The exponential store: a correct formulation for rainfall runoff modelling

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1 Hydrological Sciences Journal ISSN: (Print) (Online) Journal homepage: The exponential store: a correct formulation for rainfall runoff modelling CLAUDE MICHEL, CHARLES PERRIN & VAZKEN ANDREASSIAN To cite this article: CLAUDE MICHEL, CHARLES PERRIN & VAZKEN ANDREASSIAN (2003) The exponential store: a correct formulation for rainfall runoff modelling, Hydrological Sciences Journal, 48:1, , DOI: /hysj To link to this article: Published online: 19 Jan Submit your article to this journal Article views: 364 Citing articles: 25 View citing articles Full Terms & Conditions of access and use can be found at

2 Hydrological Sciences Journal des Sciences Hydrologiques, 48(1) February The exponential store: a correct formulation for rainfall runoff modelling CLAUDE MICHEL, CHARLES PERRIN & VAZKEN ANDREASSIAN Water Quality and Hydrology Research Unit, Cemagref, Parc de Tourvoie, BP44, F Antony Cedex, France charles.perrin@cemagref.fr Abstract Stores are widely used tools in conceptual, soil-moisture accounting rainfall runoff models. Various mathematical laws are used to describe the outflow of such stores. The exponential store has received less attention than the linear store, but it has proved to be a highly efficient tool in streamflow modelling. The aim of this study is to show that the common way of dealing with this store without integration in time is mathematically and physically flawed. This misuse can result in poorer reliability and efficiency of rainfall runoff models that include such a store. Surprisingly, this inconsistency had a significant impact on only a limited number of catchments among the 138 used to carry out the tests. However, it can have dramatic consequences for the simulation of large flood events. Therefore, an exact discrete formulation should be preferred in rainfall runoff models. Key words rainfall runoff modelling; exponential store Le réservoir exponentiel: une formulation améliorée pour la modélisation pluie débit Résumé Les réservoirs sont des constituants bien connus dans les modèles conceptuels pluie débit fonctionnant avec un suivi d humidité des sols. La vidange de ces réservoirs peut se faire selon diverses lois mathématiques. Le réservoir exponentiel n est pas d une utilisation aussi répandue que le réservoir linéaire, mais il est un excellent outil pour représenter les débits. Le but de cet article est de montrer que le traitement habituel de ce type de réservoir, sans intégration dans le temps, est mathématiquement et physiquement erroné. Ce mauvais usage peut se traduire par une perte de fiabilité et d efficacité de modèles pluie débit qui utilisent un tel réservoir. De façon surprenante, cette inconsistance n a eu un impact que sur un nombre limité de bassins parmi les 138 utilisés pour mener les tests. Cependant, elle peut avoir des conséquences dramatiques sur la simulation de forts événements de crue. Par conséquent, la formulation discrète correcte devrait être préférée dans les modèles pluie débit. Mots clefs modélisation pluie débit; réservoir exponentiel INTRODUCTION Stores are common components of lumped rainfall runoff models. Among those used as production or routing tools, the best-known and most widely used type of store is certainly the linear one, whose output is proportional to the amount of stored water. It has been used in, among others, the HBV model (Bergström & Forsman, 1973), the IHACRES model (Jakeman et al., 1990), the MODHYDROLOG model (Chiew & McMahon, 1994), the NAM model (Nielsen & Hansen, 1973), the SMAR model (see e.g. Kachroo, 1992), the Sacramento model (see e.g. Burnash & Ferral, 1982) and the Tank model (Sugawara, 1995). Sugawara (1995) gives full details of the mathematical treatment of this modelling tool. Open for discussion until 1 August 2003

