# Deep Learning at Two Timescales: Dual Neural Networks for Predicting Fast Urban and Slow Karst Floods

Peniel Adounkpe, Valentin Wendling, Alain Dezetter, Bruno Arfib, Kalil Traoré, Guillaume Artigue, Octavian Dobricean, Séverin Pistre, Anne Johannet

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To cite this version:

Peniel Adounkpe, Valentin Wendling, Alain Dezetter, Bruno Arfib, Kalil Traoré, et al. Deep Learning at Two Timescales: Dual Neural Networks for Predicting Fast Urban and Slow Karst Floods. Inżynieria Mineralna, 2025, 3 (2), 10.29227/IM-2025-02-03-27. hal-05360721

HAL Id: hal-05360721

https://imt-mines-ales.hal.science/hal-05360721v1

Submitted on 12 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Julien Yise Peniel Adounkpe<fnref n="1" />,<fnref n="2" />*), Valentin Wendling<fnref n="3" />), Alain Dezetter<fnref n="4" />), Bruno Arfib<fnref n="5" />), Kalil Traoré<fnref n="6" />), Guillaume Artigue<fnref n="7" />), Octavian Dobricean<fnref n="8" />), Séverin Pistre<fnref n="9" />), Anne Johannet<fnref n="10" />)

https://doi.org/10.29227/IM-2025-02-03-27

Submission date: 2025-06-06 | Review date: 2025-07-12

Abstract

Flash floods in urban and karst environments present major modeling challenges due to their complex hydrodynamics, characterized by a rapid urban runoff response and a delayed slower karst groundwater response. This study explores the use of artificial neural networks ANN (multilayer perceptron in particular) to predict flash floods at the downstream of the Las River in Toulon (France). The Las River is fed in a larger proportion by the nearby karst springs and in a smaller proportion by the urban drainage network. In this study, we propose an ensemble modeling strategy to address the system's double hydrological regime. The initial step was to identify rainfall events in the six-year hydrometeorological database and classify them according to their karst contributions. Two specialized models: urban runoff model (UM) and karst model (KM) were trained solely on each event type (urban runoff and karst events). These models were combined by two methods in an attempt model all events, disregarding of their event type: the first approach was to combine the outputs of the specialized models in an ANN called output combination model (OM), the second approach was to combine the structures of the specialized models and retraining the model parameters called structure combination model (SM). A third more "basic" approach, called bulk model (BM), was to optimize the ANN by selecting the inputs with the best performance improvements. As expected, the specialized models (UM and KM) performed the best on the cross-validation sets and on the test sets on their respective event types but failed to generalize across regimes. The OM was the most robust ensemble strategy across all event types with consistent accuracy on predicting both urban runoff and karst flood events. The BM was better on the karst events while having worst performance on karst events and the SM was the least accurate model. These findings confirm the added value of combining specialized ANNs to model complex hydrological systems. In addition, selecting the right inputs to the models has a bigger impact on the model's performance than choosing its structure by changing its hyperparameters.

**Keywords:** Karst flood, artificial neural networks, multilayer perceptron, dual hydrodynamic, urban runoff, ensemble learning.

1. Introduction

Flash floods in the Mediterranean region are a major source of natural disasters causing massive societal damages [1]. They are characterized by short duration and high intensity precipitations. Forecasting floods is one of the most cost-effective methods to reduce economic and life losses caused by flood disasters [2]. Hydrological models are used to perform this forecast by representing the water-related processes of the system. For example, forecasting models specialized in predicting a river's water level river can be defined as a set of equations describing the systems' characteristics to estimate the river's future water level [3].

Karst systems are natural environments where dissolving carbonate rocks create complex underground drainage networks and highly permeable flow paths. Karst flash floods is a term introduced by Bonacci referring to flash floods in karst terrains [4]. They are responsible for a sudden rise in groundwater levels causing multiple, unexpected, and intense temporary springs to flow. A good example of karst flash floods in the Mediterranean region is Nîmes, which experienced a substantial karst contribution during the intense flood event of September 2005, highlighting the importance of considering groundwater inputs in modeling this type of event [5]. The latter reasons illustrate the higher levels of complexities flash floods in karst environments.

Hydrological models can be classified into three processes given their underlying principles: empirical models, conceptual models and physically based models [6]. Empirical models are data driven models that disregard features and processes of the

<footnote n="1">¹*)HSM (Univ. Montpellier, IMT Mines Ales, IRD, CNRS), 6 av. de Clavières, 30100 Ales, France, E-mail: julien-yise-peniel.adounkpe@mines-ales.fr, 0000-0001-9326-0669</footnote>

<footnote n="2">²*)HydroSciences Montpellier, Univ. Montpellier, IRD, CNRS, UFR Pharmacie, Bâtiment HYDROPOLIS, 15 Avenue Charles Flahaut, 34090 Montpellier, France, E-mail: julien-yise-peniel.adounkpe@mines-ales.fr, 0000-0001-9326-0669</footnote>

<footnote n="3">³)HSM (Univ. Montpellier, IMT Mines Ales, IRD, CNRS), 6 av. de Clavières, 30100 Ales, France, 0000-0003-4163-9700</footnote>

