# What can be expected from a semi-distributed multi-model approach for streamflow forecasting? Tailoring the structure and size of a super-ensemble on the Rhône basin

Cyril Thébault, Charles Perrin, Sébastien Legrand, Vazken Andréassian, Guillaume Thirel, Olivier Delaigue

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Cyril Thébault, Charles Perrin, Sébastien Legrand, Vazken Andréassian, Guillaume Thirel, et al. What can be expected from a semi-distributed multi-model approach for streamflow forecasting? Tailoring the structure and size of a super-ensemble on the Rhône basin. Journal of Hydrology, 2025, 661, pp.133589. 10.1016/j.jhydrol.2025.133589. hal-05154479

HAL Id: hal-05154479

https://hal.science/hal-05154479v1

Submitted on 16 Jul 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Research papers

# What can be expected from a semi-distributed multi-model approach for streamflow forecasting? Tailoring the structure and size of a super-ensemble on the Rhône basin

Cyril Thébault<sup>a,c</sup>, Charles Perrin<sup>a,*</sup>, Sébastien Legrand<sup>b</sup>, Vazken Andréassian<sup>a,b</sup>, Guillaume Thirel<sup>a,d</sup>, Olivier Delaigue<sup>a,b</sup>

## ARTICLE INFO

This manuscript was handled by Marco Borga, Editor-in-Chief, with the assistance of Murray Peel, Associate Editor

**Keywords:**

Streamflow forecasting
Hydrological modelling
Multi-model
Semi-distribution
Super-ensemble

## ABSTRACT

Streamflow forecasting is useful for various purposes, from ensuring the safety of populations during floods to managing hydraulic structures. The aim of this work is to combine two hydrological modelling approaches widely used in streamflow forecasting in order to define their benefits and limits in a probabilistic framework: the multi-model approach (which accounts for structural and parametric model uncertainty) and the semi-distributed approach (which considers explicitly the spatial variability of precipitation and hydrological processes).

The study focuses on 12 tributaries of the Rhône River, which were modelled using 39 hydrological model configurations. Tests were carried out at an hourly time step for lead times ranging from 1 h to 120 h, considering ensemble meteorological forecasts. The results show that explicitly considering uncertainty with a probabilistic super-ensemble (meteorological ensemble chained to a multi-model approach) improves the quality of streamflow forecasts. On the other hand, there is no clear benefit from a semi-distributed approach compared with a lumped framework. This paper also explored the structure and size of the super-ensemble, showing that it is possible to reduce its complexity through model selection or combination methods without impairing predictive performance.

This study provides valuable insights into the strengths and limitations of a super-ensemble approach and how to limit its complexity, contributing to the ongoing efforts to improve streamflow forecasting for operational purposes.

## 1. Introduction

Hydrological forecasting serves several important purposes, primarily revolving around managing water resources effectively and mitigating risks associated with hydrological events such as floods or droughts. The stakes involved in forecasting are therefore numerous, including the safety of populations, access to drinking water, agriculture, energy production (hydroelectricity or cooling of thermal power plants) or river navigation (see e.g. Celie et al., 2019).

### 1.1. Relevant literature

The first scientific documents related to hydrological forecasting date back to the 1930s (e.g. Bachet, 1934; Bernard, 1939). In general, they show the socio-economic benefits of hydrological forecasting, even in the short term, although this requires considerable human expertise and hydrometeorological data, which are often incomplete and inaccurate (White, 1939). With the emergence of computer modelling and the growing availability of increasingly accurate data, hydrological forecasting has become a major research topic and is constantly evolving (Todini, 2007, 2011; Dasgupta et al., 2023). Over the past decades,

<footnote><sup>a</sup> Université Paris-Saclay, INRAE, HYCAR, Antony, France</footnote>

<footnote><sup>b</sup> Compagnie nationale du Rhône, Lyon, France</footnote>

<footnote><sup>c</sup> University of Calgary, Schulich School of Engineering, Department of Civil Engineering, Calgary, Canada</footnote>

<footnote><sup>d</sup> Univ Toulouse, CNES/IRD/CNRS/INRAE, CESBIO, Toulouse, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: charles.perrin@inrae.fr (C. Perrin).</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Map showing the Rhône tributaries studied and their spatial division into sub-catchments.</figcaption></figure>

research on this topic has expanded globally particularly in addressing the challenges of uncertainty quantification and operational forecasting (Seo et al., 2006; Bennett et al., 2014; McInerney et al., 2017; Li et al., 2017). These studies highlighted the need for a broader uncertainty framework that accounts for meteorological data and hydrological model uncertainties in a forecasting context. It should be noted, however, that residual errors persist at the end of the modelling chain in a forecasting context. These errors are generally addressed in a statistical post-processing stage (see e.g. Li et al., 2016; McInerney et al., 2020, 2021, 2022). However, we will not delve into this aspect here, as it falls outside the scope of our study.

### 1.1.1. Meteorological ensemble (input data uncertainty)

Since the estimation of future meteorological variables is inherently

uncertain, and this uncertainty increases with the forecast lead time, it is generally difficult to rely on a single forecast. Boucher et al. (2011) have shown that it is far more advantageous to use ensemble forecasts rather than deterministic forecasts despite the generally lower spatial resolution of the former. Ensemble approaches are increasingly used to provide a range of possible forecasts and therefore to better assess the associated uncertainty. Numerous methods exist for generating ensemble scenarios in hydrology (see e.g. Troin et al., 2021 for a review). We will focus here on hydrological ensemble forecasting systems based on ensemble weather forecasting from numerical models (for a detailed description of the system, refer to the guide proposed by WMO, 2012), i.e. based on real-time precipitation and air temperature forecast scenarios. Ensemble weather forecasts emerged in the 1990s (Mureau et al., 1993; Toth & Kalnay, 1993), but were only applied to hydrology a
<table><caption>**Table 1**
Main characteristics of the Rhône tributaries studied (P: precipitation, PET: potential evapotranspiration, Q: streamflow).</caption><thead><tr><th>Tributary</th><th>Downstream gauged station</th><th>Area [km²]</th><th>P [mm/y]</th><th>PET [mm/y]</th><th>Q [mm/y]</th></tr></thead><tbody><tr><td>Valserine</td><td>Lancrans</td><td>355</td><td>1654</td><td>590</td><td>1260</td></tr><tr><td>Fier</td><td>Motz</td><td>1378</td><td>1403</td><td>629</td><td>802</td></tr><tr><td>Seran</td><td>Pont de la Thuilière</td><td>239</td><td>1431</td><td>630</td><td>794</td></tr><tr><td>Guiers</td><td>Belmont</td><td>598</td><td>1484</td><td>651</td><td>967</td></tr><tr><td>Ain</td><td>Port Galland</td><td>3710</td><td>1544</td><td>642</td><td>991</td></tr><tr><td>Saône</td><td>Couzon</td><td>29,320</td><td>1063</td><td>703</td><td>449</td></tr><tr><td>Eyrieux</td><td>Pontpierre</td><td>818</td><td>1206</td><td>687</td><td>640</td></tr><tr><td>Drôme</td><td>Livron</td><td>1641</td><td>991</td><td>681</td><td>410</td></tr><tr><td>Ardèche</td><td>Sauze</td><td>2264</td><td>1359</td><td>746</td><td>821</td></tr><tr><td>Cèze</td><td>Bagnols-sur-Cèze</td><td>1113</td><td>1138</td><td>823</td><td>465</td></tr><tr><td>Gard</td><td>Remoulins</td><td>1860</td><td>1147</td><td>837</td><td>413</td></tr><tr><td>Arve</td><td>Bout du Monde</td><td>2000</td><td>1359</td><td>539</td><td>1114</td></tr></tbody></table>

decade later. Such approaches then gained much interest, as indicated by the rapidly growing global community of researchers and practitioners, called “HEPEX” (Hydrologic Ensemble Prediction Experiment), which has been contributing to scientific advances in hydrological ensemble forecasting since 2004 (Schaake et al., 2007; Ramos et al., 2018). Progressively, meteorological ensemble forecasts are being included in operational hydrological forecasting chains (see e.g. Papenberger et al., 2016, for a review).

### 1.1.2. Hydrological multi-model (model uncertainty)

Hydrological models are simplifications of real-world processes and contribute with their own uncertainty (either structural or parametric). Although these uncertainties are often considered second order in a forecasting framework, they are not negligible, and multi-model approaches (i.e. using multiple hydrological models) appear as viable solutions to account for these uncertainties. Shamseldin et al. (1997) were among the first to test the multi-model approach for streamflow simulation and showed that a single model cannot perform better than other models on every catchment and that a combination of several models (i.e. calculating a simulation by linear or non-linear weighting of individual model simulations) is more adapted to this. This multi-model approach using deterministic combinations of models was also applied in a forecasting framework and led to similar results (Ajami et al., 2006). However, although the deterministic multi-model approach provides superior performance to a single-model framework, it is generally difficult to define the optimal combination a priori because a model with average (or even poor) individual performance can make a positive contribution when combined with others (Seiller et al., 2012). In addition, such an approach does not enable explicit quantification of structural and parametric model uncertainty. Instead of combining the model outputs, Duan et al. (2007) considered the multi-model approach to be an ensemble of hydrological models in order to quantify the uncertainty associated with the choice of model structure but also in the estimation of parameters. Indeed, from three hydrological models calibrated using three different objective functions (each designed to target a different streamflow range) and using multiple weightings with a Bayesian model averaging approach, the authors showed that explicitly considering this uncertainty generates more skilful predictions.

