# Characterization of flash flood events by their Event-specific GIUH (EGIUH) in taking into account both the spatial variability of rainfall and channel topology

Hervé Andrieu, Roger Moussa, Pierre Nicolle, Olivier Payrastre

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Hervé Andrieu, Roger Moussa, Pierre Nicolle, Olivier Payrastre. Characterization of flash flood events by their Event-specific GIUH (EGIUH) in taking into account both the spatial variability of rainfall and channel topology. Journal of Hydrology, 2025, 656, pp.132885. 10.1016/j.jhydrol.2025.132885. hal-05008988

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Research papers

# Characterization of flash flood events by their Event-specific GIUH (EGIUH) in taking into account both the spatial variability of rainfall and channel topology

Hervé Andrieu<sup>a, ID</sup>, Roger Moussa<sup>b, * ID</sup>, Pierre Nicolle<sup>a ID</sup>, Olivier Payrastre<sup>a ID</sup>

<sup>a</sup> GERS, Univ. Gustave Eiffel, IFSTTAR, 44344 Bouguenais, France

<sup>b</sup> LISAH, Univ Montpellier, AgroParisTech, INRAE, IRD, Institut Agro, Montpellier, France

## ARTICLE INFO

Handling Editor: Dr Marco Borga

Associate Editor: Michael Bruen, B.E., Ph.D.

Keywords:

GIUH

Flash-flood

Rainfall-runoff modeling

Inverse method

Rainfall spatial variability

## ABSTRACT

The Geomorphological Instantaneous Unit Hydrograph (GIUH) is widely used in rainfall-runoff modeling. We are proposing herein to apply the GIUH concept to characterize flash flood events. The width function-based GIUH has been adapted by means of inverse modeling to identify the Event-specific GIUH (hereafter denoted EGIUH), which takes into account the spatial variability of rainfall and channel topology. The EGIUH can be considered as a hydrological signature characterizing the catchment response to a rainfall event. The purpose of this manuscript is twofold. The first goal is to proceed with a rigorous evaluation of the EGIUH identification method, while the second is to study the variability of the EGIUHs that characterize flash floods, which tend to strike several catchments. Applications are run on 15 catchments of the French Mediterranean Region over the period 1997–2021, i.e. encompassing a total of 233 flash flood events over which the peak flow has a return period of at least two years. This study provides a unique set of EGIUHs derived from observed flash floods. An analysis of EGIUH variability and parameters is also performed. A study of the sensitivity to EGIUH parameters as well as to the initial hyetogram of effective rainfall, as defined by the user, serves to assess the benefit and limitations of this method. A hierarchical cluster analysis is used to combine the EGIUHs of each catchment into a limited number of GIUHs, which remain significantly different from one another and representative of the catchment flash floods. A comparison of GIUHs representative of each catchment is complemented by comparing the main hydrological signatures of catchment response, as derived from these GIUHs. The GIUHs representative of the flash-flood clusters pave the way to a typology of the flash floods affecting a catchment.

## 1. Introduction

The spatiotemporal variability of a rainfall event at fine spatial and temporal scales, in combination with channel network topology, heavily impacts the genesis of flash-flood processes: runoff, antecedent soil moisture conditions, flow velocity and discharge both on hillslopes and in the channel network (Dasari and Vema, 2024; Emmanuel et al., 2017; Nguyen et al., 2024; Saharia et al., 2021; ten Veldhuis et al., 2018; Zhou et al., 2021; Zhu et al., 2018). An improved simulation and forecasting of floods at the catchment scale (i.e. several tens or hundreds of km²) requires a better understanding of the hydrological processes of runoff genesis and transfer via the channel network (Douinot et al., 2018; Le Mesnil et al., 2021). However, an analysis of the hydrograph measured at the catchment outlet may help to identify the spatial origin of the

water. At the catchment scale, runoff and flow processes can be parsimoniously modeled by the convolution product of the production function (derivation of runoff from rainfall) and a transfer function (routing of runoff to the outlet), which are not mutually independent and must be determined simultaneously. The inverse problem consists of identifying both the effective rainfall and transfer function, in knowing the hydrograph at the outlet (Andrieu et al., 2021). This exercise is complicated due to numerical instabilities and equifinality when using inverse modeling or deconvolution approaches (Ghorbanidehno et al., 2020; Fenton, 2024).

The theory behind the Geomorphological Instantaneous Unit Hydrograph (hereafter GIUH) has contributed to the success of lumped rainfall-runoff models, thanks to its parsimony and robustness. The widespread use of GIUH within the hydrology community, coupled with

<footnote>* Corresponding author.
E-mail address: roger.moussa@inrae.fr (R. Moussa).</footnote>
its development over the past few decades, attests to its relevance in
representing catchment responses when the runoff of effective rainfall
constitutes the dominant flood process (Innocente dos Santos et al.,
2023). From a conceptual point of view, GIUH theory has reinforced the
concept of Unit Hydrograph, by virtue of linking it to the geomorpho-
logical features of catchments. The founding papers on this subject
(Gupta et al., 1980; Rodriguez-Iturbe and Valdés, 1979) were followed
by a very rich scientific literature, whose initial synthesis is provided in
Chapter 7 of the textbook by Rodriguez-Iturbe and Rinaldo (1997).
Among the various expressions proposed for GIUH, it does seem how-
ever that the versions relying on the Nash cascade model of linear res-
ervoirs (Nash, 1957) and the width function-based GIUH (Rigon et al.,
2016) are the most widely used. In the first case, the GIUH is defined by
the two parameters of the Nash model. Rosso (1984) showed that the
Nash model offers a good and easy to use approximation of the model
developed by Rodriguez-Iturbe and Valdés (1979), which expresses the
GIUH as a function of Horton ratios, a scale parameter and the mean
stream flow velocity. Cudennec et al. (2004) proposed a statistical
physics reasoning to demonstrate that the probability density function of
path lengths can be expressed by a Gamma law. The Nash cascade GIUH
has enjoyed great success for ungauged catchments (Singh et al., 2014)
and is regularly the subject of papers for such applications. In the second
case, the GIUH is defined by the width function introduced by Kirkby
(1976) along with two parameters, namely: flow velocity and coefficient
of dispersion (Rinaldo et al., 1991; Rinaldo and Rodriguez-Iturbe, 1996).
In addition, it is worth noting the formulation of the GIUH derived from
the Diffusive wave Hayami linear model (Moussa, 1997). Saco and
Kumar (2002) provided an analysis of the coefficient of dispersion
resulting from the superposition of three mechanisms: geomorphologic,
kinematic, and hydrodynamic. In its various forms, the GIUH has been
the topic of many studies over the last few decades, including the recent
period (Hallema et al., 2016; Bunster et al., 2019; Chen et al., 2019;
Chong and Gironás, 2022; Achour et al., 2023). Width-function based
GIUH models have also been used in urban hydrology:

GIUH theory was developed at a time when information on catch-
ments and flow events was often limited and based on assumptions that
did not allow for parameter variability within the catchment. It thus
offers an efficient and elegant framework for a parsimonious approach
to rainfall-runoff modeling enriched by recent contributions (e.g. Cha-
van and Srinivas, 2015; Hosseini et al., 2016; Khaleghi et al., 2018;
Oppel and Schumann, 2020).

Since that time, the rapid development of distributed and process-
based hydrological models, concomitant with easier access to data-
bases on topography, land use, meteorological and hydrological data,
has led to beneficial changes. Such progress has enabled a better un-
derstanding of the basin response to rainfall forcing and moreover has
shown that distributed hydrological models are necessary for many
applications (Fatichi et al., 2016; Guse et al., 2021). GIUH-based models
have also evolved for adaptation to conditions broader than their initial
assumptions. Maidment et al. (1996) and Olivera and Maidment (1999)
derived the unit hydrograph from a digital elevation model, in consid-
ering a spatial velocity field. These two papers initiated the Spatially
Distributed Unit Hydrograph models, for which a review has been pro-
posed by Bunster et al. (2019). These authors introduced the upstream
contribution to each cell of the catchment in order to improve the local
velocity estimation. Let's note that Saco and Kumar (2002) revealed that
the dispersion coefficient of the width function-based GIUH includes a
component (the kinematic dispersion) representing the variations in
velocities between the paths of a catchment. Hallema and Moussa
(2014) subdivided the catchment into representative elementary hill-
slopes, each of which was associated with defined geometric and flow
velocity parameters; the computed GIUH was then given by the sum of
all elementary hillslope responses. Rigon et al. (2016) proposed modi-
fying the GIUH formulation in order to include evapotranspiration.
Andrieu et al. (2021) introduced the rainfall-weighted width function
into the GIUH formulation so as to consider the spatial variability of

rainfall. Yi et al. (2022) implemented a semi-analytical form of the width
function-based GIUH to account for the rainfall intensity and soil
moisture content in sub-catchments of the semi-distributed model.
Huang and Lee (2023) developed a method to dynamically estimate
partial contributing areas and then introduced them into the GIUH.
Perez et al. (2023) established the Directional Unit Hydrograph model,
which relaxes the assumption of spatially uniform rainfall by incorpo-
rating storm direction and speed into the width function-based GIUH.
(Seo and Schmidt, 2014) used a Gibb's model to generate synthetic
urban networks and realistic width function based GIUHs. (Perez et al.,
2024) formulated a parsimonious model inspired from the (WFIUH)
approach to simulate a sanitary sewer system including sanitary flow,
groundwater infiltration and runoff. These few examples illustrate the
ability of the GIUH to adapt to the evolving context of rainfall-runoff
modeling and moreover to reach a compromise between distributed
and more parsimonious global models.

The GIUH is a Unit Hydrograph, and its derivation from typical
observational data of flood events is far from simple. The process entails
identifying the impulse response (unit hydrograph) of a system whose
output (hydrogram) is known, yet whose input (hyetogram of effective
rainfall) is not measured and most often uncertain. Consequently, the
identification of the unit hydrograph's temporal components is often
unstable, as noted by Dell'Aira et al. (2022), who reported studies
conducted on the subject, as well as by Boudhraa et al. (2018), who
recalled the very early studies. Several authors have proposed identi-
fying the components of the discretized unit hydrograph. Duband et al.
(1993) developed a method to perform the simultaneous identification
of the hyetogram of effective rainfall and the unit hydrograph for a set of
flood events displaying similar characteristics. This method allows
determining the unit hydrograph representative of a particular set of
flood events. Furusho et al. (2014) used this same method to show that a
suburban basin displayed two distinct types of hydrological behavior, as
characterized by distinct unit hydrographs. Zhao and Tung (1994)
proposed two linear programming formulations to determine the
optimal unit hydrograph. In a recent paper, Dell'Aira et al. (2022)
reviewed the optimization approaches developed for unit hydrograph
derivation; they then developed a method consisting of a linear pro-
gramming algorithm complemented by monotonic constraints applied
to all unit hydrograph components featuring: a monotonically ascending
limb, a single peak, and a monotonically receding limb. In a very recent
paper, (Fenton, 2024) reviews the deconvolution methods developed to
identify the unit hydrograph. In particular, this authors details decon-
volution methods using: i) Discrete Fourier Transform, introduced by
(O'Donnell, 1960), ii) approximation by spectral methods able to
address numerical instabilities. These deconvolution methods usually
assume that the hyetogram of effective rainfall is well known, and don't
rely on the geomorphologic characteristics of hydrographic network.
These numerical methods are often intended to yield a mean unit
hydrograph representing the sample of considered flow events. Andrieu
et al. (2021) defined the Event-specific GIUH (or EGIUH) and proposed a
method to derive it from observational data of each flood event, namely
the time series of rainfall fields and hydrogram at the catchment outlet.
This EGIUH method seems promising by virtue of taking into account
both the spatial variability of rainfall events and the channel topology.
However, this method has not yet been applied to a large number of
basins and events. A wider application is required in order to both
demonstrate its robustness and assess the value of EGIUHs as "hydro-
logical signatures" to compare and classify catchment hydrological
responses.

