# Lead-time-dependent calibration of a flood forecasting model

Paul C. Astagneau, François Bourgin, Vazken Andréassian, Charles Perrin

► To cite this version:

Paul C. Astagneau, François Bourgin, Vazken Andréassian, Charles Perrin. Lead-time-dependent calibration of a flood forecasting model. Journal of Hydrology, 2024, pp.132119. 10.1016/j.jhydrol.2024.132119. hal-04717741

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Research papers

# Lead-time-dependent calibration of a flood forecasting model

Paul C. Astagneau<sup>a,b,c,d,* ,1</sup>, François Bourgin<sup>a</sup>, Vazken Andréassian<sup>a</sup>, Charles Perrin<sup>a</sup>

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

## ARTICLE INFO

This manuscript was handled by Andras Bar-dossy, Editor-in-Chief, with the assistance of Szilá-gyi József, Associate Editor.

**Keywords:**

Calibration

Forecasting

Floods

Hydrological modelling

Lead-time-dependency

## ABSTRACT

The hydrological models used for flood forecasting purposes are usually first calibrated in simulation mode and then applied for flood forecasting in combination with some data assimilation procedure allowing to update and correct the model in real-time. Such a two-step procedure may not be the best choice to train model parameters for forecasting purposes. An alternative approach is to calibrate the hydrological model separately for each target lead time in the presence of the updating procedure. However, it is unclear whether this approach can provide the most efficient and the most robust forecasts. We compared in this paper three approaches to calibrate the parameters of a flood forecasting model: calibration in simulation mode (i.e. without data assimilation), lead-time-dependent calibration (i.e. with data assimilation), and a procedure combining both approaches. An hourly flood forecasting model (a parsimonious hydrological model combined with a state updating assimilation procedure) was used to produce forecasts for three lead times on 687 catchments and 40,411 flood events in metropolitan France. The performance of the model calibrated with the three approaches was evaluated according to catchment response times and flood rise times. An analysis of parameter robustness was also carried out. The results showed that lead-time-dependent calibration improves performance for catchments characterised by slow temporal dynamics. However, for catchments with fast temporal dynamics, it leads to degraded performance at short lead times and reduces parameter robustness. We found that the combined calibration approach is the best compromise between parameter robustness and performance at all lead times and for all catchment and flood types.

# 1. Introduction

## 1.1. Errors in flood forecasts

Hydrological models are imperfect representations of a complex natural system, and flood forecasting models are no exception. Hydrological models struggle to characterise the complex interactions inducing non-linear streamflow responses to rainfall (e.g. Chappell et al., 2017), particularly in dry and arid areas (e.g. McMillan et al., 2016; Melsen et al., 2018) and during flash floods (e.g. Hapuarachchi et al., 2011) where extreme nonlinearities characterise the streamflow generation processes (e.g. Gaume et al., 2009).

While model structural inadequacy can explain poor model performance in these catchments (e.g. Astagneau et al., 2021, 2022), and is sometimes considered to be one of the main sources of uncertainty in hydrological modelling (e.g. Clark et al., 2008; Knoben et al., 2020), other sources of uncertainty contribute to errors in flood forecasting.

For example, precipitation uncertainties (e.g. Ruiz-Villanueva et al., 2012; Zhang et al., 2017) or uncertainties arising from stage measurements (e.g. Horner et al., 2018), play a major role in hydrological prediction performance. Uncertainty in parameter estimation can also have a major impact on streamflow predictions (e.g. Beven, 2016), especially when few data points are available (e.g. Perrin et al., 2007).

## 1.2. Errors from parameter estimation

The hydrological models used in the flood forecasting chain require calibration of their parameters to achieve acceptable performance. This process consists in finding the parameter set that provides the best fit between simulations and observations (usually streamflow). The advantages and disadvantages of the calibration process have been extensively discussed in the literature, as calibration issues contribute to uncertainty in streamflow predictions (e.g. Beven, 2006; Andréassian et al., 2012).

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

<footnote><sup>b</sup> WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, Switzerland</footnote>

<footnote><sup>c</sup> Institute for Atmospheric and Climate Science, ETH Zurich, Zurich, Switzerland</footnote>

<footnote><sup>d</sup> Climate Change, Extremes and Natural Hazards in Alpine Regions Research Center CERC, Davos Dorf, Switzerland</footnote>

<footnote>* Corresponding author at: WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, Switzerland. E-mail address: paul.astagneau@slf.ch (P.C. Astagneau).</footnote>

<footnote n="1">¹ Now at b, c, d.</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Difference between calibration options for flood forecasting.</figcaption></figure>

When searching for an optimal parameter set for a given catchment and time period, different types of problems can arise. For example, if the search algorithm used for calibration is not suitable for the problem at hand, the numerical optimum may be missed, leading to a miscalibration situation (Andréassian et al., 2012). Another common problem in parameter calibration is when the resulting parameter set is a numerical optimum rather than being a hydrological one. This overcalibration problem often results in parameter sets that perform well in the calibration period but poorly in extrapolation (Gharari et al., 2013), leading to low model robustness (e.g. Fowler et al., 2016). Over-calibration can result from a mismatch between model structure and data (e.g. too many parameters for the available information; Jakeman and Hornberger, 1993; Perrin et al., 2001), lack of information, which usually translates into parameter equifinality (e.g. Beven, 2006), or the use of an inappropriate objective function to reproduce different flood characteristics (e.g. Gupta et al., 2009; Brunner et al., 2021). To find a better hydrological optimum, several studies have evaluated the benefit of using hydrological signatures to calibrate hydrological models (e.g. Hingray et al., 2010; Shafii and Tolson, 2015; Huynh et al., 2023). These calibration procedures are commonly used within multi-objective frameworks to reduce the dependence of parameters to one metric (Esfstratiadis and Koutsoyiannis, 2010). Improvements in parameter estimation have also been made by integrating additional data to streamflow during the calibration process (e.g. Nijzink et al., 2018; Pelletier and Andréassian, 2022). Another way to improve calibration for flood simulation is to use metrics that place more emphasis on high flows, therefore favouring specific locations of the parameter space (e.g. Moussa and Chahinian, 2009; Mizukami et al., 2019).

