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Stepwise development of a two-parameter monthly water balance model
Journal of Hydrology 318 (2006) 200–214 www.elsevier.com/locate/jhydrol
Stepwise development of a two-parameter monthly water balance model Safouane Mouelhi, Claude Michel, Charles Perrin*, Vazken Andre´assian Cemagref, Hydrosystems Research Unit, Parc de Tourvoie, BP 44, 92163 Antony Cedex, France Received 1 March 2004; revised 18 May 2005; accepted 3 June 2005
Abstract It has recently been shown that the structures of monthly water balance models can be very parsimonious. In this paper, we describe how a new two-parameter rainfall-flow model was developed using a stepwise approach. A sample of 410 basins representing a wide range of climate conditions from semi-arid through temperate to tropical humid was used to develop the new model, on the basis of previous work on a four-parameter daily rainfall-flow model. Comparisons with several well-known models showed that the new structure is efficient and provides valuable insight into the rainfall-flow transformation at a monthly time-step. q 2005 Elsevier Ltd All rights reserved. Keywords: Hydrological modelling; Rainfall-runoff models; Water balance models; Lumped parameter models; Continuous simulation; Monthly time-step
1. Introduction Over the last century many models have been proposed to represent the transformation of precipitation into streamflow at the outlet of a basin. When the time step is large, these models have usually been called water balance models, assuming the response time to be negligible compared to the time step. Monthly water balance models are valuable tools in water resources management, reservoir simulation, drought assessment or long-term drought forecasting. These models are also very useful because, due to their inherent parsimony, they lend themselves to * Corresponding author. E-mail address: [email protected] (C. Perrin).
0022-1694/$ - see front matter q 2005 Elsevier Ltd All rights reserved. doi:10.1016/j.jhydrol.2005.06.014
regionalisation and can be further used on ungauged basins. Since they can be very simple, they are easy to handle for water resources managers. Of course, their low level of complexity means that they have to focus on the most prominent features of the rainfall-flow transformation. Developing monthly water balance models implies a threefold lumping—spatial, temporal and conceptual—of processes at work. Spatial lumping means that rainfall and evaporation are averaged over the basin of interest and are referred to as scalars. Temporal lumping means that only cumulative inputs and outputs over a month are considered; we do not know the actual sequence of rainfall events, but only the resulting monthly aggregation of all inputs recorded in a given month. Conceptual lumping
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means that processes are not individualized and some of them are assembled into simple sub-models. From a physical point of view, one might argue that, with such a large time step, the processes underlying the rainfall-flow transformation may be distorted out of recognition, leaving little hope of success. However, the performances of monthly water balance models are evidence of their efficiency. The MOSAZ model developed by Jayasuriya et al. (1991) and presented by Boughton (2005) in his review of water balance modelling in Australia, was likely the first attempt to represent the monthly rainfall-flow transformation with only two free parameters. A few years later, the model proposed by Makhlouf and Michel (1994), developed on French catchments, proved to be more successful than many other models in that climate context. Since then, this model has been widely used in France (Lavabre et al., 1999, 2002) as well as in Western Africa (Paturel et al., 1995; Niel et al., 2003; Mahe et al., 2004). More recently, Guo et al. (2002) also developed a twoparameter model that can be used in a semidistributed way when dealing with large basins. It was applied to large basins in China to predict the impacts of climate change. Vandewiele et al. (1992) developed a series of three-parameter models that were applied in several countries (see e.g., Xu and Vandewiele, 1995; Xu et al., 1996; Mu¨ller-Wohlfeil et al., 2003). Ibrahim and Cordery (1995) built a four-parameter model for Australian basins. Alley (1985) reports on an efficient four-parameter model—the abcd model—applied to several catchments in the United States. In the literature, there are also other monthly models with a much higher number of parameters: see among others the CATPRO model (Kuczera et al., 1997) with up to 12 parameters used in Australia and the model developed by Pitman (1978a,b) and later used by Hughes and Metzler (1998) in Namibia and by Wilk and Hughes (2002) in India, which also contains 12 parameters. Among the models of intermediate complexity, the model proposed by Jothityangkoon et al. (2001) was developed following a downward approach. In these intermediate models, some parameters must be derived from physical basin characteristics (such as slope, soil field capacity and
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hydraulic conductivity) while the others need calibration. In this article, we focus on lumped water balance models involving a small number of free parameters to be calibrated from rainfall and discharge data. Identifying the relationships between parameters and basin characteristics is considered here as a question to be addressed a posteriori (in regionalisation studies) rather than a priori (i.e. based on preconceived ideas of the physical meaning of parameters). Indeed, we believe that insisting on a priori connections to physical characteristics jeopardises the ability of the data to verify or disprove these conceptual links. The issue of parameter explanation is not addressed in this paper. Looking at past experience of rainfall-flow modelling at a monthly time step, it seems that very few authors have made extensive comparative tests of these water balance models in various climate conditions (see e.g. Vandewiele and Ni-Lar-Win, 1998). However, much could be gained from such assessments to improve model efficiency (Perrin et al., 2001). Therefore, one objective of our research was to try to develop a new or improved model, even more general in its applicability, making the most of recent experience in this domain and evaluating the proposed model on a large sample of catchments. It would be tedious to retrace all the stages of model development that have led to the monthly model presented in this article. The first step was to choose a model structure general enough to encompass the components of existing monthly models that seemed the most efficient. Then, we made systematic attempts to improve its performance while reducing its complexity. By testing the various model versions, we let hydrological data corroborate or invalidate hypotheses about the best way to represent the main features of the rainfall-flow transformation in a tractable model. Note, however, that leading up to the rather straightforward approach outlined here, there has been a great amount of data mining and empirical research. The next section presents the data used for model development and comparative assessment. In the third section, a Parent Model Scheme (here called PMS) with five parameters is presented. In the fourth section, we analyse this model to identify the number
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of functions and/or parameters that deserve to be present and/or calibrated. The resulting model structure including the seemingly most interesting components was further improved to give a new monthly model, called GR2M and detailed in Section 5. This model is then compared to five well-known models from the hydrological literature in Section 6. The seventh section is devoted to a discussion of the water exchange term included in the GR2M model. The last section gives the main conclusions of this study.
2. Data used for model development and model assessment methodology 2.1. Data The large sample of basins used as a basis for developing and comparing the various models spans four continents. It has recently been used to explore the role of complexity in model performances (Perrin et al., 2001) and to complete the development of the daily lumped GR4J model (Perrin et al., 2003). Most of the basins are situated in France, others in the United States, Australia, the Ivory Coast and Brazil (see Fig. 1). Basin surface areas range from 1 to 50,600 km2. The climate conditions vary from temperate through semi-arid to tropical humid, with mean annual potential evapotranspiration between 630 and 2040 mm, mean annual rainfall of 300–2300 mm and mean annual streamflow between less than 10 and 2040 mm. Inputs to the models are essentially areal precipitations (potential evapotranspiration rates are interannual mean monthly values). The average record length is about 15 years in our sample. Fig. 2 shows mean interannual values of precipitation, potential evapotranspiration and streamflow for all 410 basins of our sample. In a few cases, mean streamflow exceeds mean rainfall, but the corresponding basins have not been excluded from our test sample because of the lack of an objective criterion to do the same critical work on the opposite range of behaviours, i.e. when streamflow might be suspected of being unreasonably low.
2.2. Assessment approach The data record from each basin was divided into two periods of almost equal length. Every model was assessed in simulation mode on each period using parameter values calibrated on the other period, according to the split-sample technique (Klemesˇ, 1986). This operation was repeated after switching the two periods. At the beginning of each test period, the first year of simulated discharge was not used in the computation of evaluation criteria (i.e. the warmingup period was 1-year long). The overall assessment was made on cumulative performances from both tests in simulation mode. When only one basin was considered, the criterion of the sum of least squared errors was chosen to judge the efficiency of the model. For each basin, a single value of RMSE (root mean square error, in millimetres) was computed by combining the sum of squared errors obtained in simulation mode for each period in turn (Eq. (1)). vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ni X 12 u 1 X (1) ðQ K Q^ i;j;k Þ2 RMSEi Z t 12ni jZ1 kZ1 i;j;k where the subscript i, varying from 1 to N, indicates the basin, the subscript j, varying from 1 to ni, indicates the year in the time series of catchment i and the subscript k corresponds to the month in year j. Q stands for observed runoff, Q^ for calculated runoff. The mean monthly rainfall, Pi , and the variance of the monthly rainfalls, VARPi, were computed on the same dates: Pi Z
ni X 12 1 X P 12ni jZ1 kZ1 i;j;k
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ni X 12 u 1 X VARPi Z t ðP K Pi Þ2 12ni jZ1 kZ1 i;j;k
(2)
(3)
where P is the observed precipitation. For each basin i, a goodness-of-fit criterion Ci could be computed by: Ci Z
RMSEi VARPi
(4)
to yield the overall evaluation criterion C over the whole test basin sample:
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Fig. 1. Catchment locations in (a) France, (b) the Ivory Coast, (c) Brazil, (d) Australia and (e) the United States.
