# Using supervised machine learning for regional hydrological hazard estimation in metropolitan France

Qifan Ding, Patrick Arnaud

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Qifan Ding, Patrick Arnaud. Using supervised machine learning for regional hydrological hazard estimation in metropolitan France. Journal of Hydrology: Regional Studies, 2024, 54, pp.101872. 10.1016/j.ejrh.2024.101872. hal-04794012

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# Using supervised machine learning for regional hydrological hazard estimation in metropolitan France

Qifan Ding *, Patrick Arnaud

INRAe Hydrometeorological Research Team, UMR RECOVER, Aix-en-Provence 13100, France

## ARTICLE INFO

**Keywords:**

Hydrological hazard
Regional Flood frequency analysis
Neural Network
Random Forest
Machine learning

## ABSTRACT

**Study region:** This study is carried out for 1929 gauged catchments in France, ranging from 1 to 10,000 km², where quality hydrometric observations are available for flood frequency analysis.
**Study focus:** The regional estimation of hydrological hazards is studied for flood risk management and prevention in hydrology. For gauged catchments, flow quantiles can be estimated from observations using statistical approaches based on suitable probability distributions or simulation approaches based on rainfall-runoff transformation models. For ungauged catchments, the lack of hydrological observations means that we have to extrapolate our knowledge of hazards from gauged catchments to ungauged catchments, using regionalization methods. It is therefore necessary to combine regionalization methods with the implemented hazard estimation approach. In this paper, two popular machine learning methods, Random Forest and Neural Networks, are tested and compared as regionalization methods. A classical regionalization method using multiple linear regression is also applied as a benchmark to evaluate the performance of all configurations. All these regionalization methods are applied to a simulation-based approach (the SHYREG method) and to a statistical-based approach using generalized extreme value distribution (GEV).

**New hydrological insights:**

* Regionalization approaches based on multiple linear regression have limitations to explain parameters with environmental descriptors in Regional Flood Frequency Analysis (RFFA) domain.

* Regionalizing RFFA parameters using Random Forest allows more explanatory variables to be considered through non-linear relationships, resulting in better parameter estimation.

* Machine learning techniques can better handle environmental descriptors for regionalization, this providing a notable performance improvement, especially for the statistical approach.

<footnote>* Corresponding author.
E-mail address: qifan.ding95@outlook.com (Q. Ding).</footnote>
* The tested simulation-based approach is less sensitive to the choice of spatial interpolation method than the studied statistical approach.

# 1. Introduction

Hydrological hazard estimation involves assigning a probability of occurrence (or return period) to extreme precipitation events or flood events that could have serious consequences for the safety of property and people (Beevers et al., 2022). Possible consequences include flooding, soil erosion, infrastructure damage, etc. (Chinnasamy et al., 2020; Poussin et al., 2015; Vinet et al., 2022). Hydrological hazard studies are therefore important, for example, for flood risk prevention, hydraulic structure design or dam management (Chaleeraktrakoon and Chinsomboon, 2015; Erdlenbruch et al., 2009; Mangini et al., 2018; Sordo-Ward et al., 2013). Hydrological hazard estimation methods are calibrated on hydro-meteorological information, available at specific “gauged” catchment outlets. This information, observed over long periods in the past, is used to match the characteristics of observed flood events with their probability of occurrence. The goal of hazard estimation is to clearly define these relationships so that frequency extrapolations can be made to gauge the characteristics of extreme flood events. Related approaches can be divided into two categories: so-called "statistical" approaches and so-called "simulation" approaches.

Most “statistical” approaches are based on the extreme value theory (Coles et al., 2003), which defines adapted theoretical probability distributions for Flood Frequency Analysis (FFA), such as GEV distribution (Jenkinson, 1955; Schellander et al., 2019; Seckin et al., 2011) or GPD distribution (Sugahara et al., 2009; Wang and Holmes, 2020). Other probability distributions, such as Pearson’s distribution or lognormal distribution, may also be used (Bobée, 1975; Griffis and Stedinger, 2007; McMahon and Srikanthan, 1981). Statistical approaches aim to fit a probability distribution to the values that are sampled from observations, based on certain sampling rules (Hansen, 2020; Lombardo et al., 2019; Nadarajah and Kotz, 2003). However, considering the hydrological hazard problem as a purely mathematical problem can lead to high uncertainties related to the data sampling problem, which have been discussed in many studies (Bača and Bačová Mitková, 2007; Cai and Hames, 2010; Kidson and Richards, 2005; Merz and Thieken, 2005). To reduce these uncertainties, increasing the sample size of observations can be achieved by using historical floods as additional information (Lang et al., 2014; O’Connell et al., 2002; Reis Jr and Stedinger, 2005) or by using “regional” information, such as the observations on nearby catchments (Dalrymple, 1960).

“Simulation” approaches attempt to use rainfall information and take into account complex natural processes during the rainfall-runoff transformation (Calver et al., 2009; Hoes and Nelen, 2005; Paquet et al., 2013). They are coupled with hydrological models to simulate the rainfall-runoff transformation. These approaches can be “continuous” when they perform a continuous processes simulation (Calver et al., 1999; Cameron et al., 2000; Winter et al., 2019). They can also be “event-driven” if they only focus on simulating flood scenarios (Gericke, 2021; Kvočka et al., 2016; Muzik, 2002). In the latter case, it is important to properly estimate the initial moisture state of the catchment under extreme rainfall events (Berthet et al., 2009; Brocca et al., 2008; Manfreda et al., 2005; Seck et al., 2015).

All the above approaches are aimed at estimating hydrological hazards at gauged catchment outlets, which provide hydrometeorological information for model calibration. Hydrological observations are not available at ungauged catchment outlets or at the outlets of sub-catchments in gauged catchments. These cases require a regionalization approach, referring to Regional Flood Frequency Analysis (RFFA). Information from gauged catchments is to be spatially extrapolated to ungauged catchments. This raises the question of how to well perform a flood hazard predetermination in ungauged areas (Sivapalan et al., 2003).

For statistical approaches, regionalization consists in the direct regionalization of flow quantiles estimated locally at gauged outlets, or the regionalization of parameters from probability distributions calibrated locally at gauged outlets (Jakob et al., 1999; Malekinezhad et al., 2011; Mamun et al., 2012; N’Guyen et al., 2013; Noto and La Loggia, 2009; Vogel et al., 1999). For simulation-based approaches, regionalization consists in regionalizing parameters from linked models (hydrological models and sometimes rainfall generators) that have been calibrated at gauged outlets (Calver et al., 2004; Cipriani et al., 2012; Durrans and Kirby, 2004).

Thus, regional estimation of hydrological hazards requires using regionalization methods, regardless of the approach implemented, to spatially estimate either flow quantiles or model parameters. The regionalization method is therefore an essential part of regional approaches aimed at estimating hydrological hazards at ungauged sites.