Overall, these studies showed that the calibration process has large impacts on streamflow simulations. These differences affect model errors and therefore flood forecasting accuracy.

### 1.3. Hydrological parameters are affected by the associated data assimilation procedure

Despite the use of calibration to improve streamflow efficiency, significant errors remain when a hydrological model is used for flood prediction (e.g. Roundy et al., 2018). To further reduce errors in flood forecasting, hydrological models are often combined with data assimilation (DA) techniques (Refsgaard, 1997), such as state updating

at the time of forecast (e.g. Pagano et al., 2014; Pappenberger et al., 2016). The forecast error at (or up to) the time of forecast is used to adjust the initial states (e.g. Piazzi et al., 2021), the outputs (e.g. Li et al., 2015b), or the parameters (e.g. Ercolani and Castelli, 2017) of the model. Additional data to streamflow can also be assimilated to modify the states of the model, such as soil moisture measured by satellites (e.g. Yan and Moradkhani, 2016) or remotely sensed snow observations (e.g. Metref et al., 2023). Streamflow data are still used in most cases for DA (Piazzi et al., 2021).

For flood forecasting applications, a hydrological model is classically implemented in a two-step approach, first by calibrating the model in simulation (i.e. without DA), and second by implementing the calibrated model for forecasting using a DA procedure (Fig. 1). The rationale behind this is that the model parameters should represent intrinsic characteristics of the catchment and therefore be time-invariant in the hydrological modelling process (e.g. Moradkhani et al., 2005). The assimilation procedure will help to follow deviations from the general behaviour by correcting the model (states, parameters, or outputs). The hydrological parameters are usually calibrated according to this procedure (e.g. Pagano et al., 2011; Li et al., 2014; Mazzoleni et al., 2018; Piazzi et al., 2021). For example, Li et al. (2015a) evaluate the performance of an assimilation scheme for updating the states of a semi-distributed model applied to an Australian catchment. In their study, the parameters of the hydrological model are calibrated in simulation mode (i.e. without state updating or error correction). Thiboult and Anctil (2015) investigate the suitability of using the ensemble Kalman filter (EnKF) to update the states of 20 lumped models over 38 Canadian catchments: the tuning of the EnKF parameters is done after the calibration of the hydrological models. Another example of the two-step approach can be found in the study of Demirel et al. (2013) who investigated uncertainty in forecasting low flows with two conceptual models in the Moselle River. In a first step, they perform parameter estimation using a parameter uncertainty estimator in simulation mode. The routing stores are updated in a second step following a deterministic approach to produce low-flow forecasts.

An invariant parameter set obtained in simulation mode may, however, limit forecasting capability (e.g. Coustau et al., 2013). In other words, this two-step process may result in sub-optimal model parameters for the forecasting objective, i.e. minimising model error for the target lead time, which may result in a loss of efficiency. Therefore, when using DA methods to improve flood forecasting, it may
<figure><img src="image_4.png" /><figcaption>Fig. 2. Schematic representation of a fixed lead time forecast compared to forecasts made at three consecutive times for a given lead time (H = 2 h) and simulations without state updating. The difference between the two options is further illustrated by the calculation of a squared error objective function (OF).</figcaption></figure>

be preferable to implement a single-step approach, i.e. by estimating the hydrological model parameters for the target lead-time (Goswami et al., 2005). In other words, the model is calibrated directly for a flood forecasting objective by minimising model errors in the forecast (with DA) for a given lead time (Fig. 2), rather than in the simulation (without DA). In hydrological modelling, parameter optimisation is based on a cost function, usually emphasising a specific modelling objective (e.g. high flows). In the single-step calibration approach described above, a lead time can be considered as an additional constraint during the optimisation procedure, helping to find parameters that are sensitive to a flood forecasting objective.

When autoregressive (AR) models are used to correct streamflow simulations, the AR parameters are sometimes jointly estimated with the hydrological parameters (single-step approach), either simultaneously or sequentially (e.g. Li et al., 2016; Zhang et al., 2018a,b). Zhang et al. (2018a) state that when the hydrological parameters are calibrated before error correction is applied, simulated streamflows are more consistent with observations but forecasted streamflows are not. However, joint approaches (or single-step) are sometimes non-robust and induce interactions between AR parameters and water balance parameters (Evin et al., 2014). Li et al. (2015b) found that the parameter sets identified using a joint procedure can degrade the performance of the baseline hydrological model (i.e. without DA). As the effect of DA decreases with the lead time (Piazzı et al., 2021), this situation could result in degraded performance at long lead times. Since model errors tend to increase with lead time due to the progressive loss of persistence of the assimilated information, the drawback of the single-step approach is that it may result in parameters that are too dependent on the lead time chosen.

These studies focused only on the estimation of hydrological parameters when an AR method is used. To our knowledge, hydrological models are rarely calibrated for target lead times (i.e. in forecasting mode when state updating is activated), mostly because of computational limitations (data assimilation techniques increase computation times, e.g. to approximate the distribution of a state variable). One example of joint calibration is the simultaneous optimisation and data assimilation (SODA) method introduced by Vrugt et al. (2005). In the SODA method, posterior parameter sets are searched with a global optimisation algorithm together with the application of an EnKF state updating procedure for one-day lead time. Another example can be found in the works of Tangara (2005) and Berthet (2010): there, a choice was made to perform lead-time-dependent calibration. Their procedure is used for the calibration of the French operational flood forecasting model (Viatgé et al., 2019). However, this procedure has been found to be challenging for operational applications and its impact on parameter robustness has not been investigated. Furthermore, the usefulness of lead-time-dependent calibration needs to be evaluated

with respect to catchment and flood temporal dynamics. In addition, except for the studies by Tangara (2005) and Berthet (2010), the aforementioned studies are based on 1-day-ahead forecasts and a limited number of catchments (e.g. Vrugt et al., 2005; Evin et al., 2014; Li et al., 2015b).

