Vincent Guinot
email: guinot@msem.univ-montp2.fr

B Cappelaere

C Delenne

D Ruelland

Objective Functions for Conceptual Hydrological Model Calibration: Theoretical Analysis of Distance-and Weak Form-Based Functions

published or not. The documents may come

Towards improved criteria for hydrological model calibration : theoretical analysis of distance-and weak form-based functions

Introduction

Calibration is recognized as an essential step in the operation of conceptual, hydrological models. It is classically translated into an optimization problem, whereby an objective function expressing the goodness-of-fit of the model, must be minimized or maximized depending on the definition. Although several authors have pointed out the importance of seeing calibration as a multi-objective optimization exercise using variables and criteria of different natures [START_REF] Yapo | Multi-objective global optimization for hydrologic models[END_REF][START_REF] Gupta | Toward improved calibration of hydrologic models: multiple and noncommensurable measures of information[END_REF][START_REF] Meixner | Sensitivity analysis using mass flux and concentration[END_REF], it is still conducted as a single-objective optimization procedure in a vast majority of practical applications. The same holds for model performance assessment that is often performed using the same type of objective functions as those used in the calibration process. Such analyses are usually performed on the basis of empirical considerations, for which formal foundations are lacking. Although there is a commonly shared perception of the calibration/validation exercise in the hydrological community, this lack of theoretical bases often does not allow reliable guidelines to be derived.

The present paper focuses on two types of objective functions: so-called distancebased and weak form-based (or integral) objective functions (see Section 2 for a definition). The purpose is to analyse the behaviour of such functions and to determine under which conditions some may be better-suited than others. The choice of the model variable(s) to be used in the calibration process is also discussed. The behaviour of distance-based and weak form-based objective functions is analysed theoretically and illustrated by two simple case studies.

Distance-based objective functions represent the vast majority of objective functions used in hydrological modelling (Kavetski et al., 2006b;[START_REF] Schaefli | Do Nash values have value?[END_REF]. Distances may be defined for the original (Kavetski et al., 2006a) or transformed flow variables. In [START_REF] Hogue | A multistep automatic calibration scheme for river forecasting models[END_REF][START_REF] Hogue | A 'User-Friendly' approach to parameter estimation in hydrologic models[END_REF] and Kavetski et al (2006b), a logarithmic transformation error is presented. The Nash-Sutcliffe Efficiency (NSE) criterion [START_REF] Nash | River flow forecasting through conceptual models. Part 1 : A discussion of principles[END_REF], based on a Square Error (SE) measure of distance, is definitely the most widely used objective function in hydrological modelling. It is a normalized variant of the Least Square Estimator (LSE), and gives equivalent information to that given by the Mean Square Error (MSE), or Root Mean Square Error (RMSE). A number of theoretical justifications can be provided for the NSE. For instance, the NSE optimum corresponds to the Maximum Likelihood Estimator for a homoscedastic, gaussian distribution of model errors [START_REF] Cacuci | Sensitivity and Uncertainty Analysis[END_REF]. This justifies its use in model performance evaluation and uncertainty assessment techniques such as the GLUE approach [START_REF] Beven | The future of distributed models: model calibration and uncertainty prediction[END_REF][START_REF] Beven | Prophecy, reality and uncertainty in distributed hydrological modelling[END_REF][START_REF] Romanowicz | Comments on generalised likelihood uncertainty estimation[END_REF]. The NSE can also be seen as the sum of three indicators [START_REF] Murphy | Skill scores based on the mean square error and their relationships to the correlation coefficient[END_REF][START_REF] Weglarczyk | The interdependence and applicability of some statistical quality measures for hydrological models[END_REF] involving the correlation coefficient between the measured and modelled variable, as well as a measure of conditional and unconditional bias. [START_REF] Gupta | Decomposition of the mean squared error and NSE performance criteria: implications for improving hydrological modelling[END_REF] provided another decomposition of the NSE involving the correlation, the bias and a measure of variability in the measured and modelled signals. Such decomposition substantiates the proposal by [START_REF] Taylor | Summarizing multiple aspects of model performance in a single diagram[END_REF] that model performance should be assessed using both a measure of distance (such as the normalized standard deviation of the bias) and the correlation between observations and model outputs.

The NSE is not the only possible measure of distance. In [START_REF] Perrin | Does a large number of parameters enhance model performance ? Comparative assessment of common catchment model structures on 429 catchments[END_REF] the Mean Absolute Error (MAE) is proposed. It can be normalized into a dimensionless index such as the volumetric efficiency [START_REF] Criss | Do Nash values have value ? Discussion and alternate proposals[END_REF]. In [START_REF] Legates | Evaluating the use of « goodness of fit » measures in hydrologic and hydroclimatic model validation[END_REF], the Nash-Sutcliffe efficiency criterion is generalized by replacing the square of the deviations with a power to be adjusted by the modeller. The purpose is to balance the larger weight given to large flow values (that are often measured with the larger imprecision) by using a power smaller than 2 [START_REF] Krause | Comparison of different efficiency criteria for hydrological model assessment[END_REF]. [START_REF] Lin | A nonlinear rainfall-runoff model embedded with an automated calibration method -Part 2: The automated calibration method[END_REF] use the inverse of the SE and the NSE for computing respectively the efficiency of chromosoms and the objective function in a genetic, global optimization algorithm. Several objective functions may also be defined for low flows or peak flow events, so as to allow for multiobjective calibration [START_REF] Madsen | Automatic calibration of a conceptual rainfall-runoff model using multiple objectives[END_REF][START_REF] Madsen | Comparison of different automated strategies for calibration of rainfall-runoff models[END_REF]. Multiobjective calibration may also be carried out using variables of different natures (such as response signatures, see [START_REF] Pokhrel | Multiple-Criteria Calibration of a Distributed Watershed Model using Spatial Regularization and Response Signatures[END_REF]. A review of multiobjective calibration approaches can be found in Efstradiadis and Koutsoyiannis (2010). Conversely, multiple distance-based objective functions may be aggregated into a single one [START_REF] Madsen | Comparison of different automated strategies for calibration of rainfall-runoff models[END_REF][START_REF] Cappelaere | Hydrologic process simulation of a semiarid, endorheic catchment in Sahelian West Niger -2. Model calibration and uncertainty characterization[END_REF][START_REF] Schoups | Multicriteria optimization of a regional spatially-distributed subsurface water flow model[END_REF]. Additional information may be brought by integral criteria such as the bias [START_REF] Hogue | A 'User-Friendly' approach to parameter estimation in hydrologic models[END_REF][START_REF] Schoups | Multicriteria optimization of a regional spatially-distributed subsurface water flow model[END_REF], volume error [START_REF] Madsen | Automatic calibration of a conceptual rainfall-runoff model using multiple objectives[END_REF], also called Cumulative Error (CE) [START_REF] Perrin | Does a large number of parameters enhance model performance ? Comparative assessment of common catchment model structures on 429 catchments[END_REF]. To overcome the deficiency of the original GLUE method in reflecting modelling uncertainty, more formal derivations of likelihood functions ( eg., [START_REF] Schoups | A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and on-Gaussian errors[END_REF] or empirical adaptations of the approach (Cappelaere et al., 2003, Xiong and[START_REF] Xiong | An empirical method to improve the prediction limits of the GLUE methodology in rainfall-runoff modelling[END_REF] have been proposed.

