Water Resources Management 11: 51–67, 1997.
c 1997 Kluwer Academic Publishers. Printed in the Netherlands.
51
Application of Water Balance Models to
Different Climatic Regions in China for
Water Resources Assessment
CHONG-YU XU
Hydrology Division, Uppsala University, Villavägen 16, S-75236 Uppsala, Sweden.
e-mail: chong-yu.xu@hyd.uu.se
(Received: 1 June 1995; in final form: 16 November 1995)
Abstract. Uneven precipitation in space and time together with mismanagement and lack of knowledge about existing water resources, have caused water shortage problems for water supply to large
cities and irrigation in many regions of China. There is an urgent need for the efficient use and regional
planning of water resources. For these purposes, the monthly variation of discharges should be made
available. In this paper, a simple water balance modelling approach was applied to seven catchments
(385–2000 km2 ) for water resources assessment. Six catchments were chosen from the humid region
in southern China and one catchment from the semi-arid and semi-humid region in northern China.
The results are satisfactory. It is suggested that the proposed modelling approach provides a valuable
tool in the hands of planners and designers of water resources.
Key words: China, water resources planning and management, water balance models, calibration
and simulation
1. Introduction
A sharp increase of water consumption during the last 30 years caused by population
growth and economic development in China has changed the existing viewpoint,
i.e. water is annually renewed in the water cycle and is an unlimited gift of nature.
It appeared that fresh water reserves are far from unlimited.
The ability to meet water needs varies considerably with time and space, and
this has caused, causes, and will continue to cause (to an even greater degree)
serious problems impeding the economic and social development in many regions,
even those not located in the arid zone. These problems are generated not only by
natural factors, such as uneven precipitation in space and time, which are indeed
the major characteristics of water resources in China (Zhang, 1983), but many have
also been caused by mismanagement and lack of knowledge about existing water
resources.
There is an urgent need for the efficient use and regional planning of water
resources not only in China, but also in many other countries. This was clearly
confirmed by the UN Conference on Environment and Development (UNCED,
1992).
INTERLINIE: PC3:WARM1101: PIPS Nr.: 104744 MATHKAP
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CHONG-YU XU
A traditional way of water resources assessment was based on the long-term
average water balance equation over a basin in a form of
P = E + Q;
(1)
where P; E and Q are the long-term average annual precipitation, evapotranspiration and streamflow, respectively. To solve Equation (1), two terms must be known.
There is often little known about the actual losses in a basin. In gauged catchments,
actual evapotranspiration is commonly computed as the difference between longterm averages of precipitation and streamflow, assuming the storage variation is
negligible over a long period. In ungauged sites where Q is also unknown, annual
evapotranspiration has been calculated by using highly empirical methods. Turc’s
(1954) formula is one of them. These earlier studies typically estimated annualaverage water resources and are unable to determine seasonal or monthly values of
hydrological variables.
Knowledge of short-term storages is important where agricultural demands and
water pumping are competitive. The monthly variation of discharges and other
balance components should be made available for the purpose of planning water
resources. Complex hydrological modelling may be used to balance components
for time intervals of less than 1 week (e.g. Anderson and Burt, 1985), as fieldprocess monitoring is impractical and usually too expensive on these timescales
for anything other than very small catchments.
Simple water balance models that simulate hydrographs of streamflow on the
basis of available meteorological data and few physically relevant parameters would
without doubt be a valuable tool in the hands of planners and designers of water
resources. Study of the models typically used in water resources assessment and
planning, and specific problems related to the use of such models in water balance
computation on basin and regional scales form the focal points of the paper. Other
potential applications of such models have been reported before (e.g. Alley, 1984;
Gleick, 1987; Xu and Vandewiele, 1995).
2. Model Concept and Equations
Easy calibration and easily obtainable input variables are the conditions of an
engineer of water resources projects. The modelling approach presented here,
which has been successfully applied to several regional schemes for water resources
planning, is a single reservoir model (Figure 1) working on monthly time steps.
