Hydrological Sciences–Journal–des Sciences Hydrologiques, 50(1) February 2005
45
Assessing uncertainties in a conceptual water
balance model using Bayesian methodology
KOLBJØRN ENGELAND1, CHONG-YU XU2 &
LARS GOTTSCHALK3
1 Cemagref, 3bis quai Chauveau, CP220, F-69336 Lyon Cedex 09, France
engeland@lyon.cemagref.fr
2 Department of Earth Sciences—Hydrology, Uppsala University, Villavagen 16,
S-75236 Uppsala, Sweden
3 Department of Geophysics, University of Oslo, PO Box 1022 Blindern, N-0315 Oslo, Norway
Abstract The aim of this study was to estimate the uncertainties in the streamflow
simulated by a rainfall–runoff model. Two sources of uncertainties in hydrological
modelling were considered: the uncertainties in model parameters and those in model
structure. The uncertainties were calculated by Bayesian statistics, and the MetropolisHastings algorithm was used to simulate the posterior parameter distribution. The
parameter uncertainty calculated by the Metropolis-Hastings algorithm was compared
to maximum likelihood estimates which assume that both the parameters and model
residuals are normally distributed. The study was performed using the model
WASMOD on 25 basins in central Sweden. Confidence intervals in the simulated
discharge due to the parameter uncertainty and the total uncertainty were calculated.
The results indicate that (a) the Metropolis-Hastings algorithm and the maximum
likelihood method give almost identical estimates concerning the parameter
uncertainty, and (b) the uncertainties in the simulated streamflow due to the parameter
uncertainty are less important than uncertainties originating from other sources for this
simple model with fewer parameters.
Key words Bayesian analysis; Markov Chain Monte Carlo analysis; maximum likelihood
estimation; model uncertainty; water balance models
Estimation bayésienne des incertitudes au sein d’une modélisation
conceptuelle de bilan hydrologique
Résumé L’étude est consacrée à l’évaluation des incertitudes entachant les débits
simulés par un modèle pluie–débit conceptuel. Deux sources d’incertitudes dans la
modélisation hydrologique ont été prises en compte: les incertitudes dues aux
paramètres du modèle et celles dues à sa structure. Ces incertitudes ont été évaluées
par une approche bayésienne, et l’algorithme de Metropolis-Hastings a été utilisé pour
simuler la distribution a posteriori des paramètres. L’incertitude sur les paramètres
évalués par l’algorithme de Metropolis-Hastings a été comparée à celle que fournit la
méthode du maximum de vraisemblance sous hypothèse de normalité de la
distribution de ces paramètres et des résidus du modèle. L’étude a été menée en
appliquant le modèle WASMOD à 25 bassins versants du centre de la Suède. Les
intervalles de confiance relatifs aux débits simulés dus à la seule incertitude sur les
paramètres d’une part, et dus à l’incertitude totale d’autre part, ont été calculés. Les
résultats indiquent que (a) l’algorithme de Metropolis-Hastings et la méthode du
maximum de vraisemblance donnent des résultats quasiment identiques pour
l’incertitude sur les paramètres, et que (b) pour les débits simulés, les incertitudes liées
aux paramètres sont plus petites que celles qui sont dues à d’autres causes, pour ce
modèle simple à peu de paramètres.
Mots clefs analyse bayésienne; analyse Markov-Chaîne de Monte Carlo; maximum de
vraisemblance; incertitude de modèle; modèles de bilan hydrologique
INTRODUCTION
Conceptual catchment models are common tools in calculating the runoff dynamics
and the water balance at various scales and regions. The model parameters are usually
Open for discussion until 1 August 2005
Copyright  2005 IAHS Press
46
Kolbjørn Engeland et al.
calibrated in order to obtain a good fit between observed and simulated outputs. Since
computer models are not a perfect representation of reality, the results are uncertain. In
many cases, assessment of the uncertainties is important, e.g. in water recourses
management, where decisions have to be taken based on several uncertain factors;
climate change impact studies; and water balance calculations in ungauged basins. The
uncertainties in hydrological modelling have four important sources (e.g. Refsgaard &
Storm, 1996): (a) uncertainties in input data (e.g. precipitation and temperature);
(b) uncertainties in data used for calibration (e.g. streamflow); (c) uncertainties in
model parameters; and (d) imperfect model structure. The error sources (a) and (b)
depend on the quality of data, whereas (c) and (d) are more model specific. The
uncertainties in the input data (a) originate from the observations (e.g. Førland et al.,
1996) and the interpolation in space (e.g. Lebel et al., 1987). The uncertainties in the
streamflow observations (b) depend on the quality of the rating curve. Some examples
are given in Kuczera (1996), Clarke et al. (2000), Clarke (1999) and Jónsson et al.
(2002). In this study the uncertainties (c) and (d) were investigated.
Perhaps no studies include a full investigation of how the four error sources
contribute and interact in the total modelling uncertainty. However, there are several
papers where two or three of the error sources are evaluated simultaneously.
Andréassian et al. (2001) demonstrate that the quality of the rainfall data influences
both the simulation errors and the calibrated model parameters. Several studies
indicate that the precipitation is the most important uncertainty factor compared to the
model parameters (e.g. Thorsen et al., 2001; Refsgaard et al., 1983; Storm et al., 1988)
or the model structure (e.g. Krzysztofowicz, 1999). The papers cited above indicate
that results presented here should be interpreted carefully since the study does not
account explicitly for the error sources (a) and (b). What appears to be a parameter or
model uncertainty might partially be caused by uncertainties in the observations.
Several methods are available for evaluating the parameter sensitivity and
assessing the modelling uncertainties. Many studies focus on the formulation of the
objective function that describes the fit between observed and simulated values. In a
statistical framework, the objective function is called the likelihood function and is a
stochastic model for the simulation errors. To account for auto-correlations, stochastic
error model of AR(1) and ARMA types are used (e.g. Sorooshian & Dracup, 1980;
Alley, 1984; Xu, 2001). To account for non-constant variance, heteroscedastic error
models are used (e.g. Sorooshian & Dracup, 1980) or the data are transformed (e.g.
Kuzczera, 1983; Xu, 2001). A transformation might also account for the non-normality
of the simulation errors (e.g. Krzysztofowicz, 1999). Some papers focus on the
parameter sensitivity without aiming to estimate a probability distribution (e.g. Mein &
Brown, 1978; Spear et al., 1994; Wagener et al., 2003). The multi-objective method
(Yapo et al., 1998) is used to quantify parameter uncertainty and analyse model
structural uncertainty by generating a Pareto optimal set of parameters with respect to
different output fluxes (e.g. Beldring, 2002) or different objective functions (e.g.