### 1.1.3. Super-ensembles

Forcing a probabilistic multi-model approach with ensemble meteorological data leads to a hydrological super-ensemble. Velazquez et al. (2011) applied this approach and their findings indicate that using a super-ensemble leads to more skilful and reliable hydrological forecasts compared with predictions generated by either a single hydrological model using a meteorological ensemble or a hydrological multi-model with a deterministic meteorological forecast. Thiboult et al. (2016) also aimed to disentangle the various sources of uncertainty in

hydrological modelling processes, considering uncertainties arising from model structure (multi-model approach), meteorological forecasts (meteorological ensemble) and initial conditions (data assimilation using an ensemble Kalman filter). The authors showed that in the short term, uncertainty is predominant over initial conditions, but that its effect tends to fade with the forecast lead time. The benefit of the multi-model approach appears as soon as we move away from initial conditions, where assimilation methods are no longer able to compensate for the various sources of uncertainty. However, overall, the meteorological ensemble remains the main factor for streamflow forecast quality, and its importance increases with lead time. While the super-ensemble approach seems attractive because it makes it possible to consider the main sources of uncertainty and leads to more skilful and reliable hydrological forecasts, it has barely been applied. In their literature review focusing on ensemble streamflow forecasts, Troin et al. (2021) identified such an approach in only 2 % of the reviewed studies. Super-ensembles can be difficult to implement, since they need powerful computational resources and require processing a large amount of probabilistic information, which can hamper decision-making in an operational context.

## 1.2. Research questions and objectives

The above literature review has shown the appeal of a super-ensemble approach. However, since its application is relatively recent, there are still many areas of research to explore on this topic. For example, super-ensembles have generally been applied at a daily time step and in a lumped modelling framework. However, some hydrological processes require a finer spatio-temporal resolution, as in the case of flood generation for example. The aim of the present study is therefore to answer the following question: What can be expected from a semi-distributed super-ensemble approach for hourly streamflow forecasting?

We will also investigate how the super-ensemble should be designed and how parametric uncertainty can be considered in such an approach.

This paper is organised as follows: Section 2 presents the catchments, data and models used, as well as the super-ensemble and modelling procedure applied. Section 3 sums up the results and their analysis followed by a discussion in Section 4. Finally, the main conclusions are highlighted in Section 5.

# 2. Material and methods

## 2.1. The Rhône River catchment

The Rhône is a transboundary 812-km-long river, rising below the glacier of the same name in Switzerland and flowing into the Mediterranean Sea after crossing south-eastern France (Fig. 1). Its topographic catchment covers a total area of around 97,800 km² (with more than 90 % in France). The catchment is characterised by a wide variety of climates, landforms and complex geology. This hydrometeorological and geomorphological diversity makes the Rhône catchment a good study case for a multi-model approach. As the main river is strongly influenced by hydroelectric dams along its entire length, we will study 12 of its tributaries that have limited human influence. These 12 tributaries are divided into several sub-entities (all gauged) according to data availability and operational expectations at the Compagnie Nationale du Rhône (CNR, National Rhône Company) (Fig. 1), which is the operator of the dams along the main river course. These dams are managed mainly for hydropower, navigation and irrigation objectives, which requires forecasts with medium-range lead times given the long propagation times from upstream to downstream (see Bompart et al., 2009; Caillouet et al., 2022). Table 1 summarises the main characteristics of the tributaries studied. It should be noted that the Durance and Isère catchments (Fig. 1, dashed lines), the main Alpine tributaries, were not included in our study due to the impact of dam regulation.
<table><caption>**Table 2**
List of rainfall-runoff models used for this work.</caption><thead><tr><th>Original name</th><th>Number of free parameters</th><th>Reference</th><th>Name used in this study</th></tr></thead><tbody><tr><td>Topmodel</td><td>7</td><td>Beven and Kirkby (1979)</td><td>TOPM</td></tr><tr><td>IHACRES</td><td>6</td><td>Jakeman et al. (1990)</td><td>IHAC</td></tr><tr><td>HBV</td><td>9</td><td>Bergström and Forsman (1973)</td><td>HBV0</td></tr><tr><td>Mohyse</td><td>8</td><td>Turcotte et al. (2010)</td><td>MOHY</td></tr><tr><td>Mordor</td><td>6</td><td>Garçon (1999)</td><td>MORD</td></tr><tr><td>Simhyd</td><td>8</td><td>Vaze et al. (2011)</td><td>SIMH</td></tr><tr><td>SMAR</td><td>9</td><td>O'Connell et al. (1970)</td><td>SMA0</td></tr><tr><td>TANK</td><td>10</td><td>Sugawara (1979)</td><td>TANO</td></tr><tr><td>Gardenia</td><td>8</td><td>Thiéry (1982)</td><td>GARD</td></tr><tr><td>PDM</td><td>8</td><td>Moore and Clarke (1981)</td><td>PDM0</td></tr><tr><td>CREC</td><td>8</td><td>Cormary and Guilbot (1973)</td><td>CRE0</td></tr><tr><td>NAM</td><td>10</td><td>Nielsen and Hansen (1973)</td><td>NAM0</td></tr><tr><td>GR5H</td><td>5</td><td>Ficchi et al. (2019)</td><td>GR5H</td></tr></tbody></table>

## 2.2. Hydrometeorological data

This study was conducted at the hourly time step using actual real-time meteorological forecasts (i.e. data estimated before the events occurred). Precipitation and temperature forecasts are derived from the Ensemble Prediction System (EPS) of the European Centre for Medium-Range Weather Forecasts (ECMWF) (Molteni et al., 1996). This archive consists of 51 future weather scenarios (50 disturbed members and 1 control member, representing undisturbed weather outputs) and is available twice per day (0-h and 12-h cycles), at a resolution of 1°, for lead times of up to 15 days, at a 6-h time step. Data were uniformly disaggregated to the hourly time step. Ensemble precipitation and temperature forecasts tend to exhibit biases and dispersion errors (Hamill and Colucci, 1997). Thus, meteorological post-processing of weather forecasts was applied to the ensemble members by coupling the EMOS (Ensemble Model Output Statistics) and ECC (Ensemble Copula Coupling) approaches (as done by Bellier, 2018). Details of these methods can be found in Gneiting & Raftery (2005) for EMOS and in Schefzik et al. (2013) for ECC. It should be noted that COMEPHORE reanalysis (Tabary et al., 2012) was used to bias-correct the meteorological ensemble, using only data preceding the forecasting period (i.e., prior to 2008).

Potential evapotranspiration was calculated with the formula proposed by Oudin et al. (2005). This equation was chosen for its simplicity, as the only input required is daily air temperature and extra-terrestrial radiation (which depends only on the Julian day and the latitude). Once calculated, the daily potential evapotranspiration was temporally disaggregated to the hourly time step using a simple parabola (Lobligeois, 2014).

Streamflow time series were mostly extracted from the French national streamflow archive Hydroportail (Dufeu et al., 2022), but some gauged stations were also provided by CNR and EDF (Électricité de France, French electricity producer). All meteorological data were spatially aggregated at sub-catchment level for every tributary studied.

## 2.3. Hydrological models

For this study, the airGRplus software (Coron et al., 2022), which is an add-on to airGR (Coron et al., 2017, 2021), was used. airGR includes various versions of the GR models (Michel, 1983; Perrin et al., 2010). Here we selected the GR5H model version (Ficchi et al., 2019), which was well adapted to our needs. airGRplus includes simplified versions of 12 additional conceptual rainfall-runoff model structures running at an hourly time step (based on the works of Perrin, 2000; Mathevet, 2005;

Thèbault, 2023). A total of 13 hydrological models were therefore used for this study (Table 2). To avoid confusion with the original models, a four-letter abbreviation was used here for the simplified versions.

Each conceptual model was calibrated in simulation against observed streamflow for each sub-catchment by maximising the Kling-Gupta efficiency (KGE) score (Gupta et al., 2009), which is a criterion widely used in hydrology:

$$KGE = 1 - \sqrt{(r - 1)^2 + (\alpha - 1)^2 + (\beta - 1)^2} \quad (1)$$

where *r* is the correlation, *α* is the ratio of standard deviations and *β* is the ratio of means (i.e. the bias) between observed and simulated/ forecasted streamflow. The optimum value for KGE is 1.

To ensure the independence of calibration and evaluation in forecasting, a split-sample test was applied over the periods 2009–2013 and 2014–2018. Each sub-period was successively used for calibration and evaluation. This approach ensures that the models are tested on unseen data, allowing to better assess their generalization ability. The selected period includes significant drought (in 2011, 2017) and flood (2014, 2015 or 2018) events in various parts of the Rhône basin.

In order to consider parameter uncertainty, we used three parameterisations of each model obtained by three objective functions corresponding to KGE computed on three prior transformations applied to streamflow: KGE(Q<sup>+0.5</sup>), KGE(Q<sup>+0.1</sup>) and KGE(Q<sup>-0.5</sup>). These transformations will give more weight on high, intermediate and low flows respectively (see e.g. Pushpalatha et al., 2012; Thirel et al., 2024). This led to three parameter sets for each model on each sub-catchment. Note that parameter uncertainty could have been addressed more explicitly with Monte Carlo simulations to consider multiple parameters set for each objective function. Such an approach would be interesting to consider as a perspective for this work but will not be covered here for computation time constraints. In the remainder of this paper, a model structure associated with a given parameter set will be referred to as a “model” for the sake of simplicity.

The algorithm used for model calibration comes from Michel (1991) and is available in the airGR package. It combines global and local optimization approaches. First, a coarse screening of the parameters space is performed using either a rough predefined grid or a list of parameter sets. Then a steepest descent local search algorithm is applied, starting from the result of the screening procedure. This algorithm enables efficient model calibration on a single metric with realistic parameter exploration, using empirical bounds that have been established based on calibrations conducted on large samples of catchments around the world.

We opted for a single-objective calibration approach, in which all parameters are optimized simultaneously using the same objective function. This choice reflects a trade-off between theoretical optimality and practical feasibility, particularly in a semi-distributed multi-model context. By doing so, we avoid introducing biases that could arise from using model-specific calibration strategies (e.g. a different objective for each of the parameters or models), which may compromise the fairness of the comparison. While multi-objective methods may offer theoretically a better representation of the system, they significantly increase complexity and computational costs, potentially limiting the scope of the analysis.

