiente de escorren-
tia y duración de la tormenta.
It evaluates the hydrogram for different hypothesis of
hourly intensity of rainfall, runoff coefficient and the
duration of the storm.
144 HIDR 2 Calcula el hidrograma con intensidad y coeficiente de
escorrenti’a corriente.
It computes the hydrogram with normal intensity and
usual runoff coefficient.
145 ABC Realiza el estudio económico (análisis, beneficio y cos -
to), expresándolo en forme de corrientes monetarias as

tualizadas en función de la tasa de descuento y obtiene
la ratio
Beneficio - Gastos
costos
195
It performs the economic study (analysis, profit and
price) expressing it in actual monetary currency in func-
tion of the standard rate of deduction and it obtains the
ratio
Profit - Expenses
Prices
146 ABC 10 Programa ABC para estudio económico de varias cen-
trales.
It programmes ABC for an economical study of several
centrals.
147 ABC TV Programa ABC para estudio económico, con tasa varia-
ble.
It programmes ABC for an economical study, with varia
ble standard rate.
148 BNZ Dado el número de muestra, tiempo de lectura y número
de desintegración, obtiene la concentración de Tritio pa-
ra muestras de agua.
Given the number of the sample, lecture time and num-
ber of disintegration, it obatains the concentration of -
Tritium for sample of water.

196
149 PARAM Calcula los siguientes indices : RMG/RCA, RNA/RK,
RNA/RCA, ANA/RMG, (RCA-RMG), RALC, BR/CL,
RCL/RHCO3, RHCO3/RCL, RS04/RCL, (RCA + RHCOQ)/
/ (RCH + RS04), RHC03/RHC03 t RS04 t RCL, RALC/RCL,
RCA/RALC, SAR, ICB, ID, FI, Tipo de agua. Además,
lista los elementos en meq/l. y en 70.
It calculates the following indexes: RMG/RCA, RNA/RK,
RNA/RCA, ANA/RMG, (RCA-RMG), RALC, BR/CL,
RCL/RHC03, RHC03/RCL, RSOq/RCL, (RCA t RHCOS)/
/ (RCH t RS04), RHC03/RHC03 + RSO4 + RCL, RALC/
/ RCL, RCA/RALC, SAR, ICB, ID, FI, Type of water.
Besides, it lists the elements in meq/l. and in %.
150 HISTO Calcula los histogramas de las siguientes relaciones,
clasificándolos en clases y valores fuera de clase:
RCL/RS04, RCL/RHCO3, RALC/RCL, RNAIRCA, RCL,
RSO4, RHC03, RN03, RALC, Res. seco, T.D.S., Dureza
total. El histograma lo dibuja por impresora.
It calculates the hystogram of the following relations,
classifying them in classes and values out of class:
RCL/RS04, RCL/RHCO3, RALC/RCL, RNA/RCA, RCL,
RSO4, RHC03, RN03, RALC, Res. (dry), T.D.S., total
hardness. The hystogram is designed by the printer.
151 TUBEC Dado un muestrario de tubedas de diferentes diámetros,
con sus precios y caracteristicas hidráulicas,, determi-
na para una configuración topológica y topografica de la
red y para diversas hipótesis, la combinación de distri-
bución de tubos más económica que permita el suminis-
tro solicitado con la minima pérdida de carga.
Given a sample book of pipes of differents diameters,
with their prices and hydraulical characteristics, it
determines for a topological and topographical form of
the system and for several hypothesis, the combination
of distribution of pipes more economical, that allow the
solicited supply with the minimum loss of loading.

197
152 CANAL Definido un canal por sus secciones y pendientes en difg
rentes tramos, as: como por distintos tipos de cornpuer_
tas, el programa determina la evolución de los caudales
transportados en el curso del tiempo, as: como los cala
dos alcanzados en los distintos tramos del canal.
Permite estudiar las maniobras de apertura y cierre de
compuertas más convenientes para la explotación del cg
na1 .
Defined a canal by its sections and pendants in different
stretchs, as well as by different types of floodgates, -
this programme determines the evolution of the flows
carried in the course of time, as well as soakage reach-
ed in the different stretchs of the canal.
It allows also to study the process of opening and closing
of floodgates more convenient to exploitation to the canal.
153 SER-EL Depura los datos suministrados por las empresas hidro-
eléctricas relativos a la producción mensual de energi'a
de los diferentes saltos de cada una, recogidos en tarje
tas perforadas. La depuración se hace verificando la -
concordancia de los datos geográficos, número de horas
de utilización de los controles en función de la potencia
instalada y producción.
Una vez corregidos todos los errores detectados, pre-
para unos cuadros resúmenes estadisticos de producción
de energTa, clasificados por diferentes conceptos :
Producciones globales por cuencas hidrográficas en ca-
da mes.
Producciones globales UNESA, IN1 y otr os en cada mes.
Producciones anuales clasificadas por:
Empresa o concesionario.
Centrales por magnitud de su producción.
Centrales y cuencas por magnitud de su produc-
cion.
Centrales y rios por magnitud de su producción.
Centrales y provincias por magnitud de su produc-
ción.

198
It purifies the supplied data by hydroelectrical companies
relatives to monthly productions of energy of the diffe--
rents waterfalls of each one, collected in perforated cards.
The cleansing is done verifying the harmony of geographi-
cal data, number of hours of utilization of controls in -
function of installed power and production.
Once corrected all the detected errors, it prepares a
summary of statistical charts of production of energy,
classified by different ideas :
Total productions for hydrographical basin in each month.
Total productions UNESA, INI, and others in each month.
Annual production classified by:
Company or dealer.
Centrals by magnitude of production.
Centrals and basins by magnitude of production.
Centrals and rivers by magnitude of production.
Centrals and provinces by magnitude of production.

COMPUTATION OF RESERVOIRS SEDIMENTATION
A.V. Karaushev, I.V. Bogoliubova
State Hydrologic a 1 Institut e
Leningrad, USSR
-- ABSTRACT
Methods for the computation of sedimentation by suspended
sediments and bed load of the projected reservoirs are given,
or the first year Of the reservoir operation computation is
made according to the balance of sediments computed by the
difference between the transport capacity and the hydraulic
parameters of the current at the upper pool (transient region)
and at the dan of the reservoir. The subsequent attenuation of
the process as well as the total duration of sedimentation is
evaluated by empirical relations obtained from the observational
data on reservoirs under operation.
RESUME
Les auteurs exposent des méthodes pour le calcul de
l'envasement des barrages par les matériaux transportés en
charriage ou en suspensión. On fait, pour la première année
d'exploitation, le bilan des matériaux déposés dans la retenue
par différence entre ce qui entre à l'amont (station de mesure
dar,s la zo~e du remous) et ce qui sort par le barrage; ces
mesures sort reliées aux paramètres hydrauliques du cours d'eau.
u', extrapole les résultats dans le futur en utilisant des
relations empiriques obtenues pour d'autres réservoirs en cours
! 1 exploit at ion.

200
The construction of reservoirs in mountain areas and at the
foothills on rivers with a considerable sediment concentration
inevitably faces with the necessitg to remove or to impede sediments
transported by the river to keep the projected capacity
of the reservoir. The present paper gives methods accepted in
the USSR providing the evaluation of possible sedimentation rate
for the whole reservoir or for its individual parts during the
first year of its operation and for subsequent years.
Methods for the computation of re semoirs sedimentat ion are
based on the equation of sediments balance applied to the whole
reservoir or its parts, to the gross composition of the transported
sediments or its particular fractions. The use of this equation
makes it possible to compute the difference between sediments inflow
and its discharge out of the reservoir i.e. sediments
accumulation. The inflow of sediments is computed by observational
out of the reservoir
data or by indirect methods. The discharge of sediments is estimated
by equations of the transporting capacitg of the current at
the specified values of water discharge Q, mean depth Hm, mean
current velocity Vm, and granulometaAc sediment composition.
The computation of sedimentation during one year is reduced
by the determination of that portion of sediment discharge
which is accumulated in the reservoir. When starting computation
it is essential to establish design values of annual water
discharge, of suspended sediments and bed load, as well as typical
chronological graphs of these values for the inflow site of
the reservoir. It is recommended to divide the hydrograph of the
typical year into 3 or 4 design time intervals& and to compute
sediments accumulation according to the values o $ Q, Vm Hm, etc.,
averaged for every time interval. The computation of se Aimentation
rate is made by individual fractions i, in this case it is
sufficient to subdivide all the transported fractions into 3 or
5 categories. Then sediments are summarized according to all
categories of the fractions.
The computation of sedimentation by suspended fractions for
a design interval A tj is niade by equation:
where: Pa j is the amount of sediments of all the fractions
(tons) in the reservoir or in the design area dur A t ;
*i in J is inflow of sediments of the i-th fractio3tonsg
durmg time ~t through the initial (upper) discharge site of
the reservoir $ or its part)determined by the chronological
graph or by computation das made for the upstream area; Qter
is mean water discharge (
sec) for time at through the
terminal (downstream) discharge site of the Jeservoir ( at the
dam) or the design area; Si t is mean particular turbidity
for time At. of the i-th fractfoi at the terminal discharge site
of the resehoir (certain area) (g/m3); ~ t is j time interval
(sec).

202
Turbidity of the i-th fraction at the terminal discharge
site Si ter j is computed by equation of A.T. Karaushev:
- G"AL
(2)
where: Si 4" J Ois particular turbidity at the initial discharge
site mean or time interval B ta; S? is the turbidity
corresponding to a particular thins$of%idg capacity of the current
computed by equation (6) given below; e is the base of natural
l2gar ith;
G is dimensionless value determined by equation
where: ui is fall velocity of fraction i under consideration;
kg is a parameter having a dimensionality of velocity and which
is computed by equation
LL; ri
(4)
The value of r is the value of hydroneclnanic param?.t;er
of sediments whichi= be obtained by graphs according to the
Chezygs coefficient C and ratio of *i (Fig. 1 ). In equation (3)
AL indicates the length
-
of the reservoir (or it8 part) given
In relative units:
A L
AL =----
Hm (5)
where: AL is the length of the reservoir (or design area (m);
IL, is mean depth of the reservoir ( or some area), (m) for time
iIltel?ral A t*o
A particulAr transporting aapacj. of the current S: tr -
(for the i-th fraction of i8 CO uted with the %se of
data on bed load composition. The value of 8 tr j is computed
with the use of hydraulic elements of the current mean for time
interval A tj related to the whole reservoir or its desim area:
Here a
actual
indicates a correcting factor estimated by the ratio of
and computed turbidity at the initial discharge site:
SS4. r a= --
s cokvlvip (7)

2 02
d is composition ( in per cent) of the i-th weighted
f&&&o& in the roiling portion of bed load.
The value 0% droil i is determined by the ratio
--
IC0 , (8)
roil i - t CL bed i
where A is the portion (per cent) of the i-th fraction in
bed load &&&sition; r is gross portion ( in per cent) of the
weighted fraction in bed load composition. In this case sediments
wPth fall velocity corresponding to the condition u 4 1- are
regarded as weighted fractions, and Vi indicates maxmum value
of the vertical component of the puls%on velocitg. The latter
salue is computed by a special equation according to mean velocity
of the current and Chew's coefficient. Gross turbidity of roiling
(Sroil) is obtained by equation:
where: N is characteristic dimensionless number depending on
Chew's eoefficient C; is the ratio of velocity at the bottom
go mean veloci%y; the ! est of the symbols are given in previous
equations, *
When comguting.Si tr for the first time interval the composition
of bed load xn the reservoir is accepted according to the
averaged conposition of river alluvium in the cñannel and in the
f lood-plain; for subsequent intervals it is essential to consider
the composition of sediments obtained by computations.
If sedimentation is computed for certaih areas, then Si 8' j
estimated by equation (2) for the first (upper) area 1s use as
Si in j for the second area domstream, etc.
The computation of reservoir sedimentation by bed load is made
according to the same design intemals a8 by suspended sediments.
For an approximate evaluation it i8 possible to be confined to
the computation for flood periods when the major portion of coarse
fractions flows into the reservoir.
The amount of bed load in the reservoir is determined by the
difference:
(10)
where: Pa bed is the weight of bed load in the reservoir (tons);
n t. is time interval (sec); Rbed in
J
and Rbed ter jindicate
bed load discharge at the initial and termi.mil discharge sites
(kg/sec) nean for design time intervalat.. For bed load discharge
computation it is reasonable to recommed $he equationsof G.I.
Shamov, VaNe Gomharov, I.V. Egiaearov, K.I. Rossinski, et al. The
equation of G.I. Shamov is the most simple one providing a suffici-

203
ent coincidence of computation results with the data of measure-
ments at a wide range of fractions dimensions:
where: Rbed is bed load discharge (kg/sec); B is current width
(m); Hm is mean depth (m); a, is mean diameter of mobile particles
of bed load (m); Vm is mean velocity of the current (m/sec); vsed
is mean velocitg of the current (m/sec) when fractionswith Q
in diameter stop moving; K is coefficient consider- non-homo-
geneitg of bed load composition.
The value of is computed by equation:
WL (12)
d =c~oíz.i.ul!
i=/ L '
WL
where: d, and d. res ectively indicate percentage and mean
diameter o) a certain ?i-th) fraction; summation is made accord-
ing to all the mobile fractions, the number of these fractions
is-indicated as
Separation of
equation:
m.
immobile (coarse) fraction is made by
3
9,o 12 'yl
-
4,t -
i-
L H,
-
The value of Vsed is obtained by equation:
fis
= 3,) d, H ~
(13)
( 14)
All the computations of bed load transport are made accord-
ing to hydraulic elements mean for time interval A tj.
Annual accumulation of all the sediment fractions for the
first year of reservoir operation is determined by equation:
where: Pai is gross sediment weight for the 1st year (tons);
and Pa bed j respeCtiVely indicate the Wight of suspend-
88 g%hde.ents and bed load for the 1st year (tons)j j is the number
of design internali n is the number of intervals during a year.
If computation 1s made according to certain areas, then
summation is made for all the areas to obtain gross sedimentation
of the reservoir. The obtained value for the whole of the reservoir
is transformed into volumetric units:

204
where: Wai is the volume of sediments during the 1st ear (m 3 );
P is the weight of sediments for the 1 t year (tons5; fs is
&e volumetric weight of sediments ( t/m 3 ). After the computation
being done the initial volume of the reservoir W is corrected.
The obtained volume of the reservoir at the end of the 1st year
W, = W - Wa
is used for the computation of sedimentation for the
next year. $he computation of sedimentation for subsequent years
may be performed in the same way as for the 1st year or by
extrapolation equations considering time attenuation of sedimenta-
tion.
It is recommended to use the equation of G.I. Shamov for the
computation of chronological variations of sedimentation:
where: Wa t is the volume of Sedimentation (m3) in t years;
M'ad is sedimentation volume during the 1st year (m3) computed
by the methods described above; Wa ext is the extreme volume
of sediments in reservoir (m3) approximately computed by
e quat - ion:
3
where: W is the initial volume of the reservoir (m ); Wtis
area of river cross section (m2) when dischar e is close to
maximm;Ldp is the maximum cross section area fm2) of the upper
pool near the dam.
The method is applicable to the reservoir as a whole.
One year should be accepted as a design time interval. In
case of correct initial values the method provides variations
of reservoir sedimentation close to actual.
REFERENCES
1. Bogoliubova I.V. Resultam polevykh issledovaniy i rascheta
stoka vlekomykh nanosov r. Mzymty (The results of field
investigations and bed load discharge computation for the
Mzymta river) Trans. of the State Hydrological Inst.,
1968, ~01. 156, PP* 3943.
2. Karauschev A .V . Rechnaya gidravlika (River hydraulic 8) .
Leningrad, ñydrometeorological Publishing House, 1969,
PP. 303-778.
36 Razumikhina K.V . Primenenie formuly transportiruyushchei
sposobnosti dlia rascheta godovogo stoka vzveshennykh
nanosov (Application of transporting capacity equation
for the computation of annual discharge of suspended
sediments). Trans. of the State Hydrological Inst., 1969,
vol. 175, ppe 137-154.
4. Ukasania PO raschetu eailenia vodokhranilishch pri strof.t;elnom
proektirovanii (Instructions for the computation
of reservoirs sedimentation for engineering projects).
Leningrad, Qdrometeorological Publishing House, 1968,
54 PP.
5. Shamov G.I. Recbnye nanoqy (River sediments). Leningrad,
Hydrometeorological Publishing House, 1959, pp. 2-282.

c
Figure 1. - Computation of reservoirs sedimentation
205

ABSTRACT
CALCULATION OF RUNOFF IN IRAQ
R.K. KLIGUE, MECHDI EL SACHOB
There given a general characteristic of runoff in
Iraq and its distribution within the area of the country.
The authots analyse correlation of runoff with elevation
of the territory, river basin area and other factors with
the aim of using these relationships for regions without
runoff data. Hydrologic time series analysis of runoff
and analysis of runoff fluctuations through the territory
are cited.
RES UME
On donne la caractéristique générale d'écoulement
fluvial en Irak et sa distribution par la territoire. Sont
analisdes les relations entre écoulement fluvial, l'altitude
de lieu, l'aire de surface réceptrice et d'autres facteurs.
L'objectif de cette analyse d'est utilisation de ces
relations pour les régions avec l'absence des données
d'écoulement. On fait l'analyse des séries hydrologiques
et des variations de débits par la territoire.

208
The investigation of river flows in Iraq is of great
importance keeping in view the constantly increasing water
balance stress in the country that arouses the necessity to
design special rnultipurpose projects as well as projects
aimed at more complete use of water resources by construct-
ing irrigation systems, overyear storage reservoirs, by im-
proving crop management under irrigation and by wider use
of groundwaters.
The main rivemof Iraq - the Tigris and Euphrates -
cross the oountry by their middle and lower reaches. Con-
fluencing they form a river called Shatt-al-Arab flowing
into the Perbian Gulf. The main tributaries of the Tigris
within Iraq are the Greater Zab, Lesser Zab, Adhaim and
Diyala. The Euphrates river have no tributaries on the ter-
ritory of the country. The arid regions are characterized
by the existence of "wadi".
Water resources of Iraq are mainly determined by the
flow of the Tigris and Euphrates rivers making about 77.7 km 3
3 3
a year - about 22.2 km flows into the sea and 55.5 km
(71.4%) is used for irrigation, municipal and industrial
water supply and power generation. A considerable part of
flow is lost due to evaporation, transpiration and filtra-
tion.
The mean annual flow of the Euphrates on entering the
territory of Iraq is 928 cumecs decreasing downstream (Na -
\I
siriya) to 454 cumecs. Thus, the rate of flow changes along
the river length from 3.52 to 1.57 1/s. km'.
The Tigris river on the territory of Iraq has several

ig tributaries, which increase its flow from 587 cumecs
209
at Tusan to 1534 cumecs at Salman-Pak. Downstream the flow
decreases due to intensive withdrawal for irrigation and
makes 49.6 cumecs.at Qalat-Saleh. The mean annual rate of
flow in the Tigris basin changes from 12.7 l/s.km
2
in the
upper part to 0.26 l/s.km
2
(Qalat-Saleh) in the lower
part .
The coefficient of annual flow variation (C,) for the
Tigris and Euphrates changes from 0.26 to 0.31 decreasing
with the altitude of watershed (Haam, m). It can be ex -
pressed by a correlation:
The flow of the Tigris and Euphrates is distributed
very uneven through a yew' - the greater part of it falls on
the flood period (April-May), making about eo"/4 in the upper
reaches and 5% - in the lower reaches.
The beginning of flood period is closely dependent on
the mean altitude of watershed m) and may occur in
January-April. The mean duration of flood period in the
piedmont regions is 45 days, in the middle mountain regions -
90 days and in high mountain - 135 days. The more is the
water availability through the year, the longer is the flood
period. Por the year with mean water availability the
duration of the flood period (I, days) may be calculated
by the equation:
T = Hmean - 20.6
14.4

210
In Icaq the maximum flow of rivers occurs due to snow
melting and rain. The rates of maximum daily flows of the
Tigris river and its tributaries vary from 220 l/s.km2 (the
2
Khaair) and 135 l/s.km (the Greater Zab) in the mountain
2
regions to 1.6 l/s.hm2 (the Tigris-Amara) and 0.71 l/s.hm
(the Qalat-Saleh). in the south of the Mesopotamian lowland.
The rates of maximum daily flow of the Euphrates river vary
from 13.3 l/s.km2 (Hit) to 8.5 l/s.km
2
(downstream the dam
Hindiya). The rates of maximum monthly flow of the Tigris ri-
2
ver and its tributaries vary from 48 l/a.km (the Greater
2
Zab river at Eski-Kalek) in mountain regions to 0.6 l/s,km
(the Tigris river at Qalat-Saleh) in the south lowlands. For
the Euphrates river the rate of maximum monthly flow fluc -
tuate from 8.6 l/s.km
2
(Hit) to 4.0 l/s.km
2
(Nasiriya). As
I rule there is observed an increase in the maximum rates of
flow throughout the mountain regions a
In the mountain regions of Iraq,above 2000 m (the Tigris,
Greater Zab and Euphzlates river basins) the rate of maximum
daily river flow increase from 8.5 to 136 l/s.km2, making
the mean about 35 l/e.km
2
for every 100 m. For this region
a certain dependence between maximum rates of flow (%ax)
and mean annual rates (Mme= an. ) is traced.

211
In the middle mountain region with the altitudes 1000-
2000 m (the Lesser Zab and Diyala basins) the rates of maxi-
2
mum daily flow fluctuate within the limits 48-112 l/s.km ,
increasing averagely by 38 l/c,km2 on every 100 m of alti-
tude* The dependence of maximum and mean annual flows for
this region is expressed by the correlation
- 45.8
%ax äaiïy 0.138
In piedmont regions, lower than 1000 m (AdhaFm and
Khazir river basins) the rates of maximum daily flow being
inversely proportional to the altitude of watershed increase
from 48 to 220 l/s.km2. The phenomenon is observed in the
case of the maximum monthly flow. This can be explained by
the fact that the Khazir river has the lesser, compared to
the Adhaim basin, area of watershed but receives greater
rainfall. The dependence between the maximum and mean annual
flows is expressed by the correlation.
%ax monthly = 2.5 %ea* an.
The coefficients of maximum daily flow variation for
the rivers of Iraq fluctuate within the limits from 0.14
(the Tigris-Amara) to 0.74 (the Adhaim-Injana). The coeffi-
cients of maximum monthly flow are limited by 0.18-0.29. There
observed the inverse proportion between coefficients of va-
riation and the altitude of watershed. The correlation of
coefficients of maximum monthly flow variations (Cv max 1 and
the coefficients of annual flow variations (Cv an. ) can be
put as:
'v max monthly =(2*1 'v an. ) - 0.21

212
For the rivers of Iraq it is characteristic the increase
of maximum flow with the decrease of the watershed area
(F b2),
Barnax daily = 62 - 0.2 F + -9
Minimum flow usually occurs in autumn (September-Octo-
ber) and is caused by the groundwater depletion by the end
of the hot and dry period. Small rivers in piedmont regions
dried up as early as the beginning of summer and till the
winter rains they have no flow.
One of the most important factors of the Iraq rivers
regime is the low-water period, when the river flow is cha-
racterized by stable low levels and discharges and when the
rivers under the condition of great reduction of surface flow
or its complete cessation are recharged through groundwaters.
Low-water period usually occur in summer or autumn (oftener
from June to December). Its beginning (t days from the begin-
ning of the year) has a certain dependence on altitude,
OSI75 +
days = 64.6 (jy,ean-lOOO) - K
where, K - coefficient of water availability of the designed
period. It can vary from + 20 (for high-water period) to -20
(for low-water period). At the mean level it approaches zero.
As a rule the greater is the altitude of a watersheii, the
shorter is the period of low water. In piedmont zones it
makes averagely 171 days, in middle mountain zones - 159 days
and in high mountain - 153 days. The latter is mainly due
to more favourable conditions of humidification in the moun-
tains where the rainfall reaches 1000 mm than in piedmont re-
gions where the rainfall makes 200-300 m.

213
The rates of minimum flows of Iraq rivers have a wide
range of variations from 0.04 (daily) and O.Oí'(monthly) for
the river Adhaim (Injana) to 5,56 (daily) and 5.71 (monthly)
l/s.km2 for the Greater Zab river (Bekhma). Usually the
rates of minimum daily and monthly flows increase with the
altitude and this relationship can be expressed ass
10 3 J
'L$in = 5.8 IO-.
On the territory of Iraq there observed quite a defi-
nite reduction of the minimum flow rates with the increase
of a watershed area. For example, on the Tigris river at
2
Al-Fatha the minimum monthly flow equals 2.95 l/s.km , the
2
watershed area being 1076b0 km and downstream the Kut bar-
2
rage the flow - 1.35 l/s.km and the watershed area -
177540 b2. For the Greater Zab and Tigris river basins
(high mountain zone) the relationship will be similar,
%in monthly = 6.35 - 3.14 IÕ7F
%in daily = 6.14 - 3.08
The decrease of the minimum flow rates with the in -
crease of a watershed area can be explained mainly by great
withdrawals of water for irrigation and due to considerable
evaporation.
The mean annual flow of Iraq being studied better than
the minimum one their interrelation may arouse certain in-
terest
'min monthly = 'Os4 'mean an.
%in daily = O*** mean an. 0.15

214
These relationships do not reflect the conditions
in the lower reaches where the regime of the rivers is great-
ly distorted under the influence of anthropogenic factors.
The variations in the flow of Iraq rivers (mean, mini-
mum and maximum) occur principally at the same time. For the
investigation of the minimum flow variation of the Tigris
river (at Mosul, 1920-1970), the Euphrates river (at Hit,
1932-1970) and the Greater Zab river (1938-1970) they used
the method of the differential integral curves permitting
to bring out the succession of low-water and high-water
groups of years in the considered period. The duration of
low-water periods with minimum flow (mean coefficient of wa-
ter availability 0.78) can change within 1-18 years (the Tig-
ris river at Mosul) and high-water periods (mean coefficient
of water availability 1.24) - within 1-11 years. The coeffi-
cients of variation (C,) for minimum monthly and daily river
flows fluctuate from 0.18 (monthly, the Tigris river at Al-
Fatha) and 0.17 (daily, the Euphrates river at Hit) to 1.27
(monthly,the Aähaim river at Injana) and 2.03 (daily, the
Adhaim river at Injana). The value of the coefficient of va-
riation is inversely proportional to the altitude which is
explainable by a considerable aridity of lowland territories
in Iraq.
The values of Cv for minimum flow, definitely correlate
to Cv for mean annual flow. For the watersheds where the
withdrawal of water for irrigation is low this relationship
can be expressed by the following empirical equation,
'v min daily = 1 0 ~ 'v ~ mean 5 an.
- 0.16

In Iraq because of the drying up many rivers have no
215
flow for a considerable period (the Adhaim, Al-Wend, Galal-
Bedrah, Wadi-river, etc.). The watershed areas of drying-up
2
rivers may reach 13000 km (the Aàhaim river) and the dura-
tion of a drainless period exceed 250 days. In Iraq, especialare
ly, in its south-western part there a numerous strema of
temporal nature, "wadi", which have flow only several days
a year. The length of some of them reach many tens of kilo-
meters. This phenomena is a result of the extreme aridiky of
the region where they are met (the rainfall is less than
100 mm and evaporation - over 2500 mm).
It should be noted that the given relationships of &if-
ferent flow characteristics on the territory of Iraq despite
their approximate nature and the necessity of further preci-
sion, permit to give duly evaluation of a number of flow pa-
rameters for the insufficiently studied regions of the consi-
dered territory.

ABSTRACT
DETERMINATION OF EVAPORATION IN CASE OF THE
ABSENCE OR INADEQUACY OF DATA
P.P. Kuzmin, A.P. Vershinin
State Hydrological Institute
Leningrad, USSR
The possibilities for the determination of evaporation
from water surface and land are given in case of the absence
of data of direct evaporation measurements. The analysis and
classificatiqn of methods for the computation of evaporation
are presented. Practical recommendation for the determination
of evaporation by means of standard observational data from
hydrometeorological stations are given.
Les auteurs examinent les possibilités d'évaluation de
l'évaporation des surfaces d'eau libre lorsqu'il n'existe pas
d'observation directe. Ils analysent les différentes méthodes
utilisées et en proposent une classification. Ils font des
recommandations pour l'évaluation de l'évaporation 'a partir
des observations standards effectuées dans les stations
hydrométéorologiques.

218
The determination of evaporation under natural conditions
is of great importance for the ewtimation of the present and
future water resources, Por water resources management and
for the solution of various theoretical problems in the field
of hydrology and meteorology. Methods of direct evaporation
measurements under natural conditions are still being developed,
therefore computations are the main source of information.
The existing computation methods might be subdivided into
three groups. The first group comprises the methods based on
the physical analysis of the evaporation process. The second
group (combined ox complex methods) includes methods based on
physical principles combined with semi-empirical constants which
can be determined with the help of accurate measurements of
actual evaporation in representative regions.
Methods based on the statistical analysis using only
empirical relations, where empirical constants and coefficients
are h igw variable and depend on meteorological conditions,
make the third group.
Besides, according to the basic data (factors) included i-nto
the design schemes, it should be noted that computation methods
may be complex and simple as well as difficult and easy to be
applied in practice. In this respect the most simple and
practicable methods are those of the third and some of the second
groups, while the methods of the first group which are based
on the physical analysis, are most inconvenient in practice.
The first group includes the well-known methods of estima-
tion of evaporation from heat balance eqqtion, water balance
equation and turbulent diffusion method /ll/; the accurate
solution of these equations cannot be obtained because it is
impossible to estimate with sufficient degree of accuracy some
individual components of the above equations.
The estimation of the turbulent heat exchange between the
underlying surface (water or land) and the atmosphere is one
of the difficultiee of the solution of heat balance equation.
This component can be estimated approximately without conside-
ration of temperature stratification and horizontal gradients
of turbulent heat exchange (advection).
In particular, in the course of estimating evaporation from
the reservoir surface it is difficult to determine time
variation8 of the heat accumulated by the reservoir (heat
content) as well as heat income and losses due to all kind8
of water inflow and outflow (both surface and subsurface).
Therefore this method is applied only in research studies.
Heat b8Lance equation of the land surface is more complicated
than that of the water surface /U/#
In the 'USSR, however, 8 method of estimating evapotranspira-
tion has been developed and is widely applied,from thefollowing
equation /14/:
ß-B
( 1)

which is deduced of the heat
balance of the land with the
account of Bowen ratio:
219
Here: E is evapotranspiration, R is the measured value of the
radiation balance of the surface, B is heat income into %he
soil, II is the atmospheric pressure, P is turbulent heat
exchange with the atmosphere, C is heat capacity under
constat pressure, L is the latgnt heat of evaporation;
at and ai are respectively the differences in temperature
and water vapour pressure measured at two levels above the
ground.
Equation (l), naturally, would rather belong to the second
group of methods than to the first one; it does not include
horizontal gradients of turbulent heat exchange (advection)
and temperature stratification. It can be applied for homogeneous
areas large enough to ensure wind r u over homogeneous
top cover over plain area at the distance of 300-400 m.
The relative standard error of 10-day and monthly evapotranspiration
sums estimated from equation (1) for the re ions of
natural moistening and for irrigated fields makes f 1%.
Equation (1) cannot be recommended for the estimation of
evapotranspiration in very dry regions (semi-deserts, deserts) .
Full water balance equation is not applied in practice
since it is both difficult to determine water exchange with
the bed of reservoir (the difference between underground
water inflow and outflow in a reservoir) while estimating
evaporation from the water surface and to determine water
exchange between the upper layer of the aeration zone and the
underlying ground (upward and downward streams of moisture in
the ground) while estimating evapotranspiration from the land
surface in a river basin.
In case of deep water table ( no lesa than 3-5 m) the
simplified water balance equation is used in the USSR to
estimate evapotranspiration from non-irrigated agricultural
fields; according to this equation evaporation is estimated
from precipitation (x) and the change of moisture storage in
the upper soil layer:
E =K+(L4pW.) (2)
where8 W and W are moisture storage in soil at the
beginnid and a? the end of the design period.
Equation (2) can be used only under the condition that all
precipitation is absorbed by the soil and no surface runoff
is formed and besides that the depth of rainfall water per-
colation should not exceed the depth up to which soil moisture
content was measured and moisture content was determined. Such
conditions usually exist during the vegetation period. The

220
depth of the upper layer of soil in which soil moisture storage
should be determined is 1 m in wet areas and up to 3 m in
arid zones. In case of reliable estimation of precipitation
and moisture storage the standard error of the estimation of
monthly sums of evapotranspiration by this method makes
approximately l5-2m. Another example of a partial solution
of water balance equation is the estimation of mean annual
sum8 of evapotranspiration as the difference between precipitation
and runoff /3/.
Theore tical and experimental development of the turbulent
diffusion method which is also known as the gradient or
aerodynamic method, has not yet reached the stage which would
allow its wide application in practice /1,2,9,12,15,16/. However,
this method :is promising. Being universal, this method is
based on gradient measurements of wind speed and air humidity
and provides estimation of evaporation from any land or sea
surface irrespective of the state and character of the latter.
Accurate enough (universal) solutions of the equations
of the first group require a consideration of a great number
of factors and special observations to be made. The development
of simplified semi-empirical and empirical design schemes
will provide possibilities for the estimation of evaporation
without the data of specialized observations. At present it
seems possible that standard observational data from hydrometeorological
stations are enough far the estimation of
long term average annual and monthly evaporation sums for
a given territory and monthly evaporation sums for individual
years, as well as for the estimation of evaporation from
different surfaces - snow cover, swamps, forests, irrigated
and non-irrigated agricultural fields /1,4,5,10,17/.
Most convenient are the methods of computation of mean
annual evapotranspiration sums for the regions of natural
moistening based on the equation developed by Y.I. Puàyko /IO/.
The right part of equation /3/ includes only one parameter
taken from standard observational data, i.e. long term average
precipitation X ( cm year'l) which reflects the natural moisture
content of a region. Another parameter, reflecting the heat
regime and the character of the underlying surface - averag
annual adiation balance of moistened surface Ro (kcal cm -2
year - $ is $&en from the map prepared by N . Efimova /IO/.
L is the latent heat of evaporation (kcal, $*). The standard
error of evaporation estimated from equation (3) makes about
17% /6/0
Mean annual sums estimated from equation /3/ can be
easily distributed by months because long term mean monthly
'

221
evaporation sums given as percenta e of the annual sum, change
regularly according to geobotanic ?soil-climatic) zones. This
method is called the method of percentage ratios /7/. The
percentage ratios by months are given in tables, developed
by experimental or design ways. Table I, for Instance, presents
monthly evapotranspiration values as percentage of annual sums
for the main geobotanic zones of the European territory of the
USSR .
Table I
C onif mous
forests O 0,5 2 6 17 25 22 15 8 4 O,5 C
Mixed and
decideous forests,
fore st-steppe s 0,5 I 3 9 18 20 ia i3 9 5 3 05
Steppes 1 I 3 1 1 1 9 2 0 1 6 12 8 5 3 I
M.I. Budyko suggested a combined method for the computation
of monthly evapotranspiration sums with the use of the main
elements of heat and water balances /1/. This method can be
applied in practice since it is based on the use of the standard
observational data, i.e. precipitation (x), runoff c y), air
temperature and humiditg.
In this case it is assumed that, when soil moisture content
is less than its water holding capacitg, monthly evapotranspira-
tion ( E ) is proportional to th8 monthly sum of potential
evapora8on ( E ) and to the average monthly storage of
productive moisgure in 1-metre layer of soil WI + W2
2
( WI and W2 are moisture storage at the beginning and at the
end of the month), that is:

222
where: W is critical storage of productive moisture in
soil lay& 1 m deep at which and above which
to Eo. Equations (4) and (5) are applied to
the year. WI at the beginning of the first warn month ( Fa
spring) is estimated approximately, and later it is assumed
to be equal to W estimated for the end of each previous
month from equatzon:
or from equation:
where: y indicated runoff.
Eo and WO are taken from graphs and tables included in
publication /IO/. The value of E depends on the conventional
humkdkty deficit of the air whicf: is determined as the
difference between maximum water vapour pressure estimated
from mean monthly air temperature, and vapour pressure of the
air at the altitude of 2 metres.
This method was developed in two variants and is applied
for the estimation of long term average monthly evapotranspiration
sums of individual months of certqin years /4/.
Aver
ed areal evapotranspiration values (areas to 1000 -
3000 km2y are determined from equations (3) and (4) - (7).
Design schemes for the determination of evapotranspiration
from different kinds of surfaces include parameters which are
seldom measured at observational stations. For instance, in
the scheme developed by V.V. Romanov /13/ evapotranspiration
from a swamp is assumed to be proportional to the radiation
balance of the swamp surface ( Q =dR ); in the scheme
developed by S.F. Fedorov /18/ evapotranspiration from forests
is proportional to the potential evaporation and the proportion
coefficient is presented as the function of the radiation
index of dryness ( R/LX).
Monthly sums of evapotranspiration from irrigated fields are
estimated with the help of simplified heat balance equation /i/

223
using special observational data, the standard error bei%
l5%, or with the help of modified formulae of the complex
method /ïg/ usiq standard observational data, the standard
error being 3%. To estimate evapotranspiration from irrigated
agricultural fields empirical design schemes similar to that
of Blaney and Criddle /li/ can be used if only empirical
coefficients are tested and corrected for each point of their
application,
Most simple design schemes allowing estimation of evapora-
tion from water, snow and ice surfaces by means of standard
observational data, are the following binomial and monomial
equations :
and
(9)
where E is evaporation in mm/day, U, is the wind speed at
the height 2, above the surface in m/sec; es and e2 axe th
maximum water vapour pressure estimated from the surface
temperature and water vapour pressure at the height of 2 m
in mb; A, a and b are coefficients estimated from experiments.
Substituting a = 0.18 ab = 0.098 in equation (8) and
assuming z = IO m one can obtain the formula for the estima-
tion of evaporation from the snow surface /8,11/,
When a L 0.14, b = 0.72 and z = 2 m, equation /8/ can
be used for the estimation of evaporation from lake (reservoir)
surface. In this case in equation (8) parameters uz , e and
e2 shouid be substituted by correspondent values measure8 at
different points above the reservoir and averaged for a month
with respect to the whole water area of the reservoir,
In case of the absence of such obsemrational data one can
use the data from land meteorological stations situated in the
same climatic zone. T9e transition of the obeained above-land
coefficients &z/ ef and 4; to the corresponding above
reservoir values h ou d be carried out with respect to the
transformation of the air flux affected by the underlying SUT-
face, the topography of the environment, the rate of wind
protection of the reservoir and the average length of wind
run above the reservoir /l7/.
In conclusion it should be mentioned that the present paper

224
deals with the methods which can be used in practice and produce
relatively reliable estimates of evaporation on the basis of
standard observational data from meteorological stations.
Therefore more complicated methods of the first group
which cause difficulties being applied in practice, and
numerous empirical design schemes which produce unreliable
results, are not treated here. Penman and Turc methods are
not mentioned since they are known well enough. It should be
mentioned as well that the above classification of methods
is conventional.
All methods are closely interrelated, and their develop-
ment, particularly the improvement of methods of computation
in case of inadequate data, depends greatly on further
experimental and theore tical research on the evaporation
problem .
R E F E R E N C E S
1. Qudyko Y.I., 1956. Teplovoi balans zemnoi poverkhnosti
(Heat balance of the Earth's surface). Hydrometeorological
Publishing House, Leningrad.
2, Buãyko M.I., 1948, Isparenie v estestwenoykh usloviakh
(Evaporation under natural conditions). Hydrometeorological
Publishing House, Leningrad.
3. Wodnye resuray i wodny balans territorii Sovetskogo Sojuza
(Water resources and water budget of the USSR area).
Hydrometeorological Publishing House, Leningrad, 1967.
4. Zubenok L.I., 1968, Ob opredelenii sumaiarnogo isparenia za
otdelnye godg (On estimation of evapotranspiration
during particular years). Trans. of GGO, vol. 233,
Leningrad.
5. Konstantinov A.R., Astakhova N.I., Levenko B.A., 1971,
Metoày rascheta isparenia s selskokhoziaystvennykh
polei (Methods for the computation of evaporation
from agricultural fields), Hydrometeorological
Publishing House, Leningrad.
6. Kuzmin P.P., 1966. Teoreticheskaya skhema otsenki oshibok
rascheta isparenia s poverkhnosti sushi (Theoretical
scheme of evaluation of estimation errors of
evaporation from land). Materials of Interagency
meeting on the problem of study and substantiation
of methods of evaporation computations from water
and land. Ed. GGI, Valdai.
7. Kuzmin P.P., Zubenok L.I. Konstantinov A.R., Astakhova N.I.,
Vinogradov V .V . , 1968 . Vnutrigidivie rasprede lenie sumsuschi
na territorii SSSII (Annual distribution of
evapotranspiration from land over the USSR territon),
Trans. of GGI, vol. 151.

225
8. Kuzmin P.P., 1953. K metodike issledovania i zascheta
isparenia s poverkhnosti snezhnogo pokrova. (On
methodology of research and computation of evaporation
from snow pack surface) Trana. of GI, vol.
41 (95).
9. Leichtmap D.L., l9W. Profil vetra i obmen v prizemnom
sloe atmosfery (Wind profile and exchange in the
lowest atmosphere). Izv. AN SSSR, ser. geofis.,
No.1.
IO . Materialy mezhduvedomstvennogo sovetchchania PO probleme
izuchenia i obosnovania metodov rascheta isparenia s
vodnoi poverkhnosti i suchi. (Materials of Interagency
meeting on the problem of study and substantiation
of methods for the computation of
evaporation from water and land surfaces). Ed.
by GGI, Valdai, 1966.
11. Measurement and estimation of evaporation and evapotranspiration.
Technical Note No. 83, WO-N0.201. TP.
105, 1966, Geneva.
12. Monin A.S., Obukhov A.M., 1954. Osnovnye xakonomernosti
turbulentno o peremeshivania v prizemnom sloe
atmospgery $Principal laws of turbulent mixing in
the lowest atmosphere) . Trans. of Geophysical Inst.,
AN SSR, vol. 24 (151).
13. Romanov V.V., 1962. Isparenie s bolot Xvropeiakoi territorii
SSSR (Evaporation from swamps from the USSR
European territory). Hydromet. Publ. House, Leningrad;
14. Rukovodstvo PO gradientnym nabliudeniam i opredeleniu sostavlia
jushchikh teplovogo balansa (Guide on gradient
observations and determination of heat balance components)
* Hydromet. Publ. House, Leningrad, 1962.
15. Rusin N.P., 1959, Gradientny metod opredelenia isparenia
s sushi i ego ispolzovanoe na seti stantsiy (Gradient
method of estimation of evaporation from land and its
use on the network of stations). Trans. of III-rd
All-ünion Hydrological Congress, vol. III, Hydromet.
Publ. House , Leningrad.
16. Tbornthwaite C.W. and Holtzman B., 1942. Measurements
of evaporation from land and water surfaces. U.S.
Dept. Agr. Technical Bul. 817.
17. Ukazania PO raschetu isparenia s poverkhnosti vodoemov
(Instructions for the computation of evaporation from
reservoir surface). Hydromet. Publ. House, Leningrad,
1969
18. Fedorov S.F., 1969. O reaultatakh issledovania digrologicheskoi
roli lesa. (On the research results of hydrological
role of forest). Trans. of GGI, vol. 176.
19. Kharchenko S.I., 1968. Gidrologia oroshaemykh zemel
(Hydroìogy of irrigated areas). Hydromet. Publ.
House, Leningrad.

ABSTRACT
OBJECTIVE CRITERIA TO DECLARE A SERIES OF
DATA SUFFICIENT FOR TECHNICAL PURPOSES
by
Penta A., Rossi F.
It is supposed: that for technical purposes it is
necessary to estimate the values xo that an hydrological
variable x may assume with a given probability 6; that x can
be measured directly and that its n values have been recorded.
The series of the n values of x is'defined sufficient
if it consents to estimate xo with a reliability adequate for
technical purposes.
By referring to the usual statistical methodologies,
the authors present objective criteria to recognize whether
the series of n values is sufficient. The authors furnish some
diagrams that indicate which minimum values of n are necessary
for the series to be considered sufficient.
From the diagrams it is evident that for the same values
of n the series sufficiency is strictly linked to the variability
of x.
Particularly, the authors considere the normal, the lognormal
and the double exponential (Gumbel) distributions, the
m.ost applied laws of hydrology.
-- RESUME
On suppose qu'à l'égard du problème technique il faut
estimer les valeurs XQ qu'une variable hydrologique x peut
assumer avec la probabilité 0, que x peut être mesurée et que
n valeurs de x ont été enregistrées.
La série des n valeurs de x est définie suffisante si
par elle on peut estimer xa avec une confiance adéquate au but
du technicien.
En se rapportant aux méthodologies statistiques usuelles
on donne des criteriums objectifs pour reconnaitre si la série
des n valeurs est suffisante.
On donne des diagrammes par lesquelles on indique les
valeurs minima du nombre n qui son necessaires afin que la série
soit suffisante.
D'après les diagrammes il apparait évident que, n ayant
la même valeur, la suffisance de la série dépend de la variabilité
de x.
En particulier, les auteurs considèrent la loi normale,
la loi log-normale e la loi de Gumbel, qui sont plus fréquemment
employées en hydrologie.

228
Symbols and definitions
1: Let us indicate by I
- x , a generic hydrological variable3
-E , ax and y, respectively the mean, the standard deviation
and the coefficient of variation of the x population;
- @(XI, the distribution function of x ;
- xQ, , the value of x corresponding to the cumulated probability
@ e
Moreover, let us also indicate by :
- x , with 14isn , the n values of x registered, in each
single year,iduring the observation period i
- - x and ax, respectively the estimates of [ and a ;
- Pix) , the estimate of the distribution function (Dix);
- xppa,, the estimate of
xa ;
- y(xP,@), the sampling coefficient of variation of 5 E@
2: If x is normally distributed,the best estimate xpsio, of x
is obtained [ 1 1 by t
X pn(D = Z + u
@
where u is the value of the variable u, that in equation:
@
1 2
2
1
@(U) E -
du (2)
corresponds to the fixed value of @ .
The sampling coefficient of variation of xp could be obtained
approximately [2] by
-@
I
or whenever n is sufficiently high, the equation (3) becomes :
1 +u$/*
i i n
(1)
(3')
(D

3: If x is distributed according to the log-normal function,
having established that y = log x , we indicate by :
- Yi 9 the value of y corresponding to the generic value xi ;
I - y and s respectively the mean and the standard .deviation of
the n values of Yi.
Y'
Therefore, the best estimate x of x is obtained [a] by
P=UJ 0
(4)
log xp' E y. + U@ 8
=a>
Y
229
where the value of uUJ is deduced by means of equation (2) while the values
of y and s are deduced respectively by the equations :
' n
log xi
- 13.1
Y" n
and
r n
s =
Y
n-1
The sampling coefficient of variation of xPIUJ could be obtained
approximately [ 21 by :
2
1 + u0/2
or, whenever y* is sufficiently mall, the
I 2
- 1 (7)
equation (7) becomes :
4: If x is distributed according to the double exponentlal,namely
@umbel function, an almost correct and efficient estimate xP=@ of x UJ is
obtained [4] by :
xppUJ= + K UJ
(8)
where Ka, is the value of the variable K that'in the equation :
-- 6
1
K = (0,5772 + In In -
(9)
x a,
corresponde to the fixed vaïue of

230
The sampling coefficient of variation of x could be obtained
approximately [ 21 by :
5: When the variable x is measured in k gaging stations, lying
in a detertnined zone, there exists an hydroloRica1 similitude between the k
stations if the parameters, or some parameters al , z2 , ..... of the x
diatribution assume the same value or if they vary from one to another with a
known regression relation in function of a certain number of parameters
y1
y2 ..... '
[5].
The inter-station correlation is the correlation which exists, in
such cases, among the values of x registered in them contemporaneously (e.g.
in the same year if maximum and minimum annual values are considered).
Therefore, the information that can be derived from the k stations,
considered all together, in regard to the x distribution parameters al, a2,..
..... is the same as the information furnished by a number k of independent
stations. Such a number, known as the equivalent number, depegds both on k
and on the mean interstation correlation coefficient F , thus becoming so
smaller than k, the higher the value of F is.
So, e.g. if in the k stations the mean 6 of x would assume the
same value, the information that the complex of the data registered in the k
etatiomwould furnish in regard to would. be equal 16) to the information
furnished by an equivalent number of independent stationsequal to :
k
keE l+F(k-1)
Basic Risk. Uncertainty and Effective Risk
6: Normally, for design purposes, by referring to a given hydrolog&
cal variable x,we indicate by :
xd , the value of
(deBiRn Value) ;
N , the desinn duration.
x that Is assumed as the basis for the design
Particularly, in a flood problem, we select a value Of so that
there exists a probability of failure W that xd will be exceeded
xd
at least
once in N years.
Consêquently, xd coincides with the value xD of x whlch corre
sponda to a value of D of the cumulated probability furnished by :

E.g. when N = 25 years and W=0,025, @ is equal to 0,999.
Likewise, in a drought problem, we select a value of so
xd
that
there exists a probability of failure W that xd will not be exceeded at least
once in N years.
Consequently, instead of using equation (121, we must apply the
following equation :
E.g.
1
@ = l -
/N
(1 - w)
when N = 25 years and W = 0,025, @ is equal to 0,001 .
The basic risk is defined [ 71 as the risk that would be encountered
if, by knowing the probability distribution of x,we would assume x EX@ .Such
d
risk is measured by means of the probability of failure
In reality, however, the distribution of x is not known. Consequently,
having fixed the basic risk W and having calculated @ by means of eqpations
(12) or (131, with the use of a series of n values of x,only an estimate
x of x could be had, aiid, therefore, to assume x = xQ an error
equal toP =?x - xD ) would be made. In reality the effective risk that is
encountered ?;treater than the basic risk due to the uncertainty with which
the value of x could be estimated.
0
Sufficiency of a Single Series of Data
7: Once the basic risk W has been determined, to judge whether a
single series of data is sufficient for technical purpose4,i.t is necessary to
take into account the uncertainty with which x@ could be estimated.
Generally, by considering also the observation periods which are usually
available, a series of at least 30+40 data is defined IIlonP and it is
implicitly retained sufficient; a series with less than 30+40 data is defined - Vshort1I and is considered insufficient.
In reality, however, such criterion might be erroneous. In fact, if
the uncertainty, with which x0 could be estimate, is measured by means of
y{xp } , from
eq. (31, or eq. (7) or eq. (101, we recognize Immediately that
the sad uncertainty, beside n , depends also on :
i) the variability of the hydrological magnitude x being considered,
which can be measured by y ;
w.
231
ii) the probability Q of the design value xd , which is a function
of the basic risk W and the design duration N.
In particular, let us consider e.g. the annual rainfall depth xuh distributed generally according to the log-normal function [ 81, with a coefficient
of variation y, which varies from 0,l to 0,9 as we progressively move from
sub-humid zones to semi-arid and arid zones, the mean annual rainfall changes
from vaïues of circa i 500 mm to values of circa 50 mm [ 9 J .

232
As it could be noticed from the diagram (a) of fig. 1, if it were nec
essary to estimate the median value x 5o of x,a long series could be retained
sufficient from a technical point of v hw for each of the possible values of yx,
since in no case y{xp, would be greater than 15%.
However, wheh we fix the duration N equal to 25 years and the basic
risk equal to 2,5%, by applying eq. (12) or eq. (13) we notice that we must refer
to values of @ equal to 0,999 or 0,001. In this case, from the diagram (b)
of fig. 1 it ie evident that a long series of data would be sufficient from a
technical point of view only if y were rather low.
In fact, even for values of y, greater than 0,5, Y{X~,~) could
s be greater than 20%.
On the other hand, from the same diagrams (a) and (b) of fig. 1 ,it
can be derived that a short series of data, which is certainly insufficient
for values greater than y , could be sufficient if y, would assume too
small values.
Analagoue considerations could be made if x follows the double
exponential distribution by examining the diagrams (a) and (b) of fig. 2 in
which are represented the function of y{xp, 1 as n and
(corresponding to the distribution mode) -and fi? CD = Y,
0,999.
the Data Registered in Other Stations
for @ P 0,368
9: The regions where regular hydrological measurements have been
taken for a short period of observation, have often arid or semi-arid climate,
therefore, it becomes practically impossible to estimate from a single series
of data the values that, with a given probability, those magnitudes might assume.
It becomes therefore necessary to recognize if it is possible to improve the
estimate of xrp in a given station by using the data obtained in others. As
it is known, to render this possible, it is necessary that the different stations
considered be hydrologically similar (see pgr.5). For this to happen, it is
necessary that the values taken by x in the &d stations depend on common
meteorological and hydrological factors. Consequently, this implies that there
exists an inter-station correlation.
It is udeful to point out that from this point of view it is very
important to consider either one of the hydrological magnitude. In fact, the
mean inter-station correlation coefficient P is amaller when the daily or
weekly rainfall is considered, while it is greater when we take into account
annual rainfall [ 101 . In the case of annual rainfall, in a research conducted
from the information furnished by 1141 pluviometers installed in the Western
D.S. and in the South-West California [lOl, Caffey has shown that the mean
inter-station correlation coefficient F situated in a zone meteorologically
homogeneous varies from Q,30 to 0,SO. In a recent research on pluviometers
installed in Basilicata and in Southern Italy, we have found fn0,5 t 0,6 and

in a research on the Morocco pluviometers, being conducted at the time of this
report, r = 0,90 which is still higher.
233
10: By referring to the mean value 6 of x, in the diagram of.
fig.3, ne have repreeented equation (11) which formulates the law according which
the equivalent number ke of independent stations, defined in pgr.5, varies as
a function of r' and the number k , of statione installed in the zone.
As it can be observed from fig.3, for each value of F, ke increasel
at each increase in k ,tending asymptotically toward a maximum value
kernax= F
Consequently, the maximum increase of information that is obtained in
regard to 5 by applying the hydrological similitude criteria is inversely
proportional to F . E.g. when F = 0,5 , the information, at the most, could be
doubled; for still greater values of ? , which are often encountered in hydrology,
the advantage obtained could be almost negligible.
On the other hand, no real benefit is obtained by increasing the number
of k stations above a certain limit strictly connected to F . To prove this,
we have represented in the diaáram of fig.4 the law with which - ke
varies as
k
a function of k for different values of r. As it can be noti$eyxif we are
satisfied with the 90% of the maximum information that can be obtained, by
ke
accepting that - = 0,9, this objective could be reached with only 9
k
stations, for F = with only 4 stations for r 5 0,7.
CONCLUSIONS
11: In eome countries, systematic, reliable, homogenous measurements
have been taken for only few years and in few stations. Moreover, to render the
problem more severe, such regions have an arid, or semi-arid climate. Therefore,
due to the extreme variability of the hydrological magnitudes, with the same
number of data, the uncertainty with which the probability distribution of them
could be estimated, is greater.
Consequently, in the said regions it is particularly important to
utilize all the information that the few available data could furnish, by applying
either correct statistical methods to interpret each single series of data and/or
by defining objectively some hydrological similitude criteria that would consent
the interpretation on how the magnitude varies from one station to another.
Particularly, for a reference magnitude x , by applying the hydrolog-
ical similitude criteria, it is possible :
a) to obtain a reliable estimate of xo even for points where no
direct measurements of x were even taken ;

2 34
b) to improve the estimate of x in points where only few data
are available.
Q,
The advantages obtained in regard to point b) could be noticeably
limited by the inter-station correlation located within an hydrologically
homogeneous zone.
In any case, only when all the information available has been uti-
lized, it is possible to establish whether the data available are sufficient
or not to be used in practical applications.
12: If the available data in the region should be insufficient, a
supplementary research program would be necessary. Even in this case, it is
absolutely necessary to take into account the information furnished by all the
data available so that the research program is carried out in an adequate manner.
On the other hand, we must be well aware of the results of a short
research program.
In fact, if an appropriate localization of the stations is made, it
is useful :
i) to individualize and improve the delimitation of the region in
hydrologically homogeneous zones ;
ii) to determine the regression law of a variable x as function of
some parameters which characterize the point or the basin (e.g. the regression
relation of the mean rainfall depth vs the level of the point or the regression
relation of the mean annual runoff vs mean annual rainfall).
On the other hand, when both aims have been attained, the research
program could be useful also to estimate the probability distribution of x
in different points (or basins) only if in the region there are one or more
gaging stations functioning for a long time.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
KENDALL M.G. and STUART A., (1967). The Advanced Theory of Statistics.
London, Griffin, Vol. 2, 2nd Ed., p.54.
ROSSI F., (1 972) Distribuzione di campionatura di alcune grandezze stg
tistiche. Fac. di Ing. delltllniv. di Napoli, 1st. di Costr. Idr.,
Quad. no 5 .
AITCUISQM J. and BROWN J.A.C., (1957). The Lognormal Distribution.
Cambridge University heas.
235
LOWRaY N.D. and NASH J.E., (1970). Methods of Fitting Double Exponential
Distribution. Journal of Hydrology, 10, pp. 259 - 275.
VIPARELLI C., (1 965). Idrologia applicata all 1 Ingegneria. Parte II, Fond.
Politecnica del Mezzogiorno d'Italia, Napoli.
MATALAS N.C. and BENSON M.A., (1961). Effect of Interstation Correlation
on Regression Analysis. Journal of Geophysical Research, Vo1.66, nolo.
YEVJEVICH V., (1972). Probability and Statistics in Hydrology. Water
Resources Publ., Fort, Collins, Colorado.
MARKQVIC R.D., (1965). Probability Functions of Best Fit Distributions of
Annual Precipitation and Runoff. Hydrology Papers no 8, Fort Collins, Co-
lorado.
GARCIA-AGREDA R., RASULO G., and VIPARELLI R., (1 973) Pluviometric Zones
and the Criteria to Define their Boundaries for Regions with Scarce Data.
Simposio sobre proyectos de recursos hidraùlicos con datos insuficientes,
Madrid.
CAFFEY J.E., (1965). Inter-station Correlations in Annual Precipitation
and in Annual Effective Precipitation. Hydrology Papers no 6. Fort Collins,
Colorado.

20 40 60
OBJECTIVE CRITERIA TO DECLARE A SERIES OF DAPA SUFFICIENT I'OR
TECHNICAL PURPOSES
*O n

OBJECTIVE CRITERIA TO DECLARE A SERIES OF DATA SUFTICICNT FOX
TECHNICAL PURPOSES

10
Ke
8
6
4 I
2
O
6 - /
- I
O, 50
Y
O, 70
I
?=%O
10 20 30
1
Fig. 3
OBJECTIVE CRITERIA TO DECLARE A SERIES OF DATA SUFFICIENT FOR
TECHNICAL PURPOSES
I I
I I
1 K. 50

l,oo
0,8 O
0,60
0,4 O
0,20
- 1,oo-
/
Fig. 4
10 20 30 50
40 K
OBJECTIVE CRITERIA TO DECLARE A SERIES OF DATA SUFFICIENT FOR
TECHNICAL PURPOSES

ABSTRACT
SOME CRITERIA USED IN HYDROLOGIC STUDIES
WITH INADEQUATE DATA
Carlos Quintela Góis
In territories where the hydrologic networks are
still scarce, it is necessary to adopt simplified designing
criteria which might lead to sufficiently reliable results.
In this paper those which are normally used for the hydro-
logic characterization of the drainage basins under these
conditions are presented and example of their application
to Okavango Basin in Angola is given.
RESUME
Dans les territoires où les réseaux hydrologiques
sont encore insuffisants, il faut recourir a des procédés
de c alcul simplifiés qui puissent conduire a des résultats
dignes de confiance. L'auteur expose des méthodes normalement
utilisées pour évaluer. dans de telles conditions
les caractéristiques hydrologiques des bassins, en prenant
pour example du bassin du Cubango, en Angola.
* Civil Engineer - Member of the Working Group
of the Overseas Ministry (Portugal) for the I.H.D.

242
1. Introduction
In the framework of the hydraulic policy which has b en followed for he
last years in the Portuguese Overseas Provinces, the study of the general plans
for the development of the water resources plays a very important role. In the
two main provinces of Africa - Angola and Mozambique - this action led to the fact
that the main drainage basins are already covered by such studies; that enables
an adequate hydroelectric and hydro-agricultural overall planning to be made,
Hydrological studies are obviously the fundamental basis of such general plans
because they. determine the hydrologic characterization of the basin and from the-
re the preliminary design of the several schemes and estimate of their potentia -
lities. In this field international cooperation which was achieved with the other
territories of Southern Africa, as a result of established agreements, is also of
a great importance and it gives an idea of the value that water has got for the
common development on that part of the world.
The inhospitable characteristics of these areas together with the communica-
tion difficulties and low human occupation result usually in very scarce and
recent hydrologic networks so that on carrying out hydrologic studies one faces
the difficulty of applying the classic methods or those used for more developed
areas.
Therefore it is necessary to adopt approximative methods and special crite -
ria enabling to arrive at sufficiently correct and reliable results for the ai -
med purposes.
In this paper the methods which have been followed for carrying out the abo-
ve mentioned hydrologic studies are presented and the approximate criteria that
have been adopted as a result of inadequacy of data are pointed out; at the end
a practical example is given for the case of a drainage basin in Angola. Only
the aspects of rainfall and run-off in average terms are stressed because they
are of most interest for the hydrologic studies of general plans.
2. Rainfall
Among the hydrologic data, rainfall is commonly measured for a longer pe-
riod, even in developing territories. Although networks do not cover satisfacto-

ily the areas to be studied, they enable the characterization of the phenomenon
with enough accuracy to be achiewed.
243
Usually the daily precipitation data measured in raingauges normally loca-
ted at villages or townships are available. Record periods of twenty years ormore,
at least in some of the stations, are frequent and the use of correlation techni-
ques enables to obtain monthly rainfall all over the stations of the network, On
the other hand, uniform rainfall regime of the African subtropical regions with a
long period of four months without precipitation is well known, which makes it ea-
sier to fulfil some failures in the records.
The study of that regime is usually done by taking the annual weighed pre-
cipitations obtained from the isohyet maps drawn for the basin. The isohyet me -
thod is considered to be the most adequate when dealing with incomplete informa-
tion, because local surveys, topography, etc. may help to introduce corrections
or indicate the best drawing of the curves of equal precipitation so that a pat-
tern, as close as possible with reality, can be obtained. Once the basins have
usually a drainage area of tens of thousands of square kilometers, the used sca-
le for drawing isohyet maps is normally 1:l O00 000.
After those maps are obtained, some characteristic sections are chosen and
the weighed values are determined. These are the bases for the study of the rain-
fall regime and periods of about 20 years permit the application of stochastic me-
thods. Among these, the method of Hazen-Foster has been considered to be the most
adequate to interpretate the phenomenon. After graphical and analytical confirma-
tion of its applicability, it is possible to obtain the mean annual value and tho-
se corresponding to characteristic return periods. The probability relating to each
one of the years of the period can be obtained as well.
This analysis gives a first idea of the natural sequence of the years and
principally the occurence of dry periods and their degree of drought so that fur-
ther studies for comparison with the run-off can be done.
The study of rainfall is usually completed with a short analysis of dry and
wet seasons and mainly of the frequency with which longer dry seasons may occur.
3. Run-off
As far as flow measurements are concerned, data is always very scarce and only

few flow stations in Portuguese Africa have records available for more than 5 to
10 years. Besides, it has been verified that the study of general plans normally
shows the need and lead to the best choice and establishment of the hydrometric
networks.
Stochastic methods cannot be applied safely with such short periods and
therefore the first approximative criterium to be used is trying to characterize
the available flow record period by relating it with the similar period of the
rainfall studies. Hence it is possible, as a first approximation, to consider the
same probability of occurence for the annual flow and rainfall of a certain year.
From this it is often possible to chose certain years which can be considered
as average or with a given degree of dryness. Therefore a critical period
corresponding to an unfavourable sequence of years can be chosen in order to fix
the storage capacity of interannual reservoirs and to obtain a complete regulation
of the flows. This sequence is normally formed by an average year followed
by two or more dry years with fixed characteristics. Undoubtedly this is an approximate
approach, but experience has shown that for studies at the level of general
plans this analysis is quite acceptable and safe because the pessimism in
the reasoning compensates the.uncertainties resulting from the inadequacy of data.
Sometimes, as an exception, there exists in the basin a measuring section
with a longer period of records and for which stochastic methods can be applied.
Two ways can then be followed, (1) correlation analysis with other stations of
the basin, trying to obtain more data for those which have shorter records or
(2) characterization of the shorter period by relating it with the longer one
of that station in a similar way as mentioned in the previous paragraph for the
rainfall.
The first method is not always easy to apply, because the rivers might
show a change of regime along their course as a result of the phisiography and
correlations are no more valid.
The second one is more reliable and on applying it, it is possible to ar-
rive at safe and easily interpretable results. Normally one can obtain not only
the annual flow but also the monthly ones of the average and dry )ears of the cho-
sen critical period and therefore carry out more reliable regulation studies.

245
The study of rainfall/run-off relations has not been, as far as our expe-
rience is concerned, successful for large drainage basins as a method of enlar -
ging the available flow record period. This is probably the result of the speci-
al type of the rainfall regime of those regions - short, heavy and localized storms-
together with high temperatures and evaporation rates which affect the usual me-
chanism of transforming rainfall into run-off. Besides, this method would only
lead to global annual values and its distribution along the year is not possible
to obtain.
4. Application example
4.1 - General characterization of the problem
The Okavango is one of the three big international rivers of the
South of Angola. It springs on the central plateau of the territory and flows
more or less North-South down to the border with Southwest Africa where it
shifts eastwards, forming the border, crossing Kaprivi Strip and spreads in-
to a wide swampy area ( Figure 1).
Its drainage basin in Angola is about 150 O00 km2 from which 61 O00
km2 belong to its main tributary Cuito.
The northern part of the basin is the most rainy one and there the
altitudes reach 1 800 m, decreasing gradually southwards to 1 O00 m.Here the
climaté is semi-arid. Rainfall occur in the wet season from October to April;
the other months are dry.
From the geological standpoint, the northwest part of the basin is
formed by igneous and metamorphic rocks; sedimentary formations occur in the
rest of the basin.
The hydrographic pattern is characteristical as well, the tributa -
ries being normally parallel to each other and flowing North-South. The sha-
pe of the beds is ruled by the local geological and topographical conditions.
As far as the vegetation is concerned, it changes from the more or
less dense forest in the North into the savana in the South.
The problem was to carry out the general plan for the development of

246
the water resources and obviously the first step was the hydrological study.
In the following chapters a summary will be presented of the analy-
sis made for the study of the rainfall and run-off, according to the methods
and criteria mentioned above in this paper, once the available data was ina-
dequate.
4.2 - Rainfall studies
For the rainfall studies, the records of 28 stations for the period
1943/1970 were available. However, only from 1950/51 onwards, the number of
stations with complete records was sufficient and therefore the basical study
period considered was 20 years, from 1950 to 1970. Some shortage of monthly
records necessary for the evaluation of the annual values was easily overcome
by correlation with more complete and reliable stations.
With these annual values, the isohyet maps were drawn on a scale
1:l O00 O00 introducting the influence of altitude and other known climatical
factors and avoiding a cold interpretation of the plotted values.
After chosing some characteristic sections, the weighed annual rain-
fall was determined and analysed by applying the Foster-Hazen method. Figure
2 shows a diagram with the sequence of annual precipitation and the correspon-
ding probability graph for a section of the main course of the river where the
international border starts.
From the joint study of these graphs, some conclusions can be drawn.
First of all, the applied stochastic method can be considered adequate to interpretate
the phenomenon and therefore it is possible to determine a mean annual
precipitation of 950 mm as well as precipitations corresponding to certain
return periods. One can note the occurence of a sequence of four dry
years which might be considered as the basis of the critical period for regulation
purposes.
4.3 - Run-off studies
The basin has 19 flow measuring stations and the records started to
be obtained early in 1963. Before that date, there were some random measure-

247
ments made with floating device> but their reliability was doubtful. The net-
work is nowadays equipped with automatic level gaugings and flows are measu -
red with current meters suspended from steel cables crossing the river from
one bank to the other.
It was then possible to have flow records for a period of 7 years
consisting of maximum and minimum flows, average daily flows, and consequent-
ly monthly and annual values.
For such a short period stochastic methods are not applicable with
reliability; nevertheless the analyses made for the rainfall showed that such
period has average characteristics and therefore the mean annual flow can be
estimated by averaging the flows of those seven years for every station.
The same criteria cannot be applied to determine the dry year flow,
because in this seven years period (1963/1970) any of the years of the cri-
tical period obtained from the rainfall study is not included.
Fortunately, there is a station in the international strech measu-
red by the South African Services which has got records for a longer period
(25 years) from 1945 on, although some of its valueshave been obtained by cor-
relation. It was then possible for this station to apply the Foster-Hazen rne-
thod which showed a rather well interpretation of the phenomenon.
Figure 3 shows in the same way as for the rainfall the diagram of
annual flow sequence and the probability graph.
The former indicates a notorious resemblance with the one of the
rainfall, being characteristical the four dry year period 1966/1970. It skws
as well that the period 1963/1970 is an average one and that 1966/67 can re-
present the dry year of the critical period.
In order to obtain the annual flows in any section of the river,tk
curves showing the variation of the specific annual flow with the drainage ba-
sin for the average and dry year, were drawn ( Figure 4); these curves show
bi uniform pattern and thus one can consider them sufficiently reliable for
obtainment of the desired values.
The regulation of flows can be studied by considering the sequence

248
of an average year followed by four dry years as determined above.
5. Conclusions
Some criteria normally utilized for hydrological studies of the general
plans for the development of the water resources of rivers in semi-arid areas of
Portuguese African territories were presented and an example of their application
given. The obtained results are obviously approximate but they can be considered
sufficiently safe for the purpose and moreover when decisions would be taken for
the design of specific projects further data will be available and then a more re-
liable analysis can be made.
.........................

W
I
I-
249

nm
6 O0
LOO
200
O00
BOO
600
LOO
200
O
Pmm
500
LOO
300
200
100
O00
900
SEQUENCE GRAPH
FOSTER- HAZEN ADJUSTMENT
- a% m m
0 0 0 - N Y i 0 O O 0 0 0 0 O O y> m O b m 6
- N
- < m u > c m m m m a m r n m
FiGQRE 2 - STUDY OF ANNUAL RAINFALL
PROEABILIT Y
YEAR

IL LL
O
io6,'
10 O00
z 9000
I 3
< 8000
z
7000
6 O00
5000
.4oco
3 WO
2 O00
1 O00
O
lo6 m'
1.4 o00
IL
Y
O
z 3 O00
3
2
U
3
z U
2 ooa
1000
3 000
9 o O0
a O00
7 O00
6000
5 O00
4000
3000
2 O00
1 O00
SEQUENCE GRAPH 251
FOSTER- HAZEN ADJUSTMENT
-ri yl In "01
0 0 0 - h i - 0 0 0 0 0 0 0 0 o ~n m m o i m m
- N
m - m w p i m 01 m 0 . m m m m
PROBABILITY
FIGURE 3 - STUDY OF ANNUAL RUNOFF
i
1

252
.
3
2
a
a
W
U
:;
W
I
I
U
I-
U
a
3
n
3 .
3
O
3
0
=l
U
3
3
Y
2
6
W
II:
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AB S TRA CT
UTILIZING CLIMATIC DATA TO APPRAISE POTENTIAL WATER YIELDS
Robert L. Smith"
Precipitation and temperature measurements often represent the
only significant hydrologic data available in developing areas. Initial
assessments of potential surface and ground water supplies must build
on this limited climatic base. Early in the planning studies there is
need for an accurate estimate of mean annual streamflow, and of the
probable variance in annual flows. These determinations can be made
utilizing an empirical function relating the mean annual runoff
coefficient to the aforementioned climatic parameters. The relationships
have been tested in a wide range of environments, and their general
utility can be extended appreciably with limited surface and subsurface
observations. Applicability of the recommended relationships is
demonstrated by selected case studies involving a variety of problems.
Included are examples illustrating the calculation of: (a) mean yields
for ungaged areas, (b) the probability distribution of annual flows for
ungaged areas, (c) daily flow duration curves, (d) potential yield of
selected groundwater areas, and (e) the potential impact of precipita-
tion augmentation on surface water supplies.
RESUMEN
A menudo las medidas de precipitación y temperatura son los Úni
cos datos hidrolbgicos disponibles para áreas en desarrollo. Los esti-
mados iniciales sobre abastecimientos potenciales de aguas superficia-
les y subterráneas deben partir de esta limitada base climática. Muy -
pronto en el curso de la planificación se hace necesario un estimado -
preciso del caudal promedio anual y de la variación probable en flujos
anuales. Estas determinaciones pueden hacerse mediante la utilización
de una función empírica relacionando el coeficiente de escorrentía me-
dia anual con los antes mencionados parbmetros climáticos. Este tipo -
de relación ha sido puesto a prueba en una amplia serie de medio am--
bientes y su utilidad general puede extenderse apreciablemente con li-
mitadas observaciones sobre y bajo tierra. El éxito con que se han --
aplicado las relaciones recomendadas se demuestra por medio de casos -
escogidos que cubren una variedad de problemas. Se incluyen ejemplos -
que ilustran el cálculo de: (a) rendimientos promedios para áreas Ca--
rentes de medidas, (b) la distribución probabilística de caudales anua
les en áreas carentes de medidas, (c) curvas diarias de caudal-dura---
ción, (d) rendimiento potencial de áreas de agua subterránea escogidas
y, (e) el impacto potencial de la incrementación de precipitación so--
bre abastecimientos de agua superficial.
-
JI Deane Ackers Professor of Civil Engineering, University of Kansas,
Lawrence, Kansas, USA.

254
The water resources planner is often required to appraise the water yield
characteristics of streams for which flow data is unavailable. In these situ-
ations the initial appraisal has to be based on climatic factors supplemented
by prior experience in similar terrains. This paper presents an empirical
relationship designed to further this appraisal, and which the author has found
useful on a number of occasions.
The basic water balance equation applied to a catchment area may be
expressed as
P = R + E + AS (1)
where all temo represent units of depth over the catchment area, and P = precipitation,
R = basin outflow, E = evapotranspiration and AS = change in
storage.
For the condition of an extended time interval the AS term becomes
negligible. In this case, and after dividing all terms by P, the equation may
be rewritten as
RIP = c = 1 - E/P (2)
Thus, in the long term the runoff coefficient C is governed by climatic considerations.
Geographers and agricultural scientists have long utilized climatic
parameters in appraisinz water balance questions relating to management of soil
moisture. In 1967 Guisti and Lopez [i] proposed that the mean stream discharge
could be determined as a €unction of (a) mean annual precipitation and (b) the
basin climatic index. BCI. The latter is based on the work of Thornthwaite i21
Jan *'"I
where P is the average monthly precipitation in centimeters and T is the aver-
age monthly temperature in degrees Centigrade.
If the hypothesis presented by Guisti and Lopez has merit, it should be
possible to develop a relationship between BCI and the deviations from the mean
line dram on a sc.atter diagram of average precipitation versus average runoff.
Their initial efforts to develop such a relationship were limited to examination
of relatively short term data in Puerto Rico. Smith 131 subsequently extended
this approach by examining data from approximately 250 ca.tchments in the
United States and Puerto Rico. The resulting empirical relationship between
the coefficient C in equation (2) and the BCI is graphed in Figure 1. It dif-
fers appreciably from t.he curve initially presented by Guisti and Lopez.
available data provided firm definition of the relationship for BCI values
ranging froiri 35 to 150. Currently, extension beyond these limits is most ten-
tati.ve and i.s Eased on the following. The lower end was extended to the obvious
terminal al: the origin. Extension of the upper end of the curve was based un
concurrer.t appraisal of the nature of the 13CI vs P relationship in high rainfall
zrens, and on recognition that the change in C with X I should be such that the
i.iic.rment.al percent of precipitation which becomes runoff is constantljr
The

increasing bur never exceeds unity. One word of caution. Data utilized in
developing the re1 onship was obtained €rom catchments for which the sub-surface
outflow was negligible. Thus the ruhoff calculated by Figure 1 represents
tot91 runoff and cannot be directly equated to streamflow in those instances
wtie're a significant percentage of the yield' is discharged as sub-surface flow.
Utilization of the relationship is enhanced by conversion of existing
climatic data into a basic P vs Bdi relationship for the area in question.
Worldwide the relatioriship between BCI' and P varies markedly. Regionally It
preciably with topographie considerations. However, for a given
he relationship between BCï and P is weJ.1 defined. Figure 2 illustrates
a typ'ical relationship for a basin in the State of Kansas in the central
United States Qhere elevation changes are negqigible, and similar relations fcr
Puerto Rico where elevation is a significant factor. Note that the slope of the
relationship also varies slightly with location. Figures 1 and 2 caq be used
conjunctively to develop the mean annual rainfall-runoff relationship for the
catchment. Experience has shown that actual data will scatter about the curve
so determined because AS is seldom negligible on an annual basis. The individ-
ual curves tend to approach a 45" asymptote as evapotranspiration tends EO
255
become fully satisfied and thereby constant. For example, in Puerto Rico the
evapotranspiration demand is satisfied at all elevations when the rainfall
exceeds SOO centimeters, but the magnitude of this consumptive loss is a function
of elevation.
The basic C vs BCI relationship has been tested in several ways with
satisfactosy results. Figure 3 will seme LO illustrate. Figure 3(a) presents
a coiqparisbii of calculated versus observed discharge for thirty streams in
Puerto Rico [4]. The calculated values were determined via conjunctive use of
the appropriate curve from Figure 2 and Figure 1.
Since the qbserved records
wexe relatively short, many no longer than three years in length, the applicable
BCI was based on the average precipitation during the period of observed stream-
flow. BCI values for these streams range from 49 to 178. Figure 3(b) presents
the mean annual precipitation versus mean aqnual runoff relationship €or the
State of Kansas. The solid curve thereon was based on observed data from 122
basins [5]. The dashed cuí-ve was calculated using the Kansas curve of Figure 2
and Lhe basic coefficient chart of Figure 1. Basin BCI values for the ctndition
of mean precipitation range from 25 to 70.
The basic relationships can also be utilized to appraise possible stream
response under several yeats of above or below nomal precipitation. For example,
in recent years appretiable attention has been directed to the potential
application of weather modification tachniqves in improving water supply con-
dtionc.
Although the bulk of the research effoxt has been directed toward
seeding techniques and understanding the mechanisms of cloud physics, several
investigators in the 1Jnited States, via the use of hydrologic simulation techniques,
have attempted LO explore how streams would respond to a given increase
in precipitation. The relationships in Figures 1 and 2 can be utilized to
estimate the percent gain in runoff thaL will oc €or a given increase in
average precipi tatjon. Let the subscript I represent natural conaitioi-s, subscript
2 represent augmented conditions, and the symbol PM equal P2/P,. Then

256
Percent gain in runoff = 100 - = 100
Ri
PC-PC (PM1 c*-cl
22 113100 (4 1
plcl cl
Table 1 summarizes the impact of precipitation augmentation on water yield
as determined by hydrologic simulation and as reported by Linsley and Crawford
[6], Crawford [7], Lumb [8] and Smith [3]. The first three authors utilized
the Stanford Watershed Model and the latter utilized the Kansas Watershed Model.
In aggregate, these investigators conducted simulations on 14 separate watersheds,
13 in the United States and one in New South Wales. The last two columns
provide a comparison of the average increase in yield as determined by
simulation and as estimated by use of Figure 1.
The calculations assumed that
the slope of the BCI vs P relationship was equivalent to the typical Kansas
curve. This approximation introduces some error because the slope of this
relationship does vary slightly from watershed to watershed, Nonetheless, the
calculated and simulated values are -most comparable. Examination of the computer
simulations again reveals that year to year increases scatter about the
mean value listed in the table. - Table 1 - Comparative evaluation of the impact of precipitation aiigmentatiun
on mean yield.
-
-
,ength lbserved
of Period
'eriod ainfall -- Runoff
'ears cm/year iainf all PM
One HundredlTen Mile Creek,
Kansas -
Stranger Creek, Kansas - 11
Doniphan Creek, Kansas - i/
Black Vermi lion River,
Kansas A
Salt Creek, Kansas - 11
17
17
17
20
8
14
14
14
8
88.4
88.4
88.4
90.5
86.4
77.5
77.5
77.5
58.0
.205
.205
,205
.200
.261
.125
.125
.125
,066
1.05
1.10
1.20
1.10
1.10
1.05
1.10
1.20
1.05
S. Fk. Solomon River, Ks - 11
Beaver Creek, Kansas
Cottonwood Creek, Calif. - 21
8
8
20
21
2
58.0
58.0
53.1
45.6
40.9
.O66
.O66
.O57
.O16
,080
1.10
1.20
1.10
1.10
1.15
Wollombi Brook, 3l
New South Wales -31
5 107.7 .141 1.10
Beargrass Creek, Ky T~ 5 110.6 .403 1.10
Arroyo Seco, Calif. -
5 68.2 .386 1.10
LaBrea Creek, Calif. -4/
41
18 28.6 .O84 1.10
Dry Creek, California - 22 130.0 .472 1.10
Saxtons River, Vermont - 41
16 111.6 .499 1.10
- i/ From data presented by rn i70)
- 2/ From data presented by Linsley and Crawford (1962)
- 3/ From data presented by Crawford (1965)
- 4/ From data presented by Lumb (1969)
5/ Not calculated
-
-
-
% Gain i
Computer
#irnulatiori
16
33
74
35
30
21
41
94
23
49
107
41
62
82
35
20
19
41
18
19
Runoff
:alculated
17
34
70
32
31
22
40
87
26
52
117
53
- 51
78
40
25
24
44
20
20

257
Earlier reference was made to the fact that a plotting .of annual precipitation-runoff
values for a given basin will scatter about the mean annual relationship
one develops with Figure 1 and the basin applicable Figure 2. Also,
it was noted that year to year percentage gains in flow from precipitation
augmentation, and as determined by computer simulation, would scatter about the
average gain observed for the entire period of record. This scattering is due
to the well established phenomenon of hydrologic persistence and reflects shortterm
storage changes. Question arises, therefore, as to whether the relation-
ship can be used to determine flow characteristics other than the mean.
answer is yes but a reasonable amount of judgment is required. Determination
of the distribution of annual flows will serve to illustrate.
Available climatic data can be utilized to develop the probability
distribution of basinwide annual precipitation, and the basin average curve
for Figure 2. When working with a basin whose geologic structure is not conducive
to the development of significant baseflow components, i.e., a basin
with minimum persistence characteristi.cs, an estimate of the prcbability distribution
of annual flows can be developed by direct application of the pre-
cipitation probability function to Figures 2 and 1.
Figure 4 provides a
comparison of calculated and observed annual runoff distributions for the
Marias des Cygnes River, Kansas, USA. This basin has little natural storage
and experiences a wide range. in annual precipitation, from less than 50 an to
more than 150 a.
Experience has shown that the foregoing approach is generally applicable
to the above average years. However, where lag or persistence is expected to
be a significant factor the lower portion of the distribution function should
be handled differently. In this case, replotting of the precipitation proba-
bility function using a two year running average will provide a more appropriate
solution. The effect, of course, is to convert the naturally skewed distribu-
tion which results from direct application of the basic coefficient relation-
ship to a more normal distribution so often encountered in the annual flow
relationship. Exercise of the judgment option inherent in the alternative
approaches outlined above requires that the planner be cognizant of the nature
of typical distribution functions in basins of similar geologic character.
For areas where freeze is of minor concern mean monthly yields can be
estimated by allocating monthly values in proportion to their contribution to
the BCI as defined in equation (3). However, this calculation should be made
using the average two month running total due, again, to the problem of lag.
Extension of this concept as a means of developing a stochastic generator of
monthly yield needed for preliminary appraisal of storage-yield relations is
currently underway.
That is, monthly BCI values based on two month running
averages are being utilized to determine the regression, correlation, and
standard deviation parameters required for stochastic generation of long term
monthly yield 191.
Utility of the basic relationships can be extended to the determination
of additional flow characteristics with the acquisition of certain short-term
The

258
and miscellaneous field measurements. For example, experience has shown that a
daily flow duration curve obtained from a short-term record acquired over a pe-
riod of two to three years can be adjusted to a long-term appraisal if the ordi-
nates of the short-term record are expressed as a dimensionless ratio to the
average flow observed during the short record period. Subsequent mul.tiplication
of these ratios by the long-term mean as determined from Figures 1 and 2 will
provide a reasonable approximation of the long-term flow duration curve,
The relations described herein have also proven useful in appraising the
potential yield characteristics of coastal aquifers in southern Puerto Rico [IO'.
Historic groundwater use from these aquifers far exceeds the possib1.e direct
recharge assuming all the locally generated flow, as determined from Figure 1,
is di.verted to the groundwater aquifer. In this case the principle recharge
mechanism, excluding the recirculation effect of well irrigation, is infi.l.tration
of surface water as it flows across the alluvial plain. Figures 1 and 2
were utilized to determine the mean surface inflow from the mountainous central
core at Lne point where the water entered the coastal plain. Following the
analysis of various short-term flow duration records which were available, this
mean yield was converted to a daily flow duration curve as described above.
Local stream seepage measurements , available from the U. S. Geological Survey,
were coupled with other similar information from prior studies to develop a
channel infiltration rate as a function of channel width and slope.
Applica-
tion of the potential loss capacity of each channel to its flow duration curve
allowed subdivision of the surface flow into two components; the portion which
was infiltrated into the subsurface and the portion which escaped tcj th* sea.
An areal mass balance was then performed to determine the magnitude of trie
subsurface discharge to the sea (precipitation on the plain plus streamflow
from the mountains minus the sum of direct local runoff plus evapotranspiration
plus surface flow escaping to the sea). The recharge due to infiltration and
the subsurface discharge to the sea, both as determined above, were then incor-
porated in a subsequent mass balance of the subsurtace aquifer in which ïwliarge
was equated to ilet pumping (gross p-mpirig minus recirculation or return flow)
plu: subsurface discharge tu the sea. The calculations were repeated '01 con-
dition? other than the mean, e.g., the vondition of protracted drouth. Results
of these calculations provided a satisfactory explanation of the respons< in
aquiftr water levels that has been expeiienced during both noimal and subnormd
climatic condi t I .,ns.
One additional word of caution beforc losjxg this discussion. The
estimates obtained from Figures 1 and 2 as2uine natural catchments, and wtural
climatic ,oriditLons. Whenever man's activities have materially altered he
naturai environment, e.g., by applicatio? of irrigation water or the I UI~ ruc-
tion of appreciable impervious areas, adjustments must be niade. The fc,l 3ing
will serve L'J illustrate.
Thr lower 6500 hectares of Cherry Creek, Colorado is partially iirbanized.
Approxinial eli' 15 percent of the area is impervious surface for which hie run..ff
coefficient appioxiniates 0.9, and an addit,( nai 10 perrent is in urban awns
which are heavily irrigated (an average of ~uciut t.7 cm per year). 'Hie mem
anmal rainfall appri,. hates 38 m . nd ';c tesponding BCI Is 32. The ad-jii. ;. I:

BCI for the irrigated area approximates 87. An estimate of annual yield, with
and without consideration of the man induced changes, is summarized below.
-
Percent Moisture Applied Weighted Runoff
Area (a.) C cm.
Natural Conditions
Modified Conditions
100 38 .O25 .97
Natural
Impervious
Irrigated
55
15
30
38
38
104
.O25
.goo
.320
.52
5.12
- 10.00
15.64
The observed mean annual discharge from this 6500 hectare portion of the basin
for the four calendar years 1966-69 was approximately 16 centimeters.
The foregoing example is of interest on two counts. First, it illustrates
the procedure required for adjusting the appraisal of mean yield to accommodate
significant man induced changes. Second, it provides a relatively unique exam-
ple of the impact of urbanization on flow response.. In many areas the effect of
urbanization is to reduce the opportunity for recharge and thus diminish baseflow
contributions. The reverse is true for the example cited above. .Here, the im-
pact of lawn irrigation in a relatively dry climate has created a substantial
baseflow contribution to the stream and a significant increase in overall yield.
In summary, precipitation and temperature measurements often represent the
only significant hydrologic data available to the water resources planner. AE
empirical function relating the mean runoff coefficient to the aforementioned
parameters hac been developed. The relationship has been tested in a wide
range of environments, and has proven most useful in undertaking preliminary
assessments of water availability. The general utility of the relationship can
be extended appreciably with limited field data and the application of basic
hydrologic concepts. Continued exploration of the utilization of climatic data
in the preliminary appraisal of water yield characteristics must be encouraged.
Acknowledgements
A portion of the material described herein was deve.loped during conduct of
a research project sponsored by the Kansas Water Resources Research Institute
and the Office of Water Resources, U. â. Department of the Interior. The
author is also indebted to Black & Veatch, Consulting Engineers, Kansas City,
Missouri; R. A. Domenech & Associates, Hato Rey, Puerto Rico; and the Puerto
Rico Aqueduct and Sewer Authority for permission to cite information developed
by these several offices in their analysis of water availability in Puerto Rico.
References Cited
1. Guisti, E.V. and Lopez, M.A., (1967). Climate and streamflow of Puerto
Rico, Carribbean Journal of Science, Vol. 7, pp 87-93.
259

260
2. Thornthwalte, C. W., (1931). The climates of North America according to a
qew classification, Geographic Review, Vol. 21, pp 633-55.
3.
4.
5.
6.
7.
8.
9.
10.
Smith, RL., (1970). Water utilization aspects of weather modification in
Kansas, Contribution No. 46, Kansas Water Resources Research Institute,
Lawrence, Kansas.
Black & Veatch - R. A. Domenech & Assoc., (1971). Water Resources of
Puerto Rico, phase 2, surface water appraisal, Puerto Rico Aqueduct
and Sewer Authority, San Juan, Puerto Rico.
Furness, L.W., (1959). Kansas streamflow characteristics, part 1, flow
duration, Kansas Water Resources Board Technical Report No. 1, Topeka,
Kansas.
Linsley, B.K. and Crawford, N.H., (1963). Estimate of the hydrologic
results of rainfall augmentation, Journal of Applied Meteorology,
Vol. 2, NO. 3, pp 426-427.
Crawford, N.H., (1965). Hydrologic consequences of weather modification:
case studies, Human Dimensions of the Atmosphere, University of Chicago
Press, Chicago, Illinois, pp 41-57.
Lumb, A.M., (1969). Hydrologic effects of rainfall augmentation, Tech.
Report 116, Dept. of Civil Engineering, Stanford University, Palo
Alto, Calif ornia.
Thomas, KA., Jr. and Fiering, M., (1962). Mathematical synthesis of
streamflow sequences for the analysis of river basins by simulation,
Design of Water Resource Systems, Chapter 12, Harvard Press, Cambridge,
Mass achuse t t s.
Black & Veatch - R. A. Domenech & Assoc., (1970). Water Resources of
Puerto Rico, phase 1, ground water appraisal, Puerto Rico Aqueduct
and Sewer Authority, San Juan, Puerto Rico.

z
9
t- U
o -7
O .6
o -5
k
0
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E 0.4
n.
z
U
w
2
I*
k 0.3
O
z
3
K
z
Q
w
E 0.2
0.1
o .o
COORDINATES
COEFFICIENT
110 .430
I20 .470
140 .5 35
I GO ,583
I RO .624
20 o .655
I I I
O 40 80 120 I60 200
BASIN CLIMATIC INDEX
Figure 1 - Basic climatic index related to the ratio of mean runoff
divided by mean precipitation
261

262
300
200 -
PUERTO RICO CURVES
ME& BASIN ELEVATION
1500 METERS
1000 METERS
x
Li1
O
z
o100 -
i=
a
z - J
o
5
cn
a 50m
TYPICAL
CURVE
SOO METERS
-
20 20 50 100 200 500
MEAN ANNUAL PRECIPITATION - CENTIMETERS
Figure 2 - Selected examples of the relationship between precipitation
and basin climatic index

‘5 1.0
w
E
w
2 0.5
J
3
o -1
a
O
o. 2
OBSERVED MEAN DfSCHARGE IN CMS
MEAFJ ANNUAL RAINFAL.LA
RUNOFF RELATION FOR
STATE OF KANSAS, USA.
BASED ON 122 DATA POINTS
0
(FURNES)
CALCULATED USING STANDARD COEFFI-
CIENT CHART
I I I
40 60 80 100 I20
MEAN ANNUAL PRECIPITATION - CENTIMETERS
Figure 3 - Sone canparative rec4ults obtaiiied with the basic BCI vs C
relationship
-
263
40.
2 o.
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IO. E
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T.
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rr(
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z
a
0.05
\ \
-++H-OBSERVED ANNUAL RAINFALL
O000 OBSERVED ANNUAL STREAMFLOW
CURVE FITTED TO RAINFALL DATA
---- CALCULATED STREAMFLOVJ CURVE
\
O ooo\
\
o \
O\
\
\
O L
__
.O5 .I5 .30 .50 .70 .85 .95
PROBABILITY OF EXCEEDANCE
Figure 4 - An example plot of calculated versus observed probability
distribution of annual flows
O

ABSTRACT
DETERMINATION OF HYDROLOGICAL CHARACTERISTICS
IN POINTS WITHOUT DIRECT HYDROMETRIC DATA
S i 1 v i u S t an e s c u Jc
In Colombia, hydrological information is very scarce.
Consequently no direct hydrometric data are available for most
of the sites of projected hydrotechnical works and exploitation
of water. Therefore one must generally apply methods of generalization,
transfer of direct information from observed points
to points of interest, and indirect estimation of the hydrological
characteristics. In relation with this, there are several
proceedings of indirect determination of mean, maximum and
minimum runoff, as well as of other characteristics of the
hydrological regime which are applied to the concrete conditions
of Colombia. The examples, which are included to illustrate the
application of the methods pointed out, are selected from complex
hydrological studies, elaborated or in the process of elaboration,
within the frame of the activities of interpretation and hydrological
calculations worked out in the Colombian Service of Meteoro -
logy and Hydrology.
RESUMEN
En Colombia, la información hidrológica es muy escasa. Co-
mo consecuencia, la mayoría de los sitios de proyectos de obras -
hidrotécnicas y aprovechamientos de agua no disponen de datos hi-
drométricos directos. De tal manera, se deben aplicar ampliamente
métodos de generalización, de transferencia de información direc-
ta desde puntos en que se dispone de observaciones y mediciones -
hacia puntos de interés práctico, así como métodos de calculo in-
directo de las características hidrológicas. Relacionado con esto,
se presentan varios procedimientos de determinación indirecta de
caudales medios, máximos y minimos, así como de otras caracteris-
ticas del régimen hidrológico, aplicados a las condiciones concre
tas de Colombia. Los ejemplos que se incluyen para ilustrar la -
aplicación de los métodos indicados son seleccionados de estudios
hidrológicos complejos, elaborados o en curso de elaboración den-
tro del marco de labores de interpretación y cálculos hidrológi--
cos desarrollados en el Servicio Colombiano de Meteorología e Hi-
drología.
* Hydrology Expert - World Meteorological Organization and United
Nations Development - Program. Bogotá, Colombia - Servicio Co--
lombiano de Meteorología e Hidrología.
-

266
Introduction and generali ties
In Colombia, owing to the scarcity of direct hydrometric
information, in most of tile specific cases, the sites of hydro-
technical works, water uses or other works which come in direct
or indirect contact with the rivers, do not coincide with the
sites of the hydrometric stations. In such cases, the necessary
hydrological parameters must be determined by methods of indirect
estimation. The indirect nydrologic estimations are not however
considered as final values, but only used in an approximate way
or for guidance. The magnitude of these estimations is verified
by means of hydrological field activities, which are generally
grouped into two categories:
a) Temporary hydrometric stations.
b) Expeditional hydrological activities.
The temporary hydrometric stations with intensive and complex
programs of observations and measurements have been widely used
in Colombia, and in particular for dam and reservoir projects
for hydroenergetic purposes, drinking and industrial water uses,
for the most important towns in the country, irrigation and
drainage districts of national importance, and, to a lesser
extent, for navegation, construction arid protection of bridges,
and also for preservation of hydrographic basins.
The expeditional hydrological activities have also encountered
a very wide field of application, especially in aqueduct projects
for medium and small towns, road bridges and sewers, land use
programming, reforestation programmes,.prefeasibility studies for
work projects which come in direct or indirect contact with the
water currents, and also when planning the use of hydric resources.
Regarding this latter aspect, the expeditional hydrological
activities not only act as a means of verifying indirect Iiydrological
estimations, but tiicy moreover constitute almost UE only
really acceptable and reliable way of assessing the scope of the
hydrological parameters in over half the national territory (in
other words 60ú- 000 km2) , where the stationary hydrological
activities are almost completely missing.
The use of temporary hydrometric stations is very similar to
the use of what are normally called secondary stations. As this
is generally a known method, no details on this question will be
given. Only the most important features of the problem will be
presented.
More emphasis however will be made when describing
the most usual methods of the expeditional hydrological
activities, since in Colombia, these vast fields of application
are not only found in the past, but also in the present and the
future.

