jffydr&i&gmMl Seismees -Journal™ des Sciences Hydrologiquesf39,2, April 1994
1*5 /
Sensitivity of monthly rainfall-runoff
moiels to Input errors and data length
C.-Y. MJ & G. L. VANDEWIELE
Laboratory of Hydrology, Free University Brussels, Pleinlaan 2,
B-1050 Brussels, Belgium
Abstract Two problems are addressed which arise when using monthly
water balance models as an aid to making decisions in water resources
engineering: what is the influence of data errors on model performance,
and what is the data length required in order to obtain reliable models?
Two previously defined types of models are used: in PE type models the
input series are precipitation and potential évapotranspiration; in P type
models the only input is precipitation. The main conclusions are:
(1) random errors in precipitation data, when great enough, affect model
performance significantly; (2) systematic errors in precipitation data are
less important for the estimation of river flow; and (3) a data length of
10 years is necessary and sufficient for a reliable calibration of monthly
water balance models of humid basins.
Sensibilité de modèles pluie-débit à l'échelle mensuelle aux
erreurs de mesure concernant les précipitations et à la durée
des chroniques de mesures utilisées
Résumé Deux problèmes se posent lors de l'utilisation de modèles de
bilan d'eau en tant qu'outils d'aide à la décision en matière de
planification des ressources en eau: quelle est l'influence des erreurs de
mesure sur les performances du modèle et quelle est la durée des
chroniques nécessaires à l'obtention d'un modèle fiable? Deux types de
modèles, définis auparavant, ont été étudiés. Les données d'entrée des
modèles PE sont les précipitations et l'évapotranspiration potentielle. Les
précipitations constituent la seule entrée des modèles P. Les principales
conclusions sont les suivantes: (1) des erreurs aléatoires supérieures • à
10 % sur les précipitations affectent de façon significative la qualité des
résultats du modèle; (2) des erreurs systématiques sur les précipitations
revêtent moins d'importance, en particulier pour les modèles de type P;
et (3) une chronique de mesures d'une durée de dix ans est nécessaire et
suffisante pour un calage fiable.
INTRODUCTION
Monthly rainfall-runoff models (or water balance models) are useful tools in
the hands of engineers in charge of water resources projects. Such models are
helpful in computing forecasts and in generating arbitrarily long runoff series.
The latter can be used to estimate return periods of relatively rare hydroîogical
events such as droughts. More details of these applications are given, for
example, by Xu & Vandewiele (1993) and Vandewiele et al. (1993).
For engineering use, models have to fulfil two conditions: the data
Open for discussion until 1 October 1994
158
C.-Y. Xu & G. L. Vandewiele
necessary for calibration have to be readily available, and calibration must be
easy. The latter condition leads to the requirement that only a few parameters
(unknown basin characteristics to be estimated) are used: one should be
parsimonious with respect to the number of parameters. In the past a number
of such models have been defined (Alley, 1984; Vandewiele & Xu, 1991;
Vandewiele et aï., 1992; Xu & Vandewiele, 1993).
An important question concerning such engineering models is their
sensitivity to errors in the input data (especially precipitation data), and to the
length of data to be used during calibration. Errors in areal precipitation have
two important sources (apart from gross errors): measurement errors in point
precipitation and Thiessen errors. The present paper aims at giving the engineer
an indication of the influence of such errors on model performance. At the
same time the paper tries to give an indication as to the question how long data
series have to be for obtaining a reliable model, or if the improvement of
model performance is worth the extra effort involved in data collection and
preparation.
The Thiessen error has been studied in a number of works like Sutcliffe
(1966), Herbst & Shaw (1969), and Singh & Birsoy (1975), and the influence
of input errors by Singh & Woolhiser (1976). Model reliability in general has
been treated by O'Donnell & Canedo (1980) on a daily model for the humid
case, and Gorgens (1983) on a monthly model for a semiarid catchment. The
present paper treats these reliability problems in relation to the new class of
water balance models defined by Xu & Vandewiele (1993) and Vandewiele
et al. (1992, 1993) applied to humid catchments in Belgium and southern
China. Vandewiele et al. (1992) showed that the latter models perform much
better than a number of previous models, which were studied by Alley (1984).
First the model structure used here will be explained by means of its
equations. In the subsequent section some details of statistical analysis are
discussed, especially the question of model performance. Two sections are then
devoted to input errors and data length respectively. In the final section
conclusions are drawn.