In a semi-distributed context, calibration was carried out sequentially, i.e. in each sub-catchment from upstream to downstream, considering the simulations from upstream catchments as input for calibrating the model at the downstream gauge. This approach allows downstream catchments to benefit not only from meteorological data but also from upstream streamflow information. The sub-catchments are linked together with a simplified routing scheme. For this study, a single routing scheme was used for streamflow propagation from upstream to downstream. Specifically, a time-lag approach, as implemented by Lobligeois et al. (2014), was adopted. To reduce computation time, the authors propose calibrating a single additional lumped parameter, C₀, in
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Presentation of the different forecast approaches tested for the event on May 18–22, 2013, on the Drôme at Livron within a lumped modelling framework. The dotted black line represents the observed streamflow, and the various hairs represent the forecast streamflow. The vertical line indicates the time of forecast issue.</figcaption></figure>

each sub-catchment that incorporates upstream information. Given the known hydraulic lengths $d_i$ (i.e. the distance between the sub-catchment outlet and the nearest upstream sub-catchments outlets), the time-lag - $T_i$ is calculated accordingly:

$$T_i = \frac{d_i}{C_0} \qquad (2)$$

This approach offers performance comparable to more complex hydraulic routing schemes such as lag and route methods (with linear or quadratic reservoirs) that account for peak-shaving phenomena (results not shown for the sake of brevity). The division in sub-catchment was designed in collaboration with CNR’s operational experts (see Fig. 1).

Since some of the studied catchments are mountainous, the snow-accounting CemaNeige module (Valéry et al., 2014) was applied along with the hydrological models. This module was chosen because it was extensively studied over Alpine catchments and can be applied with a fixed parameterisation without degrading performance in this area (Valéry et al., 2014).

## 2.4. Super-ensemble modelling procedure

As a reminder, a super-ensemble can be obtained by forcing multiple models with ensemble meteorological data. For this study, we used 50 meteorological members as inputs for the 39 hydrological models (13 conceptual model structures calibrated in three different ways), which leads to a 1950-member super-ensemble. A secondary objective of this work is to reduce the dimension of the super-ensemble to ease the operational implementation, without degrading the forecasting quality. As the 50 meteorological members are considered equally likely, we have chosen to keep them all. Thus, the only option for reducing the dimensions of the super-ensemble is to exclude some of the 39 hydrological models. Two reduction methodologies were developed and applied in this study. The first is based on a static pre-selection of hydrological models according to their simulation performance (Section 2.4.1), and the second is based on a dynamic combination of hydrological forecasts for each model structure depending on the forecasted streamflow range (Section 2.4.2). In the end, seven forecasting
<table><caption>**Table 3**
Model pre-selection for each catchment.</caption><thead><tr><th>Catchment</th><th>Model pre-selection</th></tr></thead><tbody><tr><td>Ain</td><td>TOPM (Q<sup>+0.1</sup>)</td></tr><tr><td>Ardèche</td><td>GR5H (Q<sup>+0.1</sup>)</td></tr><tr><td>Arve</td><td>IHAC (Q<sup>+0.1</sup>)</td></tr><tr><td>Cèze</td><td>GR5H (Q<sup>+0.1</sup>)</td></tr><tr><td>Drôme</td><td>MORD (Q<sup>+0.1</sup>)</td></tr><tr><td>Eyrieux</td><td>SMA0 (Q<sup>+0.5</sup>)</td></tr><tr><td>Fier</td><td>SMA0 (Q<sup>+0.5</sup>)</td></tr><tr><td>Gard</td><td>GR5H (Q<sup>+0.5</sup>)</td></tr><tr><td>Guiers</td><td>MORD (Q<sup>-0.5</sup>)</td></tr><tr><td>Saône</td><td>TOPM (Q<sup>+0.5</sup>)</td></tr><tr><td>Séran</td><td>IHAC (Q<sup>+0.1</sup>)</td></tr><tr><td>Valserine</td><td>GR5H (Q<sup>+0.5</sup>)</td></tr></tbody></table>

approaches were tested and compared (Fig. 2).

### 2.4.1. Static pre-selection of forecasts (multi-structure approach)

Static pre-selection of forecasts aims to define a priori a subset of hydrological model structures from the available set. This pre-selection is based on the models' performance in simulation and their ability to work together when combined. It should be noted that the assumption made here is that the structures constituting the best combination of models in simulation will be the same in a forecasting context. In the end, a pre-selection of three to four hydrological models, depending on the sub-catchment, was defined to reduce the size of the super-ensemble. Table 3 shows the models pre-selected over each of the 13 tributaries in a lumped context. Details of this methodology are presented in Appendix A.

### 2.4.2. Dynamic combination of forecasts (multi-parameter approach)

It may seem somewhat inappropriate to use a model calibrated to represent low flows (resp. high flows) to predict a flood event (resp. a low-flow event). With this in mind, we set out to develop a simple method applicable in a multi-model framework to overcome this problem. This work is inspired by the findings of Oudin et al. (2006), who demonstrated in a simulation framework the benefits of using a seasonal index to weight hydrographs from two distinct calibrations, one

favouring low flows and the other high flows, depending on the filling rate of the model production store. The seamless dynamic combination developed here consists of applying a variable weight to the three different forecasts issued with the three parameter sets available for a given model structure, depending on the forecast streamflow range (Fig. 3) over each sub-catchment. Details of this methodology are presented in Appendix B.

## 2.5. Forecasting methodology

Hydrological forecasting can be defined as “the translation of meteorological observations and forecasts into estimates of future river streamflow” (Ramos, 2018). For this study, hydrological forecasts are calculated once a day (considering only the meteorological forecast cycle of 0 h UTC) at an hourly time step with lead time up to 120 h (5 days), over the period 2008–2018. These choices were made to match operational conditions applied by CNR (to optimise the operation and maintenance of hydropower plants, ensure hydraulic safety, and sell the energy) while limiting computing time issues and ensuring streamflow data availability for each sub-catchment (less than 10 % gaps per year).

A common practice in forecasting is to exploit the information contained in the last few time steps in observed streamflow data before the forecast time. The underlying assumption is that the latest streamflow observations carry (partial) information about the future response of the catchment. The autocorrelated nature of streamflow tends to support this assumption. Data assimilation techniques, typically based on modifications of parameters and/or states of the model, are usually used for this purpose. While Thiboult et al. (2016) have shown the benefits of applying such approaches to a large sample of lumped models, Royer-Gaspard et al. (2024) highlighted the difficulties encountered when implementing them in a semi-distributed framework. We therefore decided not to apply a classic data assimilation step to states or parameters, but instead an output correction step—aiming to update the forecasts a posteriori according to the error observed at the forecast time—to each sub-catchment. Therefore, the corrected forecast starts from the streamflow observation at the time of forecast issue, and then progressively returns to the uncorrected forecasts following a linear function that acts over a duration based on the catchment response time

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Representation of the dynamic combination for a forecast made on September 5, 2010, at 0 h on the Gard at Remoulins. The dashed black line represents the observed streamflow. The three grey lines represent forecasts made with the CRE0 structure, calibrated according to three distinct transformations. The orange curve represents the dynamic combination. The horizontal lines indicate observed streamflow quantiles. The red arrow indicates that, in the dynamic combination, most of the weight is attributed to the model favouring low flows, i.e. calibrated with KGE(Q<sup>-0.5</sup>). The blue one indicates a greater weight attributed to the model favouring high flows, i.e. calibrated with KGE(Q<sup>+0.5</sup>). Q90, Q75, Q50, Q25 and Q10 correspond to the different streamflow quantiles. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Performance distribution (MAE) of the 39 lumped model configurations (columns) over the 12 catchments for lead times of 6, 24, 72 and 120 h (rows). The box plots represent the 10 %, 25 %, 50 %, 75 % and 90 % quantiles. The dotted red line represents the optimum MAE value. Each colour represents a hydrological model structure, and each geometric pattern represents the power transformation applied to the streamflow during the calibration phase to obtain different parameter sets. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

(calculated here by maximising the cross-correlation between past precipitation and streamflow time series). The correction step is defined by the following equation:

$$ 
\begin{array}{l}
\text{If } H < T_c \qquad Q'_{t+H} = Q_{t+H} + \epsilon_t \\
\text{If } T_c \le H \le 2.T_c \qquad Q'_{t+H} = Q_{t+H} + \epsilon_t \times \left(2 - \frac{H}{T_c}\right) \\
\text{If } H > 2.T_c \qquad Q'_{t+H} = Q_{t+H}
\end{array}
\qquad (3) $$

where $Q'_{t+H}$ and $Q_{t+H}$ respectively are the corrected and uncorrected forecast streamflow at time t for lead time H, $\epsilon$ is the difference between the simulated and observed streamflow in the last observed time step (corresponding to the forecast time) and $T_c$ is the catchment response time.

In the semi-distributed framework, we chose to keep a single hydrological member for upstream catchments. This single hydrological member corresponds to the combination of forecasts issued from 1 to 4 models (selected for their combined simulation performance). This decision was made to avoid a divergence in the number of members within the super-ensemble, which would otherwise grow exponentially from upstream to downstream since mosaic of models are allowed (models can be different for each sub-catchment). Meteorological members were preserved (1 for the deterministic forecast and 50 for the meteorological

ensemble), as they must be applied consistently across the entire catchment. Using a mosaic of meteorological members—i.e., assigning a different member to each sub-catchment—would not be physically meaningful.