Verification of hydrological estimations by means of
temp o r a ry hy d rometric stations,
The hydrometric activity of one or several temporary
stations with complex and intensive observations and measurements
programmes, verifies and completes the hydrological estimations,
riot only by means of direct data it provides during its operation,
but also through the possibility of spreading its series of direct
data, by means of correlatioris with data of other reference
stations, which have been working for over 15 years, in zones
of similar hydrological characteristics.
In most cases, preference is to install one of the temporary
hydrometric stations exactly in or near the sites of the projected
works. The stretch of river corresponding to the work site does
not always however meet satisfactory conditions to install a
hydrometric station. In such cases, various hydrometric stations
must be installed in the hydrographic basin where the site of the
works is found, and later deduct the hydrological parameters
by interpolation, balance of discharges or relation with physiographic
or morphometric characteristics of the basin,
In many cases, although a temporary hydrometric station can
be installed in the very site of the projected works, it proves
preferible to install several more stations in the corresponding
hydrographic basin, in order to have possibilities of
controlling the activity developed in the station of the works
site, complete the eventual gaps in observations and measurements,
avoid errors and confirm the results,
The working duration of the temporary hydrometric stations
is variable, pursuant to the specific conditions. When the
temporary station is right in the river as .the reference station,
and the areas of its hydrographic basins differ by less than lo%,
the correlation is generally established in one year alone.
\then the two stations are in the same river, but the areas of
their basins differ by over lo%, the time needed to establish
a reliable correlation takes various years, In short, when the
temporary station is not in the same river as the reference one,
the operation of the first one does not end when an acceptable
mathematical correlation is obtained, but as soon as this is
physically verified, during a period which contains sufficient
humid and dry average years, in other words, rather representative
for the average nultiannual situation, Otherwise the correlation
obtained, although good from a mathematical point of view, may
only express a temporary situation, which would lead to great errors
Besides the above cases, situations have occasionally been
found where the reference station was too far from the site of
the projected works. Hence a direct correlation was practically
impossible to establish. The problem could however be solved
with various intermediary stations, which facilitated the transfer
of data, by means of chain correlations.
267

268
The decision to suspend the running of a temporary hydro-
metric station has always constituted a great difficulty.
Generally, only in a very few cases can the duration of operation
of these stations be really considered sufficient. Therefore,
even after the works execution has commenced, it is considered
preferible to continue running temporary stations near these
work sites, and these stations sometimes remain in operation
even after the corresponding exploitation has started. The
supplementary data supplied by these stations prove highly useful
to complete and confirm the hydrological estimations, and also
as guidance for eventual improvements in both the works and in
the water uses programmes,
Verification of indirect hydrological estimations by
cxpeditional methods
The verification of hydrological estimations by means of
temporary hydrometric stations with intensive and complex
programmes of observations and measurements, constitutes a superior
method, from a qualitative point of view, in relation with the
verification of estimations by means of expeditional hydrological
activities. It is not always however possible to install and
operate stations in the sites of the projected works and water
uses, or in their hydrographic basins. In scarcely populated
regions, it is difficult to operate hydrometric stations, but
the estimation of the main hydrological parameters of these
areas is essential to plan the uses of hydric resources on a
long term basis and also for prefeasibility studies. This
situation is due to certain specific conditions, In Colombia,
in more than half the national territory, the land communication
lines are completely or partly missing, or are in a deficient
state, such that in rainy periods, penetration is rarely possible.
In these areas moreover, it is very difficult to find
satisfactorily qualified people to act as hydrometric observers,
and the installation of limnigraphs, in areas which have no
watchkeepers, generally proves a hazard and failure.
The lack of hydrometric networks, and the difficulty of
organizing stationary hydrological activities in almost half
the national territory, constitute conditions which favour the
wide use of expeditional hydrological activities. Although
these cannot give well defined determinations of the hydrological
system characteristics, as the Stationary systematic hydrometry
cannot be substituted, they constitute essential work to verify
hydrological estimations or to obtain approximate or guide
indications in isolated regions of difficult access, where the
installation and operation of hydrometric stations fail to
encounter satisfactory conditions.
The hydrological measurements in campaigns are frequently
applied in Colombia to determine the following factors:

a) Maximum discharges, duration of floods and
time of wave propagation;
b) Minimum flows and duration of low waters;
c) Sediment charges;
d) Water temperatures;
e) Physical, chemical and biological
characteristics of the water;
f) Overall hydrological characteristics of
the currents.
üetermination of maximum discharges and flood characteristics
by means of hyd rological expeditions
269
In most of the concrete situations (except the case of
flood sweeping), the aim is to determine the maximum runoff,
pursuant to information on maximum historic levels and maximum
floods known in the region, which supposes the almost total
absence of evident traces of maximum waters in the beds of the
currents.
Thus, the information obtained on the field, acquires
decisive importance. It can be classified into two categories:
a) Information supplied by the river bank dwellers,
b) Microphysiographic analysis in the largest beds
of the rivers and on their banks.
The information solicited from the river-bank dwellers
refers to the following factors:
a) The maximum level of the greatest flood known,
b) The year and eventually the date when the flood
came about,
c) The time the flood waters took to reach their
maximum level,
d) The time the waters took in dropping to their
normal levels,
e) Eventual artificial influences on the maximum
runoff system,
Pursuant to the specific possibilities, the information
on maximum waters is solicited from river-bank dwellers who
have lived on the premises for over 30 years and the questions
are put to various people, in order to have a chance to compare
replies,
The microphysiographic analysis in the largest beds of
the currents and on their banks, consider the possibility
of finding certain traces about the maximum water levels,
These analysis generally refer to the following factors:
a) Geomorphological aspect,
b) Alluvial material and grounds;
c) Vegetation and organic vegetable material

270
The geomorphological or morphohydrographic aspect of the bed
constitutes the first sign on the possibility of the river waters
overflowing. The indicative details are micromorphological
aspects of very recent age, traces of erosion processes, alluvial
formations scarcely fixed by the vegetation etc, Sometimes
one can even determine lines of separation between the lowest parts
of the banks, characterized by very recent morphogenetic
processes, and the upper parts of same, relatively fixed and of
more advanced evolution. All the information resulting from
the detailed analysis of the bed micromorphology, cannot lead
to an exact determination of the maximum level of the waters, but
it does offer a first and very useful general guidance, on the
extension of the maximum flood and its possible lines of demark-
ation on the banks or in the largest bed.
The micromorphological analysis is completed with observ-
ations on the alluvial materials and the soil. The fine sediments,
coming from the smaller bed, found on the banks, are a sure sign
of flooding. The secondary soil, discontinuous on surface, and
those which are scarcely at the beginning of the formation
processes, also indicate the overflow of the waters, Finally,
the mineralogical analysis of the fine sepry recent sediments
of the largest bed, may indicate the presence of materials which
are not of that place but come from upstream in the section under
study, which indicates the flooding of the larger bed. The
vegetation can also indicate the overflow of the waters, On the
one hand, the discontinuity of the vegetable formations indicates
the approximate limit of the flood. On the other hand, the
detailled inventary of the vegetable species of the area can
constitute a highly important piece of information, because all
vegetable material which is different, in the larger bed, may
have been brought by floods from upstream. The detailed
laboratory analysis of the vegetable content of the sample
sediments aiid soils may lead to decisive results, when pollen
particles are found in them which do not belong to the vegetable
species of the area under study, but to others from upstream zones.
During the .land activities, the river-bank dwellers'
information is always completed with microphysiographic analysis
made in the largest beds of .the currents and on the banks of same.
Without these analysis, the riversiders' information cannot be
verified, and can consequently i d to very great mistakes, The
errors arise from subjective reasons which make the riversiders
hide the truth or merely offer information on unknown events.
The microphysiographic information, although unable to fix the
maximum level of the waters, indicates essential approximations,
as general guidance and verification factors.
Besides finding information on the characteristics of the
maximum runoff (iiiformation frorii river-side dwellers and micro-
physiographic information), the following main operations are
carried out in each section studied:

271
Survey of three cross-sectional profiles, spaced at
equal distances or more, of the river width, and
continuing for no less than 1 m, above the maximum
historic level of the waters. During the survey, the
maximum levels are markeù on the profiles, and any
lithological sign of soil or vegetable removed from
the place;
Survey of the longitudinal profile of the current,
with a length equal to or at least 5 times the width
of the river.
Execution of at least one gaging (if the natural
conditions so permit)
Approximate drawing of the river span, including the
largest bed, the marking of the cross-sectional profiles;
indications on the types and sizes of lithological
materials and vegetation of the largest and smallest bed,
the lines defining the maximum flood, certain reference
elements, etc.);
Sampling of sediments of the smaller bed and of alluvial
material, and eventually soils from the larger bed and banks,
along the cross-sectional profiles made;
Inventary of the vegetable species of the area and eompiling
of vegetable remains differing to the local species,
The litliological samples of soil or vegetables are suitably
packed, and all the necessary references are marked on the
packages to establish the site from which they have been taken.
Afterwards, pursuant to possibilities, the samples are analysed
in the laboratory.
The above activities are made in various representative
sections of the hydrographic basin studied, in order to have
sufficient data available to permit a comparison of values,
an analysis of the territorial variation of same and generalization
of runoff maximum. During the field work, the information from
different sections are permanently compared, bearing in mind the
territorial continuation of the processes, the variation of
the magnitudes, the periods and dates on which the events have
come about, etc,
Once the field and laboratory activities have beencompleted,
the following factors are determineo during office work:
a) Maximum discharges of homogeneous probability (generally 1%).
b) Main flood characteristics;
c) Eventually, time of wave propagation.
The discharges corresponding to the maximum historic levels
are calculated by hydraulic methods. The measurements made
during the expeditions help to determine the hydraulic formula
factors which contain the rugosity coefficient. These values
are not used directly when estimating the maximum discharges,
but merely offer comparison criteria. Once the maximum historic

2 72
discharges corresponding to a certain frequency have been
calculated (for example 3% if they have been produced in
30 yars), the values should be increased, in accordance with
the coefficients, which permit one to pass from larger
frequencies to rare occurrences. Thus a homogeneization of data
is made (the 1% probability is convenient), essential for
comparisons and generalization. In order to change values
of various probabilities into 1% probability values, it is
preferible to use coefficients established with base on the
direct hydrometric data available in the same zones or in
regions which are hydrologically similar. If these completely
fail, coefficients will then be used estimated with base on the
theoretic frequency curves, considered adequate for the region
under survey,
A final verification of the 1% maximum probability discharges
- is made through generalizations of various forms. The most
comfipn are the type: Qmax = f (A); lg qmax = f(1gA); qmax =
f ( ); etc, where Qmas = maximum discharge, in m3/s; A area
of R e basin in km2; qmax = maximum yield, in l/s/km2, or mm;
Hm = average elevation of the basin, in m; n = a subunit exponent,
specific for the natural conditions of a given zone; f = a
different function for each zone.
The chief flood characteristics (swelling time and total
duration of same) are also verified by comparison of data and
generalizations. These latter are determined by reason of
various morphometric and physiographic factors of the hydro-
graphic basins (length of currents, gradients of same, etc,)
Likewise, the time of wave propagation is also verified and
defined. The generalizations are generally determined in
relation with the lengths and gradients of the currents,
The final verification of the results is made by comparison
with the direct hydrometric data available in the region. Thus,
it is not acceptable that the 1% probability maximum discharges
estimate? by expeditional methods, be less than the discharges
measured in hydrometric stations, during short intervals,
Determination of minimum discharges and duration of low waters
by expeditional methods.
In most of the concrete situations, the minimum low water
characteristics are determined during the expeditions made to
find the maximum runoff, subject to the condition that these be
made during low waters.
The activities developed on the field have three categories:
a) Compiling of information froin the riversiders.
b) Hydrological and topographic work in the bed of the current.
c) Observations on the lithology and freatic layers of the region,
The reports from the riversiders refer to the following aspects

273
a) Eventual interruption of the runoff,
b) Minimum historic levels;
c) Year and month when the runoff was interrupted or when
the minimum level came about,
d) Low waters and duration of same,
e) Eventual artificial influences on the minimum runoff system.
Preferibly the information is requested from various people
who have lived near the river, for ovcr 30 years. The most
marked sections €or analysis are those which pertain to spans
of current where ancient floodgate openings are found, and also
the sections near to irrigation land. The existence of
derivations, upstream from the section under survey, must be
considered, to avoid considering the minimum discharges in
influenced state as minimums in natural state.
Tile hydrological aiid topographic work in the bed refer to
the following:
a) Execution of measurements;
b) Topographic survey of loiigi tudinal prof iles,
The measurements are generally made by wading. After making
the measurements, the wet section is drawn and on this, the line
of the surface of the water corrcsponding to the lowest water
(in accordance with the information on minimum historic levels).
The topographic survey of longitudinal profiles is made on
the water surface, and spreads for at least three times the
width of the lower bed. These operations are made in various
sectioiis representing the basin or area under survey, which
are generally assimilated in the main confluences, The information
is permanently compared, bearing in mind the territorial
continuity of the hydrological phenomena.
Throughout the hydrological expeditions , the lithology of
the region is continually observed. If geological maps are
available, they are taken to the field, to have prior indications
on the areas where there are permeable rocks Any discontinuity
in the runoff during low waters should be explained either as
a result of human activities or due to lithological influences,
Research on the depth of the freatic layers, in existing wells,
is also made, and also on possible contacts of these layers with
the flows, which could constitute an important additional inform-
ation for estimating the minimum runoff characteristics.
The estimations, interpretation, verification and generalization
of data are made at a later stage, at the office. The minimum
discharges are estimated by means of hydraulic methods, For the
factor containing the rugosity coefficient, the values are used
which result from the measurements made, The discharges of
diverse statistical probabilities are transformed into 97%
probability discharges (three times in 100 years) to obtain liomogeneous
values which can be compared. The coefficients used to

274
change values of greater frequency into values of lesser
probability shauld be determined based on direct hydrometric
data of the zone or regions with similar hydrological system,
If these fail, determinate coefficients may be used based on
the theoretic frequency curves, considered adequate for the
region studied.
In the event of intermittent runoff flows once in 30 years,
all the minimum discharges with probabilities above 959. may
be considered the same or zero.
The 97% probability minimum discharges are firstly analysed
in relation with the areas of basins and by means of balances of
discharges. The yields are analyzed by means of generalization
relations, which may be of type qmin = f (iim) for mountainous
areas and qmin = f (B%) or qmin = f (Ud) for flat areas, In
these relations qmin = minimurn yield, in l/s/km2, or mm; Iim =
average elevation of the basin, in m; B% = forestal covering
coefficient of the basin, in %; Ud = drainage density or density
of the hydrographic network in km/km2; and f = a different function
in each zone.
If maps are available with monthly mean isohyets, the minimum
yields may be compared with the monthly mean precipitations of
the driest month, by means of relations of type qmin = f(Pm),
where Pm = mean precipitation of the driest month, in the basin
correspoiiding to each section, in mm.
The duration of the low waters is analysed from the point of
view of territorial continuation of the hydrological phenomena
aiid moreover, in relation with the distribution of the precipit-
ations within the year, When special meteorological maps are
available, indicating the average duration of the drought periods,
this data can be used to verify the maximum low water durations,
by means of relations of type Te = f(Ts), where Te = time or
duration of the drougnt period, in days; and f = a different
function in each zone.
Determination of sediment charges by expeditional methods
The hydrological campaigns to determine sediment charges
are made during high waters periods and after floods of certain
importance have occurred, The expeditions organized during high
waters periods try to determine sediment charges in suspension,
whereas the others refer to haulage volumes.
The main activities to determine sediment charges in suspension
are as follows:

a) Sampling of waters with suspensions;
b) Execution of measurements;
c) Geomorphological observation of the land.
275
The water samples are taken during tlie execution of
measurements, in tlie same points where tlie speeds of the
water are measured. The measurements and water sampling
are made in various characteristic sections of the hydrographic
basin or study zone, where bridges or other facilities
are available to execute the gagirigs.
iluring these runs, permanent geomorphological observations
are made on the existence of erosion processes, land degradation,
lithological conditions ardvegetation, and also their relation
with washing of the soils, etc. All these factors help to
explain the abrupt changes in the territorial variation of the
sediment concentration. To make our work easier and as general
guidance, it is convenient to take to the field the lithological
or geological, general geomorphological and special geomorphological
maps (gradient, fragmentation of the relief, erosion, etc) if
they exist.
Once they have been estimated, the sediment charges in
suspension are analysed, bearing in mind the territorial
continuity of the hydrological processes, Any discontinuity
ihould be explained by tlie different contribution of any of
the affluents, or by evident morpholithological changes in the
basin, which determine changes in the erosion and the transport
of sediments. The concentration of sediments in suspension may
moreover be analysed in function of the territorial variation of
tlie corresponding runoff, within the hydrographic basin or area
studied.
In order to establish the magnitude of the averages of sedi-
ment concentration in suspension, various hydrological campaigns
are made, until relations between discharges and sediment
charges, in various characteristic sections can be determined.
Thus, the verification and generalization of concentrations or
sediment charges in suspension is made through the flow and run-
off magnitudes,
Finally, the verification of the magnitude of the average
values of eoncentretion of suspensions is made, by morpholitho-
logical zones, in function of the variation of the gradients,
coefficients of covering with forestal vegetation, etc.
To find voluniEs of dragged sediments, certain activities are
carried out, during low water periods, and the following are the
most important among these:
a) Set up marks and fixed reference points.
b) Topographic surveys of alluvial accumulations in flow beds;
c) Set up and recover traps for sediments and measure accumula
t i on s .

276
These operations are performed in various characteristic
sections , generally downstream of important confluences regard-
ing the drags contribution. To obtain a general idea on tlie
size of the sediment charges dragged along, various campaigns
are made. In the first, the marks and fixed reference points
are set up on the banks, in the larger lied, and sometimes even
iii tile smaller bed of the currents, and also the sediment traps
in the smaller and larger river beds. In later campaigns,
topographic surveys are made of the alluvial accumulations;
the sediments accumulated in the traps are removed; the marks
are repaired and also the reference points that have been
damaged during floods, and the traps are again set up for bottom
sediments.
lhe dragged sediment charges are analysed in relation with
the magnitude of the discharges and suspension charges, and also
in function of the morpliolitliolpgical local conditions (litho-
logical complexes, erosion and gradient processes, etc.)
Finally, coeffients may be established which, for each zone
of specific morpnolithological conditions, indicate the magnitude
of tlie proportion that the sediment charges dragged along
represent , in relation with the suspension charges.
Determination of water temperatures by expeditional methods
The hydrological expeditions which determine the water
temperatures, refer to the following operations:
a) Measurement of air temperatures.
b) Measurement of water temperatures,
c) Observations on the land lithology.
The air temperatures are measured in order to have values
available to determine correlations between these and the water
temperatures. Once the correlations have been established,
characteristic values and the variation in space and time of
the water temperatures can be determined, based on the values
of the former, Naturally, in such cases, maps with isotherms
of the air in tlie surveyance regions are available,
‘She water temperatures are measured in various characteristic
sections, parallel with those of the air temperatures. The
variation in their values is analysed bearing in mind the territorial
continuation of the hydrological processes, Aiiy jump in
the water temperatures, throughout a flow, should be explained
either by confluences with different temperature flows, or by
imp0 r t ant prouiid water contributions,
Observations oii the lithology of the region are made to
detect possible substantial ground water contributions, and related
to this, explain the sharp changes in temperature experienced by
the waters throughout the flows,

277
in order to compile representative data not only from
the territorial variation point of view, but also regarding
tiic temporary variation, expeditions are made tliroughout all
the seasons of the year.
Determination of physical, chemical and biological character-
istics of the water by means of expeditional methods
The campaigns to determine the quality of the waters are
organized during low water periods, when the physical , chemical
and biological characteristics of the flow waters are most
stable.
Iii cases of waters whose quality is unchanged by human
activities, the characteristic sections for expeditional work
are the confluences. To the contrary , the conf luences with
drainage and sewerage are also taken into account,
The main activities carried out during the campaigns arc
as follows:
a) Compiling of water samples;
b) Execution of measurements;
c) Analysis of water samples , and eventually, preservation
and packing of same;
d) Geological observations.
The water samples are analysed on the field, if mobile
laboratories are available (the most suitable]. When there
are no possibilities of making complete analysis on the field,
the samples are preserved, and at least the analysis of the easily
changeable characteristics are made, anù which can be only
determined in fresh tests. ‘Iiie samples sent to the laboratory
are suitably packed, and all tiic indications regarding the site,
and date of collection are noted on the packets.
The measurements are made to find the discharges to which
the characteristics measured correspond, anù also the amounts
of waters available for dilution of chemical coiicentrations,
of vital use, especially in cases of pollution tipping.
The geological observations are similar to those made during
expeditions to find the miiiimum runoff characteristics. In the
case of physical, chemical and biological qualities of the flow
waters, any sharp change should be explained either by artificial
influence (tipping of pollutions), or by natural influence, due
to confluences with flows of different biological, chemical and
physical characteristics, or due to an abundant food of ground
waters from different lithological zones,
As soon as thc physical, cliemral and biological character-
istics of the water have been determined, they are analysed,
bearing in mind the territorial continuity of the hydrological

278
processes, and they are verified, according to litliokgical
zones, in relation with the discharge and runoff magnitudes.
Determination of whole hydrological characteristics of the
Flows by means of observations and measuremeiits on campaigns
In practice, the caso very frequently turns up of there
being no direct hydrometric data available in certain Iiydro-
graphic basins, or estimations of various hydrological
characteristics must be verified.
In such situations, complex expeditional hydrological
activities are developed, based essentially on tiie following
p r i nc ipk s :
a) iixccution of simultaneous measurements hydrologically,
in various sections;
b) Periodicity of campaigns, in accordance wi tli the hydro-
logical method phases;
c) Installation of recorder apparatus, with long duration,
autonomous operation;
Ci) General Observations o11 the genetic characteristics of
the hydrological sys tem.
The measurements may refer to most of the hydrological
characteristics (levels, discharges, sediment charges, temperature
and physical, chemical arid biological characteristics of the
waters, etc,) and they are made in various representative
sections of the basins under survey, and also in nearby
hydrometric stations, located iii areas with similar hydric
system.
The principle of Iiydrological simultaneity should be strictly
respected during the measurements, in order to compare and
correlate the results. From an operational point of view, this
supposes a need to execute work by means of various teams of
hydrologists working parallel, in accordance wi tli strictly
established programs regarding the sections and measurement hours,
The periodicity of the campaigns, iri functinn of the hydro-
logical system, is irnposcd as a compulsory condition, in order
to establish the variation ranges of the characteristics measured
and the correct correlations between the data of nrious sections
arid those of the nearby liydrometric stations.
"lie installation of recording apparatus, of long duration
autonomous operation, is convenient when the periodicity and
frequency of the campaigns cannot be assured on a satisfactory
level, and also when oiie is trying to complete the correlations
between the data of the sections studied and the reference
hydrometric stations. tiowever, in most specific cases, the land
difficulties prevent execution of works for installation of
recorder apparatus (lack of roads and labour; the maintenance and

279
periodic inspection of the installations and apparatus cannot
be assured, etc.)
The measurement of tlie hydrological characteristics is
organized in accordance with the indications given in the '
above paragraphs.
The data analysis is made bearing in mind the territorial
continuity of the hydrological processes and the correlations
between various sections and the reference hydrometric stations,
arid also in terms of the local physiographic conditions
influencing the variation of the hydrological system factors,
Naturally, the most important thing is to properly determine
tile correlations so as to extend the series of data of the
sections studied, in terms of the long series of data available
in the reference hydrometric stations. It is therefore
convenient for the expeditioiiai hydrological activities to be
developed in each zone, at least during two complete years.
The verification, analysis and interpretation of the data
is made before suspending the field activities. Pursuant to
tlie results, the initial programmes can be changed and the
work intensified, to define the processes which have not yet
been satisfactorily determined. The total suspension of the
expeditionai hydrological activities in the study area can only
be macle after conclusive results have been obtained, or,
exceptionally, when the sure conclusion is reached that the
methods used are sufficient to deterriiine or verify the hydro-
logical characteristics which must be known.
bibliography
1.
2.
3.
4.
5.
6.
7.
Uiaconu, C., Lazarescu D, (1965). Iiidrologie, Bucuresti.
Irjorld Meteorological Organization (1970). Hydrometeorological
Practices Guide, OMM-No. 168 TP. 82, Geneva.
Roche, M. (1963). llydrologie de suÏface, Paris,
Stanescu, S. (1969). Chief prcsent day problems of the
national network organization of hydrological stations
in Colombia, Aperiodic Publication 1 , SCMII, Bogota,
Stanescu, S. (1971). Expeditional Iiydrological Activities.
Aperiodic Publication 22, SCMiI, Bogota.
Vircol, Al. (1960). Calculul debitelor maxime folosind
cercetarile expeditionare, Studii de liidrologie 1, Bucuresti.
World Meteorological Organization (1972). Casebook on
Hydrological Network Design Practice, WMO- No. 324 , Geneva.

280
1100
260
E STAC I ON ME TE ORO LOG IC A I NGE N I O MAN U EL I TA
PROMEDIO 1901 -1970
ESTACION HIDROMETRICA CAUCA - JUANCHITO
PROMEDIO 1934-1970
24 O
1935 1940 1945 1950 1955 1960 1%5 1970
COMPARACION DE PROMEDIOS MULTIANUALES SUCESIVOS (GLISANTES) DE PRECIPITA -
CION CON EL PROMEDIO DEL PERIODO 1901-1970 EN LA ESTACION METEOROLOGICA
INGENIO MANUELITA (A) Y DE CAUDAL CON EL PROMEDIO DEL PERIODO 1934.1970
EN LA ESTACION HIDROMETRICA CAUCA- JUANCHITO (8)
FIGURA I
I GRAFICO PARA CURVA DE FRECUENCIA I
COMPARACION DE CURVAS DE FRECUENCIA DE CAUDALES MEDIOS ANUALES
DEL PERIODO 1934-1970 Y i951 -1970 EN LA ESTACION HIDROMETRICA
CAUCA - JUANCHITO
FIGURA 2
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ABSTRACT
NEW MODELS OF FREQUENCY LAW OF RUNOFF
STARTING FROM PRECIPITATIONS
J.R. TEMEZ
Professor
1.T.O.P.College-Madrid
Two interesting applications of a new hydrometeorological
method are developed with scientific rigour.
From the frequency law of annual precipitations, we can
deduce symply the law of runoff, and with precission, proved
in all tne esperimental verifications. To do it, we must know
or estimate the potential evapotranspiration on the basin (ETP)
and the minimum effective precipitacion (Po).
An analogous reasoning, yet simplier, allow us to convert
the frequency law of maximum precipitations in the law of
volumes of superficial runoff in floods. The only necessary
datum is the minimum effective rainfall PL, analogous to the
Po *
We can simplify the calculations with special paper of
double scale. In them, the statistic function of runoffs is
the same as precipitations if we read each one in the correspondent
scale.
The profit of this methodology, evident in bassins without
data of discharge is also important when we know the registers
of flow because it makes easy models of adjustment more reasonable
than the classic one of Galton, Goodrich and so on, and
in this way we avoid nonsensical extrapolations in the intervals
of large and small values.
--
RE S U ME N
Dos interesantes aplicaciones de un nuevo método hidrometeo
rológico se desarrollan con rigor científico.
A partir de la ley de fre'cuencia de las precipitaciones --
anuales, se deduce la de aportaciones de manera sencilla, y con
precisión, demostrada en todas las comprobaciones experimentales.
Para elio solamente se necesita conocer o estimar la evapotranspiración
potencial en la cuenca (ETP) y la lluvia mínima eficaz
(Po).
Un razonamiento análogo, aún más simple, permite convertir
la ley de frecuencia de máximas precipitaciones en la de volumenes
de escorrentía superficial en avenidas. El Gnico dato necesa
rio es la lluvia mínima eficaz PL, análoga a ia Po.
Los cálculos se simplifican con papeles especiales de doble
escala. En ellos, la función estadística de aportaciones es la -
misma de precipitaciones con tal de leer cada una en la escala -
correspondiente.
La utilidad de esta metodologia, evidente en cuencas sin da
tos de aforo, es también importante cuando existen registros fo-
ronómicos, pues facilita modelos de ajuste más racionales que --
los clásicos de Galton, Goodrich, etc., y se evitan así absurdas
extrapolaciones en los intervalos de grandes y pequenos valores.

2 88
1. MOTIVATION
The precipitations, phenomenon of general type, are adjusted
correctly to the also general classical frequency laws: Gumbel (maximum
rainfalls), Gauss (annual precipitations*), and so on, altering the mean value
and the dispersion from some places to others.
These precipitations are transformed partially in runoffs, but is
fundamentaly a deterministic process, peculiar of each basin in relation to
their edafogeological and climatical characteristics. Therefore the regime of
runoffs can not be defined with the statistic functions, at present in use, which
ignore the concrete parameters of each basin, significatives of the hydrological
cycle. These functions, by virtue of the liberty that their indetermined coef-
ficients give them, are able to adjust to the experimental points only in the -
interval of the intermediate values, but they point their inadequate conception
out to represent the hydrological phenomenon in the band of small and large
values where the disadjustments are significative and many times nonsensical,
such as what happens when the runoff of low frequency are negatives and the
high ones exceed notably the cipher of the precipitations. Their inadequateness
is manifested more clearly in basins of irregular regime than in the regular
ones, since these last are easy to adjust any curve to the reduced range of
variation of the registers not indicating the erroneous extrapolations which are
done about the extreme values.
On the other hand, in the basins without registers of flow, we
must define correctly from the precipitations, not only the mean flow, but also
the other parameters which are characteristical of their hydrological regime
like functions of frequency of runoffs floods and so on.
These considerations have moved the author to develop a new hydro
meteorological method of precision and scientifica1 base, though of
simple application.
The article shows two interesting applications of this metodology,
which will be the object of an exhaustive treatment in a later publication.
* The author will propose in a later publication a modified Gauss law.

2. TYPE OF FREQUENCY LAW OF THE ANNUAL
RUNOFFS PROPOSED BY THE AUTHOR
289
When we treat of regulating high percentages of the mean flow of
a river, the decisive fact is the frequency law of their annual runoffs, lossing
importance in the study, the precise knowledge of the monthly variations, more
sensible to the geological characteristics of the basin, on the other hand deci-
sives when the volumes of water to regulate are low.
Therefore it has a big interest to determine the said law starting
from the precipitations of the basins without registers of flow, or by the
convenient adjustment to the experimental points if there are registers of
flows. The method that is exposed later on, solves the probleme in both cases.
The balance of water in a period like the water year,
permits to establish the relation
A = P - E where A = annual total runoff
P = annual precipitation
E = annual and actual evaporation
This equation applicable to the individual values of one or several
years, suggest us also about the relative configuration of the frequency of run-
offs and rainialls laws.
For very big values of P, the actual evaporation will be identified
with the potential one, practically constant from some years to others (in the
humid years is smaller than in the dry ones). On the other hand, in the extre-
mely dry years all the precipitation will evaporate from its own basin E = P, and
A = O. In intermediate conditions E and A will increase with p.
The qualitative sight of the precipitations and runoffs laws is
represented by the figure 1. We can calculate the frequencies of A from the
frequencies of P if we know the values of 6 (P) that in this way it has the
significance of a middling evaporation to that precipitation.
That relation 6 = 6 (P) according to the considerations done previoly,
will be represented in the figure 2 and different from some basins to others
in relation to the capacity of retention of water on their soil R and the regime
of temporal distribution of their climatical variables.
The method proposed consists in establishing the family of curves
i=$
(- P
ETP ETP)

290
among which we must select the most suitable in each concrete case.
We can make that selection in relation to - R , but the curve,
according to what was said before, is also conditioned E by the temporal
distribution of the precipitation and the evapotranspiration, which is variable from
a meteorological zones to another. Therefore we think more practical to
represent the influence as a whole with all these variables by their inmediate
effect Po (figure 2), minimum effective precipitation from which frequency
corresponds a null runoff.
Having present the definition of 6 , the figure 2 is transformed in
the figure 3.
The calculations of the frequency laws of precipitation and runoffs
in many basins with registers of flows were made, and in all of them we found
out that the values, correspondent to a same frequency, are combined realy
by a relation of the type schematized in the figure 3 arid has the following
expression:
A = O para P < PO y para P > PO
that permits to transforme the frequency law of precipitations in the frequency
law of runoffs.
The figure 4 contrasts the results obtained by this process and
the usual ones at the basin of the Guadalmellato river. In front to the good
adjustment of the author’s law, Galton gives nonsensical high values, higher
including to the precipitations, in the interval of high frequencies, while Good
rich in the interval of small values, decisive in the studies of regulations of a
river, recomends negatives ciphers including for frequencies higher to O, 10.
The functions of Goodrich, and specially Galton, ignoring the physical sense
of the hydrological phenomenon, are not capable of simultaneously adapting
to the total range of values.
The adjustment of the author is repeated in the figure 5 with
double scale of ordinates: the normal and their transformation according to
formula (2). In this way the law of frequency of runoffs must be the same as
that precipitations provided that to read each one in the correspondent scale;
effectivelly we can verify that the experimental points as much as the preci-
pitations as the runoffs are confounded and are distributed in the grapht round
about to an only curve.

291
Guadalmellato is an example of the many empirical cornprobations
carried out, which demostrate the big precision of the method proposed here.
1)
2)
3)
4)
The procedure of calculation consists in the following steps:
Calculation of the frequency law of precipitations,
Determination of the potential evapotranspiration value of the
basin (ETP) deductible from the evaporimetrical measures (or
charts) existing in the zone.
Valoration of the minimum ei'fective precipitation Po. The data of
flows, if they exist, must orientate the computations to choose
theparameter Po more convenient. On the contrary the valoration
of Po must be guided by the values obtained in other basins gauged
of that zone and based in the capacity of retention of water in their
soil. Po is equal to the capacity of retention of the soil plus a
function of the climate which sign can be positive or negative
depending on the cases. The author prepares an orientative table
of values of Po, though a hydrologist can estimate sufficiently
exact that cipher with so dear physical sense, based only in their
experience and in a superficial knowledge of the characteristics
of the basin.
In humid climates where the precipitations of low frequency are
much higher than PO, is enough a gross approximation of this
last parameter.
Once that data are known, the runoff A with frecuency F is obtain-
ed in relation to the precipitation P of that same frequency by the
formula (1) or their equivalent (2).
It must not be forgotten that the knowledge of the frequency law
determines automatically the value of the mean runoff. Inversely we can
choose the parameter Po with the condition to proporcionate a mean runoff
equal to the previous valuation by other procedure.
In any case, if there are registers of flow the values of ETP and
Po will be needed to get the best adjustment with the experimental points.
To programme the method to the use of computers will yet be
easier the calculation.

292
3. CORRECTIVE RUNOFF
Let us imagine that after a serie of years of intermediate
characteristics, a year so dry is produced that all their precipitation is
evaporated and not producing any runoff. In spite of that circunstance, the
flows of the river will be not necessarily null, since they can feed from the
underground reserve (or superficial) of the basin decreasing in agreeable to
the curve of exhaustion of the base flow.
The corrective runoff, or variation of the reserve from the end
of a water year to the end of the following one, will be an analogous function
to the represented one in the figures 6 and 7: with high values of theprecipi-
tation, the reserve will increase and discharge of the river, diminish in this
quantity; on the contrary with low values it will decrease, to feed the super-
ficial flows; in mean raining years the reserve will not change.
For smaller frequency than the correspont one to the frequency
of minimum effective precipitation F (PO), the total runoffs are identified with
the corrective ones, responding to terms fundamentally different so far mention
ed. Neither the potential evapotranspiration, ETP, nor the Po has now any
incidence in the phenomenon, as neither the value of P; the law is determined
by the curve of exhaustion of the reserves of the basin, as well as by the
frecuencies of the initial state of said reserves and of PO.
The classical functions of frequency are not either capable of
being adapted simultaneously to the interval of usual values, prevailing the
direct runoff of the year, and to the different interval of small values corres-
pondent to the variation of the reserve. These functions treat them indiscrimi
nately with an intermediate dull adjustment in which ignoring this real duplicity
to get out of orbit the dry runoffs, which are precisely the decisive ones in
the regulation process of a river.
The article will not extend in the detail of this corrective runoff
which in several cases is necessary to have present, while in the others, on
the contrary, it has little importance, like what happens in impermeable basins
with little variation of their reserves of a year to the next one and of which
mean value and pluviometrical regularity are at least moderated, in this way
the probability of precipitations close to the minimum effective one PO is
extremely small not participating on the computations. The substractive term,
that according to the graph of the figure 6, must be applied also to the zone of
strong precipitations, represents a percentage very small of the total runoff
in these dates which is not worthwhile considering.

4. MAXIMUM ACTUAL EVAPOTRANSPIRATION
ON A DRY CLIMATE
293
As it has been exposed previously, the precipitations increase
the availabilities of water for the evaporation and this one will increase up to
the potential evapotranspiration, or more exactely to the potential evapotrang
piration of the humid years which is about O, 9 times their mean value. The
previous affirmation is evident to humid climates, but not so to the dry ones.
In climates like the mediterranean one, there is a season of the
year (Summer), when their potential evapotranspirations are maximum
while the precipitations are practically null as much in dry years as in the
plenty ones. It exists in these dates a permanent deficit of precipitation; the
actual total evaporation of the year does not reach ever to the value of the
potential one and its maximum value will be the potential evapotranspiration
of the period of precipitations (ETP) p increasing in the capacity of retention
of water on the soil (R) evaporating in posterior dates.
Is very important that in these cases the ETPwhich intervenes in
the formulas be replaced by the maximum actual evapotranspiration ETP*
where (ETP)* = (ETP)p t R.
It could be said that the ETP of the formula will in any case be
the least of the following values:
1)
2)
potential total evapotranspiration of the year
potential evapotranspiration in the period of rains increased
in the retention of the soil
5. FREQUENCY L AW OF VOLUMES OF MAXIMUM FLOODS
The relation between the total rainfall P’ and the volume of
superficial runoff A’ is of the type schematized in the figure 3 for P - A, but
now, treating of a phenomenon of short duration and strong concentration of
rainfall, exist the following differences:
. The evaporation in so short time and in an atmosphere of big
relative humidity is worthless and not altering the process.

2 94
The essential element is the quantity of water that can be retain
ed in the soil, characterized by a minimum actual rain Pó,
similar in idea to the Po of the annual runoffs but with ciphers
much smaller.
According to the documentation of the Soil Conservation Service
of EE. UU. and verified with several studies of the author, the relation is:
If P’ GPO , A’= O andif P’ >Pó
universal law in relative values to Pó, which is their only indetermined
parameter (figure 8).
The previous one, suggests the creation of a new special paper
(figure 9) with scale of frequency according to Gumbel and double scale of
ordinates: the normal one and their transformed as:
deducted from the equation (4).
x = It (it JTx)
If we draw the frequency law of maximum rainfalls on the mention
ed paper, that same straight will define the frequencies of the volumes of flood
A’ reading them in the correspondent scale. The method can not be simpler.
The figure 9 shows an example of application to the basin of the Cheliff at
Algerie.
The hydrologists defenders of the analytical and non-graphical
adjustment, only have to transform the law of maximum precipitations accord-
ing to formula (3) or their equivalent (4).
There upon these volumes are related only to surface runoff and
to obtain the total ones is necessary to increase them in the correspondent
groundwater runoff, worthless however in the interval of the high values, the
most interesting to the calculations.
In summary, once the frequency law of maximum rainfalls is
defined, the process of calculation of the volumes of flood only need the
estimation of the value of the minimum effective rainfall PA,

295
The book "Design of Small Dams" of the Bureau of Reclamation
take in the information of the Soil Conservation Service and facilitates a tables
which suggests values of the Pó, principally in relation to the nature and thick-
ness of the soil, although modified by the type of cultivation; this book explains
that each concrete case will depend naturally on the humidity of the soil in the
initiation of the rainfall and the values of the formula are considered to an
intermediate conditions in the dates of presentation of the floods. The author
in keeping with the theory of the Soil Conservation Service but precisely by
that remarkable influence of the initial humidity of the soil, the Pó of the
formula has to change also in relation to the climate and in this manner, other
things being equal, it will be higher in a dry one than in other humid one, where
there is a big probability that at the beginning of the rainfall, the soil would be
in proximate conditions to the saturation of the water.
If data of flows could exist, the experimental points of the volume
of superficial runoff in the maximum flood of each year of register, could
advice the Pó to choose to obtain the best adjustment.
The conditioning factors of Pó and of Po are basically the same
and it must not be forgotten, since any information that w e can orientate in the
estimation of one (for example the tables of the Soil Conservation, Service) can
be used in the determination of the other.
In the order of magnitude it can be said that Po is aproximately
fifteen times greater than Pó ; a study directed in establishing with greater
precision this relation would be interesting.
6. CORRELATIONS PRECIPITATIONS-RUNOFFS
The relations:
2
(P - Po) 2 (P'- Pó)
A' =
A = Pt ETP - 2 Po Y P't 4Pó
are rooted in the essence of the hydrological cycle and have a big physical
signification. Besides obtaining its specific end in the transformation Of
frequency law, it also makes clear the types of correlation more adequate
to the individual values of these variables.

296
7. LIMTS OF THE METHOD
This method, as any other hydrometeorological one, is not
strictly applicable to a singular basin with appreciable captures of water from
other zones or leakages towards them, since their flows are conditioned also
by precipitations outside the said basin.
It is conceived for regimes fundamentally rainy and it has not
been studied for any possible adaptation to the snowy ones.

Fig. I . ESQUEMA DE SITUACION RELATIVA DE LAS LEYES DE FRECUENCIA DE "P"Y "A':
RELATIVE SITUATION SCHEME Op FREQUENCY LAWS OF "P" AND "A",
i
4_
ET P
Fig.2 . ESQUEMA DE VARIACION CE 6.
VARIATION SCHEME OF 6.
P = Precipitacidn anual de frecuencia F.
A= Aportacion especifica onud dr lo misma frecunieia
F
5 :Diferencio entre P y A de ia misma frecuencia F.
ETP: Evapotranspiracidn pciencid.
R,= Precipitacidn a cuya frecuencia F ( Po) corresponde
uno aportación nula.
P= Annwl pr.cipitol)an at treginncy F.
297
A= Annwl rprcific totalrunoff Or the SQIY fp.p-y
F.
P I
ETP
6 : DiffWlnCe beîueon P and A of iha saw m
ETP x Patentid ewpotranspira tion.
y F.
PO= Precipitation ta which frequrnoy F( Po) Oorrewponh
o null iotalrunoff.

!
-f
I
--- Ley da Goodrich.
Ley d. ßolton
---
4
I
0 1
5
O Puntos experimentales de atoro.
Puntos rperimntoler de lluvia.
-7-
l !
i i
Fig. 4 r 5 - AJUSTE EN LA CUENCA DEL GUADALMELLATO
ADJUSTEMENT AT GUADALMELLATO BASIN.
-
--- Qoodrich's low
--- Galton's br
o Annual runoff miperiiaentol peints.

AA
P A
299
Fip. 6 y 7 - ESQUEMA DE APORTACION CORRECTIVA DE LA LEY DE FRECUENCIA DEBIDA A LA VARIACION
DE L& RESERVAS.
CORRECTIVE CONTRIBUTICW SCHEME OF FREOUENCY LAW WING TO VARIATION OF RESERVES
P =Procipitacibn anual de frecuencio F , P =Annual precipitation of frequency F
A= Apartacidn especifica M U O I de la misma frocww ' A =AnnuOJ aQocific t#alrUnolf of the s- tre-
cia F. qurncy F
ETP = Evapotranipiroci6n potwcial, 1 ETP- Patrntlal rwpotronspiration
PO = Procì~itoci~ 0 cuyo frccunicia F(Po) corres- , PO = Proclpitotion to which frequency F(Pe) carrespande
una apwtación nula. , pondi a null totalrunoff.
AA = Aportación carroctiva. AA = Corroctlve totalrunoff
&,= Máxima valor de la aportaci6n correctivo , AO = Maximum valu# at correctivo totalrunoff

30G
- A'
PA
4
3
2
i
O
a -RELACION ENTRE P' Y A' DE UNA MISMA FRECUENCIA.
RELATION BETWEEN P' AND A' OF THE SAME FREQUENCY.
- P'
?A ~
6 T-
5 1
44
.- . ~- . -- -
Bassin of the Cheliff river ( ALGERIE 1
Pk = 35 IlMn.
3 1
I P'with frequency F
I- -- - -- -
ol , i c
. ~~ -
Experimental pointa P'
FIP. 9. GRAFICO ESPECIAL CON DOBLE ESCALA. EN EL UHA MISMA RECTA REPRESENTA LAS LEYES DE
FRECLÆNCIA DE P' Y A'.
SPECIAL GRAPHIC WITH DOUBLE SCALE.ON WHICH THE SAME STRAIGHT LINE REPRESENTS THE
FREQUENCY LAWS OF P' AND A'
P' = Precipitación toto1 de un aguacero de frecuen- P'vTotal precipitation of o rainfall with frequency F.
cia F
A' = Volumen de escorrentia auperficio( de b misma A', surface runoff volume of the rame frequency F
frecuencia F en mdrimaa avenidar.
in maximum floods.
Ph= Precipitación de un opuacero do frecuencia - PA= Precipitotlon of o rainfall with frequency F( Po 1 to
F ( PO ) o lo que correrpondo una escorreniio which corresponds o null rurtoce rurloff.
superficiai nulo

ABSTRACT
TRAITEMENT OPERATIONNEL DES DONNES PLUVIOMETRIQUES
ENTACHEES D'ERREURS OU INSUFFISANTES
R. Trendel - Der Megreditchian - Mme Rulliere
The Bureau of Water of the National Meteorology, at present
applies a method that allow us, under certain hypothesis, to obtain
the equation of lineal multiple regression, which permits to
calculate the theoretical values of the monthly rains. The
application of this formula is possible by the existance of base
data, corresponding to each season an index actual value/theore-
tical value that is useful to correct.
In this way, we can calculate the values of theoretical rain,
that allow to correct and complete the series, and also the rainy
periods in season without data or with inadequate data. It is
carried out an analysis of correlation to establish the degree of
guaranty of this method and to choose the parameter to use in the
different possible hypothesis and, particulary, the iterative
method of Van Isacker.
-- RESUME
Le Bureau de l'Eau de La Météorologie Nationale applique
actuellement une méthode opérationnelle, découlant sous certaines
hypothèses de l'équation de regression linéaire multiple, permettant
de calculer des valeurs dites "théoriques" des pluies mensuelles.
L'application de cette formule est rendue possible grâce à
l'existence d'un important fichier de normales. A chaque estation
correspondant un indice (valeur réelle/valeur théoriqye), il est
alors aisé de repérer les valeurs qui divergent trop a l'intérieur
d'une même zÔne d'homogénéité.
On calcule ainsi les valeurs de la pluie "théorique"
permettant de combler les données manquantes et de pallier les
erreurs les pius grossières.
De même, pour les précipitations, on calcule les valeurs
mensuelles et s'il y a lieu, celles des episodes pluvieux, pour
les poster fermés ou insuffisants.
On effectue une analyse de corrélation pour étayer le degré
de validité de la méthode opérationnelle et effectuer le choix des
paramètres à utiliser. On examine les possibilités offertes dans
ce domaine par la méthode des composantes principales, en parti-
culier sous la forme itérative de Van Isacker.
Différents critères sont également testés pour déceler les
valeurs douteuses, éventuellement entachées d'erreurs.
Certaines indications sont fournies sur la répartition
rationnelle du réseau pluviométrique.

302
I - DETECTION AUTOMATIQUE DES ERREURS :
1 - Pl~~e_theoriq~e_'encuelle
Le Bureau de L'Eau de la Météorologie Nationale a mis au point
une méthode permettant la critique automatique dos données pluvio-
métriques. Elle est appliquée dans le domaine relativement peu
étendu d'un département français; elle utilise une formule empi-
rique permettant de calculer 1a"pluie théorique" mensuelle pour
une station donnée, en fonction de la somme pondérée des valeurs
réelles de la pluie mensuelle aux autres stations du département.
considéré, les coefficients de pondération étant le rapport du
seuil de référence de cette station 5 celui des autres stations.
La formule proposée est de la forme :
O0 .~
Pth - p luie théorique mensuelle de la station à étudier
m - seuil de référence de cette station
ms - seuil de référence de La station s
Pr(s)-pluie réelle mensuelle de la station s
n - nombre de stationssans données manquantes utilisées
pour.l'interpolation
La formule précitée découle de la formule utilisée en analyse
objective pour l'interpolation de Gandine.
Les seuils de référence des stations pluviométriques ont ete
obtenus en partant des normales mensuelles publiées par la Météoro
log i e Franc a i se C.13
______________---_--
Indice d'homogénéité
On appellera indice d'homogénéité le rapport "pluie réelle
mensuelle/pluie théorique mensuelle" calculé pour une station
donnée. IL est bien evident que ce rapport sera nul pour une sta-
tion ne possédant aucune donnée pendant le mois étudié, et qu'il
sera inférieur 5 la valeur réelle si la station possède des don-
nees manquantes.
Lorsque l'on porte les valeurs des indices de toutes les sta-
tions du département sur une carte, i l apparait des zÔnes d'homo-
généite bien délimitees 5 l'intérieur desquelles ces valeurs sont
très voisines; ce qui veut dire que, dans ces zÔnes, Les pluies
sont fortement corrélées. Les stations ne s'inscrivant pas dans
La répartition spatiale des indices sur le département sont jugees
douteuses; telle est la base de la critique proposée.
_______ ~
~
2 - Pluie mensuelle _______ -_----
estimée
Pour I estimation de Ta pluie mensuelle des stations possbUdi~t
une série incomp!&te, nous utilisons la somme pondérée des indjccs
d'homogénéite relatifs aux trois stations complètes lec. plus ~r'oches
multiplice par la pluie theorique de la station en question.
. . ./

Ob
La formule est de La forme :
'est - pluie mensuelle estimée de,la station à étudier
KS - facteur de pondération relatif à la station s
CS - indice d'homogénéité de la station s
303
Pth - pluie théorique mensuelle de ta station à étudier
Le facteur de pondération utilisé ici est fonction de l'inverse
des distances entre stations.
3 - ------------ Décalages et --------- anomalies -
Pour rechercher des anomalies ou décalages éventuels, souvent
dus à des erreurs de transcription, nous appliquons le principe
suivant :
On considere qu'une valeur journali&re (nous la noterons flk)
est décalée ou anomale si :
a) O, alors que la valeur journalière pour chacune des
trois stations les plus proches est supérieure
ou égale à 1 mm.
b) vk
supérieur à 3 mm. avec une valeur journlière nulle
aux trois stations les plus proches.
Dans Les deux cas, i l faut que les conditions suivantes soient
vérifiées pour un jour donné;
nombre de stations of = O 1
nombre total de stations
G?T
4 - ------ Cumuls
nombre de stations où ,>
nombre total de stations
3 mm. 1

O0 Cjlest(j) - p luie journalière estimée au jour j a la station
étudiée
Fil-(s,j> - pluie réelle journalière La station s Le jour j
- pluie cumulée de la station étudiée
n,
"1
- ler jour des données cumulées
"2 - dernier jour du. cumul ("1 \< j < n2)
Dans la formule (I), nous utilisons un seuil de référence établi
a partir des normales établies par Angot en 1913 pour la periode
1850-1900. Pour les stations n'existant pas à cette époque, ce
seuil est obtenu par la méthode du tracé des isohyètes. Pour avoir
des valeurs aussi précises que possible, nous corrigeons reguli&rement
ce seuil de reference au fur et 3 mesure du développement
du fichier.
Pour cela, nous calculons les moyennes mensuelles des données
du fichier, en tenant compte s'il y a lieu, des observations manquantes.
Ces moyennes considérées comme pluies réelles dans la
formule (1) permettent de calculer les indices d'homogénéite qui
devraient être voisins de 1. Si les coefficients appartiennent à
l'intervalle (0,90 ; 1,îO) la normale est acceptée, sinon elle
est modifiée.
II - REMPLACEMENT DES DONNEES MANQUANTES OU ABERRANTES PAR DES
VALEURS C ALC ULEES.
------ 1Pre méthode - ----- :
En cas de donnees manquantes, la formule (2) permet de calculer
la pluie estimée. Si la différence "pluie estimée-pluie réelle"
est négative, les données manquantes de la station ne sont pas recherchées
car, dans la plupart des cas, elles correspondent a des
traces ou 5 des cumuls oubliés.
Pour calculer ces données, nous utilisons toujours le même
principe de pondération, ce qui conduit A la formule :
Test(j) - pluie estimée du jour j de la station étudiée
V%(s,j) - pluie réelle journalière de la station s le jourj
R m - pluie réelle mensuelle de la station à étudier
- nombre de pkriodes distinctes de donnees manquantes
"1 1 - ler jour de donnees manquantes de la lierne pbriode
n2(
- dernier jour de cette période(nll

305
--------
REMARQUE
Cette méthode ne donne pas toujours des résultats acceptables.
a) Lorsque la station dont on veut calculer la pluie estimee se
trouve à la frontière séparant deux zônes de répartition spatiale
différentes des indices d'homogénéité, la pondération
utilisée relative aux trois stations les plus proches, traduit
une distribution particuliere des poids qui peut s'éloigner de
la réalité.
b) Alors que les indices d'homogénéité caractérisent les précipi-
tations mensuelles aux stations, ils entrent dans le calcul de
la pluie journalière estimée.
Pour toutes ces raisons, i l a été nécessaire de réduire l'échelle
du temps. La critique automatique des données pluviométriques
est maintenant appliquée aux épisodes pluvieux. L'efficacit6 de ce
procédé a déjà éte vérifié par l'étude de certains mois ne présentant
qu'une seule période pluvieuse.
Pour pallier ces difficultés, nous avons mis au point une deuxième
méthode permettant d'orienter Le choix du météorologiste.
------------
2ème methode :
Elle est basée sur la recherche d'un rapport de proportionnalité
moyen entre la somme des précipitations correspondant aux périodes
des données manquantes et la différence du total mensuel complet
avec cette somme.
I Nous utilisons ici les trois stations les plus proches sans
données manquantes.
Appelons Pr(l), Pr(2), Pr(3) les totaux mensuels respectifs de
la première, seconde et tro.isikme stations.
DI, D2, D3 la somme des précipitations tombées respectivement
à ces trois stations durant les périodes considérées.
P/,(s), la différence Pr(s) - Ds (s variant de 1 3 3).
Nous calculons : 3
K=Z K s L (5)
s=' P;(q
OU Ks est un facteur de pondération, fonction de l'inverse de la dis-
tance séparant la station étudiée de la station s.
Connaissant P:, le total mensuel incomplet de la station étudiée,
et La valeur de K d'après la formule (5), nous pouvons ecrire :
ob D représente la somme des précipitations correspondantes aux
jours des données manquantes pour la station étudiée.
Pour calculer les quantités journalibres manquantes, i l suffit
de reprendre la formule (4) utilisée pour la première méthode en
remp1,açant par D.
(6)

0
Y I
. 0
I-
I-
C 2
i1
4
W
Ii
.I- U
c
I-
- .e W
N N N
n n n
I I
c .
o
4 L
O
e.
I
52
306

1 - 1 1 I
~:ooooooooooooooooooooooooo 00000
n~ooooooooooooooooooo~~oooo 00000
t
Q:000000000000000O0000Q0000 00000
t
yI toe, 00.0 oo oooo 0-0 oo o oo a a o o o oo.oee
.I
*~00O0O00000000000000000000 00000
t
s-~ooooooooooooooooooooooooo ooooa
t
Q~OOOOOOOOOOOOOOOOOOOOOOOOO O0000
*
m80000000000000000000000009 O0000
e
w a
æ
-i1 O.
W O
-
a
Y
w
a
=-
Y
O
v-
-a 23
cn
v-
a
m
o >
z
T
a

308
I I I - CALCUL A PARTIR DES STATIONS EXISTANTES DES VALEURS MENSUELLES
ET DES EPISODES PLUVIEUX POUR LES POSTES FERMES OU INSUFFISANTS
Les développements précédents nous permettent de :
1 - calculer le seuil de référence d'une station dont les données
n'existent que pour une période très courte (2 ou 3 ans)
sous réserve cependant qu'elles ne soient pas erronées.
2 - déterminer d'après la formule (1) La pluie théorique d'une
station B l'aide de son seuil de référence et des données
disponibles pour les autres stations du département
3 - calculer la pluie estimée d'après la formule (21, (Le météo-
rologiste pouvant éventuellement l'obtenir B partir de la car-
te des indices d'homogénéité) '
4 - rechercher les pluies journalières en utilisant les formules
(1) (2) et (4) dans Le cadre des épisodes pluvieux.
Les impératifs de l'élaboration d'un fichier valable de pluvio-
métrie avaient déterminé l'adoption dans la pratique opérationnelie
de la méthode simple de critique des données, que nous venons d'ex-
poser.
Parallèlement B cela, Le Bureau de L'Eau a poursuivi des recher-
ches théoriques afin d'elucid.er le degré de validité de la méthode
adoptée et les améliorations qu'il convenait de lui apporter.
IV - 'LE PROBLEME DES DONNEES MANQUANTES C8,3,43
Quatre methodes ont été vérifiées sur un fichier donnant les
hauteurs des pluies journalieres en 15 stations des Côtes du Nord
pour les mois de Janvier de 1'1 années consécutives (de 1961 2 1971)
1 - Analyse en composantes principales
__________-I_---------------------
Le principe de La méthode est le suivant :
On passe des données initia1es:r;ii valeur de la pluie le jour
i B la station j aux données centrees'réduites =;i 3c.i , où
m
ni % SJ
x.j:A Xij , Sj '2 2 (lu- .
i:4 b-4
les valeurs manquantes étant remplacées par la moyenne mensuelle
de la station concernée. On calcule les valeurs propres A; et Les
vecteurs propresci de la matrice de corrélation. Les composantes
principales sont alors déterminées par la transformation linéaire
des données initiales 3 l'aide de la matrice des vecteurs propres.
On effectue une reconstitution approchée du fichi'er initial en
ne conservant que les premibres composantes principales. Les va-
leurs manquantes sont alors remplacées par les valeurs ainsi re-
constitu6es.
L'efficacité de la méthode est mesurée B l'aide du coefficient

30 9
ry
où x est ta vraie valeur, ta valeur reconstituée,gx L'écart
moyen quadratique pour La station concernee, N Le nombre des "trous"
supplémentaires introduits dans le fichier de façon aléatoire afin
de tester la methode, La sommation étant etendue à toutes les va-
leurs de x correspondant aux trous supplémentaires.
Une étude expérimentale a montre que le nombre optimum de com-
posantes principales retenues pour reconstituer le fichier initial
était n = 4.
---_--_-__-
--___- ----------
2 - Analyse des correspondances
La méthode est analogue 3 la première mais, au Lieu de passer
aux variables centrées réduites =Li -3C.L on utilise La trans-
6J
formation classique en analyse des correspondances
I *
où xi.=- L.r..est la te jour i pour L'ensemn
jz4 Y
1
ble du département, x.j=-g x~j est La moyenne mensuelle à la
1 m ir4
station j et x..=-P =.j=L 2 ~~.
n j:d
rn ir4
Les valeurs propres de la matrice de corrélation des nouvelles
V ariables tij sont très voisines: 1,24 . 10-2

v -
310
I -4
Pour maximiser {()o on minimise La forme quadratique x V X .
La condition de minimum est ainsi
X'V'1dX = O
si t'on pose XI, =Z,--- ~n les valeurs connues et "&+d j - - - -1
7t, les valeurs inconnues (manquantes) la condition de minimum
s'-écrira
.-
On obtient n-k équations à n-k inconnues pour determiner les
valeurs inconnues x p+4 > ---- =,.
b) On ne connait pas Vxx. On calcule alors les covariances, couple
par couple, en effectuant la sommation pour les valeurs simultnnement
non-manquantes pour chaque couple donné. On obtient ainsi une
matrice Vxx, qui n'est pas nécessairement definie positive étant
donné que les indices de sommation =ont étendus à des ensembles
differents. On diagonalise ensuite Vxx, on range Les valeurs.propres
par ordrededécroissance et on ne conserve que celles d'entre
elles qui sont supérieures 5 un seuil positif donne. Un seuil optimal
semble exister qui est d'.autant plus grand que la taille de
l'échantillon est petite.
On a presenté dans le tableau 1 les valeurs du coefficient 6'
(c-f 7) en fonction du pourcentage de trous dans le fichier pour
les trois premikres méthodes retenues.
Le tableau Z permet de comparer, pour la quatrième méthode, Le
gain obtenu en remplaçant la .donnee manquante, non pas par La valeur
moyenne mais, par la valeur reconstituee à l'aide de cette
méthode, les crit&re_c de qualité étant respec.tivement,E(r-;E)',
E(~c-2)~
et -4-t*Cr,~).
RECHERCHE DES ZONES HOMOGENES DE PLUVIOMETRIE PAR UNE METHODE DE
VISUALISATION DE MATRICE D'INTERDISTANCE
Soit N points dans l'espace RP . On construit La matrice symetrique
NxN dont les termes sont les distances euclidiennes entre
points, mesurés dans l'espace R
P ;
On recherche une représentation plane des N points XI, Xz, ...,
XN deRP à l'aide de N points images YI, Yz,-Yp de Rz, de façon
à ce que la distance entre deux points images Yi et Yj soit La
plus proche possible de la distance entre Xi et Xj. ,
Pour cela, on part d'une configuration arbitraire des Yi et on.
deplace ces points de façon a minimiser un critere de type Xa
entre distances reelles et distances images. Cette methode ne neces
site pas de diagonalisation de la matrice de distanceset peut être
aisement mise en oeuvre. La pr8cision de La visualisation diminue
quand le nombre de points augmente. I l semble toutefois possible
de traiter des matrices 15Ox15C avec une précision de reppesen'
tation satisfaisante.

Pourcentage de trous
dans le fichier
Méthode des composantes
principales
Analyse des
correspondances
Méthode de
regression
15,19 19,6 24,3 29,4 32,9
o,33
31 1
0,34 0,26 0,35 0,51
O ,33 0,36 0,20 O ,33
0,30 O ,30 0,21 0,32 0,46
Tableau I - Valeurs des 6% en fonction des pourcentages de
trous dans le fichier
tations
1 o1
161
471
1131
1211
1271
1581
1681
1711
2101
21 51
2231
2281
2 62.1
--
E(X -%y
381
335
755
343
797
359
5 79
253
43 4
391
327
41 6
390
761
3621
51 2
T'ablcau 2
JANVIER
E(*- ;y 1 - r2
2795 0,14
1454
2864
1318
1844
649
3453
1632
21 59
1236
1477
2533
2791
2418
1699
O ,23
0,26
0,26
0,43
0,55
0,16
0,15
0,20
0,32
0,23
0,16
0,14
O ,31
0,30
:(x -ry
319
177
179
1 o1
191
97
409
145
220
160
3 85
2 88
186
161
221
JUILLET
E(X-%)=
1542
Y65
992
1126
1027
257
1311
1262
2285
822
1 3 2'ï'
1413
1448
1511
85 1
0,21
0,18
0,18
0,09
0,18
O ,38
0,31
0,11
0,10
0,19
0,29
o ,20
0,13
0,1Î
0,26

312
A partir de 24 stations notées A,B,C,-, X dans la région Sud-
Ouest de la France (Fig 2), on a construit 12 matrices de distances
(une par mois) B partir des hauteurs de pluie journali6res rele-
vées dans chaque station. Ces douze matrices ont été visualisées;
on trouvera en exemple (Fig 3)les deux visualisations Décembre et
Mai. Il a eté possible de trouver quatre g'roupes de stations voi-
sines qui se retrouvent sur chaque visualisation.
Ces groupes ont été reportes sur la carte et correspondent A
des zbnes pluviométriques homogènes en ce sens que deux stations
d'un même groupe cont"proches" vis 3 vis de la pluviométrie.
Ces th6mes de recherche ont été développés sous l'egide du
Bureau de L'Eau et rendus opérationnels avec La participation
active de Mrs. B.RAMBALDELL1, J.F. ROYER, J.C. BARESCUT, J.ZIRPHILE.
Notons en conclusion.que ces recherches se poursuivent au
Bureau de L'Eau et que d'autres methodes d'analyse des données
sont également étudiées dans le but de parvenir 3 une meilleure
connaissance du champ de pluviométrie.
REFERENCES BIBLIOGRAPHIQUES
CI1 ANGOT A. (1911-1914) Annales du Bureau Central Meteorologique
de France
C21 BUCK S.F. (1960) A method of estimation of missing values in
multivariate data suitable for use with an electronic computer.
Journal of the Royal Stat'istical Society, Series 8.22
pp. 302-306
C31 AFIFI A.A., Elashoff R.H. (1966). Missing observations in
multivariate statistics. Journal of the American Statistical
Association. 61 pp. 595-604.
C41 KELEJAN H.H. (1969). Missing observations in multivariate
regression : e fficiency of a first order methods. American
Statistical Association Journal. 65 pp 1609-1616.
C51 SAMMON J.U. (1969). A nonlinear mapping for data structure
analysis. IEEE Transactions on computers - Vol C - 18, N' 5
pp. 401-409.

I
Fig. 1 - ndiccs d'homogéneitE
5'
I
ARNE
I
31 3

314
Fig.2 -
12 BIS
Fig.3 - Exemple de visualisation des matrices d’interdis-
tances.

INFLUENCE OF INADEQUACY OF HYDROLOGICAL DATA
ON PROJECT DESIGX AND I'ORMULATION
GENERAL REPORT
by
Leo R. Beard ( 1)
NATURE OF DATA INFLUENCE ON PROJECT DESIGN
In evaluating the effect of data inadequacy on water resources project
design, it is important to recognize that a moderate error in project size that
might result is not necessarily accompanied by a proportional over-all loss in
project net benefits. As a matter of fact, the difference between benefits
derived from almost any water resources project and the costs of that project
changes very little over a relatively large range of project size in the vicinity
of optimum project size. However, it is in this range that added uncertainty
in design reduces project net benefits on the average, beca:ice net benefits
decrease in both directions from the optimum, and, even thoiigh increased expected
cost due to uncertainty is usually a smaìl fraction of the total project cost, it
can be large enough to justify care and extra cost in obtaining data for more
reliable design.
When project design level is quite different from tconomic optimum
(and this can occur because of financial constraints, political constraints, and
other factors) , then the net project benefits change vei *J rapidly with errors in
design magnitude, but these errors tend largely to carit-el in the expectation
computation. Hence, in general but not always, errors in determining over-all
project size have far less than a proportional effect on project net benefits,
provided that the project operation can be modified as necessary to make effective
use of the project facilities under conditions different from those anticipated
during design.
On the other hand, rather minor inadequacies in data can have an unex-
pectedly large effect on the over-all project size. In flood control design, for
example, errors due to data inadequacies can cause differences as great as a
factor of 2 or 3 in estimating extreme flood sizes corresponding to specified
exceedence probabilities. In the case of drought regulation (water supply) , a
change in magnitude or duration of a prolonged drought can result in differences as
great as a factor of 2 or 3 in the amount of supplementary supply (usually storage)
(l)Technical Director, Center for Research in Water Resources, The University of
Texas, Austin, Texas, USA.

316
that must be provided for. Here again, though, project design magnitude
does not necessarily respond linearly to changes in flood or drought magnitude,
because cost and benefit considerations have a strong dampening or stabilizing
influence.
INFLUENCE OF DATA INADEQUACY ON PROJECT SIZE
Considering then, that there is no simple relationship between data
inadequacy and project net benefits, it is safe to say that evaluation of the
effects of data inadequacies on design requires a detailed study of the
inadequacies and all of the interrelated factors that influence project design.
Such detailed studies have been demonstrated in rather simplified applications
in work cited by Mr. lames and used as a basis of the studies described by
Mr. James.
In his paper, “Data Requirements for the Optimization of Reservoir
Design and Operating Rule Determination ,” Mr. James develops the theory
and some practical demonstrations for determining the optimum length of
stream gaging stations where their value for reservoir design and operation
alone is considered. In effect, the question to be answered is, how soon
should gaging records be started if a project will be constructed at some
distant time in the future. His basic solution is first for a known future
construction time, and then he considers uncertainty in the time of construction.
Benefits of gaging records are a function of increased efficiency of design and
operation.
Although much simplification of the design and operation problems is
assumed, the concepts developed by Mr. James are of fundamental importance.
It is interesting to note that optimum record periods are in the order of 25 to
50 years, but there is insufficient information in the paper to determine whether
the basis of these results is real or largely assumed. Perhaps the author could
elaborate on this.
Stream gaging records are of value for many things other than project
design. It would be helpful if the author could express some opinions on whether
other benefits exceed these or are rather minor. It would seem off-hand that our
great heritage of hydrologic data could not have been justified many years ago on
such grounds alone, and yet w e know Chat the body of data that now exists is
invaluable.

INFLUENCE OF DATA INADEQUACY ON METHODOLOGY
317
In addition to affecting the size of a project, data inadequacies can
greatly influen..e the methodology used in planning and designing a project.
Professor Reid o in his paper, "Tiie Design of Water Quality Management
Projects with Inadequate Data, " points out that mathematical models must
be built with availability of data in mind, that there is never as much data as
needed, and that the only defense against inadequate data is judgment. He
describes a number of water quality models very briefly in the form of mathe-
matical equations, but does not attempt to describe their purpose or application
or to delineate the need for data in each case. Perhaps he could elaborate on
this. He expresses some thoughts on the cost of waiting for more data before
designing a project, and points out that an important element is the zost of
postponing the stream of net benefits from the project.
Dr. Reid suggests 8 quality parameters that are commonly measured in
the U.S. with adequate reliability and accuracy and at reasonable cost. It
would be helpful to discuss these in relation to the models described, with
particular attention to data gaps that would exist if only these paranieters are
measured.
Dr. Reid also suggests a time scale for a progressive pollution abate-
ment program, showing abatement of lake eutrophication by 1980, reuse by
1990 and recycling by 2000. This is apparently for the United States, but
would be of interest to other countries. It would help if some of the abbrevia-
tions used would be explained, if distinction between reuse and recycle is
explained, and if the basis for or origin of the table were stated.
METHODS USABLE WITH INADEQUATE DATA
Two other papers prepared for this session describe specific methodology
that should be used when data inadequacies exist.
In the paper, "Designing Projects for the Development of Ground Water
Resources in the Alluvial Plains of Northern India on the Basis of inadequate
Data, " Sarherwal describes generalized ground-water yield criteria, developed
for guidance in developing ground-water supplies in the Punjab until such time
as systematic data on ground-water reservoirs becomes available. The develop-
ment of high-yield crops has occasioned a marked increase in ground-water
exploitation as an assured supply for critical irrigation needs. In order to
further increase the use of ground water effectively, studies based on such
criteria are essential.

318
The criteria described are based on approximating the pertinent
components of the hydrologic cycle. Of primary concern are those components
associated with replenishment of the ground-water supplies. Formulas are
given for the amount of rainfall that contributes to deep percolation, seepage
from lined and unlined canals, recharge from water courses, and return
seepage from irrigated fields. It was found that horizontal movement of ground
water is very small compared to vertical recharge and could therefore be
ignored in this set of approximate criteria. Water withdrawal criteria consist
of generalized values for evaporation from water-logged areas and draft from
various types of wells.
Planning of new wells is based on a water balance study using these
generalized criteria and a safety factor dependent on the region. An example
of criteria application is given for the Bist Doab Tract.
Mr . Sarherwal supports his paper with an abundance of background
material indicating the importance of this subject to the economy, to the
ecology and to social conditions in India. It would appear that some elaboration
on the role of surface water development in conjunction with ground water
management would also be very useful in such an outstanding paper.
In their paper, "Design of Water Resources Projects with Inadequate
Data in India - General and Particular Case Studies," Banerji and La1
describe a variety of methods used in India for the estimation of design
floods and for monthly and seasonal runoff quantities.
Rough approximations of seasonal runoff from monsoon rainfall are
obtained with Strange's Table of runoff ratios for apparently arbitrary
categories of good, average and bad catchments. A less arbitrary method
of estimating runoff from rainfall uses Khosla's Formuld, which simply
substracts monthly evaporation and transpiration loss from monthly precipi-
tation. The loss is a universal, unique function of average monthly temperature.
A modified formula is given for calculating annual loss from annual temperature
in order to compute annual runoff from annual rainfall.
A third technique for obtaining runoff from rainfall is the correlation
of short runoff records with rainfall on an annual or monsoon-season basis
and then estimating runoff for all the years of rainfall reconl. The fourth
technique uses the standard unit-hydrograph method for relating runoff to
rainfall.

319
Methods of estimating peak flows include empirical formulas
relating maximum observed floods to size of catchment area , envelope
curves of maximum floods and regional flood frequency analysis. Criteria
are given for obtaining probable maximum precipitation and unit-hydrographs
for ungaged catchments. An interesting exponential recession technique
in lieu of the unit-hydrograph technique is described.
The authors do not discuss the degree of development or of flood
protection that is needed for various types of structures, so it is difficult
to visualize how their criteria would be applied for a great variety of
structures such as culverts, levees, dams and spillways, where there is
a great range in the degree of safety needed. Also, they do not indicate the
degree of adequacy of the methods and whether further development of
methodology or increased amounts of data would substantially improve the
reliability of project design. It appears that they are in an excellent
position to render judgment in this matter, and perhaps they would do so
in their discussion at this session.
MINIMUM DATA REQUIREMENTS
There is always the question as to the minimum data required for any
design, and, of course, this varies with the type of project and level of
development. In a paper, "Minor water Resource Projects Formulation
on Micro Hydrological Data for Standardization and Quicker Execution
in Developing Areas: Guidelines, " received through written communication,
Mr. Sikka discusses the problems of data needs in reldtion to development
of projects of moderate size. He supplies a list of miriimum data requirements,
which should be of value to countries outside of India as well as to India.
These include topographic and soil mapping, many types of hydrologic data
and data on irrigation efficiency. Special emphasis is placed on the fact
that past drought periods can be exceeded in the future and that this should
be taken into account in design.

320
Mr. Sikka discusses environmental impacts and the needs for indices
of environmental conditions and for value weights that can be related to
economic efficiency benefits and costs. He lists water and air quality,
wilderness and scientific areas, esthetic features and wildlife habitats
as environmental elements of principal concern. He stresses the conservation
of water resources through more efficient application of irrigation water. He
also discusses the conjunctive development of surface and ground waters and
related data needs.
Mr. Sikka brings up the question of the adequacy of basin-wide studies
of surface waters in conjunction with ground-water aquifers that extend beyond
the boundaries of river basins. This is a rather common circumstance, and
it is apparent that the scope of surface water studies must be extended where
ground water is a substantial element and where horizontal movement of ground
waters across the river basin boundaries is significant. This emphasizes the
importance of obtaining surface and ground-water data extending far beyond
river basin boundaries in some studies.
INFORMATION CONTENT OF DATA
Many of the effects of data inadequacies discussed thus far in this
general report are direct influences that are relatively easy to understand.
There are some subtle effects that are little understood and yet can have
major impact on project design.
Weber, Kisiel and Duckstein in their paper, "Maximum Infonnpon
Obtainable from inadequate Design Data: From Multivariate to Bayesian
Methods," discuss some theoretical aspects of a subiect that is critical in
the use of inadequate data and has considerable impact even where substantial
data exists in many applications. They examine the effect of possible
inapplicability of theoretical assumptions underlying techniques such as linear
regression, discriminant functions, canonical correlation, principal component
analysis, factor analysis and cluster analysis. In many cases, departure of
data from underlying assumptions such as linearity or normality will cause
erroneous results. More markedly and more generally, confidence estimates
will be in error.
The authors discuss the complexity that is introduced into Bayesian
analysis by uncertainties in the basic assumptions and cite some degree of
success in applications to discriminant analysis.

321
Remarks relative to the interpretation of the results of principal
component analysis are interesting. Attempts to identify the physical
significance of the components or to use the components in subsequent
regression analysis bring up serious questions. The general reporter feels
also that such attempts would constitute a misapplication of the technique
(as the authors may feel also).
This paper does not attempt to answer the problems but simply identifies
them. It should be of great value if it occasions attempts by the authors or
others to find answers to these problems. It would be useful if the authors would
comment on the effects that inapplicability of assumptions may have on the
stability of maximum-likelihood solutions. The general reporter has witnessed
cases where highly erratic results were obtained through use of maximum-likelihood
parameters that were apparently sensitive to the form of a distribution function
and where the data were not known for sure to fit the assumed distribution.
GEOGRAPHIC CONS IDERATIONS
None of the papers in this session discuss the differences in the
various geographic regions that affect the adequacy of data. It is known that
many rivers are very stable and that a relatively small amount of data can be
adequate for fairly reliable hydrologic determinations. On the other hand,
there is almost never sufficient data for evaluating the runoff potential of
some highly erratic streams where flows some years may be ln0 to 1000 times
as great as flows in other years.
Also, there is a difference in the nafure of data transfer potential in
various geographic regions. In regions where genera! storms or general snow-
melt floods predominate to produce high conelations among hydrologic events
within the region, data at a long-record site may be used to effectively extend
data at a short-record site. In this manner, short records can be made to serve
for long records to a large extent. It should be noted, however, that this permits
estimates whose reliability is limited to that obtainable with the longest records
of the region.
On the other hand, where great hydrologic heterogeniety exists, such
as where small-area storms predominate, information might be transferred if
the rainfall-runoff process can be modeled accurately. In this case, there is
a virtually unlimited amount of information in a region that might be assembled
to yield estimates far more reliable than those obtainable from the longest
records. At present, the technology does not exist for effectively assembling
such data, but the potential certainly is there.

322
SUMMARY
In summary, it is not at all obvious how data inadequacies can affect
design without making a detailed. study and without a thorough understanding
of the factors Involved. Errors due to data inadequacies can accidentally
improve a design, but the expectation is that better data will produce better
designs, as long as sound policies and technology are employed.
Many important contributions are contained in the papers for this
session, and the authors are to be congratulated on their efforts. They have
studied the need for and value of basic data and the impacts of data deficiencies
on techniques and on design adequacy, and have defined new problem areas
where special considerations are required in the use of small data samples.