Regionalization methods are tools that have been developed for any type of application involving the prediction of certain spatially variable values. Examples can be found for regionalizing physical variables such as temperature (Yu et al., 2018), rainfall intensity (Bharath and Srinivas, 2015) and soil moisture (Liang et al., 2020). In hydrology, regionalization methods are generally based on the assumption that similar physical characteristics in catchments lead to similar hydrological responses, and therefore flood quantiles or hydrological parameters (Guo et al., 2021; Kokkonen et al., 2003; Merz and Blöschl, 2004; Porcheron, 2018). The main question involves defining this similarity.

Of course, we can assume that neighboring catchments will have similar hydrological behaviors. This is known as geographic proximity (Aubry, 2000; Parajka et al., 2005). It is an acceptable assumption if considering the existence of spatial autocorrelation of the catchment characteristic across neighboring catchments (Anselin and Bera, 1998; Le Gallo, 2002). Having information about the physical characteristics of gauged and ungauged catchments makes it possible to explain the hydrological behavior of catchments by these characteristics. This is known as physiographic proximity (Oudin et al., 2008; Sefton and Howarth, 1998). This is the basic assumption of classical regionalization methods, which explain hydrological behavior variability by regressions with representative
environmental characteristics (Bao et al., 2012; Gibbs et al., 2012; Swain and Patra, 2017). In this way, it is possible to establish an
explanatory relationship between an unknown feature related to hydrological behaviors (flows, hydrological parameters) and known
features related to catchment characteristics (topography, land use, geology, etc.).

The search for an explanatory relationship between hydrological and environmental characteristics of catchments raises two questions: which environmental characteristics are truly representative of hydrologic behaviors, and what are the quantitative relationships? We can systematically consider all available environmental characteristics at the same time and attempt to calibrate methods that best explain the variability of observed data such as flows. In this case, searching for relevant regionalization could be very complicated, as there would be too many variables retained numerically without any real hydrological relevance. On the other hand, we could preselect characteristics to reduce the number of explanatory variables in the relationship. However, this could result in omitting important characteristics. We thus need techniques to extract relevant information from an exhaustive data set. This might be achieved using machine learning techniques.

Generally, in hydrological and environmental fields, some of the most popular categories of artificial intelligence (Ongsulee, 2017; Samuel, 1959) include Classifiers and Machine Learning approaches, Fuzzy Sets, Evolutionary Computation, and Wavelet models (Yaseen et al., 2015). They are used to explain complex natural process and uncover mechanisms from a data-science perspective (Cai et al., 2022; Evrendilek, 2014; Herath et al., 2021; Jiang et al., 2022; Liu and Lu, 2014). This creates a “black box” relationship between input and output data with less constraints related to physical processes (Mohammadi, 2021). It allows algorithms to “learn” patterns within provided data.

Machine learning is a subcategory of AI approach. This technique has been utilized across various fields of hydrology, including water management and flood protection (Chadalawada et al., 2020; Ghorpade et al., 2021; Kisi et al., 2016; Lu and Ma, 2020). Some applications aim to establish an empirical association between flows and explanatory factors (Cannon and Whitfield, 2002; Tyralis et al., 2021). Other applications integrate machine learning techniques directly into conceptual or even physical modeling to better account for natural hydrological processes (Adnan et al., 2021; Okkan et al., 2021). They can be used as predictive methods for streamflow or water level prediction/forecasting, or as extrapolation methods for a regional approach.

In the context of regional hydrological hazard estimation, it is appropriate to use supervised learning methods since the input data (catchment characteristics) and output data (flow quantiles or hydrological parameters) are well defined (Jiang et al., 2020; Uddin et al., 2019). Our study will test two types of supervised learning algorithms.

The first method is based on neural networks (NN). These are suitable for both regression and classification tasks (Kumar, 2005; Wang et al., 2019). This particular method has been used for a diverse set of objectives, including the reconstruction of past floods (Bomers et al., 2019) or the prediction of daily flows (Singh and Deo, 2007). In the RFFA context, the number of studies remains relatively limited (Aziz et al., 2015, 2014; Filipova et al., 2021; Jingyi and Hall, 2004; Kordrostami et al., 2020; Muttiah et al., 1997; Seckin et al., 2013; Shu and Burn, 2004; Shu and Ouarda, 2007).

The second method is Random Forest (RF) (Breiman, 2017, 2001). It is also extensively used in various fields. Unlike regression-based algorithms, such as multiple linear regressions or neural networks, which seek a numerical explanation, this method relies on classifications. This algorithm is increasingly being used to study hydrological hazards (Desai and Ouarda, 2021; Esmaeili-Gisavandani et al., 2023; Schoppa et al., 2020).

Studies that have used machine learning methods to analyze regional flood frequency are usually based on statistical approaches. They aim to estimate a direct statistical correlation between hydrological hazards and environmental characteristics, while leaving aside natural processes related to hydrological phenomena (Dombry, 2015; Kidson and Richards, 2005; Singh and Strupczewski, 2002). However, since hydrologic data are derived from complex natural processes, it is preferable to simulate them through hydrologic modeling, especially when studying the complexity of hydrologic scenarios and extrapolating to extreme events. In this scenario, instead of regionalizing flow quantiles directly, the parameters of the hydrological model are regionalized, in order to derive flow quantiles from simulated events (Wang et al., 2023).

Currently, few studies discuss performance among machine learning methods for RFFA. (Prieto et al., 2019) have regionalized the parameters of a statistical approach with a focus on continuous runoff prediction. (Kiran and Srinivas, 2021) introduced a Regression Forests approach for RFFA within a probabilistic methodology. Even fewer studies address the use of machine learning methods in RFFA using simulation approaches. (Golian et al., 2021) attempted to regionalize parameters of lumped hydrological models to estimate target flow characteristics. We are interested in comparing the performance of **NN** and **RF** as regionalization methods for both simulation-based approach and statistical approach.

The paper addresses this subject through the study of a simulation-based RFFA approach (the SHYREG method), which links the variability of a hydrological hazard to the variability of both precipitation and a hydrological parameter (Arnaud et al., 2014a). To regionalize this hydrological parameter, the **NN** and the **RF** methods are tested and compared to a classical regionalization method using multiple linear regression (**LR**). The above methods are also applied to a statistical approach (GEV type), which is commonly used in RFFA researches. The objective of this article is not comparing the SHYREG method to the GEV approach as this work has already been studied (Arnaud et al., 2016; Kochanek et al., 2014), but to valuate the improvement that can be obtained by using machine learning methods compared to classical regionalization methods.

**2. Methodology**

Estimating hydrological hazards at a regional scale first requires having a good estimate of hazards at the local scale. The accuracy of “local” hazard estimates can be assessed by comparison with empirical distributions of observed values. As illustrated in Fig. 1, frequency distributions of annual maximum observations (peak flow and daily flow) in a catchment are presented. The points represent
maximum values recorded each year, with peak flows in black and daily flows in white. The curves represent frequency distributions obtained by calibrating two approaches: in red, the application of a GEV-type probability distribution (the “statistical” approach), and in blue, the result of calibrating the SHYREG method (the “simulation” approach explained later).