The input data required are monthly areal precipitation and monthly potential
evapotranspiration (or pan evaporation). The model outputs are monthly river flow
and other water balance components, such as actual evapotranspiration, slow and
quick components of river flow, and the state of soil moisture condition. Snow
routine has been eliminated from the model, since it is not a remarkable process in
the region under studying.
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CHONG-YU XU
with parameter a1 constrained by a1 0. This equation is physically sound too
and is based on the assumption that the actual evapotranspiration is a fraction
of the available water, a fraction which is an increasing function of potential
evapotranspiration. Larger values for a1 will produce increased evapotranspiration
losses at all moisture storage states. Both Equations (4) and (5) fulfil condition (3).
The number of runoff components to be analysed depends on the characteristics
of the basin and on the objective of the separation which also includes the time
base to be considered (Dyck, 1983). For the models being developed in this study,
a distinction was made between the slow component and fast component of the
streamflow. There was no further separation between these two components. This
is mainly because the time base considered is one month. Both single linear and
single nonlinear reservoir concepts were tested and they were found to be sufficient.
Being similar to baseflow, slow flow depends essentially on the storage in the
catchment during the month considered. The relationship between slow flow and
the storage can be linear or nonlinear depending on the value of b1 . The general
form of slow flow equation is
st = a2 (sm+t 1)b ;
(6)
1
where a2 and b1 are nonnegative parameters. In practice, however, a2 and b1 are
highly correlated, resulting in difficult calibration and high imprecision of the
estimates. By considering the physical meaning, b1 was given one of three standard
values, viz. b1 = 1/2 or 1 or 2, of which at least one value suits any given river basin.
In the case when b1 equals 1, Equation (6) becomes
st = a2 sm+t 1:
(7)
If there is no precipitation for a considerable length of time, the total river flow is
only groundwater discharge and, in this case, one will have
dt = a2sm+t 1 ;
(8)
dt+1 = a2 (sm+t 1 a2 sm+t 1);
(9)
or
dt = 1 ;
(10)
dt+1 1 a2
where dt is the total monthly discharge. This is the well-known baseflow recession.
Thus b1 is a discrete parameter, while a2 is a continuous one (free parameter)
to be determined by the optimisation procedure.
Fast flow depends on precipitation pt , on other meteorological conditions as
reflected by ept , on the state of the basin as measured by storage smt , and on
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APPLICATION OF WATER BALANCE MODELS
55
the physical characteristics of the basin, which are taken into account by the
introduction of parameters.
An useful quantity is the ‘active’ rainfall, defined as
nt = pt ept (1
exp(
pt =ept )):
(11)
A most efficient equation for fast runoff proved to be
ft = a3 (sm+t 1)b nt ;
(12)
2
where a3 and b2 are nonnegative parameters. For the same reason as in the case of
slow flow, b2 was given one of three standard values, viz. b2 = 1/2 or 1 or 2.
Equation (12) can be seen as a translation of the variable source area concept:
the greater the storage sm+
t 1 , the wetter the catchment, the greater the ‘source’ of
fast runoff, the greater the part of the ‘active’ rainfall running off rapidly.
Since smt varies slowly in contrast with nt , Equations (6) and (12) really can
represent slow and fast flow, respectively.
The total computed discharge is
dt = st + ft :
(13)
The soil moisture storage at the end of the month t is updated by a water balance
equation
smt = smt
p
1+ t
et dt :
(14)
By specifying values of b1 and b2 and by the choice of the evapotranspiration
equations (4) and (5), one finds what will, henceforth, be called a particular model.
Since there are three possible values of b1 and b2 , and two possible evapotranspiration equations, there are 2 3 3 = 18 possible models defined by Equations (2)
through (14), which reflect a large flexibility.
From the above description, one sees that the models are parsimonious with
respect to the number of free parameters (three in the present form). As a consequence, the calibration is easy and proceeds by automatic optimisation. A programme is available which leads to automatically finding the ‘best’ model for a
given catchment based on the criteria given in the next section. Thanks to the simple
structure and monthly time step, the amount of computer work is very small. The
whole procedure for optimising the 18 possible models needs only about 5 minutes
of execution time on an IBM PC-486 computer.
Another advantage of this type of model is that the parameters have close relationships with the physical characteristics of catchments. Application to ungauged
catchments is possible (Vandewiele et al., 1991).