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Fig. 2. (a) Mean streamflow versus mean precipitation and (b) mean potential evapotranspiration versus mean precipitation for the 410 basins. N 1 X C Z C N iZ1 i
(5)
Often, RMSE is compared to the variance of observed streamflow, as in the widely used Nash and Sutcliffe (1970) criterion. In Eq. (4), we used the variance of observed rainfall to standardise model error and emphasise the role of the model in transforming these inputs into streamflow. However, since the common approach is to use the Nash and Sutcliffe criterion, we also give its mean value over the whole test sample (hereafter called NS) along with the C-criterion value. Given the size of our test sample, any comparison or model ranking is bound to be sensitive to the choice of the overall criterion C. However, there is little chance of greatly modifying our conclusions on the respective value of the different model components tested here. To confer even greater robustness on our splitsample test approach, the parameters were calibrated by using the least square errors computed on the square root transforms of the discharges as the objective function, while the validation criteria were computed without prior transformation on the discharge. Many authors advocate the use of the square root of the discharges before evaluating the square errors to make the most of the information imbedded in the data. In addition, the change of criterion between the calibration and validation
assessment penalizes any overparameterization phenomenon. Model parameters were calibrated by a local optimisation scheme described by Edijatno et al. (1999).
3. The parent model scheme (PMS) In order to determine which elements are important in a monthly rainfall-flow model, we started by adopting a parent model structure (PMS) that was later altered in a number of ways. Starting from the PMS, our approach is close to that described by Boughton (2005) as a ‘parameter reduction’ process. This is a very common approach in hydrological modelling, used among others by Boughton (1984), Jayasuriya et al. (1991), Chiew and McMahon (1994), Perrin et al. (2003), Boughton (2004), whose objective is ‘to reduce the number of variable parameters in a model by either assigning a constant value or else relating the value of a parameter to that of another, such that one value determines two or more of the original parameters’ (Boughton, 2005). The parent model structure used here is derived from the experience gained during the development of the monthly model proposed by Makhlouf and Michel (1994) and the daily GR4J model (Perrin et al., 2003), whose structures already appeared satisfactory in comparison with the existing ones. The functions
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incorporated in the PMS model were selected after a trial-and-error process showing their relevance and efficiency. The GR4J model has only four free parameters, which severely limits the amount of complexity in modelling the monthly rainfall-flow transformation. Given the temporal lumping from daily to monthly time-steps, it should be possible to model the rainfall-flow transformation with only two or three parameters. The PMS model is not overly complicated and still accommodates many of the most features commonly found in the model structures. Since we wanted a general model whose most important components could then be identified, five degrees of freedom were initially introduced to this model structure. Fig. 3 outlines this model. The loss function (responsible for the determination of effective rainfall) in the PMS model is very similar to that of the daily GR4J model. It is based on a soil moisture accounting store. Due to the rainfall P, the soil moisture storage, S, becomes S1 obtained by:
S1 Z
S C X1 4 ; 1 C 4 XS1
where 4 Z tanh
P X1
(6)
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where parameter X1, the maximum capacity of the store, is positive and expressed in millimetres. It yields excess rainfall P1: P1 Z P C S K S 1
(7)
Due to evapotranspiration (the calculation of actual evapotranspiration depends on the potential value E), S1 becomes S2: S1 ð1 K jÞ E S2 Z where j Z tanh (8) S1 X 1 1 C j 1 K X1 Then, the soil moisture storage releases water P2 and takes its new value, S, ready for the next month: S2 SZ
X2 1=X2 1 C XS21 P2 Z S 2 K S
(9)
(10)
where X2 is a positive parameter. The sum of P1 and P2 is the net rainfall, P3, that enters the routing part of the model. A fraction, X3 (0%X3%1), is a first discharge component, Q1. The complementary part is
Fig. 3. Diagram of the Parent Model Scheme (PMS) used as a basis for developing a parsimonious rainfall-flow model at the monthly time step.
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routed by a quadratic reservoir, with storage R. The store capacity, (parameter X4) is a positive number expressed in millimetres. The store releases a discharge Q2. The calculations are as follows: R1 Z R C ð1 K X3 ÞP3
(11)
R21 R1 C X4
(12)
Q2 Z
R Z R1 K Q 2
(13)
The new storage R, is ready for the next month’s computations. The sum of Q1 and Q2 is multiplied by the positive parameter X5 to give the actual discharge, Q, from the studied basin: Q Z X5 ðQ1 C Q2 Þ
(14)
On the one hand, X5 can be interpreted as a water exchange term. If X5 is greater than 1, there is a water supply from the outside of the basin; otherwise there is a loss. To use such a parameter is the easiest way to model a water exchange with an outside environment other than the atmosphere. On the other hand, X5 can also be interpreted as a catchment area correction factor.
4. Stepwise search for an optimal version of the monthly water balance model After selecting the PMS model as an a priori reliable starting point to build our monthly model, we tried to reduce its complexity and identify its most useful components. Here, the modelling task is carried out as a fact-finding operation with little allowance made for any prior knowledge of what a monthly model should look like. The spatially lumped approach together with a large time step excludes most lines of physically based reasoning. Consequently, we adopted an empirical search, to discover which structure is most faithful to the actual rainfallflow transformation. With the proposed structure of the PMS model, all versions of the model arising from all combinations of free parameters were tested on the large data sample described above. In order to investigate prospective parameter values, we first calibrated all parameters. This calibration produced five distributions of parameter
values over the 410 basins, one for each parameter. The median value for parameter i was noted Mi. The values found for the test sample and other characteristics of parameter distributions are shown in Table 1. Then, each parameter in turn was set at its median value, Mi, and five versions involving four parameters were calibrated and tested against the same data. The following step involved setting two parameters, Xi and Xj, at their median values Mi and Mj, which yielded 10 models with three free parameters, and so on, until we reached the version with no free parameters, i.e. the model with fixed parameters M1,., M5. Results of model tests are shown in Table 2. In order to explore the possible range of C-criterion values, we also tested the very simple Ol’dekop (1911) model in the same calibration-simulation pattern. This model has no free parameters and computes monthly flow as: P Qk Z Pk K Ek tanh k (15) Ek where the subscript k indicates the month. This model was originally developed for an annual time step and includes no memory device, contrary to the PMS model versions tested here. The C-criterion value for the Ol’dekop model is 0.628 and the mean Nash– Sutcliffe criterion NS is K1.490. It is comforting to discover that the version of the PMS model with no free parameters (version #32 in Table 1) is far more efficient than the Ol’Dekop model since the C-criterion is 0.286 instead of 0.628 Table 1 Percentiles of parameter distributions obtained for the Parent Model Scheme (PMS) with, respectively, five and two optimised parameters (the other parameters of the two-parameter version were fixed at median values) and for the final GR2M model on the whole sample of catchments Values of distribution percentiles PMS model (5 parameters)
PMS model (2 parameters) GR2M Model
X1 X2 X3 X4 X5 X1 X5 X1 X5
(mm) (K) (K) (mm) (K) (mm) (K) (mm) (K)
0.05
0.50
0.95
162 1.38 0.00 0.0 0.02 84 0.02 140 0.21
330 3.19 0.00 53.0 0.85 360 0.86 380 0.92
1670 39.6 0.57 469 1.62 1940 1.73 2640 1.31
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Fig. 4. Plot of the C-criterion values of the 32 PMS model versions as a function of the number of free parameters (the line links best performance for each number of free parameters).
