Water Resources Management 15: 75–92, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
75
Statistical Analysis of Parameters and Residuals of
a Conceptual Water Balance Model – Methodology
and Case Study
CHONG-YU XU
Department of Earth Sciences, Hydrology, Uppsala University, Villavagen 16, S-752 36 Uppsala,
Sweden, e-mail: [email protected]
(Received: 30 May 2000; in final form: 24 May 2001)
Abstract. Statistical analysis of parameters and residuals of conceptual hydrological models has
received little effort in the hydrological research, certainly by orders of magnitude less than on
many other problems like development and comparison of automatic calibration methods, optimisation algorithms, etc. Much more work is required than is presently undertaken to investigate the
properties of model residuals. There is a need of an easily understandable and applicable statistical
analysis scheme. In this article, a procedure is presented through which two basic issues of model
evaluation are accounted for. First, different techniques used for parameter analysis are discussed.
Second, methodology of residual analysis is discussed and the general behaviours of residuals are
examined. To illustrate the procedure, a simple water balance model was applied to the Stabbybäcken
River Basin in central Sweden.
Key words: calibration, conceptual models, parameter analysis, residuals analysis
1. Introduction
Conceptual catchment models have been formulated to varying degrees of complexity (e.g., Thornthwaite and Mather, 1955; Crawford and Linsley, 1966;
Bergström, 1976, 1992; Xu and Singh, 1998). All these models accord with the
definition given by Moore and Clarke (1981), namely: (1) they describe conceptually land-based hydrologic processes which are spatially averaged or lumped, and
(2) some of their parameters are estimated by fitting to observed hydrologic data
such as rainfall, potential evapotranspiration, and streamflow.
There are two basic approaches to estimation of model parameters: manual
and automatic. In recent years a great deal of research has been devoted to the
development of later approach, which has two major components: (1) the estimation criterion and (2) the optimisation algorithm. The choice of estimation criterion has been discussed extensively (e.g., Sorooshian and Dracup, 1980; Sefe
and Boughton, 1982; Servat and Dezetter, 1991). The selection of an automatic
parameter optimisation algorithm has also been studied extensively (e.g., Dawdy
and O’Donnell, 1965; Nash and Sutcliffe, 1970; Clarke, 1973; Pickup, 1977; So-
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CHONG-YU XU
rooshian and Gupta, 1983). Until recently, virtually all calibration methods belong to ‘local-search’ procedures. The convergence problems encountered by local
search algorithms have been well documented in the literature (e.g., Gupta and
Sorooshian, 1985; Hendrickson et al., 1988). Research into optimisation methods
has led to the use of population-evolution-based search strategies (e.g., Wang,
1991; Duan et al., 1992). In this regard the use of the shuffled complex evolution
(SCE-UA) global optimisation algorithm has been reported in a number of studies
(e.g., Duan et al., 1992; Sorooshian et al., 1993; Kuczera, 1997, Yapo et al., 1996;
Gupta et al., 1998). Notwithstanding the success mentioned above, Vertessy et al.
(1993) showed that the gains of using the global optimisation algorithm can be at
a substantial computational price.
Regardless of the means employed to calibration, optimisation methods are
based on a philosophy that involves some assumptions (hypotheses), which often
are not satisfied. Typically, model parameters for gauged catchments are usually estimated by ordinary least squares (OLS), which involves solving the minimisation
problem:
2
(1)
min sum of squares = min nt=1 qt,obs − qt,com (xt ; A)
where qt,obs and qt,com are observed and computed discharge, respectively. The
difference of which is called model error or residual εt . The xt is a vector of
inputs (such as rainfall and evaporation), and A, a parameter vector about which
inference is sought. Probably the main reason for the popularity of this criterion,
as stated by Sorooshian and Dracup (1980) has been its direct applicability to any
model. However, the use of Equation 1 as an objective function to be minimized for
parameter optimisation implies certain assumptions about the residuals εt (Clarke,
1973):
(a) that the εt have zero mean and constant variance σε2 (i.e., E(εt ) = 0, E(εt2 ) =
σε2 ;
(b) that the εt are mutually uncorrelated (E(εt , εt −k ) = 0 for all k =0).