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CHONG-YU XU
3. Optimisation and Evaluation Criteria
3.1. OPTIMISATION
There are different objective functions in use in the literature, depending on the
hypotheses postulated relating to the nature of residual ut . It is common to suppose
that
ut = qt dt ;
where qt is the observed monthly river flow and dt
(15)
is the computed flow. For
statistical analysis, it is convenient to have homoscedastic deviations (i.e. common
variance 2 for all deviations). If they are not, a transformation is usually needed.
Previous studies (Xu, 1992; Vandewiele et al., 1993) show that taking a square root
transformation is a good hypothesis, i.e.,
pq
t =
p
dt + ut ;
(16)
with
ut N (0; 2 );
(17)
i.e., ut is normally distributed with zero expectation and common variance 2 , the
so-called model variance. Moreover, deviations are assumed to be independent,
i.e., for all t,
EXPEC (ut ut
1) =
0;
(18)
where EXPEC is the expectation operator.
The independence of the ut has been discussed by Xu (1992) and Vandewiele
et al. (1993), and it turned out to be a good hypothesis when compared with other
transformations.
To estimate the parameters, the maximum likelihood method was used. Because
of the hypotheses in Equations (16) through (18), maximising the loglikelihood
with respect to the continuous (free) parameters is equivalent to minimising the
sum of squares
SSQ
=
X p
i
qt
p 2
dt ;
(19)
where the sum is extended over all months for which output qt as well as input data
pt and ept are available.
The quality of minimisation was checked by plotting SSQ versus each of the
filter parameters. In that way, it was possible to see whether a global minimum
was reached. This is done for every model-basin combination. Illustrations of this
procedure can be found in Xu (1992).
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APPLICATION OF WATER BALANCE MODELS
3.2. RESIDUAL ANALYSIS
The resulting best model for each catchment was tested in numerous ways by
applying statistical methodology including residual analysis. The general behaviour
of the residuals is judged by graphs of the residuals versus time, the input variables
and computed runoff dt . The residuals versus time graph is used for checking the
absence of trend and also homoscedasticity. The scattergrams of residuals versus
the other variables pt , ept and dt have to be symmetric with respect to the horizontal
axis (zero expectation), and the conditional standard deviation has to be constant
(homoscedasticity).
3.3. MODEL EVALUATION CRITERIA
Model quality: Evidently, standard deviation , as given by Equation (20)
SSQ
= minimum
N K
(20)
R2 = 1 UU
0
(21)
(where N is the number of terms in Equation (19), and K is the number of
model parameters) is an inverse measure of the quality of model performance.
As a criterion of model performance, it is subject to the same objection as SSQ
of Equation (19), i.e. it is a dimensioned quantity, adequate for comparing the
accuracy of various models on a single catchment, but unsuitable for comparing
different models on different catchments.
What is required is a standardisation of the residual variance whose expected
value will not change with the length of the record or the scale of the discharges.
The criterion R2 of Nash and Sutcliffe (1970) is one of the criteria which fills this
need by comparing the residual variance with the initial variance
with
U0 =
XN
q = N1
q
(
t=1 t
XN
q )2 ;
U=
XN
t=1
q
( t
dt )2 ;
q;
t=1 t
where qt and dt are observed and computed discharges, respectively.
Seasonality: Some formal tests are performed. It is checked whether
p
u N
std u
K (N
K; 5%)
(22)
where std u is the standard deviation of ut series; N is the number of terms, and K
is the number of free parameters (three in this case); t(N
K , 5%) is the critical
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58
CHONG-YU XU
value of the student-distribution with N
K degree of freedom and a significant
level of 5%. The same test but restricted to residuals belonging to each season can
be used to check whether there remains a seasonal component in the residuals.
The ‘seasons’ are: autumn (September, October, November), winter (December,
January, February), spring (March, April, May) and summer (June, July, August).
The best model (among the 18 possible models) for each catchment was determined in such a way that the model has the highest R2 value among those with
minimum residual seasonality (number of seasons – out of four seasons – with
significant residuals).
