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Modelling catchment inflows into Lake Victoria:
uncertainties in rainfall–runoff modelling for the Nzoia
River
a
b
Michael Kizza , Allan Rodhe , Chong-Yu Xu
a
b c
& Henry K. Ntale
d
School of Engineering, Makerere University, PO Box 7062, Kampala, Uganda
b
Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36, Uppsala,
Sweden
c
Department of Geosciences, University of Oslo, PO Box 1047 Blindern, N-0316 Oslo,
Norway
d
Faculty of Technology, Makerere University, PO Box 7062, Kampala, Uganda
Available online: 19 Oct 2011
To cite this article: Michael Kizza, Allan Rodhe, Chong-Yu Xu & Henry K. Ntale (2011): Modelling catchment inflows into Lake
Victoria: uncertainties in rainfall–runoff modelling for the Nzoia River, Hydrological Sciences Journal, 56:7, 1210-1226
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Hydrological Sciences Journal – Journal des Sciences Hydrologiques, 56(7) 2011
Modelling catchment inflows into Lake Victoria: uncertainties in
rainfall–runoff modelling for the Nzoia River
Michael Kizza1 , Allan Rodhe2 , Chong-Yu Xu2,3 & Henry K. Ntale4
1
School of Engineering, Makerere University, PO Box 7062, Kampala, Uganda
michael.kizza@hyd.uu.se; michael.kizza@gmail.com
2
Downloaded by [Uppsala universitetsbibliotek] at 07:46 19 October 2011
Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36 Uppsala, Sweden
allan.rodhe@hyd.uu.se
3
Department of Geosciences, University of Oslo, PO Box 1047 Blindern, N-0316 Oslo, Norway
c.y.xu@geo.uio.no
4
Faculty of Technology, Makerere University, PO Box 7062, Kampala, Uganda
hntale@vala.biz
Received 18 June 2009; accepted 29 March 2011; open for discussion until 1 April 2012
Citation Kizza, M., Rodhe, A., Xu, C.-Y. & Ntale, H. K. (2011) Modelling catchment inflows into Lake Victoria: uncertainties in
rainfall–runoff modelling for the Nzoia River. Hydrol. Sci. J. 56(7), 1210–1226.
Abstract Climate and soil characteristics vary considerably around the Lake Victoria basin resulting in high spatial
and temporal variability in catchment inflows. However, data for estimating the inflows are usually sparsely distributed and error-prone. Therefore, modelled estimates of the flows are highly uncertain, which could explain early
difficulties in reproducing the lake water balance. The aim of this study was to improve the estimates of catchment
flow to Lake Victoria. The WASMOD model was applied to the Nzoia River, one of the major tributaries to Lake
Victoria. Uncertainty was assessed within the GLUE framework. During calibration, log-transformation was performed on both simulated and observed flows. The results showed that, despite its simple structure, WASMOD
produces acceptable results for the basin. For a Nash-Sutcliffe efficiency (NS) threshold of 0.6, the percentage of
observations bracketed by simulations (POBS) was 74%, the average relative interval length (ARIL) was 0.93,
and the maximum NS value was 0.865. The residuals were shown to be homoscedastic, normally distributed and
nearly independent. When the NS threshold was increased to 0.8, POBS decreased to 54% with an improvement
of ARIL to 0.49, highlighting the effect of the subjective choice of likelihood threshold.
Key words rainfall–runoff modelling; GLUE; Nash-Sutcliffe efficiency; Nzoia River; Lake Victoria; WASMOD; uncertainty
Modélisation des apports de bassin au Lac Victoria: les incertitudes dans la modélisation pluie–
débit du Fleuve Nzoia
Résumé Les caractéristiques du climat et du sol varient considérablement dans le bassin du Lac Victoria, résultant
en une forte variabilité spatiale et temporelle des apports du bassin. Cependant, les données pour l’estimation des
entrées sont généralement clairsemées et sujettes à erreur. Par conséquent, les estimations modélisées des débits
sont très incertaines, ce qui pourrait expliquer les premières difficultés à reproduire le bilan en eau du lac. Le but
de cette étude était d’améliorer les estimations des apports de bassin au Lac Victoria. Le modèle WASMOD a été
appliqué au fleuve Nzoia, l’un des principaux affluents du Lac Victoria. L’incertitude a été évaluée dans le cadre
de GLUE. Pendant le calage, une transformation logarithmique a été réalisée sur les débits à la fois simulés et
observés. Les résultats ont montré que, malgré sa structure simple, WASMOD produit des résultats acceptables
pour le bassin. Pour un seuil de 0.6 sur l’efficacité de Nash-Sutcliffe (NS), le pourcentage d’observations compris
entre des simulations (POBS) a été de 74%, la longueur de l’intervalle relatif moyen (ARIL) était de 0.93, et
la valeur maximale du NS a été de 0.865. On a montré que les résidus étaient homoscédastiques, normalement
distribués et presque indépendants. Lorsque le seuil sur NS a été augmenté à 0.8, POBS a diminué à 54% avec une
amélioration de l’ARIL à 0.49, soulignant l’effet du choix subjectif de seuil de vraisemblance.
Mots clefs modélisation pluie–débit; GLUE; efficacité de Nash-Sutcliffe; Fleuve Nzoia; Lac Victoria; WASMOD; incertitude
ISSN 0262-6667 print/ISSN 2150-3435 online
© 2011 IAHS Press
http://dx.doi.org/10.1080/02626667.2011.610323
http://www.tandfonline.com
Modelling catchment inflows into Lake Victoria
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INTRODUCTION
In studies on water balance of Lake Victoria, strong
emphasis is usually placed on accurate estimation of
rainfall and evaporation over the lake surface (e.g.
Sene and Plinston 1994, Yin and Nicholson 1998,
Tate et al. 2004). This is because these components are the largest input and output of the lake
water balance respectively. However, the higher relative variability of catchment inflow compared to
lake rainfall plays an important role in the variability of the net basin supply (Tate et al. 2004).