If it were known that either assumption (a) or (b), or both, were invalid, then
Equation (1) would not be the most sensible objective function; estimates of model
parameters would, of course, still be obtained by minimising Equation (1), but their
interpretation would be fallacious.
Clarke (1973) also stated that if approximate confidence intervals are to be given
for the estimated model parameters, a further assumption must be made about the
probability distribution of the residuals, that is:
(c) that the εt are distributed normally.
The above assumptions need to be tested. The success or otherwise of the fitted
model as a description of the relation between rainfall and streamflow from the
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
77
catchment is illustrated by the model residuals, which also give evidence of the
validity or invalidity of the assumptions (such as (a), (b) and (c) above) made
in the model formulation. However, in the field of hydrological modelling, few
writers examine and describe any properties of residuals given by their models
when fitted to data, although most present diagrams comparing the observed and
fitted hydrographs, together with a measure of ‘goodness of fit’ such as R2 , the
analog of the coefficient of determination. The lack of such studies was pointed out
by, for example Aitken (1973), Clarke (1973), Sorooshian and Dracup (1980), and
Kuczera (1983), among others. The point is that the whole set of residuals requires
study, and not just a single descriptive measure calculated from them. One of the
reasons might be that there is a lack of an easily understandable and applicable
statistical analysis procedure for the average hydrologist, or for that matter any
model user to use.
The purpose of this article is not to discuss and compare which optimisation
methods are preferable to the others. Instead, this article addresses two basic issues of model evaluation, such as what is the basic requirement for the statistical
analysis of the parameters and residuals, and how to perform such analysis with a
simple procedure. To illustrate the procedure, a simple water balance model was
applied to the Stabbybäcken River at Stabby in central Sweden.
2. The Model and Data
The water balance model used in this study was developed for water balance computation for the NOPEX region (Xu et al., 1996). The input data are areal precipitation, long-term average potential evapotranspiration and air temperature. The
model outputs are river flow and other water balance components, such as actual
evapotranspiration, slow and fast components of river flow, soil-moisture storage
and accumulation of snowpack, etc. The time step used in the study is 10 days. The
model works as follows: precipitation pt is first split into rainfall rt and snowfall st
by using a temperature-index function, snowfall is added to the snowpack spt (the
first storage) at the end of the time step, of which a fraction mt melts and contributes
to the soil-moisture storage smt . Snowmelt is calculated by using a temperatureindex method. Before the rainfall contributes to the soil storage as ‘active’ rainfall,
a part is subtracted and added to interception evaporation loss. The soil storage
contributes to evapotranspiration et , to a fast component of flow ft and to slow
flow bt . The model concept is shown in Figure 1 and the main equations are shown
in Table I.
The hydrological data of the Stabbybäcken river at gauge station Stabby are
used in the study. The basin has an area of 6.6 km2 and is situated 4 km SW of
Uppsala. The area is dominated by forest with abundant bog areas. Eleven years
(1981–1991) of ten-day precipitation, air temperature and runoff data were used in
the study to calibrate the model parameters, of which the first year 1981 was used
as a ‘warming-up’ period.
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CHONG-YU XU
Figure 1. Schematic computational flow chart of the NOPEX water balance model.