<footnote n="4">⁴)HydroSciences Montpellier, Univ. Montpellier, IRD, CNRS, UFR Pharmacie, Bâtiment HYDROPOLIS, 15 Avenue Charles Flahaut, 34090 Montpellier, France, alain.dezetter@umontpellier.fr, 0000-0003-3728-0467</footnote>

<footnote n="5">⁵)Aix Marseille Univ., CNRS, IRD, INRAE, CEREGE, Aix-en-Provence, France, arfib@cerege.fr, 0000-0001-9345-6813</footnote>

<footnote n="6">⁶Synapse Informatique, 140, rue Clément François Prunelle, 34790 Grabels, France, kalil.traore@synapse-info.com, 0000-0002-6230-2026</footnote>

<footnote n="7">⁷HSM (Univ. Montpellier, IMT Mines Ales, IRD, CNRS), 6 av. de Clavières, 30100 Ales, France, guillaume.artigue@mines-ales.fr, 0000-0002-4610-584X</footnote>

<footnote n="8">⁸Synapse Informatique, 140, rue Clément François Prunelle, 34790 Grabels, France. octavian.dobricean@synapse-info.com</footnote>

<footnote n="9">⁹HydroSciences Montpellier, Univ. Montpellier, IRD, CNRS, UFR Pharmacie, Bâtiment HYDROPOLIS, 15 Avenue Charles Flahaut, 34090 Montpellier, France, severin.pistre@umontpellier.fr, 0000-0002-3666-4613</footnote>

<footnote n="10">¹⁰HSM (Univ. Montpellier, IMT Mines Ales, IRD, CNRS), 6 av. de Clavières, 30100 Ales, France, E-mail: anne.johannet@mines-ales.fr, 0000-0001-5717-9871</footnote>
hydrological system by deriving input-output relationships from mathematical equations not originating from the physical system. Conceptual models describe the main hydrological components of the physical system by interconnecting reservoirs with specific characteristics and by semi empirical equations. Physically based models are process-based models that ideally represent the physical system's hydrodynamics as a function of both time and space [7].

Modeling karst systems is a challenging task because of the difficulty of solving complex hydrodynamics behavior and the lack of information regarding the system's properties [8,9]. The parameters of conceptual and physically based models cannot be measured directly and therefore must be determined by calibration and their prediction uncertainty increases with increasing model complexity. An obvious example are distributed karst models that generally have high data requirements (such as spatial data of study area's hydraulic properties of the matrix and the karst conduits, aquifer and conduit geometry, ...) only available on well monitored test sites [9].

Artificial Neural Networks (ANN), particularly multilayer perceptrons (MLP), are statistical models in the empirical category of models. Considering that the MLP complies with the universal approximation and parsimony properties, it is able to identify non-linear and dynamic relationships linking input and output variables. Indeed, Funahashi proved that a MLP can approximate any continuous mapping with at least one hidden layer whose output functions are sigmoid functions [10] and Barron established that the error in the approximation of functions by ANN is bounded by the number of input variables and neurons in its hidden layer [11]. MLP are particularly useful in hydrology because they can establish the rainfall-runoff relationship without assuming any prior behavior of the karst systems while remaining shallow in its architecture [12,13]. They are used as the simpler and effective way to model flash floods in karst systems [14,15]. Moreover, it has been proven that informing the MLP with expert knowledge by adjusting its architecture accordingly improves the model's flash flood predictions [16]. Ensemble learning in machine learning refers to a meta-learning method in which multiple learners are trained and combined to solve real-world problems [17]. The fundamental principle is that weak learners can be grouped to form a strong learner to enhance the overall predictive performance, as illustrated by Darras [18].

The study area of this work is the Las River: a river located at Toulon in the south-east of France. This river, as many others around the Mediterranean Sea, is fed by karst springs and urban runoff, and its behavior is thus controlled by two very different dynamics. This complexity makes it interesting to develop new design methods, investigating for example the potentiality of ensemble learning of ANNs for karst flash flood prediction. To build a neural network able to represent the complexity arising from this multi-dynamic behavior, we propose to build two MLPs whose structures respectively correspond to the double influence: the first one built and trained on fast urban runoff events, and a second built and trained on the events with slow karst contribution. Three modeling approaches were proposed to train on both event types. The first approach just combined the model outputs, the second approach fixed the model's structure and retrained it. The last approach was the most "basic" one which consists of using only one model, the input selection method maximizing the model's performance. At each step of the model's design, its performance will be evaluated using persistency (measuring how the model outperforms the naive forecast) and peak criteria with a lead time of 30 minutes (two time-steps). A comparative analysis will determine the most effective approach for forecasting extreme events of the Las River in test situation.

