HYDROLOGICAL PROCESSES
Hydrol. Process. 27, 304–312 (2013)
Published online 19 March 2012 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/hyp.9258
Uncertainty issues of a conceptual water balance model for a
semi-arid watershed in north-west of China
Zhanling Li,1* Quanxi Shao,2 Zongxue Xu3 and Chong-Yu Xu4,5
1
3
School of Water Resources and Environment, China University of Geosciences (Beijing), Beijing 100083, China
2
CSIRO, Mathematics, Informatics and Statistics, Private Bag 5, Wembley WA 6913, Australia
Key Laboratory of Water and Sediment Sciences, Ministry of Education, College of Water Sciences, Beijing Normal University, Beijing 100875, China
4
Department of Geosciences, University of Oslo, Oslo, Norway
5
Department of Earth Sciences, Uppsala University, Uppsala, Sweden
Abstract:
Hydrological models are useful tools for better understanding the hydrological processes and performing the hydrological
prediction. However, the reliability of the prediction depends largely on its uncertainty range. This study mainly focuses on
estimating model parameter uncertainty and quantifying the simulation uncertainties caused by sole model parameters and the
co-effects of model parameters and model structure in a lumped conceptual water balance model called WASMOD (Water And
Snow balance MODeling system). The validity of statistical hypotheses on residuals made in the model formation is tested as
well, as it is the base of parameter estimation and simulation uncertainty evaluation. The bootstrap method is employed to
examine the parameter uncertainty in the selected model. The Yingluoxia watershed at the upper reaches of the Heihe River basin
in north-west of China is selected as the study area. Results show that all parameters in the model can be regarded as normally
distributed based on their marginal distributions and the Kolmogorov–Smirnov test, although they appear slightly skewed for two
parameters. Their uncertainty ranges are different from each other. The model residuals are tested to be independent,
homoscedastic and normally distributed. Based on such valid hypotheses of model residuals, simulation uncertainties caused by
co-effects of model parameters and model structure can be evaluated effectively. It is found that the 95% and 99% confidence
intervals (CIs) of simulated discharge cover 42.7% and 52.4% of the observations when only parameter uncertainty is
considered, indicating that parameter uncertainty has a great effect on simulation uncertainty but still cannot be used to explain
all the simulation uncertainty in this study. The 95% and 99% CIs become wider, and the percentages of observations falling
inside such CIs become larger when co-effects of parameters and model structure are considered, indicating that simultaneous
consideration of both parameters and model structure uncertainties accounts sufficient contribution for model simulation
uncertainty. Copyright © 2012 John Wiley & Sons, Ltd.
KEY WORDS
uncertainty analysis; bootstrap; Heihe River basin; Yingluoxia watershed
Received 1 June 2011; Accepted 9 December 2011
INTRODUCTION
For the purpose of better representing the reality,
hydrological models evolve from simple conceptual
models to more complex physically based models through
the gradual introduction of more complicated and
comprehensive equations. Although the physically based
models are more popular in recent decades due to their
capability of serving a wide range of purposes besides
primarily estimating discharge from rainfall and the
meteorological information (Michaud and Sorooshian,
1993; Refsgaard and Knudsen, 1996; Carpenter and
Georgakakos, 2006), the selection of appropriate model
usually depends on the goal and specific requirements of
the study, data requirements, model parameters, model
structure and the user’s preference (Cunderlik and
Simonovic, 2007). For example, if the interest is primarily
in the discharge prediction, or only annual data in a short
period are available, then a lumped model would be a
*Correspondence to: Zhanling Li, School of Water Resources and
Environment, China University of Geosciences. E-mail: zhanling.li@cugb.
edu.cn
Copyright © 2012 John Wiley & Sons, Ltd.
suitable tool from the technical and economical points of
view and would also provide simulations as good as
complex physically based ones (Refsgaard and Knudsen,
1996; Beven, 2000).