The purpose of this manuscript therefore is to assess the relevance of
characterizing each flash flood event by its specific EGIUH, which can be
viewed as a signature of the particular flood, thus offering a new
perspective for the GIUH, in addition to its widespread use in rainfall-
runoff modeling. This assessment is based on a case study compiling
233 flood events observed on 15 catchments regularly affected by flash
floods within the French Mediterranean region during a period
exceeding 20 years. It is also worth noting that the derivation and analysis of EGIUHs from observed data over a large number of flood events is, to the best of our knowledge, an original application that complements studies often devoted to ungauged catchments. Such an assessment relies on application of the EGIUH identification method proposed by Andrieu et al. (2021), in which a robust evaluation is then performed. The procedure includes a sensitivity analysis to the EGIUH parameters and an analysis of the influence of the choice of initial hyetogram of effective rainfall on the results obtained. This sensitivity analysis provides useful information on the accuracy of the parameters for the identified EGIUHs. Lastly, the implementation of a classification method allows drawing up a typology of the EGIUHs that characterizes the flash floods of each catchment, in conjunction with drawing a comparison among these EGIUHs.

The manuscript has been organized as follows. Section 2 presents the method employed to identify GIUHs from observed data by taking into account the spatial variability of rainfall along flow paths. The case studies are described in Section 3; they comprise a set of catchments located in Southern France whose weather radar survey data are available over at least 20 years. Section 4 is devoted to the results obtained and has been divided into five parts: presentation of application conditions, presentation of results, sensitivity analysis and influence of the initialization choice on identified EGIUHs; use of a classification method to derive those GIUHs representative of the catchment floods; and lastly comparison of the GIUHs representative of each catchment. Section 5 discusses the obtained results, the advantages and limitations of the proposed method, and presents future research directions. Section 6 concludes the manuscript.

## 2. Methodology

### 2.1. The Event-specific Geomorphological Instantaneous unit hydrograph (EGIUH)

The EGIUH (Andrieu et al., 2021) has been adapted from the width function-based GIUH (D'Odorico and Rigon, 2003; Rigon et al., 2016). This adaptation process consists of taking into account the spatial rainfall variability according to the distance to the catchment outlet at the event scale. Emmanuel et al. (2015) showed that the influence of this adaptation on the hydrograph is correctly assessed by the deviation between the width function and the rainfall-weighted width function. Saharia et al. (2017b) confirmed this finding in a very different context. Emmanuel et al. (2017) proposed a method based on this result to assess the influence of the spatial variability of rainfall on modeling results. Their findings indicate that rainfall on the catchment can be represented as the product of two independent functions: the hyetogram describing the temporal evolution of rainfall over the catchment, and the rainfall-weighted width function representing its spatial evolution along the flow paths. A detailed presentation of EGIUH derivation has been provided by Andrieu et al. (2021). It is commonly accepted that the response to rainfall forcing of catchments subject to flash floods is dominated by runoff. In the following, the term runoff includes all processes contributing to the fast response of a catchment to rainfall forcing. In particular, runoff excludes baseflow. The resulting flow at the catchment outlet can then be expressed as:

$$Q_r(t) = A \int_0^t \overline{R_e(t-\tau)} G_E(U_E, D_E, \tau) d\tau \quad (1)$$

with $Q_r(t)$ denoting the runoff outflow at time $t$, $\overline{R_e(t)}$ the effective rainfall on the catchment at time $t$, and A the catchment drainage area. $G_E$, the EGIUH specific to flood event E, is expressed as follows:

$$G_E(U_E, D_E, t) = \int_0^{t_{max}} \frac{l w_E(l)}{\sqrt{4\pi D_E t^3}} \exp \left[ -\frac{(l - U_E t)^2}{4 D_E t} \right] dl \quad (2)$$

where $l$ is the hydrological distance (along flow paths) to the catchment outlet whose maximum value equals $l_{max}$. ($U_E, D_E$) are two parameters, respectively velocity and coefficient of dispersion. $w_E$, the rainfall-weighted width function of the catchment associated with flood event E, is expressed as:

$$w_E(l) = \frac{\overline{R_E}(l)}{R_E} \quad (3)$$

with $\overline{R_E}(l)$ denoting the total rainfall at hydrological distance $l$, and $R_E$ the total areal rainfall on the catchment.

Moreover, the volume of runoff ($V_Q$) is equal to the volume of effective rainfall during event E, i.e.:

$$V_Q = \int_0^{t_{max}} Q_r(\tau) d\tau = A \int_0^{t_{max}} \overline{R_e(\tau)} d\tau \quad (4)$$

with $t_E$ being the duration of rapid flow, defined as the duration of the rain event prolonged by the time of concentration of the catchment (see section 2.3).

In addition, Rodriguez-Iturbe and Rinaldo (1997) found that the mean and variance of the width function-based GIUH satisfy the following expressions:

$$E(G_E) - \frac{1}{U_E} E(L) = 0 \quad (5)$$

$$Var(G_E) - 2 \frac{D_E}{U_E^2} E(L) - \frac{1}{U_E^2} Var(L) = 0 \quad (6)$$

where $L$ is the random function representing the distribution of path lengths to the catchment outlet. $E(L)$ and $Var(L)$, the mean and variance of the distribution of flow path lengths of the catchment, are expressed as:

$$E(L) = \int_{0}^{l_{max}} w_E(l) l dl \quad \text{and} \quad Var(L) = \int_{0}^{l_{max}} w_E(l) [l - E(L)]^2 dl \quad (7)$$

The next paragraph presents the procedure used to determine the EGIUH associated with a flood event.

### 2.2. EGIUH identification method

The problem posed consists of identifying the EGIUH that best estimates the observed hydrograph at the basin outlet from observational data. This identification procedure takes place within the following framework:

- The EGIUH is defined by the rainfall-weighted width function $w_E$ and two parameters $U_E$ and $D_E$, which remain to be determined. Strictly speaking, the rainfall-weighted width function should be derived from the spatial pattern of effective rainfall. It is therefore assumed that the rainfall-weighted width function can be inferred in a representative manner from rainfall observations.

- In addition to Eq. (1), which relates the effective rainfall to the flood hydrograph, the EGIUH determination is controlled by several constraints. Equation (4) is a non-bias condition stating that the runoff volume is equal to amount of effective rainfall. The shape of the EGIUH is defined by Eq. (2) and by Eqs. (5) and (6) which constrain its two first moments.

- Available data combine the observed hydrograph at the catchment outlet, with time series rainfall fields provided by any rainfall measurement device giving access to the spatial variability of rainfall. However, this spatial variability is only needed for the total rainfall and can be derived from a sufficiently dense rain gauge network. Additional information on land use and catchment features may be used to assist with the estimation of effective rainfall.
- The outflow hydrograph is a linear function of both the EGIUH and the hyetograph of effective rainfall (Eq. (1)) yet nonetheless a nonlinear function of the two EGIUH parameters. The transformation of time series rainfall data fields into a hyetograph of effective rainfall has a non-negligible influence on the identified EGIUH. The determination of effective rainfall and EGIUH respectively are therefore not independent and must occur simultaneously, even if the goal is to determine the EGIUH. The effective rainfall is calculated by a production function to be defined by the user (e.g. SCS-CN considered herein).

- The standard approach entails coupling a “production function”, which transforms the hyetogram into a hyetogram of effective rainfall, with a “transfer function”, i.e. the EGIUH, and jointly calibrating their parameters. As part of this contribution, a variant approach has been adopted, in order to clearly distinguish the EGIUH to be identified from the hyetograph of effective rainfall, which may be called into question depending on the level of confidence placed in it.

These various elements have led us to consider the identification of EGIUHs as an inverse problem (Menke, 2012; Tarentola, 2005). The purpose herein is to gain knowledge of a physical system characterized by parameters, as combined in vector **p** through an examination of observational data, compiled in vector **d**<sub>obs</sub>, related to the parameters by means of a model denoted **m** (Menke, 2012). The main difference between data and parameters lies in the level of information. Solving the inverse problem is intended to improve the information on parameters through both data and model. The problem is initialized by the a priori information on parameters, denoted **p**<sub>prior</sub>, as well as extra information helpful to solving the problem, which proves to be crucial in solving the underdetermined problems.

In the present case, this approach is well adapted to taking into account the specific role of the hyetogram of effective rainfall, as an a priori information, thus its influence on the result is able to be controlled.

The algorithm used to identify the EGIUH parameters has been detailed in the textbooks by Menke (2012) and Tarentola (2005). In assuming that uncertainties (observations, model, a priori values of parameters) are Gaussian, then the solution minimizes the following expression:

$$ \Phi(\mathbf{d}, \mathbf{p}) = [\mathbf{m}(\mathbf{d}) - \mathbf{d}_{\text{obs}}]^T \mathbf{C}_d^{-1} [\mathbf{m}(\mathbf{p}) - \mathbf{d}_{\text{obs}}] + [\mathbf{p} - \mathbf{p}_{\text{prior}}]^T \mathbf{C}_p^{-1} [\mathbf{p} - \mathbf{p}_{\text{prior}}] \quad (8) $$

$$ \mathbf{d} = \mathbf{m}(\mathbf{p}) $$

where $\Phi$ is the likelihood function, $t$ signifies transpose, $C_p$ is the covariance matrices of residuals between truth (and unknown) and a priori parameters, and $C_d$ the covariance matrix of the errors on data.

$$ \mathbf{m}(\mathbf{p}) \approx \mathbf{m}(\mathbf{p}_{\text{prior}}) + \mathbf{M}(\mathbf{p} - \mathbf{p}_{\text{prior}}) \quad (9) $$

If model **m** can be linearized around $\mathbf{p}_{\text{prior}}$:

$$ \mathbf{m}(\mathbf{p}) \approx \mathbf{m}(\mathbf{p}_{\text{prior}}) + \mathbf{M}(\mathbf{p} - \mathbf{p}_{\text{prior}}) \quad \text{with} \quad \mathbf{M} = \left(\frac{\partial \mathbf{m}}{\partial \mathbf{p}}\right)_{\mathbf{p}_{\text{prior}}} \quad (10) $$

A. Tarentola (2005) showed that the solution $\hat{\mathbf{p}}$ is approximately given by:

$$ \hat{\mathbf{p}} \approx \mathbf{p}_{\text{prior}} + (\mathbf{M} \mathbf{C}_d^{-1} \mathbf{M} + \mathbf{C}_p^{-1})^{-1} \mathbf{M} \mathbf{C}_d^{-1} [\mathbf{d}_{\text{obs}} - \mathbf{m}(\mathbf{p}_{\text{prior}})] \quad (11) $$

This author detailed the derivations leading to the iterative algorithm using a quasi-Newton method (see Section 3.2.3, p 69):

$$ \begin{bmatrix} \mathbf{p}_{n+1} = \mathbf{p}_n - \mu_n \left[ \mathbf{M}_n^t \mathbf{C}_d^{-1} \mathbf{M}_n + \mathbf{C}_p^{-1} \right]^{-1} \\ \left[ \mathbf{M}_n^t \mathbf{C}_d^{-1} (\mathbf{m}(\mathbf{p}_n) - \mathbf{d}_{\text{obs}}) + \mathbf{C}_p^{-1} (\mathbf{p}_n - \mathbf{p}_{\text{prior}}) \right] \end{bmatrix} \quad (12) $$

in which $\mathbf{p}_n$ constitutes the result of the n<sup>th</sup> iteration, $\mathbf{M}_n = \frac{\partial \mathbf{m}(\mathbf{p}_n)}{\partial \mathbf{p}_n}$ is the matrix of (first-order) partial derivatives of the model at point $\mathbf{p}_n$, and $\mu_n < 1$ serves to control convergence.