4. A Brief Review of the Characteristics of Water Resources in China
Water resources refers generally to that part of freshwater which is renewable
annually. China, located at the eastern side of the European-Asian Continent,
Monsoons prevail very much in the country. The Chinese water resources are
characterised by uneven regional distribution, uneven time distribution both within
a year and between years, and generally high silt content.
In most regions of China, rainfall is rare in winter and spring, but abundant in
summer and autumn. This is because, in summer and autumn, the wet currents from
the Pacific Ocean and the Indian Ocean bring more rain, while in winter and spring,
the dry and cold currents from the central part of the European-Asian Continent
and the Mongolian Plateau bring less rain. In southern China, about 70–80% of
annual rainfall is measured from April to August. In the northern part, 60–80% of
annual precipitation is collected from July to September, in some years, rainfall in
24 hours of a single storm may exceed the annual mean precipitation, resulting in
an extraordinarily big flood.
Regional water resources classification is given in Figure 2. The distribution
of surface runoff is basically similar to that of the precipitation, but with an even
higher degree of variability. About 45% of the land area belongs to the arid and
semi-arid regions in China with less than 50 mm annual runoff depth.
The aforementioned characteristics combined with the inefficient comprehensive utilisation of water resources, bring about many particular problems in river
control works and water supply. The efficient use of the limited water resources and
rational water resources management are the big challenge for the hydrologists and
water resources engineers in China. This paper is intended to test and eventually
provide a operational tool for the hydrologists and engineers who worked in the
field of water resources assessment and planning.
5. Study Region and Data
Seven catchments (385–2000 km2 ) were used in this study. Six catchments were
chosen from the Pearl River Basin in southern China, and one from the Hai River
Basin in northern China.
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60
CHONG-YU XU
Table I. Hydrological characteristics of the study catchments (1971–1984)
Code
River/Station
Area
Precipitation
(km2 )
(mm/month)
Evapotranspiration
mm(month)
Discharge
2000
156.49
67.65
89.95
0.96
595
164.77
71.71
126.02
0.95
1556
135.16
71.71
79.59
0.94
1881
154.25
76.75
89.35
0.88
385
149.84
72.12
92.96
0.96
1357
120.41
84.33
68.32
0.93
429
56.69
77.25
8.99
2.36
(mm/month)
Coefficient
of variation
of runoff
Southern China (6 basins)
CWJ
CHJT
CFHS
CXGL
CJZ
CST
Wengjiang
at Wengjiang
Tongguangshwi
at Huangjiangtang
Xingzi
at Fenghuangshan
Zhenshui
at Xiaogulu
Andunshui
at Jiuzhou
Chuantonhe
at Shuntan
Northern China (1 basin)
CSFK
Shahe
at Shifuko
6. Results and Discussion
6.1. BEST MODELS
The models and statistical methodology described in Sections 2 and 3 were applied
to the seven catchments. The ‘best’ model (among the 18 possible models) for each
catchment was determined based on the criteria given in section 3. An example is
given in Table II.
It is seen that model number 4 (with evaporation Equation (4), b1 = 1 and b2 =
0.5) is the best model for the catchment, since it has the highest R2 value (0.94)
and minimum seasonality (0 in this case). In the same way, the best models for
each of other catchments were found (Table III).
6.2. SPLIT-SAMPLE TEST
The best model for each catchment is calibrated using the first 10 years (1971–
1980) of the available record, with the parameters optimised to simulate the flows
in the last 4 years (1981–1984) of the available record to investigated the ability of
the models in predicting monthly catchment yields for an independent test period.
The results are also given in Table III. This follows the procedure proposed by
Klemes (1986), although the reverse procedure of calibrating against the second
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Evap.
equation
(4)
(4)
(4)
(4)
(4)
(4)
(4)
(4)
(4)
No.
1
2
3
4
5
6
7
8
9
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
b1
0.5
1.0
2.0
0.5
1.0
2.0
0.5
1.0
2.0
b2
Model characteristics
0.934
0.915
0.912
0.940
0.924
0.915
0.945
0.936
0.922
R
2
0
1
2
0
1
2
1
0
2
Seasonality
Model quality
10
11
12
13
14
15
16
17
18
No.