Indeed, Sutcliffe and Parks (1999) noted that difficulties in explaining the historical lake level fluctuations
partly stemmed from underestimation of catchment
inflow. In the Lake Victoria basin, hydrological data
are, generally, sparse and unreliable, and simplifying assumptions are usually used in estimating the
catchment flows. As a consequence, model estimates
of catchment flows vary quite considerably, even for
studies that cover similar periods. For example, several studies carried out for the period 1950–1978 give
long-term mean values that range between 338 mm
water depth over the lake (equivalent to 23.2 ×
109 m3 ) and 420 mm (28.9 × 109 m3 ) per year, or
a 20% variation (Flohn and Burkhardt 1985, Yin and
Nicholson 1998). Discrepancies in runoff estimates
reflect inherent uncertainty in modelling of environmental systems (Kundzewicz 1995, Beven and
Freer 2001, Refsgaard et al. 2007). Uncertainties in
rainfall–runoff modelling are a result of errors in both
input and output data, and also result from the simplifications that come with mathematical representation
of the physical processes that govern flow generation
and routing. The proper recognition of, and accounting for uncertainties is currently acknowledged as an
integral part of any hydrological modelling process
(Wagener and Gupta 2005).
Estimating catchment flows into Lake Victoria
is complicated by the fact that only a small portion of the basin has been gauged consistently for
long periods of time (Yin and Nicholson 1998, Tate
et al. 2004). Out of the 20 tributaries, only Kagera,
Nzoia, Yala, Sondu and Awach Kaboun, draining
about 40% of the land basin, have relatively long
flow records. However, records from these also tend
to be patchy and unreliable over the recent past.
Some of the rivers were also gauged for brief periods during the Hydrometeorological Survey project
(WMO 1974). Therefore, most of the schemes that
have been used for estimating catchment flows into
Lake Victoria involve constructing linear regression
equations between runoff and rainfall for those basins
1211
that have long gauging records (mainly Kagera,
Nzoia, Yala, Sondu and Awach Kaboun) and using
these equations to derive the flow from other parts of
the basin. This approach has a disadvantage, in that
it results in estimates that have variances which are
similar to those of the rainfall. However, studies show
that the variability of catchment flow is up to three
times that of rainfall mainly due to varying flow generation characteristics around the lake basin (Sutcliffe
and Parks 1999).
The Nzoia River provides the second largest tributary flow to Lake Victoria after Kagera (Sutcliffe
and Parks 1999). While the area of the Nzoia basin
is only 6% of the Lake Victoria land basin, its flow
contributes about 14% of the total catchment flow.
Compared to the other major rivers that flow into
Lake Victoria (Kagera, Yala and Sondu), the Nzoia
annual flows have the highest coefficient of variation,
about 0.4 (Tate et al. 2004). The Nzoia River has also
been shown to be highly sensitive to variations in rainfall input. Kite and Waititu (1981) studied the effects
of varying precipitation and evapotranspiration on
river flow in the Nzoia River and Lake Victoria.
They showed that the rainfall–runoff process in the
Nzoia basin is highly sensitive to changes in input
data. According to the analysis of Kite and Waititu
(1981), a 10% increase in rainfall input would result
in a 40% increase in runoff, while a 30% increase
in rainfall would result in up to 100% increase in
runoff. The Nzoia basin suffers from frequent (in
most cases annual) flooding, especially in the lower,
flatter reaches. This is further exacerbated by changes
in land-use patterns within the basin. Deforestation
and intense agriculture within the upper to middle
reaches of the basin have left bare terrain with serious
soil erosion and sediment deposition, which modify
the catchment morphology.
The objective of this study is to improve the
estimation of catchment flows to Lake Victoria by
incorporating aspects of uncertainty assessment in
the runoff modelling process. The WASMOD model
(Xu 2002) is used for modelling the water balance
for the Nzoia River, and the GLUE strategy is used
for uncertainty analysis. The objective is achieved
through the following steps: (1) evaluation and comparison of four areal rainfall estimation methods,
namely inverse distance weighting, ordinary kriging,
universal kriging and kriging with external drift;
(2) quantifying the impact of sampling size in the
GLUE method on the estimated uncertainty; and
(3) quantifying the impact of threshold values in the
GLUE method on the estimated uncertainty. Model
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Michael Kizza et al.
performance and uncertainty were assessed using
multi-evaluation criteria, namely maximum NashSutcliffe coefficient, the percentage of observations
bracketed by simulation bounds (POBS) and the average relative interval length (ARIL) defined by Jin
et al. (2010).
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UNCERTAINTY ANALYSIS IN
HYDROLOGICAL MODELLING
Uncertainty sources in the hydrological modelling
process include: (1) uncertainties in input data (e.g.
rainfall, temperature, etc.); (2) uncertainties in calibration data (e.g. river discharge); (3) uncertainties in
model parameters; and (4) uncertainties due to model
imperfection. A variety of methods are available for
assessing the effect of the various sources of uncertainty on the accuracy and reliability of the estimation
of catchment hydrological variables (see, for example, Melching 1995, Tung 1996, Kuczera and Parent
1998, Montanari and Brath 2004, Romanowicz and
Macdonald 2005, Shrestha et al. 2007). Pappenberger
et al. (2005) provide a decision tree to find the
appropriate method for a given situation. The uncertainty analysis process in rainfall–runoff modelling
varies mainly in the following ways: (a) the type of
rainfall–runoff models used; (b) the source of uncertainty to be treated; (c) the representation of uncertainty; (d) the purpose of the uncertainty analysis;
and (e) the availability of resources, especially computation resources. The different uncertainty analysis
methods involve: (i) identification and quantification of the sources of uncertainty; (ii) reduction of
uncertainty; (iii) propagation of uncertainty through
the model; (iv) quantification of uncertainty in the
model outputs; and (v) application of the uncertain
information in decision making process.
Due to the fact that many parameter sets
within a given model structure may give acceptable results given the calibration data, i.e. the
so called equifinality problem, an approach called
the Generalised Likelihood Uncertainty Estimation
(GLUE) was proposed by Beven and Binley (1992)
to quantify the uncertainty resulting from the
equifinality. The GLUE approach starts from a
premise of equifinality in hydrological model structures and parameter sets. Within the GLUE framework, input and calibration data errors can be handled implicitly (Beven and Freer 2001). Parameter
interactions and nonlinearity in model responses are
also handled implicitly. Thus the likelihood measure
reflects the ability of a particular model to predict a
particular series of observations (which may not be
error free) given a particular set of inputs (which may
not be error free). There is thus an implicit assumption
that, in prediction, error structures will be “similar” in
some broad sense to those in the evaluation period.