Table I. Principal equations of the NOPEX-6 monthly snow and water balance model
Snow fall
Rainfall
Snow storage
Snowmelt
Potential evapotranspiration
Actual evapotranspiration
Slow flow
Fast flow equation
Total computed runoff
Water balance equation
st
rt
spt
mt
ept
et
bt
ft
dt
smt
= pt {1 − exp[−(ct − a1 )/(a1 − a2 )]2 }+
= pt − st
= spt −1 + st − mt
= spt {1 − exp[(ct − a2 )/(a1 − a2 )]2 }+
= (1 + a3 (ct − cm ))epm
= min[wt (1 − e−a4 ept ), ept ]
= a5 (sm+
t −1 )
= a6 (sm+
t −1 )(mt + nt )
= bt + ft
= smt −1 + rt + mt − et − dt
a1 ≥a2
0≥a4 ≥1
a5 ≥0
a6 ≥0
+
wt = rt + sm+
t −1 is the available water; smt −1 − max(smt −1 , 0) is the available storage;
nt = pt −ept (1−e−pt /ept ) is the active rainfall; pt and ct are monthly precipitation and air temperature, respectively; epm and cm are long-term monthly average potential evapotranspiration
and air temperature, respectively; ai (i = 1, 2, . . ., 6) are model parameters.
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
79
3. Methodology of Statistical Analysis
The basic issues of statistical analysis of conceptual catchment models include the
use of different techniques for the estimation and analysis of model parameters, for
evaluation and analysis of the statistical behaviour of residuals, etc.
3.1. PARAMETER ESTIMATION
The estimation procedure is defined given the validity of the assumptions (a) to (c)
presented in the introduction section. The validity or invalidity of the assumptions
(a) to (c) will be tested in the residual analysis section. A classical estimation
method is to maximise the loglikelihood with respect to the parameters. Because
of the hypotheses (a) to (c) maximising the loglikelihood with respect to model
parameters ai is equivalent to minimising the sum of squares (Equation (1)). Minimisation of Equation (1) with respect to the parameters ai results in estimates of
ai . The model standard deviation, σ , is estimated by
minimum sum of squares
SSE
=
(2)
S=
N −K
N −K
where N is the number of terms in (2), and K is the number of parameters.
An approximate standard deviation of model variance is easily found to be (e.g.,
Spiegel, 1975):
2σ 2
std(S 2 ) = √
2(N − K)
(3)
For n (i.e., N − K in this case) ≥ 100 the sampling distribution of S is very nearly
normal. If the population is normal (or approximately normal), it can be shown that
(Spiegel, 1975, Table 5.1, page 162):
std(S) = √
σ
2(N − K)
(4)
Note that (3) yields (4) in case the population is normal with
µS 2 = E(S 2 ) = (n − 1)σ 2 /n
(5)
which is very nearly σ 2 for large n (n≥30).
The fact that (N − K)S 2 /σ 2 has a chi-square distribution with N − K degrees
2
of freedom enables us to obtain confidence limits for σ 2 or σ . For example if χ0.025
2
2
and χ0.975 are the values of χ for which 2.5% of the area lies in each ‘tail’ of the
distribution, then a 95% confidence interval is
2
≤
χ0.025
(N − K)S 2
2
≤χ0.975
.
σ2
(6)
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CHONG-YU XU
From these we see that σ can be estimated to lie in the interval
√
√
S N −K
S N −K
≤σ ≤
.
χ0.975
χ0.025
(7)
Applying the large sampling theory, a 95% confidence limits for σ is approximately
(e.g., Spiegel, 1975, page 204):
(8)
S±1.96σ/ 2(N − K) .
If the model is correct, then the quantity ai /Si is distributed as a t-distribution with
N − K degrees of freedom. Thus the confidence limits on parameter ai can be
estimated from (e.g., Haan, 1977)
âi ±t1−α/2,N−K Si
(9)
where âi is an estimate of ai , α is the significance level, and Si is an estimate
for σi , the standard deviation of ai . For N − K large enough (>100), a normal
approximation can be applied for ai /Si , the half width of a 95% confidence interval
of ai thus is
H W CI (ai ) = 1.96σi .
(10)
These formulae are standard results in regression analysis.
3.2. PARAMETER ANALYSIS
In parameter analysis we should ask: are all parameters statistically significant? Are
parameters highly correlated with each other? Which parameter is more sensitive?
The first question can be studied by checking whether the zero values belongs
to the 95% confidence interval
(ai − 1.96σi , ai + 1.96σi ).