Hydrological models, regardless of their types, are
subject to uncertainties. The assessment of uncertainty of
hydrological models is one of the key objectives in
hydrological modelling. Generally speaking, there are
four important sources of uncertainty in hydrological
modelling: uncertainties in input data (e.g. observed
precipitation and temperature data), in data used for
calibration (e.g. observed streamflow), in model parameters and in model structure, all of which have attracted
many attentions recently, especially the latter two, which
are more model specific (Beven and Binley, 1992;
Seibert, 1997; Bates and Campbell, 2001; Engeland
et al., 2005; Mugunthan and Shoemaker, 2006; Gallagher
and Doherty, 2007; Yang et al., 2007a,b). Many methods
have been used widely in parameter estimation and
parameter uncertainty analysis, including first-order
approximation (Vrugt and Bouten, 2002), generalized
likelihood uncertainty estimation (GLUE) (Beven and
Binley, 1992), sequential uncertainty fitting (SUFI)
UNCERTAINTY ISSUES OF A CONCEPTUAL WATER BALANCE MODEL
(Abbaspour et al., 1999, 2004), parameter solution
(ParaSol) (Van Griensven and Meixner, 2006), automatic
calibration and uncertainty assessment using response
surfaces method (ACUARS) (Mugunthan and Shoemaker,
2006), and the Bayesian method (Bates and Campbell,
2001; Engeland et al., 2005; Gallagher and Doherty, 2007;
Yang et al., 2007a,b; Jin et al., 2010; Li et al., 2010a).
In contrast to these methods, the non-parametric bootstrap
method, to the best of our knowledge, has seldom been
employed for the analysis of parameter uncertainty in
hydrological models. Li et al. (2010b) used the bootstrap
method to assess model uncertainty for a dynamic
hydrological model. Selle and Hannah (2010) used the
bootstrap method to assess parameter uncertainty in the
abc hydrological model. In comparison with the
methods mentioned above, the bootstrap method has many
significant advantages: Firstly, it is easy to describe
and implement in arbitrarily complicated situations,
particularly where complicated data structures are common, such as censoring and missing data as well as highly
multivariate situations. Secondly, no distribution assumptions such as normality are needed. Thirdly, no extensive
programming is needed, and the distribution of a statistic
can be easily found through intensive resampling with
replacement (Efron, 1979; Davison and Hinkley, 1997;
Abrahart, 2001; Srinivas and Srinivasan, 2005).
The major goals of this study are to quantify parameter
uncertainty in a conceptual hydrological model and to
investigate the effect of parameter uncertainty, co-effects
of parameter uncertainty and model structure uncertainty
on simulation results. A well-tested WASMOD model
(Water And Snow balance MODeling system; Xu, 2002)
will be used to demonstrate the procedure and applied to
the Yingluoxia watershed. Like many well-known models,
WASMOD has different versions working for different
spatial and temporal scales (Xu, 2002; Widén-Nilsson
et al., 2007; Gong et al., 2009; Widén-Nilsson et al., 2009;
Jin et al., 2010). The monthly version at a catchment scale
will be used in this study. The bootstrap method will
be employed to quantify parameter uncertainties in the
WASMOD model. The statistical behaviour of model
residuals will also be examined in the context because
it has a connection with the estimates and interpretation
of model parameters and is the significant foundation of
evaluations for model simulation uncertainties. This article
is organized as follows: The study area and data are given
in the next section, followed by the introduction of the
WASMOD model and the bootstrap method. The results
and discussions include parameter analysis, residual
analysis, discharge analysis and comparisons with the
SWAT (Soil and Water Assessment Tool) model applied
in the same study area. Conclusions drawn from this
study are given in the last section.
DESCRIPTION OF STUDY AREA AND DATA
Study area
The Heihe River basin, covering approximately
130 000 km2, is the second largest typical inland river
Copyright © 2012 John Wiley & Sons, Ltd.
305
basin in north-west of China and geographically includes
the Yingluoxia watershed, the middle Hexi Corridor and
the northern Alxa high-plain from south to north (Qi and
Luo, 2006). The Yingluoxia watershed at the upper
reaches of the Heihe River basin is selected as the study
area in this investigation (Figure 1). It covers an area of
10 009 km2, with the elevation ranging from 1674 m
above sea level (ASL) at the lower point to about 5120 m
ASL at its headwaters. The climate is characterized
as inland with cold and dry winters and hot and arid
summers, with an annual precipitation of 400 mm and an
annual potential evapotranspiration of 1600 mm. The
annual mean discharge is 160 mm, with weak inter-annual
variability (Zhao and Zhang, 2005). The dominant land
use types are grassland and brush land, occupying nearly
50% and 12% of the total area, respectively. The main
soil types in the watershed are alpine meadow soil (49%),
alpine frost desert soil (15%) and chestnut soil (13%).