(5)
(5)
(5)
(5)
(5)
(5)
(5)
(5)
(5)
Evap.
equation
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
b1
0.5
1.0
2.0
0.5
1.0
2.0
0.5
1.0
2.0
b2
Model characteristics
0.936
0.921
0.916
0.941
0.930
0.919
0.940
0.939
0.927
R2
1
1
2
1
1
2
1
1
1
Seasonality
Model quality
Table II. Comparison of model quality (R2 ) and seasonality among 18 possible models for catchment CST (1971–1984)
APPLICATION OF WATER BALANCE MODELS
61
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CHONG-YU XU
half of the record and using the first half of the record as independent test data is not
carried out here. It is seen from Table III that (1) the optimised parameter values
and R2 values are statistically the same for a given catchment when the model is
calibrated on the entire period and on the first part of the available record. This
means that the performance of the model in fitting monthly streamflow when model
parameters are optimised using the entire period of record and using part of the
record is practically the same. (2) The performance of the modelling approach in
estimating streamflow for the calibration and verification periods is also practically
the same. The R2 values and the residual seasonality in the verification period are
almost as good as that in the calibration period except for catchment CSFK where
a relatively lower value of R2 (0.80) and a higher number of seasonality (two in the
case) were obtained for the test period. The reason could be that, for this catchment,
the verification period is much drier than the calibration period. It is found that
the mean of the observed discharge in the calibration period is 4.6 times higher
than the mean of the observed discharge in the verification period. Nevertheless,
more case studies for the semi-arid region need to be carried out before a general
conclusion could be obtained.
In general, the simulations indicate that the proposed modelling approaches can
generally be used with sufficient confidence to predict flows for another period.
6.3. A DETAILED EXAMPLE
For illustration purpose, the calibration and verification results for catchment CST
(1357 km2 ) are presented here in details. The calibration results towards the entire
period are summarised in Table IV. Parameter values with a 95% confidence interval
and the parameter correlations are given in Tables V and VI. It is seen in Tables V
and VI that model parameters are all significantly different from zero, and their
correlations are small in absolute value. A plot of the estimated and observed
monthly runoffs for the last 10 years (1975–1984) is given in Figure 3 which shows
a very good agreement between computed and observed runoff series.
Following the strategy outlined in Section 3, the residual assumptions were
checked using various diagnostic plots, two of which are now discussed.
Plot of residuals versus predicted runoff (Figure 4). It is seen that the variability
of disturbances does not display any noticeable dependence on predicted runoff.
The slight tendency of disturbances to be positive for very low runoffs was found
to be statistically insignificant.
Plot of residuals versus time (Figure 5). The plot shows that residuals were
random without trend or outlier, residuals are homoscedastic.
Comparison between simulated and observed streamflows for the independent
test period is shown in Figure 6. As can be expected from the high R2 values as
shown in Table III, this figure shows a very good agreement between these two
time series.
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8
4
5
8
CST
CXGL
CSFK
16
CHJT
CJZ
8
CWJ
14
No.
