Journal of Hydrology, 134 (1992) 315-347
315
Elsevier Science Publishers B.V., Amsterdam
[3]
Methodology and comparative study of
monthly water balance models in Belgium,
China and Burma
G.L. Vandewiele, Chong-Yu Xu and Ni-Lar-Win
Laboratory of Hydrology and Interuniversity Postgraduate Program in Hydrology, Vrije Universiteit,
Brussels, Belgium
(Received 20 March 1991; revision accepted 24 August 1991)
ABSTRACT
Vandewiele, G.L., Xu, C.-Y. and Ni-Lar-Win, 1992. Methodology and comparative study of monthly
water balance mod.~,ls in Belgium, China and Burma. J. Hydrol., 134:315-347.
A set of new monthly rainfall runoff models (water balance models) is defined, for use in river
catchments smaller than about 4000 km2, without appreciable frost or natural or artificial lakes. The input
series are areal precipitation and potential evapotranspiration. The output is riverttow. The number of
parameters, used in tbe description of the hydrological phenomena in the catchment, is in most cases three,
sometimes four. The statistical methodology used for calibrating the models of given catchments is
described; it reduces essentially to regression analysis, including residual analysis, sensitivity to calibration
period and extrapolation test. In particular, automatic calibration is used, excluding subjective elements.
The models are applied to 79 river basins in Belgium, China and Burma. The results are compare¢ with
four similar models taken from the literature. The results of applying the new models are satisfactory from
a statistical point of view and are much better than those quoted in the literature; a greater par~ of the
observed runoff is explaineo and there is no residual seasonality. This results from the different mathematical'
structure of the models, arid especially from the use, in the published models, of several storages with
maximum 'capacities', with no distinction between slow and fast runoff corresponding to baseflcw and
direct runoff, respectively.
INTRODUCTION
The models considered in thi~: r~aper have precipitation and potential
evapotranspiration as input and runoff as output. The time base is one month.
Consequently autoregressive moving average (ARMA) models, such as those
described by Weeks and Bou[hton (1987) which use precipitation as their
unique input series, are not discussed.
Monthly rainfall-evaporation-runoff models are useful in several ways (see
also Alley, 1984). Firstly, they can be used to fill in missing rm~off data when
rainfall and evaporation data are available. In many cases the series of
f
0022-1694/92/$05.00
© 1992 - - Elsevier Science Publishers B.V. All rights reserved
316
G.L. VANDEWIELE ET AL.
observed runoffs (ten years for example) is too short for constructing a reliable
pure runoff model (i.e. modelling runoff as a time series). The monthly rainfall
runoff model exploits these relatively short runoff series. After calibration, the
model can be applied to periods where the input series (precipitation and
potential evapotranspiration) are available, but the runoff is not. In this way
a longer runoff series can be obtained (several tens of years). This runoff series
can model then be used for constructing a pure runoff model, which can be used
to produce a still longer Monte Carlo simulation (a thousand years, for example).
A second application is related to ungauged basins. By building up some
experience, the model parameters can perhaps be related to physical basin
characteristics, or can be regionalized, or determined in some other way. The
resulting model is then used as described above.
A third application of monthly models is the disaggregation towards
models with a smaller time base (such as a week or a day). This seems logical
since the greater the time base, the easier the model construction and
calibration.
A fourth application has been mentioned by Alley (1984). If the model
contains a moisture time series (as is the case with the models in the present
paper), this series yields important information for agriculture (e.g. irrigation
demand).
Although monthly series could be obtained by aggregating daily series,
there are several inconveniences in doing this: (1) daily models are far more
complicated, because more details have to be modelled; (2) the database
necessary for calibrating is much greater, and data gathering becomes a time
consuming job; (3) the calibration itself is much more difficult and eventually
takes much more computer time.
Thornthwaite and Mather (1955), Palmer (1965) and Thomas (1981)
defined monthly models, which are primarily meant to be water balance
models for agricultural use (see the fourth application above). Alley (1984)
defined two varian,~s of Thc~rnthwaite and Mather's (1955) model and studied
the performances of those five models. He calibrated the models by optimization. This technique is also used in the present paper, and comparisons with
four of Alley's models are made.
In the present paper a number of new models are discussed of which the
prototype was defined by Bladt et al. (1978), and in Lowing (1987). Advanced
results were obtained by Xu (1988) and Ni-Lar-Win (1989). The general
model structure and particular model equations are first introduced. Then
statistical analysis is discussed. The methodology of estimation, model cheek
and model comparison is discussed and illustrated. The basins in Belgium,
China and Burma to which this methodology has been applied are then
described together with a brief account of their geographies. Results of the
317
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
analysis are then summarized. Models taken from the literature are very
briefly described and the results of their analysis are discussed. The results of
new models and published models are compared in some detail, and
conclusions are then presented.
MODEL FORMS
General model structure
In the model, monthly areal precipitation p, and potential evapotranspiration e, are the inputs, whereas monthly observed runoff qt is the output series
(t is time in months). The series p, and e, are thus the 'observed' factors.
Clearly this relationship is also influenced by other phenomena, the
'unobserved' factors, such as measurement errors, 'Thiessen' errors, the non~.
homogeneity of rainfall during the month, model imperfections, etc.
Therefore q, is considered to be a random variable, the result of a deterministic
function of the p, and e, series on the one hand and of a random deviation u,
on the other. The deterministic function is a rainfall runoff filter, of which the
output is the computed discharge d,. This is represented in Fig. 1.
Since for statistical analysis it is convenient to have homoscedastic
deviations, it is supposed that
=
+
(l)
u,
Writing q, simply as the sum of d, and u, resulted in heteroscedastic residuals,
whereas taking logarithms does not work well when q, is very small (nearly dry
river). Taking square roots as in (1) is a compromise between these two
extremes, and it allows
u, ~ N ( 0 , a 2)
(2)
i.e. u, is normally distributed with zero expectation and common variance a 2.
Moreover, autocorrelation is supposed to be zero, and this turned out to be a
relatively good hypothesis, since autocorrelation in the residual series was either
non-significant (at the 95% level) or very small. In consequence the u, may be
t
Ut
Pt
e t
Rainfall ] dt [Compounding i
Runoff
Rule
Filter
I ==~ l
J
Fig. !. Rainfall runoff model.
qt
==-
318
G.L. VANDEWIEI.E ET AL.
presumed to be stochastically independent. Anyway it is possible to take account
of autocorrelation by modelling u, as an autoregressive process of order 1
(AR(I)) process (or a more complicated A R M A process) (see, e.g. Alley, 1984).
Setting
z
x/q, '~' N ( x / d , , 0.2)
=
we know that
(Ez) 2 + varz
Ez 2 =
and consequently
Eq,
=
(3)
d, + 0.z ~ d,
where var means variance, and E is the expectation operator.
The Taylor expansion of z 4 around a constant c is
Z4
~-
C4 -[-
4(z - c)c 3 + 6(z - c)2c 2 + 4(z - c)3c + (z - c) 4
Replacing c by E z and taking expectations on both sides of this equation we
obtain
Ez 4 =
(Ez) 4 + 6(Ez) 2 var z + 3(var z) 2 =
d,2 + 6d,0. 2 + 30. 4
because z is normally distributed, which entails that
E(z-
Ez) 3 =
O,
Ez) 4 =
E(z-
3(varz) 2
(see, e.g. Cramer, 1963, p. 212). On the other hand
Ez 4
(Ez2) 2 + var z 2 =
'
( d , + 0.2)2+ v a r q ,
Combining the: two expressions of Ez 4,
=
var q,
0.:'(4d, + 20.2) ~ 40.:'d,
(4)
0.x/(44 + 20.2) ~ 20.x/dr
(5)
and
std q, =
cv q,
0.~/(4d, + 2tr2)
=
a, +
20.
x/a,
(6)
where std and cv stand for standard deviation and coefficient ot variation. The
latter is the standard deviation divided by the expectation; it is a dimensionless
measure of the relative influence of th~ 'unobserved' factors. The approximate
expressions in formulae (3) through (6) are valid when a 2 is much smaller than
d,, which is very frequently the case in practice. Unfortunately cv q, depends
on d,. To have a unique measure of model quality this coefficient of variation
can be computed for d, equal to the mean observed runoff (/
ecv =
ax/(4(/ + 20.2)
20.