Distance-based objective functions are well-known to introduce local minima in model response surfaces [START_REF] Freedman | Parameter identifiability for catchment-scale erosion modelling : a comparison of optimization algorithms[END_REF][START_REF] Skahill | Efficient accommodation of local minima in watershed model calibration[END_REF][START_REF] Xiong | Analysis of the response surface of the objective function by the optimum parameter curve: how good can the optimum parameter values be?[END_REF]. In [START_REF] Freedman | Parameter identifiability for catchment-scale erosion modelling : a comparison of optimization algorithms[END_REF], distance-based objective functions such as the Least Squares Estimator (LSE) and the Heteroscedastic Maximum Likelihod Estimator (HMLE) are shown to introduce local extrema in the objective functions of a sediment transport model, thus introducing the need for global optimization or objective function exploration algorithms (see e.g. [START_REF] Brazil | Optimisation of complex hydrologic simulation models using random search methods[END_REF][START_REF] Goldberg | Genetic Algorithms in Search, Optimisation, and Machine Learning[END_REF][START_REF] Nelder | A simple method for function minimization[END_REF][START_REF] Skahill | Efficient accommodation of local minima in watershed model calibration[END_REF][START_REF] Duan | Effective and efficient global optimisation for conceptual rainfall-runoff model[END_REF]. Modelling experiments where the LSE and HMLE were used to calibrate and validate different models on the same data indicated that the choice of the objective function plays a significant role on the final, calibrated parameter values [START_REF] Gan | Effects of model complexity and structure, data quality, and objective functions on hydrologic modelling[END_REF].

Weak form-based objective functions are somewhat less popular, as indicated by the inventory in Appendix A. The best-known weak form-based objective functions are the Cumulative Error (CE) [START_REF] Perrin | Does a large number of parameters enhance model performance ? Comparative assessment of common catchment model structures on 429 catchments[END_REF], also called Volume Error [START_REF] Madsen | Automatic calibration of a conceptual rainfall-runoff model using multiple objectives[END_REF], and the Balance Error (BE) [START_REF] Perrin | Does a large number of parameters enhance model performance ? Comparative assessment of common catchment model structures on 429 catchments[END_REF]. The BE is nothing but a scaled version of the CE. The flow variable used in the CE and BE is usually the discharge at the outlet of the modelled catchment. The optimal value of the BE/CE is zero. The BE/CE may be used either as a constraint (typically, CE = 0) in a single objective optimization process or as an objective function in a multiple objective calibration exercise (see e.g. [START_REF] Ruelland | Modelling the long-term impact of climate change on rainfall-runoff processes over a large Sudano-Sahelian catchment[END_REF]. That the BE/CE is only a particular case and can be generalized so as to generate a wider family of weak form-based objective functions has been little investigated in the literature. This is one of the aspects explored in this paper.

The question also arises of whether using additional variables (such as model internal variables) allows the calibration problem to be better constrained. Examples of this approach applied to conceptual models can be found in [START_REF] Seibert | On the dialog between experimentalist and modeler in catchment hydrology: use of soft data for multicriterial model calibration[END_REF], [START_REF] Werth | Integration of GRACE mass variations into a global hydrological model[END_REF] and [START_REF] Winsemius | Comparison of two model approaches in the Zambezi river basin with regard to model reliability and identifiability[END_REF].

The present paper deals with objective functions for conceptual hydrological models that can be described by first-order differential equations. The main objectives are to (i) generalize the formulation of weak form-based objective functions, (ii) analyze the respective behaviour of distance-based and weak form-based objective functions and the resulting degree of difficulty in the calibration exercise, (iii) investigate whether certain model variables (e.g. internal variables or output fluxes) bring more information than others in the calibration of model parameters.

In Section 2, distance-based and weak form-based objective functions are defined and a mathematical justification is proposed for them.

In Section 3, the behaviour of such objective functions when applied to conceptual models is analyzed. Weak form-based objective functions are shown to be more monotone and less prone to local extrema than distance-based objective functions. Simple rules for detecting parameter redundancy are given and guidelines are provided for the choice of calibration variables.

Sections 4 and 5 provide two application examples. In Section 4, a single reservoir model is considered and synthetic time series are used in order to avoid any possible sitedependent bias. In Section 5, a three-reservoir model is applied to a Western African catchment.

Section 6 is devoted to a discussion and concluding remarks.

Objective Function Definition

Introduction

Consider a model in the form

(1

)
where U is the state variable, t is the independent variable (in hydrological modelling, the time coordinate), j a parameter to be calibrated, and f is a known function of U, t and j. The standard calibration approach consists in comparing the variable U or a function F(U) of U with an observed variable V(t) over a certain domain W = [t1, t2] and adjusting j in such a way that U (or F(U)) is « as close as possible » to V. In what follows, the function F is assumed to be a monotone function of U and j. This assumption is verified by many models such as conceptual models, where F can be, for instance, taken as the discharge Q that is a function of U (see Section 4). In the general case, where F is not necessarily a physical function but any scaling function, it is chosen monotone in order to avoid several values of F(U) for a given value of U.

The question then arises of how the closeness between U (or F(U)) and V should be assessed via an objective function. If the model is perfect, the output U or F(U) reproduces exactly the variations of the measured variable V, that is, U = V or F(U, j) = V for all t over the time interval W (the issue of data accuracy and measurement precision is not considered in the present work).In practice, this is never the case and the purpose of the calibration procedure is to bring the difference (U -V) or (F(U, j) -V) as close to zero as possible. Two types of objective functions are examined hereafter: distance-based and weak form-based objective functions.

Distance-based objective function

The distance-based approach is the most widely used in hydrological modelling. In this approach, the objective function is defined as one of the following two functions

(2a) (2b)
where a and b are respectively an offset and a scaling constant, e is the modelling error, defined as the difference between the modelled and observed variable, and the operator has the properties of a norm [START_REF] Courant | Methods of Mathematical Physics[END_REF]:

(3a) (3b) (3c)
In other words, the objective function J provides a measure of the distance between the model output U or F(U, j) and the measurement V. Property (3a) provides a fundamental justification to the distance-based approach. If the model is perfect (that is, if it allows the observed variable V to be reproduced exactly), the error e is zero over W and the objective function J is a, which is the extreme possible value. Conversely, if J is a, the model is perfect. In practice, J is never equal to a because the model is not perfect.

However, J can be optimized by adjusting j suitably, hence the need for optimization procedures.

