Artificial neural networks are statistical black-box models that use input–output measurements to identify non-linear functions of a system. Basics about neural modelling can be found in Dreyfus (2005), only specific information, mandatory for a comprehensive presentation of this study, will be provided hereafter. The chosen model is the multilayer perceptron because of its properties of universal approximation and parsimony (Barron, 1993). The universal approximation is the capability to approximate any differentiable and continuous function with an arbitrary degree of accuracy (Hornik et al., 1989). In our study, the multilayer perceptron is a feed-forward model, a finite impulse response model based on Nerrand et al. (1993). Designing a multilayer perceptron consists mainly of selecting input variables and the number of hidden neurons. This determines the number of parameters mechanically; model complexity increases with the number of parameters. The general equation of the function calculated by the feed-forward multilayer perceptron is the following: yk=gNNypk-1,…,ypk-wy,uk,…,uk-wu-1,C, where the estimated value of the output at the discrete time k is yk; the observed value of this variable is ypk; the input vector is uk; the non-linear function implemented by the neural network is gNN; wu and wy are the width of windows used to apply the input time series, they are linked to the length of the vectors u and yp; and C is the matrix of parameters of the model, also called “weights”.

As statistical models, neural networks are designed in relation to a database. This database is usually divided into three sets: a training set, a stop set, and a test set. The training set is used to calculate parameters through a training procedure that minimises the mean quadratic error calculated on output neurons. The training is stopped by the stop set (cf. Sect. 2.1.2), and model quality is estimated by the third part of the database: the test set, which is separate from the training and stopping sets. The model's ability to be efficient on the test set is called generalisation. However, the training error is not an efficient estimator of the generalisation error: the efficiency of the training algorithm makes the model specific to the training set. This specialisation of the neural network on the training set is called overfitting. Overfitting is exacerbated by large errors and uncertainties in field measurements; the model learns the specific realisation of noise in the training set. This major issue of neural network modelling is called bias–variance trade-off (Geman et al., 1992). Usually regularisation methods are used to avoid overfitting; to this end, two regularisation methods were used in this study.

In the context of this study, the goal of regularisation methods is to minimise output variance. To this end, cross-validation (Stone, 1974) was used as explained in Kong-A-Siou et al. (2012) to empirically select input variables and the number of hidden neurons. Cross-validation thus minimises model complexity and therefore output variance (Schoups et al., 2008).

Another regularisation method is commonly employed: early stopping (Sjöberg et al., 1995). This method stops training before overtraining occurs. A dedicated set, called a stop set, is considered separately from the database.

Working also on the Lez aquifer but considering only underground water at the Lez spring, Kong-A-Siou et al. (2011) applied multilayer perceptron to perform forecast at Lez spring and validated cross-validation as a useful method to select the complexity of the model. Moreover, Kong-A-Siou et al. (2012), for the same basin, focussed on regularisation methods (early stopping and weight decay). They conclude that early stopping used in conjunction with cross-validation was efficient.

Nevertheless these results, obtained with a 16-year daily database cannot be applied directly in the present study because the flash-flood database is too limited to extract definitively another set from the database (the stop set). Thus, to apply early stopping without stop set, a pre-defined maximum number of training iterations were selected to stop training before the complete convergence and, by this way, avoid overtraining. Nevertheless, for this purpose, the selection of the optimal number of training iterations is done using a stop set. Then afterwards, the model is run without the stop set using this pre-defined optimal number of training iterations. In the first stage, the database, not including the test set, was divided into S subsets corresponding to flash-flood events. Training was performed on S-1 subsets with 50 different parameter initialisations. The remaining subset was used as a stop set. Each subset was used in turn as a stop set. For each trial the training iteration with the minimum mean quadratic error over the stop set is set aside. The median of these numbers of iterations was calculated for all stop sets and all initialisations and selected as the optimal number of training iterations. In a second stage, this optimal number of training iteration (12 iterations) is used in all the following without further utilisation of a stop set.

In this study, parameters are iteratively calculated using the Levenberg–Marquardt algorithm (Hagan and Menhaj, 1994).

It is also well-known that model performance depends strongly on the parameters initialisation. To define a reliable simulation independent from the initialisation, Darras et al. (2014b) proposed to establish an ensemble of 50 models trained from different initialisations. The output is calculated at each time step by the median of the 50 outputs. It is well-known that this method can smooth the output of the model; nevertheless this is not a drawback in this study as this method improves the robustness of the model, which is very important to extract information.