The data are defined by ($\mathbf{d}_{\text{obs}}$, $\mathbf{C}_d$), which collates the available observations and associated uncertainty. The a priori information, introduced in terms of ($\mathbf{p}_{\text{prior}}$, $\mathbf{C}_p$), contains the initial knowledge of the parameters to be identified, along with the confidence ascribed to this knowledge. Modeling uncertainties can be regrouped with data uncertainties in the covariance matrix $\mathbf{C}_d$.

Regarding the GIUH identification, the data and parameters are defined as follows:

The vector data $\mathbf{d}_{\text{obs}} = (\mathbf{Q}_r, V_Q, 0.0, 0.0)$ comprises the hydrogram of runoff (vector $\mathbf{Q}_r$), the volume of runoff (Eq. (4)) and the two zero values standing for the second terms of Eqs. (5) and (6). The covariance matrix is assumed to be diagonal, and the standard deviations associated with both the data and model equations have been assembled in the vector $\sigma_d = (\sigma_{Q_r}, \sigma_{V_Q}, \sigma_{EG}, \sigma_{VarG})$.

The vector parameters $\mathbf{p} = (U_E, D_E, R_E)$ comprise the velocity ($U_E$), coefficient of dispersion ($D_E$) and hyetogram of effective rainfall (vector $R_E$). The a priori velocity ($U_{\text{prior}}$) and coefficient of dispersion ($D_{\text{prior}}$) are both set by the user and accompanied by their standard deviations of $\sigma_U$ and $\sigma_D$, respectively. The a priori hyetograph of effective rainfall components ($R_{\text{prior}}$) is accompanied by its covariance error matrix $\mathbf{C}_R$.

## 2.3. Evaluation process for this EGIUH identification method

The EGIUH identification is based, first of all, on the runoff hydrograph at the catchment outlet, a convolution product of the EGIUH and the hyetograph of effective rainfall. The forcing variable, i.e. the hyetograph of effective rainfall, is imperfectly known; it is therefore required to solve an underdetermined inverse problem, often referred to in rainfall-runoff modeling by the concept of “equifinality” and non-uniqueness of parameters (Beven and Freer, 2001). The goal here is to proceed with a comprehensive evaluation of the EGIUH identification method within the framework of a case study that assembles a sizable number of flash flood events and catchments in order to: i) test the method under a wide range of conditions, in terms of catchments and rain event features and quality of data, in order to provide an overview of its robustness and limitations; and ii) highlight the variability of the EGIUHs that characterize the flash floods of each catchment. This evaluation is performed in three steps:

### 2.3.1. EGIUH assessment and EGIUH variability

Any assessment of the determined EGIUHs can only be indirect. It is therefore assumed that an EGIUH becomes even more accurate as the hydrogram simulated using it lies closer to the observed one. The widely used Nash-Sutcliffe criterion is applied for this very purpose, i.e.:

$$ \text{NSE}(x) = 1 - \frac{\sum_{i=1}^{l=n} (x_{\text{ref}}^i - x_{\text{tes}}^i)^2}{\sum_{i=1}^{l=n} (x_{\text{ref}}^i - \bar{x}_{\text{ref}})^2} \quad (13) $$

with NSE($x$) being the Nash-Sutcliffe criterion with respect to the variable $x$, which compares a series of values to be tested {$x_{\text{tes}}^i = 1, n$} to a series of reference values {$x_{\text{ref}}^i$}. In the manuscript, NSE is used to compare hydrograms, denoted NSE($Q$), GIUHs, denoted NSE($\text{GIUH}$), and hyetograms, denoted NSE($R$).

Since the GIUH represents the probability density of travel times to the catchment outlet, the distance between two GIUHs can be assessed
by means of the Kolmogorov-Smirnov distance (Massey, 1951), as widely used to compare two statistical distributions:

$$D_{ks}(f_1, f_2) = \sup_{k \in \{1, n\}} |F_1(k) - F_2(k)| \text{ and } F(k) = \sum_{l=1}^{l=k} f(k) \quad (14)$$

where $f_1$ and $f_2$ are two discrete probability density functions of n components, whose cumulative distribution functions are $F_1$ and $F_2$, respectively.

The variability in the EGIUHs of each catchment is evaluated by calculating the mean distance, denoted $\Delta_E$, from each EGIUH to all the others of the same catchment defined as follows:

$$\Delta_E(\text{EGIUH}_i) = \frac{1}{n_c} \sum_{k=1}^{k=1, n_c \text{ and } i \neq k} D_{ks}(\text{EGIUH}_i, \text{EGIUH}_k) \quad (15)$$

Where $n_c$ is the number of EGIUHs of the catchment C and $D_{ks}$ is the Kolmogorov-Smirnov distance between EGIUH$_i$ and EGIUH$_k$.

Note that the Nash-Sutcliffe criterion cannot be used to calculate the distance between two EGIUHs because, it is designed to compare a reference variable to a variable to be tested.

The EGIUH identification is applied to a large number of flash floods affecting several catchments. Some results are displayed using boxplots plotted with the R package ggplot2. Boxplot graphs summarize distribution with a few values. The box ranges from the first quartile (Q1) of the distribution to the third quartile (Q3) of the distribution. The line inside this box represents the median of the distribution. The width of the box is the interquartile range (IQR). The lines which extend beyond the box are called the “whiskers”. These lines include the values included in the intervals [Q1 – 1.5 IQR, Q1] and [Q3, Q3 + 1.5*IQR]. Finally, the “extreme” values beyond 1.5*IQR from either end of the box are represented by dots (https://r-craft.org/the-ultimate-guide-to-the-ggplot-boxplot/).

### 2.3.2. Sensitivity study

The sensitivity study is of particular importance due to the indirect nature of the evaluation. Indeed, it yields the influence of a variation in EGIUHs parameters ($U_E$, $D_E$) on the results output, hence the level of confidence that can be ascribed to these results. In addition, these results may depend on the choice of the a priori hyetogram of effective rainfall. The influence of this choice can be appreciated by comparing the results obtained with the hyetogram of effective rainfall provided by two very different production functions. Lastly, the decision to take into account or not the variability of rain along flow paths allows illustrating the “equifinality” of the problem to be solved.

## 2.4. Clustering of the EGIUHs and definition of hydrological signatures

This part of the evaluation procedure consists of comparing the EGIUHs identified for every catchment, so as to both group those that can be considered as very similar and derive a limited number of GIUHs representative of each catchment. This stage is performed by means of a hierarchical cluster analysis (Ward, 1963). (Bajracharya and Jain, 2022) used this method to assess hydrologic similarity between catchments, based on width function and hypsometry. Hierarchical clustering requires a distance criterion between the EGIUHs to be compared. The NSE criterion is not suited for such a use since it compares a reference to a test value. The Kolmogorov-Smirnov distance (Eq. (14)) is used as distance criterion. This clustering allows to characterize the floods of a catchment by a reduced number of distinct GIUHs. Finally, the set of EGIUHs of every catchment can be visually summarized by:

- the envelope of the catchment EGIUHs, which is the surface delimited by the minimum value ($GIUH_{min}$) and maximum value ($GIUH_{max}$) respectively of all GIUHs at each time step,

- A limited number of distinct GIUHs representative of the types of floods affecting the catchment.

The GIUH on every flood event over a given catchment can be characterized by different hydrological signatures (Beven, 2020; Gericke and Smithers, 2014). (Huynh et al., 2023) take into account hydrological signatures to calibrate a hydrological model. We propose to use classical hydrological signatures to compare the representative GIUHs of a catchment. For that purpose, the signatures are derived from the GIUH as follows:

- The lag time, $T_L$, normally defined as the time difference between the centroid of the runoff hydrogram and the hyetogram of the mean effective rainfall, can be written as:

$$T_{\text{lag}} = \frac{E(L)}{U_E} \quad (16)$$

- The time to peak of the GIUH, $T_{pic}$, complemented by the peak value of the GIUH, $G_{pic}$;

- The time of concentration, $T_C$. (Beven, 2020) notes that the concept of time of concentration is accompanied by multiple definitions and estimation methods. In this study, it is proposed to consider it as a measure of the EGIUH duration. It can then be defined as the time corresponding with 95 % of the EGIUH integral, that is the time when 95 % of the runoff water falling onto the catchment at t = 0 has reached the outlet:

$$\int_{0}^{T_C} \text{GIUH}(t)dt = 0.95 \quad (17)$$

- The flashiness (Saharia et al., 2021, Saharia et al., 2017a), $Fl$, adapted as the slope of the rising part of the GIUH:

$$Fl = G_{pic}/T_{pic} \quad (18)$$

- An additional descriptor, describing the shape of the GIUH has been introduced:

$$Z = \frac{U_E E(L)}{4D_E} \quad (19)$$

The parameter Z is a positive dimensionless number that characterizes the shape of the GIUH (Moussa and Bocquillon, 1996). As Z tends to 0 (i. e. Z < 0.1), the GIUH tends to an exponential decay from t = 0 (i.e. case of a linear reservoir). For high values of Z (i.e. Z > 50), the GIUH corresponds to the kinematic wave model and tends towards a simple translation of $T_L$ between the effective rainfall and the hydrograph at the outlet. The intermediate values of Z (i.e. 0.1 < Z < 50) correspond to the Diffusive Wave model.

## 3. Case study

The rivers of Southern France are known to be prone to flash floods, and various catchments in this region are regularly struck by intense rain events, which cause floods of various magnitudes, including catastrophic events. A large number of this region's catchments have been gauged, and the hydrological data can be accessed via the website Hydroportail (www.hydro.eaufrance.fr), which contains a presentation of the surveyed catchments, along with their hydrometric stations and data series; moreover, this site provides a statistical analysis of these various data series. The observational dataset used for precipitation estimates is the COMEPHORE product developed by Météo France (Emmanuel et al., 2012; Tabary et al., 2007). A specific processing chain has been implemented (Tabary et al., 2012), notably in order to address the various sources of error affecting radar data. The quality of
COMEPHORE has improved over time, along with the number of radars in the ARAMIS operational network. The weather radars surveying the Mediterranean region are S-band, weakly affected by attenuation during intense rainfall; they have been in operation since 1988 for Nîmes, 1992 for Bollène, 2001 for Opoul, 2002 for Collobrières, and 2005 for Monclar (C-Band radar). Since the paper by Dolcine et al. (2001), an increasing number of papers have used rainfall fields recorded by weather radars for the rainfall-runoff modeling of flash floods in this region, including (Anquetin et al., 2010; Coustau et al., 2012; Emmanuel et al., 2017; Le Bihan et al., 2017; Navas and Delrieu, 2018).

The catchments have been selected according to the following criteria:

- Length of the data period, allowing for a minimum of 10 flood events per catchment. The threshold for selecting a flood event has been set at a peak flow exceeding a two-year return period (denoted Qi2x);

- Distribution of the catchments over the entire French Mediterranean region subject to flash floods.

