Télédétection du manteau neigeux et modélisation de la contribution des eaux de fonte des neiges aux débits des oueds du haut atlas de Marrakech

VI.5.3 Analysis of model parameters interdependence

The traditional concept of model calibration is built on the hypothesis that a unique optimum set of parameter values exists; however there is a multitude of parameter combinations that are «equally good» for a particular objective. In most rainfall-runoff modelling exercises, calibration is performed on the only observation that is available, i.e. streamflow at the outlet. Streamflow integrates the various interactions between the different intermediate storage, loss and redistribution processes. The relative intensity of each of these processes is usually poorly known, and a satisfying match between the observed and the simulated streamflow can be achieved for a large range of parameters values. Beven & Binley (1992), Beven (1993) and Beven (2008) call this the «equifinality issue». This means that, for instance, a low value for one parameter that positively affects one redistribution process (e.g. surface runoff) combined with a high value of a second parameter governing a second redistribution process (e.g. subsurface flow) will give a similar value for the performance criteria as a higher value of the first parameter combined with a lower value of the second one. The resulting parameters values deemed as «acceptable» can span a very large range of values in the solution ensemble, even within realistic boundaries. A method to check whether such equifinality problem is to produce the contour plot of the Nash performance criteria for all pairs of parameters. If two parameters can be calibrated independently, the resulting contour plot peaks around one solution couple. This is the case for the C_r and C_s parameters of SRM in the MOD2 configuration ( Figure ýVI , top) and, in a lesser extent, in MOD1, even though both parameters contribute to surface runoff. The same conclusion can be drawn for the C_r and degree-days factor parameters ( Figure ýVI , top centre). If the contour lines are either horizontal or vertical, one of the two parameters is rather insensitive to streamflow prediction. This is the case for the critical temperature that initiates melting ( Figure ýVI , bottom centre). If the contour lines are organized diagonally, parameter dependence is high and the two parameters cannot be calibrated independently. This is partly the case for the x and y parameters that govern the recession ( Figure ýVI , Bottom).

Because the SRM model is rather simple and parsimonious, equifinality is limited to the above-discussed examples. After the analysis of the Figure ýVI , we conclude that only T_melt could be fixed at an arbitrary value, and that a narrow optimum range can be found for most parameters. Because the Nash criteria is reaching a plateau when the less sensitive parameters are successively calibrated, we decided to retain as «acceptable» all the parameter sets that produce a Nash efficiency above 90% of the overall maximum. These parameter sets will be used to generate an ensemble of streamflow time series in the validation period for each model configuration (MOD1 or MOD2). The ranges of model parameters of these subsets are shown in Table ýVI . Because (1) streamflow is not that sensitive to the parameter values of the snowmelt processes and (2) we expect that the climate is less variable from one sub-catchment to the next than the lateral redistribution of rain and meltwater, the model parameters related to temperature (a, T_f and T_c) are considered to be the same for the five sub catchments, while the parameters generating runoff or recession are catchment-dependent.

As expected, the calibrated values for the parameters that govern runoff or the redistribution of water are more variable from one catchment to the other than from one configuration to the other. Performance of the model during the calibration period (from January 1st to Mai 31, 2005) indicate good efficiencies for Nfis, Rheraya and Ourika sub catchments where E>0.79 for MOD1 and E>0.70 for MOD2 ( Table ýVI ) and satisfactory for Zat and R'dat sub-catchments where E>0.63 for MOD1 and E>0.68 for MOD2.

Figure ýVI-: Variations of Nash efficiencies depending on the model parameters (2D analysis). The model is driven by snow maps derived from SPOT- VEGETATION (MOD1, left) and simulated using the degree day method (MOD2, right).

Table ýVI-: Range of optimal model parameters giving 90% of the overall maximum of NASH.

Table ýVI-: Statistics associated to streamflows simulations.