A stochastic model of continuous hourly areal rainfall series applied to a wide range of French catchments

A stochastic model of continuous hourly areal rainfall series applied to a wide range of French catchments | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A stochastic model of continuous hourly areal rainfall series applied to a wide range of French catchments Patrick Arnaud, Philippe Cantet This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8594020/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 9 You are reading this latest preprint version Abstract This study introduces a stochastic continuous hourly areal rainfall model designed to simulate rainfall time series for hydrological risk management. The rainfall model, named SCHYPRE (Simulation of Continuous HYetographs for Predictive Risk Estimation), extends an established event-based rainfall model to basin-scale applications, integrating both extreme event modeling and continuous simulation of seasonal and long-duration rainfall patterns. The rainfall model was calibrated using 28.5 years of rainfall data set with hourly and kilometric resolution across 2,108 catchments in France, covering diverse climatic regimes from continental to Mediterranean and mountainous. The evaluation framework demonstrates rainfall model’s ability to faithfully reproduce observed rainfall statistics, including mean and extreme values, seasonality, autocorrelation, and intermittency. Frequency analyses conducted over durations from one hour to one year show strong agreement between the simulations and the adapted law, with only limited bias in the estimation of extreme values. A major advantage of rainfall modelling is its robustness in estimating extreme quantiles. Unlike traditional probabilistic methods, which are more sensitive to sampling variability, the rainfall model’s Monte Carlo approach, calibrated on large observational datasets of interne variables, ensures stable quantile estimation across all return periods, including extremes. Additionally, rainfall modelling inherently avoids quantile crossing inconsistencies, a common issue in independent duration-based probabilistic modeling. Stochastic rainfall simulation hourly areal rainfall multi-duration estimations seasonality and extremes Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. Introduction Rainfall constitutes the primary driver of numerous natural hazards, including flash floods, debris flows, urban flooding due to surface runoff, river overflow inundations, groundwater flooding, soil erosion and landslides. Effective management of risks associated with intense or prolonged heavy rainfall necessitates estimating their probability of occurrence, specifically through the determination of key characteristics, such as intensity or volume associated with predefined durations, linked to specific annual exceedance probabilities. The rainfall characteristics under investigation may vary depending on the specific hazard of interest. The analysis of rainfall-related hazards can be conducted through a range of methodological approaches. These span from frequency analysis, which facilitates the estimation of the frequency distribution of rainfall characteristics to construct design rainfall events ( Coles, 2001 ; Hosking and Wallis, 1993 ; Katz et al., 2002 ; Merz and Blöschl, 2005 ) , to more sophisticated stochastic simulation techniques for generating rainfall scenarios or extended time series (Cadavid et al., 1991 ; Onof et al., 1995 ; Cameron et al., 1999 ; Odry and Arnaud, 2017 ; Blazkova et al., 2017 ). The choice of methodologies depends on data accessibility and the defined research aims (Castellarin et al., 2012 ; Kochanek et al., 2014 ). In all cases, the overarching objective is to associate a rainfall scenario with a probability of occurrence. Such scenarios can subsequently be integrated into hydrological models to generate design floods, thereby facilitating the hydraulic computations required for designing hydraulic infrastructure (e.g., dam spillways, dikes, and reservoirs) or assessing flood risks that may impact infrastructure and populations (Myers, 1967 ; Chow et al., 1988 ; Clarke, 2008 ; Lavabre et al., 2010 ; Drobot et al., 2021 ). Stochastic simulation methods confer a distinct advantage by mitigating certain assumptions inherent in the estimation of a singular design rainfall event. For instance, they circumvent the simplistic assignment of an identical return period to both a rainfall event and the resultant flood ( Gräler et al., 2013 ; Kuczera et al., 2006 ; Vangelis et al., 2022 ). These approaches are founded on the generation of rainfall time series over extended periods, which can be transformed into long-term simulated flow time series using hydrological models (Lamb et al., 2016 ). This process facilitates the incorporation of antecedent hydrological conditions (Boughton and Droop, 2003 ). The simulated flow series can then be directly employed to perform hydraulic calculations for flood modeling or the sizing of hydraulic structures, while integrating operational management rules (Carvajal et al., 2009 ). By rejecting the conventional reliance on design rainfall and flood events, and by considering all possible flood scenarios, these methods enable a direct assessment of the frequency distribution of hydrological hazard impacts (e.g., flooded areas, dike overflows or dam water level) (Arnaud and Lavabre, 2002 ; Paquet et al., 2013 ). For these approaches to be effective, the rainfall models must exhibit robust and accurate performance in reproducing rainfall characteristics, particularly in simulating extreme events (Callau Poduje and Haberlandt, 2017 ; Evin et al., 2018 ; Nguyen et al., 2023 ). Numerous techniques exist for simulating rainfall, each addressing distinct requirements. These include for example, resampling-based approaches, e.g., Stochastic Storm Transposition (Wright et al., 2020 ), Markov chain models for simulating sequences of wet and dry periods (Gabriel and Neumann, 1962 ; Buishand, 1978 ; Mehrotra and Sharma, 2005 ; Evin et al., 2018 ), Poisson cluster models such as Bartlett-Lewis rectangular pulse models (Onof et al., 1995 ; Cameron et al., 2000 ; Ritschel et al., 2017 ) and Neyman-Scott rectangular pulse models (Neyman and Scott, 1958 ; Rodriguez-Iturbe et al., 1987 ; Cowpertwait, 1991 ), scaling factor methods based on multifractal theory(Mandelbrot and Wallis, 1969 ; Deidda, 1999 ; Langousis et al., 2009 ; Ramanathan et al., 2022 ), temporal disaggregation models which leverage the more abundant information available from daily rainfall data (Lebel et al., 1998 ; Burian and Durran, 2002 ; Hingray and Ben Haha, 2005 ; Gyasi-Agyei and Mahbub, 2007 ; Bhattacharyya and Saha, 2023 ). The emergence of advanced computational resources has driven substantial progress in the development of rainfall models leading to the creation of user-friendly tools (Burton et al., 2008 ; De Luca and Petroselli, 2021 ; Dykman et al., 2024 ). However, existing models often address only a subset of the complexities inherent in rainfall dynamics. This limitation is rooted in two fundamental challenges. The first challenge is related to temporal resolution, since many models operate at fixed time steps (e.g., daily or hourly) (Srikanthan and McMahon, 2001 ). Given the relative scarcity of sub-daily data, both in terms of recording duration and spatial density, many rainfall models have historically been developed at the daily time step (Baxevani and Lennartsson, 2015 ; Bennett et al., 2018 ; Gao et al., 2018 ). This is particularly critical when analyzing floods in relatively small catchments (< 1000 km²) or urban environments, where sub-daily time steps are essential to account for the rapid kinetics of flow processes. At the hourly scale, rainfall models are less common. The intricate structure of rainfall at fine temporal resolutions has led some approaches to focus on individual rainfall events (event-based approaches) (Rodriguez-Iturbe et al., 1987 ; Acreman, 1990 ; Cernesson et al., 1996 ; Arnaud et al., 2007 ). However, these methods encounter difficulties in accounting for antecedent rainfall conditions when coupled with hydrological modeling. In contrast, continuous approaches offer the advantage of modeling entire time series, thereby capturing temporal variability across multiple scales (Blazkova et al., 2017 ; Park et al., 2018; Grimaldi et al., 2022 ). The second major challenge stems from the inherent spatial complexity of rainfall across varying temporal scales (Papalexiou, 2018). Effectively coupling rainfall model with hydrological models requires careful attention to the spatial dimensions of rainfall. While at-site simulations disregard spatial variability and focus on reproducing time series statistically consistent with rain gauge observations, they are only representative of highly localized conditions or small surrounding areas (a few square kilometers). To address larger spatial domains, like areal rainfall or basin rainfall, spatial reduction factors must be calculated and incorporated into rainfall data to account for spatial averaging effects. Multi-site methodologies synthesize simulations from various points to better reflect the spatial patterns of rainfall. Advances in rainfall field models offer a more comprehensive approach to capturing the spatial heterogeneity of rainfall over extensive regions, such as entire catchments (Papalexiou and Serinaldi, 2020 ; Cantet et al., 2025 ). These diverse rainfall simulation methods are commonly integrated with hydrological modeling to simulate flood events within catchments. While their performance in reproducing specific rainfall characteristics varies, each method exhibits distinct strengths and limitations. These are influenced by several factors, including the degree of complexity or simplification, temporal resolution (e.g., daily vs. sub-daily), spatial resolution (e.g., uniform vs. spatially distributed rainfall), and their ability to capture seasonality and accurately extrapolate extreme values (Nguyen et al., 2023 ). The present study focuses on the development of a continuous areal rainfall model operating at an hourly time step. This model extends an existing event-based hourly at-site rainfall model, originally developed within the SHYPRE framework (Arnaud and Lavabre, 2002 ). The objective is to leverage the demonstrated strengths of this rainfall model in estimating extreme rainfall at both local (Arnaud and Lavabre, 1999 ; Arnaud et al., 2007 ; Cantet and Arnaud, 2014 ) and regional scales (Arnaud et al., 2008 ; Carreau et al., 2013 ), while expanding its applicability to areal rainfall simulation. Particular emphasis is placed on improving the representation of rainfall seasonality to accurately reproduce maximum rainfall over extended durations. Addressing rainfall characteristics across all timescales is a central focus of this work and a recurrent challenge in rainfall model development (Paschalis et al., 2013 ; Park et al., 2018). The underlying concept involves adapting the existing event based model to simulate areal rainfall events and positioning these events within a continuous time series to produce a coherent series of hourly basin-scale rainfall data. These simulated hourly rainfall are intended to serve as inputs to continuous hydrological models, with the aim of accounting for both rainfall seasonality and extreme rainfall characteristics. The remainder of this article is organized as follows. Section 2 presents the rainfall data and catchments utilized in this study. Section 3 outlines the methodological tools and developments undertaken to achieve the continuous areal rainfall model, including validation metrics. Section 4 details the results of the rainfall model’s application to the data. Section 5 provides a discussion of the results and conclusions. 2. Data The rainfall data used in this study cover the entire metropolitan French territory, which is characterized by a wide diversity of climatic regimes. These include an oceanic climate in the west, a Mediterranean climate in the south, mountainous climates in the Alps and the Pyrenees, and a continental climate in the central and eastern parts of the country. Spatially distributed rainfall products were employed to enable systematic calibration of the rainfall model over a large number of catchments. These products integrate data from weather radars and ground-based rain gauge networks, specifically the COMEPHORE precipitation dataset (Champeaux et al., 2009 ; Tabary et al., 2012 ). This dataset provides spatially resolved total precipitation at an hourly time step with a 1 km × 1 km grid resolution. COMEPHORE currently represents the most advanced spatial precipitation product available at an hourly resolution in France, with data available from January 1, 1997, to June 30, 2025, corresponding to 28.5 years of observations ( https://meteo.data.gouv.fr/datasets/ and https://doi.org/10.25326/360 ). From these gridded precipitation data, hourly areal precipitation time series were derived for individual catchments. Catchments were selected based on the availability of hydrological data, ensuring the potential for future calibration of a lumped hydrological model and its coupling with the rainfall model to determine flood flow quantiles. An area threshold was applied, retaining only catchments smaller than 10,000 km², as lumped hydrological applications beyond this scale are deemed less appropriate at the hourly time step (Arnaud et al., 2002). In addition to area, catchments were excluded based on snow occurrence, as the COMEPHORE dataset does not distinguish between liquid and solid precipitation. Catchments with a snowfall fraction greater than 20% were removed, based on SAFRAN reanalysis data for 1958–2025 (Quintana-Seguí et al., 2008 ; Vidal et al., 2010 ). In the end, a set of 2,108 catchments is selected. For each catchment, a time series of hourly areal precipitation was computed, corresponding to the temporal sequence of precipitation averages across all pixels within the catchment boundary. This yielded 28.5-year time series of hourly areal precipitation for each catchment, derived from the COMEPHORE dataset. Figure 1 illustrates the spatial distribution of the studied catchments and associated characteristics of areal rainfall. The selected catchments provide a representative sample of the climatic diversity across France, encompassing a comprehensive range of gauged catchment areas (from 5 to 10,000 km², with a median of 220 km²) and varied rainfall regimes. The figure presents the following characteristics: (graph a) catchment locations and boundaries in relation to topographic relief, (graph b) seasonality index (described below), (graph c) percentage of zero rainfall (hourly areal rainfall < 0.01 mm), (graphs d and e) quantiles of 1-hour and 24-hour maximum rainfall for a 2-year return period, estimated via GEV distribution fitting (Coles et al., 2003 ), (graph f) mean annual rainfall, (graphs g and i) relationships between the aforementioned variables and catchment area. The data reveal substantial variability in rainfall characteristics that the rainfall model must accurately reproduce. Notably, high-altitude catchments prone to snowfall have been excluded. In terms of seasonality, a seasonality index is calculated using the following formula, \(\:I=\frac{{\sum\:}_{1}^{12}\left|{x}_{i}-\stackrel{\prime }{x}\right|}{{\sum\:}_{1}^{12}{x}_{i}}\) where x i is the average monthly rainfall for month i and \(\:\stackrel{\prime }{x}\) the mean of x i . This index simply measures the variability of the monthly values relative to their mean. A value near zero indicates low seasonality, while a higher index reflects stronger seasonality. Central and eastern France exhibit relatively uniform annual rainfall distribution (index values close to 0), characteristic of a continental climate, whereas coastal regions display more pronounced seasonality, reflecting oceanic and Mediterranean climates, in the west and the south, respectively. The proportion of zero rainfall highlights a sharp contrast between oceanic zones, where rainfall occurs during up to 50% of the time steps, and Mediterranean regions, which are characterized by a much lower frequency of rainy time steps, closer to 25%. Analysis of 1-hour and 24-hour maximum rainfall quantiles confirms that the Mediterranean region is subject to significantly more intense rainfall events compared to other areas. Finally, total annual rainfall is strongly correlated with elevation, with the highest accumulations observed in the Vosges, Jura, Cévennes, and Massif Central mountain ranges. The influence of catchment area on rainfall characteristics is also noteworthy. Integrating rainfall over the spatial extent of a catchment introduces smoothing of hourly rainfall intensities, as illsutrated in Fig. 1 g, where 1-hour rainfall quantiles is negatively correlated with the area of rainfall integration. A reduction in the variability of 1-hour maximum rainfall values is also observed as catchment area increases. The effect weakens for 24-hour rainfall and is negligible for annual totals. Thus, in addition to the intrinsic variability induced from climatology, the developed basin-areal rainfall model must deal with the variability induced by the spatial integration area of rainfall. 