Madsen, 2000). Examples of statistical approaches for quantification of parameter
uncertainty include maximum likelihood methods (e.g. Xu, 2001) and Bayesian
methods (e.g. Kuczera, 1983; Engeland & Gottschalk, 2002). A variant of the Bayesian
method was introduced under the name generalized likelihood uncertainty estimation
(GLUE) by Beven & Binley (1992). In the GLUE concept, it is recognized that a
number of the requirements in the Bayesian method cannot be fulfilled in many cases.
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
47
It might be difficult to define a likelihood function, and the response surfaces in the
parameter space can be very complex due to the interacting modelling errors. A
“subjective” likelihood is therefore used in the GLUE approach instead of a more
strictly defined statistical one. The aim is to discover the whole response surface in
order to find a set of many different models that can be acceptable. The GLUE concept
is now becoming a popular method in studying the parameter uncertainty and interval
prediction in both hydrological and hydraulic models (e.g. Beven et al., 2000; Hankin
et al., 2001; Aronica et al., 2002), and a dynamic version of GLUE is developed in
Wagener et al. (2003).
In developing and implementing the Bayesian method, the Metropolis-Hastings
(MH) algorithm (Hastings, 1970), a Markov Chain Monte Carlo (MCMC) method, has
been used. The MCMC method has been especially popular in Bayesian statistics (e.g.
Geyer, 1992), and more recently the MH algorithm has been applied to hydrological
models (e.g. Kuczera & Parent, 1998; Engeland & Gottschalk, 2002; Vrugt et al.,
2003).
In this study, the Bayesian method was applied as presented in Engeland &
Gottschalk (2002). They used the physically-based Ecomag model operating on a daily
time resolution, but two properties prohibited application of the Bayesian method in a
consistent way: a complex behaviour of the simulation errors, and a complex shape of
the parameter distribution. In order to overcome these difficulties and evaluate the
Bayesian method in a consistent framework, a more simplistic model was applied in
this study. A monthly water balance model called WASMOD was chosen because it
has been carefully constructed to have well-defined, independent parameters and
simulation errors that are normally distributed (Xu et al., 1996; Xu 2001, 2002).
The aim of this study was to estimate the model parameters and to quantify the
uncertainties of the simulated streamflows that resulted from both parameter
uncertainty (c) and model structure uncertainty (d). To start with, only two of the four
error sources were chosen, to keep the calculations simple, but it is intended to include
additional error sources at a later stage. This aim was achieved by estimating the
probability density of the model parameters and the statistical parameter of the
simulation errors simultaneously. Both the maximum likelihood (ML) method and the
Bayesian method were used to calculate the parameter uncertainty, and the results were
compared. For the Bayesian method, the calculations were performed by the MH
algorithm. The Bayesian method was used to calculate confidence intervals for
simulated streamflows. To account for heteroscedastic residuals, i.e. dependence of
error variance on computed runoff, a square-root transformation on both computed and
observed runoff data was used. The resulting model error was assumed to be a white
noise process with Gaussian distribution. The later hypothesis was then tested with a
cross-basin validation approach. The robustness of the model performance and the
confidence intervals were also tested by cross-validation. The study was performed
using the WASMOD model (Xu, 2002) on 25 basins in central Sweden.
The paper is organized as follows: after this brief introduction, the study area, data
and the WASMOD model are described. The subsequent section outlines the Bayesian
method and the specific consideration in using the method in this study, which
includes the choice of likelihood function, the prior and proposal densities, the MH
simulation procedure, etc. Next, the application results including parameter estimates
and resulted streamflow simulation uncertainties are presented, then the results are
Copyright  2005 IAHS Press
48
Kolbjørn Engeland et al.
discussed and, finally, the conclusions and proposal for further investigations are
presented.
THE STUDY AREA AND THE HYDROLOGICAL MODEL
The study area
The study area is located in central Sweden (Fig. 1). The area has 30 sub-catchments
ranging in size from 6 to 4000 km2. Available meteorological data consist of daily
precipitation from 41 stations, daily temperature data from 12 stations and all with an
observation period of at least 10 years. Twenty-five of the 30 sub-catchments were used in
this study, since earlier investigations (e.g. Xu, 2003) showed that some sub-catchments
might have errors in the determination of water divides. The mean annual precipitation
and discharge are 800 and 310 mm, respectively. The land use of the area includes
5.7% lake (not including Lake Mälaren itself), 69.3% forest and 25% agricultural land.
The hydrological and land-use data of the study region are shown in Table 1.
SO
1.NOPEX region
2. Study region
Fig. 1 The study area and basins (see Table 1 for abbreviations of basins).
The WASMOD model
The monthly water balance model WASMOD was developed for water balance
computation for the NOPEX region (Xu et al., 1996; Xu, 2002). The model parameters
are related to the physical characteristics of the basins (Xu 1999; Müller-Wohlfeil et
al., 2003). The input data for using the model on gauged basins are monthly areal
precipitation, potential evapotranspiration and/or air temperature. To use the model on
ungauged basins, the land-use and/or soil distribution data are needed. The model
outputs are monthly river flow and other water balance components, such as actual
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
49
Table 1 General information of the 25 study basins in central Sweden.
Station
Abbr.
Code
Area
Mean
Mean*
precip. evap.
(km2) (cm)
(cm)
Åkesta Kv.
AK
2216 727
60.1
39.6
Åkers Krut. AR
2249 214
60.3
43.3
Bergsh.
BE
2300 21.6
55.6
40.2
Berg
BG
2218 36.5
63.9
43.0
Bernsh.
BS
1573 595
78.0
43.4
Dalkarlsh.
DL
2206 1182
76.4
42.4
Fellingsbr.
FB
2205 298
62.6
39.9
Finntorp.
FT
2242 6.96
65.9
43.9
Gränvad
GR
2217 167
59.4
40.9
Härnevi
HA
2248 312
60.2
38.1
Hammarby
HB
2153 891
73.3
43.1
Kåfalla
KF
1532 413
81.0
44.3
Kringlan
KL
2229 294
78.3
44.3
Karlslund
KS
2139 1293
69.7
43.4
Lurbob
LU
2245 122
60.8
36.3
Odensvibr.
OB
2221 110
63.6
41.7
Ransta
RA
2247 197
59.8
38.2
Rällsälv
RS
2207 298
79.3
43.1
Sävja
SA
2243 722
59.7
40.4
Skräddart.