Data description
Daily precipitation data at the Qilian, Yeniugou, Tuole,
Zhamushike, Zhangye and Yingluoxia stations;
temperature at the Qilian, Yeniugou, Tuole and Zhangye
stations; and discharge at the Yingluoxia station are
available in the period of 1985–2000. A summary of the
stations is given in Table I. All data are obtained from
the Environmental & Ecological Science Data Center for
West China, National Natural Science Foundation of China
and Digital River Basin. Besides the precipitation,
temperature and discharge data, potential evapotranspiration
data are also required in the model setup. The Blaney–
Criddle method (Brouwer and Heibloem, 1986) is used to
estimate the potential evapotranspiration. The first five years
(1985–1989) are used as a warm-up period in order to
diminish the effects of initial values on simulation, the
period from 1990 to 1996 for calibration and the period from
1997 to 2000 for validation.
WASMOD MODEL AND BOOTSTRAP METHOD
WASMOD model
Developed by Xu (2002), WASMOD is a conceptual
lumped modelling system for simulating streamflow from
both snowmelting and rainfall. The input data include
monthly areal precipitation, potential evapotranspiration
and air temperature. The model outputs are monthly
river discharge and other water balance components such
as actual evapotranspiration, slow and fast components
of river flow, soil-moisture storage and accumulation
of snowpack. The primary equations of the model are
presented in Table II. Parameters a1 and a2 are two
threshold temperature parameters, snowfall stops when
air temperature is higher than a1, and snowmelting
begins when air temperature is higher than a2; both
snowfall and snowmelting are allowed to take place when
temperature is between them. Parameter a3 controls the
potential evapotranspiration transferred to the actual
one as a function of soil-moisture storage. The slow
flow parameter a4 controls the proportion of runoff that
Hydrol. Process. 27, 304–312 (2013)
306
Z. LI ET AL.
Heihe River basin
Yingluoxia watershed
Z
$
Z
$
Z
$
Zhangye
Yingluoxia
Tuole
Yeniugou
Z
$
Zhamushike
Z
$
Z
$
Qilian
Figure 1. The location of Yingluoxia watershed with hydro-meteorological stations
Table I. Basic information of hydro-meteorological stations in Yingluoxia watershed
Annual precipitationa (mm)
Station
Yeniugou
Zhamushike
Qilian
Tuole
Zhangye
Yingluoxia
a
Latitude
Longitude
Elevation
(m ASL)
38 25′
38 14′
38 11′
38 49′
38 56′
38 49′
99 35′
99 59′
100 15′
98 25′
100 26′
100 11′
3180
2810
2787
3367
1483
1700
Annual temperaturea ( C)
Min
Max
Mean
Min
Max
Mean
275
371
349
181
77
120
602
624
573
404
195
258
405
459
419
285
128
186
3.55
–
0.59
3.04
6.91
–
1.44
–
2.18
1.05
8.51
–
2.82
–
1.18
2.34
7.58
–
Note: The annual precipitation and the annual temperature are calculated from the period of 1985 to 2000.
appears as ‘base flow,’ whereas a5 controls the fast runoff
(Xu, 2002; Engeland et al., 2005).
There are eight possible models in terms of two choices
of evapotranspiration equations, and two choices of b1
and b2. The best model is defined as the one with the
highest Nash–Sutcliffe coefficient (NS, Nash and Sutcliffe,
1970) and the lowest residual seasonality, where NS is
given by
N P
NS ¼ 1 t¼1
N
P
t¼1
2
Y obs ðtÞ Y^ ðt Þ
ð
Y obs ðt Þ
Copyright © 2012 John Wiley & Sons, Ltd.