3. Methodology and Tools The continuous hourly basin-scale rainfall model is developed based on a previously established event-based hourly point rainfall model at INRAE. (Cernesson et al., 1996 ; Arnaud et al., 2007 ; Arnaud and Lavabre, 2010 ). The event-based rainfall model, named SHYPRE, has demonstrated accuracy and stability in estimating extreme rainfall, as documented in numerous studies (Arnaud and Lavabre, 1999 ; Neppel et al., 2007 ; Carreau et al., 2013 ; Lang et al., 2014 ; Kochanek et al., 2014 ; Arnaud et al., 2016 ). It is founded on an analysis of the geometric and statistical properties of observed local hourly rainfall time series. As an event-based model, it focuses exclusively on the most significant rainfall events within a time series, selected according to a threshold-exceedance criterion (detailed in § 3.1). In the present study, this event-based model is not applied to point-scale rainfall but directly to areal rainfall, with the aim of simulating basin-scale rainfall. The first challenge is to assess whether the approach developed for point-scale rainfall remains suitable for the smoother signal characteristic of basin-scale rainfall. Furthermore, to generate continuous hourly rainfall time series, the simulated rainfall events must be positioned within a continuous time series, including dry periods as well as periods with rainfall below the event-definition threshold. The process of modeling areal rainfall events and subsequently constructing a continuous series constitutes the new continuous rainfall model, designated SCHYPRE (Simulation of Continuous HYdrographs for Predictive Risk Estimation). The implementation of this rainfall model, intended for coupling with hydrological models, comprises two phases: 1) Parameter calibration, based on a descriptive analysis of observed rainfall time series and 2) Simulation of extended time series, accompanied by comparative analysis with observations. These phases are detailed in the following methodological sections, followed by a description of the analysis variables and the parameters governing their generation. 3.1 Rainfall Time Series Analysis The initial phase aims to define the primary variables that best describe the temporal structure of rainfall. The objective is to capture the complexity of rainfall variability at hourly scales, as well as across longer timescales (daily, monthly, etc.), while also representing the full frequency distribution from common to extreme values. This analysis of observed series is conducted at daily and hourly time scales (Fig. 2 ). At the daily scale, three distinct period types are identified: “Rainy events” (in red) are defined as sequences of consecutive days with daily rainfall less than 4 mm, including at least one day with rainfall exceeding 20 mm. “Non-event” rainfall periods (in green) consist of days with non-zero rainfall that do not meet the event criteria (i.e., sequences without a day exceeding 20 mm). “Dry periods” (in blue) are days with no rainfall. Four descriptive variables are defined at this daily scale: 1) the number of rainy events in a given month (NE), 2) the empirical frequency of the cumulative rainfall of an event (TER) relative to other seasonnal events (P TER ), 3) the depth of daily rainfall for non-events in each month (DR < 20 ) and 4) the proportion of dry days (areal rainfall < 0.01 mm) during non-event periods (P 0 ). Then, the hourly scale focuses only on rainy events, building upon prior studies related to the development of the event-based rainfall model. The analysis of hourly observations concentrates on rainy events containing the most significant rainfall data, which constrain the rainfall model’s asymptotic behavior toward extreme values. Rainy events at the hourly scale are described as sequences of wet periods, with the following characteristics determined by the number of rainy periods (NP) within an rainy event, the dry durations separating rainy periods (IPD) and the number of showers per rainy period, defined as a sequence of hourly rainfall with a single relative maximum (NS). Each shower is characterized by its duration (SD), its total volume (SV) and its form defined by the position of the maximum hourly rainfall within the shower (PX) and the ratio of the shower’s maximum hourly rainfall to the shower’s total volume (RX). The remaining shower volume (1-RX)*SV is then positioned to satisfy a shower form -ie- a single relative maximum.Showers within rainy events are divided into two subsamples. The “primary” showers (denoted with subscript p ) are the most intense showers in an rainy event. Their number per event (NpS) is defined as one plus an additional primary shower for each day in the rainy event with cumulative rainfall exceeding 50 mm. The “ordinary” showers (denoted with subscript o ) encompass all remaining showers in the event. Different probability distributions are employed to describe variables based on shower type. This analysis of hourly-scale rainy event characteristics is conducted over two seasons defined for metropolitan France. Season 1 (June–November) is characterized by typically shorter and more intense rainy events, whereas season 2 (December–May) is associated with longer and more regular precipitation events. The defined variables enable a comprehensive description of hourly rainfall time series. Each variable is characterized by a probability distribution used to generate values for reconstructing time series. These distributions are listed in Table 1 . Note that, since the last published version of this rainfall model (Cantet and Arnaud, 2014 ), modifications have been implemented to test new functions for modeling showers. Table 1 Summary of variables required to simulate continuous hourly rainfall time series, including their associated probability distributions and parameter estimation methods. Variable Description of the variable Unit Probability law Parameter estimation [1] Estimated for ... Variables required to simulate rainy events NE Number of rainy event per month - Poisson θ = µ 12 months NP Number of rainy period per rainy event - Geometric θ = 1/µ 2 seasons NS Number of shower per rainy period - Geometric θ = 1/µ 2 seasons NpS Number of “principal” shower [2] - Binomial θ = f (µ) [3] 2 seasons IPD Inter rainy period duration hours Truncated geometric θ 1 = 1/µ θ 2 = Prob(IPD > 12h) 2 seasons SD p et SD o Shower duration, for the two types of shower [2] hours ( p ) Truncated Poisson ( o ) Truncated Poisson θ 1 = µ θ 2 = Prob(SD p >9h) θ 1 = µ θ 2 = Prob(SD p >6h) 2 seasons and 2 types SV p et SV o Shower volume, for the two types of shower [2] mm ( p ) Truncated Log-normal ( o ) Truncated Gamma θ 1, 2, 3 = f(µ, σ, min) θ 1, 2, 3 = f(µ, σ, min) 2 seasons and 2 types RX p et RX o Ratio of shower maximun hourly rainfall over its volume, for the two kind of shower [2] - Normal θ 1 = f 1 (SD) [4] θ 2 = f 2 (SD) [4] 2 seasons and 2 types PX For a shower, relative position of the maximum hourly rainfall in the shower - Normal θ 1 = µ θ 2 = σ 2 seasons P TER Probability of the total event rainfall - Uniform θ = median 12 months τ p and τ o Kendall’s Tau for copula between duration and volume probabilities ( p ) Franck copula ( o ) Gumbel copula Kendall’s Tau 2 seasons and 2 types Variables required to simulate “non-event” rainfall periods DR < 20 Depth of non-event daily rainfall mm Exponential θ 1 , θ 2 optimised 12 months HR0 Proportion of hourly rainfall equal to zero within a non-event daily rainfall - Uniform θ = µ 12 months Variables required to simulate dry periods DR0 Probability of days with no rain during a non-even period - Uniform θ = µ 12 months [1] µ = mean of the variable values, σ = standart deviation of the variable values and min = minimum of the variable values [2] Only the variables SD, SV, and RX are considered for the two shower types. [3] The binomial distribution parameter describing the variable NpS​ is a regionally calibrated function of the variable's mean. Using our dataset, θ = 0,15. ln(µ NpS ) for season 1 (1) and θ = 0,122. ln(µ NpS ) for season 2. [4] The mean and standard deviation of the normal distribution used to randomly sample the variable RX are estimated as functions of shower duration SD. For each randomly drawn duration, the variable RX is generated from a normal distribution N(θ 1 , θ 2 ), where θ 1 = a + b ln(SD) and θ 2 = c + d. ln(SD). The parameters a, b, c et d are regionally optimized to match the observed mean and standard deviation of RX. These four parameters are determined separately for both seasons and both shower types. The parameters derived from? the two distinct seasons are linked to the simulation of seasonal rainfall events, reflecting the event-scale modeling approach of the rain model. Conversely, the parameters established for each of the twelve months are used for the continuous part of the rainfall model. Most parameters are estimated using the method of moments, based on the mean and standard deviation of observed variable values. The large sample sizes (hundreds of observations) enhance the stability of parameter estimation. For specific variables, parameter calibration is optimized. For instance, DR < 20 is modeled using a bounded exponential distribution (between 0.01 to 20 mm), with parameters optimized via least squares method to fit the empirical monthly distribution. RX variable follows a normal distribution, whose parameters are functions of shower duration. The coefficients of this function were calibrated globally across all observed rainfall time series (from hundreds of thousands of observations). The dependence between shower duration (SD) and volume (SV) is modeled using copulas, with parameters estimated in a dimensionless manner from multiple observed time series (Cantet and Arnaud, 2014 ). Seasonality is incorporated during the calibration of probability law parameters. Some parameters are determined for the two seasons defined for the rainy event analysis, while others are calibrated monthly (see Table 1 ). 3.2 Rainfall Time Series Simulations The second phase of rainfall model implementation involves simulating continuous time series. This process comprises tree steps: 1) the simulation of rainy events, 2) the generation of a continuous series by positioning the simulated rainy events within a temporal framework and 3) simulation and positioning of non-event rainfall and dry periods. 3.2.1 Rainy events simulations per one season: Independent rainy events are generated for a given season using a Monte Carlo approach, in which descriptive variables are randomly sampled to reconstruct events. Events are constructed shower by shower following a predefined sequence. First, the number of rainy periods (NP) is generated. For each rainy period, a number of showers (NS) and a dry duration to the next rainy period (IPD) are then generated. Each shower is classified as either primary (p) or ordinary (o), respecting the observed proportion (NpS). Finally, shower duration (SD), volume (SV), and shape characteristics (RX, PX) are generated, while inter-variable dependencies, such as those between shower duration and volume, are modeled using copulas (reference Cantet & Arnaud, 2014 ). At this stage, a set of simulated rainy events is available for each season and need to be positioned inside a continuous times series. 3.2.2 Positioning rainy events inside one season: This stage consists of positioning independent events within a continuous times series by assigning rainy events to months within the season, in accordance with the observed monthly probabilities (NE). The chronological event positions in a month are conditionned by the event’s cumulative rainfall to match the median monthly frequency (P TER ). After assigning rainfall events to months in a predefined order, the total non-event duration—derived from month duration minus total event durations and a minimum separation (e.g., 24 hours)—is randomly allocated within each month to finalize the placement of events. At this stage, the time series still needs to be completed by incorporating dry periods and non-event rainfall. 3.2.3 Simulation and positioning of non-even rainfall and dry periods: For each non-event day (dry or wet), a random number u∼U(0,1) is generated. The day is assigned to zero hourly rainfall if u < DR0. Otherwise, the day is assigned non-event daily rainfall (DR < 20 ), sampled from the corresponding probability distribution. The following temporal disaggregation procedure enables to distributes, in a simple way, daily non-event rainfall across 24 hours. First, weighting coefficients, α i=1 to 24 , are generated using a first-order autoregressive process AR(1). A percentage of zero hourly rainfall is also generated via HR0. The lowest HR0 values of α i are set to 0, and the remainder are normalized to have a sum equal to 1. With these coefficients daily rainfall is disaggregate into 24 hourly values. Note that, this procedure has minimal impact on extreme rainfall statistics, since it only applies to lower-intensity rainfall (< 20 mm/day). 