Hydrological modelling of the Zambezi catchment for ... - TU Delft

Master's Thesis
Hydrological modelling of the
Zambezi catchment for gravity
measurements
A.M.J. Gerrits
February 2005

Master’s Thesis
Hydrological modelling of the Zambezi catchment for gravity
measurements
Delft, February 2005
Author:
A.M.J. Gerrits
Graduation committee:
Prof.dr.ir. H.H.G. Savenije (Delft University of Technology, CiTG, Water Resources Section)
Ir. W.M.J. Luxemburg (Delft University of Technology, CiTG, Water Resources Section)
Dr. L.L.A. Vermeersen (Delft University of Technology, L&R, Astrodynamics & Satellite Systems)
Ir. N.J. de Vos (Delft University of Technology, CiTG, Water Resources Section)
Dr.ir. M.G.F. Werner (WL | Delft Hydraulics)

PREFACE
Preface
This Master’s Thesis describes the results of my research on hydrological modelling of the
Zambezi catchment for gravity measurements. The research was carried out at the Water
Resources Section of the faculty of Civil Engineering and Geosciences of Delft University of
Technology. During this research project, I investigated several rainfall algorithms to obtain
accurately input data for the hydrological model, recalibrated the Zambezi-model and
executed a sensitivity analysis on the model, in order to obtain a hydrological model with less
error as possible. This is necessary to be able to compare hydrological data with data from
gravity measurements provided by the satellite GRACE. In theory, it should be possible to
detect variations in the water storage from gravity changes, caused by mass redistributions. In
this thesis a first qualitative comparison is made.
In relation to my research I owe gratitude to a lot of people who have encouraged and helped
me during my study. First my words of thanks go to my graduation committee. I would like to
thank professor Savenije for his helping thoughts on the improvement the Zambezi model in
Delft and via long distance communication to Australia. I would like to thank Wim
Luxemburg for his critical reading of my reports and for being a wonderful colleague during
several field trips during my study time. I would like to thank Bert Vermeersen, from the
faculty of Aerospace, for introducing me into the world of satellites and gravity. I would like
to thank Nico de Vos for being a nice room mate and giving me advise about model
uncertainties and last but not least Micha Werner, from WL | Delft hydraulics, for providing
me his GLUE code and expertise.
I would also like to thank Jeroen Aerts and Aad Versteeg from the ‘Vrije Universiteit’ of
Amsterdam for providing me and helping me with the model environment STREAM.
I would like to thank my Italian MSc.-colleague, Alessandro Ficara, for starting with
improving of the Zambezi model.
Thanks to Hessel Winsemius for being a nice study friend and helping me with Matlab and
special thanks to Hanneke Marcano Bakker, from the faculty of Aerospace, for providing me
analyzed GRACE data. I wish Hessel and Hanneke good luck with continuing this very
interesting and promising topic about gravity measurements in hydrology.
iii

PREFACE
Furthermore, I would like to thank all my colleagues of the Water Resource Section during
my work as student assistant. They provided me a nice and stimulating environment to
accomplish my Master’s Study in hydrology. They motivated me and gave me the opportunity
to learn more about hydrology than just the knowledge in the regular lecture notes of the
curriculum.
Finally I am very grateful to my boyfriend Jeroen Coenders. Besides helping me with
computer related problems, he encouraged me to learn a new programming language,
stimulated me during my research by listening and talking together about the gravity
measurements and he supported me to develop myself in a way to come to a fine research
result.
Miriam Gerrits
Delft, February 2005
iv

TABLE OF CONTENTS
Table of contents
PREFACE..............................................................................................................................................................III
SUMMARY ..........................................................................................................................................................VII
ACRONYMS & SYMBOLS .............................................................................................................................XI
2.1 ACRONYMS............................................................................................................................................XI
2.2 SYMBOLS..............................................................................................................................................XII
1 INTRODUCTION ........................................................................................................................................1
2 PROBLEM ANALYSIS ..............................................................................................................................3
2.1 GRAVITY MEASUREMENTS RESEARCH................................................................................................3
2.1.1 Problem definition of the research ................................................................................................7
2.1.2 Objective of the research.................................................................................................................7
2.1.3 System boundaries ............................................................................................................................7
2.2 APPROACH ..............................................................................................................................................8
2.2.1 Remarks..............................................................................................................................................9
2.3 PROBLEM DEFINITION AND OBJECTIVE ............................................................................................. 11
2.4 STUDY AREA DESCRIPTION: Z AMBEZI RIVER BASIN...................................................................... 11
2.4.1 Climate ............................................................................................................................................. 15
2.4.2 Land cover / Dambos..................................................................................................................... 18
3 HYDROLOGICAL MODEL INPUT DATA...................................................................................... 21
3.1 PRECIPITATION DATA.......................................................................................................................... 21
3.1.1 Microwave-Infrared Rainfall Algorithm (MIRA)...................................................................... 22
3.1.1.1 Data.......................................................................................................................................... 24
3.1.1.2 Performance............................................................................................................................. 25
3.1.2 Famine Early Warning Systems (FEWS) ................................................................................... 27
3.1.2.1 Data.......................................................................................................................................... 31
3.1.2.2 Performance............................................................................................................................. 31
3.1.3 Comparison precipitation algorithms......................................................................................... 35
3.2 POTENTIAL EVAPORATION DATA ....................................................................................................... 38
4 HYDROLOGICAL MODEL DESCRIPTION................................................................................... 41
4.1 STREAM.............................................................................................................................................. 41
4.2 Z AMBEZI SCRIPT ................................................................................................................................... 42
4.2.1 Conceptual rainfall-runoff model of the Zambezi in STREAM............................................... 45
4.2.1.1 Land process: evaporation through interception...................................................................... 46
4.2.1.2 Land process: transpiration...................................................................................................... 48
4.2.1.3 Land process: ground water flow ............................................................................................ 53
4.2.1.4 Land process: capillary rise ..................................................................................................... 56
4.2.1.5 Lake process: evaporation ....................................................................................................... 57
v

TABLE OF CONTENTS
4.2.2 Routing in spreadsheet .................................................................................................................. 57
4.2.2.1 Muskingum routing................................................................................................................. 57
4.2.2.2 Reservoir routing: Kariba ........................................................................................................ 61
4.2.2.3 Reservoir routing: Itezhitezhi .................................................................................................. 62
4.2.2.4 Reservoir routing: Cabora Bassa ............................................................................................. 63
4.2.2.5 Reservoir routing: Nyasa ......................................................................................................... 64
5 ERROR- & SENSITIVITY ANALYSIS .............................................................................................. 67
5.1 ERRORS .................................................................................................................................................. 67
5.2 M ODEL RESULTS ..................................................................................................................................72
5.3 GENERALISED LIKELIHOOD UNCERTAINTY ESTIMATOR (GLUE)............................................... 81
5.4 IMPLEMENTATION OF GLUE FOR THE ZAMBEZI-SCRIPT............................................................... 86
5.4.1 Determination of dominant parameters...................................................................................... 88
5.4.2 Objective function and likelihood measure................................................................................ 91
5.5 GLUE RESULTS .................................................................................................................................... 91
5.5.1 GLUE for parameter uncertainty................................................................................................ 92
5.5.2 GLUE for input uncertainty.......................................................................................................... 97
6 GRAVITY MEASUREMENTS AND HYDROLOGY..................................................................103
6.1 BACKGROUND ....................................................................................................................................103
6.2 RELATION SURFACE MASS TO GRAVITY..........................................................................................105
6.3 ERROR SOURCES IN GRACE DATA .................................................................................................106
6.4 DIRECT ESTIMATION OF HYDROLOGY FROM GRACE DATA.......................................................108
7 CONCLUSIONS & RECOMMENDATIONS..................................................................................113
REFERENCES ....................................................................................................................................................119
2.1 LITERATURE........................................................................................................................................119
2.2 DATA...................................................................................................................................................123
APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI........................................................ 127
APPENDIX 2: FAO SOIL UNITS DESCRIPTION............................................................... 153
APPENDIX 3: ZAMBEZI SCRIPT..................................................................................... 157
APPENDIX 4: MODEL RESULTS..................................................................................... 161
APPENDIX 5: GLUE SCRIPT......................................................................................... 167
APPENDIX 6: ROUTING MODEL .................................................................................... 173
APPENDIX 7: GRACE RESULTS ................................................................................... 177
vi

SUMMARY
Summary
Hydrologists are interested in determining each term of the water balance, so they can predict
for example floods and droughts. Besides the problems with determining the precipitation, the
evaporation and the river discharge, most difficulties occur at determining the storage stock. It
is possible to carry out ground measurements, but this will only result in point values and is
not providing spatial information. Also microwave remote sensing and modeling techniques
have their limitations. However, with the launch of the twin satellite GRACE (Gravity
Recovery and Climate Experiment) on the 17 th of March 2002 new opportunities have arisen
for measuring the water storage stock.
GRACE is able to measure tiny differences in the Earth’s gravity field. These spatial and
monthly changes are caused by variations in the mass distribution on the surface of the earth,
which are caused by processes like: ocean current changes, terrestrial water storage changes,
postglacial rebound, atmospherically changes, etc. Processes like the movement of continents
are neglected, because they only vary on geological time scales and not on a monthly time
scale. Despite the different processes, which are influencing the gravity measurements, it
should be possible to filter out the hydrological (= terrestrial water storage) signal. To find the
relation between the gravity and the storage stock, first a hydrological model will be built,
which will simulate the terrestrial water storage on a monthly basis. From these results the
spherical harmonic coefficients, which are describing the Earth’s gravity field, will be
calculated and compared to measured GRACE data.
Hydrological model
For the above described approach, the Zambezi catchment in Southern Africa is chosen as
study case, mainly because of its large basin size and the absence of tidal influences. Because
the quality of the precipitation input data is very important for the performance of the
hydrological model, different rainfall algorithms are investigated for the period after 1992.
First the MIRA-algorithm (Microwave-Infrared Rainfall Algorithm) is researched. This
algorithm calculates from IR-measurements rain rates and combines these with Passive
Microwave data in order to obtain optimized rainfall estimates. If the estimates are compared
with ground measurements it can be concluded that the performances is fair: the correlation is
high, but the rainfall is overestimated. To compensate for this deficiency the MIRA-data
needs to be corrected.
vii

SUMMARY
The second rainfall-algorithm is the FEWS RFE 1.0. This algorithm first estimates the rainfall
based on thermal Infrared data with the GPI-algorithm. Next, the estimates are corrected by
fitting the values to ground stations. Finally, an orographic lifting and humidity model is
applied on the estimates. A comparison between the RFE 1.0 estimates and the observed data
shows that the algorithm overestimates the rainfall somewhat. However, the correlation
coefficient is 0.85, which is quite low compared to the other algorithms.
The last algorithm, which has been investigated, is the FEWS RFE 2.0. This algorithm first
estimates the rainfall separately on three satellite sources: Special Sensor Microwave/Imager,
AMSU-A microwave and METEOSAT Infrared. Next the estimated are weighted to reduce
the error. The last step is to correct the weighted estimates with observed data. The
performance of the RFE 2.0 is very good, with a correlation coefficient of 0.97 and a very
little overestimation.
Finally after a comparison between the algorithms, the corrected MIRA-algorithm is chosen
for the period 1993-2000 and the FEWS RFE 2.0 for the period 2001-2003. For the period
before 1993 interpolated rainfall data from ground stations is used.
The hydrological model, which is built, consists of two sub models: a water balance model
created in STREAM and a Muskingum routing model. The water balance model is a storage
reservoir model, which contains three levels. First, the shallow soil from where interception
takes place. Second, the unsaturated zone with the transpiration process and third, the
saturated zone. From the saturated zone three ground water outflows are implemented: the
saturation overland flow, the quick flow and the slow flow.
The model is only calibrated for the western part of the catchment on the locations Lukulu
and Victoria Falls, because in the western part less influence of tides exists. Looking at the
results it can be concluded that the model is quite reasonable in simulating the discharge with
a Nash-Suttcliffe coefficient of 0.70 and 0.68 for Lukulu and Victoria Falls respectively.
However the model underestimates the discharge slightly, especially the high peaks.
GLUE
To obtain more knowledge about the sensitivity of the model, the GLUE-procedure (Beven,
1989) is used twice. First, the procedure is carried out on the uncertainty of four parameters
and second, the procedure was executed for the uncertainty in the precipitation and potential
evaporation input data of the Zambezi model.
viii

SUMMARY
The results of the GLUE for the uncertainty in the parameters showed that it is not possible to
define an optimal parameter set. Only the separation parameter cr shows an optimum around
the default value. The other parameters are not very sensitive on the discharge output. From
the difference between the 5% and 95% uncertainty bound of the total water storage it appears
that an uncertainty of 30 mm can be expected due to parameter uncertainty.
The uncertainties in the total water storage stock due to input uncertainties are much higher
with an average of about 310 mm. Because this is mainly caused by the highly sensitive
precipitation data, this emphasis the importance to obtain good rainfall data as input for a
hydrological model. From the GLUE results it also appears that the optimal precipitation data
should be multiplied by a factor of 0.95. Such an optimum could not be found for the
potential evaporation data.
GRACE
Besides errors at the field of hydrology also errors are introduced by GRACE. Next to the
measuring error, which is increasing significantly by an increasing degree, also errors occur
due to truncation, interpolation, leakage and errors due to the not correct removal of processes
like atmospheric pressure, postglacial rebound, etc.
As a first attempt GRACE data (spherical harmonic coefficients provided by GRACE) is
transformed into a hydrological signal, which is called the ‘direct estimation of hydrology
from GRACE data’-method. Comparing the calculated hydrological signal from GRACE with
the Zambezi-model results, shows that in qualitative terms a clear relation exists. Only the
amplitude from the GRACE-data is higher then the amplitude of the Zambezi-model. This
could be due to leakage error. By using averaging kernel functions this error could be
reduced.
The next step is to compare the GRACE and the hydrological data spatially and try to find the
relation by the use of the ‘inverse estimation of GRACE data’-method. This method
calculates from the hydrological model output the GRACE data in term of spherical harmonic
coefficients. However, at this stage of this research, this information was not available yet.
ix

SUMMARY
x

ACRONYMS & SYMBOLS
Acronyms & symbols
2.1 Acronyms
ATI
CPC
DEM
FEWS
GIS
GLUE
GPI
GPS
GTS
IR
ITCZ
MIRA
NASA
NDVI
NOAA
NS
PMM
PMW
RFE
RMSE
RS
SAFARI
SD
SSM/I
TRMM
WAD
Area-Time-Integral
Climate Prediction Center
Digital Elevation Model
Famine Early Warning Systems
Geographic Information System
Generalized Likelihood Uncertainty Estimation
GOES Precipitation Index
Global Positioning System
Global Telecommunication System
Infrared
Inter-Tropical Convergence Zone
Microwave-Infrared Rainfall Algorithm
National Aeronautic and Space Administration
Normalized Difference Vegetation Index
National Oceanic and Atmospheric Administration
Nash-Sutcliffe coefficient
Probability Matching Method
Passive MicroWave
Rainfall estimation
Root Mean Square Error
Remote Sensing
Southern African Regional Science Initiative
Standard Deviation
Special Sensor Microwave/Imager
Tropical Rainfall Measuring Mission
Weighted Average Distance
xi

ACRONYMS & SYMBOLS
2.2 Symbols
ρ
ave Average density of the earth (=5517 kg/m³) [kg/m³]
θ Co-latitude [-]
ρ
w Density of water [kg/m³]
P lm Legendre function [-]
φ longitude [-]
λ Spatial scale [km]
σ Surface density [kg/m²]
a Radius of the Earth [km]
blue Blue water (colors of water) [mm/month]
cap Capillary rise [mm/month]
caprise Actual capillary rise [mm/month]
C lm , S lm Dimensionless Stokes coefficient [-]
cr Separation coefficient [%]
D Interception threshold [mm/month]
dsw Open water evaporation [mm/month]
f Tree-top factor [%]
fd Flow direction [-]
green Green water (colors of water) [mm/month]
GWS Ground Water Storage level [mm]
GWSmax Threshold value for saof [mm]
GWSquick Threshold value for Qflo [mm]
int Interception by threshold method [mm/month]
intmar Interception by Marieke de Groen-method [mm/month]
k Recession constant overtop [month]
k l Elastic love number of degree l [-]
l Degree [-]
Ln_gwsmax Logarithm of gwsmax [mm]
m Order [-]
N Shape of the geoid [gal]
overtop Percolation [mm/month]
pe Potential Evaporation [mm/month]
pnet Net precipitation [mm/month]
prec Precipitation [mm/month]
qc Coefficient to determine threshold GWSquick [%]
xii

ACRONYMS & SYMBOLS
qflo Quick flow [mm/month]
qout Runoff [m³/s]
rtq Recession constant Qflo [month]
rts Recession constant Sflo [month]
runoff Discharge [mm/month]
saof Saturation overland flow [mm/month]
sflo Slow flow [mm/month]
smax Field capacity [mm]
su Storage Unsaturated zone [mm]
subsub GIS-map with the sub catchments [-]
tp Potential transpiration [mm/month]
tra Actual transpiration [mm/month]
water GIS-map with the water bodies [-]
wetland GIS-map with the wetlands [-]
white White water (colours of water) [mm/month]
W l Gausian filter weights [-]
xiii

ACRONYMS & SYMBOLS
xiv

INTRODUCTION
1 Introduction
High precipitation can result into large floods. However, the response of a catchment on
precipitation depends on more. Sometimes, the response is limited after intensive rain. At
other times, big floods can occur after an average rainstorm. Besides non-uniform appearance
of rain over a catchment in time and space, the variability in response is caused by the initial
state of the water storage in a catchment at the beginning of the rainstorm.
Because of this for discharge forecasting, it is of importance to know the actual state of the
water storage stock. Most of the stock appears as ground water, which in quantitative terms is
hard to assess for a complete catchment. However, with the introduction of the satellite
GRACE there will be new opportunities. GRACE is a satellite, which measures the local
gravity field. This gravity field is related to the amount of mass including water stock. If there
is a possibility to find a relation between GRACE data and the water stock, this would
enhance prediction of floods and droughts. For finding this relation a hydrological model will
be built and the results will be compared to the measured data of GRACE. Before making this
comparison it is important to obtain knowledge about the errors and sensitivity of the
hydrological model and GRACE.
In Chapter 2 the problem analysis of the research will be given. It describes the research
approach, the problem definition and gives a short description of the study area: the Zambezi
catchment in Southern Africa. From the problem analysis it follows that a hydrological model
is necessary. In Chapter 3 is described what kind of input data is used for the hydrological
model. Three different rainfall algorithms are investigated and assessed on there performance.
Next in Chapter 4 the used hydrological model is described and in Chapter 5 the model results
are presented. To obtain more knowledge about the sensitivity of the hydrological model the
GLUE procedure is used. This procedure is explained in Chapter 5 and also shows the results.
In Chapter 6, the theoretical relation between gravity measurements and hydrology is
described. It gives the possible error sources and it shows some preliminary results. Finally in
Chapter 7, the conclusions and recommendations are presented.
1

INTRODUCTION
2

PROBLEM ANALYSIS
2 Problem analysis
This problem analysis consists of two parts: one dealing with the larger framework of gravity
measurements and one about the MSc.-work. In Section 2.1 the analysis of the research on
gravity measurements is discussed. The approach to reaching the objective is presented in
Section 2.2. This approach results in the problem analysis of the MSc.-work, which is
described in 2.3. Last in Section 2.4 a description of the study area is given.
2.1 Gravity measurements research
Hydrologists are interested in the cycle of water on earth. They want to know how water is
moving around the world and its atmosphere to forecast for example floods, droughts, etc. For
the understanding of the water cycle the water balance is used. The water balance describes
the amount of water that is going into an area and that is going out of an area. The difference
between the incoming and the outgoing terms is the storage change.
Evaporation (E)
Precipitation (P)
A
dS
P−E A− Q=
dt
( )
River discharge (Q)
Figure 2.1: Catchment and water balance
The incoming term in Figure 2.1 is the precipitation [L/T], the main outgoing terms are
evaporation [L/T] and river discharge [L³/T]. The difference between them will result in a
change of the storage (dS).
3

PROBLEM ANALYSIS
Hydrologists attempt to determine each term of the water balance. Difficulties are:
Precipitation (flux)
We are able to measure precipitation with rain gauges. Besides the instrumentation
and installation errors, the most important error is the spatial variability. The data
from the rain gauges are ‘point values’ instead of spatial information and
interpolation techniques (e.g. Thiessen polygons, Kriging, Inverse Weighted
Distance, etc.) have to make a spatial distribution of that data. A way of reducing the
error of the spatial variability is to take a larger time step, so the spatial correlation is
higher. The use of satellites will also be helpful (e.g. TRMM or MIRA), because they
represent the spatial variability very well. On the other hand, satellites usually suffer
from inaccuracy at this moment.
Evaporation and transpiration (flux)
Evaporation also has the difficulty of spatial variability, but less then precipitation.
The largest problem in determining evaporation is that there is no possibility to
measure evaporation directly.
River discharge (flux)
At this moment, measuring the river discharge is the easiest of all terms of the water
balance. However, also on this flux large errors occur. Depending on the accuracy of
measurements and interpretation of measurements, errors of 10-20% are no
exception.
Storage (stock)
In general the amount of stored water consists of the sum of:
• Interception storage (storage on vegetation and the surface floor)
• Soil moisture storage (storage in the unsaturated zone)
• Ground water storage (storage in the saturated zone)
• Surface water storage (storage in lakes, rivers, etc)
It is very hard to determine the amount of water that is stored in a catchment, because
a large part is sub surface storage, which is the water that is stored in the ground.
Even if only is considered the change of the storage in time (which is the part we are
interested in), it is still difficult to quantify this term of the water balance.
4

PROBLEM ANALYSIS
GRACE
As explained above, the changes in sub surface storage are hard to determine. GRACE
(Gravity Recovery and Climate Experiment) offers new opportunities. GRACE is a satellite
that can measure the gravitational field of the earth on a local scale.This local gravity is a
measure for the amount of mass.
Figure 2.2: GRACE consists of a twin satellite [from: www.csr.utexas.edu/grace]
The GRACE-mission consists in fact of twin satellites (see Figure 2.2). Orbiting, they follow
with a distance of approximate 220 km (Davis, 2004). If the locally mass (i.e. the amount of
mass, which is most close to the satellite and thus has a large influence on the gravity
measurements) on earth is increasing the first satellite will accelerate and the distance
between the two satellites will increase (see Figure 2.3). This distance will be precisely
measured and together with GPS-satellites, it is possible to determine the gravity and the
position. Chapter 6 gives a more detailed description of GRACE.
5

PROBLEM ANALYSIS
≈220 km
distance
2
G⋅ m⋅M
m⋅v
Fg
= = F =
2
c
R
R
v
Fc
Fg
If M increases and R is negligibly
decreasing, then v will increase and
R
M
the distance increases.
F g
F c
= Gravitational force [N]
= Centripetal force [N]
G = Gravitational constant, 6.673E-11 m³ s -2 kg -1
m = Mass of object [kg]
M = Locally mass of earth [kg]
R = Radius from object to center of earth [m]
v = Velocity of object [m/s]
Figure 2.3: Working of GRACE, with simplified gravitation force equation, which assumes the mass as
a point mass.
Because GRACE is traveling each 30 days above each place on earth (sun synchronized),
GRACE will be able to give monthly mass variations on each location. These fluctuations in
mass are the sum of different effects/noises (all varying on a monthly time step), like
(Flechtner, 2000):
• Deep ocean current changes (reallocation of water through the oceans)
• Large-scale evapotranspiration (reallocation of water from the earth to the
atmosphere).
• Soil moisture changes (reallocation of water in the unsaturated zone)
• Mass balance of ice sheets and glaciers (melted ice has a larger density as frozen ice)
• Changes in the storage of water and snow of the continents (idem)
• Mantle and lithospheric density variations (the lithosphere is moving under the earth
mantle and vice versa)
• Postglacial rebound (through melting of the icecaps, decrease in weight, which causes
the rebound of the earth crust)
• Solid Earth's isostatic response (like the law of Archimedes)
6

PROBLEM ANALYSIS
Hence it is not possible to link an increase in gravity linear to an increase in the water amount
in a catchment. The hydrological signal has to be filtered out.
2.1.1 Problem definition of the research
For discharge predictions it is important to know the amount of water that is stored in a
catchment area. The measuring methods, which are used until now, are not sufficient to
determine the water content on the scale of a catchment.
2.1.2 Objective of the research
To find the relation between the GRACE data and water storage fluctuations.
2.1.3 System boundaries
In Figure 2.4 the total and the hydrological system boundaries are depicted. Satellites are
measuring the total system, which is represented in the figure. However, hydrology describes
just a part of that total system. Horizontally the hydrological boundaries are determined by the
catchment area. Vertically the boundaries are restricted by the canopy of the vegetation and
the deep groundwater. This implies that evaporation and river discharges are considered as
‘losses’ and that precipitation is an incoming flux.
E
P
Total system
boundaries
Q
Hydrological
system boundaries
deep gr.
deep earth
Figure 2.4: System boundaries
7

PROBLEM ANALYSIS
2.2 Approach
In order to find the relation between the GRACE data and water storage fluctuations in a
catchment, there is a need to know what the storage fluctuations were in a certain time period.
These fluctuations can be calculated with a hydrological model. In this case the model
environment STREAM is used. STREAM is a grid-based spatial water balance model, which
is able to make each time step grid-based maps with the change in the water storage. Read for
more explanation about STREAM Section 4.1.
From the obtained grid-based maps by STREAM, it is possible to translate the monthly
changes in water storage to the signal, which GRACE should have measured. GRACE
measures changes in gravity field (dG).
By comparing the calculated GRACE signals and the measured results of GRACE it may be
possible to find the hydrological signal inside the signal of GRACE. If that is successful it is
possible to directly derive the change of water storage from the GRACE-data (see Figure 2.5).
HYDROLOGY
AEROSPACE/
GEODESY
Real world
Input-data:
• Precipitation + error
• Evaporation + error
• DEM + error
• Land cover + error
• Etc.
Hydrological
model
GRACE measured (dG) +
Error
Comparison
GRACE calc signal (dG)
+ Error
Try to find
the relation
between
GRACEdata
and
water
storage
changes
dS+ Error map
Algorithm
Figure 2.5: Approach
8

PROBLEM ANALYSIS
As study case the Zambezi catchment in Southern Africa will be used (Figure 2.6). This
catchment is suitable first because its large area (1,3 x 10 6 km²), which is necessary for the
constrains in the current resolution of the GRACE-satellite. The second advantage of this
catchment is that a large part is out of the influence of tides, which affects the measurements
because they are changing within the monthly temporal resolution of GRACE. A third
advantage is the quite constant atmospheric pressure during summer, which is also a
disturbance with a monthly time step. And the last reason to choose for the Zambezi
catchment is that a hydrological model with a grid-based water balance model already exists,
which is very desirable for remote sensing applications.
Figure 2.6: Location of the Zambezi catchment.
2.2.1 Remarks
In using this approach attention should be paid to the errors introduced during the different
processes. Not having a good overview over the errors makes it difficult to do a good
comparison between the modeled gravity field and the measured field by GRACE. Figure 2.7
shows a more detailed picture of the existing errors.
9

PROBLEM ANALYSIS
Error(measurement)
Real world
Error(measurement)
Error(measurement)
River measurements
Satellite
measurements
GRACE
measurements
STREAM input maps
Error(interpolation)
Algorithm
1
Error(calculation)
STREAMmodel
dG measured
map
Error(algorithm1)
comparison
Error(model)
Error(model)
dG calculated
map
Calculated
River
discharge
Modeled
River
discharge
dS hydro map
Algorithm
2
calibration
Figure 2.7: Overview of errors. Hydrology: yellow left side. Geodesy/Aerospace: green right side.
Aerospace/Geodesy errors
Figure 2.7 shows three types of errors are shown. First there are measuring errors. GRACE
measures the distance between the two satellites. Although the equipment is very precise,
there will still be a measuring error. Also in the determination of the position of the two
satellites with GPS are deviations.
Second, there are algorithm(1) errors. GRACE measures the distance between the twin
satellites. The measured distance has to be converted to gravity field values. Some kind of
algorithm will do this, but there will be errors inside, because of unknown effects,
misunderstanding of the system, etc.
Finally, there will be errors (algorithm(2)) introduced by ‘up scaling’ the dS-maps, calculated
by STREAM, to the signal that GRACE would have measured.
10

PROBLEM ANALYSIS
Hydrological errors
Also at the side of hydrology are errors. First there are, just like at the GRACE-satellite,
measuring errors: errors in the measuring equipment. The images also have to be corrected for
the earth curvature, cloudiness, etc.
Second, there are interpolation errors. Although satellite date is quite continuous in time and
space, a certain interpolation technique needs to be applied to make the data completely
continuous (gap filling).
Last, there are model errors due to wrong model schematizations, wrong assumptions and/or
wrong parameter estimation. For example, the NDVI-maps (Normalized Difference
Vegetation Index) are used to select the wetlands and the high lands to say something about
the importance of capillary rise and the DEM (Digital Elevation Model) is used to determine
the flow direction of the water. These assumptions introduce errors. To get an impression of
the importance of the parameter errors, a sensitivity analysis is necessary.
2.3 Problem definition and objective
Resulting from the approach described in the previous section, the problem definition and the
objective of this study are:
Problem definition
In order to find the relation between the GRACE measurement and the water storage
fluctuation, there is a need to have knowledge about the magnitude of the errors introduced by
the hydrological model and about the sensitivity of the model.
Objective of the study
To obtain knowledge about the sensitivity and about the errors of the hydrological Zambezi
model in STREAM to be able to find the relation between the GRACE data and the water
storage fluctuation.
2.4 Study area description: Zambezi River Basin
The Zambezi catchment is the fourth-largest river basin of Africa, after the catchments of the
Nile, the Congo and the Niger and drains an area of 1.3*10 6 km² (Shahin, 2002). The Zambezi
basin consists of eight countries: Zambia, Angola, Namibia, Botswana, Zimbabwe, Tanzania
11