ABS TRACT
DESIGN OF WATER RESOURCES PROJECT WITH INADEQUATE
DATA IN INDIA - GENERAL Q PARTICULAR CASE STUDIES
S.Banerji" E V.B. Lal* - INDIA
India has rich experience in successful construction of water
resources projects with inadequate data. While rainfall data of
considerable length are availaBle in or around the catchment, runoff
observations are usually available for 10 years or less, Commonly
some gauge site some distance away from the dam site may be available
Data on soil moisture, infiltration, and evapotranspiration are
almost non-existent,
The paper, based on a study of several important reports rela-
ting to many projects situated in different climatological, topo-
graphical and geological regimes, describes the practices followed
in: (i) transferring rainfall data from a hydrologically similar
region to the reservoir catchment by short term correlation, Ciil
establishing correspondence between rainfall and runoff by applying
a regional empirical formula, or by first deriving a regression
equation for rainfall vs, runoff for the small period for which simul-
taneous records of both parameters are available and then applying
it to longer rainfall records for getting the discharge series (iii)
tranferring gauge discharge relationship of a distant site to the
dam site to work out time distribution of inflows, and the peak flow.
For the latter, a method evolved for estimating maximum flood in the
Narmada and Mahanadi rivers, principally based on assessing the
contribution due to different zones of catchment each extending to 1
day's flow time, has been descrìbed for the Benefit of monsoon regions.
RESUME
Les Indes ont une grande expérience dans la réalisation d'amlna-
gements des eaux h partir de données insuffisantes, L'observation
directe des débits ne porte en général que sur des périodes de moins
de 10 ans, tandis qu'on dispose souvent de données de précipitations
sur de longues périodes. D'autre part, il est courant de disposer
d'une échelle limnimétrique 3 quelque distance du sîte du barrage pro-
jeté, alors que les données sur l'humidité du sol, 1iinfPltration et
ì'êvaporation sont presque ìnexfstantes.
L'étude présentée est basée sur plusieurs rapports importants
relatifs à des projets situés dans des régions qui présentent des
conditions variées en climatologie, topographie et geolog2e. Les
auteurs décrivent les méthodes employdes pour (31 transférer les
données de précipitations d'une région hydrologiquement analogue au
bassin qui alimente le r&servol'r, Ci'il &tqblir uqe correspondance
entre les precipitations et l'écoulement par l'application d'une formule
empirique régionale ou d'une Iquatioy de r@gress.Con, calcul@es
sur la période d'observation commane des pluzes et des débits et utilisées
avec des données de prgcipitations de longue durée pour obtenir
une extension des debits, (iii) déterminer la relation hauteur-débit
au droit du barrage 1 partir d'une relation établie a une station siT
tuée 3 quelque distance pour estimer la distrìbution des d&bìts et les
pointes de crues, Les auteurs exposent a titre d'exemple une méthode
utilisée pour évaluer la crue maximale des rivibes Narmada et Mahanadi;
cette mlthode, intéressante pour les régimes de mousson, tient
compte de la contribution des différentes parties du bassin, le
découpage correspondant a un isochronisme journalier.
~
* Scientists, Secretariat of 1.H.D' National Committee
I

324
1.0 introduction
India h a a rich experience in 'successful' construction of water msources
projects with inadequate hydrologioal data. Since 1951, when tha fimt Five Year
Plan commenced, 537 major and mriàium projeots, each having a reservoir etorage of
over 6167 hectare-mtree i.e. 50,000 aumfset, ham been taken up and about 300 have
been completed (1). However, since most of the gauge ami discharge sites f a
regular observation on Indien riveru have been eet up only after independenoe in
1947, runoff observatioas or even gauge-readings, if at all available at the sito
of a proposed dam were of very short duration, sw, lesa than 10 gbm. 'phr redseeing
factor in this situati= has been tknt for most aress of the oouutry long records
of rainfall, of 50 years or more ere gen9ra;lI.y available. Aluso, ooemonlJ aoma gauge
site would be evaileble on the ccgcerned river som distema away from tkie dai site.
Data on soil moisture, infiltration end evapotranpipiratiai axe praotiedïy non-
existent.
1.1 hia mesent paper is based m e tatu* of neveral intportant reports (vide
appendix 1) mlat- to meny projects of variona siaes rituated in different
ûlbtOlOgiû& topmphioal a d @ûl~iC8l =gimes Of the Cmtrj. W pY.æOtiCeS
described relake to the following thme oetegoriee of problem:
(i) 'Jkarisferring rainfall date from an adjacent, lydrologieally nimilar
regiai to tlie reservoir oatchnte
(U) Eatciblisw oormepandeme between rainfall and runoff
(iii)Tramaferring gaw discharga relatioriehip of a distant site to dam site
Mœt of thee relate to estimation of runoff vdrmies. Eltimation of p.nk
flore has been discussed separately.
2.0 Indien Pcaotice
It meg be etated hem that 88 there is lege di.arsity not od7 in the
2.1
size and the region of lm&Aon but also in the nature of data available for
differed projeote, lhm are no 8~tau&d8 or mallar8 Wd-8 to -rocmm
problem of dafa maralty. Then, he8 ban no partioular Pmferenoe for either
the Mit i@rograph or tb etatietiad Mqoenny distribution In &te- tìm
'dosigm flod8, and, bpending upoe the gravity ni wzumxmnnoe of a likely failure
of tb stmûtm, attsmpts hem ben meâe, wimromr m-88- ad poisiblev
to arrive at the design flood by =me of both thse approdms and a for others
of regid applloatian.
1Lo 8 general guideline it h m been pmsuribed that inajar and odium groJects
2.2
should be deeigmd for Probable Msximtri Flood (m) "that wodd reriult from the

co&ination of critical msteorslsgicd and hYhOle@;iO f oonditians
ca.sidend physi.~c~llg possible in the mgid"' Ln oases of SPJQ~ wojects
with mery larga catchm;.nte whem applicaticm Of imit Wai.oS=Qh is bdVhabble
a 1OOO-yeer flood esthte ia attempted by fnqwncy onalg.eie from a dkchprgs
aeries at site cmtmted frem &ta of rainfall or other infomtim that Ippg b~
amilable. In tb CPB~ of germanent ba-8
tu
less thpn 6167 ha. m the design is to be based on the Stanbra Pr@d**t
Flood (SPF) that would msdt from Ithe mest severe combination of ~ t e ~ ~ l W i c ~
and hg&ologic omditime, considersd reasonably characteristi0 ef liha =gim
excluding extmmb ram oo?abineticars', er a 1oO-year flmd whia*r IS
For smaller projects design flood ae;y be estimated by approximate end empirioel
methods applicable locally. (2~3)-
3.0 Transferring Rainfall Data from Adjacent Catchment
In Cases where rainfall data for considerable periods are not
available for the catchment upto the damsite two tendencies are discerni-
ble. If it has been possible to construct a discharge series at the dam
site by some technique from discharge data available elswhere no
outstanding necessity has been felt for precîpitation data for estimating
the flood peaks or periodic inflow volumes e.g. Tehri Dam. Otherwise
attempts are made to work out precipitation figures for the catchment
under consideration. If the number of raîngauge of raingauge station
in the project catchment is samall, the statìons in the adjacent region
considered hydrologically similar are utilised for constructing
Thiessen's Polygon for working out weighted average figures of rainfall
in different years within the catchment e.g. Hasdeo (Bango) Project. It
is also possible to have a few years'data at some specially set up sta-
tions within the catchment and to correlate them with the observations
at some stations with longer records, lying outside the project catch-
ment but still within a hydrometeorologically similar region. The
short-term correlation thus established is then applied to the longer
records of outside stations and the series completed for the project
catchment,
4.0 Establieu C o r m s D o n m y 1 & run& f
Bainfali reoords are @rerally available for projeet-catohwnte in
the form of 24 hour ralnfail amounte obeemd at a fixed hour for moat staticne
and 88 continuous recorde for selected etations with self-moording raingaugda.
For estimating runoff data the follmlng methode am generally follmedt-
a. Regional correlations, like Strange's Table; b.Khosla's F o d a
C. Regression equations defining correlatia betweem short-term
raipfall runoff data; à. whograph application.
While PrrtboC a,b,o, yield estiiiatee of runoff volume, hydrograph application
ia good for eetiaieta of flood volume as well aa flood peaksl.
4-1
l&2&!2w C-OXUl a SWQE'S TABLE
325
It gime permntage of runoff from 10~18ocn-1
rainfalls for different
lnàian catchmanta, which were rathbr subjeotively divided into three oatagories~
'good' 'average' and 'bad'. Thus far a total monsoon rainfall of 1000 e a good
catchment will yield 37.@ runoff, an average oatchnient, 2& ami a bad eatohRIent
18.776. Inspite of the faat tìmt theee tablea am ncm very old and o m yield only

326
rough estimates, they ere often applied in the projects for assessrnt of runoff
volUries, e.g. Chambal Valley Development Scheme, where such calculations baw also
been checked againat the observed data of a few years. €&o (3) has also used Strangego
table while working out dischargea for Nagarajuriaegar aiid Srisailm projects.
4.2 sx-&a's Fornu4
Khoela (4) working 'on tb rational concept that runoff is the residual of
rainfall after deduction of evaporation and transpiratian loss' aesuiped that
'temperature can be taken to be a complete maaure of all the factore which are
responsible €or the loes of rainfall to runoff'. The formula hos no mgional lid-
- ta5cms of applicability.
1
His empirical formula is Ra Pm -Lp wbm Rm, P, Lm am rssp8otiln3ly the
runoff, rainfall and 'loss' figures for a given month in mu. Lm is taken as equal to
5 'Ern, ilkre Tm is the man monthly tempereture in centigrades and is more than
4.5"C. For Tm

327
-
aucounte for ab ut 9% of tlm anatm1 rainfall. Out of thee thee equstioriey the
momoon rainfdl-monaoon nuioff equatian gave the highemt QOrIdaticeiS coefficient
(0.869) od thia was u. ad to derive the amoal runoff from the ennuel raiafoll f m s .
4.4 &hr-Dh Applicationr The tuchnique is lairly well haai. Later diseusaions
will ahow how the design storm is selected and its tiiir, distribution obtaineà for
applylag the mcipltetion figues to th unit hydrograph.
5.0 -Disc- Belet ioliahip of a Distant site to the Barn Site
We piok up tbme o- studies Vix the Eirakuà Dam, tts ThiLTi Deia a d the
NagarJunaaagar to illustrate hou this is being done.
5-1 In the Hhalcud Dam project (1947) the &am was prapomd to be looeted at a
site near SamboLpur whr8 gauge recad8 existed sinee 1921, but there were no
COrreaPpopdine gauge dieche@ curves. Ebwever, at Earaj, a site som 230 miles downstream
froaSembalpur, gauge diaßhaqp recards existed sinue 1868. The gaugs madia@
at Sambdpur were corzhlated with the gauge readiq at NaraJ, ding due allowance
for the tinia. 1- and similar epuea, discbrge c m 8 were prepared for Sdelpur
and checked 8gainst the dally discharge obrervatians o M d eince Jm, 1946.
5-2 The Tebri Dam Project (1969j enwbws construction of a dam eor~~s thb river
BhagFratM near Tehrii tha catohnmt arsa upto tb dpn eite ia 7511 8q.h.inaluding
2328 8q.b of constantly snou bound axea. Daily rimr g&ugoa andwe8-U~ âiaaharge
observatiais at tkm damaite .ere available only fra May, 1964. This Catchment is
a part of tb Ganga cstcharnt in which, at Bairele, near Haricbrar about 105 km.
damst- Of 'pehri, deilr aid dia- dot8 Svdlabh €hail l9Olo The
catcent
up%o Raiwala t 23000 8q.b. inoludbg 8450 8q.h- is anow-
bound.The Raiwala data have been utilised to compute runoff at Tehri
in 10-day periods of the year. For this purpose the runoffs for
different 10-day periods, in the period of actual observation of
discharges at Tehri, have been compared with the corresponding 10-day
runoffs of Raiwala, assuming a one-day time lag for the flow to reach
Raiwala from Tehri. The percentages of Tehri discharges to correspon-
ding Raiwala discharges have been plotted against the relevant 10-day
periods for the period of observation, 1964-66. These percentages vary
for the same 10-day period from year to year due to variation in
precipitation, temperature, humidity, vegetation, soil moisture etc.
and for individual catchments of the tributaries of the river Ganga
above Raiwala. The required factors have been worked out as below:
AvoroRs of runoff et Tem.
ri
Ave- of runoff at Raiwtle
~eing r vaime for &ifferet 10- periodi, the -off figme at Beirala
have ben canverted to f-a for Tehri for 30 (fra 1936 to 1966).
5.2.1 'Ffiib Rairela Qata haw albo heen wed, ia ocajimotian with the ih&-tea
rseard at TekrirL, for estimating the flood peds et 'psbri, &a- Baiwda 88 th
etaticus, tfie peroeritege d tias e pcirticular noOb hae been equalla8 or
emeebd le plotted agaInet the flood 021 a sed-log paper to giw e lozig-tsra data
c m for tb inder itattian. Bor tb short-term for which data ara available both
for %hi arrd Baida, rimilar o m s u. plot- far both th6 et&ticma. Tbs IOWterm
c m for the project 8t&iOn (%-i) L then coiiltriiabd from the abow three
oms, and ths flooda of various frequencies &PB obtabd from this OPM. It is,
-

328
hmemr, only apIQ of m w w
adopted in the projeot for eetimting tha flood peek.
5.3 Ra0 (5,3) applies a different apprcmh to determim peak flooda at a section,
when discharge data are available for a diffemnt site al- the river. W
principle applied is simple: thedischage observed at a dametream aite ieequal to
the discharge at an upstream site plue the dischar@ contributed by me interniedia*
catcbnt mincis the oharial' trough' oapaieitg between the two sites. W 'trough'
capacity oan be computed ideally with the help of croes eectime of the river at
close internals, or otherwise in the absence of thie inforretian, by taking the
average width of the river flow at one end, the difference in the depth of flw on
the day of Peak flood and 24 hours before its occurmncc and the length of the river
reach into woount. Th? inflow from ths intermediate catchnt q be worked out from
the rainfall recorda using strange*s table. In this way the flood aeries at the
upstream point ia Constructd for a number of years and subjected to frequsnoy
analysis far estimeting tb design flood of a given recurrence interval.
6.0 Esthau= of Ped Floa
Nodly, peak floods are estimated by several mithoda before adopting a
design flood. Such niethods range fra empirical fofiaulae directly giving peak flows
from a oatchniant of given area to the elaborate etarm-trailspoeition and aiaximisation
mthode. "hey m y broadly be clamifiad into tuo oategoriest
(a) Non-mteorological =th& (b) Meteorologici1 methods
In the non-mteorologieal oategory we may include tb following: Empirioal
formulae, Enveloping Curves, Regional Flood Frequency analysis. In the meteorological
category we inelude æ?thoQe that proceed frcm atom analysis. They msy or mqr not we
unit hydrograph.
6.1 .O Non-ktoorolouiad Cateaory
6.1 .1 Enipiriaal F a
(a) Ths noet popul formulae link the peak flood with the ama of the basin,
like Diokm's, Q= CAY4 , for the Central and Northern India, the Byve's,
g CABB, for the south India, the &lis Q- O0O ."or faa shaped catohmmb ki
the Bombay aegica, wherm Q &vea the peak ra k- f disohaya in cusecs, A, tis ama
in sq. miles and C is a coneteat differing fron loeation to looation. &oaueß Of
their simplicity euch regional formulae
appraaimatiar of the likely flood.
still hi wide use for getting a fimt
(b) Quite often, if high flood mark6 ara avrilable with raferrtaae to old trees
or =oient atructplae, or OWE from the m ory of th looal inhabitruita, elow-ama
method is employed as (UI aid to gueoo tkm ordar of t b dieoharm. No reliable idea
c m obviously be bad of t b Seetion prevail* at fhs t- of flood fim, end thsm
is diffiaulty in estabbliehin& ttie bed slope, which i8 teken OB equal to tkm SudaCe
Slop establiaha0 from mrka at different points. Kutter's or uamih@' Coeffioient
of rugosity ie eitbr (usumd, or dete-d by t b eubmtitution of ieaemd äata
for a few flood8 In th oonoerned formula.
%Sidea th faot tht eu& formulae are ueeful only for limited regiolipl
applioation, present ri& eoope far eubjectim fagtom in choosing the valm
of tïæ constant. Also, it is not paisible to have any idea of tbie probable frequencg

329
of the flood so estimated, ao that a partlulm value of C may give a flood which
may be too high for designing a minor work, mey a culvert and too loa for &signing
a spillwey.
6.1.2 Envelope curvesi
Working on tb assumption that basins of similar hgdmlogical characteristics
should produce the sana mexlmum floods psr unit of catchment ama, Kanwar Saia and
Karpov plotted data of mm~imum floods in Indian rimiers againet the drainage amss
producing thoee floods on a log-log paper, and gave two envelope curves one enveloping
data of South Indian Basins ond the other enveloping data far northern and oentral
Indian basina. !be likely maximum flood from a catchment of given area is then
expected to be indicated by these curves. Besides tìm basic inadequaoy that these
curves relate flood potential only with the drainage area, they do not provide for the
occurrence of floods of higher magnitude than those on reoord.
6.1.3 Resi onal Flood Freauew-
Data of all the stations (points) in a statietioally homogeneoua region are
combimd to produce a flood-frequenoy oume that is asswd to be valid for the entire
region and can thye be applied to determine flood of a retuni period for an imgauged
catchment In the region. The simplified procedure recommended by tìm Central Water &
Power Commissian (2) is as follows:-
All stations in the regim with flow recorde of 10 years or more are eelected
and for each etation a frequency curva go- upto a 100-year flood Is constructedby
the Gumbel's Ethod, with a confidence-band of 9% reliability. All points am tested
for homogeneity as f ollors:
The ratio of 10-year flood to man annual flood is determined for each point!
this ratio awreged for all points is taiœn to give the mean 'lo-year ratio' for the
ama. The mturn period corresponding to the ran 'lO-year ratio' time the maen
annual flood Is detemlned from the frequency oume of each station and plotted
against the number of years of record far that etatim oq a sed-log test graph. If
tiæ pointa far ail tim etations lie between the 9% confidence limits, they are
oonsidered homogeneous.
The frequency curves of different stations in a homogsneoua region a m regarded
as different estimates of the regimal curve, and tlaey are averaged as follows:
For eaoh statim, flood ratioe (flood of a return period T over the niean
mual flood) a m computed for a number of arbitrarily selected values of T. The rean
of the flood ratioe for all stations for a particular period T is taken to represent
the flood ratio for the r egid curve. The resulting mans for different vaìws of T
are plotted a t b extreiiie value probability paper and the best fit line through them
gives tïm mquized regional frequency curve.
The application of thie curve to an wuged catchment rsquims M setimate of
the E= BIIILup1 flood for the wtchnmnt. This is dom from another e m which gives
the plot of man umual floods at different statims agai-t tbe corrsepmding
drainage amm. From this C- the value Of ttn? likely
flood W d t
the are of the naxg mgaugad oatobment can be mado
6.200 &teorOlaasop1~

330
records of all the precipitation stations in the region of, and around, the project
catchment, which may rather subjectively be ree;ardeà 88 hyäromteorologicdly homogeneous,
are studied to sglect storms of high rainfall covering an area more or
less equal to or larger than the project oatchment. Far this purpose it mu b
neceesary to carry out the umtaai Deptharea-Duration (DU> analysis of selected major
st >rp~s 6nd from there maximum one-day, maximum two-deg, maximum three-day precipitati-
are worked out. These aeleoted stom, are then transposed to the project
oatcìnrent adjusting the precipitation axis dso to an orientation that will give the
maximun runoff producing effect, if such directional change of storm axis is within
20° from the original axis. The storm is then maximised for the moisture content by
applying a moisture-adJustiPent factor (maf) defimd as the ratio of the max. precipitable
water over the catchmnt, W piex, and the precipitable water of the storm,
P
W This factor can be worlred out from consideration of the repremntatiw dew point
op the storm, and the mucimm dew point over the catchment and then finding out the
corresponding precipitable waters from the 'Pressure Vs Precipitable Water' diagram
between the pressure range 1000 mb to 300 mb. Alternatively, in the absence of
sufficient data, a multiplying factor lying between 2C$ to 5s ia assumed .
Havhg thue determined the design storm, the tina-distribution of the rainfall
has to be obtained. From DAD analysis maximtua rainfall depths for durations of 6,12,
18,24,36,48 etc. hours are obtained for each of the atorpis and expressed a8 percentof
the total rainfall, From a study of these pementages suitable distribution for
the desiga storm is arrived at. Alternativelyif a limited number of self-moording
rainges are available the ti- distributim cum be obtained from the continuous
records. If no self-recerding gauges are available time distribution based on the
experience of storma elsewhere in comparable area is adopted. Effective rainfall
for difr'emnt time incremnte is estimated by any of tb usual rays, vis,(a) tha
calculation of infiltration loes by finding the total surfme flws from actual
flood-hydrographe and commng them with corresponding rainfall vol~aes e.g.
Tenughat krojeat or (b) by simply assuming a runoff factor and applying it to the
design stozm values, e.g. Fíasdeo (Bsngo) Project. These effective rainfall values
are then arranged in the oritieal sequence which may be a mm or less sgmnietric
arrangement of valiies with the greatest value in the middle, or my be determined
by arranging the rainfall increaients against the ordinates of the design d t
hydrograph ln such a way that the longest odinate faces tke largest effectiw
rainfall and the next largest ordinate faces tb next largeet rainfall increment
and so on, and then reversing this arraageeient to give tb oritical seqrrenœ. It is
then applied to the design unit hydrograph, which can ba derivad by eriy of the
wual nieans,actual obsemtiolls ar synthetic.
6.2.1.1 A recent report (6) suggeete e new Psthod to qatimete t 3 design flood
peak (50-yem recumme) from small oatch~mnta (25 Km to 500 KID 1. It takes into
account selected besin characteristios (length and weighted .Ban BloP of the bmid 88 representative of the beeh response to tiie storm intaet and the atora P-PieterS
like areal to poNt rainfall ratio. The procedure hae been evolved from an -lysis
of short-term diaeharge data (5 to 10 yeare) for 60 drahmege basi- Of different
slopes and sim soattered all over India. It Gen be briefly summed up a8 Pollairs:
The weighiiù mm slope of the main stream, defined 88 given belw,
worked out'
L - 2 -
C
= ( Li/+ +L 21 SB 2 + .....
1

331
where Lc is the length of the mPin stream ln dles fra th@ maeuremnt site to a
point on the main stream near the centre of grevity (CG) of the catchment area,
and S1,S2 etc. are the slopes of the stream in the remhee of lengths L,,L2 etc.
into which the length Lc is divided. Lengths axe mesured from the topoaheet,ln=l mile,
From the value of s, the peak rate of flow Qt, in a tc-hour mit graph in cuuecs can
beestimated by th following formiilaer-
-
(i) Qtc I 16000 A%2'3, if s 4 0.0028
(ii) Qtc 320 A 6 , if s > 0.0028
0-9
t, is the duration of th rainfall excess given by 255/(Qtc/A)
where A is the area of the catchment in sq. miles. For estimeting the design rainfall,
a 'design storm by6tograph' table hae been pmgred giving point-rainfall volume (m)
of 5O-gear return period for durations varying from 15 minutes to 24 hours, and
these are then conmrted to arsal rainfall volume by applying ama1 to point rainfall
ratios that have been worked out earlier by analysing data of 12 àense networks. To this
areal rainfall a uniform loss rate is applied which is determimd from the empirical
relatioioohips which have been deriwd for different types of soil. This rill .determine
tb rainfall exoess in t, hours, and the Qtc value multiplied by this excess would
give the design flood peak.
6.2.2
Bowever, these formulae need to be tested further by the field events.
Meteoro l wcal cat BPON i without usina t b U.G
Banerji md Mantan (7,û) ham adopted a new approach for eStiIU8ting volume
and peak of runoff from data of atoras. They have studied flood in the Namada and
the ~hanaäi oatohmants. Studring hydrographe of a number of floods (including ia floods), thy find out the ti- base of the hydrographs after the bese flow has been
eeparated; it was 6 days in each of these oaseu. The basin is then subdivided into
zone8 of 1-day travel time each, by using the following equation (9)
1 .i5 0.38
Tc = L /7.700 ii
where Tc is the time of ooncentration, calculated for all the min tributarie8 to tirs
points of outflow in hours, L is tìm length of the remoteat point in the zone to tb
outlet point in feet ani II ia the diffemzuu in elevatia betwen the waterslmd outlet
and the moet distant ridge h feet.
It io assumed that the contribution to the flood-vol- from each ~ OPYI is
depeudent on the average ama1 depth in ewh didelm, ths scdl moiet- cmdition
and the retenticm orpmity; the proentsge contribution of flood vol- from elch
zone is th- made independent of the %d catchment characteristioB. It has ben
further aaaued that tb infiltration or retention deoleame from upatream to dom-
8tnam zoma so that if K i8 the Storage facta (considered a8 the fmtiOn Of
ecierage preci itatian &pa a p p a r ~ aa runoff ) in the zone nearest to the point
of outflcnr, l! is tb starogb faotor fm the nth diviaion m y from the outflm point.

The total daily runoff *Fit at the outlet can then be sriPiPed up as
Shew An i8 t h area and Pn-, ia t b average precipitation recorded in the nth
division. The value of K is seleoted, by trial and error, from past records of
discharge for which simultaneous precipitation data are aleo available. A graph
is then Plotted between values of K obtained for different perioda and corns-
ponding antecedent catchnt rainfall.
hydrograph, but utilises all the sante the es8ential underlying principle that the
ordinates of two hydrographs for the 8amB basin and similar tine bese are proportional
to their respective volumes of runoff. The correspondence may be effected between
tkie biggest storm on record of which only rainfall data are available and any or
all available hydrographs if the oharacteristios of stom are meteorologically
similar to tke outflow point.
Heferences
For working out peak runoff rates (7) the method does not need a unit
1. India, Irrigation and Power Projecte (Five Year Plans) 1970, Govwrzuœnt of India,
&inistry of Irrigation and Power.
2. Eetimation of Design nood, bcoimPsndeà Procedures 1972, Govt. Gf India,
Central Water 4 Power Commission.
3. bo, G.ii~. 1569 Modern Trenda in Hydrologic Computations, New Celhi
Central Water and Pmer Commission.
4. Khoela, Ad. 1949 Analysis and Appraisal of Data for t h Appraisal of water
Resources. Central Board of Irrigatia Jour. pp 410-422.
5. Ha0 G.A.H. 1967. Computation technique for Probable Maximum Flood Discharge
at place in the river while gauge dischare data is available for anotber
pïaoe with special referena to dam on Krishna river, India. Proc. Int. Sgmp.
Floods and their Computation, Aug. 1967, Leningrad, u-s-sj.~*
6. 1973 Flood Estimation Directorate, Central Water & Power Commission, New Delhi,
&sign Office Report No. 1/1973.
7. Bansrji, Sdhton, D.C. (1967). On estimating peak discharges correeponding
to heaviest redorded atom in a oatchment. Ind. Jour. Wt. and Ceoph. V01.17
Spl. N0.M 297-306.
6. Banerji, S.Manton, D.C. (1967) Determination of the distribution
of rainfall floods in large catchments using hydrometeorological
data. Unesco Int. Symp. on Floods and their Computation, Lenin-
grad.

d
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8
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333

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I

ABSTRACT
DATA REQUIREMENTS FOR THE OPTIMIZATION OF
RESERVOIR DESIGN AND OPERATING RULE DETERMINATION
James, Ivan C., Ir
U.S. Geological Survey, Washington, D.C., USA
Approaches to the design of multipurpose reservoirs have usually
assumed a given set of operating rules. Conversely, studies of oper-
ating rules have often taken reservoir size as fixed. In only the
former case have estimates of the optimal data requirements been made.
This paper gives the estimates of and compares the optimal length of
data sequences for reservoir design where operating rules are fixed,
for operating rule determination where reservoir design is fixed, and
for the combined determination of operating rules and reservoir size
for a multipurpose reservoir where the benefit is a piecewise-linear
function of storage and release. A strategy is developed for the
economically efficient design of the combined program of additional
data collection and project deferment. The shape of the benefits
foregone versus time function is such that project deferment is
usually optimal only where very short hydrologic records exist, and
the effect of an uncertain project inception date is to increase the
optimal-length of the data sequence.
RESUMEN
Métodos para el diseño de un embalse multipropósito usualmente
han asumido un juego fijado de reglas de operación. Reciprocamente,
los estudios sobre las reglas de operación frecuentemente han ini-
ciado con un tamaño fijado de embalse. Estimaciones de los reque-
rimientos Óptimos de datos se han hecho solamente en el caso ante-
rior. Este artículo presenta las estimaciones del largo Óptimo de
series de datos para el diseño de embalses con reglas fijadas de
operación, para la determinacìón de reglas de operación cuando el
embalse se fija, y para la determinación junta de reglas de operación
y tamano para un embalse multipropósito en que los beneficios son una
función contìnua por arcos de abastecimiento y descarga. Una estra-
tegia se desarrolla para el diseño eficiente economicamente de un
programa junto de aplazamiento del proyecto y recopilación de datos
adicionales. La forma de la función de beneficios renunciados contra
tiempo es tal que el aplazamiento del proyecto usualmente sea Óptimo
cuando existen solamente registros hidrológicos muy cortos. El efec-
to de una fecha incierta del comienzo del proyecto es crecer el largo
Óptimo de la serie de datos.

336
Introduction
Fundamental to any development process is an information base
for use in making planning, design, and operational decisions.
This input has measurable costs and benefits as do the other inputs
such as planning resources, capital, and site values for alternate
uses. Economic efficiency requires that the balance between the
inputs of a development process be such that the marginal returns
on all inputs are equal. These marginal returns should be equal
to their marginal costs where budget constraints are not active,
or equal to their shadow costs when they are active. Viewed as
another input, information may be conceptually handled as any
other input. As Weiner [i] succinctly states:
"Information is only one of many development inputs;
development, in turn, is but a transformation process
adopted in order to reach certain objectives. Infor-
mation is, thus, purely an instrumental objective and
not a final purpose in itself, a basic fact we some-
times tend to forget. "
Information requirements for project development include such
diverse factors as hydrology, future prices and extent of markets,
climatology, topography, soil classification, demand level, geol-
ogy, demography, and political trends. In reviewing this list of
information requirements, it can be seen that hydrologic data,
particularly those of a stochastic nature such as streamflows
have distinguishing characteristics that require a different treat-
ment than other information inputs such as economic, demographic,
political, and physiographic data. Climatic and hydrologic phe-
nomena require rather long data sequences to develop suitable
representations of their generating mechanisms. In contrast,
programs for the collection of physiographic information such as
topographic, soil, and geologic data can be deferred until shortly
before these inputs are needed in the planning process. Models
for projecting economic, demographic, and political trends into
the project life horizon heavily weight the latest inputs, thus
these data are usually collected only shortly before their use.
Planning the hydrologic data collection program requires the
longest lead time and is usually accomplished when a high degree
of uncertainty exists about other project information inputs.
Efforts devoted to hydrologic network design cannot rely on highly
formal tools in the absence of these other inputs. An alternative
is the development of heuristic rules based on generalizations
from results of pilot studies of optimal data record length for
specific planning and design situations.
The relationship between the timing of investments in data
collection and the time stream of benefits gained through the use

337
of these data in the design and o'peration of water developmecii
projects is an important consideration when these time streams of
costs and benefits are discounted to a common point in time.
Discounted marginal costs of the first years of stream gaging are
much higher than the costs of gaging just before construction.
Design and construction produce large sunk costs for which there
is little recovery from incorrect decisions unless the designs
have incorporated a high degree of option flexibility such as
through staged construction or other often costly methods of main-
taining decision liquidity.
The problem for the designer of a water development project
is to maximize net project benefits subject to exogenously
supplied constraints. The decision variables pertaining to the
water supply design of a reservoir are usually the size of storage,
release target, and operating rules for determining specific
releases. Herein, the decision variables are divided into those
physically immutable design values'such as sizing, and those
operating rule variables which could conceivably be changed to
reflect the results of new information.
Whereas design sizing requires historical data, the use of
information for defining operating rules may be another matter.
Additional data collected as normal requirements of project oper-
ation may be used to update operating policies. On a theoretical
basis, sequential decision theory provides a methodology for a
flexible and continually updated operating policy. In practice,
however, political and institutional constraints make changes in
operating policy a difficult and expensive process.
The determination of trade-offs between capital and operating
expenditures is a straightforward process when marginal benefits
are known. Planning decisions, by their very nature are further
removed from the time stream of benefits than are design decisions.
Little is known about the mix of planning process input resources
which achieve an optimal design from the viewpoint of econonic
efficiency. A large effort cannot be expended to determine the
optimal length of record at every possible site: rather, simple
guidelines are needed that answer such questions as:
a) How much streamflow data is optimal at a site €or the
expected design decisions and economic parameters?
b) What is the most efficient operation of a gaging
station when there are uncertainties in decisions and
parameters?
c) I€ the gaging station has already.been operated longer
than the optimal length, what factors would justify
à3scontinuing or retaining the station?

338
An approach to the determination of optimal lengths of gaging
for the design sizing and operating rule determination of irriga-
tion reservoirs is presented herein. This system was chosen for
several reasons. In the western United States irrigation reser-
voirs receive the majority of their inflow in the spring months
from rainfall and snow-melt runoff. Usually only small quantities
of natural flow are available during the peak demand months of
July and August. The essence of such a system can be captured by
a model in which inflows must be stored for satisfying demands in
later seasons or future years. This permits the use of annual
flows and operating rules which are nonseasonal in nature. Bene-
fits from irrigation projects are also usually more easily measured
than for other types of water supply.
The value of data in a particular situation is often limited
by constraints in the decision space. For example, if arable land
were limited because of available streamflow, reservoir design
would become sensitive to the irrigation requirements, not mea-
sures of the available streamflow. Systems which have a high
degree of operational flexibility may have low sensitivity to the
availability of data since operating policies can be changed to
consider streamflow data collected after project construction.
The Value of Data for Reservoir Design and Operation
The seminal work on the determination of optimal data lengths
for project design is that of Dawdy, Kubik, and Close [2]. This
work has been extended by considering the effect of discounting
and benefits foregone by Moss [31, Herfindahl 141, and Tschannerl
151.
For the most part, these studies have dealt with the situation
where operating rules were given and design siting was the primary
decision variable. Hydrologic data also have value for the deter-
mination of optimal operating rules.
Operating Rule Determination
The transform of state variables such as storage and inflow
into release values is accomplished through operating rules.
Perhaps the simplest and most often used operating rule for analyt-
ical purposes is the Z rule, where the release R is defined by:
Minimum { S + I, T 1 for S + I
Vm
Maximum { T, S + I - Vm for S + I > Vm
where S is the carry-over storage, I is the inflow, T is the
target, and Vm the maximum conservation storage. This rule would
Se optimal if losses were linear with the deficits below the target
value and no benefits or losses accrued from releases greater

339
than the target value. Though these may be rather restrictive
conditions, the Z. rule has proved to be a useful conceptual tool.
In recognition of nonlinear loss functions and seasonal differ-
ences in expected inflows, Maass et. al. -161 describe modifica-
tions to the basic release rule called the pack rule and the
hedging rule. Young 171 and Hall and Howells [8] used regression
analysis on optimal deterministic releases derived from dynamic
programming to define release rules. Russell 191 uses dynamic
programming to determine the form of an optimal operating policy
following a development similar to that of Gessford and Karlin 1101.
Both investigators assumed serial independence in inflow sequen-
ces. Gablinger and Loucks [ill, Loucks and Falkson 1121, and
Loucks (131 consider design and operation of storage facilities
where flows can be described by Markov transition processes.
The Z rule for release determination is not optimal when
marginal losses increase with the amount of the release deficit
or where significant benefits accrue from competing reservoir
purpo'ces such as recreation or water power. One method for devel-
oping operating rules is to select from a set of operating rules
which are defined as a function of state variables such as storage
and target draft. Parameters values of these release rules can
then be optimized in conjunction with reservoir design parameters.
The selection of a particular functional form of the release rule
will depend upon the objective function, and the nature of the
assumed mechanism which generates the inflows. For other than
extremely simple problems, this determination is non-analytic.
where
The operating rule chosen for this study was:
R = Maximum'{O.O, Minimum (aA - ßB+T, Si+J)}
A = Maximum (0.0, S+I-T'Vol
B = Maximum-{O.O, T+V -S -I}
O i
si e: Minimum {V ; s +I)
rn i-1
and CL and ß are release rule parameters, R is the release to bene-
ficial uses, and ifo is a minimum target conservation storage. In
other words, the relea'se is the target modified by a fraction a
of the end-of-season contents above minimum pool level Vo or by a
fraction 6 of the end-of-season contents below minimum pool level.
This release rule is more sensitive to project economics than the
2 rule, yet adds only the two parameters a and $ to the optirniza-
tion problem.
The previously defined parameters a, ß, T, and V , the active
storage volume above Vo, were determined by the methoa of general

340
function minimization described by Berman 1141, using a reservoir
simulation model which returned the negative of discounted net
project benefits for any set of the four parameters from the opti-
mization routine.
The selection of a project benefit function is not an easy
task. The marginal economic response of crops to marginal water
applications vary widely. For this analysis a linear benefit
function was used with the following coefficients.
C1 .O016 $/m3/ short term loss for end-of-season
storage below V
0
C2 .O004 $/m3/ short term benefit for end-of-season
c3 .o12 $/m3/
storage above V
O
short term loss for releases below
target
C4 .O04
3
$/m
short term benefit for releases above
target
C6 .O057 $/m3/yr long term benefit for target releases
Assumed marginal capital cost3 for providing reservoir storage C
5
ranged from .O017 to .O04 $/m /yr. All annual benefits and
costs were discounted at 6% to the starting point of project
benefits.
Analysis of Data Requirements for Irrigation Projects
Five gaging stations were selected which had been used in
the design of existing irrigation project reservoirs. The his-
torical data were extended with generated operational hydrology
to a length of 200 years. The reservoirs were sized and operat-
ing rules determined for all 100, 50, 33, 20, 15, and 10-year
sequences of the 200-year sequence, and on 8 and 5-year segments.
for some of the projects. For each set of determined parameters,
the reservoir was simulated for the 200-year period and discounted
values of the objective computed. These values were then averaged
across all equal-length segments of record. Average objective
function values were then plotted against segment length and a
curve smoothed in as shown for an example in figure 1. To the
values of this curve are added the present value of the cost of
gaging, and a total cost curve results. The optimal length of
record occurs at the minimum of the total cost curve.
Operating rules were held constant at their optimal values,
and the analysis repeated for reservoir sizing alone. Also,
reservoir size was fixed and the analysis repeated for operating
rules alone. For clarity, only the objective function values

and not the total cost curves are shown in the published figure
for these analyses.
These latter analyses were done to attain comparability with
previous studies on the worth of data where operating rules were
fixed and only reservoir sizìng allowed to vary.
Discussion of Results
341
The present value of the cost of a gaging station record
which is assumed to cost $2000 per year of operation is given in
table 1 assuming an interest rate of 6%. Thus, in a 50-year
gaging record, the first 10 years has a marginal cost of $580,000-
$310,000 = $270,000 as compared to the original $20,000 investment
for that record. The effect of discounting the gaging costs is
to make the marginal cost of gaging much higher than might ordi-
narily be perceived.
The results from these studies are tabulated in table 2.
For the economic parameters used, several results are apparent.
The optimal data length for both design sizing and operating-
rule determination is considerably longer than the optimal data
length for design sizing alone, but only slightly longer than
the optimal data length for operating rule determination. There
is a strong correlation between optimal data length and size of
stream, as is seen in figure 2.
If a gaging station is already in existence and has already
achieved its planned "optimal" length of record but the project
has not been yet built, then the decision problem must be viewed
from a different economic viewpoint. For this, the present value
of the marginal benefits of an additional year of record must
equal or exceed the cost of that marginal year of gaging. The
right-hand column of table 2 is the point on the marginal benefit
curve that equals the marginal cost of gaging. For example, if
the decision maker designing a reservoir on the Smoky Hill River
near Arnold, Kansas was one year away from construction, then he
should continue to gage if the total available record is less
than 55 years, even though it may exceed the apriori optimal
length of 30 years.
In previous work by James, Bower, and Matalas í151, it has
been suggested that the total variability of meeting a water
quality target was more strongly influenced by economic and
political uncertainties than hydrologic uncertainty. This was
based upon a multivariate sensitivity analysis of output measures.
Such an analysis is similar to measurement of the type A error,
which is an apparent error in design caused by incorrect economic
parameters evaluated at those incorrect parameters. The type B

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343
error, or efficiency loss, is the pertinent measure to compare
against efficiency losses from inadequate gaging records. The
type B error is the loss measured with the true economic param-
eters of a system whose desfgn was optimìzed with the incorrect
parameters o
The type A and type B errors resulting from 40% errors in
each of the cost parameters used are shown in table 3.
Table 3. Type A and type B errors resulting from 40% error
in cost coefficients for a reservoir on the Smoky
Hill River near Arnold, Kansas
Cost Coefficient Type A error Type B error
(% of net project benefits)
c1 recreation 3.1 < .1
c2 recreation < .1 < .1
c3 short run deficit .6 < .1
c4 short run release 22 2.5
c5 res e rvoi r
construction
10 < .1
C6 long run release 60 11
For comparison, on this same project the efficiency loSS for
incorrect record lengths of 5, 10, and 15 years is 0.4, 1.4, and
3.7% resPectivelY. Project feasibility is usually quite sensitive
to type A errors and to the interest rate used for discounting
project costs and benefits to a common time name. Type A-and B
errors were not estimated for errors in interest rate, however,
on this project a change in interest rate from 6% to 5% would
change the optimal record length from 30 to about 35 years.
This analysis has taken the manager's viewpoint of a project
gaging network, whose objectives are overall economic efficiency.
Budget constraints are not active. The network manager seldom has
precise information
is then to minimize
years in the future
m
C P(t)L(t--r) is a
ta0
the project will be
on the date of project inception. His problem
the losses by starting gaging at a point
such that the expected loss
minimum, where P(t) is the probability that
started t years in the future, and L(t-T) is
the total loss function as sho& in the example in figure 1.
Water resources development projects also must consider other
criteria than economic efficiency. Considerations of regional
income distribution and the degree of risk aversion in the parties
receiving the economic benefits will temper gaging network
decisions.

344
If the analysis presented herein is to be used to determine
ghe optimal sìarting time for a gaging station at a previously
yngaged site, some estimates of the flow characteristics will
bave to be made. Estimates of flow characteristics prior to gag-
ing can be made for sites in the Unfted States using regional
gelationships presented in Thomas and Benson n61. Additional
gaging information can then be incorporated into the prior ecti-
wgtes using Bayesian analysis.
cgnclusions
Optimal record lengths for determining reservoir sizing and
ekerating rule parameters can be determined by computing expected
project benefits from designs resulting from varying lengths of
design data. Optimal record lengths for the combined process of
reservoir sizing and determining operating rule parameters are
significantly longer than for reservoir sizing alone under a
fixed optimal operating rule.
The optimal length of record increases with size of stream,
ganging from 7 to 47 years for the five str ams used in this study,
9
which range in discharge from .25 to 27.1 m /sec.
Econcmic efficiency requires that the marginal worth of col-
ìecting additional streamflow data be greater than the marginal
costs when these two values are discounted to a common point in
%ime. Hence the decision problem of discountinuing an existing
gage is different than the decision problem for the optimal
gtarting time for the gage because the discount factors applied
gp the initial year are larger than those applied to the final
year.
References
Wiener, Aaron, (1972). The role of water in development,
New York, McGraw-Hill, Inc.
Dawdy, D.R., Kubik, H.E., and Close, E.R. (1970). The value
of streamflow data for project design - a pilot study,
Water Resources Research, v. 6, no. 4, pp. 1045-1050.
MOSS, M.E. (1970). Optimum operating procedure for a river
gaging station established to provide data for design of a
water supply project. Water Resources Research, v. 6, no. 4,
pp. 1051-1061.
Herfindahl, Orris C., (1969). Natural resources information
for economic development, Baltimore!, Johns Hopkins Press.
Tschannerl, G., (1971). Designing reservoirs with short
streamflow recoräs, Water Resources Research, v. 7, no. 4,
pp. 827-833.

6
7.
8.
9
lu.
1 .
IL
13
14
15.
16.
Maass, Arthur, et. al. (1962). The design of water resource
sy~tems, Cambridge, Harvard University Press.
345
aung, G.K. , (1967). Findiiig reservoir 01 crating ruler. Jour.
f the Hydrau1ic.s Div., Am Soc. of Cìvi I Enqr.. v. 93. no.
HY6, pp. 297-
Hall, Warren A., and Howell, David T., (1963 . The uptimiza-
tion of single purpose reservoir design with the appii ition
of dynamic programming to synthetic hydrology samples, Jour.
of Hydrology, v. 1, pp. 355-363.
Russell, C. Bradley, (19121. An optimal policy for operating
d multipurpose reservoir, Operations Research, v :O, no. 6,
pp. 1181-1189.
Gessford, John, and Karlin, Samuel, Optimal policy for hydro-
el-ectric operdiicns, pp. 179-200 in Arrow, Kenneth, J.. Karlin,
Samuel, and Scarf, Herbrrt (1958). Studies in the mathematical
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bablinger, Moshe, and Loucks, Daniel , (1970). Markov models
for flow regulation, JOUI. of the Hydraulics Div., Am. SOC.
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Loucks, D.P., and Falkson, L.M., (1970). A comparison of Some
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Loucks, D.P., (1970). Some comments on linear decision rules
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omputing Machinery, v. 16, n,,. 2, pp. 286-294.
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U
346

O 10 20 30 40 50
OPTIMAL LENGTH OF RECORD
347

ABSTRACT
THE DESIGN OF WATER QUALITY MANAGEMENT PROJECTS
WITH INADEQUATE DATA
George W. Reid
Regents Professor
University of Oklahoma
One of the increasingly important elements in the design of
water resource projects is, of course , the management of quality
and a technology that was almost purely hydrological and hydraulic
is now being expanded to include what might be classed as the
environmental and edological impact areas and systems, So, it is no
longer sufficient to understand the interrelationships, flows and
transports, but to this must be added the impacts on ttbe lisJrng and
nonliving water, and peripherral environments; with a need to
develop ecological models or more specifically, water quality models,
Unfortunately, there is rarely adequate data to properly describe
these interrelationships, The methodology used for hydrological
studies involving inadequate data such as the transfer of okserved
points to points of interest; short term interise studies; &,Y use of
simulation techniques, can and are being used in quality management
modeling, Perhaps more basic is an understanding of data requirements,
using the system approach, the sequence of events ar- il) problem
formulation, (2) symbol1 modeling, (3) data collectlon, (4) analysis
and (5) design. (See Figure 1) Frequently, the order is ,hanged,
particularly the entire process will start with available data.
The complexities, of course, arise due to the fa.t that the
I rocesses associated with water quality management: hydraulic,
hydrologi,al, chemical, biological and ecological -- are extremely
and imperfectly understood. So, that is a complex reality, with a
great many variables on which there is available veri poor measures
and which themselves interrelate in ways very inadequately
understood -- must be measured and appropriately related to be useful,
Certainly, one recognizes the superiority of an expli-it quantifiable
data and models over intuitive models and hurlChes. The
alternatives to such a model, based on partial knowledge, is a mental
model, based on the mixture of incomplete information and intuition
similar to those controlling most political decisions. A mathematical
mcdel deals with the same incomplete information available to an
1 1 *uitive model, but through organization of information from many
iifferent sources into a closed loop at last analyses is permitted
and data needs studied,

350
RESUMEN
Uno de los elementos altamente importantes en el diseño de pro
yectos de recursos de agua es, desde luego, el manejo de la calidad
y tecnologia que fue casi puramente hidrolögica e hidráulica y está
siendo ahora expandida para incluir lo que debe de ser clasificado -
como áreas y sistemas de impacto ambientales y ecológicos. Asi que -
ya no será suficiente entender las interrelaciones, flujos y trans--
portes, pues a éstos deben de ser agregados los impactos en las ---
aguas con y sin presencia de formas de vida y los ambientes perifi--
cos; con la necesidad de desarrollar modelos ecológicos o más especl
ficamente modelos de calidad de agua. Desafortunadamente, rara vez -
existen datos adecuados para describir propiamente estas interrela--
ciones. La metodología usada para estudios hidrológicos incluye in--
formación inadecuada, tales como el cambio de puntos observados a -
puntos de interés; estudios intensos de corto plazo; o uso de técni-
cas de simulación, pueden y han sido usadas en modelos de manejo de
calidad. En el modelado existe siempre una cierta incompatibilidad -
entre puntos de sustancia y generalidad; requerimientos de informa--
ciÓn y la representatibilidad del mundo real. El objetivo desde lue-
go, es proveer por medio de una abstracción idealizada un comporta--
miento aproximado el cual es siempre un compromiso entre simplicidad
y realidad. En años recientes una gran cantidad de modelos han sido
desarrollados, pero, desafortunadamente parece haber un alto grado -
de polarización. En un extremo, hay un elegantisimo y sofisticado mo
delo basado en técnicas econométricas requiriendo un alto grado de -
especificación de información, que en la realidad no existe, Por --
otro lado del espectro, los senarios dependen casi muy poco de info;
mación, más sobre conceptos. La necesidad básica es para modelos en
aìgfin lugar entre los dos extremos que están construidos usando in--
formación existente y que puedan ser responsables a las necesidades
de las agencias de acción, Es en esta realidad en la cual el autor -
ha desarrollado una serie de modelos de calidad de agua. Los proyec-
tos siendo modelados son generalmente de una naturaleza tal que la -
realización final ocurrirá bastante después de la partida de los di-
señadores y por tal los procedimientos de evaluación directa son im-
posibles, necesitándose de alguna forma de evaluación o integridad -
interna. El problema es que usando cuanta información esté disponi--
ble, para 50 a 100 años a la fecha y haciéndolo de manera que no sea
tan elegante que se convierta en un modelo dogmático. El autor ha -
desarrollado una serie de modelos respondiendo al desafio. La esen--
cia de la metodologia es reconocer la complejidad de un problema y -
trazar una combinación de técnicas de investigación de operaciones,
técnicas deterministicas, asi como m’etodos empîricos, fenomológicos
y analiticos. Modelos para sistemas de rios responden a la polución
organizada de cuatro maneras: bioquimica, biodegradable, sedimentos
nutricionales, incluyendo modelos adicionales para flujos urbanos y
poluciiin dispersa, así como flujos rurales. Todos los modelos usaron
información existente y ésto los sitG‘a para modelado pronosticable -
de nivei de las cuencas siendo computarizados y sistematizados y es-
tán siendo usados en problemas específicos en el Suroeste de los Es-
tados Unidos.

Problem Formulation: To arrive at a water resource project design, the number
of variables is enormous, and they are mostly nonlinear. The structure of
the system is more hierarchical than functional, and many of the parameters
and variables are unquantified at present, certainly those associated with
ecology. Nonetheless, to some degree, a merging of disciplines and the
increased use of the system approach has been taking place in the study o€
urban systems, and it is not just a matter of collecting data and figuring
out what one has.
Lf one looks at the type of models being postulated €or the design of
water quality systems today, it will be seen (Figure 2) that they fall
within a spectrum ranging from erudite mathematical models at one end of
the spectrum to scenarios at the other. In the first case, the mathe-
matical models may be rigorously developed in a mathematical sense, but
are all too often of little use in describing a real complex system in
inadequate data. On the other hand, the scenario model - little data,
numerous ideas --may accurately depict the significant elements of the
real system, but it is of little use to the engineer-planner because he
cannot manipulate it. or quantify it.
The target one should try to hit is a reasonable and useable balance
hetween the poles of intuition and selecting hard data. One would like
to be able to use the mathematical rigor of the physical scientist and.
at the same time, give equal weight to the heuristic insight of the social
scientist.
The result would be a useable model for a system design. So,
perhaps, or certainly, for planning purposes, one is dealing with the lowest
level of quantification that allows good estimates and the lowest level of
complexity which gives a reasonable picture of the real world system with
the lope of expounding in both directions.
The dpplication of mathematical modeling techniques to water quality
management can significantly aid the decisionaakers to arrive at better
decisions. Thus, modeling provides relevant facts and alternatives, the
decision-maker chooses the strategy. Operational modele are still prim-
itive, primarily becaiise of the probilistic or random nature of the
physical processes involved in waste diffusion. One is sometimes inclined
to be skeptical of the value of increasing model sophistication which
often seems to have progressed much further than our understanding of the
rmrnplex real world situation; all models currently proposed in the literature
have enormous data requirements which far exceed those data usually available,
and which, for the most part, must be derived from actual measurement.
Many parameters in the more sophisticated models are simply not known in
actual situations.
The water quality management design problem require:
1. The cause and effect relationship between pollution from any
source and the present deteriorated quality of water in the estuary.
2. Forecasting variation of water quality due to the natural and
man-made causes.
3. Methods of optimal management, including treatment and flow
regulation to control the quality in the estuary for muaicipal, industrial,
agricultural, fisheries, recreation and wild life propagation.
4. Chemical, biological, hydrological, hydraulic, at the same the,
same place, and same accuracy.
3 51

352
Models In modeling there is always a certain incompatibility and representativeness
of the real world. The aim, of course, is to provide through
an idealized abstraction an approximate behavior of the system which always
is a compromise between simplicity and reality. Water quality models can be
used to simulate, describe and predict, and programming leading to optimization
of design. Programing which leads to policy requires an explicit
set of objectives, or an objective function to maximize benefits or minimize
costs. Simulation does not require explicit results. So, simulations are
misunderstood, if one expects to use the numerical projections and values.
Using numbers is wrong if it leaves the impression that design projections
are in any way predictions of the future. It is helpful, E as a prediction
but to get one to realize how short-sighted -- how present-oriented - images
of the future ordinarily are, but extrapolltion of present trends is a time-
honored way of looking into the future.
Most people intuitively and
correctly reject extrapolations -- the point is that it provides indications
of the system's behavioral tendencies and as an analysis of current trends,
of their influence on-each other, and of their possible outcomes.
Models may be classified usefully by areal extent into national, regional
and local . At the highest, or national level, data is necessary for broad
planning purposes, such as to determine an overall level of water pollution,
to determine the total investment necessary for pollution abatement, to
determine national policies and to project the problems into the future.
At the second highest level, the regional level, all of the above information
is necessary, plus the particular information needs for the region. The
third, local level, consists usually of checking the operation of waste
treatment plants to insure compliance with regulations and statutes.
Thus,
due to the different requirements and objectives, a data program which may
be optimal at one level, is usually far from optimal at some other level.
Unless a clear objective has been set, there is no guarantee that all
critical bits and bytes of information are collected, and that the gathering
of useless data is minimized. Similar calssification classification can be
made with relation to time.
Hypothetical attempts to describe the intricate relationships between
nutrients, phytoplankton, zooplankton, fish, detritus, bacteria and maninduced
waste loads. There has resulted a great variety of models. One of
the first developed, classical Streeter-Phelps equation, describes adequately
the deoxygenation and reoxygenation in the river. The familiar form of the
oxygen sag equation is:
-
Do -
-
where: D oxygen dificit at time t
-
oxygen deficit at time zero
BOD at time zero
Lo
t = time (distance) in days
deoxygenation coefficient
kl =
k2 = reoxygenation coefficient
This equation has been expanded to provide for evection and diffusion; algae
growth, beuthal deposits, etc., into, inreality, impossible data requirements.
The basic need is for models somewhere between two poles that are built using
existing data and as such can be responsive to the needs of the action agencies.
Tt is in this realm in which the author has developed a series of water quality
models. The projects being modeled generally are of such a nature that the
ultimate realization will occur long after the departure of the designers, and
as such direct validation procedures are impossible, necessitating some form

of internal validation or internal integrity. The problem is one of using
what information is available for a 50-100 year future, and doing it in
such a fashion that it is not so elegant that it becomes a classroom make-
believe world, The essential thread in the author's methodology is that of
recognizing the complexity of a problem and drawing on a combination of OR
techniques, deterministic techniques, as well as imperical, phenomological,
and analytical methods. River system models respond to organized pollution
L,I modes.
There are suggested six categories of stream responses: biodegradable,
nutritional, bacterial, solids, persistant of slowly degradable chemicals
and thermai. The response of a given stream to these categories can be
formulated; or the reverse. given an instream criteria (RQS), allowable
effluent quality can be calculated. The specific criteria now can be
grouped under response headings; for nutritional, one might select N, P,
NIP, or AGP, etc. If primary treatment is established as a lower con-
straint on the effluent, the solids criteria can be deleted; and further,
if a public health constraint on toxic and bacterial levels can be exercised,
fosir rather than six responses can now be used leaving a four-by-four matrix
'o be examined.
TABLE I
Municipal Industrial Agricultural Recreational
Biodegradable
Nutritional
Controlled by D. O. levels
Controlled by N and P levels
Thermal Controlled by Temperature increases
Persistent
Chemical Controlled by Salt, CCE's or ABS, etc.
353
So, a response/use matrix, changing with time will set goals; based on a
matrix such as the one in Table I "d alternative socio-operated projections.
A linking technical basin model can be built and operated to provide the optimal
use of water resources, and of necessary treatments; or in pianning for
ruture population increases and the concomitant increased use of water, it
is possible to build mathematical models depicting the optimum treatments and
stream flows necessary to meet ths RQS. The one-to-one input-output relationships
f-r he four c-tegorius vf waste discharges follows with the Low Flow
Augmentation FA), associated with each treatment level (mi) , will be QL,
QN, Qp and Q,. This is a terminal flow in MO. T'Li is a fraction where i
refers to BOD, N and P.
BIODEGRADABLE MODEL (L)
Y PE or P A (P)
Q, y+ (i-Y) CS - RQsDO
(i) where:
Y = Fraction of total population in SMA's
E = Efficiency term, Point LoadIUniform Load
PE = Population Equivalent Ln millions
P = Percentage discharge to river, expressed

- as a fraction, Decision
Variable (1-TL)
Cs = DO saturation level 6 given temperature
A = 942,900 relates to stream characterk2
4 * istics
where n is essentially the number of reoxygenized volumes, Ir the reaeration
2
constant, L the reach, V the velocity -- these valués will change as the
stream Ltself is subject to management.
ACCELERATED RITROPHICATION MODEL
Z'P (1-%-1.44 (1-5) (TLL3250) (3)
Qp = 2-P (1-TLp) - .27 (1-5) (TLL 1080)
F, ROS
THERMAI. MODEL (T)
Qr =
ATw - C
AT +C
Q
AQ
= Thermal Dilution Required, MGD
where:
Qp or QN = Nutritional Dilution Required, MGD
(4)
Z = Relative portion impounded and
effected by RQS, level
- P = Population, millions
T$ or %
-
Phosphorus or Nitrogen removal level
expressed as a decimal
F or Fp BOD/, Ratio divided by optimum
N
combining ratio
RN = BOD removal level expressed as
a decimal
RQS, or RQS, = Acceptable level, RQS determined
by RQSAGp
A Lw - Allowable temperature difference between added flow and RQSt (t-RQSt)
A TQ =
-
Allowable temperature change (RQST - To)
( Ratio of K/Vx when K Geometric mean for Bowmen's ratio and V =

subsidance velocity
AQ = Waste Flow, MGD
CONSERVED OR PERSISTENT SHEMICAL MODEL (C)
These models, though’cast in terms of dilution requirements, can be
altered, given a diluted level to provide permissible loadings. The
models (2) thru (6) are based on organized (sewered) pollution. Models
for storm drainings or dispersed pollution have also developed such as:
DISPERSED POLLUTION MODEI. (D)
Y2 = 4.8 + 0.0827X2 + 0.489X8 (7)
where Y is BOD
3
Y = 2.36 - 0.188 1nX + .310 Inxl0
5
where Y is ON and Y6 is PO, in
5
Y6 = 2.90 + .OOOOSX1 - .OOOlX, - .0137X8 - .741Xll
and Xi = population
X2 =. population density
X = number of households
3
X8 = comercial establishments
Xl0 = streets
Xll = environmental index
Models (2-9) can be used to relate waste inputs to stream responses under
varying municipal stream characteristics and against varying goals (RQS).
Many technical models are available to project flows (Q), and other stream
characteristics it2, L, V, et.. but a final model is needed for evaluation of
the effects of the rural upstream watershed programa on downstream runoff to
complete the set. Such a model was developed for the Congress in 1969. ls2
For details of model. development see, THE OUTLOOK FOR WATER, Wollman and
Bonem, The John Hopkins Press, Baltimore & London, 1971, Appendix C., p. 203.
This was a special consultative report to the Secretary of the Interior,
October, 1967.
355

356
UPSTREAM USE MODEL (U)
Y=-16+XX
1 3 - u7x2
Where:
Y =i percentage of nonna1 runoff
X1 -i percentage of normal precipitation
X2 * percentage of watersheds controlled by hydraulic structures
Xj = annual above one inch precipitation
In these equation, the simple Phelps equation (1) has been reduced to:
n
Expanding this to
dL = % dO = f dO
ao E 2 E a20
-iP
- kdL - kn L" + ka (Cs -C) -
three dimension, (x, y, z,) would require:
at - xa: ++
EZa2 o
i- -
- kic , etc.
ax ax
(10)
(11)
That is to say, the load equals the capacity. Distribution factors are
added, load is put in terms of people, PE's, etc. This is useable. On the
other hand and by way of contrast, O'Connor uses a one dimensional, differentia1
equation, first involving:
(13)
Also the evaluation of E's, U, Ki, etc. in terms of velocity, solar energy,
depth, turbidity, etc. 3
The effectivehess of models is, of course, acceptance. Actually, very few
models have been used. Limitations of applying them to "real" systems are
rooted in many factors, most related to data inadequacies; the acquisition
of proper data, adjustment of non-homogenity, or inconsistency, to MIU~ a
few.
SYSTEMS ANALYSIS AND WATER QUALITY, Thoman, Environmental Science Service,
New York, 1972.

Every model, or system, is always embedded in a larger system in space
or time, so one is limited to selection of a free body cut and exogenously
determined parameters. Finally, serious factors, mostlv associated with
social values cannot, at present, be quantified.
An efficient use of models thus, argues for different models to answer
different question. For example, one for sediments, one for social costs,
etc. The systems process is iterative and continues while the models are
refined and until satisfactory results are obtained.
The flow of information for all the mested models eventually leads to the
decision process. Forward and feedback information flows take place between
models until the alternative selection and information developed is accepted
for decision-making.
As illustrated, there is no attempt to "hang" all
models together. More important, different levels of data, can be used in
each mode, providing homogenity in each model.
DATA
The data must support the models. Some of the questions for which answers
are needed are, goals, include,:
1. What significant parameters of water quality should be measured,
for an alert system, for treatment plant control, for a quality forecasting
system, for a river management system?
2. What should be the periodicity or time interval in collecting
specific data?
3. What are the cross correlations of these parameters?
4. Are there any synergisLic relationships between the parameters?
5. What is being accomplished to develop instrumentation that can
gage quantitatively those essential parameters, such as BOD, that are not
being measured automatically at the present time?
So, there are all sorts of data, much of it redundant. One needs a model
to discover needs, costs, etc. The process is shown graphically in
Figure 3.
Data has a cost, collection and deferral of decisions.
The quantity of information collected should be increased so long as the
present value of the investment opportunity (or cost savings if this is the
use to which the information is put) is increased by more than the cost of
the information.
The expected value of a decision will be low with little data available, but
will rise with more data available. With little data available. the solution
often would be overstated (resulting in unused capacity) or understated
(resulting in lost opportunity), thus reducing the expected present value of
tht opportunity. For small enough quantities of data, the expected value
will be negative.
The conclusion thatthe decision take place when the cost of getting one more vear
of information is equal to the resulting increase in expected present value.
The cost of getting one more year of data is made up of two elements, the

outlay during the during the year to get the data, k, and interest on the
expected present value of the opportunity one would experience if a year of
waiting is not included. That is, if V(t) is the basic function, one should
not wait until its rate of increase, V’(t), is equal to [rV(t) + LI, where
r is the rate of discount ( the rate of return on investment).
Several conclusions are evident. First, it never will pay to wait for
”complete” information. Second, an extremely important element of the
problem is the cost coming from postponement of the stream of net revenues
from the decision. This factor means it does not pay to accumulate data
until the increment in expected value is equal to the annual cost of the
data.
Experience in the United States has resulted in the common utilization of
only eight water quality parameters that are thought to satisfy the re-
quirements of reliability, accuracy, and low maintenance. These parameters
are dissolved oxygen, pH, turbidity, conductivity, temperature, OñP, solar
radiation intensity and chlorides. Time sequence is important. Parameters
needed today may not be the correct ones torrorrow.
TABLE II
---
TIME SCHEDULE FOR WATER POLLUTION ABATEMENT
Secondaw BOD N&P TDS Thermal
- Time
Treatment Eff Eff Eff - Ef f
1960
1970
X
X X
1980
1990
X
X
X
X
X
X X
2000 X X X X X
Criteria Fish KJlls Eutrophi- Reuse Recycle
Water Treatueur. cation
Problems
Figure 4 suggest a water pollution abatement time scale; that is, the
standard will be upgraded with time, and the resource must be used
within these constraints.
One is still concerned with the frequency with which data should be
collected, the optimum locations of collection, the provisions for data
storage and the resources for analysis of the data. The use of a shortterm
survey approach or establishment of a minimal number of permanent
stateions. An analysis of historical data will yield insight into those
parameters which require continuous analysis because of significant fluctuations
and help to identify those locations which best identify changing
conditions in the receiving water.
In contrast to the monitoring of a simgle point over a long period, studies
can be concenrrated over shorter times but more intensive. There is a
questi01 ,f manual collection versus continuous, automatic recording. All
parameters of interest can be determined on a continuous basis and the results

transmitted to a central storage facility, while water quality parametere
that can be economically and dependently measured in the field are still
somewhat limited.
CONCLUSIONS
Briefly, models to illucidate design parameters should be built with
available data in mind. By a process of separating and nesting, submodels
can overcome inconsistencies. If goals are precisely stated as
to function, various parameters can be represented by what ia available.
The author has developed a series of models using very general data,
leaving a latitude of alternative data.items to define a parameter. Data
has a cost, collection and opportunity or decision errors also cost. If
inadequacies continue, short-term intensive studies are justified, either
now or backward, for example, point reviews can be used. Manual systems
can be replaced by automatic monitors; all eight suggested parameters
handled by electrodes. Generally speaking, however, automatic monitors
tend to provide more data than are needed, because noone dares to turn
these expensive machines off or set the sampling interval to such a time
interval that meaningful deviations can be recorded.
One never has adequate data, nor can one afford to wait for it. So, models
must be made using every device available, recognizing that the final
result will still involve uncertainty and risks, and require judgement -
the only defense against inadequate data.
359

360
SELECTED REFERENCES
1. Biswas, Asit K. PROCEEDINGS, INTERNATIONAL SYMPOSIUM ON MODELLING
TECHNIQUES IN WATER RESOURCES SYSTEMS. Volumes 1 and 2. Ottawa,
Canada: Environment Canada, 1972.
2. Herfindahl, Oris C. NATURAL RESOURCE INFORMATION FOR ECONOMIC
DEVELOPMENT. Baltimore: John Hopkins Press, 1969.
3. Krenkel, Peter A. (ed.). PROCEEDINGS OF THE SPECLAZTY CONFERENCE
ON AUTOMATIC WATER QUALITY MANAÇEMENT IN EUROPE, No. 28.
University, 1971.
Vanderbilt
4. Mancy, Khalil H. (ed.). INSTRUMENTAL ANALYSIS FOR WATER POLLUTION
CONTROL. Ann Arbor, Mich.: Ann Arbor Science Publishers, Inc., 1971.
5. Public Health Service. U. S. Department of Health, Education and
Welfare. SYMPOSIUM ON ENVIRONMENTAL MEASUREMENTS, VALID DATA AND
LOGICAL INTERPRETATION. Cincinnati, Ohio: Public Health Service, 1964.
6. Public Health Service. U. S. Department of Health, Education and
Welfare. SYPPOSILJM ON STREAMFLOW REGULATION FOR QUALITY CONTROL.
Cincinnati, Ohio: Public Health Service, 1965.
7. Thomas, William A. (ed.). INDICATORS OF ENVIRONMENTAL QUALITY.
New York: Plenum Press, 1972.

#
PROBLEM
FORMULATION
DESIRED DATA
V
DATA
COLLECTION
ADEQUATE DATA
I
ANALYSIS
1-SIMUUTION
2-PROGRAMING
- L
V C
r
L
DESIGN
CRITERIA -
Figure 1.
c
t
361

362

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C
al
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ABSTRACT
DESIGNING PROJECTS FOR THE DEVELOPMENT OF GROUND WATER
RESOURCES IN THE ALLUVIAL PLAINS OF NORTHERN INDIA ON
THE BASIS OF INADEQUATE DATA,
BY
B. K, SABHERWAL
Utilization of ground water potential to develop irrigated
agriculture in the alluvial plains of Northern India through
"Push button" water wells has played a vital role to bring about
the Green Revolution for meeting country's food deficit. But the
positive development on the food front is only a phase. Continuing
population growth and the resultant increase in demand for food,
fibre and other services obtaining from water use are adding to the
water requirements thereby underlining the urgency to hasten
execution of projects capable of delivering assured water supply to
meet the demands of high yielding varieties (-HYV] crops, This can
be achieved by installing more water wells in the alluvial plains
of India rich in ground water potential. Ground water resource
though it gets replenished annually, is not an inexhaustible resource,
Ecological responsibility makes it incumbent on the planners of
ground water development projects that this precious resource, IS
not exhausted due to over exploitation, Surface waters are tangible
and their potential can be predicted upto reasonable certainity on
the basis of long term observations of flow in channels. Assessment
of ground water potential on the other hand is quite complicated.
The difficulty arises on account of the fact that ground water
relates to that invisible part of hydrologic cycle which occurs
beneath the land surface. Evaluation of ground water resource to a
high degree of accuracy is a multi discipline study involving,
collection, analysis and synthesis of hydrological, geological,
meteorological, geophysical, hydrochemical data, computing quantums
of recharge, discharge and balance of ground water in a basin or a
sub-basin and correlating the results with the changes in ground
water levels and its regime, A comprehensive study of this type is
time consuming and costly, In view of the latest developments in
ground water hydrology the available hydrological and geological
data is not adequate enough for a comprehensive and precise
assessment of ground water potential though exploitation of ground
water in India commenced quite some time back. On the other hand
preparation and execution of plans and schemes for the exploitation
of ground water cannot be held over till the completion of such a
study which may take four to five years, ît has therefore become
necessary to adopt some reasonably accurate methodology to evaluate
the ground water potential with the help of the available data and
plan ground water exploitation projects on its basis though at the
same time keeping margin for subsequent adjustments when better
data becomes available, Appraisal techniques and adopted criteria
for an approximate evaluation of ground water balance in water table
aquifers are described with particular reference to the Bist Doab
Tract of the State of Punjab-India which has an area of 9000 sq.
kilometers and where 80% of annual rainfall occurs in the months of
July to September. Significant part of the assessment study is the
recharge to ground water from the annual flow of about 1.25 M.A.F.
of; surface water thorough a net work of unlined and lined irrigation
canals and its ultimate spillage in the cropped fields. On the

366
discharge side is the drawal by approximately, 0.1 million existing
shallow and deep water water wells which are either electrically or
diesel driven. The electrically driven wells have unmetered electric
supply, the tarrif being on the basis of horse power of the electric
motor. For both the tupes of water wells log books recording the
number of hours a tubewell operates are not being maintained by the
private owners, This aspect further adds to the problem of working
out accurate drawals from and return seepage to ground water in
tubewell irrigated fields, In the absence of adequate data to
correctly evaluate ground water potential and pressing necessity to
exploit the potential for food production statistìcal or empirical
methods have been adopted to work out ground water balance and then
apply a reasonable safety factor to take care of short comings in the
approach. In the project areas water table fluctuations are also
being observed more frequently to closely watch the effect of
additional draft,
RESUME
L'utilization du potentiel des eaux souterraines pour le
développement de 1"agriculture irriguées dans les plaines alluviales
de l'Inde du Nord au moyen des puits d'eau du button-préssoir a
joué un rôle vital pour accompler la "Revolution Verte'' afin de
satisfaire les besoins deficitaires des aliments du pays. Mais le
développement positif sur le front de nourriture n'est qu'une phase.
La continuation de la croissance de la population et l'augmentation
resultante du besoin de nourriture, tissus, et des autres services
utilizant l'eau necessitent les besoins de l'eau supplementaires,
ainsi soulignant l'urgence de l'ëxecution des projets capables de
l'alimentation fourniture assuT+e d'eau pour subvenir la demande des
récolte de haute z-eqdement. On peut satisfaire cette demande en
installant plus de pults d'eau dans les plaines alluviales de l'Inde
du Nord, riches en potentiel des eaux souterraines. Des ressources
des eaux souterraines quoiqu'elles se remplissent chaque année, n'est
pas une ressource ingpuissable. La responsabilité écologique le rend
obligatoire aux planificateurs des projets des eaux souterraines de
voir que cette ressource prlcieuse ne seppuisse pas, en raison de
sur-éxplóitation. Des eaux de surface sont tangibles et on peut prédire
leur potentiel jusqu'une certitude raisonnable sur la base des obser-
vations à long terme de l'écoulament des eaux dans les canaux. L'esti-
mation du potentiel des eaux souterraines par contre est bien compli-
quee. La difficulté s'8le've en raison du fait que l'eau souterraine
se rapporte à cette partie invisible du cycle hydrologique qui se
fait au-dessous de la surface de la terre. La nature héterogene des
formations géologiques à travers lesquelles l'eau souterrasne circule
rajoute à la complzxitê du problame. YaloTisation des ressources des
eaux souterraines a une haute degr6 d'exactitude est une étude de dis-
ciplines multiples comprenant recuîl, analyse et synthese des données
hydrologiques, géologiques, méteorologiques, géophysiques et hydro-
-chemiques, calculant les quanta de récharge, d@scñarge et le bi'lan
l'eau souterraine dans un bassin ou sous-bassin et mettant en corréla-
tion les résultats avec des changements dans les niveaux d'eau soute-
rraine et son régime. Une étude detaillée de cette type demandes plus
de temps et est coûteause. En vue des plus derniers dêveloppments dans
l'hydrologie de l'eau souterraine la données hydrologiques et géologi-
ques disponibles ne sont pas assez pour une estimation complJte et

367
exacte du potentiel des eaux souterraines, bien que l'exploitation
des eaux souterraines commence il y a quelque temps dans le passé.
D'un autre cote, la préparation et l'dxecution des plans ou schemes
pour l'exploitation des eaux souterraines ne peut pas &tre arrêtées
jucqu'a la complétion d'une telle étude quì puisse prendre, 4 ou 5
ans.Donc, i1 est devenu nécessaire d'adopter une méthodologie
raisonnable exacte pour estimer le potentiel des eaux souterraines
avec l'aide des donnêes dìsponsible et planifier des projets
d'exploitation des eaux souterraines, au même temps en retenant une
marge pour les modificatìons subséquentes quand p+us de données
seront disponlbles. Les technìques d'estìmatïon, pour une valorisa-
tion approximative de balance d'eau souterraine decrite avec une
réference particulière à BIST DOAB tracte Etat de Punjab en Inde qui
a un terrain de 9000 kilometres-carres et ou 80% de pluie annuelle
arrive aux mois de Juillet.Septembre. La partie signìficative d'6tude
estimative concerne la récharge 2 l'eau souterraine de l'écoulement
annual d'environ 1.25 M.A.F. (million acre pieds) d'eau de surface
par un réseau de canaux d'irrigation alignes et non alignes, et son
utilization ultîme dans les champs cultivés, A cbtk de déchargement
1.0 million des existants puits d'eaux qui sont opérées soit par
electricit6 soit par essence. Des puits mechanizes par electricit6
assurent une alimentation d'eau sans compteur d'electrictricite, le
tarrif étant basé sur le C.V. des moteurs eléctrlques, Pour les deux
types de puits d'eaux, des carnets à régle concernant le nombre des
heures qu'un puit S opére, ne sont pas tenus par les propriétaires
prives. Cet aspect ajoute encore au problème de calculs des puise-
ments exacts de l'eau souterraine dans les champs irrigués au moyen
des puits à moteurs électriques, Dans l'absence de données de valo-
riser correctement le potentiel d'eau souterraine & la nécessité
pressante d'exploiter le potentiel pour la production de nourriture
les methodes empiriques et de statistiques ont Btd adoptées pour
retrouver la balance d'eau-souterraine et d'appliquer un facteur
raisonable de sÛréte de bein rendre compte des fautes dans la mainère
d'aborder, Dans les regions sous observation on étudie aussi tres
souvent le niveau de variabilité d'eau pour remarquer de pres
l'effect d'eau puisée en supplement,

368
1. UTROWCTICN
1.1 India i6 the seventh largeet country in tbe
World. ït's area is 328 million hectarest 3-28 million bqtlue
iiiïorn9ters) with a population of 547 million (1971)
Agricultural out put accounts for half of thr country's Gross
23: tioneï droductí GPW.
1.2 In the year 1947 wheu the country was divided,
the major irL-i;jation syskais únd %cod piav~ucLi? ?reas were
lost to rtkistm resuïtiag ILI a deficit of 4 dillions tmms of food grains. India had tbrefcm to iwort ,ILL t;.c's.us
from the major wheat producing countries of the world till
the advent of Green devolution recently brought about by
the'incrrased utilization of country(s surface water resources
for irriyqtion from 93745 priïlion ~1 m (76 million acre ft.)
in IL361 to 222000 milîion CU m ( 186 miliion acYe a.) at
Grosent ?nd tof ground water lli000 Pilllion CU m (I30 million
ecre ft.) üse í~f high yielding variety (W) seePS of
cercnls hke wheat ,rice,wiize,Jawar and Bjra hes Ris0
hastened to a great extent the tremendous increase III
'food cut put. Cevelopmgnt Of rtwarf varieties of wheat made
Possible following the introduction of valuable genetic
material from bxtco 141 1962 has alone increased the production
of thb important cereal from neerly 12 to 23 million tonnes
within a period of about five yeam.
1.3 Eilt the maximm production per unit of any
Particular variety of m d seed is the result of a set of
cultivation practices proper doses o9 ioputs prophylactic
and curative measures to check the atta& OP insects, pests
and disewes end above all adequate irrigation at proper time.
2. INDIA-PH!EICAL AND OTHE=TI FJ3AlWRES.
2.1 Physio raphially Indkais main land can be
divided into six divisfons comprishg of i-
i) the Himslayan mountains
ii) the indo-Gangetio Plains
iii) the Central Hiagi Unda
the Decm Plateau
the Eastern Coastal Belt
vi) the Western Coastal Belt
2.2 The Himla moimtains are of comparatively
recent origin. The Deccan Eteau end the CentraR Hi@ Lande
are composed of ancient rocks. The Plains are hilt up
of layers of sends, clays of molo loally very recent &te.
The metern anditestem Coastal befts comprise of deltaic
and sedimentary marine deposits.
2.3
About 7of of the country's ama is under lain by
hard rock with a thin soil cotrer at top derived fra l%o
wealtherinn of rocks. IO 1i, mainly the Indo-Gan etic Phkis and
the two deltaic Eastern and Westem Coastal dts which are
made up of alluvial solls and sedfmentwy deposits varying in
thickness from a few hundred feet ln the coastal belts to
thousands of feet in the Plains.
2.4 Ailuviai soils are suitable for agricultum
and respond well to artificial irrigation. Being generally
permeable in character and having laysrs of coarser deposits
also provide under ground storage for seepage water. NO
wonder the Indo-GangetLe Plains, 8nd the tu0 coestal belts
though accomt for on1 l./3 of the oauitry'e laad rnam ?ut
suwort atmut of tL caintrps pornlation.
2.6
The maor snow fed riveris of tu country naaiely
the triaitaries of the U&s, the cianges and the Bwhaii Putra
flow rlugyishly through the indo-Gangetic Blain. The main rivers

369
flowing to the coastal belts are the Xarkda and the Tapti on
the western side Cmd the Maha Nadi, the Godavari, the Krishana
rind the Cavery on the eastern sir%. All these rivers outfall
into sea. The rfvers also provide irririation supklies to the
vast net work of m al systems part of which was constructed
about a century back. host of the old canals are designed as
Il mn of the riveril schemes and are unlined. The unlined cana.ls
act es additicnal souY'ce of recharge to ground water besides
seepage from rivers, streams and rainfall.
2.6 In dia s clima. te ranges from con t in en ta. 1 to
oceanic, from extrems of heat to extrerns of cold, Prom high
a.ridity a.nd negligible rainfall to excessive humidity and
torrential rainfall. Sauth destemi monsoons in summer accounts
for mre than 85% of the precipitation and that too in a
short span of about 4 months. The great diversity in weather
conditions and uncertainity of rainfall results in the prevalence
of draught condition in about one third of the country.
3. GROW D :,! AT-g 2 17s
3.1 In the face of variability and tirircliability of
rsinfall and also lack of' adequate storage support for some Of
the major canal irrigation schemes, tappin:? of zround water
resource through iiells and tubewells for intensive agriciiltu re
has pla ed a yitzl role in ushering the (Green Revolution
particu Y a.rly in these parts of the country where low or badly
distributed rainfall is quickly lost thrm$l evaporation Ixit
where g-ound watnr potential is available stored in alluvial
deposits
3. 3 Cultive.tion of high yielding varieties and
intmsive cropping dernad water at the right time and of the
rewired quantity. These pre-requisits have made the
cultivators in areas with copious ground watcr supplies take
to the instcllatïon of their own', push aittontt irrigation
systems. The water scarcity during the yesr 1365-67 which
created draught conditions almost all over the country acted
as catalyist to boost up exploitation of ground water poteiitfal
thmu& diesel or electrically operated tubwells 100 feet
to 200fbet dealfor the protection of Crops.
3.3 The Govemrcent also rose to the occassion and
undertook to provide large saale loan finance to the cultivstors.
for the installation of tubewells on their farms in areas where
the ground watcr potentialities were promising. The result is
that at present an investment of about Rs.2000 crores hac Filreacbr
bean Lade in the field of ground water exploitation in the
country. Most of this investment has taken place in private
sector.
3.4 The following table, indicates the progress Of
insxellation of tubwells in the a3untry:-
(In thousands I
250 - 1965 A969 - 1971 -.- (anticipated)
Mo.of private tubewells 3 100 279 470
rio.of diesel pumps 66 471 837 1150
No.of electric pump sets 19 513 1080 1620
Total 88 1084 2196 3240

370
3.5 The spectacular development of ground water
utilization in the country has been influenced by a npmber of
factors namely the rzcognition by farmers of the importmt role
played by ground water in sustaining modern agricultural
techniques, incrceaed availability of institutional credit for
financing the ground water exploitation programme, rapid
electrification Of rural areas, local availebility of technical
how how to drill well8 with machines, and indiyenously
manufactured pumps, motors and other equipmat for the
construction of wells and above all large scale village road
deve lopmen t p rogr amme .
3.6
Heavy investments in gound water exploitati.
schemes and the involvement of the Government back institufional
credit far the purpose has made it incunibent to plan and execute
this programme of utmost natio'lal imoortance duly sumorted by
proper assessment of ground water potential.
4. HYDROLOCEIC CYCB.
4.1 All the waters in existance en be located by
what is ïmìwn as 1) hydrologic cyclen or 11 Nater Cycle". This
C cle involves total earth system comprising of the atmosphere,
d e hydro-sphere Snd the lithosphere. The activities of the
n ilater Cycle" are vast extending from an average depth of about
half a mile in the lithosphere to abcut 10 miles in the
a tmsphere .
4.2 Hydrologic cycle is greatly influenced by the
geologic history of a particular area. If the geology consists
of alluvial fmnatlons, water will occur in the openings
between granular XcFosits; EUT; if the area fOr~tiOnS are rocky,
the ground water Will be found in decomposed parts of ro&B,
freotures or in tabular openings in soluable rocks or opening
in lava formed by flow or gas expansion during solidification.
Guide lines to evaluate ground water potential in alluvial
formatias have only been discussed in this p3Per.
4.3 Ground water originates from surface water and
gets renewed or recharged with the down vard percolation Of
precipitation, flow in stream, canale, return flow fra irri,ated fields etc. Propm assessment of this valuable
resource fomd in Permeable geQlOgac formations and in motim
through the voids or pore spaces in an area requires working
out its total storage and quantities whir& are annually pumped
out or replenished into the ground water reservoir. bality Of
grOUnd water .leo requires to be known. Comprehensive studies
and explorati-ns -re necessary to evaluate the potential to a
hfgh degree of accuracy.
5. APPRbACH TO WORK OUT GRWdD WATER BULLANCE;
ON TI% &.SIS OF INADECrJATE DATA.
5.1 hthodology for the precise eva2natioB of ground
vater potential is quite complicated. The difficulty arises 691
account of the fact that ground water relates to the.t invisible

371
pert of hydrologic cycle which occurs 'beneath the land surface.
9etero~~eneoUs nature of the ,(.f?ological format ions through which
ground water moves e-dds to the coarplexity of the problem.
5.2 It ha.s been observed by pump tests that in Punjab
which is the Northern-Western part of the Indo-Gangetic Plain
alluvial materials constitute an extensive hetrogeneous and
a.nisotropic unconfined aquifers . Discharge from tubwells as
deep a$ 300' results in- draw-down of water tsóle over larye
prea. ?nd is sustained by dawrltering of surface watr,r recharne,
such condit iow jrevail through out the top aauffers ~f slluaiurn.
5.3 Planning and designing of ground vmte? development
through small and medium sized tubewells (1.10 to 200 feet
deep) in the ground water &sins and sub-basins o0 the indo-
Gmjetic plain ' cím therefore be ,compared to re:.err)ir prObLem.
This approach cal-1s for drawing iipm the fresh water table
a,quifem upto the Safe Yield which should not. exceed ^che long
term mean -annual supply or recherge involving wet and dry years.
In view of lack of complete data the genersl. i"om CJf bhe squrtie.cn
of hydrologic equilibrium in thcs project areas has been simplified
cmd suitably adjusted to arrive at worh.ble ,g:rourid ;iater hiance.
In areas having ground water quality problem Safe Yield cannot
be equated to mean annual recharger
5.4 Installation af tubewci-11s upto 300 feet for
irriz8tion is being practised in India since 19.34-36. In
edditìon, tubwells vere a.lso install.ed Por municlpal,rai%ays
Tnd indust, ia1 use. Geological Survey of India, state Zrri%ticn
&partmats and Central Gróund -.:a.ter Board have bom iminte.in1ng
oeological, hydrological, geochemical and other ground water
data of a rudimentary character. Irrigation Departnlsnts have
also meintahed record of water table fluctuations keduced to
mean sea level ( 1%.Lr) from a net work of observa.tion wells.
kmccipitatScn, racord is kept by Indian bieteorological Department.
Ifit no are?wlse systematic investi.zations and exploration to
a-ssess ground water potential were conàucted. In the absence
of adequate ùata to evaluate ground water potlontkal on the basis
of lat$st deVelOPI~ent6 in grmd water hydrology and pressing
necessity to exploit ground water potential statistica.1,
analytical and empiriel wtbods were resorted to arrive at
preliminary quantitative evaluatîon of ground water balances in
the pro,je ct areas
5.5 Ground water balance te Plan schemes was
computed on the collp,ction and malysis of the following basic
data in project areast-
1. Village -wise iocatim5 and other details of existing
tu heiaelle
2. Colleetion of reliable litholo- of tubewer-ils.
3. Iso-pstch featums E@ revealed by litho-lons and
geologid correlation of strata upto the available
depths to broadly understand t.he geometry of aquifers.

372
4.
5.
6.
7.
8.
il.
10.
11 0
12.
13.
,Sample observatiais of pumping rates o? existing tubewells
by ushg simple devices(0rifice or ri,qht Angled '6-notch) .
dimple surveys to assess pumping hours for surn'lier and
winter crops to work out present ground water draft
in the Gi'oj ect .-;.rea.
Locatim of raingaue stations m d annual ra.infel1 datn.
for the last 20 to 30 years. '
::eighted mean average annual rainfall fill;ures for
different blocks of the area by Theisson method.
Locations of existing pugln?JdischPrp sites on stresms,
drains and CO! lectfon of run off data monthwise.
Available ground water qiality data to demrcate fresh
and inferior ground water zones.
Location of existing cmel irriyation s,Ftem m d data
about their len$.hs ,. sectims,( lined/unlined) desiled
dischzcrges, actual flow time and areawise.
Yeriod-wise flow in enals Rt the point OP entry into
and exit from the project area.
Locations of existkg water table observation wells
and past data of tiatcr table fluctustbas for pre
and post monsoon periods
hater table depth data to delineate high water table
areas . Cropping pet tern, cropkning cslander and water
14
requirements for summer and winter crops.
15 . :,orking out zvera- value of specific yield of the
formations, either by pump tests or empirically.
6. TJXH1vICk.L CRITl3RI.A FOF GROTD ?dUKE PLZ./L".JJCI:
C0PîJT;RTIDN.
Technical criteria adopted to work out ground
water balance is given as under:-
6.1 Rechars oroni R&infall
Unconfined %quifers get recharwd from local
rainfall. Based on the sQiilles ccnducted in the Ganges &sin
for period 1937-78 to 19Sû-51 a relationship vas evolved to
WO& out net penetmtim of rain water to water teble in
alluvial areas
2/5
Rp t 2*O(R-15)'
'Ihere R average annual rainfall Zn inches.
Rp = annial rainfall penetration to water
table in inches.
ïhis relationship applies to areas having
annual rainfall in excess of 15".
6.2 beemge from Canals
seepage loss values from unlined canals based on

373
experimental data are given below. These figures are inclusive
of evaporation losses which form only a small proportion:-
i) In the :tete of Uttar Brzdech-ïndia it is about 8 Cs.
(GU ft/:iec.jfor ordínary clay loam to about 16 cs.for
sandy loam Per million 5;q.feet of wcttad parameter.
i,verage king 10 CS ./k.qt .
li> In the Wz1iarashtr.a State-India these sire calculated at
15 Cs./M sat. for discharse upto 250 CS. anil lO.&s!./M%ft
for hiisher discliarzes.
iïi) In FunJab and Haryana state -India =lue, of ..eepa

374
alopes of the $round water table in the $lains the quantity of
subterrainem flow g ~tthg $.h and out of project are- is
negligible 2s conipared to vertical recharge from rainfall,
cmal PB~PU~L, return flow from irrigated fields etc. Henthis
item has been omitted from computatiuns on both sides
of hydraulic equation amialy due to' the non-availability of
adequate data.
6 r? Ground wqtrr loss due to non bene3tçkl
etrapo-tranepiration in water logged areas.
EvagotreEspiration losses ars related to depth
of water table froin th@ ground surface 81.14 vegetatkq cover.
Airing July to October period when recharge to grow-d water
reservoir is maximum Qe to rainfall and high supplies in rivers
and canals etc. water table rises towards ground surface. In
the riverain md water Jogged tracts, the depth of Found water
veries between zero to 6 feet below lanu surface. in 1963,
'J.S. B.RI conducted experiments/ obervctiuns for salvaging
ground water t6i?g evawqsteü from ground LI' transpired non
beqeficially by vegetation In the central part of san-his Valley,
Ceil'tral Coloredo ( U.S+) t Graphs were plotted, correlating E.T.
loss to ground water depth E.T.loss th negligible if water table
is lowered to 12.5k
Persistance of higher water table 5r water
logged amas indicates that recharge to ground waster is equivalent
to the E.T.loss or may be $ven morg. Pending detailed studies,
it would be reasonably pod planning to draft grouad water
within the limits of water est9rnated to be lost through evapotranspirat
ion.
6 08 Draft froor the exhthg state (deep) and
Arivate (shallow ) tubewells.
The State tubewells ( 1.5 to 2r0 CS. cawcity)
ere planaad to operate at 22 hours a day for 240 days in a y6arr
Henœ dmal draft from state Tubewells varies frm 660 acre ft.
to 880 acre ft. Shallow tubeiwslls (0.2 to 0.8 cs.cepacity) for
abut 800 to i000 hours per year. Draft fra these wells cm b
taken as 15 acre ft, to acre e, per well per year. por wells
driven by animal power dralpt Is taken W 5 acre ft. while for
drinking sugply wells in villages the draft 1 acre ft. has been
adopted per mimam.
6s 9 SpaqIpg q$ shallo~ tubewells, '
Closdl spaaed tubewells muse mutual hydlaulic
interfpence due to ovQr fapphg 09 their ocrnes of depression.
In ~ thi&y populated &we&9 of the IndolGangetic Plph@ Qr
abry Basin ?mm ho&&hgs are very small < abcut 10 to 15 acres
per head ) . The tuWells aro of 0.2 to 0.3 cubic feet/ $eco
dfaeharge and 100'-190' deep. These wells do not run more than
10 to 15% of the time in a year. After tests results minimm
epacinp of such wells has been kept at abut 600 feet 6

375
6.10 Grotmd WatEr Ealmce
Computing the item of ,annual recharge and
discharge as por criteria discussed above if a balance is
struck 5 first estimation of the balance of T2cbyze potcntntlal
in a Project crea beccmes laown for planning further explcit-ticq.
A reasotinble factor of safety can be adopted to plen exploitetim
of the comnuted ground water ImiBncs which tcakes care of the
saps in the available data or the appraisal approech. The fa.ctor
will depend on area conditicns.
7. GHOTjND b~dkii BLLA.?JiCG IN B145T DCAB 'I'R.'ICT
7.1 Bict Doab is a triangu1a.r part of the Punjab
StF.te ( India ) enclosed the rivers Sutlej and the Beas on
t?:c ciäes and aivalu hi1 9 s (lover iïj.ml8yan ranyes) on thc
third side. Three districts of the state ncmely Jullundur ,
doshiarmr and Kapurthala are located in th;c tract .
7.2 The area of the trect is 9900 Fq. Kilometers
mostly mugrising of alluvial plpin except the 8 miles wide
belt of Shivalik Hills on the XGrth-eestorri ,?id&. depth of
alluvium in the plain as revealed by seismic surveys is thousends
feet.
7.3 Gmeral water table is abut 20 to c% feet in the
plains. Avcrzge ?.nnual rainfall in hilly region 1s 1200 mms
tJhils in the plains it varies between 914 mm tc 635 mm. 80% of
the rainfpll occurs during the monsoon period.
7 04 The soils are fertile and the ??SB 112s Copious
gromnd water supplies. There are abcut 0.2 million irriqatiw
tubwells clnà dugwells in the area. Quality OP grmnd water is
qood for cultivation.
7.5 The tract is also irri,gated tl-iraigh Bist mab
eanal which draws its supplies fYom the barrage on tho river
Lutkj at Rupar. Abut 1.25 M.A.F. of water is used annually
for cultiva.i;ion. The net work of canal system Measures 7,54.28
Lilcirieters out of which 34.5i) Km. is lined. mcthcr feature
of tile; area is ;iutiierous hilly utrea=( clioec) which descend
from the :,hivz~lik hills and fiord only &i.ing monsoon period
w ith flashy uischw .;es
7.6 The recharge and discharge computntions ta
work out the ground Kater bzlance in the m ea dis+;rictwise are
talxilated 88 per stateuats i & 11. .A safety factcr 0.60 has
been cùopted to the computed figures to arrive a.t the exploitable
grourd water stential. The water ttible f1uctuatir:ns end the
rainfall hi the area 8i-e being closely observed to watch the
strezses m d strains covered by ground water exb>lcitat*on on
the shallow unconfined aquifers under water table conCAtions
8. ca4crusCáu
8.1, Demographic trends indicete that the Lndia's
populatun is likely to inn,ease to 700 millions by the end of
the present dea&. On the kcis of the piiojecte8 growth rate

376
cf 1.45 per tbausand per anliuam during 1981-85 the population
wmld rise to 300.millions in the year ZOO0 A.D. i.e. about
S5$ increase over the 1371 population. Keeping in view the
expected improvement in the standard of living of the people
during the intervening period, the food and fibre requirements
will increase by abwt 100% of 1371 production. Suck an
enormous increase in the production is possible through intensive
agriculture and bringing additional areas under irrigation by
optima utilization of the water resources (Surface and qround )
8.2 Tnis will eventually result in intensified drawels
of ground waters from shallow as well as deep aquifers. Ground
water resource though it gets replenished annually, is not an
inexhaustible resource. Ecological respansi bility mkes it
incumbent on the planners of ground water development projects
that this precious resource: is not exhausted due to over
exploitatbn arid is so utilized that it also remakis ava.ilable
ïor the yencrstion to come. Therefore aremise potential of
ground wa.ter anà its safe yield both from shallower and deep
aquifers neads to te assessed as accurately as possible to
prepa1.e realistic exploitat ion plms and schemes. This aspect
has ben duly recognized and separate state level orgpnizations
cmpï-is ing of hydrologists, hydrometeorologist, geologist,
agronomist, geoph scist and drilling engineers have been set up
to carry out deta s led investigationa. These detailed studies
will however take time.
However to maIntaln the continuity of
N grow more food 1) compaign exploitation of ground water
recharge &lance may be planned on the %sis of Safe Yield worked
out with the approximations and applicatmn of safety factors
suited to each project area.
1.
mmmv CES
Report of the Irrigation Commissian, 1972, Volume-I,
Ministry of Irrigation and irower, New Delhi.
2.
3.
Krishnm, M.S.,m Geology of India and Emma
Tolm, C.J.,l) GroundWater " .
4, Ehattacharya, R.P. 1) Ground Water supplice, depletion of
water table and penetration of rain water to ground water
table in Western Uttar Pradesh ( India )IIo
5. U.S.G.S. Water-supply paper, 1608-G Anplycis of Aquîfcr
Tests in the Punjab Region of West Pakistan
6. tJ.s.';.S. :uater supkly papc'r, 1608-G '1 Ground Water
Hydrciogy of the Punjab, West Pakistan 1~1th '=mphasiS
of ?rcblems caused by Canal Irriggtion II .

7.
8.
9.
10 o
11 b"
12
33.
14
15 o.
377

37 8
ITEMS
-
ISWlhRpI
-
1.410
.7ss
b.30
-
D.310
0.oy
0.27
0.06
&IO<
o .ai
ODs'
o .41
o .Ia!
STATEMENT I

A DRAFT
CROSS DRAF T(1+2+:
1 41
II 9
6SC
7 S2
96
2s
. Il
379
0.309
o. 132

380
S E\SM\C L \ NE’S
RE F LE CTI ON
REFRACTI ON
h
8-
TEST WELL LOCATION 8
CONTOUR INTERVAL 02KM
DATUM M.s L
hLLUV\kL PU\H5 O
?LRYLbRY (SM\\uhL\KS)

MAP 15
381

ABSTRACT
IMPROVED TECHNIQUES FOR WATER RESOURCE SYSTEMS DESIGN
J R SEXTON
D G JAMIESON
WATER RESOURCES BOARD, READING, ENGLAND
Flow data inadequacy can take different forms. One extreme is
the complete lack of any information but the more usual case is
insufficient length of record since very long sequences of flow
data are required to evaluate the yield and reliability of water-
-resource systems with confidence. Using traditional concepts of
failure and reliability, all water-resource systems are being
designed on inadequate data with only the degree of inadequay
varying between schemes, The use of simulation as a design technique
has necessitated a more rigorous definition of reliability which
accepts the lack of data yet maintains a means of comparing the
reliability of different schemes both in terms of frequency and
magnitude of failure, A new definition of reservoir reliability
has been used for the hydrological design of the Wash Estuary
Storage, a proposed series of pumped-storage reservoirs in south-east
England.
RESUMEN
La insuficiencia de datos de flujo puede tomar formas distin-
tas. Ocurre el caso extremo de la falta total de información, pero
lo más usual es la duración insuficiente de registro puesto que se
necesitan cantidades inordenadas de datos de flujo para que se eva-
IÚen confianza la eficacia de sistemas de recursos hidráulicos. Em-
pleando conceptos tradicionales del fracaso y de la eficacia, todas
las instalaciones de recursos de agua se han concebido con datos de
flujo inadecuados, con grado de insuficiencia como sola variación
entre ellas. El uso de simulacibn como modo de diseñar sistemas com-
plejos de recursos de agua exige definición más riguroso de eficacia
que mientras acepta la falta de datos de flujo mantiene sin embargo
un medio de comparar la eficacia de un proyeqto con otro y en térmi-
nos de su frecuencia de ella y en grado de su fracaso. Un concepto
de esos -la frecuencia de poTcentaje cumulativo- se ha empleado en
el disefio hidrológico Ifel depósito del estuario del Washff, serie de
depÖs*itos de reserva a bomba en el sudeste de Inglaterra,

384
INTRODUCTION
The analysis and study of water resource systems can be conveniently
subdivided into three stages, planning, design and operational.
Each stage has its own specific flow data requirements and what maJr be
adequate for one stage could well be inadequate for another.
planning stage, a large number of possible combin&t&ons of sources are
evaluated but not in detail: the requirement for hydrological data is
minbal, since the yields of individual sources need only be determined
approximately. The most promising combinations of sources are subsequently
examined in considerably more detail at the design stage.
This stage is concerned with aspects such as frequency, probability and
reliability all of which make considerable demands in terms of data
quantity and quality. The requirement is for long period of flow
records which may have a time increment of a day or more.
At the
In the oper-
ational staze, the data requirement emphasis changes from long-term
flow records to shorter but more detailed flow records perhaps even on
an hourly basis.
This paper is concerned with the relationship between the assessment
of reliability, the definition of failure and flow data inadequacy
at the design stage. Flow data can be inadequate in many ways: it may
be that there is no data or Rot enough data, or the wrong data has been
collected. Data can be of inadequate quality or have too coarse a time
increment between successive values. To sunmiarise, inadequate data is
an occupational hazard to all those involved in the hydrological design
of water-resource systems. However, with traditional concepts of
reliability and what constitutes a failure, the problem of flow data
inadequacy will remain for a very long time.
In the planning of water resources for England and Wales, many
diverse types of sources such as pumped-storage reservoirs, multipurpose
reservoirs, rivers, aquifers and estuarial storage are being
considered. &ch proposed source is n? longer considered in isolation
hut as part of a much larger water-resource system.
stances the individual yield of the proposed source loses importance
since it is the yield of the system as a whole that requires evaluation.
The increase in the scale of the problem caused by consideration
of a water-resource system as a whole has outdated many of the traditional
techniques for analysing the performance of a resernoir: some of
the implicit assumptions have been made invalie by the complexity of .
modern water-resource systems, other assumptions have never been valid.
mHOD OF ANALYSIS
In these circum-
owing to the complexity of the water-resource systems currently
envisaged and the lack of theoretical techniques cspable of analysing
such systems, simulation is considered to be the only viable method of
analysis. A simulation model of a proposed water-resource system can be
constructed by joining appropriate component models of particular types

385
of reservoirs iri an ofder corresponding to the physical system.
Examples are given in Fi$ures 1 and 2 of component models for a pumpedstorqe
reservoir and a pumped aquifer. It should be appreciated that
not all the links indicated in these models need be included since in
the specific application some can be set to zero.
The simulation is structured in a general form with physical constraints
such as the capacity of the reservoir, maximum pumping capacity,
minimum residual flows in rivers etc treated as input vaxiables. The
model can then be used to find the frequency with which the system fails
to meet the specified demands and the sensitivity to changes in any of
these or other input variables in terms of frequency of failing to meet
specified iieiads. The relative importarice of each data ita! xc thus
be determined and the effect of data inadequxy can be qusntirird i?i
terms of confidence limits on the resulting reliable yield. LÅoFeovcr,
since the w a ~ in which a water-resource system is managed will dfect
the reliability of the system, different operating rules can be compared
and evaluated.
Since the design of the system is concerned with rare events,
large amounts of historic or synthetic flow data have to be routed
through the models. Consequently the component models have to be relatively
simple to keep camputing costs down and therefore they are essentially
accounting procedures with lags and attenuation built in.
Given adequate data it is possible to include both conservative and
degradable water quality parameters in the model. The build up of
pollutants in various parts of the system can be monitored in the seme
w a ~ as the quantity of water and the performance of the system can be
depicted as histograms of both quantity and quality of water (Figure 3).
In this WEIJ the interactions between water quality and quantity can be
investigated.
BPPLICBTION
The techniques described [i) are being used in the hydrological
evaluation of the Wash Storage, a pumped-reservoir scheme in the estusry
of the Great Ouse, a river in south-east England (Figure 4). The preliminary
estimate for the total capital cost of the scñeme is
2140 O00 O00 at 1971 prices. The work outlined here forms a small part
of the e2 900 O00 feasibility study though much of it will have application
even if estuary storage is rejected. A schematic diagram of the
whole system is given in Figue 5 with symbols defined in Table 1. The
complexity of the complete system has necessitated the division into
three interlinked subsystem namely, the Welland and Nene, the Great
Ouse and the Wash Storage. The first two subsystems define the potential
input to the third.