LUMPED SYSTEM WATER BALANCE MODELS FOR HUMID
BASINS
In the present section the model equations are introduced together with some
explanations and justifications. A full discussion can be found in the above
mentioned papers and also in a booklet by Vandewiele et al. (1993).
Two types of water balance models are discussed here, depending on
which input data are used: models of type PE use monthly areal precipitation
pt and monthly potential évapotranspiration et, either calculated by the Penman
method or observed by evaporation pan; models of type P use only monthly
areal precipitation^ (t is time in months). For calibration and use of type P
models no observed PET data are necessary and this can prove a necessity
when no such data are available; instead, generated periodic PET data are used.
159
Sensitivity of monthly rainfall-runoff models
General model structure
Monthly areal precipitation/?, and potential évapotranspirations, are considered
to be the inputs transformed by a so-called filter into a computed discharge d„
which should be the monthly observed runoff qt, except for a deviation ut
(Fig. 1). The input series are the "observed" factors. Clearly, discharge is also
influenced by other phenomena, the "unobserved" factors, such as measurement
errors, Thiessen errors, the nonhomogeneity of rainfall during the month,
model imperfections, etc. Accordingly, qt is considered to be a random variable
resulting from a deterministic function (rainfall-runoff filter) of the inputs on
the one hand and of a random deviation u, on the other, as represented in
Fig. 1.
deviation
v
precipitation
Pt
Rainfall-runoff
filter
computed
runoff
u,
observed
runoff
Compounding
Rule
qt
observed PET
ei
Fig. 1 General structure of rainfall-runoff models. "Compounding
rule " does not necessarily mean "addition " (see text).
There are different ways of "compounding" computed runoff dt and
deviation w, (see Fig. 1), of which the commonest is to suppose that ut =
qt - dt. However for statistical analysis it is convenient to have homoscedastic
deviations (i.e. common variance a1 for all deviations w,). Experience shows
that a square root transformation of the flows solves this problem. Therefore
it is supposed that:
(1)
^~+«,
with
(2)
2
i.e., u, is normally distributed with zero expectation and variance a , the socalled model variance. Moreover the deviations u{ are supposed to be
independent and this turns out to be a good hypothesis.
Genera! filter structure
From past and present values of the input series a new time series m, is
computed which represents the state of the catchment at the end of month t,
and is to be interpreted as a moisture index, soil moisture content or storage;
mt summarizes the memory of the catchment. This is expressed by the balance
equation:
C.-Y. Xu & G. L. Vandewiele
160
mt = mt_i+pt-rt-dt
(3)
where rt is actual évapotranspiration during month t. All quantities in the
balance equation are expressed in mm depth. Storage mt in equation (3) is
allowed to become negative (i.e. a deficit), in order to keep a real water
balance. If the storage mt is negative, it has to be refilled before a contribution
to runoff becomes possible. This will appear in the subsequent equations (5),
(13) and (15).
A distinction is made between a slowly responding component of flow
(or slow flow for short) st and a quickly responding component of flow (or fast
flow for short) ft such that:
dt = st+ft
(4)
Slow and fast flows can more or less be interpreted as baseflow and
direct runoff, although they are not identical to them, since it is impossible to
distinguish properly with a monthly model between baseflow and direct runoff.
The size of the slow response is dominated by moisture storage alone, while
the rapid response depends on the concurrent precipitation input as well as
moisture storage.
In PE models rt, st mdf; are time invariant functions of pr, mtA and et:
r, - r( p„ mlA, et)
s, = s( p„ m,.lt et)
ft=APt>
« n . et)
whereas in P models the corresponding equations are:
r, = K p„ w M )
st = s( pt, wiM)
ft=A
Pt> mt-i)
Rainfall-runoff filters differ by their functions r( • ), s(') and/(• ).
Evapotranspiration equations for PE models
In PE models potential évapotranspiration (PET) is measured by an evaporation
pan or computed by a formula such as Penman's, based on the measurement
of several variables. PET is an input series.
For computing monthly actual évapotranspiration rt, two quantities
(among others) are important: the monthly potential évapotranspiration e„
Sensitivity of monthly rainfall-runoff models
161
(evaporation for short), and the available water wt during month t defined as:
(5)
wt = Pt+m,^
where m*_x = max(w,_x,0) is the available storage.