Forecasts were produced over the study period in hindcast mode, and performance evaluation was carried out at the outlet of each catchment for fixed lead times with the mean Continuous Ranked Probability Score (CRPS) (Epstein, 1969):

$$ \text{CRPS} = \frac{1}{n} \sum_{i=1}^{n} \int_{-\infty}^{+\infty} (F_y(y) - I(y \ge x))^2 dy \qquad (4) $$

where $F_y$ is the forecast cumulative distribution function, $x$ is the observation corresponding to the variable y to be predicted, $I$ is the unit step function (equal to 0 for every $x < y$ and 1 otherwise) and $n$ is the total number of forecast times. The optimum value for CRPS is 0.

This metric, widely used for hydrological ensemble forecast evaluation, provides a way to compare the performance of the ensemble forecast (e.g. the super-ensemble) with deterministic forecasts (our benchmark) for the same catchment, since this score is equal to the mean absolute error (MAE) in the latter case. Note that in a comparative analysis, it is important to quantify the sampling uncertainty surrounding performance scores (Clark et al., 2021). To this end, we
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Evolution of CRPS according to forecast lead time for each catchment for different forecast scenarios in a lumped framework. The ribbon corresponds to the sampling uncertainty of the CRPS value for the super-ensemble. The dotted red line represents the optimum CRPS value.</figcaption></figure>

systematically applied a bootstrap method, as performed by Candille et al. (2007). Finally, the reliability of the ensemble will be assessed using two methods: the rank histogram (Talagrand and Vautard, 1997) and the alpha score (Renard et al., 2010). All metrics were calculated using the *evalhyd* R package (Hallouin et al., 2024).

# 3. Results

## 3.1. Benchmark

Operational streamflow forecast chains are usually lumped and deterministic. In order to define the benchmark, each of the 39 model configurations (13 structures with 3 parameter sets each) was compared within a lumped deterministic forecasting framework (i.e. a deterministic meteorological forecast used as an input to a single hydrological model). Fig. 4 shows the results for four lead times (6, 24, 76 and 120 h). Overall, the forecasting configurations are homogeneous, with similar performance. However, models calibrated with an objective function

favouring low flows (power transformation equal to -0.5) have poorer predictive quality (higher MAE score), whatever the lead time considered. This result is not surprising, as the absolute errors are much larger for high flows than for low flows and the models optimised towards low flows tend to have a larger bias on mean flow. Fig. 4 also highlights the benefits of the streamflow correction step at the forecast time, since the errors for the lead times closest to that forecast time are significantly reduced. However, forecast error increases with lead time, even when the streamflow correction has disappeared. This result is directly correlated with the uncertainty of weather forecasts, which increases with lead time. Finally, the CRE0 model with a high-flow parameter set was chosen as the benchmark, as it offered the highest median (and average) predictive qualities (lowest MAE score) for streamflow forecasting over all of the catchments and forecasting lead times.
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Performance distribution (CRPS score) of the various super-ensemble approaches tested over the 12 catchments for lead times of 6, 24, 72 and 120 h. The box plots represent the 10 %, 25 %, 50 %, 75 % and 90 % quantiles. The dotted red line represents the optimum CRPS value. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Rank histogram of hydrological forecasts obtained from a reduced super-ensemble approach in a lumped framework for the Valserine catchment at Lancrans. Note that the y-axis scales vary from one panel to another, to make them clearer.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Evolution of the alpha score across forecast lead time for all the catchments in a lumped framework. The box plots represent the 10 %, 25 %, 50 %, 75 % and 90 % quantiles. The dotted red line represents the optimum alpha score value. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Schematic representation of theoretical forecast benefit in a semi-distributed context. Blue sections represent periods when streamflow is observed and orange sections when streamflow is forecasted. The dotted zone corresponds to the period when the predicted downstream streamflow is influenced by the streamflow already observed upstream. This zone is of length Δt, corresponding to the propagation time from upstream to downstream. The two sub-catchments are coloured differently to reflect the spatial variability of meteorological inputs and geomorphological characteristics.</figcaption></figure>

## 3.2. Lumped (super-)ensembles

### 3.2.1. Meteorological, hydrological and super-ensembles

The aim of this section is to check whether ensemble approaches are able to outperform deterministic approaches. Fig. 5 shows the evolution of predictive quality over lead times for each catchment and for four different scenarios in the lumped framework:

- benchmark: deterministic meteorological forecasts feeding a single hydrological model, $CRE0 - KGE(Q^{0.5})$,

- hydrological ensemble: deterministic meteorological forecasts feeding all hydrological models,

- meteorological ensemble: ensemble meteorological forecasts feeding a single hydrological model,

- super-ensemble: ensemble meteorological forecasts feeding all hydrological models.

The use of ensembles improves streamflow forecasting compared with a usual deterministic approach (benchmark). This improvement can be characterised as significant, since it is larger than the confidence interval surrounding the score value, whatever the forecast horizon considered (only the sampling uncertainty for the super-ensemble is shown in Fig. 5 for the sake of clarity). The performance differences between ensemble scenarios, however, are smaller and therefore more difficult to discuss. Nevertheless, some patterns can be highlighted. For example, the super-ensemble has the best predictive quality (i.e. the lowest CRPS score), whatever the catchment and forecast lead time considered. The superiority of the meteorological ensemble over the hydrological ensemble is highly dependent on the forecast target lead time. While explicitly taking meteorological uncertainty into account tends to be more beneficial than hydrological uncertainty for distant lead times, the opposite is true for lead times closer to the forecast time, with a tipping point around 80 h. Note the presence of oscillations over several catchments, whatever the forecasting approach considered. These patterns (which can be multiple per catchment, as on the Fier River for example) are directly linked to an event that was poorly forecasted. In addition, oscillations have a 24-h frequency, which corresponds to the modelling choice of performing a single forecast per day (at 0 h).

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Comparison between lumped and semi-distributed frameworks in terms of (a) predictive performance (CRPS) and (b) reliability (alpha score) for the reduced super-ensemble approach (i.e. using static pre-selection and dynamic combination of forecasts), across lead times of 6, 24, 72 and 120 h. Each point represents a catchment. The red dashed line indicates equivalence between the two frameworks. Lower CRPS and higher alpha scores indicate respectively better performance and reliability.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. B1.** Distribution of the weights assigned to the different models according to the transformation used on streamflow when calculating the KGE during the calibration: -0.5 (red), 0.1 (green) and 0.5 (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. B2.** Scatter plot of hourly streamflow quantiles for simulations obtained with models calibrated with a 0.1 transformation vs a -0.5 transformation (left), a 0.5 transformation vs a -0.5 transformation (centre) and a 0.5 transformation vs a 0.1 transformation (right), considering all the model structures and the 150 catchments. The red line represents the bisector, i.e. there is no difference between the respective quantiles for the same time step of the simulated streamflow. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

### 3.2.2. Reduction of super-ensemble members

The super-ensemble approach achieves the highest performance compared with the other approaches tested. The aim of this section is to investigate a potential reduction in the complexity (number of members) of the super-ensemble and to increase (or at least maintain) its predictive qualities. Fig. 6 shows the distribution of performance (CRPS score) over the catchments for different forecast lead times according to the different member reduction methods implemented. While the member reduction methods did reduce the complexity of the super-ensemble approach, they did not show any significant improvement in CRPS score when compared with the initial super-ensemble.

### 3.2.3. Reliability

In probabilistic streamflow forecasting, the reliability of the ensemble is a key aspect to ensure that the forecast probabilities align with the observed frequencies. However, the oscillations observed in Fig. 5 suggest that some events have been missed, and that the probabilities provided by the super-ensemble are not always representative of the true uncertainties. Fig. 7 shows the rank histogram (Talagrand and Vautard, 1997) of the reduced super-ensemble (i.e. the super-ensemble reduced with a static pre-selection of forecasts and with a dynamic combination of forecasts) across different lead times to investigate a potential reliability issue. This graph visualises the dispersion of ensemble members, under the assumption that, in an ideal case, the observation has an equal chance of falling in any rank. In this case, the shape of the rank histogram is unusual — a sequence of x U-shapes, where x corresponds to the number of hydrological models in the super-ensemble. This pattern is characteristic of multi-model approaches (see e.g. Bellier et al., 2021) and suggests that the forecasts generated by each individual hydrological model suffer from under-dispersion. This issue is

particularly noticeable for the shortest lead times, where the correction step applied to match the observed data at the forecast time severely restricts the ensemble spread. It should be noted that similar results are observed regardless of the catchment analysed.

Given the unusual shape of the rank histogram, we complemented this reliability analysis by computing the alpha score (Renard et al., 2010). A perfectly reliable ensemble has an alpha score of 1. Fig. 8 shows the evolution of this metric across forecast lead times for all the catchments. The least reliable forecasts are observed at short lead times (in the first few hours following the forecast issuance). This can be explained by the impact of the correction step used as a data assimilation method, which strongly constrains the ensemble spread. The alpha score then gradually increases, reaching a peak around 20 h which correspond to the mean catchment response time ($t + T_c$). This peak is generally followed by a slight decrease — more or less pronounced depending on the catchment — until 40 h ($t + 2T_c$). This result suggests that the chosen approach of gradually returning to normal conditions (i.e., without any influence of the correction step) after a characteristic time is not always optimal. Reliability continues then to improve with longer lead times and reaches median alpha score values of 0.86. These values are consistent with those found in the literature for similar modelling frameworks (see e.g. Bourgin et al., 2014; Hapuarachchi et al., 2022; Zhang et al., 2025).

Ensuring the reliability of meteorological forecasts (pre-processing step with the EMOS-ECC approach; see Section 2.2) is therefore not a sufficient condition for ensuring the reliability of hydrological forecasts. To take specific account of hydrological uncertainties, a statistical post-processing step could be performed at the end of the modelling chain and greatly improve the reliability of hydrological forecasts (Bourgin et al., 2014; Zhang et al., 2025). This post-processing step can be directly
applied to the super-ensemble, as done by Bellier et al. (2021) for example.