SD
2222 17.7
66.7
41.6
Sörsätra
SO
2220 612
60.8
33.5
Stabbyb
ST
1742 6.18
56.4
36.2
Tärnsjöb
TA
2299 13.7
59.7
39.5
Ulva Kv.
UL
2246 976
61.2
43.9
Vattholma
VA
2244 293
60.6
40.8
* Actual evapotranspiration calculated by the model.
Mean
runoff
(cm)
21.6
17.6
16.3
22.2
34.9
35.2
24.6
22.1
19.8
23.3
30.9
36.9
34.2
27.0
25.2
23.3
22.3
38.4
19.5
25.3
27.3
18.7
21.8
17.1
21.0
Lake
Forest
Open
(%)
4.0
5.2
0.2
0.0
8.6
7.5
6.0
4.7
0.0
1.0
9.5
6.2
7.6
6.6
0.3
6.3
0.9
7.4
2.0
2.5
1.1
0.0
1.5
3.0
4.8
(%)
69.0
66.3
69.5
71.4
77.3
74.6
63.8
95.3
41.1
55.0
80.9
80.8
87.2
62.7
68.2
71.0
66.1
78.8
64.0
96.1
61.0
95.6
84.5
61.0
71.0
(%)
27.0
28.5
30.3
28.6
14.1
17.9
30.2
0.0
58.9
44.0
9.7
13.0
5.2
30.7
31.5
22.7
33.0
13.8
34.0
1.4
37.9
4.4
14.0
36.0
24.2
Basin
slope
3.3
2.7
3.4
4.0
3.5
1.5
10.3
7.1
1.6
1.8
Table 2 Principal equations for the six-parameter WASMOD model.
Snowfall
st = pt{1 – exp[–(ct – a)/(a1 – a2)]2}+
a1 ≥ a2
Rainfall
rt = pt – st
Snow storage
spt = spt-1 + st – mt
Snowmelt
mt = spt{1 – exp[–(ct – a2)/(a1 – a2)]2}+
Potential evapotranspiration
ept = [1 + a3(ct – cm)]epm
Actual evapotranspiration
et = min{wt[1 – exp(–a4ept)], ept}
0 ≤ a4 ≤ 1
Slow flow
bt = a5(sm+t-1)2
a5 ≥ 0
Fast flow
ft = a6(sm+t-1)2(mt + nt)
a6 ≥ 0
Water balance
smt = smt-1 + rt + mt – et – bt – ft
wt = rt + sm+t-1 is the available water; sm+t-1 is the available storage; nt = rt – ept(1 – exp(rt/ept)) is the
active rainfall; pt and ct are monthly precipitation and air temperature respectively; and epm and cm are
long-term monthly averages.
ai (i = 1, …, 6) are the model parameters.
The superscript plus means x+ = max(x,0).
evapotranspiration, slow and fast components of river flow, soil-moisture storage and
accumulation of snowpack, etc. The primary equations of the model are presented in
Table 2, and a schematic computational flow chart is shown in Fig. 2. The parameters
a1 and a2 determine the phase of precipitation and the rate of snowmelt. Parameter a3
Copyright  2005 IAHS Press
50
Kolbjørn Engeland et al.
Fig. 2 The schematic computational flow chart of the WASMOD model.
converts long-term average monthly potential evapotranspiration to actual values of
monthly potential evapotranspiration. The potential evapotranspiration is then transferred to the actual one as a function of soil moisture storage, controlled by the
parameter a4. An increase in a4 will increase the actual evapotranspiration. The slow
flow parameter a5 controls the proportion of runoff that appears as “baseflow”,
whereas a6 controls the fast runoff.
THE BAYESIAN METHODOLOGY AND PROCEDURE
The choice of likelihood function
In a Bayesian context, the model parameters are regarded as random variables, and
their distributions are estimated. Bayes’ theorem gives the posterior density π of the
model parameters θ and some statistical parameters φ conditioned on the streamflow
observations Qobs:
π (θ , φ q obs ) = f (q obs q(θ )sim , φ ) p(θ ) p(φ )
(1)
where qobs = T(Qobs) and q(θ)sim = T(Q(θ)sim), where T is a transformation to obtain
homoscedastic simulation errors; Q(θ)sim is streamflow simulated by the hydrological
model utilizing the hydrological parameters θ; φ are some statistical parameters
describing the simulation errors; p(θ) and p(φ) are prior densities for the parameters;
and f is a probability density for the observed streamflows conditioned on the
parameters. When the data Qobs are given, f is called the likelihood function,
L(q(θ)sim, φ|qobs). The likelihood function is proportional to the distribution of the
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
51
1/2
Residual (mm )
6
4
2
0
-2
-4
-6
0
2
4
6
8
10
12
1/2
Simulated streamflow (mm )
Fig. 3 Simulation errors as a function of simulated streamflow.
14
simulation errors and controls the distribution of the model parameters. In Xu et al.
(1996) and Xu (2001), the statistical properties of the simulation errors for the
WASMOD model applied to some of these 25 Swedish basins were carefully investigated. Based on this experience, the authors know it is necessary to transform the
variables to obtain homoscedastic residuals and square-root transformation is a good
choice. To verify this assumption, the simulation results presented are based on ML
optimization of model parameters. Ten years of simulation results from 25 basins were
used giving a total of 3000 data points. The simulation errors were divided by their
basin-specific standard deviations to obtain a homogeneous sample. This sample of
3000 data points indicates the average behaviour of all 25 basins. In Fig. 3, the
residuals are plotted against the square root of simulated streamflow, and the simulation errors seem to be homoscedastic. Furthermore, it was assumed that the normal
distribution truncated at zero is a good approximation:
(
)
f t q obs ,t θ, µ t , ω t = N 0 (q (θ )sim ,t + µ t , ω t )
é
æ q (θ )sim ,t + µ t
= ê1 − Φç −
ç
ωt
êë
è
öù
÷ú
÷ú
øû
−1
æ 1 (q obs ,t − q(θ )sim ,t − µ t )2 ö
÷
expç −
ç 2
÷
ω
2 πω t
t
è
ø
1
é
æ q(θ )sim ,t + µ t
= Lt (θ, µ t , ω t qobs ,t ) = ê1 − Φç −
ç
ωt
êë
è
öù
÷ú
÷ú
øû
−1
(2)
æ 1 (ε (θ )t − µ t )2 ö
÷
expç −
÷
ç 2
ω
2 πω t
t
ø
è
1
where the statistical parameters φ are given by µt and ωt; t is a time index; Φ is the
standard cumulative normal distribution; q obs ,t = Qobs ,t , q (θ )sim ,t = Q(θ )sim ,t and
ε t (θ ) = Qobs ,t − Q(θ )sim ,t is the simulation error of the hydrological model. The
normal distribution was truncated at zero since the square root of the observed
streamflow cannot be negative. This truncation has an influence only when the
simulated streamflow is very small. In Fig. 4, the empirical quantiles are plotted
against normal quantiles. In this figure, the standardized simulation errors from all 25
basins are pooled and treated as one sample. The normal distribution is a good
approximation for quantiles less than 2, which means that for 95% confidence intervals
and less, a good approximation will be obtained. The highest quantiles, however, are
Copyright  2005 IAHS Press
52
Kolbjørn Engeland et al.