Both methods perform frequency extrapolations to estimate flood quantiles for high return periods. Given that observation periods are often too short compared to the return periods of extreme events (such as 1,000-year quantiles), the estimates of extreme quantiles can vary considerably between approaches, even though the approaches may result in very similar frequency estimates (1–10 years). For this reason, we observe that the current quantiles (quantiles of 2, 5, and 10 years) estimated by the SHYREG approach are similar to those obtained by the statistical approach, but the frequency extrapolation can lead to very different estimates. The validity of frequency extrapolation methods at gauged sites will not be discussed here, as this distinct issue has already been addressed in other studies (Arnaud et al., 2016; Carreau et al., 2013a; Renard et al., 2013). This article concentrates on the application of spatial extrapolation techniques to ungauged sites.

In the following sections of this article, the “local” index will denote values derived from local observations at gauged sites. In order to focus on regional hazard estimations at ungauged sites, the performance of regional methods will be evaluated primarily on current quantiles estimations (quantiles of 2, 5, and 10 years). Quantiles resulting from fitting a GEV distribution can thus be assimilated to locally “observed” quantiles (*Qlocal_GEV*). The selection of this probability distribution is insignificant when estimating current quantiles (Carreau et al., 2013b). Note that the shape parameter of GEV distribution is bounded between 0 and 0.4 in order to control the impact of sampling on its estimation.

The regionalization procedure allows the estimation of flow quantiles without local observations. The “region” index represents values obtained from regional approaches. Regional estimates from statistical and simulation approaches are denoted as *Qregion_GEV* and *Qregion_SHYREG*, respectively.

Validation of the different approaches for estimating regional hydrological hazards will consist in comparing *Qregion* quantiles with *Qlocal* quantiles for a sample of validation catchments selected in various ways (see Section 3.2). We focus here on the accuracy of estimating two flood characteristics, annual peak flow quantiles (*QP*) and annual maximum daily flow quantiles (*QD*).

## 2.1. SHYREG method

The SHYREG (Regionalized SHYpre model) method was developed by INRAE, to estimate regional rainfall and flow quantiles (Arnaud et al., 2014b; Arnaud and Lavabre, 2002). It combines a stochastic hourly rainfall generator with a simplified hydrological model (Arnaud, 1997; Cernesson, 1993).

The rainfall generator has already been regionalized for the entire French territory (Arnaud et al., 2008; Cantet, 2010; Carreau et al., 2013b). Based on this rainfall generator, we can generate rainfall events at any point in the study area. The role of the SHYREG rainfall generator is to generate rainfall events with the same characteristics as the observed ones from a statistical point of view. The precipitation generation for one year starts with a random drawing of the total number of events during this year according to a specific probability distribution. For each event, a successive drawing of some other parameters is then performed in order to determine the number of rain showers, their total precipitation volume, the time gap which separates the rain showers, and their form (Arnaud et al., 2007; Cantet and Arnaud, 2014).

This procedure is then repeated to generate independent events at an hourly time step for a period equivalent to 10,000 years. The distribution of characteristics of all generated events follows that of the observed daily rainfall record, which is widely available in France. The calibration and regionalization of model parameters is described in (Arnaud and Lavabre, 2010).

Since the rainfall is generated at a calculating mesh, the rainfall-runoff transfer is also performed at the same scale. For each event, given the parameters of associated hydrologic model, the runoff events could be simulated at each time step. A stochastic frequency analysis is then implemented to derived elementary flood quantiles at mesh (Arnaud et al., 2016). It is also worth mentioning that the

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Runoff observation of a gauged catchment and corresponding quantile estimation.</figcaption></figure>
simulation is continuous during a rainfall event, but each event is considered independently. The catchment condition is reset to the initial state before the simulation of another event. Research on fully continuous simulations is ongoing. Various works, such as research by (Ding and Arnaud, 2022; Odry, 2017; Odry and Arnaud, 2017), have addressed the regionalization of the hydrological model.

In this study, we performed hydrological modeling at the kilometer scale using a conceptual GR-type hydrological model with reservoirs (Edijatno and Michel, 1989; Folton and Lavabre, 2007; Graff, 2004). To enable regionalization, successive studies have reduced the number of parameters, resulting in a hydrological model with a single parameter to be calibrated, denoted as S₀/A (Odry and Arnaud, 2017).

"A"[mm] refers to the total size of the production reservoir of the GR model. "S₀"[mm] refers to the initial filling depth of the same reservoir at the beginning of each rainfall event. Therefore, "S₀/A" is a dimensionless parameter representing the initial filling rate of this reservoir, ranging from 0–1. It describes the average initial hydric status of the modeled area before a rainfall event. Local calibration of the SHYREG method therefore requires finding the best spatial distribution of the S₀/A parameter to best reproduce the flow quantiles in the different calibration catchments.

This parameter, representing the initial filling level of a reservoir simulating surface flow production, can be assimilated to a nonlinear runoff coefficient during a flood. By determining the value of this parameter, “elementary” flow quantiles can be calculated for each grid cell, as demonstrated in Fig. 2.

The elementary quantiles are aggregated to the catchment outlet and adjusted by an areal reduction function that accounts for catchment area and flow quantile duration (Aubert et al., 2014). Previous research has calibrated the coefficients of the areal reduction function, for use in catchments ranging from 1 to 10,000 km² in the studied area (Aubert, 2012; Fouchier, 2010). Thus, the flow quantiles obtained by the SHYREG method depend on the catchment area, the precipitation generated on the grid cells and the value of parameter S₀/A, which must be determined for each grid cell during a calibration procedure.

For a given mapping of the hydrological parameter across a catchment, different flow quantiles can be simulated, denoted as Qsim. The purpose of calibration is to determine the hydrological parameter mapping that minimizes the weighted relative error, ERcatchment between six quantiles of simulated flows (Qsim) and observed flows (Qobs_GEV). We are interested in six specific quantiles: peak flow quantiles (QP) with return periods of 2, 5, and 10 years, and daily flow quantiles (QD) with return periods of 2.5 and 10 years. These quantiles have respective weights of 3, 2.1, and 3.2.1 for relative error calculation (formula 1).

$$ER_{\text{catchment}} = \frac{\sum_{i=1}^{n} \left[ \frac{(Q_{\text{sim}} - Q_{\text{obs}})}{Q_{\text{obs}}} \right] \times Weight_i}{\sum_{i=1}^{n} Weight_i}$$

**Formula 1.** Weighted relative error to be minimized during the local calibration procedure for the SHYREG method

We first assume that the spatial distribution of the parameter is uniform within each catchment. In this case, we only need to find an optimal value for the parameter for all grids in the calibration catchments that produce a minimal error (ERcatchment). This result is called the local calibration result. Only the spatially uniform distribution of the parameter is determined for each calibration catchment. Values determined at this stage are referred to as S₀/A local.