-!- 0.2
~ ~/O
(7)
319
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
where ecv is the estimated coefficient of variation.
General filter structure
The computed discharge d, is a function of the present and past precipitations P t , P , - ~ , P , - 2 , • • • and potential evapotranspirations e,, e,_t, e , _ 2 , • • •
Therefore the rainfall runoff filter is a 'backward looking filter', and precipitation and potential evapotranspiration are 'leading indicators' of the
discharge.
From past and present values of the input series, a new time series m t is
computed. It 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; it
summarizes the memory of the catchment. This is expressed by the balance
equation
rn, = m,_ ! + p , -
r,-
d,
(8)
where r, is real evapotranspiration during month t. All quantities are
expressed as depth in millimetres. Moreover r, and d, are time invariant
functions of p,, e,, m,_ I and m, only:
r,
=
r ( p , , e:, m , _ l ,
art
=
d(p,,
e,, m,_l,
(9)
m,)
mr)
(10)
The quantity m, is included because the mean storage during month t is
(m,_, + m,)]2
Filters differ by their functions r ( . ) and d ( . ) .
A distinction is made between slow discharge s, and fast discharge f~ such
that
a, =
s, + f,
(11)
Slow and fast discharges can perhaps be interpreted as baseflow and direct
flow, but this interpretation is questionable since it may be impossible to
distinguish properly with a monthly model between baseflow and direct flow.
The above hypotheses have important consequences concerning the applicability of the model.
(1) There must be no large natural or artificial lakes in the catchment, and
no appreciable abstraction or introduction of water. These have to be
corrected for or modelled separately and this has not been included in the
models of the present paper.
(2) There must be no direct influence of the precipitation that fell during the
preceding month. In consequence, the concentration time must be much
320
G.L. VANDEWIFLE ET AL.
smaller than one month (less than two or three days), and this limits the
applicability of the model to catchments with an area smaller than about
4000 kin2.
(3) Frost requires a separate storage, so catchments with appreciable frost
are not covered.
An important characteristic of a filter is the number of its parameters
(unknown time invariant constants to be estimated). Too small a number
leads to too rough a model, and model quality (as measured by formula (7))
will be low. Too great a number of parameters leads to imprecision and
estimation difficulties and is in conflict with the principle of parsimony.
Evapotranspiration equations
For computing monthly real evapotranspiration r,, two quantities (among
others) are important: the available water w, during month t defined as
w,
=
p, + m~+_,
(12)
where m,+_l = max (m,_~, 0) is the available storage, and the monthly
potential evapotranspiration e, (evaporation for short). The latter quantity
does not take the soil cover into account; it is the evaporation at a fi'ee water
surface.
For evident reasons, a good evapotranspiration equation must be such that
r, increases with e, and w,
(13)
r, = 0
(14)
whenw, = 0 o r e , = 0
r, ~ e, and r r ~< w,
(15)
r,
(16)
~
e,
when w, ~ oo
Two equations proved to be etticient with the data used in this paper. They
are the oniy ones which are discussed in detail in the rest of the paper. The first
is
r, = min[e,(l - , , i,,lwl/el /,w,]
(17)
where the symbol al is a parameter (i.e. an unknown time invariant constant
to be estimated), which is characteristic of the river basin under study. This
parameter a~ is constrained by
0~a~
~< 1
because of the conditions (13) through (16). Equation (17) is represented in
Fig. 2.
The second equation, based on Romanenko (1961) (see also Hounam,
321
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
rt
"as< ~
r
.al< 1
7, _/,i
W~
" ~ " " ""
e
-Ina~ .w.
\al • 1
--
,,
o
1
k
wt
et
o
et
(a)
wt
(b)
Fig. 2. The evape,transpiration eqn. (17). (a) Curves with w, = constant; the tangent at the origin is at 45 °
for all a~ values. (b) Curves with e, = constant; tangents have a slope < 42" except when an < l/e.
1971), is
r, = rain [w,(l
--
a ;e,
),e,]
(18)
with parameter a~, constrained by
0~<am~<l
Equation (18) is represented in Fig. 3. Both equations fulfil conditions (13)
through (16).
An important difference from the point of view of interpretation between
formulae (l 7) and (18) is that in (18)
r, ~ w, when e, -~ oo
(19)
w! w t
i.
wt
et
0
et
(a)
wt
(b)
rt
w,[
0
al =o
al =o
v
w~
et
(c)
0
et
wt
(d)
Fig. 3. The evapotranspiration eqn. (18). (a, c) Curves with w, = constant. (b, d) Curves with e, = constant.
(a, b) Diagrams show the general case, where~.s (c, d) are relative to the particular case an = 0.
G.L. VANDEWIELE ET AL.
322
whereas this is not the case in (17). Whether or not this condition is really a
reasonable requirement for evapotranspiration equations is doubtful.
Condition (19) in fact requires that all available water is used up in evapotranspiration when the available energy (measured by e,) is great.
The replacement of w, by the mean available storage during month t
(mr_ t + mr) +/2
did not improve the model quality and was not used.
Slow runoff equations
Only functions of m,_~ were considered as the use of (m,_~ + mr)~2 did not
improve the model.
The general form of the slow runoff equation is
+
st = a2(mt_t)
a~
(20)
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 three standard values
a2t = ~! o r l o r 2
of which at least one valae suits a given river basin. In some cases it was not
even necessary to include a slow runoff term.
Adding a constant term in expression (20)
s,
=
,,
(a2 + a2(rnLi
did not improve the model significantly in most cases.
Fast runoff equations
Fast runoff depends on precipitation Pt, on other meteorological conditions
as measured by et, on the state of the basin, as measured by the storage, 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
n, = Pt -- e,(l
--
e -pde')
(21)
In many cases it seems to translate the influence of precipitation and meteorological conditions. The n, function is represented in Fig. 4. This figure shows
that the 'active' precipitation n, decreases when potential evapotranspiration
increases. On the other hand n, increases with p, and is equal to total precipitation p, minus e, for heavy rainfall (high p,).
323
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
/ /
,eI
x,,
mt
et
(a)
Pt
0
Pt
(b)
'~t
Fig. 4. The active rainfall n, in formula (21). (a) e, is constant; (b) Pt is constant.
A most efficient equation for fast runoff proved to be
f
(22)
= a3m+lnt
This equation can be seen as a translation of the variable source concept: the
greater the storage re+l, the wetter the catchment; the greater the 'source' of
direct runoff, the greater the part of the 'active' rainfall running off rapidly.
The 'state' of the basin is translated by its storage m+_l. Again, just as in
the case of slow runoff, it seems logical to take mean storage (mt_~ + mr) +/2,
but again this did not prove to be worthwhile.
In isolated cases, three other equations were necessary to model fast runoff
f
= aant
(23)
f
= aaP, + a~m+lp,
(24)
f
= p,-
(25)
a,3e-",+-,/a3(1 _ e-t,,I,,~)
In eqn. (23) the state of the catchment has no influence on fast runoff, whereas
in eqn. (24) two terms are used, one with and one without the use of storage.
Moreover in eqn. (24) total rather than active precipitation is used. Equation
(25) is represented in Fig. 5. Equations (24) and (25) have two parameters. All
parameters have to be non-negative.
ft
p,
rot-1
(b)
(a)
Fig. 5. Fast runoff eqn. (25). (a)
m +_ i
iS constant; (b) p, is constant.
324
G.L. VANDEWIELE ET AL.
A n example: m o d e l 2
As an example, consider one of the most successful models, at least in the
set of catchments studied. The general structure is given by eqns. (1) and (2)
and by the hypothesis that the u, values are stochastically independent (s.i.):
x/qt
=
(26)
x/at + u,
u, ~ N(0, a 2) and s.i.