If the data used in the calibration process is discrete (e.g. daily, weekly or monthly hydrographs), the norm is computed using a discrete sum. If the data can be considered a

) , , ( d d t U f t U j = W W - + = + = V U b a e b a J W W - + = + = V U F b a e b a J ) , ( j 0 0 ) ( = Û W Î " = W u t t u  Π" = W W k u k ku W W W + £ + v u v u
continuous function of time, the norm is computed using an integral. In what follows, only continuous functions of time will be considered for the sake of conciseness. However, the conclusions drawn for such functions remain valid for discrete model outputs.

Examples of distance-based objective functions are given in Appendix A. The Nash-Sutcliffe Efficiency (NSE), the Square Error (SE), the Root Mean Square Error (RMSE), the Mean Absolute Error (MAE), the Volumetric Efficiency (VE) or the Generalized Nash-Sutcliffe Efficiency (GNSE) presented in the appendix can be recast in the form (2) via a proper definition of the constants a and b. The NSE gives exactly the same information as the SE, only the offset and scaling differs. The same remark holds for the MAE and VE.

Note that: (R1) Eq. ( 2a) is a particular case of Eq. (2b). Eq. (2a) provides a measure of distance between the modelled state variable U in the model and the measured one, V. It is recalled that, in most applications of hydrological models, however, the variable used in the objective function is not the state variable U itself (e.g. the water depth in one of the model reservoirs) but a function of it (e.g. the output discharge). Consequently, Eq. ( 2b) is the most widespread form of objective function used. (R2) The function F in Eq. (2b) may also include a transformation in the variables. For instance, in some applications the logarithm, or square root of the discharge, is deemed a more appropriate variable than the discharge itself because it gives more weight to low flows. (R3) The objective function may be defined for a specific range of the observed (or modelled) variables. For instance, two different values of the objective function may be computed over a given period, one for low flows and another one for high flows (see e.g. [START_REF] Perrin | Does a large number of parameters enhance model performance ? Comparative assessment of common catchment model structures on 429 catchments[END_REF]. In this case, a weighting function w(V) is used, which is nonzero only over a subset of W, and the norm can be written as

(4a) (4b)
where w(V) is a weighting function, equal to 0 or 1 depending on whether V is considered to belong to the category of low [high] flows, and i is the record number. Eqs. (4a) and (4b) are respectively the continuous and discrete versions of the norm. In what follows, the continuous form (4a) will be used for the sake of notation consistency, but the reader should keep in mind that the discrete form (4b) may be used instead without loss of generality. The conclusions derived using the formulation (4a) also hold for the formulation (4b) of the objective function.

(R4) In Eq. ( 4), any positive weighting function w may be used over W. The simplest possible case is w = Const, but non constant, positive functions of V, U, F(U) and/or t may also be considered. (R5) The distance-based objective functions presented in Appendix A are particular cases of Eq. ( 4), where the norm of the modelling error is defined as a power of its absolute value, called a p-norm. In the NSE and SE objective functions, p = 2, while p = 1 in the MAE and VE. In the GNSE, p may be set to any value, which

0 , d ) , ( ) ( / 1 > ú ú û ù ê ê ë é - = ò W p t V U F V w e p p j 0 , ) , ( ) ( / 1 > ú û ù ê ë é - = å p V U F V w e p i p i i i j
does not necessarily have to be an integer. However, other definitions may be proposed for the norm. For instance the maximum of the modelling error over the domain W also verifies the definition (3a) for a norm:

(5) Note that the power 1/p in Eqs. (4a-b) is not indispensable but allows Eq. (3b) to be verified. (R6) The calibration process is an optimization process.

Weak form-based objective function

The weak form-based approach uses the property (6)

where v(t) is a function defined over W. In what follows, v is defined as v = w|e| p-1 for the sake of similarity with Eq. ( 4). This leads to the following definition for the weak formbased objective function ( 7)

where a and b are respectively an offset and scaling parameter, and w is a positive weighting function defined over W. The Volume Error/Cumulative Error (CE) and the Balance Error (BE) presented in Appendix A are particular cases of Eq. ( 7) with p = 0 and w(t) = 1. In contrast with the distance-based approach, the objective function defined in Eq. ( 7) is not necessarily positive.

The following remarks may be made: (R7) Equation ( 7) is a particular case of (6) obtained for the specific choice v = w|e| p-1 of the weighting function. Other formulations may be considered for v. The formula proposed in (7) has the advantage that it bears similarity with familiar distancebased objective functions (only an absolute value operator needs to be modified). (R8) The calibration procedure is transformed into a root finding problem. The most desirable value for the objective function is the offset value a.

Sensitivity Analysis for Conceptual Models

Assumptions -preliminary remarks

Consider a model obeying Eq. ( 1). The specific form of (1) examined hereafter is

(8a) (8b)
where g is a known function of U and j, and R is a known function of U, t and j. In conceptual models, R represents the recharge, or inflow, and g represents the outflow.

The following assumptions are made: (A1) R and g are positive over W:

(9a) [ ] V U F e - = W ) , ( max j ) ( 0 d 0 ) ( t v t ev t t e " = Û W Î " = ò W 0 , d 1 ³ + = ò W - p t e e w b a J p p d ( , , ) ( , ) d U R U t g U t j j = - 1 1 ( ) U t U = W Î " ³ t t R 0 ) , ( j (9b) (A2)
The difference R -g is a decreasing function of U and a monotone function of j:

(10a) (10b) (A3)
There always exists a positive, steady state solution, that is, a value of U0 for which the outflow g is equal to the inflow R: t1) is positive: (12) Assumptions (A1)-(A4) are typical of conceptual, conceptual models (see Section 4). When these assumptions hold, U is positive over the domain W (see Section B.1 in Appendix B for the proof).

(11) (A4) U(
The sensitivity of U with respect to j is defined as . The governing equation for s is obtained by differentiating (8) with respect to j:

(13a) (13b)
where the initial condition s (t1) = 0 is derived considering that U (t1) = U1 is known and does not change with j.

It can be shown (see Section B.2 in Appendix B for the proof) that if keeps the same sign for all t, s has the same sign as :

(14a) (14b)

Distance-based objective function

Consider a distance-based objective function using the definition (4) for the norm of the modelling error: (15) where U(t) obeys (8), F is a monotone function of U as mentioned in Section 2.1 and w is a strictly positive weighting function over the domain W. Then the derivative of the objective function with respect to j is given by one of the following two formulae depending on whether the function F involves the parameter j:

( ) 0 g t t ³ "ÎW U g R U " £ - ¶ ¶ 0 ) ( j j " = ú û ù ê ë é - ¶ ¶ Const ) ( sgn g R ) , ( ) , , ( ) , ( , 0 0 0 0 j j j t t U R U g U " = > $ 0 ) ( 1 ³ t U / s U j = ¶ ¶ ) ( ) ( d d g R U s g R t s - ¶ ¶ + - ¶ ¶ = j 0 ) ( 1 = t s j j ¶ ¶ - ¶ ¶ / / g R j j ¶ ¶ - ¶ ¶ / / g R W Î " £ Þ ï þ ï ý ü = W Î " £ ¶ ¶ - ¶ ¶ t t s t s t g R 0 ) ( 0 ) ( 0 1 j j W Î " ³ Þ ï þ ï ý ü = W Î " ³ ¶ ¶ - ¶ ¶ t t s t s t g R 0 ) ( 0 ) ( 0 1 j j ò W - + = t V U F w b a J p p d ) , ( j (16a) (16b)
Since the sign of s is constant (see Section 3.1) and F is assumed to be monotone, the product keeps the same sign over W. Two possibilities arise:

-F is not a function of j. In this case, Eq. ( 16a) is applicable. The terms and in the integral keep a constant sign over W, while the second term F(U) -V may change sign. There is a possibility for to cancel, which is a desirable property because the purpose of the calibration exercise is to find an optimum of the function Jp. -F is a function of both U and j. Then, Eq. ( 16b) is applicable. In the case of conceptual models (see Section 4), the sign of is not necessarily constant because and may have opposite signs. Then both and in the integral may change sign.