(K0.758 instead of K1.490 for NS). If there is one parameter to calibrate (see versions #27 to #31) it should undoubtedly be X5 (version #31, which further reduces the C-criterion from 0.286 to 0.186 and increases NS from K0.758 to 0.497). This is very surprising since the corresponding process—an exchange with the outside of the basin—is generally omitted in most models. Here, on the contrary, X5 is so important that the best models with two, three or four
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free parameters, X5 excluded (versions #15, #17 and #6, respectively), are clearly less efficient (CZ 0.199, 0.203 and 0.199, respectively and K0.002, K0.021, K0.000 for NS) than the model with X5 as 0:186, the single free parameter (version #31, CZ NS Z 0:497). As to the second best parameter to calibrate, the solution with calibration of the routing 0:161, storage capacity, X4 (version #26, CZ NS Z 0:619) is slightly better than the one where the soil moisture storage capacity, X1, is calibrated 0:160, NS Z 0:617). The C(version #20, CZ criterion is further slightly improved when both X1 and X4 are calibrated in addition to X5 (version #11, 0:155, NS Z 0:635), but this improvement is CZ probably too small to justify an additional parameter. Note that two other three-parameter models (version #7 with X3, X4 and X5 calibrated and version #14 with X1, X2 and X5 calibrated) give very 0:158, NS Z 0:629 for #7; 0.160 similar results (CZ and 0.624, respectively for #14). However, since X1, X4 and X5 are the ones most frequently optimised within these three best performing structures, one can conclude that they are the most important ones in the PMS model. Fig. 4 shows the evolution of model performance when the number of free
Fig. 5. Diagram of the GR2M model.
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Fig. 6. Map of the two-parameter experimental probability density of the GR2M model. The scale indicates the probabilities multiplied by 1000.
parameters is increased. It confirms that, as soon as the model has two free parameters, very little can be gained by adding more flexibility. The provisional conclusion is that a two-parameter model is sufficient at a monthly time-step. The parameter distributions characteristics of this two-parameter PMS model version are shown in Table 1. However, given the importance of parameter X5, a more elaborate way to model the outside exchanges was investigated, as described below.
We propose to further refine the outside exchange function. It is no longer modelled as if it were the effect of a change in the effective basin area (which translated into multiplying Q1CQ2 by X5). Instead, the routing reservoir is subject to an outside exchange proportional to its content. This mathematical formulation was chosen as the most satisfactory among many others. After adding input P3 to the routing reservoir to obtain level R1, an outside exchange term is computed as follows:
5. The new model: GR2M The following conclusions were drawn from the preceding section: (i) the parameter of the percolation equation, X2, is no longer a free parameter; (ii) there is no direct discharge by-passing the routing reservoir (X3Z0) and therefore, Q1Z0; (iii) the capacity of the routing store, X4, is fixed. Here, the formulation of the soil moisture part was also kept unchanged. The two other parameters (X1, the capacity of the soil moisture reservoir and X5, the parameter of the outside exchange) were left free.
Fig. 7. Distributions of the C and Nash–Sutcliffe criteria obtained by the GR2M model over the 410 catchments.
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Table 2 Values of C and NS criteria obtained for all combinations of free parameters within the PMS model over the catchment sample Version # (number of free parameters) 1 (5) 2 (4) 3 (4) 4 (4) 5 (4) 6 (4) 7 (3) 8 (3) 9 (3) 10 (3) 11 (3) 12 (3) 13 (3) 14 (3) 15 (3) 16 (3) 17 (2) 18 (2) 19 (2) 20 (2) 21 (2) 22 (2) 23 (2) 24 (2) 25 (2) 26 (2) 27 (1) 28 (1) 29 (1) 30 (1) 31 (1) 32 (0)
Parameter values (*Zcalibrated)
X1
X2
X3
X4
X5
* 330 * * * * 330 330 330 330 * * * * * * * * * * 330 330 330 330 330 330 * 330 330 330 330 330
* * 3.19 * * * 3.19 * * * 3.19 3.19 3.19 * * * * 3.19 3.19 3.19 * * * 3.19 3.19 3.19 3.19 * 3.19 3.19 3.19 3.19
* * * 0 * * * 0 * * 0 * * 0 0 * 0 * 0 0 * 0 0 * * 0 0 0 * 0 0 0
* * * * 53 * * * 53 * * 53 * 53 * 53 53 53 * 53 53 * 53 * 53 * 53 53 53 * 53 53
* * * * * 0.85 * * * 0.85 * * 0.85 * 0.85 0.85 0.85 0.85 0.85 * 0.85 0.85 * 0.85 * * 0.85 0.85 0.85 0.85 * 0.85
C criterion
NS criterion
0.155 0.156 0.156 0.154 0.157 0.199 0.158 0.159 0.177 0.229 0.155 0.159 0.214 0.160 0.199 0.203 0.203 0.236 0.214 0.160 0.247 0.229 0.178 0.275 0.183 0.161 0.230 0.246 0.285 0.275 0.186 0.286
0.641 0.635 0.635 0.640 0.635 0.000 0.629 0.629 0.541 K0.520 0.635 0.627 K0.259 0.624 K0.002 K0.023 K0.021 K0.341 K0.269 0.617 K0.678 K0.534 0.524 K0.633 0.515 0.619 K0.321 K0.617 K0.812 K0.645 0.497 K0.758
The negative Nash–Sutcliffe criteria are in bold characters. The fixed parameters take the median value of calibrated parameters in version #1.
F Z ðX5 K 1ÞR1
(16)
where X5 (non-dimensional) is a positive parameter. If X5 is greater than 1, F represents a gain of water for the basin and a loss otherwise. Then, the reservoir level becomes: R2 Z X5 :R1
(17)
Reservoir output is computed by: Q2 Z
R22 R2 C X4
(18)
and the reservoir level is updated to the R value ready for the computation of the next month: R Z R2 K Q2
(19)
The water exchange term must act on the routing reservoir before it empties. If the sequence of calculations is changed, model performance significantly decreases. Since this improved formulation of the water exchange function substantially modified the model structure, the choice of fixed parameter values must be challenged and a new analysis similar to the one
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described in Section 4 must be made. For the sake of brevity, only the main results of this analysis are presented. The best two-parameter model, called GR2M, is the one where X1 and X5 must be calibrated. X4 is set at a fixed value of 60 mm and the X2 parameter at 3. Fig. 5 shows a sketch of the final GR2M model. The characteristics of the two parameter distributions are set out in Table 1 and the two-parameter experimental probability density function is mapped in Fig. 6. With the new specifications, the C criterion obtained in simulation mode on the 410 catchments is 0.153 and the NS criterion is 0.642, i.e. a substantial improvement on the best two-parameter version tested in Section 4. One can see in Table 1 that the previous version would have required four free parameters 0:154, NS Z 0:640) to obtain a (version #4, CZ performance similar to (though still lower than) the new version. For a more complete view of the performance range, the distributions of the C criteria (Eq. (4)) and of the Nash–Sutcliffe criteria obtained on the 410 catchments are shown in Fig. 7.
6. Comparison of GR2M with models from the literature The validity of the new version can be confirmed by comparing its performance with that of other wellknown models applied to the same large sample of
basins. To this end, we used the evaluation process applied in Section 4 to compare GR2M to five models: the two-parameter model proposed by Guo et al. (2002), the MOSAZ model (Jayasuriya et al., 1991), a three-parameter version of the Wilk and Hughes (2002) model involving linear functions of the soil moisture reservoir, the Thornthwaite and Mather T-alpha model as described by Arnell (1992), and the four-parameter model, abcd, detailed by Alley (1985). For the sake of brevity, these models are not described here and one should refer to the referenced articles for a detailed description. Table 2 summarizes the results (C and NS criteria) obtained for each tested model as well as the Ol’dekop relationship (Eq. (15)) that is used as a baseline model. The GR2M model performs very satisfactorily as compared to the other models. The good performance of two of the three two-parameter models is likely to be taken as evidence that the optimal number of parameters is two, as far as the monthly time step is concerned. The Ol’dekop and abcd models perform quite poorly. It could be argued that none of them has a functionality similar to that offered by parameter X5 in GR2M, which may be viewed as a way to force long-term water balance closure. In order to make the comparison more balanced, an additional free parameter to scale the rainfall input was introduced both in the Ol’dekop and the abcd models. The performance of both models is greatly enhanced by this modification. However, the gain is not enough
Table 3 Comparison of the performance (C and NS criteria) of several monthly water balance models in simulation mode Model
Number of free parameters
NS (best: 1)
C (best: 0)
Ol’dekop (1911) Ol’dekop (1911) with a rainfall multiplying parameter MOSAZ (Jayasuriya et al., 1991) GR2M (this version) Guo et al. (2002) Vandewiele et al. (1992) with linear relations Thornthwaite and Mather (Arnell, 1992) abcd model (Alley, 1985) abcd model (Alley, 1985) with a rainfall multiplying parameter
0 1
K1.490 K0.170
0.628 0.409
2 2 2 3
K0.336 0.642 0.278 0.325
0.220 0.153 0.194 0.188
3 4 5
0.351 K1.717 K0.003
0.216 0.406 0.339
Tentative variants of the Ol’dekop and abcd models involve a rainfall multiplying parameter.