PROBLEM ANALYSIS
and Mozambique. In Figure 2.8 the percentage of the total Zambezi catchment area of each
country is written between the brackets.
Figure 2.8: Zambezi River Basin.
The catchment is divided into eight sub catchments according to the Digital Elevation Model
(DEM) [1]. In Figure 2.9 the DEM and the sub catchments of the Zambezi are depicted. In
Table 2.1 the surface area of each of the sub catchments are written down. The names of the
sub catchments are taken from Chapagain (2000) except from the change of name of Lake
Malawi into Lake Nyasa.
Table 2.1: Surface area of the sub catchments.
Sub catchment Surface area (km²) Sub catchment Surface area (km²)
Upper Zambezi 242.985 Kafue 163.974
Barotse 96.448 Luangwa 170.955
Cuando-Chobe 147.212 Lake Nyasa-shire 198.670
Middle Zambezi 192.419 Lower Zambezi 165.488
12

PROBLEM ANALYSIS
UPPER ZAMBEZI
KAFUE
LUANGWA
LAKE NYASA-
SHIRE
CUANDO-
CHOBE
BAROTSE
LOWER ZAMBEZI
MIDDLE ZAMBEZI
Figure 2.9: DEM [1] of the Zambezi catchment and the eight sub catchments..
The Zambezi River starts from the Kalene hills in Zambia in the Upper Zambezi sub
catchment (Shahin, 2002) and crosses the border of Angola. This sub catchment is a huge
shallow basin, which consists of alluvial deposits like Karoo (or Barotse) sands. These Karoo
sands are very permeable, which causes that there is no significant surface runoff
(Bastiaansen, 1990). As can be seen from the soil classification map [2] in Figure 2.10 the
main soil types in the Upper Zambezi are arenosols 1 , gleysols 2 and ferralsols 3 .
When the Zambezi crosses again the border of Zambia, the river enters the Barotse flood plain
and turns eastwards. Just up to Lukulu the tributaries Lungue-Bungo and Kabompo
confluence with the Zambezi.
Downstream Lukulu the Zambezi enters the Barotse sub catchment. Its main tributary is the
Luanginga. Nearby Kassane the sub catchment ends after the confluence with the mouth of
the Cuando-Chobe sub basin. This system consists of the Cuando River and the Okavango-
Chobe River. The Okavango-Chobe River is the spillway between the Zambezi and the
Okavango catchment and can work in two directions (Shahin, 2002). The Cuando-Chobe is a
1 Arenosols: sandy soils with little profile development.
2 Gleysols: water saturated soils that are not salty.
3 Ferralsols: highly weathered soils rich in sesquioxide clays and with low cation exchange capacities.
13

PROBLEM ANALYSIS
large tributary, although its contribution to the main river discharge is small. Because
Cuando-Chobe is low and swampy area evaporation plays a major part (Chapagain, 2000).
Figure 2.10: Soil classification map [2] of the Zambezi catchment. In Appendix 2 a short description of
the soil types can be found.
After the confluence of the Barotse and the Cuando-Chobe basin into the Middle Zambezi the
terrain changes (Shahin, 2002). From Victoria Falls the Zambezi flows through a narrow
gorge, with waterfalls and rapids. The gorge is excavated in crushed rock produced by fault
lines.
The Zambezi follows its way to the Kariba Reservoir. This reservoir is an artificial lake
formed by the Kariba Dam, which was build in 1961 (Index of World lakes). The lake has a
surface area of about 4422 km². The Kariba Dam was mainly build for the generation of
hydropower and has an installed capacity of 1266 MW (Chapagain, 2000).
About 150 km downstream of the outlet of Lake Kariba, the Zambezi River confluences with
the tributary of the Kafue basin. In the Kafue sub basin two dams are build in the river: the
Itezhitezhi dam and the dam at Kafue Gorge. The dams were constructed to generate
hydropower.
14

PROBLEM ANALYSIS
Just before the inlet of Lake Cabora Bassa a second large tributary enters the Zambezi: the
Luangwa. The soil type in the Luangwa sub catchment is mainly classified as luvisols 4 , just as
in the Lower Zambezi sub basin.
The Lower Zambezi is the last sub basin before the Zambezi drains into the Indian Ocean.
The basin has one large reservoir: Lake Cabora Bassa. Again is the main purpose of the
reservoir the generation of hydroelectric power. With a capacity of 2075 MW (Chapagain,
2000) it is the largest barrage of Africa.
Before the Zambezi enters the ocean, the last large tributary, named Shire, confluences with
the Zambezi River. The Shire drains mainly water from Lake Nyasa (or Lake Malawi). Lake
Nyasa is the most southern of the African Rift Valley lakes. It is, especially the north and east
part, delimited by faults (Index of World Lakes). After the confluence, the Zambezi enters
through a swampy area into the Indian Ocean, which is the end of its 3000 km long trip.
2.4.1 Climate
The climate in the Zambezi catchment can be divided into three main parts as can be seen in
Figure 2.11. The northern part of the catchment has mainly a Cwa-climate, according to
Köppens climate classification system [3]. The Cwa-climate is a temperate climate with a dry
season in the winter. The Cwa-climate is the most dominant climate type in the Zambezi
catchment.
The second existing climate type is the BSh-climate, or the steppe climate. This climate is
distinguished by the fact that the annual evaporation is exceeding the annual rainfall.
The third climate type in the Zambezi catchment is the Am-climate. This tropical climate
exists in the upper northwest and the east part of the catchment. High temperatures and a short
dry period characterize the Am-climate.
4 Luvisols: soils with strong accumulation of clay in the B-horizon and not dark in color.
15

PROBLEM ANALYSIS
Figure 2.11: Climate classification according to Köppen [3]. Am=tropical, monsoon type;
Ami=tropical, monsoon type, isothermal subtype; Cw=temperate, winter dry season; Cwa=temperate,
winter dry season, hot summer; Cwb=temperate, winter dry season, cool summer; BSh=dry, steppe
climate, subtropical desert.
Seasonal variability
The movement of the Inter-Tropical Convergence Zone (ITCZ) is a very important factor,
which affects the climate of Africa. The ITCZ is a surface of discontinuity, which separates a
warm and dry air mass from a cooler and moist air mass (Shahin, 2002). Its movement during
the season is related with the position of the sun compared to earth. In January, the position of
the ITCZ in Africa is, until 30° E, parallel to latitude 5° N (Figure 2.12). Then the ITCZ has a
depression due to the orography of the region. When the ITCZ crosses the 18 th eastern
longitude it bends again to the east. By July the ITCZ is moved up to 18° N. At this position
the ITCZ has reached its most northern position and in August the ITCZ is moving back to
the south.
Figure 2.12: Position of the ITCZ during the year (from: http://www.planearthsci.com).
16

PROBLEM ANALYSIS
In the center of the ITCZ the solar radiation is maximal (Magdelyns, 2004). The warm air
rises and new air is supplied by the trade winds. The rising warm air is cooling down and is
condensing, which causes rainfall. That’s why the ITCZ is characterized by warm and wet
weather.
The movement of the ITCZ can be seen very well in
the rainfall and temperature patterns of the Zambezi
catchment. In January the ITCZ has moved to its
/////// ///////
most southern position. As can be seen in Figure
2.14 that is the moment with the highest rainfall
amounts and the highest temperatures. Also the
depression of the ITCZ near Malawi is easy to find.
ITCZ
northern
southern
trade winds
trade winds
During the season the ITCZ is moving up. In May Figure 2.13: ITCZ (after Magdelyns,
the dry season starts with its top in July when the
2004).
ITCZ is at the most northern position. After July the ITCZ is moving back to the south.
October is the end of the dry season.
January
May
September
February
June
October
March
July
November
Unit:
mm/month
April
August
December
Figure 2.14: Monthly precipitation in 2003 based on FEWS [11].
17

PROBLEM ANALYSIS
January
May
September
February
June
October
March
July
November
Unit:
°C
April
August
December
Figure 2.15: Long term mean monthly temperature [4] 1930-1960.
2.4.2 Land cover / Dambos
The Zambezi catchment is mostly covered with savannas (Figure 2.16) and in the northwest
of the catchment with woody savannas and dense forest. Furthermore a lot of cropland can be
found, especially in the center and northeast part.
Figure 2.16: Land cover [6].
18

PROBLEM ANALYSIS
Table 2.2: Land cover percentage of the catchment.
Land cover
% of basin
Forests 3.8
Closed Shrublands 0.0
Open Shrublands 0.0
Woody savannas 15.4
Savannas 57.1
Grassland 0.3
Croplands 5.3
Urban and Built-up 0.1
Croplands and natural vegetation mosaic 14.4
Barren of sparsely vegetated 1.0
Water 2.6
Along the river there are also some grasslands. These grasslands are wetlands. The wetlands
can be divided into three categories: pan dambos, river valleys or flood plains (Bastiaansen,
1990). The latter two are caused by cracks in the earth crust. Dambos are natural depressions
without a clear outlet. Likely, they are formed by strong winds. There is made a distinction
between dambos, which are permanently flooded (type I) and dambos, which are only flooded
during the wet season (type II). See Figure 2.17.
In the part, which is permanently flooded, the vegetation is mostly dense reed and the soil
contains very often loam. Near the edge where seepage occurs the dambo has a thick peat
layer. The space between the inner part and the edges is mostly covered with sandy soils and
is vegetated with grass.
A logic result of the open water in the dambos, originate from seepage, is that the evaporation
is high.
Figure 2.17: Schematization of two types of dambos. Type I is permanently flooded and type II only
during the wet season (Bastiaansen, 1990).
19

PROBLEM ANALYSIS
20

HYDROLOGICAL MODEL INPUT DATA
3 Hydrological model input data
3.1 Precipitation data
One of the most important inputs of a rainfall runoff model is the precipitation. This is the
reason why it is important to have good rainfall maps, which reflect the rainfall intensity and
the spatial variation in an accurate manner.
For the time period 1978-1992 Seyam (2002) has
chosen to use the data from rain gauges [8]. Seyam
has interpolated these point data with the weightedaverage-distance
method. A disadvantage of this
method is that it is only quite reliable if there are
enough rain stations in the catchment to reflect the
Figure 3.1: Maximum available rain
spatial variability. Because this is not the case for
stations in ‘78-‘93
1993 until present and because it is very often not
possible to obtain from every station an (almost) complete data series for the period 1993-
present, other methods to determine the precipitation have been studied.
In the next sections an investigation is made of two different rainfall algorithms: MIRA and
FEWS. First each algorithm will be described and afterwards the output will be compared
with observed rainfall data from ten rain gauges in the Zambezi catchment (Figure 3.2).
Magoye
Mongu
Figure 3.2: Location of the ten rain stations, which are used to verify the satellite precipitation data.
21

HYDROLOGICAL MODEL INPUT DATA
3.1.1 Microwave-Infrared Rainfall Algorithm (MIRA)
Due to large demand from meteorologists, climatologists and hydrologists on accurate and
high-resolution rainfall estimates on a daily base, the MIRA [10] algorithm was developed
(Todd et al, 2001). Previous algorithms, which determine the rainfall by remote sensing,
usually have a spatial resolution of 2.5° and a monthly time scale. Some algorithms have a
better spatial resolution, but this mostly means that the temporal resolution is bad and vice
versa. The MIRA algorithm avoids this constraint by combining the input of the different
satellite sources. The MIRA algorithm uses two sources:
• Passive Microwave (PMW) and
• Infrared (IR)
The PMW is a low orbiting sensor that can provide accurate rainfall estimates, but suffer from
poor temporal sampling. Conversely, the IR sensor provides high temporal data, but gives
only a very weak signal and an indirect relationship with the surface rain.
GOES Precipitation Index
The most common algorithm to derive rainfall estimates out of the IR-sensor is the GOES
Precipitation Index (GPI) (Arkin and Meisner, 1987). For large grid cells the IR-sensor
measures the temperature of the cloud top and next the rainfall is calculated by:
R = Fc
× G× T
Equation 3.1
Where:
R = rainfall per cell [mm]
F c = fractional coverage of each cell by cloud colder than 235 K [%]
G = GPI coefficient equal to 3.0 [mm/h]
T = number of hours in the integration period [h]
Atlas and Bell (1992) showed that the GPI (Equation 3.1) is essentially an area-time-integral
(ATI) approach. This ATI approach (Doneaud et al., 1984) has been used to estimate the
storm rainfall volume from the area and duration of the storm measured by radar. As the GPI
uses cold cloud area as a surrogate for rain, Atlas and Bell (1992) showed that G can be
defined as:
−1
⎛A ⎞
G = R ⎜ c ⎟ c ⎜A ⎟
⎝ r ⎠
Where:
Equation 3.2
22

HYDROLOGICAL MODEL INPUT DATA
R c
A c
A r
= climatological conditional rain rate [mm]
= average area of cold cloud [L²]
= average area of rain [L²]
Richards and Arkin (1981) found that if the GPI is calculated over a large enough domain it is
satisfied to say that the GPI coefficient is stable (mostly 3.0). This assumption is only true for
grid cells of 2.5° for periods as short as one hour. So for the objective of MIRA to obtain
high-resolution rainfall estimates, this assumption is not satisfactory. Furthermore Arkin et al.
(1994) revealed that because of the fact that R c and A c /A r vary over space and time it is
unlikely that one single G value or IR threshold will be appropriate for all regions (dominant
cloud microphysical processes). Hence it is necessary to optimize the IR threshold and the
GPI coefficient. The MIRA algorithm calculates such optimized G and IR threshold fields by
comparing the IR data with the PMW data.
MIRA
The MIRA algorithm is based on the assumption that PMW data gives accurate rainfall
estimations and that these results can be used to calibrate the IR-parameters. This will lead to
improved rainfall estimates obtained from IR data at a high temporal frequency.
To calibrate the IR-parameters, frequency distributions from both satellite estimates are made
from coincident satellite imagery. The frequency distribution of the PMW sensor contains
rain rate estimates (R MW ) and the distribution of the IR contains brightness temperature
(IR TB ). To ensure sufficient observations of the PMW and the IR estimates the imagery is
accumulated over some time and space domain. To derive the optimized IR TB /R MW
relationship the Probability Matching Method (PMM) is used. This method of Atlas et al.
(1990) compares the histograms of the coinciding R MW and IR TB observations in such way
that the proportion of the R MW distribution above a certain rain rate is equal to the proportion
of the IR TB below the associated IR TB threshold. The method is carried out from the highest to
the lowest rain rates. In this way the optimized IR TB /R MW relationship is produced.
Subsequently, this relationship can be applied on the IR images to derive accurate rainfall
estimates at a high temporal scale.
For the validation the results were compared to other rainfall estimates. As can be seen in
Table 3.1 and Figure 3.3 a large improvement between the MIRA algorithm and the GPI is
realized.
23

HYDROLOGICAL MODEL INPUT DATA
Table 3.1: Validation statistics of rainfall estimates from MIRA and GPI compared with other rainfall
estimation methods (Todd et al., 2001)
Independent validation data MIRA GPI
CC Ratio RMSE CC Ratio RMSE
SSM/I BUC [1]
0.47 1.1 2.2 0.04 1.54 2.84
SSM/I BUC [2]
EPSAT gauge data [3]
GPCP gauge data [4]
WetNet PIP-3 GPCP land based
gauges [5]
WetNet PIP-3 Comprehensive Pacific
Rainfall Data Base [6]
0.54 1.01 2.02 0.24 1.97 1.98
0.96 1.08 2.04 0.76 1.4 5.3
0.54 1.76 25.17 0.66 2.74 36.9
0.8 1.24 82.6 0.69 1.69 147.5
0.85 0.96 74.6 0.75 1.08 100.8
[1] 140°-180°E, 20°S-20°N, July 1991, 0.25°, conditional instantaneous, n= 515, mean=1.7mmhr -1
[2] 140°-180°E, 20°S-20°N, July 1991, 0.5°, conditional instantaneous, n= 228, mean=1.04 mmhr -1
[3] 2°-3°E, 13°-14°N, July 1992, 1°, daily, n=30, mean=3.5mm.day -1
[4] Selected locations (see Todd et al., 2001; Table 2), July 1991,2.5°, pentad, n= 46, mean=8.7mm.pentad -1
[5] Selected locations (see Morrissey et al., 1994), 2.5°, monthly, n=159, mean =83.7mm
[6] Selected locations (see Morrissey et al., 1994), 2.5°/monthly, n=17, mean=217.6mm
Rain rate [mm/h]
40
30
20
10
0
1
Validation Validatation of of MIRA and GPI
4
7
10
13
16
19
22
25
Time [days]
Gauge MIRA GPI
28
Figure 3.3: MIRA and GPI compared to rain gauges (Todd et al., 2001).
3.1.1.1 Data
The MIRA algorithm has calculated daily rainfall estimates in mean mm/hour for the period
from 1993 till 2002. It covers only the African continent and has a spatial resolution of 0.1° x
0.1°. The rainfall estimates from MIRA can be downloaded from the Southern African
Regional Science Infinitive (SAFARI 2000)-site:
http://ltpwww.gsfc.nasa.gov/s2k/html_pages/groups/precip/daily_rainfall_mira. html
Although the high temporal resolution of the rainfall estimates, there are unfortunately some
gaps (Table 3.2). These gaps are for example due to the fact that some data files became
corrupt by archiving on CD. To repair the data one of the following actions is executed:
24

HYDROLOGICAL MODEL INPUT DATA
1. If the missing day(s) are in the dry season (May-October) there is no precipitation
added to the month value. It is assumed that the precipitation in those days is zero.
The month sum is easily obtained by summing all the daily values.
2. If one or two days successive are lacking data, the value of the day before and/or the
day after, is filled in. Assumed is a high correlation between two successive days.
Again is the month sum easily obtained by simply summing up all the daily values
inclusive the new values.
3. If more then two days are missing and at least one rain station is available with a
complete daily range of rainfall data, the percentage (p) of the amount of rainfall of
the missing days is calculated from the rain station. The month sum is then divided
by (1-p) to obtain a repaired new month value.
4. If more than two days are missing data and there are no rain stations available, the
average percentage (p) of the amount of rainfall of the missing days is computed
from other years of that month. The month sum is then divided by (1-p) to obtain a
repaired new month value.
5. If a complete month is missing, a highly correlated year is looked for. The missing
month is filled in with the same month of the found correlated year.
A complete overview of the data transformations and gap repair is given in Appendix 1.2.6.
Table 3.2: Number of gaps in the MIRA data.
rainy
dry
rainy
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
January 0 0 0 1 5 0 9 2 10 0
February 0 0 0 10 1 0 0 0 2 0
March 1 0 0 0 1 0 1 0 0 0
April 0 0 9 0 6 2 0 0 0 0
May 2 10 0 0 0 0 2 0 0 0
June 0 0 4 0 0 5 0 0 0 0
July 0 0 31 10 0 1 0 0 0 11
August 0 0 10 0 0 0 0 0 0 21
September 0 0 0 0 0 0 0 0 0 0
October 0 0 0 0 0 0 0 0 0 21
November 0 0 8 5 1 0 0 0 0 11
December 0 0 0 9 0 4 12 0 0 31
Tot perc. of gaps 0.82% 2.74% 16.99% 9.56% 3.84% 3.29% 6.58% 0.55% 3.29% 26.03%
rainy season 0.27% 0.00% 4.66% 6.83% 3.84% 1.64% 6.03% 0.55% 3.29% 11.51%
dry season 0.55% 2.74% 12.33% 2.73% 0.00% 1.64% 0.55% 0.00% 0.00% 14.52%
3.1.1.2 Performance
To verify the rainfall estimates of the MIRA algorithm, the output is compared with ten rain
stations in the catchment. From the selected stations the average is plotted in time. The same
is done to the values of MIRA on the locations of the rain stations. See Figure 3.4. Also a
double mass curve and a scatter plot are made (Figure 3.5. and Figure 3.6).
25

HYDROLOGICAL MODEL INPUT DATA
Monthly precipitation data
(average of 10 stations)
450.00
400.00
350.00
300.00
[mm/month]
250.00
200.00
150.00
100.00
50.00
0.00
jan-93 mei-94 sep-95 feb-97 jun-98 nov-99 mrt-01 aug-02 dec-03
[month]
RAIN GAUGES
MIRA
Figure 3.4: Output of MIRA compared with rain gauges in the Zambezi catchment.
cum. Observed
9,000
8,000
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
MIRA [mm/month] y = 0.8693x
R 2 = 0.9991
0 2000 4000 6000 8000 10000
cum MIRA
Figure 3.5: Double Mass Curve of MIRA.
26

HYDROLOGICAL MODEL INPUT DATA
MIRA [mm/month]
Observed
500
450
400
350
300
250
200
150
100
50
0
0 100 200 300 400 500
MIRA
Figure 3.6: Scatter plot of MIRA with 45° line
As can be seen from Figure 3.5 MIRA over-estimates the rainfall with circa 15%. The
correlation coefficient between the satellite estimates and the rain gauges is 0.93, which is
very high.
3.1.2 Famine Early Warning Systems (FEWS)
In framework to identify problems in the food supply system a network named ‘U.S. Agency
for International Development, Famine Early Warning System Network’
(USAID/FEWSNET) has been developed. This network has the objective to provide timely
and accurate information regarding potential food-insecure conditions. Together with the
‘National Aeronautic and Space Administration’ (NASA) and the ‘National Oceanic and
Atmospheric Administration’ (NOAA) they collect and process satellite data to provide
rainfall estimations.
The FEWS rainfall estimations [11] are operational from July 1995. Since then there have
been two algorithm versions. The first version, RFE 1.0 (Herman et al., 1997), was
operational since July 1995 till December 2000 and the second, RFE 2.0 (Xie and Arkin,
1997), started at January 2001 and is still operational.
RFE 1.0
The RFE 1.0 algorithm uses for its rainfall estimates:
• METEOSAT 5 thermal Infrared (IR) satellite data,
27

HYDROLOGICAL MODEL INPUT DATA
• Global Telecommunication System (GTS) rain gauges and
• Model analyses of wind and relative humidity and orography.
The first step of the algorithm is to make a preliminary estimate of the precipitation based on
the GOES Precipitation Index algorithm (GPI) (Arkin and Meisner, 1987). The amount of
precipitation is determined by measuring the duration of cold cloud tops with the
METEOSAT 5 satellite. Each hour that the temperature of the top of the cloud is below 235
Kelvin (≈ -38° C) will result in 3 mm of accumulated rainfall.
The next step is to correct the estimates of the GPI-algorithm by fitting the estimates to the
rain gauge data. This is done by calculating the difference between the GPI estimate and the
rain gauge data and subsequently creating a gridded bias field via an optimum interpolation
scheme. This scheme assumes that in each grid square one or more observation points are
available. Based on one rainfall observation (d v ) on location (x v ,y v ) within a grid cell and the
GPI-estimates (z i,j ) a precipitation estimate (m v ) is made (Equation 3.3). For each observation
point this procedure is carried out. By minimizing the cost function J (Equation 3.4) a gridded
precipitation field is obtained.
mυ =α
υzi,j +β
υ
zi+ 1,j
+γ
υ
zi,j+ 1
+ζ
υ
zi+ 1,j+
1
Equation 3.3
where:
(xi+ 1−x υ
)(yj+ 1−y υ)
α
υ
=
(x −x )(y −y )
i+ 1 i j+
1 j
(xυ −x i)(yj+ 1−y υ)
β
υ
=
(x −x)(y −y)
i+ 1 i j+
1 j
(xi+ 1−x υ)(yυ
−y j)
γ
υ
=
(x −x )(y −y )
i+ 1 i j+
1 j
(xυ
−x i)(yυ
−y j)
ζ
υ
=
(x −x)(y −y)
i+ 1 i j+
1 j
j + 1
j
i i + 1
= z i,j (GPI estimate)
= d v (observed)
Cost function:
2
( ) ∑ ( ) 2
υ υ
J = 1/2σ m −d
υ
Equation 3.4
where:
2
σ = variance of measurements
28

HYDROLOGICAL MODEL INPUT DATA
The last step of the RFE 1.0 algorithm is to take precipitation due to orographic lifting into
account. Orographic precipitation occurs during favorable low-level wind conditions, when
moist air masses are condensing. In regions where warm cloud tops are presents (temperatures
between 275 and 235 Kelvin (2 till -38°C) an analysis of the ‘Environmental Modeling
Center, Global Data Assimilation System’ (EMC/GDAS) is executed. This analysis combines
the relative humidity, wind direction and the terrain slope. First the dot product of the wind
direction (u(x,y), v(x,y)) and the horizontal gradient of the terrain slope (grad[h(x,y]) is
defined. Subsequently, the scalar product is calculated (Equation 3.5) and multiplied by the
relative humidity to incorporate the orographic lift of low-level moisture.
⎡ ∂h
∂h⎤
⎢u
+ v
∂x
∂y
⎥
⎣ ⎦
Equation 3.5
After this multiplication, the outcome is converted to rainfall estimates and is calibrated with
the values from the rain gauge reports.
For the validation of the RFE 1.0 algorithm field data, which was not the same as the WMO
stations used to calibrate the model, was collected. The evaluation is based on 180 stations
yielding 1882 observations in the period from June to September 1995 in the Sahel region of
Africa. Comparing the estimates with the rain gauge reports, it can be concluded that the
rainfall estimates have a maximum error of 40% of the measured rainfall value with a 68.3%
assurance.
RFE 2.0
As RFE 1.0 only uses one satellite data source, RFE 2.0 uses two extra satellite sources, but
did not incorporated orographic precipitation anymore. So the successor of RFE 1.0 uses the
following data sources (NOAA CPC):
• Special Sensor Microwave/Imager (SSM/I) satellite data,
• AMSU-A (till June 2001) or AMSU-B (from June 2001) microwave satellite data,
• METEOSAT Infrared (IR) satellite data and
• Global Telecommunication System (GTS) rain gauges.
The main idea behind RFE 2.0 is to merge all the satellite data and next merge the results to
the rain gauges. Combining is advantageous, because separately each satellite input is not
complete in spatial coverage and contains significant random error and systematic bias.
Merging the data will improve the accuracy enormous.
29

HYDROLOGICAL MODEL INPUT DATA
The first step of the RFE 2.0 algorithm is to estimate the rainfall out of the separated satellite
sources. The SSM/I detects four times per day the scattering of upwelling radiation by
precipitation sized ice particles within the cloud layer. The obtained patterns are compared
with previously derived rainfall amounts to make rainfall estimates. Just as the SSM/I
algorithm converts emissivity to rain rates, the AMSU-A and AMSU-B data are converted in
same way. Over land estimates are obtained by measuring the brightness temperature
differences due to ice concentration. Over the ocean the cloud liquid water determines the
rainfall estimate. The difference between the AMSU-A and AMSU-B data is only the
increased temporal and spatial resolution.
The GOES Precipitation Index (GPI) is calculated at the same way as the RFE 1.0 algorithm
does. Based on the Cold Cloud Duration measured by the METEOSAT IR satellite the
algorithm will calculate the rain rate.
The next step is divided in two parts. In the first part all the rainfall estimates of the satellite
data are linearly combined through a maximum likelihood estimation method (Equation 3.6).
This reduces the random error significantly.
W
σ
−2
i
i
=
3
−2
∑σi
i−1
Equation 3.6
Where:
W
i
2
σ
i
= weighting coefficient
= random error (rainfall estimate – observed rainfall)
= method 1 (SSM/I), 2 (AMSU) or 3 (IR)
After the determination of the weighting coefficients the new (weighted) rainfall estimate (S)
can be made based on the coefficients and the individual satellite rainfall estimate (S i ).
3
S= ∑ WS
Equation 3.7
i=
1
i
i
The second part of the merging process deals with removing the bias as much as possible.
Comparing the output of the first part with the rain gauges can reduce the bias (Xie and Arkin,
1996). Therefore the satellite rainfall estimates reflect the shape of the precipitation and the
values of the rain gauges determine the magnitude of the rainfall intensity. This involves that
30

HYDROLOGICAL MODEL INPUT DATA
near the rain station the final precipitation is equal to the rain station value. If the distance to
the rain gauge increases, the rainfall estimates will more rely on the satellite rainfall estimates.
The validation of the RFE 2.0 algorithm is carried out by repeatedly removing 10% of the rain
stations and then run the RFE 2.0 algorithm again. Sequentially the outcome is compared to
the values of the removed rain gauges. This is repeated in such way that each station is
removed one time. The result of this cross validation is a bias of –0.15 mm/day and a
correlation coefficient of 0.501 (based on the AMSU-A).
3.1.2.1 Data
The FEWS-algorithm provides daily and decal rainfall estimates in mm for the African
continent (long: -20° to 55° and lat: 40° to –40°). The first algorithm RFE 1.0 started at July
1995 and ended in December 2000. The second algorithm RFE 2.0 started at January 2001
and is still operational. The spatial resolution of FEWS is 0.1° x 0.1°.
The rainfall estimates are downloadable from the site of FEWS/NET:
http://edcw2ks21.cr.usgs.gov/adds/ or from the website of the NOAA CPC:
http://www.cpc.ncep.noaa.gov/products/fews/data.html. In appendix 1.2.7 an overview of the
data transformations done on the raw FEWS-data is given.
3.1.2.2 Performance
To verify the satellite rainfall data with measured rainfall data in the Zambezi catchment, the
average rainfall values of ten locations (Figure 3.2) are compared with each other. In Figure
3.7 until Figure 3.9 a rainfall graph, a double mass curve and a scatter plot are plotted. In the
graphs of the double mass curve also the regression line and regression formula and R² are
depicted.
31

HYDROLOGICAL MODEL INPUT DATA
450.00
FEWS (RFE 1.0 & RFE 2.0)
Monthly precipitation data
(average of 10 stations)
400.00
350.00
RFE 1.0 RFE 2.0
300.00
[mm/month]
250.00
200.00
150.00
100.00
50.00
0.00
jul-95 nov-96 mrt-98 aug-99 dec-00 mei-02 sep-03
[month]
RAIN GAUGES
FEWS
Figure 3.7: Output of FEWS compared with rain gauges in the Zambezi catchment. Note: from January
2001 a new algorithm is used.
cum Observed
8,000
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
y = 0.9029x
FEWS [mm/month]
R 2 = 0.9799
RFE 1.0 RFE 2.0
0 2000 4000 6000 8000 10000
cum FEWS
Figure 3.8: Double Mass Curve of FEWS.
32