## 2. Materials and methods

### 2.1 Study area

The Las River is the main urban river that flows through Toulon, a city with 180,000 inhabitants located in the south-east of France (Fig. 1a). It is an 8 km long river that originates from Dardennes springs (small permanents springs and the intermittent Ragas Spring), joined by water from Saint-Antoine et Baume-de-Dardenne springs, and outlets into the Mediterranean Sea at Toulon's harbor. Dardennes springs are fed by a 70 km² impluvium on Siou Blanc karst limestone plateau around 600 meters above sea level and covered with forest and scrubland. The east limits of the recharge area are uncertain [19] (Fig. 1b). Nearly all the net precipitation on the plateau infiltrates through different karst features such as sinkholes, dolines, poljes and karrens [19]. Dardennes Springs are the main contributors to the Las River flows and a water level station was installed in Ragas's vertical opening to monitor the state of the karst aquifer (Fig. 1c). During heavy rainfall over the recharge area, the vauclusian Ragas Spring outflows the open cavity at 149 m above sea level (a.s.l.) [16,17]. The Dardennes springs waters are collected within a dam built 1 km downstream the Ragas Spring to serve as a water supply in periods of low flow. The water level in the reservoir varies from 123 m a.s.l. during winter to around 115 m a.s.l. during summer when the water demand is higher [20]. At 123 m a.s.l., a spillway evacuates the reservoir water to the Las River. The dam's spillway valves are sometimes used to manually release some stored water before intense rainfall. The second contributor to the Las comes from the urban drainage network collecting runoff from an about 12 km² urban area with high impervious surfaces and slopes. Reservoir overflow, spring and surface runoff converge to the valley, until the flow is largely confined to the downstream urban drainage network.

### 2.2 Dataset: acquisition, pre-processing, and description

Rainfall and water level data were collected respectively at seven and three sites inside or in the vicinity of the study area (Fig. 1b). The water level measurements of the springs of Ragas and Saint-Antoine and of the Las River at Lagoubran, near the Las outlet, were provided by [21,22]. The water level data ranges from October 2012 to September 2018 with a timestep of 15 minutes. The rainfall data for the same period is downloaded from the French meteorological service website (https://meteo.data.gouv.fr/). The rainfall data, originally with a 6-minute timestep, is resampled to a 15-minute timestep to match that of the water level data while preserving the quantity recorded over time. This conversion was done by resampling the rainfall data at a 1-minute interval, then aggregating these values over 15 minutes, while filling in missing data, if present, in the 15-minute time interval.

To model flood events of the Las River, periods of heavy rainfall were selected and defined by a cumulative rainfall over 24 hours exceeding 50 mm for at least one of the seven stations. The start of the event is defined as the last time the sum of the cumulative rainfall over 12 hours for all stations is lower than 0.1 mm. The end of the rainfall event is defined as the first-time sum of the cumulative rainfall over 24 h for all stations is lower than 1 mm. The end of the event was then defined based on the water level time-series at Lagoubran: it is defined as the end of the recession curve of the flood following the rainfall to ensure that the entirety of each event was considered. A total of 62 heavy rainfall events were identified over the 6 years of data. Fig. 2 shows an example of event 33 where there is a clear distinction between the start and end dates of the events and the dates at which the stations reach the cumulative threshold.
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Localization and (b) map of the study area; (c) schematic cross-section of the Las River [20]. The recharging area delineated to the east is partly uncertain.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Rainfall times-series measured at all the stations during event 33 starting on the 25th of November 2014 and ending on the 3rd of December 2014. (Left) Rainfall accumulation over 15 minutes. (Right) Rainfall accumulation over 24 hours. Date Start Cumul and Date End Cumul are respectively the first and the last time step the cumulative rainfall over 24h exceeds a 50 mm threshold. Date Start Rain and Date End Rain are respectively the starting and ending time of the event.</figcaption></figure>

Rainfall events with missing data at Lagoubran were excluded from the dataset, reducing the number of events to 47. The rainfall stations had more than 90% of the available data. There was less than 10% of missing data for each rainfall station. Missing data was filled with available series from nearby correlated stations. Indeed, the cross-correlation function between the rainfall series of each station and each event was used to obtain the correlogram between the rainfall for each pair of stations. This information is thus specialized on high water configurations. The correlation at the maximum of the correlogram was averaged over all events to obtain the correlation matrix between the rainfall series (Fig. 3). The series most correlated with each other were used to fill in their missing values. The stations linked for the interpolation of the rainfall series were as follows: Toulon rainfall was completed using mean values from Castellet SAPC, Cap Cepet, and Castellet; Cap Cepet using Castellet and Toulon; Castellet SAPC using Castellet and Toulon; Castellet using Castellet SAPC, Cap Cepet, Sainte Baume, and Toulon; Sainte Baume using Castellet; Meounes using Cuers; and Cuers using Meounes.