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Fig. 8. Observed and simulated monthly flows for the Leaf River (a) in the 1950–1968 period (with parameters calibrated on the 1969–1986 period) (Nash–Sutcliffe criterion: 80.9%) and (b) in the 1969–1986 period with parameters calibrated on the 1950–1968 period) (Nash–Sutcliffe criterion: 82.6%).
for these models to compete with the best ones in Table 3. A tentative conclusion could be that the underground water exchange parameter X5 is not just a fudge factor that compensates for bias in basin rainfall and/or potential evapotranspiration data.
However, this issue deserves further discussion (see Section 7). When the data sample is this large, it would not be fair to use only one simulation example to show the results. But since the well-known Leaf River basin
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near Collins, Mississippi (USA) is part of our test sample, we used this example to illustrate the simulation results of the GR2M model in Fig. 8. It is true that the good performance of the GR2M model on our large test sample is not a proof of its validity. But it indicates that it is likely to be more reliable than other competing models. Note that although the GR2M model appears significantly better than the other models tested here, the ranking among the best models might be slightly modified by using other criteria to measure their performance. However, the main conclusions are likely to hold true. For example, the results obtained with the C and NS criteria in Tables 2 and 3 are quite consistent.
7. Is the underground water exchange parameter (X5) a fudge factor? Questioning the validity of a parameter such as X5 is natural, because some modellers could see it as a fudge correction factor. Indeed this parameter may be interpreted both as (i) a correction parameter that could account for possible biases in precipitation or streamflow measurements, or (ii) an underground exchange parameter accounting for the actual water exchanges with the outside environment. What elements may help to decide whether we can restrict the interpretation of X5 to the first case? In the following, we attempt to answer this question. First, the introduction of a parameter to adjust rainfall in the abcd model led to an insufficient improvement of model performance (see Table 3). We interpreted this as an indication of the genuine necessity to allow for a water exchange between the basin and its environment other than the atmosphere. But there is a more compelling reason to introduce the underground water exchange parameter X5. The real reason is that it is extremely hard to imagine an absence of underground exchange in any basin. Let us return to the natural system that the models are intended to represent. Fig. 9 shows such a system that is obviously three-dimensional although the term basin often refers to the upper surface of the system. We must stress that very little is known of the vertical dimension. In Fig. 9, the underground boundary of the system is composed of two parts. First, a vertical
Fig. 9. Sketch of a basin with input precipitation P and potential evapotranspiration E. Q is the streamflow at the outlet and F (loss or gain) is the resultant of all flows crossing the lateral underground boundary (arrows are just indicative of where the fluxes originate).
cylinder on the topographical boundary of the basin forms the lateral boundary of the system. Second, the system is arbitrarily bounded at the bottom by a layer at sea level such that fluxes through this horizontal boundary are likely to have a zero sum. Since the lateral underground boundary is an imaginary surface, it is not an impermeable casing reproducing itself as we move along the river. The resulting algebraic flux through this boundary, named F in Fig. 9, cannot be equal to zero. There is no direct information by which to assess the amplitude and dynamics of F. In our lumped modelling approach, hypotheses about F can only be made indirectly, based on precipitation and outflow series. F is the resultant of local fluxes that are probably very complex. In some basins, it may be roughly assimilated to the common underflow that occurs in the valley substratum. In large artesian regions, F could be a gain for the system. In other basins, F might be small, but the hypothesis of F being equal to zero for every basin at all times is indefensible. Just imagine what occurs when the basin outlet is gradually moved upstream. With F remaining equal to zero, there is no transfer through the vertical slice of soil between the original system and the system obtained by an incremental displacement of the downstream section. If this phenomenon occurs everywhere and at every time the most physical image we can get of the system in Fig. 9 is that of a lump of impermeable material. Therefore, F does exist and cannot be ignored on the grounds that it
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is difficult to model. The hypothesis that F is negligible must be demonstrated by the fact that the models where it is not included are more efficient than the others since they are not overburdened by a useless additional component. The main result of our research is that F, far from being a negligible flux of water, is likely an important feature of water balance modelling.
8. Conclusion A new model of rainfall-flow transformation at the basin scale is proposed to answer hydrological questions which are best addressed at the monthly time step, e.g. those pertaining to reservoir management and long-term drought forecasting. It was not possible, within the scope of this article, to retrace all the development studies undertaken to finalise the model. The chosen approach was clearly empirical, based on a large bulk of data. The position taken by the authors was to trust the data in order to unveil the structure that best depicts the rainfall-flow transformation and to distinguish the components that are important from the ones to which fixed parameter can be assigned. The relative success of GR2M proves the overriding importance and complexity, at the monthly time step, of the loss sub-model to which both free parameters are allowed. The vital role of the loss model is not entirely surprising when the time step is as large as a month but it is surprising that the routing part does not require a single parameter to be calibrated. However, model development is a neverending process. It is likely that a more data-faithful model will be found in the future, possibly with new insights into underground water exchanges.
Acknowledgements The authors thank the scientists and institutions that provided data sets for model testing: Dr Francis H.S. Chiew (University of Melbourne, Australia) for data sets of the Australian catchments; Dr Eric Servat (Institut de Recherche pour le De´veloppement, France) for data sets of catchments in the Ivory Coast; Jane L. Thurman (USDA, United States) for data sets in the US (ARS data base); Dr Nilo Oliveira
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Nascimento (University of Minas Gerais, Belo Horizonte, Brazil) for data of Brazilian catchments; Dr John Schaake and the MOPEX team for US catchments; the HYDRO database of the French Ministry for the Environment and Me´te´o France for the hydrological and meteorological data sets of the French catchments. This research was partly supported by a research grant—provided by the French Embassy in Tunisia to the first author—which is also gratefully acknowledged. D.A. Hughes and an anonymous reviewer are heartily thanked for their many suggestions that helped to improve the clarity of the original version of this paper. References Alley, W.M., 1985. Water balance models in one-month-ahead stream flow forecasting. Water Resources Research 21 (4), 597–606. Arnell, N., 1992. W., Factors controlling the effects of climate change on river flow regimes in a humid temperate environment. Journal of Hydrology 132, 321–342. Boughton, W., 1984. A simple model for estimating the water yield of ungauged catchments. Civil Eng. Trans., Institution of Engineers, Australia CE26 (2): 83-88. Boughton, W., 2004. The Australian Water Balance Model. Environmental modelling and software 19, 943–956. Boughton, W., 2005. 2005. Catchment water balance modelling in Australia 1960- Agricultural Water Management 71, 91–116. Chiew, F.H., McMahon, T.A., 1994. Application of the daily rainfall-runoff model MODHYDROLOG to 28 Australian catchments. Journal of Hydrology 153, 383–416. Edijatno, Nascimento, N.O., Yang, X., Makhlouf, Z., Michel, C., 1999. GR3J: a daily watershed model with three free parameters. Hydrological Sciences Journal, 44(2), 263-277. Guo, S., Wang, J., Xiong, A., Li, D., 2002. A macro-scale and semidistributed monthly water balance model to predict climate change impacts in China. Journal of Hydrology 268, 1–15. Hughes, D.A., Metzler, W., 1998. Assessment of three monthly rainfall-runoff models for estimating the water resource yield of semiarid catchments in Namibia. Hydrological Sciences Journal 43 (2), 283–297. Ibrahim, A.G., Cordery, I., 1995. Estimation of recharge and runoff volumes from ungauged catchments in eastern Australia. Hydrological Sciences Journal 40 (4), 499–515. Jayasuriya, L.N., McMahon, T.A., O’Neill, I.C., 1991. Development of a simplified two-parameter rainfall-runoff model with potential for ungauged catchment application. International Hydrology and Water Resources Symposium, vol. 2, 498–503. Jothityangkoon, C., Sivapalan, M., Farmer, D.L., 2001. Process controls of water balance variability in a large semi-arid catchment: downward approach to hydrological model development. Journal of Hydrology 245, 174–198. Klemesˇ, V., 1986. Operational testing of hydrological simulation models. Hydrological Sciences Journal 31 (1), 13–24.
