Water Resources Management 13: 353–368, 1999.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
353
Estimation of Parameters of a Conceptual Water
Balance Model for Ungauged Catchments
C.-Y. XU
Department of Earth Sciences, Hydrology, Uppsala University, Villavägen 16, S-75236 Uppsala,
Sweden, e-mail: Chong-yu.Xu@hyd.uu.se
(Received: 14 December 1998; in final form: 15 November 1999)
Abstract. The problem of parameter estimation constitutes the largest obstacle to successful application of conceptual catchment models in ungauged catchments. This paper investigates the usefulness
of a conceptual water balance model for simulating river flow from catchments covering a wide
variety of climatic and physiographic areas. The model is a 6-parameter water balance model which
was applied to 26 seasonally snow covered catchments in central Sweden. The model was calibrated
on a group of catchments and the calibrated parameter values related to physical catchment indices.
The relationships were tested by comparing observed and simulated runoff records from 4 catchments that were not contained in the regression analysis. The results show that the model can be
satisfactorily applied to ungauged basins in the study region. In order to test the physical relevance
of the model to a wider set of conditions, the model was modified by excluding the snow routine
part. The resulting model and the same technique were tested on 24 catchments taken from northern
Belgium. The verification results were found to be satisfactory.
Key words: parameter estimation, ungauged catchment, water balance model.
1. Introduction
Both physically-based hydrological models and simple conceptual water balance
models are useful tools to address a range of hydrological problems. Physicallybased models have primarily been concerned with the aim of improving understanding of the hydrological phenomena operating in a watershed and of how
changes in the watershed may affect these phenomena (see, e.g., Freeze, 1972;
Smith and Hebbert, 1979; Beven, 1989). Simple conceptual models are useful in
the generation of synthetic sequences of hydrological data for facility design, for
water resources planning and management, and for use in forecasting (see, e.g.,
Allred and Haan, 1996; Xu and Singh, 1998). One problem in using physicallybased models is that the ability of the user to provide such models with the information that they need has not kept pace with model development (for example, the use
of SHE model of Abbott et al., 1986). The main problem in using conceptual water
balance models is related to the determination of parameter values in ungauged
catchments. Unlike complex physically-based distributed-parameter models, unitary parametric values of conceptual lumped-parameter models cannot be derived
354
C.-Y. XU
directly from strongly heterogeneous catchment characteristics. Consequently, empirical regression methods have been used for relating parameters to catchment
characteristics. A few case studies are reported by Jarboe and Haan (1974), who
used a multiple linear regression method to relate each of the four parameters of
Haan’s model to measurable catchment characteristics. Magette et al. (1976) used
Jones’ (1976) procedure to fit a subset of six parameters of the Kentucky watershed
model, and were able to obtain acceptable multiple regression equations using
indices of fifteen watershed characteristics. Weeks and Ashkanasy (1983) related
nine of the sixteen parameters of the Sacramento model to six catchment characteristics, and the resulting regional parameters were found to be satisfactory. In the
above studies, some parameters are derived through secondary relationships with
the use of other predetermined parameters. Hughes (1989), using an isolated event
conceptual model, developed relationships between eight of the twelve parameters
and physical catchment characteristics. More recently, Servat and Dezetter (1993)
related catchment characteristics to parameters of two rainfall-runoff models, one
model has seven parameters and another one has three parameters. Better results
were obtained for the three-parameter model than for the seven-parameter model.
Although ‘few hydrologists would confidently compute the discharge hydrograph
from rainfall data and the physical description of the catchment, nevertheless, this
is a practical problem which must often be faced by practising engineers’ (Nash
and Sutcliffe, 1970).
The problem of parameter estimation still constitutes the largest obstacle to the
successful application of rainfall-runoff models. Clearly, from an operational point
of view, the full benefit of a conceptual model will only be realised to the extent that
it is possible to synthesise data for ungauged catchments. It is therefore appropriate
to continue research in this aspect.
This paper presents the results of one such study. Different features of the
present work are that: (1) Two variants of a conceptual water balance model were
tested. Variant 1, with snow routine included and, variant 2, with snow routine
excluded. (2) In order to test the physical relevance of the model to a wider set of
conditions, 50 gauged catchments ranging in size from 6 to 1293 km2 were investigated. Of which 26 catchments are taken from central Sweden and 24 catchments
from Belgium. Useful and practical results were produced.