3 110 Claude Michel et al. Another efficient tool is the exponential store used, for example, by Lambert (1969), Rutter et al. (1975) in an interception model and by Beven et al. (1979, 1995) in the well-known and widely applied TOPMODEL. The exponential store is generally considered to be a tool for recession and baseflow simulation but, as part of a rainfall runoff model, it can also play an important role in the simulation of high flow events. Hydrological models are generally run at the time step of available hydrological data series. Because of this discrete accounting for time, a common approximation is to assume that the time steps are small enough to consider that continuous model formulations can be applied in a straightforward manner over the time step. However, there is an important difference between continuous and discrete representations of processes: between these two formulations, flux variables are different due to the temporal integration during the time step to which the discrete process is applied. This difference has important implications for the mathematical formulation of the stores. The objective of this study is to investigate the consequences of neglecting this fundamental difference between continuous and discrete formulations. FORMULATIONS OF THE EXPONENTIAL STORE In rainfall runoff models, volumes of water are commonly converted into millimetres, by dividing them by the catchment area. Therefore, rainfall and flow can be expressed by the same unit. It is assumed that any model part is run with the time step of the hydrological data series at hand. In the following, the mathematical treatment of the exponential store is examined in a four-step development: firstly, the mathematical characteristics of the continuous formulation are presented; secondly, the mathematical characteristics of the discrete formulation are detailed for the case of no input to the store, with possible difficulties linked to numerical treatment; thirdly, the above discussion is extended to the application of the exponential store in the case of input to the store; finally, the results that can be obtained are compared with the continuous and discrete formulations. Continuous formulation The output, Q (mm s -1 ) of the exponential store at time t can be related to the storage, S (mm) through an exponential relationship given by: S( t) Q( t) = a exp (1) b where a and b are two parameters. Parameter a has no specific meaning for this store, except that it is equal to outflow when S = 0. Parameter b controls the slope of the recession limb of the hydrograph and is equivalent to parameter m in TOPMODEL (see e.g. Beven et al., 1984). In the formulation of the exponential store, there are two apparent differences from that of the linear store:

4 The exponential store: a correct formulation for rainfall runoff modelling 111 (a) Two parameters seem to be necessary to describe the relationship instead of one (b ) for the linear store, whose output is given by: S( t) Q( t) = (2) b For the exponential store, a is positive and has the same dimension as Q, and b is positive and has the same dimension as S. For the linear store, b has the dimension of time. This is why different notations are used, to avoid confusion. (b) In the exponential store, S can take values from - to +, while, for the linear store, S has to be positive or zero. The storage value here is relative to an arbitrary origin and it is negative whenever the level in the store is below this origin. The exponential store is essentially a bottomless reservoir, which explains why the bottom (S value for which Q is zero) cannot be taken as the origin. Mathematically, the value S = 0 has no special significance for the exponential store as represented by equation (1). Discrete formulation in the case of no input A problem appears as soon as one tries to find a suitable discrete formulation for the exponential store, because at a given time step, the input to the store may be any positive function of time. For the moment, it is assumed that the input is zero but the discharge is not equal to zero. Time is denoted by t and is used for the time step duration. From t to t + a volume of water has flown out of the store and this volume is denoted by R. Computation of R requires that the differential equation describing the process be solved. This differential equation comes from associating the dynamic equation (1) and the continuity equation given by: ds = Qdt (3) Therefore, using equation (1), one obtains: S ds = a exp dt (4) b Integration between time t (beginning of the time step with storage S in the store) and time t + (end of the time step with storage S R in the store) is straightforward and gives: S R S a exp exp = (5) b b b The following output volume R is then obtained: a S R = b ln 1 + exp (6) b b Up to now, the problem of the origin (zero) on the S-axis has not arisen. This origin is arbitrary, since it has no specific role in the formulation expressed by equation (6). One can take advantage of this situation by choosing a new origin S 0 calculated as:

5 112 Claude Michel et al. a S0 = b ln (7) b The term S will be replaced by S *, with S = S * + S 0. Using this new origin, but keeping the same notation for S (instead of S * ), one obtains: S R = b ln 1 + exp (8) b This formulation is interesting because it demonstrates that the exponential store requires only one parameter, as does the linear store. Another interesting property is that the same parameter can be used irrespective of the time-step duration, since only the intrinsic b parameter remains in the relationship. A change of the time-step duration only translates into a change in the origin on the S-axis. This way of setting the origin is hydrologically significant because as soon as a recession begins, S becomes negative. This can be illustrated by computing the difference S R, i.e. the new value S of the storage one time step further on, after the release of R: S S = b ln 1 + exp (9) b The value of S is always negative. The only way of making S positive is by adding water to the store. Thus, with equation (8), the level S = 0 is no longer an arbitrary point along an S-axis but represents the level that cannot be exceeded at the end of a time step without input, irrespective of the initial level: it is the highest possible level for a time step with no input. This behaviour is reminiscent of the quadratic store, described in the Appendix, when applied for recession flows. Numerical treatment of the discrete formulation In the numerical treatment of equation (8), care must be taken to avoid numerical overflows during the computing. To this end, it may be useful to use two approximations when the absolute value of S/b is large compared to 1: If S/b > 5 then R S (10) S If S/b < 6 then R b exp (11) b These two approximations entail a relative error ( δr /R) of less than compared to equation (8). Application of the exponential store with input The previous calculations were carried out under the hypothesis of no input to the store. However, when an exponential store is run with actual rainfall time series, for example, inputs are of course not always zero and the manner of introducing them into the computations must be defined. It is only during the time steps with non-zero inputs

6 The exponential store: a correct formulation for rainfall runoff modelling 113 that the storage S in the store (computed from equation (8)) can become positive, i.e. the level in the store rises above the new chosen origin. There is no exact solution since the input distribution over the time step is not known. It should be noted that the same problem exists for all types of stores, even the linear one. When the continuous form (equation (1)) is used, the output from the store cannot be computed solely from the cumulative input, but needs the temporal distribution of rainfall. The discrete evolution of storage in the store should be obtained by solving the following differential equation: S ds = a exp + I( t) dt b (12) where I(t) is the time-varying input to the store. In a discrete framework, where only the cumulative input is known, there is no analytical solution to this differential equation. A common approximation is to assume I(t) to be constant over, as done for example by Moore & Bell (2002) for power-law and exponential stores. Another approximation is to add to S the result of integrating I(t) over (i.e. the cumulative input denoted by P) at the beginning of the time step and subsequently to apply equation (8). In the context of rainfall runoff modelling, the choice of one of these assumptions is arbitrary, but does not have much impact on the results. It may only change the value of the delay parameter between rainfall and flow. In the following, the second approximation is used to introduce input to the store. Possible consequences of the misuse of the exponential store Potential failure in case of large input Very often, when discrete time series are used at a time step, equation (1) is chosen instead of equation (8), because Q is mistaken for R, to give the following relationship: S R = a exp (13) b where parameter a has the same dimension as R or S (mm). The use of equation (13) may be detrimental in several respects. First, it can cause optimization problems since calibrating parameter a and looking for the initial condition (S for t = 0) are one and the same problem. This redundancy can lead to lengthy calibrations and poorly defined parameter values. However, there is a more serious consequence. If one starts with S = 0, equation (8) yields: R = b ln(2) One can obtain the same result with equation (13) when parameter a = bln(2). With this value of parameter a, the systems are in the same hydrological state, having the same storage, and same output in the case of no input. Assuming that an instant input P occurs, before recession, at the time when S = 0, which becomes S = P, equation (8) yields: (14)

7 114 Claude Michel et al. R b P = ln 1 + exp (15) b If P/b is very large, then, with the approximation of equation (10), R is about equal to P. On the other hand, equation (13) yields: R b P = ln( 2)exp (16) b Depending on the value of P in equation (16), R can be much larger than P, as illustrated by Fig. 1. Moreover, the larger the P value, the lower the storage S at the end of the time step, because S = P R (due to the initial zero storage). In other words, the larger the input, the drier the store becomes. This does not make sense hydrologically and can be detrimental in the case of large inputs R /b 2 1 Incorrect treatment - Eq. (13) 1:1 Correct treatment - Eq. (8) P /b Fig. 1 Comparison of values of normalized store output R/b for values of normalized input P/b, starting from the same initial conditions in equations (13) and (8). Comparison of recession curves obtained with the two methods The difference between the two store output formulations can also be seen in the recession curves when no input is added to the store. Let R/b be denoted by r and S/b by s and let n be used as a discrete time index. Using equation (13) written at two successive time indexes n and n+1 and considering the conservation equation s n+1 = s n r n, equation (13) can be written: r ( r ) n+ 1 = rn exp n (17) Similarly, from equation (8) one obtains the following expression for the correct treatment of the exponential store: [ 2 ( )] rn + 1 = ln exp r n (18)