obs
Y Þ
2
(1)
where Y is the average observed data during the
simulation period and Y^ ðtÞ and Y obs ðt Þare the simulated
and the observed values. The closer NS gets to 1, the more
accurate the model prediction result becomes. A perfect
match of simulated data to observed data occurs when
NS = 1. The formula
pffiffiffiffiffiffiffiffiffiffiffiffiffi
u N K
≤t ðN K; 5%Þ
(2)
stdðuÞ
obs
is used to check whether there is significant bias of the
residuals in each season, where std(u) is the standard
deviation of the residual series ut, N is the number of
Hydrol. Process. 27, 304–312 (2013)
307
UNCERTAINTY ISSUES OF A CONCEPTUAL WATER BALANCE MODEL
Table II. Principal equations of WASMOD model with snow module
Component
Snowfall
Rainfall
Snowmelt
Snow storage
Actual evapotranspiration
Slow flow
Fast flow
Active rainfall
Total runoff
Water balance
Equation
snt = pt{1 exp[ (ct a1)/(a1 a2)]2}+
rt = pt snt
mt = spt 1{1 exp[ (ct a2)/(a1 a2)]2}+
spt = spt 1 + sn
t mt w = maxðep ;1Þ t
t
et ¼ min
; wt or et = min{wt[1 exp( a3 ept)], ept}
t 1 a3
ep
b1
st ¼ a4 smþ
t1b
2
ft ¼ a5 smþ
ðnt þ mt Þ
t1
nt = pt ept[1 exp( pt/max(ept, 1))]
dt = st + ft
smt = smt 1 + rt + mt et dt
Note: pt and ct are monthly precipitation and air temperature; ept is the potential evapotranspiration; wt is the available water with the equation of
+
wt ¼ smþ
t1 þ pt ; b1 = 1 or 2; b2 = 1 or 2; and the superscript plus means, for example, x = max(x, 0).
terms in each season, K is the parameter numbers and t
(N K, 5%) is the critical value of the Student distribution with N Kdegree of freedom and 5% of significance
level (Xu, 2002).
Bootstrap method
The bootstrap method developed by Efron (1979) is
a resampling technique that can be used to estimate
properties of estimators such as confidence intervals (CIs)
and standard errors and is popularly used in applications (Efron, 1979; Davison and Hinkley, 1997; Abrahart,
2001; Srinivas and Srinivasan, 2005). Detailed description
of the bootstrap method can be found in Hall (1992). A
model-based bootstrap method developed by Stine (1985)
will be employed in this study and is described below.
The original data is specified as {X(t), Y obs (t)}
(t = 1, ⋯, N); where X(t), Yobs(t) and N are the set of input
data, the observed discharge and the length of observations, respectively. The hydrological model in this study is
written as Y ðtÞ ¼ f ðX ðt Þ; θÞ , where θ ¼ ðθ1 ; ⋯; θm Þ is
the parameter vector, with m being the number of
model parameters. After model calibration, the estimated
^ of parameter vector θ is obtained, as well as the
value θ
simulated value Y^ ðtÞ of model output Y(t) calculated by
^ . The model residuals are given by
Y^ ðt Þ ¼ f X ðtÞ; θ
^
(3)
ut ¼ Y obs ðtÞ Y^ ðtÞ ¼ Y obs ðtÞ f X ðtÞ; θ
where ut is generally assumed to be independent with zero
mean and constant variance when least square method is
employed for model calibration (Clarke, 1973; Xu, 2001),
t = 1, ⋯, N. Let F be the joint population distribution of
parameter vector θ, and Fi the marginal distribution of
parameter θi (i = 1, ⋯ m). Both the joint and marginal
distributions are unknown and will be estimated by the
bootstrap procedure described below. More details about
this method can be found in Li et al. (2010b).
1. Randomly resample the model residuals ut with
replacement to form a new residual series ut . The
superscript ‘*’ denotes something calculated from
bootstrap resampling.
Copyright © 2012 John Wiley & Sons, Ltd.
2. Add ut to the model output Y^ ðt Þ to form the new output
values Y ðtÞ ¼ Y^ ðt Þ þ ut .
3. Calibrate the bootstrap sample {X(t), Y*(t)} (t = 1, ⋯, N)
^ of parametervector θ
to obtain a bootstrap estimator θ
^ .
and simulated runoff data Y^ ðt Þ ¼ f X ðt Þ; θ
4. Repeat the bootstrap sampling for R times (1000 times
herein).
5. Obtain the ordered bootstrap estimates {θ i1 ; ⋯θ iR }
for θi derived from the bootstrap resampling method.
6. The two-side CI for θi at level a is then given by
[θ iRða=2Þ ; ⋯θ iRð1a=2Þ ].
RESULTS AND DISCUSSIONS
Inasmuch as there are eight possible models in WASMOD in terms of two choices of evapotranspiration
equations, and two choices of b1 and b2, a suitable model
for the catchment needs to be determined. The results of
all eight possible models for the study area are listed in
Table III. Inasmuch as the best model is defined as the
one with the highest NS and the lowest residual
seasonality, we can see that Model 5 is the best for our
study, with b1 = 1, b2 = 1, and the second evapotranspiration equation in Table II is selected. We only focus on
the best model in this article.