Note that the single-step approach differs from updating the parameters before the forecast time, resulting in time-varying parameters, which is also classically used for flood forecasting applications. As errors remain after the calibration process and as more observations become available during the forecasts, parameters are sometimes updated before the forecast starts (e.g. Yang and Michel, 2000; Calvo and Savi, 2009; DeChant and Moradkhani, 2012; Bahramian et al., 2023). In other words, the optimal parameter set in forecasting mode is different from that in simulation mode and it changes over time (e.g. Vrugt et al., 2013). A more advanced updating of parameters in forecasting mode can be performed by dual-state parameter assimilation (e.g. Moradkhani et al., 2005; Dang and Anh, 2023).

## 1.4. Scope and novelty of the paper

Calibration problems are a major contributor to errors in flood forecasting, and the use of DA to reduce errors in flood forecasting affects the estimation of the hydrological model parameters. As the effect of DA changes with lead time, the model parameters estimated without DA (two-step approach) may not always be the best choice for flood forecasting in a variety of hydrological cases. However, estimating different sets of parameters for different lead times can be seen as a kind of overcalibration. It is not clear from the above-cited literature whether lead-time-dependent calibration improves flood forecasting and parameter robustness.

The aim of the present study is to investigate this issue. Specifically, we compare three approaches to calibrate a flood forecasting model according to catchment temporal dynamics and parameter robustness. To our knowledge, the advantages and disadvantages of lead-time-dependent calibration have never been investigated on a large sample of catchments and flood events. Note that this study is not about the impact of DA on flood forecasting but about the impact of lead-time-dependent calibration on streamflow forecast efficiency and parameter robustness.

The novelty of this paper lies in the analysis of the benefits of lead-time-dependent calibration with respect to catchment and flood temporal dynamics. Furthermore, we present a new procedure that combines lead-time-dependent calibration and calibration in simulation mode. Our study addresses parameter uncertainty by (1) using a parsimonious rainfall-runoff model, which reduces the degrees of freedom during calibration, thus parameter equifinality; (2) using a large sample of catchments and flood events, which makes our results
<figure><img src="image_5.png" /><figcaption>Fig. 3. Location of the 687 French catchments selected.</figcaption></figure>

less dependent on systematic errors at particular locations; (3) using
a calibration algorithm that reduces overcalibration; (4) examining
parameter variations between calibration periods, which can be seen
as a manifestation of equifinality.

## 2. Data and methods

### 2.1. A large catchment set

Our study is based on a large set of 687 catchments located in metropolitan France (Fig. 3). These catchments were selected from an original set of 5503 hydrometric stations to select only stations where snow and human activities have limited impact on streamflow. The catchment selection was also based on criteria related to data availability and quality. Only stations with a minimum of 10 years of hourly streamflow data between 01-01-2007 and 31-12-2021 were extracted from the original set. The catchment surface areas range from 5 to 10,000 km².

The time series of hourly precipitation come from the Comepore gridded product at 1-km resolution (Tabary et al., 2012) and were aggregated at the catchment scale. The time series of potential evaporation were calculated with the (Oudin et al., 2005) formula using daily temperature time series and the disaggregation method used by Astagneau et al. (2021). The time series of daily temperature were extracted by Delaigue et al. (2020) from the SAFRAN climate reanalysis database of Météo-France (Vidal et al., 2010). Hourly time series of streamflow were retrieved from the database of the French hydrometric services (Leleu et al., 2014). The hydroclimatic time series cover the period from 01-01-2007 to 31-12-2021.

An automated procedure was used to select 40,411 flood events over the dataset. Flood events with peak flows higher than the 98th percentile were picked. The start of an event was taken as the time at which streamflow exceeds 20 % of the flood peak. The end of an event was taken as the time at which streamflow is inferior to 25 % of the flood peak. Up to 100 events were selected per catchment. The characteristics of these events are presented in Fig. 4. The flood event set encompasses a large variety of event characteristics that are representative of the dominant floods over France. The catchments associated with long response times predominantly experience long winter floods, whereas the flashy catchments experience floods with short rise times, low volumes, and high peak flows.

### 2.2. Hydrological modelling and forecasting

The GR5H-RI lumped hourly model (Astagneau et al., 2022, 2024) was used to forecast streamflows at the outlet of the 687 catchments. The GR5H-RI model derives from the GR5H model (Le Moine, 2008; Ficchì et al., 2019). GR5H-RI is based on two main modules: a production function that calculates effective rainfall (i.e. the fraction of rainfall that will theoretically reach the catchment outlet) and a routing function that transfers this effective rainfall to the catchment outlet (Fig. 5). Additionally, GR5H-RI allows an explicit account of the effect of intense rainfall on catchment response. This model has 8 free parameters to calibrate. The GR5H-RI model is combined with a procedure to update the two routing stores at the forecast time (see section 2.3.2; Astagneau et al., 2024). This procedure is deterministic and based on the flow partitioning simulated by the model without the updating procedure. At each time step, the routing store levels are modified to match the observed streamflow. A forecast is then issued from the time of forecast ($t_0$) to the target lead time ($t_0 + H$). All the forecasts of a given lead time are combined to create a time series of fixed-lead time forecasts (Fig. 2). Three lead times are considered in this study: 6 h, 12 h and 24 h. These lead times correspond to commonly used target lead times for operational flood forecasting in France, for catchments with different streamflow dynamics. Only observed rainfall time series (i.e. no rainfall forecasts were used) are fed to the hydrological model in order to reduce the interactions between parameter calibration and rainfall uncertainty.