8 The exponential store: a correct formulation for rainfall runoff modelling Incorrect treatment - Eq. (13), i.e. Eq. (17) Correct treatment - Eq. (8), i.e. Eq. (18) r n Time step Fig. 2 Comparison of two recession curves derived from equations (13) and (8), respectively. Equations (17) and (18) are very different, except in cases where r n is small compared to 1. In such cases, both equations can be approximated by: rn+ 1 = rn( 1 rn) (19) Figure 2 shows the computed time series starting for example with r 1 = 2. Clearly, the recession curves are different. The volumes of water released by the store in the two cases differ, especially in the first time steps. The difference tends to decrease with time and the recession curves tend to converge toward the same behaviour, as shown by equation (19). ASSESSMENT OF THE PRACTICAL CONSEQUENCES OF MISUSING THE EXPONENTIAL STORE IN A RAINFALL RUNOFF MODEL Objective After demonstrating that equation (13), when used inappropriately, can lead to hydrological inconsistencies, it remains to assess the practical impact of this misuse on the efficiency of a rainfall runoff model involving an exponential store. This additional check is justified by the fact that hydrological models are so imperfect, that physical or mathematical mistakes are often masked by the numerous unknown modelling insufficiencies that plague any model. This could explain why no troubles are reported in the literature with reference to the exponential store. Therefore, it was decided to test the practical significance of obvious misuse of one of the many components of a model. To this end, a simplified version of TOPMODEL (Beven & Kirkby, 1979) was used, which is one of the best known models that uses an exponential store.

9 116 Claude Michel et al. Assessment methodology The model was calibrated and verified on a large sample of 138 catchments in France at a daily time step, following the same approach as that chosen by Edijatno et al. (1999). For each catchment, the data set was split into several non-overlapping periods and the model was calibrated on one period at a time and used in simulation mode on the others. A total of 446 periods were identified and 1124 verification tests were performed on the whole sample. The set of efficiency criteria obtained in simulation (verification mode) was used to judge model performance. As in Edijatno et al. (1999), the objective function used to calibrate model parameters is the Nash & Sutcliffe (1970) efficiency criterion, F, computed on the square root of daily flows and given by: F n i= 1 = n ( O Q ) ( Oi M ) i= 1 i i 2 1 (20) 2 where O i and Q i are the observed and simulated flows, respectively, at time step i, M is defined as: M = n ( 1 n). i= 1 O i and n is the number of time steps. The prior transformation on flows avoids giving too much weight to flood peaks. The quality of the simulation was assessed with the classical formulation of the Nash-Sutcliffe criterion calculated on flows. Description of TOPMO In this study, a modified version of TOPMODEL, called TOPMO, is used, which is very similar to that presented by Edijatno et al. (1999). Figure 3 shows a diagram of the model. The infiltration store included in the original model is not used here, since it does not exist in the later versions. The main difference from the original TOPMODEL is that the distribution of the topographical index is approximated by a calibrated twoparameter probability distribution function (logistic equation) instead of being derived from catchment topography (see equation giving PS/PR in Fig. 3). Overland flow is routed with a quadratic store. The depletion of the exponential store is calculated with either equation (13) or equation (8) for the continuous and the discrete formulation, respectively. Actually, as explained in the case of equation (6), optimizing parameter a in equation (13) is equivalent to choosing the origin on the S-axis. Therefore, by an adequate choice of the origin, equation (13) can be rewritten as: S R = b exp b Hence, both the discrete and the continuous formulation depend on a single parameter, henceforth called X 1. The model has a total of eight parameters to be optimized. (21)