For evident reasons, a good évapotranspiration equation must be such
that:
rt increases with e{ and wt
(6)
r,-0
when wt = 0 or et = 0
(7)
rt < et
and rt < wt
(8)
rt -* et
when wt -» oo
Two equations proved to be efficient. The first one is:
. J
/,
w/max(e,,l)\
(9)
I
rt = mm\e^l-al
J,wJ
(10)
where the symbol at is a parameter (an unknown constant to be estimated),
which is a characteristic of the river basin under study. This parameter is
constrained by 0 <, ax <, 1 because of the conditions (6) through (9). Equation
(10) is represented in Fig. 2. The expression max(e„ 1) is used in the exponent
instead of et itself, in order to avoid occasional division by zero.
a^ f
W,
/A
-Ina^w,
&^
f
1
^7ï
1
w,
Fig. 2 Graphical representation of the évapotranspiration
equation (10). The left diagram shows curves with wt = constant,
the right diagram shows curves with et = constant.
The second equation, based on Romanenko (1961) (see also Hounam,
1971), is:
r, = min{H>,(l-e"~e'/<V,}
(11)
with parameter ax constrained by ax > 0. Equation (11) is represented in
Fig. 3. Both equations fulfil conditions (6) through (9).
162
C.-Y. Xu & G. L. Vandewiele
et
a, = o
w,
a,=o
X
X
w,
w,
Fig. 3 Graphical representation of the évapotranspiration
equation (11). The left diagrams show curves with wt = constant,
therightdiagrams show curves with et = constant.
An important difference from the point of view of interpretation between
equations (10) and (11) is that in equation (11) one has:
when e, -* <»
w,
whereas this is not the case in equation (10). This condition in fact requires that
all available water is used up in évapotranspiration when the available energy
(measured by e) is very great.
Evapotranspiration equations for P models
In P models observed PET data are taken to be unavailable. Instead a periodic
"driving force" is introduced, which for easiness is denoted by the same
symbol et and is given by:
et= ja 4 +a 5 sin ^ | ( r - Û 6 )
1
(12)
Equation (12) contains three parameters, a4, a5 and a6, which are
characteristics of the basin under study. This equation is justified by the fact
that PET varies more or less in a periodic sinusoidal way, and that errors in
PET input are less important than errors in precipitation input in a water
balance model (Xu, 1992) (this does not mean that évapotranspiration is not an
important term in the water balance). Equation (12) is to be inserted in
Sensitivity of monthly rainfall-runoff models
163
equations (10) or (11) instead of observed PET input. The plus sign at the end
of equation (12) is necessary to avoid negative values of et which otherwise
may occur in rare cases.
One remark concerning equation (12) is that it is not intended to compute
physically real potential évapotranspiration; as a consequence, the resulting e,
series here is just an auxiliary series internal in the model. The inclusion of this
equation in the model was in keeping with the original purpose of the study of
developing monthly water balance models that required a minimum of readily
available input data. No special claim is made concerning the correctness of the
internal workings of the model, as is apparent from its simplicity. This is
acceptable, since the objective here is to compute monthly streamflow series;
the et series estimated by equation (12) may lose its true physical meaning to
a certain extent.
Slow flow equations
Being similar to baseflow, slow flow depends essentially on the storage in the
catchment during the month considered.
The general form of the slow flow equation is:
st - ^{mUr
(13)
where a^ and a{ are non-negative parameters. In practice, however, a2 and a^
are highly correlated. This results in very difficult calibration and high
imprecision of the estimates. Therefore a£ was given one of three standard
values viz. a^ = lA or 1 or 2 of which at least one value suits any given river
basin. Thus a£ is a discrete parameter, whereas a^ is a continuous one.
Fast flow equations
Fast flow depends on precipitation pt, on other meteorological conditions as
measured by e„ on the state of the basin as measured by the storage mt and on
the physical characteristics of the basin, which are taken into account by the
introduction of parameters.
A useful quantity is the "active" precipitation defined as:
», =pret(l-Q-p/m&Kie-A))
<14)
where et is observed (or Penman-derived) PET or the quantity defined by
equation (12). In many cases nt seems to translate the influence of precipitation
and meteorological conditions. The nt function is represented in Fig. 4. This
Figure shows that the "active" precipitation nt decreases when potential
évapotranspiration increases. On the other hand nt increases with;?, and is equal
to total precipitation pt minus et for heavy rainfall (highp,).