The calculation of the correlograms between the rainfall series, and the water level series at Lagoubran, for each event indicated that the response time and memory effect varied greatly from one event to another [23]. The events with the longer response time and memory effect were strongly influenced by the karst networks, as is presents the higher water levels at the two springs. This led us to separate events according to the importance of the karst water contributions. Four types of events were distinguished: urban runoff events, karst events, intermediate events, and low hydrological response events. Urban runoff events are characterized by fast response times and low water levels at the karst springs. They are caused by high rainfall intensity at stations located in the south and southwest of the study area (Toulon, Cap Cepet, Castellet SAPC and Castellet rainfall stations) and due to urban drainage system. For the urban runoff events with no data at Saint-Antoine Spring, this water level time series was set to zeros. Karst events are characterized by slow response times and high water-level at the karst springs. They are triggered by water transfer from the karst system and accumulated rain on the north of the basin. Six events present intermediate behavior and five present low hydrological response; they have been removed from the dataset. Six karst events were also removed because three of them were influenced by the dam opening and three of them had no data available at Saint-Antoine station. Finally, 12 urban runoff events and 17 karst events form the dataset considered in this study. Fig. 4 presents a summary of the main characteristics of these events. The response time of urban runoff events is between 1 and 2 hours over the average of all rainfall stations (0.5 for the fastest), while that of the karst events are between 10 and 39 hours. This illustrates the double hydrological behavior of Las River depending on the urban and karst contributions. Due to their longer response time, the dataset is dominated by karst events representing 89% of the samples compared to 11% for urban runoff events samples.
<figure><img src="image_5.png" /><figcaption>Fig. 3. (Left) Median for all the events of the maximum cross-correlation for each pair of rainfall series. (Right) Spatial distribution of rainfall stations linked together for completion of missing data. The color of the link represents the value presented on the left.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Event characteristics with urban runoff events on the left and karst events on the right. (a) TIME in hour (DURATION: event total duration, MET: average of memory effect time, and RT: average of response time); (b) MAX WATER LEVEL in meters (LAGOU: Lagoubran, ST ANT: Saint-Antoine, and RAGAS: Ragas); (c) AVERAGE RAINFALL: average over all rain gauge stations (MAX INT: maximum rainfall intensity in mm/15min, MEAN INT: average rainfall intensity in mm/duration, and CUMUL: total rainfall in mm). White cells indicate in: (a) rain gauge-water level correlation below 0.2 for at least two stations; (b) missing data.</figcaption></figure>

## 2.3 Modeling with Neural Networks

Since the study area has two distinct hydrological behaviors, two independent modeling experiments were conducted for each hydrodynamic regime; these models are called “specialized models”. Three modeling experiments taking into account both dynamics are compared: two modeling experiments combining specialized models, and one directly built on the whole dataset. This section details the experiments leading to the models’ selection, training, and evaluation.

### 2.3.1. Model Structure

A multilayer perceptron is a type of ANN that consists of multiple layers of interconnected neurons. It takes input variables and feeds them through a series of nonlinear transformations and generates output variables. The experiments were conducted using feed-forward MLPs with up to five hidden layers. A general formula to express the structure of a MLP from a previous layer $l-1$ with $n_{l-1}$ neurons to a current layer $l$ with $n_l$ neurons is as follows (Equation 1 and 2).

$$ \mathbf{a}^{(l)} = g^{(l)}(\mathbf{z}^{(l)}) \qquad (1) $$

$$ \mathbf{z}^{(l)} = \mathbf{W}^{(l)}\mathbf{a}^{(l-1)} + \mathbf{b}^{(l)} \qquad (2) $$

with $\mathbf{a}^{(l)}$ [$n_l, 1$] the activation of the current layer, $g^{(l)}$ the activation function of the current layer, $\mathbf{z}^{(l)}$ the pre-activation input, $\mathbf{W}^{(l)}$ [$n_l, n_{l-1}$] the weights and $\mathbf{b}^{(l)}$ [$n_l, 1$] the biases or parameters of the neurons of the layer $l$, and $\mathbf{a}^{(l-1)}$ [$n_{l-1}, 1$] the activation of the previous layer.

At the first layer of the MLP, the previous layer is the input nodes representing the number of columns $m$ of the input dataset. The MLP’s last layer is the output layer consisting of 1 neuron with the value of the predicted target variable at discrete time $k$ (Fig. 5). All the weights and biases of each layer are the model’s parameters.
<figure><img src="image_7.png" /><figcaption>Fig. 5. Structure of a multilayer perceptron. Connections from the input nodes (squares) to hidden neurons (circles) are shown on the left, from previous hidden layer to current hidden layer in the center, and from the final hidden layer to output neuron on the right. Each line indicates a weight parameter connecting neurons across layers and the bias coefficients are represented in red by the nodes on top of each layer.</figcaption></figure>

The chosen activation function in the experiments is the hyperbolic tangent with an adjustable slope s and (Equation 3). The term identity activation function was used if no nonlinear transformation was used in the layer.

$$g(z, s) = \frac{1 - e^{-2sz}}{1 + e^{-2sz}} \qquad (3)$$

### 2.3.2. Model Training

The goal of the training is to minimize a loss function between the estimated and observed outputs by adjusting the model's parameters. The loss function considered in this study was the mean squared error. Based on this loss function, the model's parameters were calculated using the Levenberg Marquardt (LM) algorithm [24,25].

Training is performed on a set of data called “training set” which represents a major part of the database. Each forward pass of the training set is called an iteration or an epoch. At the beginning of training, the parameters of the ANN are set to random values. During training, the parameters iteratively adjusted to reduce the loss function and improve the model's predictions. At a given maximum number of epoch or any other specific condition, the training stops, and the trained parameters are used for estimations.