If the zero value is included in the confidence interval, i.e., if the hypothesis ai = 0
is acceptable, then parameter ai can be set equal to zero without diminishing the
explanatory power of the model. Regarding Question 2, the correlation matrix of
the parameters has to be computed. A correlation coefficient between two parameters very near +1 or –1, implies that perhaps a model can be found with a
smaller number of parameters and with the same explanatory power; alternatively
the parameters may have to be built into the model in a different way, so that
their explanatory effects are more dissociated, and optimisation is easier. As for
Question 3, an analysis of the parameter sensitivity is performed by plotting the
criterion function SSE versus the percentage of relative deviations of the parameter
values from the optimised parameter value, i.e., SSE versus âi â−i ãi (%). Where âi is
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
81
the optimised parameter value, ãi are the parameter values at the neighbourhood of
âi . The smallest value for SSE, i.e., S ŜE, is found at ai = âi and SSE increases
with both the increase and decrease of ai . These curves (e.g., Figure 3) must have
a parabolic aspect centred around the minimum S ŜE and according to the form of
the cup one can make a qualitative analysis of the parameter sensitivity.
3.3. RESIDUALS ANALYSIS
Residual analysis is checking whether the residuals εt behave as are required by
the model hypotheses, especially whether they are independent, homoscedastic
and normally distributed with zero expectation. The general behaviour of residuals
can be studied by various methods (e.g. Aitken, 1973; Sorooshian and Dracup,
1980; Vandewiele et al., 1992). The methods used for testing the three assumptions
of residuals (i.e., homoscedasticity, independence, and normality) as stated in the
introduction section are discussed in this section.
3.3.1. Check on Homoscedasticity
The homoscedasticity of residuals can be checked by plotting the residuals versus
important variables such as computed discharge. The scattergram has to be symmetric with respect to the horizontal axis (zero expectation), and the conditional
standard deviation has to be constant (homoscedasticity). Examples are given in
the case study section.
The homoscedasticity can also be tested by the Kruskal-Wallis statistics (Kruskal
and Wallis, 1952). The Kruskal-Wallis test, or H test enables us to test the null
hypothesis that k independent random samples come from identical populations.
It is a nonparametric test. The method assumes that the variable has a continuous
distribution, but nothing is said about the form of the population distribution or
distributions from which the samples were drawn. The test is based on the statistic
R2
12
i
− 3(n + 1) .
n(n + 1) i=1 ni
k
H =
(11)
In the test, all observations are ranked jointly, and Ri is the sum of the ranks
occupied by the ni observations of the ith sample, and n1 + n2 + . . . + nk = n.
When ni > 5 for all i and the null hypothesis is true, the sampling distribution
of the H statistic is well approximated by the chi-square distribution with k − 1
degrees of freedom. The null hypothesis of homoscedasticity will be rejected for a
2
. Examples and
given significance level, α, if computed H is bigger than χ1−α,k−1
results of the test are given in the case study section.
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CHONG-YU XU
3.3.2. Check on Independence
The hypothesis that the residuals are mutually uncorrelated can be checked by
computing the autocorrelations of the residuals, ρk , with time lag k and the corresponding confidence interval. In general, the autocorrelations ρk with time lag k
is
ρk = E[(xt − µ)(xt +k − µ)]/σ 2
(12)
where µ and σ 2 are the mean and variance of the residuals, respectively.
An estimate of ρk is
ρ̂k =
n−k
n−k
2
t =1 xt
t =1
−
1
n−k
xt xt +k −
n−k
t =1
xt
1
n−k
n−k
t =1
2 1/2 xt
n
2
t =k+1 xt
n
t =k+1 xt
−
1
n−k
n
t =k+1
xt
2 1/2
(13)
when n is large and k is small, n/(n −k)→1, a simpler estimator of autocorrelation
coefficient is
n−k
(xt − x̄) (xt +k − x̄)
(14)
rk = t =1n
2
t =1 (xt − x̄)
The confidence interval for the autocorrelation coefficient of an independent series
is given by the limits (Haan, 1977)
rk (95%) =
1 −1±1.96 (n − k − 1)
n−k
(15)
If the calculated rk falls outside these confidence limits, the hypothesis that ρk is
zero (Ho : ρk = 0 versus Ha : ρk =0) is rejected. Examples of the tests are shown
in the case study section.