Despite this assumption, the GLUE method makes
fewer assumptions about the variations of the errors
than alternative methods (Beven 2006). Whereas the
GLUE approach has been applied in some humid
catchments (e.g. Freer et al. 1996, Cameron et al.
1999, Choi and Beven 2007), applications in tropical and semi-arid catchments are still rare (Mwakalila
et al. 2001, Campling et al. 2002).
MATERIALS AND METHODS
Study area
The Nzoia River, located in western Kenya, has its
headwaters in the Cheranganyi hills, with tributaries
from Mount Elgon, and flows into Lake Victoria just
north of Yala Swamp. The Nzoia catchment has an
area of 12 700 km2 (Fig. 1). It lies in an agriculturally productive area of Kenya, the main crops being
cotton, maize and sugar cane. Demand for irrigation
water is highest during the dry season and irrigation is
carried out on a small to medium scale. Flooding is a
frequent problem, especially in the lower reach of the
catchment. However, flooding contributes to fertility
of the soils by carrying eroded sediment from the
highland areas. Catchment elevations vary between
about 1150 m a.s.l. at the outlet into Lake Victoria
to over 3500 m a.s.l. in the Mt Elgon ranges. The
rainy season lasts from March to September with
two distinct peaks: a larger peak in April–May and a
smaller peak in August, though there may be an additional peak in October–December (Kizza et al. 2009).
The diurnal, seasonal and annual rainfall patterns
are controlled by the migration of the Inter-Tropical
Convergence Zone (ITCZ) over the Equator, southeast
and northeast monsoons, Indian Ocean sea-surface
temperature, and other meso-scale systems (Mistry
and Conway 2003, Nicholson 1996).
The mean monthly rainfall in the catchment for
the period 1970–1988 varies from about 40 mm in
December and January to about 185 mm in April,
with an additional peak of 145 mm in August (Fig. 2).
There is a marked seasonal variation in the mean
monthly temperature, with a minimum of 20◦ C in July
and a maximum of 23◦ C in March. Monthly potential
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Modelling catchment inflows into Lake Victoria
1213
Fig. 1 Nzoia River basin (inset is a map of East Africa showing the location of the study area within the Lake Victoria basin).
The rainfall stations used in this study are labelled G1 to G13. Stations G7 and G12 are used for potential evapotranspiration,
and G6, G12 and G13 are used for temperature.
Fig. 2 Mean monthly values (line plots with circles) and box plots of the different input data used in the current study for
the period 1970–1988. Each box plot shows the median (dash sign), lower and upper quartiles (boxes), data within 1.5 times
the inter-quartile range (whiskers), and data that may be considered as outliers (crosses).
evapotranspiration varies from a minimum of 116 mm
in June to 175 mm in March. The intra-annual variation of discharge follows a similar pattern as that of
rainfall, except that the largest peak is moved from
April–May to August–September.
Data
Monthly data covering the period 1970–1988
(19 years) were used for the current study. Data
included rainfall, temperature and discharge, as well
as mean monthly values of potential evapotranspiration. Rainfall data were selected from a total of
35 stations in and around the catchment. The main
criteria were the quality of data and the degree of
completeness of the records, which was defined as
the ratio of months with rainfall data to the total
number of months in the study period. The minimum
degree of completeness was set at 75%; 13 stations
satisfied this criterion and were used for estimating
areal rainfall. The point rainfall data used were
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Michael Kizza et al.
available at stations marked G1 to G13 in Fig. 1, and
the basic information of the stations is summarised in
Table 1.
Monthly potential evapotranspiration data were
obtained for two stations within the basin (Bungoma –
G7 and Eldoret – G12) that were part of the monitoring network used by the hydrometeorological survey
project of the WMO (WMO 1974). The station values
used in this study were estimated during the WMO
study using the Penman method (Penman 1948). The
arithmetic mean of long-term mean monthly potential
evapotranspiration at the two stations was used as an
estimate for the areal long-term mean monthly potential evapotranspiration in the basin. The temperature
data were estimated as arithmetic means of temperature data at three stations (Mumias – G6, Eldoret –
G12 and Turbo – G13). Discharge data at Rwambwa
gauge (station number 1EF01) were used for model
calibration (Fig. 1). The time series for the discharge
data had 59 months of missing values out of a possible
total of 228 months (74% level of completeness).
Data quality tests included: homogeneity tests
and outlier detection, investigation of suspicious values and removal of clearly erroneous values. Some of
the stations had missing periods that were filled using
a forward stepwise multiple regression approach
using data from nearby stations.
Estimation of mean areal rainfall
Rainfall is the largest component in a water balance
and the most important input to hydrological models.
Rainfall data are subject to uncertainty as a result of
measurement errors, systematic errors in the spatial
interpolation, and stochastic errors due to the random nature of rainfall. There are several approaches
for accounting for uncertainties in mean areal precipitation estimates in rainfall–runoff modelling. The
two most commonly used methods are: (1) empirical methods and (2) sensitivity analysis. Empirical
analyses are generally based on the comparison of
various interpolation approaches (Creutin and Obled
1982, Lebel et al. 1987, Johansson, 2000), or on
under-sampling of relatively dense raingauge networks (Anctil et al. 2006, Balme et al. 2006, Bardossy
and Das 2008). In the current study, a comparison
of four rainfall interpolation methods using crossvalidation was adopted. The methods tested included:
inverse distance weighting (IDW), ordinary kriging
(OK), universal kriging (UK) and kriging with external drift (KED). All the four methods are based on
weighted linear combinations of the point rainfall
data. IDW is a deterministic interpolation method,
while OK, UK and KED are stochastic methods.