The LM algorithm was used to update the model's parameters after each epoch. LM is an approximation of a second-order optimization method that combines gradient descent and an approximation of the Hessian matrix (second order) [25]. In the gradient descent, the loss function is minimized by updating the model parameters in the direction of the steepest descent. The second order method assumes that a local quadratic loss function accelerates and improves the minimization [24]. The LM algorithm utilizes a “damping” parameter to control, during training, which of the methods (first or second order) is applied (Equation 4). At the start of training where the parameters are far from their optimal values (values at which the loss function is minimized), LM behaves like a gradient descent algorithm. When the parameters are closer to their optimal values, LM reacts more like a second order method. This behavior is achieved because the damping parameter ($\lambda$ in eq. 4) is used as a factor added to the diagonal of the Hessian matrix: by setting it to a high initial value to emphasize gradient descent and by reducing it as the loss function converges, allowing the algorithm to transition toward the influence of the second order method. When the loss function is not reduced over an iteration, the damping parameter is multiplied by a predefined value called “update factor”. In this study, the damping parameter is set to $10^{-3}$ at the beginning, and the update factor at 0.1 if the loss function improves and 10 if it decreases. The LM algorithm is adapted from [25] and implemented by [26].

$$\theta_{i+1} = \theta_i - (H_i + \lambda I)^{-1}\nabla f(\theta_i) \qquad (4)$$

with $\theta_i$ the weights and biases of the model at iteration i, $H_i$ the Hessian matrix evaluated at $x_i$, $\nabla f(\theta_i)$ the gradient at $\theta_i$ and $\lambda$ the damping parameter.

### 2.3.3. Regularization methods

Regularization methods are used because the bias-variance tradeoff, mentioned above, indicates that the error calculated on the training set is not a good estimator of the generalization error [27]. The generalization error is the error observed on a dataset never seen by the model, neither in training nor for the selection of the model.

Cross-validation: the “cross-validation” method proposes to estimate the generalization error on the entire training base, but in a “validation” situation [28]. Cross-validation consists of dividing the training set into P subsets. P training runs are then performed, each on P-1 subsets, and the error obtained on the subset set aside is calculated. This calculation is repeated for each subset in turn to obtain the generalization error on all subsets. This makes it possible to measure the generalization error on the entire database used for training, in a “validation” situation. The cross-validation score is then calculated based on the P errors obtained for each subset. The cross-validation score is therefore an estimator of the generalization error. It can be the average or median of the P scores obtained for each subset.

Early stopping: it stops the model's training before full convergence is reached [29]. During training, stop set errors, seen as a validation error, are observed, which allows generalization error to be measured. Generally, the generalization error first decreases, but begins to increase after a certain number of iterations, indicating that the model is becoming overfitted. Training is thus stopped at this iteration and the parameters are saved at this iteration. The stop set therefore allows training to be stopped, which means that the error measured on this set is at its minimal. The model trained in this way is therefore specialized, in a sense, on the stop set. To avoid any distortion due to this observation, Toukourou [30] proposed that the stop set chosen should be the cross-validation set having the best cross-validation score. The stop set is therefore fully consistent with the training set and does not add any additional distortion.

Ensemble modeling: in several configurations, one observes that the performance of the same architecture is strongly affected by the initialization of parameters before training. To mitigate this effect, which is particularly noticeable in the context of flash floods, we propose building an ensemble model. This model consists of several members, each with the same architecture but with different initializations. The ensemble result is calculated as the median of the outputs, at each time-step, of all members [31].

The regularization method required splitting the dataset in several groups. Moreover, some events are kept out of the learning
process to evaluate the modeling performance to forecast unseen extremes events. The events were split into three groups to form three subsets: cross validation set, early stopping set and test set. The test set contains two events: the urban and karst events with the highest water level at Lagoubran to assess the model's ability to generalize extreme values. The early stopping set contains two events: the urban and karst events leading to the best cross validation score in a preliminary phase. The cross validation set contains all remaining events (10 urban events and 15 karst events).

### 2.3.4. Performance criteria

Four metrics were used to evaluate the model's performance on the validation set. All the criteria range from $-\infty$ to 1. Here are the notations used for the criteria: $k$ is the discrete time, $h_k^o$ is the observed water level at $k$, $h_k^s$ is the simulated water level at $k$, $l$ is the lead time, $k_{max}$ is the time step where $h_k^o$ is maximum, and $n$ is the number of observations.

**Nash-Sutcliffe Efficiency (NSE):** widely used in hydrology and it measures the simulated variance error divided by the variance of the observed time series (Equation 5) [32].

$$NSE = 1 - \frac{\sum_{k=1}^{n}(h_k^o - h_k^s)^2}{\sum_{k=1}^{n}(h_k^o - \bar{h}_k^o)^2} \quad (5)$$

**Persistence coefficient (PERS):** similar to the $R^2$ but adapted for forecasting [33]. When this criterion is positive, it indicates that the model's output is better than the “naïve” forecast that assumes that the process value at the lead time is equal to the current value (Equation 6).

$$PERS = 1 - \frac{\sum_{k=1}^{n}(h_{k+l}^{o} - h_{k+l}^{s})^2}{\sum_{k=1}^{n}(h_{k+l}^{o} - h_{k}^{o})^2} \quad (6)$$

**Peak coefficient (PEAK):** expresses the model's ability to accurately simulate the water level at the time of the maximum of the observed peak (Equation 7) [34].

$$PEAK = 1 - \left| 1 - \frac{h_{k \max}^{s} - \min(h_k^{o})}{h_{k \max}^{o} - \min(h_k^{o})} \right| \quad (7)$$

**Combined criterion (PERS PEAK):** Average of PERS and PEAK, measuring the prediction's ability of both being synchronous and reflecting accurately the peak values.