3.3.3. Check on Normality
The hypothesis that residuals are distributed normally is needed if the estimated
confidence regions for the parameters are required. The normality can be tested
using different methods. The Kolmogorov-Smirnov test method is used here. The
test has several advantages. It is easy to use and the procedure is graphic; a large
number of samples can be tested on the same plot; the test is nonparametric and is
not subject to the very small sample limitation. The test is conducted as follows:
1) Let F (x) be the completely specified theoretical cumulative distribution function under the null hypothesis.
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
83
2) Let Fs (x) be the sample cumulative density 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.
3) Determine the maximum deviation, D, defined by
D = max |F (x) − Fs (x)|
(16)
4) If, for the chosen significance level, the observed value of D is greater than or
equal to the critical tabulated value of the Kolmogorov-Smirnov statistic, the
hypothesis is rejected.
Examples of the tests are shown in the case study section.
4. The Case Study
4.1. RESULTS OF PARAMETER ESTIMATION AND ANLYSIS
Optimised parameter values and their 95% confidence intervals were computed and
shown in Figure 2 (Run 1). Correlation matrix of parameter values were computed
and shown in Table II (Run 1). Results show that five of the six parameters are
statistically significant and correlation coefficients between different parameters
are ranging between 0 to 0.67. Third, an analysis of the parameter sensitivity is
performed by plotting SSE versus the parameter values at the neighbourhood of
the minima (Figure 3). In the figure, âi is the optimised value for ai , and ãi are
the values of ai at its neighbourhood of the minimum. The figure shows that the
snowmelt parameter a1 and fast flow parameter a6 are the most sensitive ones
while a3 , a parameter that is used to convert long-term average monthly potential
evapotranspiration to the monthly potential evapotranspiration, is least sensitive in
this case.
4.2. RESULTS OF RESIDUALS ANALYSIS
4.2.1. Check on Homoscedasticity
The residual’s homoscedasticity was checked using both the graphic method and
the Kruskal-Wallis test method as discussed before. The result of the graphic method
is shown in Figure 4. This plot reveals two deficiencies. First, for low runoff there
appears systematic bias in predicting runoff. Second, the residuals’ variability increases with increasing runoff. This suggests that the assumption of constant error
variance (homoscedasticity) is violated.
In performing the Kruskal-Wallis test, the following procedure is used:
1) Divide the residual time series into three groups for low (computed discharge
dt is less than 75% of the long-term average value, d̄), median (computed
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CHONG-YU XU
Figure 2. Comparison of optimised parameter values with the 95% confidence interval (line
with circle for the first run and line with plus for the second run). For the sake of plotting,
parameters a1 and a2 have been scaled down by 0.1; parameters a4 and a6 have been scaled
up by 10, respectively.
Table II. Correlation matrix of the parameters
Run 1
a1
a2
a3
a4
a5
a6
a1
a2
a3
a4
a5
a6
1.000
–0.675
–0.134
0.057
–0.507
0.387
–0.675
1.000
0.052
–0.167
0.358
–0.453
–0.134
0.052
1.000
0.114
0.219
–0.075
0.057
–0.167
0.114
1.000
0.163
0.512
–0.507
0.358
0.219
0.163
1.000
0.036
0.387
–0.453
–0.075
0.512
0.036
1.000
a1
a2
a3
a4
a5
a6
α
1.000
–0.498
–0.067
0.077
–0.335
0.360
–0.095
–.0498
1.000
0.042
–0.074
0.268
–0.265
–0.052
–0.067
0.042
1.000
–0.014
0.230
–0.026
0.013
0.077
–0.074
–0.014
1.000
–0.072
0.205
–0.115
–0.335
0.268
0.230
–0.072
1.000
–0.145
–0.003
0.360
–0.265
–0.026
0.205
–0.145
1.000
–0.063
–0.095
–0.052
0.013
–0.115
–0.003
–0.063
1.000
Run 2
a1
a2
a3
a4
a5
a6
α
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
85
Figure 3. Intersection of the hypersurface of the criterion function. (âi = optimised parameter
value, ãi = parameter values at the neighbourhood of âi ). The diagram is used to check the
sensitivity of the parameters to the criterion function value. The x-axis shows the percentage
of relative deviation of the parameter values from their optimised values and the y-axis shows
the change of the criterion function value.