Table III. Summary of calibrated results of WASMOD model
applied in Yingluoxia watershed
Model
IE
b1
b2
NS
S
1
2
3
4
5
6
7
8
1
1
1
1
2
2
2
2
1
1
2
2
1
1
2
2
1
2
1
2
1
2
1
2
0.940
0.938
0.939
0.939
0.942
0.939
0.941
0.940
0
2
0
0
0
2
1
1
Note: Model means the number of model versions; IE means the choice of
the two evapotranspiration equations; and S means the number of seasons
with significant residual.
Hydrol. Process. 27, 304–312 (2013)
308
Z. LI ET AL.
Parameter analysis
The marginal distributions of five parameters in the
WASMOD model obtained by the bootstrap method
are shown in Figure 2. Intuitively, all parameters are
approximately normally distributed, which is consistent
with the findings of Engeland et al., (2005). For the
purpose of further testing whether they are normally
distributed, the non-parametric Kolmogorov–Smirnov
(K–S) test is employed. In this test, the maximum
deviation D is defined as
D ¼ maxjF ðxÞ Fs ðxÞj
(4)
120
125
100
100
Frequency
Frequency
in which F(x) is the theoretical cumulative distribution
and Fs(x) is the sample cumulative distribution function
based on n observations; for any observed x,
Fs(x) = k/n, where k is the number of observations less
than or equal to x. If, for the chosen significance level, the
observed value of the maximum deviation D is greater
than or equal to the tabulated critical value of K–S
statistic, the null hypothesis is rejected. The maximum
deviation D for parameters a1,a2,a3,a4 and a5 are 0.034,
0.031, 0.023, 0.020 and 0.034, respectively, and the
critical values of K–S test statistic are 0.052 and 0.043 at
the 0.01 and 0.05 significance levels. Therefore, the
hypothesis that the parameters are normally distributed is
accepted at the 0.01 and 0.05 significance levels. In spite
of this, they are a little skewed, especially for parameters
a1 and a5, according to Figure 2.
In terms of their marginal distributions, the uncertainty
ranges given by the 95% and 99% CIs are obtained and
shown in Table IV, together with their optimal values
acquired from the calibration process. The optimal values
for a1and a2are 8.35 and 1.17, respectively, meaning
that snowfall stops (not begins) when air temperature
is higher than 8.35 C, snowmelting begins (not stops)
when air temperature is higher than 1.17 C, and both
snowfall and snowmelting are allowed to take place when
temperature is between them. Although the CIs at a
80
60
40
75
50
25
20
0
0
5.00 6.00 7.00 8.00 9.00 10.00 11.00
-1.80
-1.60
-1.40
a1
120
-1.00
-0.80
120
Frequency
Frequency
-1.20
a2
90
60
30
90
60
30
0
0
0.10 0.15 0.20 0.25 0.30 0.35 0.40
a3
0.02
0.03
0.04
0.05
a4
Frequency
120
90
60
30
0
0.20 0.30 0.40 0.50 0.60 0.70 0.80
a5
Figure 2. Margin distributions of five parameters in WASMOD model
Copyright © 2012 John Wiley & Sons, Ltd.
Hydrol. Process. 27, 304–312 (2013)
309
UNCERTAINTY ISSUES OF A CONCEPTUAL WATER BALANCE MODEL
0.5
Table IV. Results of normal test for five parameters and their
uncertainty ranges
a1
a2
a3
a4
a5
95% CI
99% CI
Optimal value
[6.39, 9.40]
[1.46, 0.97]
[0.18, 0.31]
[0.02, 0.04]
[0.33, 0.59]
[5.98, 9.92]
[1.61, 0.86]
[0.16, 0.34]
[0.02, 0.05]
[0.30, 0.65]
8.35
1.17
0.25
0.03
0.45
Autocorrelation
0.3
Parameter
0.1
-0.1
-0.3
-0.5
1
5
9
13
17
21
25
29
33
Time lag
Residual analysis
Since the least square method is used for model
calibration in this study, the following assumptions about
the model residuals are involved (Clarke, 1973; Xu,
2001): (1) the residuals have zero mean and constant
variance; (2) the residuals are mutually uncorrelated; and
(3) the residuals are normally distributed. If some or all
of these assumptions were invalid, then, firstly, the
estimates of model parameters would be fallacious,
and furthermore, the subsequent uncertainty analysis
for model simulation would be problematic. Therefore,
it is necessary to test the related statistical assumptions
about the residuals. Here, we mainly focus on testing
whether the residuals are independent, homoscedastic
and normally distributed with zero mean. Previous
study (Xu, 2001) has shown that a square-root transformation of discharge data is a good choice for
obtaining the homoscedastic residuals
for monthly
WASpffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
obs
^
MOD; therefore, ut ¼ Y ðt Þ f X ðt Þ; θ instead of
ut ¼ Y obs ðt Þ f X ðt Þ; ^θ is used in this study.