3 86
The Welland and Nene subsystem which comprise8 the right hand portion
of Figure 5 is a model of a pumped-storage reservoir, %pingham (now under
construction), in conjunction with a confined aquifer, the Lincolnshire
Limestone.
Water will be pumped into lbpingham from both the River
Welland and River Bene when the flows axe in excess of specified minimum
values. Rnpingham can be used for a variety of purposes including meeting
direct-supply requirements as well as regulating the lower Welland to
enable it to support downstream abstraction. Some of the water from
hpinghm will be returned to the Nene aa effluent, upstream of the
intake pumps for Ehpingham.
Water from Rnpinghem will also be used to
maintain the flow in the River Glen. The Lincolnshire Limestone is used
mainly for direct-supply in conjunction with abstractions from the Welland
but any spillage from the aquifer helps to maintain the flow in the Glen.
The possibility of artificially recharging the aquifer from the lower
Velland has been included.
The Great Ouse subsystem comprises the left hand and upper centre
portions of Figure 5. The model is a simulation of an existing pumpedstorage
reservoir, Grafham Water, in association with an unconfined
aquifer, the Great Ouse Chalk. Grafham Water is replenished by pumping
water from two points on the Bedford Ouse, a tributary of the Great Ouse.
Agairi, there is an element of recirculation since some of the water
supplied direct to a demand centre is returned as effluent upstream of
the reservoir's intake pumps. The Great Ouse Chalk aquifer has been
modelled as six interlinked unconfined aquifers. In a scheme shortly to
be promoted all the sub-aquifers are to be used for direct-supply and
river regulation. Obviously pumping water from an unconfined aquifer
will affect the natural outflow from the aquifer to the tributary.
Poreover, if the aquifer is drawn down, the possibility of seepage
through the bed of the tributary exists. Both these effects have been
incorporated in the model.
The lower centre portion of Figure 5 is a schematic representation
of the proposed first two stages of the Wash Storage which comprises the
third subsystem. Water could be pumped from both the Great Ouse and the
lower Nene.
The possibility of having sea-water recirculation schemes
on both the Great Ouse and lower Nene has been included. This enables the
low-flow constraint at the tidal limit of each river to be zero.
Sgnthetic flow data generation techniques [27 have been used for
this invastigation. Currently the historic flow record on the River
Nene has been used as the master series and all other subsidiary flow
sequences have been obtained by regreseion on the logarithmic values of
flow. hproved multisite daily data generation techniques are being
developed under contract o] and will be used when available. Prior to
being used as the master series, the Nene record was corrected for all
upstream abstractions and effluent returns to obtain the 'natural' flow
series.

INADEQUACY OF FLOiV DATA
387
A simulation model such as that used in the hydro1oglc.d design
of the T.3h Stor2.p rcyui.:>e., a coneiderable amount of information as
input data. It is inevitable that some of this data will be inade-
quate in one form or another. The usual case is where some inîorma-
tion is available but in insufficient quantity to estimate input
paxmeters reliably, m d for some parts of the system there is a
complete absence o€ data. To amplify these problems specific ex-
amples which have been encountered in the bdrological desi,m of the
Wash Storage axe given together with the way in which they have been
partly overcone.
INADESUACY DUE TO HU DATA
in modelling an unconfined aquifer such as the Great Ouse Chalk
it is evident that when punping the aquifer for either water-supply
or river regulation, the natural outflow from the aquifer to the
river will decrease. However, pumping the aquifer will have no
effect on the run-off from the non-aquifer portion of the catchment.
It is the combination of these two flow components that is measured
by the downstream gauging station. In short, if the aquifer is to
be developed by pumping, it is neaessaxy to have two inputs, the
recharge to the aquifer and the run-off from the remaining portion
of the catchment when only one measurement of the combined effect is
available. No details on the natural recharge of the aquifer were
known.
The aasumption was made that the downstream flow comprised two
flow recimes, a slow response from the aquifer itself and a fast
response from the remainder of the catchment. Having separated oyt
the base flow component, the overall 'proportion of base flow to
surface flow for the period of historic record was ascertained. The
surface flow component alone was cross-correilated with the corresponding
historic flow data for the master station on the River Nene. The
cross-correlation was performed on the logarithmic flow values which
gives weighting to the low flows and avoids the difficulty caused by
zero flows. This'relationship was then used to generate the surface
flow component direct. The base flow component could not be treated
in a similar manner since this was a measure of the output from the
aquifer rather than the input.
It was assumed that the temporal distribution of the surface
flow component was indicative of the periods when natural recharge
occurred. Therefore the surface flow component was scaled by the
overall ratio of base flow to surface flor and used as input to the

388
recharge process. This data stream was attenuated by an exponential
delay function to simulate porous-media flow prior to adding the
percolate to the water already in storage. The delay induced by this
process was made equal to the observed mean delay between rainfall
ind the resulting maximum well levels.
The aquifer above the threshold constraint defining when channel
loss occurred, was modelled as a single linear storage. Consequently
the natural outflow from the aquifer to the river is proportional to
the mount of water in storage, the storage coefficient being derived
from the base flow recession. In this W¿QJ the effect of pumping the
aquifer was to reduce the amount of water in storage thereby reducing
the natural outflow from the aquifer without interfering with the
surf ace flow component.
INADEQUACY DUE TO INJCOIJPLEZ'E MTA
Although a historic flow record was available close to the proposed
abstraction point on the Ely Ouse, it would have been of little
use for the hydrological design of the Wash Storage even if it had
'been an accurate flow record. The river acts as a source of supply to
both industrial and agricultural consumers as well ag a disposal
system for treated effluents. No detailed records have been kept of
abstractions or returns and consequently the record can not be adjusted
to obtain natural flows. Ideally it would have been fax simpler to
have used the natural flow record at this station and account for the
net changes as time progressed rather than to have to construct a
simulation model of the entire river basin. In this specific case,
however, development of the chalk aquifer necessitated a simulation of
the entire basin. Fortunately better quality flow records existed on
ail of the important tributaries which were all upstream of the.major
industrial and agricultural demands.
INADXQUACY DUX TO INSUFFICIENT DATA
Traditionally the criterion for assessing the reliability of a
reservoir system has been the mean recurrence interval between failures.
This concept of return period has generally been defined quantitatively
in one of two ways, namely, a once in T year event where T is typically
50 or 100 yeam or in terms of probability where it is said that there
is a 100 per cent chance of failure occurring in any one year. Assum-
T
ing that reservoir failures axe rare events and that the time between
failures has an exponential distribution, these two definitions are

equivalent and the probability of m failures within n years is given
by :
-0
phn> = e- Mrn
n!
consequently there is a 37 per cent chance of there not being a once
in T year event in any T year period of record.
Even in the recent past attempts have been made to isolate low
flow events with return periods of 50 or 100 years from a short
period of historic flow data. The usual lengths of these records
typically range from 20 to 50 years. These lengths of record axe
totally inadequate for isolating such rare events and consequently
very little codidence can be placed in the results obtained. For
exmple, to be 9% certain that an estimate of return period is
within 2 10 years of a 50 yeas return period would require 2000
years of data and to be 9@ certain that the estimate was within
f 5 years would require no less than 11,000 years of data. bioreover,
even to be 9% certain that the return period was in the
r,mge of 50 years to 100 years would require 1600 years of data.
These data requirements show the absurdity of tho present reliability
criterion. It infers that all water-resource systems are
designed on inadequnte data with only the degree of inadequacy
varying between schemes.
Even if a once in T yeas low flow sequence could be isolated,
there is no guarantee that this would produce a once in T year
failure rate in a reservoir system designed to withstand such an
event. Shortkves in water supply are not independent events due
to the effect of storage. If a reservoir has failed one year and
has not recovered it is more likely to fail in the follovnng ye:=
than if it had been full at the start of the year. Consequentlg
reservoir failures come in groups rather than completely random
sequences and an event less severe than a once in T year flow
sequence closely following on a similar loa-flow sequence could
ceuse the system to fail. The occurrence pattern of these extreme
low-flow events is therefore as important as their severity and
individual events should not be taken from the historic record and
used in isolation when designing a reservoir system. Unfortunate-
ly the historic flow record provides just one realisation of the
occurrence pattern at a given point and the probability of the
historic sequence being repeated in the future i3 infinitesimal.
Consequently even if the whole historic record were used and even
if it contained what were considered to be extreme events there is
no guarantee that this would enable a realistic prediction of the
reservoir's reliability to be made.
389

390
The difficulty of determining 'rare1 events from 'short' data
cannot be overcome. Recently the use of synthetic data generation has
alleviated some of the problems. The historic flow &ta is used to
estimate the parent population by modelling statistical chaxacteristics
and many synthetic samples can be generated each of which is
equally as likely to occur in the future as the historic record was to
have occurred in the past. In this way vaxious occurrence patterns
may be obtained and long perio&of synthetic data can be ?%gzìxdd as
producing a larger sample from the infinite population of possible
flows than the historic record affords. With the larger sample there
is a. correspondingly increased chance of the record containing (rare1
flow events providing a 'true' model has been used. However, the
synthetic data can only be as representative of the parent population
as the historic data. If an untypical historic record has been used
or there is insufficient data for the reliable estimation of node1
parameters then little confidence can be placed on the generated
sequences and in particulm on inferences about extremes within the
data.
in looking for a suitable design criterion we must accept the
lack of data and use a criterion that can be estimated with more
confidence from the same &ta. Rather than defining failure as a
reservoir or aquifer becoming empty, an event which would understandably
be accepted only raxely, the introduction OP rationing of
water supplies can be used as the definition of failure. This would
occw when only a certain amount of water remained in store and would
obviously be tolerated more frequently. In practice a reservoir
would not be used at normal demand until it was empty. Instead a
level of storage would be reached below,which the supply would be
rationed. If rationing could be accepted, say, every twenty year3
then this would be a more frequent event and one has a correspondingly
increased confidence in the design.
Another shortcoming of the return period criterion is that it
gives no indication of the magnitude of the shortage. For example
in figure 6 the reservoir failed only once in the first case whereas
in the second case it failed twice. Therefore although the first case
is clearly the more severe condition the concept of return period
indicates the second is worse as it has twice as many shortages.
This is only.to illustrate a point but in practice reservoir failures
do group together which poses the problem of deciding whether such
a series should be consideyed as a single failure or a number of
individual failures. Therefore return period is not ideally suited
to describe the pattern in which reservoir shortages occur. An alternative
criterion i~ required which must be a measure of both the
frequency and magnitude of failures. It must be flexible enough to
allow for a variable definition of failure (as the introduction of
rationing is somewhat subjective) and it must be simple to calculate.

3 91
The concept of cumulative percentage frequency (CPF) of a
.specified failure level being reached meets these requirements.
CPF measures the percentage of time that the reservoir is at or
below a spec.ified storage. It will not however differentiate
between sw one &y of failure every year or a one hundred day
failure every hundred years. Fibwe 6 shoirs that the first case
would have a CPF of and the second 100 (x + y1 which
V V
'correctly assigns the less severe shortage to the latter.
The CPF value for any reservoir state can easily be obteined
from the storage histopans already referred to (Figure 3). By
rerunning the model with different demands a graph showing the
CPF of vmious storage levels for different demands can be con-
structed (Fibwe 7). Given a reservoir level at which rationing
would be introduced the relationship between quantity of water
and reliability can be obtained. In this way the effect of dif-
ferent policies on reliability can be easily determined in toms
of yield and the definition of failure does not need to be pre-
judged.
CONCLUSION
Hydrological design criteria are based on rare events and
there will always be some degree of inadequacy in flow data.
Synthetic flow àata is only a partial solution because the
techniques are dependent upon the assumption that the historic
sample is representative of the infinite popultxtion of flows.
Even then, the historic data will only contain limited infoma-
tion on long-term periodicities and persistencics which are
important when examining rare events.'
Where no flow information is available there seems to be
little alternative to improvisation. This inay take the form of
transposition of data, scaling flow data or estiinating data
indirectly as in the case illustrated. Any improvisation should
always be treated with suspicion and attempts made to verify it
if possible, Failing this, simulation can be used, at a cost,
to ascertain the sensitivity of the system's performance to this
input. If the outcome is insensitive to that specific input there
is little cause for concern. If on the other hand the outcome is
sensitive to that input,at least it shows where the àata collection
effort should be concentrated.
Another way of improving the confidence in the prediction of
reliability, given a limited amount of flow data, is to choose a
better design criterion by changing the definition of failure.

392
Hence the proposal is made that the introduction of rationing should
be used as the definition of failure since this would be tolerated
more frequently than the complete emptyiw of the reservoir. Nore-
over, by changing the concept of reliability to one which is both a
measuce of frequency and magnitude of failure rather than just the
frequency of failure enables two schemes to be compared objectively
even :?hen based upon a small amount of flow data. Thus the cam-
bination of synthetic flow data generation, introduction Qf ration-
ing au the definition of failure and cumulativa percentage frequency
as a masure of reliability helps to overcorce the problem of
inadequate flow data.
ACiO-f?LEIlc~
The authors thank their Director, Sir Norman Bowtree, for
permission to publish this paper in which the views expressed are
those of the authors and not necessarily those of the Hater Resources
Board.
1. Jamieson, D.G., Radford, P.J. and Sexton, J.R. (1973).
The Hydrological design of water-resource systems.
Water Resources Boosd. (To be published)
2. Bloomer, R.J.G.B. and Sexton, J.R. (1972). The
generation of synthetic river flow data. Water Resouroes
Boad publication No. 15.
3. Weiss, G. (1973). Shot noise models for synthetic
generation of multisite àaily streamflow data. Symposium
on Desi,v of Water Resouxces Project with Inadequate Data,
Uadrid.

TâBLE 1
LIST Q SYNBOLS ASSOCIATZD WITH FIGURE 5
D hand centre
E Effluent retumi
R Naturd recharge
AR Artificial recharge
S Seepage or spill-
I Natural inflow
L "ranslational delay
P Precipitation
V Evaporation
t b P
tc- AtiPinimum-flow constraint
3 93

394
a
II minimum flow constraint
FLOW
II minimum flow constraint
A
SECOND RIVER FLOW - -
FIGURE 1 Component Model of a Pumped-Storage Reservoir
FIGURE 2 Component Model of a Pumped Aquifer

3 95
NUMBER OF DAYS RESERVOIR AT THAT STORAGE
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3 99

MAXIMUM INFORMATION OBTAINABLE FROM INADEQUATE DESIGN DATA:
FROM MULTïVARIATE TO BAYESIAN METHODS
ABSTRACT
Jean Weber1, Chester C.Kisie12 and Lucien Duckstein2
An overniew is given of some theoretical and empirical issues
involved in designing water pesource projects in the face of inade-
quate data. The primary focus is on multivariate analysis of samples
whose properties are not consistent with the assumptions of the
analysis, The multivariate models discussed include multiple linear
regression, discriminant functions, canonical correlation, principal
components, and factor and cluster analysis. Each of these models
is discussed in terms of its assumptions, data requirements and
applications in hydrologic research. The Bayesian approach to para-
meter estimation and decision making is introduced for the purpose
of considering both the uncertainty due to inadequate data and
economic losses.
RESUME
Les auteurs exposent queJques ccFsidérations générales, théo-
riques et empiriques, sur 1'élaboratio.i des projets d'am€nagement
des eaux quand on se trouve en prdsence de donnges insuffisantes.
Ils mettent l'accent sur les problèmes que pose l'analyse multiva-
ride lorsque les échantillons qui lui sont soumis ne répondent pas
aux hypotheses de base de cette analyse. Les modèles multivariés
dont il est question comprennent: les régressio?s linéaires, l'ana-
lyse discriminatoire (variable dependante discrete), la corrélation
canonique, les composantes principales, l'analyse factorielle et
l'analyse groypde. Chacun de ces modeles est examiné sous l'angle
de ses hypotheses de base, des données qu'exige sa mise en oeuvre
et de ses applications en recherche hydrologique. L'approche bayé-
sienne de liestirnation das paramètres et de la décìsion, permet
d'introduire a la fois l'incertitude due à l'insuffisance des données
et ses conséquences économiques.
lprofessor, Department of Management, University of Arizona, Tucson,
Arizona 85721.
2Professors, Department of Systems and Industrial Engineering and
Department of Hydrology and Water Resources, University of Arizona,
Tucson, Arizona 85721.

402
1 .O Introduction-
This DaDer considers L.e problem o .-recastinq o hvdroloaic variables for
water resoke projects when the data-are inadequati, thät is, when there is a
mismatch between data and model. This mismatch is considered in terms of multivariate
methods of data analysis. Mismatch implies a discrepancy between model
structure and structure suggested by the data and/or data inadequacy in relation
to model requirements. Several types of data inadequacies are considered in the
context of models frequently used in hydrologic research. The discussion is from
two related points of view; it considers limitations of a model in terms of the
assumptions on which it is based and sensitivity of the predictions of a model to
data inadequacies of various types. These considerations are inextricably related
since the more restrictive the assumptions of a model are, the more likely
it is that data obtained are inadequate for estimating the parameters of the model.
Uncertain input information for the design of water resource systems is the
result of the inability of hydrologists to model large basins in substantial detail
as projected by Freeze (1972) and a result of the "curse" of small samples in
developing space-time series models and probability density models of flow, precipitation,
temperature and evapotranspiration. Problems of extending data at a
design site and to ungaged sites are of long standing concern.
the implications of assumptions in mu1 tivariate statistical methods applied to
these problems is important to subsequent steps of coping with the consequent
assumptions and offering alternatives and decision strategies.
1.1 Model Building and Its Assumptions
An awareness-of
When data such as streamflow are obtained, it is almost always for the ulti-
mate purpose of designing or operating a structure (bridge opening, dam, drainage
structure); one intermediate step consists of predicting or forecasting future
events (floods or droughts) using a model. The sequence of events in accumulation
of knowledge for predictions can be characterized as follows: some knowledge is
obtained by observations, a preliminary theory or hypothesis (for example, log
normal probability density function (pdf) of flow) is formulated on the basis of
these observations, additional data are obtained perhaps more systematically, the
theory or hy othesis is revised and/or refined (for example, log Pearson type III
pdf of flews!, additional data are obtained, and so forth. As this interaction
between theory and data proceeds, the theory becomes more reproducible and per-
haps less general and the data required for its verification or modification also
become increasingly accurate, so that the design process may be started without
having to use large safety factors to compensate for uncertainty.
At some point, after accumulation of sufficient supporting data, a theory or
hypothesis is generally accepted and, unless subsequent theory and/or obser-
vations strongly indicate otherwise, the theory is used for prediction of a design
quantity such as the 50-year flood Q(50). By this time the theory is frequently
referred to as a model. As a theory becomes generally accepted, even tentatively,
the purpose of obtaining data gradually shifts; data are used less as a basis for
reformulating theory and more as a basis for estimating the parameters of a model
whose form has been determined, at least in most respects. Unfortunately. it is
frequently tempting to accept a theory and corresponding model prematuwly,

especially if the urgency of making predictions or forecasts is compelling (for
example, in a decision to be made at once on the building of flood control works,
a water supply reservoir, or hydroelectric power dam).
Premature acceptance of a model can have very serious consequences, particularly
since the model is likely to be idealized to the point of being unrealistic
or to hold only under very restricted conditions, such as time invariance of a
watershed (Foge1 et al., 1971). Any model is an oversimplification of reality.
This is inevitable, because the purpose of theory is to simplify reality, which
is enormously complicated, by abstracting from it those elements that explain a
large proportion of the observed phenomena. ' Watershed models certainly are in
this cateogry. Acceptance of a model thus involves a compromise between realism
(e.g., a distributed model involving all details of the water cycle) and simplicity,
represented by a lumped model. In economics and behavioral science, the
predictions of a theory or model are said to be appropriate, ceteris paribus,
that is, other things being equal (not varying). Under controlled laboratory
conditions extraneous variation can be minimized; in the real world, it generally
cannot. Thus , in using a rainfall-runoff model for forecasting streamflow,
for example, it is important to know how robust the model is to its assumptions,
including the ceteris paribus assumption which, for example, precludes urbani- .
zation. That is, it is important to know whether minor perturbations in the conditions
under which a model is applied have relatively small or relatively large
effects on its predictions. Clearly, if the predictions of a watershed model are
sensitive to changes in a variable, this variable should be included in the model.
Unfortunately, a model developed for one set of conditions is frequently used
under quite different conditions without consideration of the inadequacies of the
data obtained under those conditions. A linear rainfall-runoff model that has
been shown to yield a correct design of a culvert draining 50 km2 cannot be extrapolated
to a 500 km2 watershed. Here the predictions of the model may be quite
erroneous.
One of the major difficulties in choosing and using models for decisionmaking
in hydrology or other engineering design is the mismatch between available
models and available data. The implications of the mismatch are not clearly
understood. Most of the analytic derivations of the properties of deterministic
and statistical models (for example, sampling distributions of estimators , tests
of significance, standard errors of estimates and predictions, and so forth) are
based on assumptions that are almost always violated in applications. These
assumptions include linearity in the parameters of the system and of estimation
equations, normality of population distributions, independent random sampling,
large samples (for applicability of asymptotic results) and a variety of assumptions
concerning the covariance or correlation structure of multivariate observations
and/or their errors. In most applications at least one of the assumptions
of the model is violated; the relevant concern should thus not be with
the properties of the model when its assumptions are satisfied, but should be
with the properties of the model when its assumptions are violated in various ways
and to various extents. The research in this area is not nearly sufficient to
provide practical guidelines (Dhrymes et al., 1972).
Specific assumptions and their violations are discussed in the next sections
for several models. In addition to inadequacy with respect to these assumptions
data may be inadequate in several more general respects including sample size,
missing observations, measurement errors , secondary variables.
403

404
These types of data inadequacy occur, for example, in the method of regionalization
used by the U.S. Geological Survey (USGS). Assume that a stream flw characteristic, say the 50-year flood Q(50), is needed at an ungaged site for
design purposes. Regionalization may be used to calculate Q(50) in those cases
for which a data collection network for the region has been in operation for a
certain time. One method of regionalization, used by the USGS, relies on re-
gression analysis (Thomas and Benson, 1970).
characteristic Q(50) is regressed upon basin characteristics, such as basin area,
precipitation, channel slope, elevation, forest cover and soil index. Then,
given the basin characteristics of the ungaged design site, the regression equat-
ion is used to predict Q(50) for that site.
For most regional data collection networks, many sites have record length of
the order of 20 years or less; small sample bias thus is quite substantial, so
that distributions other than the normal distributions should be used to compute
the logarithm of the flow Q(50); see Metler (1972). The basic reason for using a
regional regression is that data are missing in space at the design location and
large scale physically based models of combinations of river basins are non-exis-
tent to help in augmenting the data. Although calibration curves of flow veysus
gage height are periodically recalculated, there are difficulties associated with
sediment flows and with recording of flow data. All the variables entering the
regression equation constitute by definition secondary data (primary data is the
flow itself).
2.0 Multivariate Models and Data Inadequacies
The follwing sections concern multivariate models frequently used in
hydrologic research and focus on model limi tations arising from inadequate data.
The multivariate models discussed include multiple regression, discriminant
functions, canonical correlation, principal components ,:and' factor and cluster
analyses. Each of these multivariate models, with its assumptions, data require-
ments and applications, is discussed in the following sections. Primary refere-
nces on these models are Anderson (1958),'Christ, C1966), Dem ster (1969) Dhr mes
(1970 and 1972), Harmon (1967), Johnston (19631, benta (1971!, Morrison (19671,
Press (1972), Tryon (1970), Zellner (1971).
2.1 Multivariate Linear Regression
First, the desired streamflow
The purpose of multivariate linear regression analysis is to obtain an
equation -%-for predicting the value of a dependent variable 1 as a linear
function of a vector of k independent variables &=[Xi ,. . , Xk]. The criterion
for obtaining the vector k=[bi, . , &] is that the sum of squared errors
(y-@) - '(Y-Xi) be minimized.
Linëarregression and analysis of variance, which can be viewed as a special
case of linear regression, are the only multivariate models for which there has
been considerable investigation of the effects of violation of the assumptions.
The assumptions. of the multivariate linear regression model ~=XB+E can be stated
as follows: E sN(0,~) where c=$I is the covariance matrix ofthe multinomial
(N) population, & Ts non-stocFastTc, and has rank k

405
The number of observations is n, so 1 is n x 1, X is n x k, is k x 1 and E is
n x 1. Thus the ässumptions are linearity, normality of errors, serial indëpen-
dence of errors, nonstochastic independent variables, and an observation matrix
of full rank with the nunber of independent variables less than the number of
observations.
The assumptions required for estimating a model are generally much weaker
than the assumptions required for inferences concerning the estimates. In
particular, normality assumptions are usually required for inference but not
for estimation (say of the 6's). This is the case for multivariate regression.
Since estimates are of little use without knowledge of their distributional
properties, the assumptions stated for mu1 tivariate linear regression and for
models discussed subsequently are those required for standard tests of significance
and confidence intervals.
The effects of violating the various assumptions of multivariate linear regression
are summarized in Table 1; also included in the table are procedures
for detecting violations of assumptions and proposed al ternatives for remedying
detected violations.
2.2 Canoni cal Corre1 ati on
The purpose of canonical correlation analysis is to study linear relationships
between two sets of variables l'=[Yi ,. . ,Y,] and L'=[XI ,. . ,Xq]. Peck
(1972) uses the analysis to determine whether 12 meteorological parameters (like
vorticity, vertical velocity, wind speed at different elevations and from various
directions, temperature differences, etc.) are sufficient to predict variations
in orographic winter precipitation patterns without the need for storm typing.
Nimnannit (1969) uses the technique to relate spring runoff at a set of stations
in the target region (where clouds are seeded) to runoff at a set of stations in
the control region (no seeding); the urpose is to assess the effectiveness of
weather modification. Torranin (19725 investigates the potential of the method
for (1) forecast of monthly precipitation of three large areas of the U.S. west
coast and (2) forecast of seasonal snowmelt ruiroff for three gaging stations in
the Flathead River Basin in Montana. The applications in hydrology have been
very few in nunber.
The analysis obtains vectors and such that the correlation between &'Yand
b'& given by
a'YX'b
. ---
r =
Ja'Y 'Y a b 'X 'Xb
------
is maximized, subject to the normalizing conditions C'L'Y~ = 1 = d'&'Xb.
Subsequent vectors are obtained such that for each successive vector %e canonical
correlation is maximized subject to normalizing conditions and the condition of
independence with respect to previous vectors.
The number of canonical correlations between r=[Yl,. . .,Y,] and X'=[Xi,. . ,Xq]
is min !p.q), although in practice usually only the first few canonic3 correlates
are of interest. For the special case when either or is scalar, that is, consists
of only one element, canonical correlation is equivalent to multiple correlation.
Except in this special case, canonical eorretatim analysis is not useful

406
for prediction but is of value only to aid in formulating a modez; this is in
contrast to hydrologic uses mentioned above.
Under the assumption that Y' and X' are jointly normally distributed, the
joint significance of sets of cänonicar correlations can be tested using a likelihood
ratio statistic. Unfortunately, the exact distribution of this statistic
is complicated. An approximate large sample distribution has been obtained, but
its convergence properties have not been studied. Thus , inferences concerning
canonical correlations can appropriately be made only on the basis of large
samples from a multivariate normal population. No information is available concerning
the nature and extent of the effects of violations of the assumption of
normality on the distribution of canonical dorrelations. Canonical correlation
analysis thus appears of limited use for building models for eventual use in
design with limited or inadequate data, in contrast with the hydroiogic uses
mentioned above.
2.3 - Discriminant Analysis
The purpose of discriminant analysis differs from the purpose of multivariate
linear regression analysis only with respect to the type of prediction
required for the dependent variable; in regression analysis the dependent variable
is continuous and its value is to be predicted, while in discriminant analysis
the dependent variable is discrete and its classification is to be predicted, for
example, classification of watersheds.
For the case of a dichotomous dependent variable, discriminant analysis can
be computed as a special case of multiple regression analysis by using a dumy
variable having values zero and one for the dependent variable and point biserial
or biserial correlations between the dependent (dummy) and independent variables.
The regression coefficients obtained by this type of analysis are proportional
to the coefficients obtained by discriminant analysis.
The follwing discussion concerns discriminant function analysis for a
dichotomous dependent variable; the discussion can readily be extended to a
dependent variable having more than two categories.
Suppose that the independent variables &=[Xi ,. . ,Xk] are jointly normally
distributed in each of two populations with mean vectors and g and connnon
covariance matrix c of full rank k. If the prior probabilities of each population
(pop) are equal ana the costs of misclassification are equal, then the probability
of misclassification is minimized by using the following rule for classification
of an observation g
classify in pop 1 if ~'~+(~1+g)'~
classify in pop 2 if K'L

are extremely complicated and its convergence properties have not been investi-
gated. The affects of nonnormality of & are not known. Thus discrinimant function
analysis is appropriate only when x is normally distributed for each population
and, in addition, its application to small samples is appropriate only if the popu-
lation mean vectors p~ and u and the comnon population covariance matrix are
known.
2.4 Principal Components
The purpose of principal component analysis is to reduce the dimensionality
of K=[Xl ,. . ,Xk] on the basis of dependence among the variables. For example.
Fiering (1964), in his work on extending the single-site streamflow synthesis
model to the multi-site case, applied the technique to a river basin with p gaging
sites each site having an n-year record of annual flaws. Craddock (1965)
applied the principal components method to monthly temperature series from 1680
to 1963 for Central England. Other applications include increases in sediment
discharge from 31 watersheds after two major floods in northern California
(Anderson, 1970), sediment network design in California to insure accuracy of
predicted sediment yield (Wallis and Anderson, 1965), establishment of the uniformity
of a hydrological region in Northland, New Zealand (Blake et al., 19-70),
derivation of a water yield model from monthly runoff data (Snyder, 19631,
identification of watershed factors from annual precipitation and runoff data of
watersheds in Coshocton , Ohio and Riesel, Texas (Diaz et al., 1968) , and shortrange
forecasts of river stage or discharge on the river Kolyma, U.S.S.R.
(Nechaeva and Mukhin, 1968).
Mathematically, principal components analysis transforms the X's to a set
of variables which are pairwise uncorrelated and of which the first has maximum
possible variance, the second has maximum possible variance subject to the condition
of being uncorrelated with the first, and so.forth. Principal components
are estimated on the basis of a random sample of n observations as follows. The
first principal component of & is denoted by Li=X a~ and g, is obtained such that
Z'1L1=g'lX1h1 is maximized subject to the normaTizing constraint g'1&1=1.
n e secona principal component Q=X- a is then obtained by determining 7uch
that g&= am2X-'X9 is maximiaed sdject to the normalizing constraint ti23=1
and the independence constraint &'I 3 = O. This procedure is repeated until the
k principal components have been obtained.
Large sample distributional prooerties of principal components have been
obtained assuming that has a multivariate normal distribution with a covariance
structure such that the covariance matrix c has k distinct characteristic roots.
The effects of nonnormality and the covergence properties of the large sample
distributions have not been investigated. Small sample distributional properties
of principal components are not known; this again limits the use of this technique
for the problems considered here.
In many cases determination of the number of principal components needed to
account for a reasonably large proportion of the variance in X is a matter of
judgment on the part of the investigator. Even if the investTgator is willing to
make this decision on judgmental rather than statistical grounds and he concludes
that a relatively small number of principal components seem to account for a
reasonably large proportion of the variance in X, there is still the problem of
interpreting the principal components in terms of the original variables.

40 8
hfortunately, pi.incipal components are not ahap interpretable and this hae
been a deterrent to the extensive use of principal components in developing
models.
The use of principal components as independent variables in regression
analysis has been suggested for the purpose of reducing the dimensionality of 5
and thus avoiding problems with degrees of freedom and for the purpose of
circumventing the problems resulting from multicollinearity in X.
applications see earlier references in this section as well as Singh's (1970a)
application for predicting infiltration in an aspen-grassland watershed in
southwestern Alberta, Canada. Al though principal components have been used as
independent variables in regression analysis by numerous investigators, there is
no generally accepted procedure for determining the number of principal com-
ponents to be included in such analyses, nor is there agreement concerning whether
it is acceptable to include one of more of the original x variables in addition
to principal components.
In spite of these shortcomings and limitations, a principal component
analysis could possibly lead to a better use of insufficient or correlated data
for hydrologic prediction. For example, it offers an alternative to the method *
of regionalization described elsewhere in this paper.
2.5 Factor Analysis
The purpose of factor analysis is to account for the covariance structure
of a set of observable random variables in terms of a minimal number of unobservable
or latent random variables referred to as factors. Among hydrologic
applications have been those that sought decision rules that resulted in reduced
inventory and survey costs for specific areas and problems, as in the study of
the chemistry of groundwater quality (Dawdy and Feth, 1967), in the design of a
hydrologic condition survey in the TVA system (TVA, 1965), in parameter screening
for watershed analysis (Shelton and Sewell, 1969), in predicting reservoir
losses in cavernous terrain (Knisel, 1970) and in reducing a set of edaphic
variables for a soil (Singh, 1970b).
Factor analysis estimates the coefficients to be us& in expressing each
response 'variable as a linear combination of a small number of unobservable
common-factor variables and a (latent) specific variable. The common factors
generate the covariances among the observable variables (responses) and each
specific term contributes only to the variance of the particular associated
response variable. The coefficients of the common factors, estimated by factor
analysis, are not required to be orthogonal and their matrix is unique only up
to multiplication by an orthogonal matrix. The observations are assumed to be
a random sample from a multivariate normal population of full rank and the nunher
of common factors is assumed to be known; both of these requirements limit the
use of the technique in hydrology. The factor analysis model can be written as
- X=nl+c where X is pxl, is pxm, 1 is mxl and is pxl, There are thus p
response variames L'=[Xi, ..., X,], m common factor variables r=[Y, ..., Y,] and p
specific-factor variables E'=[E~ ,. ..,E 1. The matrix
For hydrologic
gives the factor loadings
where aij is the loading of the ith regponse variable on the je common factor
variable. The conunon-factor variables l'=[Y1 ,. . .,Y,] are independently
distributed N(0, 1). The specific-factor variables E'=[E~ ,. are independently

distributed N(0,q~~). Factor analysis estimates the elements of the loading
matrix A. Maximuil: likelihood estimates can be obtained assuming that is
multivariate normal with covariance matrix L=@&' of full rank p.
Note that principal components can be viewed as a particular solution of
the problem of factoring the covariance matrix. The- principal components
solution ignores variance associated with a specific response variable and requires
the factors (components) to be orthogonal and of decreasing importance in
accounting for (common) variance in the response variables.
Assuming normality, the adequacy of the m-factor model can be tested for
large samples using a likelihood ratio test of the null hypothesis E=&+$'
against the alternative hypothesis that is any symmetric positive definite
matrix. In most applications the number of common factors is not known and
successively larger numbers of factors are extracted until the goodness of fit
hypothesis is accepted or the computing routine fails to converge. Successive
tests used in this procedure clearly are not independent and the statistical
properties of the result are unknown.
Factor analysis has been used since the beginning of the twentieth century
to study the covariance structure of multivari te observations. Many variations
of the model discussed above have been propose 3 and many estimation procedures
have been developed. Unfortunately, factor analysis, in any of its forms, may
be very difficult to interpret in practice. Part of the difficulty arises from
the fact that, regardless of the method of extimation used, the factor solution
is unique only up to a rotation of the axes. Various criteria, notably those
involving simple structure, have been suggested for obtaining the rotation most
readily interpreted; in practice, considerable subjectivity may be involved in
applying these criteria, even if their appropriateness is not in question.
Another difficulty in factor analysis arises from the fact that evaluation of
factor scores for use in subsequent analyses is not uniquely defined; several
intuitively appealing approaches have been suggested, but there are no apparent
criteria for choosing among them. Thus there are serious problems involved in
interpreting the results of factor analysis and using them in subsequent analyses.
In addition, relatively little is known about the sampling properties of
the estimates obtained in factor analysis (see Matalas and Rieher (1967) for
hydrologic discussions of this issue). The test for appropriateness of structure
assumes normality and large samples; unfortunately, the alternative hypothesis
for this test may not be the most interesting alternative in many applications.
There is considerable evidence that factor analysis can give meaningless results
if its assumptions are ignored.
As an example of the last point, consider Rice's (1970) use of variables
describing the physiography of experimental basins on the San Dimas Experimental
Forest in southern California. His goal was to identify variables which would
be useful in flood prediction. He notes "that the hydrologist might be better
rewarded if he turns his efforts toward developing physiographic variables
which better portray hydrologic processes rather than relying on a mathematical
artifact such Bs factor analysis to appraise the utility of various expressions
of basin physiography." This point is borne out in a non-hydrologic study by
Armstrong (1967); he finds that, while factor analysis "explains" a large proportion
of the variances, it fails to identify the known factors in the model!
409

41 O
2.6 Cluster Analysis
The purpose of cluster analysis is to group multivariate observations according
to various cri teria based on their degrees of homogeneity and .heterogeneity.
In hydrology, cluster analysis can be used to classify watersheds, flaw regimes,
and climates. Bogardi et al. (1972) have used it to group statistical properties
of monthly water levels in Lake Balaton (Hungary). Hydrologic appli-
cations are very few.
cl us i on in mu1 ti vari ate regression.
The other multivariate methods discussed above assume that the variables
belong to particular populations and that these populations have specific
(usually normal) distributions.
Cluster analysis can help to identify variables for in-
In cluster analysis the variables are not assumed
to have even the minimal structure of belonging to particular populations and
the purpose is to establish appropriate populations as a basis for structuring
the variables.
Techniques of cluster analysis have been developed, almost exclusively, not
only for computer application but also on the basis of computer analysis. Al-
though mathematical rigor is minimal and statistical inference is almost non-
existent for cluster analysis, very useful results have been obtained in appli-
cations. Because of its (lack of) assumptions concerning population structures
and distributions, cluster analysis is applicable to a wide variety of hydrologic
and other problems;. its results can be useful if they are recognized as tentative
and if even tentative conclusions are based only on results from large samples.
There are several questions or decisions that must be considered in any
cluster analysis: the number of clusters must be determined, the cluster
boundaries must be established, the method for handling correlated variables must
be specified, the technique for examining similarities must be chosen, and so
forth. Several approaches have been proposed for each of these aspects of cluster
analysis. Which criteria or rules of thunh are most appropriate depends on the
problem. Regardless of the techniques and criteria chosen for cluster analysis,
the investigator usually examines successive computer printouts and uses his
judgment to al ter apparently poorly selected cri teria and techniques. Compared
with the other multivariate analyses discussed, cluster analysis is more of an
art and less of a science, but so is engineering design under uncertainty assoc-
iated with insufficient data.
2.7 Bayesian Inference
The preceding discussion of multivariate models is entirely from the point
of view of classical sampling theory. Several of these models have been analyz-
ed from the Bayesian point of view and these results are summarized in the follo-
wing discussion. Bayesian inference incorporates, with sample infomation, the
investigator's prior information concerning the sampling distributions of the
parameters to obtain point or interval estimates. More general, however, is
Bayesian decision theory that incorporates both prior information and a loss
function with sample information in order to obtain parameter estimates or to
determine the optimal decision. Bayesian analysis is intuitively appealing; in
many applications the investigator has considerable prior data or experience as
a basis. for prior parameter distributions and in most applications he has at

least a general idea of the?(economic) loss function associated with inaccurate
estimation. Unfortunately, the results for many multivariate Bayesian methods
are complicated and at best are applicable only for large samples. However,
since many classical results also have this limitation, Bayesian methods may be
p:i ferable because of their flexibility in incorporating prior distributions and
loss functions in the estimation of parameters or in determining optimal decisions.
Also, human beings are better at estimating prior distributions than at estiniatlng
posterior distributions (Ferre11 , 1972).
There has been considerable application of Bayesian methods in mu1 tivariate
linear regression analysis. Bayesian point estimates of the regression coefficients
can be obtained with or without incorporating loss functions and Bayesian
interval estimators (credibility intervals) can be formulated.
As discussed in section 2.5, the maximum likelihood factor analysis solution
is unique only up to a rotation of the axes; the use of subjective information
in a Bayesian analysis is an intuitively appealing basis for eliminating this
ambiguity. Unfortunately, the technical difficulties involved in obtaining
numerical solutions have thus far precluded use of this approach.
The Bayesian approach has also been considered for canonical correlation
analysis; unfortunately, even for the simplest assumptions with respect to both
the prior distributions of the parameters and the sampling distributions of the
data, the Bayesian results for canonical correlation analysis are so complicated
that their applicability is extremely limited.
The most notable success of Bayesian methods in multivariate analysis thus
far has been for discriminant functions. As summarized above, a number of methods
based on the sampling theory viewpoint have been proposed for discriminant
analysis, but these results are unsatisfactory for use with small samples. The
Bayesian approach provides a useful and simple al ternative.
Consider the case of classification into one of two mutually exclusive populations.
Denote the populations by Pi and P , the vector of observations by
x'=[xl, ..., xk]. the density functions by fl(Kf and fp(&) and the prior probabi-
Tities by p1 and p2 where p +p2=1. The costs of misclassification are C(211) if
an observation from P1 is classified in P2 and C(112) if an observation from P2
is classified in Pl. The problAem is to determine a classification rule of the
following form: partition K into regions R1 and R2 such that if ZERI, the observation
is classified in Pi and if x~R2, the observation is classified in P2.
The expected cost of misclassTfication is given by
and the corresponding ,classification rule is
f+X) C(l MP2 f+x) C(112)PZ
R2:
R1:q-g 'copl
< cop1
411
This classification rule involves the densities f1(&) and f2(&) which may not be
known. Assume that fi(&) and f2(&) are multivariate normal with mean vectors PJ
and and common covariance matrix c. Then the above rule can be written
c(1 PIP, c(1 12)P2
R, :L'i.-+(IL,+q) 'L>log, R2:~'6-+(~1+4) '&

41 2
1
where 6 = E- (ply) and x'6 is the discriminant function. If p1=p2=% and
C(112)3(2ll),'-this reducësto the rule, given in the discussion of the sampling
theory appLoack to discriminant analysis. As noted in that discussion, sample
estimates x x and 2 may be used to obtain from the sample data without
knowledge d'i' -2 and c.
Bayesian aiscriminant analysis can easily be extended to other cases; for
example, fl(x) and f2(&) may have some form other than the normal distribution
or there may be more than two populations into which an observation may be
classified. Finally, for the sake of completeness, we should mention the use of
Bayesian decision theory in design to imbed uncertainty in parameters resulting
from inadequate samples into a loss function (Davis et al., 1972; Davis et al.,
1973).
3.0 Sumnary and Conclusions
In this overview, we have critiqued the current status of multivariate
methods of data analysis because of their central position in making estimates
and predictions of both hydrologic and econometric (e.g., cost) inputs to design
of water resource systems. The design implications of many of the assumptions '
in these methods remain to be evaluated - a task of importance to many profes-
sional disciplines.
models without considering the assumptions involved, we believe that use of
Bayesian decision analysis, while not the final answer, may be a viable alter-
native for anticipatinq poor design. Bayesian analysis offers flexibility in
incorporating prior (subjective) knowledge about probabil i ty distributions on
design parameters and it encourages the design engineer to invoke his general
ideas of loss functions associated with inaccurate estimation in many specific
design problems. The focus is on the consequences for a specific use and not on
a precise design estimate. The latter has a subtle linkage of probability and
utility, depending on one's value structure, but Bayesian decision analysis
encourages specific consideration of each in an open manner. With the continuing
emphasis on environmental impact evaluation, such an approach seems timely and
necessary in the face of small samples of hydrologic and other environmental
data.
4 .O References
In contrast to the current tactic of using multivariate
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41 3
Blake, G. J., A. D. Cook and D. H. Greenall. 1970. The use of.principa1 component
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41 4
Fogel, M, M., C. t. Kisiel and L. Duckstein. 1971. Space-time validation of a
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Harmon, H. H. 1967. Modern Factor Analys'is, 2nd Ed. Chicago: University of
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Johns ton, J. 1963. Econometri c Methods. New York: McGraw-Hi 11 , Inc.
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Knisel, W. G. 1970. A factor analysis of reservoir losses. Water Resources
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Matalas, N. C. and B. J. Rieher. 1967. Some comnents on the use of factor
analyses. Water Resources Res., Vol. 3, No. 1, pp. 213-223.
Metler, W. A. 1972. Bayes Risk Analysis of Regional Regression Estimates of
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Univ. of Arizona, Tucson.
Morrison, D. F. 1967. Multivariate Statistical Methods. New York: McGraw-
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Nechaeva, N. S. and V. M. Mukhin. 1968. The use of statistical methods for
short-range forecasts. IAHS Publ. 81, pp. 405-416.
Nimannit, V. 1969. Multivariate analysis of hydrologic changes. Doctoral
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Peck, E. L. 1972. Relation or orographic winter precipitation patterns to
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Rodda, J. C., et al. 1969. Hydrologic Network Design, World Meteorological
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415
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1971 Fall Annual Meeting of the American Geophysical Union, San Francisco,
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New York: John Wiley & Sons, Inc. 480 pp.

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PRATIQUES COURANTES POUR L'EVALUATION DES CRUES ET
DES DEBITS D'ETIAGES PRIS EN COMPTE DANS LES PROJETS
Rapport General
P ar
Marcel ROCHE
Pour &borer les données hyärologiqueis néceseaires à la mise au
point diun projet d'aménagamcmt des eaux, on dispose finalement de trois
graisda types d'approche :
- u'utilieer que lee observatione concernaut les débite et re-
cueillie6 au site m be de l'amhgement ou & p roat6 ;
- utili- 6gelement les données climatiquee ãieponibles BUF le
baadn, notemnent les relevée de précipitatione i
. 88 mvlr de formules r6gionalee1 ou de corr6lationa. Pour a-
trapoler des reeultate recueillis d dee statione hgdrom&riqUeE
et/ou plu~iométrigues tdtu&s BiUeure äaus ia regiou, à 1li.n-
ttieur OU d l'extbieur du basain fluvial.
Le premier typa groupe lee dthodee dites Wirscteetl, tandis qUe
lee d ew autree pmoèdemt de 1'Cvaluation indirecte. La m&hode "la plu8
aireate" consiste en lkdyse statietique d'un Bchentillon, par exemple

420
de débits maximaux annuels t elle nécessite, pour que l'évaluation ait un
sens, une infoimation riche, des m emes précises de débite portant sur de
longues périodes.
Le second type de méthodes ccmaiste en fait à l96tendrett des échan-
tillons existante de debits, portant BUT dee périodes courtee, en utilisant
les relations qu'on peut dkager entre lea dábits et les donnbe climatiques
disponibles sur une dur& beaucoup plus longue. C'est @ll'exteneion des don-
néedi, qui fait largement appel aux méthodes de r¿greesion et à l'anaïyse
multivariate, mais aussi dans certaine cas aux modèles dits conceptuels. On
peut rattacher à ce type lee m&hodes qui consistent à effectuer les trans-
formations ou à appliquer les régressions & u11 évènement climatique de pé-
riode de retour connue, ou considéré comme un maximum possible.
Le tmi&me type, enfin, rassemble les méthodes d'interpolation
ou d'extrapolation géographique. Ces méthodes vont de la simple analogie &o-
morphologique et climatique, aux raffinements de l'analyse factorielle, en
passant par les tablee et abaquee régionaux.
U paraft se dégager de cette &um&ration une id& simple de l'uti-
lisation dee m6thodee en face de l'information disponible ; il semble évident
a priori que les m6thodes de type I n'ont de sens qu'en cae d'information très
abondante, que odles du type II correspondent A une information hydrologique
réduite, mais à une informetion cïimatoïogique consistante, et que le type
III groupe les mgthodes utilie8ss lorsqu'on ne dispose de pratiquement rien.
On pourrait en conclure que la m&hodologie de type I devrait être exclue
d'un symposium tel que celui-ci.
En fait, le jugement doit être plue nuancé, car on commence à ten-
ter des régionalisatione BUF lee lois de distribution, donc à rechercher par
là des estimations indirectes, a melanger dane lee études statistiques des
dondes de conaietances t Ae différentes. C'est pourquoi nous rQerverom,
ciam notre expod, une piace à l * m & e statistique.
Chaque m&hode d'heluation fait appel à des outils de calcul qui
ne lui sont pae forchent propre. C'est aind qu'un modèle 6 structure détor-
ministe, ì'hydmgranmie unitaire par example, peut etre utilisé pour une ex-
tension des donnhe, pour u11 celcul de transformation d'una, averse de fr&
quence donnét, ou pour une extrapolation &graphique.

ûn pourrait envisager, dans ce rapport, une présentation par
"outil de ~ elcul~~ ; il nous a paru plus cmfonne au eujet du eymposiuc
d'aùmter un ordre d'exposé qui se rattache, autant qu'il mit possible,
à la quantité d'information disponible.
421
Les sujets proposés dans la question qui fait l'objet de ce rap-
port général se rappsrtent aux débits extr&nes, c'est-à-dire aux étiages
et aux crues. I1 faut bien reconnaître que le premier sujet nia guère tenté
les ~ãpécialistes, puisque un 6eUl rapport, eur les quatorze que nous avona
examinés, traite des basses eau, Finalement, cela n'est pas tellement sur-
premt ; les basses eaux sont liées de très pr&3 aux problèmes d'hydrogh-
logie ; or ce n'est que dans quelques cas trèer particuliers qu'il est possi-
ble de dégager des paraadtree morphologiques et climatiques simples qui
soient en relation directe avec ces pmblèmeo et qui permettent une tramp-
sition &,panhique suffisamment précise. Par contre, u11 auteur aurait pu
être tent6 par l'aspect "extedon des donnée^'^, qui dierpoee d'une méthodo-
logie courante sinon riche, du moina 88888 efficace ; cela ne s'est pa6
produit.
I1 était précisé enfin que les expos6s devaient se rapporter à
des llpratiques courantes", ce qui semblait exclure les sujete de recherche
et certaim procédk de calcul non totalement dégagés de leur phase exp&i-
mentaie. Aussi n'Fneieterona noua par3 tmp LNF certaine aapecte qui no-
ont été soumie ; ceci ne veut pas dire que noua lee trouviane peu dignes
d'inter&
Dana notre prbentation des rapporta, nous parlerone d'abord des
étiages, puis des crues.
Le eeul rapport traitant de ce pmblème est celui de
MM. Vladimirov et Chebotarev [I3]- Lee autemrs définiasent avec m in les

422
en
variables par lesquelles on caractÓriae lee basses eaux/G.R.S.S. :
- d&it journalier minimal de l'annw,
- d6bbit moyen meneuel minimal de l*annh.
Chacune de ce6 oarlablee est définie pour choune dee deux saisons
de basses eaux : celle d'hiver et celle d'&&-automne.
Les auteurs exposent ensuite ï'infïuence, sur ces débits de bM6e6
eaux, des caractéristiques climatiques, en insistant Bur le Ale du gel, et
des carúct&istiques phyeiographiquuee, en notant le r81e particulièrement
important des lacs, des rnaraic, du sol, du sow-sol, du karst. 110 définie
sent enmite le cadre régionel dans lequel va s'exercer la méthadologie :
les bassins sont classés en
- petits bassins : jusqu'a 1 o00 1 500 h2en zone de plaine hu-
mide, jusqu'à 2 000-2 500 h2 dan6 les zones de montagnes oc les
zones de plaines peu humides, jutSqu'8 5 OOO-10 o00 km' dans les
régiow arides ou pour les rivières soumises à un gel intense ;
- bassins inoyene : jusqu'8 75 O00 km2.
Pour lee petite bassins, on applique, dana un cadre strictement
régionali66 6 partir des caractériatiques phydographiquee et climatiques,
une formulo de la fome Q = a (A ;I. fIn oÙ Q eat le dBit meneuel minimal
moyen (dit l~omuiì~l daps le texte) en d/s, A ia eurface B.V. tan d, f un
correctif tenant compte d, la non coincidence hentueile du bassin topogra-
phique et du bassin souterrain ; a ot n srnt de6 paramdtres régionaux. Le
passage du débit % od1* 6 dee débits de fr8quences donnbs er'effectue par
deux méthode8 diffbentes.
?our lee baemns moyenne, on utilise des cartes d'isopl&thes (iflolines)
pour la détexmination du débit d'étiage meneuel ; ces cartes sont établies
pour les valeur8 moyennes (normeles) et pour la frbence 80 % de dépasscirent.
coefficient de p-.
Les étiages joumdiers sont dikìuits de6 étiages menauele par un
Il s'agit donc d'un proc6dd8 de formule @kd8gionale1@ dont les parani&
tres sont d&erminb gar un catalogue, et non l ib à dee caractéristiques

423
ghorphologiques et climatologiques meeurables. C'est pratiquement le seul
moyen de e'en eortir en matière de basaes eaux, pour los raieons que noua
avo- deja indiqubs. I1 est dommage que l'auteur n'ait pea indiqué quel-
ques valeure de cosfîiciente r6gioll~yx, ni montré sur un exemple la concor-
dance des réniltate du calcul avec des dombs observks.
Lorequ'on dit qu'on détermine dee crue8 en l'absence de données,
c'est faux. Sane donnéeel on ne caicüie ria du tout. Simplement, on cherche ici
.4 trveer des do~8es existantee ou à établir des relations entre le phé-
nomène qu'on &tudie et des données d'une autre nature. Toute L1infonnation,
et par suite +out le sérieux de6 edhations qui en dbdent, tiennent dam
Ce8 d O~h6.
C'est pourquoi nous regrettons un peu qu'il ne se mit pas trouvé
d'auteur pour exposer lea méthodee, pourtant de pratique bien courante, per-
mettant de reconetituer UTI certain nombre d'6v&nementa marquanedu passé.
Noue verrou tout 6 l'heure qu'on commence a se prkoccuper d'insérer dane
des échantillons r4guliere de dbbite d maw de crue5 annuelles, dee obser-
vations discontinues, parfoie t mnquh et souvent entachées d'erreur6 beau-
coup plus grandee que celleede l'&chantillon régulier. Lorsque lee donnéee
sur les ornes sont rame et surtout portent SUT de5 périodes très courtee,
il devient extrhment important de dispaser du plus grand nombre de telles
obaervatiom.
chi peut aauvent trouver dana les archivea, dane la preme, ~ ur
de6 télégrammes adminietratifel dee indications parfoiPs trèB préci898
oertaines 5mdee crues. On peut également mener des enquatee 6ur place,
auprès dee riverains et em a'intbrsemant aux d&hhs& ou autres marques
laiss6es par ces orue~). 11 y a 1a toute une m6thodologie dont nom no pou-
viam peu ne pas tout BU moina signaler l'existence.

424
des crues réellement observées e d traditionnellement un outil pour informa-
tion riche. Des études relativement récentes ont pourtant 6tk men688 afin
de voir s'il n'Était pas possible, en cas de données rares,
- soit de compléter l'échantillon de données rkgulierement recueillies
par des observations sporadiques et/ou tmnqubs,
- soit de faire de la transposition régionale sur les lois stutisti
ques elles-mêmes
Bien entendu, lee deux ne smt pae incompatibles.
La première question a été traitée, au moins partiellemmt, dens la
cornunication de M. Morven N. keee 183. L'auteur considère deux aspects de
la question. Dans le premier, il suppose qu'on dispose d'une série n o d e
d'observations (enregistrements par exemple) au cours de laquelle un certain
nombre de maximums annuels n'ont pas &é observés, par exemple par suite d'uno
déficience d'appareillage : on cornaft seuìernent , pour les crues concernhs,
le seuil inf&ieur du débit, seuil supposé constant eur la période. C'est le
cas pratique du stylet d'enregietreur qui sart des limites du tambour ou de
la table dbulante.
Le second aespect est celui de la station pour laquelle on dispose
d'une skie d'observations régULiBres de N années, courte mais complète,
des maximums muela de cruea. On connaît par ailleurs, sur une p aode de
M autres annha, les n d8bits maximaux de crues ayant d&ad un seuil 5 ;
et on est certain que p e a t lee N-n anahs reetantes de cette période, aucun
débit n'a atteint ou d6paee6 xh. Ce eont là des hypotheses assez restrictives
qui correspondent au cas pratique des échelles de hautes eaux Wloitées Pon-
dant u11 certein nombre d'années avant que les services se préoccupent des
basses et moyennes eaux ; ce peut (tre aussi le cas de certaines marques
relatives a des cmes historiques.
Lee deux problemes sont trait& par la méthode du maximum de vrai-
semblance, suivant la technique indicph pm Kendall aux paragraphes 32.15
et suivants de son ouvrage. L'auteur donne un exemple d'application a la ri-
vière Avon a la dation de Bath (U.K.). Le but de l'étude
eat de déter-
miner le gain d'information apport& par les observations tronquées ou les don-
nks que l'auteur qualifie d'historiques ; ce gain d'information est estimé

425
ici par la r&ction apport& à 1'eITeur etandard d'estimation pour différent;
quant i les.
Cette 6tude est fort int&resaante, bien qu'on puisse ne pas être
pleinement d'accord avec touteei les hypathèees introduites par l'auteur. Elle
est susceptible d'importants prolongements vers d'autres formes d'information
tronquée ou sporadique, mais le traitement rieque alors de ne pas être aussi
simple.
En ce qui concerne la aeconde question, la traueposition des lois
statistiques a été mainteefois tenth, souvent avec succ8s. Nous nous permet-
tona de rappeler les 6tudes de U, Oktay hanoglu sur le5 mo~ules pluviom&
trique8 de IlAfrique de l'Ouest (Cahiers de l*O.R.S.T.O.M., hérie @droiogie>.
MM. Herbst, Van Biljon, Olivier et Hai1 noue présentent une c odcation sur
la régionalisation des paramètres das lois de distribution des crues 151.
Les auteurs prennent comme m>dèle statistique la loi log-gaiinna
incomplète (log-Pearson III), tout en hoquant la possibilité d'effectuer
les mi3mee opératiom en Re basant mr une lo' de Qumbel. La méthode consiste
à
- calculer. '.es paP810dtime8 des lois pour toutes les longues séries
diaponibleB,
- apposer que chacun de ces paramatres d6pend deun certain nombre
de facteur@ gbmorpholagiques et climatiques (il cite la super-
ficie du bm&n, la pluie annuelle moyenne, la pente moyenne,
la longueur de la rivière, la pluie menquelle maximale médiane,
un facteur de fome mais n'utilise dans la suite du texte
que La surface du bassin A et la pluie annuelle moyenne R),
- appliquer pour chaque parametre de la loi une régression multiple
avec lea facteurs retanus ; les coefficients de la r6gression
sont calculés par lea moindres carrés et l'opération constitue
en fait un "lieeage gbgraphiquelt des valeurs de ces paramètres.
En r&äiité, lee auteum neiitilisent pas les param¿tres fLgurant
dana l'expresdon mathhatique de la Loi, &is la moyenne (des logarithmes
dee débite), ll&cart-type et 10 coefficient d1msiymétrie, ce dernier étant
du reste trait6 de fapon trèe diffhnte. Le pint le plus important au calcul

426
se rapporte à la variance d'estimation de tele paramètres et dee quanti-
qui en dkoulmt, toujours parr l'intermediaire de la loi log-ggnnaa.
Les auteurs donnant quelques résultats obtenue en Afrique du Sud.
Lo cadcation de MM. Davis, Ducketein, Kisiel et Fogel /2]
traite du problhe de la distribution de la pkiode de retour correspondant
au déparssement d'un maxipnrm ou diun volume de crue donni, en partant des don-
nées sur les précipitations. La méthode se rapporte plut8t au type II,
d e les techniques de calcul expos6ee relhnt enti8rement de l'analyse
atatidique. La formule de transformation pluies-débits adoptée pour les
volumes de cnie e& de la forme Q = C (R - A). Dana l'analyse de &bili-
té conduite par dmulation, les auteurs ne se préoccupent que de C et trou-
vent, come il fallait s'y attendre, une énorme influence de la variance
d'estimation de ce paramètre sur la variance d'estimation de la période de
retour. I1 n'est pas pmuvé, come semblent le supposer les auteurs, que
la variance de A n'ait qu'une influence négligeable.
2.2. - kJB-hodes_de rn-1; = *&e-d-og gee ~ o ~ . g
Dane Bon sena le plus littéral, l'extension Eee d o ~ de h d6bits
consiste en l'opération suivante.
- Une variate Y &tant définie mivant le phhomene i@rologique
qui intbesse et le probl8me qu'm a à résoudre (par emmpie
Y a d&it maximal instantané de l'année) ;
- on dispose d'un échantillon de n valeurs de Y obtenues par l'Obsemation
directe des débits ;
- on dispoee d'un échantiiìon de N > n
valeurs de une 01 ph-
sieurs oaractgristiqueim cìimatoiogiques XI, U .. . Xk(pm 8xezaple :
averse dmaìe annuelle, indice de pluies antk6dentee mtte
aveme, etock de neige sur le baasin au d h t de la fonte) N
contenant n i
(StJ - on cherehe à établir une r8grulsolon multiple (au simple) entre
11 et XI, X2 ... xk (ou seulement XI) :

427
- on appïiqtm la régreseion trouvée aux Io-n valeur8 de XI ... non
contenues dane la période commune de n années ;
- on a ainsi une nouvelle ebie de N valeurs de Y, plus longue que
la &rie originale de n valeurs, laquelle on peut appliquer
11analyse statistique.
OU BDN - on &ablit un modèle déterministe de transformation pluies-débits
(par exemple hydrogramme unitaire + fonction de ruissellement) ;
- on applique ce modele aux donn&s climatologiques XI ..., et on
procède comme pour lee données reconstitubs par régression.
On doit noter que, ni par une méthode ni par l'autre, on ne :ail;
de transposition ou interpolation gbgraphique. Les modeleis, qu'ils soient
régressionsll ou de structure déterministe, sont appliquh aux bassi116
mêmes pour lesquels ils ont et6 établis.
Le problème du gain d'information se pose dans les deux cas. I1
e~t bien évident qLe le nouvel Qchantilion de n valeurs de X observées
+ N-n valeurs de X calculés n'est pas équivalent, du point de vue quantité
d'information, à un échantillon de N valeurc de X observées, mais '1 un &chan-
tillon de NI valeurs, avec n < Nu < N. Si on a procédé par régressions et
qu'on se soit m angé pour que ces régressions répondent à peu près aux con-
ditions suivantes :
- homsc8aasticit6,
- linéarité,
- distributions marginales n odes, en opérant bei changements
de vexiables ou dea anamorphoeesp le gain d'information, c'est-d-dire la wan-
tit6 NI-n, peut être facilement &du61 en comparant les variance8 des estima-
tiom.
Si on a utili& un modèle d&erministe, cette &elmtian n'est pas
imgdiate. Ii est nécetseaire de rechercher empiriquement la loi des écarts
rkiduels, ou la corrélation entre valeurs obeervbe et vele~~W calculées

42 8
en se baeent ~ u1 lee n ann&s d'obmrvatioiiei conmnme0. il faut dire que bien
souvent on ne fait pas cette remherche et on se psisee de 1*6valuation des in-
tervallee de confiance.
L'avantage du modale déterministe, (notamment de ì'hydrolp.Emime uni-
taira, est de fournir la totalité de l'hydrogramme de amel donc simultané-
ment le débit maximal, le volume ruisselé et la fome. Taiidie que ces Biéments
(au moins débits maximaux et volumes) doivent &re étudiés séparbment par une
méthode strictement 1% régressionet'.
On peut élargir la notion d'extension des données en coneidérant non
plus In totalité de l@échantillon des Xi, mais des ensembles de valeurs de ces
données sélectionnée par des critères statistiques (averse de fr6quence donnée,
par exemple), ou bien cornidbée comme représentant dee situations partidi&-
rement défavorables eu égard au but poursuivi (précipitation maXimale probn-
ble, par exemple).
Cae&
une attitude très répandue, qui correspond bien à la con-
ception moderne des crues de projet. Mais elle ne - J~B pas s- poser quelques
problèmes, ourtout quand on procède par hydrogramme unitaire. h effet, sur
lfenser;ible des v:trLat?s Xi, une seule peut &tre introduite avec sa fréquence.
Supposons qu'on désigne par XI l'averse décennale et qu'on prenne pour X2 ...
X& des valeurs moyennes, ou médianes ; quelle probabilité peut-on attribuer
à la crue obtenue par application du modèle à l'ensemble des Xi ? On convient
souvent que la probabilité de la crue est la même que celle de l'averse.
C'est certainanent faux, mais dans quelle mesure ?
M. Beran, ciana la c odcation qu'il noua soumet IlIltente de ré-
pondre a cette question. La e.ynthèse qu'il propose pour 1'Wdrogmme e& tout
à fait classique. Le choix de l'averse est fait à partir de ia reìation
hauteur-dicrée-fréquence. La distribution des crues obtenues à partir de cette
averse est étudiée par une simulation effectub pour toutes les combinaisons
possibles d'un choix de
- 12 valeurs de 1~ durée de l@averBe,
- 3G 8Ch&~ de hyétogmrmee,
- 12 valeurs d'un indice d'humidité du bassin.

La conclusion de l'autemr eat que, ai l'on utildm dee valeurs
m8dianes pour la durée de Le pluie, la r-ition de celle-ci au esin de
l'averse, le taux d'infiltration, la crue obtenue a une fr6quence voiaine
de celle qui a 6tB choidc pour la hauteur de precipitation. La forme àu
hyéto,p-amme ne semble pas jouer un grand Ale.
429
MM. Kitmaita et Haehimoto 171 exposW?i'au Japon on part de l'ana-
lyse statistique des prboipitations de deux jours pour lee petite bassins,
ou de trois JOWS pour les grands. Lee auteurs attachent une très grande
importance à la distribution de la ?hie à l'intérieur de ces intervalles ;
l'étudc de cette ustribution sat rai'.? AU RB de temps horaire et ibdécri-
vent une méthode d'élaboration du hy6tograrame de projet qui met en jeu un
"facteur d'ag-an disse ment^^ et un hyétogramme dit lfreprésentatifl1 qui n'est
autre qu'un hyétogrme naturel observé lors d'une averse récente. Autrement
dit, on sélectionne une "formea* qu'on applique à la hauteur de pluie déter-
minée par l'analyse statistique.
Pour cette détermination, la période de retour choisie dbend en
fait du type c'e projet et des conditions économiques, sociales et politiques
dans lesquelles il est envisagé. La durée de cette période est de cinq à sent
ans pour un projet d'égout, de vi.?& ans pour un petit bassin urbdn, de
cent rms pow un projet sur une gl

430
se aont tous cantonnb daOs deux aspects particuliers du problème : la trans-
position de l'hydmgramme unitaire et l'utilisation de formules régionales
dbivées de la méthode rationnelle.
Noue rappelerone que, parmi d'autres, on peut considher la méthode
des courbes enveloppes comme une methode de tramposition géographique, quand
elle est assortie d'une "formule de r6gionalisation", come c'eet le cas
pour les abaques de I'ranmu-Rodier (Cahier6 O.R.S.T.O.I., ewe hydrologie).
D'autres méthodee pourront 8tre conatmites H partir du catalogue des crues
exceptionnelles de l'U.N.E.S.C.O., lorsque celui-ci pourra enfin voir le jour.
De tels catalogues, lorsqu'ile comportent dee descriptions sufff-
ates dee cara& Bristiquee climatiques et gbmrphologiquee du baaseiin (sans
toutefoia trop compliquer lee C~OSOE)~ constitueraient par leur sede exis-
tence un outil de tout prender choix pour l'bvaluation dee crues en l'absence
de &m&s insuffisantee. Cseat m8me à vrai dire la seule choae qui actuelle-
ment fasse vrdnent ahfaut.
Rappelons enfin que la transposition des loi8 de distributions des
crues pourrait atre traitée BOU cette rubrique. Nous avons pr8î8i.h en parler
6 propos de l'analyse etatietique,davantage pour uno queetion de m&hodologie
que dans un souci de préeentation logique.
L'hydr-ogranme unitaire paraît $tre encore, malgré ses dé-
tracteurs, un inetment de choix pour l'evaluation des crues mar lee petits
baseins. Noue n'ailone pas ici emtemer une foie de plue une äi~cuseion am
la d6finition de ce derni- terme. Noue amne d6jà parlé de e m utilisation
à un mame batwin, il s'agit maintenant ae voir comment on peut trawposer
lee résuitate.
Cette traaepdition est eseentielle dans la m6thoùologie
ConcermELllt lee cruce dee petit8 bamuins. ûn 88 bute bien qu'il n'est pas
poesible d'entretenir des rbaux de longue dude sur la totalité des petits
bassiris d'un pays. I1 n'est m8me pas toujours possible, pour chaque petit
projet, de mettre en oeuvre sur le bassin correspondant des observations
d'une densité suffisante, pendant une dur& suffieante pur l'application

de l'bydrogrme unitaire au bassin lul-m&ue.
431
Pour lee petits bassins, l'inauffisance, et mbe l'absen-
ce totale de donnbs au lieu d'utilisation, est donc la &gte. La préparation
des données hydrologiques pour les projets consiste donc à échantillonner
un certain nombre de basains, dits reprbentatifs, correepondaat B un nombre
suffisant de conditione climatiques et morphologiques. Ce erant lee résultats
recueillis sur ces baasins qui permettent la mise en oeuvre des différentes
méthodes de trampsition.
H. ~~äier, sa communication [Il] , présente ia mho-
dologie mise en oeuvre par 1'O.R.S.T.O.M. pour les paye tropicaux. Cette mé-
thodologie est bask sur les résultats obtenus par l'exploitation, pendant
des durées égales ou supérieures A trois ans, de plus de lo0 ensembles de
bassins représentatifs. Elle ee rapporte surtout aux zones mahéliemes et
tropicales, mais des résultats sont ¿galement disponible8 pour lea zones dé-
sertiques et pour lee zonee équatoriales.
Les parametree sélectionnés pour représenter la forme de
l%ydrograme eant :
- le temps dû bue (Tb) OU durée du d,esûllment,
- le temps de montée (GI,
- le rapport K du d6bit de pointe au d.bit moyen de lib-
drogranune de ruissellement.
Le volume ruisselé est évalué à partir de la hauteur totale
de l'averee par l'intermédiaire d'un coefficient de ruissellement i$.
L'analyse des rewiltats disponibles a permis de lier les
paramètres %, l& et i à certaines caractéristiques gbamorphologiques du
bassin, soit :
- la Burface du bassin,
- une clame de relief (R) d8terminb A partir d'un indice
-
de pente,
une clame de pennbbilit6 6valub 4 lleetime.

Lee relations sont prkentbee SOUS fome d'abaques. Suivant
l'&umbation ci-deesus, ces abaques ne tiennent pas compte explicitement du
rdle pourtant importaut de la couverture végétale. C'est que, dam les régions
étudiée^, cette couverture v¿&ale abend eaeentielleiacsnt de la zone clima-
tique dane laquelle se t mwe le bmsin. Come les jeux d'abaque eont établis
par Bones climatiques, loensemble tient compte simplicitemciait dea conditions
de végétation. Dane lee cas particuliers, il cmvient au epéciaìiste d'gvaluer
ie ltcoup de pouce" à donner pour tenir compte d'une anomalie de ce8 cmditione.
Le coefficient de fome K est &du6 mivant les zones
climatiques, la eurface du baesin, mu8 forme de tableau.
Les abaques fournissent des valeur6 moyennes de6 parmètres
c,iLL ALI^^..: L. 1x1 ir; q 3vc- 'iverse äécr-~qal-.
ans une communication Is] consacrée e o u t à ilinfìuence
du degré d'urbanisation eur les cme~ des petits bassine, M. Hall propose
une méthode de r6gionalisation des hyärogrmmes unitaires. I1 part d'un hydro-
grme unitaire dimension en utilisant comme paramètre d'khelle des
-- *
le temps 'r retctrd l i (la?). TL est dors exprimé en fonction du
llrapport de bessin" Z O L /p, où 2 e& en km, L est la longueur du cour6
d'eau principal, en km, et S la pente moyenne du cours principal en %.
W. Eelliwell et Chen, dans leur c odcation [4] présen-
tent 6gaìsment une m6thode de traneposition r8gionale b a h ar un bydrogrme
BBPB dimmion. Leur pmblhe e& absolument typique du cbantp dgapplication de
la m6thode de l'bydmgramme unitaire. Les rivières de la colonie de Hong Kong
sont très nombreueee pour un ai pctff territah (1 o00 kd de terres) et la
taille de leure baeeina est hidemuent td6 reduite. Il n'est pas concevable
d'utiliser dane ces conäitions la fermule classique du réseau hydrologique.
Lea hydrologues de Hong Kong ont donc s6lectionub quelques
baseins reprbentatife dont ilpl ont 6tudi6 en détail le comportement hydrolo-
gique. Le8 auteurs dbivent lee mathodee d'analyse utiliekm qui mnt d'ail-
leiir8 trh claesiques, sauf que le pa^ de tamps tde caurt nécesmire (15 am
ou mine) a créé quelques difncuitb par euite de la ree&¿ de6 enregistre-
ments pluviographiquem exploitables. L'hydmgramme aan~ dimension eet obtenu,
comme chez Hail, en multipliant les ordannies par le Lag, en divisant les

abscisses par le h g et Bll r ~ e le ~ tout t un volume unité.
433
L'analyse a conduit, pour chaque baeain btudi6, à un hydro-
graimne unitaire moy'en, dont le tempe de retard (Lag) a 6th mie en relation
avec l'indice L L, /r8, 0ii E est la longueur de la rivière principale, Lc
la distance le long de cette rivière entre l'exutoire et le point le plus
près du centre du bsrasin, 9 la pente moyenne du cour8 principal. Le Ia[:
a été mio auSei en relation avec la siarface du baeain, et cette régression
donne du rede um meilleure corrélation (0,92 contre O,%).
de LI méthode mt;.onnelle.
Touteo les formulee prbentkei par les auteurs sont d&riv&s
EIM. Jarmwathma et Pinkayan dkrivent dana leur rapport [6]
les abthode6 de d cul dee crues utiliebea en Thai'lande pour les petits bas-
si-. Apre6 avoir rappelé la pauvreté dea donnke ditiponiblee dane ce pays,
ils adreseent quelques crítiquea à ia formuìe rationnelle cl as rip^^
Q P C i A et lui préfèrent la fozmule de Mc Math Q p: A C i @/A) "' qui in-
trodult la pente du tasdn.
La codcation de M. Pereira 9 damie entre autres qual-
que8 indíaatione mur i'utilieation de la methode rationnelle au Erbil, notam-
ment den valeurs du coefficient de miesellement. L'auteur y donne &alement
dea rsnseignsme&e Bur le@ tsmpe de retour dOpt68 dane ca, pp d-raE le
type de l*amhgement et l'erivimmemnt, a r l'intensité dee pluies au &&fi,
BUF lea P.W.P.,sur l*eatimation des volumes N imelb à partir deer prkipita-
tio-
en U.R.S.S.,
(fonmile du Soi1 Conservation Service).
d e
M. Sokalov 12 hoque l'emploi de l'hydrogrmme unitaire
fait une place plue large & la méthode rationnelle, ainsi
qu'à des foruniLee empiriques de la forme

434
où sax est le débit maximal spécifique en m 3/s.km 2 , q un paramètre qui exprime
le débit spécifique extrême lorsque la gurface A du bassin tend vers zéro.
C est en fait égal à 1 ; n varie de 0,15 - 0,30 pour une crue de fonte de neige
à O,5 - O,7 pour les crues dues à de violentes averses locales.
M. Won [ 141 expose les méthodes utilisées en République de
Corée. I1 propose une formule qui procède .?i la fois de l'hydrogramme global
(sinon utilitaire) et de la méthode rationnelle :
- = CY A R/T
qo
dans laquelle g, est le débit maximal, qo le débit avant la crue, C un coeffi-
cient de forme de l'hydrogramme, 9 le coefficient de ruissellement moyen, A la
surface du bassin, R la pluie totale, T la durée de la crue. I1 propose d'autres
formules concernant le temps de concentration, la courbe intensité-durée, la
durée critique de la pluie (t =

435
M. Rendon Herrero a choisi, pour sa codcation bo] un eujet
bien particulier. Il s'agit du transport de sédiments en suspension €tudi6
à l'échelle de l'averse. L'auteur met d'abord l'accent sur l'importance des
apports latéraux de sediments (Washload), mitit par l'brosion en nappe (sheet),
soit par le ravinement (gully) par rapport aux dát&iaux du lit mis en jeu
durant le transport. Lee r6sultatcs sont interpr8t6e par des techniques ana-
logues & celles de l'hydrogranmie unitaire (s6dimentopame unitaire). Une
application est faite au bassin de Bixler Run (U.S.A.)
CONCLUSION
I1 eet certes Intbeesant de mettre au point des méthodes d'analyse
de plus en plw bborées pour essayer d'amocher le moina mal possible les
caract6riatiques des crues et des basses eaux, lorsqu'on dispoae de donnbs
rarea ou peu précisoa. Xais il ne faut pas trop se faire d'illueion BUT la
portée réelle de cette tentative, ni oublier que toute la confiance qu'on
peut attribuer d une eetimation rénide dana la quantité d'information , c'est-
&-dire finalement dane la masee et la qualité des donnéee diqmniblee. La com-
titution de cette infomation n'est pai3 te&&, il est faia de dire qu'elle
ne pose plus de probl8mes.
Ea matière de cruet3 par exsmple, ce qui fait le plus défaut danri la
plupart des paye, e out lorsque les rivihs y epnt difficiles, torrentielles
et inetables, c'est une bonne connei~aance des débits dee plus grandes crues
connues. L'organisation d'un service hydrologique efficace n'est pae une petite
affaire : elle demande une grande compbtence, un soin de tous les instants et
une certaine aportivité. Elle demande awai de l'argent et c'eet L4 que rbide
souvent la plus gnrnde difficult).
le
Notse conolwion aera donc que/meiUeur moyen de suppleer d la ca-
rence dee donube hyàrologiques est encore de s'attacher à la Buppremion, ou
tout au moins d la diminution de cette carence.

436
111 - M.A. Bleuw (kglanä)
E.timation of dedp floods and the problema of equating the
probability of rainfall and runoff -
R. DAVIS,bCKSTFZN, C. KIISIEL, N. Fo(zEL (U.S.A.)
A decision - theoretic approach to uncertainty in the return
period of maximum now volumee using rainfall data -
131 M.J. HALL (U.K.)
Synthetic dthydrograph technique for the design of flood
alïepiatlon works in urban areas -
[4] P.R. EEUWIEL, T.H. CEW (~ong-~ozy)
A dimeriPiionlEss unitgraph for Hong-Kong -
[5] P.E. HERBBT, S. VAN BiLúûN, J.P.J OLMER, J.H. HAIL (South Africa) -
Flood estimation bp determination of regional parameters from
limited data -
r
[GI D. JWATHANA, S. PINKAYAN (Thailand)
Practice8 of design flood frequency for epiall Watereheds in
!l%arland -
171 T. K0IWSITA, T. HAsHuIoTo (Japan)
-
Design diecharge derived from design rainXall -
[8] W.N. LEEBE (U.K.)
The um of asmoreci data in estimating T - y ~ar floode
-
-
L91 P.P. Pm!uzl?A (Brasil)
Amesment of deeign noode in bradl -
Eo] o. RIBIDON mmmo (U.S.A.)
-
A method for
-
the prediction of W o a d in certain d l
wateraheäm
FI] J.A. ROD= (fime)
Méthodes utilidee gour l'kaiuation dee dbits
-
de crue des
petits cour6 d'eau en r8gione tmpicalee

[la I A.A. SOICOIDI7 (U.S.S.R.)
Methods for the estimationa of meximum dischargea of snowmelt
and rainfall water vith inaàequate observational &ta -
Ilq A.M. VLADMIROV, A.1. CHEBOTARGv (U.S.S.R.)
Computation of pmbabilietic values of lot flow for ungaugeü
rivers -
T.B. WON (Korea)
A study on maximum flood discharge fondee -
437

ESTIMATION OF FLOODS BY MEANS OF THEIR SILT LOADS
ABSTRACT
Modesto Batlle Girona
Dr. Civil Engineer
An empirical and experimental formula of very simple
structure is studied, to obtain €he flows of maximum floods in
relation to the sediment loads that the floods produce, depen-
ding only of the maximum size of aridities of the channel.
This formula can be useful to study also the behaviour of the
river bed, alluvial volume, and so on.
Se estudia una fórmula empírica y experimental de es
tructura muy simple, para obtener los caudales de máximas cre-
cidas en función de los arrastres que éstas producen, depen-
diendo Únicamente del tamaño máximo de ’aridos del cauce. Esta
fórmula puede ser Útil también para estudiar el comportamiento
del lecho de los rios, volumen de acarreos, etc.

440
ESTIMATION OF FLOODS BY MEANS OF TIIliIR SILT LOADS
Based on the physical fact that every flood deposits
a mass of arids whose maximum cliametres are proportional to
the magnitude of the flood, by means of a reciprocal process
an attempt was made to find a way of estimating tne discharge,
obscrving the silt loads produced,
Accordingly a very simple network formula has been
obtained. In a series of 15 tests, the prevision of the
maximum floods that have occurred could be made (correspond-
ing to a return period between 100 and 500 years) WITH Ah'
LRKOK bELOW 13%. To do so, one merely has to know the maximum
size of the river-bed arids.
Besides being a new instrument to calculate floods,
this formula may, as indicated iii the "Summary" , open up an
interesting field of investigation regarding mobility of the
river beds, volume of bed-loads, etc,
1. - FORMULA f'llOPOSEI3
1.1. WORK SCHEME, -
On the one hand, the maximum silt load diametre
is function of the flood Jischargc.
On the other hand, the arids are moved by the
force of the silt load, wnicii is proportional to the gradient
arid to tho draft.
Considering the above two factors, a formula was
sought which related the diametre of the deposited arid, with
the draft and gradient. The mathematical deduction of this
relation is however inaccessible and a semi-empirical formula
was sought, verifying it and deducting the unknown values of
same, by neans of experimentation.
Once a relation was obtained between the diametre
of the arids, tlie.gradiarit aiid the draft, this could be
deducted from the previous ones, thus defining the maximum
level obtained by the flood waters. Since the bed-section
is also known, the discharge of the flood which has borne
along the arid through this section, depositing it immediately
downstream, can moreover be obtained.