Regionalizing the method consists in regionalizing the S₀/A parameter obtained from the local calibration, both inside and outside the calibration catchment, in an attempt to relate the spatial variability of the S₀/A parameter to the variability of the catchments' environmental characteristics. To achieve this, we will use several regionalization methods described in the next section.

## 2.2. Regionalization methods

### 2.2.1. The random forest method

The Random Forest (RF) method is essentially a classification and regression algorithm based on the principle of decision trees (Breiman, 2001; Quinlan, 1987). Training a decision tree consists in recursively splitting sampled data to different branches, in such a

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Estimating flow quantiles using a grid-scale runoff simulation.</figcaption></figure>
way as to minimize the overall error. Splitting continues until each branch reaches a minimum training data size (6 in our study) and becomes a terminal branch, then called a “leaf.” This is in fact the basic principle of CART algorithm (Lewis, 2000).

The Random Forest method is a collection of CARTs. Compared to training a single model, the Random Forest method performs better thanks to its ensemble approach, “bagging,” an acronym for “bootstrap aggregation” (Breiman, 1996; Tibshirani and Efron, 1993), which trains a set of trees with the same objective (Hansen and Salamon, 1990). Each tree is trained individually based on random resampling of the initial training data. At each split of a tree, a random selection of candidate descriptors is also performed. The ensemble of trees becomes an “RF” model. The final prediction of a trained RF model is then equal to the average predictions for each tree. This ensemble approach was developed to stabilize prediction performance and reduce overfitting (Dietterich, 1997; Sharkey, 2012; Webb and Zheng, 2004). Details of its technical implementation are well presented in (Hastie et al., 2009).

### 2.2.2. Neural networks

The term “neural network” (NN) covers a family of different models. In general, the most common types of NNs are multi-layer perceptrons (MLP), convolutional neural networks (CNN), and recurrent neural networks (RNN). CNN and RNN types are generally used in fields such as image recognition, text processing, or video descriptions (Lei et al., 2019; Peris et al., 2016; Wang et al., 2019). In hydrology, the MLP-type is the most widely used one (Kuok et al., 2010; Riad et al., 2004).

An MLP-type neural network includes an input layer, one or more hidden layers, and an output layer. Theoretically, a higher number of hidden layers can extract higher order information, but there is no clear procedure for determining the optimal layer number (Haykin, 1998; Hunter et al., 2012; Ibnu Choldun R., et al., 2020; Madhiarasan and Deepa, 2016). We will also need to define the number of neurons to be implemented in each hidden layer (Hornik et al., 1989; Stathakis, 2009). Reviewing literature from recent studies in related fields points to highly variable hidden layer numbers and neuron numbers in each hidden layer (Aziz et al., 2017; Kalteh, 2013; Lee et al., 2018; Makwana and Tiwari, 2014; Tao et al., 2016; Venkata Ramana et al., 2013; Zemzami and Benaabidate, 2016). Studies have shown that a single hidden layer is sufficient for flood frequency analysis (Filipova et al., 2021; Shu and Burn, 2004; Shu and Ouarda, 2007). Our study uses a single hidden layer with twelve neurons.

In our study, input data involves environmental catchment characteristics, including meteorological ones. The regional statistical approach (GEV) involves determining output data flow quantiles. In the regional simulation approach (SHYREG), output data are the parameter S₀/A of the associated hydrological model, which is then used to calculate flow quantiles.

Similar to classification methods, many hydrological studies on the application of neural networks have shown that an ensemble NN model can lead to improved prediction performance (Kan et al., 2015; Ouarda and Shu, 2009; Piotrowski et al., 2016). Similar to the Random Forest Bagging approach, when training an NN model, random data resampling and a random drawing of the explanatory descriptors (catchment characteristics) are performed. Only one third of the descriptors are used to train each NN model. An ensemble NN model is composed of 100 individual models and the final prediction is equal to the average predictions made by the ensemble of 100 trained models.

### 2.2.3. Multiple linear regression

To evaluate the contribution of machine learning methods, we also apply a multiple linear regression method, denoted “LR.” This method selects the best covariates using a stepwise procedure based on Akaike’s Information Criterion (AIC) between the explanatory descriptors and the variable to be explained (Akaike, 1998). According to (Odry, 2017), this method best explained the spatial variability of the hydrological parameter for the SHYREG method compared to other methods tested in this study. The regression will

<figure><img src="image_7.png" /><figcaption>Fig. 3. Regionalization procedure of flow quantiles.</figcaption></figure>
be re-optimized via a "Nelder-mead" method to explain the spatial variability of observed quantiles as best (Nelder and Mead, 1965).

## 2.3. Regionalization procedure

Calculating regional flow quantiles aims to estimate hazards at ungauged sites. The calculation of these quantiles is carried out by different approaches implemented at validation sites using information from calibration sites. Three regionalization methods are used to estimate hydrological hazards over the entire study area. For the simulation-based approach (SHYREG), machine learning methods are used to regionalize the hydrological parameter $S_0/A$, and for the statistical approach (GEV), the same machine learning methods are used to regionalize the flow quantiles estimated locally. The procedure is summarized in Fig. 3.

**For the simulation approach (SHYREG),** the simulation of flow quantiles depends on the spatial distribution of the parameter $S_0/A$. We focus on the regionalization of this parameter. The steps are as follows:

1. The hydrological model is calibrated using observed flow data from the calibration catchments to obtain local values for the hydrological parameter ($S_0/A_{local}$). At this stage, the value of the parameter $S_0/A_{local}$ is spatially uniform across each calibration catchment. It is called the local calibration step.

2. The environmental characteristics of the calibration catchments and these $S_0/A_{local}$ parameters are the input and output data for training the RF, NN and LR models. By supplying trained models with environmental characteristics available throughout the study area, we obtain a spatial distribution of the $S_0/A$ parameter, denoted **Prediction** (predicted values). It represents the parameter predicted by the RF/NN/LR models based on the environmental descriptors. This step consists in searching physiographic similarity.

3. Flow quantiles simulated from its prediction may differ from local quantiles (as shown in Section 4.1), because the predicted spatial distribution of $S_0/A$ parameter can be different from the local one. The difference between predicted parameter and optimized local value represents information that is not explained by a trained model. For each calibration catchment, the “Prediction” is corrected by a spatially uniform value (*Residual* of $S_0/A$), in such a way that the weighted error of predicted quantiles (formula 1) can be reduced as much as possible.