2. Study Region and Data
The landscape of central Sweden (Figure 1) is dominated by large lakes and plains
separated from each other by high undulating ridges and rich in faults. The geology is characterised by oldest granites in the northeastern part while sedimentary
gneisses characterise the south. Leptites and hälleflintas are found in the northwestern side together with some small granite-dominated area. Forest and agriculture are the dominant landuse. Forest is a dominant factor in the northwest and
agriculture is concentrated in the south, with meadow and grain cultivation being
ESTIMATION OF PARAMETERS OF A CONCEPTUAL WATER BALANCE MODEL
355
Figure 1. Map of Sweden with the locations of study region and NOPEX region.
predominant agriculture use. General information about catchment characteristics
is presented in Table I. An earlier investigation (Seibert, 1995) has shown that the
distribution of soil type is quite similar to the distribution of landuse, i.e., areas
with forest consist of sandy soil, whereas agricultural areas consist of clay soil.
This level of detail is more typical of situation that might be faced under ‘applied’
conditions rather than ‘research’ conditions.
Belgium (30 500 km2 ) can be divided into three regions according to several
criteria: the Polder region regained from the sea and estuaries has an altitude of
between 0 to 5 m above sea level (a.s.l.) (3000 km2 ); a more or less hilly region
(henceforth to be called northern Belgium) between 5 to 200 m a.s.l. (18 000 km2 );
and the Ardennes plateau (9000 km2 ), mostly higher than 200 m a.s.l. The 24 catchments investigated in this study are located in the northern Belgium region, and the
following discussion is restricted to this region only. As for the climate, evaporation
from a free water surface is spatially fairly uniform. In winter evaporation is very
356
C.-Y. XU
Table I. General information of the 26 study catchments in central Sweden
Station
Abbr.
Code
Meana
evap.
(mm)
Mean
runoff
(mm)
Lake
Forest
(km2 )
Mean
prec.
(mm)
(%)
Area
Basin
slope
(%)
Open
field
(%)
3.3
Åkesta Kv.
Åkers Krut.
Bergsh.
Berg
Backa Ö.
Ak
Ar
Be
Bg
Bo
2216
2249
2300
2218
1374
727
214
21.6
36.5
834
60.1
60.3
55.6
63.9
73.8
39.6
43.3
40.2
43.0
40.7
21.6
17.6
16.3
22.2
32.8
4.0
5.2
0.2
0.0
7.5
69.0
66.3
69.5
71.4
68.7
27.0
28.5
30.3
28.6
23.7
Bernsh.
Dalkarlsh.
Fellingsbr.
Finntorp.
Gränvad
Bs
Dl
Fb
Ft
Gr
1573
2206
2205
2242
2217
595
1182
298
6.96
167
78.0
76.4
62.6
65.9
59.4
43.4
42.4
39.9
43.9
40.9
34.9
35.2
24.6
22.1
19.8
8.6
7.5
6.0
4.7
0.0
77.3
74.6
63.8
95.3
41.1
14.1
17.9
30.2
0.0
58.9
Härnevi
Hammarby
Kåfalla
Kringlan
Karlslund
Ha
Hb
Kf
Kl
Ks
2248
2153
1532
2229
2139
312
891
413
294
1293
60.2
73.3
81.0
78.3
69.7
38.1
43.1
44.3
44.3
43.4
23.3
30.9
36.9
34.2
27.0
1.0
9.5
6.2
7.6
6.6
55.0
80.9
80.8
87.2
62.7
44.0
9.7
13.0
5.2
30.7
3.4
Lurbob
Odensvibr.
Ransta
Rällsälv
Sävja
Lu
Ob
Ra
Rs
Sa
2245
2221
2247
2207
2243
122
110
197
298
722
60.8
63.6
59.8
79.3
59.7
36.3
41.7
38.2
43.1
40.4
25.2
23.3
22.3
38.4
19.5
0.3
6.3
0.9
7.4
2.0
68.2
71.0
66.1
78.8
64.0
31.5
22.7
33.0
13.8
34.0
4.0
Skräddart.