In the general case, the two terms do not cancel for the same value of j, thus increasing the possibilities for the appearance of local extrema. In both cases, using a strictly positive weighting function w minimizes the number of extrema for Jp.

Weak form-based objective function

Consider a weak form-based objective function defined from Eq. ( 7) :

(17

)
with the same assumptions on F, U and w as in Section 3.2. Then (18a)

As in Section 3.2, two possibilities arise: -F is not a function of the parameter to be calibrated j. In this case, and Eq. ( 18a) is applicable. Since keeps the same sign over W, the sign of cannot change. No local extremum can appear in the objective function.

-F is a function of both U and j. Then Eq. (18b) applies. The sign of is not necessarily constant, as shown in Section 4. This may lead to local extrema in Jp. Nevertheless, the derivative of Jp as defined by Eq. ( 18b) is less prone to sign change than that defined in Eq. (16b) because the term keeps a constant sign.

[ ]

0 if d ) , ( ) , ( d d 2 = ¶ ¶ ¶ ¶ - - = ò W - j j j j F t s U F V U F V U F w bp J p p [ ] 0 if d ) , ( ) , ( d d 2 ¹ ¶ ¶ ÷ ÷ ø ö ç ç è ae ¶ ¶ + ¶ ¶ - - = ò W - j j j j j F t F s U F V U F V U F w bp J p p s U F ¶ ¶ / 2 ) , ( - - p V U F j s U F ¶ ¶ / j ¶ ¶ / p J j ¶ ¶ + ¶ ¶ / / F s U F s U F ¶ ¶ / j ¶ ¶ / F [ ] V U F - ) , ( j j ¶ ¶ + ¶ ¶ / / F s U F ò W - - - + = t V U F V U F w b a J p p d ] ) , ( [ ) , ( 1 j j 0 if d ) , ( d d 1 ¹ ¶ ¶ ¶ ¶ - = ò W - j j j F t s U F V U F w bp J p p 0 if d ) , ( d d 1 ¹ ¶ ¶ ÷ ÷ ø ö ç ç è ae ¶ ¶ + ¶ ¶ - = ò W - j j j j F t F s U F V U F w bp J p p 0 / = ¶ ¶ j F s U F ¶ ¶ / x J p ¶ ¶ / j ¶ ¶ + ¶ ¶ / / F s U F 1 ( , ) p w F U j v - -

Choice of calibration variables

In the light of the expressions derived in Subsections 3.2 and 3.3, the following remarks may be made: (R9) When calibrating a given model parameter, it is not advised to use a flow variable, the calculation of which involves this parameter. For instance, in a linear conceptual model, the output discharge is defined as kU, where U is the water level in the reservoir and k is the discharge coefficient. Using U as a calibration variable for k is advisable, while using the discharge kU to calibrate k may generate local extrema in the objective function. Conversely, the discharge kU may be used to calibrate the effective catchment area. (R10) In many situations however, the only variable available for measurement is not U but a function of U (for instance, the outflowing discharge). In conceptual models, the internal, state variable U of the model is almost never used in calibration/validation procedures, while the discharge, that is only a function of U, is used in almost all situations. Then using a weak form -based objective function minimizes the probability of finding local extrema compared to a distance-based objective function. Consequently, classical gradient-based methods may be used with a larger probability of success to find the zero of the weak form-based objective function than in finding the global optimum of a distance-based objective function. (R11) Owing to the presence of and in Eqs. ( 16b) and ( 18b), using such a function F in the objective function, rather than the primary model variable U, has a strong influence on the direction of the gradient of the objective function in the parameter space. Using several variables of different natures (or transformations of the measured flow variables), such as water levels and discharges, to calibrate the models, may be more beneficial and more helpful in removing indeterminacy than using several objective functions on the same flow variables. This was already stated in [START_REF] Beven | A manifesto for the equifinality thesis[END_REF] about the ill-posed character of the calibration exercise. (R12) Models with several reservoirs in series have a similar behaviour with respect to the objective function. Indeed, when a reservoir discharges into another, its outflow discharge g (U, j) is the recharge R (U, t, j) of this second reservoir and Assumptions (A1-4) still hold. Remarks (R9) and (R10) also hold for the calibration of parameters governing the internal fluxes between several reservoirs in a model.

Objective functions as indicators of parameter redundancy

Assume that two parameters j1 and j2 in the model are redundant. In this case, for any given variation in j1, at least one alternative value can be found for j2 such that the modelling result remains unchanged. In other words, in the parameter subspace j1×j2, there exists a relationship in the form (19)

Eq. ( 19) becomes (20)

Eq. ( 20) defines a hypersurface in the parameter space and a curve in the subspace j1×j2. On the hypersurface (20) one has

U F ¶ ¶ / j ¶ ¶ / F 0 )] ( d[ = U F t U F U F " = ¶ ¶ + ¶ ¶ 0 d ) ( d ) ( 2 2 1 1 j j j j (21)
which means that the objective function is constant along (20). Consequently, the contour lines of the distance-based objective functions (15) obtained with different values of p never intersect. The remark also holds for weak form-based objective functions ( 17).

An easy way of detecting the redundancy in two model parameters is to plot the contour lines in the subspace j1×j2 of two objective functions (15) or ( 17) defined with two significantly different values of p (for instance, p = ½ and p = 2). If the contour lines of these two objective functions do not intersect, then the parameters can be suspected to be redundant.

Plotting the contour lines of the objective function requires a systematic exploration of the parameter subspace. An alternative to this approach consists in computing the dimensionless gradients of the objective functions obtained with two different powers p and q:

(22)
where L1 and L2 are scaling factors for the parameters j1 and j2 (e.g. the typical ranges of variation of these parameters), and checking colinearity via the dimensionless determinant (23) where the operator denotes the norm of the vector. The derivatives of the objective functions with respect to the parameters may be computed empirically from two values of Jp and Jq computed using two slightly different values of the parameters. The closer D to zero, the smaller the angle between the gradient vectors Gp and Gq, the more (locally) redundant the parameters j1 and j2.