4. We can also interpolate this *Residual* to ungauged areas using the IDW (inverse distance weighting) interpolation method (Setianto and Triandini, 2015). Residuals are interpolated from the 1x1km grid, where each pixel in a catchment is associated with a residual value that will be used in the interpolation. In this case, the residuals are retained in the calibration catchments and extrapolated outside the catchments boundary. In this manner, we hope that unexplained information can also be transferred to ungauged areas through a geographic proximity approach. The mapping of extrapolated residual is denoted as the interpolated *Residual*.

5. We then obtain a map of the regionally optimized hydrological parameter ($S_0/A_{region}$), which has a spatial variability associated to environmental descriptors. Based on this regionalized parameter, flow quantiles ($Q_{region\_SHYREG}$) can be estimated using the SHYREG approach at ungauged sites (validation catchments).

**For the statistical approach (GEV),** the same regionalization procedure is applied directly to flow quantiles determined in the calibration catchments. All quantiles are independently regionalized using RF/NN/LR methods. The calculation of the *Residual* for each quantile is also independent, and theretofore the predicted quantiles should perfectly match the observed ones. Flow quantiles are regionalized without considering the direct relationship between rainfall and runoff. However, meteorological variables can still be used as environmental descriptors during regionalization.

In order to have a performance reference, a “null” test is also introduced. In this case, the spatial distribution of the variable to be

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Location of gauged catchments and distribution of their area.</figcaption></figure>
predicted is uniformly equal to the average local values in its region (see Fig. 5). Environmental descriptors are not used during regionalization. The regional variability of the hydrological hazard is based solely on geographical proximity (IDW).

The approaches presented above are applied to three types of catchment samples to separate calibration and validation data sets (Section 3.2). It should also be noted that the SHYREG method requires regionalizing a single parameter (S₀/A) to estimate all types of quantiles (regardless of duration and return period), whereas the statistical approach applied in this study must repeat the regionalization procedure for each quantile.

# 3. Application

## 3.1. Data

The study is based on information from 1929 gauged catchments in France, located on the map in Fig. 4. The catchment selection is based on a set of quality criteria (long enough recording periods, low percentage of missing values, good flood information quality, etc. (Odry, 2017). Hydrometric data for these catchments is available at http://www.hydro.eaufrance.fr/. The area of these selected catchments varies widely, from 1 to 10,000 km².

The available observation time for each catchment is also shown in Fig. 4, ranging from 10 to 95 years, with a median of 30 years. The majority of hydrometric stations have observation records that are long enough to correctly estimate the current quantiles of observed flows ($T \le 10$ years) (Carreau et al., 2013b).

The research has been conducted under the assumption of temporal stationarity of the hydrological hazard. It would be interesting to perform non-stationarity tests if we have longer observations on average (Bertola et al., 2020).

These catchments cover a very large part of France. For this large study area (national scale), catchments are under very different hydro-climatic contexts, making it impossible to achieve a single regionalization across the whole area. For this reason, catchments were divided into ten groups belonging to the ten hydrographic zones inspired by the study of (Wasson et al., 2002), as shown in Fig. 5. An independent regionalization procedure is implemented for each zone.

Forty-two environmental descriptors have been identified for hydrological hazard regionalization. These descriptors can be either direct measurements of given variables (altitude, mean temperature, etc.) or characteristics recalculated from these measurements (drainage density, evapotranspiration, moisture index, etc.). They are divided into three categories: geographical descriptors, meteorological descriptors, and hydrogeological descriptors, as shown in Fig. 6:

This data has been acquired and processed in several previous studies (Desouches and Sol, 2005; Finke et al., 1998; Odry, 2017; Organde et al., 2013; Poncelet, 2016). Climatic variables are mainly from SAFRAN data (Vidal et al., 2010), while soil data are from the European Soil Database (Panagos, 2006).

## 3.2. Hydrometric data sampling for validation

To evaluate the performance of the different regionalization methods, we calibrate these methods on calibration catchments and validate the assumptions on validation catchments. When calibrating and regionalizing the methods, validation catchments are considered as ungauged catchments and their data is not used.

Three catchment samples were taken to discuss different issues:

1. **"LB/SB" sampling:** The selection of donors (calibration catchments) and receivers (validation catchments) depends on a criterion of area and location (Ding and Arnaud, 2022). Performance validation is carried out in the smallest catchments (areas less than 70 km²) within the large catchments used for calibration. We focus on the ability of the methods to interpolate hazards at a scale finer than the calibration scale within the gauged catchments.

2. **"50/50" sampling:** The selection of donors is completely random. Receiving catchments are hence also random (remaining catchments). This type of sampling is used to test the ability to extrapolate spatially from gauged to ungauged areas across all scales.

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Mapping of the 10 hydrographic zones.</figcaption></figure>
<table><caption>Fig. 6. Environmental descriptors selected for regionalization.</caption><thead><tr><th>Name</th><th>Explanation</th><th>Name</th><th>Explanation</th><th>Name</th><th>Explanation</th></tr></thead><tbody><tr><td colspan="2">Geographical</td><td colspan="2">Meteorological</td><td colspan="2">Hydrogeological</td></tr><tr><td>Alt</td><td>Altitude (IGN) [m]</td><td>Snow<br>(3 periods)</td><td>Cumulative snowfall [mm]<br>(annual, winter, summer)</td><td>RUP</td><td>Potential usable reserve [mm]</td></tr><tr><td>Slope</td><td>Slope [m/km]</td><td>Rain<br>(3 periods)</td><td>Cumulative rainfall [mm]<br>(annual, winter, summer)</td><td>SAJ<br>(3 periods)</td><td>Soil moisture index<br>(annual, winter, summer)</td></tr><tr><td>Arable</td><td>Agricultural coverage [%]</td><td>Temp<br>(3 periods)</td><td>Average temperature [mm]<br>(annual, winter, summer)</td><td>SAJ20<br>(3 periods)</td><td>Soil moisture index before<br>rainfall events<br>(annual, winter, summer)</td></tr><tr><td>Forest</td><td>Forest coverage [%]</td><td>Duration<br>(2 periods)</td><td>Average rainfall duration [h]<br>(winter, summer)</td><td>VHcapa</td><td>High storage capacity basin<br>percentage</td></tr><tr><td>Grassland</td><td>Grassland / herbaceous<br>vegetation coverage [%]</td><td>NE<br>(2 periods)</td><td>Average number of rainfall<br>events<br>(winter, summer)</td><td>MEDcapa</td><td>Medium storage capacity basin<br>percentage</td></tr><tr><td>DDr</td><td>Drainage network density</td><td>PJ<br>(2 periods)</td><td>Average rainfall intensity<br>[mm/h]<br>(winter, summer)</td><td>Vlcapa</td><td>Low storage capacity basin<br>percentage</td></tr><tr><td>Profsol</td><td>Soil depth [mm]</td><td>Arid</td><td>Aridity (ETP/Rainfall) [%]</td><td>AHg</td><td>Average size of SHYREG<br>production reservoir based on<br>hydrogeology [mm]</td></tr><tr><td>CLC</td><td>Land use quantified by<br>imposing the impact of each<br>class</td><td>ETP<br>(2 periods)</td><td>Evapotranspiration [mm]<br>(winter, summer)</td><td>HGrock</td><td>Percentage of "rock"<br>hydrogeological class (hard<br>soils)</td></tr><tr><td>HGwet</td><td>Percentage of "wet"<br>hydrogeological class</td><td>HGsand</td><td>Percentage of "sandy"<br>hydrogeological class</td><td>HGlow</td><td>Percentage of low storage<br>capacity class</td></tr></tbody></table>