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Kuczera, G., Raper, G.P., Brah, N.S., Jayasuriya, M.D., 1997. The quest for more powerful validation of conceptual catchment models. Water Resources Research 33 (10), 2325–2335. Lavabre, J., Aranaud, P., Masson, J.M., Folton, N., 1999. Apport de la mode´lisation de la pluie en de´bit pour la connaissance de la ressource en eau et la pre´de´termination des crues. La Houille Blanche (3), 67–71. Lavabre, J., Folton, N., Fouchier, C., 2002. Connaissance re´gionale de la ressource en eau de surface: application au quart sud-est franc¸ais. Hydrology of mediterranean and semi-arid regions. IAHS Publ. n8 278, 94–100. Mahe, G., Paturel, J.E., Servat, E., Conway, D., Dezetter, A., 2004. The impact of land use change on soil water holding capacity and river flow modelling in the Nakambe River. Burkina-Faso. Journal of Hydrology 300 (1-4), 33–43. Makhlouf, Z., Michel, C., 1994. A two-parameter monthly water balance model for French watersheds. Journal of Hydrology 162, 299–318. Mu¨ller-Wohlfeil, D.-I., Xu, C.-Y., 2003. Legard Iversen. H., Estimation of monthly river discharge from Danish catchments. Nordic Hydrology 34 (4), 295–320. Nash, J.E., Sutcliffe, J.V., 1970. River flow forecasting through conceptual models. Part I - A discussion of principles. Journal of Hydrology 27 (3), 282–290. Niel, H., Paturel, J.-E., Servat, E., 2003. Study of parameter stability of a lumped hydrologic model in a context of climatic variability. Journal of Hydrology 278, 213–230. Ol’dekop, E.M., 1911. On evaporation from the surface of river basins. Trans. Metorol. Observ. University of Tartu 4, 200.
Paturel, J.E., Servat, E., Vassiliadis, A., 1995. Sensitivity of conceptual rainfall-runoff algorithms to errors in input data case of the GR2M model. Journal of Hydrology 168, 11–25. Perrin, C., Michel, C., Andre´assian, V., 2001. Does a large number of parameters enhance model performance? Comparative assessment of common catchment model structures on 429 catchments. Journal of Hydrology 242, 275–301. Perrin, C., Michel, C., Andre´assian, V., 2003. Improvement of a parsimonious model for streamflow simulation. Journal of Hydrology 279, 275–289. Pitman, W.V., 1978a. Flow generation by catchment models of differing complexity - A comparison of performance. Journal of Hydrology 38, 59–70. Pitman, W.V., 1978b. Trends in streamflow due to upstream land use change. Journal of Hydrology 39, 227–237. Vandewiele, G.L., Xu, C.Y., Ni-Lar-Win, 1992. Methodology and comparative study of monthly models in Belgium, China and Burma. Journal of Hydrology 134, 315–347. Vandewiele, G.L., Ni-Lar-Win, 1998. Monthly water balance models for 55 bassins in 10 countries. Hydrological Sciences Journal 43 (5), 687–699. Wilk, J., Hughes, D.A., 2002. Calibrating a rainfall-runoff model for a catchment with limited data. Hydrological Sciences Journal 47 (1), 3–17. Xu, C.Y., Vandewiele, G.L., 1995. Parsimonious monthly rainfallrunoff models for humid basins with different input requirements. Advances in Water Resources 18, 39–48. Xu, C-Y., Seibert, J., Halldin, S., 1996. Regional water balance modelling in the NOPEX area: development and application of monthly water balance models. Journal of Hydrology 180, 211– 236.
Stepwise development of a two-parameter monthly water balance model Safouane Mouelhi, Claude Michel, Charles Perrin*, Vazken Andre´assian Cemagref, Hydrosystems Research Unit, Parc de Tourvoie, BP 44, 92163 Antony Cedex, France Received 1 March 2004; revised 18 May 2005; accepted 3 June 2005
Abstract It has recently been shown that the structures of monthly water balance models can be very parsimonious. In this paper, we describe how a new two-parameter rainfall-flow model was developed using a stepwise approach. A sample of 410 basins representing a wide range of climate conditions from semi-arid through temperate to tropical humid was used to develop the new model, on the basis of previous work on a four-parameter daily rainfall-flow model. Comparisons with several well-known models showed that the new structure is efficient and provides valuable insight into the rainfall-flow transformation at a monthly time-step. q 2005 Elsevier Ltd All rights reserved. Keywords: Hydrological modelling; Rainfall-runoff models; Water balance models; Lumped parameter models; Continuous simulation; Monthly time-step
1. Introduction Over the last century many models have been proposed to represent the transformation of precipitation into streamflow at the outlet of a basin. When the time step is large, these models have usually been called water balance models, assuming the response time to be negligible compared to the time step. Monthly water balance models are valuable tools in water resources management, reservoir simulation, drought assessment or long-term drought forecasting. These models are also very useful because, due to their inherent parsimony, they lend themselves to * Corresponding author. E-mail address: [email protected] (C. Perrin).
0022-1694/$ - see front matter q 2005 Elsevier Ltd All rights reserved. doi:10.1016/j.jhydrol.2005.06.014
regionalisation and can be further used on ungauged basins. Since they can be very simple, they are easy to handle for water resources managers. Of course, their low level of complexity means that they have to focus on the most prominent features of the rainfall-flow transformation. Developing monthly water balance models implies a threefold lumping—spatial, temporal and conceptual—of processes at work. Spatial lumping means that rainfall and evaporation are averaged over the basin of interest and are referred to as scalars. Temporal lumping means that only cumulative inputs and outputs over a month are considered; we do not know the actual sequence of rainfall events, but only the resulting monthly aggregation of all inputs recorded in a given month. Conceptual lumping
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means that processes are not individualized and some of them are assembled into simple sub-models. From a physical point of view, one might argue that, with such a large time step, the processes underlying the rainfall-flow transformation may be distorted out of recognition, leaving little hope of success. However, the performances of monthly water balance models are evidence of their efficiency. The MOSAZ model developed by Jayasuriya et al. (1991) and presented by Boughton (2005) in his review of water balance modelling in Australia, was likely the first attempt to represent the monthly rainfall-flow transformation with only two free parameters. A few years later, the model proposed by Makhlouf and Michel (1994), developed on French catchments, proved to be more successful than many other models in that climate context. Since then, this model has been widely used in France (Lavabre et al., 1999, 2002) as well as in Western Africa (Paturel et al., 1995; Niel et al., 2003; Mahe et al., 2004). More recently, Guo et al. (2002) also developed a twoparameter model that can be used in a semidistributed way when dealing with large basins. It was applied to large basins in China to predict the impacts of climate change. Vandewiele et al. (1992) developed a series of three-parameter models that were applied in several countries (see e.g., Xu and Vandewiele, 1995; Xu et al., 1996; Mu¨ller-Wohlfeil et al., 2003). Ibrahim and Cordery (1995) built a four-parameter model for Australian basins. Alley (1985) reports on an efficient four-parameter model—the abcd model—applied to several catchments in the United States. In the literature, there are also other monthly models with a much higher number of parameters: see among others the CATPRO model (Kuczera et al., 1997) with up to 12 parameters used in Australia and the model developed by Pitman (1978a,b) and later used by Hughes and Metzler (1998) in Namibia and by Wilk and Hughes (2002) in India, which also contains 12 parameters. Among the models of intermediate complexity, the model proposed by Jothityangkoon et al. (2001) was developed following a downward approach. In these intermediate models, some parameters must be derived from physical basin characteristics (such as slope, soil field capacity and
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hydraulic conductivity) while the others need calibration. In this article, we focus on lumped water balance models involving a small number of free parameters to be calibrated from rainfall and discharge data. Identifying the relationships between parameters and basin characteristics is considered here as a question to be addressed a posteriori (in regionalisation studies) rather than a priori (i.e. based on preconceived ideas of the physical meaning of parameters). Indeed, we believe that insisting on a priori connections to physical characteristics jeopardises the ability of the data to verify or disprove these conceptual links. The issue of parameter explanation is not addressed in this paper. Looking at past experience of rainfall-flow modelling at a monthly time step, it seems that very few authors have made extensive comparative tests of these water balance models in various climate conditions (see e.g. Vandewiele and Ni-Lar-Win, 1998). However, much could be gained from such assessments to improve model efficiency (Perrin et al., 2001). Therefore, one objective of our research was to try to develop a new or improved model, even more general in its applicability, making the most of recent experience in this domain and evaluating the proposed model on a large sample of catchments. It would be tedious to retrace all the stages of model development that have led to the monthly model presented in this article. The first step was to choose a model structure general enough to encompass the components of existing monthly models that seemed the most efficient. Then, we made systematic attempts to improve its performance while reducing its complexity. By testing the various model versions, we let hydrological data corroborate or invalidate hypotheses about the best way to represent the main features of the rainfall-flow transformation in a tractable model. Note, however, that leading up to the rather straightforward approach outlined here, there has been a great amount of data mining and empirical research. The next section presents the data used for model development and comparative assessment. In the third section, a Parent Model Scheme (here called PMS) with five parameters is presented. In the fourth section, we analyse this model to identify the number
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of functions and/or parameters that deserve to be present and/or calibrated. The resulting model structure including the seemingly most interesting components was further improved to give a new monthly model, called GR2M and detailed in Section 5. This model is then compared to five well-known models from the hydrological literature in Section 6. The seventh section is devoted to a discussion of the water exchange term included in the GR2M model. The last section gives the main conclusions of this study.