The Nash–Sutcliffe efficiency is the most widely used criterion for evaluating hydrological models. It is equivalent to the R2 determination coefficient: R2=1-∑k=1nypk-yk2∑k=1nypk-ypk‾2, where k is discrete time, n the number of time steps used to calculate R2, yk the simulated discharge, ypk the measured discharge, and ypk‾ is the measured mean discharge. The Nash score is not really convenient for assessing flood simulations as it takes into account errors on the whole event and not specifically on the peak. For this reason, other criteria were proposed.

Synchronous percentage of peak discharge is especially designed for the evaluation of flash-flood modelling. It is the ratio of measured and simulated discharges at the time of the observed peak discharge: SPPD=100ykpmax⁡ypkpmax⁡, where kpmax⁡ is the time of the measured peak discharge.

The delay between simulated and measured peak discharge is calculated using Eq. (4). A positive delay means a retarded simulated peak discharge. Conversely, a negative lag means advanced simulated peak discharge. The peak delay can be expressed as Pd=kmax⁡-kpmax⁡, where kmax⁡ is the time of the simulated peak discharge.

Hourly rainfall data are available at five rain gauges: Saint-Martin-de-Londres, Prades-le-Lez, Sommières, Vic-le-Fesq and Saint-Hippolyte-du-Fort. The French Weather Forecasting Service (Météo France) manages the first two gauges, and the Flood Forecasting Service of the Grand Delta (SPCGD) manages the last three gauges. Only the Prades-le-Lez rain gauge is inside the Lez system, but as pointed out in introduction, it is essential to make use of spatialised rainfall information. In addition, no data at the considered time step is available further south than the Prades-le-Lez rain gauge. Spatial rainfall variability is thus not correctly described in the southern part of the basin. This will limit the reliability of this study regarding the southern zone of the basin. Unfortunately, it is not convenient to use weather radar information in this basin because, due to the distance of the Nîmes radar (50 km), this information is not robust from one event to another and generally underestimate the rainfall value compared to the rain gauge measurements (Marchandise, 2007; Visserot, 2012); also radar information is not available for all events in the database. Discharge data are provided by the Lavalette gauging station managed by an office of the French ministry of ecology and sustainable development (DIREN). Both rainfall and discharge data are available at an hourly time step, which is convenient for flash-flood modelling.

The data suffer from high noise and uncertainty. The uncertainties of discharge measurements have been estimated at around ±20 % for flash floods. The uncertainty of rainfall measurements, can be as high as ±10 to 20 % (Marchandise, 2007). Rainfall and discharge time series are available from 2002 to 2008. Fifteen flood events whose peak discharges exceed 80 m3s-1 were selected (Table 2). Events 7 and 8 were the most intense; contrary to other intense events, events 13 and 8 occurred on dry soils.

The postulated model describing flash-flood genesis at Lavalette station is based on the work of Kong-A-Siou et al. (2013) as the considered basin is the same (surface and underground). Remember that the primary difference is that flash floods are considered at hourly time steps at the Lavalette station in this study. Using continuous data at daily time steps at the Lez spring, Kong-A-Siou et al. (2013) showed that the north-eastern and north-western zones are the principal contributors to Lez spring discharge. To estimate the contributions of each zone to flooding at Lavalette, we distinguished both behaviours: surface (rapid if inside the impervious watershed) and underground (slower if infiltrated into karst outcrops or in faults: faults play the role of a drain in impervious parts of the basin inside and outside of the Lavalette surface watershed). Schematically, by looking at the map presented in Fig. 2 and following the previous reasoning, one can propose that the north-western zone would make a minor contribution to flash flooding at Lavalette because it is outside the surface (topographic) basin and because its underground time response is high (Table 5). The south-eastern zone would also have a minor impact because its impervious area is mostly outside the Lavalette watershed. Regarding the south-western and north-eastern zones, it is difficult to propose an a priori quantification. It is thus not easy to estimate the principal contributors to flash flooding. Application of the KnoX method would provide this quantification. The postulated model of the basin behaviour is thus composed of four branches, each corresponding to a zone of the basin, involving surface and groundwater, and feeding a complex mixing process. The postulated model is represented in Fig. 3 in a grey block diagram.