### 2.3. Calibration

The eight free parameters of GR5H-RI ( Table 1) were estimated on two independent sub-periods of equal length between 01-01-2007 and 31-12-2021. Each sub-period contains approximately the same number of hourly streamflow measurements. The calibration algorithm used in this study is based on the EGD (exhaustive gridding discretisation) method of Mathevet (2005) and Perrin et al. (2008). This algorithm consists of two steps to find an optimal parameter set given a performance criterion. First, a starting point is found from a global screening of the parameter space. Then, a local optimisation is performed. This algorithm was used in previous studies with the GR5H-RI model and
<figure><img src="image_6.png" /><figcaption>Fig. 4. Distribution of event characteristics for the 40,411 selected floods (see definitions of characteristics in Section 2.4).</figcaption></figure>

<table><caption>**Table 1**
List of GR5H-RI parameters. The 80% range was obtained from the reference calibration.</caption><thead><tr><th>Parameter</th><th>Definition</th><th>Unit</th><th>80% range</th></tr></thead><tbody><tr><td>X<sub>1</sub></td><td>Production store capacity</td><td>mm</td><td>152–663</td></tr><tr><td>X<sub>2</sub></td><td>Intercatchment exchange coefficient</td><td>mm/h</td><td>-0.14–-0.01</td></tr><tr><td>X<sub>3</sub></td><td>Indirect branch routing store capacity</td><td>mm</td><td>45–353</td></tr><tr><td>X<sub>4</sub></td><td>Unit hydrograph time constant</td><td>h</td><td>1–35</td></tr><tr><td>X<sub>5</sub></td><td>Intercatchment exchange threshold</td><td>–</td><td>0–0.35</td></tr><tr><td>i<sub>1</sub></td><td>Rainfall intensity parameter of the production store</td><td>h/mm</td><td>0–0.01</td></tr><tr><td>i<sub>2</sub></td><td>Rainfall intensity parameter of the partitioning coefficient</td><td>h/mm</td><td>0–0.05</td></tr><tr><td>K<sub>L</sub></td><td>Direct branch linear store coefficient</td><td>–</td><td>0.01–1</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Structure of the modified GR5H model (GR5H-RI; modified from Astagneau et al., 2022). E: potential evapotranspiration; P: rainfall; Q: streamflow; X<sub>i</sub> (i in [1,5]), i<sub>1</sub>, i<sub>2</sub>, K<sub>L</sub>: model parameters (see Table 1); other letters are internal state variables; in red: modifications to account for the effect of intense rainfall on catchment response.</figcaption></figure>

similar models (e.g. Astagneau et al., 2021, 2022) because of its efficiency, low computation cost and ability to reduce overcalibration for GR models. The Kling-Gupta efficiency criterion (KGE; Gupta et al., 2009) was used as objective function. A warm-up period of 3 years before the start of each sub-period was set to initialise the states of the model.

Three calibration procedures (Table 2) were performed to assess whether lead-time-dependent calibration improves flood forecasting and parameter robustness:

* the reference calibration procedure that consists in calibrating the model parameters in simulation mode, i.e. without state updating (cal_Ref);

* calibration of all the parameters at the three target lead times (cal_LT). In this case, an optimal parameter set is found by the calibration algorithm by comparing the fixed-lead-time time series of forecast values with observations. For this configuration, a specific parameter set is calibrated for each catchment, sub-period, and lead time;

* a calibration procedure combining cal_Ref and cal_LT. First, the parameters are calibrated following the reference calibration (i.e. without state updating). Then, only five of the eight free parameters are calibrated for the three target lead times (i.e. with state updating). The parameters related to the water balance equation (X<sub>1</sub>, X<sub>2</sub> and X<sub>5</sub>) are only calibrated during the first step (for details about the parameters and the model equations, see Astagneau et al., 2021, 2022). The routing parameters (X<sub>3</sub>, X<sub>4</sub> and K<sub>L</sub>) and the rainfall intensity parameters (i<sub>1</sub> and i<sub>2</sub>) are calibrated during the first and second steps. The parameters found during the first step are used as starting points for the local optimisation of the second step (calibration algorithm described above). The assumption here is that the water balance parameters are less sensitive to the forecast errors between (t<sub>0</sub>) and (t<sub>0</sub> + H), especially at short lead times when state updating has large effect on forecasts. Therefore, it seems reasonable to calibrate them without the impact of state updating. We name this procedure cal_WsRf (Water balance parameters in Simulation mode and Routing parameters in Forecasting mode). Other combinations were tested (e.g. the rainfall intensity parameters calibrated in the first step) but were found to be less efficient (not shown here).
<table><caption>**Table 2**
Calibration options tested.</caption><thead><tr><td>Option</td><td>Abbreviation</td><td>Characteristics</td></tr></thead><tbody><tr><td>Reference calibration</td><td>cal_Ref</td><td>Calibration of all the parameters in simulation mode</td></tr><tr><td>Lead-time-dependent calibration</td><td>cal_LT</td><td>Calibration of all the parameters in forecasting mode (one parameter set for each lead time)</td></tr><tr><td>Combined calibration</td><td>cal_WsRf</td><td>Calibration of X<sub>1</sub>, X<sub>2</sub> and X<sub>3</sub> in simulation mode and X<sub>1</sub>, X<sub>4</sub>, i<sub>1</sub>, i<sub>2</sub> and K<sub>t</sub> in forecasting mode</td></tr></tbody></table>

Note that, after calibration, forecasts were produced by the GR5H-RI forecasting model (i.e. combined with state updating as assimilation procedure) with the parameter sets obtained from the three calibration options.

The model runs were performed in the R environment (R Core Team, 2023) with a modified version of the airGR package (Coron et al., 2017, 2020).