HYDROLOGICAL MODEL INPUT DATA
FEWS [mm/month]
Observed
400
350
300
250
200
150
100
50
0
0 100 200 300 400
FEWS
Figure 3.9: Scatter plot of FEWS with 45° line.
As can be seen from the graphs, FEWS averagely over-estimates the rainfall with about 11%.
The correlation coefficient is 0.87. It can also clearly be seen that there is a large
improvement in the rainfall estimates with the RFE 2.0 algorithm. Splitting the double mass
curve into two curves, respectively RFE 1.0 and RFE 2.0, shows the large difference between
the two algorithms.
6,000
5,000
FEWS [mm/month]
RFE 1.0
y = 0.9624x
R 2 = 0.9504
cum Observed
4,000
3,000
2,000
1,000
0
0 1000 2000 3000 4000 5000 6000
cum FEWS
Figure 3.10: Double Mass Curve of FEWS calculated with RFE 1.0.
33

HYDROLOGICAL MODEL INPUT DATA
FEWS [mm/month]
RFE 1.0
Observed
400
350
300
250
200
150
100
50
0
0 100 200 300 400
FEWS
Figure 3.11: Scatter plot of FEWS calculated with RFE 1.0. In pink the 45° line.
FEWS [mm/month]
RFE 2.0
y = 0.9893x
R 2 = 0.9978
2,500
cum Observed
2,000
1,500
1,000
500
0
0 500 1000 1500 2000 2500
cum FEWS
Figure 3.12: Double Mass Curve of FEWS calculated with RFE 2.0.
FEWS [mm/month]
RFE 2.0
Observed
400
350
300
250
200
150
100
50
0
0 100 200 300 400
FEWS
Figure 3.13: Scatter plot of FEWS calculated with RFE 2.0. In pink the 45° line.
34

HYDROLOGICAL MODEL INPUT DATA
Looking at the graphs of the separated FEWS algorithms, it can be concluded that RFE 2.0
performs much better. Also the correlation coefficient is improved from 0.85 for RFE 1.0 to
0.97 for RFE 2.0.
3.1.3 Comparison precipitation algorithms
As can be seen from the previous sections, the different algorithms have different results if
compared with measured rainfall data. In Table 3.3 is given an overview of the performances.
Table 3.3: Summary of the performance of the precipitation algorithms.
Algorithm
Correlation
coefficient
Gradient of trendline
R² trendline
MIRA (ended)
(jan 1993 - dec 2002)
FEWS RFE 1.0 + RFE 2.0
(operational)
(jul 1995 - dec 2003)
FEWS RFE 1.0 (ended)
(jul 1995 - dec 2000)
0.93 0.8693 0.9991
0.87 0.9029 0.9799
0.85 0.9624 0.9504
FEWS RFE 2.0 (operational)
(jan 2001 – dec 2003)
0.97 0.9893 0.9978
The MIRA and the FEWS RFE 2.0 algorithm have a very high correlation coefficient. Also
looking at the gradient of the trendline, which is an indication of an over- or an
underestimation, FEWS RFE 2.0 is the preferred algorithm. In order to obtain a complete time
serie from January 1993 till present, it is also desirable to use the MIRA algorithm for the
months before the FEWS RFE 2.0 algorithm starts.
To achieve even better absolute values of the MIRA-algorithm, the algorithm has been
corrected. Each month a comparison is made between the average of the MIRA rainfall values
on the location of the rain stations (1, 2, … , n) and the average of the observed rainfall values
[7]. Subsequently, the correction α is calculated by dividing the average of the observed
rainfall by the average of the MIRA rainfall. These correction factors (see appendix 1.2.8) are
subsequently multiplied by the MIRA rainfall each month. In formula:
35

HYDROLOGICAL MODEL INPUT DATA
α =
t
n
1
∑ rainstation
n i=
1
n
1
i
∑ MIRAt
n
i=
1
i
t
Equation 3.8
MIRA
=α ∗ MIRA
Equation 3.9
corrected,t t t
In Figure 3.14 and Figure 3.15 are depicted the double mass curve and the scatter plot if
compared with the ten selected stations in Figure 3.2.
cum. Observed
8,000
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
MIRA_corrected
[mm/month]
y = 0.9819x
R 2 = 0.9982
0 1000 2000 3000 4000 5000 6000 7000
cum MIRA_corrected
Figure 3.14: Double Mass Curve of MIRA corrected [=MIRA * α].
500
MIRA_corrected
[mm/month]
400
Observed
300
200
100
0
0 100 200 300 400 500
combi MIRA_corrected
Figure 3.15: Scatter plot of MIRA corrected with 45° line.
36

HYDROLOGICAL MODEL INPUT DATA
As can be observed in the graphs the MIRA corrected performs well with a correlation coefficient
of 0,98, when compared with the ten selected rain stations. The final precipitation input for
the model will be a combination of MIRA corrected and FEWS RFE 2.0. For the time period
January 1993 to December 2000 MIRA corrected will be used and for the period January 2001 till
present is FEWS RFE 2.0 suitable.
Table 3.4: Summary of performance best algorithms.
Algorithm
Correlation
coefficient
Gradient of trendline
R² trendline
MIRA corrected
(jan 1993 – dec 2000)
0.98 0.9819 0.9982
FEWS RFE 2.0 (operational)
(jan 2001 – dec 2003)
0.97 0.9893 0.9978
Combi MIRA_cor & FEWS RFE 2.0
(jan 1993 – dec 2003)
0.98 0.9760 0.9992
The final precipitation data is plotted in Figure 3.16 against the average of the ten selected
rain stations. In Figure 3.17 and Figure 3.18 are given the double mass curve and the scatter
plot.
450.00
COMBINATION OF MIRA_CORRECTED & FEWS RFE 2.0
Monthly precipitation data
(average of 10 stations)
400.00
350.00
300.00
[mm/month]
250.00
200.00
150.00
100.00
50.00
0.00
jan-93 mei-94 sep-95 feb-97 jun-98 nov-99 mrt-01 aug-02 dec-03
[month]
MIRA_cor -----|----- FEWS
RFE 2.0
RAIN GAUGES
Combi
Figure 3.16: Output of final precipitation data compared with rain gauges in the Zambezi catchment.
37

HYDROLOGICAL MODEL INPUT DATA
Combi MIRA_cor & FEWS RFE2.0
[mm/month]
y = 0.976x
R 2 = 0.9992
10,000
cum. Observed
8,000
6,000
4,000
2,000
0
0 2000 4000 6000 8000 10000
cum combi
Figure 3.17: Double Mass Curve of a combination of MIRA corrected and FEWS RFE 2.0.
Observed
Combi MIRA_cor & FEWS RFE2.0
[mm/month]
500
450
400
350
300
250
200
150
100
50
0
0 100 200 300 400 500
combi
Figure 3.18: Scatter plot of final precipitation data with 45° line.
3.2 Potential evaporation data
The potential evaporation is necessary to calculate the open water evaporation of the lakes
and to calculate the potential transpiration on land. For the calculation of the monthly
potential evaporation the Thornwaithe (1948) formula is used. The main variable in this
equation is the temperature. Because of this simplicity, Thornwaithe is chosen instead of for
example Penman or Makkink. Thornwaithe calculates the potential evaporation with Equation
3.10.
38

HYDROLOGICAL MODEL INPUT DATA
⎧
⎫
⎪
2
⎪
− 415.85 + 32.24*T − 0.43*T ,T > 26.5
⎪
⎪
Epot
= ⎨
0 ,T<
0 ⎬*dlf
⎪
a
⎪
⎪ ⎛ T ⎞
16* 10* ,0 < T < 26.5⎪
⎪ ⎜ ⎟
⎩ ⎝ I ⎠ ⎪⎭
Equation 3.10
I
12 1.514
⎛Tm
⎞
= ⎜ ⎟
1 5
∑ ⎝ ⎠
Equation 3.11
a 0.49 1.79*10 *I 7.71*10 *I 6.75*10 *I
−2 −5 2 −7 3
= + − + Equation 3.12
Where
Epot = potential evaporation [mm/month]
T
= average monthly temperature [°C]
I
= annual heat index
a
= time dependent factor
dlf = daylight factor (see Table 3.5, chosen is latitude –15°)
Tm
= mean long term monthly temperature [°C]
Table 3.5: Daylight factors for each month (Lorente, 1961).
39

HYDROLOGICAL MODEL INPUT DATA
As can be seen from the formulas, two types of monthly temperature data are necessary:
average temperatures and long-term temperatures [4]. The latter is obtained from the IIASAdatabase
and covers the years 1930 till 1960. Because near Malawi there were some gaps, a
simple gap-repair was done. In appendix 1.2.4 is written which data transformations have
been carried out.
The average monthly temperature data is obtained from different sources. For the period
1978-1992 Seyam (2002) has made temperature maps interpolated from ground stations
(Figure 3.19). The interpolation technique he has used is the weighted-average distance
method (see Section 5.1 Figure 5.3).
Figure 3.19: Maximum available temperature stations for the period 1978-1992 (Seyam, 2002).
For the period from 1993 until 1998 gridded temperature data [5] from the SAFARI project is
used. The resolution of these temperature data is 0.5° x 0.5°. Because less temperature ground
stations are available no comparison between the observed and gridded data is executed.
For the period 1998 until 2003 completely no temperature data was available. Because the
temperature does not differ much from year to year and the spatial variability is not large, the
period July 1993 until December 1998 is used for the period July 1998 until December 2003.
40

HYDROLOGICAL MODEL DESCRIPTION
4 Hydrological model description
4.1 STREAM
For making the hydrological model the software package STREAM is used. STREAM is an
abbreviation of ‘Spatial Tools for River basins and Environment and Analysis of Management
options’ and is still under development by the Institute of Environmental Studies of the ‘Vrije
Universiteit’ of Amsterdam. STREAM enables the
user to make grid-based spatial water balance models.
By writing the code in the scripting language ‘Blaise’
the user can implement the hydrological processes.
Because STREAM is grid-based, it is very suitable for
working with GIS-maps, for example obtained from
Remote Sensing (RS).
Figure 4.1: GUI STREAM.
STREAM is based upon a ‘multi-compartment’ methodology (Aerts and Bouwer, 2003),
which means that the hydrological cycle of a catchment is described as a serie of storage
compartments and flows. Each time step (iteration) STREAM calculates per grid cell the
transformation of the input (precipitation) and the output (evaporation and runoff).
Successively, the flows are calculated by collecting the flows according to the flow direction
(Figure 4.3). The flow direction is mostly determined by the DEM of the catchment. By
dumping the calculated cell runoff in one go to the downstream cells, there is introduced an
error. In the catchment the ground water is flowing from one cell into the other. And it takes
some time to travel that distance. In STREAM this time is set to zero (Figure 4.2). This
assumption is reasonable because the model is on a monthly basis. In one month it is
presumable that the water is drained into the river.
Q Q Q Q Q Q
Q
Figure 4.2: Model assumption by accumulation of flows. Q is the runoff of that specific cell
(Ficara, 2004).
41

HYDROLOGICAL MODEL DESCRIPTION
For models with large lakes, it is possible to define lake cells (Figure 4.3). In those cells all
the processes, which take place on land (e.g. interception, transpiration, infiltration,
percolation, etc.) are set to zero. Processes, which take place in lakes, like open water
evaporation, are implemented in the lake cells. After running the complete model it is possible
to calculate the discharge of the outlet of the lake in a spreadsheet program with reservoir
routing formulas. Then, the flows of the outlet of the lake can be added to the land cells and
those can be recalculated through the catchment. This is possible with for example
Muskingum routing implemented.
Lake script:
Land processes=0
- Precipitation
- Open water
evaporation
Lake
grid
cell
Land
grid
cell
Land script:
Lake processes=0
- Precipitation
- Interception
- Transpiration
- Infiltration
- Percolation
- Soil moisture
- Ground water
- Capillary rise
- Runoff
Runoff (according
to flow direction)
Figure 4.3: Principle of calculating the water balance by STREAM.
4.2 Zambezi script
The original Zambezi script (period: 1978-1993) was developed by Seyam in 2002, but
contained some shortcomings. Seyam did some recommendations about changing the
interception and transpiration process, including the capillary rise and about implementing of
threshold values for the ground water flows. In 2004 Ficara has looked to these
recommendations and implemented them. He also extended the model for the period 1993-
2002. Because the model did not fit the observed data well, the model is further calibrated and
is extended till June 2004 to cover the time period of the GRACE-measurements. Also the
size of the maps for the period 1978-1993 to the same size as for the period 1993-2004 are
fixed (see Appendix 1.2.9). This has the advantage that it is possible to carry out one single
run for the entire time period.
42

HYDROLOGICAL MODEL DESCRIPTION
The final Zambezi script has a spatial resolution of 3x3 km and has a monthly time scale. The
model consists of two parts: pre-routing and routing. Pre-routing is modelled in STREAM and
comprises most hydrological processes. After the pre-routing is executed, the outcome is
imported in a spreadsheet program like MS EXCEL (later in a Java program, see Chapter 5).
In the spreadsheet the flows are routed with Muskingum routing and the outflow of the
reservoirs is calculated with different spillway formulas.
Next to the calculation of all the water balance terms to simulate the hydrological system, also
the Zambezi script in STREAM calculates the colours of water described by Savenije in 2000.
In Figure 4.4 is given a schematic representation of the stocks and fluxes in a hydrological
system. The abbreviations used in this figure are different from the ones in the Zambeziscript.
Figure 4.4: Global Water Resources: Blue, Green, White (Savenije, 2000).
The first and most important flux is precipitation (P), the origin of water, which can come
down on the surface or in water bodies, like rivers or lakes. If the rainfall comes down in the
water bodies it is directly in the blue-compartment. The only way for depletion is through
open water evaporation (O) or through the drainage to the sea or ocean (Q).
When the precipitation comes down on the surface it enters the white component from where
it feeds back directly to the atmosphere through evaporation from interception or bare soil (I).
From the surface the water can also drain to the water bodies (Qs). This is called Horton
43

HYDROLOGICAL MODEL DESCRIPTION
overland flow. In the Zambezi-script this process is neglected due to the well permeable soil
in the Zambezi catchment.
The water that did not quickly evaporated to the atmosphere infiltrates (F) into the unsaturated
zone. This is the green compartment, where transpiration (T) (≈ plant evaporation) takes
place. When the unsaturated zone is saturated the water will percolate (R) to the (renewable)
ground water. Through seepage (Qg) the water can flow from the ground water to the water
bodies.
For each of the resources the quantities of the fluxes and stocks can be calculated. This results
in the average residence time. In Table 4.1 the average values at a global scale are given.
Table 4.1: Global Water Resources, fluxes, storage and average residence times (Savenije, 2000).
Resource Fluxes [L/T] or [L 3 /T] Storage [L] or [L 3 ]
Residence
time
[T]
Green T 100 mm/month S u 440 mm S u /T 4 months
White I 5 mm/d *) S s 3 mm *) S s /I 0.6 days
Blue Q 46 x 10 12 m 3 /a S w 124 x 10 12 m 3 S w /Q 2.7 years
Deep blue Q g 5 x 10 12 m 3 /a *) S g 750 x 10 12 m 3 *) S g /Q g 150 years
Atmosphere P 510 x 10 12 m 3 /a S a 12 x 10 12 m 3 S a /P 0.3 month
Oceans A 46 x 10 12 m 3 /a S o 1.3 x 10 18 m 3 S o /A 28000 year
Note: transpiration and interception fluxes apply to tropical areas storage in the root zone can be
significantly less than 440 mm
*) Indicate rough estimates
In Section 4.2.1 all the hydrological processes of the Zambezi script will be described and in
Section 4.2.2 the routing will be explained. In Appendix 3 the code of the Zambezi-script can
be found, written in the scripting-language Blaise.
44

HYDROLOGICAL MODEL DESCRIPTION
4.2.1 Conceptual rainfall-runoff model of the Zambezi in STREAM
Figure 4.5 shows a schematization of the Zambezi script. In the following sub-paragraphs
each process will be described.
Prec
Int
Shallow soil
Unsaturated zone
(SU)
Cap
D
Pnet
cr x Pnet
(1-cr)Pnet
Smax
Tra
Overtop
K=3
Runoff Zambezi
River
Saof
Ground water (GWS)
0
GWSmax
GWSquick
Rtq = 4
Rts = 12
Qflo
Sflo
-25
Figure 4.5: Zambezi-model schematization. The unit is millimeters and months.
45

HYDROLOGICAL MODEL DESCRIPTION
4.2.1.1 Land process: evaporation through interception
As known, not all the precipitation will infiltrate in the subsurface. Before the rainfall can
infiltrate a part is already back in the atmosphere to account for the moisture recycling
process. This part, which is a feedback component within a monthly time scale, consists of
several processes (Savenije, 1997):
• Canopy interception
• Shallow soil interception
• Evaporation from temporary surface storage (pools)
• Immediate transpiration (within a month)
All these interception processes can be modelled with one simple threshold value D. By
subtracting the threshold D from the precipitation (Prec), the net precipitation (Pnet) will be
obtained. Because it is not possible to evaporate more than the precipitation, Pnet is minimal
zero. The net precipitation will successively infiltrate in the unsaturated zone.
Pnet = max(Pr ec − D,0)
Equation 4.1
The threshold value D is of course a calibration parameter, although it can roughly be
determined with the RAINRU-model (Savenije, 1997). Because the runoff on time step t is
dependent on the rainfall amount of that month and the rainfall of the previous months, the
runoff can be written in the following linear transfer function:
n
∑ i ( )
Equation 4.2
Q(t) = b × max Pr ec(t −i) −D,0
i=
0
The net runoff coefficient c can then defined as:
c =
m
t=
1
m
∑
t=
1
∑
Q(t)
Pr ec(t)
Equation 4.3
Mathematically, it can be demonstrated that the net runoff coefficient also follows from:
n
c= ∑ b
i=
0
i
Equation 4.4
In Table 4.2 and Table 4.3 are depicted the results of the regression analysis. As can be seen
has a threshold value D of 80 mm/month the largest R². With D=80 mm/month the runoff
coefficient c will be 0.14 for Lukulu and 0.10 for Victoria Falls.
46

HYDROLOGICAL MODEL DESCRIPTION
Table 4.2: Regression results for Lukulu.
D
[mm/month]
R² b0 b1 b2 b3 b4 b5 b6 c
80 0.72 0.01 0.04 0.04 0.03 0.02 0.01 0.14
90 0.71 0.01 0.04 0.05 0.03 0.03 0.01 0.16
100 0.70 0.01 0.04 0.05 0.03 0.03 0.01 0.17
Table 4.3: Regression results for Victoria Falls.
D
[mm/month]
R² b0 b1 b2 b3 b4 b5 b6 c
80 0.75 0.00 0.00 0.02 0.03 0.03 0.02 0.00 0.10
90 0.75 0.00 0.00 0.02 0.04 0.03 0.02 0.00 0.11
100 0.75 0.01 0.01 0.02 0.04 0.03 0.02 0.00 0.13
For the interception calculation of the model the threshold-concept is used. However in the
model also the white, green and the blue water (Savenije, 2000) are calculated. The white
water consists of the part of the rainfall that feeds back directly to the atmosphere. The white
water is calculated with the method of De Groen (2002). The difference between the threshold
method and the method of De Groen is that De Groen only considers canopy, mulch and wet
soil interception. De Groen calculates monthly interception (intmar) based on the Markov
property of daily rainfall and using a daily interception threshold D d :
⎡ ⎛ nD
r d ⎞⎤
Intmar = Pr ec ⎢ 1−exp
⎜ − ⎟
Pr ec ⎥
⎣ ⎝ ⎠⎦
Equation 4.5
Where:
Intmar
n r
D d
Prec
= monthly interception by Marieke de Groen
= number of rain days per month
= daily interception threshold
= monthly precipitation
The number of rain days follows directly from the Markov process:
n
r
n ⋅p
=
1−
C
m 01
Equation 4.6
Where:
n m = number of days in a month (30)
C = p 11 - p 01
p 01
= probability of a rain day after a dry day
47

HYDROLOGICAL MODEL DESCRIPTION
p 11
= probability of a rain day after a rain day
De Groen demonstrated that the transition probabilities can be described as:
p01
p11
( ) r
= q Prec
Equation 4.7
( ) v
= u Prec
Equation 4.8
The parameters q, r, u and v are only calibration parameters, which have to be calibrated at
different locations, although p11 is spatial quite homogenous.
For the Zambezi catchment the C can be regarded as a constant (De Groen, 2002). With D d =
4.75 mm/day, q = 0.03, r = 0.4 and C=0.6 the interception is calculated by:
⎡ ⎛
Intmar = Prec⎢1−exp⎜−4.75
⎣ ⎝
0.4⋅
Pr ec
0.9
0.6
⎞⎤
⎟⎥
⎠⎦
Equation 4.9
The white water is equal to the interception calculated by De Groen.
Interception [mm/month]
100
Interception
80
60
40
20
0
0 50 100 150 200
Int (Threshold =80) Intmar (De Groen)
Prec
Figure 4.6: Interception as a function of precipitation (prec) calculated by the threshold-method and
by De Groen’s method.
4.2.1.2 Land process: transpiration
After some water is evaporated through interception, the other part is infiltrating in the
subsurface. However, a part of the water is going directly to the ground water. This fast
component is the water that flows through macro pores and cracks. The division between the
fast and the slow component is determined by the separation coefficient cr. Seyam
investigated the relation between the separation coefficient and the slope and/or the land use.
48

HYDROLOGICAL MODEL DESCRIPTION
His first attempt was if the slope is steep the separation coefficient is high and he assumed
that the separation coefficient would be higher if the catchment is covered with for example
grass then with forest. Although his sample size was too small to draw a conclusion, he did
not find any rough correlation between cr and the slope or the land cover. Hence the values of
cr are found by trail-and-error.
During the calibration, it appeared that introducing a higher separation coefficient in the
dambos gave better results. This is explainable by the small unsaturated zone in the dambos,
which causes the quick percolation to the ground water. To model this process the cr is
increased. In Figure 4.7 the distribution of the separation coefficient can be seen.
Figure 4.7: Separation coefficient value cr.
The water of the slow component fills the unsaturated zone. This water in the unsaturated
zone is available for transpiration. Transpiration is the process of the absorption of water by
plants, usually through the roots, the movement of water through plants, and the release of the
water to the atmosphere through small openings on the leaves (from USGS-glossary).
However not all the water that is infiltrating in the unsaturated zone is available for
transpiration. Because there is a limit in the amount of water that the soil is able to hold (field
capacity, Smax), the amount of water above field capacity will percolate (overtop) to the
groundwater. Because of deeply rooted vegetation, which causes that there is still
transpiration possible during the percolation process, there is modelled a time delay through
dividing the overtop by a recession constant, K of 3 months.
The field capacity (Smax) of the soil is roughly depending on the land cover. The estimates
are shown in Table 4.4 and are depicted in Figure 4.8.
49

HYDROLOGICAL MODEL DESCRIPTION
Table 4.4: Field capacity (Smax) based on land cover (Seyam, 2002).
Land cover
% of basin
Smax
[mm]
Forests 3.8 270
Closed Shrublands 0.0 180
Open Shrublands 0.0 130
Woody savannas 15.4 280
Savannas 57.1 320
Grassland 0.3 150
Croplands 5.3 250
Urban and Built-up 0.1 170
Croplands and natural vegetation mosaic 14.4 330
Barren or sparsely vegetated 1.0 110
Water 2.6 0
Figure 4.8: Field capacity (Smax).
The field capacity (Smax) determines the amount of water that will percolate to the ground
water and which part will stay in the unsaturated zone. This water in the unsaturated zone is
stored in the pores of the soil together with air and is called soil moisture (Su). If there is
enough soil moisture the transpiration can be potential (Tp). However due to soil moisture
depletion, the actual transpiration can be less. Rijtema and Aboukhaled (1975) found the
following relation between the soil moisture and the actual transpiration:
⎛ 1
⎞
Tra = min ⎜
× Tp×
Su,Tp
( 1−p)
⋅Smax
⎟
⎝
⎠
Equation 4.10
Where:
Tra
p
Smax
Tp
= Monthly actual transpiration [mm/month]
= Amount of maximum available moisture that is readily available
for transpiration (≈ 0.5) [-]
= Water holding capacity [mm]
= Montly potential transpiration [mm/month]
50

HYDROLOGICAL MODEL DESCRIPTION
Su
= Monthly soil moisture [mm]
In Figure 4.9 is given the relation between the soil moisture and the actual transpiration.
Sub surface
Infil.
Field capacity
Su
Unsaturated
zone
Smax
(1-p)*smax
Wilting point: Su = 0
Tp
Tra
Percolation
(su>field capacity)
Figure 4.9: Relation between soil moisture (Su) and actual transpiration (Tra).
Normally, the potential transpiration (Tp) used in Equation 4.10 follows from:
Pe = Int + Tp
Equation 4.11
Where:
Pe = Monthly potential evaporation by Thornwaithe [mm]
Int = Monthly interception [mm]
Tp = Monthly potential evaporation [mm]
This equation assumes that the potential evaporation properly accounts for interception.
However the potential evaporation is in our case calculated with the Thornwaithe formula.
This formula only takes temperature into account and so it is not clear to what extent this
assumption is true. Therefore three possibilities for relating potential transpiration to the
potential evaporation are considered (Seyam, 2002).
1. Potential evaporation (pe) does not account at all for interception (int); Because the
potential evaporation is calculated based on surface temperatures and the interception
process mostly takes place at a higher level and is driven by mechanisms as wind and
air temperature, the potential transpiration is equal to the potential evaporation. A
typical example where this could happen is in dense forests.
2. Potential evaporation does account for interception;
51

HYDROLOGICAL MODEL DESCRIPTION
This is the case where interception is mainly evaporated from bare soil. The potential
transpiration is then equal to the potential evaporation minus the interception.
3. Potential evaporation does partly account for interception;
This situation happens between option 1 and 2 and depends on the land cover.
In Figure 4.10 the three possibilities for the calculation of the potential transpiration are
schematically depicted.
Option 1
Option 2
Option 3
Pe
Tp Int
Pe
Tp Int
f x Int
Pe
Tp Int
Dense forest
f = 1
Bare soil
f = 0
Partly vegetated
0 < f < 1
Figure 4.10: Three possibilities for potential transpiration.
As can be concluded from the description of the three options is that the land cover
determines the potential transpiration. If the catchment is covered with dense forest then there
can be assumed that interception is not included in the potential evaporation. The treetop
factor f is set to one. If the catchment is not covered at all (bare soil) it is likely that
interception is included in the potential evaporation and f is set to zero. In general the
potential transpiration can then be written as:
Tp = Pe −( 1− f ) × Int
Equation 4.12
Where:
Tp
Pe
f
Int
= Monthly potential transpiration [mm]
= Monthly potential evaporation by Thornwaithe [mm]
= Treetop factor (between 0 and 1) based on land cover map
= Monthly interception [mm]
In Table 4.5 and Figure 4.11 the values of the treetop factor f are depicted.
52

HYDROLOGICAL MODEL DESCRIPTION
Table 4.5: Relation between land cover and the treetop-factor.
Land cover % of basin f
Forests 3.8 1.00
Closed Shrublands 0.0 0.50
Open Shrublands 0.0 0.50
Woody savannas 15.4 0.70
Savannas 57.1 0.60
Grassland 0.3 0.15
Croplands 5.3 0.25
Urban and Built-up 0.1 0.50
Croplands and natural vegetation mosaic 14.4 0.25
Barren or sparsely vegetated 1.0 0.15
Water 2.6 /
Figure 4.11: Map of the treetop factor f multiplied with 100.
With the calculation of the actual transpiration it is now possible to determine the amount of
green water. The green water consists of the water in the unsaturated zone and is calculated
with Equation 4.13.
Green = Tra + (I nt− I nt mar)
Equation 4.13
4.2.1.3 Land process: ground water flow
The ground water is recharged with the fast component (cr*Pnet), which is the water that
flows through macro pores and cracks, and the slow component: Overtop. The Overtop is the
water that percolates from the unsaturated zone if the field capacity (= Smax) is reached.
From the ground water the depletion is separated into three components: the saturation
overland flow (saof), the quick flow (qflo) and the slow flow (sflo).
Saturation overland flow is the very fast component. This is the water that flows over the
surface, because the ground water table has reached the surface due to the saturated soil. The
Digital Elevation Model (DEM) and the bottom of the draining river (GWS 0 ) determine when
53

HYDROLOGICAL MODEL DESCRIPTION
the ground water table reaches the surface. The distance between the surface and the bottom is
called GWS dem (see Figure 4.12).
Water divide
Surface according to DEM
GWS dem
Reference level, GWS 0
Figure 4.12: Schematization of GWS dem .
The space GWS dem is not completely available for water, because of the porosity and the
compression of the soil. That’s why the empirical equation written in Figure 4.13 is used to
give the relation between the GWS dem and the actual space, GWSmax. When the ground water
table exceeds the GWSmax then the water will directly drain into the river.
GWSmax
GWSmax [mm]
180
160
140
120
100
80
60
40
20
0
( )
GWSmax = 0.025⋅
ln GWSdem
0 100 200 300 400 500
GWSdem [m]
Figure 4.13: Relation between GWS dem and GWSmax.
54

HYDROLOGICAL MODEL DESCRIPTION
The second ground water component is the quick flow (qflo). This component is the quick
ground water flow through macro pores and cracks. It is assumed that the outflow linearly
depends on the ground water level minus the threshold level, GWSquick. This level is found
by multiplying the GWSmax with the calibration factor qc. The qc-factors differ from sub
catchment to sub catchment (Table 4.6). Subsequently, the outflow is calculated with
Equation 4.14. The recession coefficient rtq is set to 4 months.
Table 4.6: qc-factors for each sub catchment.
Sub catchment qc-factor Sub catchment qc-factor
Upper Zambezi 0.70 Kafue 0.50
Barotse 0.30 Luangwa 0.30
Cuando-Chobe 0.50 Lake Nyasa-Shire 0.30
Middle Zambezi 0.30 Lower Zambezi 0.30
max(GWS−
GWSmax,0)
Qflo = Equation 4.14
rtq
The third and last component is the slow ground water flow (sflo). Again calculated as a
linear reservoir, however with a recession coefficient (rts) of 12 months.
As can be seen from Figure 4.14 there is a water loss in the recession period: the river is given
its water to the cells around the river. In the existing model this is modelled with a negative
ground water level, which causes negative slow flows. The calculation of slow flow in the
exiting model is given in Equation 4.15.
max(GWS, −15)
Sflo = Equation 4.15
rts
In STREAM this is a good manner to simulate infiltration from the river to the ground water.
However, problems occur when using negative discharges in the routing procedure. Therefore
it is recommended to limit the slow flow by requiring at least a discharge of 0 m³/s and to
look for another manner to implement the infiltration process. In this study, the model is
executed with Equation 4.15, which means that negative discharges are possible, although this
is physically wrong.
55