3.3 Performance Validation The generated hourly rainfall time series must exhibit statistical characteristics equivalent to those of observed series. Validation relies on a suite of statistical criteria to assess the relevance of simulated rainfall for substitution in long-term and frequency-extrapolation analyses. These statistical criteria are compared between observed areal rainfall time series and 10,000-year simulated areal rainfall series. Monthly variables are employed to verify the reproduction of rainfall seasonality: Relative errors in mean daily rainfall, standard deviation, and the proportion of dry days are calculated. Seasonality index, ensuring distinction between low- and high-seasonality sites is calculated. Multi-Duration variables evaluates the rainfall model’s capacity to model both average and extreme values of time-aggregated rainfall: Mean and standard deviation of rainfall are computed for aggregation time steps ranging from 1 hour to 1 year. 10-year quantiles of multi-duration rainfall are estimated. Simulated quantiles are empirically estimated with the Hazen formula, without the need to fit a probability distribution. Observed rainfall quantiles are estimated by fitting a GEV distribution to the observed maximum values for rainfall durations ranging from 1 hour to 10 days, corresponding to those illustrated in Fig. 1 . For longer durations (more than 10 days), the log-normal distribution, which is a more suitable law, is used to estimate the observed quantiles. Ensuring that these rainfall characteristics are preserved across different durations allows us to verify the accurate reproduction of temporal rainfall features at various time steps from high-intensity short-duration events (1 hour to 3 days) to cumulative rainfall over multiple days or months. The temporal structure of rainfallis assessed by calculating the first-order autocorrelation of rainfall values aggregated over different time steps. Rainfall intermittency is also analysed by calculating transition probabilities between states (e.g., "dry" to "wet," "wet" to "wet," and "dry" to "dry"). This includes examining the likelihood of transitions from low to high rainfall intensities, as well as the persistence of high or low rainfall conditions over consecutive time steps, for different thresholds. All these characteristics ensure that the generated rainfall time series can be reliably used in hydrological modeling. Finally, to assess the stability of extreme quantile estimates, the rainfall model was calibrated over two distinct 14-year periods : Period P 1 (January 1, 1997, to February 28, 2011) and Period P 2 (March 1, 2011, to May 30, 2025). For each period, quantiles of cumulative rainfall over various durations are estimated for different return periods. Then, the SPAN index (Garavaglia et al., 2011 ) is calculated. It is defined as: \(\:{SPAN}_{T}=2\cdot\:\frac{\left|\left({q}_{T}\left({P}_{1}\right)-{q}_{T}\left({P}_{2}\right)\right)\right|}{\left({q}_{T}\left({P}_{1}\right)+{q}_{T}\left({P}_{2}\right)\right)}\) where \(\:{q}_{T}\left({P}_{i}\right)\) is the T-year quantile estimated from data in period P i . The stability increases as the SPAN index value converges toward zero, indicating greater consistency between the estimates derived from the two distinct time periods. 4. Results 4.1 Parameter values Calibration of the rainfall model involves estimating the parameters of probability distributions that describe the various rainfall descriptive variables (presented in Section 3.1 and Table 1 ). These parameters are estimated from observed values. Some parameters are determined locally, meaning that each catchment have its own set of local parameter estimated from its observed areal rainfall time series. An another set of regional parameter are common to all catchments, and is estimated by pooling all available areal time series. A set of 25 local parameters is computed for the two seasons defined for simulating rainy events. Most of these parameters are statistical moments (mean and standard deviation) calculated from samples of several hundred observations. Figure 3 a) illustrates the boxplots of the means of some variables. Note that means are easier to interpret than parameters themselves and are used to estimate parameters of distributions (such as geometric, Poisson and normal). These mean values exhibit high variability, reflecting the diversity of rainfall characteristics in the studied sample, including variations in intensity, duration, and structure. For example, the following seasonal differences are observed: In season 1 (June - November),, hourly areal rainy events generally consist of more showers (higher mean values of NP and NS), with longer shower durations (higher mean SD) but lower volumes (lower mean SV) and intensities (lower mean RX) compared to season 2 (December – May). Thus, season 2 is chracterized by shorter but more intense rainy events. The catchment area also influences basin-averaged rainfall characteristics. Correlation coefficients presented in Fig. 3 b) show that as the spatial integration area increases, rainy events tend to include more showers, which are longer but less intense (with RX negatively correlated with area). This reflects the "smoothing" effect of spatial averaging, where local intensity peaks are reduced by lower intensities across the integration area. This introduces additional complexity in basin-scale rainfall modeling, since the areal rainfall signal structure varies with the size of the averaging area. Another set of 6 parameters is monthly calculated for continuous rainfall simulation (Fig. 3 c). This monthly parameterization enables to take account the variation inside a season for different rainfall characteristics. The seasonality of heavy rainfall (defined by rainy events) is captured through the monthly distribution of the number of rainy events (NE) and the relative frequency of seasonal total rainfall (P TER ). The seasonality of lower rainfall (non-event rainfall periods) is represented by the two parameters defining the DR < 20 variable, while the seasonality of dry periods is accounted for by parameters associated with DR0 and HR0. A general trend is observed, with more frequent (higher NE) and intense (higher P TER ) rainy events in autumn, while the most pronounced dry periods occur in summer (higher DR0 and HR0). Finally, a set of regional parameters is determined by analyzing all available observations across the study area. These parameters are coefficients used to establish relationships between different parameters, allowing the estimation of local values. This applies to the parameters of the RX variable, which are conditioned by shower duration (Table 1 ), and to the parameter associated with the probability distribution of NpS. 4.2 Simulation setup Once calibrated, this complete set of parameters allows generating values for the rainfall characteristic variables. These simulated variables, obtained through random sampling, are then used to reconstruct long rainfall time series by progressively generating significant rainy events composed of multiple showers with varying durations, intensities, and shapes. These events are integrated into the time series and supplemented with periods of lower rainfall and dry spells. For each of the 2,108 studied catchments, rainfall data are simulated over a long period. The simulated time series are then compared to observed series based on the comparison criteria defined in Section 3.3 . The purpose of a stochastic rainfall model is to produce a more exhaustive sample of rainfall time series than those available from the observed dataset. By simulating very long time series that preserve the same statistical characteristics as observations, but with a greater number of possible realizations, extreme values can be estimated by simply extending the simulation duration. Under the assumption of rainfall stationarity, a continuous 100,000-year of hourly rainfall is assumed enough long to empirically estimate rainfall quantiles. The simulated time series are then compared to observed series based on the comparison criteria defined in Section 3.3 4.3 Reproducing seasonality First, the seasonality of rainfall is examined by analyzing monthly variables. Figure 4 a shows, for each month, the relative error in reproducing the mean of daily rainfall (µ(DR)), the standard deviation of daily rainfall (sd(DR)), and the proportion of days with no rainfall (% zero). The relative errors are generally small, mostly within ± 10% for rainfall moments and around − 5% for the proportion of days with no rainfall. Note that, the cases where the relative errors exceed 0,4 correspond to summer months and mediterranean catchments for which rainfall is extremely rare (mean close to 0). In these cases, the absolute error remains minimal. Importantly, these relative errors are largely independent of the month, indicating no bias in the seasonality of these variables. The seasonality index is also well reproduced, with a clear distinction between sites with low and high seasonality (see Fig. 4 b). 4.4 Reproducing rainfall statistics for different time scales Next, the reproduction of rainfall depth at different time scale ranging from 1 hour to 1 year, is assessed. Tree metrics are examined: mean, standard deviation, and 10-year return period quantiles of annual maximum values (see Section 3.3 ). Figure 5 represents scatterplots comparing observed and simulated cumulative rainfall values for durations of 1 hour, 6 hours, 24 hours, 7 days, 60 days, and 365 days. The results show excellent agreement in the means of rainfall across the entire range of aggregation scales, from hourly to annual. Relative errors in standard deviations and quantiles are also small. A slight tendency to underestimate 24-hour rainfall quantiles is observed. Overall, rainfall characteristics across different durations are well preserved across all temporal aggregation scales. The simulated areal rainfall also represents the wide range of intensities and cumulative rainfall amounts of observed sample. Rainfall depth at different time scales also exhibits no bias with respect to the catchment area. For example, using the variables presented in Fig. 1 , Fig. 6 illustrates the relationship between catchment area and the three validation metrics: quantiles of 1-hour and 24-hour maximum rainfall for a 2-year return period and mean annual rainfall. demonstrating that the simulated data consistently reflect the observed characteristics described in Section 2 . 4.5 Reproducing intermittency Finally, the temporal characteristics of rainfall are analysed at different aggregation time scale (from 1h to 72h). In this way, the first-order autocorrelation coefficient and rainfall intermittency indices are calculated for both simulations and observations. The results are represented in Fig. 7 . The simulations accurately reproduce the observed trends. Indeed, autocorrelation decreases with increasing aggregation duration similar to observation, ranging from values between 0.3 and 0.7 for hourly rainfall to values near 0 to 0.2 for longer durations (72 hours). About intermittency indices, the proportion of successive dry periods (dry-dry) decreases with rainfall duration, while transitions from "dry" to "wet" increase with rainfall duration. The simulated time series closely follow the observed trends. The selected validation metrics offer a comprehensive and robust characterization of hourly rainfall features, capturing the essential properties that the model must accurately reproduce. These variables encompass average and extreme rainfall amounts, seasonality, and the temporal structure of successive time steps across a range of aggregation scales from hourly to annual. The comparison of these variables, calculated from both simulated and observed time series, demonstrates the relevance of the rainfall model's results across a wide range of areal rainfall conditions, diverse climatic contexts, and a broad spectrum of catchments sizes. The long-term synthetic areal rainfall time series generated by the model reproduce the statistical properties of observed rainfall with high fidelity. Note that performance evaluation criteria are based on variables not directly used in parameter calibration. For example, none of the calibration variables are related to rainfall autocorrelation. 5 Discussion 5.1 robustness of simulations The objective of a Frequency Flood Analysis (FFA) method is to estimate the occurrence of extreme rainfall events. This estimation is highly sensitive to sampling, since, it relies on the observation of rare and thus infrequently recorded values. To assess the stability of these estimates, Garavaglia et al. ( 2011 ) propose using the SPAN index (Section 3.3 ). This index is calculated from quantiles estimated by a probability law (GEV or log-normal distributions applied on observed data) and those estimated by the rainfall simulation (empirical quantiles of simulated rainfall time seires by the SCHYPRE method). Figure 8 represents the SPAN index across the 2,108 catchments for different rainfall durations and return periods (2, 10, and 100-years), comparing quantiles estimated via the probability laws and the SCHYPRE method. For the 2-year return period, SPAN values are low and comparable between the two methods, falling within the range of sampling variability expected for median values. However, as more extreme return periods are considered, a more pronounced increase in the SPAN index is observed, particularly for methods based on probability laws. This reflects the sensitivity of three-parameter distributions like the GEV to sampling variability, particularly the presence or absence of extreme values in the calibration dataset. In contrast, according to SPAN values, the SCHYPRE method exhibits greater stability for extreme quantiles. This stability arises because the internal probability distributions describing shower characteristics are calibrated using a large number of realizations per year of observation, rather than relying on a single annual maximum value. Despite the higher number of parameters required to calibrate the rainfall model, the stability of the simulations is significantly greater than that obtained by fitting a probability distribution to estimate extreme rainfall values. 5.2 Quantile crossing Rainfall model’s complexity can be justified, since it enables the reproduction of continuous areal rainfall time series, thereby providing comprehensive information on all statistical characteristics of rainfall without requiring additional parameter estimations. For example, using a probability distribution to estimate rainfall quantiles typically necessitates calibrating a set of parameters for each rainfall duration studied. In the case presented in Fig. 8 , the GEV distribution (for durations d 10 days) were independently fitted for each duration to estimate quantiles. Even if these calibrations are independent across durations, consistency is expected among the estimated quantiles for different durations. However, "quantile crossing case" can appear : a quantile estimated for a duration (d₁) is greater than a quantile estimated for a longer duration (d2 > d1). By construction, this case is not possible in simulated quantiles. At worst, quantiles can be equal. Figure 9 shows the percentage of quantile crossing cases resulting from fitting probability distributions for different durations and return periods. Quantile crossing cases occur even for 5-year return periods and can reach up to 20% for the largest return periods and shorter duration.. Note that, for longer period (> 30days), there is no quantile crossing cases. This may be explained by the use of log-normal distribution rather than a GEV for the shorter duration. In contrast to the log-normal distribution, the GEV is a heavy tailed probability distribution, in which the shape parameter controls the highest return levels. However, estimation of the shape parameter is challenging, and remains uncertain due to sampling effects. Temporal consistency in rainfall quantile estimation is intrinsic to the SCHYPRE method, which does not produce cases of quantile crossing. Such inconsistencies can be significant when extrapolating frequencies using probability distributions, a well-known limitation that has led to the development of multi-duration statistical methods. These methods ensure the calibration of distributions that maintain consistency in quantile estimations (You and Tung, 2018; Fauer et al., 2021 ; Roksvåg et al., 2021 ). 