The upstream part of several catchments of interest has been equipped with dams built to store water or mitigate floods, unfortunately preventing their selection. One such catchment (Cèze) has

however been preserved as an example. Some mountainous catchments, comprising a major share of spring floods and possibly influenced by snow melt, have also been excluded.

In all, 15 catchments were ultimately chosen for this case study. Fig. 1 displays these catchments and their respective locations. Their characteristics have been compiled in Table 1. The mean surface area of these catchments equals 612 km², with a standard deviation of 160 km². The number of floods exceeding the Qi2x threshold is highly variable depending on the length of the data period. The specific peak flows of the catchments display significant variability, with values ranging from 0.21 m³/s/km² to 1.22 m³/s/km² (i.e. a ratio of 1 to 6), thus confirming that the hydrological functioning of this catchment sample is indeed diverse and moreover that these catchments are subjected to extremely varied rainfall forcing.

The total number of flood events equals 233 on 15 catchments. Most of the selected flood events are of medium magnitude, while the return period of 76.5 % lies between 2 and 5 years. Nevertheless, 12 % of the flood events display a return period exceeding 10 years.

The shape of each flood hydrograph is characterized by the following flood descriptor (Le Mesnil et al., 2021):

$$S_Q = (Q_p - 0.5Q_p)/\Delta T_p \qquad (20)$$

<figure><img src="image_3.png" /><figcaption>Fig. 1. Location of the fifteen studied catchments Southern France. The area of the 15 catchments ranges from 228 km² (C13) to 960 km² (C7). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<table><caption>**Table 1**
Main characteristics of the studied catchments. Qi2x is the two years return period peak flow.</caption><thead><tr><th>Catchment – Station</th><th>Area (km²)</th><th>Slope (%)</th><th>Period of data series</th><th>Qi2x (m³/s)</th><th>Specific peak flow Qi2x/A (m³/s/km²)</th><th>Flood Descriptor F<sub>Q</sub> (m³/s/h)</th><th>Mean value S<sub>Q</sub>/A</th><th>Number of flood events</th></tr></thead><tbody><tr><td>C01 Ardèche – Vogüe</td><td>630</td><td>2.49</td><td>1997–2021</td><td>670</td><td>1.06</td><td>[27, 610]</td><td>0.30</td><td>22</td></tr><tr><td>C02 Gardon – Anduze</td><td>543</td><td>2.05</td><td>2002–2021</td><td>660</td><td>1.22</td><td>[57, 803]</td><td>0.51</td><td>18</td></tr><tr><td>C03 Doux – Tournon</td><td>620</td><td>1.48</td><td>2006–2021</td><td>250</td><td>0.40</td><td>[14, 330]</td><td>0.11</td><td>12</td></tr><tr><td>C04 Eyrieux – Ollières</td><td>638</td><td>1.70</td><td>2007–2021</td><td>360</td><td>0.56</td><td>[13, 214]</td><td>0.12</td><td>14</td></tr><tr><td>C05 Cèze – Montclus</td><td>834</td><td>1.45</td><td>2002–2021</td><td>600</td><td>0.72</td><td>[20, 340]</td><td>0.11</td><td>12</td></tr><tr><td>C06 Ouvèze – Vaison</td><td>585</td><td>2.22</td><td>1997–2002 2009–2021</td><td>125</td><td>0.21</td><td>[6, 56]</td><td>0.04</td><td>15</td></tr><tr><td>C07 Hérault – Laroque</td><td>960</td><td>1.28</td><td>1997–2021</td><td>584</td><td>0.61</td><td>[24, 323]</td><td>0.13</td><td>29</td></tr><tr><td>C08 Orbieu – Luc</td><td>620</td><td>1.11</td><td>2005–2021</td><td>265</td><td>0.43</td><td>[21, 280]</td><td>0.11</td><td>13</td></tr><tr><td>C09 Verdouble – Tautavel</td><td>325</td><td>1.66</td><td>2002–2004 2006–2020</td><td>171</td><td>0.53</td><td>[17, 143]</td><td>0.15</td><td>10</td></tr><tr><td>C10 Gapeau – Ste Eulalie</td><td>580</td><td>1.58</td><td>2002–2021</td><td>150</td><td>0.26</td><td>[9, 50]</td><td>0.04</td><td>16</td></tr><tr><td>C11 Var – Entrevaux</td><td>675</td><td>3.91</td><td>2002–2021</td><td>150</td><td>0.22</td><td>[7, 20]</td><td>0.02</td><td>16</td></tr><tr><td>C12 Siagne – Pégomas</td><td>496</td><td>2.85</td><td>2002–2021</td><td>153</td><td>0.31</td><td>[7, 91]</td><td>0.07</td><td>15</td></tr><tr><td>C13 Aille – Vidauban</td><td>228</td><td>0.84</td><td>2002–2021</td><td>151</td><td>0.66</td><td>[23, 119]</td><td>0.23</td><td>19</td></tr><tr><td>C14 Allier – St Haon</td><td>514</td><td>1.36</td><td>1997–2021</td><td>210</td><td>0.41</td><td>[12, 79]</td><td>0.08</td><td>13</td></tr><tr><td>C15 Loire – Goudet</td><td>432</td><td>1.67</td><td>1997–2021</td><td>210</td><td>0.49</td><td>[14, 193]</td><td>0.19</td><td>11</td></tr></tbody></table>

with $Q_p$ peak flow and $\Delta T_p$. time interval between 0.5$Q_p$ and $Q_p$, $S_Q$ represents the slope of the rising part of the flood hydrograph preceding the flow peak. $S_Q/A$

Table 1 shows the range of $F_Q$ values for the 15 catchments. It confirms that $F_Q$ displays a very important variability for the floods of each catchment. In order to compare the 15 catchments $F_Q$ is normalized by the catchment area, resulting in a flashiness index ($F_Q/A$) close to the one proposed by (Saharia et al., 2017b). Table 1 regroups the mean values of this flashiness index for the 15 catchments and confirms the wide variability of the flash flood severity among the studied catchments.

The a priori hyetogram of effective rainfall is estimated by the widely used SCS model (section 4.1.1). The CN values for the catchments C01 to C13 are provided by the map of CN values which covers the French Mediterranean region (Naulin et al., 2013). This map is based on the (USDA, 1986) proposed method to determine the CN values as a function of land use, soil types (Corine Land Cover data base and European Soil Database). Let's note that the CN map is not available for catchments C14 and C15. For these catchments, the Curve Number CN value is assumed to remain constant over the entire catchment area.

It is worthwhile to ascertain whether the rainfall from the rain event causing the flood strikes to a greater extent the upstream or downstream

part of the catchment. This information can be derived by calculating the first scale moment of the spatial rainfall variability on the catchment (Zoccatelli et al., 2011). This moment, denoted $\Gamma_E$ for flood event E, describes the distance of the centroid of catchment rainfall with respect to the average flow distance value (i.e. the catchment centroid):

$$ \Gamma_E = \frac{\int_0^{l_{max}} w_E(l)dl}{\int_0^{l_{max}} w(l)dl} \qquad (21) $$

with $l$ being the hydrological distance to the catchment, $l_{max}$ the maximum hydrological distance, $w$ the width function of the catchment, and $w_E$ the rainfall-weighted width function of event E.

Fig. 2 summarizes the distribution of the first scale moment of the spatial rainfall variability for the set of studied catchments. To interpret these results, it is considered that: a value of $\Gamma_E$ over the interval [0.95, 1.05] reflects a flood event with a spatially uniform or nearly uniform total rainfall; a value of $\Gamma_E$ over the intervals [1.05, 1.12] and [0.88, 0.95] represents upstream and downstream rain patterns, respectively; and values of $\Gamma_E$ above 1.12 and below 0.88 indicate rain patterns with major differences between the upstream and downstream parts of the catchment. It can be seen that the catchments reveal significant differences as regards the distribution of $\Gamma_E$. Most, if not all, of the floods considered

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Distribution of $\Gamma_E$, first scale moment of the spatial rainfall variability, for the studied catchments (see section 2.3.1 for the interpretation of boxplots). The dotted lines delimit the rainfall variability intervals: $\Gamma_E = [0.88, 0.95, 1.05, 1.12]$.</figcaption></figure>
for four catchments, namely C01 (Ardèche), C05 (Cèze), C14 (Allier) and C15 (Loire), were caused by events with greater upstream rain. The floods of six catchments, i.e., C06 (Ouvèze), C09 (Verdouble), C10 (Gapeau), C11 (Var), C12 (Siagne) and C13 (Ail), were due to events whose total rain was spatially uniform or nearly uniform. C07 (Hérault) is the only catchment for which the majority of floods are characterized by rain events with higher downstream totals, yet still not very distinct. The remaining catchments C02 (Gardon-Anduze), C03 (Doux), C04 (Eyrieux) and C08 (Orbieu) are affected by both upstream and downstream events, as indicated by the wide scattering of $\Gamma_E$. Catchment C02 is highly illustrative. The two most significant floods over the period 1997–2021, with maximum hourly peak flows of 2700 m³/s (return period of about 100 years) and 1750 m³/s (return period of approx. 15 years), display values of $\Gamma_E = 0.81$ (most downstream rainfall) and $\Gamma_E = 1.17$ (most upstream rainfall). It is interesting to note that catchment C05, equipped with a dam located upstream is mostly hit by rain events with higher upstream totals.

# 4. Results

The first subsection presents the application conditions established to reach a satisfactory functioning of the identification procedure. The second subsection describes the results obtained and discusses their reliability, while the third subsection provides a more detailed analysis of these results.

## 4.1. Application conditions

The identification procedure is based on a number of assumptions, namely: non-bias conditions, and Gaussian or at least unimodal distributions (to be satisfied to the greatest extent possible). In addition, the procedure must adapt to very different hydrological contexts, as well as to a sizable number of floods whose observational and modeling quality may vary significantly. One stage of the “trial and error” has been undertaken in order to determine the adapted application conditions. This stage has confirmed the importance of two points: i) satisfying the non-bias condition of the identification algorithm, and ii) making the procedure easy to use and highly robust.

### 4.1.1. A priori hyetogram of effective rainfall

Recent studies however have shown that flash flood outflows in Southern France include a subsurface contribution (Douinot et al., 2018) and moreover that preexisting water contributes to these floods (Bouvier et al., 2021). In this context, the term runoff includes all processes contributing to the fast response of a catchment to rainfall forcing. For the sake of simplicity, the SCS model has been used to provide a priori hyetogram of effective rainfall, in considering that this conceptual model can represent a combination of runoff processes plus their evolution with progressive soil saturation. The SCS model has in fact been successfully applied to basins subject to flash floods in this Mediterranean region (Coustau et al., 2012; Naulin et al., 2013). The Curve Number values of the SCS model (Chow et al., 2013) were mapped over the Mediterranean region. For each flood event, a very simple calibration of the SCS model was conducted to ensure that the volume of effective rainfall equals the runoff volume, thereby guaranteeing a non-bias condition. This calibration step consists of adapting a mean value of the storage capacity ($S_{max}$) of the SCS model to each flood. The base flow is first removed from the measured hydrograph; the pre-event base flow is the flow measured at the beginning of the rain event. At the end of the event, it is assumed that runoff contribution becomes negligible once the rain has stopped for a period equal to the basin time of concentration. The post-event base flow is the flow measured at this specific time. It is also assumed that the base flow varies linearly between these values during the flood event. The time of concentration is estimated by the GIUH duration. It is estimated by the time when 98 % of the runoff water falling onto the catchment at has reached the outlet, which is the time

corresponding to 98 % of the GIUH integral. Given that the GIUH is specific to each flood event, the time of concentration varies from flood to flood, the maximum value obtained for a given catchment is considered as time of concentration of this catchment. The mean calibrated value of $S_{max}$ has been spatially disaggregated based on a pre-defined spatial distribution that accounts for both land cover and soil characteristics. Initial losses were set at 18 % of $S_{max}$, provided the prior rainfall had occurred more than three days after the beginning of the event, 9 % of $S_{max}$ between one and three days, and 0 % if the flood event had closely followed a rain event.