HYDROLOGICAL MODEL DESCRIPTION
Observed discharge
4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
0
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
Discharge [m³/s]
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
Date
Lukulu
Vicfalls
Figure 4.14: Observed data of Lukulu and Victoria Falls.
With the calculation of the three ground water components, it is possible to calculate the blue
water with Equation 4.16
Blue = Saof + Qflo + Sflo
Equation 4.16
With:
Saof
Qflo
Sflo
= Saturation overland flow [m³/s]
= Quick flow [m³/s]
= Slow flow [m³/s]
4.2.1.4 Land process: capillary rise
From the ground water some water will go back to the unsaturated zone. This process of the
upward movement of water through narrow pores by cohesive forces is called capillary rise.
Capillary rise causes an increase in transpiration if the capillary zone (area just above the
ground water table) is in the root zone of the vegetation.
Capillary rise takes place everywhere in the catchment if the ground water table is not to low.
If the ground water table is 25 mm below the reference level, the capillary rise is set to 2
mm/month, which is a very small amount. Assumed is then that it is almost not possible to
refill the unsaturated zone with water from the ground water in the root zone. When the
56

HYDROLOGICAL MODEL DESCRIPTION
ground water level is higher then –25 mm, there will be more capillary rise. Especially in the
dambos and in the floodplains, where the ground water table is near the surface, the capillary
rise is high. The capillary zone is then in the root zone of the vegetation and this will cause
more transpiration.
4.2.1.5 Lake process: evaporation
The only fluxes, which cause the water level fluctuations in the lakes, are the precipitation,
the evaporation and the in- and outflow. The open water evaporation is set to the potential
evaporation calculated with Thornwaithe-formula. The evaporation is subtracted from the
precipitation to obtain the net precipitation (pnet). The inflow and the outflow are not
calculated in STREAM, but they are calculated after routing in a spreadsheet. See paragraph
4.2.2.
4.2.2 Routing in spreadsheet
After STREAM has carried out the calculation of the Zambezi-script, routing is executed for
the channel storage and the reservoir storage. This is important because of the large
floodplains and the stabilization effect of the reservoirs. In the Zambezi-script only the
reservoirs: Kariba, Itezhitezhi, Cabora Bassa and Lake Nyasa are considered.
4.2.2.1 Muskingum routing
The storage in a channel is important, because it determines the smoothing of the flood wave.
Unfortunately, it is not easy to determine the storage volume in a channel from for example
topographic maps due to the crudeness and the lack of information about the water levels.
Another way to calculate the storage is by using the simplified water balance formula:
S − S I + I Q + Q
= −
∆t 2 2
t+ 1 t t t+ 1 t t+
1
Equation 4.17
Where:
S
I
Q
= Storage [m³]
= Inflow of channel [m³/s]
= Outflow of channel [m³/s]
57

HYDROLOGICAL MODEL DESCRIPTION
When calculating the storage volume in a river channel it appears that storage is relatively
higher during rising stages than during falling stages, which can be seen in Figure 4.15.
Figure 4.15: Relation between storage volume and inflow and/or outflow (from Savenije 2000 (left) and
New Mexico Tech (right)).
In general it can be said that the storage volume is a weighted function of the inflow- and the
outflow storage (Equation 4.18).
S= x⋅ S + (1−x) ⋅ S
Equation 4.18
I
Q
I= a⋅d
I
Q
c
Q= a⋅d
S
S
c
= b⋅d
m
= b⋅d
m
Equation 4.19
Assuming power-law functions (Manning), with d = depth, for the inflow, outflow and
storage, the following relation is found when substituting Equation 4.19 in Equation 4.18:
m
c
⎛ I⎞ ⎛Q⎞
S= x⋅b⋅ ⎜ ⎟ + ( 1−x)
⋅b⋅⎜ ⎟
⎝a⎠ ⎝ a ⎠
m
c
Equation 4.20
The value of m/c indicates linearity (m/c = 1) or non-linearity (m/c ≠ 1). It is also well seen
that if x is low the channel acts like a linear reservoir (large floodplains and storage effects)
58

HYDROLOGICAL MODEL DESCRIPTION
and if x has a high value there are low storage effects and the water is transmitted very
quickly through the channel.
For a linear response, thus m/c = 1, Equation 4.20 can be rewritten as:
S= K[ x⋅ I + (1−x) ⋅ Q]
Equation 4.21
With K = b/a and unit time. K is called the storage constant and approximates the travel time
of the wave through the reach. The storage constant can be determined by calculating the
gradient of the right hand graph in Figure 4.15. The values of x vary from 0 to 0.5.
Combining Equation 4.17 and Equation 4.21 gives the Muskingum Routing equation.
Q = c I + c I + c Q
Equation 4.22
t+ 1 0 t+
1 1 t 2 t
c
0
− 2x +∆t K
=
2(1 − x) +∆t K
Equation 4.23
2x +∆t K
c1
=
2(1 − x) +∆t K
Equation 4.24
c
2
2(1 −x) −∆t K
=
2(1 − x) +∆t K
Equation 4.25
c0 + c1+ c2
= 1
Equation 4.26
This equation considers that the inflow is entering the reach at the beginning and that the
insignificant inflow is running into the reach after the inflow point. In reality, there is usually
lateral inflow from tributaries. Very often this can be neglected, but in the case of the
Zambezi catchment lateral inflow must be taken into account. This can be done in five
manners. First, the later inflow can be added to the inflow. Second, the lateral inflow can be
added to the outflow. Third, divide the later inflow (weighted) over the inflow and outflow.
Fourth, use the four-point Muskingum method from Ponce and Yevjevich (1978). Or last use
the three parameter Muskingum method from O’Donnell (1985). Seyam has chosen for the
first option. This is an appropriate manner if the lateral inflow streams into the reach near the
59

HYDROLOGICAL MODEL DESCRIPTION
inflow point. This is not everywhere case. Mostly there are several tributaries along the entire
reach, so the first option is maybe not a very good choice, which has to be investigated.
Seyam has figured out by trial-and-error which reaches have to be routed and which not. In
Figure 4.16 are depicted in blue the reaches which are routed. The routing has been carried
out in the spreadsheet program MS Excel.
Up Nduben
Luangwa up1
Kabompoo
Nduben
Up Lukulu
Luangwa up2
Kalabo Lukulu
Luangwa bc Z.
Up Cuando
Inlet Itezi.
Kafue bc Z.
Gorge
Outlet Cahora
Vic1
Outlet Itezi.
Inlet Cahora
Outlet Kariba Matundu Cais
Zambezi bc Z.
Vic2 Vic3
Sanyati
Inlet Kariba Umtuli
Zambezi bc Z.
Gwai-North
Cuando bc Z.
VicFalls Gwai-Sud
Shangani
Outlet Nyasa
Liwonde
Shire bc Z.
Mouth
bc Z. = before confluence Zambezi
Figure 4.16: The red dots are calculation points for STREAM. The blue lines are routed reaches with
Muskingum-routing.
60

HYDROLOGICAL MODEL DESCRIPTION
4.2.2.2 Reservoir routing: Kariba
Lake Kariba is an artifical lake, which was
finished in 1959 with the building of the double
curved concrete arch dam (Figure 4.17). The dam
was mainly built for the generation of
hydropower. For the routing in the reservoir
Seyam found the following equations.
Figure 4.17: Kariba dam ( http://airzim.co.zw)
Table 4.7: Reservoir characteristics of Kariba.
Minimum storage (S min)
[m³]
Maximum storage (S max)
[m³]
Q const
[m³/s]
Kariba 1,0 x 10 10 4,0 x 10 10 870
If S ≤ S min :
( )
I + di + P −900
net
= + Equation 4.27
Q 550
15
If S > S max :
( I di P )
⎧ + +
net
⎫
Qconst
⋅ 0.8+
⎪
2 ⎪
Q= max⎨ S S
⎬
⎪
−
max ⎪
⎪⎩
( 3600⋅24⋅30)
⎪⎭
Equation 4.28
If S min < S < S max :
Q
( )
I + di + P −1000
net
= Qconst
+ Equation 4.29
15
Where:
S
Q
I
di
P net
= Storage [m³]
= Outflow of the reservoir [m³/s]
= Inflow into the reservoir [m³/s]
= Lateral inflow into the reservoir [m³/s]
= Net precipitation into the reservoir [m³/s]
61

HYDROLOGICAL MODEL DESCRIPTION
Q const
= Constant outflow of the reservoir [m³/s]
4.2.2.3 Reservoir routing: Itezhitezhi
The Itezhitezhi reservoir is the smallest of the
considered reservoirs. The dam was made for the
generation of hydroelectric power and regulates the
flow of the Kafue River. For the reservoir routing of
the Itezhitezhi reservoir Seyam found the following
relations.
Figure 4.18: Itezhitezhi dam (WWF).
Table 4.8: Reservoir characteristics of Itezhitezhi.
Minimum storage (S min )
[m³]
Maximum storage (S max )
[m³]
Q const
[m³/s]
Itezhitezhi 5,2 x 10 9 7,0 x 10 9 300
If S ≤ S min :
( )
⎧ I+ di+ P
Q min
net
⎫
= ⎨ ⎬
⎩ 50 ⎭
Equation 4.30
If S > S max :
( I di P )
⎧ + +
net
⎫
Qconst
+
⎪
2 ⎪
Q= max⎨ S S
⎬
⎪
−
max ⎪
⎪⎩
( 3600⋅24⋅30)
⎪⎭
Equation 4.31
If S min < S < S max :
Q= ( I+ di+ P net )
Equation 4.32
Where:
S
Q
I
= Storage [m³]
= Outflow of the reservoir [m³/s]
= Inflow into the reservoir [m³/s]
62

HYDROLOGICAL MODEL DESCRIPTION
di
P net
Q const
= Lateral inflow into the reservoir [m³/s]
= Net precipitation into the reservoir [m³/s]
= Constant outflow of the reservoir [m³/s]
4.2.2.4 Reservoir routing: Cabora Bassa
The third large reservoir for hydroelectric power
generation is Reservoir Cabora Bassa in Mozambique. In
Figure 4.19 a picture of the double curve concrete arch
dam is given. The outflow of the reservoir is described in
Table 4.9 and in Equation 4.33 till Equation 4.35.
Figure 4.19: Cabora Bassa dam
(http://www.sweco.se).
Table 4.9: Reservoir characteristics of Cabora Bassa.
Minimum storage (S min )
[m³]
Maximum storage (S max )
[m³]
Q const
[m³/s]
Cabora Bassa 4,0 x 10 10 6,0 x 10 10 2300
If S ≤ S min :
Q = 1700
Equation 4.33
If S ≥ S max :
Q= Q +
const
S−S
max
( 3600⋅24⋅30)
Equation 4.34
If S min < S < S max :
Q= Q const
Equation 4.35
Where:
S
Q
Q const
= Storage [m³]
= Outflow of the reservoir [m³/s]
= Constant outflow of the reservoir [m³/s]
63

HYDROLOGICAL MODEL DESCRIPTION
4.2.2.5 Reservoir routing: Nyasa
Lake Nyasa (or Malawi) is the only natural lake, which is considered in the Zambezi model.
Its only outlet is the river Shire. The outflow is determined with the general spillway formula
in Equation 4.36.
Q= K⋅B⋅( h− h ) c
c
Equation 4.36
With:
Q
K
B
h
h c
c
= Outflow [m³/s]
= Spillway constant [m 2-c /s]
= width of spillway [m]
= water level [m]
= crest level [m]
= exponent
Because it is a free overflow spillway c is equal to 3/2 and K is 1,5 m 0.5 /s.
Successively, the outflow of the lake is calculated with the predictor-corrector procedure:
( ) c
Q= K⋅B⋅ h − h
Predictor:
t−1
c
( )( )
*
S = St−
1+ di + Pnet−Q 3600⋅24⋅30
h
⎛S
−S
⎞
= +⎜ ⎟
⎝ A ⎠
*
* 0
hc
Corrector:
*
* t−1
Q = K⋅B⋅ −hc
( net )( )
** *
t−1
**
c
**
( S −S0
)
**
** t−1
Q = K⋅B −hc
t−1
⎛h
⎜
⎝
+ h
2
S = S + di + P −Q 3600⋅24⋅30
h
= h +
⎛h
⎜
⎝
A
+ h
2
**
( )( )
S = S + di + Pnet −Q 3600⋅24⋅30
c
⎞
⎟
⎠
c
⎞
⎟
⎠
64

HYDROLOGICAL MODEL DESCRIPTION
With
di
P net
A
= lateral inflow into the lake [m³/s]
= net precipitation [m³/s]
= surface area [m²]
For the spillway at the end of Lake Nyasa, Seyam found B=140 m, h c =290 m and A=3,0x10 10
m².
65

HYDROLOGICAL MODEL DESCRIPTION
66

ERROR- & SENSITIVITY ANALYSIS
5 Error- & sensitivity analysis
In this chapter the errors and the sensitivity of the Zambezi-model will be analyzed. First an
overview of the existing errors will be given in Section 5.1. If possible, the errors are also
quantified. Keep in mind that the term ‘error’ does not mean error due to e.g. model
simplifications, but that the term ‘error’ is used for occurring deficiencies due to uncertainties.
In Section 5.2 the model results will be presented. In Section 5.3 the GLUE-procedure
(Generalised Likelihood Uncertainty Estimator), which will be used to give an indication of
the model uncertainties and sensitivity, will be explained. Next will be described how the
GLUE-procedure is implemented in the Zambezi-model in Section 5.4. In the last section the
GLUE-results are presented.
5.1 Errors
In Chapter 2 an overview of the existing errors is given, which is also depicted in Figure 5.1.
On the left side of the figure the hydrological model process is depicted and on the right side
the process of Geodesy/Aerospace. As can be seen, in almost every process step a new error is
introduced. Starting at the left side the first error is introduced by measuring and calculating
the river discharge. Usually, the discharge (Q) is calculated by measuring the water level (h)
in the river and the discharge is derived by a Qh-relation. Errors in measuring the water level
can occur due to wrong installation of the measuring device, due to a malfunctioned
measuring instrument, due to inaccurate reading of the observer, etc. Also in the Qh-relation
errors can exist, because the Qh-relation is usually found in an experimental way. The empiric
relation can change in time due to a lot of possibilities, for example erosion, sediment deposit
or obstacles. Overall it is difficult to quantify the magnitude of measuring and calculation
errors, but it appears from practice that errors of 10 to 20% are not an exception. This should
be reminded when using the observed data to calibrate the model output. It is an assumption
that the observed data is the ‘real world’
In the centre column again the measuring errors occur, but this time in relation with satellite
measurements. Next to the errors, which have been described in the previous paragraph, a lot
of extra errors arise. First, every received signal has to be corrected for causes like earth
curvature, cloudiness, scatter, etc. This correction is just an estimate and because of that the
corrected signal could still contain some false information. Second, the algorithm that
translates the satellite signals into for example precipitation, temperature or NDVI maps
67

ERROR- & SENSITIVITY ANALYSIS
contain some errors due to the lack of knowledge about the system. For example, for
generating precipitation maps with the MIRA or FEWS algorithm an assumption is made
about the relation between temperature and rain rate. However, the real relation between
temperature and precipitation is not known yet.
Error(measurement)
Real world
Error(measurement)
Error(measurement)
River measurements
Satellite
measurements
GRACE
measurements
STREAM input maps
Error(interpolation)
Algorithm
1
Error(calculation)
STREAMmodel
dG measured
map
Error(algorithm1)
comparison
Error(model)
Error(model)
dG calculated
map
Calculated
River
discharge
Modeled
River
discharge
dS hydro map
Algorithm
2
calibration
Figure 5.1: Overview of errors. Hydrology: yellow left side. Geodesy/Aerospace: green right side.
The second error that occurs in the centre column is the interpolation error. After obtaining
the satellite maps of our interest it is sometimes necessary to interpolate the data to obtain
continuous information in time and space. For example in the MIRA-precipitation data some
days had not a rain rate value for the complete catchment. To fill those gaps it is essential to
make assumptions, which result in errors. Besides the temporal gaps, it is also possible to
have spatial gaps like in the used temperature maps. By assuming a certain temperature value
for the missing cells (e.g. the average of the surrounding cells) presumably an error is
introduced.
68

ERROR- & SENSITIVITY ANALYSIS
To quantify the errors in the satellite input maps, the estimated satellite data is compared with
the observed data. From the residual the 95% confidence level is calculated, assuming a
normal distribution. Next, the uncertainty in the satellite data is estimated by dividing the
value of 2 times the standard deviation by the average value of the observed data. Although
this is a quite rough way of uncertainty estimation, it can be used as a first estimate.
It is important to know the uncertainty in the input maps, because the Zambezi model is a
threshold model, which causes a non-linear model reaction. A small overestimation can result
in very large discharge increase, because it could be that the threshold value is just exceeded.
Uncertainties in precipitation data
The uncertainty estimation for the precipitation satellite data is not carried out for the
complete time series from January 1978 until December 2003, because it makes only sense to
do this for the FEWS RFE 2.0-algorithm. Namely, the two other algorithms are using the
observed data as input for the algorithm, which will result in an underestimation of the
uncertainty.
Used precipitation algorithms
180 96 36
WAD
MIRA_corrected
FEWS RFE2.0
0
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
312
Time (1 = jan 78,...,312 = dec 03)
Figure 5.2: Used precipitation algorithms 1978-2003. The value in the bar is amount of months that the
algorithm is used.
For the period January 1978 until December 1992 the Weighted Average Distance (WAD)-
algorithm (or Inverse Distance-method) is used. This algorithm interpolates the observed data
over a certain area. A characteristic of this method is that the interpolated values at the same
location as the observation stations are similar. This can be seen in Figure 5.3, where the
peaks and the depressions are the original observation stations. Comparing the rainfall
estimates with the same observed data would result in a residual of zero, which is not
reflecting the real error of the entire catchment. Hence the uncertainty in the precipitation is
not based on this algorithm.
69

ERROR- & SENSITIVITY ANALYSIS
Weighted Average Distance:
n
0 ∑ i i and
i=
1
( ) = λ × Z( x )
Z' x
λ =
with:
Z’(x 0 ) = estimate at x 0
Z(x i ) = observation in point x i
λ i = weight for observation Z(x i)
d i = distance between x 0 and x i
b = distance weighting power, mostly 2
i
1d
n
∑
i=
1
b
i
1d
b
i
Figure 5.3: Weighted Average Distance-method to interpolated rainfall data from rain gauges
(Savenije et al., 2003).
Also the MIRA corrected -algorithm is not used, because this algorithm is corrected to the
available observed data (see Section 3.1.3). Although the residual will not be equal to zero, it
will be very small and will give a distorted picture of the uncertainty in the rainfall.
Hence the only algorithm that is independent of the available observed data is the FEWS RFE
2.0-algorithm. In Figure 5.4 the residual between FEWS and the observed data is plotted. As
can be seen the standard deviation (SD) is equal to 19.14 mm/month. Dividing two times the
SD by the average value of the observed data (70.22 mm/month) gives an uncertainty
estimation of 55% with 95% assurance.
Residual Bias [mm/month]
[mm]
60.00
40.00
20.00
0.00
-20.00
-40.00
-60.00
jan-01
Residual Bias Precipitation
( = FEWS RFE2.0 - Observed)
overestimation
apr-01
jul-01
okt-01
jan-02
apr-02
jul-02
okt-02
jan-03
underestimation
Date
SD = 19,14
µ = 0,56
apr-03
jul-03
Figure 5.4: Monthly residual between the FEWS RFE 2.0-algorithm and the observed
data.
70

ERROR- & SENSITIVITY ANALYSIS
Uncertainties in temperature data
The same procedure could be executed to estimate uncertainties for the temperature maps.
However, for the period January 1978 until December 1992 again the Weighted Average
Distance method is used, which makes comparison with observed data useless. Also for the
period from January 1993 until December 2003 no uncertainty calculation is carried out,
because hardly any observed data was available.
Looking back at Figure 5.1 the last errors in the centre column are the ‘model errors’. These
errors occur due to wrong model concepts and/or wrong parameter estimation. In case of the
Zambezi-model a storage reservoir-model is applied. This kind of model assumes that the
zones in the soil can be schematized as a series of reservoirs. When a storage reservoir is full,
the superfluous water will flow into a second reservoir. This means that when a storage
reservoir is not full yet, totally no water will flow into another reservoir. In reality this is not
the case. For example, when the unsaturated zone is completely dry and it starts to rain, it
takes some time before a first drop of water from the unsaturated zone will percolate into the
saturated zone. The start of the percolation process is modelled by the threshold value (= field
capacity). Of course this is explainable, because first the pores of the unsaturated zone have to
be filled up. However, the starting moment is not fixed and certainly not happening abruptly.
Hence, the storage reservoir-model makes large assumptions, however it appears to work fine
for modelling hydrological processes on a monthly time scale.
The second possibility of the model errors are due to wrong parameter estimation. To
determine the value of a parameter is difficult, because usually a range of possibilities is
present. For example the parameter ‘soil porosity’ is very depending on the soil conditions:
which soil type is present, what is the grain distribution, how dense is the soil packing, etc.
Assigning one single value for the porosity is then not easy. Especially when the parameters
are calibration parameters it is very difficult to assign a proper parameter value, because then
also the relation with reality becomes vaguely or is totally disappeared. In the Zambezi-model
almost all parameters are calibration parameters.
To quantify the model errors the GLUE-procedure is used, which will be explained in Section
5.3.
71

ERROR- & SENSITIVITY ANALYSIS
5.2 Model results
As a first approach the Zambezi-model was
only calibrated for the western part of the
catchment, because of the objective to do a
WESTERN PART
comparison with the gravity measurements
of GRACE. For this comparison it is
important to have as little noises as
possible. The main noise, which is
influencing the gravity measurements, is the Figure 5.5: Western part of the Zambezi catchment.
tide. Because in the western part this noise
is not present main focus is given on this part of the catchment. Another advantage of the
western part of the catchment is the absence of reservoirs. Because of the lack of knowledge
about the rule curves of the reservoirs, the capacity of the reservoirs, etc. it is better to keep
this out of consideration in the first approach.
For the calibration of the western part, the simulated discharges are compared with observed
data. In Table 5.1 an overview of the available discharge data is given. If a cell is coloured
grey, this means that for that specific year observed discharge data is available. However, it
does not mean that the time series is complete. If in the last column is written ‘gaps’ then the
time series is incomplete.
Table 5.1: Available observation data.
Observation
Year
River
location
78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04
Gwai-north Gwai gaps
Gwai-sud Gwai gaps
Itezithezi OUTLET Kafue
Kabompo Kabompo gaps
Kafue at gorge Kafue
Kalabo Luanginga gaps
Kariba INLET Zambezi
Kariba OUTLET Zambezi
Liwonde Shire
Luangwa up2 Luangwa
Lukulu
Zambezi
Matundu Cais Zambezi
Nduben
Kafue
Nyasa OUTLET Shire
Sanyati Sanyati gaps
Shangani Shangani gaps
Umtuli Umtuli gaps
Vic. Falls Zambezi
For the calibration it is desirable to have a time series without gaps, otherwise a comparison
should be made between assumed observed data and simulated data. As mentioned before,
focus is given on the western part of the catchment and parts without reservoirs. Remaining
calibration locations are then only Lukulu and Victoria Falls (VicFalls), also because a long
subsequently time series is available.
72

ERROR- & SENSITIVITY ANALYSIS
Lukulu
In Figure 5.6 to Figure 5.8 the model results of Lukulu are presented. In the hydrograph it can
be seen that the Zambezi model simulates the discharge reasonably. The moment of occurring
of the peak discharges are mostly similar to the observed discharges and the magnitude of the
peak discharges are usually in the same order of the measured discharges. The Root Mean
Square Error (RMSE) is 269,81 m³/s and the Nash-Sutcliffe efficiency coefficient is 0,70.
Hydrograph LUKULU
3,500.00
0
3,000.00
100
Discharge [m³/s]
2,500.00
2,000.00
1,500.00
1,000.00
500.00
200
300
400
500
600
700
Precipitation [mm/month]
0.00
800
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
jan-04
Date
Precipitation simulated observed
Figure 5.6: Hydrograph of Lukulu.
Double Mass Curve LUKULU
(routed)
200000.00
180000.00
160000.00
140000.00
MIRA corrected
Q
cum. simulated
120000.00
100000.00
80000.00
FEWS
60000.00
40000.00
20000.00
0.00
0.00 20000.00 40000.00 60000.00 80000.00 100000.00 120000.00 140000.00 160000.00 180000.00 200000.00
cum. observed
Q
Figure 5.7: Double mass curve of Lukulu. The pink line is the 45° line. are presented
73

ERROR- & SENSITIVITY ANALYSIS
Scatterplot LUKULU
3000.00
y = 0.9367x
R 2 = 0.6963
2500.00
Simulated discharge [m³/s]
2000.00
1500.00
1000.00
500.00
0.00
0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00
Observed discharge [m³/s]
Figure 5.8: Scatter plot of Lukulu with trend line.
Although the hydrograph is reasonable and the sum of the simulated discharges is equal to the
observed discharges (see double mass curve in Figure 5.7), the model generally
underestimates the discharge slightly. This can be concluded from the gradient of the trend
line in the scatter plot in Figure 5.8, which is less than one.
This overall underestimation is usually due to the underestimate of the peak discharges. In the
scatter plot it can be seen that the high discharges are more often simulated too low. This
could be due to the temporal rainfall distribution. When precipitation falls in a short period
the reaction on the discharge is higher. The model, in that case, will underestimate the
rainfall, because it assumes a uniform temporal rainfall distribution.
Another remarkable fact is that in the period before January 1992 the low discharges in the
dry season are underestimated and after January 1992 the low discharges are overestimated.
Furthermore, it can be said that the magnitude of the simulated discharge follows the
magnitude of the precipitation, except from the peak in 1989. Despite this relation is
expectable, this is not the case for the observed discharge compared to the precipitation. This
could be an indication that the used precipitation data is not correct. However, there can be
more reasons why the discharge did not follow the precipitation, like temporary storage
somewhere in the catchment, which causes a time lag in the release of the water.
74

ERROR- & SENSITIVITY ANALYSIS
Victoria Falls
The same graphs as showed for Lukulu are given in Figure 5.9 until Figure 5.11 for Victoria
Falls. Again the model results are quite reasonable, but they are worse than the results in
Lukulu, because Victoria Falls is further downstream in the catchment and thus the error
accumulates. The RMSE of the discharge is 439,11 m³/s and the Nash-Sutcliffe efficiency
coefficient is 0,68.
Hydrograph VICTORIA FALLS
Discharge [m³/s]
4,500.00
4,000.00
3,500.00
3,000.00
2,500.00
2,000.00
1,500.00
1,000.00
500.00
0.00
0
100
200
300
400
500
600
700
800
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
jan-04
Precipitation [mm/month]
Date
Precipitation simulated observed
Figure 5.9: Hydrograph of Victoria Falls.
Double Mass Curve Victora Falls
(routed)
300000.00
250000.00
cum. simulated discharge
200000.00
150000.00
100000.00
MIRA corrected
FEWS
50000.00
0.00
0.00 50000.00 100000.00 150000.00 200000.00 250000.00 300000.00
cum. observed discharge
Figure 5.10: Double mass curve of Victoria Falls. The pink line is the 45° line.
75

ERROR- & SENSITIVITY ANALYSIS
Scatterplot VICTORIA FALLS
4500.00
4000.00
y = 0.8565x
R 2 = 0.5871
3500.00
Simulated discharge [m³/s]
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
0.00
0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00 3500.00 4000.00 4500.00
Observed discharge [m³/s]
Figure 5.11: Scatter plot of Victoria Falls with trend line.
As can be seen the performance in Victoria Falls is worse than in Lukulu. First, the
underestimation is much higher and second, the total discharge volume is not exactly
following the 45° line, which can be seen in the double mass curve in Figure 5.10. The double
mass curve first constantly underestimates the discharge and then it overestimates, like a S-
curve around the 45° line. From the double mass curve and the scatter plot the seasonal
variability can also be seen. Especially in Figure 5.10 the horizontal parts indicates the
underestimation of the discharge in the wet season.
The third difference between the results of Lukulu and Victoria Falls is that the relation
between the magnitude of the precipitation and the discharge (simulated as well as observed
discharge) is less clear.
Storage
Besides the discharge in the Zambezi River also the storage stocks in the unsaturated zone
(su) and saturated zone (gws) are of importance for the gravity measurements. These two
stocks have a large influence on the mass distribution in the catchment. Figure 5.12 and
Figure 5.14 are showing the change in storage of the unsaturated and saturated zone for the
76

ERROR- & SENSITIVITY ANALYSIS
locations Lukulu and Victoria Falls. The sum of the SU and GWS are given in Figure 5.13
and Figure 5.15. Finally, in Figure 5.16 and Figure 5.17 the distributions of storage stocks are
depicted.
Storage variation LUKULU
100
80
60
40
ds/dt [mm/month]
20
0
jan-78 sep-80 jun-83 mrt-86 dec-88 sep-91 jun-94 mrt-97 nov-99 aug-02
-20
-40
-60
-80
-100
Date
dSU/dt
dGWS/dt
Figure 5.12: Storage variations in Lukulu.
Total storage variation LUKULU
100
80
60
40
ds/dt [mm/month]
20
0
jan-78 sep-80 jun-83 mrt-86 dec-88 sep-91 jun-94 mrt-97 nov-99 aug-02
-20
-40
-60
-80
-100
dS/dt
Date
Figure 5.13: Total storage variations in Lukulu (= SU + GWS).
77