441
Once the purpose of the study was specified,
a formula liad to be proposed which would relate maximum
diametre-draft-gradient. The probing was systematized,
and the proposed formula was verified and as already
mentioned, the unknown coefficients of same wcre verified
with a series of 15 samples or tests. The margin of error
obtained was found and compared with other existing methods,
The return period of the flood-waters calculated with the
formula, was sought, defining an inferior limit, The possible
limitations of the formula due to the petrography of the arids,
the morphology of the basins used or the non-existence of
certain sizes of arid, were studicd. Finally, the conclusions
drawn are summarized, All the documentation involvcd in the
tests, regarding diametres of arids and drafts observed, was
collected photograpliical ly .
1.2. - MATilEMATIC OBTENTION OF TiíE PROPOSGL) FOIMULA
Une tried to reach a formula, deducing it mathe-
matically from the silt load force equàtions, but the
influence on the larger arids cannot be defined quantitatively,
nor can the protector inter-action which the silt loads offer
between themselves, in the face of the dynamic thrust of the
current.
Various hypothesis were used, but the subsequent
elaboration did not crystallize into any practical formula.
Ori the other hand, adopting one or another hypothesis as base,
produced inadmissible differences of above 100%.
In view of the above, it was decided to employ a
semi-empirical formula, Its structure was obtained matliematically
but it has been verified and defined from the experiments made.
1.3. - UL.1)UCING A SEMI-EMPIRICAL FORMULA
To obtain the formula in question, various methods
were applied: a) .- considering the dynamic thrust on the arid;
b).- balancing the silt load forces, The liermanek aiid
Manning formulae were likewise used in one case or the other
to determine the mean velocity,
When using the Manning formula, the possibility was
considered of the rugosity of the bed 'In" b e in g prop or t i ona 1
to one sixth the power of the arid diametre. ïliis is correct
in canals, but ir1 natural beds, the most accepted formulae of
river hydraulics (Ilermanek, Christen, Wiiikel, etc.) do without
the rugosity or adopt a constant value of same.
Equations were reached through different channels,
with identical structure:
0b
Ila = -.
u. 1

442
but in which the exponents a and b varied in terms of
the velocity distribution law, adopted. The degree of
parabolic speed distribution is normally one seventh,
and it was with this value that a arid b were calculated.
The values of "a" and "b" were also deduced in the hypo-
thesis of supposing a one ninth degree distribution.
ïiie results were:
Degree of the speed
distribution parabola 1/9 1/7
E X P O N E N T a b a b
Hermane k 1,28 O,78 1,21 0,71
Manning: n = Cte
Manning: n = Cte 0
10
- Manning: 11 = Lte
Manning: 11 Cte 0 1/6
1,11 0,78 1,OS
1~11 1#11
1,13 1,OO 1,O8
1,13 1,33 1,08
0,72
1#O5
1,OO
1,33
Nevertheless, the empirical
-
formula proposed and
verified with tests, was, as will be seen later on:
00,s
. 11
u.i
The formulae obtained in the previous table (where
the later experimental definition of B) is required,)
iiave no faithful tradition in the reality of the river
beds, as they give different values to those of the
said formula where "a" =
The most approximate
In this, m = 7.
1.4, - DLFINITION OF THE PROPOSED FORMULA
The formula we are trying to verify, wkeby on
experimentam obtaining the value of B, this value
should be practically constant, was:
1/2
ii = am
13. i

II = Draft expressed in centimetres,
flm = blaximum diametre of the arid in centimetres,
i = Gradient
ki = Coefficient to be defined.
44 3
In each test, knowing the draft 11, obtained directly
or from the discharge, in a sector, wliicli was termed "control",
tlie 0 of tlie arids deposited in the area, if possible
immedyately downstream of same, was defined, calculating:
i/ 2
B = am
li. i
The series of 15 tests undertaken, permitted a verification
that coefficient B is almost constant; its value could be
calculated, and at the same time tlie exponent 1/2 of 0 was
found to be most suitable.
m
2.- RESULTS OBTAINED, -
2.1. - MET1iOL)OLOGY
First of all, a "control section" must be defined. Knowing
the maximum historic flood in a prudential period, the draft
corresporiding to H was calculated in it. This iì of the control
section, was in certain cases measured directly after some
important flood, through the traces of undergrowth and residues
that the current left in tlie river bed shrubbery 0. The gradient
i of the span corresponding to the control section was sought,
and the maximum diametre 0 of the arids deposited downstream
in this section was meacurgd. With this information, the
following was calculated: 1/2
B = k'm tl, i
Control Section:
The control section should be sited in stretches with as
uniform system as possible. Consider the influence of bridges,
etc. The arids should clearly define 6,.
Gradients:
The "i" adopted, is that of the stretch from 1 to 1,s kms.,
immediately upstream from the control section, It is obtained
from plan 1:50.000 of the Geographic and Cadastral Institute,
rounding off, if iiecessary, the excessive twists in the
longitudinal section.
- Draft:
The "H" draft is measured from the lowest reading of tlie
control section,

444
izlaximiim diametre:
The arids probe area for defining Øm will be chosen
downstream the "cnntrol section" and as ncar to this as
possible, so that the sizes observed are effectively the
largest that have passed through this section,
'To define the maximum diainetre, only those rounded
or parallel-epipedical shape arids will be suitable , whose
smallest dimension is at least 2/3 (two thirds) the largest
orie, The 0 value will be the mean of the largest two
dimensions of the arid.
The suitable arids for defining 0 should be found
with a minimum density of 1 per every pwo square metres.
Uy density, we understand the number of units in sight,
per river bed surface, In some cases, this may even drop
to 1 per 4 square metres.
The choice of arids on which the 0 is going to be
measured, demands a careful, critical judgement on same,
considering the possibility of it coming from the erosioned
sides and not upstream, that they may pertain to demolished
works, or under construction, etc, Certain geological know-
legge of the area will always prove most useful. Any kind
of rock excepting slate is suitable.
It should be emphasized that in minimum densities ,
relation 2/3 of dimensions, etc., a qualitative common sense
should always preside over ali rigorist criterion.
It is important to take photographs with scales which
will act as referenee so as to compare the field observations
at the office.
One must remember that the problem consists in a trans-
port through the control section of the arids, which will be
deposited immediately afterwards, and that they may even be
covered by other finer ones, which have settled when the flood
waters dropped.
Qu a 1 i t y :
To have a p;rudeiit judgement of the suitability of the
tests made, they have been classified into GOOD (G), MEDIUM (M)
and FAIR (F) , in accordance with the guarantee deserved by
the definition of the data obtained li, am and i.
pemarks :
1.- In low river-bed spans, with very slight gradient, it
is often difficult to define this on plans 1:50.000
and ori a smaller scale, the bottom oscillations are
excessive. In these cases, the river is also usually

very wide, and the transversal section presents siiarp
relative off-levels. In this case, the definition of 11
must be closely examined, to avoid falling into errors.
2.- In some cases, owing to the type of alluvial terrace
in wliicli the bed is fouiid, the prehistoric and millenary
arids cannot be differentiated from those corresponding
ti1 the latest historic floods.
2.2. KLSULTS OF TIIE TESTS.-
Below, the data i, 1i and id obtained in each test, anci
the resulting value of B are given:
Test Nomenclature and Site i 11 0 yual Coeff
;.i O ím) (CH) ity icient b
445
1 K. Llémana at crossroads 0,00822 2,81 18 M 1,84
Main road S,Martfn de
Llémaria
2 R. Llémana in Sta, Afra 0,00590 3,45 19 li 2,13
3 ‘ler in S.Julián de Kamis 0,00725 3,73 29 F 2,OO
4 Ter in S.Julián de Kamis 0,00725 5,63 60 I: 1,90
5
6
Ter in outlet of the Dar6
Uñar at crossroads N-II
Madri d - F r an c e li i g hw ay
0,00100
0,00300
7,93
5,42
25
10
F
G
2,OO
1,94
7 R.Uerneda at bridge Riudellots,maximum
flood 0,00237 5,lO 8 G 2,33
8 K.berneda at bridge Riudellots,
flood 11-X-70 0,00237 4,OO 5 G 2,05
9 Tordera in Sai1 Celoni 0,00835 2,52 16 G 1,90
10 Tordera in Tordera 0,00320 2,85 4 G 2,20
11 R. Rifer in San Celoni 0,00120 1,05 7.5 G 2 , 16
12 Corigost in La Carriga 0,00880 2,90 28 M 2,07
13 Cardoner in Manresa 0,00435 4,65 18 G 2,09
14 Llobregat in S.Vicente
de Castellet 0,00345 5,85 22 G 2,32
15 Llobregat in Martorell 0,00209 7,65 10 G 1,98

446
2.3.- VERIFYING THE EXPONENT OF QIm.-
In principle, the value 1 was adopteù as exponent of OmD
proposing tiie formula: z
-
0,s
am
li
r
ìiowever, the value 0,s was later esteemed and its worth
confirmed as the tests were made, owing to the scanty dispersion
presented by the values of B obtained.
The possibility of the dispersion of the resulting 13 being
less with another exponent , is consequently likely.
dased on the values i, li and 0,,, obtained in the tests, the
b coefficient has been calculated for the following exponents
of 0,,,: 0,4ü - 0,45 - 0,50 - 0,55 and 0,67.
,A The typical
E
e
0.55
o
u 0.50
derivations obtained are:
for B = 0, 0140/i H - T = 0,195
for U = 0, i 11 -G = 0,175
for B = 0m H - G = 0,143
for U = 0, li - TT = 0,189
8 0.45
for B = 0, OaU7/i H - o = 0,484
J
0.40..
- Desviación típico
. 0.2 0.3 0.4 0.5
0.35
1 7
0.1
In the figure, the aforementioned results are represented,
and it can be seen how the minimum typical deviation corresponds
to the 0,s thus verifying that this value is the most suitable
as exponent of 0,.
2.4 - CALCULATING THE COEFFICIENT U. -
Having obtained the value of U for each of the 15 tests madc,
what is proposed for the famula in question must be defined.
In order to consider the quality of each test in the determination
of B, by means of a prodent mean, the ones qualified as good
(E) are assigned weight 3, the "mediumtv (M) , are given weight 2,
and the "fair" (F) , weight 1. Tnus we get:
B = 2,08lY2,08

-
The value of the coefficient L4 adopted is:
i3 2,08
Whereby the formula proposed will be:
2,08 . i
li = maximum draft in centimetres (see 2.1)
Bm= maximum diametre in centimetres (see 2.1)
i = gradient (see 2.1)
2.5 SIGNIFICATION wrE OF TIE TEST SERIES.-
447
The "signification rate" of tlie test series made, or in other
words tlie quality of the whole ensemble of same and the mean i =
2,08 was obtained. Accordingly, tiic likiliood of another value
of B, obtained as a result of a new series of tests, being
within specific limits, was calculated.
- (mean) aiid ';r (typical deviation) be the monthly
characbgigtics of the series of Values of B obtained; tlie number
of representative tests is n = 15.
In tiie interval of possible values of B included between:
cr
and Y + tp .-
if we choose a percentage of probability P (ex. p = 1%) where
tiie value of the medn U of a new series of tests is outside these
limits, in the table of function of Student for this 1% and n-1
= 14 degrees of freedom, a value of tp is obtained with which
the above mentioned "confidence interval" is defined. The
probability p is the "signification level".
-
x = 2,OG; q= 0,143
,* -
- = 0,03823
m
According to tlie Student t8t" function table, we get:

448
sigriification Value of tp
Confidence limits
G
level p for 14 degrees t p . m Be 1 ow Above
of freedom
i¿-tp.m G - X*tp.fl=$y o-
~~~~~ ~
__ -~ ~~~~~
0,l % 4,140 0,16 1,90 2,22
1,o % 2,977 1,11 1,95 2,17
2,o % 2,624 0,lO 1,96 2,16
5,o % 2,145 0,08 1,98 2,13
Thus, the probability that the mean of a new test series
is between 1,95 and 2,17 is 99%.
Let us set this conclusion out in terms more befitting
our problem.
The confidence interval between 1,95 and 2,17 admits a
possibility below 1 %, that the 13 obtained in a new series is
1,95; this represents a discrepancy of 0,11 in respect of the
mean of 2,OG of the experimented series.
This difference of 0,11 means an error of
o 11 I
2,06 *'Oo' = 5B3 %
in the appreciation of the drafts. Let us see how much this is
?illen translated into flood water discharges:
and if the section is approximately rectangular T I1 , Whereby:
Ob7' i1l2,ki.b = 0,34 b.i 1/2 , 1i1,75
Qi = 0,34 li ,
Thus an erro,,i2 draft of 5,3% multiplies the discharge
obtained by 1,053 = 1,095 which means an error of 9,5%.
Therefore, the quality of the series of tests made and
consequently that of the B value adopted, can be defined as
follows:
The probability
-
that in a new series of tests, the value
of B defined for the formula proposed means an error in the
determination of discharges, below 9,5%, in respect of those
obtained with B 2,08, is 99%.

2.6. - MAXIMUM ERROR AND COMPARISON WITH OTIIER METIfODS. -
In practice, to determine the maximum historic flood,
applying the proposed formula, a certain number of tests
should be made in certain other control sections of the bed,
and finally obtain a mean, In this way, the inevitable errors
and discrepances will be compensated for, and which will take
place on defining the gradient (i) and in particular the
maximum diametre (e,) with which to enter into the formula.
The number of verifications will depend on the exactness
to be obtained, and on common sense, in face of the data
defined in each "control section". On an average, four tests
may prove sufficient.
Supposing that all the tests made correspond to a same
bed, we can find the maximum possible error by mixing
together the results of the whole series.
The five tests with highest B values are 2,33
2,20 -2,16 - - 2,32 -
2,13; the mean is B 02~23.
Similarly, the 5 tests with lowest B values are 1,90 -
1,34 - 1,98 - 2,OS - 2,09; the mean is B = 1,99.
In both series, those tests classed as "medium" (M) and
"fair" (F) have been omitted,
The difference between these means and the value B = 2,08
of the formula is:
2,23 - 2,08 a 0,15
2,08 - 1,99 = 0,09
Considering the most unfavourable case where the tests
have given the 5 highest values of 8, whereby their mean would
be B = 2,23 instead of U = 2,08, this means an error in draft
appreciation of ;.tW O 15 .lo0 = 7.2%.
-
As seen in 2.5, this indicates that the real discharge
has been multiplied by:
1 , 0 7 2 ~ ~ 1,129 ~ ~ 1,13
which means a 13% error,
To conclude: THE MAXIMUM ERROR OBTAINED ~ViiliN ESTIMATING
'ï1ii.i IIISTORIC FLOOD DISCHARGE, ACCORDING TO VERIFICATIONS MADG
WITH THE PROPOSED FORMULA, IS 13%.
Other calculation methods:
Wit11 t h statistical method, the errors for return periods
449

450
above 50 years, may be around 20 to 3090.
Specifically in the floods study of the Congost river
made at the liydrographic Confederation of tlie East Pyrenees ,
by the Measurerncnts Service, using the historia method, and
for a 50 year return period, a discharge of 160 m3./sec. is
obtained with the Gumbel method and 135 rn3./sec. with another
law of distribution, which means a difference of 20% between
iaoth methods. Applying the tational method to the same study,
biie gets a discharge of 305 m3./sec. for the same return period.
2.7. - RETURN PERIOD
The return period of the floods of the various tests
was also studied, to try and relate it with the discharges
foreseen with the formula, and at least obtain a lower limit
of samc.
The return period of the floods for which the formula
has been verified is:
Test 1 -----__----_-_- 400 years
Test 2 -------_-_---_- 400 years
Test 3 -_----_----_--- 95 years
Test 4 --------------- 95 years
Test 5 --_------_---_- 95 years
Test 7 -_------------- >70 years
Test 8 -__------_----- >70 years
Test 9 ---------_----- 90 years
Test Il--------------- 1000 years
Test 12--------------- 160 years
Test 13--------------- 180 years
Test 14--------------- 180 years
Test IS--------------- 180 years
In face of these figures, it would appear that the lower
limit of tlie return period is 100 years,
ilowever, it must be remembered that this conclusion is
based on a series of i3 tests.
As indicated in 2.8, the influence of wear throughout time
is very scarce and does not change the return period of the
floodwaters,
It can therefore be said that the fbod obtained generally
has a return period between 100 and 500 years.

2.3. - STUDY OF TIIE POSSIBLE LIMITATIONS
Due to the petrography of the arids observed and their
possible erosion:
There is a possibility that the determination of the
maximum diametre d has only been made with certain rock
types, as the othefs were excessively worn by the erosion,
If this has occurred, in other areas where there are no
arids of the most resistent types, false results of 0,
could be obtained,
To approach this problem, the petrographic classification
of the arids was made, based on the photos obtained
in the determination of the 0, of cach test, and which
defined 0, with the following symbolics:
A - Sandstone Co - Conglomerates Gn - gnes
B - Basalts G - Granite and P - Slate
C - Limestone
eruptive rock Q - Quarzite
The types of arid used in cach determination of the Ibrn
were:
451
‘rest 6 ------------ 2A + 2G + 3Q
Test 7 ------------- A + B + 2Co+ G + P + Q
Test 8 ------------ 3B + 2G
Test lo------------ (2A + B + CO + 2G * 34
(A + B + G + 4Gn + 2A.
Test Il------------ íG + 34
(3A + 44
Test 12- - - - - - - - - - - - 2A + 2G + Cn + 2q
Test IS------------ SA + 2C + 2G + 3Gn + Q
Test 14------------ 2A + 4C
Test 15------------ SA + 14C + 24
Test lb------------ (2A + 2C + Gn
(2C + P
Making a tctal used of: 26A + 13U + 32C + 3Co + 16G +
+ 9Gn + 22Q
In view of the above results! evidently the rock type
does not influence the determination of the diametre, since
all the classes appear as maximum arids (8 ) in considerable
number (except the slates which will be dipcussed below) and
it can therefore be supposed that the erosion, for the effects
of the method adopted whep determining the 0, iii a way affects
any type of mtk.

452
This is because these arids are not from the bed downstream
and they consequently only suffer the effects of the river
erosion with high waters or normal flooding, which take place
intermittently arid not very frequently. On the other hand, the
large adjustment obtained in the value of B makes one suppose
the influence of the erosion in tlie various arids could not be
important.
It must however be emphasized tiiat the slate, as definers
of tlie 0 only appear twice, as a logical consequence of their
greater tensitivity to the environment, as is deduced from the
above table. For this reason, what lias been said in the above
two paragraphs cannot be applied to the slate arids.
We can consequently say that the definition of Om, is
independent of the arid petrography except in the case of slates,
which should not be used for the determination.
ihe to the region studied:
All the tests made have been in the provinces of Barcelona
and Gerona. Therefore extrapolation to another type of basin
could present doubts,
In the enclosed table , the chief geographic characteristics
of the tested basins are indicated.
However, it should be stressed that the only thing that
can change the validness of the proposed formula, is the arid
which defines 0, and according to the above paragraph, it has
been considered that the formula is valid for any type of
rock (except slate).
On the other hand, although not definite, it is most
significant that in a later test made in the Guaro river of
tlie basin in Southern Spain, near Vélez-Malaga, the value of
the U coefficient obtained is:
-
B = 2,14
whereas that adopted is B 2,08 which reprecents a 3% error.
Although not conclusive, this result opeiis up a hopeful field,
awaiting an extension of the tests to other regions.
üue to the lack of arids:
Thecase may arise of there being no arids of a diametre
superior to a certain size, as they do not exist in the bed or
because they are retained by some dam or weir. The lack of such
arids does not produce any change since the floods correct the
gradient of the river according to the existing sizes. In short
stretches where the local effect of a weir modifies tlie gradient,
this should be taken into account,
b

The formula should be applied to river-beds whose
possible mobile bottom during the flood later permits the maximum
arids deposited by the flood peak to be discovered.
If a mobile bottom of considerable thickness is produced,
by means of burrows, the thickest deposits made by the flood
peak should be reached, aiid if the sediment thickness is very
important , the gradient adopted should be corrected. However ,
none of this kind have been experienced in the tests.
453

O
4
+J
VI
P,
E-
t.4
o>
>
.FI
d
x
+J
.rl
rl
O
U
a
a
fl
3
d
0
3-
V
+
45

3.- SUMMARY
To obtain the draft (H) that has been produced in the
maximum historic floods, knowing the size of the arids (Om)
which may have been hauled along by this flood-water,
across a section termed "control", and the bed gradient (li)
the following formula is proposed:
o, 5
which should be experimentally checked when the coefficient
U is defined.
455
The above formula could not be obtained mathematically;
The most reached expressions of type:
- the one nearest the proposed one is that where a 1
0,67.
and b =
To define U, a series of 15 tests was made, obtaining
B = 2,08.
There was a possibility of the proposed formula not being
correct, which would occur if the value of B obtained in the
tests was variable. However these values ali varied around 2,33
and 1,84. The typical deviation of the series was = 0,143,
The series also helped to contrast the favourability of the
exponent of 0, since the 0,s produces the minimum typical
.devi at i on.
The "signification level" of the mean of the series of
tests made, corresponding to the "confidence interval" between
B = 1,95 and B = 2,17, is 99%. Expressed in other terms, it
means that the l ikgmd of another value of B, defined by a
new series of tests, having ari error in obtaining discharges,
less than 9,5% (regarding those obtained with B = 2,08) is 99%.
THE MAXIMUM ERROR IN DETERMINING DISCHARGES, ACCORDING TO
TIIE SERIES OF 15 TESTS, IS 13%.
The formula proposed is therefore:
O m a ~ 5
li =
2,08.i
li - Draft of the maximum historic flood in crns. defined
according to 2.1.
0,- Maximum diametre of the arid in crns. defined according to 2.1.

456
i - Gradient of the bed, defined according to 2.1.
2,08 - Coefficient with dimensions L
-l/Z
,
Tlie methodology to define these figures is indicated
in greater detail in 2.1.
The return periods of the floodwaters estimated with
the formula are generally between 100 and 500 years,
The control section is that in which i, II and the
silt loads Ibm which have crossed it, are defined.
In the river bed, whose discharge one wishes to define,
a series of tests will be made in accordance with the
precise ùegree of exactitude, and the guarantee that the data Bm
and i offer. An acceptable number may be four.
The formula is valid for any type of arid, except the
slates which should not be used to define firn.
Tlie formula was applied to beds whose mobile bottom
during the flood was sufficiently scarce to permit the
maximum arids deposited by the flood peak to be later dis-
covered. If this mobile bottom leaves the arids correspond-
ing to the maximum discharge hidden, by the smaller arids,
work may be done as indicated in 2.8, but in this study it
was not necessary to experiment with buried arids.
The 15 tests were made on the Catalan slope, The formula
also appears acceptable in other regions, but it has only
been verified with a test in Malaga.
This study does not pretend to have exhausted the
subject, but merely initiatesa new field of operations.
The series of tests can be expanded, The value of B can
be adjusted more in accordance with the variations of am
and i. The case of bed with far thicker mobile bottoms
may be studied , during the flood as indicated in 2.8, analysing
the buried arids. The observation of the arids need not
be restricted to the surface, By means of burrows the
diametres of the arids of the lower layers can be obtained,
thus extending the period studied. It may be used to measure
floods, previously photographing a panorama of the river bed
arids, with sufficient detail and after the flood water to
be studied, with another new photograph, establish the size
of the silt loads contributed by it. The dimensioning of the
protection rockfills of the river bed is defined with the
proposed formula arid for the slopes, the pertinent corrections
need merely be made, In terms of the draft of each flood,
the silt laden arids cari be foreseen and with the granulometries
of the bed, the sedimentation volume can be defined.
The longitudinal section can be studied in terms of the

ADDITIONAL NOTE: Afìer the present doctoral thesis was
approved, the author continued making a series of tests
in various points in Spain, obtaining the following results:
1. - Guadalquivir basin (Jacsi)
-Kiver Guadalbullón in Mengibar; B -
2.- Ebro basin (Calatayud)
-River Jalón in Cetina;
3.- Ebro basin (Calatayud)
-River Jalón in Ateca;
2,lO
u
-
2,lO
B 2,lS
As can be seen, these values, added to the one obtained
in the basin in the South of Spain (Malaga) in the river
Velez Guaro, with B = 2,14, make solidly based hopes arise
that the formula is applicable for all types of basins.
At the same time, it brings the number of tests made
up to 19, verifying the proposed formula,
457

458

ESTIMATION OF DESIGN FLOODS AND THE PROBLEM OF EQUATING THE PROBA-
BILITY OF RAINFALL AND RUNOFF
M.A. Beran
Floods Stuies Team, Institute of Hydrology, Wallingford, Berkshire,
England.
ABSTRACT
Where data on river discharge are scarce it is a common engi-
neering design practise to concoct a design flood with the aid of
rainfall depth-duration-frequency information and a catchment res-
ponse model. Two major waknesses of this approach are (.a) the pro-
blem of the sensitivity of the design to legitimate changes in the
design assumptions and (b) the uncertainty of preserving the nomi-
nal rainfall return period in the design flood. A solution to the-
se problems is proposed which makes use of a computer simulation
investigating the sensitivity of flood magnitude to variations in
return period, storm duration, temporal rainfall intensity pattern,
infiltration loss rate, base flow and unit hydrograph shape. An es
tension to the sensitivity analysis allows an estimate to be made
of any quantile of the distribution of flood magnitude based on
sampling across all causative rainfall and antecedent conditions.
RESUME
El est courant, lorsque les données sur les débits sont insuf
fisantes, que l'ingénieuc élabore la crue de projet 'a partir de
l'information qu'il possede sur la distribution des pluies, en uti
lisant un modèle de transformation pluies-débits. Les deux inconvk
nients majeurs de ce procédé concernent (a) la sensibilité de l'am$
nagement a la variation des paramètres du projet, (b) la conserva-
tion de la période de Tetour (ou de la probabilité) lorsqu'on passe
de la ptuie de projet a la crue de projet. L'auteur propose une so
lution a ces problèmes, en utilisant une simulation pour recher-
cher la sensibilité de la grandeur de la crue aux variations de la
période de retour, de la durée de l'averse, de la configuration du
hyétogramme, de la capacité d'infiltration, du débit de base, de
la forme de l'hydrogramme unitaire. Une extension de cette analyse
de la sensibilité permet d'estimer n'importe quelle quantité de la
distribution des crues, en se basant sur un échantillonnage des
pluies et des conditions antécédentes.

460
3 . INTRODUCTION.
Modern engineercg pract

2. THE SAMPLING PROCEDURE.
461
The procedure follows closely the steps used to estimate the design flood.
(a)
(b)
(c)
Determine a nominal return period
Choose a storm duration and calculate the total depth of
rainfall from the depth-durat ion-frequency relationship.
Distribute the total rainfall within the duration to form
the gross rainfall hyetograph.
(d) Subtract from this an infiltration loss to form the net
rainfall hyet ograph.
(e)
(f)
Convolute the net rainfall hyetograph with the unit hydro-
graph to form the design inflow hydrograph.
Process the inflow hydrograph and extract the particular
flood magnitude measure of interest.
In practical engineering application an arbitrary single choice is made at each
step (a) to (f); in the procedure described in this paper, however, the choice
is made from a selection of possible values, each one with a frequency proport-
ional to its probability of occurrence. Figure 1 illustrates the procedure as a
tree diagram on which the "single choice" method would be represented by a single
pat h.
As implied in figure 1 the continuous distributions of variables such as
rainfall duration are "discretized" so that each variable is made to assume only
one of a finite number of possible values to each of which a probability weight
is attached. Twelve values of duration, 36 temporal intensity patterns and 12
values of catchment wetness index (Cm .- and index of antecedent conditions
governing infiltration loss and base flow) are used. In a separate study to
provide rainfall information (Appendix 1) no dependences were noted between the
rainfall variables and this assumption was made throughout the simulation. This
means that the weights associated with each sampled variable value was itself
invariable; for example the weights associated with each of the 12 CWI values is
the same for 3 hour as for 48 hour duration storms. This particular consequence
might represent some departure from actuality as, in the United Kingdom, both
are seasonable variables.
However assuming independence and discretizing allowed considerable simpli-
fication in the programming and allowed the associated weights of each of the
12 x 12 x 36 combinations to be calculated from the product of the weights of
each of the contributing variables. This product weight is associated with the
flood magnitude in calculating statistics or assembling data into histograms.
To summarise, let pi be the weigM(or probability) of the ith duration,
Di ; let qj be the weight (or probability) of the jth hyetograph distribution,

462
Hj; let rK be the weight (or probability) of the kth CWI, CK; and let QijK
be the flood magnitude resulting from the combination of Di , Hj and Cu. Then
under the assumption of independence the weight or probability to be associated
with QijK is Wijk = pi qj rn and the expected flood magnitude is calculated from
B E pi qjrn Qi~n , while the mean flood magnitude following al1 storms of say
Fh'e fourth duration is calculated from E C W Qijh (Figures 2A and 2B).
j K 41~
Table 1 shows the results of the simulation for the IO -year return-period
at Burbage and Grendon. The contingent distributions show the effect of different
assumed values on the peak discharge. One noticeable result is that changes to
the rainfall variables have small effect on the average peak discharge showing
that the design flood would be insensitive to variations in hyetograph pattern
or storm duration. This is not to say that floods resulting from storms following
particular combinations of duration and hyetograph pattern cannot be found that
depart from the average, but as can be seen from the low standard deviations of
peaks contingent on chosen CWI values centrally chosen rainfall variables will
introduce little bias into the design flood. It has been found that this same
effect is even more marked when the measure of flooding being investigated
involves seme element of storage.
On the other hand, small changes in the CWI have a marked effect on the
resulting flood. It happens that a CWI value chosen to be near the median of the
distribution of CWI would have yielded a peak discharge only 5% in excess of the
expected flood.
Figure 3 shows some of the histograms of flood peaks following the 100-year
storm. These are noticeably negatively skewed and the modal value is typically
20% to 30% in excess of the mean. The inference from this is that a single choice
of each of the variables is likely to yield a flood that exceeds the average flood.
The sharpness of the histograms contingent upon CWI and the discrete sampling is
responsible for the spikey nature of the other histograms.
3. RAINFALL AND DISCHARGE DISTRIBUTIONS.
It had been noted in Section 2 and Figure 3 that the probability distribu-
tion of floods following rainfalls of fixed return period is negatively skewed.
One might anticipate from this that T-year return-period storms tend on average
to give rise to more floods with return period less than T-years than floods of
return period greater than T-years.
To test this and to derive the flood distribution the simulation was gener-
alised to sample the distribution of storm depths. Instead of sampling only
storms of depth and duration such as lie on a line of equal return period the
sampling is now conducted across all combinations of storm depth and duration.
The depth-duration-frequency is again used in order to calculate the probability
of occurrence of any combination (Figure 2C).
Figure 4 shows a comparison between the flood frequency relation as derived

463
from the two simulations and from recorded flood peaks. In the case of Grendon
Underwood there is an apparent tendency for the simulated relation to underestimate
the flood discharge based on the recorded peaks. although independent
evidence from regional analyses has suggested that the distribution as estimated
from the six only annual maxima would overestimate floods quite severely. However
the agreement with Burbage Brook, a small upland catchment in the Derbyshire
pennines with 43 years of data, is rather better. At small return periods the
generalised simulation produced lower flood values than the expected flood
following storms of that same return period.
4. CONCLUSIONS.
A technique has been described whereby the solution of several problems
pertinent to hydrological design in regions of inadequate data may be approached.
In particular, the sensitivity of the design flood to design assumptions can be
assessed. Experience with the technique suggests that the size of the flood is
determined more by the total depth of the rainfall than by its temporal distribu-
tion through the storm's duration. Correct choice of loss rate is in consequence
most important.
It appears that median values of duration, temporal distribution and loss
rate yield a design flood not far removed from the overall average flood follow-
ing the T-year storm. Because of the skewed nature of the flood distribution a
random choice of duration etc. would be more likely to yield a design flood
rather larger than the overall average.
The ability of the technique to reproduce tolerably well the flood magni-
tude frequency relation could be of very great value at a site where flow data
are scarce, whilst even at a well-endowed location the simulation result may be
used with profit to augment the flow record.
While attention has been concentrated on peak discharge as the measure of
flooding it should be emphasized that the technique is suited to more complex
design criteria. The hydrograph may be treated as an inflow and routed through
the scheme and so the actual design criteria of interest mqr be calculated.
Examples are:-
(a) volume between inflow and outflow hydrographs for reservoir
freeboard design; (b) time to peak for a flood warning scheme; (c) volume over
a threshold level for a levee design.
The technique may also be adopted to use an entirely different catchment
response model such as that inherent in the rational formula, a multiple
regression equation or conceptual model although it can be expected that the
data requirements will be rather different from those of this investigation.
5. FUTURF RESEARCH.
The simulation appears promising as a tool for assessing the sensitivity of
design floods to variations in their causative factors and in estimating the
magnitude-frequency relationship for small return periods. However the technique
has not succeeded in reproducing the observed rapid growth in flood discharge
with increasing return period and it is here that further research is being

464
direct ed.
It is felt that the disparity between the definition of storms used to
determine the distribution of depth and duration (Appendix A - Introduction)
could be responsible for the "slow" growth and so long term autographic rainfall
records are to be. analysed to provide information on the distribution of the type
of storm used for the duration statistics.
Further investigation into dependencies between the variables could produce
results wkich would affect the aiscñarge distribution. For example seasonal simulation
would reduce the coincidence of winter storm types with low summer CWI's
and vice versa.
The dependence of CWI on losses and base flow is essentially statistical
and this source of variability could be preserved fn the simulation by the addition
of a random quantity to the values predicted from the best fit lines.
6. ACKNOWLEDGEMENTS.
Although few references have been cited the labours and opinions of others
have played no small part in the development of the procedure. Colleagues and
consultants of the United Kingam Floods Studies Team Dr. J.Y. Sutcliffe,
Professor J.E. Nash, Mr. M.J. Lowing, Mr. C. Cunnane, Mr. R.T. Clarke and
Mr. A.F. Jenkinson have all provided advice and encouragement. Mrs. J. Haworth
was responsible for the FORTFAN computer program and the numerical experiments
were run on the ICL 1906A of the Science Research Council's computing laboratory.
7. REFERENCES.
1.
2.
3.
4.
5.
6.
NASH,J.E. ; "Frequency of discharges from ungauged catchments".
Trans.A.G.U. , Vol. 37, No. 6, December 1956.
CHOW,V.T. and RAMASESHAN,S. ; "Sequential generation of rainfall and
runoff data". Proc. A.S.C.E., Journ. Hyd. Div. , Vol. 9, HY4, July 1965.
EVANS,T. ; "River Eden flood relief studies". Feasibility report by
Sir M. Macdonald and Partners for Kent River Authority. Chapter 4,
September 1971.
DYCK,S. and KLUGE,C. ; "Investigations on the structure of frequency
distributions of floods". I.A.S.H. Warsaw, Vol. 3 , July 1971.
EAGLESON,P.S. ; "The dynamics of flood frequency". Trans. A.G.U.,
Water Resour. Res..Vol. 8, No. 4, November 1972.
LECLERC , G. and SCHAAKE , J. C. ; "Derivat ion of hydrologic frequency
curves from rainfall". Water Resour. Res. (in print).

TABLE 3
FLOOD DISCHARGE FOLLOUNG 3 O-YEAR RETURN PERIOD STORMS.
GFiENDON UNDERWOOD
MEAN STANDARD
DEY
3
m3/s m Is
Overall 5.9 2.0
Constant duration storms.
1 hour
3 hour
6 hour
9 hour
12 hour
15 hour
18 hour
21 hour
24 hour
30 hour
36 hour
48 hour
4.5
5.7
6.2
6.2
6.2
6. o
5.9
5.7 .
5.0
4.4
4.0
4.1
I .6
2.0
2.1
2.0
2.0
1.9
1.8
1.7
1.5
1.3
1.2
1.2
Constant Quartile Type
I 5.8 2.0
II 6. o 2.0
III 5.9 2.0
IV 5. a 2. o
Constant CWI
i5
35
50"
60
1.6
2.8
55 70 3.9
70 ao 4.7
80 go 5.2
90 100 5.6
100 i10 6.1
i10 120 6.6
120 130 7.1
130 140 7.7
i40 150 8.3
150 165 9.4
0.2
0.3
O. 4
0.5
0.6
0.7
O. 7
o. 8
o. 8
0.9
o. 9
1.0
Recorded data
Graphical '-
fit 12.0 I
MaX.
Likelihood1 3.1 3. O
MAXIMUM MINIMUM
DISCHARGE
m3/ s *3/s
11.3
7.7
10.0
33.3
11.3
11.1
11.0
10.9
9.8
9.1
8.4
8.9
13.3
10. 5
10.5
11.1
1.9
3.4
4.8
5.7
6.4
6.9
7.5
8.0
8.6
9.3
10. 1
11.3
1.0
3.0
3.2
1.4
3.6
1.6
3.5
1.6
3.6
1.4
1.2
3.1
1.2
1.0
1 .O
1.0
1 .o
1.0
I. 8
2.1
2.3
2-5
2-7
2.8
3. O
3.2
3.4
3.8
4.6
MEAN
3
m /s
6.7
4.8
6.4
7.3
7.3
7.0
6.8
6.5
6.3
5.8
5.2
4.7
5.3
6.2
6.7
6.8
6.9
1.3
2.0
2.9
3.7
4.4
5.1
5.8
6.5
7.3
8.1
9.0
11.0
BURBAGE BROOK
STANDARD
DEY
3
m /s
3.9
3.4
3.8
2.0
3.9
1.8
1.7
1.7
1.6
1.5
3.4
Y .2
3.4
1.8
1 .8
1.9
2.0
0.2
0.3
0.4
0.5
o. 6
o. 6
0-7
0.8
0.9
1.0
3.1
1.2
--
8.6
8.2 1.6
* First figure refers to assumed CWI at Grendon, second to Burbage Brook.
46 5
MAXIMUM MINIMUM
DISCHARGE
3
m /s
14.0
8.8
13.5
13.4
14.0
13.9
13.7
13.5
13.4
12.5
33.7
31.0
12.3
12. i
12.7
12.4
14.0
1 .6
2.7
3.7
4.7
5.6
6.6
7.5
8.4
9.4
10.4
13 -6
14.0
m3/5
0.6
0.6
0.9
1.3
3.1
1.2
1.1
1.1
1 .O
0.9
0.8
0.8
0.9
o. 6
0.6
0.6
0.6
0.6
1.2
1.7
2.2
2.4
2-7
2.9
3.2
3.5
3.9
4.4
5.9

466
APPENDIX A - DATA REQUIREMENTS.
INTRODUCTION.
Statistical distributions were required for the three modes of rainfall
variability: depth, duration and temporal variability- for each catchment invest-
igated. In order not to predetermine any of the variability modes it was necess-
ary to define a storm in a manner unlike that of the customary rainfall depth-
duration-frequency diagram. The definition was expressed k terms of the condi-
tions for starting and ending a storm: a storm was considered to begin at the
onset of rain and to end when in the preceeding Y hours not more than X mms of
rain occurred. X and Y were chosen to represent the conditions under which a
flood hydrograph would return to near base flow and allowed sbrt spells of zero
rainfall to occur within a storm event.
Hourly analysis of catchment average rainfall was available from three
catchments; Grendon Underwood, Coalburn and Plynlimon (Wye). Sufficient records
were available to permit an investigation hto statistical distribution of storm
durations and temporal patterns but not to conduct an investigation into storm
depth. For this element of the simulation, results of a depth-duration-frequency
analysis of the entire country were available from A.F. Jenkinson (Ref. Al).
The catchment response model is one currently under investigation by the
Floods Study Team. A relation between catchment wetness index (CWI) and total
storm losses, and CWI and base flow was used. A unit hydrograph based on recorded
unit hydrographs from the catchments were convoluted with the gross rainfall less
losses.
DETAILS OF THE SIMULATION DATA.
(a) Rainfall depth: The basic equation used to relate the T-year return period
rainfall of any duration (MT) to that of the five-year return period rainfall
(~5) is
MIM5 = (T/5)'
where c is the "growth factor" and is related uniquely to M5 which is mapped
for the entire United Kingdom. Other necessary information required by the
simulation and provided in Ref. A1 concerns areal reduction factors to convert
point to areaì rainfall.
(b) Rainfall duration: This distribution is dependent upon the storm definition
and for the values X = 2 m s, Y = 5 hours used for both Grendon Underwood and
Burbage Brook simulation is given below
STORM DURATION 1 3 6 g 12 15 18 21 24 30 38 48
(HOURS )
RELATIVE FREQUENCY 5 12 26 20 13 30 8 1 1 2 1 1
(PER CENT)

It was found that the distribution was very similar for hoth upland and lobiland
rainfall stations and varied slowly with changes to X and Y, longer storms
becoming commoner as the conditions for ending a storm were relaxed.
467
(c) Temporal distribution of storm rainfall: Several alternative schemes for
describing the hydrograph shape were investigated. The one chosen was due to
F. Huff (Ref. A2) in which four quartile types are recognised depending upon in
which of the four quarters of the storm duration tiïe largest rainfall fell. The
fine detail of the hyetograph is sampled by plotting all curves of the same
quartile type on a graph showing accumulating fraction of storm depth against
fraction of total storm duration. Composite storms can then be constructed by
connecting points which are exceeded by lo%, 202, 30% etc. of all storms. Sampl-
ing from these composite storms is analogous
from a distribution function in order to sample a variable in proportion to its
frequency of occurence,and were used by the sfmulation. The shapes of the campos-
ite storms were found to be insensitive to changes to the storm definition and
were nearly indistinguishable between upland and lowland catchments. The percent-
age frequency of the four quartile types were 12% type I, 32% type II, 35%
type III, 21% type IV.
(d) CWI distribution: CWI is calculated in nuns. from the soil moisture deficit
(SMD) as computed by the Meteorological Office (Ref. A3) and a five day anteced-
ent precipitation index (API5) using a daily decay constant of 0.5. The formuïa
usedwasCWI = 125-SMD+API5. It had been observed in a recent study (Ref. Ab)
that wet day rainfall and SMD were statistically independent and SO the end of
month values were adopted as representative of all cw? values. Oxford data nr&S
used to provide the distribution for Grendon Underwood and Buxton for Burbage
Brook. In the simulation a linear relation with CWI was used to calculate total
storm losses and the reciprocal of the temporal variation of CWI as the storm
progresses (assuming no evaporation to increase SMD) determined the loss rate
curve. An exponential relationship with CWI determined the base flow.
REFERENCES.
Al
A2
A3
A4
to sampling at regular intervals
JENKINSON,A.F. ; "Meteorological office progress report. January 1972".
Report prepared for Floods Study Steering Committee.
HUFF,F.A. ; "Time distribution of rainfall in heavy storms". Water Resour.Res.
Vol. 3, No. 4, fourth quarter 1967.
GRINDLEY,J. ; "Estimation and mapping of evaporation". 1970 I.A.S.H.
symposium, Reading, I.A.S.H.
BEM,M.A. and SUTCLIFFE,J.V. ; "An index of flood-producing rainfall based
on rainfall and soil moisture deficit". Journ. of Hydrology, Vo1.17, 1972
pp 229-236.

46b
FIGURE 1

DuraîK
M 1 etc.
j
~
~
FIGURE 2A FIGURE 2B
WïES
1 1 Consider case where two variables only aFFect discharge Q, for example storm duration
and CUI (Figure 24).
For each Combination of duration and CWI a value OF Q md B probability of occurrence CU
be calculated. For exsmple combining the duration in the Fourth interval, Dy, with the
CWI in the second interval. C2, a discharge q(H) and a probability p(H) - p(D,,)xp(ci) arc
ïoud
'-1 Summing ail the probabilities in each discharge interval a discharge distribution [Fimi
20) may be COOStmCted.
) This concept can be generalised to sample From Further variables.
I
9
Discharge den<
/ / / /
wm0. Durat ion
a) Depth uld duration are plotted on the base plane ( Piwe 2c)
b) Each coibtiatim ia assoeiited nith a probability of occurmce u) givm by thr depthduration-frequency
diagni.
c) Contingent on each depth duration ccmbination B diatribution of diachugai like Pipure
OB CM be visudiaed on the vertical discharge arii.
d) Integrating such densities above all points on the bue m e 011 locu. or =qual retur,,
period yields the results of Section 2.
0) Integrating over the entire base plane Yields the dihributim of diichuge of Section 3.
FIGURE 2C
469

470
>-
u
æ
W
x
E
s
i=
0,
W
œ
20-
18.
+
l
i 4
I
!
I
4
I
l
l I
PEAK DISCHARGË- M"/S
Burbage Brook-Floods following 100-year Rainfalls
Distribution of all floods
Floods from storms of given duration
Floods from storms of given CWI ---
FIGURE 3

Q/C
2.2
2.0
1.5
1.0.
O. 5
O
Return period-years
I I I I I I
2 33 5 IO 20 50 100
Mean annual
flood
Most likely peak following storms of given return period
\
/'Simulated flood peaks
following storms of given
Peaks have been standardised by the arithmetic mean of
the recorded annual maxima 5.39 mYs.
Plotting position corresponds to expected value of order statistic.
Graphical fit to plotted points so'recorded'line misses (1,l).
1 1 I I I I
O 1 2 s 4 5
Reduced variate- y
FIGURE 4
471

ABSTRACT
A DECISION - THEORETIC APPROACH TO UNCERTAINTY
IN THE RETURN PERIOD OF MAXIMUM FLOW VOLUMES
USING RAINFALL DATA
Donald R. Davis('), Lucien Duckstein(t:) Chester C. Kisiel('),
and Martin M. Fogel
The maximum seaaonal rUno#f YolUw Q #or an ungaged atream site is
derived using (1) an event-based rainfall mode1 for thunderstorma, and
(2) a linear rainfall-runoff model. Major emphasis is placed on effect
of uncertainty in parameters of rainfall inputs on the return period of
maximum runoff volumes in a season. The event-based rainfall model, derived
previously by the coauthors and others, has the following features:
(1) the distribution of the number of events per season N is Poisson
with mean m; (2) the d' 1s t ribution of point rainfall amount R per
event is exponential with mean llu; (3) N and R are independent. More
explicitly, we obtain a correct distribution function for the return pe
riod T (x) under the uncertainty in m and u, and demonstrate the necessity
0 P following this approach for a decision-theoretic analysis of a
water resource design problem. The approach enables us to design structures,
relying only on rainfall data, on watersheds with ungaged
streams by taking into account uncertainty of design site parameters.
Also, we cari tailor the design to a specific problem rather than use a
pre-specified design flood, such as the magical lOO-year flood.
RESUME
Le volume d'écoulement maximum est calculé 2 un site non instrumenté,
en utilisant: (1) un modele de pluie d'orage construit par événe
ment; (2) un modèle pl,uie-débit linéaire. La maniere dont l'incertitudë
sur les paramètres du modèle de pluie affecte la période de récurrence
TQ(x) du volume $'écoulement maximum Q est analysée d'une manière quantitative,
Le modele de pluie d'orage a les caractéristiques suivantes:
(1) se nombre d'événements par saison N suit une distribution de Poisson
a moyenne m; (2) la quantité de pluie ponctuelle R par événement
suit une distribution exponentielle de moyenne l/u; (3) N et R sont des
variables aléatoires indépendantes, Nous obtenons la fonction dti distri-
bution de TQ(x) tenant compte
de l'incertitude sur m et u et montrons
l'utilité de cette méthode pour une application correcte de la théorie
de la d6cision à un problème de planification de ressources en eau.
Nous pouvons ainsi de conceyoir des ouvrages sur des bassing déversants
sans données d'écoulement, a l'aide de données pluvincgtriques, tout en
tenant compte de l'incertitude sur les paramètres. Par ailleurs
pouvons spécialiser la conception & chaque cas d'e;tpèce au lieu'd:ItTli
ser une crue standard, telle la magique crue de pêriode de retour centë
naire.
1 Respectively, Assistant Professor and Professors, on joint appointment,
Departments of Hydrology and Water Reaources and Sistems and fndustrial
Engineering, University of Arizona, Tucson, Arizona 85321,
2 Professor, Department of Watershed Management, Same address as in.(l),

474
1.0 Introduction
Fle,ods or stream discharges are properly described by their durations and
volumes above a certain flow level and their instantaneous peak flows. Of
~I1cs.e three properties, this paper is concerned with the uncertainty in the
return period of maximum flow volumes which is a design parameter for flood pro-
tection and other structures. In particular, we consider the uncertainty due tc
inadequate data on small watersheds (up to 500 lon2).
Jt is well known that there is a good chsnce that a flow event Q.with a
large return period TR may be exceeded at least once in an R-year design period.
Typically, however, calculated risk diagrams (Gilman, 1964) do not consider the
uncertainty in the return periods of rainfall and flow events. TO a design
engineer, the uncertainty of inadequate rainfall or flow data cari result in either
overinvestment (overdesign) or underinvestment (economic losses) in the design of
flood retarding or retention structures or of water storage facilities (farm
ponds or water supply reservoirs for small towns or industries). The Bayesian
framework presented in this paper allows
logic uncertainty as noted above and for
for an explicit
a methodology
consideration of hydroto
evaluate potential
losses associated with that uncertainty.
Approaches takea to arrive at estimates of the return period of hydrologie
f'low properties include:
(a) tipirical fitting of probability density functions to historical data;
in particular, the Soi1 Conservation Service (1965) fitted Pearson
Type III distributions ta flow volumes for various time periods in
Arizona. This approach disregards any available information in precipitation
records or any knowledge about the rainfall-runoff prccess.
(h) Use of phenomenological relations such as a linear trensformation of
rainfall volume to flow volume as a basis for obtaining probability
density functions (pdf) of flow. The pdf of rainfall volume may be
denrribed empiri~ally (with its consequent uncertainty) or from a
procesc viewpoint .,herein individual rainfall events are modeled as
(c)
a stochastic process along the time axis
Use of detailer! dynemical flow equations
(Duckstein
to relate
et al. 1972).
pdf of rainfall
psoperties to pdf of flow properties (Ragleson, 1972).
In this paper we use the second approach. Herein we build on previaus work
(Davis et al. 1972) where we evaluated the ucertainty in the return period of
point rainfall amounts from summer thunderstorms. We define an event-based
process in this case as a sequence of thunderstorms in tine. The return period
TR(k) of maximum point rainfall e (with k the rainfall smount or value of the
random variable 5) is derived by considering the following elements of the
event-based nrocess:
(a) l?hë number 1; of events per season is Poisson distributed with met-a m
(of number of events per season):
b)
(cl
(d)
Rainfall events 53, R,,..., are independent identically distributed
random variables.
The amo\‘Jit 5 of point rainfall per t\sent is exponentially distributed
with parsmeter u (equal to reciprocal of mean emount rainfall per event):
fR(klu) = ueBuk
N and 5 are indepenaent.

Then, the return period of k units of rain in a season, given the event-based
parameters m and u, is
Because m and u are uncertain due to small sample size, T is uncertain.
475
To encode the uncertainty, the posterior distribution of m and u represents
the likelihood of the values of m and u which produced the data. This posterior
is given by the conjugate distributions for the exponential and Poisson distributions
(de Groot, 1970, Chapt. 9). The distribution that is conjugate to both
of these is the gamma:
a a-1 -bx
b x e
gX(xla,b)
-. = r(a) (4)
For the Poisson distributution,
x = m , the parameter of the Poisson and estimated as m.
b = n , the number of seasons.
a = &-I , the total number of rainfall events in n seasons.
For the exponential distribution,
x = u , the parameter of the exponential and estimated as Û.
a = &-I , the total number of rainfall events in n seasons.
b = h/û, the total amount of rainfall for the mn events.
The resulting F (x)s in each case are posterior distributions and represent
z
the likelihood that various values of m and u axe the values describing the rainfall
process that we are observing, after getting the data. These posterior
distributions are used in a computer simulation to develop the posterior distribution
of TR(k). The mean of this distribution is the expected return period
E[T (k)Tl fo; a k-inch rainfall.
R
Computer results' given by Davis, et al. (1972)
indicate that the return period of point rainfall is subject to considerable
uncertainty even with 20 years of data. The design and operational implications
are obvious for flood control, dry farming with irrigation, and water supply.
Next, we extend the procedure to uncertainty in return periods of seasonal flow
volumes on small watersheds.
2.0 Extension to Seasonal Flow Volumes
If m is the total number of runoff producing rainfall events in a summer
season, then the exact expected return period T (y) of the maximum seasonal
runoff volume Q is, under our previous hypotheses,
T& (ylm,u) = [i-exp I-m + m F (ylu)~~-l (5)
9
where F (ylu) is the distribution function of runoff per event CJ which we will
9
9 Q
write F (y) for simplicity. Our approach is to obtain F (y) from the distribution
function F (x) of rainfall 5 per event, using the linear rainfall-runoff relation-
R
where A are the initial abstractions depending on the watershed and c is a coefficient
depending on the rainfall characteristics for a given watershed, in
particular, a time factor such as the maximum 15-minute intensity (Duckstein
et al. 1972).
--
Q
R

476
If we let
p = !-A for R > A I
= o for 5 < A
then Equation (6) becomes = CP -- or y = cx; the distribution function of P - is
Fp(x) = 1 -exp (-u(x+A)) for x > O (7)
and thit of €j (Feller, 1967, Chapt. 2) is
m
FQ(y) = I,"p($) fC(c) dc
because c is a random variable as noted in previous work by the coauthors
(Duckstein et al. 1972). Since, physically, we cannot obtain more runoff than
rainfall, then O 5 c 51, and a beta distribution for c seems to be most appro-
Driate:
The uncertainty on a,b will not be considered in the present study.
Equations (7), (8) and (9) may be combined to obtain
(9)
To sum up,
Equations (10) and (11) are now substituted into Equation (5) to obtain an explicit
expression of Tg (ylm,u). Because we have the sufficient statistics,
fi and a, our knowledge of m and u can be expressed as a pdf !Tiao and Box, 1973).
Hence, this encoded uncertainty results in a pdf on T (ylm,u).
3. O Met hodolopy
To obtain the pdf of the return period on hand, T (ylm,u,n) is a problem of
transformation of random variables, where a closed form is beyond reach.
Thus, a simulation approach is used as follows: (a) consider a fixed
yearly maximum flow volume Q = yo, and (b) sample values m,u are drawn from the
conjugate pars.
-
%(mlfi,n) and gu(uli,n), respectively, as noted in our discussion
of Equation (b), (c) these sample values are substituted into T (y,Im,u) to
obtain one value of the return period T
8
and (a) the process is repeated to
0'
obtain pdf of T for a fixed y (for example yo = Q = 0.7 inch in Table 1).
9
A similar procedure is then used to calculate the pdf of (T )-', which is
9
the probability of exceedance of y . The design parameter of interest may be
O
either T (for sizing a small dam) or (T (estimating long-range replace-
B 9
ment costs of structures.)
Finally, to be considered in a later study is the pdf of maximum seasonal
flow Q that corresponds to a fixed return period. Such a pdf may be of interest
for flood plain insurance purposes and can be calculated by the same simulation
procedure as above.
Q
9

ABSTRACT
A DECISION - THEORETIC APPROACH TO UNCERTAINTY
IN THE RETURN PERIOD OS MAXIMUM FLOW VOLUMES
USING RAINFALL DATA
(1) (1)
Donald R. Davis"), Lucien Duckstein (2j Chester C. Kisiel ,
and Martin M. Foge1
The maximum seas.ona1 Trino$$ YolYge for an ungaged styeam site is
derived using (1) an eyent-based rainfall podel $or thunderstorms, and
(2) a linear rainfall-runoff model. Major empñasis is placed on effect
of uncertainty in parameters of rainfall inputs on th-e return period of
maximum runoff volumes in a season. The event-based rainfall model, de-
rived previously by the coauthors and others, has the following featu-
res: (1) the distribution of the number of events per season N is Pois-
son with mean m; (2) the distribution of point rainfall amount R per
event is exponential with mean 1Lu; c3) N and R are independent, More
explicitly, we obtain a correct distribution function for the return pe
riod T (x) under the uncertainty in m and u, and demonstrate the neces-
sity 09 following this approach for a decision-theoretic analysis of a
water resource design problem. The approach enables us to desigr, struc-
tures, relying only on rainfall data, on watersheds with ungaged
streams by taking into account uncertainty of design site parameters.
Also, we can tailor the design to a specific problem rather than use a
pre-specified design flood, such as the magical 100-year flood.
Le volume d'écoulement maxjmum est calculé i un site non instru-
menté, en utilisYnt: (1) un modele de pluie d'orage construit par 'evéne-
ment; (2) un modele pluie-débit linéaire. La maniere dont l'incertitude
sur les paramètres du modèle de pluie affecte la période de récurrence
TQ(x) du volume d'écoulement maximum Q est analysée d'une maniere quan-
titative. Le modèle de pluie d'orage a les caractéristiques suivantes:
(1) le nombre d'événements par saison N suit une distribution de Pois-
son à moyenne m; (2) la quantité de pluie ponctuelle R par événement
suit une distribution exponentielle de moyenne l/u; (3) N et R sont des
variables aléatoires indépendantes. Nous obtenons la fonction de distri
bution de TQ(x) tenant compte de l'incertitude sur m et u et montrons
l'utilité de cette m'ethodf pour une application correcte de la théorie
de la décision à un probleme de planification de ressources en eau.
Nous pouvons ainsi de conceroir des ouvrages sur des bassing déversants
sans données d'écoulement, a l'aide de donn'e$s pluviométriques, tout en
tenant compte de l'incertitude sur les parametres. Par ailleurs, nous
pouvons spécialiser la conception à chaque cas d'espèce au lieu d'utili
ser une crue standard, telle la magique crue de période de retour cent:
naire.
'Respectively, Assistant Professor and Professors, on joint appointment,
Departments of Hydrology and Water Resources and Sìstems and Industrial
Engineering, University of Arizona, Tucson, Arizona 85721.
2Professor, Department of Watershed Management, Same address as in (1).

474
1.0 Introduction
Floods or stream discharges are properly described by their durations and
volumes above a certain flow level and their instantaneous peak flows. Of
briese three properties, this paper is concerned with the uncertainty in the
return period of maximum flow volumes which is a design parameter for flood protection
and other structures. In particular, we Consider the uncertainty due to
inadequate data on small watersheds (up to 500 h2).
It is well known that there is a good chance that a flow event 9 with a
large return period T may be exceeded at least once in an N-year design period.
Typically, however, calculated
R
risk diagrams (Gilman, 1964) do not consider the
uncertainty in the return periods of rainfall and flow events. To a design
engineer, the uncertainty of inadequate rainfall or flow data can result in either
overinvestment (overdesign) or underinvestment (economic losses) in the design of
fiood retarding or retention structures or of water storage facilities (farm
ponds or water supply reservoirs for small towns or industries). The Bayesian
framework presented in this paper allows for an explicit consideration of hydrologic
uncertainty as noted above and for a methodology to evaluate potential
losses associated with that uncertainty.
Approaches taken to arrive at estimates of the return period of hydrologic
flow properties include:
(a) Rupirical fitting of probability density functions to historical data;
in particular , the Soil Conservation Service (1965) fitted Pearson
Type III distributions to flow volumes for various time periods in
Arizona. This approach disregards any available information in precipitation
records or any knowledge about the rainfall-runoff process.
(b) Use of phenomenological relations such as a linear transformation of
rainfall volume to flow volume as a basis for obtaining probability
density functions (pdf) of flow. The pdf of rainfall volume may be
described empirically (with its consequent uncertainty) or from a
process viewpoint wherein individual rainfall events are modeled as
a stochastic process along the time axis (Duckstein et al. 1972).
(c) Use of detailed dynamical flow equations to relate pdf of rainfall
properties to pdf of flow properties (Eagleson, 1972).
In this paper we use the second approach. Herein we build on previaus work
(Davis et al. 1972) where we evaluated the uncertainty in the return period of
point rainfall amounts from summer thunderstorms. We define an event-based
process in this case as a sequence of thunderstorms in time. The return period
T
R
(k) of maximum point rainfall (with k the rainfall amount or value of the
random variable FJ) is derived by considering the following elements of the
event-based process:
(a) The number N of events per season is Poisson distributed with mean m
(of number of events per season):
(b) Rainfall events R -1, G2, ..., are independent identically distributed
random variables.
(c) The amount of point rainfall per event is exponentially distributed
with parameter u (equal to reciprocal of mean amount rainfall per event):
fR(klu) = ue -Uk
(a) $ and R are independent.

477
4.0 Results
The results of the computer simulation are summarized in Tables 1 and 2 and
Figures 1 and 2. In these we consider the variance of c, representative of conditions
on the watershed, and the variance in our knowledge about rainfall
parameters m and u.
Table 1 shows that u, the average rain per event, is much more important
than m, the average number of storms per season, as judged by the variance of
T ,(Var !? ), for different values of Var c. We also note the following
9 9
(a As Var c increases, EL?,? and Var ? decrease, Thus by not randomizing
9
C the estimated return period of Q = 0.7 is much higher. By varying C
the variable effects of rainfall intensity and watershed behavior on
the return period are anticipated;
(b) Var T increases dramatically when Var $ = O for joint uncertainty in
9
m and u;
(c) The mean reciprocal return period (= exceedance probability = p) and
,. -1
Var T increase rapidly as Var c increases. This result is shown
9
because p is commonly used as the design parameter in hydrologic risk
analysis.
These patterns hold for all values of runoff volume used in the sensitivity analysis
(Q = 0.5, 0.7 and 0.9 inches of runoff) as shown in Table 2.
As expected, the Var T decreases with doubling of available data (10
9
to 20 years used in the simulation) as summarized in Table 2. The E[id is only
slightly changed. A more general manifestation of the simulated process is evident
in Figure 2 where the posterior pdf (of return periods for 0.7-inch runoff) based
on 20 years of data has a much sharpy modal value than the posterior pdf based
on 10 years of data; note that mear, T is just to the right of the mode. While
9
not shown, the posterior pdf's become more peaked as Var c increases.
The effect of increasing runoff Qolume is to increase E[Td, Var T and
9
coefficient of variation CV(T ) as shown in Table 2. The latter result about
9 I
CV(TQ) also implies that a Var T increases more rapidly than E[Td. It is
9
intriguing to note the dramatic effect that the introduction of Var has on the
parameters.
The results in Table 2 for n = 10 years are shown in Figure 1, a plot on
Gumbel extreme value paper. As previously noted, as Var c increases the smaller
"Liil. From the tabulated results we note that so-called confidence limits for
ea& line would get wider as T increases because Var T increases with runoff
61 9
volume. These confidence limits are narrower for n = 20 years of data as is
evident from Table 2.
Of interest is the modest computer time (maximum of 25 seconds for 20 years
of data) per simulation run on the CDC-6400. Given the number of uncertain
parweters in this problem, it does not appear feasible to prepare charts and
graphs for routine design use unless more exhaustive computer studies are performed.
4.1 Comments on Results A
In contrast to the classical empirical frequency approach in deriving T 9,

478
the event-based approach outlined here results in evaluation of uncertainty in
.
T from physically meaningful parameters like m and u. This is a much more
Q
efficient use of the available data on rainfall and runoff.
We have seen how the design would depend on the uncertainty in m and u and
on the interaction between*uncertainty in m and u and Var C. The end result, a
posterior distribution on T is of value to inference on hydrologic stochastic
9’
processes as discerned from limited data of value to the next important step
of invoking Bayesian decision theory for evaluating design decisions and for
judging if better designs are possible by waiting for-additional cata.
It would be desirable to express the moments (ErTJ and Var T ) of the
Q
posterior pdf in terms of m, u, Q and C, but this is intractable. -The next
approach for thinking about our results in simpler terms is to consider the
mean and variance of 5 = cp:
ECQI = EC~J CIFI
Var = E2[C] Var P + E2[P]
- Var C + (Var C) - (Var P) -
as given by Benjamin and Cornel1 (1970, p. 169).
equations become E[g = CE[?] and Var 9 = C2 Var p.
When C is not random, these
The variance of Q (and
thus its frequency of exceedance and its return period) is dramatically affected
by randomization of C. It is common in hydrologic design to choose a “frequency
factor” z (or standardized variate) in the relation 9 = Ere] + z (Var Q) 1/2 .
.
To contrast properly this classical approach to finding a design flow Q with the
method outlined in this paper wou1.d require a full-fledged decision theoretic
analysis for a specific design problem. The evaluation would have to be repeated
for each design use of the posterior pdf. Much work remains to be done in this
direction.
4.2 Relationship of results to Bayesian decision theory
Let the loss function for the design of a flood protection structure, say
a dike, be L(h,T) where h is the height of the dike and T is a design return
period such as T or an exceedance probability (T )-l. The result of our
a 9
investigation was to determine the posterior pdf f (t) as given in Figure 2.
Thus, we are now able to calculate Bayes risk, which corresponds to the optimum
design h*
+m
BR(h*) = min L(h,t) fT(t)dt (14)
.
h o
We can also calculate the worth of sample information to sharpen the estimate
of T (Davis et al. 1972) for each intended use of the data. Such studies are
9
left for the sequel. It is very important to emphasize that the worth of data
discerned by this methodology is based on the economic loss function associated
with a particular design use of the data; the results are not in terms of the
variance of the return estimate (return period in this case).
5.0 Conclusions
It is important to keep in mind when judging the results of the research
reported here that we are dealing with maximum flow volumes generated by a sequence
of thunderstorms during a season. Additional work is necessary to extend the
T

479
approach to other runoff-producing precipitation events (including snow) during
the year. The use of the Gumbel distribution in this paper goes beyond its
classical use for the instantaneous rainfall and flood maxima during the year.
We thus have found the following points in our theoretical and simulation
arialy,is :
The approach enables us to design structur'es, relying only on rainfall
data, on watersheds with ungaged streams by taking into account
uncertainty of the site parameters,
Using this approach we can tailor the design to a specific problem
rather than use a pre-specified design flood, such as the magical
100-year flood.
Simulation is an appropriate method for evaluating uncertainty in
estimates of physically-meaningful parameters arising in the eventbased
approach.
Return period varies with record length rainfall, and watershed events,
etc. We have given an event-based approach to evaluate this variation.
The sensitivity analysis demonstrates the dramatic importance of uncertainty
in the average amount of rainfall per event and the importance
of considering variability in the rainfall and watershed parameter
called C in this paper.
The resats, if encoded in the posterior pdf of the return period
TI, allow the user to exercise inference or to find sensitivity of the
analysis to design decisions in the face of inadequate data. Bayesian
decision theory is the framework suggested for undertaking the decision
analysis.
The results have implications for design of a variety of hydraulic structures
in both urban and rural watersheds, in temperate and arid climates, and in
regions of the world confronted with inadequate hydrologic data. In the face
of changing watershed conditions, as reviewed by Fogel, et al. (1972), the
approach offered in this paper permits exercise of judgment on the effects of
lack of knowledge and of nonstationary meteorologic and hydrologic parameters
such as m, u and C. In o w jument, classical empirical frequency methods do
not provide such a clear basis for evaluation. Extension to nonlinear water-
shed models are possible as noted by Duckstein, et al. (1972) and Fogel, et al.
(1972).
6. O Acknowledgments
The work was supported in part by U.S. National Science Foundation Grant
GK-35791 and by a matching grant (Decision Analysis of Watershed Management
Alternatives) from the U.S. Office of Water Resources Research. !Che computer
programming skills demonstrated by Joel Friedman have contributed substantially
to the realization of the results.
7.0 References
BenJanin, J.R. and C.A. Cornell. Probability, Statistics and Decision for Civil
Engineers, McGraw-Hill Book Co., New York, 1970.
Davis, D.R., L. Duckstein, C.C. iíisiel, and M. Fogel. Uncertainty in the return
period of maximum events: A Bayesian a.pproach. Proceedings, International
Symposium on Uncertainties in Hydrologic and Water Resource Systems,
University of Arizona, Tucson, Arizona; 1972, pp. 853-862.

480
Davis, D.R., C.C. kïsiel and L. Duckstein. Bayesian decision theory applied
to design in hydrology, Water Resources Research, Vol. 8, No. 1,
February 1972, pp. 33-41.
de Groot, M.H. Optimal Statistical Decisions. McGraw-Hill Book Co., New York,
1967.
Duckstein, L., M.M. Fogel and C.C. Kisiel. A stochastic model of runoffproducing
rainfall for summer type storms, Water Resources Research,
Vol. 8, No. 2, April 1972, pp. 410-421.
Eagleson, P.S. Dynamics of flood frequency, Water Resources Research, Vol. 8,
NO. 4, August 1972, pp. 878-898.
Feller, W. An Introduction to Probability Theory and its Applications, Vol. 2.
John Wiley, New York, 1967.
Fogel, M.M., L. Duckstein and C.C. Kisiel. Choosing hydrologic models for
management of changing watersheds, Proceedings, National Symposium on
Watersheds in Transition (American Water Resources Association) , Fort
Collins, Colorado, June 1972, pp. 118-123.
Gilman, C.S. Rainfall (Section 9). In "Handbook of Applied Hydrology,"
Edited by V.T. Chow, McGraw-Hill Book Co., New York, 1964, pp. 9-59.
Soil Conservation Service, Runoff volume-duration-probability analyses for
selected watersheds in Arizona. Central Technical Unit, Hydrology
Branch, SCS, U.S. Dept. of Agriculture, April 1965.
Tiao, G.C. and G.E.P. Box. Some comments on "Bayes" estimators, The American
Statistician, Vol. 27 (i), February 1973, pp. 12-14.

Table 1: Sensitivity anaiysikon return period moments as
function of uncertain parameters for rural water-
shed with only 10 years of data.
Jncertain
)arameters Var C
l&U O
.O005
.O05
* O5
)nly m .O005
.O05
O5
nly u .O005
.O05
* O5
flean T (years
?
~
A
Moments of return period T
cv( TQ)**
.,
41.82
39.00
26.33
6.30
35.36
23.41
6.20
37.61
24. i4
6.52
Var I
Q -
538.
442.
153.
2.35
8.30
3.89
t 19
383.
110.
1.96
.555
.539
.470
.243
.o81
.O84
.O70
.521
.435
.215
~~ ~
Reciprocal
return period -
nean variance
.O00299
.o00318
.o00328
.o01809
* Conditions for the analysis: A = 0.4 inches, mean C = 0.3 for beta
distribution, Q = 0.7 inches on the average; rainfall is
distributed on basis of an exponential distribution for
amounts above 0.3 inches with an average of 14.0 storms/
season and an average of 0.39 incheslevent.
*+ Coefficient of variation of T s-
481

Average
runoff
volume Q
0.5
Table 2: Sensitivity analysis on return period moments
asa fun&ion of rainfall P, length of record n
ànd variance of C; both m and u are uncertain;
watershed is rural; conditions are as noted in
Table 1.
n
(years of
data)
Var <
I
Moments of return period 'f
9
&lean (years
I
~ ~-
10 .o01 11.70
58.07 .650
.O05
O5
6.97
2.94
14.90
- 27
.553
* 177
20 .O05 6.25 4.38 ,335
10 O
.O005
.O05
O5
41.82
39.00
26.33
6.30
20 37.44
10 O
.O05
O5
24.36
271
103
6.41
14.29
~~~
538
441
153
245
68
48,085
3 , 602
20 .O05 95.95 1,721 I
2.35
1.09
21.25
.432
.555
.539
.470
,243
.418
.339
.163
.809
.582
.323

I .o
O. 8
v,
w
I
o
z 0.6
-
-
w
z
3
-I
$ 0.4
LL
IL
O
Z
3 0.2
O
2 5 IO 20 50
RETURN PERIOD, YEARS
100 200
Figure 1: The effect of the varlance of C on tha Teturn
period of punoff volume.
483

w
3
o
w
CE
LL
484
.25 -
.20 -
.I5 -
.IO -
.O5 -
RETURN PERIOD, YEARS
Figura 2: Posterior prooabilitr densiTy function of return periods
f o 0.7-i’nch ~
runoff of record length.
O

ABS TRACT
SYNTHETIC UNIT HYDROGRAPH TECHIQUE FOR THE
DESIGN OF FLOOD ALLEVIATION WORKS IN URBAN AREAS
by
M.J. Hall
Lecturer in Civil Engineering, Imperial College
of Science and Technology, University of London
The development of rural land for urban, suburban or industrial
purposes can radically alter the flow regime of the catchment area WL
thin which such changes take place. The volume of surface runoff tends
to increase, the lag time of the flood hydrograph to decrease and the
peak rate of flow to increase. These ch-anges should be anti’cipated in
the design of flood alleviation works for catchment areas undergoing
urbanisation, but in general, little quantitative information is avai
lable on the magnitude of the effect at different stages of urban de-
velopment. If flow records are available from several catchment areas,
each of which has reached a different stage of urban development, the
finiteperiod unit hydrographs derived from these data can be used as
an index to the influence of urbanisation. The application of a syn-
thetic unit hydrograph technique to flow records from both urban and
rural catchment areas within the headwaters of the River Mole near
Crawley, United Kingdom, has confirmed the feasibility of the
approach but has shown that more thought is necessary in choosing cai
chment characteristics which reflect the character of the urban deve-
lopment.
RESUME
L’utilisation des espaces ruraux pour le développement urbain et
industriel peut changer radicalement le régime hydrol2gique des bas-
sins concernés. Le volume du ruissellement tend a croitre, le temps
de réponse du bassin à décoitre et les pintes de crues s’amplifient.
Lors de l’élaboration des projets, ces modifications devraint être
prévues et on devrait chercher à atténuer l’effet des crues par des
travaux appropigs, mais on ne dispose en général que d’une informa-
tion très succincte sur l’importance de cet effet aux différents sta-
des du développement urbain. Si on dispose de relevés de débits sur
plusieurs bassins atteints à des degrés différents par le développe-
ment urbain, on peut utiliser les hydrogrammes unitaires tirés de ces
données pour constituer des indices concernant l’influence de l’appli-
cation d‘u?e technique d’hydrogramme unitaire synthétique aux débits
observés, a l’issue de bassins urbains et ruraux, dans le bassin sup5
rieur de la Mole, pres de Crawley (Royaume Uni), a confirmé les possi
bilités de cette méthode; elle a montré aussi que le choix des caracy
téristiques du bassin reflétant l‘influence du d6velo;pement urbain
demandait une sérieuse réflexion.

486
I. INTRODUCTION
According to Toynbee [I], almost half the World's population had become
urban by 1969. This increase in the urban population has been accompanied
by an even more marked expansion in the area occupied by streets and buildings.
The development of rural land for urban, suburban and industrial purposes is
characterised by two important physical changes, both of which may have a profound
effect on the hydrological cycle of the area within which such urbanisation
takes place.
Firstly, the area covered by relatively impervious surfaces increases,
thereby increasing the proportion of storm rainfall which becomes surface run-
off. Owing to the concomitant decrease in soil moisture recharge, dry weather
flows are reduced.
Secondly, the natural surface water drainage system of the area is invariably
subjected to a variety of changes, ranging from realignment of channels to the
installation of stormwater sewerage. Since the flow velocities in the modified
drainage network are generally higher than those observed in the orlginal
natural channel system, both the time-to-peak and the length of the recession
of storm hydrographs tend to decrease as a catchment is urbanised.
The increased
volume of runoff, and the shorter time within which that volume is discharged,
inevitably produce peak rates of runoff that are markedly higher than the flow
records from a catchment in its previous rural state would tend to indicate.
Although the effects of urbanisation on the flow regipie of a catchment
area have been appreciated qualitatively for over a decade [2], relatively
little information has been available on the magnitude of the changes brought
about by different forms of urban development. Of particular importance to
the engineer concerned with the design of flood alleviation works for urban
areas are
i)
ii)
the frequency distribution of peak rates of flow ; and
the shape of the flood hydrograph.
The changes in the magnitude of the parameters of the frequency distribution
of annual floods caused by urbanisati)onhave been studi,ed by Carter [3],
Martens [4] and Anderson 151, each of whom approached the problem by means of
regional analysis. Much of the work on changes in the shape of flood hydrographs
has employed a similar treatment, with the finite-period unit hydrograph
(TUH) being used as an index to catchment response. Flow records for
catchment areas in different stages of urban development within the same
hydrologically homogeneous region have been used to derive WH's of a predetermined
duratiqn. Selected parameters of these !CUH's have then been expressed
in terms of pertinent catchment Characteristics using multiple linear regression
analysis. The relationships so obtained may then be employed to derive TUHts
for both ungauged catchments and gauged catchments in a more advanced state of
development. For example, Espey et al [6] used 5 hydrograph parameters : peak
rate of flow ; time-of-rise and base length of the hydrograph ; and hydrograph
widths at 50 and 75 per cent of the peak discharge. Since the catchment
characteristics selected by those Authors did not ipclude any parameter reflecting
changes in the surface water drainage system, an empirical coefficient (

L
ref. river
no.
1 Mole
- Mole
2 Gatwick Stream
3 Ifield Brook
4 Crawters Brook
5 Crawters Brook
487
that simple one and two-parameter linear conceptual models can be used to
advantage in characterising the changes iq catchment response caused by urbanisation.
However, a prerequisite to either approach is the availability of
hydrometric data for a sufficiently large number of urban and rural catchment
areas to effect a regional analysis. Where the number of flow records is
limited, techniques which employ as few hydrograph parameters as possible are
an obvious advantage.
In the following paper, a dimensionless unit hydrograph technique which
involves the use of only one parameter is outlined. The method of approach
is illustrated by means of data from an area in the south-east of England.
The paper begins in Section (2) with a brief description of the area and the
available hydrometric data, and continues in Section (3) with an outline of
the method by which TüH's were derived. The regionalisation of these TüH's
is discussed in Section (4). The paper concludes in Section (5) with a brief
diqcussion of the existing data inadequacies in Urban Hydrology.
2. DATA PRXPARATION
2.1 Data inventory
Between 1949 and 1969, the population of Crawley, a town situated some
30 miles to the south of London, increased from 5,000 to 68,000. The
developed area i? drained by the headwaters of the River Mole, a south-bank
tributary of the River Thames. !Che western side of the town drains to Ifield
Brook, whereas the centre and eastern sides are served by Crawters Brook and
Gatwick Stream respectively (see Figure 1). A major part of the urban area
lies on Weald Clay overlying Tunbridge Wells Sand, the latter outcropping to
the south of the area. The average annual rainfall in the Crawley region
(1916-1950) ranges from 750-850 rnrn.
There are 6 gauging stations within the area of interest, the details
of which are summarised in Table 1. Both the gauging statipns on the River
Mole and that on Gatwi.ck Stream are operated by the.Thames Conservancy ;
Crawley Urban District Council maintain the records at the remaining 3 sites.
For the purposes of the present study, data were available for all sites apart
from the River Mole at Gatwick Airport.
TABU 1
: Details of gauging stations within the Crawley region.
Horley Weir
stat ion cat chment records
from
Gatwick Airport
Tinsley Sewage Works
Ifield Mill
Hazelwick Roundabout
Woolborough Road
89.8
31 -8
31 .o
12.3
4.7
2.2
Nov., I961
Nov., 1967
Jul., 1952
Dec., I958
May, 1954
Sew.. 1952

488
The positions of the 3 principal autographic raingauges located within
the headwaters of the River Mole are indicated on Figure 1 along with the
gauging stations. 2 of the 3 raingauges have been in operation since before
the first regular streamflQw measurements were taken at Tinsley Sewage Works
and Woolborough Road, and records from all 3 raingauges are available from
before 1961 when the two gauging stations on the River Mole were brought into
use.
2.2 Selection of storm events
The first stage in the analysis of the available data involved the
preparation of a short-list of suitable storm events for each of the £ive
catchment areas. The criteria used in choosing these events were somewhat
arbitrary, but in genera1,an attempt was made to confine the analysis to
hydrographs with well-defined peaks having both a smooth rising limb and a
smooth recession. Rainfall data for each of the selected storm events were
then abstracted.
The raingauge at Broadfield was taken to be representatjve
of the rainfall patterns over the catchment areas draining to gauging statipns
3, 4 and 5 (see Table I>, and the arithmetic mean of the catches at Broadfield
and Gatwick Airport was taken for the areas commanded by gaugipg stations 1
and 2. The records from Tinsley Sewage Works were only used when no information
was availab1.e at either of the other gauges.
The above selection procedure produced 8 storm events at gauging statipn
3, 11 at gauging station 1, 12 at gauging station 4 and 16 each at gauging
stations 2 and 5, the majority of whi,ch were associated with rainfall totals
exceeding 12 mm.
2.3 Baseflow separation
The second stage in the analysis consisted of the separation of the baseflow
component from each of the recorded streamflow hydrographs, The procedure
adopted involved the plotting of the recession limb of each hydrograph on
semi-logarithmic graph paper with discharge on the logarirchmjc scale. A
straight line was then fitted by eye to the lower portion of the curve, the
point at which the recession departed from this straight line being taken to
mark the time at which surface runoff effectively ceased. The variatipn of
baseflow with time during the storm was then represented by a straight line
joining this point on the recession limb to the beginning of the rising limb
of the hydrograph.
The above method of baseflow separation, which is both straightforward
in use and less subjective than the majority of the alternatgve procedures
was applied to each of the 63 recorded hydrographs selected for analysis.
The ordinates of the resultant surface runoff hydrographs were then abstracted
at I-h intervals for all events at gauging stations 1-4 and at 30-min intervals
at gauging station 5.
These data were subsequently transferred on to 80-
column punched cards along with the total recorded rainfalls witpin the same
time incrementso

3. DERIVATION OF UNIT H!¿DROGRAPHS
489
There are two distinct methods of approach to determining the instantaneous
unit hydrograph (IW) or finite-period unit hydrograph (TUH) of a
catchment area from rainfall and streamflow data [g]. The first of these
methods of approach involves the fitting of a linear conceptua1,model to the
records of rainfall excess and surface runoff.
The 4mpulse response functi.on
of the fitted mode1,is then taken to approximate the IUH of the catchment.
This indirect synthesis approach may be contrasted with the more direct methods
of analysis which operate on the rainfall excess and surface runoff data to
yield an IUH or TLTH wi,thout the need to postulate a model. The harmoni? method
for defining the TUH of a catchment [9], which was adopted for the purposes
of the present study, falls into the latter category.
3.1 The harmonic method of unit hydrograph derivation
In order to apply the harmonic method, the volumes of raiyfall excess
and the ordinates of both the surface runoff hydrograph and the TIM are defined
in terms of harmonic series. For example, if the equally-spaced ordinates of
the surface runoff hydrograph are given by yi, i = 1, 2, ...., n,
+ = + C[A~ P cos j - 2xi sin j &]
Yi.
n j n
j=l
If n is an odd number, p = (n-1)/2 and
B = 2 c y
sink -;i
n k
j k= 1
The volumes of rainfall excess, xi, i = 1, 2, ..., m, within the same
equal time increments may also be expressed as a harmonic series with the same
fundamental period and number of terms if (n-m) zeros are added to represent
the terms xi, i = m+l, m+2, ...., n.
This series will be identical in form
to equation (I), but with n harmonic coefficients a, b whose values can be
obtained by substituting rainfall excess volumes for surface runoff ordinates
in the equations (2). If the TITH is also assumed to have n equally-spaced
ordinates, Le. the same fundamental period as that of the rainfall excess
and surface runoff data, O'Donnell 191 has shown that the harmonic coefficients
a, ß, of the harmonic series which defines the ordinates of the TUH can be
calculated directly from the harmonic coefficients A, B, a, b usipg the linkage
equations
a.A. + b.B - -
a = - but cio
n
-
j a. +b
~j
- 2 w
Pj - n 2
aj2+bj
1 %
n a
O
eq. (3)

49 O
Substitution of the aj, ßj in a series expansion of the form of equation
(1) then gives the successive ordinates of the TUH, ui, i = 1, 2, ...., n
dir e c t ly .
The application of any of the established methods of analysis, such as
the harmonic method, is liable to produce TUH's which are distorted by highfrequency
oscillations of varying amplitude. PhiiliFpee and Wiggert [IO] who
applied the harmonic method to data from 38 storms on 4 drainage basins in
thr vicinity of Detroit, encountered this problem but offered no explanation
as to its cause. More recent studies by Blank et al [Il], who used the Fourier
transform approach, which bears some relatipnship to the harmonic method, have
indicated that such oscillations can result from errors in the data and are
not necessarily caused by the inherent non-linearity of the rainfall-runoff
relationship. Blank et al [Il] also show that oscillatory TUH's can be avoided
by applying a low-pass digital filter to the rainfall excess and surface runoff
data prior to the derivation of the TUH.
One of the principal advantages of the harmonic method is its flexibility
in dealing with storm events which produce such oscillatory "UH's without the
need to use digital filters. This property of the method stems from the form
of the linkage equations (3) by which the aj, ßj of the harmonic series representation
of the TKH depend only on the harmonic coefficients of the rainfall
excess and surface runoff data for the same frequency. Individual harmonics
may therefore be omitted from the series representation of the TUH without
affecting the calculation of other üj, ßj.
In particular, if the ordinates
of the TUH obtained by using all the aj, ßj exhibit high-frequency oscillations,
truncation of the series representation may help to eliminate these
oscillations. However, the amount of truncation applied should not be suffic-
ient to cause the hydrograph obtained by convolving the smoothed Tw with the
distribution of rainfall excess to depart significantly from the original
surface runoff hydrograph.
3.2
Appliiation to Crawley area data
A computer program was written to derive TUH's using the harmonic method
described ip Section (3.1) above. The computation began with the determination
of the distribution of rainfall excess using the @-index method. The total
volumes of both rainfall and runoff were calculated and their difference
averaged over the number of time intervals with non-zero rainfall. This
average t'losstt was then subtracted from the recorded volumes of rainfall w5thin
each time interval, any negative differences being set to zero. The whole
procedure was repeated until the difference between the total volumes of
rainfall and runoff was less than 0.25 mm. Having obtained the distribution
of rainfall excess, the derivation of the TUH was carried out according to the
method outlined in Section (3.1) above. The surface runoff hydrograph was
then reconstituted by convolving the derived TUH with the distribution of
rainfall excess.
The data from all 63 storm events were processed using the full number
of harmonic coefficients in determining the ordinates of the TUH. The results
obtained were then plotted and compared. The majority of the derived TUH's
were found to exhibit high frequency oscillations of varying amplitude. The
storm events which gave rise to such behaviour were therefore re-processed
using fewer harmonic coefficients in determining the TUH ordinates.

The choice of the most appropriate number of harmonic coefficients to
use for any given storm event is largely subjective. Truncating the harmonic
series representation of the TLTH may remove the high-frequency oscillations,
but the hydrograph obtained by convolving that TUK with the distribution of
rainfall excess should not depart markedly from the original surface runoff
hydrograph, particularly in regard to the magnitude and timing of the peak
flows. The amount of computer time that would have been involved in
systematically reducing the number of harmonic coefficients in the series
representation of the !ì'üñ until the fit provided by the reconvolved surface
runoff hydrograph was no longer acceptable would have been excessive. A pilot
study using a restricted number of truncated,,series,each having a predetermined
proportion of the full number of harmonics, was therefore carried out. For the
majority of the storm events, halving the number of harmonics successfully dampened
the high-frequency oscillations, and gave rise to a reconvolved hydrograph
whose maximum ordinate was generally within 5 per cent of the peak of the
original surface runoff hydrograph.
491
As a result of the above analysis, 8 TiJH's were obtained for gauging
station 2, 6 each for gauging stations 4 and 5, 4 for gauging station 1 and
3 for gauging station 3. The changes in flow regime which had occurred at
gauging stations 2, 4 and 5 during the period of record were immediately
obvious, and TiJH's for each of these sites were therefore grouped according
to the dates of occurrence of the storm events from which they were derived.
For convenience, these different groupings will be referred to by the letters
IfAt1 (for the earlier storms) and "BI1 (for the later storms). Of the 8
separate sets of TUH's, none consisted of less than 3 hydrographs. The TUH's
within each set were then plotted together using a common starting time, and
an "average" TiiH obtained by drawing in a smooth curve through the plotted
points, care being taken to ensure that the area under the curve was equivalent
to 25 mm over the catchment area.
Figure 2 shows the smoothed TITH'S obtained for Crawters Brook at Woolborough
Road, and is indicative of the change in flow regime which has taken place as
the town centre of Crawley has developed over a period of some 15-20 years.
4. REGIONALISATION OF UNIT KYDROGRAPHS
One of the simplest assumptions that can be made in regiqnaliTi9g a group
of unit hydrographs is that all TLTH'S of a common duration are reducible to
the same dimensionless shape. The scaling parameters that are required to
describe the dipensionless hydrograph (of which there are generally two) are
expressed in terms of catchment characteristics by means of a multiple linear
regression analyses. The appljcation of this approach,$ the present study
is complicated by the necessity to include independent variables which reflect
thg man-made changes within the catchment areas affected by the development of
Crawley.
Previous authors who have applied a dimensionless unit hydrograph approach
have differed widely ip their choice of scaling parameters, For example,
Commons u23 developed a l'basic hydrograph" with a time base of 100 arbitrary
units, a height of 60 arbitrary discharge units and an area of 1196.5 square
units. The scaling parameters required were peak rate of runoff and total
volume of runoff. In common with many similar pairings, these parameters

49 2
are not entirely independent, and as Diskin [I31 has recently pointed out, a
choice of parameters which satisfy the constraint of unit area under the TLTH
is to be preferred.
A review of previously-published work on the hydrological consequences
of urbanisation showed that, of several possible time scaling parameters, the
lag time TL, defined as the time interval between the centroid of rainfall
excess and the centroid of surface runoff, has been found to show a consistent
variation with the length and slope of the main channel for both rural and
urban catchment areas 13-53. If, however, the constraint of unit area under
the TUH is observed, making the time scale of each TüH dimensionless by expressipg
the timing of all ordinates as a proportion of TL also determines the ordinate
scale. Hence, the dimensionless unit hydrograph can be specified in terms of
only one parameter, the functional form of the curve being
ut.TL = f (t/TL) eqe (4)
where ut is the ordinate of the actual TUH at time t.
The above method of producipg a dimensionless unit hydrograph was applied
to the data obtained from the 5 catchments in the Crawley area. The lag time
of the I-h TUH for each catchment was obtained by computing the tjme interval
between the origin of the TüH and its centroid and subtracting O.5h. Following
Carter [3) and Anderson [5], a double-logari4hmic plot of lag time against
basin ratio was prepared (see Figure 3). Basin ratio is defined by the quotient
L@, where L is the length of the main channel between the gauging station
and the watershed (km) and S the main channel slope.
S is defined by the alti-
tude difference between points located 10 and 85 per cent of the main channel
length upstream from the gauging station divided by their distance apart [14].
Values of L and S were obtained from 1 : 25000 scale maps for all catchment
areas apart from that of gauging station 5 for which 1 : 500 longitudinal
sections were available.
In preparing Figure 3, the number of data was increased by the inclusion
of TUH's for 2 gauging stations on the River Wandle in the southern suburbs
of London to the north of Crawley (see Table 2). These hydrographs, which
were obtained by Nash [l5], relate to conditions before and after the execution
of channel improvement works. The data tabulated by Nash (loc. cit. Table 3,
p.323) were assumed to relate to a duration of 30 min. The reduction in lag
time shown by the Itpost-works" hydrographs was estimated by Nash from the
records for 3 adjacent catchment areas whose channels were considered to be
in an equivalent state to the improved conditions on the River Wandle. The
"post-works" hydrographs were therefore synthetic and not derived directly from
recorded data. Nevertheless, the data were considered to be useful in providing
an independent measure of the influence of channel improvement works without
a simultaneous growth in the impervious area within a catchment.
TABLE 2 : Details of gauging stations within the River Wandle catchment area
(from Nash [IS] 1
river station catchment

493
Also plotted in Figure 3 are the lag timebasin ratio relationships
obtained by Anderson [5] for catchment areas in three different stages of
development. Drainage class N refers to natural (rural) areas. Drainage class
B includes areas in which the impervious cover ranges from 20 to 30 per cent,
the tributary streams are sewered but the main channels are retained in their
natural state. Drainage class U refers to fully developed urban areas having
more than 30 per cent impervious cover and all stream channels completely
sewered or improved and realigned. The majority of the gauging stations used
by Anderson in deriving these relationships were situated within the Washington
D.C. metropolitan area.
Examination of Figure 3 shows that data for gauging stations 2 and 3
exhibit markedly longer lag times than would be predicted from Anderson's class
N relationship. Since gauging station 3 is situated at the outfall of a mill
pond covering an area of some 8-10 ha, such behaviour is to be expected. The
presence of a lake of similar size within the headwaters of Gatwick Stream
catchment has a similar if not as pronounced an effect on lag time. In contact,
the data for gauging stations 1, 4A, 5A, 6A and 7A all show reasonable agreement
with the Anderson class N relationship, although bearing in mind the
channel improvements above gauging station 5 and the existing urban development
above gauging station 6, the broken line drawn above and parallel to Anderson's
equation is perhaps a better approximation to natural catchment conditions
within the Crawley area.
Figure 3 shows that at gauging station 5, an increase in the proportion
of impervious cover (as estimated from data supplied by Crawley Urban District
Council) from 5 to 26 per cent is associated with a reduction in lag time of
23 per cent, whereas at gauging station 4 an increase in impervious area from
18 to 27 per cent apparently causes a reduction in lag time of 72 per cent.
The latter anomaly is thought to result from extensive renewal of sewerage
within the catchment which took place concurrently with the increase in paved
area. The data from gauging stations 6 and 7 indicate that channel improvement
(not including installation of sewerage) can cause a 30-40 per cent reduction
in lag time. The results obtained at gauging stations 4 and 5 are therefore
not as inconsistent as they might at first appear.
Figure 3 also shows that the lag times for Anderson's developed (class B)
and fully developed (class U) catchments are markedly shorter than those
observed in the Crawley area. The lower broken line, drawn parallel to and
immediately above Anderson's class B relationship, is probably the best approximation
to the behaviour of developed catchments with approximately 30 per cent
impervious cover and improved channel systems within the Crawley area that the
available data will allow.
Having obtained the relationship between the chosen scaling parameter and
two readily-computed catchment characteristics, only the form of the dimension-
less curve is required to construct the I-h TUH for an ungauged catchment
within the Crawley area. Accordingly, the 8 observed TITH'S whose derivation
was described in Section (3) above were reduced to the form of equation (4)
using the appropriate observed values of lag time (see Figure 4). A single
dimensionless hydrograph was then fitted by eye to the plotted points, care
being taken to ensure that the area under the curve was unity.

494
In practice, application of the method to produce a I-h TUH for an ungauged
catchment may be summarised as follows :
i)
ii)
iii)
measure the length and slope of the main channel of the catchment area
from a 1 : 25000 Ordnance Survey map, and compute the basin ratio ;
use Figure 3 to estimate the lag time of the catchment for a particular
stage of urbanisation ; and
given the lag time, use the dimensionless unit hydrograph of Figure 4
to construct the I-h TLTH of the catchment.
5. CONCLUDING REMARKS
Since the procedure outlined above uses the lag time as the only scaling
parameter, there is an obvious analogy with the single linear reservoir model
whose storage constant is equivalent to the lag time as defined in the present
study. The major difference between the two approached lies in describing
the TUH by means of a series of plotted points, rather than an equation.
peak of any TUH constructed from Figure 4 is therefore constrained to occur
at a specific proportion of the lag time rather than at a time equivalent to
the duration of the unit hydrograph.
According to Rao et al 181, the single linear reservoir model provides
an adequate description of the behaviour of both urban and rural catchments
less than approximately 13 km2 in area. Those Authors obtained an expression
for lag time in terms of volume and duration of rainfall excess and proportion
of impervious cover. The results of the present study (in particular, Figure 3)
tend to indicate that, for the area under study, proportion of impervious cover
provides a less than adequate description of the man-made changes within a
drainage basin. In the absence of additional parameters relating to changes
in the channel system, and perhaps distribution of impervious cover with respect
to the outfall of the catchment, the engineer concerned with the design of
flood alleviation works must rely on diagrams such as Figure 3 whose construction
is unfortunately largely subjective and highly dependent on local knowledge
of the area.
The urbanisation of a catchment area provides one of the most dramatic
examples of man's interference with the hydrological cycle. Whereas the
expansion of any conurbation creates an increasing water demand for domestic,
industrial and recreational purposes, the very presence of the urban area
accelerates the processes by which locally stored ana precipitated water is
returned to the sea. Despite the major changes in the flow regime of a catchment
area which urbanisation can bring about, relatively little attention has
been given to the quantification of such changes when compared with other land
use changes,such as that of forest to grassland. Bearing in mind the large
sums which have been and are being devoted to flood protection schemes for
urban areas, the available information can justifiably be labelled as inadequate.
The

ACKNOWLEDGEMENTS
The study described above was carried out on behalf of the Resources
Group for West Sussex County Council. The author wishes to thank Dr. T.M.
Prus-Chacinski, partner, C.H. Dobbie and Partners, for his encouragement
to prepare and permission to publish this paper. The assistance received
from the Chief Engineer, Thames Conservancy, Mr. E.J. Brettell, and the
Engineer and Surveyor, Crawley Urban District Council, Mr. H.J. Lumley, in
providing hydrometric data was also greatly appreciated.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
IO.
11.
12.
Toynbee, A. (1970).
Cities on the move, Oxford Univ. Press, 257 pp.
Savini, J., Kammerer, J.C. (1961). Urban growth and the water regime,
U.S. Geol. Survey, Water-Supply Pap. 159l-A, 43 pp.
Carter, R.W. (1961). Magnitude and frequency of floods in suburban
areas, U.S. Geol. Survey, Prof. Pap. 424-B, pp. B9-SII.
Martens, L.A. (1968). Flood inundation and effects of urbanisation in
metropolitan Charlotte, North Carolina, U.S. Geol. Survey, Water-Supply
Pap. 1591-C, 60 pp.
Anderson, D.G. (1970). Effects of urban development on floods in
Northern Virginia, U.S. Geol. Survey, Water-Supply Pap. 2001-C, 22 pp.
495
Espey, W.H., Morgan, C.W., Masch, F.D. (1965). A study of some effects
of urbanisation on storm run-off from a small watershed, Centre for Res.
in Wat. Resour., Univ. of Texas, Teoh. Rept. KYD 07-65OI,.CRWR-2, IO9 pp.
Espey, W.H., Winslow, D.E., Morgan, C.W. (1969). Urban effects on the
unit hydrograph, in Moore, W.L., Morgan, C.W. (eds.), Effects of watershed
changes on streamflow, Proc. Wat. Resour. Symp. no. 2, Centre for Res.
in Wat. Resour., Univ. of Texas, Univ. of Texas Press, pp. 215-228.
Rao, R.A., Delleur, J.W., Sarma, B.S.P. (1972). Conceptual hydrologic
models for urbanising basins, Proc. Am. Soc. Civ. Engrs., J. Hydraul.
Div., 98 (KY71, pp. 1205-1220.
O'Donnell, T. (1966). Methods of computation in hydrograph analysis and
synthesis, Recent trends in hydrograph synthesis, Proc. Tech. Meeting
no. 21, T.N.O., The Hague, pp. 65-102.
Philippee, J.T., Wiggert, J.M. (1969). Instantaneous unit hydrograph
response by harmonic analysis, Wat. Resour. Res. Centre, Virginia
Polytechnic Institute, Bull. 15, 36 pp.
Blank, D., Delleur, J.W., Giorgini, A. (1971). Oscillatory kernel
functions in linear hydrologic models, Wat. Resour. Res., 7, pp. 1102-1117.
Commons,, G,G. (1942). Flood hydrographs, Civ. Engrg. (New York), 12,
pp. 571-5720

496
13. Diskin, M.H. (1972). The role of lag in a quasi-linear analysis of the
surface runoff system, paper presented at the 2nd Internat. Hydrol. Symp.,
Fort Collins, Colorado.
14. Benson, M.A. (1959) . Channel-slope factor in flood-frequency analysis,
Proc. Am. Soc. Civ. Engrs., J. Hydraul. Div., 85 (Kyk), pp. 1-9.
15. Nash, J.E. (1959). The effect of flood-elimination works on the flood
frequency of the River Wandle, Proc. Instn. Civ. Engrs., 13, pp. 317-338.

W
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497

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PLOT OF LAG TIME AGAINST BASIN RATIO FOR THE CRAW!.EY AWEA.

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500

A DIMENSIONLESS UNITGRAPH FOR HONG KONG
P. R: HELLIWELL
Department of Civil Engineering, University of Southampton.
ABSTRACT
T.Y. CHEN
Royal Observatory, Hong Kong
The large number of individual catchments in Hong Kong makes it
impracticable to measure stream flows on all but a small proportion
of streams. Rainfall characteristics and topography are similar over
much of the area.
Using data for several storms at ea-h of the seven stream gau-
ging stations, a mean dimensionless unitgraph was derived. Basin lag
was used in the conversion of both time and discharge scales. For un-
gauged catchments basin lag can be estimated either as a simple func-
tion of catchment size, shape and slope.
This work was based on records collerted in 1964 and 1965, and
was one of the first studies made possible by the installation of a
network of hydrometric stations in Hong Kong.
RESUME
Le grand nombre de bassins fluviaux du territoire de Hong Kong
fait qu'il n'est possible d'effectuer des mesures de débit que sur un
faible pour centage d'entre eux. La pluviométrie et la topographie
présentent des caractéristiques semblables sur la plus grande partie
du territoire.
En s'appuyant sur les données recueillies à .ept stations de jaz
geage au cours d'un certain nombre d'averses, on a mis au point un h l
drogramme unitaire moyen sans dimension. Le temps de réponse du bas-
sin intervient dans les conversions à la fois pour l'échelle des temps
et pour celle des débits. Pour les bassins qui ne tont pas l'objet de
m sures des débits, le temps de réponse peut être estimé soit simple-
ment en fonction de la surface du bassin, soit en fonction de sa tai-
lle, de sa forme et de sa pente.
La présente étude est basée sur des observations recueillies en
1964 et 1965; ce fut une des premières qui aient été [,endues possibles
par l'installation d'un réseau hydrométrique dans le territoire de
Hong Kong.

502
Introduction
The British Crown Colony of Hong Kong is located on the coastline
of China, just inside the Tropic of Cancer at latitude 220N and longitude
114O. The land area of the Colony is approximately 1000km2, comprising
a section of the mainland, the islands of Hong Kong and Lantau, and a
large number of very small islands. The total area, including sea, is
approximately 2 500km2.
It is an area of high relief, the highest point being over 1OOûm
above sea level. Drainage lines are short, usually less than lokm, giving
a large number of small steep catchment areas draining to the very long
coastline. In some areas, mast notably in the northwest, there is an area
of almost flat land between the hills and the sea shore, formed by silting
up .If shallow bays, and subsequent uplift of the land relative to sea level.
These areas are intensively cultivated, with vegetable crops replacing the
traditional rice cultivation where levels are high enough to be clear of
sea water intrusion, and fish ponds starting to replace brackish water rice
cultivation near sea level.
Much of this flatter land, particularly near
the larger stream channels, is natural floodland, which makes stream
gauging at high flows very difficult.
The vegetation of the upland areas is of coarse grasses or mixed scrubland.
Gathering wood for firewood and traditional seasonal burning of hillside
vegetation tend to degrade the cover. Soils on the hills are coarse,
thin, and poor. Gullying, sometimes severe, occurs mainly in the west of
the Colony. In the East, the grass cover is complete, and sediment loads
are very low. The geology is predominantly granitic. Soil moisture and
groundwater storage are small.
Climate is seasonal. Winters are cool and dry, although periods af
li-ght rain do occur. Summers are warm (daily maximum temperature up eo 35W,
fvith very little diurnal variation) and wet. The mean summer half year rainfall
at the Royal Observatory is 1850mm and the mean winter half year rainfall
is 350mm. Observation-day rainfalls in excess of 250mm occur in most
years. Frosts can occur at levels above 600m, but snow does not fall.
Annual rainfall elsewhere in the Colony varies between 1250mm and 3000m.
Water supply has always been a major problem in Hong Kong. The small
size of catchment areas, the seasonal nature of rainfall, and occasional
severe droughts have presented a major challenge to the water engineers(l1.
Of necessity, reservoirs with small direct catchments have been built, and
water has been broughtin from much larger areas by systems of catchwater
channels or tunnels, intercepting many small streams which would otherwise
discharge to the sea. More recently, arms of the sea have been converted to
freshwater storage, at Plover Cove and at High Island.