6
Normal quantiles
4
2
0
-2
-4
-4
-2
0
2
4
6
Empirical quantiles
Fig. 4 Empirical quantiles of the simulation errors vs the normal quantiles.
Auto-correlation
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29
Lag
Fig. 5 Autocorrelation functions for all 25 basins.
underestimated by the normal distribution. This was expected to have minor influence
on the results presented herein.
It was assumed that all the simulation errors were independent, and the likelihood
obtained was conditioned on the data qsim,1,…, qsim,T by multiplying Lt in equation (2)
over all time steps:
é
æ q(θ )sim ,t + µ t
L(θ ,φ q obs ) ∝ ∏ ê1 − Φ ç −
ç
ωt
t =1 ê
è
ë
T
öù
÷ú
÷ú
øû
−1
æ (ε (θ ) − µ t )2 ö
÷
expçç t
÷
2
ω
ωt
t
è
ø
1
(3)
In Fig. 5, the auto-correlations are plotted as a function of time-lag for all 25 basins.
The lag-one autocorrelation is insignificant for 12 of the basins. For the remaining 13
basins, the lag-one autocorrelation is relatively small (less than 0.5 for all basins). The
assumption about independent simulation errors was therefore reasonable. Finally a
decision had to be made on whether the mean and variance (µ and ω in equation (3))
are time dependent or not. The mean value µ was required to be zero since the hydrological model was punished for bias in the simulations. The variance ω might depend
on the hydrological processes that take place. Therefore values were inspected, both
visually and by performing the MH simulations with different error models, to
ascertain whether the variance of the residuals depends on the observed temperature,
precipitation and evapotranspiration. No significant structures of the simulation errors
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
53
were found. This is in accordance with previous studies (e.g. Xu, 2001). Therefore a
constant variance was chosen for the residuals. All the parameters including ω, were
estimated independently for each catchment.
The prior densities
Since the variance has to be positive, flat priors above 0 were used for ω
( p(ω ) ~ Uniform(0, ∞ ) ). For the WASMOD model parameters, uniform priors
between the limits listed in Table 3 were applied. The parameter ranges shown in the
table are consistent with the constraints given in Table 2, but in a practical situation the
ranges are much smaller.
Table 3 The prior intervals for the WASMOD parameters.
Parameter
a1
a2
a3
a4
a5
a6
Uniform prior interval
a2
–∞
–∞
0.00
0.00
0.00
∞
a1
∞
0.02
∞
∞
The MCMC simulations
The Metropolis Hastings (MH) algorithm (Hastings, 1970), a Markov Chain Monte
Carlo (MCMC) methodology, was used to estimate the parameters. The MH algorithm
generates a Markov chain that converges to the distribution of interest. After removing
the initial “burn-in”, this chain might be used as a dependent sample from the
distribution. In this implementation of the MH algorithm, the parameters were updated
for each iteration in a random order. The vector of all parameters (here a1–a6 and ω), is
denoted by x, their posterior densities by π, their proposal densities by d, the number
of iterations by m, and the number of parameters by n (here n = 7). The algorithm
involves the following steps:
– For L = 1, 2, ..., m, let x(L) be the current state of the chain.
– Draw I randomly from 1, 2, ..., n (only once for each iteration; this is actually a
random permutation).
– Draw a new value for xI from a specified irreducible proposal distribution d:
(
)
x I* ~ d x ( L ) , x * where x *j = x (jL )
–
∀j ≠ I
Compute the acceptance probability:
(
ax
( L)
( )[
( )[
]
]
ì π x * d x * , x (L ) ü
, x = min í1,
( L)
(L) * ý
î π x d x ,x þ
*
)
ìx*
let x I( L +1) = í I( L )
îxI
(
(4)
(5)
)
with probability a x ( L ) , x *
with probability 1 − a x ( L ) , x *
(
)
(6)
Copyright  2005 IAHS Press
54
Kolbjørn Engeland et al.
As the ratio of two posteriors densities is calculated in equation (5), this algorithm
does not require the normalization constant of the posterior probability density.
The proposal densities
A random walk proposal density (equation (4)) was used, i.e. the proposal distribution
was dependent on the current state of the chain. For the ω parameter, uniform proposal
centred at the current parameter value was used. However, no negative values were
proposed because the acceptance probability then will always be zero.
[ (
)
(
d ω ~ Uniform max 0, ω ( L ) − s ω , max ω ( L ) + s ω ,2 s ω
)]
(7)
For the WASMOD parameters, uniform proposals with the limitations specified in
Table 3 were used.
The acceptance probabilities
For all parameters, the uniform proposal densities were constructed to have a constant
variance. As a result, d x ( L ) , x * = d x * , x ( L ) , and the acceptance probability (equation
(5)) could be simplified to calculate only the ratio of the new and old posteriors:
(
) (
( )
( )
ì π x* ü
a x ( L ) , x * = min í1,
(t ) ý
î πx þ
(
)
)
(8)
Tuning the MH algorithm
In order to obtain a chain that converged fast, the amplitude of the uniform proposal
densities was tuned. If the variances of the proposals are too large, only a few of the
new parameters will be accepted, thus the chain will easily get stuck in one point. If
the variance is too small, the chain will need many iterations to explore the entire
parameter space. According to Chib & Greenberg (1995), the acceptance rate for a new
point in the chain should be about 45%. The amplitudes were tuned to have an
acceptance rate between 40 and 50% for each parameter.
Convergence of the chain
The convergence of the chain was studied by looking into the lagged auto-covariances
for each parameter. For a chain that has converged, the auto-correlations decrease
rapidly towards zero. The convergence was also visually inspected by plotting the
chains as functions of iteration.