code
CFHS
Model
Basin
1971–1984
1971–1980
1971–1984
1971–1980
1971–1984
1971–1980
1971–1984
1971–1980
1971–1984
1971–1980
1971–1984
1971–1980
1971–1984
1971–1980
Period
Calibration
0.627
0.653
0.213
0.212
0.326
0.319
0.775
0.800
0.601
0.560
0.753
0.762
0.564
0.543
10
10
10
10
Parameters
a1
2
2
2
2
0.191
0.237
0.135
0.141
0.122
0.118
0.160
0.145
0.178
0.165
0.083
0.077
0.068
0.081
a2
10
10
10
10
10
10
10
10
3
3
3
3
2
2
3
3
0.122
0.135
0.049
0.050
0.301
0.303
0.098
0.102
0.050
0.051
0.265
0.272
0.153
0.172
a3
10
10
10
10
10
10
10
10
2
2
2
2
2
2
2
2
10 2
10 2
0.92
0.92
0.93
0.95
0.92
0.93
0.87
0.88
0.94
0.95
0.92
0.92
0.90
0.90
Quality
R2
1
1
1
1
0
0
0
0
0
0
0
0
1
1
Seasonality
Table III. The model quality and optimised parameter values of the best models for the 7 study catchments
1981–1984
1981–1984
1981–1984
1981–1984
1981–1984
1981–1984
1981–1984
Period
Verification
0.80
0.91
0.92
0.85
0.91
0.88
0.90
Quality
R2
2
0
1
1
0
0
1
Seasonality
APPLICATION OF WATER BALANCE MODELS
63
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APPLICATION OF WATER BALANCE MODELS
Table IV. Summary of calibration results
Catchment
Calibration period
Model characteristics
Chuantonhe River at Shuntan
1971–1984
Evap. Equation (4), b1 = 1, b2 = 0.5
Efficiency coefficient R2
Seasonality
nonsignificant parameters
parameter correlation
0.94
none
none
small
Mean observed runoff (mm/month)
Mean computed runoff (mm/month)
Contribution of base flow
Contribution of quick flow
92.96
92.87
29%
71%
Table V. Optimised parameter values and the half
width of 95% confidence
interval
a1
a2
a3
0.601
0.178
0.050
0.050
0.040
0.005
6.4. MEAN ANNUAL WATER BALANCES
The monthly values of hydrological variables were calculated for each of the seven
catchments. To summarise the hydrological environment of these catchments, the
long-term mean annual water balance components for all seven catchments are
presented in Table VII, which provide valuable information for planning the water
resources.
7. Summary and Conclusions
A generalised water balance modelling approach has been applied to different
regions in China for water resources assessment and the following general conTable VI. Correlation matrix of
parameters
a1
a2
a3
a1
a2
a3
1
0.15
0.14
0.15
1
0.28
0.14
0.28
1
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66
CHONG-YU XU
Table VII. The computed mean annual water balance for the 7 catchments in percentage of
precipitation (1971–1984)
Catchment Area
Mean prec.
2
Mean evap.
Mean runoff
Base flow
Fast flow
Code
(km ) mm
%
mm %
mm
%
mm %
mm %
CWJ
CHJT
CFHS
CJZ
CST
CXGL
2000
595
1556
385
1357
1881
100
100
100
100
100
100
773
543
681
784
711
652
1111
1439
948
1078
1114
800
59
73
58
58
62
55
704
933
680
605
788
563
38
47
42
33
44
39
407
507
269
473
327
236
22
26
17
26
18
16
CSFK
429
76 11
26
4
1878
1977
1622
1851
1798
1445
680 100
41
27
42
42
40
45
580 85
101 15
clusions hold. (1) The models provide accurate estimates of surface runoff when
compared to measured streamflow, accurate estimates of relative changes in soil
moisture, reliable evapotranspiration estimates under different climatic regimes; (2)
optimised parameter values are significantly different from zero at the 5% level;
correlations between parameters are sufficiently low; (3) residuals are homoscedastic, (4) seasonality is well explained in most cases. A less good result obtained in
the verification of the model on the CSFK catchment is an indication that more
catchments need to be studied for the semi-arid region in China. This was not done
in this investigation because of the limitation of data.
The study shows that these models are both flexible and understandable – two
characteristics that are attractive in any hydrologic modelling technique. Another
significant advantage of the water balance models is their ability to evaluate hydrologic variables on a reduced time-step: from annual to seasonal and to monthly
values. A principal drawback to many of the earliest attempts to evaluate regional
hydrologic variables is the difficulty in determining seasonal or monthly values.
An additional, modern advantage of the water balance models is that detailed water
balance calculations can be done on the existing generation of microcomputers
with higher speed and memory capabilities.
Given these considerations, using of water balance models is a promising
method for evaluating regional water resources not only in China but also in
other regions of the world.
Acknowledgements
I would like to thank Prof. G. L. Vandewiele of Free University Brussels for his
helpful comments and advice in developing the models. Hydrological data used in
the study were provided by the Pearl River Water Resources Commission and the
Ministry of Water Resources of China.
warm1032.tex; 17/02/1997; 12:16; v.7; p.16
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67
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