The autocorrelation of model residuals as a function of
time lag together with the 95% CI is plotted in Figure 3. It
can be seen clearly that the lag-one residual autocorrelation is not significant, indicating that the assumption
about independent residuals are reasonable.
The non-parametric Kruskal–Wallis statistics test
(Kruskal and Wallis, 1952) is used to check the
homoscedasticity of model residuals. This test is based
on the statistic
Figure 3. Autocorrelation of model residuals for Yingluoxia watershed
where Ri is the sum of the ranks occupied
by the ni
Pk
observations of the ith sample, and
n
¼
n (k = 3
i¼1 i
in this study). The sampling distribution of the H statistic
is well approximated by the chi-square distribution with
k 1degrees of freedom when ni > 5 for all i. The null
hypothesis of homoscedasticity will be rejected for a
given significance level a if computed H is bigger than
w21a;k1 . According to Equation (5), we get H = 0.80 in
this study, less than the value of w20:95; 2 for 2 degrees
of freedom 5.99; therefore, the null hypothesis of
homoscedasticity cannot be rejected, meaning that the
residuals can be regarded as homoscedastic.
The K–S test mentioned in the section of ‘parameter
analysis’ is also used to check whether the residuals are
normally distributed. The theoretical normal probability
distribution function values and the sample probability
distribution function values are plotted in Figure 4 for
better visualization. According to the calculation, the
maximum deviation D between the theoretical line and
the sample line on the probability scale is about 0.08, and
the critical values of K–S test statistic are 0.18 and 0.15
at the significance levels of 0.01 and 0.05 for n = 84;
therefore, the hypothesis that the residuals are normally
distributed is not rejected at the significance levels of
0.01 and 0.05. All these mean that the assumptions of
independent, homoscedastic and normally distributed
residuals are valid in the study. That is, the above estimates
of parameter and the subsequent uncertainty analysis in
model simulation are reasonable and creditable.
1.00
Cumulative distribution
certain significance level based on parameter distribution
cannot be used to compare the parameter uncertainties
directly, it can be used for an approximate comparison in
general. The 95% CI for a1 ranges from 6.39 to 9.40 and
for a2 ranges from 1.46 to 0.97. Parameter a3,
controlling the relationship between potential evapotranspiration and actual evapotranspiration, has a 95% CI of
0.18–0.31. The smaller the value for a3 is, the greater the
evaporation losses at all moisture storage states. Parameter a4 controls the proportion of runoff that appears as
‘base flow,’ with a 95% CI of 0.02 to 0.04. Parameter a5,
controlling the proportion of fast runoff, has a 95% CI of
0.33 to 0.59.
0.75
Sample
distribution
0.50
Normal
distribution
0.25
0.00
-1
H¼
12
nðn þ 1Þ
k
X
R2i
i¼1
ni
-0.5
0
0.5
1
1.5
Residual
3ðn þ 1Þ
Copyright © 2012 John Wiley & Sons, Ltd.
(5)
Figure 4. Comparison of cumulated probability distribution of residuals
with the theoretical normal distribution function values
Hydrol. Process. 27, 304–312 (2013)
310
Z. LI ET AL.
Discharge analysis
By using the WASMOD model, NS reaches 0.942 and
0.928 for calibration and validation periods, respectively. It
can also be seen from Figure 5 that the base flow is
simulated well during both periods, and the peak flow also
shows agreeable simulation results except for those in
1993 and 1998. The predicted hydrograph is acceptable
compared with the observed one as a whole, and the
simulation error is relatively small. These results correspond to the optimal parameters obtained from the
calibration process. However, due to the uncertainties in
hydrological modelling, it is necessary that not only the
simulation results but also the simulation uncertainty be
taken into account for the purpose of deeply and accurately
assessing the model performance and simulation reliability.