Confidence intervals for streamflow
To calculate confidence intervals for streamflow, 100 000 streamflow values for each
month were obtained by running the model with all the 100 000 parameter sets from
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
55
the MH sample. The 95% confidence intervals due to parameter uncertainty were
estimated from these streamflow samples. The 95% confidence intervals for both the
parameter uncertainty and the model structure uncertainty were calculated by adding
the model residuals in the form of a random uncertainty with zero mean and sample
dependent standard deviation ωi to each of the 100 000 streamflow values that were
available for each time step:
(
Qt ,i ,samp = æç Qt ,i ,sim (θ i ) + rnorm 0 , ω i
è
)
− Qt ,i ,sim (θi ) ωi
ö÷
ø
2
(9)
where t is a time index, i is a sample index, samp is an index for streamflow sample
with the random term added, sim is an index for streamflow calculated from the
WASMOD model, and rnorm is a random number from a truncated normal
distribution with the specified parameters. The truncation assured that no negative
values were simulated.
MAXIMUM LIKELIHOOD ESTIMATION
To provide a comparative result with the MH algorithm, ML estimation method was
also performed in the study. Maximum likelihood estimation was carried out by
maximizing equation (3) without the truncation term, zero mean and constant variance.
The optimization was performed with the help of the VA05A computer package
(Hopper, 1978; Vandewiele et al., 1992) and the program EOX4F (NAG Fortran
Subroutine Library, 1981). In the ML method, it is assumed that the parameters are
normally distributed. Based on this assumption the covariance matrix is estimated.
Derivation of covariance matrix for the model parameters is shown in Vandewiele et
al. (1992).
CROSS-VALIDATION
Three cross-validation tests were carried out to test the quality of at-site simulations. In
all tests, blocks of data from two years were successively left out in the parameter
estimation. Then the streamflow and the confidence intervals were calculated for the
independent data. The cross-validated Nash-Sutcliffe coefficient Reff and the crossvalidated bias were calculated using all the independent simulations. To test the quality
of the estimated confidence intervals, the number of observed streamflow values that
fall outside the confidence intervals for the independent simulations were counted.
RESULTS
Convergence of the MCMC routine
The parameters were estimated independently for each basin, thus a total of 25 chains
with 110 000 iterations were calculated by the MH algorithm. The first 10 000
iterations in each chain, the “burn-in”, were discarded and the remaining 100 000
iterations were used as samples from the distributions. Visual inspection of the
trajectories of the chains indicated that the MH algorithm had converged. They seemed
Copyright  2005 IAHS Press
56
Kolbjørn Engeland et al.
stationary and no obvious trends or periodic variations could be observed. The autocorrelations of each chain were insignificant after about 40 iterations. It was therefore
concluded that all the 25 chains had converged, and the samples were used for
inference about the parameters.
The a4 parameter required special attention. This parameter controls the
relationship between potential and actual evapotranspiration. When this parameter is
large enough, the actual evapotranspiration equals the potential one. A further increase
in the value does not change the likelihood. It was therefore difficult to obtain
convergence for four of the wettest basins (Finntorp, Kåfalla, Kringlan and Rällsälv).
Based on previous experience with the model, a very strict prior [0.0–0.02] was put on
this parameter so that the MH algorithm converged.
Parameter and streamflow estimates
The estimated parameters for all 25 basins are shown as box-plots in Fig. 6, and the
estimated variances of the residuals are shown in Fig. 7. Figure 8 shows the marginal
posterior parameter densities calculated by the MH algorithm (based on histogram
values) together with the ML estimates of the density for the station Åkesta Kvärn,
12
a1
10
0
10
a2
-28
8
-46
6
-64
4
0
0
-10
a3
0.2
0.010
0.1
0.005
0
0.000
0.0006
0.0005
a5
AK
AR
BE
BG
BS
DL
FB
FT
GR
HA
HB
KF
KL
KS
LU
OB
RA
RS
SA
SD
SO
ST
TA
UL
VA
0.015
AK
AR
BE
BG
BS
DL
FB
FT
GR
HA
HB
KF
KL
KS
LU
OB
RA
RS
SA
SD
SO
ST
TA
UL
VA
0.3
0.0007
a4
0.020
0.00005
a6
0.00004
0.00003
0.0004
0.00002
0.0002
0.0001
0.00001
0
0
AK
AR
BE
BG
BS
DL
FB
FT
GR
HA
HB
KF
KL
KS
LU
OB
RA
RS
SA
SD
SO
ST
TA
UL
VA
0.0003
AK
AR
BE
BG
BS
DL
FB
FT
GR
HA
HB
KF
KL
KS
LU
OB
RA
RS
SA
SD
SO
ST
TA
UL
VA
0.4
AK
AR
BE
BG
BS
DL
FB
FT
GR
HA
HB
KF
KL
KS
LU
OB
RA
RS
SA
SD
SO
ST
TA
UL
VA
-82
AK
AR
BE
BG
BS
DL
FB
FT
GR
HA
HB
KF
KL
KS
LU
OB
RA
RS
SA
SD
SO
ST
TA
UL
VA
2
Fig. 6 Estimated parameters for all 25 basins.
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
57
ω
2.5
Var
2.0
1.5
1.0
0.5
AK
AR
BE
BG
BS
DL
FB
FT
GR
HA
HB
KF
KL
KS
LU
OB
RA
RS
SA
SD
SO
ST
TA
UL
VA
0.0
1
10
100
1000
Catchment area (km2 )
10000
Fig. 7 (a) Estimated variance for all 25 basins, and (b) estimated variance vs the size
of the catchments.
1.2
Density
MH
1
ML
0.8
0.6
0.4
0.2
a1
0
1.55
20
2.32
3.08
3.85
4.62
Density
ML
10
5
a3
0
0.02
(a)10000
0.07
0.12
0.16
0.21
Density
8000
Density
700
600
500
400
300
200
100
0
MH
ML
a2
-2.81
MH
15
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2.21
-1.62
-1.02
-0.42
Density
MH
ML
a4
4.13E-03
5.45E-03
6.78E-03
8.10E-03
MH
(b) Density
250000
MH
ML
200000
ML
6000
150000
4000
100000
2000
50000
a5
0
1.15E-04 2.15E-04 3.15E-04 4.15E-04
a6
0
5.27E-06 8.52E-06 1.18E-05 1.50E-05
Fig. 8 The posterior marginal densities of the WASMOD parameters a1–a6 estimated
from the MH sample (bars) and the conditional densities calculated from the ML
method (lines) for the station Åkesta Kvärn.
which is located in the middle of the study area. For the ML densities, conditional
distributions are shown (conditioned on the ML estimates). The marginal distributions
will be somewhat different due to the correlation between the parameters. Table 4 lists
the correlation between the parameters as estimated from the MH sample, and by the
ML method for the same station. For comparative purposes, the observed monthly
Copyright  2005 IAHS Press
58
Kolbjørn Engeland et al.