Inverse distance weighting assigns weights based
on the inverse of the distance to every data point
located within a given search radius centred on the
point to be estimated. In contrast, OK, UK and
KED are generally known as geostatistical interpolation methods, and assume that spatial continuity
depends on a statistical distance defined using a semivariogram. The difference in the methods is based on
the assumptions made about the variation (or trend) in
the data (Goovarts 2000). Ordinary kriging assumes
a constant unknown trend in the neighbourhood of
the estimation point, UK assumes a linear or higherorder trend that is a function of the coordinates, while
KED uses auxiliary information (for example elevation in the current study) to estimate the trend.
The performance of the methods was compared using
widely recognized and commonly used error measures, root mean squared error (RMSE) and mean
absolute error (MAE), applied to the monthly station
data. The RMSE gives relatively high weight to large
errors, while MAE gives equal weight to all individual
differences (Teegavarapu and Chandramouli 2005).
Hydrological model
The WASMOD model is a conceptual lumped
modelling system for simulating streamflow from
both snowmelt and rainfall; it can be operated at different time scales. For the current study, a version
without the snowmelt routine was used due to the
nature of the catchment under investigation, and the
data were at a monthly time step. The present version of WASMOD was developed by Xu et al. (1996)
and detailed description of the model can be found in
Xu (2002). The different variants of WASMOD have
been applied to global and regional water resources
assessment (Xu and Vandewiele 1995, Gong et al.
2009, Widén-Nilsson et al. 2009), and for catchment
water balance calculations in more than 200 catchments located in European, Asian, American and
African countries (Xu 2002). The concept of the
model is that the actual rainfall is split into a fraction that evaporates and a fraction that is active
rainfall, which contributes to the fast flow and the
slow flow (“baseflow”). WASMOD has three to six
parameters depending on the availability of input data
and climate of the study region. The adopted model
version has four parameters (Table 2). For the current study, the inputs included monthly values of
rainfall, temperature and mean monthly potential
Modelling catchment inflows into Lake Victoria
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Table 1 Stations used in estimating areal rainfall and their summary data.
Station ID
(in Fig. 1)
WMO code
Name
Latitude
(o N)
Longitude
(o E)
Altitude
(m a.s.l.)
Mean annual
rainfall (mm/year)
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
8835039
8934008
8934028
8934060
8934072
8934133
8934134
8934139
8934140
8935010
8935016
8935133
8935170
Leissa Farm
Kitale
Kakamega
Kimilili
Kaimosi
Mumias
Bungoma
Bunyala
Kadenge Yala
Kaptagat
Soy Kipsomba
Eldoret
Turbo
1.17
0.90
0.23
0.80
0.15
0.37
0.58
0.08
0.03
0.43
0.77
0.57
0.63
35.03
34.92
34.87
34.72
34.93
34.50
34.57
34.05
34.18
35.50
35.18
35.30
35.05
1968
1968
1804
1804
1902
1401
1509
1232
1256
2624
2099
2296
2001
952
1263
2135
1482
2050
1995
1517
1063
1124
1198
1033
1025
1334
Table 2 WASMOD parameters and their range.
Parameter
a1
a2
a3
a4
o
-1
(C )
(-)
(month-1 )
(mm-1 month)
Variable controlled
Initial sampling range
Final sampling range
Potential evapotranspiration
Actual evapotranspiration
Slow flow
Fast flow
0–1
0–1
0–1
0–1
0–1
0–1
0–0.02
0–0.02
evapotranspiration, which were readily available for
the catchment. Monthly runoff and other water balance components were the outputs.
The potential evapotranspiration at time t, ept , is
estimated by adjusting the monthly long-term potential evapotranspiration, epm , using the monthly mean
temperature, ct , and the monthly long-term mean
temperature, cm :
ept = [1 + a1 (ct − cm )] epm
(1)
where a1 is a model parameter.
Actual evapotranspiration, et is computed as a
function of available water and potential evapotranspiration:
et = min[ept (1 − a2 t / t ), wt ]
w ep
(2)
where a2 is a model parameter, and wt is available
water defined as:
wt = rt + smt−1
(3)
where rt is the rainfall in a given month, and smt–1 is
the available soil moisture storage from the previous
month, and has a minimum of zero.
The slow flow depends on the soil moisture
storage in the catchment:
st = a3 (smt−1 )2
(4)
where a3 is a model parameter.
The fast flow depends on rainfall amount, rt ,
other meteorological parameters represented by ept ,
the state of the basin as measured by smt–1 and on the
physical characteristics of the basin represented by the
parameter a4 :
ft = a4 smt−1 nt
(5)
where nt is the effective rainfall after the subtraction
of the interception loss and defined as:
nt = rt − ept 1 − e−rt /ept
(6)
The total runoff is the sum of the slow and fast
components:
dt = s t + f t
(7)
Finally, the soil moisture can then be updated using a
water balance equation, using:
smt = smt−1 + rt − et − dt
(8)
Application of the model requires that the initial
moisture content is fixed by allowing for a warm-up
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Michael Kizza et al.
period that is long enough to ensure that moisture content is independent of the starting value. In the current
study a 3-year warm-up period was provided for.
N
NS = 1 −
[log(obst ) − log(simt )]
t=1
N log(obst ) − log(obst )
2
(10)
2
t=1
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Model perfomance evaluation
The performance of the model was evaluated within
the GLUE framework in order to carry out parameter
estimation using set performance criteria and assess
predictive uncertainty for the study catchment using
the WASMOD model. The aim was to examine those
parameters to which the model simulation results
were most sensitive and to assess the probability of
simulated discharge being within a certain prediction
interval. Model calibration was based on identified
acceptable (behavioural) parameter sets using Monte
Carlo simulation. Initially, broad parameter ranges
were set using information from prior applications of
the WASMOD (Table 2, column 3), and preliminary
runs were used to set the feasible sampling range for
the study catchment (Table 2, column 4).
Within a Monte Carlo framework, we can use
either importance sampling or uniform sampling
techniques. In importance sampling, the shape of the
response surface is represented by the density of sampling and each parameter set is given equal weight
in forming a distribution of predictions (Kuczera and
Parent 1998). In uniform sampling, the model simulation is reflected by the shape of the response surface.