ERROR- & SENSITIVITY ANALYSIS
Storage variation VICTORIA FALLS
150
100
ds/dt [mm/month]
50
0
jan-78 sep-80 jun-83 mrt-86 dec-88 sep-91 jun-94 mrt-97 nov-99 aug-02
-50
-100
Date
dSU/dt
dGWS/dt
Figure 5.14: Storage variations in Victoria Falls.
Total storage variation VICTORIA FALLS
150
100
50
ds/dt [mm/month]
0
jan-78 sep-80 jun-83 mrt-86 dec-88 sep-91 jun-94 mrt-97 nov-99 aug-02
-50
-100
-150
Date
dS/dt
Figure 5.15: Total storage variations in Victoria Falls (= SU + GWS).
78

ERROR- & SENSITIVITY ANALYSIS
January
May
September
February
June
October
March
July
November
April
August
December
Unit
mm
Figure 5.16: Water distribution in the unsaturated zone (SU) in 2003.
January
May
September
February
June
October
March
July
November
April
August
December
Unit
mm
Figure 5.17: Water distribution in the ground water zone (GWS) in 2003.
79

ERROR- & SENSITIVITY ANALYSIS
As can be seen in all figures the variations in the northern part of the catchment are much
higher than in the southern part. This is partly due to the higher rainfall intensity in the
northern region. When more rainfall is available in the system it is logical that more water
will be stored in the soil. Hence the red spot in the unsaturated zone just beneath Lake Nyasa
in Figure 5.16 can be clarified by this factor.
Looking at the saturated zone in Figure 5.17 it can be seen that also a large difference
between the sub catchments occurs. This is the second reason why in the northern part larger
variations in the ground water zone occur according to the model. Namely, most northern sub
catchments have a large qc-value (see Table 4.6), which determines the threshold value
GWS quick . A large threshold value means that less water will runoff and more water will be
stored. The reason why the northern sub catchment Barotse has not a high ground water stock
variation is thus because the qc-value is small compared to the other sub catchments.
Colours of water
Figure 5.18 and Figure 5.19 are showing the average resource of the Zambezi catchment. In
Table 5.2 the average monthly and yearly percentages of the white, green and blue water are
depicted. As can be seen the percentage of white water (evaporation through interception) can
be up to 50% in the wet season. Average over the year 41% is white water. This is quite
much, especially because the white water is calculated with the formula of De Groen (see
Equation 4.5), which only considers canopy, mulch and wet soil interception. De Groen
calculates the monthly interception based on the Markov properties of the daily rainfall.
Colours of water
(average of the entire catchment)
200
180
160
140
120
100
80
60
40
20
0
jan-78
jan-79
jan-80
jan-81
jan-82
Resource [mm/month]
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Date
WHITE GREEN BLUE
Figure 5.18: Average resources of the Zambezi catchment (absolute values).
80

ERROR- & SENSITIVITY ANALYSIS
Colours of water (%)
(average of the entire catchment)
100%
80%
Resource [mm/month]
60%
40%
20%
0%
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
Figure 5.19: Average resources of the Zambezi catchment (relative values).
Date
WHITE GREEN BLUE
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Table 5.2: Yearly and monthly water resources.
[mm/month]
yearly
%
jan
%
feb
%
mar
%
apr
%
may
%
jun
%
jul
%
aug
%
sep
%
oct
%
nov
%
dec
%
Precipitation 870 204 178 133 39 6 2 1 1 5 33 92 175
WHITE 351 41 70 48 65 46 53 39 22 29 4 13 1 8 1 7 1 6 4 20 20 49 44 53 65 52
GREEN 437 51 64 44 60 43 67 50 47 62 27 76 15 80 12 84 15 89 16 78 21 50 37 46 55 44
BLUE 64 8 12 8 15 11 15 11 7 9 4 11 2 12 1 9 1 5 1 3 0 1 1 1 5 4
W + G + B 852 147 140 135 76 35 19 15 17 21 42 82 124
In Appendix 4 other model results of the Zambezi catchment can be found. In this appendix
are shown the hydrographs of other locations, the magnitude of each of the water balance
terms and the amount of the water resources (colours of water).
5.3 Generalised Likelihood Uncertainty Estimator (GLUE)
In the Chapter 4 the Zambezi-model has been described. All the parameters are manually
calibrated to obtain satisfactory model results. The Zambezi-model is further calibrated to
make statements of the uncertainty and sensitivity of the model. For the calibration the
automated model calibration procedure ‘Generalised Likelihood Uncertainty Estimator’
(GLUE) by Beven (1989) is used. The procedure is further described by Binley and Beven
(1991) and Beven and Binley (1992).
81

ERROR- & SENSITIVITY ANALYSIS
The GLUE procedure is different from most calibration procedures because it assumes that it
is not likely that one optimal parameter combination exist. Whereas common procedures are
searching for the global optimum, GLUE realizes that because of the errors in the (complex)
model structure and in the observed data, there does not exist a global optimal parameter set.
It is possible to find equally satisfactory simulation results with different parameter values.
This is called ‘equifinality’.
The idea of the GLUE-procedure is that it randomly generates a large number of runs by
selecting parameter sets from specified parameter distributions. The output of the runs will be
compared with the observed data and is assigned a likelihood. This likelihood is a measure for
how well the model conforms to the observed behaviour of the system.
The first step of the GLUE-procedure is to select parameters and to define suitable parameter
ranges with a distribution function. Choosing the upper- and lower bound of the parameter
range in an objective way is difficult (Rientjes, 2004). Field measurements and available
maps can help to find appropriate boundaries. For the definition of the distribution function
mostly a uniform or Gaussian distribution function is chosen if less knowledge about the
appearance of the parameter is available. The chosen distribution function is called ‘prior
likelihood distribution function’.
After the parameter ranges and distribution functions are defined the GLUE-procedure
generates by Monte Carlo sampling lots of parameter sets. The necessary amount of generated
parameter sets is not easy to determine, because it depends on factors like number of degrees
of freedom, number of parameters, parameter inter-relation, etc. (Werner, 2000). The
generated parameter sets are successively put into the model and the output is compared to the
observed data. The performance of the parameter set is defined by the ‘likelihood measure’.
There are many possibilities for defining the likelihood measure. The only restrictions are that
the measure should approximate zero when the behaviour is not satisfactory and that the
likelihood should increase when the behaviour is increasing. However, the chosen likelihood
measure is important for the results of the GLUE-procedure. Some examples of likelihood
measures are:
82

ERROR- & SENSITIVITY ANALYSIS
N
2
⎛ σ ⎞
e
= ⎜1−
2 ⎟
(Franks et al., 1996) Equation 5.1
⎝ σo
⎠
2
−
( ) N
e
= σ
(Binley and Beven, 1991) Equation 5.2
N
2
⎛ σ ⎞
e
= ⎜−w
2 ⎟
(Lamb et al., 1998) Equation 5.3
⎝ σo
⎠
Where:
σ
e
σ
o
= likelihood measure
= variance of residuals
= variance of observations
N = likelihood shape factor (mostly set to 1)
w
= weight
It is also possible to define a rejection threshold, which determines if the simulation is good
enough to be further analyzed. If the likelihood is below the threshold value the parameter set
is termed ‘non-behavioural’ and is rejected. The value of the rejection threshold is important
for the GLUE-analysis, but is difficult to define in an objective way. Regularly, only the best
10% likelihoods are further analyzed. However sometimes also threshold values of 50% are
presented in literature (Rientjes, 2004). Hence it depends on the judgement of the modeller
which threshold value is chosen.
After the lowest likelihoods are rejected the remaining likelihoods are rescaled in such a way
that the sum of the (behavioural) likelihoods will become equal to one. To obtain the marginal
likelihood distribution function for each parameter, the likelihoods and the parameter values
should be plotted in a diagram, like in Figure 5.20. The marginal distribution function tells
something about the sensitivity of the model to parameter changes. If the distribution function
is steep, it means a high sensitivity and the parameter can be well defined. Conversely, gentle
slope implies a low sensitivity and a hardly identifiably parameter.
83

ERROR- & SENSITIVITY ANALYSIS
Figure 5.20: Marginal likelihood distribution functions (Rientjes, 2004).
It is also possible to use the rescaled behavioural likelihood measures for uncertainty
estimations of the model. By treating the distribution function of the likelihoods as a
probabilistic weighting function for the simulated variables (e.g. discharge) an impression can
be obtained about the uncertainty of the model due to non-uniqueness of parameter sets. For
example in Figure 5.21 the simulated discharges, obtained by running the model with
different parameter sets, are ranked on the x-axis with the accompanying likelihood value.
From this graph statistical features can be derived, like the 5% and 95% bounds, the mean,
etc. Also the uncertainty bound can be derived of hydrographs, which can be seen in Figure
5.22.
Figure 5.21: Distribution function of predicted discharges (Beven and Binley, 1992).
84

ERROR- & SENSITIVITY ANALYSIS
Figure 5.22: Hydrographs of different simulations. The upper figure is the output of the model and the
lower figure is showing the 5% , the 95% bounds and the mean (Beven and Binley, 1992).
Updating likelihood measures
When new observation data becomes available, it is possible to update the likelihood values
by applying the probability theory of Bayes (Equation 5.4).
( i
Y,M)
Θ =
N
∑
i
( Y Θi,M ) ( Θi,M)
( Y Θi,M ) ( Θi,M)
Equation 5.4
Where:
M
( Θ ,M i )
= expression for the applied model concept
= prior likelihood of parameter set i
( Θ i
Y,M)
= posterior likelihood of parameter set i
( Y Θi,M)
= likelihood calculated by use of additional time series Y
Remarks
Although the GLUE-procedure is generally accepted as a calibration tool (Rientjes, 2004),
some remarks about the method exist. First, GLUE examines the over-all uncertainty of the
model. Hence the likelihood values are determined by the sum of the errors in the model
concept, the errors in the parameter estimation, the input- and boundary errors and the errors
in the observed data. Second, the GLUE-procedure does not deal with the inter-relation
between the parameters. GLUE just generates parameter sets by selecting random parameter
values and treats the parameters as completely independent. The last criticism is about
85

ERROR- & SENSITIVITY ANALYSIS
choosing the rejection threshold, the appropriate likelihood measure and the prior distribution
function. This is difficult to determine in an objective way as mentioned before.
5.4 Implementation of GLUE for the Zambezi-script
For executing the GLUE-procedure Werner developed in 2002 a MATLAB procedure. This
program can be linked with different kinds of models, on the condition that the model can run
via the batch-mode. In Figure 5.23 the flow chart of the procedure is given and in Appendix 5
the Matlab code with the input parameters can be found.
Main
settings.txt
Generate
Parameter set
1..n
experimental
parameter
distributions
Run model
n-times
Parameter
set i (i=1,n)
Parameter
Template file
Determine
fitness
run i (i=1,n)
Results run i
(i=1,n)
Parameter file
Model
Executable
Determine
likelihoods
Uncertainty
bounds
Parameter
analyses
distributions
for variables
(e.g. Qpeak)
experimental
parameter
distributions
Figure 5.23: Flow chart of MATLAB uncertainty estimation procedure (Werner, 2000).
The first step (‘main’) of the procedure is to select an objective function and to define a prior
distribution function. This prior distribution function can also be the results of previous runs
(‘experimental parameter distribution’). In the second step the procedure generates parameter
86

ERROR- & SENSITIVITY ANALYSIS
sets by Monte Carlo sampling. It saves the generated values of the parameters in a ‘parameter
file’. In the third step the model is run n-times. In Figure 5.23 the ‘Model Executable’-box is
drawn as one single box. However, in our case consists the ‘Model Executable’-box of two
sub-models. The first sub-model is the water balance calculation in STREAM and the second
sub-model is the routing in MS EXCEL. Because the latter sub-model is not able to run in the
batch-mode, a program is written in the object oriented programming language JAVA to
execute the routing (Appendix 6). Based on the output of the second sub-model the fitness is
determined (step 4) by the Nash-Sutcliffe efficiency. Successively, the likelihoods are
calculated in step 5. After this step several analysis- and plotting tools are provided to obtain
insight into the GLUE-results.
Approach
The objective of the GLUE-procedure is in the Zambezi case to obtain knowledge about the
uncertainties in our model output due to uncertainties in the parameters and in the model
concept. For the gravity measurements the storage stock is the most important output. Thus
the GLUE-procedure should be carried out for the state variables SU (Storage Unsaturated
zone) and GWS (Ground Water Storage). However, the GLUE-procedure requires measured
data to determine the performance of the parameter set, which is not available for the state
variables of interest. The only available observed data is discharge data. Hence, the approach
is to calculate the likelihood of a parameter set, which is based on the goodness-of-fit of the
simulated discharge compared to the observed discharge and next assume that the goodness of
fit of the discharge is equal to the goodness of fit of the storage stocks. In Figure 5.24 this
approach is schematically depicted.
SU i
i
Input
STREAM-model
+
ROUTINGmodel
GWS i
i
Q i
R i
i
Q obs,i
Parameter set i, generated
by Monte Carlo sampling
Figure 5.24: Schematization of implementation of GLUE for run i. With SU (Storage Unsaturated
zone), GWS (Ground Water Storage), Q (Discharge), Q obs (Measured discharge), R (Nash-Sutcliffe
efficiency) and (likelihood).
87

ERROR- & SENSITIVITY ANALYSIS
5.4.1 Determination of dominant parameters
One of the disadvantages of GLUE is the large calculation time. Especially when the
calculation time of one model run is long, when a lot of runs are executed or when many
parameters exist, a lot of patience is needed to get good stable GLUE-results. One solution to
reduce the calculation time is to make a calculation cluster, which processes the calculation
parallel on several computers. Another way to save calculation time is to reduce the amount
of parameters. If less parameters exist in the model, less parameter sets are necessary to cover
the complete range of parameter values. Therefore a quick sensitivity analysis is carried out to
determine which parameters will not be assessed by GLUE. If from the sensitivity analysis
appears that a certain parameter has a low sensitivity this parameter will not be considered by
GLUE. A parameter with a low sensitivity implies that the parameter does not contribute
significantly to the uncertainty in the model output.
The quick sensitivity analysis is carried out on the storage stock outputs SU (Storage
Unsaturated zone) and GWS (Ground Water Storage), because those are the state variables of
interest. For each parameter four model runs are done with different parameter values. The
default parameter value, which is determined by manual calibration, is decreased with -40%
and -20% and increased with +20% and +40%. Next, the outcome of the model run is
compared with the output from the default parameter value. Because the sensitivity analysis is
carried out for two state variables (SU and GWS), two outputs need to be compared with the
default ones. Because the Nash-Sutcliffe-criterion gives scaled efficiencies, which makes the
comparison easier, this criterion is used to determine the sensitivity of the parameter.
F − F F
= = − Equation 5.5
0
Reff
1
F0 F0
n
F
( )
2
0
= ∑ SVi
−SV
Equation 5.6
i=
1
n
F=
∑ ( SV )
2
i −SVi
Equation 5.7
i=
1
SV is the state variable, which is SU or GWS.
With:
R eff
= coefficient of efficiency
88

ERROR- & SENSITIVITY ANALYSIS
SV i
SV
SV i
= output of the state variable at time step i when using the default parameter
value
= average output of the state variable between time step i and n when using
the default parameter value
= output of the state variable at time i when using the modified parameter
value
If the efficiency approximates one this means that the difference between the output of the
default parameter values and the output of the modified parameter values is almost zero. If the
efficiency approximates zero or even becomes negative, the performance of that modified
parameter value is not satisfactory.
By plotting the efficiencies on the y-axis and
the parameter modification factors on the x-
axis, a good insight in the sensitivity of
parameters is given. When the angle nearby
modification factor equals one is sharp, this
means that the parameter is very sensitive and
when the angle is less sharp the sensitivity is
low (see Figure 5.25).
NS-efficiency
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Sensitivity of parameters
Low sensitivity
High sensitivity
0.4 0.6 1 1.2 1.4
Parameter modification factor
Figure 5.25: Sensitivity of parameters.
Results
The Zambezi-script in STREAM contains twelve parameters (see Table 5.3), which are all
assessed by a quick sensitivity analysis to select the four most sensitive parameters. The
parameters of the Muskingum routing are not considered in this analysis.
Table 5.3: Parameters in the Zambezi-script.
Symbol Description Unit
D Interception threshold [mm/month]
cr Separation coefficient [%]
qc Coefficient to determine threshold GWSquick; [GWSquick = qc * GWSmax] [%]
k Recession constant overtop [month]
Gws25 Multiplication factor to determine GWSmax; [GWSmax = gws25 * ln_gwsmax] [-]
rtq Recession constant Qflo [month]
Cap25 Level which below only 2 mm/month of capillary rise occurs [mm]
rts Recession constant Sflo [month]
smax Field capacity [mm]
f Tree-top factor [%]
cap Capillary rise [mm/month]
Gwsmin Minimum level of slow groundwater flow; [sflo = max(gws, -15)/ rts] [mm]
89

ERROR- & SENSITIVITY ANALYSIS
The sensitivity analysis is carried out on the storage in the unsaturated zone (SU), on the
ground water storage level (GWS) and on the average of SU and GWS. Of course, it could be
expected that some parameters are more sensitive for SU and other parameters more to GWS.
This appears also from the results, which are shown in Figure 5.26 and Figure 5.27. For both
storage stocks cr and D belong to the four most sensitive parameters (Figure 5.26), but
parameter f and smax are only important for SU and gws25 and cap only for GWS.
Because both storage stocks are of importance for the gravity measurements, only the four
most sensitive parameters of the average analysis are selected. These are: D, cr, cap and
gws25.
Sensitivity parameters Zambezi-script on SU
Sensitivity parameters Zambezi-script on GWS
1.00
1
0.95
0.9
0.90
0.8
R Nash Sutcliffe)
0.85
R (Nash Sutcliffe)
0.7
0.80
0.6
0.75
0.5
0.4
0.70
0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1
0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40
Parameter multiplication factor
Parameter multiplication factor
D cr qc k gws25 rtq cap25 rts smax f cap gwsmin
Figure 5.26: Sensitivity analysis results for the storage in the unsaturated zone, SU (left) and for the
ground water storage level, GWS (right).
Sensitivity parameters Zambezi-script (average SU & GWS)
1
0.95
0.9
0.85
R (Nash Sutcliffe)
0.8
0.75
0.7
0.65
0.6
0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40
Parameter multiplication factor
D cr qc k gws25 rtq cap25 rts smax f cap gwsmin
Figure 5.27: Sensitivity analysis results for the average of SU and GWS.
90

ERROR- & SENSITIVITY ANALYSIS
5.4.2 Objective function and likelihood measure
For the assessment of the model output an objective function has to be chosen. Some often
used evaluation functions are the Root Mean Square Error (RMSE), the Nash-Sutcliffe
efficiency, the coefficient of determination, the transformed RMSE, etc. Because the Nash
Sutcliffe-efficiency is widely applied and accepted this criterion has been chosen (Werner,
2000).
F − F F
= = − Equation 5.8
0
Reff
1
F0 F0
(
ˆ
)
n 2
F = ∑ Q −Q
Equation 5.9
0 i
i=
1
(
ˆ
i i)
n 2
F= ∑ Q −Q
Equation 5.10
i=
1
With:
R eff
ˆQ
i
Q
Q i
= coefficient of efficiency
= observed discharge at time step i
= average of observed discharge between time i and n
= calculated discharge at time i
On basis of the Nash-Sutcliffe efficiency (R eff ), the likelihood measure is calculated. The
chosen likelihood is just scaling the efficiencies between the minimum and maximum
efficiency and weights the likelihood in such way that the sum is equal to one (Equation
5.11).
i
=
n
∑
j=
1
Reff ,i
− min( Reff
)
( Reff , j
− min( Reff
))
Equation 5.11
5.5 GLUE results
The GLUE procedure is carried out twice with no rejection threshold applied and with Latin
Hypercube Sampling. First, 300 runs are executed to determine the sensitivity of the model on
the parameters. These results are described in Section 5.5.1. A second GLUE-process was
carried out for the determination of the sensitivity of the input data. The amount of runs is 40
and the results are analyzed in Section 5.5.2.
91

ERROR- & SENSITIVITY ANALYSIS
5.5.1 GLUE for parameter uncertainty
In Figure 5.28 and Figure 5.29 are shown the uncertainty bounds of the hydrographs of
Lukulu and Victoria Falls.
Figure 5.28: Uncertainty bounds of the hydrographs of Lukulu (t = 1 is December 1978).
Figure 5.29: Uncertainty bounds of the hydrographs of Victoria Falls (t = 1 is December
1978).
92

ERROR- & SENSITIVITY ANALYSIS
A remarkable fact of the uncertainty bounds are the negative discharge values for Victoria
Falls, which are not possible but ‘accepted’ for this research as mentioned in Section 4.2.1.3.
Because this is not occurring in Lukulu, it can be said that this is caused by a high capillary
rise value, maybe together with a high interception threshold value. A high capillary risevalue
means a high depletion of the ground water storage and a large D-value will cause that
less water takes part of the discharge process. Both can cause negative values of the ground
water storage level, which will result in negative discharge values.
Sensitivity of the parameters
In Figure 5.30 the cumulative density function for the four parameters are plotted. As can be
seen only the parameters cr and D show some non linear reaction on the discharge.
Apparently, the parameters cap and gws25 are chosen in such a way that they do not have a
large influence on the discharge. Each selected parameter value results in the same likelihood.
As can be seen the discharge is sensitive for the value of cr and also a little for the
interception threshold D. Especially around the cr-value of 1, the model reaction is relatively
high.
Figure 5.30: Cumulative Density Functions of the assessed parameters.
93

ERROR- & SENSITIVITY ANALYSIS
Optimal parameter value
The same picture as shown in Figure 5.30 is found in the dotty plots in Figure 5.31. It is not
possible to define an optimal parameter value from the GLUE-procedure, hence the
parameters are not identifiably. Only for the separation parameter, cr an optimum around the
default value can be noticed.
Figure 5.31: Dotty plots of the parameters D, cr, cap and gws25.
From the parameter set with the highest likelihood value the hydrographs of Lukulu and
Victoria Falls can be plotted. These hydrographs are depicted in Figure 5.32.
94

ERROR- & SENSITIVITY ANALYSIS
Figure 5.32: Optimal hydrographs for Lukulu (blue) and Victoria Falls (red). The crosses are the
observed data of Lukulu and Victoria Falls (t = 1 is December 1978).
Sensitivity on soil moisture and ground water storage
For gravity measurements it is important to know the uncertainty in the storage stocks.
Because it would not make sense to know the uncertainty on the stock-level in Lukulu and
Victoria Falls, the average stock value of the western part of the catchment is assessed.
The storage stocks are divided in the soil moisture content (Figure 5.33), the ground water
storage (Figure 5.34) and the sum of the soil moisture content and the ground water storage in
Figure 5.35. As can be seen, the uncertainty in the total water storage stock due to
uncertainties in the model parameters is average about 30 mm, according to 90% of the runs.
95

ERROR- & SENSITIVITY ANALYSIS
Figure 5.33: Uncertainty bounds for the soil moisture content (SU) (t = 1 is January1978).
Figure 5.34: Uncertainty bounds for the ground water storage level (GWS) (t = 1 is January1978).
96

ERROR- & SENSITIVITY ANALYSIS
Figure 5.35: Uncertainty bounds for the total water storage stock, which consists of the soil moisture
content and the ground water storage level (SU + GWS) (t = 1 is January1978).
5.5.2 GLUE for input uncertainty
In Section 5.1 is roughly estimated the uncertainty in the precipitation data. This uncertainty
is about 55% with 95% confidence level. For the potential evaporation this estimation was not
possible. Hence for the GLUE analysis is assumed that both input data have the same
uncertainty. The uncertainty is applied on the input data, by multiplying the entire time series
with a factor between 0.45 and 1.55.
In Figure 5.36 and Figure 5.37 are shown the uncertainty bounds for Lukulu and Victoria
Falls.
97

ERROR- & SENSITIVITY ANALYSIS
Figure 5.36: Uncertainty bounds of the hydrograph of Lukulu due to uncertainty in input data (t =
1 is December 1978).
Figure 5.37: Uncertainty bounds of the hydrographs of Victoria Falls due to uncertainty in the
input data (t = 1 is December 1978).
98

ERROR- & SENSITIVITY ANALYSIS
Sensitivity of the input data
As can be seen by the steepness of the function in the right hand graph of Figure 5.38, the
precipitation is very sensitive on the discharge as expected. The potential evaporation on the
other hand is badly identifiable. This result proves that the effect of using temperature data
from other years (see Section 3.2), is not that high. Hence it is acceptable to do this
assumption as a first estimate.
Figure 5.38: Cumulative Density Functions of the precipitation and potential evaporation.
Optimal input data
The dotty plots of the input data in Figure 5.39 show that a clearly optimal parameter exists
for the precipitation data. Again it appears that the rainfall is overestimated in the Zambezi
model. This overestimation was also found in the comparison between the precipitation
algorithm and the observed data from ground stations. From the dotty plot it appears that the
rainfall data should be lowered with about 5% to obtain better model results. For the potential
evaporation data no optimal input multiplication factor can be found.
Figure 5.39: Dotty plots of the input data: precipitation (prec) and potential evaporation (pe).
99

ERROR- & SENSITIVITY ANALYSIS
The optimal hydrographs for Lukulu and Victoria Falls, with the highest likelihood, are
plotted in Figure 5.40.
Figure 5.40: Optimal hydrographs for Lukulu (red) and Victoria Falls (blue). The crosses are the
observed data (t = 1 is December 1978).
Sensitivity on soil moisture and ground water storage
Because of the importance of the uncertainty in the storage for the gravity measurements, the
uncertainty bounds for the soil moisture content, the ground water storage and the sum of both
components are given in Figure 5.41 to Figure 5.43.
The deviation of the 5% bound in Figure 5.41 looks initially strange. However, this behaviour
can be clarified by the fact that if the ground water storage level (gws) is between zero and -
25 mm (compared to the reference level), then the capillary rise has a large influence on the
ground water level. If this occurs simultaneously with a small or no net precipitation, the
ground water level is not able to recover soon. However, if the ground water level drops
below -25 mm the influence of the capillary rise will decrease and the ground water level will
stabilise.
100

ERROR- & SENSITIVITY ANALYSIS
Figure 5.41: Uncertainty bounds for the soil moisture content (SU) due to uncertainties in the input
data (t = 1 is January 1978).
Figure 5.42: Uncertainty bounds for the ground water storage level (GWS) due to uncertainties in the
input data (t = 1 is January 1978).
101

ERROR- & SENSITIVITY ANALYSIS
Figure 5.43: Uncertainty bounds for the total water storage stock, which consists of the soil moisture
content and the ground water storage level (SU + GWS). These uncertainties are due to uncertainties
in the input data (t = 1 is January 1978).
As can be seen is the variance much higher for the uncertainty in the input data then for the
uncertainty in the parameters. The difference between the 5% bound and the 95% bound is
average about 310 mm. This is mainly caused by the spreading in the soil moisture.
102

GRAVITY MEASUREMENTS AND HYDROLOGY
6 Gravity measurements and hydrology
6.1 Background
As mentioned before it is important to know the water storage stock at a certain place and
time, although it only constitutes about 3.5% of the total water in the hydrological cycle (see
Figure 6.1 and Rodel and Famiglietti, 1999). The soil moisture stock is important for its
relation with transpiration and the ground water stock has its importance for providing base
flow to rivers and provides water to deep-rooted plants in periods of droughts. Both determine
the availability of water for agriculture of domestic use and are important to predict floods
and droughts.
Figure 6.1: The global hydrological cycle, illustrating storages in 10 6 cubic kilometers (boxed) and
fluxes in 10 6 cubic kilometers per year (Dickey, 1997).
However, it is not that simple to monitor the water storage stock. By carrying out ground
measurements it is possible to obtain information, but this information is labor intensive and
provides only point information. Microwave remote sensing techniques could help to
determine the soil moisture content, but this technique is limited to the upper soil and is not
measuring the deep soil moisture and the groundwater. A third option is modelling. However,
the performance of the model results is constrained by the quantity and quality of the model
input.
103

GRAVITY MEASUREMENTS AND HYDROLOGY
A new solution, which has arisen for determining the storage stock, are gravity measurements.
The Earth’s gravity field varies in time and space and defines an irregular ellipsoid, the geoid
(see Figure 6.2). This geoid, which primarily originates from mass distribution irregularities
near the surface of the earth, consists of a static and a time variable part. The static part is
mainly due to mass distributions that only vary on a geological timescale, like continents,
mountains and depressions in the crust. The time variable part occurs due to processes like
terrestrial storage water redistribution, ocean tides, atmospheric changes, post glacial rebound,
etc. Hence the hydrological signal is included in the gravity signals.
Figure 6.2: Earth's geoid of 2003 (from http://www.csr.utexas.edu).
The first time researchers noticed that hydrology could be one of the causes for the temporal
gravity variations, was with the LAGEOS satellite. Yoder et al. (1983) believed that the
changes in the orbit of the satellite were primary caused by redistribution of ground water and
air mass and changes in sea level. Gutierrez and Wilson (1978) have tried to calculate the
disturbances in the satellite’s orbit due to seasonal redistribution of air mass and terrestrial
water storage. They concluded that they were able to roughly predict the perturbations of the
satellite orbit caused by seasonal variations in terrestrial water storage.
After several other studies about the time variable component of the geoid Dickey et al.
(1997) helped to accomplish the Gravity Recovery and Climate Experiment and mentioned
the possibilities for the field of hydrology.
104