### 3.3. Semi-distributed super-ensemble

From a theoretical point of view, semi-distribution should generate higher predictive qualities, as this method explicitly considers the spatial variability of meteorological inputs and geomorphological catchment characteristics. Another theoretical benefit should appear for lead times close to the response time at the outlet because part of the total streamflow is already observed over the head sub-catchments (Fig. 9).

The aim of this section is to verify these assumptions. Fig. 10 highlights a slight deterioration in both predictive performance and reliability when using a semi-distributed framework compared with a lumped one. While a semi-distributed approach theoretically benefits “in advance” from upstream observations and forecasts, a side effect is that the quality of downstream forecasts becomes highly dependent on the accuracy of forecasts issued further upstream. Although hydrological dynamics are generally faster in the upper parts of catchments, they are also often more difficult to predict. As a result, a forecast error in one sub-catchment can propagate and impact all downstream sub-catchments. Fig. 10a shows larger performance deterioration at short lead times with the semi-distributed framework, compared to the lumped approach. This once again illustrates the influence of the correction step, which dominates meteorological and hydrological uncertainties during the first few lead times. This result is not entirely surprising, since the method applied to the lumped catchment is virtually “perfect” over the initial lead times — assuming the observations represent the truth (a strong assumption, given that these observations are derived from stage-discharge transformations using rating curves). In the semi-distributed framework, issues such as over-correction or misestimation of upstream observations may occur, disrupting very short-range forecasts (see discussion in Section 4.1). Fig. 10b highlights a decline in reliability for all lead times with the semi-distributed framework compared with the lumped approach. It should be remembered that we did not allow the number of members in the super-ensemble to diverge from upstream to downstream. Testing all possible mosaics of hydrological models for each sub-catchment may improve the reliability of the semi-distributed approach. However, this strategy is extremely demanding in terms of computational resources, as it leads to an exponential increase in the size of the super-ensemble from upstream to downstream.

## 4. Discussion

The results showed an improvement in performance with probabilistic approaches for streamflow forecasting when compared with the deterministic approach classically used for operational purposes. This conclusion is not new, but confirms results produced by the HEPEX community (Schaake et al., 2007). The highest performance was obtained with the super-ensemble approaches, which also confirms the conclusions reached in recent studies on this topic (Bellier et al., 2021; Dion et al., 2021). We therefore propose to focus the discussion section on the most innovative aspects of this work, that is the use of a super-ensemble in a semi-distributed framework, and also the implementation of a methodology to reduce the number of members. Note that the reliability aspect of streamflow forecasts will also be discussed here.

### 4.1. What can potentially go wrong with the semi-distributed modelling framework for streamflow forecasting?

While the fact that a semi-distributed modelling framework does not generate any benefit over a lumped framework may seem illogical at first glance, it is nevertheless not surprising given the conclusions reached in the literature in a simulation framework (Ajami et al., 2004; Khakbaz et al., 2012; Lobligeois, 2014; de Lavenne et al., 2016). Indeed,

authors agree that the benefits of a semi-distributed framework are limited and highly catchment-dependent. On the other hand, it is generally rather difficult to explain why taking explicit account of spatial variability, even without inside information (upstream gauging stations), does not improve forecasts compared with a lumped framework. A first explanation for this must be the opposition between complexity and uncertainty. In fact, meteorological forecasts are by nature highly uncertain, whether quantitatively, temporally or geographically. This last point is particularly important here, since an error in the spatial dimension of meteorological forcings will have a strong impact in a semi-distributed hydrological modelling framework, while the lumped approach will tend to smooth out these errors. Hydrological forecasts are therefore highly dependent on how the catchments have been divided into sub-catchments. We have also seen in Fig. 10 that large deteriorations appear for the shortest lead times. There are several possible reasons for this phenomenon:

* Errors in observed streamflow data impact the forecast. Indeed, the method used here is based on the principle of placing blind trust in observations, with no regard for what has been simulated. As a result, although streamflow records have been visually checked for this study, erroneous upstream observations are inevitable and will propagate through the modelling chain.

* The routing scheme used may show limitations in a forecasting context. Indeed, the simple lag corresponds to a fixed delay of the hydrograph from upstream to downstream. However, the propagation time is not constant and evolves with the streamflow range. In addition, the simple lag is a conservative function while there are many cases where water can be lost (e.g. exchange with aquifer). Thus, if too much water has been stored upstream, the propagation function used will not be able to limit this amount at the outlet.

* The correction step is too simple and not suitable for a semi-distributed context. Indeed, the choice of propagating the latest observations together with the corrected upstream forecasts, then correcting the streamflow forecast at the outlet again, may lead to over-correction, with undesirable interactions between the update of the internal points and that of the outlet.

If these three points can lead to short-term forecasting problems, adapting the updating method to a semi-distributed context seems to be the preferred approach. Royer-Gaspard et al. (2024) recently showed that even with a more complex data assimilation procedure, several difficulties remain for a spatially robust approach to deal with initial conditions and research must be carried out in this direction.

### 4.2. What is the optimum size for a super-ensemble?

Ensemble size is a widely discussed issue in meteorology (Richardson, 2001; Khan et al., 2014; Leutbecher, 2019). Although the optimal size is still an active research topic, meteorologists seem to converge on climate forcing ensembles consisting of 20–80 equiprobable members. Hapuarachchi et al. (2022) showed that using ensemble meteorological forecasts from different climate models is beneficial for streamflow prediction. The authors also explored the size of this ensemble and demonstrated that, in a framework considering multiple probabilistic meteorological sources, the best trade-off between performance and complexity is achieved with approximately 200 members randomly selected from the full ensemble. However, this study only considers a single hydrological model. The size of a probabilistic hydrological multi-model is a problem that has not yet been addressed, to our knowledge. While it may seem of secondary order in a hydrological multi-model approach with a deterministic meteorological scenario, given the generally small size of the generated ensemble, it is nevertheless relevant in a super-ensemble framework where the number of members can quickly become large. The methods used here to reduce the number of members in the super-ensemble are promising, since they
enable us to maintain the performance score close to the one of the
complete super-ensemble. However, they have been developed under
the strong assumption that the behaviours observed in simulation will
remain the same in forecast. This is particularly relevant to the static
pre-selection of models, which could be conducted in a forecasting
framework by testing all possible model combinations on each catch-
ment if the meteorological forecast archives are deep enough. The dy-
namic forecast combination method would also need to be studied in
more detail, including the use of different evaluation criteria focusing on
different streamflow ranges or various events. While this method en-
ables a reduction of the super-ensemble, it was initially designed to
improve the representation of a wider range of streamflow during
forecast.

5. Conclusion

We summarise here our main findings in a few key points:

1. Given the many uncertainties in the modelling chain for streamflow forecasting, it is important to consider future scenarios in a probabilistic framework rather than in a deterministic one, especially for the most distant lead times. The information provided by an ensemble, whether meteorological or hydrological, enables uncertainties to be better considered.

2. Considering hydrological uncertainties has a greater impact in the short term, and meteorological uncertainties in the long term. Considering both sources of uncertainty (super-ensemble approach) generates the highest performance whatever the lead time considered.

3. Methods for reducing the size of the super-ensemble have shown encouraging results; they make it possible to simplify the complexity of the super-ensemble without degrading predictive performance.

4. Although more complex, the semi-distributed approach does not improve forecast quality, and even tends to degrade it, especially in the short term (performance in a longer term is closer to the lumped framework).

While this last point may seem disappointing, it nevertheless opens
the door to numerous research and operational perspectives to exploit
the potential of a semi-distributed super-ensemble. In particular, we see
many opportunities in studying the sensitivity of forecasts to the spatial
division of catchments or in developing data assimilation methods
adapted to the semi-distributed framework. From an operational point
of view, it seems important to apply a post-processing step to the fore-
cast streamflow to ensure the reliability of the super-ensemble. Finally,
although the Rhône catchment has a wide range of hydroclimatic and
geomorphological characteristics, it would be interesting to extend this
study to a larger number of French watersheds and perhaps even
worldwide.

6. Open research section

Streamflow data are freely available on https://hydro.eaufrance.fr
(Dufeu et al., 2022). Meteorological ensembles containing precipitation
and temperature information used in this study are available on https
://www.ecmwf.int/en/forecasts/datasets/set-iii (Molteni et al., 1996).
The COMEPHORE reanalysis (Tabary et al., 2012), which was used for
bias correction of the meteorological ensemble, is freely available for
academic research in France but cannot be publicly distributed due to
commercial restrictions. To access this data, please refer to https://doi.
org/10.25326/360 (Caillaud, 2019). Summary sheets containing the
structural scheme of the different models, the pseudo-code, and the table
of free parameters are available on request or can be found in the PhD
paper of the first author (Thébault, 2023).

CRediT authorship contribution statement

**Cyril Thébault:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Charles Perrin:** Writing – review & editing, Supervision, Software, Resources, Project administration, Methodology, Funding acquisition, Conceptualization. **Sébastien Legrand:** Writing – review & editing, Supervision, Resources, Project administration, Methodology, Funding acquisition, Data curation, Conceptualization. **Vazken Andréassian:** Writing – review & editing, Supervision, Resources, Project administration, Methodology, Conceptualization. **Guillaume Thirel:** Writing – review & editing, Supervision, Software, Resources, Methodology, Conceptualization. **Olivier Delaigue:** Writing – review & editing, Software, Resources, Data curation.

Declaration of competing interest

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

Acknowledgements

This work was supported by CNR and INRAE that provided the PhD
grant of the first author. The authors wish to thank Météo-France and
ECMWF for making the meteorological data used in this study available.
SCHAPI, CNR and EDF are thanked for providing the hydrological data.
The authors thank Isabella Athanassiou for copy-editing an earlier draft
of this paper. The editor, Marco Borga, the associate editor Murray Peel,
and the two anonymous reviewers are also thanked for their feedback
and comments on the manuscript, which helped to improve its overall
quality.