(27)
The evapotranspiration is given by eqns. (12) and (17)
rt =
(P,+m+_l)let]
e,[l -- al
(28)
Computed runoff is given by eqns. (1 I), (20) with a~ = 2, (21) and (22)
d, =
a2(m+_,
)2
+ a3m,+,[pt -
et(l - e-p'/e')]
(29)
The first term is slow runoff and the second term is fast runoff. Finally the
balance equation (8) has to be added:
m, = m,_l + p , -
r,-
d,
(30)
Equations (26) through (30) completely define the model. Equations (28) and
(29) define the particular filter structure of model 2.
This model is one of many models which can be obtained by combination
of the possibilities mentioned in the above sections on evapotranspiration,
and slow and fast runoff. An alternative choice only affects eqns. (28) and (29).
Model 2 has only three filter parameters. Time series m, has to be computed
recursively by the balance eqn. (30).
STATISTICAL METHODOLOGY
Estimation
Because of the hypotheses in eqns. (1) and (2), maximizing the loglikelihood
is equivalent to minimizing the sum of squares:
~'. (x/q, -- x/dt) 2
(31)
where the sum is extended over all months for which output data qt as well as
input data p, and e, are available. As a consequence, the runoff sequence may
show data gaps, but not the p, and e, data sequences. Moreover, the computation of the first d, in the series needs a knowledge of the initial storage m0.
Therefore the calibration period in (31), has to be preceded by a 'take off'
period where Pt and e, are known; this enables the compatation of m0, starting
WATER BALANCE MODELS IN BELGIUM. CHINA AND BURMA
325
with an arbitrary storage. This take-off period has to be long enough to obtain
a good value of m0. In practice, six years was found to be necessary, although
in many basins two or three years are sufficient.
The minimization of (31) was performed with the help of the VA05A
computer package (Hopper, 1978), and its quality was checked by plotting the
sum of squares (31) versus each of the filter parameters. In that way it was
possible to see whether a global minimum was reached.
Minimizing (31) with respect to the filter parameters a~ results in estimates
of a~. The model standard deviation is estimated by
x/(minimum sum of squares)
a
=
N-K
where N is the number of terms in (31), and K is the number of filter
parameters (K = 3 in model 2).
The half width of a 95% confidence interval for a is approximately
1.96a
1.38a
x / [ 2 ( N _ K)] = x / ( N - - K)
HWCI(a) =
(32)
The covariance matrix of the filter parameters a~ is
A =
20 " 2 H -
I
(33)
where H is the Hessian of the sum of squares (31) at its minimum. The Hessian
H is approximated by using difference quotients, and, as for the size of the
increments, a suitable compromise has to be found between rounding-off
errors and truncation errors. From this covariance matrix the correlation
matrix of the a~ can be computed. The standard deviation a~ of a~ is the square
root of the corresponding diagonal element in the covariance matrix, and the
half width of a 95% confidence interval of a~ thus is
HWCI(a,) =
1.96a~
These formulae are standard results in regression analysis.
A number of methodological remarks can be feund in Alley (1984) and
Sorooshian and Dracup (1980).
C h e c k s on f i l t e r p a r a m e t e r s
Are all the parameters really necessary? To answer this question,
for example in the case of model 2, the hypothesis that parameters a~,
a2 and a3 are significantly different from zero has to be tested. This can
be done by checking whether the zero value belongs to the 95% confidence
ir~terval
(ai -
1.96ai, ai + 1.96ai)
326
G.L. VANDEWlELE ET AL.
The correlation matrix of the parameters has to be checked also. A correlation coefficient between two parameters very near + 1 or - 1, implies that
perhaps a model can be found with a smaller number of parameters and with
the same explanatory power; alternatively the parameters may have to be built
into the model in a different way, so that their explanatory effects are more
dissociated, and optimization is easier.
Residual analysis
Checks are performed to determine if the residuals u, behave as is required
by the model hypotheses, especially whether they are independent, homoscedastic and normally distributed with zero expectation.
Independence is checked by computing the observed autocorrelations Pk
with time lag k and the corresponding half width of a 95% critical interval
HWCI (Pk) (see, e.g. Box and Jenkins, 1976). The hypothesis Pk = 0 is true
at the 5% significance level when
Ipkl ~ HWCI (Pk)
The general behaviour of the residuals is judged by graphs of the residuals
versus important variables such as time itself, the input variables precipitation
p, and evaporation e,, and computed runoff d,. The residuals versus time
graph is used for checking the absence of trend and also homoscedasticity.
The scattergrams of the residuals versus the other variables Pt, et and art have
to be symmetric with respect to the horizontal axis (zero expectation), and the
conditional standard deviation has to be constant (homoscedasticity).
Some formal tests are also performed. It is checked whether
I~1" ~/(N - K)
~< 1.96
std u
(34)
where fi is mean residual and std u is residual standard deviation. This test is
at the 5% significance level. When N - K < 30 Student's table has to be
consulted. The same test, but restricted to residuals belonging to one season,
can be used for checking whether there remains a seasone,l component in the
residuals. The 'seasons' used in the present study are (except in Burma where
the seasons are chosen differently owing to climate): winter w January,
February, March; s p r i n g - April, May, June; s u m m e r - July, August,
September; autumn - - October, November, December. This check on the
seasonality of residuals turns out to be a most severe test of the models°
Sensitivity to calibration period
The results of application of a given model to two different calibration
WATER BALANCE MODELS IN BELGIUM. CHINA AND BURMA
327
periods for the same catchment have to be the same (i.e. not significantly
different), at least if the input and output data can be supposed to be time
homogeneous. This requires no important changes in land-use, the same
methods of measurement, etc.
Since, in practice, long observational time series risk being non-homogeneous
in time, it is advisable to define two different calibration periods by taking, for
example, the even years in the one and the uneven years in the Other calibration period.
To test whether the two calibrations on the same basin result in the same
parameter values, let A' = (a~, a2,.
" a2,.
" • • , aT)
" be the
' • • , a~) and A" = (al,
estimates of the vector A = (a~, a2, • • •, aT) of filter parameters cc,rresponding to the respective calibration periods, where K is the number of such
parameters (K = 3 in model 2). Then A' and A" are stochastically independent and approximately normally distributed
A' ~ N(A~, A')
A" ,~ N(A'~, A")
where A' and A" are covariance matrices to be computed by formula (33).
Their difference then is also normally distributed:
A'-
A" ~ N(A~ - Aft, A' + A")
and under the nuU hypothesis A~ = A'o'
A'-
A" ~ N(0, A' + A")
and
(A'-
A")(A' + A")-'(A' - A") r - ~ X~
where T means transposition, and XzKis chi-square distributed with K degrees
of freedom. With K = 3, a level 5% test on the identity of the two models is
(A'-
A")(N + A")-'(A' - A") v <. 7.815
Equality of model standard deviations e' and tr" can similarly be tested at
the 5% level by checking whether (see formula (32))
~[
a, 2
2(N'-
Icr' -
tr"l
a,,2
K) + 2(N"-
] ~< 1.96
K)
where N' and N" are the numbers of terms in (31) for the two calibration
periods. Again ~bi~ is an approximate test.
Beside these formal tests the general behaviour of the residuals is compared.
l'est by extrapolation
The model is first ca!ibrated with part of the data (the calibration period).
328
G.L. VANDEWIELE ET AL.
The parameter values obtained are then used in the time period covering the
rest of the data (the test period). Results of the latter computation (which is
a kind of 'extrapolation' toward other time periods) are compared with
observation. A good model will not show significant differences. Calibration
period and test period are defined in the same way as in the previous section
to avoid the influence of time heterogeneity in the data.
The test bears now on the extrapolated residuals ut and is performed by
checking whether
lalx/N ~ 1.96
O"
This test is also performed on residuals restricted to one season. Furthermore
the general behaviour of the residuals is compared.
Comparison of different models for the same basin and model quality
When the standard deviation a of the random component is small, the
unexplained part of runoff is a,~o small, and the model fits observed runoff
very nearly; therefore a is an inverse measure of quality. Moreover two models
of the same basin with different a can be compared by means of the confidence
intervals (32) to check whether the difference is significant. A formal test is
difficult to construct because the two standard deviations are not stochasfi.
cally independent, but if the two values of ~ belong to each others' confidence
intervals, it can safely be assumed that there is no difference in quality.
