Identification of the elementary processes of the hydrological cycle in a drainage basin, and the comprehensive description of each of them, lead to hydrological models with a complex structure including a high number of relatively inaccessible parameters. Moreover these processes, even when simplified, are generally non-linear. Using models with a smaller number of parameters, in order to cope with non-linearity, is therefore necessary.
In this perspective, we propose an artificial neural network for rainfall-runoff modeling. Performances of this method in non-linear modeling have been already demonstrated in several scientific fields (biology, geology, chemistry, physics). In the present work, we use the error back-propagation algorithm with a three-layer neural network. The transfer functions belong to the sigmoidal type at each layer. To predict the runoff at a given moment, the input variables are the rainfall and the runoff values observed for the previous time period. The structure of the network (number of hidden nodes, learning coefficient and momentum values) is optimized to guarantee a good prediction of the runoff, using a set of test data (validation set) not used in the training phase.
Data compiled in our model are a ten year set of rainfall-runoff values collected by the Rabat hydraulic administration (September 1983 to April 1993) in the Beth Wadi catchment. In this study, we develop two types of models according to two different time steps (daily and weekly). The data are subdivided into two sets: a first set to train the model (training set) and a second set to test the model (validation set). For the daily timestep model, we used data of the last two years: April 1991 to April 1993. The initial 365 data (April 1991- April 1992) constitute the training set and the 365 remaining data constitute the validation set. For the weekly data (Monday to Sunday averages), we have 502 pairs of values. We worked by preserving the last 120 values as the validation set and trained the neural network with the remaining data, i.e. 382 pairs of values of weekly rainfall-runoff.
Three types of estimation have been carried out:
The step time is daily for the at instant prediction and weekly for one step ahead and multistep predictions. The choice of input variables is determined by autocorrelation function (ACF) and partial autocorrelation function (PACF) analyses on runoff values, and cross-correlation function (CCF) analysis between rainfall and runoff values. For the at instant prediction, the input vector is composed by runoff values of the four days preceding day t, and rainfall values for the three last preceding days as well as its value on day t. For the one step ahead prediction, the input vector is composed of runoff values of the five weeks preceding week t, and rainfall values for the three preceding weeks (without considering the rainfall at time t). Finally, for the multistep prediction, the input vector is the same as for the one step ahead prediction but rainfall values include time t. The runoff values for the week t-jh+1, as well as for the following weeks, are computed by feed backing to the input vector the runoff value predicted for the preceding week.
The rainfall-runoff models allow a good estimation for one or several timesteps, daily as well as weekly. In the validation set, correlation coefficients between observed and estimated values are high. In the at instant prediction, we obtain the Pearson correlation coefficient R=0.772 and the Spearman correlation coefficient CR=0.958. The weak value of R as compared to CR is explained by a few extremely high values of error of prediction. In the one step ahead prediction (R=0.887 and CR=0.782) and multistep prediction (R=0.908 and CR=0.727), the R coefficients are higher that CR. This confirms that predicted values are in good agreement with the peaks of observed values (absence of large exceptional errors). In all cases, the results obtained are better than those obtained with linear methods. The neural network models can thus be recommended for time series studies in environmental sciences.
Modeling, neural networks, retropropagation, rainfall-runoff, time series.
S Lek, Labo Biologie Quantitative, UPS 4R3, 118 route de Narbonne, 31062 Toulouse Cedex, FRANCE