Table 4 Correlation between the parameters estimated by the Metropolis Hastings (MH) and maximum
likelihood (ML) methods.
a1
a2
a3
a4
a5
a6
a1
MH
1.00
–0.49
–0.04
–0.05
–0.36
0.10
ML
1.00
–0.62
–0.02
–0.04
–0.43
0.16
a2
MH
–0.49
1.00
0.02
0.05
0.35
0.03
ML
–0.62
1.00
–0.00
0.04
0.46
0.00
a3
MH
–0.04
0.02
1.00
–0.06
0.01
–0.08
a4
MH
–0.05
0.05
–0.06
1.00
0.36
0.43
ML
–0.02
–0.00
1.00
–0.11
–0.02
–0.11
a5
MH
–0.36
0.35
0.01
0.36
1.00
0.42
ML
–0.43
0.46
–0.02
0.37
1.00
0.46
a)
Q(mm)
160
ML
–0.04
0.04
–0.11
1.00
0.37
0.49
a6
MH
0.10
0.03
–0.08
0.43
0.42
1.00
Qobs
ML
0.16
0.00
–0.11
0.49
0.46
1.00
Qsim
120
80
40
0
1
13
25
37
49
Q(mm)
160
61
Month
73
85
97
b)
109
Qobs
Qsim
120
80
40
0
1
13
25
37
49
61
Month
73
85
97
109
Fig. 9 95% confidence intervals for simulated streamflow due to (a) parameter
uncertainty and (b) total uncertainty, for the Stabbybäcken basin.
Q(mm)
100
80
a)
Qobs
Qsim
60
40
20
0
1
100
80
13
25
37
49
Q(mm)
61
Month
73
85
97
b)
109
Qobs
Qsim
60
40
20
0
1
13
25
37
49
61
Month
73
85
97
109
Fig. 10 95% confidence intervals for simulated streamflow due to (a) parameter
uncertainty and (b) total uncertainty, for the Ulva Kvarndamn basin.
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
59
runoff together with 95% confidence intervals including only the parameter uncertainty and the total uncertainty are plotted for two catchments with different residual
variances. Figure 9 shows the result for the Stabbybäcken basin, which has a large
residual variance and Fig. 10 illustrates the result for the Ulva Kvarndam basin that has
a small residual variance.
The observed long-term mean monthly runoff together with 95% confidence
intervals including only the parameter uncertainty and the total uncertainty for the
above mentioned two catchments are shown in Figs 11 and 12, respectively. The 95%
confidence intervals are obtained from a sample of 100 000 simulated monthly
streamflow series for which monthly averages were calculated.
a)
Q(mm)
80
60
Q(mm)
80
Qobs
Qsim
b)
Qobs
Qsim
60
40
40
20
20
0
0
1 2 3 4 5 6 7 8 9 10 11 12
Month
1 2 3 4 5 6 7 8 9 10 11 12
Month
Fig. 11 95% confidence intervals for average monthly simulated streamflow due to
(a) parameter uncertainty and (b) total uncertainty, for the Stabbybäcken basin.
Q(mm)
60
Qobs
Qsim
a)
40
60
Q(mm)
b)
Qobs
Qsim
40
20
20
0
0
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Month
Month
Fig. 12 95% confidence intervals for average monthly simulated streamflow due to
(a) parameter uncertainty and (b) total uncertainty, for the Ulva Kvarndamn basin.
Cross-validation
1
1
0.8
0.8
F(Bias)
F(Ref f )
For cross-validation purposes, the cross-validated Nash-Sutcliffe coefficient Reff and
cross-validated bias are shown in Fig. 13. The Reff is larger than 0.6 for all basins and
larger than 0.8 for 16 of them. The bias is smaller than 5% for all basins. The cross-
0.6
0.4
0.2
0.6
0.4
0.2
0
0
0.6
0.7
0.8
Ref f
0.9
1
-0.05
0
Bias
0.05
Fig. 13 (a) Cross-validated Reff and (b) bias for all 25 basins.
Copyright  2005 IAHS Press
60
Kolbjørn Engeland et al.
validated confidence intervals show that 42% of the observed streamflow values falls
inside the 95% confidence intervals when only the parameter uncertainty is taken into
account. When the simulation residuals are added, however, 95% of the observed
values were inside the confidence intervals. These cross-validation tests indicate that in
at-site applications WASMOD gives robust estimates of monthly water balance, longterm average water balance and confidence intervals.
DISCUSSION
Comparison of MH and ML estimates
The estimates from the MH routine indicate that the parameter densities are only
slightly skewed (Fig. 6). The standard deviations estimated by the MH and ML
methods are almost identical (Fig. 8). This shows that the shape of the likelihood
function at the optimum value and the normal assumption in the ML method are
reasonable approximations. The estimated densities are only slightly different. The two
methods indicate similar patterns for the correlations, but the MH-estimated
correlations are somewhat lower than those obtained from the ML method (Table 4).
Similar results are also found for the other 24 stations. All these results indicate that
the ML estimates give a good approximation to the parameter density for the monthly
WASMOD model applied in Sweden. For models operating on daily time steps, more
complex parameter distributions are reported (e.g. Duan et al., 1992; Kuczera &
Parent, 1998; Engeland & Gottschalk, 2002; Vrugt et al., 2003), and the normal
distribution will not necessarily be a good approximation.
Parameter estimates
Both the median values and the variance of the WASMOD parameters depend on the
basins (Fig. 6). The sign of the skewness is identical for all basins. The a1 and a2
parameters that control the snow cover formation and snowmelt processes are basindependent. The a3 and a4 parameters that control the evapotranspiration have the
smallest variation between basins. The Sörsätra basin has an exceptionally high a3
parameter. This basin has a high mean runoff compared to its neighbours (Åkesta
Kvärn, Gränvad and Härnevi, see Table 1), and should be treated carefully in a
regionalization context. The parameters that control the runoff, a5 and a6, show the
largest variation between basins. In particular, the a5 parameter is significantly
different between several basins. Xu (1999) estimated relationships between the model
parameters and basin characteristics for this dataset using multiple linear regression
analysis. He found that a1 is positively correlated to the lake and the forest percentages
whereas the a2 is negatively related to the lake and open field percentages. No
correlations were found between the basin characteristics and the parameters a3 and a4.