2. Data used for model development and model assessment methodology 2.1. Data The large sample of basins used as a basis for developing and comparing the various models spans four continents. It has recently been used to explore the role of complexity in model performances (Perrin et al., 2001) and to complete the development of the daily lumped GR4J model (Perrin et al., 2003). Most of the basins are situated in France, others in the United States, Australia, the Ivory Coast and Brazil (see Fig. 1). Basin surface areas range from 1 to 50,600 km2. The climate conditions vary from temperate through semi-arid to tropical humid, with mean annual potential evapotranspiration between 630 and 2040 mm, mean annual rainfall of 300–2300 mm and mean annual streamflow between less than 10 and 2040 mm. Inputs to the models are essentially areal precipitations (potential evapotranspiration rates are interannual mean monthly values). The average record length is about 15 years in our sample. Fig. 2 shows mean interannual values of precipitation, potential evapotranspiration and streamflow for all 410 basins of our sample. In a few cases, mean streamflow exceeds mean rainfall, but the corresponding basins have not been excluded from our test sample because of the lack of an objective criterion to do the same critical work on the opposite range of behaviours, i.e. when streamflow might be suspected of being unreasonably low.
2.2. Assessment approach The data record from each basin was divided into two periods of almost equal length. Every model was assessed in simulation mode on each period using parameter values calibrated on the other period, according to the split-sample technique (Klemesˇ, 1986). This operation was repeated after switching the two periods. At the beginning of each test period, the first year of simulated discharge was not used in the computation of evaluation criteria (i.e. the warmingup period was 1-year long). The overall assessment was made on cumulative performances from both tests in simulation mode. When only one basin was considered, the criterion of the sum of least squared errors was chosen to judge the efficiency of the model. For each basin, a single value of RMSE (root mean square error, in millimetres) was computed by combining the sum of squared errors obtained in simulation mode for each period in turn (Eq. (1)). vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ni X 12 u 1 X (1) ðQ K Q^ i;j;k Þ2 RMSEi Z t 12ni jZ1 kZ1 i;j;k where the subscript i, varying from 1 to N, indicates the basin, the subscript j, varying from 1 to ni, indicates the year in the time series of catchment i and the subscript k corresponds to the month in year j. Q stands for observed runoff, Q^ for calculated runoff. The mean monthly rainfall, Pi , and the variance of the monthly rainfalls, VARPi, were computed on the same dates: Pi Z
ni X 12 1 X P 12ni jZ1 kZ1 i;j;k
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ni X 12 u 1 X VARPi Z t ðP K Pi Þ2 12ni jZ1 kZ1 i;j;k
(2)
(3)
where P is the observed precipitation. For each basin i, a goodness-of-fit criterion Ci could be computed by: Ci Z
RMSEi VARPi
(4)
to yield the overall evaluation criterion C over the whole test basin sample:
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Fig. 1. Catchment locations in (a) France, (b) the Ivory Coast, (c) Brazil, (d) Australia and (e) the United States.
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Fig. 2. (a) Mean streamflow versus mean precipitation and (b) mean potential evapotranspiration versus mean precipitation for the 410 basins. N 1 X C Z C N iZ1 i
(5)
Often, RMSE is compared to the variance of observed streamflow, as in the widely used Nash and Sutcliffe (1970) criterion. In Eq. (4), we used the variance of observed rainfall to standardise model error and emphasise the role of the model in transforming these inputs into streamflow. However, since the common approach is to use the Nash and Sutcliffe criterion, we also give its mean value over the whole test sample (hereafter called NS) along with the C-criterion value. Given the size of our test sample, any comparison or model ranking is bound to be sensitive to the choice of the overall criterion C. However, there is little chance of greatly modifying our conclusions on the respective value of the different model components tested here. To confer even greater robustness on our splitsample test approach, the parameters were calibrated by using the least square errors computed on the square root transforms of the discharges as the objective function, while the validation criteria were computed without prior transformation on the discharge. Many authors advocate the use of the square root of the discharges before evaluating the square errors to make the most of the information imbedded in the data. In addition, the change of criterion between the calibration and validation
assessment penalizes any overparameterization phenomenon. Model parameters were calibrated by a local optimisation scheme described by Edijatno et al. (1999).
3. The parent model scheme (PMS) In order to determine which elements are important in a monthly rainfall-flow model, we started by adopting a parent model structure (PMS) that was later altered in a number of ways. Starting from the PMS, our approach is close to that described by Boughton (2005) as a ‘parameter reduction’ process. This is a very common approach in hydrological modelling, used among others by Boughton (1984), Jayasuriya et al. (1991), Chiew and McMahon (1994), Perrin et al. (2003), Boughton (2004), whose objective is ‘to reduce the number of variable parameters in a model by either assigning a constant value or else relating the value of a parameter to that of another, such that one value determines two or more of the original parameters’ (Boughton, 2005). The parent model structure used here is derived from the experience gained during the development of the monthly model proposed by Makhlouf and Michel (1994) and the daily GR4J model (Perrin et al., 2003), whose structures already appeared satisfactory in comparison with the existing ones. The functions
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incorporated in the PMS model were selected after a trial-and-error process showing their relevance and efficiency. The GR4J model has only four free parameters, which severely limits the amount of complexity in modelling the monthly rainfall-flow transformation. Given the temporal lumping from daily to monthly time-steps, it should be possible to model the rainfall-flow transformation with only two or three parameters. The PMS model is not overly complicated and still accommodates many of the most features commonly found in the model structures. Since we wanted a general model whose most important components could then be identified, five degrees of freedom were initially introduced to this model structure. Fig. 3 outlines this model. The loss function (responsible for the determination of effective rainfall) in the PMS model is very similar to that of the daily GR4J model. It is based on a soil moisture accounting store. Due to the rainfall P, the soil moisture storage, S, becomes S1 obtained by:
S1 Z
S C X1 4 ; 1 C 4 XS1
where 4 Z tanh
P X1
(6)
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where parameter X1, the maximum capacity of the store, is positive and expressed in millimetres. It yields excess rainfall P1: P1 Z P C S K S 1
(7)
Due to evapotranspiration (the calculation of actual evapotranspiration depends on the potential value E), S1 becomes S2: S1 ð1 K jÞ E S2 Z where j Z tanh (8) S1 X 1 1 C j 1 K X1 Then, the soil moisture storage releases water P2 and takes its new value, S, ready for the next month: S2 SZ
X2 1=X2 1 C XS21 P2 Z S 2 K S
(9)
(10)
where X2 is a positive parameter. The sum of P1 and P2 is the net rainfall, P3, that enters the routing part of the model. A fraction, X3 (0%X3%1), is a first discharge component, Q1. The complementary part is
Fig. 3. Diagram of the Parent Model Scheme (PMS) used as a basis for developing a parsimonious rainfall-flow model at the monthly time step.
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routed by a quadratic reservoir, with storage R. The store capacity, (parameter X4) is a positive number expressed in millimetres. The store releases a discharge Q2. The calculations are as follows: R1 Z R C ð1 K X3 ÞP3
(11)
R21 R1 C X4
(12)
Q2 Z
R Z R1 K Q 2
(13)
The new storage R, is ready for the next month’s computations. The sum of Q1 and Q2 is multiplied by the positive parameter X5 to give the actual discharge, Q, from the studied basin: Q Z X5 ðQ1 C Q2 Þ
(14)
On the one hand, X5 can be interpreted as a water exchange term. If X5 is greater than 1, there is a water supply from the outside of the basin; otherwise there is a loss. To use such a parameter is the easiest way to model a water exchange with an outside environment other than the atmosphere. On the other hand, X5 can also be interpreted as a catchment area correction factor.