Appendix A Static pre-selection of forecasts (multi-structure approach)

This pre-selection is based on the models’ performance in simulation
and their ability to work together when combined. The multi-model
combination method is identical to that used by Thébault et al.
(2024). Each combination of two to four model configurations among
the 39 available was tested and applied to each sub-catchment. The
authors showed that there were no longer any gains observed for a
higher number of models considered in the combinations. These com-
binations consist of a simple average of the streamflow outputs (the
authors considered also weighted average but it generates similar per-
formances in simulation). Among all the combinations tested, only the
“best”, defined as the one generating the highest performance over the
evaluation period, was selected. The performance score applied to
evaluate the different combinations is a composite criterion based on the
KGE and considering the three transformations applied to the stream-
flow (Q + 0.5, Q + 0.1 and Q–0.5), namely the average of the three
KGEs obtained.

Appendix B Dynamic combination of forecasts (multi-parameter approach)

The dynamic combination developed here consists of applying a
variable weight to the three different forecasts issued with the three
parameter sets available for a given model structure, depending on the
forecast streamflow range (Fig. 3). These weights have been determined
within a simulation framework based on a large sample of French
catchments (the 150 catchments used by Thébault et al., 2024).

In order to obtain a gradual variation of the weights, the streamflow
time series were divided into 100 intervals separated by the different
percentiles (quantiles 1 to 99) and calculated on observed data. Thus, for
each catchment, the time steps corresponding to each observed streamflow range were retrieved. On the basis of this information, we used the data simulated by each model structure on these same time steps, taking care to separate the simulations from the three different objective functions. The weights were then optimised by minimising the absolute error obtained between the observed streamflow and the combination of streamflow simulated with the three distinct objective functions. Fig. B1 shows the weights assigned to the models derived from different objective functions, depending on the streamflow range targeted. Polynomial functions were fitted using the least-squares method for each transformation, ensuring that the total weight remained equal to 1 and avoiding discontinuities. We can see that, as expected, the high-flow (resp. low-flow) transformation will dominate for high (resp. low) quantiles. Models based on a KGE calculated with the 10th root of streamflow show greater consistency in their contribution to the combination (particularly for quantiles above 20), although this remains rather low (average weight of 0.17).

However, observed information is not available in a forecasting framework. The assumption made here is that for a given model structure, the percentile (defined in relation to what has been simulated in the past) corresponding to the forecasted streamflow for a given time step will be the same whatever the objective function considered. Fig. B2 verifies this a priori by comparing the quantiles obtained on all simulated time steps according to the transformation applied during the calibration. Indeed, the various points (time steps) follow the bisector almost perfectly. Note that 98 % of time steps have a quantile difference less than or equal to 2.

Thus, the choice of forecast selected for quantile calculation becomes arbitrary, and for this work we have used models derived from a calibration performed with the KGE calculated on the 10th root of the streamflow to determine the forecast streamflow range. It should be noted that in a forecasting context, it is possible to find a value outside of the quantiles predefined with past simulations. In this case, the weightings corresponding to the extremes are applied (quantile 0 or 100 for respectively a low-flow event or a flood event of a magnitude never simulated). Finally, the dynamic combination of forecasts according to the objective function used for calibration reduces the number of members by three for each (sub-)catchment.

## Data availability

Streamflow and meteorological ensemble are freely available. Re-analysis are available only for research in France but cannot be deposited (commercial constraints). More details in the manuscript.

## References

* Ajami, N., Gupta, H., Wagener, T., Sorooshian, S., 2004. Calibration of a semi-distributed hydrologic model for streamflow estimation along a river system. *J. Hydrol.* 298 (1-4), 112–135. https://doi.org/10.1016/j.jhydrol.2004.03.033.

* Ajami, N.K., Duan, Q., Gao, X., Sorooshian, S., 2006. Multimodel combination techniques for analysis of hydrological simulations : application to distributed model intercomparison project results. *J. Hydrometeorol.* 7 (4), 755–768. https://doi.org/10.1175/JHM519.1.

* Bachet, N., 1934. A note on flood propagation and warning. *Annales Des Ponts et Chaussées* 3 (34), 409–465. https://heritage.ecoledesponts.fr/ark:/12148/bpt6k134237j/f146 item.

* Bellier, J., 2018. Probabilistic Hydrological Forecasting in a Multivariate Framework: Which Methods to Ensure Calibration and Spatiotemporal Coherence? Taking into Account Spatio-Temporal Coherence. Application to Flood Forecasting of the Upper Rhone River Basin. PhD Thesis (in French). Université Grenoble Alpes. https://theses.fr/2018GREAU029.

* Bellier, J., Bontron, G., Zin, I., 2021. Selecting components in a probabilistic hydrological forecasting chain : The benefits of an integrated evaluation. *La Houille Blanche* 107 (1), 1938352. https://doi.org/10.1080/27678490.2021.1936825.

* Bennett, J.C., Robertson, D.E., Shrestha, D.L., Wang, Q.J., Enever, D., Hapuarachchi, P., Tuteja, N.K., 2014. A system for continuous hydrological ensemble forecasting (SCHEF) to lead times of 9 days. *J. Hydrol.* 519, 2832–2846. https://doi.org/10.1016/j.jhydrol.2014.08.010.

* Bergström, S., Forsman, A., 1973. Development of a conceptual deterministic rainfall-runoff model. *Nordic Hydrol.*, 4, 147–170. https://doi.org/10.2166/nh.1973.0012.

* Bernard, M., 1939. Hydro-meteorological aspects of flood control and forecasting problems. *Bull. Am. Meteorol. Soc.* 20 (4), 160–165. http://www.jstor.org/stable/26260302.

* Beven, K., Kirkby, M.J., 1979. A physically based, variable contributing area model of basin hydrology. *Hydrol. Sci. Bull.* 24 (1), 43–69. https://doi.org/10.1080/02626667909491834.

* Bompart, P., Bontron, G., Celie, S., Haond, M., 2009. An operational hydrometeorological forecasting chain for CNR's hydroelectric production needs (in French). *La Houille Blanche* 95 (5), 54–60. https://doi.org/10.1051/lhb/2009056.

* Boucher, M.-A., Anctil, F., Perreault, L., Tremblay, D., 2011. A comparison between ensemble and deterministic hydrological forecasts in an operational context. Adv. Geosci. 29, 85–94. https://doi.org/10.5194/adgeo-29-85-2011.

* Bourgin, F., Ramos, M.H., Thirel, G., Andréassian, V., 2014. Investigating the interactions between data assimilation and post-processing in hydrological ensemble forecasting. *J. Hydrol.* 519, 2775–2784. https://doi.org/10.1016/j.jhydrol.2014.07.054.

* Caillaud, C. (2019). *Meteo-France, Radar COMEPHORE hourly precipitation amount composite* [Dataset]. doi: 10.25326/360.

* Caillouet, L., Celie, S., Vannier, O., Bontron, G., Legrand, S., 2022. Operational hydrometeorological forecasting on the Rhône River in France : Moving toward a seamless probabilistic approach. *La Houille Blanche* 108 (1), 2061312. https://doi.org/10.1080/27678490.2022.2061312.

* Candille, G., Côté, C., Houtekamer, P.L., Pellerin, G., 2007. Verification of an ensemble prediction system against observations. *Mon. Weather Rev.* 135 (7), 2688–2699. https://doi.org/10.1175/MWR3414.1.

* Celie, S., Bontron, G., Ouf, D., Pont, E., 2019. Providing expertise in the operational hydrometeorological forecast (in French). *La Houille Blanche* 105 (2), 55–62. https://doi.org/10.1051/lhb/2019015.

* Clark, M.P., Vogel, R.M., Lamontagne, J.R., Mizukami, N., Knoben, W.J.M., Tang, G., Gharari, S., Freer, J.E., Whitfield, P.H., Shook, K.R., Papalexiou, S.M., 2021. The abuse of popular performance metrics in hydrologic modeling. *Water Resour. Res.* 57 (9), e2020WR029001. https://doi.org/10.1029/2020WR029001.

* Cormary, Y., & Guilbot, A. (1973). Rainfall-runoff relationship study for three investigated catchments. *IAHS Madrid Symposium*, Red Books Series, Vol. 108, 265–279. https://iahs.info/uploads/dms/4246.265-279-108-Cormary-opt.pdf.

* Coron, L., Delaigue, O., Thirel, G., Dorchies, D., Perrin, C., Michel, C. 2021 *airGR: Suite of GR Hydrological Models for Precipitation-Runoff Modelling*. R package version 1.6.12. Portail Data INRAE. doi: 10.15454/EX11NA.

* Coron, L., Perrin, C., Delaigue, O., Thirel, G. (2022). *airGRplus: Additional Hydrological Models to the «airGR» Package*. R package version 0.9.14.7.9001.

* Coron, L., Thirel, G., Delaigue, O., Perrin, C., Andréassian, V., 2017. The suite of lumped GR hydrological models in an R package. *Environ. Model. Softw.* 94, 166–171. https://doi.org/10.1016/j.envsoft.2017.05.002.

* Dasgupta, A., Arnal, L., Emerton, R., Harrigan, S., Matthews, G., Muhammad, A., O'Regan, K., Pérez-Ciria, T., Valdez, E., van Osnabrugge, B., Werner, M., Buontempo, C., Cloke, H., Pappenberger, F., Pechlivanidis, I.G., Prudhomme, C., Ramos, M.-H., Salamon, P., 2023. Connecting hydrological modelling and forecasting from global to local scales : perspectives from an international joint virtual workshop. *J. Flood Risk Manage.*, e12880 https://doi.org/10.1111/jfr3.12880.