A most efficient equation for fast runoff proved to be:
/, = a^mt^n,
( 15 )
164
C.-Y. Xu & G. L. Vandewiele
where a3 and a^ are non-negative parameters. For the same reason as in the
case of slow runoff a^ was given one of four standard values: a'3 = 0 or lh or
1 or 2.
Fig. 4 The active rainfall nt in equation (14). In the left diagram
et is held constant, whereas in the right diagram pt fs constant.
Equation (15) can be seen as a translation of the variable source area
concept: the greater the storage m*_u the wetter the catchment, the greater the
"source" of fast runoff, the greater the part of the "active" rainfall running off
rapidly.
Since mt varies slowly in contrast withp, and nt, equations (13) and (15)
really can represent slow and fast flow respectively.
By specifying values of a£ and a$ and by the choice of the
évapotranspiration equations (10) or (11), one finds what will be called
henceforth a particular model. Since there are three possible values of a^, four
possible values of 0$, and two possible évapotranspiration equations, there are
2 x 3 x 4 = 24 possible models. A program is available for finding
automatically the best model among the 24 possible models for a given
catchment.
From the above description one sees that although P type models have
six parameters, on the other hand they need only monthly mean rainfall data
as input. This is a big advantage when a model is going to be used in
developing countries where evaporation data are rarely available.
Another advantage of the models reported herein is that the parameters
have close relationships with the physical characteristics of catchments, and can
be derived from the lithological characteristics of the catchments, at least in the
Belgian context. Application to ungauged catchments is possible at least for PE
type models (Vandewiele et ah, 1991).
BRIEF DESCRIPTION OF THE STUDY CATCHMENTS AND
The models described in the paper have been tested on 91 catchments, of which
65 catchments are from northern Belgium, 20 from southern Belgium and six
from southern China.
165
Sensitivity of monthly rainfall-runoff models
Northern Belgium (excluding the polder area) is a more or less hilly
region with elevation between 5 to 200 m above sea level (a.s.l.). Southern
Belgium (the Ardennes plateau) is mostly higher than 200 m a.s.l. (except in
the deep valleys of the main rivers) and rises up to 700 m a.s.l.
As for the climate, the potential évapotranspiration is spatially fairly
uniform. In winter it is less than 10 mm per month; in summer it amounts to
100 mm per month on average. Precipitation does not show a strong
seasonality. The mean is 60 to 70 mm per month in northern Belgium and 80
to 100 mm per month in southern Belgium. Snow and frost are not important
on a monthly scale. Practically the whole of northern Belgium is arable land
and grass. In southern Belgium one third is covered by forest and two thirds
by arable land and grass.
The six Chinese catchments under study belong to the Pearl River basin,
which lies in the subtropical zone. Frontal-type precipitation and typhoon-type
precipitation are the two most important phenomena in this area. It has fairly
good vegetation cover and plenty of rainfall, varying from 1400 to 2000 mm
per year. The precipitation occurs all year round, but seasonal differences are
very large. In the six months of the wet season (from April to September)
about 80% of the total precipitation occurs. Characteristics of the basins are
shown in Table 1.
Table 1 Characteristics of basins studied
Area (km2)
Mean precipitation
(mm month )
Coefficient of variation of
monthly precipitation
Mean PET (mm month -1 )
Coefficient of variation of
monthly evaporation
Runoff coefficient (%)
Northern Belgium
Southern Belgium
Southern China
16-3190
60-70
68-3626
80-100
385-2000
120-160
0.5-0.6
0.5-0.6
0.75-0.95
50-60
0.7-0.75
50-60
0.7-0.75
70-85
0.3-0.5
18-53
26-59
54-77
STATISTICAL ANALYSIS
Statistical analysis was confined to nonlinear regression analysis. In the present
section only a limited number of items are discussed. A full account is given
by Vandewiele et al. (1992, 1993) and Xu (1992).
To estimate the parameters the maximum likelihood method was used.
Because of the hypotheses in equations (1) and (2), maximizing the
loglikelihood with respect to the continuous parameters is equivalent to
minimizing the sum of squares:
SSQ= E (^T-f7Î
(16)
t
where the sum is extended over all months for which output qt as well as input
C.-Y. Xu & G. L. Vandewiele
166
data are available. The runoff sequence may show data gaps, but not the input
data series, because of the recursive nature of the balance equation (3).