Application Example 1: Linear Single Reservoir Model

Governing equations

Consider a single reservoir, rainfall-runoff model with a linear discharge law:

(24a) ( 24b
)
where A is the catchment area, c is the effective infiltration coefficient, k is the specific discharge coefficient, P(t) is the precipitation rate, U is the volume of water stored in the model and kU is the outlet discharge of the model. Note that Eq. (24a) can be written in the form (8) by defining g(U, j) = kU. Classically, A is known and c and/or k must be calibrated. This model verifies Eqs. (9-11) of assumptions (A1)-(A4), consequently U(t) is positive for all t.

The governing equation for the sensitivity is

p U F U F J J p p " = ¶ ¶ = 0 )] ( d[ ) ( d ú ú ú ú û ù ê ê ê ê ë é ¶ ¶ ¶ ¶ = ú ú ú ú û ù ê ê ê ê ë é ¶ ¶ ¶ ¶ = 2 2 1 1 2 2 1 1 , j j j j q q q p p p J L J L J L J L G G ÷ ÷ ø ö ç ç è ae ¶ ¶ ¶ ¶ - ¶ ¶ ¶ ¶ = 1 2 2 1 1 f f f f q p q p q p J J J J D G G kU c t AP t U - = ) ( d d 1 1 ( ) 0 U t U = ³ (25a) (25b)
Two possibilities arise:

(1) The parameter to be calibrated is the effective infiltration coefficient c. In this case, j = c and Eq. (25a) simplifies into (26) Eqs. (25a, 26) verify Eqs. (9-11) with AP>0, therefore s is positive for all times.

(2) The parameter to be calibrated is the discharge coefficient k. In this case, j = k and Eq. ( 25a) becomes ( 27)

Eqs. (25a, 27) verify Eqs. (9-11) with -U < 0, therefore s is negative for all times.

Distance-based objective functions

Assume that a distance-based measure is used for the objective function. If the model is to be calibrated against field measurements (or estimates) of the volume U of water stored in the catchment, then F(U) = U and Eq. ( 16a) becomes (28) As shown in the previous subsection, the sign of s is constant over W. The particular case of the NSE or SE objective functions, (p = 2, w = 1) yields the following formula (29) Assume now that the model is to be calibrated against the discharge Q = kU at the outlet of the catchment. Then F(U) = kU and . If the parameter to be calibrated is the coefficient c, then j = c and . If the parameter to be calibrated is the discharge coefficient k, then j = k and . Applying Eqs. ( 16) yields (30a) (30b) Note that the second equality in Eq. ( 30b) is obtained from Eq. ( 27). Since s is not monotone over W in the general case, both Q -V and ds/dt are liable to cancel over W and the objective function Jp in Eq. (30b) may have more than one extremum. This is an illustration of Remark (R10). In the particular case of the NSE or SE objective functions (p = 2, w = 1), the following formulae are obtained

j j ¶ ¶ + ¶ ¶ - - = ) ( ) ( d d AcP kU ks t s 0 ) ( 1 = t s c ks AP t s = - = j if d d k ks U t s = - - = j if d d ò W - - - = t s V U V U w bp J p p d ) ( d d 2 j ò W - = t s V U b J d ) ( 2 d d 2 j k U F = ¶ ¶ / 0 / = ¶ ¶ j F k Q U F / / = = ¶ ¶ j c t ks V Q V Q w bp J p p = - - = ò W - j j if d ) ( d d 2 k t t s V Q V Q w bp t k Q ks V Q V Q w bp J p p p = - - - = + - - = ò ò W - W - j j if d d d ) ( d ) / )( ( d d 2 (31a) (31b)

Weak form-based objective function

Assume now that the objective function is defined using the weak form-based approach.

If the model is to be calibrated against field measurements (or estimates) of the volume of water stored in the catchment, then F(U) = U and Eq. ( 18) becomes (32) Since s keeps the same sign over W, the derivative of Jp cannot cancel if w is nonzero. Then the points for which Jp = a (a being the optimum value for the objective function) define a line in the parameter space (c, k). In the particular case (p = 2, w = 1), one has

c t ks V Q b J = - = ò W j j if d ) ( 2 d d 2 k t t s V Q b t k Q ks V Q b J = - - = + - = ò ò W W j j if d d d ) ( 2 d ) / )( ( 2 d d 2 ò W - = t V U w bp J p p d s d d j ò W - = t V U b J d s ) ( 2 d d 2 2 j k U F = ¶ ¶ / c t ks V Q w bp J p p = - = ò W - j j if d d d 1 k t t s V Q w bp t k Q ks V Q w bp J p p p = - - = + - = ò ò W - W - j j if d d d d ) / ( d d 1 1 d d d if d p J bA ks t bAk s t c j j W W = = = ò ò k t s t s b t t s b J p = - - = - = ò W j j if )] ( ) ( [ d d d d d 1 2
result is confirmed by the following analysis. The ratio of the gradients computed via Eqs. (35a, b) gives

(36a) (36b)
where is the average value of s over W. The numerator in K remains bounded. If the input time series is sufficiently long to be assumed stationary, the average value of s does not depend on the length of the time interval t2 -t1. Consequently the denominator is proportional to the time interval t2 -t1. Therefore K tends to zero as the length of the calibration interval increases. This means that the CE and BE indicators are insensitive to the value of k when long time series are used. Therefore, they cannot be used to calibrate k when the variable is the outflowing discharge.

Numerical experiments

The properties of distance-and weak form-based objective functions are investigated using the following numerical experiment. An artificial time series for the observed (measured) variable V is generated using a nonlinear conceptual model with artificially randomized input time series. The water level in the nonlinear reservoir and the outflowing discharge of the nonlinear model are considered as « reality », against which a linear conceptual model is to be fitted. The steps in the generation of the times series are the following.

(1) An artificial rainfall time series is generated at a daily time step using the following model:

(37

)
where Pmax is a constant, Ran is generated randomly from a uniform probability density function between 0 and 1, and a is a threshold value between 0 and 1. Ran is generated every time step independently from the realization at the previous time steps. The probability for a rainfall Rn to be nonzero over a given day n is 1 -a.