The matrix of error covariance of the effective rainfall hyetograph is defined by:

$$ 
\begin{gathered}
\text{cov}(R_{prior}^i, R_{prior}^j) = \sigma_{Ri} \sigma_{Rj} \exp \left[ - \left( \frac{|i-j| \Delta t}{T_R} \right)^2 \right] \\
\sigma_{Ri} = \max \left( \alpha_R R_{prior}^i, \sigma_{Rmin} \right)
\end{gathered}
\qquad (22) $$

with i and j being time indices, $\sigma_{Ri}$ and $\sigma_{Rj}$ the error standard deviation of $R_{prior}^i$ and $R_{prior}^j$, respectively. $\alpha_R$ is the coefficient of variation in the error characterizing the hyetograph of effective rainfall, $\sigma_{Rmin}$ denotes the minimum value assigned to $\sigma_R$, and $T_R$ is the decorrelation time controlling the temporal structure of the errors affecting successive components of the prior hyetograph of effective rainfall. It is assumed herein that the prior hyetogram of effective rainfall is not very accurate, according to (Andrieu et al., 2021). The coefficient of variation for each net hyetograph component is then given by the value $\alpha_R = 0.2$ and $\sigma_{Rmin} = 0.2 \text{ mm/h}$. In addition, $T_R = 2 \text{ h}$ controls the temporal structure of the error on the a priori net hyetograph.

### 4.1.2. A priori GIUH parameters ($U_{prior}$, $D_{prior}$)

The tests performed have indicated that the results might depend on the a priori values adopted for ($U_{prior}$, $D_{prior}$) and moreover that the identification algorithm is more sensitive to $U_E$ than to $D_E$. The choice has therefore been made to perform several identifications, with prior parameter values covering the range of variation, i.e.: 1 m/s, 1.3 m/s, and 1.5 m/s for $U_{prior}$; and 1600 m²/s, 3800 m²/s, and 6000 m²/s for $D_{prior}$ respectively. The applicable standard deviation values are: $\sigma_U = 0.3 \text{ m/s}$ and $\sigma_D = 1300 \text{ m²/s}$, in order to leave a wide interval of variation with respect to the a priori values. The initialization of $U_E$ and $D_E$ by this scanning step seems to offer the most robust means for reaching the optimum of Eq. (7).

### 4.1.3. Data and model

The standard deviations of data errors have been combined into the vector $\sigma_d = (\sigma_{Q_r}, \sigma_{V_Q}, \sigma_{EG_r}, \sigma_{VarG_r})$, with:

$$ \sigma_{Q_{ri}} = \max[\alpha_Q Q_{ri}, \sigma_{Q_{min}}] \qquad (23) $$

$$ \sigma_{V_Q} = \alpha_V V_Q $$

$$ \sigma_{EG} = \alpha_E E(G_E) $$

$$ \sigma_V = \alpha_V Var(G_E) $$

The flow data are assumed to be of good quality. The coefficient of variation of the flow measurement error has been set at $\alpha_Q = 0.10$, with a minimum value $\sigma_{Q_{min}} = 0.1 \text{ mm/h}$. The GIUH model is assumed to be exact and free of any error. Eq. (4), which expresses the budget equation, is assumed to be associated with a weak error characterized by its small coefficient of variation, $\alpha_V = 0.02$. Eqs. (5) and (6), which describe the first two GIUH moments as a function of ($U_E, D_E$) are assigned the coefficients of variation $\alpha_E = 0.05$ and $\alpha_V = 0.1$ which reflect the importance ascribed to the GIUH theory.
The identification algorithm aims to minimize the expression $\Phi(d.p)$ (Eq. (8); an additional constraint has been added. An iteration is considered successful if the convergence criterion is reduced and furthermore if $NSE(Q)$ between simulated and observed hydrograms increases compared to the previous iteration.

## 4.2. Results presentation

Fig. 3 shows the distribution of $E(Q)$, indicator of the EGIUH accuracy, for the set of 233 flood events for which the EGIUH identification method has been applied. The accuracy is poor for a very reduced number (4) of flood events ($NSE(Q) < 0.49$). The analysis of the hyetogram and hydrogram for these 4 floods indicate rainfall and flow data are not consistent for each of these cases. The accuracy of the EGIUH identification is average for 23 flood events ($0.50 < NSE(Q) < 0.75$). These results may be due to various causes, including errors in observed data, inappropriate a priori hyetogram of effective rainfall, and inappropriate consideration of rainfall variability as regards flood dynamics. As a precaution, the acceptance threshold has been set at $NSE_{min} = 0.75$, which includes a population of 206 flood events. Fig. 4 displays the identified GIUHs for these 206 flood events split over the 15 catchments. It is also worth noting that the number of flood events for each catchment ranges from 8 (C12-Siagne and C14-Allier) to 25 (C6-Hérault). Therefore, the set of observations is quite representative of the floods affecting the studied catchments.

Fig. 4 illustrates the obtained results for 8 rain events observed on various catchments. Once defined the a priori hyetogram of effective rainfall, various a priori EGIUHs are tested to run the identification algorithm. The solution is that which minimizes both the expression (Eq. (8)) and the Nash criterion $NSE(Q)$. The scan of a several values of ($U_E, D_E$) allows to initialize the algorithm in favourable conditions. The improvement of the $NSE(Q)$ range from 0.01 (flood event not shown) to over 0.9, for the flood event (C07: 2015-091100-091523). The flood event (C02: 2020-061100-061323) illustrate a case for which the solution is less than the threshold $NSE_{min}$. It is worth noting that the identified EGIUH for the extreme flood (C02: 2002-090700-091223) does not reach the strong flashiness which would be required to simulate the extreme peak flow over $2700 \text{ m}^3\text{/s}$. It also worth noting that the algorithm is well adapted to varied rainfall hyetograms, and doesn't need short and intense rainfall hyetograms close to a Dirac impulse.

### 4.2.1. Variability of the EGIUHs characterizing the floods of the studied catchments

Fig. 5 illustrates the diversity of the GIUHs for the various catchments; the result is quite normal although the catchments display very distinct characteristics in terms of catchment area, shape hydrological functioning, etc. Fig. 3 clearly highlights the variability in the GIUHs of each catchment, with this variability being more or less pronounced depending on the catchment. The variability is assessed by the distance $\Delta_E$, from each EGIUH to all the others of the same catchment (Eq. (15)). Fig. 6 depicts the distribution of $\Delta_E$ for all EGIUHs of each catchment; it clearly shows that the catchments behave differently with respect to this criterion. The catchments can be combined into three classes, sorted by EGIUH variability. The first class includes catchments [C03, C04, C06, C11, C14], which display weak variability in the associated EGIUHs ($\Delta_E^{median} < 0.08$). The second class comprises catchments [C01, C07, C08, C10, C13], all of which display an intermediate EGIUH variability ($0.08 < \Delta_E^{median} < 0.12$). Lastly, the third class features catchments [C02, C05, C09, C10, C12, C15], as they display the most highly variable EGIUHs. The catchment C05 displays very variable EGIUHs while it is essentially affected by upstream rain events. This functioning of the upstream dam which equips the catchment might contribute to this variability.

The EGIUH associated with a given flood event depends on both the rainfall-weighted width distance and the values of parameters ($U_E, D_E$).

The EGIUH associated with a given flood event depends on both the rainfall-weighted width distance and the values of parameters ($U_E, D_E$).

### 4.2.2. Variability of parameters ($U_E, D_E$) over the studied catchments

Fig. 7 summarizes the statistical distribution of parameters ($U_E, D_E$) obtained for all events on the 15 catchments, while Table 2 complements this figure by providing the mean and standard deviation of these two parameters for the flash flood events of all 15 catchments. From these results, the following comments can be drawn.

With regard to velocity $U_E$:

- The median value of most catchments lies within the narrow interval $[0.9 \text{ m/s}, 1.1 \text{ m/s}]$. Three catchments stand out, primarily C09 (1.45 m/s), and to a much lesser extend C11 (1.18 m/s), C14 (0.87 m/s) and C15 (0.9 m/s). Adjacent catchments often display proximate values, for instance C01 and C02, C03 and C04, C14 and C15.

- The scattering of $U_E$ values appear to be highly variable from one catchment to the next: weak (less than 0.3 m/s) for four catchments (C03, C06, C11 and C14), moderate (less than 0.5 m/s) for three

<figure><img src="image_5.png" /><figcaption>Fig. 3. Cumulative distribution function of $NSE(Q)$ for the 233 flood events. The horizontal dotted line indicates the threshold $NSE(Q) = 0.75$ exceeded by 206 flash flood events (vertical dotted line).</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. The EGIUH identification to a flood event is illustrated by three graphs on the same line. The left graph shows the a priori hyetogram of effective rainfall and the final one, the middle graph shows the envelope of a priori EGIUHs and the resulting one, and the right graph compares simulated a priori and resulting hydrograms to the observed one. NSE(Q) is provided above the graph. The base flow has been removed. The name of flood events is : catchment:year-date and hour of start-date and hour of end. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Identified EGIUHs for the fifteen catchments of the case study. The discretization time step of EGIUHs is 1 h. The EGIUH is unitless. The colors are random, in order to clearly distinguish the EGIUHs.</figcaption></figure>

catchments (C04, C13, C15), and more pronounced for the remaining catchments. A couple of “atypical” values are to be noted (2.19 m/s for C02 and 1.92 m/s for C13). Let’s also point out that a second flood of C02 yielded a high value ($U_E$ = 2.3 m/s), which was discarded, while the Nash-Sutcliffe coefficient between the observed and simulated hydrograms equals 0.72, i.e. below the threshold.

- Attempts to explain the variations in $U_E$ for each catchment by the peak flow or other variables remain inconclusive. Nevertheless, a significant relationship ($R^2$ = 0.40) does exist between median velocity and catchment slope, provided catchment C09 (whose value appears to be “atypical”, see Table 2) is not taken into account.

With regard to the coefficient of dispersion $D_E$:

- The median value of most catchments lies within the interval [6500 m²/s, 8600 m²/s]. Three catchments are characterized by median values lying below this interval, namely C14 (6215 m²/s), C13 (3680 m²/s) and C15 (4300 m²/s).

- As for velocity, the scattering of the coefficient of dispersion is highly variable from one catchment to the next. The range is fairly weak however, less than 3500 m²/s for 4 catchments (C03, C04, C11, C15). This range does not exceed 6100 m²/s for catchments C01, C02, C06,
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Distribution of $\Delta_E$, distance of an EGIUH to the others) for the studied catchments. (see section 2.3.1 for the interpretation of boxplots). The dotted lines, $\Delta_E = [0.08, 0.12]$ delimit the three classes of EGIUH variability.</figcaption></figure>

C07, C10, C13, C14 and is above 7200 $m^2/s$ for the 4 remaining catchments (C05, C08, C09, C12).