Skällnorab
Stabbyb
Tärnsjöb
Ulva Kv.
Vattholma
Sd
Sn
St
Ta
Ul
Va
2222
1843
1742
2299
2246
2244
17.7
58.5
6.18
13.7
976
293
66.7
55.0
56.4
59.7
61.2
60.6
41.6
39.9
36.2
39.5
43.9
40.8
25.3
16.2
18.7
21.8
17.1
21.0
2.5
10.4
0.0
1.5
3.0
4.8
96.1
44.5
95.6
84.5
61.0
71.0
1.4
45.1
4.4
14.0
36.0
24.2
2.7
3.5
1.5
10.3
7.1
1.6
1.8
a Actual evapotranspiration calculated by the model.
b Catchments used for independent testing of the regression equations.
low and in summertime it averages about 100 mm per month. Precipitation does
not show an important seasonality. The mean is 60–70 mm per month. Snow and
frost are not important on a monthly scale. From the geological point of view, 57%
of the subsoil in the region is sandy, 30% is clayey and less than 13% of the subsoil
357
ESTIMATION OF PARAMETERS OF A CONCEPTUAL WATER BALANCE MODEL
Table II. General information of the Belgian catchments
River
Station
Code
Area
Runoff
coeff.
Molenbeek
Molenbeek
Heulebeek
Zwalm
Markebeek
Massemen
Geraadsb.
Heule
Nederwalm
Etikhove
A24/2
A32
A523a
A527
A528
44.8
0.36
19.1
0.33
92.7
0.33
114.0 0.34
50.8
0.29
Bellebeek
Aa
Dijle
Mark
Warmbeek
Essene
Poederlee
S’t Joris
Vian
Hamot-A.
A529
A531/2
A535
A565
A578
Ede
Mangelbeek
Grote Nete
Grote Nete
Grote Nete
Maidegem
Lummen
Varendonk
Hulshout
Itegem
Klein Nete
Dijle
Klein Molenb.
Grote Molenb.
Dijle
Gete
Molenbeek
Grote Kemmel
Jeker
LIT
(%)
MBS
FOC
(%)
DD
30.95 1.32
27.69 2.47
27.41 0.47
51.32 1.26
38.64 0.20
6.39
1.76
4.99
4.75
7.44
2.22 11.31
2.25 7.11
1.40 9.63
2.10 8.33
1.85 10.46
89.0
0.36
204.0 0.33
645.0 0.27
171.0 0.28
57.4
0.42
37.70 0.85
90.03 0.15
91.37 0.54
13.72 0.76
91.58 0.49
5.29
15.16
13.58
10.39
22.47
2.07 10.12
0.74 9.97
0.24 8.63
1.91 4.35
0.88 8.42
A816a
A879/4
A94
A98
AZ002
46.0
109.0
304.0
468.0
532.0
0.36
0.40
0.45
0.44
0.43
80.36
90.31
93.33
81.92
82.19
0.17
0.56
0.43
0.27
0.14
13.20
14.54
18.75
12.38
13.22
0.06
1.32
1.09
1.97
0.51
Grobbend.
Bierges
Liezele
Malderen
Wijgmaal
AZ005
D0099
D0361a
D0371
D0931
526.0
314.0
33.9
66.4
89.0
0.53
0.27
0.30
0.31
0.25
91.96
92.13
0.00
20.26
86.38
0.09
0.74
1.29
1.02
1.21
22.65
17.22
6.76
6.87
26.15
0.20 8.04
0.96 5.71
2.06 11.46
2.10 11.00
1.85 13.62
Halen
Iddergem
Vlamert
Kanne
D1521
D2941
D4933
D5531
810.0 0.23
15.9
0.34
23.8
0.30
463.0 0.19
2.32
7.77
0.59
9.64
0.77
2.49
2.25
1.53
83.63 0.26
52.76 1.76
17.46 1.83
93.53 0.65
URB
7.02
9.68
6.67
10.68
7.50
4.97
19.00
1.94
6.47
LIT = lithologic index is defined as percent of the catchment occupied by well permeable soil which
is the sum of three lithological variables, percent of sand, percent of gravel and percent of chalk.