10 The exponential store: a correct formulation for rainfall runoff modelling 117 Interception store P o te n tia l evapotranspiration E X 7 T ER R a in fa ll P PR PS PR ES ER = = e 1 e 1 S X 6 X 3 S X 2 X 5 ES PS PR-PS Exponential soil store 0 S R Quadratic routing store QB ( X 1 ) QR R² = R + X 8 Delay X 4 Fig. 3 Diagram of the TOPMO model. Streamflow Q As initial conditions at the beginning of the period, the interception store content, T, is set at X 7 (capacity of the interception store), the baseflow QB is set at one tenth of the mean annual flow (with the level in the soil store, S, set accordingly), and the level in the quadratic store, R, is set at half of its capacity X 8. To avoid errors caused by the rough initial conditions, the first year of each period was used for model warming up and excluded from the criteria calculations. The parameters were optimized with a local optimization technique that had proved efficient in the case of this model (see Edijatno et al., 1999). Before optimization, the initial parameter values for any calibration of TOPMO are: X 1 = 60 mm; X 2 = 2; X 3 = 100 mm; X 4 = 1 day; X 5 = 300 mm; X 6 = 2; X 7 = 2 mm; and X 8 = 30 mm Results To demonstrate the effect of misusing the exponential store, the model was run twice in calibration and control on the whole sample of catchments alternately with the continuous (equation (21)) and the discrete (equation (8)) formulations of the exponential store depletion equation, everything else being kept unchanged. The graph in Fig. 4 compares model performances (Nash criteria) obtained in the control in each case and illustrates the consequences of different treatments of the exponential store. It should be noted that the periods where one model formulation obtained a performance of less than 20% do not appear on this graph.

11 118 Claude Michel et al. 100 Nash criteria (%) using Equation (21) : Nash criteria (%) using Equation (8) Fig. 4 Correlation graph between the efficiency ratings (Nash criteria) obtained in control with the correct (equation (8)) and incorrect (equation (21)) treatments of the exponential store in TOPMO. The average value of the Nash efficiency criterion increases from 60.0% with the usual equation (21) to 67.5% with equation (8). The misuse of the exponential store therefore significantly reduces mean model performance. The loss of performance when equation (21) is used means reduced model robustness and reliability, but this degradation does not affect the catchments evenly: the loss of efficiency exceeds 2.5% for only 6% of the simulation tests, as shown in Fig. 5. The use of the correct store formulation even seems to be both beneficial and detrimental: it produces better performances than the incorrect formulation in 39.2% of tests, but poorer ones in 32.9% of the simulations, those of the remaining cases remain unchanged. Figure 5 illustrates this quite surprising balance between gains and losses in model performance. It should also be noted that the absolute differences in model efficiency between the two formulations are lower than 0.5% for 75.2% of the simulations, meaning almost unchanged performances in a great majority of cases. The main difference between the two formulations is in a higher percentage of strong model failures with equation (21), which can be almost eliminated with the correct treatment (equation (8)): when the correct formulation is replaced by the incorrect one, the drop in model performance exceeds 10% for 31 simulation tests (the drop is greater than 50% for half of them) whereas the improvement exceeds 10% for only two simulation tests. For these two simulations, the model does not perform satisfactorily with either store formulation, whereas among the 31 other simulation tests, performances may range from more than 80% for the correct formulation to much less than zero with the incorrect formulation. These results show that, even though the formulation of the exponential store has a significant impact on only a few cases, this impact can be dramatic.