* de Lavenne, A., Thirel, G., Andréassian, V., Perrin, C., Ramos, M.H., 2016. Spatial variability of the parameters of a semi-distributed hydrologic model. 7th International Water Resources Management Conference of ICWRS 373, 87–94. https://doi.org/10.5194/piaths-373-87-2016.

* Dion, P., Martel, J.-L., Arsenault, R., 2021. Hydrological ensemble forecasting using a multi-model framework. *J. Hydrol.* 600, 126537. https://doi.org/10.1016/j.jhydrol.2021.126537.

* Duan, Q., Ajami, N.K., Gao, X., Sorooshian, S., 2007. Multi-model ensemble hydrologic prediction using Bayesian model averaging. *Adv. Water Resour.* 30 (5), 1371–1386. https://doi.org/10.1016/j.adwwatres.2006.11.014.

* Dufeu, E., Mougin, F., Foray, A., Baillon, M., Lamblin, R., Hebrard, F., Chaleon, C., Romon, S., Cobos, L., Gouin, P., Audouy, J.-N., Martin, R., Poligot-Pitsch, S., 2022. Finalisation of the French national hydrometric data information system modernisation operation (Hydro3) (in French). *La Houille Blanche* 108 (1), 2099317. https://doi.org/10.1080/27678490.2022.2099317.

* Epstein, E.S., 1969. A scoring system for probability forecasts of ranked categories. J. Appl. Meteorol. (1962-1982) 8 (6), 985–987. https://doi.org/10.1175/1520-0450 (1969)008-0985:ASSFPF-2.0.CO;2.

* Ficchi, A., Perrin, C., Andréassian, V., 2019. Hydrological modelling at multiple sub-daily time steps : Model improvement via flux-matching. *J. Hydrol.* 575, 1308–1327. https://doi.org/10.1016/j.jhydrol.2019.05.084.

* Garçon, R., 1999. Overall rain-flow model for flood forecasting and pre-determination (in French). *La Houille Blanche* 7–8, 88–95. https://doi.org/10.1051/lhb/1999088.

* Gneiting, T., Raftery, A.E., 2005. Weather forecasting with ensemble methods. *Science* 310 (5746), 248–249. https://doi.org/10.1126/science.1115255.

* Gupta, H.V., Kling, H., Yilmaz, K.K., Martinez, G.F., 2009. Decomposition of the mean squared error and NSE performance criteria : implications for improving hydrological modelling. *J. Hydrol.* 377 (1), 80–91. https://doi.org/10.1016/j.jhydrol.2009.08.003.

* Hallouin, T., Bourgin, F., Perrin, C., Ramos, M.-H., Andréassian, V., 2024. EvalHyd v0.1.2: A polyplot tool for the evaluation of deterministic and probabilistic streamflow predictions. Geosci. Model Deve. 17 (11), 4561–4578. https://doi.org/10.5194/gmd-17-4561-2024.

* Hamill, T.M., Colucci, S.J., 1997. Verification of Eta-RSM short-range ensemble forecasts. Monthly Weather Rev. 125 (6), 1312–1327. https://doi.org/10.1175/1520-0493(1997)125<1312:voersr>2.0.CO;2.
Hapuarachchi, H.A.P., Bari, M.A., Kabir, A., Hasan, M.M., Woldemeskel, F.M., Gamage, N., Sunter, P.D., Zhang, X.S., Robertson, D.E., Bennett, J.C., Feikema, P.M., 2022. Development of a national 7-day ensemble streamflow forecasting service for Australia. *Hydrol. Earth Syst. Sci.* 26 (18), 4801–4821. https://doi.org/10.5194/hess-26-4801-2022.

Jakeman, A.J., Littlewood, I.G., Whitehead, P.G., 1990. Computation of the instantaneous unit hydgraph and identifiable component flows with application to two small upland catchments. *J. Hydrol.* 117 (1), 275–300. https://doi.org/10.1016/0022-1694(90)90097-H.

Khabbaz, B., Imam, B., Hsu, K., Sorooshian, S., 2012. From lumped to distributed via semi-distributed : Calibration strategies for semi-distributed hydrologic models. *J. Hydrol.* 418–419, 61–77. https://doi.org/10.1016/j.jhydrol.2009.02.021.

Khan, M.M., Shamseldin, A.Y., Melville, B.W., 2014. Impact of ensemble size on forecasting occurrence of rainfall using TIGGE precipitation forecasts. *J. Hydrol. Eng.* 19 (4), 732–738. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000864.

Leutbecher, M., 2019. Ensemble size : how suboptimal is less than infinity? *Q. J. R. Meteorolog. Soc.* 145 (S1), 107–128. https://doi.org/10.1002/qj.3387.

Li, M., Wang, Q.J., Bennett, J.C., Robertson, D.E., 2016. Error reduction and representation in stages (ERRIS) in hydrological modelling for ensemble streamflow forecasting. *Hydrol. Earth Syst. Sci.* 20 (9), 3561–3579. https://doi.org/10.5194/hess-20-3561-2016.

Li, M., Wang, Q.J., Robertson, D.E., Bennett, J.C., 2017. Improved error modelling for streamflow forecasting at hourly time steps by splitting hydrographs into rising and falling limbs. *J. Hydrol.* 555, 586–599. https://doi.org/10.1016/j.jhydrol.2017.10.057.

Lobligeois, F., 2014. *Can We Improve Streamflow Modeling by Using Higher Resolution Rainfall Information? Diagnostic test on 181 French Watersheds*. PhD Thesis (in French). AgroParisTech, Paris. https://theses.fr/2014AGPT0013.

Lobligeois, F., Andréassian, V., Perrin, C., Tabary, P., Loumagne, C., 2014. When does higher spatial resolution rainfall information improve streamflow simulation? An evaluation using 3620 flood events. *Hydrol. Earth Syst. Sci.* 18 (2), 575–594. https://doi.org/10.5194/hess-18-575-2014.

McInerney, D., Thyer, M., Kavetski, D., Laugesen, R., Tuteja, N., Kuczera, G., 2020. Multi-temporal hydrological residual error modeling for seamless subseasonal streamflow forecasting. *Water Resour. Res.* 56 (11), e2019WR026979. https://doi.org/10.1029/2019WR026979.

McInerney, D., Thyer, M., Kavetski, D., Laugesen, R., Woldemeskel, F., Tuteja, N., Kuczera, G., 2021. Improving the reliability of sub-seasonal forecasts of high and low flows by using a flow-dependent nonparametric model. *Water Resour. Res.* 57 (11), e2020WR029317. https://doi.org/10.1029/2020WR029317.

McInerney, D., Thyer, M., Kavetski, D., Laugesen, R., Woldemeskel, F., Tuteja, N., Kuczera, G., 2022. Seamless streamflow forecasting at daily to monthly scales : MutTHRE lets you have your cake and eat it too. *Hydrol. Earth Syst. Sci.* 26 (21), 5669–5683. https://doi.org/10.5194/hess-26-5669-2022.

McInerney, D., Thyer, M., Kavetski, D., Lerat, J., Kuczera, G., 2017. Improving probabilistic prediction of daily streamflow by identifying Pareto optimal approaches for modeling heteroscedastic residual errors. *Water Resour. Res.* 53 (3), 2199–2239. https://doi.org/10.1002/2016WR019168.

Mathevet, T. 2005. *Which Lumped Rainfall-Runoff Models at the Hourly Time Step? Empirical Developments and Model Comparisons on a Large Sample of Watersheds*. Ph.D. Thesis (in French). ENGREF, Paris. https://hal.inrae.fr/tel-02587642.

Michel, C., 1983. Que peut-on faire en hydrologie avec modèle conceptual à un seul paramètre ? *La Houille Blanche* 1, 39–44. https://doi.org/10.1051/lhb/1983004.

Michel, C. (1991). Hydrologie appliquée aux petits bassins ruraux, *Hydrology handbook* (in French), Cemagref, Antony, France.

Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T., 1996. The ECMWF ensemble prediction system: methodology and methodological models. *Q. J. R. Meteorolog. Soc.* 122 (529), 73–119. https://doi.org/10.1002/qj.49712252905.

Moore, R.J., Clarke, R.T., 1981. A distribution function approach to rainfall runoff modelling. *Water Resour. Res.* 17 (5), 1367–1382. https://doi.org/10.1029/WR017005p01367.

Mureau, R., Molteni, F., Palmer, T.N., 1993. Ensemble prediction using dynamically conditioned perturbations. *Q. J. R. Meteorolog. Soc.* 119 (510), 299–323. https://doi.org/10.1002/qj.49711950010.

Nielsen, S.A., Hansen, E., 1973. *Numerical simulation of the rainfall-runoff process on a daily basis*. Hydro. Res. 4 (3), 171–190. https://doi.org/10.2166/NH.1973.0013.

O'Connell, P.E., Nash, J.E., Farrell, J.P., 1970. River flow forecasting through conceptual models part II - The Brosna catchment at Ferbane. *J. Hydrol.* 10 (4), 317–329. https://doi.org/10.1016/0022-1694(70)90221-0.

Oudin, L., Andréassian, V., Mathevet, T., Perrin, C., Michel, C., 2006. Dynamic averaging of rainfall-runoff model simulations from complementary model parameterizations. *Water Resour. Res.* 42, W07410. https://doi.org/10.1029/2005WR004636.

Oudin, L., Hervieu, F., Michel, C., Perrin, C., Andréassian, V., Anctil, F., Loumagne, C., 2005. Which potential evapotranspiration input for a lumped rainfall-runoff model for rainfall-runoff modelling. *J. Hydrol.* 303 (1), 290–306. https://doi.org/10.1016/j.jhydrol.2004.08.026.