Wl'en the coefficient of variation of observed runoffs is small in a given
basin, model standard deviation will also be small, i.e. model quality will be
high. This leads to the definition of an 'absolute' quality measure.
Q
observed cv of runoff
estimated model cv for mean runoff
ocv
ecv
(35)
where ecv is taken from formula (7). The ocv value is compared with ecv (and
not with a directly) to obtain a dimensionless measure. This quality is almost
inversely proportional to a for different models on the same basin and thus
can also be used for such comparisons.
An example: model 2 and the Viroin river at Treignes station
This basin is situated in the Ardennes in southern Belgium (see also the
following section) and has an area of 554 km 2. Precipitation, evaporation and
runoff data are used for calibration in the period 1966-1981. Moreover,
precipitation and evaporation data in the period 1960-1965 are used in the
W~TER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
329
TABLE 1
P a r a m e t e r estimates o f model 2 on the Viroin river
al
a2
a3
Estimate
HWCI
0.422
4.53 × 10 -4
3.25 x 10 -a
0.044
1.3 × 10 -4
0.54 x 10 -a
take-off period (see earlier section on estimation). Mean runoff (/is 35.14 mm
month -~ . The runoff coefficient is 44.4% and the coefficient of variation of
observed runoff ocv is 0.881.
Calibration results are as follows: model standard deviation tr = 0.856
(formula (2)); HWCI(~) = 0.086 (formula (32)); ecv = 0.284 (formt~la (7));
quality Q = 3.10 (formula (35)); mean computed runoff = 33.5mm
month -~', mean expected runoff = 34.2mm m o ~n t, h - (formula (3)); mean
computed slow flow = 10.6mm month-~; mean computed fast flow =
22.9mm month -~. Filter parameters and their correlations are given in
Tables 1 and 2, tests on residuals in Table 3. Graphs of the sum of squares
versus parameters provide a check of whether the m;nimization is performed
properly (see earlier section on estimation) (Fig. 6).
Graphs of the observed time series Pe, et and qt~ and ofcomputed time series
r,, d,, m,, s,,f, and u, are plotted in Figs. 7 and 8. The auto~orre,advn ~"'~*~'~Pk of the residuals and the corresponding critical intervals are plotted in Fig. 9.
Scattergrams of residuals versus p,, e, and d, are plotted in Fig. 10.
The filter parameters are all significantly different from zero, and their
conelations have small absolute values. The residuals do not show a seasonal
pattern according to Table 3. Residuals are homoscedastic according to
Figs. 8 and I0, and ne trend and no outlier is present in Fig. 8. Residual
autocorrelation is not significant according to Fig. 9.
TABLE 2
Correlations between parameters o f model 2 on the Viroin river
al
a2
a3
al
G2
u3
I
0.018
- 0.355
--0.018
1
0.260
- 0.355
0.260
1
--
330
G.L. VANDEWIELE El" AL.
TABLE 3
Tests on residuals of model 2 on the Viroin river according to formula (34)
Residuals
Number
Mean
Standard deviation
T value
Sign
Year
Winter
Spring
Summer
Autumn
192
48
48
48
48
0.047
0.111
0.220
- I).057
- 0.088
0.848
0.790
0.810
0.860
0.892
0.765
0.976
1.887
- 0.456
- 0.683
No
No
No
No
No
THE DATA
Geographical and climatologicai description
Models were constructed for four regions: northern Belgium (65 basins),
Ardennes (6 basins), southern China (6 basins) and Bu~:ma (~: basins).
Belgium (30500km 2, 10million inhabitants)can be divk~ed into three
regions according to several criteria: a flat region below 5 m ~bove sea level
(a.s.l.) (3000 km2); a more or less hilly region (henceforth to be called northern
Belgium) between 5 and 200m a.s.1. (18000km2); and the Ar~tennes plateau
(9000 km2), mostly higher than 200m a.s.l. (except in the deep valleys of the
main rivers) and up to about 700m a.s.L The flat region is not ~onsidered in
the present study, because it is partly polder and is criss-crossed by canals, so
that the water balance has to be described by other kinds of models.
As for the climate, the evaporation from a free water surfa,~:e is spatially
fairly uniform. In winter it is very low and in summertime it averages 100 mm
per month. Precipitation does not show an important seasonality. The mean
is 60-70 mm per month in northern Belgium and 80-100 mm per month in the
Ardennes. Snow and frost are not important on a monthly scale.
Soil cover is nearly completely agricultural crops and grass in the north. In
the Ardennes, one-third is covered by forest and two-thirds by arable land and
grass. Except for the major towns (which are not in the catchments studied),
population density is 300 inhabitants per square km in the north, and 100
inhabitants per square km in the Ardennes.
The six Chinese catchments under study belong to the Pearl River basin,
which lies in the subtropical zone. Front-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 2000mm
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)
331
WATER BALANCE MODELS IN BELGIUM, C H I N A A N D B U R M A
f
l ....
'l ....
• i ....
i ....
! ....
I ....
l;
/
155
150
0
(11
tn
145
140
•350
.,375
.,i. O0
.425
.,~-50
.475
.500
PAP~MET':~ 1
.
.
.
.
~
.
.
.
.
i
-
"
-
i
.
.
.
.
i
.
.
.
.
' T
160
150
140
•300
.400
.500
PARAMETER 2
.600
1 0 -3
.700
-.
160
2
A
/
150
1 40
2.~0
`3.00
`3.50
4.00
1 0 -3
PARAMEFER
Fig. 6. The sum of squares SSQ of formula (31) versus each parameter in the neighbourhoo6 of the
parameter estimates (Viroin river, model 2).
332
G.L. VANDEWIELEET AL.
RAINFALL
,
i
,
,
i
,
.
|
.
.
.
.
250
Dm
200
pot
month
150
100
I
i
67
68
,
f
I
69
70
.,:.~.
71
I
i
i
72
73
74
75
7~
77
7~1
79
80
81
TIME :N YEARS
~'VAPORATION AHD EVAPOTRANSPIRATION
150
rnw
;oo
j
50
I
86
87
per
month
68
69
70
i
i
!
7t
72
7.1
0
"/4
7b
75
7"."
78
79
a0
81
OBSERVED AND CALCULATED DISCHARGES
w
,
,
,
i
I
200
qt
i1~llYl
dt
~o p e r
month
103
66
67
68
69
70
71
72
73
74
75
76
77
78
70
80
81
TIME :,~ YEAi~$
Fig. 7. Precipitation p,, evaporation e,, observed runoffq,, real evapotranspiration r, and computed runoff
d, versus time { v'iroin river, model 2).
about 80% of the total precipitation occurs. Free water evaporation is mainly
measured by using the type ~-80 evaporation p:m (which is one of the most
commonly used evaporation pans in China, with a diameter of 80cm).
Two Burmese basins, Yin and Yenwe, are studied in the present paper. The
Yin River is a tributary of the Irrawaddy near Meiktila about 480 km north
of Rangoon. The region has a semi-arid tropical monsoon climate characterized by a sc~,rce and erratic wet season and an arid dry season, with daytime
temperature often exceeding 38°C. Rainfall averages 915 mm per year and
333
WATER BALANCE MODELS IN BELGIUM. CHINA AND BURMA
STORAGE
,
i
|
i
i
i
200
tOO
,
66
67
~
,,i
68
i
I
t
!
69
70
71
72
,.L.
73
i
I
i
!
i
i
i
I
74
75
76
77
78
7~
80
81
TIME IN YEARS
FAST AND SLOW DISCHARGE
200
mm
so p e r
1
II
month
^
/
/ ,DO
0 5o
66
67
68
69
70
7~
72
73
74
75
76
77
78
79
,
w
i
i---
,
i
i
80
81
TiME IN VEARS
RESIDUALS
~ - i
,
,
,
l
J
-2
66
i
i
i
I
i
i
i
i
i
I
f
I
I
f
!