URB = urbanization index in percentage is defined as the ratio of urbanized area to total catchment
area. DD = drainage density is defined as the length of all order streams that drain the basin per
unit catchment area and has dimension (L/L2 ). MBS = mean basin slope is defined as the ratio
of maximum elevation difference to the square root of the catchment area. FOC = forest cover in
percentage.
a Catchments used for independent testing of the regression equations.
358
C.-Y. XU
Table III. Principal equations of the MWB-6 monthly snow and water balance model
Snow fall
Rainfall
Snow storage
Snowmelt
Potential evapotranspiration
st = pt {1 − exp[−(ct − a1 )/(a1 − a2 )]2 }+
rt = pt − st
spt = spt −1 + st − mt
mt = spt {1 − exp[(ct − a2 )/(a1 − a2 )]2 }+
ept = (1 + a3 (ct − cm ))epm
Actual evapotranspiration
Slow flow
Fast flow equation
Total computed runoff
Water balance equation
et = min[ept (1 − a4 t t ), wt ]
2
bt = a5 (sm+
t −1 )
+ 2
ft = a6 (smt −1 ) (mt + nt )
dt = bt + ft
smt = smt −1 + rt + mt − et − dt
w /ep
a1 >a2
06a4 61
a5 >0
a6 ≥0
+
Where: wt = rt + sm+
t −1 is the available water; smt −1 = max(smt −1 , 0) is the available
storage; nt = rt − ept (1 − e−rt /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.
is covered by hard rock, alluvial and gravel, etc. General information about the
Belgian catchments characteristics is presented in Table II.
3. Water Balance Model
MWB-6 (Xu et al. 1996) is a typical monthly water and snow balance model. It was
originally developed for investigation of water balance of the NOPEX (Halldin
et al. 1999) area and Nordic region. The principal equations of the MWB-6 are
presented in Table III. The input data to the model are monthly values of areal
precipitation, long-term monthly average potential evapotranspiration and air temperature. Precipitation pt is first divided into rainfall rt and snowfall st by using
a temperature index function. Snowfall is added to the snowpack spt at the end
of the month, of which a fraction mt melts and contributes to the soil moisture
storage smt . Parameter a1 and a2 are threshold temperature which determine the
form of precipitation and the rate of snowmelting. Before rainfall contributes to
the soil storage as ‘active’ rainfall, a small part is subtracted and added to evapotranspiration. The latter storage contributes to evapotranspiration et , to the fast
component of flow ft , and to slow flow st . Parameter a3 is used to convert long-term
average monthly potential evapotranspiration to actual values of monthly potential
evapotranspiration. It can be eliminated from the model if potential evapotranspiration data are available or calculated using other methods. Parameter a4 determines
the value of actual evapotranspiration that is an increasing function of potential
evapotranspiration and available water. The smaller the values for a4 , the greater
the evaporation losses at all moisture storage states. The slow flow parameter a5
controls the proportion of runoff that appears as ‘base flow’ and higher values of
ESTIMATION OF PARAMETERS OF A CONCEPTUAL WATER BALANCE MODEL
359
Table IV. Principal equations of the simplified MWB-3 monthly water
balance model
Actual evapotranspiration
Slow flow
Fast flow equation
Total computed runoff
Water balance equation
w /ep
et = min[ept (1 − a1 t t ), wt ]
2
bt = a2 (sm+
t −1 )
+ 2
ft = a3 (smt −1 ) (nt )
dt = bt + ft
smt = smt −1 + rt − et − dt
06a1 61
a2 >0
a3 >0
+
Where: wt = pt + sm+
t −1 is the available water; smt −1 = max(smt −1 , 0) is
−r
/ep
t
t
the available storage; nt = pt − ept (1 − e
) is the active rainfall; pt
is monthly precipitation; ai (i = 1, 2, 3) are model parameters.
a5 produce a greater proportion of ‘base flow’. It seems likely then that higher
values are expected in forest area than in open field and in sandy soil than in clayey
soil. The fast flow parameter a6 will increase with the degree of urbanisation,
average basin slope, and drainage density, and lower values should be expected
for catchments that are dominated by forest.