Pappenberger, F., Pagano, T.C., Brown, J.D., Alfieri, L., Lavers, D.A., Berthet, L., Bressand, F., Cloke, H.L., Cranston, M., Dammela, V., Demuth, N., de Saint-Aubin, C., Feikema, P.M., Fresch, M.A., Garçon, R., Gelfan, A., He, Y., Hu, Y.Z., Janet, B., Jurdy, N., Javelle, P., Kuchment, L., Laborda, Y., Langsholt, E., Le Lay, M., Li, Z.J., Mannessiez, F., Marchandise, A., Marty, R., Meißner, D., Manful, D., Organde, D., Pourret, V., Rademacher, S., Ramos, M.H., Reinbold, D., Tibaldi, S., Silvano, P., Salamon, P., Shin, D., Sorbet, C., Sprokkereef, E., Thiemig, V., Tuteja, N. K., van Andel, S.J., Verkade, J.S., Vehviläinen, B., Vogelbacher, A., Wetterhall, F., Zappa, M., Van der Zwan, R.E., Thielen-del Pozo, J., 2016. Hydrological Ensemble Prediction Systems Around the Globe. Springer, Berlin, Heidelberg, pp. 1–35. https://doi.org/10.1007/978-3-642-40457-3-47-1.

Perrin, C., 2000. *Towards an Improved Version of a Lumped Rainfall-Runoff Model Through a Comparative Approach*. PhD Thesis (in French). Institut National Polytechnique, de Grenoble. https://doi.archives-ouvertures.fr/tel-00006216.

Perrin, C., Michel, C., Andréassian, V., 2010. A set of hydrological models (Chapter 16). In: Tanguy, J.M. (Ed.), Environmental Hydraulics. Volume 2 Mathematical models. ISTE Ltd, John Wiley & Sons, Paris, pp. 493–509. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1848211546.html.

Pushpalath, A., Perrin, C., Moine, N.L., Andréassian, V., 2012. A review of efficiency criteria suitable for evaluating low-flow simulations. *J. Hydrol.* 420–421, 171–182. https://doi.org/10.1016/j.jhydrol.2011.11.055.

Ramos, M.H., 2018. *Quality and value of hydrological ensemble predictions*. Habilitation Thesis (in French). Sorbonne Université. https://hal.inrae.fr/tel-02608754v1.

Ramos, M.-H., Pappenberger, F., Wood, A., Wetterhall, F., Wang, Q., Verkade, J., Pechlividanis, I., Thielen-del Pozo, J., Buizza, R., Schaake, J. 2018. *The history of HEPEX - a community of practice in hydrologic prediction*. 10137. EGU General Assembly Conference Abstracts.

Renard, B., Kavetski, D., Kuczera, G., Thyer, M., Franks, S.W., 2010. Understanding predictive uncertainty in hydrologic modeling: the challenge of identifying input and structural errors. *Water Resour. Res.* 46 (5). https://doi.org/10.1029/2009WR008328.

Richardson, D.S., 2001. Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. *Q. J. R. Meteorolog. Soc.* 127 (577), 2473–2489. https://doi.org/10.1002/qj.4971275717.

Royer-Gaspard, P., Bourgin, F., Perrin, C., Andréassian, V., De Lavenne, A., Thirel, G., Tilman, F., 2024. Benefits of upstream data for downstream streamflow forecasting : Data assimilation in a semi-distributed flood forecasting model. *La Houille Blanche*. https://doi.org/10.1080/27678490.2024.2374081.

Schaake, J.C., Hamill, T.M., Buizza, R., Clark, M., 2007. HEPEX: the hydrological ensemble prediction experiment. *Bull. Am. Meteorol. Soc.* 88 (10), 1541–1548. https://doi.org/10.1175/BAMS-88-10-1541.

Schefzik, R., Thorarinsdottir, T.L., Gneiting, T., 2013. Uncertainty quantification in complex simulation models using ensemble copula coupling. *Stat. Sci.* 28 (4), 616–640. https://doi.org/10.1214/13-STS443.

Seiller, G., Anctil, F., Perrin, C., 2012. Multimodel evaluation of twenty lumped hydrological models under contrasted climate conditions. *Hydrol. Earth Syst. Sci.* 16 (4), 1171–1189. https://doi.org/10.5194/hess-16-1171-2012.

Seo, D.-J., Herr, H.D., Schaake, J.C., 2006. A statistical post-processor for accounting of hydrologic uncertainty in short-range ensemble streamflow prediction. *Hydrol. Earth Syst. Sci. Discuss.* 3 (4), 1987–2035. https://doi.org/10.5194/hessd-3-1987-2006.

Shamseldin, A.Y., O'Connor, K.M., Liang, G.C., 1997. Methods for combining the outputs of different rainfall runoff models. *J. Hydrol.* 197, 203–229. https://doi.org/10.1016/S0022-1694(96)03259-3.

Sugawara, M., 1979. Automatic calibration of the tank model / L'étalonnage automatique d'un modèle à cisterne. *Hydrol. Sci. Bull.* 24 (3), 375–388. https://doi.org/10.1080/02626667909491876.

Tabary, P., Dupuy, P., L'Henaff, G., Gueguen, C., Moulin, L., Laurantin, O., Merlier, C., Soubeyoux, J.-M., 2012. A 10-year (1997-2006) reanalysis of quantitative precipitation estimation over France : Methodology and first results. *IAHS AISH Publ.*, Red Books Series 351, 255–260. https://iahs.info/uploads/dms/15900.047-255-260-351-73-IAHS_Tabary_et_a.pdf.

Talagrand, O., Vautard, R. 1997. *Evaluation of probabilistic prediction systems*. ECMWF, Workshop on Predictability, 20-22 October 1997, 25 p. https://www.ecmwf.int/en/ library/76596-evaluation-probabilistic-prediction-systems.

Thèbault, C., 2023. *What Contributions of a Semi-Distributed Multi-Model Approach for Streamflow Forecasting? PhD Thesis (in French)*. Sorbonne Université, Paris. https://theses.fr/2023SORUS649.

Thèbault, C., Perrin, C., Andréassian, V., Thirel, G., Legrand, S., Delaigue, O., 2024. Multi-model approach in a variable spatial framework for streamflow simulation. *Hydrol. Earth Syst. Sci.* 28 (7), 1539–1566. https://doi.org/10.5194/hess-28-1539-2024.

Thiboult, A., Anctil, F., Boucher, M.-A., 2016. Accounting for three sources of uncertainty in ensemble hydrological forecasting. *Hydrol. Earth Syst. Sci.* 20 (5), 1809–1825. https://doi.org/10.5194/hess-20-1809-2016.

Thiéry, D. 1982. Using a lumped model to identify multiple influences due to various human activities in groundwater level (in French). IAHS Publ. 136, Reb Books Series, 71-77, https://iahs.info/uploads/dms/5615.71-77-136-Thiéry.pdf.

Thirel, G., Santos, L., Delaigue, O., Perrin, C., 2024. On the use of streamflow transformations for hydrological model calibration. *Hydrol. Earth Syst. Sci.*, 28, 4837–4860. https://doi.org/10.5194/hess-28-4837-2024.

Todini, E., 2007. Hydrological catchment modelling : Past, present and future. *Hydrol. Earth Syst. Sci.* 11 (1), 468–482. https://doi.org/10.5194/hess-11-468-2007.

Todini, E., 2011. History and perspectives of hydrological catchment modelling. *Hydrol. Res.* 42 (2-3), 73–85. https://doi.org/10.2166/nh.2011.096.

Toth, Z., Kalnay, E., 1993. Ensemble forecasting at NMC : the generation of perturbations. *Bull. Am. Meteorol. Soc.* 74 (12), 2317–2330. https://doi.org/10.1175/1520-0477 (1993)074<2317:efang>2.0.co;2.

Troin, M., Arnaault, R., Wood, A.W., Brissette, F., Martel, J.-L., 2021. Generating Ensemble Streamflow Forecasts : A review of methods and approaches over the past 40 years. *Water Resour. Res.* 57 (7), e2020WR028392. https://doi.org/10.1029/2020WR028392.

Turcotte, T., Fortier Filion, T.-C., Lacombe, P., Fortin, V., Roy, A., Royer, A., 2010. Hydrological simulation of the final days of the spring flood: the problem of the missing snow (in French). *Hydrol. Sci. J.* 55 (6), 872–882. https://doi.org/10.1080/02626667.2010.503933.
Valéry, A., Andréassian, V., Perrin, C., 2014. 'As simple as possible but not simpler': What is useful in a temperature-based snow-accounting routine? Part 2 – Sensitivity analysis of the Cemaneige snow accounting routine on 380 catchments. *J. Hydrol.* 517, 1176–1187. https://doi.org/10.1016/j.jhydrol.2014.04.058.

Vaze, J., Chiew, F.H.S., Perraud, J.M., Viney, N., Post, D., Teng, J., Wang, B., Lerat, J., Goswami, M., 2011. Rainfall-runoff modelling across southeast Australia: datasets, models and results. *Austra. J. Water Resour.* 14 (2), 101–116. https://doi.org/10.1080/13241583.2011.11465379.

Velazquez, J.A., Anctil, F., Ramos, M.H., Perrin, C., 2011. Can a multi-model approach improve hydrological ensemble forecasting? A study on 29 French catchments using

16 hydrological model structures. Adv. Geosci. 29, 33–42. https://doi.org/10.5194/ageeo-29-33-2011.

White, G.F., 1939. Economic aspects of flood-forecasting. *Trans. Am. Geophys. Union* 20, 218–233. https://doi.org/10.1029/TR020i002p00218.

WMO. 2012. Guidelines on Ensemble Prediction Systems and Forecasting (Guidelines 1091; p. 32). https://library.wmo.int/idurl/4/48473.

Zhang, J., Li, W., Duan, Q., 2025. Quantifying the contributions of hydrological pre-processor, post-processor, and data assimilator to ensemble streamflow prediction skill. *J. Hydrol.* 651, 132611. https://doi.org/10.1016/j.jhydrol.2024.132611.