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
A
TIME IN YEARS
Fig. 8. Storage m,, slo~, runoff s,, computed runoff d, and residuals u, versus time (Viroin river, model 2).
mostly occurs as scattered, localized storms during the rainy season from May
through October. The winter season starts in November with the lowest mean
monthly minimum temperature of 13°C occurring in January. The hot season
takes place from mid-February to mid-May with the highest mean monthly
maximum te:r~perature of 37°C occurring in April. Virtually no rainfall occurs
from November through April. The area is composed mainly of sandstone,
shale and interbedded sandstone and shale. Terrace alluvium is silty, finegrained sand, whereas the channel alluvium is mainly medium to coarsegrained sand. Seventy-five per cent of the area ~s arable.
334
G.L. V A N D E W I E L E ET AL.
AUTOCORRELATiON
0.2
-0.2
I
t
_i
6
12
,
_
t
_
I
24
f
36
LAG ~N MONTHS
Fig. 9. Residual autocorreiation with level 5% critical band versus time lag in months (Viroin river,
model 2).
The Yenwe river is a tributary of the Sittang near Toungoo about 200 km
north of Rangoon. It lies in a tropical monsoon zone with an average rainfall
of 2000mm per year. The mean monthly temperature varies from 31°C in
April to 23°C in January. The lowest annual temperature can be as low as
14°C, normally occurring in January. The main hydroclimatological data are
given in Table 4.
Hydrological data
The hydrological data concerning the basins studied in this paper are
summarized in I'able 5. It is clear from Table 5 that even in northern Belgium
alone, the characteristics vary within wide ranges, as is brought out in Table 6.
There are a number of different rivers with the common name 'Molenbeek',
which means mill brook.
RESULTS WITH THE NEW MODELS
Best models
Many tens of models were tried, of which the best ones will be discussed.
They are defined in Table 7. This table shows that four out of the eight models
are applicable to the two Burmese basins alone. The other four models
335
WATER B A L A N C E MODELS IN BELGIUM, C H I N A A N D B U R M A
RESIDUALS VS C A L C .
"
"
"
i
.
.
.
.
i
.
.
.
.
l
.
.
.
.
RUNOFF
i
.
.
.
.
i
.
.
.
.
i
.
.
.
.
i
-
o
UP
i
UP
• ,°up
it
lib
l/
°
up
up =* t,° u p % ° °
up, j ~ ,=lp,~tP
° °°
Io=
.,.0
S
i
°=
0°
,
up
-T'~II*~
*_
.
,
o
"d~eal, °
,
~..
I
up*
UP"°°
•
ill*
* °
--.:....;.:.:.,....
;
UP
"up
..,
°
0
*
.
i
J
i
i
i
-2
.
.
.
,
~
20
i
-
-
-
!
40
.
.
.
.
i
60
.
.
.
.
~
80
.
.
.
.
!
1 O0
.
.
.
.
!
120
.
140
RE$ibUALS V° EVAR
i
.
.
.
.
J
°
*,
.
.
.
J
'.
.
.
.
*
*°e*
°°3 '~
~.%
°'~
,
° °*
.
.
.
.
.
i
.
.
.
.
|
-
o
w
*
J
,
*
°~
,
.
5,* .
*
"~
.
up
.
UP,
°
°
°
.
UP
qb
"
"up-
"
°
*
j° • o,,O. ~ o
* . , up
.'...
UP
•
"
: . . . . .
-2
.
.
.
.
o
.
.
.
.
20
!
.
.
.
.
i
40
.
.
.
.
60
!
.
.
.
.
!
80
.
.
.
.
i
100.
,
|
=
120
o
|
|
140
RESIDUALS VS RAINFALL
/
1
°
t
I 2
,
up
t an~
° Q~"
°
'~',,~'%
-
°•°
-'---dl~p--~Z~r a
°
i,,,°°°e
°
~
°
~'
0
°" ° e " " i, , ~
~. - ~
J-
° .as ~4~. BeNpIjo
°
0
up
'.'e
UP
il
•
up,
=up
-r'- ~
•
°
.
Oqp
_
- "-~" ~
Q
.,
-~-" " ~"
°
0
•
°
~
0
_
~'
°
up
0
up
~
•
*
UP
UP
25
50
75
100
125
150
175
200
Fig. 10. Scattergrams of residuals u, versus precipitatio~ p,, evaporation e, and computed runoff d, (Viroin
river, model 2).
G.L. VANDEWIELE ET AL.
336
TABLE 4
Some hydroclimatological data
Northern
Belgium
Ardennes
Chinese
basins
Burma
Yin
basin
Mean precipitation
(mm month-~
Coefficient of
variation for monthly
precipitation
Mean evaporation
(mm month-I )
Coefficient of variation
for monthly
evaporation
60-70
80-100
120-160
0.5-0.6
0.5-0.6
0.75-0.95
50-60
50-60
70-85
0.7-0.75
0.70-0.75
0.3-0.5
Yenwe
basin
62
240
186
120
(models 1 through 4) are sutficient for modelling 77 Belgian and Chinese
basins. All models in Table 7 have o~dy three filter parameters. This seems to
be a sufficient number for describing monthly water balance, since trials using
models with more parameters did not work significantly better, except in very
few basins. As has already been indicated in an e~:dier section, a number of
alternative evapotranspiration equations can also be used.
General description of results with the best models
Except in extremely few cases:
(1) parameters are significantly different from zero at the 5% level;
(2) correlations between parameters are smaller than 0.9 in absolute value,
and most correlations are much zmaller;
(3) estimated mean runoff is nearly equal to observed mean runoff;
(4) residual autocorrelation is either non-significant at the 5% level or small
(of the order of 0.2 or 0.3). It is not worthwhile to include an autocorrelatlon
parameter in the model.
(5) residuals do not show trends or heter0scedasticity;
(6) results are not sensitive to calibration period;
(7) extrapolation is satisfactory (extrapolated residuals are non-significant).
The question whether seasonality is completely explained by the model can
be answered by testing whether residuals still contain a significant seasonal
component. There are 79 basins with four seasons each, so 79 x 4 = 316
tests were performed, following the residual analysis methodology described
337
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
TABLE 5
Data for the basins studied
No.
River
Runoff
,,station
Area
(km 2)
Runoff
data
(years)
Runoff
coefficient
(%)
Coefficient of
variation for
runoff
(ocv)
Northern Belgium (65 basins)
l
2
3
4
5
6
7
8
9
l0
I1
12
13
i4
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
Sint-Jansbeek
Poperingevaart
leperlee
Gr. Kemmelbeck
Heulebeek
Mandel
Leie
Leie
Kerkebeek
Ede
Hertsbergebeek
Rivierbeek
Rivierbeek
Eekl. Watergang
Poekebeek
Maarkebeek
Zwalm
Molenbeek
Bellebeek
Molenbeek
Mark
Molenbeek
Dender
Molenbeek
Zenne
Zuun
Zuur,~
Zuun
Ki. Molenbeek
G r Molenbeek
Zuid Mark
Grote Nete
Grote Nete
Grote Nete
Kleine Nete
K!eine Nete
Laakebeek
Aa
Merkem
Oo~,tvieteren
Zuidschote
Vlamertinge
Heule
Oostrozebeke
St-Baafs-Vijve
St-Baafs-Vijve
Loppem
Maldegem
Oostkamr~:
Oostkamp
Oostkamp
St-Laureins
Nevele
Etikhove
Nederzwalm
Aalst
Essene
Geraardsbergen
Viane
Iddergem
Denderbeile
Massemen
Buizingen
St-Pieters-L.
St-Pieters-L.
St-Pieters-!,.