12 The exponential store: a correct formulation for rainfall runoff modelling Percentage of tests D < < D < < D < < D < < D < 0 D = 0 0 < D < < D < < D < 2.5 D = Nash [Incorrect treatment] - Nash[Correct treatment] 2.5 < D < 10 Fig. 5 Distribution of differences of model simulation results, between correct treatment (equation (8)) and incorrect treatment (equation (21)) of the exponential store. D > 10 A CASE STUDY OF THE IMPACT OF EXPONENTIAL STORE FORMULATION Considering the quite small effects of the obvious mathematical mistake presented above, one cannot help worrying about the magnitude of the hydrological mistakes, which may partly explain the poor results usually achieved by current rainfall runoff models. However, regarding the few catchment periods where a severe drop in efficiency ratings was observed, it would be natural to look for the type of disorder responsible for these failures. As an example, the case is examined of the catchment for which one of the simulation tests produced the flagrant outlier in Fig. 4 with coordinates 77.9 and 10.4 %. This catchment is the small (36 km 2 ) headwater catchment of the Brèze River at Meyrueis, situated on the northern slopes of Mont Aigoual in southern France. The mean annual streamflow is about 930 mm and the mean annual precipitation is 1400 mm. Rain storms producing over 200 mm day -1 can occur on this catchment, especially in the autumn. TOPMODEL has already been used in similar sub- Mediterranean climate conditions in southern France, for example by Durand et al. (1992) in the Mont-Lozère catchments. These authors reported some discrepancies in simulating large flood peaks, but related the problems to the model representation of the infiltration and soil wetting dynamics. Twenty-one years of daily data from 1970 to 1990 were available for the Brèze catchment and four periods of equal length were chosen. All four periods include storm events with rainfall depths exceeding 100 mm in one day. The main flood event occurred in the third period with 390 mm of rain in three days. The four parameter sets obtained in calibration and the corresponding calibration-control performances are shown in Tables 1 and 2 for the usual (incorrect) and correct treatment of the exponential store, respectively. Each line corresponds to one calibration on a given period and each column to a control on a period.

13 120 Claude Michel et al. Table 1 Calibration (in bold) and control performances and parameter sets obtained on the four test periods for the usual (incorrect) treatment of the exponential store ( indicates large parameter values that reached the domain boundary during optimization). See Fig. 3 for parameter signification. Calibration period Nash (%): Parameters: Verification period X 1 X 2 X 3 X 4 X 5 X 6 X 7 X (mm) (-) (mm) (day) (mm) (-) (mm) (mm) Table 2 Calibration (in bold) and control performances and parameter sets obtained on the four test periods for the correct treatment of the exponential store ( indicates large parameter values that reached the domain boundary during optimization). See Fig. 3 for parameter signification. Calibra tion period Nash (%): Parameters: Verification period X 1 X 2 X 3 X 4 X 5 X 6 X 7 X (mm) (-) (mm) (day) (mm) (-) (mm) (mm) Incorrect treatment Model performances The results shown in Table 1 demonstrate that model failures in the control occur in period 3. With parameters calibrated on period 1, the Nash criterion in simulation is 158.9% for period 3 (out of bounds in Fig. 4) and with parameters from period 2 the Nash criterion is 10.4%. However, no problem occurred when parameters from period 4 were used. The lowest Nash criterion ( 158.9%) is essentially due to one flood event that was grossly overestimated, as shown in Fig. 6. Model parameters One might have expected that the problem associated with the incorrect treatment of the exponential store would have translated into unusual X 1 parameter values. Actually, all estimates of parameter X 1 fall within a close range in Table 1 and the problem must have spread to the remainder of the parameter set. It should be noted that the delay parameter X 4 is the same for all periods and therefore not concerned with model robustness. It seems a priori difficult to identify the bad parameter set among the four in Table 1. When the model was run successively with the four parameters sets in all four periods, mean simulation results indicated that the more robust parameter set is the third one, closely followed by the fourth one. Parameter sets 1 and 2 are very close to each other and show very similar behaviours by the different model components (evaporation, determination of flow component and routing). The high values of parameters X 5 make the determination of the actual evaporation rate almost insensitive to level variations in the exponential store. This behaviour is also observed for parameter set 3. The main difference between parameter sets 1 and 2 and parameter sets 3 and 4 lies in the parameterization