GRAVITY MEASUREMENTS AND HYDROLOGY
Finally, the twin satellite GRACE was launched for a mission period of 5 years on the 17 th of
March 2002. As shortly described in Section 2.1, the distance between the twin satellites will
change due to mass variations. GRACE measures precisely this distance between the satellites
with a microwave ranging system. An accelerometer will measure the non-gravitational
accelerations (e.g. atmospheric drag) so that only the acceleration caused by gravity is
considered. And GPS is used to determine the exact position of the satellites (NASA, 2003).
6.2 Relation surface mass to gravity
The shape of the geoid (i.e. the equipotential surface corresponding to mean sea level over the
oceans) describes Earth’s global gravity field. The shape of the geoid is commonly the sum of
spherical harmonics (Chao and Gross, 1987):
∞
l
( θφ ) =
lm ( θ) ( lm ( φ ) +
lm ( φ)
)
∑∑ Equation 6.1
N , a P cos C cos m S sin m
l= 0 m=
0
Where:
N
a
θ
φ
C lm , S lm
= geoid shape
= radius of the Earth
= co-latitude
= east longitude
= dimensionless Stokes coefficient
P lm
= Legendre function (see Wahr et al., 1998 for equation)
l
m
= degree
= order
The coefficients C lm and S lm are the variables, which will be provided by GRACE. For
measuring time-dependent changes in the shape of geoid ( ∆ N ), Equation 6.1 can be
expressed in terms of ∆ Clm
and ∆ Slm
. The changes in the spherical harmonic geoid
coefficients are caused by surface density redistributions ( ∆σ ), which is defined as mass
divided by area.
The changes in the spherical harmonic geoid coefficients consist of two parts (Equation 6.2).
The first part describes the contribution to the geoid from the direct gravitational attraction of
the surface mass. Because the surface mass also loads and elastically deforms the underlying
solid earth, a second part is added (Wahr et al, 1998). This results in Equation 6.3, which
includes both contributions.
105

GRAVITY MEASUREMENTS AND HYDROLOGY
⎧∆Clm ⎫ ⎧∆Clm ⎫ ⎧∆Clm
⎫
⎨ ⎬= ⎨ ⎬ + ⎨ ⎬
⎩∆S ⎭ ⎩∆S ⎭ ⎩∆S
⎭
lm lm surf. mass lm solid E.
Equation 6.2
( l )
( )
( φ)
( φ)
⎧∆ Clm
⎫ 31+
k
⎪⎧cos m ⎪⎫
⎨ ⎬= ∆σθφ× ( , ) P lm ( cosθ)
⎨ ⎬sinθθφ
d d
∆Slm
4πρ a sin m
ave
2l+ 1∫
Equation 6.3
⎩ ⎭ ⎪⎩ ⎪⎭
Where:
∆ C lm
, Slm
∆ = change in dimensionless coefficient
k l
= elastic love number of degree l
ρ
ave
= average density of the earth (=5517 kg/m³)
∆σ
= change in surface density
This formula can be used to derive from the hydrological signal ‘synthetic’ GRACE data.
This procedure is called ‘inverse estimation of GRACE data’.
Rewriting Equation 6.3, with
∞ l
aρ 2l + 1
ρ
w
as the density of water, results into Equation 6.4:
ave
( , ) Plm ( cos ) ( Clm cos( m ) Slm
( m ))
∑∑ Equation 6.4
∆σ θ φ = θ × ∆ φ + ∆ φ
3 1+
k
l= 0 m=
0
l
If ∆σ is divided by ρ
w
, the change in surface mass expressed in equivalent water thickness
is obtained. This equation can be used to calculate the hydrological signal from the measured
data of GRACE. This method is referred as ‘direct estimation of hydrology from GRACE
data’.
6.3 Error sources in GRACE data
Besides the errors, which occur at the side of hydrology, error sources exist at the side of
GRACE. The first error source is the measurement error, which consist of uncertainties in the
orbital parameters and errors in the microwave ranging and accelerometer measurements. The
measurement error decreases with an increasing catchment area and an increasing averaging
period (see Figure 6.3). For the Zambezi model, with a basin area of 1.3*10 6 km² and a
monthly time scale the uncertainty will be less than about 1 mm/month.
106

GRAVITY MEASUREMENTS AND HYDROLOGY
Figure 6.3: Measuring (instrument) errors (millimeters of water equivalent) versus area of the
region for monthly, seasonal and annual averaging periods (from: Rodell and Famiglietti, 1999).
The measurement error will also increase with an increase in the degree as can be seen in
Figure 6.4. At the stage of this research Figure 6.4 is quite optimistic, but in the future the
measuring errors will approximate more to the plotted function. Because GRACE provides
spherical harmonic coefficient up to degree 70-100, filtering methods are required to reduce
the impact of measuring errors.
Figure 6.4: Estimates of the square root of the contribution to the variance of the inferred surface
mass anomaly due to GRACE satellite measurement error, as a function of spherical harmonic degree
(from: Swenson and Wahr, 2002).
107

GRAVITY MEASUREMENTS AND HYDROLOGY
The second error source of the GRACE data is the truncation and interpolation error. In
theory, the geoid can be described with an infinitely high degree, which will result in a perfect
description of the Earth’s gravity field. However, in reality, the geoid coefficients will only be
described by GRACE to a maximum degree of 70. This will cause a truncation error.
The truncation induced by the fact that GRACE provides data to a maximum degree of 70
will cause that the data needs to be interpolated. The spatial scale of the gravity data, λ , is
approximated by the relation (Swenson and Wahr, 2002):
20000 km
λ= Equation 6.5
l
A degree of 70 will result in a spatial resolution of about 300 km. To make the data suitable
for catchment scale, the data need to be interpolated.
The third error source is the leakage error in regional calculations. Because the GRACE
measurements are also influenced by mass changes outside the catchment, an error is occurs.
Especially for small catchments this error is significant (Wahr et al, 1998).
Finally, the last error source arises from problems of removing the effects of atmospheric
mass redistribution and postglacial rebound. Commonly, the atmospheric mass redistribution
effect is removed by the use of modelled data of atmospheric surface pressure. For removing
the effects of post glacial rebound also modelled data is used. The uncertainties in these
models are assumed to be uniformly 20% (Rodell and Famiglietti, 1999). However, this effect
of postglacial rebound can be neglected on monthly to yearly time scales in the Zambezi
catchment.
6.4 Direct estimation of hydrology from GRACE data
As a first approach Marcano Bakker has calculated in 2005 the hydrological signal from
spherical harmonic coefficients by using Equation 6.4. The coefficients, which are provided
by GRACE, are corrected for all known processes in such way that the coefficients should
only consider the static and the hydrological component. To obtain anomalies, she subtracted
from the monthly data a mean geoid, which is based on 14 months of GRACE data. This
mean gravity field consists of all static components and an average hydrological signal.
Marcano Bakker edited the GRACE-data with two methods. First, she calculated the
hydrological signal up to degree 15 and did not apply a filter. The same thing Marcano
108

GRAVITY MEASUREMENTS AND HYDROLOGY
Bakker did for the second method, but this time she executed the calculation up to degree 50.
Because for this degree the measurement error is too high (see Figure 6.4), Marcano Bakker
applied a Gaussian filter with a half radius of 600 km. For the application of a Gaussian filter,
Equation 6.4 is adjusted to Equation 6.6. The results for the second method are depicted in
Figure 6.5.
∞ l
aρ 2l + 1
ave
( , ) 2 WlPlm( cos ) ( Clmcos( m ) Slm( m ))
∑∑ Equation 6.6
∆σ θ φ = π θ × ∆ φ + ∆ φ
3 1+
k
l= 0 m=
0
l
Where:
W l
= Gaussian filter weights
Figure 6.5: Equivalent water height difference (in cm) calculated with the difference between March
2003 harmonic coefficients and 14-month mean coefficients, using maximum degree 50 and a Gaussian
filter with half radius 600 km (from: Marcano Bakker, 2005).
109

GRAVITY MEASUREMENTS AND HYDROLOGY
If zoomed in on the Zambezi catchment Figure 6.6 is obtained. The results of the other
months can be found in Appendix 7. In this appendix also the results of the first method,
which does not apply a Gaussian filter, are shown.
Figure 6.6: Equivalent water height difference (in cm) calculated with the difference between March
2003 harmonic coefficients and 14-month mean coefficients, using maximum degree 50 and a Gaussian
filter with half radius 600 km. Zoomed in to the Zambezi catchment (from: Marcano Bakker, 2005).
Because at the moment of this research no spatial comparison has been made yet between the
GRACE data and the hydrological model output, only the average catchment values are
compared.
110

GRAVITY MEASUREMENTS AND HYDROLOGY
Equivalent water height based on global GRACE data
25
20
Equivalent water height difference [cm]
15
10
5
0
-5
apr-
02
mei-
02
jun-
02
jul-
02
aug-
02
sep-
02
okt-
02
nov-
02
dec-
02
jan-
03
feb-
03
mrt-
03
apr-
03
mei-
03
jun-
03
jul-
03
aug-
03
sep-
03
okt-
03
nov-
03
dec-
03
jan-
04
feb-
04
mrt-
04
apr-
04
mei-
04
jun-
04
jul-
04
-10
Hydrology (monthly mean) [cm] GRACE (monthly mean l=50, GF=600) [cm] GRACE (monthly mean, l=15, GF=0) in [cm]
Figure 6.7: Equivalent water heights of GRACE data (with and without filter) and hydrological
modeled data. Both are the average values for the complete catchment.
In Figure 6.7 the equivalent water height of the GRACE data and the hydrological data are
plotted. Because both signals are relative to a certain datum (the 14-month mean geoid for
GRACE and the zero-ground water level for the hydrological signal) this graph is just
showing in qualitative terms the relation between gravity measurements and hydrology. Better
insight can be obtained by plotting the differences, like in Figure 6.8.
111

GRAVITY MEASUREMENTS AND HYDROLOGY
Equivalent water height changes based on global GRACE data
10
Equivalent water height changes [cm/month]
5
0
-5
-10
apr-
02
mei-
02
jun-
02
jul-
02
aug- sep- okt-
02 02 02
nov- dec- jan- feb- mrt- apr- mei- jun-
02 02 03 03 03 03 03 03
jul-
03
aug- sep- okt-
03 03 03
nov- dec- jan- feb- mrt- apr- mei- jun-
03 03 04 04 04 04 04 04
jul-
04
-15
Hydrology (monthly difference) in [cm] GRACE (monthly difference, l=50, GF=600) in [cm] GRACE (monthly difference, l=15, GF=0) in [cm]
Figure 6.8: Equivalent water height changes of the average catchment.
In general it can be said from Figure 6.8 that the modeled hydrological signal shows the same
pattern as the calculated GRACE data. It only appears that the amplitude of the GRACE data
is higher. One of the reasons for this effect could be that for this calculation global GRACE
data is used. When zooming in from global data to a certain region also changes in mass
outside the catchment are taken into account. This error is referred as ‘leakage error’.
A second remarkable effect is the appeared time lag between the hydrological data and the
GRACE data. This time lag can be caused by the moment of measuring. The hydrological
model calculates its output at the end of the month. However the GRACE measurements are
averaged over a certain time period (up to a month).
As mentioned in the previous paragraph, leakage errors are introduced when using global
GRACE data. To compensate for this effect Swenson and Wahr (2002) investigated some
averaging techniques to calculate regional variations in surface mass density based on
variation in spherical harmonic coefficients. In general, three types can be distinguished:
exact averaging, approximate averaging and optimal averaging kernel. The averaging kernel
functions differ in the balance between measurement error and leakage error. Although the
regional, instead of global, calculation of the hydrological signal from GRACE data is more
accurate, results can not be shown at the moment of this research. Also the results of the
‘inverse estimation of GRACE data’ were not available yet.
112

CONCLUSIONS & RECOMMENDATIONS
7 Conclusions & recommendations
In this study research is carried out on a hydrological model of the Zambezi catchment in
Southern Africa. The model, which has been investigated, consists of two GIS-based sub
models: a water balance model created in the model environment STREAM and a routing
model, which first was executed in MS EXCEL and later has been transformed to a JAVA
program a more sophisticated model.
Model input
Because the main input of a hydrological model is the precipitation, special attention has been
given to different kinds of precipitation algorithms. For the period 1978 until 1992 the
traditional Weighted Average Distance-method is used, because for this time span no accurate
satellite data was available and therefore interpolated rainfall values from ground stations is
one of the best solutions to obtain spatial information. For the period from 1993 several
algorithms, although with different time spans, were available. In this study the MIRA, FEWS
RFE 1.0 and FEWS RFE 2.0 algorithms were investigated. The estimates of the precipitation
algorithms were compared to ten ground stations. The results are depicted in Table 7.1. From
this comparison is concluded that for the period 1993-2000 the MIRA corrected -algorithm (MIRA
algorithm multiplied by a correction factor) gives the best estimates with a correlation
coefficient of 0.98 and a small cumulative overestimation of about 2%. For the period 2001
until present the FEWS RFE 2.0 performs best with a correlation coefficient of 0.97 and a
very small cumulative overestimation of about 1%.
Table 7.1: Overview of the performance of the precipitation algorithms compared to ten ground
stations.
Algorithm
Correlation
coefficient
Gradient of trendline
R² trendline
MIRA (ended)
(jan 1993 - dec 2002)
0.93 0.8693 0.9991
FEWS RFE 1.0 (ended)
(jul 1995 - dec 2000)
0.85 0.9624 0.9504
FEWS RFE 2.0 (operational)
(jan 2001 – dec 2003)
0.97 0.9893 0.9978
MIRA corrected
(jan 1993 – dec 2000)
0.98 0.9819 0.9982
113

CONCLUSIONS & RECOMMENDATIONS
The second important input of the Zambezi model is the potential evaporation. The
evaporation estimates were calculated with the, mainly temperature dependent, Thornwaithe
equation, because its simplicity for Remote Sensing applications. For the period 1978-1992
interpolated temperature values from ground stations were used for the Thornwaithe
calculation. For the years 1993 until 1998 gridded satellite temperature data was found as
input for the Thornwaithe equation. Unfortunately, for the period after 1998 no temperature
data was found. Therefore the potential evaporation estimates for 1993-1998 were used to
complete the time span. This is assumed because the temporal and spatial variation of
temperature is not high. Although the GLUE-procedure showed that the impact on the model
results by the evaporation is less, it is still recommended to look for accurate satellite
temperature data. For example the CRU TS 2.0 dataset [12], which provides monthly
temperature data with a spatial resolution of 0.5° x 0.5°, could help to obtain better model
results. Looking to other methods to calculate potential evaporation values (for example
SEBAL) could also be useful.
Model uncertainties
The errors in the hydrological model mainly consist of:
• Errors in the observed data (measuring and calculation), which is assumed as the ‘real
world’. It appears from practice that errors of 10-20% are not an exception.
• Errors in the model input data. These consist of measuring errors (instrumental, signal
scatter, lack of knowledge, etc.) and interpolation errors (temporal and spatial gaps).
This uncertainty is about 55% with 95% confidence level for the precipitation data.
The error in the temperature data is not determined in this study, due to the lack of
observed data.
• Model errors, which consist of errors due to wrong model assumptions and errors due
to wrong parameterization.
With the GLUE-procedure the impact of these errors on the model results were analyzed.
Model results
The model results were only compared to the observed discharge data of the locations Lukulu
and Victoria Falls in the western part of the Zambezi catchment. From the results appears that
the Zambezi model performs quite reasonable, with a Nash-Suttcliffe coefficient of 0.70 and
0.68 for Lukulu and Victoria Falls respectively. However for both locations the discharge is
underestimated. Especially the high peaks are showing this reaction. One of the reasons for
this deviation could be the temporal rainfall distribution. In the existing model the monthly
rainfall is considered to fall uniformly in time. In reality it is possible that a large amount of
114

CONCLUSIONS & RECOMMENDATIONS
the monthly rainfall falls in a short period. This will result in higher discharge values then
when the precipitation falls more gradually. It is recommended that this possible cause will be
investigated in more detail.
As also mentioned in Chapter 4 negative discharge values are allowed in the existing Zambezi
to simulate the feeding of the ground water from the river, despite it causes problems for the
routing procedure. Although the impact on the model results is less (maximal -15/12
mm/month per cell), it is recommended to remove the concept of negative discharges from
the Zambezi model and to look for another manner to simulate the depletion of the river in
times of drought.
To overcome deficiencies due to wrong precipitation estimates in general, it is recommended
to keep looking for accurate precipitation estimates on a high spatial resolution. One of the
algorithms, which appear to be very prospective, is the TRMM-algorithm [13]. TRMM is an
abbreviation of Tropical Rainfall Measuring Mission and is not considered in detail for this
study, because the spatial resolution of the data was to low (for 3B43: 1° x 1°) at the moment
of this research.
Another recommendation can be made in relation to the seasonal variability. As can be
noticed from the double mass curve of Victoria Falls (Figure 5.10) seasonal effects appear,
especially in the last few months. In the wet season the discharges are underestimated and in
the dry season the discharges are overestimated. It is advised to investigate this cause.
Furthermore it is recommended to look closer to some parameters of the Zambezi model.
Especially the qc-value (coefficient to determine the threshold GWSquick) needs some
attention, because this parameter causes the abrupt borders in the storage stock changes, while
a more graduate pattern would be expected in reality. Until now the value of qc is completely
determined during the calibration process, however it is recommended to look for a more
physical based relation (for example a relation with the soil conditions).
In general it can be said about the hydrological model, that too much precipitation (rainfall is
overestimated) goes into the system and too little water will drain into the river (discharge is
underestimated). This means that either the evaporation is too high or that the amount of
water that is stored in the system is too high. For the gravity measurements the last reason is
most important. Therefore it is advised to refine the hydrological model.
115

CONCLUSIONS & RECOMMENDATIONS
GLUE-results
After a quick sensitivity analysis on the parameters of the water balance model it is
determined that the most sensitive parameters on the total storage stock are the inception
threshold D, the separation coefficient cr, the amount of capillary rise, cap and the
multiplication factor of the GWSmax, gws25. These four parameters are assessed by the
GLUE-procedure.
As can be concluded from the results, it is not possible to determine a parameter set, which
has a significant better performance on the discharge output. Only the separation parameter cr
has a small optimum around the default value of one. To obtain better insight in the sensitivity
of the parameters it is recommended to enlarge the parameter range for D, cap and gws25.
This will cause that the amount of runs needs to be increased. Also the application of a
rejection threshold will enlarge the differences between the outcomes of the parameter set.
From the uncertainty bounds of the total water storage stocks it can be concluded that the
uncertainty in the storage stocks are about 30 mm due to uncertainties in the parameter values.
For the uncertainties due to the input data this value is average about 310 mm. This large
value is mainly caused by the spreading in the soil moisture content and the highly sensitive
precipitation input. According to the GLUE analysis it can be concluded that the optimal
precipitation is about 0.95 times the present precipitation value. For the potential evaporation
data it is not possible to define such an optimum. Furthermore it can be concluded from the
GLUE results that it is, as expected, more important to research the input data then refining
the model concept.
Gravity measurements and hydrology
In general it can be concluded from the preliminary results of Marcano Bakker that the pattern
of the hydrological signal is the same as the GRACE signal when using the method ‘direct
estimation of hydrology from GRACE data’. However, the amplitude of the GRACE data is
larger then the amplitude of the hydrological data. Because this could be caused by leakage
errors, it is recommended to calculate the hydrological signal again, but then with the use of
an averaging kernel, which reduces the leakage error.
Besides the magnitude of the amplitude also a time lag has occurred. One of the reasons for
this time lag can be found in the measuring moment. The Zambezi model calculates the
storage stocks at the end of the month. On the other hand, GRACE measurements are
averaged over a certain time period (up to a month). To overcome this difference in the future
it is recommended to synchronize the measuring moments of both signals.
116

CONCLUSIONS & RECOMMENDATIONS
General recommendations
The existing Zambezi model is only calibrated for the western part of the catchment. This part
is chosen, because in the western part less influence of ocean tides exist and because the nonexistence
of reservoirs. As can be seen in Appendix 4 the model behaves badly downstream
of reservoirs. Therefore it is suggested that more realistic rule curves for the reservoirs are
found.
Mainly in the western part of the catchment routing is important, because of the existence of
large flood plains. Although the routing is implemented in the Zambezi model, it is advised to
look closer to the used method in relation to the lateral inflow. In the existing model the
lateral inflow is added to the begin location of the reach. This is an appropriate assumption if
most of the water will drain into the river close to the begin location. This is not always the
case for the Zambezi catchment. Therefore it is recommended to add not all the lateral inflow
to the begin location, but maybe divide it between the begin location and end location.
Furthermore it is recommended to decrease the spatial resolution of the model and transform
it to the latitude/longitude projection. Although the model resolution is already lowered from
1 x 1 km to 3 x 3 km by Seyam in 2002, it is advised to decrease the resolution more to about
0.1° x 0.1° (≈ 10 km). The model input with the highest resolution is namely the precipitation
(except from the flow direction map) with a resolution on 0.1° x 0.1°. Modeling with on a
higher resolution then the input will not result in better results and is only increasing the
calculation time. The advantage of changing the latitude/longitude (latlong) projection to the
Lambert Oblique Azimuthal Equal Area (Lazea) projection is that almost every satellite data
is stored in the latlong-projection. Changing the projection every time is unnecessary time
consuming, although you should take into account that in the latlong projection not every grid
cell will have the same surface area. However, keep in mind that changing the model
resolution and projection will cause that the model has to be recalibrated.
117

CONCLUSIONS & RECOMMENDATIONS
118

REFERENCES
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[6] Land cover
Africa Land Cover Characteristics database. Version 2 (GLCC v2). These data are distributed
by the Land Processes Distributed Active Archive Center (LP DAAC), located at the U.S.
Geological Survey's EROS Data Center http://LPDAAC.usgs.gov.
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[7] Rain station data 1993-2001 (monthly)
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[8] Rain station data 1978-1993 (monthly)
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U.S.A
[9] Runoff data
Luxemburg, W., 1999: WRS Surface Water Runoff Database System. Ministery of Rural
Resources & Water Development, Department of Water Development, Hydrological Branch.
[10] MIRA
Layberry, R., Kniveton, D., Todd, M., 2001: MIRA daily rainfall estimates for Southern
Africa (1993-2001). University of Sussex, UK.
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http://ltpwww.gsfc.nasa.gov/s2k/html_pages/groups/precip/daily_rainfall_mira. html
[11] FEWS
Herman, A., Kumar, V.B., Arkin, P.A., Kousky, J.V.: CPC/Famine Early Warning System
Dekadal Estimates. http://edcw2ks21.cr.usgs.gov/adds/
[12] CRU TS 2.0
Mitchell, T.D., Carter, T.R., Jones, P.D., Hulme, M., New, M., 2003: A comprehensive set of
high-resolution grids of monthly climate for Europe and the globe: the observed record (1901-
2000) and 16 scenarios (2001-2100). Journal of Climate: submitted.
http://www.cru.uea.ac.uk/~timm/grid/CRU_TS_2_0.html.
[13] TRMM: 3B-43
Huffman, G. J. and D. Bolvin. 2004. SAFARI 2000 TRMM 3B-43 Monthly
Precipitation, 1-Deg, 1999-2001. Data set. Available on-line [http://www.daac.ornl.gov]
from Oak Ridge National Laboratory Distributed Active Archive Center, Oak Ridge,
Tennessee, U.S.A.
125

REFERENCES
126

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Appendix 1
Data transformations in Idrisi
1.1 General transformations.......................................................................................... 129
1.1.1 Dimensions and projection of the maps used in STREAM ............................. 129
1.1.2 Masking ........................................................................................................... 130
1.1.3 ASCII-raster -> Idrisi....................................................................................... 131
1.1.4 *.BIL-files -> Idrisi.......................................................................................... 131
1.2 Data transformation of used maps .......................................................................... 132
1.2.1 DEM [1]........................................................................................................... 132
1.2.2 Soil map [2] ..................................................................................................... 132
1.2.3 Climate-Köppen [3] ......................................................................................... 133
1.2.4 Climate-Temperature (long term) [4] .............................................................. 133
1.2.5 MIRA-precipitation [10].................................................................................. 134
1.2.6 FEWS-precipitation [11] ................................................................................. 150
1.2.7 MIRA-corrected............................................................................................... 151
1.2.8 Resize GIS-maps 1978-1993 ........................................................................... 151
127

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Data transformations in Idrisi
On the Internet a lot of Geographic Information System (GIS) maps can be found.
Unfortunately, they are stored and compressed in very different ways. Because STREAM can
only read IDRISI-files, conversion needs to take place. In this appendix these conversion
steps will be described. The conversions to obtain Idrisi-raster maps are made with different
programs:
• Idrisi Kilimanjaro or Idrisi32 R2 [http://www.clarklabs.org]; GIS package, to display
raster maps and vector files, to do format conversions (real -> integer -> byte vs (see
Table 1)), to do projection transformations and other (batch) data processing
transformations. The Idrisi-maps are the only maps, which can be read by the
program STREAM.
Table 1: Data ranges.
Data format From Till
Byte 0 255
Integer -32768 32767
Real 1E-38 1E38
• ArcView 3.1 [http://www.esri.com]; GIS package, to display raster maps and vector
files, to do projection transformations and other data processing transformations.
Only used for conversion purposes.
• IDL (Interactive Data Language) [http://www.rsinc.com]; Data analyzing and
visualization program. With a scripting language many different file formats can be
read and converted to the desired format.
• WinDisp 5.1 [http://www.fao.org/giews/english/windisp/windisp.htm]; Simple GIS
package to display satellite images.
• DivaGIS 4.1 [http://www.diva-gis.org]; Simple GIS package to display certain
satellite images.
Next to the conversion steps, there is also described which other steps are carried out, like gap
filling, scaling, multiplications, etc.
128

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
1.1 General transformations
1.1.1 Dimensions and projection of the maps used in STREAM
Most GIS maps downloaded from the Internet are stored in an un-projected latitude/longitude
grid. The specification of the appropriate reference-file in Idrisi is given in Figure 1.
C:\Idrisi\Georef\LatLong.ref
ref. system : Geodetic Coordinates (Latitude/Longitude)
projection : none
datum : WGS 1984
delta WGS84 : 0 0 0
ellipsoid : WGS84
major s-ax : 6378137.000
minor s-ax : 6356752.314
origin long : 0
origin lat : 0
origin X : 0
origin Y : 0
scale fac : 1.0
units : deg
parameters : 0
Figure 1:Reference file of latitude/longitude.
The maps used in STREAM are in the Lambert Oblique Azimuthal Equal Area projection
(Lazea), with the following specifications:
C:\Idrisi\Georef\Lazea.ref
ref. system : USGS Lambert Azimuthal Equal Area
projection : Lambert Oblique Azimuthal Equal Area
datum : NAD27
delta WGS84 : -8 160 176
ellipsoid : Clarke 1866
major s-ax : 6378206.40
minor s-ax : 6356583.80
origin long : 20
origin lat : 5
origin X : 0
origin Y : 0
scale fac : na
units : m
parameters : 0
Figure 2: Reference file of Lazea.
In Idrisi the user is able to change the projection:
• REFORMAT -> PROJECT
• Fill in the appropriate file name and projection (lat/long) of the old map and fill in file
name for the new map with the desired projection (lazea).
129

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
• Fill in the following settings to obtain the right size of your maps. Remark: for the
period 1978-1993 the settings are written in the left box and for the period from 1993
fill in the right box. This is because the maps of the run 1978-1993 are not completely
cover the entire catchment. Some parts of the west and south side are missing. In
2004 Gerrits repaired the old and wrong GIS-maps to the correct size, which are
written down in the right box. See for this transformation paragraph 1.2.9.
Period: 1978-1993
Period: 1993-present
Figure 3: Size settings of the maps used in STREAM.
1.1.2 Masking
To save calculation time for STREAM it is possible to calculate only those cells, which are
inside the catchment. To achieve this all the cells, which are outside the catchment should
have a value of –9999. This is called masking.
For masking you need two maps:
• Mask_bool: cells inside the catchment have a value 1 and cells outside the catchment
have a value of 0.
0 0 0 0
0 1 1 0
0 1 1 0
0 0 1 1
• Mask_9999: cells inside the catchment have a value 0 and cells outside the catchment
have a value of –9999.
-9999 -9999 -9999 -9999
-9999 0 0 -9999
-9999 0 0 -9999
-9999 -9999 0 0
130

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
To obtain a masked map use the IMAGE CALCULATOR or the OVERLAY function and
execute the following formula:
[Masked_map] = ([map] * [mask_bool]) + [mask_9999]
-9999 -9999 -9999 -9999 v v v v 0 0 0 0 -9999 -9999 -9999 -9999
-9999 v v -9999 v v v v 0 1 1 0 -9999 0 0 -9999
= (
*
) +
-9999 v v -9999
v v v v 0 1 1 0
-9999 0 0 -9999
-9999 -9999 v v v v v v 0 0 1 1 -9999 -9999 0 0
1.1.3 ASCII-raster -> Idrisi
Some GIS-data is stored in a simple ASCII-raster file. They can be read in IDRISI by using
the GRASS-import function:
• Add the GRASS header (or change the existing header) to the ASCII-file in the first
lines by typing in a text-editor:
Proj: 3
Zone: 0
North: 40N
South: 40S
East: 180E
West: 180W
Cols: 360
Rows: 80
Depends on the size of the original map.
Projection 3 in the GRASS header means lat/long in degrees. Other settings:
0=unreferenced; 1=UTM; 2=state plane.
Most common text-editors do have difficulties with large files. Text-editors like
UltraEdit and Context are capable to deal with large files.
• Save the ASCII file with the extension *.asc.
• FILE -> IMPORT -> SOFTWARE SPECIFIC FORMATS -> GRASSIDR and select
‘real’ under ‘output image data type’.
1.1.4 *.BIL-files -> Idrisi
Although Idrisi has some import functions to import BIL-files (Band Interleaved by Lines),
they do not always work. Another way to import them is by using ArcView:
• Be sure that the Spatial Analyst Package is switched on:
FILE -> EXTENTIONS -> tick Spatial Analyst.
• Open the *.bil file in the View-window by clicking on the [Add Theme] button.
Select under ‘Data Source Types’ the type ‘Image Data Source’ and select the
appropriate *.bil file.
• Select THEME -> CONVERT TO GRID and choose a grid name (e.g. NwGrd1).
• Select THEME -> SAVE DATA SET and select as calculation for example Calc1.
131