Liezele
Malderen
Merksplas
Hulshout
Huishout
Itegem
Grobbendonk
Lichtaart
Lille
Poederlee
;6
~8
,O
24
93
243
3190
3190
56
46
73
147
64
46
! 16
53
115
27
89
19
167
16
1244
45
634
72
66
79
34
66
41
458
468
532
526
299
28
222
20
18
9
!3
17
12
22
9
18
16
12
6
13
21
7
14
21
15
12
17
10
17
35
18
8
10
8
6
14
1I
19
8
!l
35
35
5
14
7
33.8
29.6
47.7
30.1
32.9
33.2
32.3
33.1
35.0
36.4
31.0
31.2
29.3
20.9
27.0
28.7
33.7
27.2
35.6
32.6
28.4
33.8
29.0
36.2
20.0
30.6
26.7
34.6
30.3
31.4
24.7
43.8
42.6
43.4
53.4
50.3
42.4
33~6
1.109
1.097
0.731
0.938
0.943
0.857
0.758
0.540
I. 124
1.012
0.912
0.864
1.021
1.005
0.992
0.744
0.780
1.018
0.743
0.896
0.915
0.733
0.943
0.827
0.660
0.846
0.868
0.706
0.910
0,796
1.183
0.459
0.460
0.549
0,559
0,516
0.451
0.637
338
G.L. VANDEWIELE ET AL,
TABLE 5 Continued
No.
Rive.r
Runoff
station
Area
(kin2)
Runoff
data
(years)
Runoff
coefficient
(%)
Coefficient of
variation for
runoff
(ocv)
Northern Belgium (65 basins)
Poederlee
39
Aa
40
Molenbeek
Pulle
41
Aa
Turnhout
42
Grote Nete
Varendonk
43
Dijle
Wavre
44
Dijle
St-Joris-Weert
45
Laan
Terlanen
46
Dijle
Wilsele
47
Dijle
Haaeht
48
Demer
Hasselt
49
Demer
Bilzen
50
Demer
Diest
51
Demer
Diest
52
Demer
Kuringen
53
Zwartebeek
Lummen
54
Mangelbeek
Lummen
55
Herk
Wellen
56
Demer
Aarschot
57
Gete
Halen
58
Grote Gete
Hoegaarden
59
Velp
Ransberg
60
Warmbeek
Hamont-Achel
61
Dommel
Overpelt
62
Jeker
Mai
63
Jeker
Kanne
64
Herk
KemR
65
M6haigne
Moha
222
42
99
378
314
645
127
890
3160
278
110
2045
1904
343
102
1132
100
2163
810
208
99
57
93
335
463
280
345
9
19
11
19
1!
12
14
12
35
8
15
17
17
1I
lg
14
14
16
10
18
18
9
12
!6
12
5
15
32.8
51.2
23.5
46.6
26.7
26.9
32.3
24.7
29.8
34.7
21.0
25.8
24.4
32.9
40.3
40.1
23.1
28.1
23.5
22.6
25.9
41.5
37.0
20.7
18.6
23.9
27.5
0.543
0.791
0.634
0.401
0.338
0.324
0.261
0.320
0.545
0.436
0.5~43
0.547
0.540
0.441
0.513
0.469
0.452
0.540
0.520
0.393
0.779
0.430
0.547
0.338
0.371
0.739
0.676
Ardennes (6 basins)
Ambl6ve
Hoegne
Lesse
Ourthe
Semois
Viroin
1044
68
1314
1597
1235
554
16
8
13
12
16
16
52.0
58.6
41.9
45.2
53.1
44.4
0.837
0.894
0.852
0.960
0.865
0.881
2000
1556
595
16
18
17
55.3
57.5
76.1
i .033
0.987
0.935
66
67
68
69
70
71
Martinrive
Polleur
Gendron
Hamoir
Membre
Treignes
Southern China (6 basins)
72
Eongjiang
Wengjiang
73
Xingzi
Fenghuangshan
74
Tongguangshwi
Huangiingtang
339
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
TABLE 5 Continued
No.
River
Runoff
station
Area
(km 2)
Runoff
data
(years~
Runoff
Coefficient of
coefficient variation for
(%)
runoff
(ocv)
Southern China (6 basins)
75
Andunshui
Jiuzhou
76
Chuantonhe
Shuntan
77
Zhenshui
Xiaogulu
385
1357
1881
15
14
19
57.9
62.0
54.0
0.895
0.964
0.954
Burma (2 basins)
78
Yin
79
Yenwe
1100
790
5
6
! 3.5
55.2
1.793
i .401
Phongone
Myogya ung
earlier. The significance level of all tests was taken at 5%. Out of these 316
tests, less than 5% gave a negative result. Consequently it may be concluded
that seasonality has been accounted for in the models considered.
Results on quality Q (see eqn. (35)) are presented in Table 8. In this table
the quality of the model (listed in Table 7) that best suits the individual basin
is considered. If the six worst and the six best cases among the basins in
northern Belgium are left out, the quality range ~s 2.12--2.75. More detailed
results are given below in the comparative results section and Table 10.
SOME WATER BALANCE MODELS FP,.OM THE LITERATURE
Short description of the models
Four models previously defined in the literature were considered here so
that their performance could be compared with the new models described in
the previous section. They all are discussed by Alley (1984), who gives a full
description and who applied them to a number of basins.
The first one is Thornthwaite and Mather's (1955) T-model. It is a model
TABLE 6
Range of basin characteristics in northern Belgium
Area
Runoff coefficient (%)
Coefficient of variation for
runoff (ocv)
Minimum
Maximum
16
i 8.6
0.261
3190
53.4
1. i 83
340
G.L. VANDEWIELE ET AL.
TABLE 7
Definition of eight models
Model
number
Evapotranspiration
equation
Slow runoff
equation
Fast runoff
equation
Application to
(20)
(20)
(20)
(20)
1)
2)
l)
2)
(22)
(22)
(22)
(22)
Belgian and
Chinese basins
(at
(at
(at
(at
=
=
=
=
l
(17)
2
3
4
(17)
(18)
(18)
5
6
(17)
(17)
(20) (a; - l)
(23)
(25)
Yin (Burma)
No
7
(17) (al = l/e)
(18) (a I = O)
(20) (at = 1)
(20) (at = i)
(24)
(24)
Yenwe (Burma)
8
Equations (1), (2), (8) and (11) are common to all models.
with two storages: 'soil moisture index' m , and 'water surplus' v , . The model
also has two filter parameters: soil moisture capacity at and storage constant
a 2 for v,. The second model is Alley's (1984) T~-model. This model is a
modification of the preceding model in that a fraction a3 of the precipitation
is immediately transformed illto direct runoff. The rest of the precipitation
then enters the system as before. The third model is Thomas' (1981) abcdmodel. There are also two storages in this model: groundwater storage and
soil moisture storage. The fourth model is Palmer's (1965) and Alley's (1984)
P-model. Palmer (1965) used a water balance model to develop an index of
meteorological drought. The model is vnusual ~n that it uses a 'root constant"
concept for calculating evapotranspiration. The procedure consists of
dividing the soil into two layers: the upper layer roughly equivalent to the
plough layer and the underlying layer in which the available capacity depends
on the depth of the effective root zone and on the soil characteristics in the
TABLE 8
Quality of best models
Region
Northern
Belgium
Ardennes
Southern
China
Burma
Number of
basins
65
6
6
2
Range in
quality Q
i.85-3.12
2.37-3.47
3.40-4.50
3.!3-3.72
341
WATER BALANCE MODELS IN BELGIUM. CHINA AND BURMA
TABLE 9
Parameter values of the models from the literature for the basins mentioned in Table 10
Model
Parameter
Mean
Range
T
al
a,
70
0.68
1-230
0.4-0.9
Ta
a~
a2
a3
! 20
0.59
0. i 4
30-380
0.3-0.9
0.04-0.30
abcd
a
b
c
d
0.986
475
0.270
0. l I
0.96-0.999
260-1900
0.t)4-0.70
0.0003-0.415
P
al
a,
66
0.66
1-190
0.4-0.90
area under study. It is assumed that moisture cannot be removed from
(recharged to) the lower layer until all of the available moisture has been
removed from (replenished in) the upper layer. A similar approach was used
by Haan (1972~ and Zhao et al. (1980) in their monthly and daily rainfall
runoff models respectively. Runoff is assumed to occur if and only if the soil
moisture storages in both layers reach the2r moisture capacity. The P-model
was later modified by Alley (! 984) in his comparative study of monthly water
balance models. He considered a lag between moisture surplus and streamflow
runoff by using the same procedure as fbr the T-model. As a consequence
there are three storages in the model. The three parameters are: a~ and a~',
which are moisture capacities in the upper .and lower layer, respectively;
al = a~ + al, a2, which is a storage constant.