- Attempts to explain the variations in $D_E$ for each catchment also remain inconclusive. Additional investigations on a limited number of flood events would be required, beginning with the paper by Saco and Kumar (2002), who analysed the processes contributing to the coefficient of dispersion.

A few papers have dealt with the application of GIUH or a routing function, defined by its velocity within the hydrographic network, to catchments C01 (Ardèche-Vogüë) and C02 (Gard-Anduze). (Moussa, 1997) obtained a value of $U = 1.3 m/s$ for two flood events that occurred on C02 between 1977 and 1984. (Bouilloud et al., 2010) adopted a value of $U = 1.1 m/s$ to model four floods occurring on C01, C02 and C05 from October 2001 to September 2025. These results cannot be compared flood event-by-flood event to the $U_E$ values obtained for these catchments in the present study, yet they are of the same order of magnitude, which does lend credibility to the results reported herein. Regarding parameter $D_E$, (Moussa, 1997) obtained a value $D_E = 6300 m^2/s$ for the two flood events that occurred on C02. This value is fully consistent with the median value of $D_E = 6400 m^2/s$ characterizing the 13 events on C02. Surprisingly, the scientific literature on observed $D_E$ values is rather limited; to the best of our knowledge, no reference assesses the realism of the values obtained across our 15 studied catchments. The variations of the coefficient of dispersion might be related to the displacement of rainfall fields on the catchment, or to temporal and spatial variability of the flow velocity field within the hydrographic network.

The next section presents the sensitivity study and the effect of some of the application conditions defined by the user. These tests prove to be

helpful in identifying possible limitations of the method.

### 4.3. Sensitivity study and influence of application conditions

#### 4.3.1. Sensitivity to ($U_E$, $D_E$) variations

This sensitivity study examines the influence of ($U_E$, $D_E$) variations on the simulated hydrograms and EGIUHs, respectively. The identified EGIUH, $G_E(U_E, D_E, t)$, has been modified into $G_E(U_E \pm \Delta U, D_E \pm \Delta D, t)$, which is then used to simulate a modified hydrogram. The criterion $NSE(GIUH)$ is introduced to compare the modified EGIUH to the identified one. The criterion $NSE(Q)$ serves to compare the hydrogram simulated using the modified EGIUH to the observed hydrogram. The range of $\Delta U$ variations extend from $[-0.3 m/s, 0.3 m/s]$, while the range of $\Delta D$ variations is $[-3000 m^2/s, 3000 m^2/s]$. Fig. 8 shows the mean values of these criteria for the total set of 200 flood events.

These results yield a number of interesting findings, namely:

With regard to hydrograms, Fig. 8A combines the influence of $U_E$ and $D_E$ changes on the accuracy of the simulated hydrograms. At first, it appears that the sensitivity of simulated hydrograms is more heavily influenced by $U_E$ variations than $D_E$ variations. A decrease in $U_E$ of 0.1 $m/s$ and 0.2 $m/s$ with respect to the optimum results in a deterioration of the mean $NSE(Q)$ by respectively 0.007 and 0.027. In contrast, an increase in $U_E$ of 0.1 $m/s$ and 0.2 $m/s$ with respect to the optimum deteriorates the $NSE(Q)$ value by 0.006 and 0.021. A decrease in $D_E$ of 1000 $m^2/s$ and 2000 $m^2/s$ with respect to the optimum brings down the $NSE(Q)$ mean value by respectively 0.003 and 0.013. Surprisingly, an increase in $D_E$ of 1000 $m^2/s$ has a quasi-negligible impact on $NSE(Q)$ while a $D_E$ increase of 2000 $m^2/s$ produces a small decline in $NSE(Q)$ of

<caption>**Table 2**</caption>

Mean and standard deviation values of the EGIUH parameters ($U_E$, $D_E$) for the 15 catchments.

<table><thead><tr><th>Catchment</th><th><i>U</i><sub>E</sub>(m/s) Mean/Standard deviation</th><th><i>D</i><sub>E</sub>(m<sup>2</sup>/s) Mean/Standard deviation</th></tr></thead><tbody><tr><td>C01</td><td>1.16/0.30</td><td>8350/1530</td></tr><tr><td>C02</td><td>1.12/0.23</td><td>7020/2060</td></tr><tr><td>C03</td><td>0.98/0.05</td><td>8180/1040</td></tr><tr><td>C04</td><td>0.89/0.10</td><td>7440/800</td></tr><tr><td>C05</td><td>1.18/0.20</td><td>6150/2500</td></tr><tr><td>C06</td><td>1.01/0.04</td><td>7750/1050</td></tr><tr><td>C07</td><td>0.96/0.15</td><td>7940/1720</td></tr><tr><td>C08</td><td>1.10/0.19</td><td>6880/1780</td></tr><tr><td>C09</td><td>1.35/0.28</td><td>6870/2440</td></tr><tr><td>C10</td><td>1.03/0.19</td><td>6310/1780</td></tr><tr><td>C11</td><td>1.15/0.11</td><td>8740/1190</td></tr><tr><td>C12</td><td>0.93/0.19</td><td>6660/2600</td></tr><tr><td>C13</td><td>1.01/0.26</td><td>3640/1220</td></tr><tr><td>C14</td><td>0.85/0.09</td><td>5800/1170</td></tr><tr><td>C15</td><td>0.90/0.16</td><td>4440/1150</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Distribution of the EGIUHs parameters: $U_E$ (left figure) and $D_E$ (right figure). See paragraph 2.3.1 for the interpretation of boxplots.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8. Sensitivity of EGIUHs and hydrograms to $U_E$ variations (figure A) and to $D_E$ variations (figure B).</figcaption></figure>

0.005.

As for the EGIUHs, Fig. 8B combines the influence of $U_E$ and $D_E$ changes on the deviation of $G_E(U_E \pm \Delta U, D_E \pm \Delta D, t)$ with respect to $G_E(U_E, D_E, t)$. A variation in $U_E$ of $\pm 0.1 \, m/s$ results in a slight decrease of the NSE(GIUH) value to about 0.975. A decrease in $U_E$ of $0.2 \, m/s$ winds up dropping the NSE(GIUH) value to about 0.93, while an $U_E$ increase of $0.2 \, m/s$ yields a more pronounced deterioration of NSE(GIUH) to 0.90. A variation in $D_E$ of $\pm 1000 \, m^2/s$ results in a very modest change of NSE (GIUH) to about 0.987. A decrease in $D_E$ of $2000 \, m^2/s$ results in a deterioration of NSE(GIUH) to about 0.935, while a $D_E$ increase of $2000 \, m^2/s$ causes a smaller deterioration of NSE(GIUH) to just 0.97.

These results pertain to the set of 200 flash floods and moreover should certainly be refined as a function of the ($U_E, D_E$) couple values. Nevertheless, they do lead to the following conclusions, which are key to the EGIUH identification:

- The influence of $U_E$ variations on simulated hydrograms is much more pronounced than that of $D_E$ variations, which also means that the identification of $D_E$ will be less accurate than that of $U_E$.

- Assuming that the identification procedure remains close to the optimum provided the NSE(GIUH) criterion varies by less than 0.5 % compared to its maximum, then the range of uncertainty is [−0.09 $m/s$, +0.09 $m/s$] for $U_E$ and [−1200 $m^2/s$, +2000 $m^2/s$] for $D_E$. These intervals are only indicative, yet they do provide a useful order of magnitude of the accuracy that could be expected given the identified values of $U_E$ and $D_E$.

- The NSE(GIUH) criterion between two EGIUHs, whose parameters differ by a value lying within these ranges of uncertainty, is greater than 0.96, which implies that these two EGIUHs are close to each other and that an uncertainty on $U_E$ and $D_E$ does not significantly affect the resulting EGIUH.

### 4.3.2. Influence of the a priori hyetogram of effective rainfall

Results have been obtained by considering that the widely used SCS production function allows for an accurate estimation of the hyetograph of effective rainfall from the observed hyetograph. Since this hyetograph relies on a number of the parameters to be identified, it would be helpful to verify the influence of the a priori hyetograph of effective rainfall on the resulting GIUHs. In this aim, a second production function including a constant runoff coefficient (CR) has been introduced. The initial losses of this CR model are similar to those of the SCS model, and the runoff coefficient has been estimated such that the effective rainfall volume equals the runoff volume. The CR model is not used for flash flood modeling in the Mediterranean region, which suggests that such an a priori hyetogram of effective rainfall based on the CR model is far from the optimum and that the choice of such a priori hyetograms of effective rainfall represents a severe test. This test compares two cases:

- Case 1: The EGIUH identification performed using the a priori SCS hyetogram of effective rainfall;

- Case 2: The EGIUH identification performed using the a priori CR hyetogram of effective rainfall.

This comparison concerns: a) the a priori SCS and CR hyetograms of effective rainfall, criterion NSE(R0), b) the hyetograms identified from results in Case 1 and Case 2, criterion NSE(R12), c) the EGIUHs identified from results in Case 1 and Case 2, criterion NSE(GIUH12) and d) the simulated hydrograms obtained in Case 1 and Case 2, criterion NSE (Q12).

These comparative results have been collated in Fig. 9, which shows the cumulative distribution function of the four comparison criteria for the set of flash flood events. The results displayed in Fig. 9A indicate that:

- The a priori SCS and CR hyetograms of effective rainfall are indeed very different (curve A), since nearly 75 % of NSE(R0) values are less than 0.9.

- The identified hyetographs of effective rainfall of the two cases get much closer and consistent, since 66 % of NSE(R12) values are above 0.9. This result confirms the capability of the identification method to improve the overall consistency of both the data and parameters describing the catchments.

- The proximity of the simulated hydrograms based on the two production, functions (curve D) is very high since 88 % of NSE(Q12) are over 0.95, and 75 % over 0.97. The mean value of $|U_{scs} - U_{CR}|$ is 0.11 $m/s$, and the mean value of $|D_{scs} - D|$ is 1190 $m^2/s$, thus lying within the range of uncertainty provided by the sensitivity study to ($U_E, D_E$) variations (Section 4.3.1).

- The closeness of the identified GIUHs based on the two production functions is good, despite very different a priori hyetographs, since 78 % of NSE(GIUH12) are over 0.90, and 61 % over 0.95.

Fig. 9B compares the accuracy of the simulated hydrograms, based on the a priori SCS hyetogram of effective rainfall, criterion NSE(Q1) and on the a priori CR hyetograms of effective rainfall, criterion NSE(Q2) respectively. It shows that Case 2 is less accurate for the flash flood events which are less well simulated, but that most of the results are of similar accuracy.