4. Stepwise search for an optimal version of the monthly water balance model After selecting the PMS model as an a priori reliable starting point to build our monthly model, we tried to reduce its complexity and identify its most useful components. Here, the modelling task is carried out as a fact-finding operation with little allowance made for any prior knowledge of what a monthly model should look like. The spatially lumped approach together with a large time step excludes most lines of physically based reasoning. Consequently, we adopted an empirical search, to discover which structure is most faithful to the actual rainfallflow transformation. With the proposed structure of the PMS model, all versions of the model arising from all combinations of free parameters were tested on the large data sample described above. In order to investigate prospective parameter values, we first calibrated all parameters. This calibration produced five distributions of parameter
values over the 410 basins, one for each parameter. The median value for parameter i was noted Mi. The values found for the test sample and other characteristics of parameter distributions are shown in Table 1. Then, each parameter in turn was set at its median value, Mi, and five versions involving four parameters were calibrated and tested against the same data. The following step involved setting two parameters, Xi and Xj, at their median values Mi and Mj, which yielded 10 models with three free parameters, and so on, until we reached the version with no free parameters, i.e. the model with fixed parameters M1,., M5. Results of model tests are shown in Table 2. In order to explore the possible range of C-criterion values, we also tested the very simple Ol’dekop (1911) model in the same calibration-simulation pattern. This model has no free parameters and computes monthly flow as: P Qk Z Pk K Ek tanh k (15) Ek where the subscript k indicates the month. This model was originally developed for an annual time step and includes no memory device, contrary to the PMS model versions tested here. The C-criterion value for the Ol’dekop model is 0.628 and the mean Nash– Sutcliffe criterion NS is K1.490. It is comforting to discover that the version of the PMS model with no free parameters (version #32 in Table 1) is far more efficient than the Ol’Dekop model since the C-criterion is 0.286 instead of 0.628 Table 1 Percentiles of parameter distributions obtained for the Parent Model Scheme (PMS) with, respectively, five and two optimised parameters (the other parameters of the two-parameter version were fixed at median values) and for the final GR2M model on the whole sample of catchments Values of distribution percentiles PMS model (5 parameters)
PMS model (2 parameters) GR2M Model
X1 X2 X3 X4 X5 X1 X5 X1 X5
(mm) (K) (K) (mm) (K) (mm) (K) (mm) (K)
0.05
0.50
0.95
162 1.38 0.00 0.0 0.02 84 0.02 140 0.21
330 3.19 0.00 53.0 0.85 360 0.86 380 0.92
1670 39.6 0.57 469 1.62 1940 1.73 2640 1.31
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Fig. 4. Plot of the C-criterion values of the 32 PMS model versions as a function of the number of free parameters (the line links best performance for each number of free parameters).
(K0.758 instead of K1.490 for NS). If there is one parameter to calibrate (see versions #27 to #31) it should undoubtedly be X5 (version #31, which further reduces the C-criterion from 0.286 to 0.186 and increases NS from K0.758 to 0.497). This is very surprising since the corresponding process—an exchange with the outside of the basin—is generally omitted in most models. Here, on the contrary, X5 is so important that the best models with two, three or four
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free parameters, X5 excluded (versions #15, #17 and #6, respectively), are clearly less efficient (CZ 0.199, 0.203 and 0.199, respectively and K0.002, K0.021, K0.000 for NS) than the model with X5 as 0:186, the single free parameter (version #31, CZ NS Z 0:497). As to the second best parameter to calibrate, the solution with calibration of the routing 0:161, storage capacity, X4 (version #26, CZ NS Z 0:619) is slightly better than the one where the soil moisture storage capacity, X1, is calibrated 0:160, NS Z 0:617). The C(version #20, CZ criterion is further slightly improved when both X1 and X4 are calibrated in addition to X5 (version #11, 0:155, NS Z 0:635), but this improvement is CZ probably too small to justify an additional parameter. Note that two other three-parameter models (version #7 with X3, X4 and X5 calibrated and version #14 with X1, X2 and X5 calibrated) give very 0:158, NS Z 0:629 for #7; 0.160 similar results (CZ and 0.624, respectively for #14). However, since X1, X4 and X5 are the ones most frequently optimised within these three best performing structures, one can conclude that they are the most important ones in the PMS model. Fig. 4 shows the evolution of model performance when the number of free
Fig. 5. Diagram of the GR2M model.
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Fig. 6. Map of the two-parameter experimental probability density of the GR2M model. The scale indicates the probabilities multiplied by 1000.
parameters is increased. It confirms that, as soon as the model has two free parameters, very little can be gained by adding more flexibility. The provisional conclusion is that a two-parameter model is sufficient at a monthly time-step. The parameter distributions characteristics of this two-parameter PMS model version are shown in Table 1. However, given the importance of parameter X5, a more elaborate way to model the outside exchanges was investigated, as described below.
We propose to further refine the outside exchange function. It is no longer modelled as if it were the effect of a change in the effective basin area (which translated into multiplying Q1CQ2 by X5). Instead, the routing reservoir is subject to an outside exchange proportional to its content. This mathematical formulation was chosen as the most satisfactory among many others. After adding input P3 to the routing reservoir to obtain level R1, an outside exchange term is computed as follows:
5. The new model: GR2M The following conclusions were drawn from the preceding section: (i) the parameter of the percolation equation, X2, is no longer a free parameter; (ii) there is no direct discharge by-passing the routing reservoir (X3Z0) and therefore, Q1Z0; (iii) the capacity of the routing store, X4, is fixed. Here, the formulation of the soil moisture part was also kept unchanged. The two other parameters (X1, the capacity of the soil moisture reservoir and X5, the parameter of the outside exchange) were left free.
Fig. 7. Distributions of the C and Nash–Sutcliffe criteria obtained by the GR2M model over the 410 catchments.
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Table 2 Values of C and NS criteria obtained for all combinations of free parameters within the PMS model over the catchment sample Version # (number of free parameters) 1 (5) 2 (4) 3 (4) 4 (4) 5 (4) 6 (4) 7 (3) 8 (3) 9 (3) 10 (3) 11 (3) 12 (3) 13 (3) 14 (3) 15 (3) 16 (3) 17 (2) 18 (2) 19 (2) 20 (2) 21 (2) 22 (2) 23 (2) 24 (2) 25 (2) 26 (2) 27 (1) 28 (1) 29 (1) 30 (1) 31 (1) 32 (0)
Parameter values (*Zcalibrated)
X1
X2
X3
X4
X5
* 330 * * * * 330 330 330 330 * * * * * * * * * * 330 330 330 330 330 330 * 330 330 330 330 330
* * 3.19 * * * 3.19 * * * 3.19 3.19 3.19 * * * * 3.19 3.19 3.19 * * * 3.19 3.19 3.19 3.19 * 3.19 3.19 3.19 3.19
* * * 0 * * * 0 * * 0 * * 0 0 * 0 * 0 0 * 0 0 * * 0 0 0 * 0 0 0
* * * * 53 * * * 53 * * 53 * 53 * 53 53 53 * 53 53 * 53 * 53 * 53 53 53 * 53 53
* * * * * 0.85 * * * 0.85 * * 0.85 * 0.85 0.85 0.85 0.85 0.85 * 0.85 0.85 * 0.85 * * 0.85 0.85 0.85 0.85 * 0.85
C criterion
NS criterion
0.155 0.156 0.156 0.154 0.157 0.199 0.158 0.159 0.177 0.229 0.155 0.159 0.214 0.160 0.199 0.203 0.203 0.236 0.214 0.160 0.247 0.229 0.178 0.275 0.183 0.161 0.230 0.246 0.285 0.275 0.186 0.286
0.641 0.635 0.635 0.640 0.635 0.000 0.629 0.629 0.541 K0.520 0.635 0.627 K0.259 0.624 K0.002 K0.023 K0.021 K0.341 K0.269 0.617 K0.678 K0.534 0.524 K0.633 0.515 0.619 K0.321 K0.617 K0.812 K0.645 0.497 K0.758
The negative Nash–Sutcliffe criteria are in bold characters. The fixed parameters take the median value of calibrated parameters in version #1.