The model used to apply the KnoX method is based on the multilayer perceptron; it follows the postulated model represented in Fig. 3 with four zones contributing to discharge at Lavalette station. As suggested by the KnoX method, to be able to identify the contribution of each zone to the discharge, a linear hidden neuron is added between the inputs and the layer of sigmoid neurons. These neurons are intended to represent rain that falls on each zone; they facilitate the estimate of the time response of water falling in each zone.

Inputs are mean rainfalls for each zone. These rainfalls are calculated using the Thiessen polygon method. Table 2 shows the weight of each rain gauge for each zone. It highlights the sparse spatial distribution of rainfall information in the south of the basin. Nevertheless, taking into account the importance of the stakes in this zone, and as the goal of this study is to better understand the behaviour of the basin in order to develop well-suited monitoring strategy, we consider the rainfall information sufficient to carry out this study.

As presented in Sect. 2.1.2, model selection is done using cross-validation and pre-definite number of training iterations. Ranges of investigation and chosen values of various window width and hidden neurons numbers are provided in Table 3. One can note that the complexity of the model is moderate (small number of hidden neurons). To make the model assessment more reliable on the most intense events 7 and 8, model selection was done without these events (blind assessment).

The database presented in Table 4 shows seven flash-flood events. Because of the small number of events and their heterogeneity it seemed necessary to estimate modelling quality on all events. We thus decided to train seven models, testing each on one event (training performed on the six following events). The model tested on event n is noted as Tn. This is a cross-test operation. Table 4 shows the performance of the seven models in terms of R2, synchronous percentage of peak discharge (SPPD), and peak delay (Pd). After training, we compared the quality of the models: aside from model T2, R2 and SPPD scores of model T13 are the worst, respectively 0.71 and 138 %. The other models show satisfactory R2 and SPPD scores: R2 from 0.79 to 0.96 and SPPD from 87 to 99 %. Regarding the Pd, only model T2 performed badly. The models T4, T7, T8, and T14 are efficient regarding the three performance criteria. Model T13 overestimates the flood peak; note that event 13 is the sole event that occurred on dry soils, except event 8 when extremely intense rainfall was observed.

Looking at hydrographs presented in Fig. 4 for the two most intense events and taking into account the scores presented in Table 4, one can suggest that the models are efficient enough to be used for knowledge extraction. In addition, as it will be shown in Sect. 4.3.1, knowledge extraction is independent of outliers as it takes into account all events of the training database.

After training, the median of absolute values of the parameters for 50 different initialisations is calculated. It is noted as MCij for the parameter Cij linking the neuron (or input) j to the neuron i. The rainfall contribution of zone z to output at time step k-d (k is the discrete time and d a delay) is denoted as rz(k-d). It is calculated according to the chain of parameters linking one input: rz(k-d), to the output y(k). As it is shown in Fig. 3, we have three layers of parameters between the input rz(k-d) and the output y(k); therefore, there are three terms in the numerator; denominator corresponds to normalisation terms in order to estimate the specific contribution of the input rz(k-d) relative to the sum of all other parameters of the same layer. There is also three normalisation terms because there are three layers of parameters. The following notations are reported in red in Fig. 3. The contribution is calculated as Przk-d=MCHzrd∑d=0wzMCHzrd∑HN=1NcMCHNHz∑Hz=1lMCHNHz+∑d=1w1-1MCHNqdMCoHN∑HN=1nMCoHN, where Hz(Hz=1,4) is the subscript of the first hidden layer of linear neurons, HN(HN=1, Nc) is the subscript of the second hidden layer (of Nc non-linear neurons); qd is the subscript of the previously measured discharge inputs yq, and o is the subscript of the output layer.

The contribution of an entire zone can be expressed as the sum of the contributions of the considered zone at different time steps: Pz=∑d=0wzPrzk-d. This contribution calculus is done for each exogenous input: rainfall or measured discharge, and for each designed model (Tn, n=1, 7). The contributions of the previous measured discharges used as input to the model ranges from 21 to 30 % (79–70 % for total rainfall) depending on the considered model Tn (n=1, 7). Nevertheless, only rainfall contribution values are considered (for a total of 100 %) because the measured input of discharge plays the role of state variable (Artigue et al., 2012). Rainfall contribution medians for the seven models are provided in Table 5. Values obtained by Kong-A-Siou et al. (2013) are also reported; they show the difference between contributions of the same zones to very different processes (flash flood at Lavalette station for this study, and daily aquifer discharge at the Lez spring in the 2013 study).