In this study, we adopted the uniform sampling strategy which is less efficient, but is easy to implement
and requires minimal assumptions about the shape of
the response surface (Beven and Freer 2001).
The acceptable parameter sets were defined on
the basis of two likelihood criteria: the volume error
(VE) measure and the Nash-Sutcliffe (NS) (Nash and
Sutcliffe 1970) efficiency measure applied to logtransformations of measured and simulated flows.
These two performance measures are commonly used
in hydrology and are defined as shown in equations (9)
and (10). Log-transformations of measured and simulated flows were necessitated by the fact that initial
runs using untransformed flows of the model showed
that the resultant hydrographs for the study basin are
greatly influenced by high flows with the fit between
measured and simulated low flows being poor.
N
VE =
simt −
t=1
N
t=1
N
t=1
obst
obst
(9)
where obst and simt are the observed and simulated
flow values at time t, respectively.
A two-stage approach was used to select the
acceptable parameter sets under calibration using the
two criteria. The first stage was to select only those
parameter sets whose volume error (VE) was less than
10% during calibration. The next step was to select
all parameter sets having NS ≥ 0.6 as behavioural or
acceptable under calibration.
The likelihoods that were used in the current
study are based on derived NS coefficients that were
rescaled so that any value less than the set threshold value was given zero likelihood and the sum of
all likelihoods was one. For each parameter set, the
rescaled likelihood measure was used to calculate the
prediction quantiles using:
P(Ẑt < z) =
i=N
B
L M(i )|Ẑt,i < z
(11)
i=1
where Ẑt,i is the estimate of variable z at time t
using parameter set i; NB is the number of acceptable parameter sets; and M()i is the ith Monte Carlo
sample having parameter set i . The computed probabilities were then used to compute the prediction
bounds for the acceptable parameter sets with a 95%
confidence interval. The resulting quantiles are conditioned on the inputs to the model, the model responses
for the particular sample of parameter sets used, the
choice of likelihood measure and the observations
used in the calculation of the likelihood measure
(Beven and Freer 2001). The prediction limits that
are obtained have a disadvantage in that, unless a formal error model is used, they will not provide formal
estimates of the probability of estimating any particular observation conditional on the set of model runs.
Yet they have the advantage that they help us make
inferences about the response of the system after conditioning on past data, as non-stationarities in the
residual errors and model failures are more clearly
revealed (Beven 2006).
In summary, the following multi-step approach to
model evaluation was adopted for the study:
(a) The first 3 years of data (1970–1972) were used
for warming-up to get rid of the influence of
Modelling catchment inflows into Lake Victoria
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the subjectively selected value of the initial soil
moisture.
(b) The next 16 years of data (1973–1988) were
used for model conditioning to select acceptable
parameter sets in calibration by:
(i) running 500 000 ensembles of the model
and selecting all parameter sets that resulted
in VE < 10%; and
(ii) from the parameter sets in (i) above, all
parameter sets corresponding to a minimum
NS value of 0.6 were retained as acceptable under calibration on the basis of the set
threshold NS value.
RESULTS
Comparison of rainfall interpolation methods
Table 3 shows bias, mean absolute error and root
mean square error for the four methods that were
tested for the estimation of mean areal rainfall. There
are only slight differences between the performances
of IDW, OK and KED. While IDW has the largest
bias, it outperformed the other three methods with
respect to MAE and RMSE; meanwhile UK had
the worst performance. The inclusion of elevation in
interpolation using KED did not result in significant
improvements in the estimation process. Therefore,
IDW was used for the estimation of the mean areal
rainfall for all subsequent sections of this paper.
Parameter behaviour
When VE was used as a criterion, on average 12,
9 and 6% of the simulations resulted in VE values
of less than 10, 5 and 1%, respectively (Fig. 3(a)).
The acceptable values for parameters a1 , a3 and a4
were scattered throughout the sampling space, while
the values for a2 mainly varied between 0 and 0.5,
though a few were still located outside this range.
Figure 3(a) also shows that using VE alone as a criterion cannot constrain the parameters. When NS was
1217
used as the criterion, on average 13, 3 and 0.6%
of the simulations gave NS values greater than 0,
0.5 and 0.7 respectively (Fig. 3(b)). Parameter a1 was
the least well-confined among the parameters, with
acceptable values scattered throughout the parameter range (Fig. 3(b)). The other three parameters
(a2 , a3 and a4 ) were better constrained, showing
a higher degree of peakedness within the response
surface. Previous studies have also shown that parameter a1 , which is not needed when monthly potential
evapotranspiration values are available, is the least
sensitive among the parameters (Xu 1999, 2001). The
NS measure had a peak of about 0.865, though the
parameter sets covered wider ranges as shown in Fig.
3(a). The number of acceptable parameter sets satisfying the different performance criteria shows a more
dramatic increase with increasing NS limit values
compared with the VE limits (Table 4). This is related
to the higher model conditioning effect of NS as a
performance measure compared to VE.
Figure 4 shows the relative sensitivity of the
model results to changes in the different parameters
in a more comparable way by showing the sensitivity to relative changes in the parameter values. The
figure demonstrates how the model performance, as
reflected by the value of NS, deteriorates from the
maximum NS value with a small increase or decrease
in the value of a given parameter keeping all other
parameters constant. The maximum NS value was
0.865 and the corresponding parameter values are
0.146, 0.561, 0.055 and 0.0012 for a1 , a2 , a3 and a4 ,
respectively. While the figure confirms that parameter
a1 was the least sensitive, it also shows that the model
output was more sensitive to percentage changes in
a2 than to parameters a3 and a4 , a fact which is not
obvious from Fig. 3(b).
Behavioural parameter sets
As the NS likelihood threshold value was increased
from 0 to 0.8, the percentage of observations
bracketed by simulations was reduced by 27%
Table 3 Performance of the rainfall interpolation methods (MAE: mean absolute error, RMSE: root mean square
error).
Interpolation method
Bias (mm/month)
MAE (mm/month)
RMSE (mm/month)
Inverse distance weighting (IDW)
Ordinary kriging (OK)
Universal kriging (UK)
Kriging with external drift (KED)
4.3
2.0
1.3
2.1
39.2
41.5
44.1
41.6
50.1
51.9
55.2
52.6
Fig. 3 Scatter plots based on 500 000 runs showing the parameter variations with the two likelihood measures used in the current study: (a) VE variation and (b) NS
variation against uniformly sampled parameter values.