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
• Select ANALYSIS -> PROPERTIES and click under ‘Analysis Extent’: ‘Same As
Zambas.shp’. This Zambas.shp-file is a shapefile with the appropriate map
dimensions (length x height) of the catchment.
• To export the map select FILE -> EXPORT DATA SOURCE and select ‘ASCII
Raster’ and click OK. Select the appropriate calculation file (e.g. Calc1) and choose a
name for the ASCII-grid.
• Change the ArcView header to the GRASS header and import the ascii-file in Idrisi
as described in paragraph 1.1.3.
1.2 Data transformation of used maps
1.2.1 DEM [1]
Spatial: 1km * 1km
Temporal: -
Unit: m
Extension: *.bil
• Import the BIL file in ArcView as described in paragraph 1.1.4. Only add one step
after you have converted the map to a grid (step 3):
• Select ANALYSIS -> MAP CALCULATOR and type the following formula:
([NwGrd1] >= 32768).Con ([NwGrd1] – 65536.AsGrid, [NwGrd1])
This is because the downloaded maps are word-maps (0 till 65535) and Idrisi can
only read integer-maps (-32768 till +32767).
• After you have converted the file into Idrisi execute a pit removal, because the DEM
contains small gaps: GIS ANALYSIS -> SURFACE ANALYSIS -> FEATURE
EXTRACTION -> PIT REMOVAL
• Change the spatial resolution to 3 km * 3 km: REFORMAT -> CONTRACT and
choose ‘Aggregate’ x=3 and y=3.
• Mask the map
1.2.2 Soil map [2]
Spatial: 2 minutes grid
Temporal: -
Unit: -
Extension: *.img & *.doc
• Convert Idrisi16 files to Idrisi32 files by FILE -> CONVERT.
132

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
• Change projection of the map from lat/long to lazea by REFORMAT -> PROJECT.
• Reclass the soil classification as desired.
1.2.3 Climate-Köppen [3]
Spatial: 0.5° x 0.5°
Temporal: -
Unit: -
Extension: *.ida
• Use WinDisp to convert the files to Idrisi.
• Change the projection from lat/long to lazea.
1.2.4 Climate-Temperature (long term) [4]
Spatial: 0.5° x 0.5°
Temporal: monthly
Unit: centigrade degrees
Extension: *.bil
• Use DivaGIS to convert the files to Idrisi.
• Change the projection from lat/long to lazea.
• Divide map by ten to obtain °C with the SCALAR function
• Fill the gaps with the RECLASS function. Change all values from 0 to 1 to:
January => 24.0 July => 18.5
February => 24.0 August => 19.0
March => 23.5 September => 23.0
April => 23.5 October => 24.0
May => 21.0 November => 25.0
June => 19.5 December => 25.0
{these values are the average value of the cells around the gap}
1.2.5 Climate-Temperature (1993-1998) [5]
Spatial: 0.5° x 0.5°
Temporal: monthly
Unit: centigrade degrees
Extension: none (ascii-files, zipped with *.gz)
• Unzip files
• Convert ArcInfo raster ascii to Idrisi with the import function in Idrisi.
• Change the projection from lat/long to lazea.
133

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
• Mask the maps with mask_bo in such way that the areas outside the catchment get the
value zero.
• Reclass all the values between -9999 and 0 to zero (gap repair).
• Divide map by ten to obtain °C with the SCALAR function
• Mask the map with Mask_9999.rst
1.2.6 MIRA-precipitation [10]
Spatial: 0.1° x 0.1°
Temporal: daily
Unit: mm/hour
Extension: ascii
For reading the MIRA precipitation data you will need the program IDL (Interactive Data
Language). The precipitation data is in average daily mm/hour, so for our hydrological model,
we have to sum the daily values to monthly values. Secondly, the IDL-program has to export
the data into ASCII-files with a GRASS-header. But first a gap repair on the data has to be
carried out.
To repair the data one of the following actions is executed:
1. If the missing day(s) are in the dry season (May-October) there is no precipitation
added to the month value. It is assumed that the precipitation in those days is zero.
The month sum is easily obtained by summing all the daily values.
2. If one or two days successive are lacking data, the value of the day before and/or the
day after, is filled in. Assumed is a high correlation between two successive days.
Again is the month sum easily obtained by simply summing up all the daily values
inclusive the new values.
3. If more then two days are missing and at least one rain station is available with a
complete daily range of rainfall data, the percentage (p) of the amount of rainfall of
the missing days is calculated from the rain station. The month sum is then divided
by (1-p) to obtain a repaired new month value.
4. If more than two days are missing data and there are no rain stations available, the
average percentage (p) of the amount of rainfall of the missing days is computed
from other years of that month. The month sum is then divided by (1-p) to obtain a
repaired new month value.
5. If a complete month is missing, a highly correlated year is looked for. The missing
month is filled in with the same month of the found correlated year.
134

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
In the following table the final percentage of the missing days are written down. If the
percentages of one month are summed up and subtracted from 100% and next divided 100%
by the result, you will get the multiplication factor of that specific month (option 3, 4 and 5).
If a date is written, the rainfall value of the day then is written in that box used to fill in the
gap of the missing day (option 2). If there is written ‘dry=0’ in a cell then there is nothing
added, because it is the dry season. If there is written ‘SE=0’, it means that there is very less
precipitation so assumed is that also in the missing days, the missing rainfall amounts are
small (option 1).
135

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Table 2: Overview of final multiplication factors of the gap repair.
Date 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
01-jan
02-jan
03-jan
02-jan
04-jan
05-jan
14.18%
06-jan
07-jan
08-jan
07-jan
09-jan
10-jan
11-jan
12-jan
13-jan
14-jan
15-jan
16-jan
17-jan
18-jan 17-jan 17-jan
19-jan 2.82% 18-jan 20-jan
20-jan 21-jan 21-jan
21-jan
20-jan
22-jan
23-jan
24-jan
25-jan
26-jan
27-jan
28-jan
24.49%
29-jan
30-jan
31-jan
01-feb
02-feb
03-feb
02-feb
04-feb
05-feb
06-feb
07-feb
08-feb
09-feb
10-feb
11-feb
12-feb
13-feb
14-feb
15-feb
14-feb
16-feb
17-feb 16-feb 16-feb
18-feb
19-feb
19-feb
20-feb
21-feb
22-feb
23-feb
24-feb
25-feb
26-feb
27-feb
28-feb
29-feb - - -
14.95%
28-feb - - - -
136

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Date 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
01-mrt
02-mrt
03-mrt
04-mrt
05-mrt
06-mrt
07-mrt
08-mrt
09-mrt
10-mrt
11-mrt
12-mrt
13-mrt
14-mrt
15-mrt
16-mrt
17-mrt
18-mrt
19-mrt
20-mrt
21-mrt
22-mrt
23-mrt
24-mrt
25-mrt
26-mrt
27-mrt
28-mrt
29-mrt
30-mrt
31-mrt
01-apr
02-apr
03-apr
04-apr
05-apr
06-apr
07-apr
08-apr
09-apr
10-apr
11-apr
12-apr
13-apr
14-apr
15-apr
16-apr
17-apr
18-apr
19-apr
20-apr
21-apr
22-apr
23-apr
24-apr
25-apr
26-apr
27-apr
28-apr
29-apr
30-apr
02-mrt
SE = 0
03-mrt
06-apr
13-apr
16-apr
SE = 0
07-apr
09-apr
10-mrt
137

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Date 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
01-mei
02-mei
03-mei
04-mei
05-mei dry = 0
06-mei dry = 0
07-mei dry = 0
08-mei
09-mei
10-mei
11-mei
12-mei
13-mei
14-mei
15-mei dry = 0
16-mei dry = 0
17-mei
18-mei
19-mei
20-mei
21-mei
22-mei
23-mei dry = 0
24-mei dry = 0
25-mei dry = 0
26-mei dry = 0
27-mei dry = 0
28-mei dry = 0
29-mei dry = 0
30-mei dry = 0
31-mei dry = 0
01-jun
02-jun
03-jun
04-jun dry = 0
05-jun
06-jun
07-jun dry = 0
08-jun dry = 0
09-jun dry = 0
10-jun dry = 0
11-jun
12-jun
13-jun
14-jun
15-jun
16-jun
17-jun
18-jun
19-jun dry = 0
20-jun dry = 0
21-jun dry = 0
22-jun
23-jun
24-jun
25-jun dry = 0
26-jun
27-jun
28-jun
29-jun
30-jun
138

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Date 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
01-jul dry = 0
02-jul dry = 0
03-jul dry = 0
04-jul dry = 0
05-jul dry = 0
06-jul dry = 0
07-jul dry = 0
08-jul dry = 0
09-jul dry = 0 dry = 0
10-jul dry = 0
11-jul dry = 0
12-jul dry = 0
13-jul dry = 0
14-jul dry = 0
15-jul dry = 0
16-jul dry = 0
17-jul dry = 0
18-jul dry = 0
19-jul dry = 0
20-jul dry = 0
21-jul dry = 0 dry = 0
22-jul dry = 0 dry = 0
23-jul dry = 0 dry = 0
24-jul dry = 0 dry = 0
25-jul dry = 0 dry = 0
26-jul dry = 0 dry = 0
27-jul dry = 0 dry = 0
28-jul dry = 0 dry = 0
29-jul dry = 0 dry = 0
30-jul dry = 0 dry = 0
31-jul dry = 0
01-aug dry = 0
02-aug dry = 0
03-aug dry = 0
04-aug dry = 0
05-aug dry = 0
06-aug dry = 0
07-aug dry = 0
08-aug dry = 0
09-aug dry = 0
10-aug dry = 0
11-aug
12-aug
13-aug
14-aug
15-aug
16-aug
17-aug
18-aug
19-aug
20-aug
21-aug
22-aug
23-aug
24-aug
25-aug
26-aug
27-aug
28-aug
29-aug
30-aug
31-aug
dry = 0
139

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Date 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
01-sep
02-sep
03-sep
04-sep
05-sep
06-sep
07-sep
08-sep
09-sep
10-sep
11-sep
12-sep
13-sep
14-sep
15-sep
16-sep
17-sep
18-sep
19-sep
20-sep
21-sep
22-sep
23-sep
24-sep
25-sep
26-sep
27-sep
28-sep
29-sep
30-sep
01-okt
02-okt
03-okt
04-okt
05-okt
06-okt
07-okt
08-okt
09-okt
10-okt
11-okt
12-okt
13-okt
14-okt
15-okt
16-okt
17-okt
18-okt
19-okt
20-okt
21-okt
22-okt
23-okt
24-okt
25-okt
26-okt
27-okt
28-okt
29-okt
30-okt
31-okt
dry = 0
140

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Date 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
01-nov
02-nov
03-nov
04-nov
05-nov
06-nov
07-nov
08-nov
09-nov
10-nov
11-nov
12-nov
13-nov
14-nov
15-nov
16-nov
17-nov
18-nov
19-nov
20-nov
21-nov
22-nov
23-nov
24-nov
25-nov
26-nov
27-nov
28-nov
29-nov
30-nov
01-dec
02-dec
03-dec
04-dec
05-dec
06-dec
07-dec
08-dec
09-dec
10-dec
11-dec
12-dec
13-dec
14-dec
15-dec
16-dec
17-dec
18-dec
19-dec
20-dec
21-dec
22-dec
23-dec
24-dec
25-dec
26-dec
27-dec
28-dec
29-dec
30-dec
31-dec
02-nov
03-nov
0.00%
0.00%
08-nov
0.00%
28-nov
13.68%
13.70%
10.20%
25-dec
10.21%
2.49%
15-dec
1.91%
30-dec
25.31%
29-nov
141

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Table 3: Option 3.
STATION 676330 677530 677430 676410 675510 675310 676650 AVERAGE
Place Mongu Choma Livingstone Kaoma Solwezi Zambezi Lusaka
long 23.16 27.06 25.81 24.8 26.38 23.11 28.45
lat -15.25 -16.85 -17.81 -14.8 -12.18 -13.53 -15.32
01-nov-95 0 0.00% 2 3.24% 0 0.00% 10.9 7.70% 0 0.00%
02-nov-95 0 0.00%
03-nov-95
04-nov-95
05-nov-95 0 0.00% 0.00%
06-nov-95 0 0.00% 0.00%
07-nov-95 0 0.00% 0 0.00% 0 0.00% 0.00%
08-nov-95 0 0.00% 0 0.00% 0 0.00% 0 0.00% 0 0.00%
09-nov-95 0 0.00%
10-nov-95 0 0.00% 1 0.71% 0 0.00%
11-nov-95 0 0.00%
12-nov-95 7.1 5.01% 0 0.00%
13-nov-95 0 0.00%
14-nov-95 3 7.14%
15-nov-95
16-nov-95 0 0.00% 0 0.00% 0 0.00%
17-nov-95 0 0.00% 0 0.00% 1 0.71% 0 0.00%
18-nov-95 0 0.00%
19-nov-95
20-nov-95 15 35.71%
21-nov-95 0 0.00%
22-nov-95 10.9 25.95%
23-nov-95 0.5 1.19%
24-nov-95 0 0.00% 0 0.00% 0.00%
25-nov-95 0 0.00% 0.00%
26-nov-95 0 0.00% 0.00%
27-nov-95
28-nov-95 2 4.76%
29-nov-95 7.1 15.40% 0 0.00% 0.5 0.81% 27.9 66.43%
30-nov-95
55.3 46.1 77.9 61.8 212.1 141.6 42 0.00%
01-feb-96 0 0.00% 1 0.66%
02-feb-96 3 1.55% 3 1.63% 7.1 4.69%
03-feb-96
04-feb-96
05-feb-96
06-feb-96 0 0.00%
07-feb-96 0 0.00% 17 6.36% 0 0.00%
08-feb-96
09-feb-96 7 4.54% 81 53.50%
10-feb-96 7.1 4.69%
11-feb-96
12-feb-96 19 9.08% 11.9 6.15% 3 1.95% 6.1 2.28% 5.1 3.37%
13-feb-96 0.5 0.27%
14-feb-96 5.1 3.37%
15-feb-96 7.9 5.22%
16-feb-96 5.1 3.37%
17-feb-96
18-feb-96 13 7.05% 0 0.00% 3.52%
19-feb-96 1 0.54% 0 0.00% 0.27%
20-feb-96 0 0.00% 0.00%
21-feb-96 0 0.00% 0 0.00% 0.00%
22-feb-96 41.9 21.64% 0 0.00% 10.82%
23-feb-96 0 0.00% 0.00%
24-feb-96 0.5 0.33% 0.33%
25-feb-96 6.1 3.96%
26-feb-96
27-feb-96
28-feb-96 0 0.00%
29-feb-96
209.2 193.6 154.2 267.2 202.4 184.5 151.4 14.95%
03-nov
28-nov
16-feb
142

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
STATION 676330 677530 677430 676410 675510 675310 676650 AVERAGE
Place Mongu Choma Livingstone Kaoma Solwezi Zambezi Lusaka
long 23.16 27.06 25.81 24.8 26.38 23.11 28.45
lat -15.25 -16.85 -17.81 -14.8 -12.18 -13.53 -15.32
01-nov-96 0 0.00%
02-nov-96
03-nov-96
04-nov-96 0 0.00%
05-nov-96 0 0.00% 0.00%
06-nov-96 0 0.00% 0.00%
07-nov-96
08-nov
08-nov-96 0 0.00%
09-nov-96 0 0.00% 0 0.00% 0 0.00%
10-nov-96
11-nov-96
12-nov-96 0 0.00% 0 0.00% 0 0.00% 0 0.00% 0 0.00%
13-nov-96 0 0.00% 0 0.00% 0 0.00%
14-nov-96 0 0.00% 0 0.00% 0 0.00%
15-nov-96
16-nov-96
17-nov-96
18-nov-96
19-nov-96 6.1 4.99% 2 1.46% 4.1 9.76%
20-nov-96 0 0.00%
21-nov-96
22-nov-96 0 0.00% 0 0.00%
23-nov-96 0 0.00%
24-nov-96
25-nov-96
26-nov-96
27-nov-96 18 26.16% 0 0.00% 0 0.00% 0 0.00% 14 33.33%
28-nov-96 7.1 5.81%
29-nov-96 0 0.00% 0 0.00% 10.9 7.42% 10.9 7.97% 7.9 18.81% 6.84%
30-nov-96 6.84% 29-nov
68.8 122.2 146.9 136.8 68.5 179.1 42 13.68%
01-dec-96
02-dec-96
03-dec-96
04-dec-96
05-dec-96
06-dec-96 0 0.00% 0 0.00%
07-dec-96
08-dec-96
09-dec-96
10-dec-96 0 0.00%
11-dec-96
12-dec-96
13-dec-96
14-dec-96
15-dec-96
16-dec-96
17-dec-96
18-dec-96 0 0.00%
19-dec-96 0 0.00% 0.00%
20-dec-96
21-dec-96
22-dec-96
23-dec-96 27.9 18.85%
24-dec-96
25-dec-96
26-dec-96
27-dec-96 306.1 206.82%
28-dec-96 2 1.35%
29-dec-96 0 0.00%
30-dec-96
31-dec-96
107.2 137 5.6 190.2 308.8 133.2 148
143

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
STATION 676330 677530 677430 676410 675510 675310 676650 AVERAGE
Place Mongu Choma Livingstone Kaoma Solwezi Zambezi Lusaka
long 23.16 27.06 25.81 24.8 26.38 23.11 28.45
lat -15.25 -16.85 -17.81 -14.8 -12.18 -13.53 -15.32
01-jan-97 0 #DIV/0!
02-jan-97 0 0.00% 0 0.00% 0 0.00% 0 #DIV/0!
03-jan-97 4.1 1.75% 1 0.41% 1 0.49% 20.1 #DIV/0!
04-jan-97 3 #DIV/0!
05-jan-97
06-jan-97 0.5 0.20% 1 0.49% 5.1 #DIV/0!
07-jan-97 1 0.36% 0.3 0.12% 1 #DIV/0!
08-jan-97 0 #DIV/0!
09-jan-97 0 #DIV/0!
10-jan-97 1 #DIV/0!
11-jan-97
12-jan-97 26.9 #DIV/0!
13-jan-97
14-jan-97 0 #DIV/0!
15-jan-97 1 #DIV/0!
16-jan-97
17-jan-97 4.1 #DIV/0!
18-jan-97 24.9 #DIV/0! 17-jan
19-jan-97 7.9 2.82% 5.1 #DIV/0! 2.82%
20-jan-97 0 #DIV/0! 21-jan
21-jan-97
22-jan-97 23.1 #DIV/0!
23-jan-97 0 0.00% 1 #DIV/0!
24-jan-97 9.9 #DIV/0!
25-jan-97 7.9 #DIV/0!
26-jan-97 4.1 1.68% 4.1 #DIV/0!
27-jan-97 0 #DIV/0!
28-jan-97 0 #DIV/0!
29-jan-97 9.9 #DIV/0!
30-jan-97
31-jan-97
233.8 280 250.3 243.6 205.8 257.6 2.82%
01-dec-99 1.7 1.08%
02-dec-99 0 0.00%
03-dec-99 2.2 1.40%
04-dec-99 0.5 0.32% 0.32%
05-dec-99 0 0.00% 0.00%
06-dec-99 0 0.00% 0.00%
07-dec-99 1.8 1.15% 1.15%
08-dec-99 1.6 1.02% 1.02%
09-dec-99 0 0.00% 0.00%
10-dec-99 0 0.00%
11-dec-99 54.6 34.84%
12-dec-99 0.2 0.13%
13-dec-99 0 0.00%
14-dec-99 4.5 2.87%
15-dec-99 26.3 16.78%
16-dec-99 4.1 2.62%
17-dec-99 1.1 0.70%
18-dec-99 0 0.00%
19-dec-99 0.2 0.13%
20-dec-99 31 19.78%
21-dec-99 9.7 6.19%
22-dec-99 0 0.00%
23-dec-99 0 0.00%
24-dec-99 4.6 2.94%
25-dec-99 3 1.91% 1.91%
26-dec-99 0 0.00% 0.00%
27-dec-99 0 0.00% 0.00%
28-dec-99 0 0.00% 0.00%
29-dec-99 9.6 6.13%
30-dec-99 0 0.00%
31-dec-99 0 0.00%
156.7 78.6 77.2 108.8 - 151.3 33.3 4.40%
144

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
STATION 676330 677530 677430 676410 675510 675310 676650 AVERAGE
Place Mongu Choma Livingstone Kaoma Solwezi Zambezi Lusaka
long 23.16 27.06 25.81 24.8 26.38 23.11 28.45
lat -15.25 -16.85 -17.81 -14.8 -12.18 -13.53 -15.32
01-jan-01 10.4 6.17% 6.17%
02-jan-01 0 0.00% 0.00%
03-jan-01 0 0.00% 0.00%
04-jan-01 4.6 2.73% 2.73%
05-jan-01 5.2 3.09% 3.09%
06-jan-01 3.7 2.20% 2.20%
07-jan-01 0 0.00% 0.00%
08-jan-01 0 0.00% 0.00%
09-jan-01 0 0.00% 0.00%
10-jan-01 0 0.00% 0.00%
11-jan-01 1.3 0.77%
12-jan-01 0 0.00%
13-jan-01 0.4 0.24%
14-jan-01 0 0.00%
15-jan-01 0 0.00%
16-jan-01 19.2 11.39%
17-jan-01 6.6 3.92%
18-jan-01 15 8.90%
19-jan-01 13.5 8.01%
20-jan-01 16.1 9.55%
21-jan-01 3.4 2.02%
22-jan-01 0.2 0.12%
23-jan-01 0 0.00%
24-jan-01 0 0.00%
25-jan-01 4 2.37%
26-jan-01 16.2 9.61%
27-jan-01 10.2 6.05%
28-jan-01 16.5 9.79%
29-jan-01 0.2 0.12%
30-jan-01 21.8 12.94%
31-jan-01 0 0.00%
168.5 174.2 28.4 - 408.6 - 167.5 14.18%
01-nov-02 0 0.00%
02-nov-02 0 0.00%
03-nov-02 0 0.00%
04-nov-02 1.6 1.25%
05-nov-02 2.4 1.88%
06-nov-02 12.3 9.61%
07-nov-02 20.7 16.17%
08-nov-02 0 0.00%
09-nov-02 0 0.00%
10-nov-02 10.1 7.89%
11-nov-02 0 0.00%
12-nov-02 28.6 22.34% 22.34%
13-nov-02 0 0.00% 0.00%
14-nov-02 2.7 2.11% 2.11%
15-nov-02 0 0.00% 0.00%
16-nov-02 1.1 0.86% 0.86%
17-nov-02 0 0.00% 0.00%
18-nov-02 0 0.00% 0.00%
19-nov-02 0 0.00% 0.00%
20-nov-02 0 0.00% 0.00%
21-nov-02 0 0.00% 0.00%
22-nov-02 0 0.00%
23-nov-02 11 8.59%
24-nov-02 0 0.00%
25-nov-02 23 17.97%
26-nov-02 14.5 11.33%
27-nov-02 0 0.00%
28-nov-02 0 0.00%
29-nov-02 0 0.00%
30-nov-02 0 0.00%
128 25.31%
145

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Table 4: Option 4.
1993 1994 1995 1997 1998 2000 2001 2002 average
1 25-31 Jan 1999 49.32 64.60 33.04 84.12
259.03 0.19 234.78 0.28 165.71 0.20 267.41 0.31 0.24
2 1-5 dec 1996 19.95 30.85 11.46 25.04 44.71 9.04
234.78 0.08 181.42 0.17 167.68 0.07 165.03 0.15 289.08 0.15 47.07 0.19 0.14
3 7-10 dec 1998 31.31 10.53 19.57 19.40 28.98 4.08
234.78 0.13 181.42 0.06 167.68 0.12 165.03 0.12 289.08 0.10 47.07 0.09 0.10
4 19-21 dec 1996 28.03 6.21 24.68 13.10 42.51 3.99
234.78 0.12 181.42 0.03 167.68 0.15 165.03 0.08 289.08 0.15 47.07 0.08 0.10
5 1-31 dec 2002
Table 5: Option 5.
2002 1993 1994 1995 1996 1997 1998 1999 2000 2001
jan 146.80 149.44 7.00 121.65 632.22 120.64 684.29 147.93 1.28 143.19 12.98 167.16 414.81 149.22 5.85 135.82 120.58 123.18 557.86
feb 124.43 123.53 0.80 105.24 368.14 109.80 213.95 133.47 81.78 111.60 164.48 130.76 40.13 140.69 264.62 138.70 203.69 137.51 171.27
mrt 131.34 125.70 31.79 109.17 491.60 106.37 623.60 129.60 3.01 116.57 218.15 130.13 1.45 117.66 187.03 134.38 9.23 154.20 522.81
apr 96.32 77.88 340.07 77.67 347.68 52.95 1881.17 56.89 1554.76 88.70 58.04 84.76 133.64 82.13 201.35 82.33 195.64 98.20 3.52
mei 38.00 42.76 22.61 31.24 45.74 45.67 58.74 49.64 135.37 49.00 120.93 50.96 167.98 34.38 13.11 56.70 349.70 33.74 18.16
jun 22.88 19.59 10.81 17.93 24.55 3.97 357.72 9.36 182.69 34.52 135.38 16.76 37.44 18.96 15.35 19.26 13.13 21.33 2.42
jul 15.81 11.77 16.36 19.16 11.21 0.00 249.98 11.63 17.51 17.50 2.85 13.46 5.53 14.42 1.93 17.69 3.53 21.75 35.30
aug 12.11 21.71 92.22 24.13 144.55 18.96 47.00 17.72 31.53 22.81 114.59 22.88 116.16 29.61 306.45 20.49 70.24 21.83 94.64
sep 53.40 27.77 657.26 29.15 588.23 30.29 534.07 30.59 520.39 34.36 362.49 43.70 94.20 41.22 148.45 37.59 250.19 39.91 182.05
okt 22.75 60.63 1435.42 69.58 2193.08 65.35 1815.36 53.81 965.11 77.67 3016.93 54.26 992.97 59.66 1362.92 69.41 2177.15 70.00 2232.50
nov 86.26 91.41 26.54 84.81 2.11 59.14 735.67 84.40 3.47 85.83 0.18 81.09 26.74 100.46 201.62 110.28 576.97 101.48 231.63
dec 106.18 123.17 111.05 96.70 89.91 124.27 84.47 127.22 138.16
RMSE 14.83 20.10 24.50 17.07 18.72 13.01 15.02 18.19 18.38
min 14.83 1993
For the gap repair the following IDL-code was written.
pro gap_repair
;Author: A.M.J. Gerrits, Delft University of Technology
workpath = 'd:\temp'
a=fltarr(401,341)
save_before=0
save_after=0
b=fltarr(3)
d=fltarr(3)
numb=0
for year=1993,2002 do begin &$
ye=strcompress(string(year),/remove_all)
openr,1,'original/'+ye+'_rain_norm.gz',/compress
openr,2,'original/'+ye+'_information_err'
openw,3,'repaired/'+ye+'_rain_norm_repair.gz',/compress
openw,4,'repaired/'+ye+'_information_err_repair'
readf,2,b
for month=1,12 do begin
mo=strcompress(string(month),/remove_all)
for day=1,31 do begin
da=strcompress(string(day),/remove_all)
if(b(0) eq year and b(1) eq month and b(2) eq day) then begin
readu,1,a &$
print,year, month, day
;Gaps_day before: e.g. 3 March is missing, 2 March has to be counted double
if(b(0) eq 1993 and b(1) eq 3 and b(2) eq 2) then begin
146

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
save_before=1
endif
.
.
.
;Gaps_day after: e.g. 27 November is missing, 28 March has to be counted double
if (b(0) eq 1995 and b(1) eq 11 and b(2) eq 28) then begin
save_after=1
endif
.
.
.
;Block_gaps
if (b(0) eq 1996 and b(1) eq 2) then begin
a = swap_endian(a)
tmp=fltarr(401,341)
blas_axpy,tmp,1.1757,a
a = swap_endian(tmp)
endif
.
.
.
;Save_after
if (save_after eq 1) then begin
printf,4,year, month, (day-1)
writeu,3,a
save_after=0
endif
printf,4,year, month, day
writeu,3,a
;Save_before
if(save_before eq 1) then begin
printf,4,year, month, (day+1)
writeu,3,a
save_before=0
endif
if (not eof(2)) then begin
readf,2,b
endif
endif
endfor
endfor
close,1
close,2
close,3
close,4
endfor
print,'End of gap repair'
end
In the following IDL-code the daily repaired MIRA-files are summed up to monthly values.
Also the data is converted to an ASCII-raster file, which is easy to open in Idrisi.
147

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
pro MIRA_read
;Procedure is based on the script obtained from the University of Sussex, UK
;Author: A.M.J. Gerrits, Delft University of Technology
workpath = 'd:\temp'
a=fltarr(401,341)
year_begin = 1993
year_end = 2002
byear=strcompress(string(year_begin),/remove_all)
eyear=strcompress(string(year_end),/remove_all)
month_begin = 1
month_end = 12
day_begin = 1
day_end = 31
day_aantal = (day_end-day_begin+1)
month_aantal = (month_end-month_begin+1)
year_aantal = (year_end-year_begin+1)
c =fltarr((day_aantal*month_aantal*year_aantal)+1)
;0.1 degree
;lonmin=10
;lonmax=50
;latmin=-34
;latmax=-0
b=fltarr(3)
numb=0
window,1,retain=2
window,2,retain=2
window,3,retain=2
window,4,retain=2
openw,5,'daily/GRASSIDR_daily.iml'
openw,6,'monthly/GRASSIDR_monthly.iml'
openw,7,'locations/'+byear+'_'+eyear+'_Choma.txt'
openw,8,'locations/'+byear+'_'+eyear+'_Kabombo.txt'
openw,9,'locations/'+byear+'_'+eyear+'_Kafue.txt'
openw,10,'locations/'+byear+'_'+eyear+'_Kalabo.txt'
openw,11,'locations/'+byear+'_'+eyear+'_Kaoma.txt'
openw,12,'locations/'+byear+'_'+eyear+'_Livingstone.txt'
openw,13,'locations/'+byear+'_'+eyear+'_Mongu.txt'
openw,14,'locations/'+byear+'_'+eyear+'_Magoye.txt'
openw,15,'locations/'+byear+'_'+eyear+'_Solwezi.txt'
openw,16,'locations/'+byear+'_'+eyear+'_Zambezi.txt'
for year=year_begin,year_end do begin &$
ye=strcompress(string(year),/remove_all)
openr,1,'repaired/'+ye+'_rain_norm_repair.gz',/compress
openr,2,'repaired/'+ye+'_information_err_repair'
readf,2,b
for month=month_begin,month_end do begin
mo=strcompress(string(month),/remove_all)
if (strlen(mo) eq 1) then begin
mo='0'+mo
endif
month_sum = fltarr(401,341)
for day=day_begin,day_end do begin
148