## 2.4. Modeling Experiments

Due to the very short response time of urban floods (half an hour at the shortest), it was not possible to design a rapid flood forecasting model with a time horizon greater than two steps (30 minutes). This is therefore the forecasting horizon that was chosen. However, it is obvious that this horizon is very short for karst events. Nevertheless, the quality of the modeling of slow events can be appreciated by focusing on persistence, which makes it possible to differentiate between models that simply copy their water-level input to the output (naive forecasting) and those that perform true forecasting. Five different experiments were conducted to construct the ideal model to predict the water level of this double-dynamic river. For each experience, a set of operations were conducted to obtain the best model. The first step to operate is the input selection where an initial structure is proposed and only the model inputs are sequentially changed to maximize the model's selection criteria. The structure selection is the second step done by sequentially changing to model's structure or hyperparameters such as its number of hidden layers, its activation functions and its activation function slopes to maximize the model's selection criteria. For all stages of selection, the selection criterion is the cross-validation score, as this ensures maximum generalizability.

The urban runoff model (UM) and the karst model (KM) are specialized models. They are trained respectively on the urban runoff events and the karst events. They have their own early stopping and testing sets and go through input selection and structure selection. Resulting from this selection, UM is a one-layer MLP whose input variables were the best correlated to the water level at Lagoubran to train the surface runoff process of the urban part of the study area. The sliding windows of each input variable corresponded to the response time deducted from rainfall-water level correlogram. The UM model was run with an ensemble of 10 initializations [31] while all the other models used an ensemble of 5 initializations because KM is less sensitive to the problem of initialization. The selection was made on PERS PEAK cross-validation to focus both on the quality of the peak, and on persistency. Past study focused on selecting the best optimizer for modeling the urban runoff events of the Las River and showed that the Levenberg-Marquardt rule was more effective than the ADAM rule [35]. KM is a three-layer MLP with its first layer neurons solely linked to rainfall inputs and second layer neurons partly connected to the water level inputs. This model structure has been selected in a previous study [36]. The goal of the structure is to represent the state of the basin based on the water-level stations on one branch, and to better represent the role of karst aquifer which combines rainfall non-linearly in another branch.

The three other models: output combination model (OM), structure combination model (SM) and bulk model (BM) were trained using both event types. These three models were trained using the same structure as their last two layers. OM is a two-layer MLP that simply joins the outputs of the specialized models (UM and KM) and uses their outputs as inputs. SM is a deeper five-layer MLP that combines the specialized models' structures and retrains all of its parameters on the training set. BM is a two-layer MLP that is designed with a new input selection. Even though the models trained on both events don't have an early stopping set, the early stopping event are not used in their training set. Tab. 1 summarizes the training information about each experience.

<table><caption>Tab. 1. Training information of modeling experiences.</caption><thead><tr><th>Model</th><th>Training Events</th><th>Early Stopping Method</th><th>Selection Criteria</th><th>Input Selection</th><th>Structure Selection</th><th>Initialization Number</th></tr></thead><tbody><tr><td>Urban Runoff</td><td>Urban runoff</td><td>Early stopping set</td><td>PERS PEAK</td><td>Yes</td><td>Yes</td><td>10</td></tr><tr><td>Karst</td><td>Karst</td><td>Early stopping set</td><td>PERS</td><td>Yes</td><td>Yes</td><td>5</td></tr><tr><td>Output Combination</td><td>All</td><td>15 epochs</td><td>PERS</td><td>No</td><td>No</td><td>5</td></tr><tr><td>Structure Combination</td><td>All</td><td>15 epochs</td><td>PERS</td><td>No</td><td>No</td><td>5</td></tr><tr><td>Bulk</td><td>All</td><td>15 epochs</td><td>PERS</td><td>Yes</td><td>No</td><td>5</td></tr></tbody></table>
### 3. Results

The early stopping and test events were respectively events 45 and 27 for the urban runoff model, and events 21 and 33 for the karst model. The initial parameters of the model consist of biases set to 0.1 and weights initialized within a uniform distribution between -0.1 and 0.1 given a random seed. Fig. 6 presents the selected structure of each experiment. In the conducted first and second experiments (UM and KM), the cross-validation results were influenced more by the selection of input variables than by the selection by the model's structure. This implies that higher scores can be achieved by adding new input variables to the existing ones than by selecting a model structure. The experiments with input selection revealed that the stations located in the southwest area of the study area were the most contributing to improving the model's targeted performance criterion. Toulon and Castellet SAPC were the stations that contributed the most to the training of specialized models. However, Cap Cepet, despite its potential to provide valuable coastal rainfall insights, did not contribute meaningfully to the model's construction. Similarly, Cuers and Sainte Baume, located at the easternmost and northernmost areas of the study area, were not as relevant for the neural network training. The KM and the BM had the same input variables obtained throughout input selection. BM has inputs with shorter sliding time windows, probably due to having a smaller structure in its layer.