## 2.4. Evaluation

All results were evaluated using cross-evaluation forecasts. Cross-evaluation consists, for a given catchment, in using the parameter set calibrated on one sub-period to forecast streamflows of the other sub-period and vice versa. First, the forecasts obtained from the three calibration options were compared by evaluating the overall streamflow performance (performance in forecasting the entire streamflow hydrograph) with the KGE index. Then, forecast performance was evaluated using metrics calculated on individual flood events. For a given lead time and a given event, all the forecasts within the flood event temporal window were considered to calculate a specific criterion. Two flood performance criteria were calculated in this study. A bounded version of the Nash–Sutcliffe efficiency criterion (NSE; Nash and Sutcliffe, 1970; Mathevet et al., 2006) was first calculated. This criterion takes values between -1 and 1. Negative values of NSE indicate that the catchment mean flow is a better predictor than the model in forecasting a flood event. The optimal value of this criterion is 1. The use of the bounded version avoids a too large impact of strongly negative values when calculating average performance on a large set of catchments. Finally, we calculated the ratio between peak flow errors for two of the calibration options: cal_LT and cal_WsRf. We used an adjusted version of the peak flow error criterion to make it relevant for fixed-lead-time time series analyses. The reason for this adjustment is that when the lag time between the observed peak flow and the predicted peak flow equals the lead time, it means that the peak flow was predicted at the forecast time of the observed peak flow. This value should not be taken into account in the calculation of the peak flow error but would be in the original formulation of the criterion. To mitigate this issue, the peak flow error of an event j at lead time H was calculated as follows:

$$ \Delta Q_{max_{j,H}} = \frac{\max[Q_{P_{j,H}}]}{\max[Q_{obs_{j,H}}]} - 1 \quad (1) $$

where $\max[Q_{P_{j,H}}]$ is the maximum predicted flow between $t_{max} - H/2$ and $t_{max} + H/2$, where $t_{max}$ is the time of the observed peak flow. The peak flow error ratio is then expressed as follows:

$$ RQ_{max_{j,H}} = \frac{|\Delta Q_{max_{j,H}}^{cal\_WsRf}|}{|\Delta Q_{max_{j,H}}^{cal\_LT}|} \quad (2) $$

$RQ_{max}$ values range from 0 to +∞. When $RQ_{max} = 1$, both calibration options have the same performance in forecasting the peak flow of a given event and a given lead time. $RQ_{max} > 1$ indicates that the peak error of cal_WsRf is larger than the peak error of cal_LT and vice versa.

The results were assessed according to two variables describing catchment temporal dynamics: the catchment response time (Tr) and the flood rise time. Tr was defined as the lag time that maximises the correlation between rainfall and streamflow time series (Ficchi et al., 2016), therefore relying solely on streamflow and rainfall observations. The flood rise time was defined as the time between the beginning

of a flood event and the time of observed peak flow. The response time quantiles were used to define three groups of catchments with approximately (because of catchments with the same Tr value) the same number of catchments (254, 211 and 222 respectively; Fig. 3).

We performed the Wilcoxon rank test (Wilcoxon, 1945) to detect statistically significant changes in performance between the three calibration options (at significance level 0.05; non-paired; two-sided).

To evaluate the impact of the different calibration options on the parameters of the GR5H-RI model, we studied the parameter variations between calibration periods for each calibration option. To that end, the parameters obtained on all catchments and sub-periods were first normalised by their minimum and maximum values over the dataset. The normalised value of a given parameter for a given catchment and sub-period was calculated as follows:

$$ X^{norm} = \frac{|X - min(X)|}{|min(X) - max(X)|} \quad (3) $$

where $min(X)$ and $max(X)$ are the minimum and maximum values, respectively, of the parameter X for one period, one calibration option (and one lead time for cal_LT and cal_WsRf) over all catchments. Then, for a given parameter and a given catchment, the Euclidean distance between the normalised value of sub-period P1 and the normalised value of sub-period P2 was calculated as follows:

$$ d(P1, P2) = \sqrt{(X_{P1}^{norm} - X_{P2}^{norm})^2} \quad (4) $$

## 3. Results

### 3.1. Overall performance

We first examine the forecasting performance of the three calibration options over the whole streamflow time series using the KGE index (Fig. 6). In Fig. 6, each column corresponds to a lead time and each row corresponds to a group of catchment response times. The reference calibration corresponds to the calibration without state updating (cal_Ref in green).

As expected, the forecast performance decreases with increasing lead time for all calibration options and all catchment groups, which is due to the decreasing effect of DA. The differences in forecast performance between the calibration options vary greatly between catchment response times and lead times. For catchments with low response times (flashy catchments), the forecast performance of cal_LT is similar to that of cal_Ref at all lead times ($p$-values > 0.05; see Table 3). The forecast performance of cal_WsRf is significantly higher than the forecast performance of cal_LT at 6-h ($p$-value < 0.05) and similar to cal_LT and cal_Ref at 12-h and 24-h lead times ($p$-values > 0.05). Moving to catchments with intermediate response times, the performance ranking changes. cal_LT performs better than cal_Ref at 6-h and 12-h lead times ($p$-values < 0.05) and similarly at 24-h lead time ($p$-value > 0.05), equally to cal_WsRf at all lead times ($p$-values > 0.05). cal_WsRf performs better than cal_Ref at 6-h lead time ($p$-value < 0.05) and equally at 12-h and 24-h lead times ($p$-values > 0.05). For the catchments with the longest response times, the performance ranking between calibration options does not change between lead times. The difference in performance between cal_Ref and cal_LT is significant at all lead times ($p$-values < 0.05), with cal_LT performing better than cal_Ref. The performance differences between cal_LT and cal_WsRf are non significant ($p$-values > 0.05). cal_WsRf performs better than cal_Ref
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Overall forecast performance (cross-evaluation values) of GR5H-RI over 687 catchments for the three calibration options. The results are presented at three lead times (LT) and given three groups of catchments derived from catchment response times (Tr). Each group contains approximately the same number of catchments. The distributions are presented between the 5th and 95th percentiles. Note that the y-axis changes between catchment types for visualisation purposes. See explanations in the text and Table 3 for significance of the results.</figcaption></figure>

at 6-h and 12-h lead times (p-values < 0.05) and similarly at 24-h lead time (p-value > 0.05). When all the catchments are grouped together, cal_LT performs better than cal_ref at all lead times (p-values < 0.05) and equally to cal_WsRf at all lead times. cal_WsRf performs significantly better than cal_ref at 6-h and 12-h lead times (p-values < 0.05) and similarly at 24-h lead time (p-value > 0.05).

Overall, the results show that there are large differences between the calibration options in terms of performance over the whole streamflow time series, particularly at short lead times where cal_LT largely improves performance for slow catchments but not for flashy catchments. The combined calibration option yields the best performance across all catchment types and lead times. Note that this analysis does

not depend on the choice of performance metric, as similar results were obtained with the root mean square error (RMSE; see fig. S1).