5. Conclusion This study presents a stochastic model for continuous hourly areal rainfall designed to address hydrological risk management needs in France (SCHYPRE method). Building on a robust event-based approach, previously validated for point-scale rainfall, the SCHYPRE rainfall model extends these capabilities to areal rainfall modeling. By combining an event-based framework for extreme intensities with a continuous approach for seasonal and long-duration rainfall, the rainfall model faithfully reproduces rainfall characteristics across temporal scales ranging from hourly to annual. This avoids restrictive assumptions inherent in traditional methods, such as direct linkage between design rainfall and flood events, by generating long, realistic time series. Stability tests demonstrate that the rainfall model is less sensitive to sampling variability than parametric distributions (e.g., GEV), thanks to its calibration on a large number of realizations describing rainfall variables. Unlike methods that independently fit probability distributions for each duration, rainfall modeling ensures no quantile crossing between durations, thus maintaining the physical consistency critical for hydrological applications. Performances were assessed across 2,108 catchments, spanning diverse climates (oceanic, Mediterranean, continental, mountainous) and various catchments area (up to 10,000 km²). Results confirm accurate reproduction of observed statistics for both mean and extreme rainfall, as well as seasonal and temporal properties (autocorrelation, intermittency). The SCHYPRE rainfall model represents a significant advancement in stochastic hourly areal rainfall modeling, providing a robust alternative to traditional rainfall frequency analysis approaches. Its ability to generate long, coherent, and realistic time series enable their direct use as inputs to hydrological models, making it a valuable tool for hydrological studies, natural risk management, and infrastructure design. By avoiding issues such as quantile crossing or assumptions inherent in the estimation of a singular design rainfall event,, this method explicitly incorporates spatio-temporal complexity of precipitation into hydrological modeling enabling more reliable operational applications tailored to contemporary water management challenges. Future applications will include coupling with spatial disaggregation models to generate high-resolution rainfall fields (e.g., Cantet et al., 2025 ) to drive distributed hydrological models. Another application involves calibrating the rainfall model using climate model outputs to assess the impacts of climate change on extreme rainfall events following approach proposed by Cantet et al (2011). Declarations Author Contribution P. A. : Conceptualization, Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing, Supervision, Project administration.P.C. : Conceptualization, Methodology, Investigation, Formal Analysis, Review & Editing. Acknowledgement We gratefully acknowledge financial support for this study provided by Direction Générale des Prévisons des Risques (DGPR). Data Availability Comephore dataset is freely available on request at https://doi.org/10.25326/360 or at https://meteo.data.gouv.fr/datasets/669e23a7ce052a9e8521b75e References Acreman MC (1990) A simple stochastic model of hourly rainfall for Farnborough, England. Hydrol Sci J 35:119–148 Arnaud P, Lavabre J (1999) Using a stochastic model for generating hourly hyetographs to study extreme rainfalls. Hydrol Sci J -J Sci Hydrol 44:433–446 Arnaud P, Lavabre J (2002) Coupled rainfall model and discharge model for flood frequency estimation. Water Resour Res 38:1075–1085 Arnaud P, Lavabre J (2010) Estimation de l’aléa pluvial en France métropolitaine, 158 pages Arnaud P, Fine JA, Lavabre J (2007) An hourly rainfall generation model applicable to all types of climate. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8594020","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":577430189,"identity":"c5b593b0-3df0-48d8-bfd9-b969b379d877","order_by":0,"name":"Patrick 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16:11:39","extension":"xml","order_by":30,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":174589,"visible":true,"origin":"","legend":"","description":"","filename":"550a776fbf334333914a0910de1173b11structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/59c557fced547cf824214e5a.xml"},{"id":100813945,"identity":"065db54d-da5a-49d0-906c-24638ac3a121","added_by":"auto","created_at":"2026-01-21 16:11:42","extension":"html","order_by":31,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":185776,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/6507f88b082336235e62ddcd.html"},{"id":100813937,"identity":"e15868e7-70bf-4d8c-a725-309e8cc6567b","added_by":"auto","created_at":"2026-01-21 16:11:40","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":700922,"visible":true,"origin":"","legend":"\u003cp\u003eLocation of the 2,108 studied catchments and characteristics of their associated areal rainfall.\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/3cf961d925f1212ac1cdc821.png"},{"id":100813889,"identity":"b87439e2-2688-46d2-8305-b0ef8971c707","added_by":"auto","created_at":"2026-01-21 16:11:28","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":248887,"visible":true,"origin":"","legend":"\u003cp\u003eRainfall-based calibration of the continuous hourly areal rainfall model in the SCHYPRE Method: definition of rainfall descriptive variables and associated parameters.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/d326c01f82988f624f5fe011.png"},{"id":100814029,"identity":"d929cfeb-4b3f-472b-a9c6-837b3c5847ee","added_by":"auto","created_at":"2026-01-21 16:11:59","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":555346,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of parameters derived from observed basin rainfall time series: a) means of event-based variables, b) correlation between means and basin area, and c) distribution of monthly parameters.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/702bc386de6b37af226865d2.png"},{"id":100813943,"identity":"f52d5466-9e32-4194-823c-63f46b1ee778","added_by":"auto","created_at":"2026-01-21 16:11:41","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":543464,"visible":true,"origin":"","legend":"\u003cp\u003eSeasonality evaluation:a) monthly distributions of relative error in reproducing mean daily rainfall, standard deviation of daily rainfall, and proportion of dry days; b) reproduction of the seasonality index.\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/ab141981cb8abfeab6c52499.png"},{"id":100813935,"identity":"4425e407-f0ce-4dde-8174-56096810eca7","added_by":"auto","created_at":"2026-01-21 16:11:39","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":437923,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of mean, standard deviation, and 10-year return level quantiles: observed rainfall (x-axis) vs. simulated rainfall (y-axis) across hourly to annual aggregation scales.\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/34c9be9a1482e3a17d9658a2.png"},{"id":100813875,"identity":"c45b7621-3c90-4c7e-adee-2d7633185f8c","added_by":"auto","created_at":"2026-01-21 16:11:24","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":599010,"visible":true,"origin":"","legend":"\u003cp\u003erelationships between the basin area and tree variables of validation: quantiles of 1-hour and 24-hour maximum rainfall for a 2-year return period, and the mean annual rainfall.\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/c7e2bea9da814ed8edaaa08f.png"},{"id":100813961,"identity":"df274e41-d084-40a7-bc6e-25af0037f2f8","added_by":"auto","created_at":"2026-01-21 16:11:52","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":471863,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of first-order autocorrelation and intermittency metrics between observed rainfall (x-axis) and simulated rainfall (y-axis) across hourly to annual aggregation scales.\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/06aa8cb1782ca52357cc0a87.png"},{"id":100813954,"identity":"fa09f770-e542-49af-95bd-1523609e9a77","added_by":"auto","created_at":"2026-01-21 16:11:50","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":182683,"visible":true,"origin":"","legend":"\u003cp\u003edistributions of SPAN criteria for different rainfall durations calculated from quantiles estimated by probability laws and by SCHYPRE method.\u003c/p\u003e","description":"","filename":"image8.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/54238376473b6cd4e60f472f.png"},{"id":100813908,"identity":"f3b2f554-f5e6-4729-8492-4bb6f74d7f25","added_by":"auto","created_at":"2026-01-21 16:11:37","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":173155,"visible":true,"origin":"","legend":"\u003cp\u003ePercentage of quantile crossings due to independent fitting of probability laws per duration.\u003c/p\u003e","description":"","filename":"image9.png","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/3580a8416d8e0f97292bebba.png"},{"id":100859762,"identity":"9c8e51e2-59a8-473e-8296-fb32c4aa6966","added_by":"auto","created_at":"2026-01-22 07:32:49","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4520589,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8594020/v1/de2240ff-f371-4a77-a17c-903d55dc9f9c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A stochastic model of continuous hourly areal rainfall series applied to a wide range of French catchments","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eRainfall constitutes the primary driver of numerous natural hazards, including flash floods, debris flows, urban flooding due to surface runoff, river overflow inundations, groundwater flooding, soil erosion and landslides. Effective management of risks associated with intense or prolonged heavy rainfall necessitates estimating their probability of occurrence, specifically through the determination of key characteristics, such as intensity or volume associated with predefined durations, linked to specific annual exceedance probabilities. The rainfall characteristics under investigation may vary depending on the specific hazard of interest.\u003c/p\u003e \u003cp\u003eThe analysis of rainfall-related hazards can be conducted through a range of methodological approaches. These span from frequency analysis, which facilitates the estimation of the frequency distribution of rainfall characteristics to construct design rainfall events \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e(\u003c/span\u003eColes, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Hosking and Wallis, \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e1993\u003c/span\u003e; Katz et al., \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Merz and Bl\u0026ouml;schl, \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e2005\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e)\u003c/span\u003e, to more sophisticated stochastic simulation techniques for generating rainfall scenarios or extended time series (Cadavid et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e1991\u003c/span\u003e; Onof et al., \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e1995\u003c/span\u003e; Cameron et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Odry and Arnaud, \u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Blazkova et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). The choice of methodologies depends on data accessibility and the defined research aims (Castellarin et al., \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Kochanek et al., \u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). In all cases, the overarching objective is to associate a rainfall scenario with a probability of occurrence. Such scenarios can subsequently be integrated into hydrological models to generate design floods, thereby facilitating the hydraulic computations required for designing hydraulic infrastructure (e.g., dam spillways, dikes, and reservoirs) or assessing flood risks that may impact infrastructure and populations (Myers, \u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e1967\u003c/span\u003e; Chow et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e1988\u003c/span\u003e; Clarke, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Lavabre et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Drobot et al., \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eStochastic simulation methods confer a distinct advantage by mitigating certain assumptions inherent in the estimation of a singular design rainfall event. For instance, they circumvent the simplistic assignment of an identical return period to both a rainfall event and the resultant flood \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e(\u003c/span\u003eGr\u0026auml;ler et al., \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Kuczera et al., \u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2006\u003c/span\u003e; Vangelis et al., \u003cspan citationid=\"CR74\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). These approaches are founded on the generation of rainfall time series over extended periods, which can be transformed into long-term simulated flow time series using hydrological models (Lamb et al., \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). This process facilitates the incorporation of antecedent hydrological conditions (Boughton and Droop, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2003\u003c/span\u003e). The simulated flow series can then be directly employed to perform hydraulic calculations for flood modeling or the sizing of hydraulic structures, while integrating operational management rules (Carvajal et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). By rejecting the conventional reliance on design rainfall and flood events, and by considering all possible flood scenarios, these methods enable a direct assessment of the frequency distribution of hydrological hazard impacts (e.g., flooded areas, dike overflows or dam water level) (Arnaud and Lavabre, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Paquet et al., \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2013\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eFor these approaches to be effective, the rainfall models must exhibit robust and accurate performance in reproducing rainfall characteristics, particularly in simulating extreme events (Callau Poduje and Haberlandt, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Evin et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Nguyen et al., \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Numerous techniques exist for simulating rainfall, each addressing distinct requirements. These include for example, resampling-based approaches, e.g., Stochastic Storm Transposition (Wright et al., \u003cspan citationid=\"CR76\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), Markov chain models for simulating sequences of wet and dry periods (Gabriel and Neumann, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e1962\u003c/span\u003e; Buishand, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1978\u003c/span\u003e; Mehrotra and Sharma, \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Evin et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), Poisson cluster models such as Bartlett-Lewis rectangular pulse models (Onof et al., \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e1995\u003c/span\u003e; Cameron et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Ritschel et al., \u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) and Neyman-Scott rectangular pulse models (Neyman and Scott, \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e1958\u003c/span\u003e; Rodriguez-Iturbe et al., \u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e1987\u003c/span\u003e; Cowpertwait, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e1991\u003c/span\u003e), scaling factor methods based on multifractal theory(Mandelbrot and Wallis, \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e1969\u003c/span\u003e; Deidda, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Langousis et al., \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Ramanathan et al., \u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), temporal disaggregation models which leverage the more abundant information available from daily rainfall data (Lebel et al., \u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; Burian and