discharge dt is great than 75% and less than 125% of the long-term average
value, d̄) and high (computed discharge dt is great than 125% of the long-term
average value, d̄) flows, respectively. In this way, we get n1 = 206, n2 = 64,
n3 = 90, and n = n1 + n2 + n3 = 360.
2) Ranking these residuals jointly from smallest to largest. Each variable in the
residual series occupies a rank number.
3) The sums of the ranks are computed to give R1 = 39770, R2 = 10703, R3 =
14507, for low, median and high flows’ groups, respectively.
4) Applying Equation (11) we get:
397702
107032
145072
12
+
+
H =
360×(360 + 1)
206
64
90
−3×(360 + 1) = 7.139 .
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CHONG-YU XU
Figure 4. Plot of residual versus computed runoff for catchment Stabby (Run 1). Two deficiencies are revealed: systematic bias for low flows and variability of the residuals increases
with increasing predicted runoff.
Figure 5. Autocorrelation of residuals for the catchment Stabby (Run 1). Correlated case.
2
5) Since H = 7.139 exceeds 5.991, the value of χ0.95
for 2 degrees of freedom,
the null hypothesis of homoscedasticity must be rejected.
4.2.2. Check on Independence
The residual autocorrelations computed by Equation (12) together with the 95%
confidence interval by Equation (13) are plotted in Figure 5. It is seen that the
residuals are correlated significantly for small time lags. Therefore, the hypothesis
of independence of residual is not fulfilled in this case study.
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
87
Figure 6. Comparison of cumulated probability distribution of residuals with the theoretical normal distribution function values. The maximum deviation between the theoretical
line (solid) and the sample line (dashed) on the probability scale is about 0.17 at residual
εt = 0.53. The critical value of the Kolmogorov-Smirnov test statistic for α = 0.05 and
N − K = 360 − 6 = 354 is 0.08. This figure shows that the hypothesis that the residuals are
normally distributed is rejected at the significance level α = 0.05.
4.2.3. Check on Normality
The Kolmogorov-Smirnov test is 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 6. The maximum deviation between the theoretical line and the sample line on the probability
scale is about 0.17 at residual εt = 0.53. The critical value of the KolmogorovSmirnov test statistic for α = 0.05 and N − K = 360 − 6 = 354 is 0.08.
Therefore the hypothesis that the residuals are normally distributed is rejected at
the significance level α = 0.05.
4.3. CORRECTIVE ACTIONS
Because all three assumptions discussed in the introduction section appear to be
violated, i.e., residuals are heteroscedastic (variance is increasing as calculated
runoff increases, Figure 4), residuals are significantly correlated (time dependent, Figure 5), and residuals are not normally distributed at the significance level
α = 0.05 (Figure 6), the following corrections were tried in this study:
1) To remove heteroscedasticity, i.e., dependence of error variance on computed
runoff, a square root transformation on runoffs qt,obs and qt,com , as suggested
by Vandewiele et al. (1992), was used as a first try. Therefore, it is supposed
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CHONG-YU XU
that
√
qt,obs −
√
qt,com = εt .
(17)
2) To handle the correlated errors case, a first-order autoregressive scheme was
assumed to represent the transformed residuals. The same procedure has also
been used by Sorooshian and Dracup (1980) and Alley (1984):
εt = αεt −1 + βt
(18)
with
βt ∼N(0, σ 2 )
and βt independent
where α is a so called autocorrelation parameter.