Simulation uncertainties caused by sole parameters
are estimated from the discharge samples obtained by
running the model with 1000 sets of parameter values
from the parameter distribution. The result is shown in
Figure 6. About 41.7% and 52.4% of the observed
discharge fall inside the 95% and 99% CIs of simulation
results due only to parameter uncertainty, meaning that
parameter uncertainty in the WASMOD model has a great
effect on model simulation uncertainty but still cannot be
used to explain all the uncertainties in this study. This
result is close to those of the WASMOD model applied in
the Stabbybäcken and Ulva Kvarndmn basins in central
Sweden by Engeland et al. (2005) (45% of the observed
0
100
Discharge(mm)
100
60
Observed
Simulated
150
200
40
250
20
Precipitation(mm)
50
80
discharge falling inside the 95% CI in their study).
Compared with the results of the SWAT model applied in
the Thur River basin (7.2%) by Yang et al. (2007a) and
the Chaohe river basin (10%) by Yang et al. (2007b), the
percentage of the observations falling inside the 95% CI
(41.7%) is higher in our findings, indicating that the
effects of parameter uncertainty on simulation uncertainty
in the WASMOD model are greater than those in the
SWAT model in this case.
Inasmuch as it has been proven that the statistical
assumptions on model residuals (independent, homoscedastic and normally distributed) are valid, simulation
uncertainties caused by co-effects of parameter uncertainty
and model structure uncertainty can be calculated by
adding model residuals in the form of a random uncertainty
with zero mean and sample dependent standard deviation to
each of the 1000 discharge values (Engeland et al., 2005).
For each time step,
qffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
(6)
Y ðtÞi;samp ¼ ð Y^ ðt Þi þ rnormð0; oi ÞÞ2
where oi is the sample dependent standard deviation of
the residuals, i is a sample index, samp is an index for
discharge sample with the random term added, Y^ ðtÞ is the
discharge calculated from the WASMOD model, and
rnorm is a random number from a normal distribution
with the specified parameters. Figure 7 gives the 95%
and 99% CIs of simulations from co-effects of both
parameter and model structure uncertainties. It shows that
the CIs become wider, and more observations fall inside
the 95% (95.2% of the observations) and 99% (98.8% of
the observations) CIs of simulations. Therefore, it is
important to consider the uncertainties due to both model
parameters and model structure. Additionally, the coverage
of observed data is very close to the range of CI of
simulations, indicating that the WASMOD model gives
robust estimates of monthly discharge in the Yingluoxia
watershed.
300
0
Comparisons with SWAT model applied in the same
study area
350
0
20
40
60
80
100
120
Time in month (1990.1-2000.12)
Figure 5. Simulation results of monthly discharge using WASMOD model
in Yingluoxia watershed (the column on the top indicates the corresponding
monthly precipitation; the calibration and validation periods are partitioned
by the dashed reference line)
Compared with another study that discussed parameter
uncertainty in the SWAT model applied in the same study
area of the Yingluoxia watershed by using the bootstrap
60
60
Simulated(95%CI)
40
30
20
10
0
Simulated(99%CI)
50
Discharge(mm)
Discharge(mm)
50
Observed
Observed
40
30
20
10
1
12
23
34
45
56
67
Time in month(1990.1-1996.12)
78
0
1
12
23
34
45
56
67
78
Time in month(1990.1-1996.12)
Figure 6. The 95% and 99% CIs for simulated discharge caused by parameter uncertainty in Yingluoxia watershed
Copyright © 2012 John Wiley & Sons, Ltd.