## 3.2. Flood performance

The overall performance of the calibration options might hide important differences when floods occur. Therefore, we examine the performance in forecasting 40,411 flood events for all calibration options, at 6-h lead time, where the lead-time-dependent calibration (cal_LT) does no improve overall streamflow forecasting performance. In Fig. 7, each column corresponds to a group of catchment response times, from fast to slow responses.
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Distribution of flood performance (cross-evaluation values) of GR5H-RI over 40,411 floods (one criterion value per flood event) for the three calibration options. The results are presented at 6-h lead time and given three groups of catchments derived from catchment response times (Tr). Each group contains approximately the same number of catchments. The distributions are presented between the 5th and 95th percentiles. Note that the y-axis changes between catchment types for visualisation purposes. See explanations in the text and Table 4 for significance of the results.</figcaption></figure>

While the distributions were similar in terms of overall performance (Fig. 6), cal_LT significantly degrades performance (*p*-value < 0.05) compared to cal Ref for flood events occurring in flashy catchments (i.e. catchments with small response times). Additionally, for these catchments, cal_WsRf performs better than cal Ref (*p*-value < 0.05). This trend changes for catchments with intermediate response times, where both cal_LT and cal_WsRf perform significantly better than cal Ref (*p*-values < 0.05). This trend is even more pronounced for catchments with long response times. For these catchments, cal_LT performs better than cal_WsRf and cal Ref (*p*-values < 0.05). On average, cal_WsRf performs better than cal Ref and cal_LT for all catchments (*p*-values < 0.05) and cal Ref and cal_LT perform equally (*p*-value > 0.05). Note that this analysis does not depend on the choice of performance metric, as similar results were obtained with the root mean square error (RMSE) calculated for flood rises (see fig. S2).

We now investigate when the performance in forecasting peak flows differs between cal_WsRf and cal_LT (Fig. 8). We examine the differences in the peak error ratio between the two calibration options for all lead times (3 panels) and as a function of flood rise times (4 colours). When the peak error ratio is below 1, the combined calibration option performs better than cal_LT. The largest differences between the groups of flood events are observed at 6-h lead time. At 24-h, there are very few differences between the groups. At 6-h lead time, 61% of the peak flows of the events with the smallest rise times are better predicted using the combined calibration approach. For the largest rise times, the median of the peak error ratio is 50%, meaning that the same number of peak flows are better predicted by one option or the other. For 6-h and 12-h lead times, this median is close to 50% for all flood groups. Overall, the main differences between cal_WsRf and cal_LT in terms of peak flow prediction are found at short lead times, where flash flood peaks are better predicted when cal_WsRf is applied.

### 3.3. Parameter variations between periods

Improved forecast performance in extrapolation does not necessarily imply improved parameter robustness. To investigate the variation in parameter robustness between calibration options, we examine the

variations of parameters between calibration periods (Fig. 9). High values of the median distance of parameters between calibration periods indicate low parameter robustness. The reference calibration option has only one value for each parameter as the parameter sets do not vary with lead times.

Looking at the water balance parameters ($X_1$, $X_2$, and $X_5$), we can see that the variation values of cal_WsRf are equal to the variation value of cal Ref, as these parameters were not calibrated for each lead time but in simulation mode. For $X_1$ and $X_2$, the parameter variation values of cal_LT are greater than that of cal Ref (and therefore cal_WsRf), especially at 6-h lead time. For $X_5$, the variations are close between the calibration options. cal Ref induces the least variations of $X_3$ and $i_1$. For these parameters, the variations for cal_LT and cal_WsRf decrease with increasing lead time. cal_WsRf induces smaller variations of $X_3$ and $i_1$ than cal_LT at all lead times. For $X_4$, cal_LT induces more variations than cal Ref and cal_WsRf at 6-h and 12-h lead times. cal_WsRf induces the largest variations of $i_2$ at all lead times. These variations decrease as the lead time increases. Very different trends are found for the variations of the $K_L$ parameter. The largest variations in $K_L$ are found for cal Ref. The $K_L$ variations increase with increasing lead time for cal_LT and cal_WsRf. cal_LT induces less $K_L$ variations than cal_WsRf at all lead times.

Overall, the calibration with state updating degrades parameter robustness for most parameters (except for $i_2$ and $K_L$), especially for short lead times and for two of the water balance parameters. The combined calibration approach induces higher parameter robustness than calibration with state updating (except for $i_2$ and $K_L$), as it is often close to the reference calibration. There is a clear inverse trend for one of the routing parameters ($K_L$).

## 4. Discussion

### 4.1. Usefulness of updating techniques in light of catchment temporal dynamics

The results showed that the lead-time-dependent calibration of GR5H-RI significantly improves flood forecasting for catchments with
<figure><img src="image_10.png" /><figcaption>Fig. 8. Distribution of the peak error ratio between the lead-time-dependent calibration (cal_LT) and the combined calibration (cal_WsRf) for three lead times (LT). 40411 floods divided into four groups derived from flood rise times are considered here.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Median variations in GR5H-RI free parameters between calibration periods. The variation values are presented for the three calibration options and at three lead times.</figcaption></figure>

slow temporal dynamics, especially at short lead times (this is shown by the higher performance of cal_LT compared to cal Ref in Fig. 10 for catchments with high 6-h autocorrelation of streamflow). For these catchments, the autocorrelation of streamflows can be very high at short lead times (between 0.982 and 1, as shown in Fig. 10 that presents flood performance according to the 6-h autocorrelation of streamflow).