Durran, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Hingray and Ben Haha, \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Gyasi-Agyei and Mahbub, \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Bhattacharyya and Saha, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe emergence of advanced computational resources has driven substantial progress in the development of rainfall models leading to the creation of user-friendly tools (Burton et al., \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; De Luca and Petroselli, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Dykman et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). However, existing models often address only a subset of the complexities inherent in rainfall dynamics. This limitation is rooted in two fundamental challenges. The first challenge is related to temporal resolution, since many models operate at fixed time steps (e.g., daily or hourly) (Srikanthan and McMahon, \u003cspan citationid=\"CR72\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Given the relative scarcity of sub-daily data, both in terms of recording duration and spatial density, many rainfall models have historically been developed at the daily time step (Baxevani and Lennartsson, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Bennett et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Gao et al., \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). This is particularly critical when analyzing floods in relatively small catchments (\u0026lt;\u0026thinsp;1000 km\u0026sup2;) or urban environments, where sub-daily time steps are essential to account for the rapid kinetics of flow processes. At the hourly scale, rainfall models are less common. The intricate structure of rainfall at fine temporal resolutions has led some approaches to focus on individual rainfall events (event-based approaches) (Rodriguez-Iturbe et al., \u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e1987\u003c/span\u003e; Acreman, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1990\u003c/span\u003e; Cernesson et al., \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Arnaud et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). However, these methods encounter difficulties in accounting for antecedent rainfall conditions when coupled with hydrological modeling. In contrast, continuous approaches offer the advantage of modeling entire time series, thereby capturing temporal variability across multiple scales (Blazkova et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Park et al., 2018; Grimaldi et al., \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). The second major challenge stems from the inherent spatial complexity of rainfall across varying temporal scales (Papalexiou, 2018). Effectively coupling rainfall model with hydrological models requires careful attention to the spatial dimensions of rainfall. While at-site simulations disregard spatial variability and focus on reproducing time series statistically consistent with rain gauge observations, they are only representative of highly localized conditions or small surrounding areas (a few square kilometers). To address larger spatial domains, like areal rainfall or basin rainfall, spatial reduction factors must be calculated and incorporated into rainfall data to account for spatial averaging effects. Multi-site methodologies synthesize simulations from various points to better reflect the spatial patterns of rainfall. Advances in rainfall field models offer a more comprehensive approach to capturing the spatial heterogeneity of rainfall over extensive regions, such as entire catchments (Papalexiou and Serinaldi, \u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Cantet et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2025\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThese diverse rainfall simulation methods are commonly integrated with hydrological modeling to simulate flood events within catchments. While their performance in reproducing specific rainfall characteristics varies, each method exhibits distinct strengths and limitations. These are influenced by several factors, including the degree of complexity or simplification, temporal resolution (e.g., daily vs. sub-daily), spatial resolution (e.g., uniform vs. spatially distributed rainfall), and their ability to capture seasonality and accurately extrapolate extreme values (Nguyen et al., \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe present study focuses on the development of a continuous areal rainfall model operating at an hourly time step. This model extends an existing event-based hourly at-site rainfall model, originally developed within the SHYPRE framework (Arnaud and Lavabre, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). The objective is to leverage the demonstrated strengths of this rainfall model in estimating extreme rainfall at both local (Arnaud and Lavabre, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Arnaud et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Cantet and Arnaud, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) and regional scales (Arnaud et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Carreau et al., \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), while expanding its applicability to areal rainfall simulation. Particular emphasis is placed on improving the representation of rainfall seasonality to accurately reproduce maximum rainfall over extended durations. Addressing rainfall characteristics across all timescales is a central focus of this work and a recurrent challenge in rainfall model development (Paschalis et al., \u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Park et al., 2018). The underlying concept involves adapting the existing event based model to simulate areal rainfall events and positioning these events within a continuous time series to produce a coherent series of hourly basin-scale rainfall data. These simulated hourly rainfall are intended to serve as inputs to continuous hydrological models, with the aim of accounting for both rainfall seasonality and extreme rainfall characteristics.\u003c/p\u003e \u003cp\u003eThe remainder of this article is organized as follows. Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the rainfall data and catchments utilized in this study. Section \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e3\u003c/span\u003e outlines the methodological tools and developments undertaken to achieve the continuous areal rainfall model, including validation metrics. Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e4\u003c/span\u003e details the results of the rainfall model\u0026rsquo;s application to the data. Section \u003cspan refid=\"Sec16\" class=\"InternalRef\"\u003e5\u003c/span\u003e provides a discussion of the results and conclusions.\u003c/p\u003e"},{"header":"2. Data","content":"\u003cp\u003eThe rainfall data used in this study cover the entire metropolitan French territory, which is characterized by a wide diversity of climatic regimes. These include an oceanic climate in the west, a Mediterranean climate in the south, mountainous climates in the Alps and the Pyrenees, and a continental climate in the central and eastern parts of the country.\u003c/p\u003e \u003cp\u003eSpatially distributed rainfall products were employed to enable systematic calibration of the rainfall model over a large number of catchments. These products integrate data from weather radars and ground-based rain gauge networks, specifically the COMEPHORE precipitation dataset (Champeaux et al., \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Tabary et al., \u003cspan citationid=\"CR73\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). This dataset provides spatially resolved total precipitation at an hourly time step with a 1 km \u0026times; 1 km grid resolution. COMEPHORE currently represents the most advanced spatial precipitation product available at an hourly resolution in France, with data available from January 1, 1997, to June 30, 2025, corresponding to 28.5 years of observations (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://meteo.data.gouv.fr/datasets/\u003c/span\u003e\u003cspan address=\"https://meteo.data.gouv.fr/datasets/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eand\u003c/span\u003e \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.25326/360\u003c/span\u003e\u003cspan address=\"10.25326/360\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e).\u003c/span\u003e\u003c/p\u003e \u003cp\u003eFrom these gridded precipitation data, hourly areal precipitation time series were derived for individual catchments. Catchments were selected based on the availability of hydrological data, ensuring the potential for future calibration of a lumped hydrological model and its coupling with the rainfall model to determine flood flow quantiles. An area threshold was applied, retaining only catchments smaller than 10,000 km\u0026sup2;, as lumped hydrological applications beyond this scale are deemed less appropriate at the hourly time step (Arnaud et al., 2002). In addition to area, catchments were excluded based on snow occurrence, as the COMEPHORE dataset does not distinguish between liquid and solid precipitation. Catchments with a snowfall fraction greater than 20% were removed, based on SAFRAN reanalysis data for 1958\u0026ndash;2025 (Quintana-Segu\u0026iacute; et al., \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Vidal et al., \u003cspan citationid=\"CR75\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). In the end, a set of 2,108 catchments is selected.\u003c/p\u003e \u003cp\u003eFor each catchment, a time series of hourly areal precipitation was computed, corresponding to the temporal sequence of precipitation averages across all pixels within the catchment boundary. This yielded 28.5-year time series of hourly areal precipitation for each catchment, derived from the COMEPHORE dataset.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates the spatial distribution of the studied catchments and associated characteristics of areal rainfall. The selected catchments provide a representative sample of the climatic diversity across France, encompassing a comprehensive range of gauged catchment areas (from 5 to 10,000 km\u0026sup2;, with a median of 220 km\u0026sup2;) and varied rainfall regimes. The figure presents the following characteristics: (graph a) catchment locations and boundaries in relation to topographic relief, (graph b) seasonality index (described below), (graph c) percentage of zero rainfall (hourly areal rainfall\u0026thinsp;\u0026lt;\u0026thinsp;0.01 mm), (graphs d and e) quantiles of 1-hour and 24-hour maximum rainfall for a 2-year return period, estimated via GEV distribution fitting (Coles et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2003\u003c/span\u003e), (graph f) mean annual rainfall, (graphs g and i) relationships between the aforementioned variables and catchment area.\u003c/p\u003e \u003cp\u003eThe data reveal substantial variability in rainfall characteristics that the rainfall model must accurately reproduce. Notably, high-altitude catchments prone to snowfall have been excluded. In terms of seasonality, a seasonality index is calculated using the following formula, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:I=\\frac{{\\sum\\:}_{1}^{12}\\left|{x}_{i}-\\stackrel{\\prime }{x}\\right|}{{\\sum\\:}_{1}^{12}{x}_{i}}\\)\u003c/span\u003e\u003c/span\u003e where x\u003csub\u003ei\u003c/sub\u003e is the average monthly rainfall for month i and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\stackrel{\\prime }{x}\\)\u003c/span\u003e\u003c/span\u003e the mean of x\u003csub\u003ei\u003c/sub\u003e. This index simply measures the variability of the monthly values relative to their mean. A value near zero indicates low seasonality, while a higher index reflects stronger seasonality. Central and eastern France exhibit relatively uniform annual rainfall distribution (index values close to 0), characteristic of a continental climate, whereas coastal regions display more pronounced seasonality, reflecting oceanic and Mediterranean climates, in the west and the south, respectively. The proportion of zero rainfall highlights a sharp contrast between oceanic zones, where rainfall occurs during up to 50% of the time steps, and Mediterranean regions, which are characterized by a much lower frequency of rainy time steps, closer to 25%. Analysis of 1-hour and 24-hour maximum rainfall quantiles confirms that the Mediterranean region is subject to significantly more intense rainfall events compared to other areas. Finally, total annual rainfall is strongly correlated with elevation, with the highest accumulations observed in the Vosges, Jura, C\u0026eacute;vennes, and Massif Central mountain ranges.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe influence of catchment area on rainfall characteristics is also noteworthy. Integrating rainfall over the spatial extent of a catchment introduces smoothing of hourly rainfall intensities, as illsutrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eg, where 1-hour rainfall quantiles is negatively correlated with the area of rainfall integration. A reduction in the variability of 1-hour maximum rainfall values is also observed as catchment area increases. The effect weakens for 24-hour rainfall and is negligible for annual totals. Thus, in addition to the intrinsic variability induced from climatology, the developed basin-areal rainfall model must deal with the variability induced by the spatial integration area of rainfall.\u003c/p\u003e"},{"header":"3. Methodology and Tools","content":"\u003cp\u003eThe continuous hourly basin-scale rainfall model is developed based on a previously established event-based hourly point rainfall model at INRAE. (Cernesson et al., \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Arnaud et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Arnaud and Lavabre, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). The event-based rainfall model, named SHYPRE, has demonstrated accuracy and stability in estimating extreme rainfall, as documented in numerous studies (Arnaud and Lavabre, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Neppel et al., \u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Carreau et al., \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Lang et al., \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Kochanek et al., \u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Arnaud et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). It is founded on an analysis of the geometric and statistical properties of observed local hourly rainfall time series. As an event-based model, it focuses exclusively on the most significant rainfall events within a time series, selected according to a threshold-exceedance criterion (detailed in \u0026sect;\u0026nbsp;3.1). In the present study, this event-based model is not applied to point-scale rainfall but directly to areal rainfall, with the aim of simulating basin-scale rainfall. The first challenge is to assess whether the approach developed for point-scale rainfall remains suitable for the smoother signal characteristic of basin-scale rainfall. Furthermore, to generate continuous hourly rainfall time series, the simulated rainfall events must be positioned within a continuous time series, including dry periods as well as periods with rainfall below the event-definition threshold. The process of modeling areal rainfall events and subsequently constructing a continuous series constitutes the new continuous rainfall model, designated SCHYPRE (Simulation of Continuous HYdrographs for Predictive Risk Estimation). The implementation of this rainfall model, intended for coupling with hydrological models, comprises two phases: 1) Parameter calibration, based on a descriptive analysis of observed rainfall time series and 2) Simulation of extended time series, accompanied by comparative analysis with observations. These phases are detailed in the following methodological sections, followed by a description of the analysis variables and the parameters governing their generation.