14 The exponential store: a correct formulation for rainfall runoff modelling 121 of the topographical index (determination of PS/PR in Fig. 3) and the quadratic routing store. Both functions act directly on the determination and routing of the quick flow component. The values of the corresponding parameter (especially X 6 and X 8 ) for sets 1 and 2 therefore seem to confer less robustness on the model. Correct treatment These results are now compared with those (Table 2) obtained with the correct treatment of the exponential store. Model performances As opposed to model performances in Table 1, there is no model failure (low performance) here and all parameter sets seem almost equally robust. Therefore, the use of the discrete formulation of the exponential store output instead of the continuous one clearly confers more robustness on the rainfall runoff model in the case of this catchment. The previously cited problem, noticed with some large rainfall events, no longer occurs (see Fig. 6). Model parameters Strikingly, the parameters are very different from those in the first case, except for the delay parameter X 4. Although the fourth parameter set does not change much, there are great changes, especially in the values of parameter sets 1 and 2; these two parameter sets were identified as the less reliable ones when the incorrect treatment of the exponential store was used. These parameter modifications indicate a change in the behaviour of the production and transfer functions of the model. Not only parameter X 1, concerned with the depletion of the exponential store, but a majority of the parameters contributed to Streamflow (mm/d) Rainfall Observed streamflow Simulated with incorrect treatment Simulated with correct treatment Rainfall (mm) /10/ /10/ /11/ /11/ /12/1982 Fig. 6 Example of differences in flow simulation with the two treatments of the exponential store for a flood event on the Brèze River, France. 600

15 122 Claude Michel et al. compensate for the incorrect treatment of the store. Therefore, the change of the mathematical formulation of the exponential store influences most of the model components. It should also be noted that, whereas one might have expected a betterdefined X 1 parameter with the correct formulation of the exponential store, the opposite is observed and the parameter values are more variable (Table 2) than with the usual treatment (Table 1). However, one explanation may be that this parameter also compensates for the errors induced by the store misuse and therefore it is being optimized within a narrow range of values to avoid any large discrepancy in the flow simulation. Similar conclusions could be drawn for the other catchments presenting the same phenomenon. The above results show that problems generated by using an incorrect treatment of the exponential store are indirect and not confined to this store: the calibration process adapts the whole model structure to the conditions generated by the different mathematical formulations. The main effect is an erratic overestimate of some flood events, which is a consequence of physical inconsistencies inherent in the misuse of the exponential store, as discussed above. CONCLUSION The exponential store is a very efficient tool for modelling baseflows as well as flood recessions. It is commonly used in rainfall runoff modelling by applying straightforwardly the continuous formulation of the depletion equation over the time step. However, integration of this equation over a time step leads to significantly different analytical equations. A mathematical analysis of both the discrete and the continuous formulations indicates that the usual way of dealing with the exponential store contains clear physical inconsistencies and can produce great overestimates of flow, especially in the case of large inputs to the store. To the authors knowledge, this fact has not previously been identified in the literature, although it may potentially detract from the efficiency that could be obtained with this useful model component. The consequences of this misuse were assessed in the case of rainfall runoff modelling, by means of a model structure that contains an exponential store for baseflow modelling. A comparison of model performances based on 138 catchments spread across France shows that the misuse of the exponential store can significantly degrade the mean performance of the rainfall runoff model. However, the clear mathematical and physical flaw lowers the quality of the results in only a few cases. In particular, when its contribution to the total flow is small, misuse of the exponential store rarely causes a severe drop in model efficiency. Closer examination of the cases where an incorrect store formulation produces very bad results indicates that the change in the mathematical equation influences not only the exponential store parameter but also most of the other calibrated model parameters. It is considered important to avoid the usual and incorrect treatment of the exponential store in order to avoid the occasional catastrophic behaviour. This is all the more true as rainfall runoff models are now widely used, e.g. to estimate floods (see for example Cameron et al., 1999; Lamb, 1999), where their extrapolation capacities must be well founded.