(2) The generated rainfall signal is used as an input for a nonlinear conceptual model obeying the following equation:

(38a) ( 38b
)
where A is the catchment area, C is an infiltration constant, K and b are predefined constants (b ≠ 1) and W is the amount of water stored in the model. Eq. ( 38a) is solved numerically using an explicit formula at a daily time step:

(39

)
where Pn is the average value of the rainfall rate between the time levels n and n + 1. The explicit approach corresponds to the most widespread implementation of conceptual models available in the literature.

c J K k J p p d d d d = s k t t t s t s t s k t s t s K ) ( ) ( ) ( d ) ( ) ( 1 2 1 2 1 2 - - = - = ò W s ) 0 , Ran max( 1 max a a - - = P P n ( ) b KW CP A t W - = d d 0 ) 0 ( = W ( ) t KW CP A W W n n n n D - + = + b 1 (3)
The numerical solution Wn is used as the observed variable V in the computation of the objective function. Two possibilities are considered hereafter:

(40a) (40b)
Equation ( 40a) corresponds to the situation where the state variable W can be fieldestimated or measured and where the variable U in the linear model ( 24a) is considered to bear a physical meaning. Equations (40b) correspond to the more widespread calibration technique where the flow variable used in the computation of the objective function is the discharge at the outlet of the catchment. Note that the governing equation (24a) for the linear reservoir model is also solved numerically using an explicit formula ( 41)

where c and k are the infiltration coefficient and the specific discharge coefficient for the linear conceptual model. The main motivation behind the choice of a nonlinear conceptual model to generate the reference time series is that no combination of the parameters c and k in the linear model (24a) allows the solution W of Eq. ( 38a) to be reproduced exactly, which is precisely the case when real-world time series are dealt with.

The parameters used in the present experiment are summarized in Table 1.

Figure 1 shows the contour lines obtained for two types of distance-based objective functions. The first objective function Jp,U is defined using the amount of water in the reservoir as in Eqs. (40a):

(42)

The second objective function Jp,Q is computed from the outflow discharges as in Eqs. (40b): (43) where the overbar denotes the average over W. Eqs. (42-43) are nothing but the Generalized Nash-Sutcliffe Efficiency (GNSE) presented in Appendix A. In hydrological modelling, it is more customary to use the discharge as a calibration variable than the volumes stored in the reservoirs. In Figure 1, the values used for p are ½, 1 and 2.

Comparing the contour lines obtained using Eq. ( 42) (Figure 1a, 1c, 1e) and those obtained using Eq. ( 43) (Figure 1b,1d,1f) illustrates Remark (R11). The objective functions based on the water depth (or volume) and the objective functions based on the discharge have radically different contour line shapes. Using two different variables in the definition of the objective functions brings in more information than using the same

þ ý ü = = U U F W V ) ( ( ) V AKW F U AkU b ü = ý = þ ( ) t kU cP A U U n n n n D - + = +1 ò ò W W - - - = t W W t W U J p p U p d d 1 , ò ò W W - - - = t KW KW t KW Q J p p Q p d d 1 , b b b
variables and changing the value of the power p in the GNSE. Obviously, the optimal values of the objective functions depend on the variable used in the calibration process and, to a lesser extent, on the value of the exponent p. Remarks (R9, R10) are illustrated by the zero contour line in Figure 1d. It is visible from Figure 1d that the derivative cancels for small values of k around c = 1.0. This corresponds to the change in curvature of the zero contour line next to the c-axis. The objective function is not monotone with respect to k in this region of the parameter space. In the present case, however, this is not too serious a problem because (i) the extremum correponds to a minimum in the objective function, and (ii) it is located far away from the maximum of the objective function. However, in the general case, this is a potential source for local extrema in the objective function.

Figure 2 shows the contour lines obtained for two types of weak form-based objective functions. The first objective function Jp,U is defined using the state variables U and W as in Eqs. ( 40a):

(44)

The second objective function Jp,Q is computed from the outflow discharges as in Eqs. (40b):

(45) where the overbar denotes the average over W. Note that the denominator in Eqs. (44,45) is similar to that in Eqs. (42,43) and therefore the scaling is the same. In contrast with Eqs. (42,43), the best model fit is achieved for Jp,U = 0 and Jp,Q = 0. Also note that for p = 1 (Figure 2d), Eq. ( 45) gives an information similar to the Cumulative Error (CE) or Balance Error (BE).

Remarks (R9, R10) are illustrated by the zero contour line in Figure 2b. Indeed, the derivative cancels for (k = 0.15, c = 0.2). This corresponds to the curved contour in the bottom right corner of the Figure . The objective function is not monotone with respect to k in this region of the parameter space.

As in the case of the distance-based objective functions (42, 43), comparing Figures 2a, 2c, 2e andFigures 2b, 2d, 2f illustrates Remark (R11) on the complementary character of the information brought by objective functions defined using different model variables. However, contrary to distance-based objective functions, it is possible to find parameter combinations for which the objective functions defined by both equations ( 44) and (45) are optimal.

Remark (R13) on the strictly monotone character of the weak form-based objective functions is also confirmed.

Figure 2d confirms Remark (R14) that the specific discharge coefficient k cannot be calibrated using CE or BE because the CE and BE indicators have identical values for all k. At the same time, the CE or BE indicators are extremely useful in calibrating the infiltration coefficient because there is only one possible value of c for which J1,Q = 0.

Moreover, the intersection of all the zero contour lines in Figure 2 are very close to the optimum values for the distance-based objective functions using the discharge as a

k J Q ¶ ¶ / , 2 / 1 ò ò W W - - - - = t W W t W U W U J p p U p d d ) ( 1 , ò ò W W - - - - = t KW KW t KW Q KW Q J p p Q p d )d ( 1 , b b b b k J Q ¶ ¶ / , 2 / 1
calibration variable (Figure 1b,1d,1f). This indicates that the set of distance-based objective functions may yield a calibration result close to that given by the commonly admitted distance-based approach, while eliminating the local extremum problem associated with distance-based objective functions.

This example also illustrates the fact that optimum values obtained from distancebased functions are often mutually exclusive, which is not the case with weak form-based functions.

Application Example 2: a Multiple Reservoir Model with Threshold Function

Model presentation

The Medor model [START_REF] Hreiche | Calage d'un modèle conceptuel pluie-débit journalier à partir de bilans annuels[END_REF] is a three-reservoir model (Figure 3) initially designed for hydrological modelling over arid or semi-arid regions. A variation of this model has been applied recently to the modelling of karst catchments in the Mediterranean area [START_REF] Fleury | Sources sous-marines et aquifères karstiques côtiers méditerranéens[END_REF][START_REF] Tritz | Modelling the behaviour of a karst system catchment using non-linear hysteretic conceptual model[END_REF]. The top reservoir accounts for production. The input to this reservoir is the precipitation rate P, the outputs are the evapotranspiration rate E and the net precipitation rate I. E may be set equal to the potential rate, computed from standard evapotranspiration formulae [START_REF] Fleury | Sources sous-marines et aquifères karstiques côtiers méditerranéens[END_REF], or interpolated from monthly data [START_REF] Tritz | Modelling the behaviour of a karst system catchment using non-linear hysteretic conceptual model[END_REF]. E is limited only when the reservoir is empty (H = 0) and precipitations are insufficient (P < E). I is zero until the water level H in the production reservoir reaches the maximum value Hmax. In such a case, the net input P -E is transferred instantaneously to the other two reservoirs. From a conceptual point of view, the production reservoir represents the soil layer, from which previously precipitated water may be restituted to the atmosphere via evapotranspiration. Note that when the soil reservoir is empty, it remains so until P -E becomes positive again.