The slow flow parameter a5 was positively related to the lake and forest percentages,
whereas the fast flow parameter a6 was negatively related to lake percentage and
positively related to forest percentage. The study not only verified the physical
interpretations about the model parameters, but also established quantitative parameter
estimation equations in the study area. Similar regression equations between the
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
61
WASMOD model parameters and catchment physical characteristics were established
for a Danish catchment by Müller-Wohlfeil et al. (2003).
The success of such a regionalization procedure depends also on the quality of the
input data and the observed streamflow data used for calibration as biased data might
introduce biased parameters (Andréassian et al., 2001).
Uncertainties in streamflow simulations
The uncertainties in the model parameters cannot account for all the uncertainties in
the simulations. About 45% of the observations were inside these confidence intervals
(Figs 9(a) and 10(a)). When the total uncertainty was included, the confidence
intervals got much wider (Figs 9(b) and 10(b)). This indicates that the parameter
uncertainty is less important than the uncertainties caused by the model structure. The
WASMOD model simulated the average intra-annual variations well (Figs 11 and 12).
The parameter uncertainty is getting more important compared to the results in Figs 9
and 10. This is due to the way these two error sources were represented. The
parameters were assumed to be constant in time and they were therefore selected
randomly from their distribution for the whole simulation period. However, the
simulation error part was assumed to be independent between the months, and it was
therefore selected randomly for each month in the simulation period. This part of the
uncertainty will therefore decrease by a factor of 1 n , where n is the number of
years used for averaging.
There are several indications that the variance of the simulation errors was caused
not only by an imperfect model structure, but also by uncertainties in the observed
data. Based on the results presented in this study it is not possible to tell to what extent
errors in the observations contributed to the total simulation errors. Lebel et al. (1987)
showed that the estimation variance of interpolated area rainfall decreases with
increasing basin size. A similar pattern is seen in Fig. 7(b): the residual variance
decreases with increasing basin size (streamflow values in mm were used here for all
calculations, so the direct effect of basin size on variance of the residuals should have
been removed). It may also be seen that the high residual variance for the
Stabbybäcken basin was caused by one flood-event that is not captured by the model
(Fig. 9(a)). This is the smallest basin with an area of 6.18 km2 and there are no
meteorological observations within the basin borders. The quality of the interpolated
precipitation in this basin was therefore very sensitive to local rainfall events. The
quality of the observed streamflow data might also have influenced the simulation
errors. The variance marked as a filled square in Fig. 7(b) represents the Sörsätra basin,
which has the highest variance for basins with an area larger than 100 km2. The
observed streamflow from this basin might be doubtful (see subsection “Parameter
estimates”).
CONCLUSIONS
The WASMOD model was applied to 25 basins in central Sweden. A Bayesian
methodology combined with the MH algorithm was used to estimate the model
Copyright  2005 IAHS Press
62
Kolbjørn Engeland et al.
parameters and their uncertainty. The Bayesian estimates were compared to the ML
estimates. This study allows one to draw some conclusions that will be important for
further investigations, i.e. in estimating regional parameters.
All the assumptions in the application of the Bayesian method have been tested.
The investigation of the simulation residuals showed that the square-root transformation is sufficient to obtain homoscedastic residuals. The residuals are approximately
normally distributed, the variance of the residuals is as good as independent of the
climatic conditions and the residuals are nearly independent. These results are only
valid for the 25 basins that were used in this study and they have to be checked for
each new application.
The authors were able to identify parameter densities for all 25 basins and the ML
estimation was shown to give a reasonable approximation to the distribution of the
model parameters as the MH algorithm and ML method gave almost identical
estimates of the parameter uncertainty.
Ninety-five percent confidence intervals were calculated around the simulated
streamflows. The confidence intervals around the simulated streamflows indicate that
parameter uncertainty is less important than the other uncertainty sources in the
streamflow estimations. It is anticipated that this conclusion is valid for simple conceptual models with few well-defined parameters. The cross-validation tests showed
that water balance calculations and the confidence intervals are reliable.
Most of the model parameters depended on the basin. Further work will be to carry
out a full regionalization of the model parameters by relating them to basin characteristics. Bayesian hierarchical modelling (Gelman et al., 1995) then offers suitable
tools for such estimation.
Another topic for further investigation is to include the uncertainties in meteorological data and in streamflow observations in the uncertainty assessment. This would
help one to understand how all the four error sources contribute in the total modelling
uncertainty.
REFERENCES
Alley, W. M. (1984) On the treatment of evapotranspiration, soil moisture accounting and aquifer recharge in monthly
water balance models. Water Resour. Res. 20(8), 1137–1149.
Andréassian, V., Perrin, C., Michel, C., Usart-Sanchez, I. & Lavabre, J. (2001) Impact of imperfect rainfall knowledge on
the efficiency and the parameters of watershed models. J. Hydrol. 250, 206–223.
Aronica, G., Bates, P. D. & Horritt, M. S. (2002) Assessing the uncertainty in distributed model predictions using observed
binary pattern information within GLUE. Hydrol. Processes 16(10), 2001–2016.
Beldring, S. (2002) Multi-criteria validation of a precipitation–runoff model. J. Hydrol. 257, 189–211.
Beven, K. J. & Binley, A. M. (1992) The future of distributed models—model calibration and uncertainty prediction.
Hydrol. Processes 15, 509–711.
Beven, K. J., Freer, J., Hankin, B. & Schulz, K. (2000) The use of generalised likelihood measures for uncertainty
estimation in high order models of environmental systems. In: Nonlinear and Nonstationary Signal Processing
(ed. by W. J. Fitzgerald, R. L. Smith, A. T. Walden & P. C. Young), 115–151. Cambridge Univ. Press, Cambridge, UK.
Chib, S. & Greenberg, E. (1995) Understanding the Metropolis-Hasting Algorithm. The American Statistician 49(4), 327–
335.
Clarke, R. T. (1999) Uncertainty in the estimation of mean annual flood due to rating-curve indefinition. J. Hydrol. 222,
185–190.
Clarke, R. T., Mendiondo, E. M. & Brusa, L. C. (2000). Uncertainties in mean discharges from two large South American
rivers due to rating curve variability. Hydrol. Sci. J. 45(2), 221–236.
Duan, Q., Sorooshian, S. & Gupta, V. (1992). Effective and efficient global optimisation for conceptual rainfall–runoff
models. Water Resour. Res. 28(4), 1015–1031.
Engeland, K. & Gottschalk, L. (2002) Bayesian estimation of parameters in a regional hydrological model. Hydrol. Earth
System Sci. 6(5), 883–898.