16 The exponential store: a correct formulation for rainfall runoff modelling 123 Acknowledgements The authors thank the anonymous reviewers for their useful comments on the earlier version of the text. REFERENCES Bergström, S. & Forsman, A. (1973) Development of a conceptual deterministic rainfall runoff model. Nordic Hydrol. 4, Beven, K. J. & Kirkby, M. J. (1979) A physically-based variable contributing area model of basin hydrology. Hydrol. Sci. Bull. 24(1), Beven, K. J., Kirkby, M. J., Schofield, N. & Tagg, A. F. (1984) Testing a physically-based flood forecasting model (TOPMODEL) for three U.K. catchments. J. Hydrol. 69, Beven, K., Lamb, R., Quinn, P., Romanowicz, R. & Freer, J. (1995) TOPMODEL. In: Computer Models of Watershed Hydrology (ed. by V. P. Singh), Water Resources Publications, Highlands Ranch, Colorado, USA. Burnash, R. J. C. & Ferral, R. L. (1982) A systems approach to real time runoff analysis with a deterministic rainfall runoff model. In: Applied Modeling in Catchment Hydrology (ed. by V. P. Singh), Water Resources Publications, Highlands Ranch, Colorado, USA. Cameron, D. S., Beven, K. J., Tawn, J., Blazkova, S. & Naden, P. (1999) Flood frequency estimation by continuous simulation for a gauged upland catchment (with uncertainty). J. Hydrol. 219, Chiew, F. & McMahon, T. (1994) Application of the daily rainfall runoff model MODHYDROLOG to 28 Australian catchments. J. Hydrol. 153, Durand, P., Robson, A. & Neal, C. (1992) Modelling the hydrology of submediterranean montane catchments (Mont- Lozère, France) using TOPMODEL: initial results. J. Hydrol. 139, Edijatno, Nascimento, N. O., Yang, X., Makhlouf, Z. & Michel, C. (1999) GR3J: a daily watershed model with three free parameters. Hydrol. Sci. J. 44(2), Jakeman, A. J., Littlewood, I. G. & Whitehead, P. G. (1990) Computation of the instantaneous unit hydrograph and identifiable component flows with application to two small upland catchments. J. Hydrol. 117, Kachroo, R. K. (1992) River flow forecasting. Part 5. Applications of a conceptual model. J. Hydrol. 133, Lamb, R. (1999) Calibration of a conceptual rainfall runoff model for flood frequency estimation by continuous simulation. Water Resour. Res. 35(10), Lambert, A. O. (1969) A comprehensive rainfall/run-off model for an upland catchment area. J. Instn Water Engrs 23(4), Moore, R. J. & Bell, V. A. (2002) Incorporation of groundwater losses and well level data in rainfall runoff models illustrated using PDM. Hydrol. Earth. System Sci. 6(1), Nash, J. E. & Sutcliffe, J. V. (1970) River flow forecasting through conceptual models, Part I: A discussion of principles. J. Hydrol. 27(3), Nielsen, S. A. & Hansen, E. (1973) Numerical simulation of the rainfall runoff process on a daily basis. Nordic Hydrol. 4, Rutter, A. J., Morton, A. J. & Robins, P. C. (1975) A predictive model of rainfall interception in forests. II. Generalization of the model and comparison with observations in some coniferous and hardwood stands. J. Appl. Ecol. 12, Sugawara, M. (1995) The development of a hydrological model TANK. In: Time and the River, Essays by Eminent Hydrologists (ed. by G. Kite), Water Resources Publications, Highlands Ranch, Colorado, USA. APPENDIX The output of the quadratic store is defined by the following continuous formulation: 2 S( t) Q( t) = (A1) b where b is the model parameter (can be expressed in mm s). The differential equation is similar to that of the exponential store: 2 S ds = dt (A2) b and can be integrated to give the following expression: 1 1 = + (A3) S R S b

17 124 Claude Michel et al. Noting B = b / (mm), one obtains: 2 S R = S + B and the storage S at the end of the time-step is then given by: (A4) SB S = (A5) S + B Here, one can see that S can never exceed B, which could be called the maximum capacity one time step ahead of the quadratic store. This capacity depends on the time step value, in the same way as the suitable origin on the S-axis for the exponential store also depends on the time step duration. Received 31 May 2002; accepted 28 October 2002