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Rainfall Time Series Analysis\u003c/h2\u003e \u003cp\u003eThe initial phase aims to define the primary variables that best describe the temporal structure of rainfall. The objective is to capture the complexity of rainfall variability at hourly scales, as well as across longer timescales (daily, monthly, etc.), while also representing the full frequency distribution from common to extreme values. This analysis of observed series is conducted at daily and hourly time scales (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAt the daily scale, three distinct period types are identified:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e\u0026ldquo;Rainy events\u0026rdquo; (in red) are defined as sequences of consecutive days with daily rainfall less than 4 mm, including at least one day with rainfall exceeding 20 mm.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e\u0026ldquo;Non-event\u0026rdquo; rainfall periods (in green) consist of days with non-zero rainfall that do not meet the event criteria (i.e., sequences without a day exceeding 20 mm).\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e\u0026ldquo;Dry periods\u0026rdquo; (in blue) are days with no rainfall.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eFour descriptive variables are defined at this daily scale: 1) the number of rainy events in a given month (NE), 2) the empirical frequency of the cumulative rainfall of an event (TER) relative to other seasonnal events (P\u003csub\u003eTER\u003c/sub\u003e), 3) the depth of daily rainfall for non-events in each month (DR\u003csub\u003e\u0026lt;\u0026thinsp;20\u003c/sub\u003e) and 4) the proportion of dry days (areal rainfall\u0026thinsp;\u0026lt;\u0026thinsp;0.01 mm) during non-event periods (P\u003csub\u003e0\u003c/sub\u003e).\u003c/p\u003e \u003cp\u003eThen, the hourly scale focuses only on rainy events, building upon prior studies related to the development of the event-based rainfall model. The analysis of hourly observations concentrates on rainy events containing the most significant rainfall data, which constrain the rainfall model\u0026rsquo;s asymptotic behavior toward extreme values. Rainy events at the hourly scale are described as sequences of wet periods, with the following characteristics determined by the number of rainy periods (NP) within an rainy event, the dry durations separating rainy periods (IPD) and the number of showers per rainy period, defined as a sequence of hourly rainfall with a single relative maximum (NS). Each shower is characterized by its duration (SD), its total volume (SV) and its form defined by the position of the maximum hourly rainfall within the shower (PX) and the ratio of the shower\u0026rsquo;s maximum hourly rainfall to the shower\u0026rsquo;s total volume (RX). The remaining shower volume (1-RX)*SV is then positioned to satisfy a shower form -ie- a single relative maximum.Showers within rainy events are divided into two subsamples. The \u0026ldquo;primary\u0026rdquo; showers (denoted with subscript \u003cem\u003ep\u003c/em\u003e) are the most intense showers in an rainy event. Their number per event (NpS) is defined as one plus an additional primary shower for each day in the rainy event with cumulative rainfall exceeding 50 mm. The \u0026ldquo;ordinary\u0026rdquo; showers (denoted with subscript \u003cem\u003eo\u003c/em\u003e) encompass all remaining showers in the event.\u003c/p\u003e \u003cp\u003eDifferent probability distributions are employed to describe variables based on shower type. This analysis of hourly-scale rainy event characteristics is conducted over two seasons defined for metropolitan France. Season 1 (June\u0026ndash;November) is characterized by typically shorter and more intense rainy events, whereas season 2 (December\u0026ndash;May) is associated with longer and more regular precipitation events. The defined variables enable a comprehensive description of hourly rainfall time series. Each variable is characterized by a probability distribution used to generate values for reconstructing time series. These distributions are listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Note that, since the last published version of this rainfall model (Cantet and Arnaud, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2014\u003c/span\u003e), modifications have been implemented to test new functions for modeling showers.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSummary of variables required to simulate continuous hourly rainfall time series, including their associated probability distributions and parameter estimation methods.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVariable\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDescription of the variable\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eUnit\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eProbability law\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eParameter estimation \u003csup\u003e[1]\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eEstimated for ...\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003eVariables required to simulate rainy events\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNE\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of rainy event per month\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePoisson\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ = \u0026micro;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e12 months\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of rainy period per rainy event\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGeometric\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u0026thinsp;=\u0026thinsp;1/\u0026micro;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNS\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of shower per rainy period\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGeometric\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u0026thinsp;=\u0026thinsp;1/\u0026micro;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNpS\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber of \u0026ldquo;principal\u0026rdquo; shower \u003csup\u003e[2]\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBinomial\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u0026thinsp;=\u0026thinsp;f (\u0026micro;) \u003csup\u003e[3]\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIPD\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInter rainy period duration\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ehours\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eTruncated geometric\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1/\u0026micro;\u003c/p\u003e \u003cp\u003eθ\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;Prob(IPD\u0026thinsp;\u0026gt;\u0026thinsp;12h)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSD\u003csub\u003ep\u003c/sub\u003e et SD\u003csub\u003eo\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eShower duration, for the two types of shower \u003csup\u003e[2]\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ehours\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(\u003cem\u003ep\u003c/em\u003e) Truncated Poisson\u003c/p\u003e \u003cp\u003e(\u003cem\u003eo\u003c/em\u003e) Truncated Poisson\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u003csub\u003e1\u003c/sub\u003e = \u0026micro;\u003c/p\u003e \u003cp\u003eθ\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;Prob(SD\u003csub\u003ep\u003c/sub\u003e\u0026gt;9h)\u003c/p\u003e \u003cp\u003eθ\u003csub\u003e1\u003c/sub\u003e = \u0026micro;\u003c/p\u003e \u003cp\u003eθ\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;Prob(SD\u003csub\u003ep\u003c/sub\u003e\u0026gt;6h)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons and 2 types\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSV\u003csub\u003ep\u003c/sub\u003e et SV\u003csub\u003eo\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eShower volume, for the two types of shower \u003csup\u003e[2]\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003emm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(\u003cem\u003ep\u003c/em\u003e) Truncated Log-normal\u003c/p\u003e \u003cp\u003e(\u003cem\u003eo\u003c/em\u003e) Truncated Gamma\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u003csub\u003e1, 2, 3\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;f(\u0026micro;, σ, min)\u003c/p\u003e \u003cp\u003eθ\u003csub\u003e1, 2, 3\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;f(\u0026micro;, σ, min)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons and 2 types\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRX\u003csub\u003ep\u003c/sub\u003e et RX\u003csub\u003eo\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRatio of shower maximun hourly rainfall over its volume, for the two kind of shower \u003csup\u003e[2]\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNormal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;f\u003csub\u003e1\u003c/sub\u003e(SD) \u003csup\u003e[4]\u003c/sup\u003e\u003c/p\u003e \u003cp\u003eθ\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;f\u003csub\u003e2\u003c/sub\u003e(SD) \u003csup\u003e[4]\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons and 2 types\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFor a shower, relative position of the maximum hourly rainfall in the shower\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNormal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u003csub\u003e1\u003c/sub\u003e = \u0026micro;\u003c/p\u003e \u003cp\u003eθ\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;σ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP\u003csub\u003eTER\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eProbability of the total event rainfall\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eUniform\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u0026thinsp;=\u0026thinsp;median\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e12 months\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eτ\u003csub\u003ep\u003c/sub\u003e and τ\u003csub\u003eo\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eKendall\u0026rsquo;s Tau for copula between duration and volume probabilities\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(\u003cem\u003ep\u003c/em\u003e) Franck copula\u003c/p\u003e \u003cp\u003e(\u003cem\u003eo\u003c/em\u003e) Gumbel copula\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eKendall\u0026rsquo;s Tau\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2 seasons and 2 types\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003eVariables required to simulate \u0026ldquo;non-event\u0026rdquo; rainfall periods\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDR\u003csub\u003e\u0026lt;\u0026thinsp;20\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDepth of non-event daily rainfall\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003emm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eExponential\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ\u003csub\u003e1\u003c/sub\u003e, θ\u003csub\u003e2\u003c/sub\u003e optimised\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e12 months\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHR0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eProportion of hourly rainfall equal to zero within a non-event daily rainfall\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eUniform\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ = \u0026micro;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e12 months\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003eVariables required to simulate dry periods\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDR0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eProbability of days with no rain during a non-even period\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eUniform\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eθ = \u0026micro;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e12 months\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cem\u003e[1]\u003c/em\u003e\u003c/sup\u003e \u003cem\u003e\u0026micro;\u0026thinsp;=\u0026thinsp;mean of the variable values, σ\u0026thinsp;=\u0026thinsp;standart deviation of the variable values and min\u0026thinsp;=\u0026thinsp;minimum of the variable values\u003c/em\u003e\u003c/p\u003e \u003cp\u003e\u003csup\u003e\u003cem\u003e[2]\u003c/em\u003e\u003c/sup\u003e \u003cem\u003eOnly the variables SD, SV, and RX are considered for the two shower types.\u003c/em\u003e\u003c/p\u003e \u003cp\u003e\u003csup\u003e\u003cem\u003e[3]\u003c/em\u003e\u003c/sup\u003e \u003cem\u003eThe binomial distribution parameter describing the variable NpS​ is a regionally calibrated function of the variable's mean. Using our dataset, θ\u0026thinsp;=\u0026thinsp;0,15. ln(\u0026micro;\u003c/em\u003e\u003csub\u003e\u003cem\u003eNpS\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e) for season 1 (1) and θ\u0026thinsp;=\u0026thinsp;0,122. ln(\u0026micro;\u003c/em\u003e\u003csub\u003e\u003cem\u003eNpS\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e) for season 2.\u003c/em\u003e\u003c/p\u003e \u003cp\u003e\u003csup\u003e\u003cem\u003e[4]\u003c/em\u003e\u003c/sup\u003e \u003cem\u003eThe mean and standard deviation of the normal distribution used to randomly sample the variable RX are estimated as functions of shower duration SD. For each randomly drawn duration, the variable RX is generated from a normal distribution N(θ\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e ,\u003cem\u003eθ\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e), where θ\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;a + b ln(SD) and θ\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;c + d. ln(SD). The parameters a, b, c et d are regionally optimized to match the observed mean and standard deviation of RX. These four parameters are determined separately for both seasons and both shower types.\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe parameters derived from? the two distinct seasons are linked to the simulation of seasonal rainfall events, reflecting the event-scale modeling approach of the rain model. Conversely, the parameters established for each of the twelve months are used for the continuous part of the rainfall model.\u003c/p\u003e \u003cp\u003eMost parameters are estimated using the method of moments, based on the mean and standard deviation of observed variable values. The large sample sizes (hundreds of observations) enhance the stability of parameter estimation.\u003c/p\u003e \u003cp\u003eFor specific variables, parameter calibration is optimized. For instance, DR\u003csub\u003e\u0026lt;\u0026thinsp;20\u003c/sub\u003e is modeled using a bounded exponential distribution (between 0.01 to 20 mm), with parameters optimized via least squares method to fit the empirical monthly distribution. RX variable follows a normal distribution, whose parameters are functions of shower duration. The coefficients of this function were calibrated globally across all observed rainfall time series (from hundreds of thousands of observations). The dependence between shower duration (SD) and volume (SV) is modeled using copulas, with parameters estimated in a dimensionless manner from multiple observed time series (Cantet and Arnaud, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Seasonality is incorporated during the calibration of probability law parameters. Some parameters are determined for the two seasons defined for the rainy event analysis, while others are calibrated monthly (see Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Rainfall Time Series Simulations\u003c/h2\u003e \u003cp\u003eThe second phase of rainfall model implementation involves simulating continuous time series. This process comprises tree steps: 1) the simulation of rainy events, 2) the generation of a continuous series by positioning the simulated rainy events within a temporal framework and 3) simulation and positioning of non-event rainfall and dry periods.