This test yields two useful pieces of information. The GIUH identification procedure can be adapted to very different a priori hyetographs of effective rainfall and provides results that are moderately influenced by this strong constraint. It can be presumed that initialization of the procedure by two more similar production functions will result in even closer GIUHs, thus confirming the major importance of outflow data for the identification.
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Influence of the a priori hyetogram of effective rainfall. Figure A regroups the cumulative distribution of NSE comparing: the a priori SCS and CR hyetograms of effective rainfall (blue line), the identified hyetograms of effective rainfall (green line), the identified EGIUHs (red line), the simulated hydrograms (black line). Figure B displays the cumulative distribution of NSE comparing the observed hydrograms to the simulated ones in Case 1 (black mine) and Case 2 (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

### 4.3.3. What are the consequences of not considering the spatial variability of rainfall?

An EGIUH takes into account the spatial rainfall variability on the catchment by replacing the width function by the rainfall-weighted width function. However, EGIUH identification can also be performed using the width function, i.e. without explicitly considering the spatial rainfall variability. A comparison of these two ways of identifying GIUHs is illustrated in Fig. 10, which presents the scattergram between the velocity deviation $\Delta U_E = U_E^{C3} - U_E^{C4}$ and $\Gamma_E$ and the scattergram between the deviation on the coefficient of dispersion $\Delta D_E = D_E^{C3} - D_E^{C4}$ and $\Gamma_E$ respectively, with E denoting the flash flood event. Case 3 takes into account the spatial variability of rainfall, while Case 4 does not take it into account. Let's recall that $\Gamma_E$ describes the position of the centroid of catchment rainfall with respect to the catchment centroid. A value $\Gamma_E > 1$ means that the rainfall amount is greater upstream, and value $\Gamma_E < 1$ that the rainfall amount is greater downstream. The range of $\Gamma_E$ is [0.64, 1.21] for the 200 retained flood events. It is worth noting that Case 3 performs slightly better than Case 4 (i.e. the mean value of NSE(Q) on the hydrograms is 0.90 for Case 3 and 0.89 for Case 4). Fig. 10 shows that the identification procedure compensates for not taking into account the spatial rainfall variability by adapting the values of parameters ($U_E, D_E$). Thus, for upstream rainfall ($\Gamma_E > 1$), the velocity decreases ($\Delta U_E > 0.0$) in order to reflect the longer transfer time in the hydrographic network, while for downstream rainfall, the velocity increases ($\Delta U_E < 0.0$) to reflect the shorter transfer time in the network. It is interesting to underscore the linear relationship between velocity

deviation and position of the rainfall centroid on the catchment. This observation also appears to be valid for the coefficient of dispersion, a finding that confirms the benefit of taking the spatial variability of rainfall into account when defining the EGIUH, even if the accuracy is quite similar. Indeed, taking the spatial variability of rainfall into account yields parameters ($U_E, D_E$) and application conditions (rainfall-weighted width function and $\Gamma_E$) with a better physical meaning.

A careful study of the GIUHs specific to flood events in the studied catchments shows that the variability of these EGIUHs is highly heterogeneous from one catchment to the next. The sensitivity study highlights the limitations of the adopted identification method as well as the importance of the method's application conditions, as defined by users. It also suggests that a portion of the EGIUHs associated with each catchment may not actually differ. The following section will propose a method for combining the EGIUHs of each catchment so as to achieve a typology of the EGIUHs representative of each catchment.

### 4.4. Clustering of the GIUHs

The proposed approach has been inspired from hierarchical cluster analysis (Ward, 1963). The goal here is to combine the EGIUHs of a given catchment into clusters such that the EGIUHs within each cluster are similar to one another, whereas the EGIUHs in different clusters are quite different. Each cluster is represented by its mean GIUH, which is the average of all EGIUHs it comprises. This approach makes it possible to characterize the floods of a catchment using a reduced number of

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Influence of not considering the spatial variability of rainfall on the parameters ($U_E, D_E$). Figure A displays the scattergram between the velocity deviation $\Delta U_E$ and $\Gamma_E$ (first scale moment of rainfall variability). Figure B displays the scattergram between the coefficient of dispersion deviation $\Delta D_E$ and $\Gamma_E$. Each cross represents a flood event. The blue dashed lines are the regression lines, and $R^2$ is the coefficient of determination. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
distinct GIUHs. The proposed approach works as follows:

- At first, each EGIUH is treated as a separate cluster. The two following steps are then repeated:

- Step 1: Each cluster is compared to the other clusters. The selected distance is derived from the Kolmogorov-Smirnov test, as classically used to compare two statistical distributions (Eq. (14)). The distance between two clusters equals the distance between the average GIUHs of these clusters. The result of Step 1 is a distance matrix.

- Step 2: The two closest clusters, as measured by matrix distance, are merged. The new cluster is then characterized by its average GIUH. The number of clusters therefore decreases by one.

- Steps 1 and 2 can be repeated until all clusters have been merged into a single cluster.

In practice, a threshold distance is set, from which two clusters are considered as significantly different and can no longer be merged. This iterative procedure is stopped once the distance between all remaining

<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Envelopes of EGIUHs and representative GIUHs of the clusters for the fifteen catchments of the case study. The grey zone represents the envelopes of all EGIUHs of the catchment. The color of the curves indicates the number of regrouped EGIUHs by each clustered GIUH (blue: one or two, green: three to five, red: over five). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
clusters exceeds this threshold. Each cluster can also be characterized by its mean GIUH, as well as by the associated information: mean parameters ($\overline{U}_E$, $\overline{D}_E$), mean rainfall-weighted width function, mean first-scale moment of rainfall variability.

The distance threshold is of particular importance as it determines the number of clusters for each catchment. The threshold to adopt does not rely on the decision criteria associated with the classical Kolmogorov-Smirnov distance (since the goal is not to compare two empirical statistical distributions). For purposes of illustration, a threshold $D_{ks} = 0.11$ has been adopted.

Fig. 11 illustrates the results of the GIUH clustering for all 15 studied catchments. Several comments can be drawn from these results, namely:

- The clustering exercise yields representative GIUHs that appear different depending on the criterion: time to peak, peak value, recession length, etc.

- Clustering is an effective way to summarize the identified GIUHs by means of a reduced number of representative unit hydrographs highlighting the main features of each catchment. It is interesting to note that some catchments can be characterized by a single GIUH (C03, C04, C11), others by two GIUHs (C06, C08, C11, C12), while the remaining catchments by three or four. This finding must be viewed with caution since the number of GIUHs differs tremendously from one catchment to the next (ranging from 8 to 27). However, results suggest that catchments with a single cluster are affected by floods of the same type, while catchments with different clusters are potentially affected by several types of floods.

<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Comparison of the hydrological signatures of the GIUHs representative of the 12 catchments regrouping several clusters. The color of the curves indicates the number of regrouped EGIUHs by each clustered GIUH (blue: one or two, green: three to five, red: over five). The hydrological signatures are: lag time ($T_{lag}$), time to peak ($T_{pic}$), time of concentration ($T_C$), peak value ($G_{pic}$), shape parameter ($Z$), flashiness ($Fl$), defined in Section 2.4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
- For several catchments, a cluster comprises a large majority (over 2/3) of GIUHs (C03, C04, C05, C06, C08, C11, C13, C14). However, the flash floods combined into this cluster display very different peak flows, thus confirming that peak flow is not an explanatory variable of the deviations between GIUHs. In addition, this finding may partially support the assumption often made that the GIUH is invariant of a catchment.

Six catchments out of the 15 (C01, C02, C05, C06, C13, C14) include clusters encompassing only two, or sometimes one, types of floods. For these catchments, one of the two GIUHs represents a “fast response” (peak more marked, occurring early) of the catchment and the other GIUH represents a “slower response” of the catchment (peak less marked, occurring later). Sometimes, an “intermediate response” for catchments characterized by 3 representative GIUHs (C02, C05, C07, C14). Sometimes, we note the presence of “atypical” GIUHs that differ from the usual floods occurring on these catchments. Such is the case for example with the clusters of unique flash-flood events: C01 with a slightly localized downstream rainfall pattern in a catchment, where rainfall is typically greater upstream; C02, with rainfall patterns highly localized downstream; and C14 highly localized upstream. Some of the “atypical floods” are characterized by very high values of U (see Fig. 6, C01: 2.19 m/s, C13: 1.42 m/s), which appears surprising and necessitates a careful analysis in order to detect possible data anomalies or inaccuracies in the model employed. The signatures introduced in Section 2.3 allow a comparative analysis of the EGIUHs representing the clusters of the studied catchments.

## 4.5. Comparison of GIUH signatures

This comparison is performed by using the GIUH signatures introduced in Section 2.3.4. The couple ($T_{lag}, Z$) stands for the position and shape of the GIUH. The rising part of the GIUH is represented by the triplet ($T_{pic}, G_{pic}, Fl$), while $T_C$ indicates the length of the GIUH. The comparison among GIUH signatures is displayed using radar graphs, as combined in Fig. 12. Each signature is presented on a radial axis, in accordance with a scale ranging from 0 to 100, with 100 being the maximum value assigned to the signature for the given catchment. This maximum value is indicated on the radar graph. Fig. 12 is a convenient way to visually compare the GIUHs of a given catchment. Fig. 12 illustrates that the GIUHs representative of each catchment are indeed significantly different, according to these signatures. The differences between GIUH patterns are clearly visible for the catchments with 2 representative GIUHs. The GIUH characterising a “fast response” displays a larger value of the flashiness ($Fl$) a larger value of $G_{pic}$. The GIUH characterizing a “slower response” by larger values of time parameters: time of concentration ($T_C$), time to peak ($T_{pic}$), and time lag ($T_{lag}$).

A GIUH is defined by two parameters ($U_E, D_E$), to which we can add the spatial variability of rainfall, summarized by $\Gamma_E$, defined in Section (Eq.21) It appears that GIUHs representative “slower response” display larger values of the signature Z, which depends on the ratio $U_E/D_E$. In addition, GIUHs associated to a “fast response” of catchments display larger velocities $U_E$, larger values of $D_E$, while GIUHs associated to a “slower response” of catchments display weaker values $U_E$, and lesser values of $D_E$. Finally, GIUHs associated to a “slower response” display larger values of $\Gamma_E$ than GIUHs associated with “fast response”. This indicates “slower responses” are associated with flood events for which upstream rainfall total is larger than downstream rainfall total. It is important to note that these findings concern the variations of the mean values of $U_E, D_E$ and $\Gamma_E$ of the representative GIUHs of the studied catchments. They are interesting while they allow to express the classical time signatures of GIUHS in terms of variations of the physical parameters that are used to model these GIUHs. Finally, it is worth noting that “intermediate” GIUHs are associated with intermediate values of these variables, and to recall that “atypical” GIUHs are

associated with extreme values of these parameters, namely very important velocity, or very low or high value of $\Gamma_E$.

# 5. Discussion

The spatial variability of rainfall and the channel network topology play a key role in the formation and the understanding of the flood hydrograph at the catchment outlet. In the literature, two approaches are often used for modeling flood events. The first approach is “lumped” that considers the entire catchment as only one unit. This approach has the advantage of being simple with few parameters to calibrate, but does not allow to represent the spatial variability of rainfall or the shape of the channel network. The second is a “spatially distributed” approach that subdivides the catchment into elementary units articulated around the channel network. This approach is more complex, requires many parameters that are often difficult to acquire on natural basins with parameterization and calibration difficulties, but allows to better represent the physical processes and the spatial variability of rainfall.

A major question confronts hydrology modelers: What is the optimal level of spatialization about rainfall and what time step should be chosen when the objective is only to simulate the hydrograph at the outlet? How can major transfer processes be taken into account without going to total spatialization, requiring a lot of data, costing in computation time.

Recently, spatialized rainfall data as well as the channel network structure are increasingly available. One may ask how to introduce these sources of information into flood modeling approaches. The EGIUH is a modeling approach that can be considered “semi-lumped” or “semi-distributed”, in between “lumped” and “spatially distributed”. It has the advantage of introducing the maximum of easily available spatial information (here the spatial variability of rainfall and the channel network) while keeping a simple modeling structure with few parameters. Compared to classic lumped models, taking into account spatialized rainfall (even the total rainfall on an event) represents a “minimal” spatialization of rainfall. Likewise, the width-function is also a “minimal” representation of the channel network topology. The EGIUH is a simple and parsimonious approach that does not introduce additional parameters; we thus seek to introduce into the model a minimum of useful information without going further in spatialization and complicating the model as in the case of distributed models that are difficult to parameterize. This is a first step to go into the semi-distribution of rainfall.