Calibration results with literature models
To compare easily with Alley's results, a summacy of the fitted parameter
values is shown in Table 9, and some other details are mentioned in Table ~0.
The results in Table 9 are relative to the catchments mentioned and optimized
in Table 10. In many ways the results are similar to those reached by Alley
(1984).
The average value of parameter al in ,he T-model is 70 ram, but a wide
"o'~'-"
.,.~,. has been found, which is comp~_~_,~,,~
. . . . . ~.i~ to the results obtained by Mather
(1981) and Alley (1984). A!so the a? values are similar to Alley's. Values of al
342
G.L. VANDEWIELE ET AL.
TABLE 10
Quality Q and number of seasons with significant residuals. 'Good' models for each basin are
underlined
Basin
Model
2
3
4
T
Tot
abed
P
2.68-2
2.46- I
2.21-1
2.16-0
2.08-0
2.97- l
2.67-0
2.49-2
3.09- I
2.07-1
2.79-1
2.52-0
2.29-0
2.22- l
2.12-0
3.17-0
2.61-0
2.38-1
2.82- !
1.90-1
2.70-0
2.43- I
2.27-0
2.19-0
2.07- I
2.97-0
2.61-0
2.64--i
3.04- !
2.38-1
2.73-1
2.45-0
2.34-0
2.22- I
2.12-1
3.10-0
2.64-0
2.58-1
2.90-2
2.28-2
2.02-2
1.98-0
1.99-1
1.85-1
1.76-0
2.14-3
2.01-0
! .57-3
1.67-2
i .23-3
2.34-1
2.25-0
2.16-2
2.04-0
1.91-0
2.39-3
2.25-1
1.64-4
1.73-2
1.47-3
2.59-1
2.48-2
2.34-2
2.40-0
2.1 !-0
3.1n-I
2.8~-1
2.58-2
2.87-1
1.99-2
1.96-0
1.98-1
1.84-1
1.75-0
2.13-3
Z95-0
2.35-0
2.75-0
3.24-0
2.61-0
2.93-0
3.08-0
2.37-0
2.93-0
3.47-0
2.82-0
3.10-0
2.82-0
2.31-0
2.63-0
2.99-0
2.61-0
2.83-2
2.92-1
2.35-0
2.70-1
3.07-1
2.71-0
2.89-1
2.39-1
3.46-2
2.75-0
3.15-2
2.39-1
2.50-2
2.58-1
2.59-2
2.34-2
2.84-0
2.28-0
2.71-0
2.86-2
2.71-2
2.67-1
4.31-I
3.69- l
4.08-3
3.40-0
4.27-0
3.66-2
4.33-2
3.88-0
4.20-1
3.30-2
4.17-1
3.50-2
4.31-1
3.86-2
4.50-1
3.41-1
4.29-1
3.57-2
3.20-1
3.02-2
3.82-2
3.13-2
2.76-2
3.58-1
3.13-0
3.92-1
2.99-3
4.24-2
4.01-0
4.63-3
3.42-2
4.34-0
3.69-0
Model 6:
• ao n
~,1...~..!
o..
l¥1qL~qd[~l O
3.72-0
-, 1,-,. ~ - - ~,,1
.,,~.
I
Northern Belgium
6
l0
20
21
24
30
33
56
57
58
1 ,=
1.69 "*
1.22-3
Ardennes
66
67
68
69
70
71
Southern China
72
4.27-2
73
74
75
76
77
3.69-1
3.83-1
3.24-1
4.03.-I
3.57-0
3.32-1
2.46-2
2.51-2
2.54-2
2.31-2
3.19-1
3.02-2
2.76-2
3.58-1
3.41-~
Burma
78
79
Model 5:
Model 7:
•. Y , ~ J I ~ X . F
in the Ta-model were larger than those in the T-model. This phenomenon was
also pointed out by Alley (1984).
The result for the abed-model confirms the experiences obtained by
Thomas et al. (1983) and Alley (1984). Optimized values of parameter a
always exceeded 0.96. Very high values of parameter b were found, consistent
~i~h the results of Alley and Thomas et al. Parameters c and d were found to
be statistically non-significant for many catchments. A similar result was
WATER BALANCE MODELS IN BELGIUM. CHINA AND BURMA
343
obtained by Alley (1984). He suggested that the abed-model ~hould have been
calibrated simply by setting e = 0 and d = 1 for at least some stations. In
Tables 9 and 10, all four parameters are optimized. In another calibration run,
parameter d was fixed at a value of 0.2, inspired by experience; the procedure
did not significantly alter model quality.
Two calibrations for the P-model were performed in this study. Firstly, the
model was calibrated by fitting moisture capacity for each of the two layers.
The results show that parameters a'~ and a~' are not only highly correlated but
also statistically non-significant. As Alley (1984) has pointed out "it appears
that a~ and a~' should not be optimized simultaneously". A second calibration
was then performed by fixing a~. Three distinct a~ values were taken: 25 mm
was used for northern Belgium, whereas values of 5 mm and I mm were used
for the Ardennes and the Chinese catchments, respectively, since a preliminary study showed that the total moisture capacity (represented by parameter
a~) was much smaller in these two regions. Tables 9 and 10 relate to the latter
calibration.
Some general remarks may be made concerning the four models from the
literature. Frequently these models are difficult to optimize: there are high
correlations between parameters, wide confidence intervals and even nonsignificance of the parameters and sensitivity with respect to initial parameter
values. Moreover, in some cases parameter values have the tendency, during
optimization, to overstep the physical constraints. The blanks in Table 9 are
due to this phenomenon'. State variables simulated by these models may be
quite different in different catchments, as has been pointed out by Alley
(1984), and take unrealistic values. For example soil moisture was, more often
than not, at its capacity a~ in the T-model. This fact, together with the fact that
the capacity takes unrealistic values (down to 1 mm), makes the model less
interesting.
COMPARATIVE RESULTS
Table 10 allows a comparison between the models introduced in the present
paper and some models from the literature. From ?he 65 basins in northern
Belgium, a representative sample of 10 basins was taken. The numbering of
the basins is given in Table 5. Two critical aspects are retained in Table 10:
the quality Q according to eqn. (46), and the number of seasons (out of lbur
seasons) with significant residuals.
In the table, 'good' models for each basin are underlined. A model is
considered to be 'good' if two conditions are fulfilled: (1) the number of
seasons with significant residuals is equal to the minimum number among the
eight models considered for each basin; (2) the quality Q belongs to the 95%
344
G.L. VANDEWIELE ET AL.
confidence interval of the quality of the model with highest Q fulfilling the first
condition.
Taking basin 69 as an example, the minimum number of seasons with
significant residuals is zero, so model~ 4, T, T0c and P are not 'good'. Among
the remaining models, model 2 has the highest quality: Q = 3.47. Its 95%
confidence interval is (3.12-3.92). Model 3 has a quality Q = 2.99 not
belonging to the latter interval, so it is not 'good' either. Only models 1 and
2 are 'good' models for basin 69. The half width of the 95% confidence
interval of Q is roughly 10% of Q itself.
In model comparison, it has to be borne in mind that different models
sometimes have different numbers of free parameters. The new models l, 2,
3 and 4, and also model T~, have three parameters, whereas models T and P
only have two free parameters. The abed-model has four parameters. Model l
is different from model 2 only by another choice of 'parameter' a~ in formula
(20). So those two models together form a model with four parameters. The
same may be said of models 3 and 4.
Since one parameter of model P has been fixed (see previous section on
calibration results), model P nearly reduces to model T; in Table 10, the results
of models P and T are very similar. On the other hand, model T is a simplified
version of T~, so that T~t is systematically better than T. Comparing the
models in the literature, model abed is always best in terms of the quality Q;
however, it,, five cases out of 18, model T0~ has less residual seasonality,
whereas the inverse is true in six cases.