Measurement of Streamflow
Measurement of streamflow in all catchments in Hong Kong is obviously
impossible, The approach adopted has been to make measurements in a
relatively small number of basins spread through the Colony, and to transfer
data from these to other basins. This paper describes the method used to
generalise flood hydrograph data, and presents the resulting dimensionless
unitgraph.
Streamflow is measured by fixed structures. Sharp-edged and crump
weirs of various compound profiles, triangular and trapezoidal flumes,
Parshall flumes and broad-crested diversion weirs are all used. Data are
published annually(2).
Selection of Records for Study
Examination of records from streamflow stations and of autographic
rainfall records for sites in or near to gauged catchments showed that three
or more storms suitable for analysis were available at seven stations. There
were seven more catchments which had been or were being gauged, but no suitable
records for this analysis were found among them, the commonest problem
being submergence of the measuring structure at high flow.
notes on all streamflow stations, and Figure 1 is a map showing stations used
in this study. From the table it will be seen that all catchments are small.
Method of Analysis
Table 1 gives
The method used was that described in USBR Design of Small and
by Linsley, Kohler and Pa~lhus(~).
Studies of base flow had shown that depletion could be assumed to be
of the type qt = qokt.
arithmetic graph paper.
503
Base flow separation was achieved by plotting on log-
No attempt was made to separate interflow.
Duration of effective storm rainfall was found by applying the$-index
technique to the hyetograph, having found the total storm runoff by
integration of the storm runoff hydrograph. Storms with up to five unit
periods of excess precipitation were used.
Successive approximation procedures were used to find the unitgraph
ordinates.
In order that the period should be less than one third of the rise time
of the unitgraph (to avoid instability in the computations), it was necessary
to use a unit period of 15 minutes, except in the case of the smallest catch-
ment where 74 minutes was used.

504
With such short periods, the accuracy of timing of the chart records
of streamflow and rainfall is critical. In the cases of Hok Tau and
C’iung Mei, unshielded, tilting bucket rain gauges were sited on the stream
recorder house roof in order to ensure correct relative timing. Unfortunately,
the mechanism transferring the tipping bucket record to the chart proved
u.iieliable, and some records were lost. With the other stations, it was
hoped that timing errors would average out, but there is no evidence on this.
From experience, errors in long-period chart records using chart drives of
ZVknlday can be kept to less than five minutes by making corrections based
on check observations.
On the other hand, standard daily autographic rainfall
recorders in the hands of all but the most careful observers can often show
fluctuations from correct time of ten to 15 minutes.
The measure of agreement between various unitgraphs for a given catchment
varied. Two examples, one showing consistent behaviour, and another showing
rather poor agreement, are shown in Figure 2. Average unitgraphs for each
catchment are shown in Figure 3. These were formed in the usual way by
averaging time to peak, magnitude of peak and total duration, sketching in
a mean shape, and adjusting the area under the curve.
The variation of the mean unitgraphs can be seen in Figure 3. A means
of unifying these was required. They were made dimensionless in terms of the
time to the centroid of the unitgraph and the volume of unit rainfall excess.
Time-axis values were divided by time to centroid of unitgraph and
discharge values were multiplied by time to centroid of unitgraph and divided
by the volume of unit depth of runoff over the catchment area.
The seven dimensionless unitgraphs and the mean dimensionless unitgraph
found from them are shown in Figure 4. The ordinates of the dimensionless
unitgraph are listed in Table 2.
Application to ungauged catchments
The size range of individual catchments included in the analysis
adequately covered the sizes of catchments found in Hong Kong. Similarly,
geographical distribution was quite good, only the eroded area in the west
being excluded.
Catchment and stream slope variability was not so well
covered. The Tai Po Tau catchment included some lowland area, as did the
Kam Tin catchment. The other five were upland in type.
The time to the centroid of the unitgraph for the seven catchments is
the basin lag (i.e. time from centre of area of excess rain to centre of
area of hydrograph of excess runoff) plus half the unit period, (lag + 9).

505
The basin lag for the mean unitgraph of each catchment was plotted
against catchment area and against-where L is the length of the main
s
stream projecked back to the catchment divide, as measured on 1:25000scale
maps, Lc is the distance along the stream from the gauging station
to a point on the main stream nearest to the catchment centre of area,
and S is the stream slope as estimated by the difference in elevation of
the main stream at the catchment divide and the gauging station divided
by L). The correlation coefficients were 0.92 and 0.86 respectively. It
was significant that the Kan Tin value fell close to the regression line
when slope was included, and off the line where area alone was used.
However, in all work using the hydrograph, catchment area alone has been
used. Figure 5 6 show the relationship. Figure 6 also shows data from
Linsley et ad4j and Design of Small Dams(3).
The equations for estimation of catchment lag in ungauged basins are:
lag = 0.47 areaoss4
with lag in hours,area in km2
lag = 0.36 (3)0.40 with lag in hours and lengths in km
si
To apply the unitgraph to any particular storm it is necessary to
estimate a @-index value, or loss rate. Studies of this for Hong Kong
conditions showed wide fluctuations between storms, ranging from 2.5 to
80m/h, with values commonly between 10 and 40mm/h. Judgement must be
used in selecting a suitable value. When reservoir spillway studies are
in hand, a very low value is appropriate. For drainage design, a value
nearer the mean would be used.
This dimensionless unitgraph has been used in conjunction with studies
of probable maximum precipitation over Hong Kong(5~6) carried out by the
staff of the Royal Observatory, to check capacity of existing reservoir
spillways and in the design of new dams in Hong Kong.
Flood frequency
analysis has also been used but in the absence of long records this is
thought to be less reliablef7).
Conclusions
The dimensionless unitgraph derived by procedures developed in the
U.S.A. is useful as a design tool. Hydrographs from the seven catchment
areas, covering the range of sizes and types found in Hong Kong were unified
to an acceptable degree of accuracy using time to centroid of unitgraph in
converting scales into dimensionless values.

506
Whether area alone or a more complex parameter should be used for
predicting basin lag is uncertain. Area alone appeared to be adequate
except in the case of catchments with extensive lowland area.
The volume of data available for this study was small both in terms
of length of records used and number of stations.
Acknowledgements
Thanks are due to the Director of Public Works, Hong Kong Government,
for permission to publish this paper; to Mr. J. Forth, who later extended
the work on floods to other methods of approach; and to Mr. Wong Shiu Ming,
present holder of the post of EngineerIHydrologist, for his valued ascist-
ance in checking the data in this paper and providing information.
References
1.
2.
3.
4.
5.
6.
7.
Robertson, A.S. and La Touche, M.C.D., Assessing the Yield of Hong Kong's
Reservoirs, J. Institution Water Engineers, 23, (1969), 8, 507-519.
Hong Kong Rainfall and Runoff (Annually from 1965), Hong Kong, Water
Authority, Public Works Department.
United States Bureau of Reclamation. Design of Small Dams, (1960),
Washington, U.S. Govt. Printing Office.
Linsley, R.K. , Kohler and Paulhus, Hydrology for Engineers , (1958) ,
New York, McGraw-Hill.
Bell, G.J. and Chin, The Probable Maximum Rainfall in Hong Kong.
R.O. Tech. Mem. 10, (1968). Government Printer, Hong Kong.
Cheng, S. and Kwok, (1966) A Statistical Study of Heavy Rainfall in
Hong Kong. Tech. Note 24, Hong Kong, Royal Observatory.
Design Flood for Hong Kong, HS7, (1968), Water Authority, Public Works
Department, Hong Kong.

-
Altitude Catchment
Station of Crest Area
Name - m -@
I_
Tai Lam Chung 15 16.2
60ft. weir
Sham Tseng
Tai Lam Chung
'A'
Tai Lam Chung
'B'
30
75
63
2.0
0.8
1.2
Contro 1
Compound weir with ogee crest,
loft. low flow section.
30ft. compound weir with ogee crest,
3ft. low flow section.
Compound V and rectangle sharp-
crested suppressed weir.
Compound V and rectangle sharp-
crested suppressed weir.
Instrument
Staff gauge.
Locally-made float level recorder.
Staff gauge.
George Kent float level recorder.
Sloping brass staff gauge.
Munro vertical drum
Sloping staff gauges.
Streamflow Stations in Hong Kong, to 1966
Ob s erving Programe
Frequent staff gauge readings
before Jan. 1950, thereafter
continuous recording with
daily observations.
Continuous recording with
daily observations.
Frequent staff gauge readings
before June 1963, thereafter
continuous reading with daily first
and then weekly observations.
Frequent staff gauge readings
before June 1959, thereafter daily
observations.
Readings Record
Commenced Quality Remarks
Apr. 1948
Fair
Discont inued
May 1955
Jul. 1952 Fair Discontinued
June 1956
Jun. 1958
Good
Jun. 1958 Poor
Shek Pi Tau L
41.6
102ft. long with 4ft. wide broadcrested
weir.
Sloping staff gauge.
Munro vertical drum float level
recorder.
Daily observations before June 1964,
thereafter continuous recording
with bi-daily observations.
May 1960 Poor
Ho Sheung Heung 5
16.9
40ft. long broad-crested weir.
Sloping staff gauge.
Munro vertical drum float level
recorder.
Daily observations before June 1964,
thereafter continuous recording with
di-daily observations.
May 1960 Poor
Tai Po Tau
9
15.2
Broad-cres ted weir.
Sloping staff gauge.
Munro horizontal drum float
recorder.
eve1
Continuous recording from July 1961
to April 1963, daily observations
at other times.
Sha Tin
100
1.2
Compound sharp-crested rectangular
weir without separating walls. 90°
V notch upstream for low flows.
Staff gauge.
Munro horizontal drum float eve 1
recorder.
Continuous recording from Jan. 1961
to Jan. 1963, daily observations at
other times.
Nov. 1960
Hok Tau 85
6.0
Compound sharp-cres ted rectangular
weir, without separating walls.
Sloping brass staff gauge.
Munro horizontal drum float level
recorder before May 1964, thereafter
Leupold & Stevens A-35 recorder.
Daily observations before June 1961,
thereafter continuous recording with
daily first and then weekly observations.
Dec. 1960 Good
Chung Mei 13 9.1 Compound crump weir with -90° V notches
upstream for low flows.
Sloping brass staff gauge.
Munro horizontal drum float level
recorders before April 1964, thereafter
Leupold & Stevens 2A-35 recorder.
Continuous recording with daily first
and then weekly observations.
May 1962 Good
Siu Lek Yuen 74 2.1 Compound sharp-edged rectangular
weir without separating walls.
'Vertical brass staff gauge.
Munro horizontal drum float levei
recorders before June 1964, thereafter
Leupold & Stevens A-35 recorder.
Continuous recording with weekly
observations.
May 1964 Good
Tsak Yue Bu 41
1.6
Compound V and rectangle sharpcrested
suppressed weir.
Sloping brass staff gauge.
Leupold & Stevens A-35 recorder.
Continuous recording with weekly
obscrvations.
Jul. 1964 Good
Lo Shue Ling 3
10.8
Parshall flume, 15ft. throat.
Staff gauge. Leupold & Stevens
:A-35 recorder.
Continuous recording with weekly
observations.
Jul. 1964 Poor
Kam Tin 3
11.7
Parshall flume, 25ft. throat.
Staff gauge. Leupold & Stevens
2A-35 recorder.
Continuous recording with weekly,
observations.
Jul. 1964 Fair
Oct. 1960 Fair Dis continued
Aug. 1963
Table 1
Good Discontinued
March 1963

TABLE 2
Ordinates of the 15-Minute Dimensionless Unitgraph
Time + (Lag+:)
o. 20
0.30
0.40
0.45
0.50
0.55
0.65
O. 70
O. 80
0.95
1.00
1 .O5
1.30
1.50
2.00
2.20
2.75
3.40
3.90
5.13
U
lag+2
Discharge x - V
0.05
o. 10
0.19
0.32
0.57
O. 71
1.00
1.02
1.01
O. 73
O .64
0.56
0.38
0.30
O. 18
O. 15
o .o9
0.05
O .O3
0.00
509

Fig. 1
Q
51 O

O
/
I-
.-
E
I
.-
t
3
-e
oa N
L
.-
B

O
- - t 2
E .-
I-
+ .-
t
3
I
cv
O

cc ;v
--i--
-+

ABSTRACT
STUDY ,OF MAXIMUM FLOODS 1.N SMALL BASïNS OF TORRENTIAL TYPE
Rafael HERAS
Dr. Civil Engineer
Angel LARA
Civil Engineer
The methodology of study is summarized for small b=
sins of torrential character and it is applied to one of the gu-
llies of the Gran Canaria island, considering the geological and
geomorphological conditions of the basin and also the principal
physical characteristics of the same one. In relation to all these
physical characteristics and of a statistical complete study of in-
tensities, the hydrogram is established for different hypothesis
and the type of hydrogram is studied more unfavorable in relation
in relation to the duration-intensity-frequency curves of maximum
precipitations in 24 hours.
RESUMEN
S:e resume la metodologia de estudio para pequeñas
cuencas de carácter torrencial y se aplica a uno de los barrancos
de la isla de Sran Canaria, teniendo en cuenta las condiciones geo-
lógicas y geomorfológicas de la cuenca y también las principales c z
racteristicas físicas de la misma. En función de todas estas carac-
terísticas y de un estudio estadístico completo de intensidades, se
establecen los hidrogramas para distintas hipótesis y se estudia el
hidrograma tipo más desfavorable en función de las curvas duración-
intensidad-frecuencia de precipitaciones máximas en 24 horas.

518
1. Generalities
This method has been applied to the Tirajana gully, which is
one of the most important in the south zone of the Gran Canaria island. The
high part of their channel is formed by a big number of gullies which have its
origin to an altitude of about 1,700 m., following the receiver basin a direction
sensibly north-west-southeast. The maximum longitude of the channel is 27 km.
and the total area of the basin is 71,4 km2. Its location in the island is reflected
in the graph number 1.
2. Geology of the basin of the Tirajana gully
The region where the Tirajana gully is located is the southeast
of the Gran Canaria island.
For its location, it participates of the geological characteristic
of the half south of this island, appearing on the surface the most ancient
complex which have taken part in its formation, such as are the Ancient Basalt
of the Serie I, of basaltic alkaline-olivinical composition and formed by sub-
parallel running out with pyroclasts intercalated, the Trachysienite complex
with ignimbrites associated, of rhyolithical, panthelithical and trachyphenol-
ithical compositions, and the Phonolithical serie, composed in this zone by
running out, pius, end ignimbrites, frequently with laminar parting parallel
to the direction of the flow.
All these series are located principally in the middle zone of
the gully, existing also in form of little cropping out, principally phonolithic
and trachysienite, in the high part of the basin. On the other hand, all the
mentioned series are practically impermeable in the process of infiltration
from the surface and, particularly, the series of Basalts I and Trachysienite
Complex, forming the majority of the substratum on which are seated the most
recent superficial formations.

519
The series Pre-Roque Nublo and Roque Nublo, which have its
maximum power in the interior of the island appearing largely disseminated
in the Tirajana gully, specially in its middle and high zones.
The said series are composed lithologically by angular
fragments constituting xenolithical agglomerate with intercalations of
tephrithical lavas, basaltical running out and sediments. In these series, of
moderate permeability, a quick fall in the level of water is produced, being
therefore, its pondage coefficient very law.
The most modern basaltical serie that appear on the surface,
is the correspondent to the Basalts II, of basaltical olivinical composition and
constituted by aa and pahoehoe lavas, more permeable than the previous
formations. These lavas cover principally the northcast zone of the fully
disseminating also in smaller proportion in its middle zone.
Finally, this basin present a genuine characteristic which is
distinguished from the contiguous ones, since that a big part of its surface
occupied by sedimental formations, of which, the avalanches of various ages
constitute the principal cropping out of the zone of heading, while the low zone
of the basin is covered by deposits of recent alluviums, with bigger porosity
and higher permeability.
3. Physical Data
In order to know the characteristics of the basin to use them
fundamentally in the estimation of its velocity of propagation of maximum
flood, it has been calculated for the same one, the following characteristics:
, longitudinal section
. surface
. perimeter
. equivalent rectangle
, hypsometrical curve
. index of compactness
. index of slope

520
surface perimeter
17.4 km2 57.5 k m
4. Maximum floods
4. 1. General planning
The values obtained have been the following:
compac tnes s equivale nt
index rectangle
1.90 L = 26. 10
1 = 2.74
slope
index
O. 263
The principal probleme presented is the absolute lack of
direct data of gauging with sufficient extension and guarantee, as much in
the studied basin as in the rest of the island, therefore it is not possible to
study the flood from the direct data of maximum flows neither by comparison
with others basins, affinitive basins hydrologically. Therefore, using the
maximum available data, it has been performed the complete study of floods
by empirical and hydrometrical methods, constrasting each one of the
estimated parameters with data obtained by direct procedures in the Gran
Canaria gully.
4. 2. Empirical methods
In the formation of maximum floods intervenes multiple
causes, whose possibility of coincidence characterizes the risk. The surface
of the receiver basin is one of the causes among the principal ones, since
there exists a good correlation between the basin area and the maximum flood.
Using formulas that could tie directly the flows of floods with
the surface of the basin and others in which intervene others hydrological
parameters.
Among the existing formulas it has been used those which are
in the joined chart; these formulas has been selected in relation to the hydro-
logical characteristics of the basin in the present study. In the mentioned

chart it has been given, in the same way, the values which are the result of
its application.
SANTI
GREAGER
FORTI
ZAPATA
423 (Tr = 500 años) KUICKLING 255 (Tr = 100 anos)
520 (Tr = 500 años) TURAZZA 820 (Tr = 500 anos)
626 (Tr = 500 años) HERAS 780 (Tr = 500 anos)
272 (Tr.= 100 años) G. QUIJANO 292 (Tr = 100 anos)
The big dispersion of the results obtained of the same ones
can be observed.
4. 5. Hydrometrical method
521
This method consists in trying to reproduce the meteorological
phenomenon and, in this case, we will use the method of the isochronal curves,
to which it is necessary to discompose the surface of the basin in some zones
(si, s2, . . . sn) limited by lines (isochrones) in which the water fallen in one
of these ones delays in arriving to the point in wich we estimate the flood,
sucesive times of value t, 2t, . . . , being our case t half hour.
The velocity of the water if fixed by experimental and
empirical methods, in relation to the physical data, fundamentally of the
longitudinal section and index of slope, and other characteristics peculiar of
the basin (vegetation, geology and so on). In our case, we have fixed as
velócity 6 km/hour in the low zone of the basin, up to an altitude of 600 m.,
above sea level, and 7 km/hour in the high part. Once fixed this one, the pointe
are obtained from which delays in arriving the water to the place studied a
same time and with which, as contour line of a topographical elevation, we
can draw the isochronical lines obtaining simultaneously the concentration
time, that in our case is of 4.3 hours.

522
If we contrast this time with the one given by any of the
empii ical formulas existing (for example, Giandotti), we obtain a difference,
by an excess of about 1 hour. This appreciable difference is justified by the
quantity of sediment load which carry the floods in this type of gullies, and
produce a disminution in the mean velocity of propagation. The incidence of
the considered velocity in the flood peak is small, and so is only influenced
by the concentration time.
The isochrones once obtained, multiplying the area encircled
among the same ones by the intensity of precipitation and the supposed runoff
coefficient, the flow is obtained in the studied point due to the precipitation in
each one of the zones.
They are, therefore, necessary the data of maximum
intensities of precipitations in the basin for a determined period of recurrence.
To realize the statistical study of the intensity we will use from among the
several laws of distribution of frequencies which are applied in hydrological
problems, Gumbel’s law, which is used principally for distributions of
maximum values. This law has been applied to the usable series of the interior
stations of the basin and to a series of stations of lap. All of them can be seen
in the graph number 1. The maximum annual values of precipitation in 24 hours
for several periods of recurrence are reflected in the charts numbers 1, for
the stations of the interior of the basin, and number 2, for the exterior ones.
In order to adjust the distribution of the values of maximum
precipitation in 24 hours and considering the probability of coincidence of said
values, the Gumbel’s law has been applied to the monthly data in all the
stations, for October, november, december, january, february and march,
resulting to be the months of October and november the most unfavourable in
relation to the floods, as it is deducted of the observation of the chart number 3.
Also, to contrast the distribution of maximum values in the
basin, the isomaximum curves has been designed with the values of maximum
precipitation in 24 hours to times of recurrence of 50, 100 and 500 years for
the maximum maximorum annual values and for the maximum values of
October (which is the month of maximum intensity). These isomaximum curves
can be seen in the graphs numbers 5 up to 10 and have served like contrast of
the values obtained by Gumbel and also to adjust the mean intensity of
precipitation and its variation with the time.
With regard to the runoff coefficient, there is hardly no data
for maximum maximorum flows, therefore considering the impermeability of
the middle and high zone and the greater permeability of the low zone, we
estimate some runoff coefficients of O. 85, O. 80 and O. 50, respectively, for
each one of the three considered zones. To estimate these coefficients, which
could be reached in strong floods which would be produced after several days

523
of considerable precipitation, it has been realize studies with all usable data
and considering the physical, geological and geomorphological characteristics
of the basin, detached in high, middle and low zones, it has been obtained
mean runoff coefficient of O. 78 that seems to be reasonably adjusted to the
characteristics of this basin.
The duration of the storm is an important factor in the
determination of the maximum flood, the maximum value of the peak flow is
used to obtain with durations of storm about the concentration time. In our
case, we have supposed durations of storm of 1, 2, 3, 4, 5, 6 and 8 hours.
The precipitation for the several hypothesis has been estimated in relation to
the distribution of the maximum precipitation in 24 hours for smaller periods,
obtained from the short available data, which have been contrasted with direct
measures, obtaining the following values:
Duration of the storm (hours) 1 2 3 4 5 6 8
Precipitation in percentage of
the precipitation in 24 hours 35 43 57 69 75 ao 86
Although it is considered little representative the compiled
data of maximum intensities in several stations of the basin in study, the said
values are kept, putting us in security side. At the same time and in order to
procure greater aproximation to the actual phenomenon, we can consider three
stretch of different mean intensity coincident with the high, middle and low
zones, previously mentioned in the estimation of the runoff coefficients.
For the different hypothesis of duration of the flood, the
intensity, together, is distributed in the time in such a manner that in the
hydrograms of duration 1 and 2 hours it is considered all the unitary intensity
estimated and for 3 or more hours it has been supposed uniform intensity
during the two first hours and decreasing in a 20% each hour more of duration
until reaching a minimum of a 20% in the storm of 6 or more hours.
The isochrone curves used in the calculation of the hydrograms
as well as the different zones considered can be seen in the graph number 11,
and the hypothesis that the storm i s produced simultaneously in all the basin
has been made since that the hypothesis that began in the head waters and goes
displacing in the direction of the gully appears excessively unfavourable for
the climatological conditions of the basin. Once the runoff values, intensity
of precipitation and duration of the storm, are fixed, we can obtain the flows
due to each zone and the accumumulated of these ones give the flows that would
reach the sea in each moment, supposing an infinite time of rain. Displacing
horizontally this curve in the time of duration of rain and calculating the curve
difference of the two, we obtain the actual flows that reach in each moment.

In relation to the study realized it has been considered the
hypothesis (i), applying in all of them some runoff coefficients of O. 80, O. 85
and O. 50 for each one of the different zones considered and the distribution of
intensities already cited for durations higher than two hours.
As summary of the hydrograms obtained, in the graphs
numers 12 up to 15 figure the correspondent to a duration of storm of 4 hours
and periods of recurrence of 100 and 500 years, the same for the maximum-
maximorum values of precipitation, as for the maximum of October. In the
chart number 4 appears the distribution of intensities in space and time for
these hypothesis.
CONCLUSIONS
As a result of the calculations realized by the different methods
and considering that the hydrograms obtained must be affected by a reducent
coefficient in relation to the hypothesis of calculation, in which it has been
considered some maximum values of the runoff coefficient and some maximum
intensities which must be reduced due to the non-coincidence of the distribution
in the space and time of maximum values in all the stations, we obtain the
results that can be seen in the annexed chart.
(1)
The hydrogram type estimated figure in the graph number 16.
The statistical study of maximum precipitation in 24 hours has been
realized in the period of 21 years, 1949-50 - 1969-70 and the data of
the usable stations has been contrasted and, generally, it appears to
have enough guarantee, but by the extension of the period used, resulted
as a risk to extrapolate for times of recurrence higher than 100 years.
The results obtained are conditioned by the empirical-theorical
methods used, due to the absolute lack of series of maximum flows,
although we estimate that the maximum difference with the actual
values will not exceed 15%.
The study is only related to maximum values of flood and in it has not
been considered the effect of the solid flows.
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
ABSTRACT
P.H. EERBST, S. VAN BILJON, J.P.J. OLIVIER AND J.M. HALL
A regionalized study of maximum annual flows of different short dura-
tions (including peaks) has been carried out. In view of the limited
length of record available at most of the gauging stations in the region,
an attempt has been made to develop a technique to strengthen the data
available at any particular point of interest, by using all available per
tinent flow data in the region. Having chosen the extrema1 dislribution
best suited to the region, the moments of the sample (after adjustment)
are correlated with various catchment characteristics. This allows estima
tion of flood magnitude frequency curves at any site of interest within
the region, with associated confidence bands. Such frequency curves are
determined for various suitable time intervals which then allows the syn-
thesis of characteristic flow hydrographs, with a specific probability of
occurrence attached to each, along Mith associated enyelopes correspon-
ding to specific confidence limits, Comparison with hydrographs derived
from rainfall input depths with specified probabplities, subtracting los?
ses, and then using unitgraph methods, leads to the conclusion that a bet
ter relation between probability of occurrence of a specific hydrograph,
and its magnitude, can usually be obtained by direct statistical methods,
than by more indirect deterministic techniques.
RESUMEN
Fia sido ejecutado un estudio regionalizado de gastos máximos anuales
de duraciones cortas y diferentes (.incluyendo valores máximosr. En vista
de la limitación de información disponible para la mayoria de las estacio
nec de aforo de la región, se ha intentado desarrollar un método que permita
reforzar dicha información para cualquier punto de interés usando to
dos los registros existentes de la región. Habiendo elegido la distribución
extrema que mejor acomoda a la región, se han correlacionado los mementos
estadísticos de muestre0 (ajustando valores) con ias características
de diferentes hoyas. Esto permite la estimación de curvas “magnitudfrecuencia”
de riadas para cualquier punto de inter’es dentro de la región,
asociadas con bandas de confiabilidad. Tales cur~as de frecuencia
se han determinado para convenientes intervalos de tiempo las cuales permiten
la sfntesis de hidrógrafos de flujo caracteristicos, relacionados
con probabilidades específicas de ocurrencia, junto con envolventes que
corresponden a limites específicos de confiabilidad. Comparación con hidrógrafos
derivados de precipitaciones de ocurrencia especifica, substrayendo
pdrdidas y usando luego métodos de gráfico unitario, lleva a la
conclusión que una mejor relación entre probabilidad de ocurrencia de un
hidrógrafo determinado y su magnitud, puede obtenerse generalmente medi’an
te métodos estadísticos directos en vez de técnicas deterministicas indirectas.

542
INTRDDUCTIOIY
There is no need to stress the importance of reliable flood magnitude frequency estimates in mter resource development.
Whilst this problem is especially highlighted in developing regions, even so called developed
countries frequently suffer from a limitation of data on which to base reliable flood flow estimates.
An estimate of a flood with a specified recurrence interval (determined by specified design consideration)
should be accompanied by information concerning the reliability of such an estimate, but this has not
usually been the case in the past.
This paper outlines a methodology by means of which all or most of the available flood flow information
in a region can be rationally analysed and assembled, by taking into account quantifiable parameters of
characteristics of the various catchments in the region and relating these to the moments of the frequency
distr, but ion assumed.
Methods are described by means of which a flood hydrograph with a specified recurrence interval can be
estimaied and a quantitative statement on the reliability of such an estimate can be made.
lhe method was
drvtloped partly due to uncertainty about the validity of use of the method whereby rainfall intensity - dura:
::on curves are applied to a transformation function (e.g. a I - hour Unitgraph) due to the many assunptions
necessary in the latter approach.
It was felt that where some limited flow information does exist, an approach as outlined woulo provide better
estimates of flood frequencies and flood hydrographs, including also information on the reliability of such
estimates. Avoidance of any mention of the degree of uncertainty in any such estimate does not remove the
uncertainty, it only serves to diminish consideration of the fact that such uncertainty not only exists but
may be considerable.
FREOUtNCV DISIRIBUTIONS
In developing a methodology for flood frequency estimates (for annual extreme flows of various -hart dura=
tions) on a regional basis, it is essential to decide which frequency distribution should be assumEd to apply
throughout the rsgion. This is so, firstly for the reason that if a reasonable correlation between moments of
the oistribution assumed (whether the variables be transformed or not) and characteristics of the catchent can
be found, the same distribution must by force also be used for estimation purposes at some new site of interest
in the region.
Secondly it was considered that if a distribution is used which has a third parameter, this would provide
the necessary flexibility (adaptability) for the distribution to be ‘Iraiaxefi so as to fit that particular
region; the third parameter thus being a constant throughout the region (for every duration).
Furthermore, the possibility exists that such a third parameter may show some sensible variation if adjacent
regions are analysed in turn, thus promising the possibility of a “smoothing” thereof, providing there are no
gross geographic discontinuities. The coastal zone, consisting of rivers draining to the south eastern sea=
board of the Republic is considered suitable for such further analysis, a similar but less comprehensive study
having been carried out for those rivers mainly draining via the Orange, Limpopo and Komati river systems [i] .
The region chosen for use as a pilot study which this paper srnarizes, consisted of the north eastern part
of the zone mentioned and is shown on the locality map narked figure 1.
Data from wme of the gauging stations with a reasonable length of record in this region were used to com-
pare the log Gumbel and log Pearson Type III distributions. In the latter case the data were plotted on
specially made graph paper on which a distribution with a skew equal to that calculated from the sample concerned,
plots as a straight line. Camparison of the plots led to the conclusion that no particular superiority
of the one above the other was evident. The log Pearson Type III distribution was therefore chosen for the
reasons mentioned above. It should be stated however, that the techniques described in this paper could be
applied equally well to the Gunbel or log Gmbel distributions.
Moreover, the basic supposition that nature would be so kind as to ensure that the distribution of flou
extremes would follow some definite (simple) statistical distribution, should always be remembered for the
fallacy which it is. Ihis is especially true where two distinctly separate flood producing factors may pertain;
and may operate either separately or conjunctively.
The authors feel, along with Harter 121 that there can be IM finality about the recommendations made by
the U.S. Water Resources buncil 131 concerning the log Pearson Type III distribution, but for the various

543
reasons stated, and the availability of the tables provided by Hartar, the exact form of the distribution
postulated is of lesser importance than is the proper utilization and assembly of all the available
flow data in the region, in a rational manner, so as to ohtain the best possible flood flow estimates
and concomitant reliability estimates.
The various durations of extreme flows in the region that was investigated in the pilot study sunniarized
herein were: peak flow, 1 day, 2 day, 4 day and 6 day average extreme flows. The logarithms (to base 10) of
these extremes were found to have a skew of 0,3 for peak flows, 0,4 for 1 day average extreme flows and 0,5
for 2 day, 4 day and 6 day average extreme flows. It would appear that there may be a relationship between
the skew and the duration and if this is also found to be the case in other regions this could cnnveivabìy
be used to obtain more stable estimates of the skew.
Special graph paper was developed for the skew values of 0,l (0,l) 1,O and 1,5. An example of this is
the paper on which the graph shown as figure 3 appears. The ordinate has both logarithmic and linear scales,
and the abscissa consists of both the emulative probability of exceedence (e.g. of a certain flow magnitude)
and a linear scale, the units of which are essentially in the number (and decimal fraction) of standard
deviations from the population mean p, corresponding to the probability of exceedence for the particular skew
value in question. This scale is identified as the K - scale (K being analogous to Gumbel's reduced variate).
On the assumption then, that the logarithms of the annual extreme flows for the various durations are distributed
according to a Pearson Type III distribution with the applicable regional skew values, the N year
flood can be obtained from the expresfion:
X =X . KS . (i)
Here and S are the mean and ssandard deviation of the logarithms of the individual extreme annual flows.
X, is the logarithm of the N year extreiae flood magnitude, and K is the number of standard deviations from the
population man y that corresponds to the exceedence probability for the skew value in question (presented
in detail in Harters tables).
If the logarithms of the extreme flows are distributed according to the Log Pearson III distribution with
a skew ofï= 1 say, then if probability paper designed forö= 1 is used, a straight line draw hereon for
specific values of X and S, will yield a flood magnitude - frequency curve of XIversus K. As K is uniquely
related to the probability of exceedence, X, can be read and transformed to yield the flou value estimated to
be equalled or exceeded for any specified return period within the range.
ESTIMATION Of IHE M@KIIIS OF THE DISTRIBUTION
The problem therefore reduces to estimation of the values of i and S for a specific catchent. This is done
by correlation of all availab!e and pertinent measured flow data to catchment characteristics, so as to be able
to obtain best estimates for X and S. The variables investigated depend upon factors considered either as
possibly causal, or as possible contributary factors towards the occurrence of extreme flows. Although the
authors are aware of the possible application of factor analysis (or principal cuœponent analysis) here, it has
not been used during this study for various reasons [4'J .
The various independent variables considered were the following: area, mean annual rainfall, average
slope, river length, monthly rainfall with a tvo year recurrence interval (log normal distribution assumed) and
a shape factor. The data used in the present study are presented in Tables 1 and 2.
The regression nodels used were-all of the general form:
X . a . b log A+ c log R t ................. (2)
It may be noted that, as 1 is the mean of the logarithms of the extreme flous, the above formula using only
= 2 Ab RC where og,m,iS the geometric mean of the extreme flows at
A and R is equivalent to the mdel Q
a specific site. g.m.
In obtaining a further estArnate of i (1 )by simple or multiple linear regression, a value for the variance
of such a further estimate of X is always optained. This variance depends not only upon the degree of variance
explained by the regression model, but also by the extent of the deviation of any of the independent variables
from its mean. In the case of equation 2 above the expression for the variance will be of the form:
VAR (ie) = Residual Variance[l + 7 1 + cZ2 (Log Ae - m)' + cj3 (Log Re - v)'
+ZcZj(Log A e - m > (Log R e - W ] .................. (3)

544
The statistical theory applied here is very clearly set outlin text books on statistics [5,6] .
In short, for every regression equation used for estimation of X or S an accompanying equation is developed
e e
for the calculation of VAR (le) and VAR (Se).
The variables mentioned earlier were used in regression models to determine the regression equations
that would explaln the highest proportion of the variance of the dependent variables I and S, for the five
durations considered.
from some 150 regression models tested, the 10 equations that were selected as the best are presented in
lable 3. Values of ~22, C J ~ and C ~ J are also presented for !se in an equation of the type represented by
equations 3 and 4, in order to calculate the variance of the X *s and the S 's. E.g. the equation for the
variance of a further estimate of X for peak flow, X by uS"e of equatiog 3 is as follows:
1 P t e 2
VAR ($,e) E Residual Variance [ 1- + 0,1191 (Log Ae-Log A) + 11,6819 (Log R e - W I 2
n
+ 2~0,4545 (Log A e - W ) (Log Re- WR)]
= 0,0454
6 2
(this is for the brgenstond Dam site which has a catchment area of 528x10 m and a mean annual rainfall of
900 x 10-3m).
In this analysis it was hoped that the monthly extreme rainfall would be a more representative parameter
of the flood producing characteristic of rainfall than the mean annual rainfall. However, as both of these
parameters explained an approximately aqua1 amount of additional variance, it was considered advisable to
select the annual rainfall for use in the prediction equation. In view of the availability on magnetic tape,
on a large scale, of such monthly rainfall data, and the understandable hope that an extreme value rainfall
parameter would yield better results, this is a very disappointing result. It is however intended to invati.
gate this aspect further.
Hawing estimated the value of i and Se, a straight line flood magnitude-frequency curve can be drawn on
the graph paper with the apprcpriate skew, and X for any value of N within the range can then be read from
N
the graph. Ihis is done for the peak flow and for the various durations for which formulae have been developed,
thus allowing for the synthetization of a balanced hydrograph. Ihis is a hydrograph constructed symmetrically
around the peak. It can then be adjusted along the time axis (but with retention of the properties derived)
to its proper shape, either by means of information an unitgraph shapes [7] or by actual measurement of one
or two reasonably large floods at the site in question. Such measurements muld be arranged for at an early
stage of a feasibility study involving a specific site, if no data are available. It should be noted here that,
according to Nash [8,14 estimation of a unitgraph shape, even from only one good sized flood in a season will
yield more reliable results than that whlch can be obtained synthetically.
RELIABILITY OF ESTIMATES
In constructing the flood magnitude-frequency relationship we have:
XN=ie + KSe . . (1)
If and Se are not independent, the problem of estimation of the covariance term arises, for which a
formula tuch as for VAR (1 and VAR (Se) has not been developed.
This problem was solves by use of an artificial population, distributed according to Pearson Type III,
withy = O, Q = 1. Skew, 8 was varied from O to 1,5 i.e. a different artificial population for each skew.
For each skeu an artificial population consisting of 10 O00 terns was prepared, by use of Harters tables.
Samples of size W ranging from 2 to 20 were then drawn. Every individual value drawn was, however transfor:
med by addition of unity so that in fact an approximation to a universe with O- = 1 andy = 1 was used.
for each sample size N, an adeguate number of samples were drawn to define-to a sufficient degree of accuracy,
the variance of i, s and cov h,s). for each sample drawn, the-values of x and s were calculated and then an
adequate number of such samples drawn so as to calculate var ( x), var (s) and cov (x,s). for the smaller
sample sizes the number of sanples drawn were simply increased indefinitely until it was clear that the result

545
:.J& converying. In this way curves were obtained showing how var (i), var (s) and cov (x,s) varies with saw=
ples size N, ranging from 2 to 20.
The results are presented in figure 2. Not all the curves developed are shown, but all the data obtained
was used in order to achieve an integrated matching set of curves.
lhe use of these curves, in order to solve the problem encnuntered with the existence of the covariance
term in equation 4 is eqlained as follows:
From the regression equations, values are obtained for Xe and VAR (ie) and also for Se and VAR (S 1.
(See Table 3).
Therefore
PutX = k ; = k andS =ks-k
e _ i i e 2-2
(but x = s = 1)
2
VAR (x,)=VA!? (k X)=kl var (S )= VAR fk2s)=k

546
Iherefore
and
COY (EL,S:) = k3k4 cov (;,SI
let i' and S* represent the best pooled estimates of these parameters,
then we have:
Nie + N'ile
i* = ............... (6)
N +Nt
s =
(N - l)Se + (N' - 1) Se
.v N+fi'-Z
!if unbiased estimators are used)
Then we have:
xi .
A T
1' .
........... (7)
KSI . (8)
and
VAR (xi) = VAR (2.1 + Kz VAR (S')
. 2K CDV (i*,S*) . . (9)
As before
Therefore
j* =kiandS':ks
5 6
VAR (i*) = k2 var (i)
5
2
VAR(S*) = k6 var (5)
and
COV (i', S') = k k COY (g,s)
56
As i = s = 1, k and k can be calculated. By putting N* = N + W1 and entering figure 2 with this calculated
value for N*,
6
"hues for var (i), var(s) and w v (x,s) can be rea$
These latter three values can then be used to calculate VAR (X,) for any specified K value thus making
possible the calculation of the confidence limits.
An example of a flood magnitude - frequency curve (for peaks) is shown as Figure 3 (brynstond site),
along with the one Standard deviation confidence bands.
BALANCED HYDROGRAPHS AND DESIGN HYDROGRAPHS
Using the approach outlined above, an estimate of peak flow with a specified probability of exceedance and
concomitant upper and lower confidence limits corresponding to one standard deviation (or for any other confi.
dence level required) [Io3 may be calculated. lhe same can be done for each of the five durations, resulting
in a llbalanced hydrographll (i.e. symmetric about the peak) together with upper and louer envelopes wrrespon=
ding to the desired confidence level. In other words, for a confidence level corresponding to one standard de=
viation this implies that there is a 1 in 6 (or 16%) chance that the true hydrograph could be as big, or bigger
than the upper envelope.
lhe shape of this hydrograph can then be suitably adjusted along the tiw, axis, but preserving its derived
characteristics, so as to obtain an estimate of the hydrograph with the required probability of occurrence,
but incorporating the shape which is unique to the particular catchment.
CONCL US IOW
Results of work similar to that described here have in the past been used in the Department of Water Affairs
in a somewhat different manner [i] first for estimation of peak flou only and lately [il] also for ertime
average flows over durations of somewhat longer periods. It is intended to extend the study to the whole of
the eastern and south eastern coastal zone of the Republic of South Africa.
The possible existence of medium term (e.g. from 3 to 7 years) non-stationarity of the various flood
magnitude-frequency curves due to such medium term variations in sea temperatures, will also be investigated

4.
5.
6.
7.
8.
9.
10.
11.
12.
13-
547
,: series of recent flood disasters in the coastal zone mentioned makes such a study virtually imperative,
the controlling conditions may still not have reverted to normal).
grouping approach will be followed to attempt to improve reliability.
for smaller catchments a stratified
More work is also intended to attmpt to determine the particular rainfall characteristic most closely
related to the flood producing attribute thereof.
lhe possibility of using an analogous approach to that described herein for estimation of other hydroloqical
parameters is not overlooked, e.g. low flow sequences with specified probabilities, mean annual runoff, etc.
This method also holds promise in rational hydrologic network plannirig [i31 or adaptation thereof.
Preliminary comparison of this method with older methods used by the Department, in some of which the
probability of the causative rainfall is put equal to the probability of the resulting runoff hydrograph,
s e m to inuicate that this method is preferable, not only from the point of view of accuracy of estimation but
also due to the frank admission and quantification of the reliability of estimation, and the extent of the
probable errors.
ACKNOWLEDGEME NT
The permission granted by the Secretary for Water Affairs to publish this paper is acknowledged.
The assistance provided by J. de Beer of the Computer Centre and by A.J. Muller, J. Botha, S. Fitchet and
L. Eskell in the preparation of the paper is greatly appreciated.
lhe guidance given by W.J.R.
ledged.
Alexander during the course of preparation of the paper is gratefully acknoum
RE FERE NCES
1. Herbst, P.H. (1968). flood estimation for ungauged catchments, Technical Report No. 46, Department of
Water Affairs, Republic of South Africa.
2. Harter, H.L. (1969. A new table of percentage points of the Pearson Type III distribution, Technometrics,
II(I), pp.177-186.
3.
U.S. Water Resources Council, (1967). A uniform technique for determining flood flow frequencies, Bull.
No. 15, Water Resources Council, Washington, D.C.
Haan, C.T. and Allen, D.M. (1972). Comparison of multiple regression and principal component regression
for predicting uater yields in Kentucky, Water Resour. Res., 8(6), pp. 1593 - 1596.
Ostle, B. (1963). Statistics in Research, Second Edition, Iowa State University Press, Ames, Chapter 8.
Ezekiel, M and Fox, K.A. (1959). Methods of correlation and regrassion#analysis, John Wiley & Sons, Inc.,
New York, pp.320-321.
Midgley, D.C., Pullen, R.A. and Pitman, W.V. (1969). Design flood determination in South Africa,
Report No. 4/69, Hydrological Research Unit, University of the Witwatersrand, Johannesburg.
Nash, J.E. and Shaw, B.L. (1%6). Flood frequency as a function of catchment characteristics,
Symposium on River Flood Hydrology, Inst. of Civil Engineers, London, Session C 6, pp. 115 - 136.
Beard, L.R. (1962). Statistical methods in hydrology, U.S. Army Engineer District, Corps of Engineers,
Sacramento, California.
Mode, E.B. (1961). Elements of statistics, Prentice - Hall, Inc., Nw Jersey.
van Blljon, S. (1972). flood volume frequency analysis - Vaal Dam. Internal Report, Department of
Water Affairs, Republic of South Africa.
Thorne, R.B. (1966). River Engineering and Water Conservation Uorks, Buttervorths, London.
Herbst, P.H. and Shaw, E.M. (1969). Determining rain gauge densities in England from Limited data to
give a required precision for monthly areal rainfall estimates, Journal of the I.W.E., 23(4),
pp. 218 - 230.

548

549
GAUGE
SEOUENCE
NUMBER ON
ûFFICIAL RAINFALL
GAUGE AVERAGE MONTHLY
NLMBER ANNUAL WITH
AREA
Of
CATCH-
SLOPE
Of MAIN
WATER
LENGIH
OF
MAIN
NUMBER
Of
YEARS
MAP OVER 2-YEAR MENI COURSE WATER OF
CAT CHME NT RE CUR R E NCE
INIERVAL
COURSE RECORD
10+m 10-3n i06$ m/mxiû3 10% YEARS
1
2
3
4
5
6
7
8
9
10
11
17
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
XlMOl
x2Mo1
XZMOZ
XZM08
Xx109
X2M10
X2Mll
X2M12
X2M13
Xx114
X2M15
XYOI
XW6
w4Mo2
W4M3
m4
wsMo5
W
W5M7
WW8
w5Mo9
WW12
w6MOI
XlMo6
m3
WY10
W13
888
1142
1074
1163
1070
1187
849
833
907
1237
936
1492
985
889
924
929
920
947
868
862
899
905
1003
1233
921
953
1099
204 5444 5.50
254
104 l3;49
231
176 9,47
289
181 35J1
264
280 18,40
289
127 19,42
183
401 16,37
192
88 10,03
208
1502 12,45
302
251 19,19
213 1538 1256
368
174 33,03
226
76 1 17,37
200 7122 5,05
204 5843 7,41
210
448 3,91
221
751 3,65
213
176 14,95
191
536 2,68
203
119 4,79
206 2805 12,29
209 12769 8,02
238
694 12,15
279
585 18,Ol
205
218 4,73
213 2201 5,81
263 1155 10.93
-
TABLE 2
144.0
26,6
32,2
25,7
28,2
15,8
26,6
13,7
86,9
x),6
76,4
17,4
53,6
232-6
162,5
24-9
5690
1593
46,7
27,4
9197
184,3
85,3
49,l
25,7
120,7
88.5
61
19
19
23
12
22
15
14
10
11
12
23
12
18
21
10
21
21
16
18
10
12
12
12
13
13
10
Average of Log of Standard Standard Multiple
Prediction Equation
(Log A=Y ; log R.2)
10 10
= 0,802Y+l,1372-3,880
P
Area Mean An.
in Rainfall
d m 2 10-3,
2,790 2,991
Error of
Estimate
0,207
Deviation:
Dependent
Variable
0,511
‘22 ‘33 ‘23
0,119 11,681 0,454
Correlation
Coefficient
0,92
F
Value
67,5
S D =-O,119Y-O,9692+3,611
il= 0,798’f+0,7262-2,921 2,775
0,115 0,133
0,517 0,134 14,105 0,608
O,%
o,%
5,5
1%,8
S1=-0,092Y-0,4302+1,855
il2= 0,832Y+0,947Z-3,7%
S2=-0,08OY-O,4502+1,865
14’ 0,867V+1,1172-4,479
+O, 070Y -O, %32+1, 559
j6= 0,891V+1,2522-5,026
Sk=-0,081 V-O,3922+1,674
2,775
2,775
2,715
0,073
0,530
0,063
0,545
0,059
0,556
0,075
0,134 14,105 0,608
0,134 14,105 0,608
0,134 14,105 0,608
0,73
0,97
0,74
0,98
0,68
0,98
0.63
11,6
187,8
11,9
248,2
8,8
296,3
6.5

550
KM 20 K) O 20 40 60 80 100 KM
SCALE I I I SCALE
MAP SHOWING POSITIONS OF GAUGING STATIONS
FIGURE 1

551

œ
O
LL
u)
3
5-
LI
E
P
a

PRACTICES OF DESIGN FLOOD FREQUENCY FOR SMALL WATERSHEDS IN THAILLAND*
ABSTRACT
Damrong Jaraswathana
Director of Hydrology Division
Royal Irrigation Department, Thailand
and
Subin Pinkayan
Associate Professor
Asian Institute of Technology
Bangkok Thailand
Based on the fact that adequate hydrologic data do not exist and
development of water resources projects cannot be kept waiting until
data are made available. Thailand shares this fact with the other de-
veloping countries. The hydrologic data conditions in Thailand can be
categorized as follows. These are: (1) none of any kind of data avai-
lable in the catchment area; (2) some data available within neighbou-
ring areas; (3) some data with short period of record; and c4) consl-
derable data available with low reliability and accuracy.
The purpose of this paper is to present the general practices of
hydrologic analyses in Thailand particularly on design flood frequen-
cy in small watersheds. The method which was the common practice for
assessing design floods was based on the concept of rational formula,
the unit distribution graph and the design storm obtained by the con-
ventional procedures of frequency analysis.
RESUME
Les données hydrologiques sont insuffisantes, mais l'aménagement
des eaux ne peut attendre. C'est une situation que la Thaïlande parta
ge avec d'autres pays en voie de développement. En Thaïlande, on peut
classer comme suit la nature des données hydrologiques: (1) il n'y a
rien; (2) on dispose 4e quelque chose dans des bassins voisins; (3)
on dispose de données sur une courte période; (4) on a une grande
quantité de données qui n'inspirent pas confiance et sont peu préci-
ses.
Le but de cette communication est de présenter les méthodes d'ana
lyse htdrologique habituellement utilisées en Thailande, notamment
pour l'évaluation des crues de projet sur les petits bassins. Les modes
de calcul les plus fréquents sont basés sur la méthode rationnelle,
l'hydrogramme unitaire et la recherche de l'averse de projet par
les procédés classiques de l'analyse fréquentielle.
* Submitted for presentation at the International Sympsium on the De-.
sign of Water Resources Project with Inadequate Data, June 4-9,
1973, Madrid, Spain.

554
INTRODUCTION
As far as the existing hydrological networks and its operation in Thailand
are concerned, it can be considered that the design of water resources pnojects
in Thailand are based on inadequate data. While the water resources developments
cannot be kept waiting, it is the main function of hydrologist to modify
the conventional approaches to the hydrological assessment in planning of
water resources projects. Application and modification of conventional
approaches may have certain degrees of complication depending upon the availability
and limitation of the information. Inadequacy of hydrological data
within the country may be catagorized as follows: (i) none of any kind of
data available in the catchment area; (2) some data available within neighbouring
areas; (3) some data with short and/or broken period of record; and
(4) considerable data available with low reliability and accuracy. Hydrologist
has to make a great attempt and utilized his own experiences in the country in
drawing up the basic fact, assumption and related knowledge to justify the
hydrological assessment of each case under study. With such attempts the
justification of a project could be made with only a fair degree of accuracy.
Most of the small storage reservoirs in the country were planned by applying
the modified Conventional approaches to the hydrological assessment, in which
the only available information is the rainfall data in the neighbouring areas.
HYDROLOGICAL DATA PROCUREMENT
It can be stated that hydrological observation has been initiated in
Thailand since 1831, when a staff gage had been installed at Ayuthya to observe
annual flood inundation within the Central Plain areas. The nature of flood
inundation had been a major factor reflecting the crop yield during the last
century.
Such maximum water levels were indications of water condition in term
of good wateryear, too high flood or drought conditions which, in turn, would
help predicting the annual rice harvest.
Later in 1905 the streamflow measurement by surface float was introduced
to measure the discharge of the Chao Phraya river for the purpose of planning
of water conservation, diversion and irrigation control works. Development
at that time comprised of diversion schemes in the river valleys and tidal
irrigation in the delta area. Connected channels between rivers in the estuary
were also excavated to conserve water for irrigation and navigation from and
to Bangkok. Scientific approaches applied to planning and implementation of
irrigation and drainage schemes were carried out by the Royal Irrigation
Department since 1915. Development of water resources was gradually extending
towards headwater and slowly progressed.
Until 1952, when the storage work began, the modern methods and scientific
standards to be introduced in the hydrological investigation were recognized.
A network of comprehensive streamflow gaging stations was set up in major
tributaries where damsites for possible large reservoirs were found. Meanwhile,
numbers of stations were added consecutively in the accessible remote areas to
examine the runoff and flood yield from watersheds. At present there are over
230 rating stations operated by various government agencies. Among them there
are small number of stations that the drainage area is less than 100 square
kilometers. Many common problems of hydrological investigation and data procurement
still exist in the country hence they limit the expansion of the net-

work particularly into the smaller watersheds. Among those common problems,
the limitation of financial support and lack of well-trained personnel are considered
to be the main factors. Lack of popularity of work is another important
factor leading to have less fund allocated for hydrological investigation.
5 55
Besides the large multiple-purposes storage reservoir projects, the
surface reservoir of comparatively small storage volume called "tank" irrigation
projacis were commenced in 1951 in the Northeastern Region of Thailand.
Several small watercourses in the undulated topography were formed and appeared
to be good sites for storage tanks. The reservoirs range in capacity from
around 40,000 cubic meters up to 18 million cubic meters. Of course, the hydrological
investigation of such small basins has never been practiced in the
region as well as in the other parts of the country.
PRACTICES OF DESIGN FLOOD FOR SMALL WATERSHED
To present the general practices of design flood for small ungaged water-
shed, a case design by Royal Irrigation Department (1963) of the Sattaheep
Tank Project, Thailand, is described below.
It was the requirement of the Sattaheep Naval Station in 1963, to construct
a small reservoir of 2 million cubic meters capacity for domestic supply. The
proposed damsite has a drainage area of 10.9 square kilometers. None of any
kind of data is available except the rainfall data at Sattaheep, located about
8 kilometers west of the basin. Hence, the rainfall data at this station were
used in assessment of the design flood. Such adoption was based on the generally
practices that it was applicable where rainfall characteristics were similar.
Trials had been made by applying the empirical formula to determine the
maximum discharge. The rational formula, Q = C i A, was found to be less
applicable as its coefficient. C, could not be determined correctly. The McMath
formula, Q ACi(S/A)1/5, was then introduced because it seems to be
more applicable as the formula involves the basin slope which is one of the
major factors governing the peak rate.
The frequency of design flood cannot directly be determined. In this case
it was assumed to be similar to that of one-day rainfall. From 24-year period
of daily rainfall record at Sattahepp, the maximum one-day rainfall amount
of 302.7 mm was observed on 6 October 1957. The computed frequency of occurrence
of this one-day storm rainfall is once in 40 years. From the conventional
frequency analysis, the 50-year frequency one-day rainfall amount of 320 mm
was adopted in the assessment of the inflow design flood. Such frequency was
assumed to be that of the design flood.
Careful inspection of the catchment had been carried out in order to
examine the basin characteristics and to estimate the concentration time. It
is apparent that the time of concentration of such small basin is very short
and usually is much less than one-day.
The percentage of rainfall as a fraction
of one day was obtained from the graph of rainfall recorder. In this case
several storm events were examined and the envelope curve was used. The
rainfall amount falling within the time of concentration was calculated and
converted into rainfall intensity which is to be used in the McMath formula.
The basin coefficient, C, was estimated based on basin characteristics as

556
inspected. The basin slope, S, was determined from the available topographic
map of the basin. The design peak discharge obtained in this case study was
43 cubic meters per second.
Other means of assessments were also made for comparison. The unit hybograph
procedure was applied. The assumption of the base time of unit hydrograph
is important as it will result in varying peak rate. The storm runoff coefficient
was carefully assumed and flood volume was computed. Peak flow rate was,
therefore, obtained by applying triangular distribution hydrograph to the flood
volume. The second comparison was made with the specific yields of flood
flows obtained from the actual streamflow measurements observed in larger watersheds
by the Royal Irrigation Department (1965). The flood yield per unit area
computed from those stations were plotted against their respective drainage
areas. The possible maximum flood yield from smaller watersheds, in term of
cubic meter per second per square kilometeramay be read from the logarithmic
extrapolation of the envelope curve of specific yield. Such technique will be
one of the most reliable indirect approaches if the flood yields of small
streams are available with longer period of record. After several trials were
made, the design flood of 43 cubic meters per second were adopted in this study.
The assigned frequency was 50-year. The specific yield of flood flow was around
4 cubic meters per second per square kilometers, which is believed to be
adoptable in the area easily affected by tropical depression storms.
CONCLUSIONS
Several modifications of conventional approaches were used in planning and
design of water resources projects in small watersheds in Thailand. The results
obtained by such methods would be satisfied up to a certain degree. New concepts
and statistical techniques which give more reliability are needed to
design of small water resources projects in Thailand.
REFERENCES
1. Royal Irrigation Department (1963). Sattaheep Tank Project, Assessment of
Water for Storage, Hydrology No.137/63, Royal Irrigation Department, Bangkok,
Thailand.
2. Royal Irrigation Department (1965). Mean Annual Discharge vs. Drainage Area,
Envelope Curves of Maximum Recorded Peak Discharge, Specific Yield of Flood
Flow for Rivers in Thailand and Malaya, Hydrology No.186/65, Royal Irrigation
Department, Bangkok, Thailand.

ABSTRACT
DESIGN DISCHARGE DERIVED FROM DESIGN RAINFALL
Takeo KINOSITA
Takeshi HASHIMOTO
A design discharge for flood control in Japan is in general
derived from a design rainfall since discharge data are not suffL
cient for designing. The procedure of derivation and its merits
and demerits will be explained in this report according to fallo-
wing four steps. (i) A design rainfall in a certain return period
is determined by a probability process. (2) Design rainfall dis-
tribution are obtained by enlargement of rainfall distributions
of recent representative storms. (3) A simulation model for runoff
is decided by rainfalls and runoffs of recent representative
storms. (4) A design discharge is determined by the simulation mo
del with enlarged rainfall distributions.
RESUME
Au Japon, les données concernant les débits ne sont pas SUL
fisantes pour évaluer les crues de projet; on procède donc généra
lement par l'intermédiaire de l'averse de projet. Les auteurs ex-
posent le procédé utilisé, ses mérites et ses inconvénients; il
se décompose en quatre étapes. (1) On détermin: par analyse fré-
quentielle une averse de projet correspondant a une certaine pê-
riode de retour. (-2) Cette averse est distribuée dans le temps en
s'appuyant sur des hyétogrammes d'averses récentes considérées
comme représentatives. (3) On choisit un modèle de transfoTmation
pluies-débits élaboré à partir d'observations de pluies et de dé-
bits effectuées récemment au cours d'averses représentatives. (4)
On applique ce modele au hyétogramme de projet élaboré en (2).

558
I. Introduction
Japan is located in the temperate and humid zone. A river in this country
is comparatively small and its gradient is steep. Floods have occurred
very often since the prehistric age and been serious constraints against development
of the nation for a long time. Flood control is one of the major items
of water resource development works.
It is necessary to collect and analyze discharge data for design of flood
control projects. Authorized discharge gauging stations are 330 in 120 rivers
in Japan. There are many other non-authorized discharge gauging stations.
However, land developments and river improvement works have remarkably succeeded
and hydrological eituations of river badins are rapidly changing. This fact
induces that the discharge cannot be used for design purpose directly and used
only for verification of a runoff simulation model, and the rainfall which is
not affected by human activity is used for design purpose.
Mot only the peak discharge but also the flood hydrograph are necessary
for channel improvement, design of multipurpose reservoirs and soon. The procedure
to obtain the design hydrograph will be discussed in this report.
2. Probability Analysis
Since discharge is originally derived from rainfall, the design discharge
for water resource system is determined by the design rainfall through the runoff
simulati on model.
At the first step of this procedure, the total amount of the design rainfall
within a certain period should be computed by means of the probability
analysis. The important assumption of this section is that the time series of
rainfall are produced by some stationary stochastic process.
The procedure of this analysis is divided into two.
(9 Sampling from observed rainfall data.
cn) Frequency analysis.
The latter has been discussed by some hydrologists, 80 the authors intend
to focus their attention on the practical phase of the former. The series of
annual extreme values of rainfall within a certain dulation is selected from
the historical data. The duration in this paper is a period which is significant
to the design for the water resource system in the definite basin, and
cannot be so freely chosen. The rainfall within an adequate duration has the
closest relation to the magnitude of the flood discharge, and the rainfall with-
in a comparatively short or long duration has less relation to it.
the design duration must be appropriately Chosen according to the basin characteristics,
for instance the drainage area, the channel length, the slope and so
on.
The net work of daily rainfall observation covers all over Japan, and
daily rainfall data have been recorded for more than thirty years, at some stations
a hundred years. On the other hand, the network of hourly rainfall observation
is sparser than that of the daily rainfall. The hourly rainfall
data have been recorded for twenty years on an average.
Therefore
The credibility of
statistical estimations is dependent on the sample size, that is to say the
length of the series of observed data. Therefore the statistical analysis is
hardly applied to hourly rainfall data. Daily rainfall data are used for de-
termining the amount of design rainfall by means of the statistical analysis.

5 59
Then, the design duration must be an integer multiple of a day. As noted
above, the design duration muet be selected ae a time in which the rainfall
has a close relation to the peak discharge. Since the time scale corresponds
to the space scale in natural phenomena, the duration for a smaller basin must
be a day, and that for a bigger basin must be three days in this country.
A daily rainfall in Japan is defined as a rainfall observed from nine
a.m. to nine a.m. the next morning. If a storm stretches over this boundary
of observation, a daily rainfall cannot represent a storm rainfall. 'Two
days'' seems a minimum design duration even for a small basin. The fact that
a big storm in Japan tends to continue more than a day requires this limitation
of the minimum design duration. The design duration for the statistical analysis
is two days for a small basin and a medium basin, and three days for a big
basin.
The return period for design purpose is not determined by a mathematical
way, but by consideration of economical, political and social situations on the
basin. The sewerage system design claims for five to seven years as a return
period. For an urban basin, some period above twenty years is selected as a
return period. A big river basin in Japan requires almost a hundred years'
return period. For spillway design of a dam, about two hundred years' return
period is commonly used.
3. Enlargement of Observed Hyetographs
The amount of the probability rainfall for design purpose is determined
as shown in the above section. A careful attention should be paid to the
procedure for distributing the amount of the probability rainfall to the time
axis, because an hourly distribution of flood runoff, a hydrograph, is indispensable
for a flood control project, and a hydrograph is derived from an
houry distribution of rainfall, a hyetograph, through a runoff simulation
model. A hyetograph for design is deduced from that of a recent representative
storm. A part of the hyetograph observed during the representative
storm is selected aiming at the time of occurence of the maximum amount of
rainfall, where the time is taken equally to the design duration.
Suppose N is the number of hours within the duration, Rpis the amount
of probability rainfall and Roi ( for i=1,2, ... ,N is the observed rainfall
depth in the i-th hour. The enlargement factor R? is defined by the following
equation.
Then the enlargement factor is multiplied to each R
of design hyetograph.
to get the time series
EX'-%, EF'sR, a. ,EF.Rou
This procedure is called enlargement of observed hyetograph. Several hyetographs
are derived from several representative observed hyetographs in this
way.
If the amount of the representative rainfall is almost same as that of
the probability rainfall, This procedure is very successful. If not, the
enlarged hyetograph sometimes shows an unexpected pattern. In order to
avoid such an unexpected pattern, there must be some limitation for enlarge-

560
ment. Several proposals were given for this limitation, but there's no
praiseful one. For this limitation is to be deduced not theoretically but
merely empirically. One of the proposals is presented in the following
paragraph.
A certain domain is set up including the basin concerned, From all
the rainfall gauging stations in the domain, maximum point rainfall values
are selected about various periods shorter than the duration. The enlarged
hyetograph is compared with these values. If the enlarged amount during
some period exceeds the mimum point rainfall value during the same period,
the enlarged hyetograph must be abandoned because of the rareness of occurrence.
But this proposal raises another question. What region is appropriate
as the domain? For instance, if we replace Japan with the world, the
selected value of maximum poin rainfall becomes greater at any period. In
spite of this question, this proposal seems reasonable. Because there must
exist a realistic upper bound on the amount of rainfall that can occur on the
basin within a certain period. An example is adduced. On the upper Kiso
River basin from 1951 to 1971, there were 29 representative storm in which the
maximum rainfall amount durig48 hours was greater than 100 mm. The maximum
point rainfall values are made into Table 2. As a result of the comparison,
the exceedance is noted by symbol 'E* in Table I. If the exceedance has occurred
at some domein, it also occurs at any narrower domain. And yet, in this
case, the exceedance is apt to occur in a shorter period than a longer period.
This fact suggests us that the hyetograph of a heavy storm is uniformer in its
time distribution than that of a common storm.
4. Runoff Simulation Model and Effective Rainfall Analyeis
In this section, a runoff simulation model is determined, and simultaneously
an empirical rule is derived-on the separation of rainfall excess from observed
rainfall.
Among the great number of rainfall-runoff convertion schemes, 'Storage Func-
This method is expressed by the follow-
tion Method' is commonly used in Japan.
ing two equations.
where sr is the storage in the basin, qL is the outflow from the basin, r excemsive
rainfall, K and p are empirical constants dependent on the basin, and suffix
denotes delayed variable by a lag time Ta. These constants are neceseary
for the runoff simulation of the storage function, so they are previously
determined by sets of rainfall and runoff data of the recent representative
floods. As is seen in Fq. (2) this method contains a nonlinear process.
'Tank Model Method, aïso aseumes a nonlinear process, and is sometimes
used as a runoff simulation model. Unit hydrograph method has been improved
in this country, and is put to practical use today. But the use of unit hydrograph
method ia restricted to the basine where the assumption of linearity is to
a certain degree appreciable.

The separation of excessive rainfall from the observed one is an important
but an awfully suffering work. In Japan, the soil moisture of the basin general-
ly shows a violent variation from dry to wet according to the weather condition.
The soil moisture antecedent to the storm strongly governs the rising part of the
runoff hydrograph, sometimes even the crest. So the effective rainfall analysis
must be carried out carefully for the identification of model parameters. How-
ever, at the simulation for a design flood, a common value of the parameter repre-
senting the soil moisture in the basin is used.
5. Design Discharge
Finally, the design discharge is determined in this section. The enlarged
hyetographs of representative storms are used for the runoff simulation. A hydrograph
corresponding to each design hyetograph is computed by the runoff simulation
model.
Table 1: Comparison of Enlarged Hyetographs with Maximum Point Rainfall Values
tom
NO.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Ra
mm 1
-
137.1
137.6
125.1
117.4
1 IO. 1
114.9
147.7
125.2
106.3
195.9
117.2
107.6
143.6
118.7
173.6
227.5
119.1
148.0
114.3
112.3
144.8
123- 2
169.5
156.3
118.1
136.6
148.1
E3
-
2.19
?.i8
2.40
2.56
2.72
2.61
2.03
2.40
2.82
1.53
2.56
2.79
2.09
2-53
1.73
1.32
2.52
Upper
Liso River
1 3 6
E
E E E
E E E
E
Whole
Liso River
E
E E
E
E
Chubu
District
1 3 6
E E
Japan
1 3 6
E E
561
The
World
1 3 6
2.03
2.62 E
2.67
2.07
2.44
1.77 E
E
E
1.92 E E E E
E
E
2.54
2.20
2.03
263.9 1.14
- - 225.6 - 1.33
Symbols
EF : Enlagement Factor ( Rf/Ro
Rp : Amount of Two Days' Probability Rainfall at 200 Years, Return Period
RD : Maximum 48 Hours' Rainfall Amount
E : ERLarged Amount exceeds Maximum Point Rainfall Value in this Domain
1*3 and 6 are periods in houds). The estimated value of Rp is 300 mm.

I The
562
Table 2: Maximum Point ñainfall Values ( mm )
World 350 600 1488900
Although areas of these hydrographs are almost same, peak discharges are
different each other. The reasom are (i) variety of hyetograph, (n)nonlinearity
of runoff model, and (ai) loss factor Among these, (i) is the
most predominant. Owing to this fact and to M e it possible to obtain various
hydrographe, enlargement was applied to various representative storm hyetographs.
From the above hydrographs, one is selected as a design discharge. The
election of a design discharge itself brings up a brand new problem, but the
authors shall leave the discussion to another occasion.
6. Conclusion
The derivation of a design discharge is explained in thie re art. It is
composed of (1) design rainfall in a certain return period, (27 enlargement,
(3) a simulation model and (4) design discharge derivation. The procedure is
not fixed today. It will be improved everyday with development of hydrology.
Takeo KINOSITA, Takeshi HASHIMOTO : Public Works Research Institute,
Ministry of Construction,
Government of Japan.

AB ST RACT
THE USE OF CENSORED DATA IN ESTIMATING T-YEAR FLOODS
Morven N. Leese
Institute of Hydrology, Wallingford, Berks, U.K.
Types of incomplete data to be found in connection with
flood series are described, and it is shown how samples contai-
ning such data may be used to estimate the parameters of a dis-
tribution describing annual maximum flows. Formulae for the
standard errors of the resulting estimates are also given.
Examples are taken from a river for which censored data exist.
Preparatory data standardization is described, and the parame-
ters estimated using this data are compared uith these estima-
ted using the complete sample only. The marginal value of using
censored data in this context is assessed by means of the subse-
quent reduction in the standard errors of the estimates of
T-year floods for various values of T, and this is related to
the effort required to collect and standardize the data.
RESUME
L'auteur traite d'une catégorie de données incomplètes
qu'on peut rencontrer dans l'étude d'une série chronclqgique COZ
cernant les crues. I1 montre comment des échantillons contenant
de telles données peuvent être utilisés pour estimer les param;-
tres d'une loi de distribution des maximums annuels. I1 donne
également des formules pour calculer les erreurs types des esti-
mations qui en résultent. I1 prend comme exemple un fleuve pour
lequel existent de telles données tronquées c'est-à-dire défi-
nies par un suil auquel elles sont égales ou supérieures. I1 in-
dique comment on peut parvenir à une normalisation de ces bonnées
et compare les valeurs ainsi estimées pour les parametres a ce-
lles qu'on puet obtenir à partir du seul échantillon des donnees
régulières. L'appréciation du gain marginal d'information dÛ a la
prise en compte des données tronquées revient à évaluer la réduc-
tion de l'écart type qui en résulte pour les estimations des crues
de période de retour T, pour différentes valeurs de T. Ce gain
d'information est comparé 'a l'effort nécessaire pour collecter et
normaliser de telles données,

5 64
1;JTFODUCTION
The design of hydraulic structures for a water resources project depends
in part on estimates of the floods which the structures may be required
to withstand during the project's economic life. The feasibility of the
project may be examined by comparing the cost of building each structure
to the requirements of its design flood with its anticipated benefits.
The latter may be realized in terms of reduced damage to the structure
itself as well as to surrounding property, and in more effective flood
plain use.
Precision in the estimation of a design flood conveys a monetary benefit
by mitigating the costs which arise from over or underdesign.
Nevertheless, the use of additional data to increase precision will have
a marginal cost which may be greater than its marginal benefit.
circumstances it is necessary to quantify the increase in precision, and
if possible to express this increase in financial terms.
hydraulic structures thus involves both hydrologic and economic
considerations which require for their formulation: estimates of floods
with given return periods; values to be placed on the precision of the
estimates; cost and benefit curves for the structure.
In these
The design of
The standard form of data for the estimation of floods consists of a series
of annual maxima derived from a continuous flow record. It is proposed to
show how data which is not of the standard form may still be used for this
purpose, and that the use of additional data of non-standard form'hcreases
the precision of estimation.
It is not proposed to discuss in detail
the economic implications of the increase, but the order
of magnitude of
the resulting cost-reduction is indicated by means of a simple example.

ESTIMATION OF T-YEAR FLOOIX - STANDARD DATA
565
Jn order to estimate the flood with return period T, sey, (or 'T-year flood'),
it is first necessary to choose a probability distribution representative
of the annual maxima. The parameters of the distribution may then be
estimated from past records by one of many estimation procedures. The
Gumbel , or double-exponential, distribution is often used to represent
annual maxima because of a supposed validity on theoretical grounds and
although these grounds have been questioned (2), its extensive use justifies
- (=)
its further study in this context. It has the following form:-
F(r) = exp [-e -J , - o

566
-1/& E N
i = N
+ L
i = 1
P o
where y the so-called ‘reduced variate’ a is given by:-
i’
The flood of return period T, %, is then given by:-
so that
Pr (x 5 %) = 1 - 1/T,
- %-u = 1
-e-
- 1/T.
e a
m y T = %-u.
-9
a
an estimate oí’ 3 is then
A
A 3 = Q+ a YTa
A A
where

the values of whoee elements are substituted into the following:-
h
V mw be approximated by replacing a by a. A pull derivation of the
above quantities may- be fomd in Gumbel ( 1) and Kimball (4).
NON-STANDARD DATA
Those concerned with maximum flood estimation will be familiar with at
567
least two types of non-standard data found in connection with flood series:
missing peaks in continuous chart records and historic flood marks.
Hissing peaks occur when the flow is so high that the recording pen runs
off the eàge of the chart; whilst it should be possible to estimate a
missing peak discharge from h knowledge of the length of time the chart
limit is exceeded, the properties of such a method require further
investigation.
The approach used here is to assume no more than that a
flood which has exceeded a chart limit has a peak discharge greater than
the flow corresponding to the flov at the chart limit.
Historic flood marks are usually to be found in waìls,bridges or on
specially coktructed flood stones.
which have risen above a fixed point
certain circumstances, it mey be assumed that all such floods have been
marked, and that floods in the intervening years for which no marks exist
have failed to reach the fixed point.
These are the two types of data to be considered.
They indicate the levels of floods
during some historic period. In
values are only specified if they lie on one side of a given threshold.
Samples which exhibit this property are known as censored samples, the
threshold being called the censoring point.
(6)
They have this in common;
If the threshold is fixed,

568
as it is in these caces, and the proportion of censored events is a random
variable, the censoring is type I; if the threshold is a random variable,
but the proportion of censored events is fixed, the censoring is type II.
A fui1 discussion of censoring is given by Kendall and Stuart(5),
further references are given.
The incorporation of a. 'missing peak' or 'historic record' into a
where
standard sample is clearly a type I censoring problem. The general form of
the likelihood function Lc for a sample of n + k values of which n are below
the censoring point x and are specified 8s x 1, x2 ... x and k ari? above
C n'
x and arë unkna~n, b BS follows:-
C
where f (x) is the appropriate probability distribution function. A similar
expressiori rr,ay be obtained for censoring above a censoring point, and maximum
likelihood equations m4y be obtained from either expression in the USUEL manner.
ESTIMATION OF T-YEAR FLOODS - NON-STANDARD DATA
Censoring above a threshold (Missing Peaks).
(assumed
n
independent) and k missing peaks which are knm to be above the chart
Suppose a sample consists of n annual maxima xl, 5, . . . x
limit xc.
-
then :
The likelihood hction L and maximum likelihood equations are
C

-e JC
where yi = Xi - Y i Y, = X - u ; W = e .
o a
The variance-covariance matrix Y of ac U the pumeters estimated
Ca
from equations (9) is then given by the inverse of Rc, whose elements ri
are as follows:-
where c = eTC, and J and K are integrals which require to be evaluated
numerically. They are given by:
J 3 5

5 70
L
i=N+r i=N+r
- l/a (N+r) - E y. + E YieYi
1
i= 1 i= 1
+ leT%d = O;
i=N+r
- i/a r-(N+r) + C eyi +
-
-
i= 1
where yi = x.-u ; yh %-u ; u= e - 1 -
a a
(13)
A
The variance-covariance matrix $ for a and B, the parameter estimates
estimated from equations (13) is then given by the inverse 'of #,
whose eisments r5 are 88 follows:-
- rI2h = - E[a2LogI l/a2 {(N+M (0.4228) + M u(yh - 1
aaau i
where J and K, are expressicm of the form of ( 1 1) with h 5 e*
lower limit of integration.
-q 1,
as the
Equations (9) and (13) are thus the modified equations to be used for the
estimation of parameters f im the na-standard floods data described.
have been derived in the context of reliability theory, and are given in
(6). Similar expressions may be obtained for distributions other than the
Gumbel distributirm by substituting the appropriate p.d.f in (7) or its
equivalent fon censoring belar a threshold.
They

An iterative technique for solving the equations (3) for a standard sample
may be found in Jenkinson (71, who gives a worked example. This technique
my be used with slight adaptation for the solution of equations (9) and
(13); satisfactory results are obtained if the iteration matrix is left
unchanged, i.e. given values appropriate to an uncensored sample. However,
care should be taken in the choice of initial values if the proportion
of censored values is high.
THE AVON AT BATH - AN APPLICATION OF THE EQUATIONS
The most satisfactory applications of these modified equations has been in
the extension of records to achieve significantly greater precision in the
resulting estimates.
Avon at Bath, where a set of historic flood marks had been recorded during
a historic period prior to a fairly long continuous chart record.
parametex estimated from the recent record alone were compared with these
estimated frcm a combined sample consisting of the historic floods and the
recent records. The data used is shown in t'able 1, 1940 being the date of
the beginning of the recent record.
One such application was for data from the river
It is necessary to perform a number of checks on historic data before
entering it into equations ( 13). for instance, the stage-discharge
The
571
relaticnship derived for the recent record may require adjustment before it
is applied to the historic flood marks, whose site may be at some distance
from the modern gauging station. The assumption that floods are marked if
(and mly if) they have risen above the threshold makes it necessary to
investigate the circumstances surrounding each flood mark.
time-consuming exercise, but it is one which could to a large extent be
carried out by local library or museum staff who have to hand contemporary
evidence such as old newspaper reports.
This mey be 8

572
TABLE 1. Annual Maximum Flooäa Used in the Estimation of
a and u in Gumbel's Extreme Value Distribution
(In Cumecs).
* Water
Year
1865
i866
1874
1875
1879
1882
1888
Flood
206
228
12 1
218
264
362
204
375
154
239
302
i86
255
148
Water
Year
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
195 1
1952
1953
1954
Flood
84
149
' 73
118
128
282
98
i04
1 q3
229
136
116
96
296
Water
Year
1955
1956
1957
1958
1959
1960
196 i
1962
1963
1964
1965
1966
1967
1968
- !he following values were obtained for the data of table 1:-
N 0 32; M m 58; 1 = 48; r= 10; 5 200,
and when these values, and the data of table 1, were substituted into
equations (13), the estimates sham in table 2 were obtained.
Flood
128
107
138
169
169
352
12 1
103
277
110
178
172
31 1
A
Floods with various return periods were then estimated from the values of a,
and ;h obtained from the enlarged sample, by substitutitm in equation (41, and
their large-samgle stenâard errore were elso calculated from equation (6).
These are shown in teble 3, where values obtained from the original srmgle am ale0 shown.
125

TAñLE 2. Estimates of Paretem of Gumbel's Extreme Value
(1) Estimated Flood:
Large-s ample
standard error:
(2) Estimated ' Flood:
Large-sample
standard error:
573
Distribution Using (1) Historic Flood Marks and Recent Data
and (2) Recent Data Alone. (In Cumecs).
Parameter
(1) Estimate:
Large -sample
standard error:
(2) Estimate:
Large-sample
standard error:
a U Sample size
47 128 29 recent values
+ 13 historic values
2.5 27 + 48 censored values
48 128 29 recent values
27 29
TABLE 3. Estimates of Floods with Various Return Periods Using (1)
Historic Flood Marks and Recent Data and (2) Recent
Data Alone. (In Cumecs).
1 Reluni Period I 2.33 (Mean) 10 25 50 100 1000 I
THE AVON AT BATH - THE VALUE OF ADDITIONAL DATA
155 234 278 311 344 , 453
t7 - +12 +i6 219 223 i33
156 236 281 314 348 458
- +Il - +20 - +26 231 - t36 +51
It will be seen from table 3 that the sampling error in the 50year flood
estimate was reduced from 10% to 6% by the use of historic flood marks. Was
this reduction worthwhile in view of the effort required to standardize the
data?
type.
A number of approaches mw be taken ì.n answering questions of this
Ultimately they involve the formulation of expressions for the benefits
and c,osrs which arise from acquiring the data, and since these c m never be
fully known, the problem of evaluating the worth of stream flow data are far
from etrai@t-ionrard.
I

574
Reasonable attempts have been made, however, by E~RS of simplifying assumptions.
une such attempt has been made by Wilson (81, whose method allows the
estimation of the reduction in cost of a small structure consequent upon an
increase in the precision of its desis flood estimate. His approach is now
applied to data from the A vm at Bath.
It is assumed that the total cost C mey be written as C = $+Z2 where C1 is
associated with construction costs and has the form:-
failure and has the form:-
S
c2 = K2 p,
c1 = XTm> (15)
and C2 is associated with the probable future damage resulting from structural
where i is the optimum design flood with return period T, K1 and K2 are
constants, and m and s are indices dependent an the particular structure.
For smll structures m and 8 may be hssumed eausl.
i 16)
Wilson's formda depends partly on the fact that floods with r etm periods
between 5 and 50 )ears may be represented by a power law of the following form:-
5 = A$
( 17)
where A is a constant. p is an index which Wilscm suggests may be estimated
>
as the ratio of the 50- to the S-year flood.
following formula gives the reduction in cost (Ec) of a structure, given the
precision of the estimate of the design flood (Ex):-
Ec = E E'
2 x'
Writing n = lb - 8,
_-
the
It should be noted that this formula applies only to small structures, with
design floods of moderate return periods (ie. between 5 and 50 years).
For the river Avon at Bath, p was found to be 0.2; taking FS= 0.75 88 a
typical value, an increase in precision from 10% to 6% may be seen to lead
to a decrease of 1% in the cost of a structure with a 50 years design flood.
While not a high percentage, this would represent in absolute terms a sum of
money considerably in excess of the cost of obtaining and standardizing

the data.
More importantly, a similar cost-reduction, by this analysis,
575
wouïd require a further 20 years of streamflow data from continuous records,
which might be impractical and would certainly be expensive.
Since the incorporation of the other type of nan-standard data considered,
Le, chart censoring resulting in missing peaks, entails no extra cost, and
bearing in mind that flood estimates are likely to be of interest in a
variety of contexts (not only one as in the above example), it may be
concluded that it is on the whole worthwhile to use additional data of the
types described.
Acknarle dgement
The author wishes to thank the following for their assistance:
Robin T. Clarke, who initially suggested this study and provided
valuable guidance during its progress; Con Cunnane, who made available
compu+.er programs,'adaptstions of which were used in this work> and
Dr Malcoiz D.Newson who collated and helped to standardize the historic data
used in the eyample. This paper is presented by permission of the Director,
Institute of Hydrology, Wallingford, Berkshire, U.K.
1. Gumbel, E.J. (1960) Statistics of Extremes Columbia University Press
Iiew York (1959).
2. Moran, P.A.P. (1959) !he Theory of Storage. Methuen and Co. London (1970).
3. Lowery, M.D. and Nash, J.E. (1970) A comparison of methods of fitting the
double exponential distribution. Journal of Hydrology IO, 259-275.
h. Kimball, B.F. (1949) An approximation to the sampling variance of an
estimated maximum value of given frequency based on fit of doubly exponential
distribution of m&mum values. Ann. Math. Stat., 110-1 13.
5. Kendall, M.G. and Stuart N. (1961) The Advanced Theory of'statistics
Vol.11. Charles Griffin and Co., Ltd, London.
6. Harper, H.L. and Moore, A.H. (1968) Maximum-likelihood estimation, from
doubly censored samples, of the parameters of the first asymptotic distribution
of extreme values. her. Stat. Assoc. Jour. 63. 889-901.
7. Jenkinson, A.F. ( 1969) Estimation. of Maximum Floods. Chapter five of
W Technical Report 98, 193-227.
8. Wilson, K. C. ( 1972) Benefit-accuracy relationship for small structure
design floods. Weter Resources Research 8(2), 508-512.

ABSTRACT
ASSESSMENT OF DESIGN FLOODS IN BRAZIL
Paulo Poggi Pereira
The techniques utilized by the Departamento Nacional de Obras de
Saneamiento for computing the caracteristics of flood to be used for
designing works against inundations are described. Very seldom trus-
tworthy river flood discharge measurements are obtained. In most ca-
ses design flood discharges are estimated with a basis on topographic
data which can be gathered quickly. Until thirty years ago the contri
buting basin area was multiplied by a standard unit discharge in or-
der to get the design flood discharge. Later on, the rational method
was addopted, mainly for designing small canals. This system was con-
siderably improved by the execution of .a statistical study of heavy
rains observed in the Country. The choice of the heigth of some dikes
was based on the high water levels attained during ancient floods ob-
served and still remembered by local people. It has been found neces-
sary to perform more elaborate and time-consuming hydrological obser-
vations and studies for designing dams. The use of mathematical models
is still Incipient but promising. Design floods of different standard
periods of iecurrence are addopted according to the type of the work,
the size of the river and the utilization given to the area to be pro
tected.
RES UME N
Son descritas las técnicas empleadas por el Departamento Nacional
de Obras de Saneamiento en la determinación de las caracteristicas de
las crecidas a ser consideradas en el proyecto de obras contra inunda
ciones. Raramente se consiguen datos de mediciones fidedigna de las
descargas de crecidas de los cursos de agua. En la mayoria de los ca-
sos estimanse descargas de crecidas para el proyecto, con base en da-
tos topográficos que pueden ser obtenidos rápidamente. Hasta treinta
años atrás, el método utilizado consistia en multiplicar el área de
la cuenca hidrográfica contribuyente por una descarga especifica pa-
dronizada para obtener la descarga de crecida para el proyecto. De
ahí en adelante, fue adoptado el método racional, principalmente para
proyectar pequefios canales. Este sistema fue considerablemente mejora
do por la ejecución de un estudio estadístico de las lluvias intensas
observadas en el país. La altura de algunos diques fue escogida en ba
se de los niveles de agua alcanzados por antiguas crecidas, cuyos ves
tigios perduran todavia y son indicados por los moradores del lugar.
Para el proyecto de represas ha sido necesario realizar observaciones
y estudios hidrologicos más precisos y demorados. El uso de modelos
matemáticos es aún incipiente, no obstante, promisor. También, adop-
tanse crecidas de proyecto con diferentes periodos de recurrencia coz
forme el tipo de la obra, el caudal del curso de agua y los intereses
en juego de las comunidades vecinas.