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Michael Kizza et al.
Modelling catchment inflows into Lake Victoria
1219
0.90
NS
0.85
a1
0.80
a3
0.75
a2
a4
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0.70
–100% –80% –60% –40% –20% 0% +20% +40% +60% +80% +100%
Percent difference from parameter value with maximum NS
Fig. 4 Sensitivity of each of the parameters around parameter set having maximum NS. The x-axis shows the relative
deviation of the parameter values from their optimized values and the y-axis shows the value of the NS criterion.
Table 4 Number of acceptable parameter sets for different
NS and VE criteria based on 500 000 model runs.
Criterion
NS
VE
Limiting values
≥0.7
≥0.5
≥0
≤1%
≤5%
≤10%
Number of acceptable
parameter sets
3 129
15 480
63 891
32 050
44 788
61 568
Table 5 Percentage of observed discharge bracketed
by computed discharge (POBS) at different likelihood
threshold values and the resulting number of parameter
sets during the calibration stage.
NS threshold value
POBS (%)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
80.5
79.9
79.9
79.3
78.7
78.1
73.4
69.8
53.8
(Table 5). Table 6 shows the statistics of the resultant
acceptable parameter sets for a NS threshold of 0.6,
while Fig. 5 shows their distributions. All parameters
possess well-defined and unimodal posterior distributions. From such distributions, parameter estimates
can be unambiguously inferred as modal values (Jin
et al. 2010, Li et al. 2010), while the shape of the distributions indicates the degree of uncertainty of the
Table 6 Statistics for the acceptable parameters based
on 500 000 model runs for a likelihood threshold of
NS ≥ 0.6. The parameter units are shown in Table 2.
Statistic
a1
a2
a3
a4
Mean
Median
Minimum
Maximum
Standard
deviation
Skewness
P0.025
P0.05
P0.95
P0.975
0.403
0.375
0.000
0.996
0.256
0.545
0.528
0.441
0.811
0.072
0.078
0.074
0.009
0.192
0.039
0.0013
0.0012
0.0000
0.0032
0.0007
0.344
0.026
0.045
0.857
0.915
0.966
0.451
0.458
0.690
0.725
0.386
0.018
0.022
0.145
0.161
0.3714
0.0002
0.0003
0.0024
0.0027
estimates. Sharp and peaked distributions (parameters a2 , a3 and a4 ) are associated with well identifiable
parameters, while flat distributions (a1 ) indicate more
parameter uncertainty. It can be seen that only parameter a1 retains elements of the prior uniform distribution. The distributions of parameter a3 and a4 are
closer to a normal distribution, while that of a2 has
a clear negative skew. The correlations between the
parameters are moderate to small (Table 7).
Effect of model conditioning
In most cases, the aim of hydrological modelling is to
provide a means for making predictions about future
flows for purposes of water resources assessment and
management. As such, it is important to analyse how
a model constructed using one period of data will
operate for another period of data as a proxy to how
the model will perform in predicting future flows.
1220
Michael Kizza et al.
15
Frequency (%)
0
Frequency (%)
10
5
a4 (x10–4)
15
10
5
0
0.02
0.03
0.05
0.06
0.08
0.09
0.11
0.12
0.14
0.15
0.17
0.18
Frequency (%)
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5
a3
15
0
10
0.46
0.49
0.52
0.55
0.58
0.61
0.64
0.67
0.70
0.73
0.77
0.80
5
15
0.14
0.41
0.68
0.95
1.22
1.49
1.76
2.03
2.30
2.57
2.84
3.11
10
0
a2
20
0.04
0.12
0.21
0.29
0.37
0.46
0.54
0.62
0.71
0.79
0.87
0.95
Frequency (%)
a1
Fig. 5 Distribution of parameter values for a NS threshold of 0.6.
Table 7 Correlation matrix of the parameters based
on 500 000 model runs for a likelihood threshold of
NS ≥ 0.6. The parameter units are shown in Table 2.
a1
a2
a3
a4
a1
a2
a3
a4
1.000
0.078
1.000
–0.172
–0.644
1.000
–0.078
–0.311
–0.416
1.000
In the current study, the data period was split into
two periods, namely 1973–1982 and 1983–1988. The
data covering the period 1973–1982 were used to calibrate the model. The model was then run using data
covering the period 1983–1988 using the acceptable
parameter sets. Parameter sets that continued to be
acceptable were retained and the resultant simulations
were compared with those of the 1973–1982 set using
three measures, namely: maximum Nash-Sutcliffe
coefficient, the percentage of observations bracketed
by simulations (POBS), and the average relative interval length (ARIL). The ARIL is a scaled summation
of the prediction limits over the simulation period (Jin
et al. 2010):
1 LimitUpper,t − LimitLower,t
ARIL =
n
Qobs,t
(12)
where LimitUpper,t and LimitLower,t are the calculated
upper and lower confidence limits, respectively, for
the tth month, Qobs,t is the measured flow, and n is the
number of time steps. The goodness of the simulation
was judged on the basis of the closeness of ARIL to
0 and the percent bracketed to 100%.
The maximum NS value during model calibration was 0.88 but the NS value for the same parameter
set reduced to 0.79 during model conditioning. For
the model conditioning period, the maximum NS
value was 0.84 though the same parameter set yielded
a NS value of 0.81 during model calibration. This
shows that the optimum parameter set, as defined by
the maximum NS, may change depending on input
data and other factors. Table 8 shows the number
of parameter sets for different likelihood thresholds
and shows that the percentage of parameter sets
dropped increases with increasing threshold values.
Figure 6(a) shows that the conditioning period has
Table 8 Number of parameter sets dropped during model
conditioning for different NS likelihood thresholds.