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e3.2.1 Rainy events simulations per one season:\u003c/h2\u003e \u003cp\u003eIndependent rainy events are generated for a given season using a Monte Carlo approach, in which descriptive variables are randomly sampled to reconstruct events. Events are constructed shower by shower following a predefined sequence. First, the number of rainy periods (NP) is generated. For each rainy period, a number of showers (NS) and a dry duration to the next rainy period (IPD) are then generated. Each shower is classified as either primary (p) or ordinary (o), respecting the observed proportion (NpS). Finally, shower duration (SD), volume (SV), and shape characteristics (RX, PX) are generated, while inter-variable dependencies, such as those between shower duration and volume, are modeled using copulas (reference Cantet \u0026amp; Arnaud, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). At this stage, a set of simulated rainy events is available for each season and need to be positioned inside a continuous times series.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e3.2.2 Positioning rainy events inside one season:\u003c/h2\u003e \u003cp\u003eThis stage consists of positioning independent events within a continuous times series by assigning rainy events to months within the season, in accordance with the observed monthly probabilities (NE). The chronological event positions in a month are conditionned by the event\u0026rsquo;s cumulative rainfall to match the median monthly frequency (P\u003csub\u003eTER\u003c/sub\u003e). After assigning rainfall events to months in a predefined order, the total non-event duration\u0026mdash;derived from month duration minus total event durations and a minimum separation (e.g., 24 hours)\u0026mdash;is randomly allocated within each month to finalize the placement of events. At this stage, the time series still needs to be completed by incorporating dry periods and non-event rainfall.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e3.2.3 Simulation and positioning of non-even rainfall and dry periods:\u003c/h2\u003e \u003cp\u003eFor each non-event day (dry or wet), a random number u\u0026sim;U(0,1) is generated. The day is assigned to zero hourly rainfall if u\u0026thinsp;\u0026lt;\u0026thinsp;DR0. Otherwise, the day is assigned non-event daily rainfall (DR\u003csub\u003e\u0026lt;\u0026thinsp;20\u003c/sub\u003e), sampled from the corresponding probability distribution. The following temporal disaggregation procedure enables to distributes, in a simple way, daily non-event rainfall across 24 hours. First, weighting coefficients, α\u003csub\u003ei=1 to 24\u003c/sub\u003e, are generated using a first-order autoregressive process AR(1). A percentage of zero hourly rainfall is also generated via HR0. The lowest HR0 values of α\u003csub\u003ei\u003c/sub\u003e are set to 0, and the remainder are normalized to have a sum equal to 1. With these coefficients daily rainfall is disaggregate into 24 hourly values. Note that, this procedure has minimal impact on extreme rainfall statistics, since it only applies to lower-intensity rainfall (\u0026lt;\u0026thinsp;20 mm/day).\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Performance Validation\u003c/h2\u003e \u003cp\u003eThe generated hourly rainfall time series must exhibit statistical characteristics equivalent to those of observed series. Validation relies on a suite of statistical criteria to assess the relevance of simulated rainfall for substitution in long-term and frequency-extrapolation analyses. These statistical criteria are compared between observed areal rainfall time series and 10,000-year simulated areal rainfall series.\u003c/p\u003e \u003cp\u003eMonthly variables are employed to verify the reproduction of rainfall seasonality:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eRelative errors in mean daily rainfall, standard deviation, and the proportion of dry days are calculated.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eSeasonality index, ensuring distinction between low- and high-seasonality sites is calculated.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eMulti-Duration variables evaluates the rainfall model\u0026rsquo;s capacity to model both average and extreme values of time-aggregated rainfall:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eMean and standard deviation of rainfall are computed for aggregation time steps ranging from 1 hour to 1 year.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e10-year quantiles of multi-duration rainfall are estimated. Simulated quantiles are empirically estimated with the Hazen formula, without the need to fit a probability distribution. Observed rainfall quantiles are estimated by fitting a GEV distribution to the observed maximum values for rainfall durations ranging from 1 hour to 10 days, corresponding to those illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. For longer durations (more than 10 days), the log-normal distribution, which is a more suitable law, is used to estimate the observed quantiles.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eEnsuring that these rainfall characteristics are preserved across different durations allows us to verify the accurate reproduction of temporal rainfall features at various time steps from high-intensity short-duration events (1 hour to 3 days) to cumulative rainfall over multiple days or months.\u003c/p\u003e \u003cp\u003eThe temporal structure of rainfallis assessed by calculating the first-order autocorrelation of rainfall values aggregated over different time steps. Rainfall intermittency is also analysed by calculating transition probabilities between states (e.g., \"dry\" to \"wet,\" \"wet\" to \"wet,\" and \"dry\" to \"dry\"). This includes examining the likelihood of transitions from low to high rainfall intensities, as well as the persistence of high or low rainfall conditions over consecutive time steps, for different thresholds. All these characteristics ensure that the generated rainfall time series can be reliably used in hydrological modeling.\u003c/p\u003e \u003cp\u003eFinally, to assess the stability of extreme quantile estimates, the rainfall model was calibrated over two distinct 14-year periods : Period P\u003csub\u003e1\u003c/sub\u003e (January 1, 1997, to February 28, 2011) and Period P\u003csub\u003e2\u003c/sub\u003e (March 1, 2011, to May 30, 2025). For each period, quantiles of cumulative rainfall over various durations are estimated for different return periods. Then, the SPAN index (Garavaglia et al., \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) is calculated. It is defined as: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{SPAN}_{T}=2\\cdot\\:\\frac{\\left|\\left({q}_{T}\\left({P}_{1}\\right)-{q}_{T}\\left({P}_{2}\\right)\\right)\\right|}{\\left({q}_{T}\\left({P}_{1}\\right)+{q}_{T}\\left({P}_{2}\\right)\\right)}\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{q}_{T}\\left({P}_{i}\\right)\\)\u003c/span\u003e\u003c/span\u003e is the T-year quantile estimated from data in period P\u003csub\u003ei\u003c/sub\u003e. The stability increases as the SPAN index value converges toward zero, indicating greater consistency between the estimates derived from the two distinct time periods.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Results","content":"\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Parameter values\u003c/h2\u003e \u003cp\u003eCalibration of the rainfall model involves estimating the parameters of probability distributions that describe the various rainfall descriptive variables (presented in Section \u003cspan refid=\"Sec4\" class=\"InternalRef\"\u003e3.1\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). These parameters are estimated from observed values. Some parameters are determined locally, meaning that each catchment have its own set of local parameter estimated from its observed areal rainfall time series. An another set of regional parameter are common to all catchments, and is estimated by pooling all available areal time series.\u003c/p\u003e \u003cp\u003eA set of 25 local parameters is computed for the two seasons defined for simulating rainy events. Most of these parameters are statistical moments (mean and standard deviation) calculated from samples of several hundred observations. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea) illustrates the boxplots of the means of some variables. Note that means are easier to interpret than parameters themselves and are used to estimate parameters of distributions (such as geometric, Poisson and normal). These mean values exhibit high variability, reflecting the diversity of rainfall characteristics in the studied sample, including variations in intensity, duration, and structure. For example, the following seasonal differences are observed: In season 1 (June - November),, hourly areal rainy events generally consist of more showers (higher mean values of NP and NS), with longer shower durations (higher mean SD) but lower volumes (lower mean SV) and intensities (lower mean RX) compared to season 2 (December \u0026ndash; May). Thus, season 2 is chracterized by shorter but more intense rainy events.\u003c/p\u003e \u003cp\u003eThe catchment area also influences basin-averaged rainfall characteristics. Correlation coefficients presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb) show that as the spatial integration area increases, rainy events tend to include more showers, which are longer but less intense (with RX negatively correlated with area). This reflects the \"smoothing\" effect of spatial averaging, where local intensity peaks are reduced by lower intensities across the integration area. This introduces additional complexity in basin-scale rainfall modeling, since the areal rainfall signal structure varies with the size of the averaging area.\u003c/p\u003e \u003cp\u003eAnother set of 6 parameters is monthly calculated for continuous rainfall simulation (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec). This monthly parameterization enables to take account the variation inside a season for different rainfall characteristics. The seasonality of heavy rainfall (defined by rainy events) is captured through the monthly distribution of the number of rainy events (NE) and the relative frequency of seasonal total rainfall (P\u003csub\u003eTER\u003c/sub\u003e). The seasonality of lower rainfall (non-event rainfall periods) is represented by the two parameters defining the DR\u003csub\u003e\u0026lt;\u0026thinsp;20\u003c/sub\u003e variable, while the seasonality of dry periods is accounted for by parameters associated with DR0 and HR0. A general trend is observed, with more frequent (higher NE) and intense (higher P\u003csub\u003eTER\u003c/sub\u003e) rainy events in autumn, while the most pronounced dry periods occur in summer (higher DR0 and HR0).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFinally, a set of regional parameters is determined by analyzing all available observations across the study area. These parameters are coefficients used to establish relationships between different parameters, allowing the estimation of local values. This applies to the parameters of the RX variable, which are conditioned by shower duration (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), and to the parameter associated with the probability distribution of NpS.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Simulation setup\u003c/h2\u003e \u003cp\u003eOnce calibrated, this complete set of parameters allows generating values for the rainfall characteristic variables. These simulated variables, obtained through random sampling, are then used to reconstruct long rainfall time series by progressively generating significant rainy events composed of multiple showers with varying durations, intensities, and shapes. These events are integrated into the time series and supplemented with periods of lower rainfall and dry spells. For each of the 2,108 studied catchments, rainfall data are simulated over a long period. The simulated time series are then compared to observed series based on the comparison criteria defined in Section \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eThe purpose of a stochastic rainfall model is to produce a more exhaustive sample of rainfall time series than those available from the observed dataset. By simulating very long time series that preserve the same statistical characteristics as observations, but with a greater number of possible realizations, extreme values can be estimated by simply extending the simulation duration. Under the assumption of rainfall stationarity, a continuous 100,000-year of hourly rainfall is assumed enough long to empirically estimate rainfall quantiles. The simulated time series are then compared to observed series based on the comparison criteria defined in Section \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Reproducing seasonality\u003c/h2\u003e \u003cp\u003eFirst, the seasonality of rainfall is examined by analyzing monthly variables. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea shows, for each month, the relative error in reproducing the mean of daily rainfall (\u0026micro;(DR)), the standard deviation of daily rainfall (sd(DR)), and the proportion of days with no rainfall (% zero). The relative errors are generally small, mostly within \u0026plusmn;\u0026thinsp;10% for rainfall moments and around \u0026minus;\u0026thinsp;5% for the proportion of days with no rainfall. Note that, the cases where the relative errors exceed 0,4 correspond to summer months and mediterranean catchments for which rainfall is extremely rare (mean close to 0). In these cases, the absolute error remains minimal. Importantly, these relative errors are largely independent of the month, indicating no bias in the seasonality of these variables. The seasonality index is also well reproduced, with a clear distinction between sites with low and high seasonality (see Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Reproducing rainfall statistics for different time scales\u003c/h2\u003e \u003cp\u003eNext, the reproduction of rainfall depth at different time scale ranging from 1 hour to 1 year, is assessed. Tree metrics are examined: mean, standard deviation, and 10-year return period quantiles of annual maximum values (see Section \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e). Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e represents scatterplots comparing observed and simulated cumulative rainfall values for durations of 1 hour, 6 hours, 24 hours, 7 days, 60 days, and 365 days. The results show excellent agreement in the means of rainfall across the entire range of aggregation scales, from hourly to annual. Relative errors in standard deviations and quantiles are also small. A slight tendency to underestimate 24-hour rainfall quantiles is observed. Overall, rainfall characteristics across different durations are well preserved across all temporal aggregation scales. The simulated areal rainfall also represents the wide range of intensities and cumulative rainfall amounts of observed sample.