The quality of the minimization was checked by plotting the sum of
squares equation (16) versus each of the filter parameters. In that way it was
possible to see whether a global minimum was reached. Several models have
to be tried in order to find the best values of the discrete parameters and the
best évapotranspiration equation.
The model standard deviation a, which is an inverse measure of the
s
20
°
-]
9S
r
120
time in m o n t h s
E
160
120 —
ë
80
"i—r
96
120
time in m o n t h s
°
144
observed
estimated
400-
time in m o n t h s
Fig. S Precipitation, evaporation, actual évapotranspiration,
observed runoff and estimated runoff versus time diagrams for the
Xingzi River at Fenghuangshan station (1967-1984) using a PE
model.
Sensitivity of monthly rainfall-runoff models
167
quality of model performance, is estimated by:
a =
minimum SSQ
where N is the number of terms in equation (16), and K is the number of filter
parameters (regression coefficients),
The best models were tested in numerous ways by applying statistical
methodology including residual analysis, checks on confidence intervals of
individual parameters, checks on parameter correlations, split sample testing,
216
216
-5.00
96
120
time in months
216
Fig. 6 Storage, slow runoff, estimated total runoff and residuals
versus time diagrams of the Xingzi River at Fenghuangshan station
(1967-1984) using a PE model.
168
C.-Y. Xu & G. L. Vandewiele
etc. All these tests gave satisfactory results (Vandewiele et al., 1992; Xu,
1992).
These two model types were applied to 85 river basins in Belgium and
six Chinese basins and proved to perform equally well (Xu, 1992; Xu &
Vandewiele, 1993). Moreover they perform much better than a number of
models previously defined in the literature (Vandewiele et al., 1992).
As an example of model output, the case of the Xingzi river at
600
T
I
96
120
time in months
'
i
144
i
r
168
160driving function
actual
120-
500-
Fig. 7 Precipitation, driving function, actual évapotranspiration,
observed runoff and estimated runoff versus time diagrams of the
Xingzi River at Fenghuangshan station (1967-1984) using a P
model.
Sensitivity of monthly rainfall-runoff models
169
Fenghuangshan station (1556 km2) in the Pearl river basin in southern China
is presented here. The calibration results are summarized in Table 2. It is to
be kept in mind that the parameter values of PE type and P type model are not
fully comparable in this case, since the discrete parameters 4 and a3 are not
the same. Graphical outputs of the PE type model are shown in Figs 5 and 6.
For the purpose of comparison, similar outputs from the P type model are
shown in Figs 7 and 8.
500
400-
300-
200-
100-
1
0
!
,
24
!
,
48
!
,
72
!
,
!
96
120
time in months
,
!
,
144
r
168
400
E 300
S 200-
o 100
T
216
96
120
time in months
5.00
-3.00
-5.00
-,
0
!
24
,
!
48
|
j
72
,
!
,
J — , j
96
120
time in months
144
,
!
168
,
]
192
r-
216
Fig. 8 Storage, slow runoff, estimated total runoff and residuals
versus time diagrams of the Xingzi River at Fenghuangshan station
(1967-1984) using a P model.
170
C.-Y. Xu & G. L. Vandewiele
Table 2 Summary of calibration results for thé Xingzi River at
Fenghuangshan station
PE type model
Calibration results
Evapotranspiration equation
Discrete slow flow parameter a£
Discrete fast flow parameter a^
Model standard deviation a
Half width of 95 % confidence interval of a
Mean computed total flow (mm month )
Contribution of slow flow (%)
Contribution of fast flow (%)
P type model
ai)
ai)
i
2
1.134
0.107
78.6
25.6
74.4
1.128
0.107
76.8
27.9
72.1
Observed quantities
Mean precipitation (mm month"1)
Mean evaporation (mm month )
Mean runoff (mm month'v
Runoff coefficient (%)
135
71
77.5
57.4
SENSITIVITY TO INPUT DATA ERRORS
Areal rainfall may be incorrect both randomly and systematically. Therefore,
in order to examine how rainfall errors influence model performance, the
original data series was corrupted by adding a normal white noise with given
mean and standard deviation. The "error" was generated by Monte Carlo
simulation. The mean of the errors was successively taken equal to 0%, 5%
and 10% of the observed mean monthly rainfall in the basin (only positive
means were considered in order to avoid too many negative values in the
corrupted rainfall series). The standard deviation of the errors was successively
taken equal to 0%, 5%, 10%, 15%, 20%, 25% and 30% of the standard
deviation of the original rainfall series. In this way 21 different rainfall data
series were generated for each catchment. The practicing engineer is postulated
to have one of the 21 data series for his catchment. The purpose of the study
now was to give an idea to the user how the data error affects model quality.