NS threshold
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Analysis period:
1973–1982
1983–1988
4972
4753
4450
4124
3707
3168
2449
1549
613
3720
3467
3216
2906
2551
2102
1495
851
154
Parameter sets
dropped (%)
25
27
28
30
31
34
39
45
75
Modelling catchment inflows into Lake Victoria
(b)
Percent bounded
90
1973–1982
1983–1988
80
70
60
50
0
0.1
0.2
0.3 0.4 0.5
NS Threshold
0.6
0.7
0.8
1973–1982
1983–1988
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3 0.4 0.5
NS Threshold
0.6
0.7
0.8
Fig. 6 Variation of (a) percent of observations bracketed by simulations (POBS), and (b) average relative interval length
(ARIL) for different NS limits when defining acceptable parameter values.
higher values of POBS for lower likelihood thresholds compared to the calibration period. However, for
higher likelihood thresholds, the calibration period
has higher values of POBS. Figure 6(b) shows that the
model generally produces a tighter fit to the observed
data in the conditioning period compared to the calibration period. This implies that it is those parameter
sets that simulate flows that are close to the upper
and lower tails of the flow distribution which are
dropped during model conditioning. The analysis also
shows that the model continues performing well during model conditioning under the assumed simulation
conditions, and may be used for flow prediction.
Optimum sample size
In the current analysis, the parameter ranges were
sampled randomly and only the parameter sets satisfying the selected performance criteria were retained.
The optimum sample size is one that captures most
of the observed flow features in some (optimum) way
at relatively low cost in terms of time taken to carry
out the calculations. The optimum sample size for
the current study was investigated by carrying out
computations for seven different sample sizes and
computing, for each sample size, three measures that
describe the model performance, namely: maximum
Nash-Sutcliffe coefficient (NS) value, POBS for different NS thresholds, and ARIL (see equation (12))
for different NS thresholds. Figure 7 shows that the
maximum NS value increases with number of model
simulations; however, as soon as the number of sample simulations increased to 2.5 × 105 , the influence of number of sample simulations on the model
simulation results became negligible. Table 9 shows
0.87
0.865
Maximum NS
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40
1.6
Average Relative Interval Length
(a)
1221
0.86
0.855
0.85
0.845
0.84
0
1
2
3
Number of simulations
4
5
x 105
Fig. 7 Variation of maximum Nash-Sutcliffe coefficient
(NS) against number of model simulations.
that there are generally no appreciable gains in
POBS and ARIL by increasing the number of model
simulations.
Residual analysis
To get an idea of the model performance, an analysis of the residuals was carried out. The residuals were computed as the difference between the
log-transformed median simulated flow ensembles
and the log-transformed observed flow. The match
between observed flow and median simulated flow
was reasonably good with most of the flow variations well reproduced (Fig. 8(a)). The NS value
for the median flow was 0.75, while that of the
mean simulated flow was 0.82. The residuals had
a constant variance (Fig. 8(b)). The residuals followed a normal distribution as shown in Fig. 8(c)
(the Kolmogorov-Smirnov test also resulted in nonrejection of the normal distribution assumption). The
autocorrelation plot showed that the residuals were
1222
Michael Kizza et al.
Number of
model runs
POBS:
NS ≥ 0
NS ≥ 0.5
NS ≥ 0.7
ARIL:
NS ≥ 0
NS ≥ 0.5
NS ≥ 0.7
40 000
70 000
100 000
150 000
200 000
250 000
500 000
81.1
79.3
80.5
81.1
79.9
79.9
80.5
76.9
76.3
76.9
77.5
78.1
76.3
78.1
71.0
69.2
69.8
71.0
70.4
70.4
69.8
1.225
1.176
1.220
1.186
1.177
1.190
1.198
1.051
1.027
1.059
1.036
1.043
1.048
1.045
0.755
0.730
0.751
0.784
0.772
0.764
0.761
independent except for time lag of 1 (Fig. 8(d)). The
results demonstrate that WASMOD model and the log
Nash-Sutcliffe likelihood produce reasonable results
for the Nzoia basin.
threshold value is increased from 0.6 to 0.8, the confidence interval becomes tighter and there is more
chance of observed flows being outside the simulation limits. By increasing the threshold value from
0.6 to 0.8, POBS decreased from 74% to 54%, while
ARIL decreased from 0.93 to 0.49. These were large
changes in model fit and reflect a common criticism
of the subjective choice of threshold likelihood values
in GLUE. However, there have been several attempts
at reducing the subjectivity in selecting the thresholds. For example Blasone et al. (2002) proposed an
alternative strategy to determine the cut-off threshold based on an appropriate coverage of the resulting
uncertainty bounds. The choice of threshold cut-off
point could also be based on the intended application
Sensitivity to threshold likelihood value
Selection of the threshold value of the likelihood
function is a key step in applying the GLUE method.
The effect of threshold value on model results is
dependent on, among others, the model, the likelihood function and the sample size. The effects of
two NS threshold values on the model results are
compared in Fig. 9. It can be seen that when the
(b)
80
Observed
Simulated
60
1
Residuals
Runoff depth (mm/month)
(a)
40
20
0
–1
0
1973 1975 1977 1979 1981 1983 1985 1987
Years
Residuals=Log(Simulated Flow) – Log(Observed Flow)
0
(c)
40
60
20
Simulated Flow (mm/month)
80
(d)
1
Residuals
Normal distribution
Autocorrelation
Cumulative probability
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Table 9 Percentage of observations bracketed by simulations (POBS) and average relative interval length (ARIL) for
different sample sizes and Nash-Sutcliffe thresholds.
0.5
0
–1
–0.5
0
Residuals
0.5
1
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
(d)
0
12
24
36
Lag (months)
48
60
Fig. 8 Residual analysis based on log transformations of both observed and median simulated flow: (a) comparison of
monthly observed and median simulated flow, (b) residuals versus median simulated flow, (c) cumulative distribution functions for the residuals and the normal distribution, and (d) autocorrelation plot of residuals. Dashed lines in (d) represent
95% confidence interval about zero.