The objective function (1) then becomes
2
min sum of squares = min nt=1 εt − αεt −1
(19)
(20)
The above function when minimised with respect to the unknown parameters should
result in the parameter values that when used in the model for forecasting purposes
will produce the hydrograph with the highest probability of being nearest to the true
hydrograph. Again, hypotheses (18) and (19) have to be checked in the new calibration together with parameters analysis. Some of the results of the new calibration
are discussed as follows.
4.3.1. Check on Homoscedasticity
As in the first run, the plot of residual versus computed discharge was first checked
(Figure 7). Comparing Figure 7 with Figure 4 reveals that the variability of the residuals does not display dependence on predicted runoff. The tendency of residuals
to be positive for very small runoff, as shown in Figure 4, was removed (Figure 7).
Again, the Kruskal-Wallis test was applied. Following the same procedure as
described in Section 4.2.1, the computed H statistic equals to 2.89 < 5.991 = the
2
for 2 degrees of freedom. The hypothesis that three samples are from
value of χ0.95
an identical distribution cannot be rejected.
4.3.2. Check on Independence
The residual autocorrelation computed by Equation (12) together with the 95%
confidence interval by Equation (13) are plotted in Figure 8. The plot reveals that
the strong time dependence (as measured by residual autocorrelation in Figure 5)
was removed by fitting the AR (1) model to the transformed residuals.
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
89
Figure 7. Plot of residual versus computed runoff for catchment Stabby (Run 2). A case of
homoscedastic residual.
Figure 8. Autocorrelation of residuals for the catchment Stabby (Run 2). Uncorrelated case.
4.3.3. Check on Normality
The theoretical normal probability distribution function values and the sample
probability distribution function values are plotted in Figure 9. The maximum
deviation between the theoretical line and the sample line on the probability scale is
about 0.06 at residual εt = 0.50. The critical value of the Kolmogorov-Smirnov test
statistic for α = 0.05 and N −K = 360−6 = 354 is 0.08. Therefore the hypothesis
that the residuals are normally distributed is not rejected at the significance level
α = 0.05.
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CHONG-YU XU
Figure 9. Comparison of cumulated probability distribution of residuals with the theoretical normal distribution function values. The maximum deviation between the theoretical
line (solid) and the sample line (dashed) on the probability scale is about 0.06 at residual
εt = 0.50. The critical value of the Kolmogorov-Smirnov test statistic for α = 0.05 and
N − K = 360 − 6 = 354 is 0.08. This figure shows that the hypothesis that the residuals are
normally distributed is not rejected at the significance level α = 0.05.
Together with residual analysis, the parameter analysis techniques as discussed
in Section 3.2 are applied in the new calibration. For the sake of comparison,
the optimised parameters and parameter correlation matrix are shown in Figure 2
(Run 2) and Table II (Run 2), respectively. Comparing with the first run, this particular case study shows that parameter values are not significantly different as the
95% confidence intervals overlapped (Figure 2). Correlation coefficients between
parameters are slightly smaller in the second run (Table II).
5. Conclusions
A procedure to analyse parameter significance and sensitivity and to evaluate residual behaviour in conceptual catchment models is discussed and exemplified.
To illustrate the procedure, the NOPEX-6 water balance model (Xu et al., 1996)
was applied to the Stabby catchment. Through a case study it is shown that the
proposed approach is practical and simple to implement and can also provide useful
information that helps to identify and remove the violations of model hypotheses.
It might be said that the proposed methodology can also be used in other model
applications of the same type.
STATISTICAL ANALYSIS OF A CONCEPTUAL WATER BALANCE MODEL
91
Acknowledgements
The data used in this investigation was provided from the SINOP (System for
Information in NOPEX) database. The Swedish Meteorological and Hydrological
Institute (SMHI) provided most of the data to SINOP. The referees comments are
gratefully acknowledged.
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