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
if (b(0) eq year and b(1) eq month and b(2) eq day) then begin
da=strcompress(string(day),/remove_all)
if (strlen(da) eq 1) then begin
da='0'+da
endif
readu,1,a &$
a = swap_endian(a)
maxa = max(a)
c((day-day_begin)+((month-month_begin)*day_aantal)+((year-year_begin)*month_aantal*day_aantal))=maxa
blas_axpy,month_sum,24,a
print,year,month,day
openw,3,'daily/'+ye+'_'+mo+'_'+da+'_rain_output_daily'
printf,3,'proj: 3'
printf,3,'zone: 0'
printf,3,'north: 0S'
printf,3,'south: 34S'
printf,3,'east: 50E'
printf,3,'west: 10E'
printf,3,'cols: 401'
printf,3,'rows: 341
printf,3,a
close,3
printf,5,'GRASSIDR x
1*1*'+workpath+'\daily\'+ye+'_'+mo+'_'+da+'_rain_output_daily*'+workpath+'\daily\prec_daily_'+ye+'_'+mo+'_'+da+'.rst*3*N*
classes'
wset,1 & tv,bytscl(a,0,2)
wset,2 & plot,a
wset,3 & plot,c
WRITE_JPEG,'daily/images/'+ye+'_'+mo+'_'+da+'_rain_output_daily.jpg',bytscl(a,0,2)
if (not eof(2)) then readf,2,b
endif
endfor
openw,4,'monthly/'+ye+'_'+mo+'_rain_output_monthly'
printf,4,'proj: 3'
printf,4,'zone: 0'
printf,4,'north: 0S'
printf,4,'south: 34S'
printf,4,'east: 50E'
printf,4,'west: 10E'
printf,4,'cols: 401'
printf,4,'rows: 341'
printf,4,month_sum
wset,4 & tv,bytscl(month_sum,0,48)
close,4
WRITE_JPEG,'monthly/images/'+ye+'_'+mo+'_rain_output_monthly.jpg',bytscl(month_sum,0,48)
printf,6,'GRASSIDR x
1*1*'+workpath+'\monthly\'+ye+'_'+mo+'_rain_output_monthly*'+workpath+'\monthly\prec_monthly_'+ye+'_'+mo+'.rst*3*N*cl
asses'
printf,7,month_sum(171,172)
printf,8,month_sum(143,204)
printf,9,month_sum(180,182)
printf,10,month_sum(128,191)
printf,11,month_sum(149,192)
printf,12,month_sum(159,162)
printf,13,month_sum(132,188)
printf,14,month_sum(177,179)
printf,15,month_sum(164,218)
printf,16,month_sum(132,205)
endfor
close,1
close,2
endfor
149

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
close,5
close,6
close,7
close,8
close,9
close,10
close,11
close,12
close,13
close,14
close,15
close,16
wshow,1,0
wshow,2,0
wshow,3,0
wshow,4,0
print,'end of program'
end
• Import the ascii-files with GRASSIDR in Idrisi
• Transpose the maps, because IDL stores the data in a different way, then Idrisi reads the
data:
Idrisi way
IDL way
1 2 3 4 13 14 15 16
5 6 7 8 9 10 11 12
9 10 11 12 5 6 7 8
13 14 15 16 1 2 3 4
• REFORMAT -> TRANSPOSE -> ‘reverse order of rows’
• Change the projection from lat/long to lazea.
1.2.7 FEWS-precipitation [11]
Spatial: 0.1° x 0.1°
Temporal: decadal
Unit: mm/decade
Extension: *.img
• Open WinDisp and run the following script to convert the data to Idrisi16 format. Write
following code in a text-editor and save it with the extension *.cmd.
Batch Variable Prompt, "Year, Enter year desired, 95"
Batch If Begin, "((%Year%>=01) & (%Year%

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
Process Export IDRISI Image, "D:\MIRIAM\FEWS\rfe%Year%\sa%Year%%MonthOK%%Dekad%.IMG,
D:\MIRIAM\FEWS\idrisi\sa%Year%%MonthOK%%Dekad%.IMG"
File Close, ""
Batch For End
Batch For End
Batch If Else
Batch For Begin, "Month, 1, 12, 1"
Batch For Begin, "Dekad, 1, 3, 1"
Batch If Begin, "%Month%

APPENDIX 1: DATA TRANSFORMATIONS IN IDRISI
maps twice, which saves disc space. The maps, which are corrected, are the precipitation
maps and the potential evaporation maps.
• Delete the legend of the old maps
• Change in the documentation file the reference system from ‘unspecified’ to ‘plane’.
• Convert the maps from integer to real. REFORMAT -> CONVERT
• Make an Idrisi-map with the correct size (649x422), which only contains zeros. DATA
ENTRY -> INITIAL
• Concatenate the old maps to the ‘zero-map’. REFORMAT -> CONCAT
Use the following settings:
Figure 4: Concatenation settings.
The first image must be the old map and the second image is the ‘zero-map’.
• Change with a text-editor the min- and max-size of the map to:
Xmin: -178500
Xmax: 1768500
Ymin: -2812500
Ymax: -1545500
• Mask the maps.
152

APPENDIX 2: FAO SOIL UNITS DESCRIPTION
Appendix 2
FAO Soil Units Description
From: http://ag.arizona.edu/OALS/IALC/soils/fao.html
153

APPENDIX 2: FAO SOIL UNITS DESCRIPTION
Acrisols: Acidic soils with a layer of clay accumulation. Under the modified legend this class
consists only of clays with low cation exchange capacity.
Alisols: A new soil class of formerly Acrisols with clays with high cation exchange capacity.
Andosols: Dark soils formed from volcanic materials with little horizon development.
Anthrosols: New class of soils formed by human activities.
Arenosols: Sandy soils with little profile development.
Calcisols: New class of soils with accumulation of calcium carbonate.
Cambisols: Soil with slight profile development that is not dark in color.
Chernozems: Dark soils rich in organic matter.
Ferralsols: Highly weathered soils rich in sesquioxide clays and with low cation exchange
capacities.
Fluvisols: Alluvial and floodplain soils with little profile development.
Gleysols: Water saturated soils that are not salty.
Greyzems: Dark soils rich in organic matter.
Gypsisols: New class of soils with an accumulation of calcium sulfate (gypsum).
Histosols: Soils very rich in organic matter (>14%).
Kastanozems: Dark soils rich in organic matter.
Leptosols: New class of soils that are shallow in depth and with weak profile development.
Lithosols: Thin soils over rock. Removed from the revised legend.
Lixisols: New soil class, formerly Luvisols, with clays with low cation exchange capacity.
154

APPENDIX 2: FAO SOIL UNITS DESCRIPTION
Luvisols: Soils with strong accumulation of clay in the B-horizon and not dark in color.
Under the revised legend these soils have clays with high cation exchange capacity.
Nitisols: New class of soils with shiny surfaces on structural faces (peds) of the soil.
Nitosols: Acid soils with a very thick layer of clay accumulation. Removed from the revised
legend.
Phaeozems: Dark soils rich in organic matter.
Planosols: Soils with a light colored layer over a soil layer that restricts water drainage.
Plinthosols: New class of mottled, clayey soils that irreversibly harden after repeated drying.
Podzols: Soils with a strongly bleached layer and a layer of iron or aluminum cemented
organic matter.
Podzoluvisols: Soils similar to both Podzols and Luvisols.
Rankers: Shallow, dark soils rich in organic matter and formed from siliceous material.
Removed from the revised legend.
Regosols: Surface layer of rocky material. Coarse texture Regosols have been incorporated in
the Arenosols under the revised system.
Rendzinas: Dark soils rich in organic matter over calcareous material. Removed from the
revised legend.
Solonchaks: Salty soil with little horizon development.
Solonetz: Salty soil with a high concentration of sodium.
Vertisols: Clayey soils that form deep (>50 cm), wide (>1 cm) cracks when dry.
Xerosols: aridic soils. Removed from the revised system.
Yermosols: aridic soils. Removed from the revised system. Yermic has been added to the
management phases.
155

APPENDIX 2: FAO SOIL UNITS DESCRIPTION
156

APPENDIX 3: ZAMBEZI SCRIPT
Appendix 3
Zambezi script
157

APPENDIX 3: ZAMBEZI SCRIPT
Seyam, Ficara and
Gerrits1, prec - pe, prec - int)
# Calculating soil moisture, transpiration and overtop (deep soil)
#*************************************************************************
# PARAMETERS CR and QC
cr= mif (subsub==10 , 0.26 , 0.25)
cr= mif (subsub==11 , 0.22 , cr)
cr= mif (subsub==12 , 0.15 , cr)
cr= mif (subsub==13 , 0.28 , cr)
cr= mif (subsub==14 , 0.33 , cr)
cr= mif (subsub==8 , 0.13 , cr)
cr= mif (wetland==1, 0.636,cr)
cr=1.1*cr
qc= mif (subsub==10 , 0.50 , 0.30)
qc= mif (subsub==11 , 0.30 , qc)
qc= mif (subsub==12 , 0.70 , qc)
qc= mif (subsub==13 , 0.50 , qc)
qc= mif (subsub==14 , 0.30 , qc)
Su = mif(water>1, 0 , (Su + (1 - cr) * pnet))
k=3
overtop = max(0, ((Su - Smax)/k))
Su = Su - overtop
tp = mif(water>1, 0, pe - (1 - (f/ 100)) * int)
tp = max(0, tp)
tra = mif(water>1, 0, (tp* min(1,((2/smax) *Su))))
tra=min(tra,su)
Green=tra+(int-intmar)
White=intmar
destroy(tp)
su = su - tra
destroy(tra)
# monthly balance of water bodies in mm/month
#*************************************************************************
dsw = mif(water>1, pnet, 0)
dsw = zonalsum(dsw, water)
dsw = dsw * 0.003471391
158

APPENDIX 3: ZAMBEZI SCRIPT
# Calculation of saturation overland flow, quick flow and slow flow in mm/month.
#*************************************************************************
gwsmax=25*ln_gwsmax
gws = mif (water>1,0, gws + (cr * pnet) + overtop)
saof = mif (water>1,0, mif(gws>gwsmax, gws-gwsmax, 0))
gws = gws-saof
destroy(pnet)
destroy(cr)
destroy(overtop)
gwsquick = gwsmax * qc
destroy(qc)
rtq=4
qflo = mif(water>1, 0, max((gws-gwsquick),0) / rtq)
gws = gws - qflo
cap1=mif(gws1,0,cap1)
gws = gws - caprise
su = su + caprise
#TSummary(su, %OutputDirectory)
#exporttimer(su,%outputdirectory,'input\pallettes\pre.pal')
timeout(su, %DischargePoints, %OutputDirectory)
rts=12
sflo = mif(water>1, 0, max(gws, -15) / rts)
gws = gws - sflo
#TSummary(gws, %OutputDirectory)
#exporttimer(gws, %outputdirectory, 'input\palettes\gw.pal')
timeout(gws, %DischargePoints, %OutputDirectory)
runoff2 = saof + qflo + sflo
blue=runoff2
destroy(sflo)
destroy(kflo)
destroy(saof)
destroy(rtq)
#TSummary(runoff2, %OutputDirectory)
# Accumulating discharge from above lying cells in mm/month.
#*************************************************************************
qout = accu(fd, runoff2,'Input\')
QOUT = QOUT * 0.003471391
#export(qout, %OutputDirectory)
timeout(QOUT, %DischargePoints, %OutputDirectory)
Destroy(QOUT)
]]>
159

APPENDIX 3: ZAMBEZI SCRIPT
160

APPENDIX 4: MODEL RESULTS
Appendix 4
Model results
1.1 Results of Lukulu.....................................................................................................162
1.2 Results of Victoria Falls...........................................................................................163
1.3 Hydrographs of other locations................................................................................164
161

APPENDIX 4: MODEL RESULTS
1.1 Results of Lukulu
In- and outgoing fluxes LUKULU
30000
25000
Accumulated flux upstream of Lukulu [m³/s]
20000
15000
10000
5000
0
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
IN Prec OUT Int OUT Tra OUT Runoff
Figure 1: Water balance terms of Lukulu.
Runoff LUKULU
4,000.00
3,500.00
3,000.00
2,500.00
Runoff [m³/s]
2,000.00
1,500.00
1,000.00
500.00
0.00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Date
OUT Runoff OUT Saof OUT Qflo OUT Sflo
Figure 2: Runoff of Lukulu.
162

APPENDIX 4: MODEL RESULTS
1.2 Results of Victoria Falls
In- and outgoing fluxes VICTORIA FALLS
60000
50000
Accumulated flux upstream of Victoria Falls [m³/s]
40000
30000
20000
10000
0
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Figure 3: Water balance terms of Victoria Falls.
IN Prec OUT Int OUT Tra OUT Runoff
Runoff VICTORIA FALLS
6,000.00
5,000.00
4,000.00
Runoff [m³/s]
3,000.00
2,000.00
1,000.00
0.00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Date
OUT Runoff OUT Saof OUT Qflo OUT Sflo
Figure 4: Runoff of Victoria Falls.
163

APPENDIX 4: MODEL RESULTS
1.3 Hydrographs of other locations
NDUBEN
600.00
500.00
400.00
Discharge [m³/s]
300.00
200.00
100.00
0.00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Date
simulated
observed
Figure 5: Hydrograph of Nduben.
Itezhitezhi outlet
2500.00
2000.00
Discharge [m³/s]
1500.00
1000.00
500.00
0.00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Date
Figure 6: Hydrograph of Itezhitezhi outlet.
simulated
observed
164

APPENDIX 4: MODEL RESULTS
confl. Luangua to Zambezi
4500.00
4000.00
3500.00
3000.00
Discharge [m³/s]
2500.00
2000.00
1500.00
1000.00
500.00
0.00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
Date
simulated
observed
Figure 7: Hydrograph of confluence Luangua to Zambezi River.
Lake Nyasa
1,200.00
1,000.00
800.00
Discharge [m³/s]
600.00
400.00
200.00
0.00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
-200.00
Date
simulated
observed
Figure 8: Hydrograph of Lake Nyasa outlet.
165

APPENDIX 4: MODEL RESULTS
Liwonde
1200.00
1000.00
800.00
Discharge [m³/s]
600.00
400.00
200.00
0.00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
jan-91
jan-92
jan-93
jan-94
jan-95
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
-200.00
Date
simulated
observed
Figure 9: Hydrograph of Liwonde.
Lake Kariba
4500,00
4000,00
3500,00
3000,00
Discharge [m³/s]
2500,00
2000,00
1500,00
1000,00
500,00
0,00
jan-78
jan-79
jan-80
jan-81
jan-82
jan-83
jan-84
jan-85
jan-86
jan-87
jan-88
jan-89
jan-90
Figure 10: Hydrograph of Outlet Kariba.
jan-91
simulated
Date
jan-92
jan-93
jan-94
jan-95
observed
jan-96
jan-97
jan-98
jan-99
jan-00
jan-01
jan-02
jan-03
166

APPENDIX 5: GLUE SCRIPT
Appendix 5
GLUE script
167

APPENDIX 5: GLUE SCRIPT
% This m-file is the application interface between the GLUE environment and a specific application
% this is the only MATLAB-file where amendments are required
%
% Micha Werner
% Last Update: 12-August-2002
% Reason: Comments line added as documentation
%
%
% clear all open figures and variable from workspace
clear all;
close all;
% START INPUT SECTION
% Input1
% cell arrays of parameters - one line for each parameter considered
% note the use of braces rather than brackets!!
%
% Syntax Par{#} = {'Name of parameter', }
Par{1} = {'D' , 70 , 80, 100};
Par{2} = {'cr' , 0.5 , 1, 1.5};
Par{3} = {'cap' , 0.5 , 1, 1.5};
Par{4} = {'gws25' , 20 , 25, 30};
%Par{5} = {'\alpha_2' , 0.50 , 1.50, 4.00};
%Par{6} = {'\beta_2' , 0.50 , 1.00, 4.00};
%Par{7} = {'fc_2' , 100 , 193, 400};
%Par{8} = {'maxbas_2' , 1 , 1.04167, 2.00};
%Par{9} = {'\alpha_3' , 0.50 , 2.00, 3.20};
%Par{10} = {'\beta_3' , 0.50 , 2.00, 4.00};
%Par{11} = {'fc_3' , 100 , 153, 400};
%Par{12} = {'maxbas_3' , 1 , 1.54167, 2};
% Input 2
% Number of runs
% Number of runs to carry out
% Note: in case of interval sampling, the nPar root is used along each axis!
nRun = 300;
% Dimensions of outputs and measured value array
nStep = 286; % Number of steps in each run
nLoc = 2; % Number of locations considered in analysis
% ASCII file with observations
% This MUST include at least nStep rows of values and nLoc columns of values
ObservedFile = 'observed.txt'; % file name and location
ObsStartRow = 1; % Row to start reading values from
ObsStartCol = 1; % Column to start reading values from
% (Note: This is the position in the string - not the data column!)
% Settings used for communication with model through ModelMex.dll
SettingsFile = '.\settings.txt'; % name of file for settings (do not change)
CommandLine = 'run_stream'; % command line to run model
AddRunID = 1; % Option to add run ID as argument to command line (0=No, 1=Yes)
% Template file for parameters (delimited by semicolon for multiple files)
ParameterTemplate = 'final.tpl';
% Parameter file (delimited by semicolon for multiple files)
ParameterFile = '.\input\scripts\final.mxl'; % Model file for parameters
TemplateDelimitor = '$'; % Delimiter used for identifying parameters
OutputFile = 'routing_out.txt'; % Name of ASCII output file from model
OutStartRow = 12; % Row to start reading values from
OutStartCol = 0; % Column to start reading values from
% (Note: This is the position in the string - not the data column!)
OutputBinary = 'calcval.bin'; % name of binary interchange filw with results (do not change)
MexType = 0; % Option whether to run EXE or DLL (0=Use ModelEXE (default) 1 = Use ModelMex (for
Matlab function evaluations))
WorkDir = '.\'; % Working directory for running model
168

APPENDIX 5: GLUE SCRIPT
%
FitnessFunction = [1 1]; % Array of fitness funtions - should be same size as nLoc
FitWeight = [1 1]; % weigth of fitness funtions - should be same size as nLoc
SampleType = 2; % Interval= 0; Uniform samplin = 1; LHS = 2; MLHS = 3
%u_interval = [0.01 0.025 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.975 0.99]; % Uncertainty intervals
u_interval = [0.05 0.50 0.95 ]; % Uncertainty intervals
cutoff = [] ; % cuttof (depends on fitness function
FitOpt = ['[1]'] ; % options for fitness function - see specific function for definition - should be a string!!!
cutopt = 0 ; % use cutoff or not - if 0 then not, if 1 then use
batch_run = 1 ; % if batch_run is set to 0, then run goes automatically
run_type = 0 ; % run_type used in combination with batch run)
% 0=Generate New random parameter set from uniform prior
% 1=Run only from run_index (loads parameter vector)
% 2=Run once with given parameters (set in Par Cell array defined here!)
% 3=Load results for analysis
% 4=Run new MC Analysis using loaded parameter distributions
% 5=Run using previously determined parameter set likelihoods
pardistfile = '.\'; % directory where to obtain priors!
% Options for running on mutlipe machines
% settings with (Windows) or (Linux) in brackets are for each system
UseParallel = 0 ; % execute run on multiple processors (0=no (default), 1=Yes)
ParallelType = 1 ; % Type of parallelrun - 0 is TU Windows Cluser; 1 is PVM on Linux cluster
ParallelDir = 'D:\ParallelTasks\';
ParallelPut_File = 'put_LisfloodOurthe.txt';
ParallelGet_File = 'get_LisfloodOurthe.txt';
ParallelDir_File = 'D:\network.txt';
ParallelDir_Local = '.\';
ParallelClearOpt = 1;
ParallelBlock = 1;
ParallelCommand = 'D:\pcraster\apps\pcrcalc -f lf_light30012002g.mod D:\Ourthe $1$ $2$ $3$ $4$ $qsim####.###$ thsim.tss
uzsim.tss lzsim.tss';
ParallelCopyTo = cell(0); % empty cell - ie no file to copy from
ParallelCopyFrom{1} = 'qsim####.###';
ParallelRemoteDir = []; % empty cell - ie no file to copy from
ParallelLocalDir = '\\cti025\D\ParallelTasks\'; % location on local computer to copy results files to
ParallelResults = 'qsim####.###';
% Options for running with input ensembles
UseEnsemble = 0 ; % (0=No,1=Yes)
EnsembleOption = 0;
% Option in running ensembles
% 0 = Run each ensemble member with all behavioural parameter sets
% 1 = Randomly sample ensemble member to run (not yet operational)
EnsembleNumber = 8;
EnsembleType = 0 ;
% Ensemble is either a file or a directory - Type 0 is file, Type 1 is directory
EnsembleOrigin = '..\Ensemble\BenchQ.###' ; % name of file or dir to copy files from
% # is replaced with ensemble number - the number of # determines
% width of string to be searched for
EnsembleDestin = '.\BenchQ.txt'; % name of file or dir to copy files to
EnsembleWeight = 0;
% Feature to be added (perhaps)- determines weight of each ensemble member
% when set to 0, wheight is uniform
UseTSS = 0;
Tss_Cell = cell(2,5);
% GA_Options
UseGA = 0; % use GA as a sample preselector
GApop = 24; % number of GA individuals in population
GAgen = 10 ; % number of GA generations
GAcross = 0.8; % cross-over rate
GAmute = 0.1; % mutation rate
GAnicherad = 0.05; % Niching radius
GAnicheopt = 1 ; % Options for Niching
NBits = 8 ; % number of bits per parameter
SplitPerc = 0.8; % Split sample percentage
SkipGARun = 1; % 0 to run all - 1 to skip GA but import files; 2 to run until files are to be imported - then abort!
UseSTD = 1; ; % 0 to not use 1 to use std
% Sample selector options
169

APPENDIX 5: GLUE SCRIPT
SampleSel
SampleFile
= 0 ; % 0=no sample selector (default), 1=Load samples to run from file;
= 'nn_output.txt'; % file name
% END OF INPUT SECTION
% DO NOT MAKE ANY CHANGES AFTER HERE!
if (UseTSS == 0)
tsscell = [];
end
% settings cell
Settings_Cell{1} = SettingsFile ;
Settings_Cell{2} = CommandLine ;
Settings_Cell{3} = AddRunID ;
Settings_Cell{4} = ParameterTemplate ;
Settings_Cell{5} = ParameterFile ;
Settings_Cell{6} = OutputFile ;
Settings_Cell{7} = OutStartRow ;
Settings_Cell{8} = OutStartCol ;
Settings_Cell{9} = OutputBinary ;
Settings_Cell{10} = TemplateDelimitor ;
% ensemble cell definition
Ensemble_Cell{1} = EnsembleOption;
Ensemble_Cell{2} = EnsembleNumber;
Ensemble_Cell{3} = EnsembleType;
Ensemble_Cell{4} = EnsembleOrigin;
Ensemble_Cell{5} = EnsembleDestin;
Ensemble_Cell{6} = EnsembleWeight;
% Parrallel Cell
Parallel_Cell{1} = ParallelPut_File;
Parallel_Cell{2} = ParallelGet_File;
Parallel_Cell{3} = ParallelDir_File;
Parallel_Cell{4} = ParallelDir_Local;
Parallel_Cell{5} = ParallelClearOpt;
Parallel_Cell{6} = ParallelBlock;
Parallel_Cell{7} = ParallelDir;
Parallel_Cell{8} = ParallelType;
Parallel_Cell{9} = ParallelCommand;
Parallel_Cell{10} = ParallelCopyTo;
Parallel_Cell{11} = ParallelCopyFrom;
Parallel_Cell{12} = ParallelResults;
Parallel_Cell{13} = ParallelLocalDir;
Parallel_Cell{14} = ParallelRemoteDir;
% GA Cell
ga_cell{1} = UseGA;
ga_cell{2} = GApop;
ga_cell{3} = GAgen;
ga_cell{4} = GAcross;
ga_cell{5} = GAmute;
ga_cell{6} = GAnicherad;
ga_cell{7} = GAnicheopt;
ga_cell{8} = SampleSel;
ga_cell{9} = SampleFile;
ga_cell{10} = NBits;
ga_cell{11} = SplitPerc;
ga_cell{12} = SkipGARun;
ga_cell{13} = UseSTD;
% construct the run options array
run_options{1} = FitnessFunction;
run_options{2} = SampleType;
run_options{3} = u_interval;
run_options{4} = cutoff;
run_options{5} = FitOpt;
run_options{6} = cutopt;
run_options{7} = batch_run;
run_options{8} = run_type;
run_options{9} = pardistfile;
170

APPENDIX 5: GLUE SCRIPT
run_options{10} = WorkDir;
run_options{11} = FitWeight;
run_options{12} = MexType;
run_options{13} = Settings_Cell;
run_options{14} = UseEnsemble;
run_options{15} = Ensemble_Cell;
run_options{16} = UseParallel;
run_options{17} = Parallel_Cell;
run_options{18} = UseTSS;
run_options{19} = Tss_Cell;
run_options{20} = UseGA;
run_options{21} = ga_cell;
obs_options{1} = ObservedFile;
obs_options{2} = ObsStartRow;
obs_options{3} = ObsStartCol;
% clear variables
clear FitnessFunction SampleType u_interval cutoff maxfit cutopt batch_run run_type pardistfile ParallelRun;
% call the run!
if mc_gluerun(nRun, nStep, nLoc, Par, SettingsFile, obs_options, run_options) ~= 0
disp('Run failed or exited by user')
else
disp('run was succesful');
end
% end of run-file
171

APPENDIX 5: GLUE SCRIPT
172

APPENDIX 6: ROUTING MODEL
Appendix 6
Routing model
173

APPENDIX 6: ROUTING MODEL
The routing model, which is programmed in the object orientated language JAVA, will be
explained by the use of UML-diagrams (Unified Modeling Language).
Meta model (Figure 1)
The routing model is divided into three sub models: Input/Output, River system and Data. By
the use of a User Interface the program is operated. In the User Interface is asked for loading
the input file of STREAM, calculating the routed discharge of the river system and last save
the output (= routed discharge for certain locations) in a text file.
Interval
Tributary
River
Reach
InterReach
SourceReach
Location
DataSet
QDataSet
QponceDataSet
DiDataSet
DataPoint
QponceDataPoint
DiDataPoint
QDataPoint
Timeframe
InputFile
InputLine
OutputFile
OutputLine
Figure 1: Sub models of the routing model and their relations..
Sub model ‘Input/Output’ (Figure 2)
The input file, which is the output file of STREAM, consists of several lines and columns.
The columns are the not routed discharge values for different locations in the Zambezi
catchment. The same composition exists for the output file. Each line is a time step and each
column a location. In this case only the locations Lukulu and Victoria Falls are saved in a text
file.
Figure 2: Sub model 'Input/Output'.
174

APPENDIX 6: ROUTING MODEL
Sub model ‘River System’ (Figure 3)
The sub model ‘River System’ describes, in this case, the river system of the Zambezi River.
However, it will be the same for each other river system in the world. The river system is
schematized as a tree structure. It consists of a River with Tributaries and is segmented into
Reaches. A Reach can be a InterReach or a SourceReach. A InterReach has only a end
location and obtains the begin location(s) from its upstream reach(es). On the other hand, a
SourceReach is the most upstream reach of River or Tributary and is not possible to get a
begin location from an upstream reach. Therefore a SourceReach has a begin location and a
end location. As well as a River, a Tributary and a Reach are an Interval. In the class Interval
is calculated the lateral inflow (di), which is necessary to calculate Qponce with Muskingum
routing. After calculating di, all the required information is available and Qponce can be
calculated. All the calculated values are stored in dataset. In the QDataSet are stored the
output values of STREAM and in the QponceDataSet are stored the routed discharge values.
Both are linked to a location in the river system. The lateral inflow is stored in the DiDataSet
and because the lateral inflow is calculated between one end location and at least one begin
location this dataset is linked to a Reach.
Figure 3: Sub model 'River System'.
175

APPENDIX 6: ROUTING MODEL
Sub model ‘Data’ (Figure 4)
As mentioned in the last paragraph some data values are stored in a DataSet. In the routing
model three dataset are implemented: a QDataSet for storing the discharge values of
STREAM, a QponceDataSet for storing the routed discharges and a DiDataSet for storing the
lateral inflow value. The first datasets are linked to a Location and the last dataset is linked to
a Reach.
A DataSet consists of one or more DataPoints, which have a time value and a discharge or
lateral inflow value. In the routing model three types of DataPoints exists: QponceDataPoints,
DiDataPoints and QDataPoints.
Figure 4: Sub model 'Data'.
176

APPENDIX 7: GRACE RESULTS
Appendix 7
GRACE results
177

APPENDIX 7: GRACE RESULTS
Remark: the legends do not have the same upper and lower values.
YEAR: 2002
Month Global, l = 50; GF = 600 Global, l = 15; GF = 0
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
178

APPENDIX 7: GRACE RESULTS
Oct
Nov
Dec
YEAR: 2003
Month Global, l = 50; GF = 600 Global, l = 15; GF = 0
Jan
Feb
Mar
179

APPENDIX 7: GRACE RESULTS
Apr
May
Jun
Jul
Aug
Sep
180

APPENDIX 7: GRACE RESULTS
Oct
Nov
Dec
YEAR: 2004
Month Global, l = 50; GF = 600 Global, l = 15; GF = 0
Jan
181

APPENDIX 7: GRACE RESULTS
Feb
Mar
Apr
May
Jun
182

APPENDIX 7: GRACE RESULTS
Jul
Aug
Sep
Oct
Nov
Dec
183