578
-1 . INTRODUCTION
The Departamento Nacional de Obras de Saneamento - D.N.0.S.of
the Brazilian Ministry of Interior, has been building flood
control works for almost40 years.
Such works include channel improvements, dredging and lining
of canals, building of levees, dams, conduits and tunnels.
The first basic step in the design of these works is the de-
termination of the features of the floods to be controled or taken
into account. The main methods that have been used for this purpo-
se are presented in the following subtitles. It should be noted
however that not every method reported is still in use.
There is a generalized lack of good reliable hydrometric
observations and measurements. As a consequence, indirect hydrologic
methods have been used as a rule with very few exceptions.
2. RATIONAL METHOD
The rational method is the most widely adopted for designing
canals and condui te.
it gives the descharge - Q - through the equation Q = CIA ,
the elemerits of which are determined as follows:
The area of the drainage basin - A - is obtained from maps
or aerial photographs. When none is available, field surveys are
made.
The runoff coefficient - c - depends primarily on land use.
As an e)rample the following table was copied from (i), a recent
D.N.0.S.- O.A.S. publication:
Downtown areas, densely built, with paved streets and sidewalks
C = 0.70 to 0.90
Neighborhood areas, less densely built, with paved streets
and sidewalks ,C 0.70
Residential areas densely built, with paved streets C -
0.65
Residential areas averagely inhabited C = 0.55 to 0.65
Suburban residential areas, sparsely built C = 0.35 to 0.55
Residential areas with gardens and unpaved streets C = 0.30
Vegetated areas, parks with gardens, unpaved sport fields
c = 0.20
The value of the runoff coefficient for the drainage basin
is obtained by adding the products of the fractions of total drainage
area occupied by each land use, multiplied by the corresponding
coefficient.
The determination of the rain intensity - I - is made through
the following steps:

57 9
a) A recurrence interval is chosen, usually obeyine, the
following criteria (1 and 2):
Rural area Urban area
Small canal (no levees) 5 years 10 years
Large canal (no levees) 10 years 25 years
Small canal with levees 25 years 50 years
Large canal with levees 50 years 100 years
Small conduits for urban drainage 3 or more years
b) The time of concentration is computed by adding the time
n eeded by the rainwater fallen on the remotest part of the watershed
e o reach the canal or conduit, to the travel time necessary for the
water to flow to the point under study. The travel time is computed
by dividínp the length of the canal or conduit by the averape
flow velocity.
c) A total depth of rainfall is determined taking into
account the chosen recurrence interval and a duration of rain equal
to the time of concentration. (3) is resorted to for this purpose.
The ratio rainfall depthtrain duration gives rainfall intensity I.
3. INTENSE RAINS IN BRAZIL
In 1957 D.N.O.S. edited Otto Pfafstetter's "Intense Rains in
Brazil" prepared mainly for applications of the rational method (3).
This book presents the results of frequency analysis of rain
fall vhlues recorded in 98 stations of the Brazilian Departamento
Nacional le Meteorologia.
Rainfall corresponding to several duration periods of rain
(5, 15 and 30 minutes, 1, 2, 4, 8, 14, 24 and 48 hours, 1, 2, 3, 4
and 6 observation days) were analysed separately for each station.
Recurrence intervals of the precipitations - T - were carac-
terized by the equation T = n/m, being n the total period of obser-
vation and m the number of order occupied by the rainfall in a
series where all observed intense precipitations were placed in de-
creasing order of magnitude.
This book presents diagrams, tables and formulas that allow
the determination of design rainfall for the 98 studied stations up
to 1000 years of recurrence intervals. Values pertaining to the
station nearest to the place for where the design is being prepared
are usually utilized. For checking representativeness, rainfall
frequency curves of dayly precipitations of this station are some-
times compared with similar curves prepared with data from a non re
cording raingage instaled at the actual place of the contemplated
works.
4. STANDARD UNIT DISCHARGES
According to (4) the rational method was addopted when D.N.QS.
began its activities many years ago reclaiming swamps in the neiFh-