In other words, the observed streamflow value at $t_0$ can be a good predictor of the streamflow a few hours ahead, especially when the effect of a storm event on streamflow variations becomes small (e.g. recession of the flood hydrograph). Since updating the model

initial conditions at the time of the forecast implies a better fit to the observations, the prediction is improved. Calibration of the hydrological model parameters while taking into account the effect of the state update on model errors increases the benefit of DA for these catchments. These results are consistent with those reported by Evin et al. (2014), who found that accounting for autocorrelation errors in residual error models improved the characterisation of prediction uncertainty. The benefit of calibrating hydrological parameters for each lead time in the case of slow catchment dynamics could theoretically
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Distribution of flood performance (cross-evaluation values) of GR5H-RI over 40,411 floods (one criterion value per flood event) for the three calibration options. The results are presented at 6-h lead time and given three groups of catchments derived from 6-h autocorrelation of streamflow (A_6 h). Each group contains approximately the same number of catchments. The distributions are presented between the 5th and 95th percentiles. Note that the y-axis changes between catchment types for visualisation purposes.</figcaption></figure>

be achieved by an efficient bias correction procedure. Indeed, if the calibration procedure improves performance in the presence of systematic biases in streamflow prediction, then the performance should also be improved by a bias correction procedure. Examples of post-processing methods can be found in the relevant literature (e.g. McInerney et al., 2022). Nevertheless, this issue is beyond the scope of our work and the selection of an appropriate bias correction method can be challenging.

For catchments with opposite temporal dynamics (i.e. flashy catchments), calibration with state updating clearly leads to worse flood forecasts at short lead times (this is shown by the lower performance of cal_LT compared to cal_Ref in Fig. 10 for catchments with low 6-h autocorrelation of streamflow). In these catchments, especially when flood rise times are short, the autocorrelation of the streamflows is low and therefore the assimilation of errors has less impact on forecast quality. For catchments characterised by floods triggered by intense summer rainfall events, the variation in streamflow values can be very high, ranging from very low values typical of the summer season to flood peaks. In this case, if the forecast time is prior to the start of an event, the state updating procedure will not be able to correct model errors during the flood rise. This effect is likely to be greater if the catchment response time is shorter than the lead time. In this case, the effect of rainfall on streamflow variations will not be detected by the state updating procedure during the flood rise times, when model errors are usually largest. Furthermore, data assimilation in this case can sometimes be disinformative. For example, if the hydrological model overestimates low flows before the start of an event but underestimates the flood peak, the data assimilation process may correct the next forecasts towards lower values and lead to larger underestimation of the flood peak. As a consequence, making the parameters dependent on the state updating procedure, when the base hydrological model cannot be improved by DA, showed clear disadvantages. The lead-time-dependent calibration induces parameter sets that are likely to be too different from the initial hydrological parameter sets, which becomes problematic when predictive capabilities are solely dependent on the base hydrological model. This effect was also found by Li et al. (2015b).

## 4.2. Parameter variations: routing parameters vs water balance parameters

This decrease in flash flood forecasting performance when the hydrological model parameters are made lead-time dependent could be the result of the low sensitivity of the water balance parameters to forecast errors at short lead times. An indication of this low sensitivity is the greater variation in water balance parameters between periods at short lead times. As mentioned above, the state update for flashy catchments benefits less from the autocorrelation of streamflows, and thus the forecast quality depends more on the performance of the stand-alone hydrological model. For short lead times, and especially for short flood rises, the model error at the time of forecast does not provide any information on the ability of the model to reproduce the runoff coefficient of the flood event. However, calibrating the production store capacity with this error signal in the objective function implicitly means that the catchment runoff coefficient should change between lead times. This leads to performance degradation, which is unlikely to be caused by the lead time dependence of the routing parameters, as the combined calibration approach outperforms the reference calibration for fast catchments and at short lead times. For slow catchment dynamics, the water balance parameters are also likely to have low sensitivity to forecast errors at short lead times. However, for these catchments the DA has a greater impact on the quality of the forecast, especially at short lead times. In other words, the values of the water balance parameters are likely to have a limited effect on forecast quality at short lead times (compared to the effect of the routing parameters). The close performance at short lead times between cal_LT and cal_WsRf is an indication of this effect.

While lead-time-dependent calibration enabled an improved forecast performance for a significant number of catchments and flood events, it also induces lower parameter robustness, especially for short lead times and for two of the water balance parameters. This effect has been shown to some extent by Evin et al. (2014), who found that the joint estimation of model parameters and the parameters of the autocorrelation error model leads to low robustness. However, we have found that the combined calibration option introduced in our study leads to a smaller decrease in parameter robustness compared
with the reference calibration, and to improved performance for most catchments and flood types. In particular, the combined calibration has highlighted the benefit of making the routing parameters of the GR5H-RI model lead-time dependent. The robustness of one of the routing parameters ($K_L$) is even improved for cal_LT and cal_WsRf. The benefit of making the routing parameters lead-time dependent could be the result of a better assimilation of streamflows at the time of the forecast.

## 5. Conclusion

The objective of this study was to determine whether lead-time-dependent calibration of a flood forecasting model improves forecast performance and parameter robustness. The main reason why we wished to address this issue is that DA changes the structure of model errors. Moreover, these errors change with lead times. To answer this question, the GR5H-RI flood forecasting model was used to forecast streamflow for 687 catchments and 40,411 flood events. The model parameters were calibrated in three different ways: calibration of the model parameters in simulation mode (i.e. without data assimilation), lead-time-dependent calibration (i.e. with data assimilation), and the calibration procedure combining both approaches (i.e. water balance parameters calibrated according to the first option and routing parameters calibrated according to the second option). The forecast performance of the three options was evaluated by considering the temporal dynamics of the catchments. The robustness of the parameters was assessed by calculating the variations in parameter values between two calibration sub-periods.

The results of our experiments showed that the benefit of making the parameters lead-time dependent varies with the temporal dynamics of the catchments. For slow-reacting catchments, lead-time-dependent calibration improves flood prediction, particularly at short lead times. This is probably due to the resulting parameter sets, which improve the usefulness of the state updating procedure when the streamflow autocorrelation is high. However, this calibration option clearly degrades forecast performance at short lead times for fast-reacting catchments, where the state updating procedure has less impact on forecast quality. In this case, moving away from the baseline hydrological model leads to a drop in performance. This drop in performance is very much related to the differing effects of routing parameters and water balance parameters when a state updating technique is used. When only the routing parameters are made lead-time dependent, the forecast performance increases compared to the reference calibration. We also found that one of the main drawbacks of lead-time-dependent calibration is that it reduces the robustness of the parameters, especially in the case of the water balance parameters. The combined calibration procedure introduced in our study is a good compromise between parameter robustness and performance for all catchment and flood types.