3. “50/50_spa” sampling: Donor selection follows a geographic criterion based on the idea of nested (Graeff et al., 2012; Penna et al., 2017). In this case, catchments are clustered into nested catchment groups (common final outlet) and half of the groups are randomly selected to calibrate the methods. The remaining catchment groups are used for validation. All validation catchments are therefore geographically outside the calibration catchments. No information is available upstream or downstream of the validation catchments. This type of sampling allows us to assess the spatial extrapolation capability of our methods.

<figure><img src="image_10.png" /><figcaption>Fig. 7. location of calibration and validation catchments outlets.</figcaption></figure>
The locations of calibration and validation catchment outlets are shown in Fig. 7.

### 3.3. Evaluation criteria

The performance of the different regionalization methods is evaluated according to validation catchments. Deviation criteria are calculated between “local” quantiles (Qlocal_GEV) and “regional” quantiles calculated using different methods for the two approaches (Qregion_GEV and Qregion_SHYREG). It is assumed here that local quantiles, calculated for each maximum flow duration using a GEV distribution, are the most “accurate” hazard estimates for current return periods ($T \le 10$ years) to be associated with “observed” quantiles.

Index i is used to represent one of the n catchments, and index j represents one of the m quantiles of interest. Thus, Qlocalᵢⱼ represents quantile j (for example, 2-year peak flow) of catchment i. For the validation catchments, we have a set of “local” flow quantiles from observation, denoted Qlocalᵢⱼ, ..., Qlocalᵢⱼⱼ, which can be compared with a set of “regional” flow quantiles from regional estimation, denoted Qregionᵢⱼ, ..., Qregionᵢⱼⱼ. The following criteria are used to describe the differences between these two estimates:

1. Mean relative bias (Bias) between Qlocal and Qregion for each catchment using Formula 2. For a catchment i, its Bias is equal to the average of the relative error for all quantiles of interest, denoted Biasᵢ. This criterion is used to verify the existence of a systematic bias, as well as the direction of this bias (overestimation or underestimation in general).

$$ \text{Bias}_i = \frac{1}{m} \times \sum_{j=1}^{m} \frac{Q_{\text{local}}^j - Q_{\text{region}}^j}{Q_{\text{local}}^j} $$

Formula 2: Formula for the Bias criterion

2. Mean absolute relative errors (ER) using Formula 3. This criterion makes it possible to individually evaluate the overall magnitude of error for each catchment.

$$ \text{ER}_i = \frac{1}{m} \times \sum_{j=1}^{m} \frac{|Q_{\text{local}}^j - Q_{\text{region}}^j|}{Q_{\text{local}}^j} $$

Formula 3: Formula for the ER criterionThe Bias and ER criteria describe the similarity between the Qregion and Qlocal quantiles of each catchment. Boxplots are used to present their frequency distribution. In each boxplot, the values at the 5th, 25th, 50th, 75th, and 95th percentiles are displayed.

3. We can also evaluate estimation deviations for each quantile. An NSEj criterion can be calculated for a given quantile j.

$$ \text{NSE}^j = 1 - \frac{\sum_{i=1}^{n} (Q_{\text{local}}^i - Q_{\text{region}}^i)^2}{\sum_{i=1}^{n} (Q_{\text{local}}^i - \overline{Q_{\text{local}}})^2} $$

Formula 4: Formula for the NSE criterion

where Qlocalj is the mean value of Qlocalj from all n catchments

Formula 4 produces a Nash-Sutcliffe Efficiency (NSE) criterion ranging from -∞ to 1 (Nash and Sutcliffe, 1970). For the sake of clarity, we calculate the C2M criterion (Mathevet et al., 2006), which is a bounded version of the NSE criterion that ranges from -1 to 1.

$$ C2M^j = \frac{NSE^j}{2 - NSE^j} $$

Formula 5. : Formula for C2M based on the NSE criterion

Since tests are conducted on catchments with surfaces ranging from 1–10,000 km², the criteria can be greatly by the flow quantiles of large catchments, while the performance of the methods on small ones is poorly expressed (Krause et al., 2005). Therefore, these criteria are calculated on pseudo-specific flows, using a transformation Formula 6 to make them uncorrelated to catchment scales.

$$ Q_{\text{Specific}} = Q/S^{0.8} $$

Formula 6. : Formula for specific flow quantiles
where *S* is catchment area

The main difference between *ER* and *C2M* is the type of error that is better presented.

Assuming that a catchment with large specific flows is called a “productive” catchment, the *C2M* criterion is actually very sensitive to the estimation error of these catchments. For example, a 10 % estimation error of a very productive catchment has much more influence compared to that of a less productive catchment, since this criterion is based only on the global observation fluctuation. From this perspective, we can describe the *C2M* criterion as one that reflects the total amount of error.

On the other hand, the *ER* (or *Bias*) criterion has an opposite feature. It always reflects the percentage error. For a less productive catchment, since the observed runoff is already small, an extremely high percentage error (such as 200 % in extreme circumstances) may represent a negligible amount of error in terms of its actual value. Therefore, it is preferable to examine the distribution of the *ER* criteria rather than simply averaging them.

# 4. Results and discussion

## 4.1. Quantiles estimation in calibration catchments

The SHYREG simulation approach is interesting in its ability to provide comprehensive information on hydrological hazards from a single parameterization based on the S₀/A parameter. To assess the viability of this methodology, we look at its ability to reconstruct flow quantiles during the local calibration phase (when *S₀/A_local* is calculated). The graphs in Fig. 8 show the *C2M*, *Bias* and *ER* criteria for the SHYREG method calibrated with all gauged catchments. These correspond to the quantiles simulated by the SHYREG approach calibrated with local data (*Qlocal_SHYREG*). The referenced quantiles are *Qlocal_GEV*, considered to be the most “accurate” quantile estimation in gauged catchments for low return periods.

Despite simplification assumptions made in the approach, it appears that all flow quantiles are properly estimated using the calibration of a single parameter S₀/A. Daily flows appear to be better reproduced than peak flows. Local estimates are not particularly biased, as flood quantiles are sometimes underestimated and sometimes overestimated by small amounts. The mean absolute relative error (*ER*) is of the order of 0.10, while the *C2M* is between 0.8 and 0.9.