Copyright  2005 IAHS Press
Assessing uncertainties in a conceptual water balance model using Bayesian methodology
63
Førland, E. J., Allerup, P., Dahlström, B., Elomaa, E., Jonsson, T., Madsen, H., Perälä, J., Rissanen, P., Vedin, H.,
Vedin, H. & Vejen, F. (1996) Manual for Operational Correction of Nordic Precipitation Data. DNMI Klima
Report no. 24/96, Norwegian Meteorological Institute, Oslo, Norway.
Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. (1995) Bayesian Data Analysis. Chapman & Hall, London, UK.
Geyer, C. J. (1992) Practical Markov Chain Monte Carlo. Statist. Sci. 7(4), 473–483.
Gottschalk, L., Jensen, J. L., Lundquist, D., Solantie, R. & Tollan, A. (1979) Hydrologic regions in the Nordic countries.
Nordic Hydrol. 10, 273–286.
Hankin, B. G., Hardy R., Kettle, H. & Beven, K. J. (2001) Using CFD in a GLUE framework to model the flow and
dispersion characteristics of a natural fluvial dead zone. Earth Surf. Processes Landf. 26(6), 667–687.
Hastings, W. K. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrica 57, 97–109.
Hopper, M. J. (1978) A Catalogue of Subroutines. Harwell Subroutines Library, HMSO, London, UK.
Jónsson, P., Petersen-Øverleir, A., Nilsson, E., Edström, M., Iversen, H. L. & Sirvio, H. (2002). Methodological and
personal uncertainties in the establishment of rating curves. In: XXII Nordic Hydrological Conference (ed. by
Ǻ. Killingtveit) (Nordic Association for Hydrology, Røros, Norway, 4–7 August 2002). NHP Report no. 47, vol. 1,
35–44.
Krzysztofowicz, R. (1999) Bayesian theory of probabilistic forecasting via deterministic hydrologic model. Water Resour.
Res. 35(9), 2739–2750.
Kuczera, G. (1983) Improved parameter inference in catchment models 1. Evaluating parameter uncertainty. Water
Resour. Res. 19(5), 1151–1162.
Kuczera, G. (1996) Correlated rating curve in flood frequency inference. Water Resour. Res. 32(7), 2119–2127.
Kuczera, G. & Parent, E. (1998) Monte Carlo assessment of parameter uncertainty in conceptual catchment models: the
Metropolis algorithm. J. Hydrol. 211, 69–85.
Lebel, T. Bastin, G., Obled, C. & Creutin, D. (1987) On the accuracy of areal rainfall estimation: a case study. Water
Resour. Res. 23(11), 2123–2134.
Madsen, H. (2000) Automatic calibration of a conceptual rainfall–runoff model using multiple objectives. J. Hydrol. 235,
276–288.
Mein, R. G. & Brown, B. M. (1978) Sensitivity of optimized parameters in watershed models. Water Resour. Res. 14,
299–303.
Müller-Wohlfeil, D. I., Xu, C.-Y. & Iversen, H. L. (2003). Estimation of monthly river discharge from Danish catchments.
Nordic Hydrol. 34(4), 295–320.
NAG Fortran Subroutine Library (1981) E04XAF NAG Fortran Library Routine Document, NP1490/13.
Refsgaard, J. C. & Storm, B. (1996) Construction, calibration and validation of hydrological models. In: Distributed
Hydrological Modelling (ed. by M. B. Abbott & J. C. Refsgaard), 41–54. Water Science and Technology Library,
vol. 22, Kluwer Academic Publishers, Dordrecht, The Netherlands.
Refsgaard, J. C., Rosbjerg, D. & Markussen, L. M. (1983), Application of Kalman filter to real-time operation and to
uncertainty analyses in hydrological modelling. In: Scientific Procedures Applied to the Planning, Design and
Management of Water Resource Systems (ed. by E. Plate & N. Buras), 273–282. IAHS Publ. 147. IAHS Press,
Wallingford, UK.
Sorooshian, S. & Dracup, J. A. (1980) Stochastic parameter estimation procedures for hydrologic rainfall–runoff models:
correlated and heteroscedastic error cases. Water Resour. Res. 16(2), 430–442.
Spear, R. C., Grieb , T. M. & Shang, N. (1994) Parameter uncertainty interaction in complex environmental models. Water
Resour. Res. 30(11), 3159–3169.
Storm, B., Jensen, K. H. & Refsgaard, J. C. (1988) Estimation of catchment rainfall uncertainty and its influence on runoff
predictions. Nordic Hydrol. 19, 77–88.
Thorsen, M., Refsgaard, J. C., Hansen, S., Pebesma, E., Jensen, J. B. & Kleeschulte, S. (2001) Assessment of uncertainty
in simulation of nitrate leaching to aquifers at catchment scale. J. Hydrol. 242, 210–227.
Vandewiele, G. L., Xu, C.-Y. & Ni-Lar-Win (1992) Methodology and comparative study on monthly water balance
models in Belgium, China and Burma. J. Hydrol. 134, 315–347.
Vrugt, J. A., Gupta, H. V., Bastidas, L. A., Bouten, W. & Sorooshian, S. (2003) A shuffled complex evolution Metropolis
algorithm for optimization and uncertainty assessment of hydrologic model parameters. Water Resour. Res. 39(8),
39, doi:10.1029/2002WR001642.
Wagener, T., McIntyre, N., Lees, M. J., Wheater, H. S. & Gupta, H. V. (2003) Towards reduced uncertainty in conceptual
rainfall–runoff modelling: dynamic identifiability analysis. Hydrol. Processes 17, 455–476.
Xu, C.-Y. (1999) Estimation of parameters of a conceptual water balance model for ungauged catchments. Water Resour.
Manage. 13(5), 353–368.
Xu, C.-Y. (2001) Statistical analysis of parameters and residuals of a conceptual water balance model—methodology and
case study. Water Resour. Manage. 15, 75–92.
Xu, C.-Y. (2002) WASMOD—The Water And Snow balance MODelling system. In: Mathematical Models of Small
Watershed Hydrology and Applications (ed. by V. P. Singh & D. K. Frevert), Ch. 17. Water Resources Publications,
LLC, Chelsea, Michigan, USA.
Xu, C.-Y., Seibert, J. & Halldin, S. (1996) Regional water balance modelling in the NOPEX area: development and
application of monthly water balance models. J. Hydrol. 180, 211–236.
Yapo, P. O., Gupta, H. V. & Sorooshian, S. (1998) Multi-objective global optimisation for hydrologic models. J. Hydrol.
204, 83–97.
Received 21 August 2003; accepted 5 November 2004
Copyright  2005 IAHS Press