This can be done by calibrating the above mentioned models on each of the 21
data series and to find the "best" model for every pattern of corruption. The
resulting model standard deviations of the "best" model for each pattern can
then be compared. It is to be noted that the discrete parameters and the
évapotranspiration equation can be different in the "best" model for different
corruption patterns.
The test was performed on several river basins using both PE and P
models. The results are summarized in Table 3. As a typical example, results
on the Xingzi basin in southern China and the Viroin basin in southern Belgium
are shown in Figs 9 and 10 for PE models, and in Figs 11 and 12 for P
models. In these Figures the horizontal solid line corresponds to the result with
the original data (error free case). It is drawn just for comparison. The upper
and lower dashed lines correspond to confidence limits with 95% confidence
level.
It is seen from Figs 9 to 12 and from Table 3 that: (1) random errors in
rainfall data affect the model performance adversely to more or less the same
111
Sensitivity of monthly rainfall-runoff models
1.6
O 1.4 —
0-8 ~ |
0
' mean of error 0»
* * * * * mean of error 5*
11 I I I mean of error 10x
95K conf. limits
error free case
| | | | | | | | | | | | 1 | | 1 | | | |
5
10
15
| I I | | I I I I
20
25
30
S t a n d a r d d e v i a t i o n of e r r o r (%)
Fig. 9 Influence of rainfall errors on model performance as
inversely measured by the model standard deviation for the Xingzi
River at Fenghuangshan station (CFH) using PE models.
i l i i i i l i i i i l i
5
10
15
Standard deviation of
Fig. 10 Influence of rainfall errors on model performance as
inversely measured by the model standard deviation for the Viroin
River at Treignes station (VI) using PE models.
magnitude for both type PE and type P models; (2) significant effects on model
performance are expected when the standard deviation of these random errors
goes up to 10% or 15% of the standard deviation of the original rainfall series
for Chinese catchments. For Belgian catchments significant effects occur when
the standard deviation of errors goes up to 15% or 20%. The reason is
probably that the coefficient of variation of the observed rainfall series is
smaller in Belgian catchments (see Table 3); (3) systematic error in rainfall data
C.-Y. Xu & G. L. Vandewiele
172
1.6
• * • • * mean of error 0»
»«•«« raeqn of error 5K
I I I I I mean of error 10*
95s» conf. limits
error free case
5
i i i i i i i i i i i r
10
15
20
Standard deviation of error
(%)
Fig. 11 Influence of rainfall errors on model performance as
inversely measured by the model standard deviation for the Xingzi
River at Fenghuangshan station (CFH) using P models.
1.2
***** mean of error Ox
* * > « * mean of error 5x
I I 11 I mean of error 10»
95» conf. Iimit3
error free case
c 1-1
o
a
XI 1.0
-
o 0.9 —
O 0.8
0.7
I I I 1| 1I 1 I | I I 1 I | I I I
| I i I 1 | 1 ] I 1
5
10
15
20
25
30
Standard deviation of error (%)
Fig. 12 Influence of rainfall errors on model performance as
inversely measured by the model standard deviation for the Viroin
River at Treignes station (VI) using P models.
hardly affects model performance. This is perhaps an unexpected result and
different from what has been found in the literature previously. What happens
is that parameter values change significantly when rainfall input is corrupted.
This was checked by the authors on continuous parameters (fixing the discrete
parameters) of PE models. In such cases other water balance terms (except
runoff) change significantly too.