The net precipitation is routed to a fast response and a slow response reservoir via a distribution coefficient x, 0 £ x ≤ 1. Each of these two reservoirs obeys a linear discharge law. The output hydrograph is the sum of the output discharges from the fast and slow reservoir. From a physical point of view, the two reservoirs may account for different flow routing paths over the catchment. From the point of view of the transfer function of the model, such a structure allows both the rapid recession part of hydrographs and the slower fluctuations of the base flow to be accounted for via linear laws.

The governing balance equations are the following:

(46a) (46a) (46c)
where H1 and H2 are respectively the water levels in the fast and slow reservoirs. I, Q1 and Q2 are given by (47a)

(47b) (47b) î í ì < - = > - - = 0 and 0 if 0 0 if d d E P H H I E P t H A Q xI t H 1 1 d d - = A Q I x t H 2 2 ) 1 ( d d - - = î í ì = - < = max max if if 0 H H E P H H I 1 1 1 H Ak Q = 2 2 2 H Ak Q =
where k1 and k2 are the specific discharge coefficients of the two reservoirs. The total outflowing discharge is computed as the sum of Q1 and Q2.

Principle of the parameter redundancy test

It is first noticed that the total amount of water flowing to the outlets of the reservoirs is strongly conditioned by the maximum water level Hmax in the production reservoir. Indeed, I is nonzero only when H reaches Hmax. The larger Hmax, the smaller the time during which I is nonzero, the smaller the amount of water flowing to the reservoirs H1 and H2.

Consider now the configuration of the model where the specific discharge coefficient of one of the two reservoirs (say, k2) is zero. In this case, the water may accumulate indefinitely in this reservoir, while the outflowing discharge from this reservoir remains zero. In other words, if k2 = 0, the reservoir H2 acts as a loss and only a fraction xI of the total infiltration rates participates to the outflow. Therefore, x also controls the outflowing discharge to some extent.

Clearly, Hmax and x exert a similar influence on the outflowing discharge when k2 = 0. In other words, they are redundant with respect to total discharge, for k2 = 0. Consequently, plotting two different objective functions (18a-b) with two different values of p in the parameter space (Hmax, x) should yield non-intersecting contour lines for the two objective functions.

Catchment and modelling data

The Medor model was used to simulate daily discharge at the outlet of the Bani catchment (Figure 4). This large west-African catchment is characterized by a monsoon climate with a strong north-south rainfall gradient, and considerable rainfall variability since the mid-20 th century. As a result, the flow at the Douna gauging station (Figure 4) fell by 68% from 1952-1970 to 1971-2000, with a decrease in the deep water recharge and in base runoff contribution to the annual flood [START_REF] Ruelland | Modelling the long-term impact of climate change on rainfall-runoff processes over a large Sudano-Sahelian catchment[END_REF]. Some of the low-water periods were severe to the point that river flow at Douna stopped at times during the 1980s.

The Medor model may seem too simple at first sight for an operation at a daily time step given the dimensions of the catchment and the time scale of the discharge signal. However, experiments carried out using models of varying complexity have shown that complex transfer functions involving signal delay (such as the unit hydrograph convolution approach) do not contribute to improve model performance significantly [START_REF] Ruelland | A comparison of two conceptual models for the simulation of hydro-climatic variability over 50 years in a large Sudano-Sahelian catchment[END_REF]. In a similar fashion, using spatially distributed rainfall inputs and model parameters was not seen to improve the quality of the simulated hydrographs significantly. The inertia of the linear reservoirs in the Medor model are seen to be sufficient to model the rainfall-discharge transformation in a satisfactory way. This simpler model is thus retained for the analysis.

The model was applied to the 1967-1985 period, for which the discharge record was continuous at the Douna station. Daily rain series were derived from 72 rain gauges covering the area (Figure 4). An average of 70 gauges (with a minimum of 66) were used to interpolate daily rainfall maps by the inverse distance method, which proved to perform best [START_REF] Ruelland | Sensitivity of a lumped and semi-distributed hydrological model to several modes of rainfall interpolation on a large basin in West Africa[END_REF]. Potential evapotranspiration forcing consisted of monthly maps produced by the Climatic Research Unit (University of East Anglia, UK) from ~100 stations spread over West Africa, using Penman's method and spline interpolation [START_REF] New | Representing twentieth century space-time climate variability. Part II: Development of 1901-1990 monthly grids of terrestrial surface climate[END_REF]. Since potential evapotranspiration varies slowly over the year, monthly data were disaggregated evenly to the daily time step within each month. The first five years of simulation were used as model warm-up, to eliminate the influence of initial conditions. This five year period was determined from the order of magnitude of the specific discharge coefficients in the model. Indeed, as shown in Appendix B (section B.3), a model with a specific discharge coefficient k requires a warm-up period of at least a few times 1/k for the influence of initial conditions to be eliminated. The parameters used in the experiment are given in Table 2. The catchment area A is known from previous studies [START_REF] Ruelland | Sensitivity of a lumped and semi-distributed hydrological model to several modes of rainfall interpolation on a large basin in West Africa[END_REF], while the specific discharge coefficient k1 is taken equal to the value that allows the NSE index to be maximized.

Redundancy test results

Figure 5 shows the contour lines obtained for the distance-based and weak form-based objective functions. The powers used in Eqs. ( 43) and ( 45) are p = ½ and p = 2.

Figure 5e is a superimposition of Figures 5a and5c, while Figure 5f results from the superimposition of Figures 5b and5d. In Figures 5e-f, the dashed and solid lines are respectively the contour lines of the objective functions for p = ½ and p = 2. Except in the upper left part of the diagram, the contour lines do not intersect, which confirms that the parameters Hmax and x become redundant over most of the parameter space when k1 = 0 or k2 = 0.

Conclusions

Model performance assessment and objective functions classically used in hydrological modelling may be classified into distance-based and weak form-based objective functions.

Distance-based objective functions have the advantage that the calibration problem is transformed into a straightforward, single-criterion optimization problem. Their drawback is the possible appearance of local extrema in the response surface of the model, thus triggering the failure of classical gradient-based methods and requiring the use of more computationally demanding global optimization algorithms.

Weak form-based objective functions transform the calibration exercise into a root finding problem.

The theoretical considerations in Sections 2-3 and the application examples in Sections 4-5 lead to the following conclusions. (C1) Weak form-based objective functions are more monotone than distance-based objective functions when applied to conceptual hydrological models. Monotony can be proved mathematically for models verifying Assumptions (A1-4) in Subsection 3.1. (C2) The subset of zero values of weak form-based objective functions form hypersurfaces in the parameter space. A model with N parameters can be calibrated by defining N weak form-based objective functions and finding the intersection of the corresponding N hypersurfaces in the parameter space. Since the weak formbased objective functions are monotone, the intersection is unique. This allows classical gradient-based algorithms to be used without the need for more sophisticated optimum search techniques. The N different objective functions may be defined by using (i) different observation variables, (ii) transformations of these variables, (iii) different values for the power p used in the formulation of the weak form-based function (see equation ( 7)). Note that the need for a number of independent criteria matching the number of parameters to be calibrated was already pointed out by [START_REF] Gupta | Reconciling theory with observations: elements of a diagnostic approach to model evaluation[END_REF]. 49)) with two different values of p yields two families of contour surfaces in the parameter space. Non-intersecting families of contour surfaces in the parameter space indicate redundancy between two or more parameters. (C6) The theoretical analysis in Section 3 and the application example in Section 4 show that using the volume stored in the reservoir as a calibration variable for the discharge coefficient may be more appropriate than using the discharge.