Hydrol. Process. 27, 304–312 (2013)
311
UNCERTAINTY ISSUES OF A CONCEPTUAL WATER BALANCE MODEL
60
60
Simulated(95%CI)
40
30
20
10
0
Simulated(99%CI)
50
Discharge(mm)
Discharge(mm)
50
Observed
Observed
40
30
20
10
1
12
23
34
45
56
67
78
Time in month(1990.1-1996.12)
0
1
12
23
34
45
56
67
78
Time in month(1990.1-1996.12)
Figure 7. The 95% and 99% CIs for simulated discharge caused by co-effects of parameter uncertainty and model structure uncertainty in Yingluoxia
watershed
method (Li et al., 2010b), we can obtain the following
differences and similarities. Firstly, all parameters are
approximately normally distributed in the simple conceptual WASMOD, whereas most of the parameters (six
of nine) are not normally distributed in the complex semidistributed SWAT model. As a result, the parameter
estimation and inference based on normal distribution
would be feasible for the lumped monthly WASMOD,
whereas it would be more or less problematic for the
more complex SWAT. Secondly, the contributions of
parameter uncertainty to simulation uncertainty in the
WASMOD model (41.7% of the observations falling
inside the 95% CI of simulations) are greater than those in
the SWAT model (only 12%–13% of the observations
falling inside such CI). It is probably due to that,
compared with the lumped conceptual model, that the
distributed or semi-distributed model is more physically
based and takes more consideration of spatial and
temporal differences in watershed characteristics and
climatic conditions, resulting in a better identification of
model parameter coupled with a narrower uncertainty
range, and therefore, the effects of parameter uncertainty
to simulation uncertainty are relative small. This implies
that, especially for the simple lumped model, reducing
parameter uncertainty and in turn reducing the effects
of parameter uncertainty to simulation uncertainty are
of great importance in hydrological modelling. This can
be done through improving data quality, selecting a
satisfactory optimization algorithm in model calibration,
etc. Finally, for WASMOD, the statistical assumptions
(independent, homoscedastic and normally distributed)
on model residuals are proven to be satisfied through a
square-root transformation of discharge data, whereas for
SWAT, it is hard to be satisfied even with a square-root
or other transformations of discharge data. However, the
validity of statistical assumptions on residuals made in
the model formation is of great importance because it is
the base of simulation uncertainty evaluation.
CONCLUSIONS
In this study, some uncertainty issues in the WASMOD
model were explored in the semi-arid Yingluoxia
watershed in north-west of China. The non-parametric
Copyright © 2012 John Wiley & Sons, Ltd.
bootstrap method was employed to quantify the uncertainties in model parameters. Model simulation uncertainties
due to sole parameters and due to co-effects of both
parameter and model structure were examined. The
following conclusions could be drawn from this study.
The WASMOD model was capable of giving a
satisfactory simulation result of monthly discharge in
the study area, with NS value reaching 0.942 and 0.928,
respectively, for calibration and validation periods, and
the whole hydrograph showed agreeable simulating result
in total.
All the model parameters were approximately normally
distributed based on both their marginal distributions and
K–S test and had different uncertainty ranges according
to their 95% and 99% CIs. Model residuals were
proven to be independent through the checking of the
plot of lag-one residual autocorrelation, homoscedastic
after a square-root transformation of discharge data, and
approximately normally distributed in terms of a K–S test.
Results for model simulation uncertainty indicated that
parameter uncertainty had a great effect on simulation
uncertainty, with 41.7% and 52.4% of the observations
falling inside the 95% and 99% CIs of simulations;
however, it still could not be used to explain all the
uncertainties in hydrological modelling. The CIs became
wider, and about 95.2% of the observations fell inside
the 95% CI and 98.8% of the observations fell inside
the 99% CI when both parameter and model structure
uncertainties were considered. The findings indicated that,
in comparison with the uncertainty in model parameters,
uncertainties in both model parameters and model
structure had larger contributions to model simulation
uncertainties, further demonstrating that the model
structure uncertainty along with its effects on simulation
uncertainty cannot be neglected in hydrological modelling.
The comparison with another study that discussed
parameter uncertainty in the SWAT model applied in the
same study area showed that the parameter estimation and
inference based on normal distribution would be feasible
for the lumped monthly WASMOD, whereas it would
be problematic for the complex SWAT. The statistical
assumptions on model residuals were proven to be satisfied
for WASMOD but not for SWAT. The contributions
of parameter uncertainty to simulation uncertainty in
WASMOD were greater than those in SWAT, meaning
Hydrol. Process. 27, 304–312 (2013)
312
Z. LI ET AL.
that reducing parameter uncertainty is of great importance
especially for the simple lumped model.
ACKNOWLEDGEMENTS
This study is supported by NSFC(41101038), the Fundamental Research Funds for the Central Universities
(2010ZY13, 2011YXL038) and the CSIRO Water for a
Healthy Country Flagship Program. Thanks are given to the
Environmental and Ecological Science Data Center for
West China (http://westdc.westgis.ac.cn/), National Science
Foundation of China and Digital River Basin (http://heihe.
westgis.ac.cn/) for providing all the datasets. We also thank
two anonymous referees for their useful comments.
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