Sensitivity of monthly rainfall-runoff models
173
Table 3 Results of the sensitivity of model performance to rainfall
errors
Basin
code
Area
(km2)
Belgian catchments
D2941
16
VI
554
TP
3626
Mean
Standard
precipitation, deviatipn.of
(mm month ) precipitation
64.3
79.1
83.6
Chinese catchments
CHL
595
163.1
CXG
1881
119.1
CFH
1556
134.8
Variation
Upper bound:
coefficient of
precipitation PE model P model
35.5
39.7
41.6
0.55
0.50
0.50
20%
20%
20%
20%
15%
20%
130.6
91.9
101.4
0.80
0.77
0.75
10%
10%
15%
10%
10%
15%
"upper bound" means the upper bound of the percentage of the standard deviation of the
original rainfall series beyona which model performance is seriously affected.
SENSITIVITY OF MODEL PERFORMANCE TO LENGTH OF
CALIBRATION PERIOD
This investigation has been limited to type PE models. Ideally one should use
a long record of simultaneous precipitation, PET and runoff data. As such long
data series were not available the following solution was adopted for two basins
in Belgium:
(A) rainfall: the Ukkel rainfall station (near Brussels) has a 158-year record
(1833-1990). Areal rainfall in the basins has a 35-year record. It has
been shown that those rainfalls are statistically indistinguishable except
for a correction factor for each month. By using these corrections a
158-year areal rainfall record became available for each basin.
(B) PET: Ukkel PET data were used in order to compute monthly means.
These means were then used as a periodic input series in all steps of the
investigation. This procedure is justified by the fact that errors in PET
input are much less important than errors in rainfall input (Xu, 1992);
and
(C) runoff: a record of 16 years of runoff data together with the above
mentioned areal rainfall data was exploited to find the best PE model for
each basin. This model together with the above mentioned corrected
rainfalls then was used to generate 158 years of runoff (including Monte
Carlo simulation of residuals).
The procedure of investigation was as follows:
(a) Take 10 independent calibration samples of 2, 5, 10, 15 and 20 years
length (small overlapping in the case of 20 years) from the 158 years
obtained above. The "best" type PE model is then determined on each
of these 50 samples; and
(b) apply each of these 50 models (including the optimal parameter values
obtained under (a)) to compute 158 years of runoff d„ and compute for
each the root mean squared error (RMSE):
C.-Y. Xu & G. L. Vandewiele
174
1
RMSE =
N
N t=i
where qt is the "true" runoff obtained under (C). This quantity was used as a
criterion.
This procedure was applied to two Belgian catchments: the Lesse river
at the Eprave station (419 km2), and the Viroin river at the Treignes station
(554 km2). The results are shown in Figs 13 and 14. The solid line in these
figures is the mean RMSE as a function of calibration period length.
20# » * * • value of individual samples
mean of 10 individual values
\
o
E 15E
LU
to 1 0 -
I I I I | I I I I | ! TT
0
r~[~l
i | i i i i
5
10
15
20
Calibration period in years
25
Fig. 13 Influence of length of calibration period on model
performance as inversely measured by the root mean squared error
(RMSE) for the Lesse catchment.
25|
* * * * * value of individual samples
mean of 10 individual values
"£ 20o
\
E
E
E 15LU
i
in
j
10-
I I 1 I | I I l l | l l
0
5
10
[ i i i i i I i i i i
15
20
25
Calibration period in years
Fig. 14 Influence of length of calibration period on model
performance as inversely measured by the root mean squared error
(RMSE) for the Viroin catchment.
It is concluded from these figures that: (1) the mean as well as the
variability of RMSE decreases with increasing length of calibration period; and
Sensitivity of monthly rainfall-runoff
models
175
(2) there is virtually no gain in quality (mean and variability) beyond 10 years
length of calibration period.
O'Donnell & Canedo (1980) found five years as a sufficient length of
calibration period. However their model was a daily one and consequently they
were interested especially in flood problems. Gorgens (1983) found 15 years
to be needed with Pitman's (1973) monthly rainfall runoff model on a river in
a semiarid area. Apart from the relative imprecision of such limits, the
differences in these required calibration durations can be due to a greater
flexibility of the models in the present paper.
CONCLUSIONS
When using the class of models considered in this paper for modelling basins
in a humid region the following conclusions can be drawn:
(1) random errors in rainfall data influence model performance adversely
and significant influences occur as soon as the standard deviation of the
rainfall errors were greater than 10% or 15% for Chinese catchments
and greater than 15% or 20% for Belgian catchments. These are
percentages of the standard deviation of the original rainfall series.