In case snowfall and frost are not a significant factor, the MWB-6 model as
described in Table III can be simplified by eliminating the snow routine part. The
simplified version of the model, MWB-3, is presented in Table IV and this variant
of the model was used for the Belgian catchments.
4. Methodology of Parameter Estimation for Gauged Catchments
Parameters of conceptual hydrological models can be inferred by either subjective trial-and-error fitting (e.g., Pitman, 1976) or by using automatic optimisation
routines (e.g., Ibbitt and O’Donnell, 1971; Kuczera, 1983). James (1972) argued
that only rigid adherence to a standard optimisation procedure would enable compilation of a sufficiently comprehensive data base for use in regression studies
relating model parameters to catchment characteristics. In this study, optimisation
was performed with the help of the VA05A computer package (Hopper, 1978;
Vandewiele et al., 1992). The VA05A program was supplemented with a program
called EOX4F (NAG Fortran Subroutine Library, 1981). The results relating to
the statistical regression of formulation, such as the variance-covariance matrix,
the correlation matrix, the variance and the confidence interval for the individual
parameters, could then be calculated.
360
C.-Y. XU
5. Results and Discussion
5.1.
THE SWEDISH CATCHMENTS
The 26 catchments as shown in Table I were divided into two groups. One group
(22 catchments), denoted as calibration catchments, was used to develop regression
equations for predicting the model parameters. The other 4 catchments were used
in testing the established regression equations. The 4 test catchments are LU, SN,
ST and TA (see Table I for details).
The optimised parameter values for the six parameters of the MWB-6 model
were obtained for each catchment by using the technique suggested by Xu et al.
(1996). These optimum parameter values can be found in Table V. It is seen that
parameter a3 is nearly constant with a standard deviation of 0.026. As was discussed in Section 3, this parameter will be eliminated from the model if potential
evapotranspiration data are available or calculated using other methods. Parameter
a5 varies most from catchment to catchment, and a standard deviation of 0.129 was
found for this parameter. The regional variability of other parameters is between
these two extremes. The parameter values for the 22 calibration catchments became
the dependent variables, and the catchment characteristics became the independent
variables in the regression analysis. In this study, for each catchment an estimate
was made of the percentages of the catchment covered by each land use type (i.e.
lake %, forest (sandy soil) %, and open-field (clayey soil) %).
Multiple linear regression models of the form y = m + n1 x1 + n2 x2 + . . . +
nk xk , where y is the model parameter, x1 , x2 , . . ., xk are catchment characteristics,
and m, n1 , n2 , . . ., nk are constants, were investigated. Variables (x1 , x2 , . . .) which
had the highest partial correlation with the dependent parameter (y) were added to
the prediction equation in a stepwise manner. The final multiple linear equations
relating the optimised parameters and catchment characteristics are the following:
a1 = 0.106 + 0.0229×Lake % + 0.000695×Forest (sandy soil) %,
where the multiple correlation coefficient is 0.80 and the standard error of the
estimate is 0.057.
a2 = −0.0395 − 0.0340×Lake % − 0.00297×open-field (clayey soil) %,
where the multiple correlation coefficient is 0.66 and the standard error of the
estimate is 0.099.
a5 = 0.0949 + 0.0350×Lake % + 0.000999×open-field (clayey soil) %,
where the multiple correlation coefficient is 0.76 and the standard error of the
estimate is 0.088.