Conversely, the outlet discharge may be a more appropriate variable to calibrate an infiltration (or net rainfall) coefficient via a weak form-based objective function. (C7) Assuming that weak form-based objective functions are to be used, the calibration problem is best posed when the hypersurfaces as defined in (C2) are as orthogonal to each other as possible. Therefore, it is advisable to define such objective functions using as many different model state variables as possible (e.g. discharges between various reservoirs in the model, volumes stored in the various reservoirs, etc.). (C8) A necessary condition for (C5-6) to be applicable, however, is that the internal variables and fluxes in the model bear a physical reality and be field-measurable. Due to hydrological/hydraulic variability, translating field measurements (e.g. of soil moisture, or piezometric head in aquifers) into variations of model internal variables is not an easy task. This most probably calls for the definition of a different kind of objective functions. For instance, the trends (rising or falling; increasing or decreasing) of the measured and model internal variables over certain periods may be used in the form of indicators. Note that internal variables may not necessarily be useful only to the calibration of a model. Even if not incorporated into the objective function, they may be used to discriminate between different models (or different parameter sets within the same model) giving similar values of objective functions computed from the outflowing fluxes. (C9) The result of the calibration/validation process may be biased if the model has not been run over a sufficiently long warm-up period for the influence of possibly inaccurate initial conditions to be eliminated (see section B.2 in Appendix B). The minimum length of the warm-up period is a function of the parameters of the model. Consequently, it should not only be defined a priori: an a posteriori check is needed once the model has been calibrated. The theoretical considerations presented in this paper are valid for hydrological models obeying first-order differential equations. Whether such conclusions also hold for models involving delay functions (e.g. the GR3J model [START_REF] Edijatno | GR3J: a daily watershed model with three free parameters[END_REF] that embeds a unit hydrograph transformation) or other functions not verifying exactly the governing assumptions in Section 3 is the subject of ongoing research. Future research should also focus on (i) the robustness of weak form-based objective functions compared to the well-established distance-based approach and (ii) the effects of data uncertainty on the behaviour of the objective function. Although only conceptual models were considered in this study, the approach might be applicable to the calibration of some physically-based hydrological models, e.g. when backwater effects are insignificant, however such extrapolation should be subject to further theoretical investigation. Considering that the sensitivity of U with respect to the initial condition is equal to 1 for t = 0, the solution of equations (B.2a, B.4a, B.4b) is a decreasing exponential:

(B.6)

The sensitivity of U (and therefore of any function F(U)) becomes negligible after a simulation period equal to a few times 1/k. The warm-up period, that is necessary to eliminate the influence of a possible wrongly defined initial condition, should therefore be taken equal to a few times 1/k. For instance, s(3/k) = 4.98 %; s(4/k) = 1.8 % and s(5/k) = 0.7 %. In other words, only 0.7 % of the initial sensitivity to the initial conditions remains after a simulation period 5/k. Efstratiadis A., Kousoyiannis, D. 2010. One decade of multi-objective calibration approaches in hydrological modelling: a review. Hydrological Sciences Journal, 55(1), 58-78.

  is to be calibrated using measurements of the outlet discharge Q = kU, then F = kU, interesting, particular case is that of the Cumulative Error (CE), or Balance Error (BE) indicators (see Appendix A), obtained for (p = 1, w = 1is visible from Eqs. (32, 34) that the derivative of the weak form-based objective function cannot be zero, except in the trivial case U = V " t Î W. If the objective function is based on the outlet discharge Q=kU, the only possibility for its derivative to cancel occurs for k = 0, which is meaningful only in the trivial case kU = V = 0 " t Î W. (R14) Comparing Eqs. (35a) and (35b) indicates that the CE and BE indicators are extremely useful in calibrating the runoff coefficient c but are almost useless in calibrating the discharge coefficient k of the conceptual model. This well-known

  (C3) In contrast, distance-based objective functions yield mutually exclusive optimal parameter sets when different calibration variables are used. Using weak form-based objective functions allows this drawback to be eliminated. This allows for multi-objective calibration without the inconvenience of multiple solutions. (C4) Distance-based objective functions being widely recognized in the field of hydrological modelling, they could also be used in combination with weak formbased objective functions in the framework of multi-criteria optimization algorithms. A typical multicriteria optimization problem may then consist in maximizing the distance-based objective function under the constraint that all the weak form-based objective functions are zero. (C5) Using the same type of objective function (distance-based, see Eqs. (48) or weak form-based, see Eqs. (

(

  is to study the influence of initial conditions, the parameter j is In this case, a and b in equation (B.2a) are given by (B.4a) (B.5a)

Acknowledgements

This work was supported by an Internal Project financed by the Laboratory HydroSciences Montpellier (HSM, UMR5569, CNRS, IRD, UM1, UM2).

Appendix A. Classical Objective Functions

The functions are classified in Table A.1 into to Distance-based (D) and Weak formbased (W) objective functions. a, b and p in Table A.1 are respectively the offset constant, the scaling constant and the power used in equations ( 4) and ( 7).

(Table A.1 here)

Appendix B. Proofs

B.1 Sign of the solution U of Eq. ( 8) Assume that Eq. ( 8) holds, with assumptions (A1-4) verified. From Assumption (A3), at any time t there exists U0(t) > 0 such that dU/dt = 0 for U = U0. U0 verifies Eq. ( 11), that is:

If U(t) is smaller than U0, then dU/dt = R -g is positive because of Assumption (A2), and U can only increase at time t. Hence, U(t) is either greater than U0 or increasing, and thus can never be negative for t > t1 since U(t1)³0 (Assumption (A4)).

B.2 Sign of the sensitivity s in Eqs. (13)

The sensitivity equations ( 13) are rewritten as

From assumption (A2), a is negative and b has a constant sign. The case b = 0 leads to the trivial solution s = 0 and is not considered hereafter.

Consider first the case b > 0. In this case, there exists an equilibrium value for the sensitivity, with ds/dt = 0 for s = s0 in Eq. (B.2a). Since a is negative and b is positive, s0 is always positive. If s < s0, ds/dt = as + b > 0. Hence, s is either larger than the positive value s0 or increasing. Since s(t1) = 0 (Eq. ( 13b)), ds/dt > 0 at t = t1 and s(t) > 0 for all t > t1.

Reasoning by symmetry leads to the conclusion that s(t) < 0 for all t > t1 when b < 0.

B.3 Sensitivity to initial conditions and model warm-up period

The purpose is to study the sensitivity of model output to the initial conditions. Consider the linear model (B.3a)