These results may have implications for the usefulness of data assimilation and calibration techniques for flood forecasting in France, as models similar to GR5H-RI are used by operational forecasters (e.g. Viatgé et al., 2019). It appears that the signal from the hydrological model has a greater impact on the quality of forecasts for the flashy catchments, which in France are mainly located in the Mediterranean area. These catchments tend to be affected by extreme precipitation events, which are expected to increase with climate change. Our results suggest that operational flood forecasting for these catchments could benefit greatly from the application of a combined calibration procedure.

In this study, a single hydrological model combined with a deterministic state updating procedure was used on a large set of catchments to answer our research questions. Our calibration methods could be tested with distributed physically-based models and machine learning models, although the large number of parameters required to use these models could lead to high parameter uncertainty. Additionally, there is a need to further investigate the pros and cons of estimating lead-time-dependent parameters in comparison with more sophisticated

data assimilation approaches and formal likelihood-based methods for accounting for autocorrelation. Investigating the sensitivity of model parameters to data uncertainty would also provide more general and robust results, for example by applying bootstrapping methods or the generalised likelihood uncertainty estimator (GLUE; Beven and Binley, 1992), but this was beyond the scope of our analysis.

## Open research section

Streamflow data were obtained from https://hydro.eaufrance.fr/ using the hydroportail package (https://gitlab.irstea.fr/HYCAR-Hydro/hydroportail). Precipitation data can be downloaded from Caillaud (2019). Temperature data were provided by Méteo-France and are subject to third party restrictions. The GR5H-RI model was used with a modified version of the airGR R package (version 1.6.9.27), which will be available in future versions of the package.

## CRediT authorship contribution statement

Paul C. Astagneau: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. François Bourgin: Writing – review & editing, Supervision, Methodology, Conceptualization. Vazken Andréasian: Writing – review & editing, Supervision, Methodology, Conceptualization. Charles Perrin: Writing – review & editing, Supervision, Methodology, Funding acquisition, Conceptualization.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Paul C. Astagneau reports financial support was provided by Ministry of Ecological and Solidary Transition. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

The authors do not have permission to share data.

## Acknowledgements

This work was supported by the French Ministry of Environment (DGPR/SNRH/SCHAPI). The authors thank Météo-France and Banque HYDRO for providing the climatic and hydrological data, respectively. The model runs were performed on the MESO@LR-Platform, hosted by Montpellier University. We thank the editor and the three anonymous reviewers for their comments on the manuscript.

## Appendix A. Significance tests

Table 3 reports the *p*-values resulting from the significance test performed for the KGE distributions reported in Fig. 6.

Table 4 reports the *p*-values resulting from the significance test performed for the flood event NSE distributions reported in Fig. 7.

## Appendix B. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jhydrol.2024.132119.
<table><caption>**Table 3**

p-values for Fig. 6. LT stands for lead time and Tr for catchment response time.</caption><thead><tr><th>cal Ref /cal LT</th><th>cal Ref /cal_WsRf</th><th>cal LT /cal_WsRf</th><th>LT</th><th>Tr</th></tr></thead><tbody><tr><td>0.397</td><td>0.067</td><td>0.008</td><td>6 h</td><td>[1,9] h</td></tr><tr><td>10<sup>-4</sup></td><td>0.001</td><td>0.833</td><td>6 h</td><td>[10,18] h</td></tr><tr><td>10<sup>-16</sup></td><td>10<sup>-12</sup></td><td>0.168</td><td>6 h</td><td>[19,128] h</td></tr><tr><td>10<sup>-7</sup></td><td>10<sup>-9</sup></td><td>0.549</td><td>6 h</td><td>All catchments</td></tr><tr><td>0.363</td><td>0.309</td><td>0.055</td><td>12 h</td><td>[1,9] h</td></tr><tr><td>0.02</td><td>0.144</td><td>0.389</td><td>12 h</td><td>[10,18] h</td></tr><tr><td>10<sup>-9</sup></td><td>10<sup>-5</sup></td><td>0.057</td><td>12 h</td><td>[19,128] h</td></tr><tr><td>10<sup>-4</sup></td><td>0.001</td><td>0.728</td><td>12 h</td><td>All catchments</td></tr><tr><td>0.514</td><td>0.608</td><td>0.885</td><td>24 h</td><td>[1,9] h</td></tr><tr><td>0.13</td><td>0.54</td><td>0.356</td><td>24 h</td><td>[10,18] h</td></tr><tr><td>10<sup>-4</sup></td><td>0.069</td><td>0.063</td><td>24 h</td><td>[19,128] h</td></tr><tr><td>0.003</td><td>0.129</td><td>0.131</td><td>24 h</td><td>All catchments</td></tr></tbody></table>

<table><caption>**Table 4**

p-values for Fig. 7. LT stands for lead time and Tr for catchment response time.</caption><thead><tr><th>cal Ref /cal LT</th><th>cal Ref /cal_WsRf</th><th>cal LT /cal_WsRf</th><th>LT</th><th>Tr</th></tr></thead><tbody><tr><td>10<sup>-47</sup></td><td>10<sup>-4</sup></td><td>10<sup>-70</sup></td><td>6 h</td><td>[1,9] h</td></tr><tr><td>10<sup>-7</sup></td><td>10<sup>-9</sup></td><td>0.490</td><td>6 h</td><td>[10,18] h</td></tr><tr><td>10<sup>-29</sup></td><td>10<sup>-18</sup></td><td>0.006</td><td>6 h</td><td>[19,128] h</td></tr><tr><td>0.061</td><td>10<sup>-18</sup></td><td>10<sup>-24</sup></td><td>6 h</td><td>All catchments</td></tr></tbody></table>

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