Systematic errors (within 10% of the observed mean rainfall at least) are
less important for the estimation of streamflow, but they have a
significant influence on model parameter values and consequently on the
estimation of other water balance terms like évapotranspiration (Xu,
1992); and
(2) for adequate calibration of these models, a 10 year calibration period
seems to be necessary and more or less sufficient. Significant
improvement in the ability of the models to obtain stable results can be
effected by expanding a calibration sample from, say, afive-yearrecord
to a ten-year record; further increments of calibration sample length from
10 years to 15 or 20 years does not give appreciable improvement.
Acknowledgements The authors thank G. Demaree and F. Bultot (Royal
Meteorological Institute, Brussels, Belgium) and R. Verhoeven (State
University Ghent, Belgium) for help in data gathering. The Ministry of the
Flemish Region and the Government Office for Developing Countries (ABOS)
partly funded the present research. The authors are indebted to A. Van der
Beken (Interuniversity Postgraduate Program in Hydrology, IUPHY, Brussels)
for his friendly collaboration.
REFERENCES
Alley, W. M. (1984) On the treatment of évapotranspiration, soil moisture accounting and aquifer
recharge in monthly water balance models. Wit. Resour. Res. 20(8) 1137-1149.
Herbst, P. G. &. Shaw, E. H. (1969) Determining raingauge densities in England from limited data to
give a required precision for monthly rainfall estimates. J. Wit. Engrs 23(4) 2(s).
176
C.-Y. Xu & G. L. Vandewiele
Hounam, C. E. (1971) Problems ofevaporation assessment in the water balance. Report No. 13 World
Meteorological Organization/International Hydrological Decade Projects, Geneva, Switzerland.
Gorgens, A. H. M. (1983) Reliability of calibration of a monthly rainfall-runoff model: the semi-arid
case. Hydrol. Sci. J. 28(4), 485-498.
O'Donnell, T. & Canedo, P. (1980) The reliability of conceptual basin model calibration. In:
Hydrological Forecasting (Proc. Oxford Symp.), 263-269. IAHS Publ. no. 129.
Pitman, W. V. (1973) A mathematical model for generating monthly river flows from meteorological
data in South Africa. Hydrological Research Unit Report no. 2.73, Univ. of Witwatersrand,
Johannesburg, South Africa.
Romanenko, V. A. (1961) Computation of the autumn soil moisture using a universal relationship for
a large area, Proc. Ukrainian Hydrometeorol. Res. Inst. No. 3 , Kiev, Russia.
Singh, V. P. & Birsoy, Y. K. (1975) Comparison of the methods of estimating mean areal rainfall.
Nordic Hydrology 6, 222-241.
Singh, V. P. & Woolhiser, D. A. (1976) Sensitivity of linear and nonlinear surface runoff models to
input errors. / . Hydrol. 29, 243-249.
Sutel iffe, J. V. ( 1966) The assessment of random errors in areal rainfall estimation. Bull. Int. Assoc. Sci.
Hydrol. 11(3), 35-42.
Vandewiele, G. L. & Xu, C.-Y. (1991) Monthly water balance models for different input information
levels. In: Modeling and Control qfWiter Resources Systems and Global Changes, eds E. J. H.
Kerckhoffs, A. Van der Beken & G. C. Vansteenkiste. (Proc. European Simulation Symposium),188-191. Ghent, Belgium.
Vandewiele, G. L., Xu, C.-Y. & Huybrechts, W. (1991) Régionalisation of physically based water
balance models in Belgium: Application to uneaueed catchments. Wit. Resour. Manage. 5,
199-208.
Vandewiele, G. L., Xu, C.-Y. & Ni-Lar-Win (1992) Methodology and comparative study of monthly
water balance models in Belgium, China and Burma. / . Hydrol. 134,315-347.
Vandewiele, G. L., Xu, C.-Y. & Ni-Lar-Win (1993) Methodology For Constructing Monthly Witer
Balance Models on Basin Scale. Booklet (2nd ed). Laboratory of Hydrology, Free University
Brussels, Belgium.
Xu, C.-Y. (1992) Monthly water balance model in different climatic regions. PhD Thesis, Laboratory
of Hydrology, Free University Brussels, Belgium.
Xu, C.-Y. & Vandewiele, G. L. (1993) Parsimonious monthly rainfall-runoff models for humid
basins with different input requirements. Submitted for publication in Adv. Wit. Resour.
Received 19 July 1993; accepted 23 January 1994