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eRainfall depth at different time scales also exhibits no bias with respect to the catchment area. For example, using the variables presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the relationship between catchment area and the three validation metrics: quantiles of 1-hour and 24-hour maximum rainfall for a 2-year return period and mean annual rainfall. demonstrating that the simulated data consistently reflect the observed characteristics described in Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e4.5 Reproducing intermittency\u003c/h2\u003e \u003cp\u003eFinally, the temporal characteristics of rainfall are analysed at different aggregation time scale (from 1h to 72h). In this way, the first-order autocorrelation coefficient and rainfall intermittency indices are calculated for both simulations and observations. The results are represented in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. The simulations accurately reproduce the observed trends. Indeed, autocorrelation decreases with increasing aggregation duration similar to observation, ranging from values between 0.3 and 0.7 for hourly rainfall to values near 0 to 0.2 for longer durations (72 hours). About intermittency indices, the proportion of successive dry periods (dry-dry) decreases with rainfall duration, while transitions from \"dry\" to \"wet\" increase with rainfall duration. The simulated time series closely follow the observed trends.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe selected validation metrics offer a comprehensive and robust characterization of hourly rainfall features, capturing the essential properties that the model must accurately reproduce. These variables encompass average and extreme rainfall amounts, seasonality, and the temporal structure of successive time steps across a range of aggregation scales from hourly to annual. The comparison of these variables, calculated from both simulated and observed time series, demonstrates the relevance of the rainfall model's results across a wide range of areal rainfall conditions, diverse climatic contexts, and a broad spectrum of catchments sizes. The long-term synthetic areal rainfall time series generated by the model reproduce the statistical properties of observed rainfall with high fidelity. Note that performance evaluation criteria are based on variables not directly used in parameter calibration. For example, none of the calibration variables are related to rainfall autocorrelation.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Discussion","content":"\u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e5.1 robustness of simulations\u003c/h2\u003e \u003cp\u003eThe objective of a Frequency Flood Analysis (FFA) method is to estimate the occurrence of extreme rainfall events. This estimation is highly sensitive to sampling, since, it relies on the observation of rare and thus infrequently recorded values. To assess the stability of these estimates, Garavaglia et al. (\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) propose using the SPAN index (Section \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e). This index is calculated from quantiles estimated by a probability law (GEV or log-normal distributions applied on observed data) and those estimated by the rainfall simulation (empirical quantiles of simulated rainfall time seires by the SCHYPRE method). Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e represents the SPAN index across the 2,108 catchments for different rainfall durations and return periods (2, 10, and 100-years), comparing quantiles estimated via the probability laws and the SCHYPRE method.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFor the 2-year return period, SPAN values are low and comparable between the two methods, falling within the range of sampling variability expected for median values. However, as more extreme return periods are considered, a more pronounced increase in the SPAN index is observed, particularly for methods based on probability laws. This reflects the sensitivity of three-parameter distributions like the GEV to sampling variability, particularly the presence or absence of extreme values in the calibration dataset. In contrast, according to SPAN values, the SCHYPRE method exhibits greater stability for extreme quantiles. This stability arises because the internal probability distributions describing shower characteristics are calibrated using a large number of realizations per year of observation, rather than relying on a single annual maximum value. Despite the higher number of parameters required to calibrate the rainfall model, the stability of the simulations is significantly greater than that obtained by fitting a probability distribution to estimate extreme rainfall values.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e5.2 Quantile crossing\u003c/h2\u003e \u003cp\u003eRainfall model\u0026rsquo;s complexity can be justified, since it enables the reproduction of continuous areal rainfall time series, thereby providing comprehensive information on all statistical characteristics of rainfall without requiring additional parameter estimations. For example, using a probability distribution to estimate rainfall quantiles typically necessitates calibrating a set of parameters for each rainfall duration studied. In the case presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, the GEV distribution (for durations d\u0026thinsp;\u0026lt;\u0026thinsp;10 days) and the log-normal distribution (for durations d\u0026thinsp;\u0026gt;\u0026thinsp;10 days) were independently fitted for each duration to estimate quantiles. Even if these calibrations are independent across durations, consistency is expected among the estimated quantiles for different durations. However, \"quantile crossing case\" can appear : a quantile estimated for a duration (d₁) is greater than a quantile estimated for a longer duration (d2\u0026thinsp;\u0026gt;\u0026thinsp;d1). By construction, this case is not possible in simulated quantiles. At worst, quantiles can be equal.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e shows the percentage of quantile crossing cases resulting from fitting probability distributions for different durations and return periods.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eQuantile crossing cases occur even for 5-year return periods and can reach up to 20% for the largest return periods and shorter duration.. Note that, for longer period (\u0026gt;\u0026thinsp;30days), there is no quantile crossing cases. This may be explained by the use of log-normal distribution rather than a GEV for the shorter duration. In contrast to the log-normal distribution, the GEV is a heavy tailed probability distribution, in which the shape parameter controls the highest return levels. However, estimation of the shape parameter is challenging, and remains uncertain due to sampling effects.\u003c/p\u003e \u003cp\u003eTemporal consistency in rainfall quantile estimation is intrinsic to the SCHYPRE method, which does not produce cases of quantile crossing. Such inconsistencies can be significant when extrapolating frequencies using probability distributions, a well-known limitation that has led to the development of multi-duration statistical methods. These methods ensure the calibration of distributions that maintain consistency in quantile estimations \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e(You and Tung, 2018;\u003c/span\u003e Fauer et al., \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Roksv\u0026aring;g et al., \u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThis study presents a stochastic model for continuous hourly areal rainfall designed to address hydrological risk management needs in France (SCHYPRE method). Building on a robust event-based approach, previously validated for point-scale rainfall, the SCHYPRE rainfall model extends these capabilities to areal rainfall modeling. By combining an event-based framework for extreme intensities with a continuous approach for seasonal and long-duration rainfall, the rainfall model faithfully reproduces rainfall characteristics across temporal scales ranging from hourly to annual. This avoids restrictive assumptions inherent in traditional methods, such as direct linkage between design rainfall and flood events, by generating long, realistic time series.\u003c/p\u003e \u003cp\u003eStability tests demonstrate that the rainfall model is less sensitive to sampling variability than parametric distributions (e.g., GEV), thanks to its calibration on a large number of realizations describing rainfall variables. Unlike methods that independently fit probability distributions for each duration, rainfall modeling ensures no quantile crossing between durations, thus maintaining the physical consistency critical for hydrological applications.\u003c/p\u003e \u003cp\u003ePerformances were assessed across 2,108 catchments, spanning diverse climates (oceanic, Mediterranean, continental, mountainous) and various catchments area (up to 10,000 km\u0026sup2;). Results confirm accurate reproduction of observed statistics for both mean and extreme rainfall, as well as seasonal and temporal properties (autocorrelation, intermittency).\u003c/p\u003e \u003cp\u003eThe SCHYPRE rainfall model represents a significant advancement in stochastic hourly areal rainfall modeling, providing a robust alternative to traditional rainfall frequency analysis approaches. Its ability to generate long, coherent, and realistic time series enable their direct use as inputs to hydrological models, making it a valuable tool for hydrological studies, natural risk management, and infrastructure design. By avoiding issues such as quantile crossing or assumptions inherent in the estimation of a singular design rainfall event,, this method explicitly incorporates spatio-temporal complexity of precipitation into hydrological modeling enabling more reliable operational applications tailored to contemporary water management challenges.\u003c/p\u003e \u003cp\u003eFuture applications will include coupling with spatial disaggregation models to generate high-resolution rainfall fields (e.g., Cantet et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2025\u003c/span\u003e) to drive distributed hydrological models. Another application involves calibrating the rainfall model using climate model outputs to assess the impacts of climate change on extreme rainfall events following approach proposed by Cantet et al (2011).\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eP. A. : Conceptualization, Methodology, Investigation, Writing \u0026ndash; Original Draft, Writing \u0026ndash; Review \u0026amp; Editing, Supervision, Project administration.P.C. : Conceptualization, Methodology, Investigation, Formal Analysis, Review \u0026amp; Editing.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eWe gratefully acknowledge financial support for this study provided by Direction G\u0026eacute;n\u0026eacute;rale des Pr\u0026eacute;visons des Risques (DGPR).\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eComephore dataset is freely available on request at https://doi.org/10.25326/360 or at https://meteo.data.gouv.fr/datasets/669e23a7ce052a9e8521b75e\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAcreman MC (1990) A simple stochastic model of hourly rainfall for Farnborough, England. 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J Hydrol 585:124816. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1016/j.jhydrol.2020.124816\u003c/span\u003e\u003cspan address=\"10.1016/j.jhydrol.2020.124816\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"stochastic-environmental-research-and-risk-assessment","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"serr","sideBox":"Learn more about [Stochastic Environmental Research and Risk Assessment](https://www.springer.com/journal/477)","snPcode":"477","submissionUrl":"https://submission.nature.com/new-submission/477/3","title":"Stochastic Environmental Research and Risk Assessment","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Stochastic rainfall simulation, hourly areal rainfall, multi-duration estimations, seasonality and extremes","lastPublishedDoi":"10.21203/rs.3.rs-8594020/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8594020/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study introduces a stochastic continuous hourly areal rainfall model designed to simulate rainfall time series for hydrological risk management. The rainfall model, named SCHYPRE (Simulation of Continuous HYetographs for Predictive Risk Estimation), extends an established event-based rainfall model to basin-scale applications, integrating both extreme event modeling and continuous simulation of seasonal and long-duration rainfall patterns.\u003c/p\u003e \u003cp\u003eThe rainfall model was calibrated using 28.5 years of rainfall data set with hourly and kilometric resolution across 2,108 catchments in France, covering diverse climatic regimes from continental to Mediterranean and mountainous. The evaluation framework demonstrates rainfall model\u0026rsquo;s ability to faithfully reproduce observed rainfall statistics, including mean and extreme values, seasonality, autocorrelation, and intermittency. Frequency analyses conducted over durations from one hour to one year show strong agreement between the simulations and the adapted law, with only limited bias in the estimation of extreme values.\u003c/p\u003e \u003cp\u003eA major advantage of rainfall modelling is its robustness in estimating extreme quantiles. Unlike traditional probabilistic methods, which are more sensitive to sampling variability, the rainfall model\u0026rsquo;s Monte Carlo approach, calibrated on large observational datasets of interne variables, ensures stable quantile estimation across all return periods, including extremes. Additionally, rainfall modelling inherently avoids quantile crossing inconsistencies, a common issue in independent duration-based probabilistic modeling.\u003c/p\u003e","manuscriptTitle":"A stochastic model of continuous hourly areal rainfall series applied to a wide range of French catchments","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-21 16:10:14","doi":"10.21203/rs.3.rs-8594020/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-19T18:08:14+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-19T17:53:48+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-05T06:50:18+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"144536842719152045960987391465543908245","date":"2026-01-20T11:15:31+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"58254964171833755940454442023595542951","date":"2026-01-19T17:53:48+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-19T11:51:01+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-15T16:51:21+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-13T16:36:51+00:00","index":"","fulltext":""},{"type":"submitted","content":"Stochastic Environmental Research and Risk Assessment","date":"2026-01-13T15:55:23+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"stochastic-environmental-research-and-risk-assessment","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"serr","sideBox":"Learn more about [Stochastic Environmental Research and Risk Assessment](https://www.springer.com/journal/477)","snPcode":"477","submissionUrl":"https://submission.nature.com/new-submission/477/3","title":"Stochastic Environmental Research and Risk Assessment","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"dab0e7ce-34df-46d1-ba6f-6666b87ed49e","owner":[],"postedDate":"January 21st, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-18T15:55:23+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-21 16:10:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8594020","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8594020","identity":"rs-8594020","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}