To provide a visual understanding of how residual can impact quantile estimations, Fig. 9 depicts a closer look at parameter predictions for a mountainous region in France (zone 3, Fig. 5).

Boxplot in Fig. 9 indicates that the **RF** method requires very few additional residuals to find local values for calibration basins. This implies that the spatial variability of the parameter described by this method is precise enough to reproduce the calibration catchments quantiles. The regression-based models (**NN** and **LR**) have notable residuals on the calibration catchments and thus require attention. Moreover, we note that the **LR** method tends to underestimate the parameter for this area, ultimately leading to an underestimation of quantiles if these residuals are not accounted for.

The most important environmental descriptors of this region are presented in Table 1. The **RF** method determines descriptor importance by measuring the average decrease in accuracy with off-sample data (data not used during tree training) (Bylander, 2002). The **NN** method assesses descriptor importance based on the average relative importance of neuron parameters (Garson, 1991; Goh, 1995). The **LR** method selects descriptors based on the AIC principle (Akaike, 1998). Descriptor definitions can be found in Fig. 6.

In this section, a mapping of hydrological model parameters is presented to demonstrate that different models trained with the same data set can lead to very different regional estimation, which is likely to affect the performance in validation catchments. The same can be observed for the GEV approach.

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** C2M, *Bias* and *ER* criteria between *Qlocal_GEV* and *Qlocal_SHYREG* for all gauged catchments (1929 catchments).</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 9. Parameter prediction using different regionalization methods for a mountainous region in France (50/50 sampling).</figcaption></figure>

<caption>**Table 1**</caption>

Most important environmental descriptors for each method using a 50/50 sampling of the related region.

<table><thead><tr><th>RF</th><th>NN</th><th>LR</th></tr></thead><tbody><tr><td>Duration</td><td>Snow</td><td>SAJ</td></tr><tr><td>PJ</td><td>Slope</td><td>PJ</td></tr><tr><td>AHg</td><td>Altitude</td><td>Grassland</td></tr><tr><td>Slope</td><td>NE</td><td></td></tr><tr><td>Arable</td><td>PJ</td><td></td></tr></tbody></table>

## 4.2. Quantiles estimation in validation catchments

One may question the influence of residuals on validation performances. Theoretically, adding residuals may help to interpolate information that is not explained by the learning method to ungauged areas. To verify this, Fig. 10 shows C2M criteria calculated on Qregion with and without residuals for both calibration and validation catchments using the SHYREG approach.

The term “Pred” refers to the prediction performance without residual, while “Pred+Res” refers to the prediction performance with interpolated residual. “(C)” indicates that criteria are calculated for calibration catchments, while “(V)” indicates that criteria are calculated for validation catchments.

Taking into account residual is important to correct quantile simulation of calibration catchments. The LR method, which is based on a relatively simple assumption, relies heavily on residual for both calibration and validation sides. The explanation provided by this method is so limited that results are close to those of the IDW method. Remaining information in the form of residual provides extra gains through interpolation. It demonstrates the weak point of classical regionalization methods, whereas a machine-learning method can extract more information from exhaustive and complex data.

Residual interpolation for the RF and NN methods tends to decrease their performance in validation. One can argue that it is possible to keep residuals only for calibration catchments, but those residuals will not be interpolated (by IDW algorithm) towards ungauged areas, so that the performance of validation catchments might be less influenced. This is partially true if we had only considered NN method for the SHYREG approach, but the residual interpolation can have positive impact for other configurations (like for the LR method). A further discussion is presented in Section 4.4. For the sake of simplicity and in order to focus on the main topic, interpolated residual has been left for further analysis.

We now examine the performance of the different regionalization methods with residual by comparing the flood quantiles determined regionally (Qregion_SHYREG and Qregion_GEV) with the flood quantiles determined locally (Qlocal_GEV) in the validation catchments. We begin with the C2M criteria shown in Fig. 11, which focus more on the estimation of larger specific quantiles.

Performance differs between samplings. The simulation-based SHYREG approach provides reliably consistent results, while the
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** C2M criteria between Qregion and Qlocal using the RF/NN/LR/IDW method for both calibration and validation catchments with or without intrapolated residual for the SHYREG approach.</figcaption></figure>

statistical-based GEV approach also provides good results, especially for the **50/50** sampling, but these are more variable.

The results of the IDW_SHYREG and IDW_GEV configurations enable us to evaluate the contributions of environmental variables. Both configurations are based only on geographical similarity, since the distance between catchments explains their similarity.

For the SHYREG approach, the contribution of environmental variables is not obvious. Although improvements can be seen for some samples and learning methods, results with geographic similarity alone are often equivalent. This is due to the fact that locally calibrated parameters remain a good descriptor for neighboring catchments, especially when rainfall information is known regionally.

For the GEV approach, environmental variables make a more significant contribution as they offer precipitation information that is only partially available through information on flow quantiles.

For regionalization under the more complicated conditions, like LB/SB or 50/50_spa sampling, the RF method always demonstrates good stability for both approaches. This indicates that when there is limited information, the RF method is a better choice, since it can generate a sounder explanation for parameter variability.

We then look at relative error criteria *Bias* and *ER* in Fig. 12, which focus more on the estimation of smaller specific quantiles.
<figure><img src="image_14.png" /><figcaption>Fig. 11. C2M criteria of all tested methods for validation catchments.</figcaption></figure>

The performance of the SHYREG approach always seems better. It is quite hard to distinguish performances for different regionalization methods implemented for the SHYREG approach, which is consistent with observations based on C2M criteria.

The GEV approach tends to overestimate flow quantiles. In most cases, the LR and IDW methods have the worst performances. Bias criteria are either more scattered or not as close to zero, and ER criteria have larger median values, which means a more significant average error. This demonstrates that the performance of statistical approaches relies very much on regionalization methods. Choosing the regionalization method is therefore crucial for statistical-based regional flood frequency analysis, and machine learning methods appear to be a more suitable solution, especially for very difficult scenarios (50/50_spa sampling), where no runoff observation at all is available over a large area.

Since the study is carried out on all 1929 catchments with very different surface areas, we are interested in surface effects on regionalization methods. Validation catchments, obtained through 50/50 and 50/50_spa sampling, are categorized into four classes based on their surface area. Each group comprises approximately 25 % of the validation catchments respectively. ER criteria are then calculated for each class, as shown in Fig. 13.

We find that the estimation of flow quantiles is easier for largest catchments than smaller ones, with poorer criteria for the latter. This is linked to the fact that the variability of flow quantiles in large catchments could be better explained by regional "indicators" (environmental variables or geographical proximity). Regional approaches struggle to capture the specificity of small catchments and

<figure><img src="image_15.png" /><figcaption>Fig. 12. Bias and ER criteria of all tested methods for validation catchments.</figcaption></figure>
tend to find average values over a large area. This conclusion is valid for both simulation and statistical approaches, regardless of the implemented regionalization method.