The discussion focuses on the actual limitations and advantages of the EGIUH, the potential applications of the EGIUH, and future research directions.

## 5.1. Actual limitations and advantages of the EGIUH

The EGIUH is based on a “spatial representation of the catchment” around the iso-distance zones to the outlet (or the width function) where an average rainfall is calculated. A “production function” (here SCS-CN) separates the rainfall into runoff and infiltration. A “transfer function” (here Diffusive Wave Hayami equation) routes the runoff to the outlet. In the applications the EGIUH has two parameters which are identified by an inverse approach. The main limitations and advantages of the actual version of the EGIUH concern the model hypotheses, the complexity of the methods, and the accuracy of the model.

### 5.1.1. Model hypotheses

The spatial representation of the catchment is based on assumptions such as the rainfall-weighted width function that can be inferred from rainfall observations. This is indeed a simple consideration of rainfall variability which nevertheless represents an advance compared to the classic GIUH using a mean rainfall over the whole catchment. Our results show the importance of this rainfall variability which provides a GIUH whose parameters are all the more different as the rainfall variability on the catchment is strong. However, this assumption is a
simplification of the catchment into iso-distance zones. This assumption may not fully capture the complexity of real-world hydrological processes, especially in diverse catchment types. To improve this, more sophisticated spatial representations of the catchment could be explored, such as incorporating detailed topographic and land use data into the model.

One originality of this work is the inverse modeling algorithm which has a Bayesian inspiration, considering that any type of information is affected by an error, and then it is important to define the credibility to be given to each information (data, parameter). This inverse modeling algorithm is widely used in geosciences (Menke, 2012; Tarentola, A., 2005) but remains very little used by the hydrological community which favors the classical calibration of model parameters. The algorithm may also seem complex because we chose to consider the components of the hyetogram of effective rainfall as additional parameters, but the user can do without this option.

Finally, iterative algorithms may require significant computational resources, especially when dealing with large-scale datasets, which may limit their application in resource-limited settings. Unlike classical methods, the algorithm used in EGIUH requires few computing resources and has the advantage of converging quickly (3 to 10 iterations). However, other parameter optimization algorithms can be easily implemented to replace the existing algorithm.

### 5.1.2. Accuracy and robustness of the model

GIUH identification can be viewed as the deconvolution of the convolution product $Q(t) = G_E(t)*R_e(t)$, $Q(t)$ being measured, with $R_e$ being imperfectly known. Like with any inverse modeling application, the major difficulty concerns the “equifinality” and non-uniqueness of the parameter estimation, where multiple parameter sets can yield similar results (Beven and Freer, 2001). This can lead to uncertainty in the model parameters. Future work could focus on developing more advanced parameter estimation techniques, such as Bayesian inference or machine learning approaches, to better constrain the parameter space and reduce uncertainty.

The accuracy of the EGIUH model is highly dependent on the quality of input data, including rainfall and runoff measurements. We use published and validated data: hydro database radar database (Tabary et al., 2007) in order to minimize measurement errors. In areas where high-quality data are scarce, the model's performance may be limited. Future research could explore the use of alternative data sources, such as remote sensing data or satellite-based precipitation estimates, to improve data coverage and quality.

Application of the EGIUH method to a large number of Rainfall-Runoff instances shows, first and foremost, the robustness of the approach in the absence of numerical instabilities. This method yields good performance (i.e. NSE(Q) > 0.75 for 86 % of the studied flash flood events). This robustness can be attributed to the choice of both GIUH model and inverse algorithm; moreover, this robustness does not deter from the flexibility of method implementation. The identification appears as an efficient way of synthesizing all information on flash flood events. Furthermore, a poor result might indicate anomalies in the data or modeling, in addition to suggesting a careful analysis of the event in question.

The EGIUH identification depends in particular on a priori values of parameters, including the production function, as well as on the confidence ascribed to the observed data and parameters to be determined. These choices are made by the user according to knowledge held of the catchment functioning and of the flash floods under study. The sensitivity analysis has confirmed that the GIUH approach is more sensitive to $U_E$ than to $D_E$ which is similar to the results already known in the literature. This analysis serves to define a reasonable margin of uncertainty on the resulting model parameter values. A comparison between the use of spatialized rainfall and uniform rainfall along flow paths shows nearly similar accuracy in both cases, obtained with different parameter values. The deviation between these parameters is well

correlated with the distance from the rainfall center of gravity to the catchment outlet. However, the set of parameters calibrated in the case of uniform rainfall offers an example of a model that outputs good results for the wrong reasons. The EGIUHs obtained are moderately sensitive to the choice of production function, thus confirming the robustness of the proposed method. While the study conducts a sensitivity analysis, further validation of the model against independent datasets or through comparison with other hydrological models would strengthen its credibility. This could include cross-validation with different datasets or benchmarking against established models to assess the EGIUH's predictive accuracy and reliability.

## 5.2. Potential applications of the EGIUH

The EGIUH makes it possible to characterize a flood or a range of flood events on a catchment by a GIUH taking into account the spatial variability of rainfall (i.e., upstream vs downstream). The results obtained provide a good overview of the variability of EGIUHs for the 15 studied catchments. A hierarchical cluster analysis has been applied so as to combine EGIUHs similar to each other into clusters. The Kolmogorov-Smirnov test serves as a distance criterion, while the threshold for grouping two EGIUHs is set by the user. This clustering allows deriving a reduced number of GIUHs (from 1 to 4 in this study) characterizing the floods of each catchment. Several variables are usually considered as “hydrological signatures” of catchment floods. The fact that all these signatures can be derived from GIUHs leads to consider those GIUHs representative of clusters as hydrological signatures of catchments, perhaps in order to achieve a typology of flash floods affecting a catchment. These characteristics of the EGIUH can be added to other characteristics of the catchment (morphometric properties, land use, soil properties, rainfall typology, etc.) and hence can be easily used for regionalization and applications on ungauged catchments. Rain fields are not available everywhere but things are changing. NASA provides the IMERG product (https://gpm.nasa.gov/data/imerg) which is an estimate of precipitation over the majority of the Earth's surface over two decades at a spatial resolution of 0.1° × 0.1° (or roughly 10 km × 10 km) with a 30-minute temporal resolution (Pradhan et al., 2022). The existence of global spaceborne DEMs, NRCS curve number global data set (Zeng et al., 2017), and IMERG data suggests a wide range of applications for many catchments, to determine the EGIUH on catchments everywhere (a little larger than ours anyway).

## 5.3. Future research directions

### 5.3.1. Model structure

The EGIUH identification work has proceeded correctly, and its implementation allows the user to choose the application conditions according to personal knowledge held on the catchments and flash floods under study. However, a clear explanation for the variations in model parameters has yet to be found, except for a statistical link between median velocity and slope of catchments. The sensitivity study has revealed that both the uncertainties and non-uniqueness of parameter values can affect the search for explanatory variables. Several paths can be viewed so as to better understand the variability of EGIUHs representative of the flash floods affecting a catchment. Future research directions concern the improvement of the model structure and model application and validation.

The EGIUH has a modular structure around four functions: the “spatial representation of the catchment”, “the production function”, “the transfer function” and “the inverse modeling procedure”. Each of these functions can be easily replaced by the user depending on available data and information or the objective of the study. Other spatialization approaches can be used such as the catchment subdivision into elementary subcatchments or pixels. The current EGIUH model primarily focuses on runoff processes. To provide a more comprehensive understanding of flood events, future models could incorporate
additional hydrological processes, such as infiltration, evapotranspiration, and subsurface flow, which are crucial in determining the overall catchment response. For example, the production function can be easily replaced by a simple threshold, a proportional ratio of rainfall, a simple infiltration law (Horton, Holtan, etc.), a reservoir or any physically based infiltration equation. The transfer function can be replaced by a simple lag time transfer, exponential or gamma laws, Nash cascade, etc. The inverse modeling procedure can be replaced by any classical calibration procedure such as the greatest gradient, Monte Carlo, etc. The choice and number of parameters to be calibrated as well as the constraints can also be easily modified. Moreover, the model parameterization can be also improved by using spatiotemporal variable parameters of U and which enables to take into account the variability of the velocity as a function of the discharge and morphometric properties such as the slope and the cross-section shape. This modular structure of the EGIUH allows the user to adapt each module to the hydrological knowledge of the catchment, the available rainfall/runoff data and the objective of the study. Finally, the EGIUH could be easily adapted for solute transfers or other conservative variables by replacing the spatialized rain model input with any other variable where the mass is preserved.

### 5.3.2. Model application and validation

The study is primarily focused on catchments in the French Mediterranean region. This first step consisted in applying this new methodology of the EGIUH to a limited number of catchments. The results have shown that each can be characterized by various GIUHs. This result presents new insights compared to the majority of previous works which often implicitly consider that the GIUH of a catchment is unique and represents all the floods of the catchment. To enhance the generalization of the EGIUH method, it should be applied to a wider range of catchments with varying climatic conditions, sizes, and topographies. This would help validate the model's robustness and adaptability in different hydro-meteorological contexts.

Recently the GIUH approach starts to be used on sewer networks in urban hydrology (Achour et al., 2023; Perez et al., 2024) and the application of the EGIUH identification can be also easily transposed in urban areas. A wider range of application would be useful to: i) strengthen the identification method, ii) adapt it to various contexts, and iii) collect information on both EGIUHs and their variability.

## 6. Conclusion

Recently, (Andrieu et al., 2021) adapted the width function-based GIUH and proposed a protocol to determine the GIUH specific to each flood event from observational data (denoted EGIUH). This EGIUH is a parsimonious easy-to-use model, defined by two parameters, namely velocity $U_E$ and coefficient of dispersion $D_E$, as well as by the rainfall-weighted width function, which summarizes both the structure of the river network and the spatial distribution of the forcing rain along this river network. The EGIUH is identified using an inverse algorithm that allows the user to provide a current state of knowledge on the catchment. The production function used to derive the hetogram of effective rainfall is among the parameters to be determined in order to provide the user with flexibility and specify the level of reliability.

This paper has presented a more comprehensive assessment of the identification method for event-specific GIUHs. Such an assessment has been conducted on a case study combining 233 flash floods that had occurred on 15 Mediterranean catchments in Southern France during the period 1997–2021. These floods are characterized by highly intense rainfall events and sometimes by a strong spatiotemporal variability of rainfall, inducing Hortonian surface runoff. We selected catchments exhibiting an operational challenge for flood forecasting with an expectation of good quality flow measurements and time series of radar rainfall fields.

## CRediT authorship contribution statement

Hervé Andrieu: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Formal analysis, Conceptualization. Roger Moussa: Writing – review & editing, Validation, Methodology, Formal analysis. Pierre Nicolle: Writing – review & editing, Visualization, Validation, Software. Olivier Payrat-astre: Writing – review & editing, Validation, Methodology, Formal analysis.

## 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

Discharge data were uploaded on the Hydroportail site, Rainfall data (Comepore Product) were provided by MétéoFrance, and DTM data were obtained from Institut Géographique National (IGN). The authors thank the two anonymous reviewers, and the Associate Editor for providing very useful comments, which greatly improved the manuscript.

## Data availability

The elaborated data concerning the whole set of flood events and the source code are available from the authors.

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