5 80
bn,irhood of Rio de Janeiro.
The reasons for this choice were the absence of discharge
~~~.,?i~~ements, the frequent inexistence of defined streams in the
swam-is and because it was feared that the drainage canals to becons
tructcd would change so much the hydraulic caracteristics of the
watersheds that the measurements would not provide a reliable basis
f o r designs .
On the other hand, there were no recording rain gage charts
fiom ~t.ich rainfall intensities for different rainfall durations
and recurrence intervals could be deducted.
There were only rainfall measurements performed with non re-
rording rain gages for a relatively short period which showed a ma-
ximum precipitation of 120mm for l day (24 hours).
To get rainfall intensity, this observed depth of precipitation
wds supposed to be uniformly distributed through the 24 hours
of observation.
So, the rain intensity addopted was always the same, regard-
less of the time of concentration of the various basins. As a con-
sequence, the discharge became directly proportional to the drain-
age basin area.
The runoff coeficient addopted €or rural basins was 0.7,
obviously for compensating the weak rainfall intensity used. For
these
3
values, the rational met9od equation gives a discharge of
100 m /s fLt a basin of 100 km .
As a matter of fact this application of the rational method
was only a means of justifying the standard unit discharge of 1 m3/
1s km2 that was an addopted rule of thumb. The behaviour of the u~
lined rural canals designed accordingly has been good. Some ocasion
al flooding has occured but without excessive resulting damage.
Another standard unit discharge is 0.5 m Is km2. It reeult-
ed from a design especification asking for pumping rainwater out of
polders within a few days for avoiding the breeding of mosquitoes.
Here again rain intensity was not related to the concentration time
of the drainage basin.
5. DESIGN FLOODS FOR DAM SPILLWAYS - TRIANGULAR UNITGRAPH
The design OP dam spillways is usually based on flood hydro-
graphs. The triangular unitgraph presented in (5) has been used
very often because It presents the advantage of doin without hydro-
metric data.
It is believed that peak discharges obtained by this method
are exagerated but flood volumes are correct. Therefore this method
is considered good for cases where the dam reservoir retains much
of the flood volumes.
3

The following example is based on recent design computations
of a dam spillway for Northeastern Brazil.
a) The time of concentration was estimated by the equation
of the “California Highways and Public Works” adapted for metric
units :
3
5 0.95 x (L /
TC
Tc = time of concentration in hours
I, = length of watercourse in km measured from divide to
spillway site.
581
H = difference in elevation in meters between spillway site
and divide.
In our example L = 17 km, H = 400 m and
Tc = 0.95 (lì3 / 4 0 0 ) ~ ’ = ~ 2.5 ~ ~ hours
b) The time in hours from start to peak rate of unitgraph
(T ) was computed
-
as follows for excess rains of 1 and 6 hours
periods (D)
-
T D/2 + 0.6 Tc
P
For D = 1 hour, T 112 + 0.6 (2.5) = - 2 hours
P
For D = 6 hour6, T = 612 + 0.6(2.5) 4.5 hours
P
c) The time in hours from peak rate to end of unitgraph
triangle (T,) was computed as follows:
Tr =
-
1.667 T
P
- For D = 1 hour, Tr 1.667(2) = 3.3 hours
For D = 6 hours, Tr 1.667(4.5) = 7.5 hours
d) Peak rates of unitgraphs for 1 mm exceas rainfall of 1
and 6 hours duration periods were computed as follows:
A I
‘p 1.8(T + Tr)
P
= peak rate in m 3 /e
4P
- A = drainage area in km2. In the example A = 97 km2
97
For D =
-
1 hour, 9
10.2 m3~s
‘p 1.8(2 + 3.3)
For D = 6 hours, œ
97
4.5 m3/s
qp 1.8(4.5 + 7.5)
e) The excess rainfalls and corresponding runoff hydrographs
-unitgraphs - are represented schematicaly in the annex figurestogether
with lists of unitgraph discharges corresponding to the
middle of consecutive one hour time intervals.

6. PROBABLE MAXIMUM PRECIPITATIONS
The spillway of the example would be located uptream of a
large town and the failure of its dam by flood overtoping would
cause great property damage and seriously jeopardize human life in
large numbers. Therefore it vas considered appropriate to utilize
the maximum probable precipitation for computing the design flood.
There were not enough storm data for estimating directly
the values of such precipitation. The indirect approximated method
proposed in (6) was used. It is based on suppos ing that maximum
probable precipitation values are identical to those of a region of
the United States where rainfalls of 10 years recurrence interval
are the same as those observed in the watershed under study.
(3) was used for obtaining 10 years recurrence interval prg
cipitôtions from a station nearby the spillway site and (7) permil
ed to locate the area in the United States with equivalent rain-
falls and also furnished the probable maximum 6-hour precipitation
for a 10-square-mile area: 686 mm.
By using charts from (5) values of probable maximum preci-
pitations were computed for the drainage basin under study, which
has an area of 97 km2 = 37.5 square-mile, for the following listed
periods of duration.
duraticm period
houss
6
12
1
2
3
4
5
computation
88% x 686
107% x 686
50% x 604
65% x 604
76% x 604
85% x 604
93% x 604
rainfall
rnm
604
7 34
302
39 2
460
513
562
Rainfall increments disposed in descending order of intensity
were calculated as follows:
interval duration
hours
rainfall increments
mm
1 P1 = 302
392 - 302 = 90
P2
460 - 392 68
p3
513 - 460 = 53
P4
P5 = 562 - 513 = 49
P6 604 - 562 42
P12= 734 - 604 1130

583
For obtaining the design precipitation, rainfall increments
were tabulated in the following order as suggested in (5): P6, Pq;
P3, P1, P2, l 5 nad P (see annex table).
12
7. RUNOFF FCTIMATION AND COMPUTATION OF THE DESIGN FLOOD HYDRC
GRAPH
The computation of the design flood hydrograph of the exam-
ple is presented in the annex table and was made through the follo_w
ing steps:
a) Rainfall increments obtained as of the preceding cub-
title were added in order to obtain accumulative precipitation.
b) Accumulative runoff or excess rainfall was estimated by
means of the equation of the "Soil Conservat ion Servi ce" presented
in (5):
(P
2
- R =
0.2 S)
P + 0.8 s
R = runoff in mm
P = accumulative precipitation in mm
S = maximum potential difference P - R at time of rain's
begining.
S was estimated as 100 mm.
c) Increments of runoff were computed by subtracting the
accunulative runoff obtained for the preceding interval from the
accumulative runoff obtained for the interval under consideration.
d) Increments of runoff were compared with rainfal.1 incre-
ments. The difference between them should attain at least lmm for
each interval hour. As this did not happen at the last tabulated
time interval the increment of runoff for that interval was recal-
culated by subtracting 6 mm from Phe rainfall increment.
e) Increments of runoff for each interval were multiplied
by the unitgraph discharges listed in the annex figuresand the pro
ducts were tabulated in the corresponding time intervals.
f) The average discharge of the design flood in each time
interval was obtained by adding the products resulting from the
previous step for that tf.me interval.
minal.
g) The base flow was not taken into account for being no-
a. STATISTICAL METHODS
Frequency analysis is applied whenever records that allow
its use are available, for reasons of better precision and reliability.
Gumbel's and/or Hazen's methods are the most favored.
D.N.O.S. files keep reports of classical hydrological studies
mainly based on frequency analysis of water level observations
and discharge measurements.

5 84
One of them is an outstandingly interesting example: the de
termination of the heigth of levees for protection of the city of
Porto Alegre against floodings of the Guaiba River.
In that reach the Guaiba River forms an estuary and its
water levels are dependent not only on the river discharges as well
as on the water level ocurring in the lagoon where it flows to ,
which can be strongly influenced by winds.
There were little knowledge of the elements involved and
their effect.
On the other hand the Guaiba River water levels had been sys
tematicaly observed since 1899 by means of a staff gage installed
near downtown Porto Alegre. The data so obtained was frequency an=
lysed and, according to Gumbel's method the biggest recorded flood,
which occured in 1941, was found to have a recurrence interval of
about 370 years.
Local people remembered which places had been flooded and
which levels had been attained by the water in different places of
the town during the 1941 flood. With these informations it was
possible to draw a water-surface profile, which was confirmed later
by a hydraulic model of the estuary.
It was decided to set the crest of the levees 1.20 m above
that water-surface profile. No discharge considerations were taken
into account although discharges were estimated by making use of
the above mentioned model.
9. MATHEMnTICAL MODELS
Up to present time almost no use has been made of mathematic
al hydrological models for determination of design flood caracteristics.
Recently, the Streamflow Synthesis and Reservoir Regulation
(SSARR) Model began being used for forecasting the behaviour (flood
and low water levels as well) of the Paraguay River and some tributaries.
This model was develloped by the U.S. Army Corps of Engineers
which addapted it for the Paraguay River basin as part of the
activities of the "Project of the Hydrological Studies of the Upper
Paraguay River Basin" - a UNDP/UNESCO technically assisted project
for which D.N.O.S. is the responsible Brazilian counterpart agency.
The potentiality of SSARR model for evaluating the caracte-
ristics of design floods of large rivers is obvious and it is ex-
pected be much used for this purpose in the future.
10. CONCLUSION
D.N.O.S. has always used addapted foreign feekiikques for
assessing design floods. On the other hand, local data has been
used as extensively as possible. Methods that did not allow easy

585
'use of this data have not enjoyed preference. Such is the case of
empirical formulas for rainfall intensity and flood discharge which
were used only in a few instances.
Elaborate methods have not been much addopted. The main
reason for this may be the rather vague effect of high accuracy
assessment of design flood caracteristics upon the economics of
flood control works in most cases, a fact that does not encourage
too many efforts for refining design flood assessment.
REFERENCES
a
1. D.N.O.S. e Organizaqao dos Estados Americanos (1972). Relató-
rio do Estudo para Controle da Erosao no Noroeste do Estado do
Parana, Rio de Janeiro, DNOS.
2. Poggi Pereira, P. (1967). Controle de cheias: custos e benefi
cios, SANEAMENTO, Rio de Janeiro, DNOS.
3. Pfafstetter, O. (1957). Chuvas Intensas no Brasil, Rio de Ja-
neiro, DNOS.
4. Arauja Goes, H. (1942). A Baixada de Sepetiba, Rio de Janeiro,
DNOS.
5; U.S. Department of the Interior, Bureau of Reclamation (1960).
Design of Small Dama, Washington, U.S. Government Printing
Office.
6. Pfafstetter, O. (1967). Floods for Spillway Design, Neuvieme
Congres des Grands Barrages, Comission Internationale des
Grands Barrages.
7. U.S. Weather Bureau (1963). Rainfall Frequency Atlas of the
United States for Durations from 30 Minutes to 24 Hours and
Return Periods from 1 to 100 Years, Technical Paper NQ 40 ,
Washington, Weather Bureau, U.S. Department of Commerce.

586
FIGURES
c.ü 5 1 hour
L-
or runoff
-- r
L-

5 87
M
--l
I
N
N
I
d
d
-
-
-
1
O
-
Io
ci
O
c
G
a
rl

ABSTRACT
A METHOD FOR THE PREDICTION OF
WASHLOAD IN CERTAIN SMALL WATERSHEDS
by
Oswald Rendon-Herrero
Present knowledge on the prediction of washload reveals that with
the exception of the universal soil-loss equation, and sediment-rating
techniques, a rational method does not exist that can accomplish this
task. A method is presented thar is analogous to Sherman's unit-hydro-
graph method of hydrograph analysis. The ordinates of a sediment dis-
charge graph are divided by the excess runoff that mobilized it, prod:
cing a unit sediment discharge graph. When this is done for many storm
events, unit sediment discharge graphs are generated that vary conside-
rably in peak value and shape. The ordinates of the latter graphs are
then plotted logarithmically against their respective excess runoff,
yielding data points that can be fitted by straight lines. Predictions
pf sediment discharge or the generation of a sediment discharge graph
for a given excess runoff can be accomplished using the resulting
graph, Bixler Run Watershed, Pennsylvania, having a drainage area of
15 square miles, was selected as a data source. Granulometric tests
and otti~r related information disclosed that the suspended sediment in
Bixler Ruri is predominantly washload. Prediction of washload utilizing
the propose? method yielded errors that were considerably less than
that reported using available sediment transport formulae and techni-
ques.
RE C U ME N
Actualmente los conocimientos con respecto a la predicción de
"washload" son bastante limitados. Con las excepciones de la ecuación
universal de pérdida de suelo y técnicas sedimentarias (sediment-ra-
ting) todavía no existe un método racional para resolver esta tarea.
El procedimiento presentado es análogo al método de Sherman (Unit hy-
drograph) o sea un análisis hidrográfico. Las ordenadas de la gráfica
de descarga sedimentaria divididas entre el volumen del derrame excesi
vo producen una gráfica unitaria de descarga sedimentaria. Al comple-
tarse este procedimiento para muchas lluvias, gráficas unitarias de
descargas sedimentarias son obtenidas y se notarán las diferencias de
los cambios de valores máximos. Las ordenadas de estas Últimas gráfi-
cas son trazadas logaritmicamente versus sus respectivos volúmenes de
derrame excesivo rindiendo diferentes puntos de dato, los cuales pue-
den unirse con líneas rectas. Predicciones de descargas sedimentarias
dado cierto derrame excesivo pueden observarse en la gráfica obtenida.
El área seleccionada de 15 millas cuadradas, donde los datos fueron ad
quiridos queda situada en Bixler Run, Penns.ylyania. Pruebas granulomé-
tricas y otras informaciones relacionadas indican que la aescarga de
sedimentos en Bixler Run es casi todo "washload". El método presentado
rindió errores de magnitud mínima en comparación con los errores repor
tados por otras técnicas y fórmulas de transporte sedimentarias.

590
INTRODUCTION
Relationships have been developed whereby the sediment transport of materials
which are native to a channel can be computed with varying degrees of
accuracy. When the sediment transport is primarily composed of the lateral inflow
of particulate matter eroded from the land surface (washload) in a basin,
the relationships derived are no longer velid(lg2). Heretofore, the leteral
inflow component of sediment discharge was predicted via the universal soil
loss equation(*), and sediment rating techniques.
these methods are subject to large errors. The universal soil loss equation
has the disadvantage of providing only annual predictions, The need for quantitative
evaluation of washload is of paramount importance at the present time.
A method is presented which is applicable to certain small watersheds and
which can enable the prediction of sediment discharge on a storm basis. By
"small" is meant those watersheds where the spati ribution of the rainfall
is uniform over the watershed area. Some authors
Predicted quantities using
$3 ,w
define a small water-
shed as being less than 161.0 or as much as 3219.0 square kilometers in area.
"Certain" refers to the sediment discharge graph's locus (sedimentgraph) dependency
on the soil type. For general stream conditions, fine-grained and
colloidal materials transported in suspension will yield. a sedimentgraph that
appreciably parallels the shape of its associated hydrograph; under similar
stream conditions, coarser particles in transport will not result in parallelshaped
discharge graphs. The applicability of the series graph method depends
on the para!lel nature of the sedimentgraph and hydrograph for a given excess
runoff. Use ~f the adjective "series" is explained in the Analysis of Da
section of this paper. The series graph method is analogous to Sherman's F%>
unit hydrograph prc,zedure for the analysis of a direct discharge hydrograph.
The series graph method is demonstrated using Bixler Run Watershed, a
monitored drainage basin 38.9 square kilometers in area near Loysville, Penn-
sylvania (Figure 1). Granulometric measurements made of the bed, bank, and
suspended sediment, has established the sediment transport in Bixler Run as
being predominantly washload. Sediment sampling in the Bixier Run Watershed was
begun on February 1, 1954, using a U.S.D-43, and a DH-48 depth integrating
hand sampler(7).
The series graph method is used where the quantitative analysis of wash-
load is necessary for the prediction of sediment discharge and/or variation
with time. The prediction of total sediment discharge is required for example
where the rate of sedimentation can become problematic. This consideration is
particularly important in the allocation of storage volumes in new reservoirs.
WASHLOAD
Due to a series of rainfall-induced erosive processes, particulate matter
eventually reaches a stream course after being transported through a great
variety of distances in a drainage basin. Depending on such characteristics
as, for example, land slope and length, topography, and availability of trans-
portable surficial soils, various-sized particles can, given ample time, reach
the main waterways in a basin.
Depending upon the streamflow character, some of the eroded materials that
reach the stream course as lateral inflow combine with sediments native to the
channel proper and continue to be transported downstream by the prevailing flow.
The lateral inflow of Sediment is known as washload. Sediment transport in the

s t-ct'arn may be accomplished by four generally accepted modes depending primarily
upon particle diameter and stream transport capability. The transport modes are
known as contact, saltation, suspended, and solution load. The saltation load
in combination with the contact load is generally assumed to comprise the bed
load. The sum of the suspended, bed, and solution loads is called the total
load. Of particular note here is the fact that there is no sharp line of demarcation
between m terials tifried as bed load ot as suspended load. Some
authors (e.g., Graf ?I), Shen
the washload may comprise from 90 to 95 percent of the total sediment load.
The scope of this paper is limited solely to washload. Bed load, and
suspended sediments mobilized from the bed, are not considered within the con-
text of this paper.
591
, Chow(3)) have indicated that in many instances
THEORY
Of the numerous sediment transport equations that have been presented,
none have been derived which account for the lateral inflow of water-soil mixtures
(washload) originating from sheet and gully erosion of land surfaces in
a drainage basin. Shen(*) points out, "Finally, none of the equations for predicting
suspended load account for the washload of the stream." Shen(') also
indicates that application of the available suspended sediment transport
equations to stream give rise to substantial error.
The existing sediment transport equations are based solely on the mobilization
of fine particulate concentrations (sediment suspensions) and coarse
layered masses of the bed, which are native to the stream channel. Of importance
here Is the fact that in most instances, the quantity of washload derived
from lateral inflm can be substantially greater than the suspended sediment
native to the bei. Several authors (e.g., Graf (11, Shen(') y Chow(3)) estimate
that the bed load contribution to the total sediment load is usually on the
order of five percent, and may in some cases be 'neglected from total load calculati
ms.
Given the flow condition and composition of materials native to the bed,
several relationships have been developed that .provide a general relationship
for the rate of sediment transport. It is not the intent of this paper to
present a development of the available sediment transport (bed load, suspended
load, or total load) equations, since their basis of derivation places them
outside of the realm of washload phenomena and, therefore, the scope of this
study.
The reader is referred to Graf (1) y Shen(') , and Nordin and McQuivey(8)
for a general development and assessment of available sediment transport
formulae.
COMPILATION OF DATA: BIXLER RUN WATERSHED
Storm events were chosen according to accepted hydrograph analysis criteria
and which appreciably satisfied certain analogous sedimentgraph analysis con-
ditions. The storm events were primarily classified according to the degree to
which the locus of fhe sedimentgraphs were defined by sampling. In many instances
sampling in the region of the crest of the sedimentgraph was not accomplished;
the Bixler Run project hydrologist therefore estimated the peak's shape from
the relative positions of the rise and recession sample points and from know-
ledge of previous sedimentgraphs where the peak was known. The latter class-
ifications yielded 63 storm events, which were grouped on the basis of runoff
derived during winter (October to March) and Sumner months (April to September).

592
UNIT GRAPH DEVELOPMENT (WATER AND SEDIMENT DISCHARGE)
Processing of the stage hydrograph and sedimentgraph for inidividual storm
s!i>ents involved as a first step the separation of base flow from the total dis-
charge. In the case of the stage hydrograph, baseflow was assumed to comprise
both groundwater flow and interflow. Base flow for the sedimentgraph was
assumed to be the sediment flow prior to the beginning of the rise of a sedi-
mrntgraph for a particular storm event. The base flow separation technique
WJS identical for both the stage hydrograph and sedimentgraph (see Figure 2).
Point A (or A') on Figure 2 is defined as the point where the rise of the dis-
charge graph begins and is determined by inspection. Line AB (or A'B') is a
tangential straight line projection, continuous with the base flm curve pre-
ceding it, emanating from point A ,(or A') and bisecting a vertical line drawn
through the peak. Generally, the points B and B' were appreciably in phase
for most of the storm events considered in the analysis. On the average, where
such was not the case the hydrograph peak lagged the sedimentgraph peak by one
hour. The point C (or Cl) is determined by drawing tangents on the recession
and base flow portions of the curves; the bisector of the tangents intersects
at a point assumed to be at the termination of surface runoff C (or C').
Although the separation technique utilized in this analysi bitrary , the
important feature is the consistency of its use throughout $3fay3f the data
process ing.
The resulting direct flow discharge graph data was then processed by
computer to derive the unit hydrograph, unit sedimentgraph, excess rainfall,
and associated sediment mobilized.
Hyetogrqhs were constructed for the selected storms in order to determine
duration for thc derived unit graphs. This was donefor winter and Sumner
rainfall storms ordy.
GRANULOMETRIC MEASUREMENTS OF SUSPENDED,
CHANNEL-BED, AND CHANNEL-BANK MATERIALS
Grain-size analyses of suspended-load , channel-bed , and channel-bank
materials were conducted by the USGS District Office, Surface Water Quality
Branch, Harrisburg, Pennsylvania.
The compiled data serves as a basis for comparison of the materials transported
during storm events and as a basis for reiative classification of the
prevailing transport mode (washload, bed load, etc.).
Results of 115 granulometric tests conducted on the bed, bank, and suspend-
ed sediment samples are plotted in Figure 3. Granulometric distributions obtained
from the tests generally plot as three distinct bands of points, with a
minor degree of overlapping. For clarity, only the arithmetic mean curves are
presented on Figure 3.
These were determined by sumning the percents finer than
by weight at a given particle size diameter and material source (bed, bank, or
suspended), and obtaining an arithmetic average.
Figure 3 corroborates verbal communication between the project hydrologists
of the USGS, Harrisburg, Pennsylvania, and this worker, to the effect
that materials encountered in the bed are primarily coarse-grained. In many
instances, bedrock is exposed at the surface. The suspended material, therefore,
can only have had as its primary source of origin the watershed's land
slopes. The distinctness with which the individual mean curves plot on Figure
3 is also indicative of significantannoring of the bed. Armoring is the time-

wise removal of fine particulate matter from the bed.
Sed imcntgrriph Analysis
ANALYSIS OF DATA
The original premise proposed in this study was that the unit hydrograph - .
concept as applied to a direct runoff hydrograph was directly analogous in the
iinalyiiis of d sedim@negraph, A fami of a tirlit gedinientgraph idas indeed develop.
ed whose standard unit was 1.0 kilogram for a given duration, distributed over
the watershed area, analogous in unit-hydrograph analysis to 1.0 centimeter of
excess (effective) rainfall over the same area. The shape of the resulting
unit sedimentgraphs varied only slightly for different rainfall events of a
given duration, as is anticipated in unit-hydrograph analysis. In order to
utilize such a unit sedimentgraph in generating a sedimentgraph for a particular
storm event, the total amount of sediment mobilized during the event would have
to be known or estimated. A relationship has been determined between total
sediment mobilized and excess runoff for single storm events. This is shown
on Figure 4, for runoff events resulting from winter and summer storms.
The latter approach would, therefore, entail the estimation of total
sediment mobilized on the basis of a known or predicted runoff excess as an
initial step, followed by the selection, based on duration, of an appropriate
unit sedimentgraph. The latter can then yield a sedimentgraph by multiplying
the individual unit sedimentgraph ordinates by the total sediment mobilized.
A simpler approach, however, was adopted which has the advantage that consideration
of duration of runoff excess may be neglected altogether; the relationship
developed, as will be shown, is independent of duration.
Once the observed total discharge hydrographs and sedimentgraphs were
graphically convzrted to direct discharge graphs by deducting the base flow,
the following calculations were performed:
DDi = DTi - DBi (1)
Where DD. is direct water discharge in cubic meters per second (hereinafter
designated as crns)
DT is total water discharge in crns
i .
DB. is base water discharge in crns.
The subskript "i" refers to the time at which water discharge values (e.g.,
DTi) are measured on the hydrograph's abscissa. For this analysis the hydrograph's
time base was divided into "n" two-hour increments.
Similarly,
SDi = STi - SBi ( 2)
where SD, is direct sediment discharge in parts per million (hereinafter
designated as ppm)
ST. is total sediment discharge in ppm
SB: is base sediment discharge in ppm.
Thelmagnitude of the ppm units is equivalent to mg/A units (milligrams per
liter) as long as the sediment concentration does not exceed 15,900 ppm (9).
Concentrations greater than 15,900 ppm have to be multiplied by a factor (9) in
order to convert ppm to mg/Q units. The units of.the direct sediment discharge
(SDi) are then converted from ppm units to kilograms per day, thusly,
593

594
Where Si is direct sediment discharge in kilograms per day, 86.56 is a factor
for converting the sediment discharge to kilograms per day.
Excess runoff and the associated sediment mobilized are determined as
follows:
i-1
i=l
Where ER is excess runoff in centimeters per square kilometer of drainage basin,
A is the watershed area in square kilometers,
0.1157 is a factor for converting the remaining elements of Equation 4 to
centimeters per square kilometer,
ES is sediment mobilized in kilograms per square kilometer.
Individual unit sedimentgraph ordinates are determined thusly,
usoi =
'i
Yhere USO. is the individual unit sedimentgraph ordinate in units of square
kilometer per day. Multiplying USOi by kilograms per squre kilo-
Eaters , yields kilograms per day.
Equation 6, as was previously pointed out, cannot directly be used in the
fashion of a unit hydrograph ordinate. The method, therefore, requires the
following operation,
SGO.i =
'i
Where SGOi is an individual "series" graph ordinate in units of kilograms per
day per centimeter of excess.runoff per square kilometer. By
"series" is meant that in contrast to a unit sedimentgraph ordinate
which approximately superpose each other for a given duration, a
series of graphs are obtained which vary considerably in shape and
peak. For the purpose of discussion, Figure 5 will hereinafter be
referred to as a series graph.
The series graph lines were developed by plotting SGOi for a given excess
runoff. This entailed some judgment in the selection of coordinate points. The
latter procedure is analogous to selecting a mean unit hydrograph curve from a
number of curves, which in practice generally do not overlap for a given
duration. The judgment used was partly justified by the fact that the least
squares line fit of the selected coordinate points (p, p 2 2, etc.) have a
distinct tendency to plot approximately parallel to each other. This is indicative
of a prevailing trend.
Series graphs were constructed for winter including rainfall and snowmelt,
and for s mer months.
coordinate points were plotted in time groups referenced to the peak discharge
(p). Thus p + 2 for example, refers to the direct discharge ordinate two hours
(7)
These are shown on Figure 5. The SGO. versus "ER"

after the peak; in all, the time increments considered were p,+ 2, p + 4, and
fir the summer events only, p + 6. Generally p + 6 represents a negligible
discharge quantity, very frequently zero, and was therefore assumed to be
zero for winter rainfall and snowmelt events.
This writer is of the opinion that for this particular analysis a great
part of the data scatter on the series graph and Figure 4, can be explained by
the mnnncr in which the sedimentgraphs were defined by sampling. The data
points are not shown since some overlapping exists as refers to p 5 n lines.
The sedimentgraphs selected for analysis did not have continuously defined loci.
As a result, graphical interpolation and judgment by the USGS, based on experience
and knowledge of sediment behavior, were incorporated in drawing the sedimentgraphs
between measured points. The observed sediment concentration points
were used as guides. It may be possible, therefore, to considerably reduce the
scatter of points by adequately defining sedimentgraph loci for a given storm
event by more frequent sampling. Most of the sedimentgraphs considered herein
generally had from four to six, and at times as many as 10 observed sample
points defining th graphs; in many of the cases the USGS estimated the magnitude
and location of the peak in its entirety. Consideration of scatter, at
least in this study, would suggest, that an attempt at explaining the variation
due to watershed soil types, vegetative cover, slope, etc., would be meaningless.
This worker would, however, opine that the loci of well-defined
sedimentgraphs would lead to the development of series graphs prossessing
less scatter.
Exsmples of sedimentgraphs predicted on the basis of season and runoff
excess art. shown on Figure 6. Table I lists comparisons between predicted and
actual eroded sediment quantities in Bixler Run as shown in Figure 6. To
illustrate the ;ange of applicability of the series graph method to Bixier Run,
variations in excess runoff for snowmelt or rainfall are included in Table I.
For the four storm events considered in Table I, the average error of estimate
for washload ranges from 16.1 to 16.5 percent as determined by the series graph
method and the ES versus ER graphical relationships, respectively. This is
based on comparisons with ac'tual conditions observed in the field. The errors
of estimate computed are all considerably below that reported for similar suspended
sediment load predictions, which may in some cases by greater than 100
percent.
TABLE I
Comparison of Predicted versus Computed Sedimentgraphs
595
Sediment Mobilized Percent Error
Date Excess
Runoff Source of
(tons/sq. km.)
Actual Predicted
Total Sediment
Bases (%)
centimetersf sq.
h.
Runoff BY BY BY BY
Series ES vs. Series ES vs.
Graph ER Graph ER
Method Curves Method Curves
10/19/68 . O35 Win ter-Ra inf a 11 19.6 27.9 27.7 29.6 29.1
031 10/67 .343 Winter-Snowmelt 1523.0 1293.0 1330.0 17.8 12.6
10/04/62 .572 Winter-Rainfall 2710.0 2694.0 2555.0 0.50 5.7
05f07f 56 .O55 Summer-Rainfall 73.8 61.7 91.0 16.5 18.7

596
CONCLUS IONS
This study discloses two important findings for Bixler Run Watershed.

597

598
1.50
1.25
I 1.00
1.75
; peak
\
HYDROGRAPH
?noon 6pm i2pm Gain 12nocn 6pm l2gin 6am 12noon
I_ -I.-.--I-
TIME IN HOURS
FIGURE 2 : TYPICAL STAGE HYDROGRAPH AJW SEDI?KNTGRAPW
(s-roiwi OF MARCH 13, 1963, BIXLER BUN WATERSHED )
5

599

600
3502
a
w
I-
w
I
O
4
Y
W
a
4
350.
u)
\
cn
z
a
a:
W
O
1
y.
- z
.
cn
hl
e
c3
W
N
4 35.0
m
O
z
I-
z
Id
2
I
a
Ill
o)
3.5
(
BIXLER RUN WATERSHED
w
;o098 0.0098 0.098
EXCESS RUNOFF ( ER ) , IN CENTIRIETERS / SQUARE I(Il.Ol\r;ETER
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
- EIXLER RUN WATERSHED
- legend:
- p peak value.
II I,
p+n n hours before (-1 or after (el peak.
- --summer roinf al I.
- ______winter rainfall. - - winter snowmelt.
0.72C I I I I 1 1 1 1 1 1 I I I 1 1 1 1 1
I
0:00098 0.0098 0.098
6 O1

602
ò A v a p-6 p-4 P-2 P P+2 pt4 pt6
TIME IN HOURS
FIGURE 6 : PREDICTED GRAPH OF SEDIMENT DISCHARGE VERSUS
TI ME

ABSTRACT
METHODES UTILISEES pour 1'EVALUATION des DEBITS de CRUE
des PETITS COURS d'EAU en REGIONS TROPICALES
par J. A. RODIER
For most of the tropical small streams studied by the author,
the floods result from surface runoff, field of application of unit
graphs. Hydrometric networks are useless for the floods of these basins
(less than 500 km2). Two methods are described:
For area without cyclonic precipitations: the depth of the
storm of the frequency choosed for the project is computed assuming the
other characteristics equal to the more frequent values for the big
storms. The transformation of rainfall into discharge is made in two
steps: computation of runoff coefficient and flood volume, computation
of characteristics of the hydrograph. This incorrect method gives good
results if used with judgement. Empirical graphs and rules have been
deduced from systematical researches on representative basins for comp~
tation of the elements of the flood from physiographical data. A gene-
ral synthesis will permit a better characterization of the basins.
In area with cyclones: the precipitation depth are estimated
from the observations and high values of runoff coefficient are choosed
in rel&tion with observations or envelope curves are drawn from obser-
ved data 5.n the world.
RESUME
Pour la plupart des petits cours d'eau tropicaux étudiés par
l'auteur, les crues résultent du ruissellement superficiel, domaine
d'application de l'hydrogramme unitaire.
Les réseaux hydrométri ues sont sans utilité pour les crues
de ces bassins (moins de 500 km 9 ), Deux méthodes sont décrites:
Pour les régions non affectées par les cyclones: on détermine
l'averse de fréquence égale à celle de la crue du projet, les autres
caractéristiques étant les plus fréquentes pour les tres fortes aver-
ses. La transformation en débit est faite en deux temps: calcul du coe
fficient de ruissellement et du volume de crue, calcul des caractéris-
tiques de l'hydrogramme. Cette méthode non rigoureuse fournit de bons
résultats si elle est employée avec discernement. Des diagrammes ou
des règles empiriques sont déduits de recherches systématiqyes sur bac
sins représentatifs, pour calculer les éléments de la crue a partir
des données physiographiques. Une synthèse générale permettra de mieux
caractériser les bassins.
Dans les régions de cyclones: on détermine les averses d'après
les valeurs observées et on suppose des coefficients de ruissellement
tres élevés en rapport avec ces observations, ou on établit directement
les courbes enveloppes a partir des données observées dans le monde.
Chef du Service Hydrologique de l'office de la Recherche Scientifique
et Technique Outre-Mer
Conseiller Scientifique à Electricité de France (DAFECOI.

604
L'étude des ouvrages utilisant les eaux des petites rivières
tropicales ou méditerranéennes présente de très sérieuses difficultés
dès que loon aborde la dbtermination des conditions hydrologiques de
réalisation et d'exploitation des ouvrages, en particulier celle des
d6bits moyens annuels et surtout celle des ddbits de crues.
Les donnhes sur le régime hydrologique sont dans ce cas
inexistantes : la densit& des réseaux hydronktriques est faible, le
nombre de stations amdnagées est nul ou dérisoire. LEh outre, les varia-
tions temporelles des débits sont si rapides que les donn&es de ces sta-
tions sont souvent difficiles à exploiter. Enfin, contrairement & ce qui
a lieu pour les grandes rivières, les crues de faible frequente ne lais-
sent aucun souvenir dans la mémoire des habitants les plus proches.
I1 est très sou-ntinipsible de procéder 2 une étude hydrologi-
que sérieuse sur le terrain pour un seul ouvrage car elle durerait long-
temps et son prix atteindrait ou dépasserait même celui de l'ouvrage lui-
même
Tout ce que l'on peut faire c'est organiser une telle étude à
l'occasion de la r8alisation d'une série importante de tels ouvrages,
par exempie pour la construction de tous les ponts d'une longue voie
ferrée (chemin de €er transcamerounais), ou d'un grand axe routier, ou
lorsqu'o, ariiinage i la fois 30 ou 50 petits barrages comme cela a 6t6 le
cas en I!AUî.Q-VOLTA il y a quelques années.
Autrement, on est conduit & utiliser les résultats de synthèses
& caractère g6ogr;pliique.
Nos hydrologues ont souvent rencontré ce problème en Afrique
Tropicale, en Am6rique du Sud, dans les fles du Pacifique et de l'Océan
Indien et ils ont mis au point différentes rnbthodes pour la détermination
des débits moyens annuels et des dLbits de crue, premiers 6léments que le:
ingénieurs demandent aux hydrologues.
Dans ce qui suit, nous ne traiterons que le probleme de la dé-
termination des d&its de crue dans le cas de cours d'eau dont le bassin
versant couvre une superficie inférieure 200 km2 et plus souvent infé-
rieure à 50 km2. Au-delà de ces surfaces, les m6thodes ne sont plus les
memes. Ellos correspondent souvent en effet la limite d'emploi de l'hy-
drogramme unitaire et des modèles globaux.
Pour ces petits bassins, on considérera deux cas différents sui-
vant la genbse des crues exceptionnelles. Dans le premier cas, elles sont
dues a des orages convectifs avec prhcipitations intenses mais d'assez
courte durbej dans le second cas, il s'agit de précipitations cycloni-
ques ?ì iiitensité plus faible mais de plus longue durde, les derniers 616-
ments de l'hpisode pluvieux arrivant sur un sol pratiquement saturé.

1. Cas de crues provoquées par des orages convectifs,
605
La mise au point de méthodes pratiques nous a demandé quinze
ans de recherches fondamentales. Nous passerons rapidement sur ces
recherches pour insister plus particulièrement sur la m6thodologie
proposGe aux ing6nieurs, cette méthodologie n'étant guère applicable
que pour des p6riodes de retour de 10 ou 20 ans. L'idée de base est
IQutilisatian de l'information pluviomhtrique existante et plus parti-
culièrement des sc'ries chronologiques de précipitations journalières
et la transformation des hauteurs de prGcipitations en débits de ruis-
sellemelit superficiel. Pour des averses de ce type et d'assez faible
frcquerice, en réginns tropicales et méditerranéennes, il se produit
g(!nbralernent du ruissellement superficiel, ce qui permet l'emploi de
la m6thode de l'hydrogramme unitaire.
1.1. Recherches fondamentales entreprises.
Les plus importantes ont ét6 les suivantes :
1.l.l.Etudes g&n&rales statistiques des pluies journalières. En
Afrique Occidentale OU elles ont été le plus poussées, elles ont port6
sur 1 O00 stations environ. L'étude simultanée pour un grand nombre de
statioils a conduit a des valeurs assez sûres des paramètres des lois de
distribuLion pour des p6riodes de retour de 10 à 20 ans. Eh particulier,
elle u cona.iit ?i abandonner, pour cette région du monde, la distribution
de GALTOW tro2 pessimiste,pour une distribution de PEARSON III. Sur le
plan pratique, on en a déduit une série acceptable de précipitations
journali&res de période de retour 10 ans ou 20 ans.
1.1.2.Etude de l'abattement (Inverse du rapport pour une méme frdquen-
ce, eiitre la hauteur de précipitations en un point et la hauteur de prcci-
pitations sur une surface donnée entourant ce point). Les études menées à
partir de données recueillies sur bassins représentatifs ont conduit a
des ordres de grandeur acceptables pour la pratique.
1.1.3.Etudes des courbes intensité-durée : ces études faites surtout
à partir des pluviographes des bassins représentatifs ont permis, pour
l'Afrique Occidentale, de donner des courbes-types.
1.1.4.Etude des relations pluies-débits : celles-ci ont été etudibes
averse par averse pendant plusieurs années sur une centaine de bassins
reprGsentatifs, qui ont également 6té utilisés pour les recherches vi-
sées aux points 1.1.2 et 1.1.3. La méthode des r6sidus a permis de dé-
teniiiner dans chaque cas la hauteur d'eau bcoulée HR ou le rapport KR
entre HH et la hauteur de précipitation P eii fonction de P, des condi-
tions d'humidité du sol avant l'averse et de la durée de l'averse.
1.1.5.Etude de la forme des hydrogrammes. Sur les mgmes bassins re-
prdseiitatifs, on a pu appliquer la mgthode des liydrograrimes unitaires
et dbterminer la forme des hydrogranines-types. On en a retenu trois

606
éléments caractéristiques : le temps de montée tm , la durée de ruis-
sellement tg et le rapport k entre le débit de pointe de l'hydrogramme
unitaire et le débit moyen pendant la durée du ruissemment.
1.2. bisthode de détermination des débits de pointes de crue et de leur
volume.
Le cas des fréquences décennales ou de fréquences voisines
est assez différent de celui de la crue maximale probable. Dans ce qui
suit nous traiterons le cas des crues de périodes de retour de 10 ans
ou 20 ans.
De façon générale, on a cherché à mettre au point des méthodes
simples qui puissent &tre utilisées sans ordinateur. Ces méthodes sont
probablement très différentes de celles qui sont élaborées actuellement
et qui seront vulgarisées dans quelques années, mais de nombreux pays en
voie de développement ne disposent pas, à l'heure présente, de moyens de
calculs suffisants et n'ont pas assez de personnel bien entraîné pour les
utiliser pour des fins hydrologiques.
C'est pourquoi, dans ce qui suit, on adoptera les principes sui-
vants, dont certains sont discutables, mais qui permettent aux ingénieurs
d'arriver à des résultats utilisables avec les moyens dont ils disposent.
1.2.1.Principes du calcul : Le point de départ est la série d'obser-
vations de précipitations journalières au poste le plus proche de l'ou-
vrage que l'on a & étudier ou un poste pluviométrique correspondant aux
mdmes conditions pluviométriques si la qualit6 des données du pluviomè-
tre le plus proche est insuffisante.
Des bassins de 50 km2 sont généralement assez homogènes, mais,
dans le cas de forte différence d'altitude, le poste pluviométrique choi-
si devra se trouver à peu près à l'altitude moyenne du bassin et non pas
au niveau de l'esutoire, ce qui rend le choix beaucoup plus difficile.
On étudie la distribution statistique des précipitations journalières ce
qui, en région tropicale, correspond à peu près à la distribution des
averses orageuses et on détermine l'averse correspondant ?I la fréquence
de la crue (période de retour 10 ans, 15 ans, 20 ans, etc...). On recher-
che, en Ltudiant lee 'enregistrements disponibles, quel est le schéma lo
plus courant des répartitions des intensités pour une averse donnée, on
examine également quelles sont les conditions moyennes d'humidité préa-
lables que rencontrent généralement les fortes crues. Enfin, on trans-
forme la hauteur de précipitation en un point par la hauteur de préci-
pitation moyenne sur une surface en la multipliant par un coefficient
d'abattement inférieur à 1.
Au moyen du modele de transformation des pluies en débits, on
transforme la pluie décennale en crue décennale en veillant bien à ce
que la distribution des intensités de l'averse, l'index représentant
l'humidité préalable, le mois de l'année lorsque celui-ci intervient,
correspondent aux conditions les plus fréquentes pour les fortes préci-
pitations. Sur le plan statistique, ceci est très contestable : la

607
v8ritable solution consisterait 5 appliquer le modèle de transformation
pluie/débit à la totalité des averses observées au poste de référence,
sur 40 ans par exemple, et à étudier la distribution statistique de
l'échantillon de crues reconstituées sur 40 ans. Mais cette méthode
serait peu réaliste pour beaucoup de pays en voie de développement
parce qu'il est beaucoup plus difficile de mettre au point un modèle
valable pour toutes les averses qu'un modèle uniquement valable pour
les fortes averses et parce qu'ensuite la reconstitution des crues de
petits bassins pour 40 ans ne peut se faire qu'avec l'ordinateur.
La transformation pluie/débit se fait en deux temps :
lo - calcul du volume de crue par la détermination du facteur KR (voir
1 .I .4.) ;
2* - à partir de ce volume, détermination du débit de pointe par la forme
de l'hydrogramme (voir 1 .1 .5.).
Autant que possible, on a cherché à ramener ce calcul à des
opérations très simples dans un certain nombre de pays où le nombre de
bassins représentatifs était suffisant.
1.2.2.Pratique du calcul pour l'Afrique Occidentale :
Le5 études systématiques visées en 1.1.1. et 1.1.3. fournis-
sent des éléments pluviométriques permettant de déterminer l'averse de
fréquence cherchée avec son diagramme de distribution temporelle, pour
la majeure partie de l'Afrique Occidentale. Des études beaucoup plus
partielles effectuées dans d'autres r6gions du monde ont fourni les me-
mes données.
On réduit ces valeurs ponctuelles à des valeurs moyennes sur
une surface donnée en les multipliant par un facteur qui décroît de 1
pour une svface S inférieure à 25 km2, à 0,8 pour une surface comprise
entre 150 et 200 km2. Ces chiffres qui ne sont valables que pour les
orages convectifs des régions tropicales africaines, sont peut-$tre un
peu forts. Ils seront probablement diminués à la suite de recherches en
cours.
On dispose donc de la hauteur de précipitation P,.
Pour déterminer la valeur de Ks, on a établi des séries d'aba-
ques pour deux types de couvertures végetales naturelles (liées au cli-
mat), savane et savane boisée d'une part, steppe et savane à épineux
d'autre part. I1 n'a pas encore été possible d'établir d'abaques conve-
nables pour la forat tropicale.

608
Les autres facteurs pris en considération pour la détermina-
tion de KH sont : la superficie du bassin, la perméabilité globale
du sol P et la pente R. A défaut d'index quantitatif pour R et surtout
P, on a établi deux classifications : R correspond à des plaines très
plates, RG à des pentes de montagne (pentes longitudinales supbrieures
à 5 $, pentes transversales supérieures à 20 $)o
Pl correspond à un sol rigoureusement imperméable, P5 à un
sol très perméable (sable ou carapace latéritique très disloquée. Le
graphique 1 d m e un exemple de ces abaques pour des sols imperméables
(Pl - P2) et des pentes variables de R2 à R4. Ces abaques ont été &ta-
blies & partir des données des bassins représentatifs. Les valeurs de
KR correspondent des pluies de fréquence décennale (Pm vaciant de
&o à IO5 mm) dans ces régions, tombant dans des conditions d'humidité
du milieu de la saison des pluies.
Bien entendu, au cas OU des facteurs secondaires tels que le
réseau hydrographique, présenteraient des caractéristiques anormales,
par exemple lit marécageux, on devrait rectifier les valeurs de KR en
conséquence.
Le volume de ruissellement de la crue :
A ce volume il convient d'ajouter le volume correspondant au
d6bit de base do1.t on peut avoir une idée sur le terrain, sans &tude
hydroiagique très difficile.
Pour la forme de l'hydrogramme,des abaques ont été également
mis au point;. On en trouvera un exemple au graphique 2 qui donne le
temps de base ou duróe du ruissellement en fonction de la surface du bas
sin et de l'index de pente pour les m8mes conditions de végétation que
le graphique no 1.
La connaissance du temps de base TB permet de calculer le débit
moyen de ruissellement :
M
'ruis
e
s eli emen t
TB
M est obtenu en m 3 /s.
K (K P F)
Pour trouver le débit de pointe s, on utilise un coefficient
étudibi pour les mêmes rbgions sur bassins représentatifs.

609
Pour la couverture végtitaïe steppe ou savane à épineux avec
des valeurs de KR pas trop &levées, on trouve des valeurs de K variant
entre 2,5 pour 25 km2 à 3,t pour 100 kmz. Si ces valeurs de KR sont
supérieures à 50 - 60 $ K varie entre 3 pour 2 km2 et 4,5 pour 50 km2.
de base.
On obtient QM en multipliant $1 par K et on ajoute le débit
Bien entendu, si le diagramme de répartition temporelle des
intensités et si la superficie du bassin sont tels que la crue n'est pas
unitaire, il existe des abaques complémentaires donnant le temps de base.
Dans ce qui précède, c'est volontairement que nous n'avons pas
Gtabli de formules pour repr6senter les courbes des graphiques 1 et 2
auxquelles nous voulons garder un caractère provisoire.
l.Z.3.Limitations de la méthode :
Elle ne s'applique bien en Afrique tropicale que pour des su-
perficies inférieures & 50 - 100 km2.
Comme nous venons de le dire, nos courbes sont provisoires et
on met ali point des modèles plus &labor& pour revoir les bases de nos
abaques quL nécessitent encore un sérieux effort d'homogénéisation des
données et des proc6dés de calculs.
Les problèmes de forêt tropicale exigent encore un effort im-
portant de recherches sur le terrairi.
%fin, il n'est pas très facile de classifier un bassin en caté-
gorie P2 ou P . Des recherches de physique du sol sont en cours pour arri-
ver ?i des règ3es simples permettant de le faire. C'est certainement 1&
le point le plus difficile.
Pour définir quantitativement des index R une bonne combinai-
son des facteurs géomorphologiques courants doit donner satisfaction.
Actuellement, cette méthode est très souvent employée en Afrique,
mais, dans bien des cas delicats, il serait plus prudent que les bassins
soient examinés auparavant par un hydrologue confirmé. Elle présente l'im-
mense avantage d'éviter toute véritable étude hydrologique sur le terrain.
1.3. Crue de période de retour supérieure à 20 ans.
C'est là un problème très difficile car la documentation plu-
viométrique est tout à fait insuffisante. Pour des périodes de retour
de l'ordre de 100 ans, une minutieuse étude critique des relevés de nom-
breux postes pluviométriques permet d'aboutir à un ordre de grandeur.

61 O
En Afrique tropicale, les averses journalières centenaires
de caractère convectif sont peut-8tre de l'ordre de 200 ?i 3-400 mm en
24 heures, suivant les régions.
I1 reste ensuite à choisir une valeur de KR qui n'est plus
celle des abaques mais qui doit en tenir compte, car tous les bassins
pour de telles averses ne parviennent pas à la limite de O,&5 - O,9O.
Enfin, généralement, l'averse dure au moins 5 ou 6 heures et parfois
20 heures, elle n'est donc plus unitaire. On utilise donc les abaques
tels que ceux du graphique 2 pour établir les différents hydrogrammes
élehentaires qu'on ajoute après avoir découpé l'averse centenaire.
Enfin, s'il s'agit de la crue maximale probable, il ne reste
plus qu'8 appliquer la formule de FIERSHFIELD OU l'on ajoute à la valeur
moyenne de la précipitation journalière maximale annuelle 15 fois 1°é-
cart-type de la distribution de cette précipitation maximale. Mais il
faut d'abord partir d'une série de précipitations journalières de quali-
té suffisante pour en déduire une valeur correcte de l'écart-type.
D'autre part, si cette formule paraft excellente pour l'Afrique du Nord,
les régions soumises à des cyclones tropicaux, elle semble conduire à
des chiffres trop élevés pour les orages convectifs d'Afrique tropicale.
Bans ce cas également on revient à l'application de la méthode de l'hydro-
gramme unitaire pour des averses élémentaires successives, mais le choix
de la distribution temporelle des intensités est délicat. Cequi arrive
souvent clest que d'un bout à l'autre de l'estimation, on arrive à de
telles cascadea de marges de sécurité qu'il est facile de fournir des
chiffres trop élovés.
2. Crues dues à des averses cycloniques :
2.1. Crues décennales :
L'averse décennale est plus difficile & définir que dans le
cas précédent, la distribution statistique est plus difficile à étudier
et les donnkes de base sont plus mauvaises (en cas de cyclone une bonne
partie des pluviomètres débordent), mais dans beaucoup de pays du monde,
on arrive 8 d6finir une valeur à peu près convenable de l'averse décen-
nale, on doit alors découper l'averse en averses élémentaires comme au
point 1.3. et on transforme ces averses en crues par la méthode des hy-
drogrammes unitaires. Très souvent, pour les dernières averses élémen-
taires rC, est voisin de 0,9O si la pente est notable, que la couverture
soit forestière ou non. Après calcul, il est bon de comparer le résultat
aux crues maximales connues dans le monde en utilisant des diagrammes
tels que le diagramme FWCOU-RODIER.

2.2. Crue maximale probable.
Dans ce cas, on ne peut donner que des indications gén6rales.
Pour l'averse à prendre en considération, on pourra se référer, dans
les pays à fortes averses, aux valeurs maximales mondiales telles qu'el-
les sont données dans le Guide des Pratiques Hydrométéorologiques de
l'OMM ou aux résultats de la formule de HERSHFIELD, mais il sera encore
plus difficile que plus haut d'aboutir à une valeur convenable de l'écart-
type, ceci nécessitera une sérieuse étude critique des rares données
pluviométriques disponibles, en tenant compte du débordement 8ventuel
des pluviomètres. Le reste est plus facile car le coefficient de ruis-
sellement KR est de l'ordre de 0,gO.
Si la région est COMUB pour avoir des averses exceptionnelle-
ment fortes, il est normal de prendre en considération des valeurs de
précipitations supérieures aux maximums mondiaux connus car, en pays de
cyclones tropicaux, la connaissance des averses de durée inférieure à
48 heures est très incomplète et les maximums mondiaux connus doivent
&tre considér6s comme piut8t provisoires.
ïb générai, dans les cas graves concernant les cyclones tropi-
caux, l'hydrologue arrive a la conclusion un peu décevante qu'il serait
prbférable que l'ingénieur prévoie son barrage de telle façon qu'il puis-
se être submergé par n'importe quelle crue.
Qua1 que soit le cas étuùié, un examen du terrain orienté vers
la recherche L;ss traces laissées par de fortes crues est n&cessaire.
611
I1 résulte de tout ce qui précède que les ñydrologues ont
encore de nombreuses recherches à faire pour aider efficacement les
constructeurs dans leur tâche.
Quelques références utiles pour les petits bassins de ces régions :
1. 0.M.E.i. (1965). Guide des Pratiques Hydromét&orologiques, no 168
T.P. 82, Genève.
2. RODIER J., AWRGY C. (1965). Estimation des débits de crues décen-
nales pour les bassins versants de superficie inférieure à 200 km2
en Afrique Occidentale, ORSTOM, Paris.
3. HERSHFIXLD D.M. (1963). Estimating the probable maximum precipita-
tion. Am. Soc. of Civil hhgineers Transactions, Vol. 128, Part I,
PP. 534-556.

61 2
4. FRANCOU J., RODBR JO (1967), Essai de classification des crues
maximales observées dans le monde. Cahiers d'Hydrologie OHSTOM
vol. IV, ne 3, pp. 19-46. Paris.
5. BENSON M.A., (1968), Measurement of Peak Discharge by Indirect
Methods, OMM ne 225. TP. 11gP Genève.

613
I
O
h
I
O
YI
N
E
Y
C
a
I
I
O
-N
-
-0
-0
-
O
-0
OI
-0
W
.o
h
O
-a
O
-YI
O
-*
O
-m
-0
N
-0
-YI
-*
-m
-N
i ;
C
- c C
4.
*QL)c
.-
.-

2 5-
u)
?!
a
Q
C
2 o-
C
Q
v)
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- f
15-
I o-
5-
O- +
0
Ø'
' 4r
Ø
Ø'
I 2 3 4 5 6 7 IO 20 30 40 50 60 7080 loo 2c
S en km2
Fig: 2
Temps d e base en fonction d e R et de S
REGIMES SAHELIENS - SUBDESERTIQUES
I

METHODS FOR THE ESTIMATION OF MAXIMUM DISCHARGES OF SNOW
MELT AND RAINFALL WATER WITH INADEQUATE OBSERVATIONAL DATA
ABS TRACT
Prof. A.A. Sokolov
State Hydrological Institute
Leningrad, USSR
The problem of floods computation was accepted by the UNESCO
Co-ordinating Council for the IHD as one of the most important
problems. For its solution the Working Group on Floods and their
ccmputation was established; it has realized several projects on
the THD programme essential for future research on floods, deve-
lopmen: and improvement of methods for floods computation. The pc
per give;: an evaluation of the up-to-date state of this problem
as means fLr its solution.
RESUME
Le Conseil de Coordination de l'UNESCO pour la DHI a estimé
que le problème du calcul des crues était tres important. Pour
l'examiner, il a créé un groupe de travail sur les crues et leur
évaluation. Ce groupe a réalisé, dans le cadre du programme de la
DHI, un certain nombre de travaux très importants pour l'avenir
de la recherche sur les crues, pour la mise en oeuvre et l'amélig
ration des méthodes de calcul qui le concernent. Dans la présente
communication, l'auteur fait le point de la situation actuelle,
ainsi que sur les moyens de parvenir la solution du probleme.

61 6
At present triere are two uifferent approac,ies to t.ie complitatlon
of fiood discilarge of ungauged rivers cievelopec to a certain
dcbree quite independently. The first approach is based on the
s~atistical analysis and generalization of field data on
flood runoff; the second approach is based on the genetic
analysis and synthesis of flood hy&ograph. As was: ctateu
by J. Nemec and M. Moudry (Czechoslovakia) at; the Leningrad
Symposium (1967) these two approaches ase gradually being
brought together and at; present they are used simultaneously
supplementing each other Lis,-/.
Humerous design schemes have been proposed to estimate maximum
flqod runoff of ungauged and poorly gauged rivers.
According to the principles of approach and the scope of
flood computation these schemes may be divided into 2 main
groups:
3. Empirical or semi-empirical formulae for flood discharge
computation based on the account of some of its most important
factors ( e.g., drainage area or maximum precipitation rate),
providing maximum water cìischarge only.
2. Methods considering flood genesis and providing the
possibility of plotting the whole hydrograph on the basis of
time inflow of snow melt and rainfall water and its transformation
into runoff as a result of losses by infiltration,
suriace retention, lag along the slopes and channel network.
At k-esent, methods of maximum flood discharge computation
on the basis of the use of di€fereat design formulae are
widel applied. The most important formulae are as follows:
(a? formulae of extreme intensity, or the so-called
rational formulae, basea on the account of maximum or extreme
rainfall intensity during lag-time or flood flov concentration
in its general form:
Qmax =/Cpa,d,A. (1)
where : gp is coefficient of dimensionality; ar is maximum
intensity of rain or snow melt during lag time ( c ); dris
runoff coe ficient during this time interval; A is drainage
area in km 5 .
Pormula (1) is usually applied for the computa-t;ion of
maximum runoff for relatively small basins (less than 200 km ).
To determine design values of' dc curves of maximum precipitation
increase %, are plotted with the increase of time
interval Z' ? given as percentage of daily precipitation of the
same probability of exceedence ( p )
The maximum mean precipitation rate Jcp for any time
interval is estimated by formula:

617
Mean velocity and lag-time down the channel and slope6
are computed by simplified formulae of Chezy-Manning or
C he zy -Baz in.
The principal disadvantage of formula (1) consists in some
uncertainw and inaccuracy of the determination of lag time
c-
or flood concentration c, is interpreted by iiiJividiia1
scientists in different ways, therefore it causes iii-
accuracy of determination of principal parameters az and d,-
appearing in formula (1). Besides, formulae of type (1) do not
take into account the remaining flood elements (duration rise
and fall duration ratio) and do not provide the plotting of
%he vihole ïlood hydrograph essential for the determination of
maxima transformation in ponds and reservoirs;
(b) empirical or semi-empirical reduction formulae of the
general type:
is maximum specific discharge, m'/sec per 1 km';
9. is parameter comprising the extreme specific discharge
if A-O and C = 1.O;cis addition to the drainage area
considering non-lineariky of dependence 4 ~mp,aj@@~
within the range of small areas of the basin; nis the
eqonent of reduction of maximum specific discharge3 with %he
incrsase of basin area and varying according to experimental
and bheoretical data from n= 0.15 - 0.30 for runoff maxima of
snow mel? water or caused by prolonged frontal rainfalls, to
n = 0.5 - 3.7 for maxima caused by short heavy local storms.
Parameter +,,may be estimated according to the extreme rate of'
snow melt or rainfalls for minimum time interval, e.g. 1 hour,
or for snow melt water according to depth of runoff during a
flood.
In the first case when C = 1.0 formula (4) may be presented
as fo1lov;s:
where: Q,,,~
where: % is coefficient of dimensionality; % ds maximum
hourly rato of snow melt or rainfall; d, is overland flow
coefficient .
Since does not exceed 10-15 m/hr for snow melt water
and 300-400 mm/hr for rainfall water ( on the basis of computation
of heat balance of snow melt), then the extreme value of
qe in fo mula (i+) if do= 1.0 and k$= 0.23, may not exceed
2.8 -4.2 m 3 /sec pe 1 lun2 for snow melt water and up to 84-
112 mj/sec per i bS for raidail water.
On the basis of these elementary considerations it is possible
to conclude %hat many empirical formulae of type (4) with
parameter 9. exceeding the mentioned limits have no physical
substantiation.
Due to some uncertainty of C value in formula (4) when
A -0, this Lormula is sometimes used as follows:

61 8
w ei'3: Cp.6 is parameter (maxim specific discharge in
ì$/sec if &ainage area B=2ûû by:
a is tiie exponent of reduction dcterrnined uy regional
=fy/A/.
dependences ep
For practical computations on the basis of generalization
of empirical data a map of regional boundaries is prepared
with similar exponents of I2 .
The advantage of formula (6) consists of the fact that the
value of parameter c,, slightly depends on and
therefore the mapping of cp,6
possible e
in the form of isolines is
In case of eo determination according to the depth of runoff
of snow melt water during flood hrnformula (4) may be present-
ed as follows:
where: & is t,ie coe€ficient considering a number cif otlier factors,
in particular, duration and siiape of flood.
Formula (7) is used as the basis for the computation of
maxinum snow melt water äischarge for the whole USSII territory
L4L
Duriw recent years the reduct'on scheme is seldom used as
simple regional dependences =*[JI plotted by dependences
enveloping empirical points usually related to larger basin
areas.
In this case it is essential to take into account flood
runoff probability of oxceedence Its parameters are
differentiated according to climatic zones.
Along with the basin area which was previously accepted as
almost the only maximum runoff factor, numerous important
clinlatic factors are also taken into account, i.e. depth and
sate OP precipitation, snow melt rate, flood runoff depth;
and morphological factors as well, i.e. basin topography, lakes
and swamps areas, river network density, soils and subsoils
composing the basin mantle, etc.
7iith the increase of hydrological information and reliable
runoff data from small basins the reduction scheme has greatly
consolidated its positions since ibs basic parameters i.e.
reduction coefficient anCr maximum runoff of elementary (small)
basins have gainea a reliable substantiation. This particular0
concerns maximum runoff of snow melt water;
(c) the so-called volumetric formulae considering flood
shape and duration, besides maximum ordinate of flood, may be
given as follows:

61 9
where: H is depth of precipitation or snow melt water;
&is total or volumetric coefficient of runoff during flood;
Ttn ndicate general duration of flood or its rise phase;
4 and$ , are coefficients of flood shape, i.e. ratio of maximum
discharge to mean discharge.
Design formulae as (8) or (9) are preferable compared with
formulae of type (1) or (4) since all flood elements are co-ordinated
but they ara applied only for simple one-peaked
floods for which general or volumetric coefficient of' runoff
is applicable.
Despite the existing numerous formulae the computation of
rainfall flood runoff for ungauged rivers is not reliable.
Every design scheme proposed is characterized by certain a6vantages
and disadvantages. There exist no accepted design
schemes until now.
formulae is in the cornputation of individual flood elements
without their co-ordination and without the account of genesis
anrid type of flood; the latter is essential for hydraulic engineering
projects to make a correct estimation of the flood transformation
rate in ponds and reservoirs and the amount of
discharte through spillways. These disadvantages never occur
in genetx computation methods for floods of the 2nd group
based on the account of time variations of inflow of rain and
snow melt waters and their transformation into runoff hydrograph
as a result of non-simultaneous water lag from different basin
areas.
These methods are based on the plotting of runoff transit
curve showing the distribution of areas of simultaneous runoff
over time intervals.
The following methods may be mentioned for the computation
of floods:
(a) isochrone method based on the determination of ordinates
of the curve of unit areas distribution (transit curve) by
means of plotting the lines of equal transit (isochrones) on a
topographic map o€ river basin;
(b) unit hydrograph method, based on the determination of
transit curve by the ordinates of the observed unit flood
hydro .raphs caused by individual storms;
(cy method of mathematical floods simulation.
The method of isochrones is mainly applicable for floods
computation on small water courses with surface flow prevailing.
&/
A general disadvantage of the majority of empirical design
Nhen the method of isochrones is applied it is essential to
use topographic map of the basin of sufficiently large scale

620
anu tile data on time variations of rain or snow melt water
Inflow, moreover, for small water courses it is necessary to
have data on such variations within a day and this causes certain
restrictions in the sphere of its application.
Unit hydragraph method is based, as it was mentioned, on the
plotting of curve of unit areas distribution according to the
ordinates of unit floods observed.
Unit hydrograph method is based on the lag theory expressed
by the so-called genetic formula of runoff:
where: Qdt indicates discharges at the outlet at the moment t ;
ht-r is effe tive precipitation per time unit Af at the
moment k-r; #c indicates ordinates of the curve of unit
areas distribution.
Tne unit ,iydrograpli metiiod is cnaracterized by its visuality
and.pa.ysica1 substantiation, it provides the plotting of design
flood nyúrograpii according to preciFitation; tiiis resulted in its
wide application in many countries of tile world despite some
draNoacks.
Its sppircasion is aiII1cuII; mainly ow- ‘GO m e inadequacy
of the mekhods used for averaging the observed individual floods,
separation cf multi-peaked floods and methods for infiltration
rate and flow coefficient determination, There is also some
uncertainty as to the applicabili- of the unit hydrograph method
to different drainage areas; also problematic is the relationship
between an individual storm duration and flood rise duration
or the time of peak shifting relative to rainfall maximum during
the flood.
All these factors limit the application of the unit hydrograph
method.
Lately the method of mathematical floods simulation has been
more and more widely used.
For instance, in one of the variants of trie method of mathematical
floods simulation applied in the USSR the analogy between
equation (10 ), describing flood formation resulting from
water lag and summation of individual discharges from different
parts of river basin, and the eq,uation describing the change of
current in the electric circuit, was used,
To apply this method practically, it is essential to develop
investigations connected with determining parameters set for the
specific electric analog computer to estimate flood runoff of
ungauged rivers.
The importance of research, computation and prediction of
floods for many countries of the world necessitates the international
scientific co-operation on the problem of flood flow
computation.

621
This co-operation is in particular exercised under the auspices
of UNESCO and WMO within the framework of the IIID programme.
For this purpose the Co-ordinating Council €or the IHD
established the Ciorking grou:, on floods and their compuation.
This Working group has studied and generalized, to some extent,
the international experience in the field of research and
computation of flood flow.
A great contribution was made by the International Symposium
on floods and their computation held in Leningrad at the initiative
of UNESCO and with the participation of \YMO and IAkIS.
The proceedings of this Symposium were published, therefore
there is no need to cite and consider them here /18, 26/.
The review made by WMO on meteorological aspects in computing
flood flow is rather useful. ‘ïiiic review made up a technical
note on this problem /25/.
The results of processing of observation data on floods made
at the network of IHD stations also appear to be valuable. Taking
into account that in some large areas covered by the network
of IIID stations outstanding floods, may always occur. The
UNESCO IHD Working group on floods and their computation prepared
and published the technical note on collection and processing
of data on floods /27/.
The Working group also developed the programme for the
World Catalogue of very large floods, according to which a
riiimber of countries were entrusted to collect, process and
publish the data on large floods.
Besides, the Working group considered it necessary to study
and generalize the international experience in the field of
floods covputation and is now preparing for publication the
Technical Note (Casebook), wiiich would reflect a great experience
in computation of flood flow, gained in many countries.
These pub lications together with the Proceedings of the Lenin
grad Symposium on floods will serve as a good basis to improve
methods of flood flow computation, which in its turn will
contribute to a more rational and economical selection of
parameters for hydraulic structures on rivers, their durability
and resistance to floods.
R E F E R E N C E S
Alexeev G.A. (1 966). Sklema raschetov maximainyh dozhdevyh
raskhodov vody PO formule predelnoy intensivnosti stoka,
(Computation of maximum rainfall discharge by means of
the formula of extreme runoff intensity). Transactions
of the GGI, vol. 134.
Befani A.N. (1958) Osnovy teorii protsessov stoka i puti
dalneishykh issledovaniy. (Theory of runoff processes and
directions of further research). Transactions of OGMI,
vol. 15.

622
3. Velikanov M.A. (1931) GidromekhanichesQ analiz poverhnostnobo
stoka. (Hyaromechanical analysis of the surface runoff).
Geophysics Nos. 1-2.
4. Voskresenski K.E. (1956) Gidrologicheskie raschety pri proektirovmii
sooruzheniy na malyh rekah, ruchjah i vremennyk
vodotokakh (Method osn. i prakt) (Hydrological computations
for structures on small rivers and temporary water
courses), Leningrad, GIMIZ.
5. Kalinin G.P., Milukov ?.I. (1958) Priblizhennyi raschet neust
anovivshe go sy a dvi zhenia v o w h mass . (Approximate
estimation of the unsteady water motion), Transactions of
TsW, vol. 66.
6. Kovzel A.G. (1951) Opyt projektirovania hydrografa vesennego
stoka dlya malogo vodosbora. (The designing of the hydrograph
of spring runoff on small watersheds) .Tr.of GGI,v.31(85)
7. Kuzmin P.E. (1961) Protsess tayania snezhnogo pokrova.
(Snow cover melting) Hydrometeorological Publishing House.
8. Lvovich M.I. (L34-ö) Protsessy formirovania pavodkov.(Flood
formation), Transactions of GGI, vol. 10.
9. Moklyak V.I. (1965) Pormirovanie maximalnyh raskhodov ot
talyh vod i ih raskhoày (Formation of maximum snowmelt
discharges), Kiev.
10. l'rotoãyakonov M.M. (1966). Opredelenie maksimalnogo stoka
poverhnostnyh vod s malyh vodosborov. (Determination of
maximum surface runoff on small watersheds) Hydrometeos
raological Publishing House Leningrad.
11. Sokoiov A.A. (1963) Maximalnyi stok talyh vod elementarnyh
bassainov i priroda ego reduktsii. (Maximum snowmelt
runofi on elementary basins and the nature of its reduction).
Transactions of GGI, vol. 107.
12. Sokolov A.A. (1966) Metodika rascheta maximalnyh raskhodov
talyh vod pri otsutstvii ili nedostatochnosti gidrometricheskikh
dannykh (Computation of maximum discharges of
snowmelt water in case of the absence or inadequacy of
hydrometric data) Transactions of GGI, vol. 134.
13. Sokolovsky D.L. (1937) Normy maximalnogo stoka vesennikh
pavodkov rek SSSH i metodika ikh rascheta. (Norms of
maximum spring flood runoff of the USSR rivers and the
technique of their computation). Hyarometeorological
Publishing House.
14. Sokolovse D.L. (1948) Metodika postroenia hydrografa liv-
nevogo stoka PO osaäkam (Plotting of rainfall runoff hydro-
graph on the basis of rainfall aata). Transactions of
GGI, vol. 14.
15. Sokolovsky D.L., Shiklomanov I.A. (1965). Haschety hydrografov
pavoàkov s primeneniem elektronnyh modeliruyshchih
UStroiStV. (Computation of flood hydrographs by means of
electronic modelling devices). Transactions of LGMI,
voi. 23.
16. Sokolovslry D.L. (1968). Hechnoi stok, (River flovi). 3rd
edition, Hydrometeorological Publishing House, Leningrad.

623
17. Stroitelnye normy i pravila. (19661,Chast II, rasàel II,
glava 7. Raschetnye maximalnye raskhody vody pri proektirovanii
gidrotehnichesmh sooruzheniy. (Norms and
instructions for civil engineering. Part II, section
II, chapter 7. Maximum design discharges for hydrotechnical
structures). Normy proektivania (CH i II
II - 4. 7-65). MOSCOW.
18. Mezhdynarodnyi simposium PO pavocikairi i ili raschetam.
(1969). (International symposium on floods ana their
computation) I and II, Leningrad, 15-22, August, 1967.
Hyàrome teorological Publishing House, Leningrad.
19. Ukasania po opredeleniu raschotnyh maximalnyh raskhodov
talyh vod pri otsutstvii ili nedostatochnosti gidrometricheskih
nabludeniy. (1966) (Instructions for
the determination of maximum design snow melt discharce
in case of the absence or inadequacy of hyarometric
data). CH 356-66. Hydrometeorological Publishing
House, Leningrad.
20. Ukasania po opreaeleniu raschetnyh hydrologicheskih kharac-
teristik, CH 435-72 (1972). (Instructions for the esti-
mation of the hydrological design values CH 435-72).
Hydrometeorological Publishing House, Leningrad.
21. Ukasatel literatury PO pavodkam i ih raschetam (1967)
(Bibliography on floods and their computation)
Hydrometeorological YuDlishing House.
22. Chegodaev N.N. (1953). Haschet poverhnostnogo stoka s
mlyh vodosborov. (Estimation of surface runoff from
su.311 watersheds). Tranzheldorizdat.
23. Shiklomanov I.A. (1964). Kaschet transformatsii pavodkov
vodokhranilishchami i prudami pri pomoshchi electron-
nogo modeliruyushchego ustroistva (Computation of
flood transformation by ponds and reservoirs by means
of electronic modelling devices). Transactions of LGMI,
vol. 26.
24. Alexeev G.A. ana Sokolov A.A. General principles and
methods for the computation of flood discharges applied
in the USAR. Atti del convegno internazionale (Roma,
23-30 November 1969). Roma, ANDL! pp. 735-747.
25. Estimation of maximum floods. Technical Note No. 98.
LNO v No. 233, TP. 126, T;MO, Geneva (1969).
26. Floods and their computation. Proceedings of the Leningrad
Symposium. August, vol. 1 and 2. UNESCO/USH (1969).
27. Flood studies: an international scuiae for collection ~ and ~~-..
processing of data. Technica1"papers in hydrology No.8,
UNUSCO. Paris (1971).
28. Gray D-M. -Synthetic-Unit-Hydrographs for Small Watersheäs.
Proc. Am. Soc. Civ. Eng., vol. 87.
29. Linsley R.X., Kohler M.A. and Paulhus L.N. (1949). Applied
Hydrology McGrow Hill Book Company N.Y.
30. Morgan and Johnson (1962). Analysis of Unit-Graph Method.
Journal or Hydraulic Division, 88, NY-5.
31. Sherman L.R. (1932) Stream-Flow Rainfall by Unit- Graph
Method ¡in . Mews Record.
32. Snyder F.F. f1938) Synthetic Unit-Graphs. Trans. Am.
Geophys. Union, vol. 19.

ABSTRACT
COMPUTATION OF PROBABILISTIC VALUES
OF LOW FLOW FOR UNGAUGED RIVERS
Vladimirov A. M., Chebotarev A. I.
State Hydrological Institute
Leningrad, U.S.S.R.
The main characteristics of low flow (minimum daily, mon-
thly and seasonal flows) are investigated. The computation methods
are Sased on a combined use of geographical interpolation and pro-
bability analysis and considering the main factors affecting the
volume aiid regime of low flow. Principal characteristics of low
flow for mkdium-size rivers are determined by maps of flow isoli-
nes, those for small rivers are determined by regional empirical
correlation. Design flow is established by means of transition
coefficients. The principal computation methods discussed are deve
loped for U.S.S.R. rivers.
Les auteurs analysent les caractéristiques principales des
débits de basses eaux (journaliers? mensuels, minimum saisonnier).
Les méthodes $e calcul font appel a la fois à l'interpolation géo-
graphique et a l'analyse statistique, compte tenu des facteurs
principaux qui influencent l'abondance et le régime des débits de
basses eaux. Les principales caractéristiques des débits d'étiages
des rivières moyennes font l'objet d'une représentation cartogra-
phique; pour les petites rivières on les traduit par des relations
empiriques régionales. Les débits correspondant à la fréquence
choisie pour le projet sont établis en utilisant des coefficients
de transfert. Les auteurs présentent les principales méthodes de
calcul utilisées en URSS.

2 6
Low flow is one of the principal phases of the hydrological river
regime. During the dry periods, when precipitation is usually at its
lowest, rivers have rather stable and relatively small discharges. Their
variations in the flow hydrograph tend to approximate a horizontal line.
The lowest flow observed during a certain period is generally called the
minimum flow during that period.
Separation of low-flow period in river flow hydrographs
On rivers with distinctly expressed spring snowmelt floods and
autumn floods the period of low flow is observed during winter and summerautumn
seasons. Its beginning in summer is determined by the end of
spring high-water period, i.e. when the intensive rate of decrease of
discharge tends to become smaller. The summer period of low flow ends with
the arrival of the autumn floods or the appearance of ice in the river.
In the latter case the low-flow period is called the summer-autumn period.
The winter low-flow period begins at the appearance of ice events
in the river and continues until spring high-water period begins. In
case of no ice phenomena in the river the winter low-flow period is
assumed to be a period from the average data of air temperature falling
down contii:vously through O°C and below it up to the beginning of the
spring high-wcter period.
The low-flow period includes also floods if the volume of each
of them does not exceed 10-15 per cent of the flow volume for preceding
and subsequent lol.:-flow periods, without taking into account the volumes
of floods already included. If the flow hydrograph has the form of a
saw-like curve (frequent floods of various magnitudes), the period of
low-flow includes floods with maximum discharges that are 3-5 times
greater than preceding daily minimum discharges (depending on the
volume of the flood peak).
These criteria facilitate the plotting of
river low-flow periods, although they slightly overestimate the volume
of low flow.
In the U.S.S.R. low flow may be expressed as minimum daily,
minimum monthly (30-day) or minimum seasonal flow.
Seasonal flow is the average value of discharge (specific discharge)
for winter or summer-autumn seasons. The minimum monthly flow
is the average during the lowest calendar month in the given season.
On rivers with flood regime during winter or summer-autumn seasons when
the low-flow period is of less than two month duration or is interrupted
by large floods, the smallest of the average discharges during a calendar
month may appear 1,5-2 times bigger than the minimum discharge. In such

627
a case it is necessary to introduce a temporary correction taking into
account not a calendar month, but a 30-day period with the lowest flow.
If frequent and considerable floods make it difficult to find out the
30-day period of minimum flow, it may be reduced to 25-23 days in order
to exclude the influence of floods.
This secures the genetic homogeneity
of the minimum flow of years with varying water volumes, which is impor-
tant in the determination of minimum flows for ungauged rivers.
Physiographic factors of low flow
The duration and volume of low flow depend on physiographic
factors which may be divided into two groups: (1) climatic conditions
and (2) factors of underlying surface.
lhe water resources of a certain basin depend on the climatic
conditions prevailing in that basin. Precipitation contributes to the
increase of ground water supply, while evaporation decreases its recharge
and supply. In winter low-air temperatures cause a considerable freezing
of soils and subsoils and contribute to the decrease of underground
flow into rivers. Climatic factors determine areal low-flow distribution
in accordance with the low of geographical zonation.
1.i some cases and for small rivers particularly, low flow is
greatly intluenced by local (azonal) factors of the underlying surface,
i.e. the surface and underground flow contributions (lakes, swamps, soils
and subsoils, karst, etc.).
The influence of these factors may be so considerable and
exceeds the influence of the climatic conditions.
The most essential factor is the permeability of soils and subsoils.
They serve as underground flow reservoirs, detaining water during
high-water periods and releasing it during low-water periods. The
capacity of underground storage is determined by the geological structure
of the area and its hydrogeological conditions. Loose and porous or
crevassed deposits (sandstone, limestone, shingle, and the like) create
favourable Conditions for underground storage of water and far its subsequent
release during the low-flow periods in rivers. Solid clay or
monolithic crystal rocks (granite, gneiss) near the surface decrease the
regulating capacity of the storage and reduce the low flow. The influence
of karst-affected rocks on the regime and volume of low flow is determined
by their absorption capacity und the rate of water yield - the bigger
it is, the less is their influence an the low flow.

628
The contribution of underground water reservoirs to the flow in
rivers is a factor of extreme importance in the study of low flows. In
this respect due consideration should be given to the number water con-
tent and regime of aquifers contributions to river flow and the dynamics
of underground flow into rivers. These factors determine the contribu-
tion of underground aquifers to the flow in rivers.
The study of factors affecting the volume and regime of low flow
is a necessary prerequisite for the successful development of the com-
putation methods.
The analysis of the influence of main factors on conditions of
low flow formation necessitates the division of rivers into small and
middle-size rivers when developing computation methods since the process
of low flow formation is different for the two types of rivers. Large
rivers are not considered here.
Differentiation of small and middle-size rivers
The quantitative characteristics of a small river may be assumed
to be the value indicating the extent of aquifers discharge contribution
to total flow, i.e. the erosion depth of river channels. The determination
of its îharacteristic is the ratio between the erosion channel depth
and the aquifeis depth feeding the river along its length up to the outlet.
Th.e quantitative estimation of the influence of main hydrogeological factors
on low flow is difficult to make, while developing low flow computation
methods involve additional characteristics: correlations between
the capacity of underground storage and drainage densities. In similar
regions there is a definite relationship between the volume of underground
storage, river channel erosion depths, watershed boundaries and
drainage areas.
Therefore the value of river basin area provides an
integrated indicator of morphological and hydrological conditions of low
flow.
In this case, a criterion for the term "small river" may be the
largest (critical) area of the basin responsible for the complete drainage
of aquifers feeding the river and with the enlargement of which no varia-
tion of low flow modulus is observed. The value of the critical area is
established by graphs of relationship of minimum 30-day flow modulus with
river basin area for the physiographically similar regions.
For the U.S.S.R. rivers, the critical area of the basin ranges
from 1, O00 to 1,500 km2 in flat wet regions and in all mountain regions.
In semi humid zones it rises to 2, 000-2, 500 km2 due to the lower depths
of uquifers drained by rivers. In semi arid areas rivers with 5,,000-
10, O00 km2 basin area are classified as small rivers.

Computation of normal low flow of small rivers
In the U.S.S.R. computation practice for determining mean low
flow of small ungauged rivers, the following equation relating the dis-
charge to the river basin area is most widely used:
where Q is discharge (seasonal minimum) in m3/sec; A is river basin
area in km2; f is either a regional impermeable mean area or a
permeable contributing area outside the drainage basin. In the first
case, the parameter f has the sign minus (-), in the second case it has
the sign plus (+). Under usual conditions and permanent flow available
f = O. a, n are regional parameters characterizing conditions of low
flow formation.
629
The determination of the parameters of design equation (1) is made
for the regions selected on the base of a careful study of hydrogeological
conditions of the basins under study and on the analysis of principal
physiographic conditions. For instance, while dividing the territory of
the U.S.S.R. into regions the following were used:
water beating formations by rivers, hydrological descriptions of conditions
favouiing the formation of underground flows of regions, ground flow
map of the intcnsive water exchange zone, map of underground flow in percentage
of the total river flow and coefficients of underground flow in
percentage of precipitation.
precipitation for warm and cold seasons, data on evaporation, air temperature
for the winter season in ice melt regions, topographic map of the
U.S.S.R., hydrological regionalization of the U.S.S.R., map of physiogra-
phic regionalization of the U.S.S.R.
account as much as possible all the characteristic features under which
low flow of selected regions is formed. The boundaries of regions with
similar low-flow conditions during winter and summer-autumn, were plotted
along the boundaries of sharp change of hyd2logical conditions. For
instance, when in some river basins the change of hydrogeological and
other conditions take place, the change in the volume of river flow will
not be observed immediately, but gradually while the most notable change
will take place at the confluence of two rivers.
map of drainage of
Also used are maps of annual river flow,
All these allowed to take into
In this case the
region boundary follows the watershed divide between these river catchments
across the point of their confluence.
Formula (1) may be used for the computation of flow of flat and
semi-mountainous rivers with the average accuracy of 152% (for 1 500
points on the U.S.S.R. rivers the deviation of computed minimum mean
long-term 30-day discharge was 17-2w of the actual flow for the summer-
autumn season, and for 750 points in winter it was 15%). Taking into

630
account the accuracy of determining the actual data, use of formula (1)
may be recommended for the computation of low flows of rivers of basin
areas not less than 20 km2 for humid zones not less than 50 km2 for
semi-arid zones, where the low flow volume is rather small and the in-
fluence of various local factors is most evident.
In regions with very
complicated conditions of low-flow formation the area should be not less
that 100 km2.
A wide use of formula (i) in the designing practice (5, 6) paoved
it reliable.
In high mountain areas the altitude of the catchment may be of
a great significance as the factor reflecting the influence of vertical
zonation upon the conditions of low flow formation. Therefore, the low
flow modulus for regions similar in hydrogeology etc., is related to
mean basin altitude.
Determination of low flow for middle-size rivers
The low flow volume of middle-size rivers, i.e. those with area
larger than the above stated critical area, but not more than 75 O00 km2,
is formea under principal influence of zonal factors. The flow modulus of
these rivers varies smoothly and in accordance with geographical zonation
(1-atitudinal or vertical) over the area. Therefore, low flow of middlesize
rivers can be determined by maps of flow isolines, made for a certain
characteristic of low flow. The flow modulus relates to the catchment
centre, the interval between isolines is given in accordance with the map
scale and the value of flow variation over the area. In mountain regions
the average catchment altitude is taken into account; flow isolines may
not be closed, but end on the side of the mountain ridge without passing
over to the other side (due to a great difference in wetness of slopes).
Maps are plotted both for the mean and for flows of various frequencies.
For instance, for the U.S.C.R. territory there are plotted maps of the mean
and S-frequency of minimum. 30-day winter and summer-autumn flows,
which allows to determine the flow with the average accuracy of 10-20$.
Computation of low flow of different frequencies
For designing purposes the characteristics of low flows of different
frequencies are of the highest importance. In the U.S.S.R. design
flow of 75-97$ frequency is mainly used. Necessary values may be determined
with the help of three parameters: mean flow La), coefficient
variation of !Cv ' and skewness coefficient (CS).

631
The second way is by the use of a transition coefficient from one
fixed frequency (e.g. 75 or 8%) to another. This method has been lately
more and more widely used in the U.S.S.R., especially in its application
to low flow, since it is more accurate and simple than the method of three
parameters, and since available hydrometric data allow to generalize for
almost the whole territory of the U.S.S.R.
The advantage of the transition coefficients method is proved
by the mere fact thót in this case the total mean square root error will
consist of the error of the flow of fixed frequency (u 1 and the error
'P
of transition coefficient A , i.e.
terms:
i.e.
When using three parameters, the same error will consist of three
standard error (Qn), error of Cv (c($ and error of Cs (, Cc,),
Tt is evident that the error in the second instance will be
greater, and if we take into account unreliable methods for determining
coefficients C,, and Cs for any rivers, then the advantages of the method
of transition coefficients become quite obvious. It is the more so,
as in the range of frequencies under consideration (7597%) the curves of
low-flow frequencies are rather stable, gently sloping and quite reliable
in most cases.
This stipulates a rather small (for the given frequency)
variability of transition coefficient for the area and season and, con-
sequently, its high reliability. Thus, for the U.S.S.R. rivers the value
of transition coefficient from the minimum 30-day discharge of 8% fre-
quency to the discharge of 75% frequency varies from 1.03-1.06, and for
transition to the discharge of 9056 frequency - from 0.83-0.91, i.e. the
value of coefficient xchanges only by 5-1056 and may be averaged for the
given frequency over a large area. Its value varies significantly only
for episodically drying or freezing rivers.
The flow of fixed frequency is established by formula (1) or by
maps of flow isolines, plotted for this frequency. Thus, for the U.S.S.R.
territory there are plotted maps of the minimum 30-day flow of 8056 frequency
(for winter and summer-autumn periods separately) and the maps of
flow of limiting season of 75% frequency.
Also determined are the
parameters in formula (i) for the 30-day discharge of 8w and 7546 fre-
quency.

63 2
To determine flow of other frequencies, a table of transition
coefficients ;I has been prepared.
Computation of minimum daily flow is made by relationship with the
value of minimum 30-day flow (normal or fixed frequency flow for selected
regions) :
Q = K , Q
P p,30
where Op is minimum daily discharge of design frequency. Qp30 is minimum
30-day discharge of corresponding frequency, determined by maps of isolines
or by formula (1). k is the regional transition coefficients for the
given season.
For the U.S.S.R. territory the value of coefficient k, when
determining the minimum daily discharge of 8C$ frequency, varies from
0.59 to 0.90 in winter and from 0.45 to 0.86 in summer-autumn seasons.
Its value depends on the degree of river flow depletion for the period
under study and on the volume of runoff during low-flow periods.
Determination of minimum daily flow of design frequency is made
by using ti:s above-mentioned coefficients A , since the frequency curve
of daily and 30-day discharges vary practically in the same manner.
The stated methods for the computation of probability values
of river low flow are given in Gosstroy Standards of the U.S.S.R.
/5,6/ and are widely used by designing organizations of the Soviet
Union.

REFERENCES
1. Vladimirov, A. M.: Minimalny stok rek SSSR (Minimum flow
of the U.S.S.R. rivers) Hydrometeorological Publishing
House, Leningrad, 1970, p. 214.
2. Vladimirov, A. M.: Raschetnye minimalnye raskhody vody
(Design minimum discharges) Trans. of GGI, v. 188,
Hydrometeorological Publishing House, Leningrad,
1972, p. 244-272.
3. Kudelin, B. I. (ed.): Podzemny stok na territorii SSSR
(Underground flow in the U.S.S.R. territory), MGU
Publishing House, 1966, p. 303.
4. Popov, O. V.: Podzemnoe pitanie rek (Underground river
recharge) Hydrometeorological Publishing House,
Leningrad, 1968, p. 291.
5.
6.
633
Ukazania PO opredelenia raschetnykh minimalnykh raskhodov
vody rek pri stroitelnom prooktirovanii (Instructions
for determination of design minimum discharges of
rivers in engineering projects). CH 346-66, Hydrometeoxdogical
Publishing House, Leningrad, 1966, p. 17.
Ukazania PO opredeleniu raschetnykh gidrologicheskikh
’ kharakteristik (Instructions for determination of
design hydrological characteristics), CH 435-72.
Hydrometeorological Publishing House, Leningrad, 1972,
p. 18.

ABSTRACT
A STUDY ON MAXIMUM FLOOD DISCHARGE FORMULAS
Tae Sang Won, PhD.CE., Dr. En.${
This paper describes a new formula for the calculation of
approaching velocity of rain water, and a number of new formulas
for the estimation of maximum flood discharge which have been deve-
loped by the author.
Many empirical formulas, which have limited application,
exist. However, in devising his formulas, the author derived theo-
retically the form of the basic maximum discharge formula for the
case of rivers with no tributaries, and determined stochastically
the value of the coefficients in his basic formulas using the re-
cords of observed measurements. Then the author derived theoretic2
lly many differe,it formulas for the case of rivers with tributa-
ries to fit in the actual localities of the site under considera-
tion, besides the basic formulas. So the author's formulas would
be widely applicable for rivers or sewer nets, and also for any
regions, countries, with different locality. The author could con
firm these facts through the numerical examples. The author's fo:
mulas may be used not only for estimating the design flood, but
also in flood routing. The author believes that his formulas would
be very helpful in the planning of water resources development pro
jects -specially for those with inadequate data.
RESUME
L'auteur présente une nouvelle formule pour la vitesse de con
centration d'un bassin et en suggere d'autres pour le calcul du dg bit de la crue maximale.
On trouve de nombreuses formules empiriques dans de nombreux
manuels, mais ces formules sont d'une application limitée. L'auteur
parvient cependant à asseoir la forme de sa formule sur des bases
théoriques, lorsqu'il s'agit de cours d'eau sans affluents; il pro
cede à l'évaluation des paramètres qu'elle contient par ajustement
statistique aux données d'observation disponibles. I1 generalise
ensuite à différents cas de cours d'eau avec affluents. Les formu-
les proposées de'vraient pouvoir être appliquées n'importe où,
aussi bien pour les cours d'eau naturels que pour les réseaux
d'assainissement; c'es ce que l'auteur peut confirmer par des
applications numériques. Les formules peuv:nt servir non seulement
au calcul des crues de projet, mais aussi a celui de la propagation
des crues. L'auteur pense que ses formules devraient rendre de
grands services dans la planification de l'aménagement des eaux,
spécialement lorsque les données disponibles sont insuffisantes.
fg Professor of Civil Engineering, Seoul National University, Seoul,
Korea.
1

636
I e XNTR001';TION
a
Charles F. Ruff defin& Bmximm probable floodn as follows.
"The maximum probable flood does not mean the largest flood possible
but a flood so large that the chance of its being exceeded is no
greater than the hazards normal to all of man's activities." The
author will use here the term of 'maximum flood discharge" with the
same meaning of "maximum probable flood" as defined by Ruff.
It is very important to calculate maximum flood discharge cor-
rectly, and also it i5 a very difficult problem theoretically and
practically. It m y be impossible to establish a plan for flood con-
trol an8 water resources development or sewer nets projects without
reckoning correctly the maximum flood discharge or the design flood.
There are many methods for calculation of maximum flood dis-
charge, and we have to adopt the most suitable method in accordance
with the completeness of the data. However,the method of calculation
by the maximum flood discharge formulas,especially for the case of
those with inadequate data, is easy and simple for practicing engi-
nßers. There are many empirical formulas devised by many authors
such 88 Kuichling,Mead,KresnikeDickens,Metcalf and Eddy,Brix,Lauter-
burg,Possenti,Buerkli-Ziegler,Dr.Hisanaga,Kajiyama,and many others.
These old fosmulas have been devised empirioally and have limited
application. It will be clear that one may be unable to apply them
generallj.. Aliso it will not be strange to obtain results which may
be 10 or liio time8 of the correct values, according to selection OP
the coefficieqts in these formulais when these formulas are actually
applied to practical problems.
Generally speaking,the flood discharge depends upon the shape
of catchment,drainage area,amount of rainfall and the position at-
taoked by the heavy rainfall,pemneability,slope of the catchment,
shape of the water cours6,status of the surface,geological status,
etc. Strictly epeaking,such statua of catchment differs from others
from se68on to season,for every floo8,even in the same catchment as
well as in different draimge basins. In other words,flood disoharge
üepends also upon the inteneity of Fainfall which causes the flood,
duration of the rainfall and the position of the oater of the lows,
or statua of the ground in case of heavy rainfall,vie.,dry ground or
saturated qround,etc.
As the maximum flood discharge depends upon many factors, as
stated above, it may be very difficult to express it in a formula.
However,if we can consider theoretioally correct value of approaching
velocity of rain water and intensity of rainfall, we may deduct
the maximum probable flood by getting the rainfall for a certain districtc
The principle of derivation of the author's formulas belongs
to this process, and it may be said that this is an approach differe.it
from many scholars who had derived the old formulas.
~
* Ruff,Charles F.;nMaximuni probable floods in Pennsylvania Streamst'
Transactions ,American Society of Civil Engineers .Vol .i06,1g4l ,p . 11 53

637
In the first step, the author thought out a method to ascertain
correctly the approaching velocity of rain water. At the same time,
the author found that the Rizha's (Germany) formula,the only complete
one for this purpose, could not be applicable to solve practical
problem8 as it gives too small values. In the second step, the author
studied the rainfall intensity curve comprehensively, and found out
theoretically when the maximum flood discharge may occur. In the
third step, the author has theoretically derived the maximum dia,
charge formulas for rivers with many tributaries by appljing the
general rules which he has determined by the first and second step.
In the fourth step, the author determined the discharge coefficient
in his formula from the actual records. The author was then able to
calculate the value of the discharge coefficient, with a great degree
of acouracy, of the rivers in Korea and Manchuria.
II. APPROACHING VELOCITY OF RAIN WATER
The approaohing velocity of rain water (U) is defined as the
mean velocity of rain, water approaching from the farthest point F in
a river basin to the point O where the maximum flood discharge is to
be ascertained,in other words, the mean velocity of flow between F
and O (Fig-1). There is only one formula to find such approaching
velocity so far expressed in equation, given by Rizha,Germany, and a
table given by Kraven,Germany.
1) RIZYA'S FORMULA
0" 72 So'' (1)
where
a= Approaching velocity of rain water (km/hr)
s = H/L
H = Difference of elevation of height between O and F
L = Distance of OF (Length of water course)
2) KRAVEN'S TABLE
- C above 0.01 0.01 0.005 below 0.005
w (kln/hr) 12.6 10.8 7.56
Kraven had expressed only about approximate limite of Cd , the
author tried to formulate his table, to pass through the medium
points as follows.
3) THE AUTHOR'S FORMULA
The author succeasfully devised a new method to determine the
approaching velocity of rain water e3 ,theoretically which may be
applicable for rivers where the hydrographic surveying was completed.
Neglecting the principle and the process of derivation here, the result
of the author's formula is illustrated as follows.
(2)

638
The value of for rivers in Manchuria calculated using eq(3) is
shown in Table-i. As we see Table-1, the value of lies between 133
and í77, and we may be able to recognize that the Rizhass formula wil
not be of practical use because of the reason that the salue of iCt in
his formula is too smll,apparently,compared with the normal salues.
ide can also see from Table-1 that the value of K represents the
bottom slope OP the hydraulic siope of the river, and this fact coin-
cides with practice. Also it can be seen that the value of &, de-
ureases gradually according aa approaching the downstream of a river,
and this fact skok's that the bottom slope or the hydraulic slope of
a rivep generally decreases gradually as we approach the downstream.
III. RAPFALL INTENSITY CURVE
h'e can express the rainfall intensity by the following equation.
Table-I. The values of & and others fop rivers in Manchuria
Name of Rivers
/c Range of S
(400)
Tumen 8.
277 4.82 - 8,06 33,400 487.6 68.50. .LW
Whancheng R.
207 6.72 - 9.64 4,000 160.7 24.92 .l52
Rohe H.
177 3.54 - 4.22 31,455 444.0 70.84 -159
Seasamorin R.
171 2.81 - 2.97 29,927 412.0 72.64 .i76
Sealeog R.
149 8.37 - 3.48 510165 767.0 66.71 .O86
Tongleog 3.
207 1.70 - 3.16 10,318 33345 313.13 -093
The upatrean of the 150 1.83 - 2.30 178,699 . 1040.5 171.74 .O65
min Lacg R.
The middle of the Leog 158 1.62 - 1.77 187,250 1199.0 157.49 0132
GhsnF: R.
159 3.90 - 5.51 4,958 163.0 30 . 42 .i86
Icwaslg R.
137 4-36 - 5.26 2,129 94.0 22.65 231
Van R.
149 7.07 -18,66 1,072 102.5 10.46 .lo2
Pa B.
150 3.68 4.06 2,361 178.0 13.26 e 074
Csnkai R.
134 1.43 - 1.96 515 51.0 10.10 .198
F:catchmtmt area
I = ß/(t + 1 ( 4) L:length of main water oourse
wher0
t= Bwatim
I = Average intensity of rainfall during duration t
OC,,^ = Any-constant
Eq(4) represents a kind of hyperbola, and the constants a ar3.B can
be found by eq(5) by the principles of the method of the least squame
n(12t) - (ï)(ït)
d= LI)' - n(I')
B=
(Il(P2t) - (It)(12)
(I)~ - n(I')
E= nwnber of observations
Next let R be total. amount of rainfall aurin$ the buration t,
R = It = p t/(t + QL 1 (4)

639
T~ble-2 illustrates the values of the constants d and fi in eq(4)
for various regions. In this table, those for the regions of Korea
and Manchuria show the absolute maximum rainfall intensity curves
during those periods, Those for the regions marked with the asterisk
(*I were calculated by the author himself by the records of the re-
cording gauges.
IV. THE AUTHOR'S MAXIMUM FLOOD DISCHARGE FORMULAS
1) FUNDAMENTAL FORMULA FOR THE CASE OF A RIVER WITH NON-TRIBUTARY
(a) Retardation of Run-off
Table-2. The values of cc and P in eq(4)
Region d a
(min) (hour)
P b the records (minutes)
Period taken Range of t
Seoul ,K 59 0.938 7,860 131 1905 - 1920 5-60 min
Inchon , K 37.5 0.625 8,640 144 ditto 5-240
PymgYanR , K 41 0.683 6,000 100 1914 - 1920 ditto
Pusan , K 106.1 1.77 14,015 233.6 1914 - 1953 10min-24hr
Wonsan , K 75 1.250 7,740 129. 1914 - 1920 5-240 min
Taegu,K * 40.2 0.67 8,711 145.2 1929 - 1953 10min-24hr
Chonj:i,K * 81.1 1.35 15,160 252.7 1918 - 1954 ditto
Kwangju,K * 90.4 1.51 10,866 181.2 1938 - 1954 ditto
Mokpo,K * 101.8 1.70 11,398 190 1916 - 1953 ditto
ChmgCheng,): * 40.5 0.675 5,929 98.8 1937 - 1943 10min-48hr
Sping,r? * 45.1 0.752 8,487 141.4 i934 - 1944 ditto
Tokyo, J 50 0.833 5,500 91.6 1891 - 1911 5-60min
where a d/60 b P 1'3 /60 KæKorea M=Manchuria JsJapan
Prior to deecribing the flood discharge formula, the definition
of "retardationR must be understood. Now let O and F be the point
under coneiäeration and the farthest point of a catchment respective-
ly, 1 be the length of water course between O and F , CL) the approaoh-
ing velocity of rain water flowing from F to O, tc the time of con-
centration,i.s.,the time necessary for reaching O from F, tr the du-
ration OP rainfall,i.e.,the perlob between the beginning and enâing
of a reinfall (see Fig-i), then,
t, = l/Ld (79
T tr + t c * tr + l/W í 8)
where T P The time of the period between the beginning of a rain-
fall and the ending of the run-off due to the rainfall
at the point O
It may be better to use the author's formula for determining tc
When it becomes tr

640
falling at F reached O, the rainfall would have ceased. In other
words, the rainfall causing the maximum flood discharge is the rain-
fall which falls in a part of the catchment.
(b) Fundamental Formula for the case of non-Retardation
The fundamental principles of the author's maximum flood dis-
charge formulas have already been described. The author adopted de-
ductive and inductive theories for the derivation.
Since it is unable to derive the rational equation of the flood
discharge hydrograph, the author expressed the peak of the flood
discharge hydrograph for the case of non-tributary and non-retardation
by the following equation.
- T = Duration of flood tr+ tc
qm- 9. a CfA R / T (9)
where
qm = Peak discharge in flood time
qo = Discharge of run-off in normal time
tr = Duration of rainfall
tc = Approaching time or time of concentration
R = Total amount of rainfall during duration of tr
A = Catchment area
9 = Average run-off factor
C P A coefficient depending upon the shape of the flood
discharge hydrograph
Now let Fig-,? show a discharge hydrograph during a flood period.Then
the, peak disoharge qm-qO may be represented by eq(9)and the product
AR of eq(9) shows the total run-off during the period of flood
T. As this also represents the area of DMED of Fig-2 geometrically,
we may affirm that the authorls fundamental formula is reasonable
anal tically or graphically. Replacing the value of R of eq(6) into
ea(9 9 ,
qm - qo=C 9 AB / T = C 9 A b tr /(tr + t, )(tr+ a (10)
We h ow through eq(1O) that the peak discharge is a function of tr,
anä it will take a limiting value to make the peak discharge maximum.
So differentiating eq(l0) with respect to tr
and.
tr = (unit in hours) (11)
T=t,+ tG= 6 +t, (12)
Hence we know that the maximum flood discharge will occur when the
duration of rainfall t r satisfies eq(l1). Up-to-datesue have taken tr
generally without definite reason as follows: 5 or 10 minutes for de-
sign of sewers, 3 or 4 hours for small rivers flowing the vicinity of
a city, 24 home or more for big rivers. However aceording to the
author's theory, the value of tr must satisfy eq(l1) to cause the
maximum flood discharge. Substituting eq(l1) into eq(lO),

641
If we express in metric units,i.e. ,A(km2),R(~) ,t (hr) ,q (cms), (13)
becomes,
qm-q,= 0.2778C 9 b a A / (t,+fic)( a+
1
where (14)
a,b = any constants depending upon rainfall (see Table-2)
The value of tc can be found from eq(3) .(see Table-1)
(c) The value of the coefficient C
As stated above, the coefficient C in eq(9) depends upon the
8hape of the discharge hydrograph of the region . The relation be-
tween the kinds of the cuPve consisting the discharge hydrograph and
the value of C is illustrated deductively as fOllOW6.
Table-3. The value of C found by deduction
Kind of curve C kind of curve C
parabol a 1.5 cosine curve 2.0
triangle(strai ht 2.0 probability 2.394
1 ins? curve
Also the value of C can be calculated inductively from a dis-
charge hydrograph by using eq(91,whioh gives,
C 0 (qhi-Qo)T /$ARa(Qm=Q,)T/ V
(15)
nhere
V = The volume of run-off represented by the area DMED of Fig-2.
The value of C for the rivers in Manchuria found by the author using
eq(l5) are given in Table-4.
Table-4. The value of C for Manchrian rivers found by induction
Name of river Site of Duration of flood taken Value Value
measurement from the records
of' T of .c
( hr , day-hr ,day, month, r
Tongleog Ho Tidathergtse 15,3rd-21 ,4th,Aug,l9ii 30 1.664
n
Sankankeu 12,10 .e 5,21 ,Sep,1939 257 1 .697
Whan Ho P eidakeng 15,24. -1 9 , 27, AUg,l94O 76 1.543
Main stream C hengs enkong 19,2nd- 6,6th,Sep,1939 83 2.090
of Leog Ho
ditto
ditto 797th- 7,103 Se~r1939 72 1.966
Taitse Ho Whelongbo 12,31Jul-3,3~,Aug,1940 63 1.754
n
n
9,4th-i7,6th,A~g,l 940 56 1 0975
I
n
16,6th-l9,8th, I
51 1.087
n
n
17,2nd- 8 s 5thSSQp ,1939 63 2.137
n
H
9,jth-l6,9th, " " 103 1.806
n
5,6th-l3,9th, Jul , 80 2 . 204*
* show6 the value oalculated by estimation because of non-measurement
at the vicinity of the peak discharge.
(d) The fundamental formula for the case of retardation of flow

642
The basic formula for the case Of non-retarclation ,mentioned
above, is applicable for the case of retardation of flow,too. But it
is necessary to multiply the coefficientp due to retardation, viz.,
q,-qo=yCpbf& A /(tc+GL )(a+ Gc 1 (16)
p = f(tc/tr) (17)
It is clear that the value of the Coefficient /3 equals to 1 for the
case of non-retardation, but it beoomes less than 1 for the case of
retardation. The value of p varies inversely with that of tc / t, .
It is necessary to find out a general form of f(tc/tr) for
practical calculation. So the author tried to find out the general
form of the function f(tL/ty) atoohastically using some data ob tained for rivers in Korea by some other methods. The author would
like to assume the general form of the function ofp as follows.
J)= (1 + k 1 / ( tc/t,+ k 1
where k = Any constant
Finding the value of k in above equation by the method of the least
squares, we get k = 4.802 . Accordingly,
p= 5.802 / (tc /tr+ 4.802) (18)
2) 'THE MAXIYUM FLOOD DISCHARGE FORMULAS FOR THE CASE OF RIVERS
WITH TRIBCTARIES
(a) The maximum flood discharge st the confluence of a trlbutary
The author found that existence of tributaries affect greatly
the peak discharge of flood flow at the proposed site of the main
stream. Su the author derived many different formulas of maximum dis-
charm for the case of rivers with tributaries, besides the basic
formula for the case of thoee with non-tributary. Therefore it would
be said that this is a great approach different from many scholars
who never considered the Influence of tributaries in their tradition-
al formulas.
KOW assume one of the simplest case as Fig-3. The discharge
hydrograph for this case may be illustrated as Fig-&. The value of q,
in ~ig-4 shows the peak discharge of the triùutaryíI), and the value
of q2 shows that of the main river (II) alone,excluding that of the
tributary(1) , also Qm shows that of the composed maximum discharge
to be occurred at the proposed site. The rational equation of the
curve ,i.e.,the true shape of the discharge hydrograph is unknown.
But the author would like to discuss about the shape of the curve in
the following. Let us consider two cases, one of them the simplest
case,i.e.,the case assumed that the discharge hydrograph consista of
an isosceles triangle, and the other the case assumed that it consists
of a parabolio ourve, to seek the effect of the nature of the
discharge hydrograph which influences on the peak disoharge Qn

(i) 'The case of an isosceles triangle
In this case, it evident from Fig-5,
TE
Um" qp+q,(2 --1
T,
(ii) The case of a parabola
(19)
643
Since it is evident as the nature of the parabola,at Fig-6,
q = 4qot/T - 4q,(t/TI2 í a)
we can get the following equation for Fig-?,
and by dQ/dt = O Q,/TI + q2 /T2
to= 2 (q,/T,' + q,/T:)
Accordingly, substituting eq( 201 into eq( b) , we get
NUMERlCAL EXAMPLE
(20)
An ilìwtration is given here to compare the degree of accuracy
of the two cbses mentioned above.
Given T2 = 26 hr, TI = 20 hr, = 5000 cms, q = 3000 cms . Then since
Te/T,= 26/20 = 1.3 from eq81) , the case of assuming as parabolic
curve, &TA= (5000 + jOOOx1.3
/( 5000 + 3000x1.3x1.3 1 -
7865 cms
Next from eq(19), the straight line formula,
Qm = 5000 + 3000x(2 - 1.3) = 7100 cms
Hence,we h ow that there is not any remarkable difference on the re-
sults of calculation of the maximum discharge whether we assume the
discharge hydrograph as straight lines or a parabolic curve through
this numerical example. Aliso we can imagine that we shall obtain the
similar results with this numerical example even in the cases we
adopt Borne other ourves else than parabola for the discharge hydro-
graph,e.g.,cosine or probability curve. But adopting the oase as-
sumed as a parabolic curve is safer,easier to handle,& reasonable.
So the author would like to suggest those of the parabolic curve as
the general formula in this paper.
(b) The maximum flood discharge formula at the confluence for the
oaee of a river where n-1 tributaries flow into the confluence
(F ig-8 1
If we assme the discharge hydrograph consists of a parabolic
curve, by the srne priciple with that in the previous paragraph, we
(0) The maximum flood discharge at the proposed site which is
located the downstream of a tributary

644
Now let O is the proposed site, O' the confluence of the tributary
in Fig-10, and t, is the necesssry tirne for reaching of rain
water from O' to O. If we assume the discharge hydrograph consists
of a parabolic curve.FiR-11 ,then
Q = Q;,+ qiez4q, (t-t, 1 /TI - 4 qi( t - t , l2 / TF+4 qr t/T+ - 4 kt2 /Ti (
(d) The maximum flood discharge at 8 proposed site where n-1 tribu-
taries join to the main river at its upstream side.(Fig-12)
if we assume the discharge hydrograph consists of 8 parabolic
curve, by the same principle with that in the previous paragraph, we
(e) The ma>;Smum flood discharge at a proposed site where m tribu-
taies flow into this site and n-1 tributaries join to the
main river at its upetream side. (Fig-14)
This is the most general case. If we assume the discharge hydro-
graph consists of a parabolic curve, by the same principles, we get
V. CONCLUSION
The maximum flood discharge generally increase toward âown-
stream, a6 the result of increment of the drainage area. But as the
approaching time also increases approaching down stream,in other
words,ss the nearer approaching downstrem,the greater effect of re-
tardatlon. Accordingly the rate of increment of the peak discharge
decreases generally approaching downstream; and sometimes,i.e.,in
such oa8e18 where the approaching time remarkably increases compared
with the inorement of the drainage area, not only the rate but also
the actual absolute value of the peak discharge decreases at the dom
stream than those of the upstream. These fsots are experienced some-
times in practice, In such cases,it was impossible to expreee this
fa& by the old formulas. However by the author's formulas, it is
ìGYi

645
easy am2 theoretically sound to express this fact. Because as we see
the author's basic formulas-eq( 9)-(14) , which represent the drainage
area A in the numerator and the factor of the approaching time tc in
the denominator. So it may also be said that the author's formulas
are very theoretical from the point of view of this fact.
As mentioned above,the author derived theoretically,i.e.,ration-
ally or stochastically many formulas of maximum flood discharge- the
basic formulas for the case of a river with non-tributary and many
other different formulas for the case of rivers with tributaries. Be-
cause the author found that the existence of tributaries affect great-
ly not only the peak discharge but also the entire shape of the dis-
charge hydrograph at the proposed site of the downstream. Consequent-
ly it may be posaible,by applying the author's formulas,to find the
real shape of the discharge hydrograph at the point under consider-
ation to be occurreti in some flood time.
Some scholars advocate that the actual shape of the flood dia-
charge hydrograph resembles to ~ig-16. On the other hand,some other
scholars insist that it should be resembled to Fig-17. But the author
should say that these theories both advocated by the traditional
scholars are those have not been touched to the core of the true theo-
ries. The real shape of the discharge hydrograph depends upon the lo-
cality Qf the point under consideration,in other words, it depends on
the relat4ve position of the proposed point and those of the conflu-
ences of th? tributaries on the mainstream under consideration. Conse-
quently it is resembled to ~ig-16 in some cases, and also it takes a
shape resembleel to Fig-17 in some casessin accordance with the locali-
ty of the point under consideration. As stated above,it would be able
to show the real shape of the discharge hydrgraph just fitted in the
locality of the proposed site by applying the author's formulas.
The author's formulas also would be applicable not only for the
purpose of reckoning of the design flood, but also for that of esti-
mation of the flood routing for some floods. In the case of flood
routing,it would be possible to obtain more correct results by taking
the real value fop tr instead of that calculated from eq(l1) in some
cases,i.e.,the real value of tr is greatly different from that calcu-
lated from eq(i1).
The author's formulas would be widely applicable for rivers or
sewer nets,& also for any regions,countries with different locality,
and it would be possible to obtain correct and accurate results by
selecting or assuming the values of the coefficieuits in his fQrUIUlaS
appropriately. Aceoräingly the author should like to suggest that the
author's formulas shall be applied in practioe in many regions and
also for many purpose8 as far as possible.

(p
646
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e
c, 3
O
Fig- 7
e
t
-+
O
Fi 9 -2
Fig- 8
Y
ot P
O
Fìg -9
Fij-3
t

647

THE COST-EFFECTIVENESS OF WATER RESOURCES SYSTEMS
CONSIDERING INADEQUATE HYDROLOGICAL DATA
Nathan Buras, Ph.D.
The Lowdermilk Faculty of Agricultural Engineering
Technion - Israel Institute of Technology, Haifa, Israel
Introduction.
The question of how much hydrological information is
necessary for the design of water resources systems has not
been answered satisfactorily as yet. Perhaps this question
does not admit of a unique answer, but rather of a range with-
in which the specific solution to a given situation may be
found .
In general, one can state intuitively that the cost of
a water resources project decreases with the amount of avail-
able hydrological data. For example, a longer hydrological
trace at a given reservoir site will yield improved estimates
of mean annual discharges and of extreme flows, so that the
dimensions of the dam and of the spillway may be reduced for
a given probability of failure during the same period of time.
On the other hand, additional hydrological data irivolve in-
creased cost, not only in terms of more gauging stations and
of the attendant manpower, but also in terms qf 'osts incurred
to the society by delaying the design and the covistruction of
the project until more data is collected and processed. Schem-
atically, one can show these two cost functions as two curves
intersecting in the data-cost space (Figure 1). However, of
practical importance are not the individual cost curves, but
the parabola which is the sum of the two functions. We shall
define, therefore, as adequate hydrological data the amount
of hydrological information corresponding to tho niinimum
ordinate of the total cost curve. This definition impli-s
that hydrological data in excess of this amount are as '.nade-
quate as those which are short of it: indeed, the effort put
in obtaining this additional information may increase the total
cost of the project. For this reason, we recommend the use of
the terms insufficient data for the information less than ade-
quate, and redundant data for the information in excess of the
point of adequacy.
The problem of adequate hydrological data is part of the
broader issue of planning water resource; s@erns. Within this
enlarged context, the hydrological data is but one of the
several planning variables, the others being socio-economic
considerations, organizational and i.nstitutiona1 structures,
political constraints, and so on. The role of the hydrological
data in a complex water resources system was investigated relative
to the water quality in the Potomac estuary [I]. In this
analysis, four planning variables were considered: (a) hydrological
inputs; (b) models of the dissolved oxygen fluctuations

650
in the estuary; (c) economic projections of the region serviced
by the water resources system; (d) water quality objectives in
the estuary. Under the specific conditions of the Potomac, it
was found that the performance of the planned water resources
system was most sensitive to the economic projections, and
least sensitive to the hydrological planning variable (10-
year and 50-year sequences of hydrological data).
Sufficiency of hydrological data.
It does not seem that there is today a generally accepted
method for the evaluation of the amount of hydrological data
with respect to their adequacy for planning water resources
systems. However, the problem was recognized for some time and
several approaches toward its solution were developed. One such
approach, based on the concept of information content of the
observed data [2], is oriented toward the determination of an
opt,imal .letwork of hydrological stations in a region. A some-
what similar approach is based on minimizing the sum of variances
of the estimates of the mean flows at gaging stations in a hydro-
logical network subject to a budgetary constraint [3]. All these
approaches attempt, in fact, to devise optimal strategies of
hydrc logical sampling.
However, when considering the L msequences of inadequate
h,droiogical data on the cost and effectiveness of water resources
Ftrur?l,!res anfi FroJects, it secas thnt the scope of the analysis
Iza> to be broadened. This analysis takes into account not only
ali pcc:;ible sample results, but also computes the expected
worth 3r expected opportunity loss) of a strategy which assumes
that the best decisions (regarding the various components of a
water resouI-ces system - to construrit or not to construct) are
dependent upon the information content of the observed sample.
This approach is called preposterior anal sis [4], because,
&hough carried out before the sample + in ormation is obtained,
it attempis + assesserior probabilities derived on a particular
sapl e rutcome.
A simple example will illustrate the preposterior analysis.
Suppose that the Development Authority of region Aleph is considering
thF construction of a major dam. However, the Authority
wants ils~ to appraise the advisability of obtaining additional
hydrological data thus delaying the planning and implementation
schedule by a few years. It is estimated that total costs involved
in obtaining the additional data, including costs generated by the
non-availability of water and water derivatives at the dam site

651
6
during the additional time period, are 15 x 10 Monetary Units
(in short, 15 MMü). The contemplated structure needs an invest-
ment of 160 MMU, while the present value of the stream of net
benefits generated by it would add up to 200 W.
The Authority has two options:
al: build the dam
a2:
do not build the dam
with the possible outcomes
el: the project is successful
9,: the project is a failure.
On the basis of past experience and with the help of a
firm of consulting engineers, the Authority reaches the con-
clusion that the prior probabilities of success or failure are
p(el) = 0.25
p(e2) = 0.75.
On the basis of the existing data the prior expected
opportunity losses (EOL) can be computed as follows:
Table 1.
Calculation of Prior Expected 0pportimj.ty Losses
a,: build the dam
Probability Opportunity Loss, Wej-giited Oppor-
Out come p(e; - MMu tunity Loss, MNRT
el: success O
û2: failure 150
O
120
m
EOL (u,) -- 120 MMU
a,: do not build the dam
Outcome
Probability
p(e4 - )
Opportunity Loss,
ndMu
Weighted Opportunity
LGSS, MMU
el: success
û2: failure
200
O
50
O
EOL (a,) = 50 MMü
opt EOL = EOL (a,) = 50 MMü
5u

652
Thus, with no additional information, the best decision
would be not to build the dam. In this way, region Aleph would
forfeit only 50 NIMU, the expected opportunity loss.
Now the Development Authority turns to its Hydrological
Service asking its advice regarding the nature and usefulness
of the additional information which may be obtained& the cost
of 15 MMU. The attitude of the Hydrological Service is that. by
and large the additional data would yield one of the following
three types of indications regarding the effectiveness of the
reservoir (in terms of streamflow regulation, hydropower gener-
ation, flood contral, etc.):
X1: increase in effectiveness
X2: no change
X3: decrease in effectiveness.
These variables could have been measured only when projects were
constructed, whether successful or not. Thus, the Hydrological
Service had in its records a set of joint probabilities P(X.1)gi)
as follows:
J
out co1iie
Table 2.
Joint Probabili ties
P( XJW; )
'i x1 x2 x3
0,: project successful 0.20 0.05 0.05
û2: project unsuccessful 0.05 0.10 0.55
- - 7
To taï
To tal 0.25 O. 15 0.60 1 .o0
Of course, the column totals represent the mar inal proba-
bilities of .the usefulness of the additional data: PTX,) = 0.25,
P(X,) = 0.15, P(X3) = 0.60.
The expected value of the information which may be obtained
by the additional hydrological data is reached by means of' a dia-
gram, as shown in Figure 2. The set of probabilities appearing
in the last branches of the decision tree are conditional probabi-
lities p(eiJxj),

653
The amount of 28.9 MMU appearing at the node (a) in the
decision tree represents the expected opportunity loss if it is
decided to obtain additional hydrological information and if
optimal decisions would be made on the basis of the new data.
Comparingbhis amount with the 50 MMU obtained under Itno additional
data" policy (Table I), it appears that it is worth spending
50.0 -'.2&9 = 21.1 MMU in getting more hydrological information.
The difference between the outcomes of the two policies is called
the expected value of sample information. The expected net gain
of sample information is 21.1 - 15 = 6.1 MMU, i.e., the expected
value of the sample information exceeds the costs incurred in
obtaining it. The Development Authority concludes, on the basis
of preposterior analysis, that it is worthwhile to get the addi-
tional hydrological information.
Cost-effectiveness.
Cost-effectiveness is, in fact, engineering economics
[5]. It is concerned with evaluation of a system worth, before
the decision is made to construct the system. Thus cost-effectiveness
is future oriented, and because of it its mode of
expression is in terms of probabilities and expectations.
With reference to the situation represmted by Figure 1 ,
one can relate cost-effectiveness with the reciprocal of Cost,
i.e., l/(cost). In this way, the lower the cost of a project,
the higher would be its cost-effectiveness (which wculd also be
a measure of its worth).
Now, the cost-effectiveness of a system (as measured by
its worth) increases with the amount of information available
at the time when the system is designed. In other words, this
is a re-statement of the truism that the more we know about the
universe within which we design a system, the better the chances
to produce a good design. The increase in the cost-effectiveness
can then be observed with respect to two major aspects.
(a) Flexibility in plarmin . Because of hydrological,
economic, an-gacing the planner of a water
resources system, it is desirable to produce a flexible system.
In this context, byYlexibilitytl is understood one or more of
the following attributes:

654
(i) The possibility of increasing the capacity of the
system by adding additional components of the same kind (e.g.,
pwnp stations in a pipeline network).
(ii) The possibility of altering operating policies so
that the system may respond to a broader range of demands.
(iii) The possibility of modifying the system when the
nature of the demand changes, e.g., when there is a traasition
from irrigation to domestic and industrial uses of water.
(iv) The possibility of constructing the system in
stages, so as to respond to increases in the demand for water.
(b) Reversible vs. irreversible decisions. The design
of a system or of a component is a one-stage decision process:
the size and dimensions are established. If the system is im-
plemented, the design decision may have irreversible effects
upon the emironment, such as the transformation of a canyon
of unique scenic beauty into a man-made lake of doubtful
esthetic value. The decision to delay implementation is
reversible, since it keeps open the alternative to construct the
system. In addition, until the first decision is reversed,
additional information may affect several planning details, ar-d
also technologies may be improved in the interim.
As an example of the introduction of these two aspects
in the planning process, one can indicate thi. Israel Water
Scheme. The planning process was oriented toward increasing
the cost-effectiveness of the system, especially with respect
to flexibility in design and to the reversibility oî decisions [6].
The development of water resources progressed from local ground
water schemes, to regional groundwater projects, finally to the
construction of the major component related to surface water
resources - the National Water Carrier.
Planning and design with inadequate data.
The adequacy of hydrological data as defined by Figne 1
represents one aspect of the general problem of the consequences
of inadequate hydrological dataon the cost and effectiveness of
water resources structures and projects. Although this aspect
can be quantified, it is still somewhat mechanistic.
Another aspect would stress the linkage between data
and decisions in the planning process. Although this aspect

655
also lends itself to'quantification, at least as far as the
data are concerned, it seems that it reflects also the quality
of the ensuing design (and operating) decisions.
It would be perhaps beyond the scope of this paper to
survey the state of the art in the planning and the design of
water resources systems with inadequate data. However, it
would be instructive to mention two of the more recent contributions
to this problem: one dealing primarily with surface
water, and another related to groundwater.
Wallis and Matalas [7] consider the problem of deter-
mining the capacity of a surface reservoir such that a given level
of demand be satisfied. Observed hydrological data were used to
generate synthetic flow sequences, using two different sequence-
generating mechanisms: (a) a well-known model based on the
Markovian process; (b) a model developed recently [8] which
assumes the process to have a finite memory length M and the
Hurst coefficient h; this is called the filtered type 2 process.
It seems that for streamflow regulation of up to 80$ of the mean
annual flow, the Markovian model may be quite useful for the
determination of the minimum necessary storage; for higher degrees
of streamflow regulation, the filtered type 2 model with h 7 2
should be used.
Maddock [g] used mixed integer programming methods for
evolving a planning and management model of a groimd water
development project. The model is oriented toward deterqining
three components of the overall system: (a) least cost operation
of existing wells; (b) least cost spatial and temporal schedule
for installing new weììs; (c) least cost transport system to.
convey the pumped water to a central demand point. The methodo-
logy developed is tested on a hypothetical sample problem in
which ground water development has to satisfy the demands for
water of a town. The concept of expected value of opportunity
loss (similar to the expected opportunity loss encountered in
the preposterior analysis) is used as a measure of how much
the errors inherent in estimating the model parameters will
affect the cost of the project in terms of overdevelopment or
underdevelopment. The results of the analysis indicate that the
reduction of uncertainty for the purpose of decreasing the
expected value of the opportunity loss should be a balanced
activity, i.e., beyond F- given point, further reduction of the
hydrological uncertainty will not improve the decision-making
process unless the economic uncertainty is alao diminished.

656
Concluding; remarks.
The consequences of inadequate hydrological data on the
cost and effectivepeas of water resources structures and pro-
jects were assumed to have a parabolic shape in the data-cost
space. The abscissa of the minimum point of thìs vertical
parabola defines the adequacy of data.
There are several methods for the evaluation of hydro-
logical data with respect to their adequacy for planning. One
such method using the preposterios analysis is presented in
some detail. This method enables the calculation of expected
opportunity loss generated by a program designed to obtain
additional hydrological data, as well as the expected value
of the sample information. If the expected net gain of sample
information is positive, it is an indication that the existing
hydrologic& data are insufficient.
Cost-effectiveness of projects is briefly discussed,
with some emphasis on its aspects regarding the flexibility in
planning and the irreversibilityof some design decisions. As
an example of these asepcts, the Israel Water Scheme illustrates
a planning process oriented toward increasing the cost-effectiveness
of the system.
Fi,nally, how to plan and design water resources systems
with less th&? adequate hydrological data was illustrated by
two examples. In the first example, synthetic hydrology was
used to determine the capacity of the reservoir, but the design
was sensitive to the type of model used to generate the synthe-
tic sequence. The second example related to a ground water
development project.
Ref erences .,
1. James, II, I.C., Bower, B.T. and latalas, N.C. (1969)
Relative importance of variables in water rescurces planning,
Water Resources Research, 5( 6), pp. 1165-1173.
2. Matalab, N.C. (1968) Optimum gaging station location,
Proceedings, IBM Scientific Computing Symposium, Water and
Air Resource Management, White Plains, N.Y., pp. 85-94.
3. Fiering, P.B. (1965) An optimization scheme for gaging,
Water Resources Research, 1(4), pp. 463-469.

4. Hamburg, M. (1970) Statistical analysis for decision
making, N.ew York, Harcourt, Brace & World.
657
5. English, J.M. (1968) Cost-effectiveness, New York, Wiley.
6. Buras, N. (1971) Utilization of ground water resources in
Israel, Atti, Convegno Internationale sulle Acque Soterranee,
Palermo, pp. 674-680.
7. Wallis, J.R. and Matalas, N.C. (1972) Sensitivity of
reservoir design to the generating mechanism of inflows,
Water Resources Research, 8( 3), pp. 634-641.
8. Mataïas, N.C. and Wallis, J.R. (1971) Statistical pro-
perties of multivariate fractional noise processes, Water
Resources Research, 7( 6), pp. 1460 - 1468.
9. Maddock, III, T. (1972) A ground-water planning model
basis for a data collection network, International Symposium
on Uncertainties in Hydrologic and Water Resources Systems,
Tucson, Arizonp, pp. 6.3-1 - 6.3-26.

658
cost,
Monetary
Units
~~ ~ ~~~
Amount of hydrological data
Figure 1. The data-cost space.

\
3
R. A
Figure 2.
o. 20/0
659
Decision diagram for preposterior analysis, MMU.

OPTIMIZATION OF WATER RESOURC ES DEVELOPMENT PROJECTS
ZN CASE OF INARE2UATE HYDROLOGIC VATA.
A. Filetti' ) , G. Faank ' 1, C. Pahvuleb eu' ' I
Bucuhebti, Romania
----
I n t h o d u c t i o n .
The phoblemb which have to be bolved by hydhaulic
engineeh4 ahe 06 g hed diveaitq and deeibionb in theih
dieldb o 6 actiuity o@en imply a conbidehable hen ponb abi-
lity, not only in hebpect to the economic conbequenceb,
but albo to the bocial and ecologic eddectb 06 buch de-
cibionb. Being heeded to the mabtehing 06 cehtain natuh-
al phenomena, mobtly huled by btochabtic lam, the con-
ception as well a¿ the opehation od wateh hebouhceb de-
velopment dthuctuheb depend on the deghee od knowledge a-
vailable on natuhal data, ebpecially on thobe helated to
hydhologic euenth. The hydhologic hecohdb necebbahy to
hydaaufic engineehs ahc not condined to data &elated to
liquid blow, though data 06 thib type ahe ebbenaal doh
theih acLluity, but ah0 concehn bed-load phocebbeb ,hiveh
and bank dynamics, wintea phenomena etc. Euidently,due to
the aleatohq chahactes ob mobt hydhologic occuhenceb,even
long ~recohdb 04 pat hydhologic phenomena cannot oddeh an
abbolute baáety a¿ hegahdb avoiding ehhohb and deviation
@om the altehnaZiue which could be phoved ab optimal. lt
,i¿ neuehthelesb unanimoubly accepted that, ab the volume
and quality 06 in6ohmation on tlecohded hydhologie events
incheas eh, in conditionb o 6 i tlb cohhect intehphetation,
the phobability 06 cohhect ebtimation od dutute occuhencc
ob hydirologic phenomena inchease4 and, a2 the bame time,
the hhkb abbumed in taking deCihion6 decaease.
~
'1 Vocto~-Engineeh,ChCed Engineeh od the Rebeahch and Ve-
bign Inbtitute doh Wateir RebOuhCeb Engineehing.
I') Engineea, Team leadeh at the Inbtitute doh tlydsoetec-
thic SXudied and Vebign.
11') Voctak-Engineeh, Section leadeh at the Rebeakeh and
Debign Indtitute doh Wateh Reb ouhceb Engineehing .

66 2
It muAt be undeiraned that the indluence 06 incom-
plete hydirologic data on the pobbibieity 06 coirirectlg de-
teamining the technological, duncL¿onal and economic pa-
irameteu 06 wateir heb ouirceb development piroject¿ depend¿
in gheat meabuhe to the hydirologic chahacteh 06 the iriuez
bain, on the natuire 06 wateir Ubeb, on the type 06 bthuC-
tuheb etc. Theaedohe,any opL¿mizaL¿on method mubt be con-
bidehed in the fight od the condition4 in which thib me-
thod ib apptied; the bpecidic 4itUafiOnb and the tenden-
cieb in bolving the phoblemb menaoncd in thib papeh must
be looked at only ab typical exampleb.
Undeh conditionb o6 incomplete hydhologic indoirmat-
ion, the methodb Ubually appfied ahe no longeir clbe~jul and
it i8 necebbahy eitheh to adapt thebe methodb to the a-
vailable data oh to adopt bimpleh phoceduheb which aire
conbibtent with thebe data. In belecting buch methodb,
6oÆlowing itemb mubt be taken into account:
- the methodb mubt mahe integhal ube 04 the auaila-
ble volume 06 indohrnaL¿on.At the bame L¿me it mubt
be kept in mind ithat no phocebbing id able to cire-
ate quantitatively new in~ohmation and thehedoh it
lb ubetebb to thy genehafing in6ohrnat.ion not con-
tained in the basic data;
- bebideh hydhologic data irecoirded in the aiueir ba-
din bubject to analybib, it i4 pobbible to take
advantage 06 additional indohmaL¿on irecoirded in
neighbouhing hiveir babinb. Indihect hydirology may
be ubed ebpecially in ohdeir to obtain quafitufive
indohmation hegairding the beabona1 oh annual dib-
thibufion 06 dlow, the pobbibifity od occuirence 06
cehtain hydhologic phenomena in vaaiou4 pehiodb od
the yeah, etc;
- the method mut not lead to an ampfi6icaLLon od
ehhou od the hydhologic basic data but,ab much ab
pobbible to theiir attenuation.
A gheat diveuity 06 bituationb exibtb conceirning a-
vailable hydhologic data, covehing the whole dietd dirom

663
total lack to an acceptable volume od in6okmation. Theke-
dohe, tkying to bet up cefitain methodb o6 gcneaat appbic-
ation ib haadly to be hecommended. In painciple, it
would be make cokhect to talk about gkoupb ok typed 06
methodo having a common p~ncipîe. Thebe have to be adap-
ted to concirete conditionb and objectiveb 06 each inveb-
tigated pkoject. The methods which can be appLied in cabe
06 inadequate hydkological data may be ceabbidied ab dol-
LoWb :
- methodb baoed on the genekation 06 bynthetic hy-
dkologic bequenceb, btakting ('ron1 given btatibtic-
al pazameteu od hydkologic phenomena [ Monte Cak-
to methods);
- methodo based on the genekation 06 hydhologic be-
quenceb by comelation with kaindall data;
- method¿ based on the genehalization o6 kebultb od
watea he6 oukceo engineeking computation4 ;
- methods based on the theoky od gameb.
Thib kepoht intend6 to pkebent the wayd in which
thebe method4 can be appaed'in dome key pkoblemb 06
watek ire4 ouaced engineeking .
Dimenbioning 06 kivek blow hegulating
wokkh debigned dok rneefing ~atek
demandb.
Une od the mobt ubual paobtemb, cohkebponding to an
incipient phue 06 watek keboukceb development, i¿ to ed-
timate the capability 06 unhegulated UVC CM to meet u b e ~
watek demand.
The bolution od thib type 06 pkoblemb id based lebb
on the detekmination 06 avekage @ow and dependendb gnea-
tty on tow watehb and on chakactehibtic minimum valued 06
dibcchakgeb; thebe valued can pkebent a gkeat benditivity
to the quantity 06 available in6okmatiav1, to the methodb
06 dikect oh indikect detekmination od dtow valued and to
the degkee to which ba~ic data have been extkapolated.
Theke6oke, the tack 06 adequate hydkotogic data and

664
pahticulahly 04 batib dactohy hydhomethic hecohdb can be,
in thh cue, the bouhce od impohtant ehhoh4. The w e 06
cohhelative methoda d6 hibky, as the helative valueb 06
low wate~ depend on the individual bupply 06 each wateh
couue. On the otheh hand, the u4e o6 bimulated 6low beqU12MCe4,
wually applied doh genehating mean monthly
dib chahgeb, cannot be batib dactohy, bince daily valueb
od minimum dibchahge can
monthly valueb.
conóidehably did6eh dhOm mean
The dub-unitahy Ratio 06 minimum daily dibchahge to
mean monthly dibchahge i4 wually bmalleh id the phobability
o6 exceeding the dibchahge incheaeb and the buhdace
od the hiuek basin decheaeb.
The ehhoM which can be made in buch cabe4 can be
one 06 the dollowing:
- oueh-evalua-ting available low dlow,which can lead
to a bmalleh phobability od being able to meet
wateh demand od ue~; thib phobabifity might not
be acceptable becaube 06 the excebbive lobbeb due
to dhequency and bevehity od wateh bhohtage. It
i4 Wohth mentionning that in bome counthieb (as in
the U.S.S.R. ,Czechoblouahia, Romania and othehs 1
the phobabifity 04 being able to meet demand A
ebtablibhed by btandahdb and A ,thehedohe, compulb
ohy;
- undeheualuating available low dlow,which can lead
to a da&e conclubion, that the hiVeh i4 not able
to meet demand without blow hegutation and that
a stohage hCbehU0ih Oh a diuehbion dhom otheh excedentahy
hiueu h a to be budX.Thib would imply
u4eleb4 expenbeb Oh, in the bebt Ca4e,UbelQbb immobifibation
od capital.
Ab wateh demand ghowb in compahibon to available
wakeh hebOUhCQ6, the neCCbbahY deghee od blow hegula-tion
incheu eb and mutationb emehge concehning the bignidicance
06 vahiou categohieh od hydhologic indohmation. In
thib benbe, the data helated to the auehage inalow doh

665
longeh L¿me pe&¿odb: monthb, beabonb, yeah4 oh even be-
quenceb ad yeau begin to play an ebbential pmt in de-
tehmining the pahametehb 06 btohage hequihed.
The indtuence 04 genehat hydhologic data on the va-
tue 06 thebe puhameteu i4 heuealed by the geneaal phac-
-Lice 04 wateh hebouhceb engineehb a4 well a4 by home
Apecial hebeahch phoghamb. Thib inbluenee iA made evid-
ent by invehtigating :
- the genehat hetations between the chahacteaibtic
blow panameteu and the bpecidic deghee 06 deue-
topment 06 wateh hebouhceb;
- the ben&¿tiVitLj od hebUltb concehn.¿ng blow hegut-
dion at vahiou4 degheeb 06 apphoximation od the
inadequate hydhologic data.
Thub, a diut aspect 06 thib anatybib concehnb ihe
cohhelaXion bemeen the main puhametehb chahactehib.tiC
doh dtow dL¿thLbution: the vahiation coeddicient Cu, 2he
coeddicient 06 bkewnebb Cb and the coe6dicien.t od sehial
cohhelation ir on one hand and the net volume od necebba-
hy ótohage hebehV0iJr.b and theih opehating policieh on
the otheh.
Vahiou4 sepohth conceaning genehatized Jab ultb on
the connection befween thebe patameteu and the magnit-
ude 06 the ove&-annual component 04 the nequihed btohage
doh a given bade yield have been pubfibhed 111, 121 .
Rebula 06 buch hebeahch ib beAt made evident by
diagrramb phebenting the comelation between the coedtjic-
iena a, p and B, 06 which an example ib shown in diguhe
i. Following bymboeb have been uded :
- a, the blow aegulation coeddicient 04 the deve-
lopment, deiined ab the ha.t.io 06 the bade yield
od the dtohage hebehvoih to the auehage dibchahge
06 the wateh COUILA~ ;
- p id the phobabieity od meeLing wateh demand, de-
dined ab the &mit 06 the hatio 06 the numbeh 06
yeau in which no wateh bhohtage appeau to the
total numbeh 06 yeau invebtigated;
- 0 c6 the btohage coeddicient, dedined ah the hu-

666
tio 06 the oveh-annual component 06 Atohage to
the avesage yeaaly dibchahge 06 the hegulated ti-
Veh.
lt mUbt be kept in mind that the bame avehage
dib-
chahge 2 could occuh in vahioub beqUQnceb 06 bingutarr
dibchahgeb 06 the hivet; thib can be made evident by an
analybd 06 hecohded beqUenCeb 06 dibchahgeb 06 hivehb
phebenting vahioub Valued 06 the pahameteu cv,Cb and 4.
Folîowing conc~ubionb may be dhawn dhom hebeahch deveto-
ped in thib 6ield:
- the vatiadon coed{i&ent Cv ha4 a dihect inblu-
ence on the volume 06 the btohage hebehvoih, the
Atohage coe6bicient B being the gheateh, the m o u
the vahiation coe6,jicien.t incheabeb ; genehaîly
the helaAive ghowthb 06 ß ahe gheateh, sometimes
even benbibly ghedek, than the gaowth 06 Cv, i6
the valuta 06 a a m high. On the con;titahy,6oh low
valued 06 a the gaowth 06 the necebbahy volume 06
btoaage in lebb hapid than the ghowth 06 the va-
hiaiSon coe66icient [ 6iguhe 2) ;
- the coe{,jicient 06 bkewnebb cb ha an inveme in-
ence ce on the volume 06 the btohage hebehvoih;
Zhu, i6 cb incheabeb the volume decheaseb coh-
hebpondingly. Genetally, pehcentual heduciSonb 06
B avre Amalleh than the pehcentual ghowthb 06 Ch;
- the coedbicient 06 betial cohhelation h has a di-
heet in6luence on the value 06 the btoaage; thub,
the incheabing 06 h leadb to ghowthb 04 B, peh-
centual ghowthb 06 both parrametem being 06 the
bame ohdeh 06 magnitude.
Similah conbidehationb can a do be made in connect-
ion to the beabonal component lyemly component) 06 the
heqdhed btohage. Evidently, in thib cabe, the coe66ic-
Leni2 06 vatiation, o6 bhewnebb and 06 behiat cohhelat-
ion mubt be based on daily, decadal oh monthly avehage
dib chahgeb .
A beeond inteaesting aspect concehnb the inbluenee
06 the intehval taken in account 60k debign iday,decade,

667
month etc) oh the Lime intehval doh which hydhologic data
ahe ebtimated and wateh balance computations ahe undehtaken
on the hequihed btohage volume.
Vehy 06ten,waZeh balance id ebtablibhed on a monthly
bad&, taking into account a bequence 06 mean monthly
dlowb COVehing a pehiod 06 ¿evehat qeah4, doh which hecohded
oh indihectly detehmined hydhologic dda ahe a-
vailable. Thib way 06 dealing with the phoblem imptieb
the assumpLion that the dhchahge od the Jbiveh and the
demand o{ the ueh ahe baihey conbtant duhing a month.
Thib asbumption ib neveh abbolutely cohhect doh the di&chahge
od the hive&, nOh bometimeb {oh the Watch demand.
ZnvutigaLionb undehtaken in thib 6ieÆd bhowed that
in cehtain cabe6 the ,time pehiod taken into account hab
a gheat indluence on the hebultb obtained concehning the
volume 06 hequihed btOhUge.lt has been pobbible to utabtibh
cohheîationb bemeen btohageb cohhebponding to
,time pehiodb od a month oh a day ubed in wateh balance
calculations. An example 06 buch an intehdependence ib
bhown in biguhe 2.The genehat conclubion od the tedeahch
i& tha.t bhoht pehiodb, o6 a day, mubt be ubed 46 badie
ame intehvaÆ only i6 the heqLUhed btohage,hebulted dhom
pheaminahy computation4 ubing monthly UVehUge valueb c4
Amall. foh gheateh 4tOhage volume4,the indluence 06 bhoht
Lime pedo& id negtigeable.
16 the invebtigation 06 lahge-bcale phoject.4 i4 undehtaken,
u6e od 6 yntheLic stheam- {low bequenceb had
btahted to impobe itbet6 even in cae4 in which the volume
06 availabÆe hydhologic indohmation would have been
conbidehed adequate. Stheam-dlOW genehation methodb,
oh Monte Cahlo techniqueb, btah-t dohm cehtdn comphehenhive
hydhologic PaharneteM buch ab avehage dibchahge,
coeddicient 06 VahiatiOn, coe66icien.t 06 behiat cohhelation
etc, ebtimated on the babib od a minimum 06 dihect
hecohdb oh by genchalizing hebultb od hydhologic inveb-
LigatiOnb in bimilah (Meu. Ahtidici& time behieb o6
hundhed and even thouband4 od hydhologic yeau ahe genehated;
thebe include a multitude 06 pobbible bequenceb

66 8
06 day, wet and aveirage yemb, which condeh a high heliability
on the hebultb 06 watch balance
calculationb e
and btohage
Such methodb, babed on bynthea2c btneam-6low bequenceb,
a m applied on a lahge ¿cale in the U.S.S.R.
l4I,l5/, the U.S.A. 161 and otheh counthieb. ln Romania,
1101 thib method .d being applied doh the btudy o6 the
development 06 lahge hive& babinh.
A hecent hebeahch 131 ib conceancd with the iniluence
06 the length 06 the heal hecohd on the hequihed
stotrage,in 06 a seasonal oh yeuly dlow hegulation.
The teseahch phogham covehed a gheat vahiety od bituationh,
including di66ehent typeb 06 dlow dibthibufion
and bevehal VatUeb 06 berrial cohhelaZLon and btoirage
coeddicienfi.
The hebultb o6 thih hebeahch Lead to the conclubion
(bee diguhea 3 and 4) that the hequihed btohage incheabeb
with the length 06 the bequence. The inchease .&
dabteh doil higheh valued 06 the behial
stohage cae66icienh.
cohhelation and
Though thib invehtigation w a ~ concetned only with
yeahly (low rregulation, the hebula obtained doh high
valueb o6 the btohage coeddicient (maximum 1.0) phove
that the length od the dihect hecohd and the bequence
06 wet and day pehiod~ have a bignidicant indluence alho
in the cabe 06 oveh-annual atohage.
A bthiking example in this ben~e was uphedented by
the evolution od btohage hequihed 60s the wateh bupply
06 a big induthial plant in Romania. Wateh balance calculationb,
6ihbt undextaken in the eahly 1960-ieb Wehe
based on dihect dlow hecohdb covehing only a pehiod od
15 yeah6, btmfing @om 1947148. Extending thib bequence
by cohhelation with hecohdb in neighbouhing babinb genehated,
intek alia, an exthemely dhy hydhoîogic yeair
(/942/43). 76 thib yeah wab included in the hecohd ued
a~ babib doh htonage calculationb, the hequihed btoaage
wab neahly the double (220 million cubic meteu) 06 the
btoxage which would have deemed necebbahy i6 thib yeah

669
had not been included into the hecohd (120 million cub-
ic meteu). Ab a matteh 06 dact, the additionat hecohdb
06 the 6ottowing ten yeau beem to indihm the indihect
data obtained doh the yeah 1942143; it wad thehedohe de-
cided not to include thib yeah into the invebtigated be-
quence when btohage caÆcutaL¿onb wehe again undehtaken
on the basi4 06 a .tongeh dcquence 06 dihect hecohdb.
In buch bituationb , behideb the ub ual methodb 06 ex-
tending the hydhotogic time behieb by bimilahity with
otheh hiveh badinb, modehn techniques can be applied doh
detehmining the optimum decndionb. A botution might be
dound i6 the theohy 06 game4 id applied; the hecommended
de&ion lead& in thib cade to minimum heghet, taking
into account on one had the cobtb ad the btohage hueh-
voi& and on the otheh hand the pobbibte damageb. Thib ib
a cla6bica.î cade 06 a game againbt natuhe. Natuke’b se-
action ib not indluenced by the phevioub dechion4 con-
cehning the management od wateh hebouhceb and ib eitheh
aleatohy oh huled by an asbumed pkobabilibtic law 06
dib t&ib utio n .
06 couue, the methodb based on the theohy 04 gameb
do not bolve the inadequacy 06 hyditotogic data. Ubing
them maheb howeveh the minimization 06 adveue e66ec.tb
06 inadequate data po4bible. Theh~,joke, buch methods
bhould not be phebented in oppobifion to thobe based on
the genehation 06 new data oh on the genehatizaaSon o6
cehfain heb uttb od watek heb ouhceb engineehing calculat-
ionb. Both technique4 can be bimultaneoubly applied.
One 06 the advantageb o6 the theoky 06 gameb ib the
pobbibifity 06 taking into account any inadequate data,
not only od hydhologic chairacte& (do& example, data &e-
dcilning to the dUtUhQ development o{ watek ue~).
Noticing the ub e o 6 cohhelaLionb between hydirologic
pahameteu and Othe& chahacteJùbtic elemenX.4 06 the hi-
ve& basin: aveaage haindall, altitude, adtjoheAta,tion,
btope etc. bome engineeu thied to ebtablihh genehatized
helaZionbhipb be&een wateir heb ouhceb engineehing paha-
meteu, buch ah the kequiheb btohage, and geo-meteoholo-
gic elemena. Such hebeahch has been cashied out in Po-

670
land. Methodb 06 thib type have, nevehthelebb, a limited
appficability, geneaalizationb being pobbible only at a
&regional beale. They may, evidently, give a view on the
bize 06 necebbahy Wdeh hebOUhCeib development WOhkb and
can be 06 help doh the phefiminahy debign 06 ceht&n
bmall btohage damb. Theih ube, without a camparribon with
othes mom elaboaate methodb id howeveh not to be hecorn-
mended in the inwedtigation 06 impohtant btohage. hebeh-
VOihb.
Uimenbioning od hydhoelecthic Noirkb.
The utilization 06 wateh poweh id, evidently, hela-
ted to the cohhect knowledge 06 the natuhal potential
and o6 the conditionb 06 developing it. in thib conned-
ion, hydhologic data ahe necebbahy not only in ohdes to
detekmine the genehal e66ect and edbiciency 06 poweh
plana2 but albo t o ebtablhh the development bcheme, the
enehgetic parrameteu and the charractexibtics 06 the
bthuctuhes. Fah plana luith no btohage oh having a low
deghee 04 6low heguldtion a6 well c~6 {oh the intake4 06
becondahy watch divehhionb it can be parrticulahly impoh-
tant to ebtimate c~hhectly the daily, and dometimed even
the momentarry blow hégime. Condthuction 06 irégime and
dlow dukation cuhveb , chahacte&ibtlc doh kelatively long
pehiod6,may be necebbahy. Such cuhueb alre uded in ohdeh
to ebtablibh the deghee 06 utilization 06 the avehage
yeahly dh chahge and to detekmine the inbtalled capacit-
ieb
The value4 06 the hydhologic pahameteu mentionned
in the phevioub chapteh: auehage d.ib chahge, coe66icient
06 vahiation, coe6dicien.t 06 bkewnebb and coe6dicient 06
behial comelation, ahe necebbahy in ohdeh to ebtablibh
the inbluenee 06 vadoub Atohage volumed on the magnitude
and quality 06 electtic po~eh phoduction and on the
enehgetic conditions 06 btohage hydhoelectaic plan&. On
thib bahib, the opfimization o6 the volume 06
age hebehv~ikb cb aho podbible.
the btoh-

FOR. mote advanced dlow conthol, the avehage multian-
nual dibchakge i6 the hydaologic etement whobe indluence
on the economic eddiciency 06 hydhoelecthic plant6 i4
gheatest. In paht, thi6 ib due al60 to the opehation 06
hydhoeÆecthic plana2 within 6taong poweh dydtemb , genehal-
ly with inteknational linkd. In buch 6Ybtem4, the e66ect
o 6 individual hydaologic 6ituationd i6 attenuated and the
whole hydhoelecthic poweh available can actually be uded
in the dybtem, even id only doh the dilling o6 the hebeh-
voim 06 pumped 6tohage powek plana.
Re6eahch on the indluence 06 the length 06 the hecoird
on the value 06 the aveirage dibchahge hevealed that,
604 m o ~ t eutopean wateh couh6e6, time behie6 covehing a
dequence 06 30 yeam ahe u6ually batihbactohy. Conclubiond
Reached at in inve6tigating data concehning the a-
veaage didchaqe doh Recoaded Lime behie6 06 vahiou6
length6 at the OIL6ova gauge on the Danube can be bahen
a6 an example. On the basib od 133 yeah long hecoad6
(1838 - 19701 the avehage value 604 di66ehent 64 . ~ e b 06
a given Length, covehing 10 - 40 con6ecuL¿ve yeau wehe
calculated and the highebt and Lowe62 value6 o6 thehe a-
vehage6 wehe examined. The irebutRb alre phedented in the
dollowing table:
10 124 6090 4520 12.5 -16.5 0.36
15 119 5910 4760 9.2 -12.0 0.28
20 114 5770 4850 6.7 -10.2 0.22
25 109 5660 5040 4.6 -6.8 0.17
30 104 5670 5200 4.1 -3.9 0.13
40 94 5650 5230 4.4 -3.3 0.11
Thehedohe, i6 the hydhotogic data ahe not adequate
doh the de6ign 06 hydhoelecthic developemntb, a6 a huÆe,
thebe data dhoutd be extended to a 25-35 yeah6 long time

672
behie, by analogy with othek watek couhdeb ok with kain-
ball. longeh extenhionb 06 hecohdb by bimulation ate
veky kcvrely applied.
Debign and opehation 06 blood
conakol developmenth .
Flood canx7ca.t developmen;td ake kelated not only to
economic bene6itb but aedo to the becuhity 06 bocial
Lide in wide axeab. At the same time, ab álood conthol
bthuctuheb ane debigned to {ace hydhologic bituatioptb
exceeding extheme hibtohicaî hecohdb, the ube 06 cxthe-
mely accuhate data ib, in phinciple, necebbahy.
A ptoblem ahibing most dhequently id helated to the
debign 06 outle& o6 6low hegulating bthuctuheb. Othe&
impoatant phoblemb, buch ab the debign ,od bhidgeb oh 06
othes btkuctuheb chobbing hivetb, 06 dykeb and 06 blood
detenfion hehehvoitb axe aedo helated to dlood hydhol-
OgY *
Except vehy hahe cab eb, debign hydhologic conditionb
have nevek been hecohded and have to be ebtablhhed
by bpecLal computationb. In thih hebpect, ,two tendencieb
may be kemairhed:
- the extkapolation 06 phobabifity dibtkibufion
cuhvcb 06 maximum aecokded ,$toodb ubing mathematical
btatibticat methodb; thib extkapolation has
to keach imposed debign paobabilitiea . Othe& methodb,
ubed in o.theh hydhologic pkoblemb 6011 the
extension 06 tecohded fime bekieb ,buch ab bynthefie
blow genekation,ubing Monte-Cakto techniqueb.
MQ applied at a bah amalleh bcale bok 6lood conak0l;
- the geneha~on 06 valueb 06 maximum dibchakgeb
and 06 ,$laod waveb by hain6altlhun-od6 cohhelation;
thib apphoach thiea to compenbate the lack
06 hydhologic data by uning hain{all oh Othe& metheohologic
magnituded doh which, UA ually, longea
hecohdb ake available. ln theih dimplest 60hm,

67 3
thebe methodb hephe4enZ ha.in6attlhun-od6 depen-
dencie6 which have been wed doh a long time in
hydhologic indihect btudie6 .Duhing the lut yeah6
thebe methodb have been extended, 6ohmeh 6impte
dependencieb being developed into phy6ioghaphic
haintjalllhun-066 mode&. Thebe mode& have been
ubed ebpecially in the U.S.A. and in Fhance.Theih
eidiciency wab pkoved e6pecially doh the htudy 06
dlood conthot phoblem6 .
The majoh did6iculty haAed by applying genetic mo-
de& conbi6t.h in a pheviou6 detehmknation 06 the pahame-
teu 06 each model. FOh thi6 pukpo6c,aecohded dl00d6 ahe
houted thhough the model. Theiredom, ube o6 h&n6all/
hun-066 modeh implie6 the exhtence 06 a minimum o6 hy-
dhologic hecohdb; it ib, howeveh, ju6t data on dlood6
which ahe dhequently mi6bing @om hecohd6.
ln the abbence 06 adequate hydhologic data, oveh6i-
zing most dtood conthol 6thuctuhe6 i6 to be hecommended,
even ifj it’6 con~equence i6 a ubete66 inve~tment o6 ca-
pital, in ohdeir to avoid the hibh 06 ovehtopping and 06
eventual bucceeding 6ailuhe od 6thuctuhQ6 181. Thi6 po-
&cy h juhtidied by the exponential ghowth 06 the den6-
ity 06 economic and 6ocia.t objecaXve6 placed in the ama
in Which dtood COnthOt ~66e~tb 06 thebe bthUCtUhC6 i6
he6ented. Eventual damage6 due to exceedence 06 de6ign
pahameteh6 will Zhu6 be exponentially incheahed in compahibon
Stood.
to actual damage6 cohhehponding to the 6ame
Thub, adteh the hydirologic excedentahg peiriod 06
the la62 yeah6 and pahticu.tah.ty aóteh the 7970 and 7972
iloodh, the conclubion wa6 dhawn in Romania that 6lood
detention volume6 hebehved in 6tohage lahel, which, a6 a
hule, Wehe phteViOU6ly 06 the ûhdeh 06 b - 12 % 06 the a-
vehage annual blow a m knbuddicient; incireahing the6e
volume6 in butuhe up to 20 8 06 the aveaage annual dlow
i4 hecommended. Except 4 ome i6 olated ea6 e6, the execut-
ion 06 hubrneuible dyke6 has been completely abandoned
bivice 1960.

674
In the debign 06 dyke6, the gkowth 06 blood levea
due to eliminating the natukal deterifion o6 Blood in
the dtood plain and to kiVeh bed dynamic¿, pahticulahlg
the haióing od the base 04 ehobion, obbehved on many wu-
tek couhbeb 06 the plain hegiOMb. At the tail watekd 06
cehtain btohage kebehvoihb located in countkied with de-
vem climate, inadequate knowledge 06 wintek phenornena
may lead to undehebtimafing the backwatek phoduced by
the mas4 06 ice and to undokbeen dlooding, .i6 no phe-
cauLion4 ate taken.
Finally, bok the cased whehe data on the genebis 06
6loodb ahe lacking, it i4 pobbible to apply methodb od
the theohy 06 gamed. Such methodb have been applied in
Romania 604 bevekal pkojech 171.
Some aspecÁ2 o 6 multi-puhpob e
wutek ked ouhceb development.
One 06 the chahac$ehib.i.ics 06 the contempokahy wa-
tek he6 Oukceb engineehing conbibtb in the multipuhpob e
and comphehen6ive development 06 hiWh buinb, within
an ebtablibhed development bcheme. VaGoub pko jectb ake
phomoted a¿ bepahate deÜeÆopment btages 06 the genehat
bcheme, all phojecÁ2 being based on unitaky methodolog-
ieb and being conceived bo as to meet the kequikemenÁ2
06 d l wateh ubeIL4 u well as 04 the COnthOl 06 deb-
thucfive eddech 06 wateh. The btep by step development
06 a hiveh basin i4 impohtant to the phoblem dibcubbed
in thib hepoht because 06 it'd pobitive conbequenceb he-
dlected in the pobbibility 06 cokhecfing emou, 06 he-
ducing exaggehated hibkb and even 06 attenuafing bail-
ukeb by the way 06 conceiving Bututre phOjUÁ2 developed
in the bame hiveh basin. Thió i4 helated to the 6act
that the unitahg management 06 the watek kebouhceb ob
a hiveh basin cheateb a lebb 4thLc.t dependence 06 each
phoject on the dibffkibufion 06 the dlow 06 the main hi-
veh among vahiou4 thibutahieb and heduceb the benbitiv-
ity to emohb in evaluating hydhaulic hCbOUhCe6 in each

675
~pecibic bite. Thib i.4 impohtant, ab global ebtimation
od the hCAOUhCeA 06 a kiveh bain i4 UbUal.& leAb hub-
ject to ehhohb than the ebamation od the heAouhceb 06
the thibutahieb. On the otheh hand,the btep by btep de-
velopment o 6 bthuctuheA doh wateir heb ouhced management
makes changed in the pahametea od ultehioh phojectb
pobbible, Ao ab to COhJLect ~46ectb 06 Oveh oh undeh-
hizing the 4-thuctuheA built at phevioub development
A tag eb .
In the cae 06 pkojecth doh which decibionb ahe
made undeh condifionb od inadequate basic data, inveb-
tigai2on 06 the pobbibifitieb 06 butuhe sxtenbion od
bh’iUC~UReb d od gheat crteirebt.Such podbibilitied eliminate
the necedbity 06 immobilizing capital doh the
development 06 initially oveh~izc! d bthuctuheb, deaigned
in thib way in ohdeh not to loo¿e the pobbibifitieb 06
an advantageou dite.
EhhohA committed due to inadequate hydhologic
data ahC not fimited to the design btage o6 hydhaufic
AthUCtUheb. FOh the conception od buch bthUCtuheb
the lack 06 dihecz hydhologic data can obten not
be avoided. 76 the implementation od an adequate hydhometeOhOlOgic
6ohecabting nekwohk, including the meaAuhing,
thanhmibbion and data ph0Cchbing equipment d advibable
in Ohdeh to Opehate a CQhtLn hebChVOih, .¿A puhely
an economic phoblem. In thib benbe, the technical
oh economic analyAib 06 opehating conditionb 06 vahiOUA
lahge Acate pkojech and the indluence 06 a good doheca6i2ng
aybtem on theiir opehation Achedule4 lead invatiably
to the conclubion that thebe Aybtemb
culahly e 4 bicient.
ahc pahti-
Thib L¿ not only the cabe whehe hydhaufic Aybtemb
debigned 604 dtood canthot oh doh bade yield ahe addected
by the lack od adequate indohmation oh dohecabzb
at Auch exteint, that theih opehation accohding
to b chedule d phactically impohbible without imphovcng
the indoirmational bybtem. The advantage id evident

67 6
ado when a betteh knowledge 06 the phobable pahame-
tehb o6 butuhe hydhologic even22 leadb only to opehat-
ional imphovemenX.6. Thib cb, doh inbtance, the cabe 06
hydkoelectk-ic plana.
An illwthative example 06 an in6OhmatiOnal and
donecabX.ln9 netwohk concehnb khe lhon Gate¿ hydhoelec-
thiC plant on the Danube. The wateh level 06 the bfoh-
age hcbehVOih at the dam vahiable, opehating bchedu-
&A phoviding doa an ab conbtant ah pobbible wateh le-
vel & the tail 06 the Atohage hebChVOih, Upbtheani 06
the IhOn Gates gohgeb. Thib policy cb due to the con-
cenakation in thib ahea o6 bthuctuheb debigned doh the
photection 06 hipak-ian land and otheh development¿,
bthuctuheb which would be oventopped at higheh leve&.
Conbcquently, the hydhoelecthic plant bhould hegul&e
the wateh level & the dam bon an expected inblow, bo
UA to obtain the maximum u.ti.lizable head, to avoid, U
much ah pobbible, the 0vehhpit.ling o6 Waxeh and,& the
bame time, to avoid the exceedence 06 the badety level
in the photected aheu. Vue to a good dohecabtin9 netwohb
on the Danube and on the main thibttahiebp upbtlream
o6 the phojecf, it w u pobbible,even duk-ing the
dihbt WO yeahb od opehaL¿ng, to hegutate the daily
powea genetation in buch mannet that the deviaLion
6hOm the theohea2ca.t optimu did not exceed 1.2 %.
The utility 06 the exPendituaeh intended to bet
up and maintain an adequate 6onecasLing bybtem, pahticutaaly
within hiveh bain6 whehe dlood deten.tion heb
eh
oiha ahe located has not any mohe to be demonsthated.
Situation¿ may be met in which an inadequate opehation
od OUtkkX.6 dhom Atohage hebehVOihb clln OvehpObe
nohmally buccebbive blood waved, leading to an aggnavakion
06 the bituation which would
unhegulded b&eamb.
have occuhed in

C o n c t u b i o n b .
677
The accukacy o6 the dolutionb 06 debign and opekat-
ion 06 hydkauîic wokkb depend not only on hydhologic
in6akmation, bu.2 albo on the degtee o6 cokkect ebtimat-
ion 05 a multitude 06 otheir 6actou, 60ir inbtance the e-
conomic and conjectukal condixXonb ,the wateir demand etc.
Thebe iactok obten aire much m o u uncektain than hydholo-
gic evenh. in thib context, in cae 06 ceirtain ubeb 06
hydkautic bthuctukeb, Auch ab hydkoelecakic powea pko-
ducfion, watea tkanópokt, low dlow kegulaaon boa watek
ueb, the hydhological in6oirmatian bhouLd be consideked
in the hame way a6 othen uncektain basic data, the qua-
îity 06 which h a a gtobaî inbluenee on the p04bibiti-
ty 06 op~mizing bOLUtiOn4. in buch cabeb, the necebbaky
accukacy 06 hydiroLogical data mubt be Looked at in cok-
helaZion with the accukacy o6 othek elemenh kelevant to
the decibion. Them ake howevea alho othek typed 06 hy-
dkaulic b&uc.twreb,buch a6 thobe debigned 60k dlood con-
thol, 60ir which global analybib 06 accukacy 06 basic da-
ta i~ tebb impoatant and hydkologic data have to be ta-
hen bepaately into account.
R E F E R E N C E
1.- PLESHKOV, 1.F. Reguliirovanija kechnogo btoba -
GWdkometeoizdat, Uobkow, I96 1.
2.- DYCK,S.; SCHRAMM,M. Stochabtibche Methoden 6Ük die
Bemebb ung deb Wasb eupeichekhaumeb.
- Mitteilungen deb in-
AZifUteb {Ük W a b eir~ikt.6 cha@,
Nk.28, BekÆLn, 1968.
3.- STEGARUZU, P. CokectiiLe de debite zilnice in
calcuîeÆe de gobpodairike a apeloir.
- Hidkotehnica, Nk. 111972.
4.- SVANIDZE, G.G. ûbnovy k a chety keguîikovanija
aechnogo btoka metodom i\lonte -
Kaalo.- Tbiîibbi, L964.

67 u
5. L’EZNlKOVSKll, A.A.
Vodnoenehgetichebhie ha6 chety
metodom Monte-Kahlo - Enehgija,
idobbow, 1969.
Stheamdtow Synthedib . - MacMil-
6. FlERlNG, M.B.
7. - VORVEA, A. ; Fl LOTT i, A.
lan, London-Melbouhne, 1967.
PhObleme de gOdpOdahihe a apeloh
CU aplicatic? la bazinul
Bahlu&. -1nbtitutul penthu Planuhi
de Amenajahe bi Consthue-
8.- FILOTTZ, A.
tii Hidtotehnice ílPACHl.15 ani
de activitate. Bucutebti, 7968,
pp. 85 - 96.
Dib cubbiòn deb happohh concehnant
la photcefion de4 eaux en
9. - TEODORESCU, 1.
ea6 de cûtabthûpheb. -GeWabbehbchutz
im Katasttophen~all.Sympobnum
vom 23 - 26 Ohtobeh in
Flohenz. Födehafion Euhopäib cheh
Gewabbehbchutz, Vol. 15, Ziihich,
1969, pp- 92 - 96.
Gobpodahihea Apeloh. - Ceheb,
FILOTTI ,A.; CHlRl AC V.%ucuhebti, 19 73.
10. -S?MON, A; Vl LAN. A GenehcZhea bihUhil0h hidhalogice
daha aUtOcOhe&ltie. - Studii de
Economia Apeloh, Vol. 1. lnbtitutul
de Studii bi Cehceta~
penau Zmbunatatihi Funciahe
bi Gobpodahihea Apeloh, BuCuhebti,
1971, pp.311 - 366.

ß, 1
IO
Fig.1.
n
0.5
. .
0,s
C"
ß
20
LO
1.0
cv
as 0.5
. [r=45; c,=2c,
- 1
I5
ZD
10
CV Ci I
, ß
C"
2. o
LO
ß
20
ZO
'n
cv
2.0
%O
C"
ß
28
10
679
C" CV
0.5
ß
2. o
W
CV
ß
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0;s
cv
r ia I I I
cv

6 80

O /O0
pigc4. Average required etorage a8 a function
of record,for
-
different degrees of
regulation8
c= 0,5 I r 0.3
681

ItRELATIONS BETWEEN PROJECT ECONOMICS AND HYDROLOGICAL DATA"
by
A. Pobedimsky
Economic Commission for Europe
Introduction
In e::ce;j:;, water ilhich is very often considered as a source o;" wealth, m y
cause considerable damage or äisaster and impose a heavy burden on a country's
economy. sorn&jimez it may affect a goup of borderine countries (as for example,
those locate6 in the Danube and Rhine river basins).
The accelzrating rate of population eowth in the EC3 countries and the
economic proLpcss in Lechnologicrì changes, during recent years, causing the
depletion of natural resources have all rapidly increased the importance of
water resources development which, curing the last few decades, hac, become one
of the doninnting factors in the national economy of most stoLintries. All this
has obliged countries to improve their water resources nanagement so as to achieve
proper flexibility a d effectiveness corresponding to modern requirerncnts of national
e conomie s.
Mater Phnagement, deals o-lher things with a verj importent component -
hydrological data which define the available water resources to me& national or
regional demnds.
The mtcliing of the bdznce or water resources md. neecls, as vel1 as the
planning and implementation of eppropriatc masures to provide the nececjsa.0 liater
cupply for a region, have become &ieslionr of hi& priority,
The belances oi' water resources and needs mentioned above, which serve to
elaborate the measures io be taken to avoid ne,r;etive c.onsequences for ths populaLion
and the regional economy, are now Peing used as a effective tool in mqv LC3
countries. The first internationa!. l4anue.l for the compilation of these balances,
now beiny: completed by the ZCg Conrmi-kLeé on idater Problems g d groups of national
experts, emphasizes that in regions with I.imited va-ler re::ouxces a high degree of
accuracy in their assessment is an essential condition ror e. rational economy.
The E.'mual enphasizes the importtince of the earliest possible organization of
hydrological studies in a river basin or 8 region where an intensive growth oi
uater needs is in prospect. ?he Phniisl defines in the following WP.~ the economic
impact of reliable hydrological daCC. on ua-ter resources dcvelopnent in particular
cnd on the national evonoq in genernlc ?he more reliable the iiatter zupply, thi:
smaller i.iill be the damage resultin, from cutr in periods of water shortagetf. By
cornparin: losses anU expenditure, it vili, in prlnciple, be poosi'ule to determino
the economic optimum.
Sufficient and accurate hydrological data promote effective vater management and
the prevention or didnuation of damage caused by such hydrological phcnomna as
severe floods , ice jm, rnuàflotis, intemive oedirnents, dangerous va.ter pollution etc.

684
On the other hand, the intensification of human activities in river basins
rnl :,Lcii, watersheds including the increased anount of untreated effluents
uic .liarged FnLo the water couse, during recen.'c decades, uraently calls for the
re1iai;l.e esse:;smnt of available wa-tor resources. The ECS Nanual mentioned above
states that the consequences of h w n activities make advanced hydrolo$cal
research imperative.
considered important.
A relevant improvement of hydrological methods is
5Je understand that this topic is of considerable interest to hydrological
services which must strike a balance between tho cost of gauging stations and
the probable futvre benefit that will result from the information to be obtained.
TakTng into account the special importance of hydrological data for water
resources management, various aspects of the development of hydrological networks
were discussed thoroughly at the ESE Seninar on Selected Water Problems in Southern
Europe convened in Zagreb, Yugoslavia, in October 1971. Certain conclusions
concerning the design of hydrological networks and their improvement were
reflected in the recommendations adopted by the ECE Codttee on Water Problems.
Taking all this into account it is generally recognized that hydrological data
and well planned hydrological networks are prerequisites for efficient and sound
water resources planning.
The purpose of this paper is to appraise the possible economic effect of
insufficient hydrological data on the effectiveness of water planning and the
design of hydraulic engineering structures and their operation.
It is suggested to consider the following main aspects of the subject:
The economic consequences of a deficiency of hydrological data on water
planning, construction and operation,
"he impact of a deficiency of hydrological data on main water users.
The economic consesuences of a deficiency of hydroloaical data on water
P ~ construction ~ P and operation
The following questions could be raised in connexion with this aspect:
what is the extent of the economic impact of insufficient hydrological
records on a project and its subsequent operation?
in particular, what is the possible effect on investment in economic
development if hydrological data are not accurate enough and the records
are insufficient?
is the predominantly quantitative character of hydrological data sufficient
for modern intensive water resources development?
is it reasonable to postpone the initiating of water project planning and
the construction of water projects if the hydrological observations are
insufficient?
The available information on the experience and research in the ECE region
shows the following facts which could be emphasized in an attempt to answer the
above questions.

Economic -acts Qf insufficient hydrological data and difficulties caused at
the key stages of water resources development:
Mater planning and desim
Generally speaking, poor hydrological data and forecasts made on this basis
can lead to inappropriate proposals for investment in water engineering works and
the economic development of the region concerned.
The importance of sufficient
data at the following staget of planning and designing can be pointed out.
The elaboration of schemes for niltipurpose development of water resources
is greatly dependent on accurate hydrometrical data.
according to which a scheme is to be designed cannot be derived from incorrect
hydrological data.
The appropriate conclusions
It should be underlined that possibilities of considerable miscalculations
exist in preinvestment studies as well as in further stages of planning and design
of engineering structures.
characteristic river discharges, approximake methods and empirical fornulas
are used and the length of time of hydrological observation is relatively short
(considerably less than 30-40 years).
They may be especially acute when, to estimate
Taking this into account, approximate
methods are being limited to the preliminary estimations for the elaboration of
schemes of development, but are not reconmiended for the design of water structures.
Appropriate attention to this matter has been given in the ECL i.ianual
mentioned above.
Different research during recent years has also analysed the importance of
hydrological data for large scale, long-term investment for general economic
development.
The conclusions of some of these studies may be summarised as follows:
It is considered that, if the period during which hydrological records have been
kept is not sufficiently long or the data are not sufficiently accurate, then the
investment in the economic development will be larger than is necessary, or the
possible production of the plant will be less, due to its smaller size. In both
cases the result will be a loss to the overall econon&.
Experience shows that various hydrologic parameters could have economic
importance for different stages and purposes of water resources development.
For example - flow variability and drought occurence for the design of storage
reservoirs; flood occurences for the design of spillways and other control works;
lesser importance is attached to the mean flow.
one a uthod
However, as can be cited after
r/ D. Johanovic. Abstract. Vhe Role of I-Qrdrologg and Hydromteorolow in the
Economic Development of Africat! Ma, No. 301, 1971.
2/
K.C. Wilson Cost-benefit approach to hydrometric network planningtt Water Res.
Research. October 1972.
685

686
Wariation of mean annual flow, depending upon hydrologic data, and length of
observation leads to overestimation or underestimation in determination of
sizes of water engineering structures, determination of regimes, capacities
of plants, irrigated areas" etc.
In fact the following implication of an underestimation of the mean annual
flow in the planning of water resources developnient might ba indicated:
(a) Smaller sizes of water storage capacity resulting in limited possibilities
of regulation of flows with subsequent adverse impact on:
(i) possibilities of self purification of water;
(ii) availabilities of water supply for drinking and industrial purnoses;
(Xi) potential of production by hydro-electrical plants;
(iv) quantity of water for irrigated areas;
(v) navigationaï capaciw of rivers;
(Vi) inadequacy of water storage size:: requires additional investments for
increasing dans, canals, etc. at a later stase.
On the other hand, over-estimtion of inem annui flow can lead to the following
implications :
(i) oversizing of iiriter engineering strictures, low efficiency of their
operation, larger investments in conparison with normal;
(ii) insufficiency of water for designated irrigation area3
(iii) energy production below planned target;
(iv) lesser dilution of effluents discharged and slower processes of self
purification of water.
Studies of the value of hydrological data are being carried out in several
countries. Comprehensive studies were conducted jointly by the United States Corps
of Engineers and the Geological Survey (1970) with the purpose of evaluaiing the
iiorth or" hydrological data for determination of the optimum water storage conserva.tion
capacity. Some American researchers have developad a mathematical relation betireen
the llworthll of data and the length of periods during which they were recorded.
Some research has been carried out to estimate the possible loss due to
inperfect information. The research, for instance considering optimal reservoir
design, analytically defines the opportunity loss as the difference between net
benefits associated with different hydrological data depending upon streamflow
record length. Reservoir designs are obtained by simulating flovs and selecting
that combination of storage capacity and target yield which gives the greatest
net benefits. Graphic functions obtained by this research shou that opportunity
losses decrease rapidly due to increasing the length of streamflow observation,
achiedng very small magnitude beyond thirty years of observation; this conforms
to many practical observations. Various types of reseaxch and observations show
that the cost of obtaining addition& hydrological data i.e. increasing the length
of observation is insignificant in relation tothe reduction of the esrected

opportunity lox.
687
Some CanaCZan research developing generalized computer programmes to relate the
costs of operating and intensif'ying a hydrometric network to the resulting increases
in the accurecp oi: the three parameters mentioned above should be noted.
The available experience of differenct ECL countries confirms to an extent the
conclusions of the research mentioned.
-
-
As ior example in the USSR, it is considered:
the economic benefits of the hydrometeorological service due to which the
hydrological and meteorological observations can be obtained are a high as
one billion roubles, i.e. 4-5 tines the amount that is spent on maintaining
this semice;
the introduction of the use of hydrological forecasts in national planning
made it possible to raise by 10-15 per cent the efficiency of water
installations and to obtain correspondingly higher profits2/.
In the UnLted Kingdom and France, the relrvant benefits are estimated to exceed
the national hydrometeorological budgets at least 20 timed.
However, in many countries, especially in the developing ones, the povement
agencies responsible for h@rological observations do not have sufficient funds
available for the development of adequate nationwide hydrological network$. This
insufficiency of financial resources for the collection of hydrological data was
also emphasized at the ECE Seminar on certain uater problems, convened in Zagreb
in 1971.
The USSR experiences show that in the absence of hydrometeorological observation
data in an area selected for construction, a special hydrometeorological investigation
should be conducted with an expenditure of 2-3 thousand roubles for each million
roubles invested. This amount is considered to be an econom if, as a result,
sufficient hydrometeorological data proved to be available. Experience in other
countries confirms that the expense of acquired adãitional hydrological data
is much less than the losses involved in water engineering construction,the design
of which is based on inaccurate data. In this connexion as a positive experience,
a considerable extension of national and regional hydrological networks is taking
place. It should also be noted that in some countries the application of automatic
monitoring stations to control the quality and quantity of a river flow has
considerably extended during recent years. This modernization increases the
reliability of recorded data indispensable for accurate planning and design.
2/ E.J. Tolstikov, Vhe Benefits of Hydroirieteoroloefc Services in the UcsRtt
'dl40 Vhe Economic Benefit of National Meteorologic Services'World Weather
Watch No. 27 - i968
i/
Richard D.A. Kill IXydrological and Hydrometeorological data as essential
parameters for design or economic development projectsf! - ld4l Eo. 301

68 8
In spite of the sound persuasiveness of all the research mentioned above,
armther trend of research should be pointed out. Several studies and research
have been devoted to Solve the problem of whether it is desirable and profitable
to postpone the development of water resources or the construction of certain water
cngineering projects if the hydrological data for the river or particular site
investigated are insufficient. It is considered by some of the authors that, by
postponing the construction, the realization of the net benefits of that construction
is also being postponed.
The available results of research conclude that the risk of such postponement
must be carefully evaluated. Nore than that, some research clearly rejects
postponement as a protitable course of action. It is considered that only an
exceedingly loid discount rate makes the minimum total cost of postponement
e.g. for one year, equal to the cost without postponement.
Taking into account the definite discrepancy between these conclusions and
those previously mentioned we feel that further research should be continued,
emphasizing not only purely economic aspects, but also taking into account social
and technical aspects, including first of all the problem of safety of structures
and subsequently of the population. The lack of unaniixi~iy, even in a purely
economic approach, concerning the postponement of projects is to be noted.
Some authors concludeb/:
tlSince demand for project outputs is presumably growing over time, the more a
project is deferred, the more quickly it is likely to be used, and the greater the
benefits generated per time period of project life will be.”
The conclusion seems to he very sensible.
Pldn,? and desiminn of flood protection
This cspect deserves special attention, bearing in mind that economic losses
due to floods in the river basins of the world continue to be very high. Noreover,
the further extent of the economic development of regions being threatened by high
flows leads to further growth of economic damage from floods. Just one example, fron
the experience of the United States, shows that the frequency of floods, causing
major roperty damage of $50 million or more were increased aimst three times since
19L&d Based on the currßnt status of flood control works and project conditions
of flood pldm use and development the total annual floodcbage potential for the
nation is anticipated to increase from $1.7 billion in 1966 to $5.0 billion in 20Zoz/.
It is considered that, for elaboration of efficient national flood proLection
policies, the data on distribution of river flow durine a year, asml1 as the exact
characteristics of floods including maximum discharge, duration, tlme of flood end
volume of the flood flow are of considerable importance.
However, the determination of these characteristics becomes verydifficult if
hydrological observations are insufficient, as was mentioned above.
m e s
Geophysical Union, bJashington D.C. 1971
IfThe Nations Water Resources11 W.R.S. United States, 1968, 5-2-2.
1/
\J. IIowe naenefit-Cost Analysis for Water System PlanningIl - American

Experience shows that the frequency of peak floods, determined on the basis
6 89
of a short ceriod of observations cm be several times lower than the adequate
vdue, and that leads to considerable damage and to catastrophiee. The foll-owing
consequences of inaccurate flood forecasting should be pointed out as regards
water resources development and planning and designing of flood protection.
(a) Overestimation of flood discharge - leads to unnecessary expenses in
weter engineering construction;
(b) Underestimation of flood discharge - leads to full dektruction of the
sl.r.ucture, resulting in damage and possibly in loss of human lives in
the region.
Thus, proper estimation of possible flood peak, depending upon the quality
and accuracy of hydrological data is very important. At the samc tine the
experience of many countries still shows that: the continuing groi.rth of economic
damage to individuals and to the national economy in many regionu of the world,
caused by river floods, can be explained not oniy by the spontaneity of the flood
phenomena, but also by the lack of adeguate organization, insufficient hydrologic
observation and the necessary financial means, which very often are considerab1.y
less than the value of the damage caused. Talcing this into account the ECZ
Cohtiee on Water Problems has recently initiated studies on the available
experience in rational methods of flood control planning in river basin
development with the purpose of extending this experience to al1 ECd countries.
In spite of the concept generally adopted that the development of sufficient
hydrological observation is very important for the planning of proper flood
protection and the organization of adequate operational measures to prevent
considerable damage, nevertheless the available infornation from some countricc
indicates :
- a lack of reliable data regarding rainfall intensities and corresponding
stream flow;
-
-
insufficient studg of floods based on a detailed analysis of recorded flows
so that water management authorities in some river basins or their parts are
unable to arrive at more realistic estimates of expected floods;
the insufficiency of the empirical and rather arbitarg methods used up to
now in some countries for estimating peak floods; inadequecy of such methods
has been proved and should no longer)%cceptable, in dew of the magnitude
and importance of the projects undertaken.
It is also indicated in some countries that from the point of view of
safety and also from the economlc angle, the necessity for current study and
evaluation of the magnitude and frequency of the occurence of floods in
connexion with the economic development of Tiver basins has become essential.
The existence of dams in the vicinity of populated areas necessitates the closest
study of the anticipated probably floods, in order to provide adequate spillway
capacity for the safety of the dams, and the downstream areas.

690
In some countries flood forecasts are not yet of the desired accuracy, thus
increasing Lhe danger of economic damage or reducing the efficiency of water storages
down the river, while accurate flood forecasting can increase the overall potentialitie:
of a multi-purpose project.
The disoytrous floods which occurred in several regions of the world in the recent
decade, causing considerable loss of life an9 treinendous economic damage, pointed to
the need to extenä flood forecasting and warning syotems in many countries, especially
in their m.ìt vulnerable river basins. Relevant units established by national kjdro-
meteorological services or by central water and power agencies confirm their
effectivenes: It is considered that ths expenses for the maintenance of these unita
is only of mciest cost compared with the peins achieved by timely forece-sting. As
an example, considerable economic benefits are accrued in mqv countries when flood
forecast? nre usea to enable protective measures to be taken against the affects of
floods. In these cases, the economic benefits through iorecasts are caisulô.ted as
the differenw between profits from proteoted zoner or areas.
The use of flood forecasts in the IJSCR reduces the cost of damage äue (* floods
by 20-30 per 'cent. merience in the Uni'reä -:-cates sonfirms that, the redudion of
flooe .iamagt aione would far outweigh the tow?. cost of .the hydrolo@.cai forecasting
serviqe, including the proposed network.:-'.
8,
A:$ participants were informed at the
Uniteu Nations inter-regional Seminar .,r, '?loo6 DamP-Ze Prevention Measures and
Management, convened in Saptenber 1969 i,. Tbidsi, ITSSR, the annual savings due
to the exlsting flood warning systems in the United States exceed $30 million a year.
Experience in India also confirms that the early expenditure on the maintenace of
the unit in the central water and power commission as well as the cost of the necessary
equipment - is only a very moderate investment compared with the benefits which have
been brought by forecasting.
Importance of accurate data to show flow formation regardinp human activity at
watersheds of rivers
Attention should be drawn to the importance of strengthening research to analyse
the influence of human activity in watersheds on river flow. It is considered in some
countries that the euccessful design and operation of water engineering structures
could be possible if sufficiently accurate data to show the hydrological characteristic
of river basins under natural conditions of flow formation, with regard to the scale
and direction of changes caused by humans, were available. The importance of this
aspect has been emphasized among other publications by the ECE Manual mentioned in
the first part of this paper.
8/
s/
M.A. Kohïer (United States National Weather Service) Wasebook on Hydrological
Network Design Practicet' - WMO No. 324, 1972
Sh.M. Manshard (Central Water and Power Codssion) tTlood Forecasting and
Flood Warning in India". Seventh Symposium - The Civil and Hydraulic Engineering
Dept. Water Resources. May 1971.

II. The inpact of a deficiency of i1ydrolopicai data on main water users
The impact on the following water users is considered:
(a) 1)onestic water supply
(b) mdustrial water supply
(c) Xydro-power generation and thermal power plant water supply
(d) irrigation
(e) 1knigation
(f) Fisheries
(g) in-ter pollution control
Ex'üen3.ve research was carried out in i?ariy AC3 countries devoted to t'ne appreisal
of the impact of a deficient river flow aroused, among other reasons, by the lack of
hydrological obsorvations and insufficiently accurate forecasts. One such research
project has been carried out during -the recent decade 'by the central research institute
on water problems in the USSR (btinsl~)~. Analysing dieierent nethodx of estimation
for the appraisal of the impact of doficient water flow, the authors indicate that, in
sonle case:, estbation becomes difficillt by reason of too 5hoi-t a period ol 1iyarologica.l
obsemration:;. It is aïs0 pointed out that special estimation techniqucr to be applied
in cases of short available periods oi hydrological observations, which could be applied
for economic analysis, have not yet been created.
However available results of research carried out in the same corntry suggest
some methoas for use under conditions of deTiCient hydrological dataJ.
11
They atter-pt
to find a relztion between the extent of limitation of the water siipply and the
reduction of economic activities (e.g. the industrFal output) of a certain enterprize
in order to asress the economic damage caused by water shortages.
Domestic water supplx
It is generally adopted that domestic uater supply does not allow in-Lerruptions
i.e. the pyaranteed water supply for these consumers must be doze to 100 per cent.
Industrial water siipply
It is considered that the magnitude of economic damage connected. with stoppage
or partial reduction of water supply is considerably influenced by the quality of
hydrological prognosis.
Some researchers in countries with a centrally planned econoqv suggest to
subdivide the dmages, as followsw:
o/ 1.1. I-fechetov, V.N. Pluzhnikov, L.J. Popov IfBalance of Water Resources and Needs -
the method of optimal planning of water ressurces dsvelopmentl~ Multiqurpose water
resources development and conservation. Minsk. 196û.
Ir/ V. Andreyanov IiInternal distribution of river flowl'. Gidrometizdat. USSR.
Centrcrl Research Institute of Water F'roblecis. USSR. Minsk. "Miilti-purpose
development and quality conservation of water resources" - 1968.
6 91

692
(a) direct damages (expressed by direct cost of unproductive forms of enterprises
due to water stoppage or shortace);
(b) indirect - measured by the loss of profit during the time of stoppage or
reduction of water supply.
The lack or insufficiency of research to define a relation between -Lhe extent
of the limitation of the water supply and the relevant economic damago to clifrerent
tLJhnological processes is being noted in some branches of production.
Generation of electric enerpy by hydro-electric plants and themai power plants
Proper flow predictions are important +o obtain optimum utilization of *dro-electi
Timely 8n.d accurate ïlood forecasting in periods of lieais.
and tharml. power plants.
rains increase:. tlie overall potentiality of a multi-purpose project. Accurate
forecasting is especially important bearing in nind tho fact that hyciro-power
resources, depcnd on neteorological m u hydrological conditions vhich

693
Il, is emphasized that not onlx the entent of reduction o€ water supply is important
here but also .the seasonal period of this reduction which could be -

69b
pollutant to the river in a period uith lower flow could create dangerous pollution
concentrzìions in the lower reaches of the -river. %u hydrological forecasting
has therefore aezome more iraportant fÒP watcr quality control in the river basin.
The other nspect - is the importahce of better organization and modern
instmntation of water quality contrbl trg hyd.rolo~&cal sedces. The ECE Manual
mentioned above gives particular at4ention to water- qdity/data and research
simultaneously with hyàrological resáárch, reqtliring the extehsion and hprovement
of the system of observations on water”qua1ity in streams and reservoirs.
Information from the ECE countries, posticulariy diseussed at the ECE Sendnar
in Zagreb, 1971, points out khat considerable improvement in these functions is
being iqîemented in mmy countries,
As regards the assessment of economic danage caused by water quality
deterioration due to flow reductionsin rivers, including reasons caused by
untimely and inaccurate hydrological forecasts, there are as yet no methods
of assessment generally recognized or adopted. In some of the ECE countries
it is considered that the infringemant of sanitary water conditions in a river
basin, due to low flow, and the relevant damage dee economi=. assessment.
Further comprehensive studies seem to be necessary in tlis field.
The same is said of the economic assessment of the impact of water quality
deterioration on water recreation. Correspondingly it 1s considered that it is
difficult to substantiate economically permissible standards of concentration of
polluters in a river basin.
However, there is no unanimity on this point. Research in some other ECE
countries show Lliat the increase in the present value of direct quantifiable
recreation benefits of inproved water quality, for example in the Delaware River
bacin could be as high as $300-350 niillion for the highest water quaïity clase
adoptadw.
Sediment control
In e:cicnding and further improving hydrological services and their forecasting,
sediment control shodd not be neglected.
Further water resources development and growth of human activity on the watersheds
m e s quantitative and qualitative sediment data very important as a part of
hydrologic controls for different periods of the year. The iniportance of this
problem could be seen fromthe experience of q ECE countries.
It has been recognized by nany countries, that accumulation of sediments in
niagy cases ohortcns the effective economic life of water engineering structures such
as water-storagc ’, causes ii tensive wearing of pumps and turbines, requiring in all
cases new capiti investments. As Is known, the problem can be grcatly mitigated
by cidensive and complex control measures, including inproved forecasting.
12/ A.B. Iïneese and B.T. Bower. AnaJYSeS conducted by the University of Pennsylvenia.
Wanaging water quality’t. John Xopkins Press, Baltimore, 1968.

Experience in the United States shows that the damage from sediments to water
management and national econow reaches more than $500 million a nnuala.
Data on ice and slush ice conditions
Inaccurate forecasting and warning ice and ice slush conditions cause
considerable damage to hydro-electric plants and the energy consumers,
causing reduction of industrial production in some of the regions of the middle
climatic and rmuntainous zones.
1.
Conclueions
In conclusion we would Uke to emphasise the following:
Available experience and research being carried out in ECE countries clearly
demonstrate the considerable impact of insufficient hydrological data and
forecasting in all phases of planning of water resources development and use,
which in their turn could have an *act on economic development of certain
regions.
2.
695
assess the value of hydrological data already have achieved certain positive effects,
for example the development OP mathematical relations between the present value of
the value of data and the length of their record, as well as development of some
methodology to assess the impact on main water users.
3. Available research clearly shows that the cost of additional hydrological data -
i.e. increasing the length of observation - is imd@ficant in relation to the
expected losses due to insufficient data.
in many countries, especially developing countries, are very often evaluated as
inadequate for proper development of hydrological networks as was revealed and
emphasized by different conferenoes.
4. In spite of considerable research being carried out in ECE and other countries,
as veli as various studies by specialized organizations, the following main
deficiencies in hydrological observations and research are noted in different
countries, which render a certain negative impact on water resources development
-
Available attempts in ECE countries to create and improve &.sting methods to
and effective use:
However, insufficient funds available
the lack of reliable data on rainfall intensities and corresponding streamflowsj
practical usage of empirical and sometimes rather arbitary methods for estimating
peak floods; the lack of scientific studies on evaluation of the magnitudes
and frequency of the occurrence of floods; insufficient studies and research to
assess t h impact of human activity in watersheds on waterflow; lack of
special estimation techniques to be applied for economic analysis in cases
of available short periods of hydrolo,.Lcal observations; deficiencies of
u "Xnvironmental Problems". Monograph presented by the United States Government
to ECE. January 1970.

696
available methods of long-tem hydrologi.cal forecasts; insufficiency of
research to define a relationship between the extent of the limitation of
,he water supply and the relevant economic damage to different technological
processes; lack of generally recognized methods of assessment of economic
damage caused by water quality detexioration.
Strengthening of the research regarding the deficiencies listed above Is
considered as important and urgent.