<figure><img src="image_8.png" /><figcaption>Fig. 6. Best model architectures of each experiment. For each modeling experiment, the squares on the left represent the input nods of the current value (k) of each input series and duplicated for previous values (k-j), circles represent neurons, last neuron on the right represents the model output, forecasting the water level at Lagoubran at the lead time (k+2). Tanh is hyperbolic tangent function and Iden is for identity function. The number inside the parenthesis after the activation function is for the slope of the activation function.</figcaption></figure>

<table><caption>Tab. 2. Median of cross-validation scores for the best model of each experiment. In bold red are the main performance criteria for comparison of each model.</caption><thead><tr><th>Model</th><th>Event Type</th><th>NSE</th><th>PERS</th><th>PEAK</th><th>PERS PEAK</th></tr></thead><tbody><tr><td rowspan="3">Urban Runoff (UM)</td><td>Urban Runoff</td><td>0.725</td><td>0.427</td><td>0.739</td><td>0.596</td></tr><tr><td>Karst</td><td>0.390</td><td>-28.292</td><td>0.466</td><td>-13.894</td></tr><tr><td>All</td><td>0.500</td><td>-10.475</td><td>0.508</td><td>-4.983</td></tr><tr><td rowspan="3">Karst (KM)</td><td>Urban Runoff</td><td>0.581</td><td>0.062</td><td>0.424</td><td>0.203</td></tr><tr><td>Karst</td><td>0.991</td><td>0.369</td><td>0.983</td><td>0.658</td></tr><tr><td>All</td><td>0.970</td><td>0.175</td><td>0.887</td><td>0.488</td></tr><tr><td rowspan="3">Output Combination (OM)</td><td>Urban Runoff</td><td>0.727</td><td>0.308</td><td>0.592</td><td>0.473</td></tr><tr><td>Karst</td><td>0.991</td><td>0.328</td><td>0.980</td><td>0.639</td></tr><tr><td>All</td><td>0.972</td><td>0.328</td><td>0.920</td><td>0.573</td></tr><tr><td rowspan="3">Structure Combination (SM)</td><td>Urban Runoff</td><td>0.702</td><td>0.244</td><td>0.629</td><td>0.371</td></tr><tr><td>Karst</td><td>0.989</td><td>0.222</td><td>0.944</td><td>0.569</td></tr><tr><td>All</td><td>0.969</td><td>0.230</td><td>0.898</td><td>0.514</td></tr><tr><td rowspan="3">Bulk (BM)</td><td>Urban Runoff</td><td>0.726</td><td>0.408</td><td>0.676</td><td>0.520</td></tr><tr><td>Karst</td><td>0.991</td><td>0.260</td><td>0.962</td><td>0.567</td></tr><tr><td>All</td><td>0.974</td><td>0.289</td><td>0.930</td><td>0.567</td></tr></tbody></table>

To quantify the performance of the different strategies in general, the cross-validation scores are presented in Tab. 2. The specialized models (UM, KM) were the most performing models in each category of event. For example, the UM achieved a PERS PEAK of 0.59 for urban runoff events, while the KM achieved a PERS of 0.36 for karst events highlighting the superiority of
specialized models over general models. Their major drawback is that they have bad performance on the other type of event which makes them unreliable for operational forecasts, especially when it's raining all over the basin. The models using both event types as training sets performed better performance on all validation sets with the OM showing the best performance in predicting both event types. OM's overall performance will certainly improve if input selection is performed, possibly making it the best model trained on all events. The BM had surprisingly a high PERS PEAK score on urban events while maintaining a better PERS score compared to the SM. Combining the specialized model's structure was the less effective modeling method.

The evaluation of models for extreme events can be performed using the scores of the test events (Table 3). The ranking of the models on the selected test events remains the same, showing how insightful the cross-validation scores can be. The BM remained above the curve for predicting the exceptional urban runoff with a water level 66% greater than what was observed for the urban runoff events in the training set. Just like on the cross-validation scores, the specialized models performed better than the combination models. The test event selected for the karst events is harder to anticipate with the very short lead time translating to bad PERS scores. The OM still stood out as the best model for the extreme test events.

<table><caption>Tab. 3. Performance of the best model of each experiment on test events. In bold red are the main performance criteria for comparison of each model.</caption><thead><tr><th>Model</th><th>Event Type</th><th>NSE</th><th>PERS</th><th>PEAK</th><th>PERS PEAK</th></tr></thead><tbody><tr><td rowspan="2">Urban Runoff (UM)</td><td>Urban Runoff</td><td>0.728</td><td>0.806</td><td>0.612</td><td>0.709</td></tr><tr><td>Karst</td><td>0.058</td><td>-86.289</td><td>0.311</td><td>-42.989</td></tr><tr><td rowspan="2">Karst (KM)</td><td>Urban Runoff</td><td>-0.177</td><td>0.160</td><td>0.143</td><td>0.151</td></tr><tr><td>Karst</td><td>0.988</td><td>-0.155</td><td>0.941</td><td>0.393</td></tr><tr><td rowspan="2">Output Combination (OM)</td><td>Urban Runoff</td><td>0.256</td><td>0.469</td><td>0.375</td><td>0.422</td></tr><tr><td>Karst</td><td>0.985</td><td>-0.383</td><td>0.892</td><td>0.255</td></tr><tr><td rowspan="2">Structure Combination (SM)</td><td>Urban Runoff</td><td>0.234</td><td>0.453</td><td>0.316</td><td>0.385</td></tr><tr><td>Karst</td><td>0.975</td><td>-1.278</td><td>0.843</td><td>-0.218</td></tr><tr><td rowspan="2">Bulk (BM)</td><td>Urban Runoff</td><td>0.569</td><td>0.432</td><td>0.594</td><td>0.451</td></tr><tr><td>Karst</td><td>0.976</td><td>-1.218</td><td>0.832</td><td>-0.193</td></tr></tbody></table>