a6 = 0.085 − 0.0116×Lake % + 0.000889×Forest (sandy soil) %,
361
ESTIMATION OF PARAMETERS OF A CONCEPTUAL WATER BALANCE MODEL
Table V. Optimized parameter values and land-use data for the 22 Swedish catchments used in
regression analysis
Basin
Code
Optimised parameters
a1
a2
a3
a4
a5
a6
Land use data
Lake Forest
Open
field
(%)
AK
AR
BE
BG
BO
0.309
0.305
0.095
0.169
0.489
–0.151
–0.270
–0.244
–0.144
–0.523
0.120
0.074
0.089
0.097
0.073
0.567
0.434
0.508
0.560
0.634
0.274
0.238
0.114
0.056
0.445
0.126
0.039
0.149
0.070
0.066
4.0
5.2
0.2
0.0
7.6
68.1
66.3
69.5
71.4
68.8
27.9
28.5
30.3
28.6
23.7
BS
DL
FB
FT
GR
0.253
0.349
0.294
0.328
0.110
–0.382
–0.397
–0.192
–0.146
–0.234
0.060
0.070
0.114
0.034
0.062
0.560
0.653
0.663
0.481
0.534
0.475
0.365
0.337
0.107
0.110
0.074
0.051
0.078
0.126
0.125
8.6
7.5
6.0
4.7
0.0
77.3
74.6
63.8
95.3
41.1
14.1
17.9
30.2
0.0
58.9
HA
HB
KF
KL
KS
0.167
0.361
0.260
0.290
0.318
–0.234
–0.406
–0.480
–0.254
–0.489
0.061
0.065
0.046
0.064
0.096
0.648
0.488
0.511
0.449
0.481
0.172
0.413
0.375
0.294
0.234
0.141
0.083
0.046
0.074
0.047
1.0
9.5
6.2
7.6
6.6
55.0
80.8
80.8
87.2
62.7
44.0
9.7
13.0
5.2
30.7
OB
RA
RS
SA
SD
UL
VA
0.245
0.168
0.312
0.172
0.270
0.232
0.240
–0.132
–0.175
–0.359
–0.196
–0.141
–0.186
–0.154
0.097
0.082
0.100
0.095
0.020
0.030
0.063
0.602
0.637
0.625
0.468
0.582
0.333
0.457
0.229
0.233
0.481
0.316
0.243
0.188
0.468
0.063
0.161
0.035
0.173
0.163
0.090
0.117
6.3
0.9
7.4
2.0
2.5
3.0
4.8
71.0
66.7
78.8
63.5
96.1
60.6
71.4
22.7
32.4
13.8
34.5
1.4
36.4
23.8
Mean
St. dev.
0.261
0.091
–0.268
0.127
0.073
0.026
0.540
0.087
0.280
0.129
0.095
0.044
4.6
3.0
71.4
12.5
24.0
14.0
St. dev. = standard deviation.
Parameters a1 and a2 have been scaled down by 0.1; parameters a5 and a6 have been scaled up
by 1000 and 10 000, respectively.
362
C.-Y. XU
where the multiple correlation coefficient is 0.72 and the standard error of the
estimate is 0.032.
The parameters that did not have good correlation with land-use data are a3 and
a4 . As discussed previously, parameter a3 is nearly constant. As for a4 , a previous
study (Xu et al., 1995) has shown that this parameter is highly correlated with
the mean basin slope. The mean basin slope data are available only for the 10
NOPEX catchments as indicated in Table I, which are too few to apply regression
models. Therefore, the regionalised values (average values) calculated from the 22
calibrated catchments will be used for these two parameters, i.e., a3 = 0.073 and a4
= 0.54 (see Table V).
5.2.
THE BELGIAN CATCHMENTS
The 24 catchments as shown in Table II were divided into two groups. One group
(21 catchments), denoted as the calibration catchments, was used to develop the
regression equations for predicting model parameters. The other 3 catchments were
used for testing the established regression equations. The 3 test catchments are
A816, A523 and D0361 (see Table II for details).
The optimised parameter values for the three parameters of the MWB-3 model
were obtained for each catchment by using the same technique (Table VI). Using
the same regression procedure, the following equations are obtained:
a1 = 0.348 + 0.0052×LIT − 0.00225×FOC + 0.0373×DD,
where the multiple correlation coefficient is 0.92 and the standard error of the
estimate is 0.065.
a2 = 0.117 − 0.000836×LIT + 0.00481×URB − 0.0121×DD,
where the multiple correlation coefficient is 0.42 and the standard error of the
estimate is 0.058.
a3 = 0.101 − 0.000909×LIT + 0.0131×MBS − 0.00185×URB,
where the multiple correlation coefficient is 0.90 and the standard error of the
estimate is 0.018.
The quality of regression equations is better for the MWB-3 model. This is
logical, since ‘the small number of parameters increases the information content
per parameter and therefore allows both a more accurate determination of the
parameter and a more reliable correlation of the values obtained with catchment
characteristics’ (Dooge, 1977).