When the number of basins for which a model is 'good' (underlined results)
is considered, the new models l, 2 and 3 are far better than any of the models
it. the literature. There is only one basin (no. 67) where a published model is
s~gnificantly 'better' than any of the new models. This can also be seen in the
percentage of seasons with significant residuals: the abed-model has 24 in 76
or 32% such seasons, whereas the combined model 1-2 (see above) only has
seven in 88 or 8% such seasons. This i3 to be compared with the 5% limit
allowed by the 5% significance level used. As opposed to the calibration
difficulties encountered with the abc~-model and the original P-model (see
previous section on calibration results)~ calibration of the new models is easy.
In the T-, T~- and P-models the calculated moisture or storage series is
often unrealistic (see previous section on calibration results); this seems not to
happen in the new models, as can be seen in the graphs of the m, function (see
e.g. Fig. 8). Moreover, this has been checked by comparing the calculated
storage time series with measured groundwater levels. Untbrtunately ~nly
very few time series of groundwater levels are available for such comparisons.
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
34.~
SUMMARY AND CONCLUSIONS
A methodology essentially based on regression analysis is developed for
constructing monthly rainfall runoff models of river basins up to 4000 km 2
without important natural or artificial lakes and without important snow
storage or frost. The input time series are monthly areal precipitation and
monthly potential evapotranspiration. The methodology uses automatic calibration which excludes subjective elements.
A ~umber of new models are defined and compared with models taken
from the literature. These models are applied to 79 river basins in different
regions in the world The following conclusions ake relevant to the basins
studied.
~'
(1) Transforming the runoff time series beforehand is necessary to obtain
homoscedastic deviations. The square-root tran~f0rmation seems to solve that
problem (see eqn. (1)).
(2) Three filter parameters seem to be sufficient. They are nearly always
related to real evapotranspiration, slow runoff and fast runoff. Introduction
of other parameters does not result in significant improvement in nearly all
basins.
(3) In the new models, residual autocorre!ation is non-significant or very
small. It is thus not necessary to model it explicitly.
(4) When modelling a particular basin, several models have to be tried.
There is no universal model which is best for all basins.
(5) As can be expected, the form of the evapotranspiration equation in the
new models is less critical than of the runoff equation itself; several
expressions for the evapotranspiration prove to be useful.
(6) The models taken from the literature perlbrm less well than the models
proposed in the present paper; in many cases tile former are of poor quality,
and an important part of the seasonality is left unexplained.
(7) By analysing the mathematical structure nf the new models on the c,ne
hand and the models from the literature on the other, one can try t~ ,-w.--.-"""~'"
the huge differences in explanatory power of the two classes of models.
(8) In the new models there is only one stor;~ge, i.e. the memory of the
catchment is summarized in one number. In the :~ublished models, there are
two (T, Ta and abcd) or three storages (in the original P-model). In the
P-model, it was even impossible ~o distinguish between two of them (the two
ground layers). It seems superfluous to put the complication of multiple
storages into a model that ha~ a monti~l), time step. (With a smaller time step,
important lakes or significant snow packs each winter, things will be
different.)
(9) Another characteristic of the T-, T~- and P-models is the con~traints on
-
346
G . L VANDEWIELE ET AL.
storages, i.e. the soil moisture capacities (at least for part of the storages used).
This is an attractive idea from ~he conceptual point of view, because the
parameters introduced in that way have a 'direct' physical meaning. In fact
this idea turns out to be a nuisance, because this capacity is reached more
often than not by the storage (see section on calibration resu~La).
'~-" Because a
catchment is a complicated heterogeneous system it is better to avoid such
thresholds. Such thresholds are not used in the new models.
(10) Another feature in the new models is the distinction between slow and
fast response (runoff). This corresponds to the well-known distinction
between baseflow and direct runoff (because of the great time step, a storage is
to be included only for baseflow). In the models from the literature (T and P)
no such distinction is made.
(11) Although the parameters in the new models do not have 'direct'
physical meaning, they translate general intuitive ideas about the process (see
the section discussing the evapotranspiration equation, and many other
elements in the new models). Moreover the parameter values are related to
physical characteristics of the basins (see Vandewiele et al., 1991).
ACKNOWLEDGEMENTS
Thanks are due to the Belgian Royal Meteorological Institute (K.M.I.) and
especially to Dr. G. Demaree and Dr. F. Bultot, and also to the Ministry of
the Flemish region for their friendly collaboration in data gathering. The
latter, and the government Office for Developing Countries (ABOS), partly
funded the present research. Professor A. van der Beken (Interuniversity
Postgraduate Program in Hydrology, IUPHY, Brussels) is thanked for his
friendly collaboration and support.
REFERENCES
Alley, W.M., 1984. On the treatment of evapotranspiration, soil moisture accounting and
aquifer recharge in monthly water balance models. Water Resour. Res., 20(8): ! 137-1149.
Bladt, A., Demaree, G. and van der Beken, A., 1978. Analysis of a monthly water balance
model applied to two different watersheds. In: G.C. Vansteenkiste (Editor), Modeling,
Identification and Control in Environmental Systems. North Holland, Amsterdam,
pp. 759-771.
Box, G.E.P. and Jenkins, G.M., 1976. Time Series Analysis. Forecasting and Control. Holden
Day, San Francisco, 575 pp.
Cramer, H., 1963. Mathematical Methods of Statistics. Princeton University Press, Princeton,
N J, 575 pp.
Haan, C.T., 1972. A water yield model for small watersheds. Water Resour. Res., 8(1): 58-69.
Hopper, M.J., 1978. A Catalogue of Subroutines. Harwell Subroutine Library, UK Atomic
Energy Authority, Harweli, 69 pp.
WATER BALANCE MODELS IN BELGIUM, CHINA AND BURMA
347
Hounam, C.E., 1971. Problems of evaporation assessment in the water balance. Report No. 13
on World Meteorological Organization/International Hydrological Decade Projects,
Geneva.
Lowing, M.J. (Editor), 1987. Casebook of Methods for Computing Hydrological Parameters
for Water Projects. Studies and Reports in Hydrology No. 48, UNESCO, Paris, pp.
151-155.
Mather, J.R., 1981. Using computed streamflow in watershed analysis. Water Resour. Bull.,
17(3): 474-482.
Ni-Lar-Win, 1989. Monthly rainfall runoff models for Belgian and Burmese catchments.
Master's Thesis, Laboratory of Hydrology, Vrije Universiteit, Brussels.
Palmer, W.C., 1965. Meteorologic drought. Res. Pap. U..S. Weather Bur, 45, 58 pp.
Romanenko, V.A., 1961. Computation of the autumn soil moisture using a universal relationship for a large area. Proceedings of the Ukrainian Hydrometeorological Research Institute,
No. 3, Kiev.
Sorooshian, S. and Dracup, J.A., 1980. Stochastic parameter estimation pr.~cedures for
hydrologic rainfall-runoff models: correlated and heteroscedastic enor cases. Water
Resour. Res. 16(2): 430-442.
Thomas, H.A., 1981. Improved methods for national water assessment. Report, Contract
WR15249270, US Water Resource Council, Washington, D.C.
Thomas, H.A., Marin, C.M., Brown, M.J. and Fiering, M.B., 1983. Methodology for water
resources assessment. Report to US Geological Survey, Rep. NTIS 84-124163, National
Technical Information Service, Springfield, VA.
Thornthwaite, C.W. and Mather, J.R., 1955. The water balance. Pubi. Climatol. Lab. Climatol.
Drexel Inst. Technol., 8(!): !-104.
Vandewiele, G. L., Xu, C.Y., and Huybrechts, W., 1991. Regionalisation of physically based
water balance models in Belgium. Application to ungauged catchments. Water Resour.
Manage, 5: 199-208.
Weeks, W.D. and Boughton, W.C., 1987. Tests of ARMA model formr for rainfall runoff
modelling. J. Hydrol., 91: 29-47.
Xu Chong-Yu, 1988. Regional study of monthly rainfall runoff models. Master's Thesis,
Laboratory of Hydrology, Vrije Universiteit, Brussels.
Zhao Renjun, Zhuang Yilin, Fang Lerun, Liu Xinren and Zhang Quansheng, 1980. The
Xinanjiang model. International Association of Hydrological Sciences, Publ. No. 129,
pp. 351-356.