Modelling catchment inflows into Lake Victoria
(a)
(b)
100
Runoff Depth (mm/month)
1223
95% Confidence Interval
Observed Flow
100
80
80
60
60
40
40
20
20
0
1974
1976
1978
1980
1982
1984
1986
1988
95% Confidence Interval
Observed Flow
0
1974
1976
1978
1980
1982
1984
1986
1988
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Fig. 9 Comparison of the 95% confidence interval due to parameter uncertainty of monthly flow for likelihood thresholds
of (a) 0.6 and (b) 0.8.
of the model and the environment in which the model
will be applied. A very strict threshold value (for
example a high NS value) may result in failure to predict future flows in situations where uncertainties in
input data are significant.
DISCUSSION
We set out to develop an approach to improve the
estimation of catchment inflows into Lake Victoria
that handles issues of parameter uncertainty and also
can account for runoff variability around the basin.
We also tested the sensitivity of input areal rainfall estimates to interpolation method and showed
that inverse distance weighting provides estimates that
are comparable to, if not better than, the stochastically based kriging methods (ordinary kriging, universal kriging and kriging with external drift). The
WASMOD model was applied to Nzoia River and the
resulting model performance on the basis of NashSutcliffe coefficient was considered good with values
that were higher than 0.865. WASMOD has only four
parameters and is not very computationally intensive, meaning that many simulations can be carried
out (Xu 2002). This facilitated the exploration of the
model space and therefore the identification of the
acceptable parameter sets.
All four WASMOD parameters were shown to
possess well-defined and unimodal posterior distributions: the posterior distribution of parameter a1
retained elements of the prior uniform distribution,
the posterior of parameter a2 was negatively skewed,
while the distributions of parameters a3 and a4 were
transformed into more normally distributed forms.
Parameter a2 was shown to be the most sensitive,
followed by a3 and a4 , with parameter a1 the least sensitive. Parameter a1 converts mean monthly potential
evapotranspiration to monthly potential evapotranspiration using monthly temperature values and has also
been shown to be the least sensitive in other studies
(Xu 1999, 2001). Analysis of the residuals showed a
good fit between measured flow and median of the
simulated flows. The residuals, computed as the difference between log-transformations of the median of
simulated flow ensembles and log-transformations of
observed flows, had a near constant variance as well
as a normal distribution (i.e. the actual residuals had a
lognormal distribution). A plot of the autocorrelation
function showed only one value slightly outside the
95% confidence interval, implying mutual independence between the residuals. The results show that the
chosen model and likelihood function are applicable
to the study basin.
When using the GLUE method, the uncertainty
estimates are highly dependent on the selected threshold value of likelihood function. This was demonstrated by using thresholds of 0.6 and 0.8 for the NS
coefficient. Assessment of the two thresholds showed
that the percentage of observations bracketed by simulations decreased from 74% to 54%, a decrease of
27 units of percentage. This difference is large and
illustrates that the subjectivity in the choice of likelihood threshold is a major drawback in the GLUE
approach. However, the decisions are usually based on
the modellers’ understanding of the data limitations
and the intended use of the modelling results. In addition, there are ongoing research efforts aimed at
proposing objective methods of selecting the thresholds (Blasone et al.2002, Freni et al. 2008, Liu et al.
2009).
The simulated mean annual flows from the Nzoia
River to Lake Victoria ranged from 195 to 420 mm
(as water depth over the Nzoia catchment) with an
average of 291 mm, corresponding to a discharge
1224
Michael Kizza et al.
of 76–163 m3 /s and an average of 113 m3 /s. These
estimates compare favourably with other documented
estimates, such as 79 m3 /s (Tate et al. 2004),
87.5 m3 /s (WMO 1974), and 116.7 m3 /s (LVBC
2006). The annual rainfall totals over the Nzoia
catchment varied between 1102 and 1855 mm, with
an average of 1433 mm, while annual evapotranspiration varied between 944 and 1283 mm, with an
average of 1122 mm.
Downloaded by [Uppsala universitetsbibliotek] at 07:46 19 October 2011
CONCLUSIONS
The capability of WASMOD in simulating inflow to
Lake Victoria was investigated in the study using
the Nzoia River as the representative catchment. The
modelling uncertainty was analysed with the GLUE
method and the areal rainfall uncertainty resulting
from the use of different interpolation methods was
also studied. The following conclusions can be drawn
from the study:
– The WASMOD model proved to be successful
at simulating the historical streamflows in the
Nzoia River. The model performance was quite
high with a maximum Nash-Sutcliffe coefficient
of 0.865.
– Four areal rainfall estimation methods (inverse
distance weighting, ordinary kriging, universal
kriging and kriging with external drift) gave similar results with only small differences in performance, as measured by bias, mean absolute
error and root mean square error. The inverse distance weighting method was selected in the study
due to its slightly better performance in terms
of mean absolute error and root mean square
error. Inclusion of elevation in the interpolation
does not result in significant improvements in the
estimation.
– In the GLUE method, the results from an assessment of the size of the samples needed for optimum results were mixed. The maximum NashSutcliffe efficiency value showed an increase with
the sample size; however, as soon as the number
of sample simulations increased to 2.5 × 105 , the
influence of number of sample simulations on the
model simulation results became of minor importance. The percentage of observed flows bracketed
by simulations (POBS) and the average relative
interval length (ARIL) showed no appreciable
gains with the increase of simulation sample size.
– The study confirmed some previously published
results that the measures of model uncertainty are
highly sensitive to the selected threshold likelihood value in the GLUE method. The proportion
of observed flows that are bracketed by simulation dropped by 27 percentage units when the
threshold value was increased from 0.6 to 0.8.
While this study proved that the WASMOD
model, with the calibration scheme used in this study,
is suited for modelling inflow to Lake Victoria, as
demonstrated in the study catchment, other issues
including the performance of the model in other
gauged catchments and regionalization of the model
on ungauged catchments are to be investigated in a
future study.
Acknowledgements This work is part of the PhD
programme of the first author, with sponsorship of
the Swedish International Development Cooperation
Agency (Sida) through the Department for Research
Cooperation (SAREC). The third author was partially supported by the Research Council of Norway
with project number 171783 (FRIMUF). We are
pleased to acknowledge this financial support.
Acknowledgement is also extended to Prof. Keith
Beven for introducing the first author to the concept of
dealing with uncertainty in environmental modelling,
which formed the basis for this work.
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