### 4.3. Robustness of regionalization methods

Frequency extrapolations provided by the two tested approaches (quantile estimations for return periods greater than 50 years) are specific to each approach and may differ. As stated in the methodology section, frequency extrapolation is already an issue for gauged sites, and it is difficult to determine reference values to validate the regional estimation according to different methods. Unlike for current quantiles, we will not assume that the extrapolated local quantiles provided by the statistical method (*Qlocal_GEV*) are reference quantiles. The purpose of this section is to assess regionalization methods for extrapolating extreme quantiles. To achieve this, we will compare local estimations using one approach (*Qlocal_SHYREG* or *Qlocal_GEV*) with regional estimations using the same one. In this manner, validation criteria will be calculated based on their own local values provided by each approach. This will allow to understand the soundness of different regionalization methods. Fig. 14 presents the *C2M* criteria.

In the context of frequency extrapolation, different regionalization methods lead to very similar results for the simulation approach. It shows that extreme quantiles provided by the regionalized approach are very close to those provided by the local approach. This finding particularly holds for high return periods (such as the 1000-year return period). In this case, quantile estimation become essentially dependent on precipitation information.

In fact, for extreme rainfall events, associated with long, heavy cumulative precipitation, a reservoir is rapidly saturated to such an extent that generated extreme floods depend less on the value of this hydrological parameter than on modeled extreme rainfall values. el. In this context, *RF* and *NN* methods seem to be slightly better than the *LR* method.

The direct regionalization of extreme frequency quantiles using a statistical approach is much more complicated. Due to classic problems associated with data sampling and greater uncertainty in extreme quantile estimations, the spatial variability of local quantiles increases with return periods. Therefore, regionalizing these quantiles is challenging, as their spatial variability is more reliant on soundness issues in local estimations than on the actual spatial variability of natural processes leading to hydrological hazards. This makes regionalization methods more prone to failure and less sound in their predictions. In this context, the *RF* method appears to be the most resilient solution to the issue. Regionalizing GEV distribution parameters instead of regionalizing flow quantiles would also minimize this issue.

<figure><img src="image_16.png" /><figcaption>Fig. 13. *ER* criteria for validation catchments of each surface area class.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 14. C2M of 2, 5, 10, 50, 100 and 1000-year quantiles estimation for validation catchments.</figcaption></figure>

## 4.4. Overall performance

All test results are summarized in Fig. 15. For each method, we calculate the median for **C2M**, **ER**, and **Bias** criteria for the validation basins of each sampling. This also provides an opportunity to reconsider the question of residual, which was put aside at the beginning of the discussion. Criteria are thus also calculated for versions “without residual interpolation,” meaning that the interpolation of residuals was not carried out at the end of the calculation. Red indicates a criterion’s worst performance, and green is for the best performance.

The choice of a regionalization method is less important for a simulation approach, since rainfall explains much of the variability in hydrological hazards. Nevertheless, implementing machine learning techniques, particularly the **RF** method, makes the approach more stable for all samples. For a statistical approach that cannot directly link rainfall information to a hydrological hazard, the choice of a given regionalization method is more important. Moving from the classic method (pure geographic proximity **IDW** or **RL**) to the machine learning method (**RF** or **NN**) clearly improves prediction accuracy. These more complex methods take into account rainfall information more effectively, resulting in better performances. This also reminds us of the importance of including precipitation variability for regional hydrological hazard estimations.

The application of residual spatial interpolation seems necessary for methods that poorly explain hazard variability by environmental variables, such as the **RL** method. Geographical proximity can then offer supplementary information. On the other hand, methods that provide a good explanation of the parameter variability do not require the use of residual interpolation, as in the case of the **RF** and **NN** methods. These methods require significantly smaller residuals than the **RL** method, as shown in Fig. 9. Additionally, they allow for more explanatory variables to be considered, including non-linear relationships, which can better account for parameter variability during calibration. However, this does not always guarantee good parameter explicability, as demonstrated by our validation performance.

## 5. Conclusion

This paper introduces two supervised machine learning techniques (**RF** and **NN**) and a traditional regionalization method (**RL**), implemented for the purpose of Regional Flood Frequency Analysis (RFFA).

We regionalized a hydrological parameter associated with a flood predetermination approach based on process simulation (the SHYREG approach). Regionalizing this single parameter allows the approach to provide a regional estimation of flood quantiles for different durations and return periods.

These three regionalization methods are also used to directly regionalize flood quantiles obtained by fitting a probability distribution (GEV). The statistical approach involves estimating local flood quantiles that are regionalized independently, leading to more precise local quantile estimation than the simulation approach, due to a greater number of calibration parameters.

These learning methods explain spatial variability of RFFA parameters by finding relationships between locally calibrated variables and the environmental characteristics of catchments. However, it is worth noting that such methods may not always produce optimized local values. In such cases, a residual, which represents the difference between the explained value (by model) and the optimal local value, is computed. This residual is interpolated to ungauged areas via geographic proximity (IDW) and added to the variability explanation provided by the learning procedures.

Supervised machine learning methods seem to be better suited for the regionalization of RFFA parameters compared with more classical multiple linear regression methods. They result in better estimation of regionalized values and, above all, in a better determination of their relationships with environmental descriptors.

The various samplings employed offer further insights into the capabilities of different learning methods. Differences arise between methods when tests are conducted on more “challenging” samples, such as for LB/SB and 50/50_spa. When information is limited, the RF method performs better, as it can find a stronger explanation for the spatial variability of the parameters/variables.
<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Validation performance considering only physiographic similarities or adding geographic proximity for both approaches using different learning methods.</figcaption></figure>

When regionalizing quantiles with significant spatial variability (especially rare frequency quantiles), the performance of the **NN** method is not as stable as that of the **RF** method. This is likely due to the nature of the **NN** method, which is essentially an adjusted complex combination of nonlinear regressions.

In addition, we observed that using residual interpolation via the IDW method as complementary regional information for ungauged areas is not always necessary. This is particularly evident when the regionalization method can already provide a good
regional explanation, as is the case for *NN* and *RF* applications.

Choosing the right regionalization method is crucial for statistical-based RFFA, especially for very difficult scenarios, where no runoff observation at all is available over large areas. For the SHYREG approach, the contribution of environmental variables is not as obvious as for the statistical approach. Although improvements can be seen for some samples and learning methods, results with geographic similarity alone are often equivalent. This is because locally calibrated parameters are still good descriptors for neighboring catchments, especially when rainfall information is known regionally.

In general, according to these analyses, the Random Forest machine learning method shows interesting performances because it manages to consider non-trivial information from environmental variables. This becomes more apparent when calibration information is less spatially informative, which is the case when spatially extrapolating approaches outside of gauged catchments. The Neural Network method may perform well in some cases, but it lacks stability. This may be due to its multiple nonlinear regression mechanism.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data Availability

Data will be made available on request.

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