Fig. 7 compares the predictions of all the models to the observation. In this graph, we clearly see the difference between the urban runoff events and the karst events. The urban runoff events have a very brusque ascending slope with a peak generally reached in less than an hour after the water level starts to rise. Their recession is also as fast as their rising limb. In contrast, the karst events have slower responses with smaller peaks caused by the urban runoff observed on the karst water contributions. The most evident observation of this graph is that the UM is the only model to predict the urban test and cross-validation events on time while having mediocre performance on the karst events. Despite the low estimation of the urban peaks of the karst events, their response comes mostly on time. The KM shows the best performances on karst events and is worst on urban runoff events.

<figure><img src="image_9.png" /><figcaption>Fig. 7. Observed versus forecasted water levels of the Las River at Lagoubran with a lead time of 30 min (k+2) on test events and key cross validation (CV) events. The full length of the events is cut to focus on periods of high water levels.</figcaption></figure>
## 4. Discussion

The results of this study confirm the need for specialized models to address the dual hydrologic dynamic of the Las River. UM and KM respectively captured the fast and the slow flood responses but lacked the capacity to generalize outside their dynamic. The idea of using ensemble models with different configurations addressed successfully the previous flaw observed. The simplest combination of the specialized model's outputs (OM) showed the most promising results despite not integrating the input selection method and confirms the benefits of using ensemble modeling on complex hydrosystems. The BM, initially seen as the most "basic" method to model this hydrosystem, came in second proving the ability of the MLP to achieve universal approximation and underlying the importance of input selection and the learning abilities of LM rule applied to the MLP.

The model combinations had to disable the early stopping to allow a better training convergence. This adjustment was necessary due to the greater variety of training sets and dynamics, thus requiring a larger training epoch to capture these complex patterns.

The lower PERS values on karst events reflect a challenge in the choice of adequate lead time depending on the forecast horizon. Using a lead time of 30 minutes (k+2) on karst events makes the model over relying on groundwater water levels instead of mapping less obvious patterns. A major improvement to this study would be to investigate the proper lead time of karst model to obtain better PERS scores.

Zhou studied the use of ensemble models to forecast karst spring discharge using machine learning models [37]. His findings support that the use of ensemble learning models not only outperforms individual base models by capitalizing on their strengths and overcoming their weak generalization capacity. He suggests that a linear combination of deep learning models is heavily affected when one or more members are poorly trained. Meanwhile, he only concatenated the deep learning model outputs with an identity activation function instead of using a nonlinear activation and did not tune their hyperparameters. Khalil forecasted groundwater level by comparing linear and MLP model to a bagging MLP ensemble model [38]. He arrived at the same conclusion that ensemble modeling ensures model robustness and reduces the model's variability. These results are in the same conclusion as the present study.

## 5. Conclusion

This study investigated the impact of ensemble learning to predict the water level of a karst hydrosystem with a dual hydrodynamic response. By distinguishing between each the fast response resulting mostly from the urban runoff with negligible karst contributions and the slow response with major karst contributions, the importance of specialized models was proven, and their assembling was explored. As expected, the specialized models, trained only on each event type, performed the best on the events they were trained on but performed poorly or even disastrously on other types of events. For instance, the urban runoff model (UM) performed the best on urban runoff events but terribly on karst events and the karst model performed the best on karst events but had a persistency of zero on urban events. To overcome this major flaw, the specialized models were combined given to their outputs (OM) and structure (SM) and trained on both event types. A third model called the bulk model (BM) was trained on both event types joint with the input selection method. Among them, the OM emerged as a promising candidate for operational forecasting under diverse flood conditions, offering the best trade-off between performance and generalization.

Our results show that the choice of inputs, rather than the complexity of the structure, was the key to improving the accuracy of the models. Even though the early stopping sets were crucial in training the specialized models, they stopped the training of the dual regime models before convergence. To improve the usefulness of forecasting in operational conditions, it is essential to increase the lead time of the karst models and to find a more clever method to combine models with different lead time.

## Acknowledgments

We are grateful to the French ANR for funding this research (ANR-21-LCV1-0001) on the Laboratory of Hydrological forecasting by Artificial Intelligence (Hydr.IA) project and to Météo-France for 24 the availability of rain gauge data. We would also like to thank Abdou Ali, Vincent Bailly-Comte, Pierre Camps and Patrick Lachassagne for their helpful discussions during this research.

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