363
ESTIMATION OF PARAMETERS OF A CONCEPTUAL WATER BALANCE MODEL
Table VI. Optimized parameter values and catchment data for the 21 Belgian catchments
used in regression analysis
Basin
Code
Optimised parameters
a1
a2
a3
Catchment characteristics
LIT
MBS FOC
DD
URB
A135
A24/2
A32
A527
A528
0.4920
0.6250
0.5529
0.6268
0.5746
0.0565
0.1640
0.1376
0.1376
0.0782
0.9459
0.8491
0.8733
0.6001
0.3096
20.95
30.95
27.69
51.32
38.64
0.52
1.32
2.47
1.26
0.20
7.14
6.39
1.76
4.75
7.44
1.75
2.22
2.25
2.10
1.85
9.05
11.31
7.11
8.33
10.46
A529
A531/2
A535
A565
A578
0.6584
0.6469
0.8883
0.4315
0.7758
0.1042
0.1125
0.0073
0.0816
0.1158
0.3292
0.2794
0.0058
0.8864
0.1818
37.70
90.03
91.37
13.72
91.58
0.85
0.15
0.54
0.76
0.49
5.29
15.16
13.58
10.39
22.47
2.07
0.74
0.24
1.91
0.88
10.12
9.97
8.63
4.35
8.42
A879/4
A94
A98
AZ002
D0099
0.8325
0.8244
0.8425
0.7492
0.8815
0.0509
0.1279
0.0878
0.2087
0.0090
0.0918
0.2047
0.1402
0.3571
0.0078
90.31
93.33
81.92
82.19
92.13
0.56
0.43
0.27
0.14
0.74
14.54
18.75
12.38
13.22
17.22
1.32
1.09
1.97
0.51
0.96
9.68
6.67
10.68
7.50
5.71
D0371
D0931
D1521
D2941
D4933
D5531
0.5975
0.8476
0.8261
0.6322
0.5170
0.9021
0.1090
0.0073
0.0134
0.1654
0.0979
0.0027
0.5011
0.0072
0.0178
0.5454
1.2751
0.0012
20.26
86.38
83.63
52.76
17.46
93.53
1.02
1.21
0.26
1.76
1.83
0.65
6.87
26.15
2.32
7.77
0.59
9.64
2.10
1.85
0.77
2.49
2.25
1.53
11.00
13.62
4.97
19.00
1.94
6.47
Mean
St. dev.
0.7012
0.1458
0.0893
0.0590
0.4005
0.3771
61.33
31.05
0.83
0.62
10.66
6.78
1.56
0.66
8.81
3.57
St. dev. = standard deviation.
Parameters a1 and a2 have been scaled down by 0.1; parameters a5 and a6 have been
scaled up by 1000 and 10 000, respectively.
LIT, URB, DD, MBS and FOC have the same meaning as in Table II.
5.3.
PREDICTIONS ON UNGAUGED CATCHMENTS
The parameters calculated from the prediction equations were used to simulate
runoff on the test catchments. The average monthly observed runoff, the average monthly calibrated runoff and the average monthly simulated runoff were
calculated. The error of prediction, in percent, was defined as 100 times the differ-
364
C.-Y. XU
Table VII. Mean monthly observed, calibrated and simulated runoff for
the test catchments
Catchments
Qobs
runoff
(mm)
Qcal
runoff
(mm)
Relative
error
(%)
Qsim
runoff
(mm)
Relative
error
(%)
Swedish catchments
ST
SN
LU
TA
18.74
16.22
25.20
21.76
19.48
16.01
25.63
21.83
3.94
–1.29
1.70
0.32
18.88
16.62
21.78
22.80
0.75
2.44
–13.57
4.83
Mean
20.48
20.74
1.26
20.02
–2.25
Belgian catchments
A523
A816
D0361
20.07
22.95
19.45
19.92
22.01
19.94
–0.75
–4.10
2.52
16.95
21.05
18.20
–15.54
–8.28
–6.43
Mean
20.82
20.62
0.96
18.73
–10.00
Qobs , Qcal , Qsim , are mean monthly observed runoff, model calculated
runoff using calibrated parameter values, and model calculated runoff
using parameter values that are estimated from catchment characteristics,
respectively.