F Z ðX5 K 1ÞR1
(16)
where X5 (non-dimensional) is a positive parameter. If X5 is greater than 1, F represents a gain of water for the basin and a loss otherwise. Then, the reservoir level becomes: R2 Z X5 :R1
(17)
Reservoir output is computed by: Q2 Z
R22 R2 C X4
(18)
and the reservoir level is updated to the R value ready for the computation of the next month: R Z R2 K Q2
(19)
The water exchange term must act on the routing reservoir before it empties. If the sequence of calculations is changed, model performance significantly decreases. Since this improved formulation of the water exchange function substantially modified the model structure, the choice of fixed parameter values must be challenged and a new analysis similar to the one
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described in Section 4 must be made. For the sake of brevity, only the main results of this analysis are presented. The best two-parameter model, called GR2M, is the one where X1 and X5 must be calibrated. X4 is set at a fixed value of 60 mm and the X2 parameter at 3. Fig. 5 shows a sketch of the final GR2M model. The characteristics of the two parameter distributions are set out in Table 1 and the two-parameter experimental probability density function is mapped in Fig. 6. With the new specifications, the C criterion obtained in simulation mode on the 410 catchments is 0.153 and the NS criterion is 0.642, i.e. a substantial improvement on the best two-parameter version tested in Section 4. One can see in Table 1 that the previous version would have required four free parameters 0:154, NS Z 0:640) to obtain a (version #4, CZ performance similar to (though still lower than) the new version. For a more complete view of the performance range, the distributions of the C criteria (Eq. (4)) and of the Nash–Sutcliffe criteria obtained on the 410 catchments are shown in Fig. 7.
6. Comparison of GR2M with models from the literature The validity of the new version can be confirmed by comparing its performance with that of other wellknown models applied to the same large sample of
basins. To this end, we used the evaluation process applied in Section 4 to compare GR2M to five models: the two-parameter model proposed by Guo et al. (2002), the MOSAZ model (Jayasuriya et al., 1991), a three-parameter version of the Wilk and Hughes (2002) model involving linear functions of the soil moisture reservoir, the Thornthwaite and Mather T-alpha model as described by Arnell (1992), and the four-parameter model, abcd, detailed by Alley (1985). For the sake of brevity, these models are not described here and one should refer to the referenced articles for a detailed description. Table 2 summarizes the results (C and NS criteria) obtained for each tested model as well as the Ol’dekop relationship (Eq. (15)) that is used as a baseline model. The GR2M model performs very satisfactorily as compared to the other models. The good performance of two of the three two-parameter models is likely to be taken as evidence that the optimal number of parameters is two, as far as the monthly time step is concerned. The Ol’dekop and abcd models perform quite poorly. It could be argued that none of them has a functionality similar to that offered by parameter X5 in GR2M, which may be viewed as a way to force long-term water balance closure. In order to make the comparison more balanced, an additional free parameter to scale the rainfall input was introduced both in the Ol’dekop and the abcd models. The performance of both models is greatly enhanced by this modification. However, the gain is not enough
Table 3 Comparison of the performance (C and NS criteria) of several monthly water balance models in simulation mode Model
Number of free parameters
NS (best: 1)
C (best: 0)
Ol’dekop (1911) Ol’dekop (1911) with a rainfall multiplying parameter MOSAZ (Jayasuriya et al., 1991) GR2M (this version) Guo et al. (2002) Vandewiele et al. (1992) with linear relations Thornthwaite and Mather (Arnell, 1992) abcd model (Alley, 1985) abcd model (Alley, 1985) with a rainfall multiplying parameter
0 1
K1.490 K0.170
0.628 0.409
2 2 2 3
K0.336 0.642 0.278 0.325
0.220 0.153 0.194 0.188
3 4 5
0.351 K1.717 K0.003
0.216 0.406 0.339
Tentative variants of the Ol’dekop and abcd models involve a rainfall multiplying parameter.
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Fig. 8. Observed and simulated monthly flows for the Leaf River (a) in the 1950–1968 period (with parameters calibrated on the 1969–1986 period) (Nash–Sutcliffe criterion: 80.9%) and (b) in the 1969–1986 period with parameters calibrated on the 1950–1968 period) (Nash–Sutcliffe criterion: 82.6%).
for these models to compete with the best ones in Table 3. A tentative conclusion could be that the underground water exchange parameter X5 is not just a fudge factor that compensates for bias in basin rainfall and/or potential evapotranspiration data.
However, this issue deserves further discussion (see Section 7). When the data sample is this large, it would not be fair to use only one simulation example to show the results. But since the well-known Leaf River basin
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near Collins, Mississippi (USA) is part of our test sample, we used this example to illustrate the simulation results of the GR2M model in Fig. 8. It is true that the good performance of the GR2M model on our large test sample is not a proof of its validity. But it indicates that it is likely to be more reliable than other competing models. Note that although the GR2M model appears significantly better than the other models tested here, the ranking among the best models might be slightly modified by using other criteria to measure their performance. However, the main conclusions are likely to hold true. For example, the results obtained with the C and NS criteria in Tables 2 and 3 are quite consistent.
7. Is the underground water exchange parameter (X5) a fudge factor? Questioning the validity of a parameter such as X5 is natural, because some modellers could see it as a fudge correction factor. Indeed this parameter may be interpreted both as (i) a correction parameter that could account for possible biases in precipitation or streamflow measurements, or (ii) an underground exchange parameter accounting for the actual water exchanges with the outside environment. What elements may help to decide whether we can restrict the interpretation of X5 to the first case? In the following, we attempt to answer this question. First, the introduction of a parameter to adjust rainfall in the abcd model led to an insufficient improvement of model performance (see Table 3). We interpreted this as an indication of the genuine necessity to allow for a water exchange between the basin and its environment other than the atmosphere. But there is a more compelling reason to introduce the underground water exchange parameter X5. The real reason is that it is extremely hard to imagine an absence of underground exchange in any basin. Let us return to the natural system that the models are intended to represent. Fig. 9 shows such a system that is obviously three-dimensional although the term basin often refers to the upper surface of the system. We must stress that very little is known of the vertical dimension. In Fig. 9, the underground boundary of the system is composed of two parts. First, a vertical
Fig. 9. Sketch of a basin with input precipitation P and potential evapotranspiration E. Q is the streamflow at the outlet and F (loss or gain) is the resultant of all flows crossing the lateral underground boundary (arrows are just indicative of where the fluxes originate).
cylinder on the topographical boundary of the basin forms the lateral boundary of the system. Second, the system is arbitrarily bounded at the bottom by a layer at sea level such that fluxes through this horizontal boundary are likely to have a zero sum. Since the lateral underground boundary is an imaginary surface, it is not an impermeable casing reproducing itself as we move along the river. The resulting algebraic flux through this boundary, named F in Fig. 9, cannot be equal to zero. There is no direct information by which to assess the amplitude and dynamics of F. In our lumped modelling approach, hypotheses about F can only be made indirectly, based on precipitation and outflow series. F is the resultant of local fluxes that are probably very complex. In some basins, it may be roughly assimilated to the common underflow that occurs in the valley substratum. In large artesian regions, F could be a gain for the system. In other basins, F might be small, but the hypothesis of F being equal to zero for every basin at all times is indefensible. Just imagine what occurs when the basin outlet is gradually moved upstream. With F remaining equal to zero, there is no transfer through the vertical slice of soil between the original system and the system obtained by an incremental displacement of the downstream section. If this phenomenon occurs everywhere and at every time the most physical image we can get of the system in Fig. 9 is that of a lump of impermeable material. Therefore, F does exist and cannot be ignored on the grounds that it
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is difficult to model. The hypothesis that F is negligible must be demonstrated by the fact that the models where it is not included are more efficient than the others since they are not overburdened by a useless additional component. The main result of our research is that F, far from being a negligible flux of water, is likely an important feature of water balance modelling.
8. Conclusion A new model of rainfall-flow transformation at the basin scale is proposed to answer hydrological questions which are best addressed at the monthly time step, e.g. those pertaining to reservoir management and long-term drought forecasting. It was not possible, within the scope of this article, to retrace all the development studies undertaken to finalise the model. The chosen approach was clearly empirical, based on a large bulk of data. The position taken by the authors was to trust the data in order to unveil the structure that best depicts the rainfall-flow transformation and to distinguish the components that are important from the ones to which fixed parameter can be assigned. The relative success of GR2M proves the overriding importance and complexity, at the monthly time step, of the loss sub-model to which both free parameters are allowed. The vital role of the loss model is not entirely surprising when the time step is as large as a month but it is surprising that the routing part does not require a single parameter to be calibrated. However, model development is a neverending process. It is likely that a more data-faithful model will be found in the future, possibly with new insights into underground water exchanges.
Acknowledgements The authors thank the scientists and institutions that provided data sets for model testing: Dr Francis H.S. Chiew (University of Melbourne, Australia) for data sets of the Australian catchments; Dr Eric Servat (Institut de Recherche pour le De´veloppement, France) for data sets of catchments in the Ivory Coast; Jane L. Thurman (USDA, United States) for data sets in the US (ARS data base); Dr Nilo Oliveira
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