Relative error (%) = (Qcal − Qobs )/Qobs ×100 or (Qsim −
Qobs )/Qobs ×100.
ence between observed and calibrated, and between observed and simulated mean
monthly runoff divided by the average observed monthly runoff. These values are
presented in Table VI.
For the four Swedish test catchments, the minimum relative error between mean
observed and predicated runoff is 0.75%, the maximum error is –13.57%, and the
average relative error is –2.42%. Given an uncertainty of 5–10% in the observed
runoff values themselves, this test suggests that the model is capable of estimating
monthly runoff for ungauged catchments within acceptable margins of error. For
the three Belgian test catchments, the range of the absolute value of the relative
error is 6.4–15.9%, which is also acceptable for practical applications.
The observed, calibrated and simulated hydrographs were plotted at a monthly
time step for all 7 test catchments. Two examples (a best case and a worst case)
of plots are given in Figures 2 to 5 for Swedish and Belgian catchments, respectively. They again indicate that reasonable hydrograph estimates can be obtained for
ungauged areas for both model variants and in both countries.
ESTIMATION OF PARAMETERS OF A CONCEPTUAL WATER BALANCE MODEL
365
Figure 2. Comparison of observed (solid line), calibrated (line with plus) and simulated (line
with spot) monthly hydrographs for the catchment SN in central Sweden (best case).
Figure 3. Comparison of observed (solid line), calibrated (line with plus) and simulated (line
with spot) monthly hydrographs for the catchment LU in central Sweden (worst case).
Figure 4. Comparison of observed (solid line), calibrated (line with plus) and simulated (line
with spot) monthly hydrographs for the catchment D0361 in northern Belgium (best case).
366
C.-Y. XU
Figure 5. Comparison of observed (solid line), calibrated (line with plus) and simulated (line
with spot) monthly hydrographs for the catchment A523 in northern Belgium (worst case).
Compared with a few similar studies reported in the hydrological literature
(Jarboe and Haan, 1974; Weeks and Ashkanasy, 1978; Hughes, 1989; Servat and
Dezetter, 1993), the results shown in the present study are promising in several
aspects. First, the average error of simulation (–2.25% for Swedish test catchments,
–10% for Belgian test catchments) was among the lowest. Second, all earlier studies were limited to single climatic and physiographic zones. Larger catchment
samples are tested in the present study, which include the boreal forest region
(central Sweden) and temperate region (northern Belgium). Third, all earlier studies were carried out in snow and frost-free catchments. Seasonally snow covered
catchments are investigated in this study.
6. Conclusions
Two variants of the NOPEX water balance model were studied in terms of their
applicability to ungauged catchments. Variant 1 of the model (MWB-6) was tested
on 26 seasonally snow-covered catchments in central Sweden. Variant 2 of the
model (MWB-3) that excludes snow routine part was tested on 24 catchments
in northern Belgium. In both cases, the optimum parameters were regressed on
a group of catchment characteristics. Predicted equations were derived and used
to calculate model parameters from the catchment characteristics for independent
test catchments. Using these calculated parameters, simulations of the runoff record
were made and compared with observed runoff. Good agreement between observed
and calculated runoff exists not only for the long-term mean values but also for the
monthly hydrograph.
This study demonstrates that the parameters of the NOPEX water balance model
are indeed physically relevant and that these optimum parameters can be estimated
from catchment characteristics. It should also be emphasised that both study regions (central Sweden and northern Belgium) belong to humid climate. Therefore,
the model and the technique need to be tested in arid and semi-arid regions. This
ESTIMATION OF PARAMETERS OF A CONCEPTUAL WATER BALANCE MODEL
367
has not been done due to the lack of data. Anyhow, results obtained are encouraging
and further research in this respect is warranted.
Acknowledgements
The research is partially financed by NFR (Swedish Natural Science Research
Council). The data used in this investigation was obtained from the SINOP (System for Information in NOPEX) database and the Belgian Royal Meteorological
Institute. The Swedish Meteorological and Hydrological Institute (SMHI) provided
most of the data to SINOP. Professor V. P. Singh (Louisiana State University)
carefully read the revised version of the paper and made a number of corrections.
I am also grateful to the referees who provided helpful suggestions that led to an
improved manuscript.
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