# Statistical behavior of temporally compounding multi-peak flood events

Greta Cazzaniga, Fabiola Banfi, Carlo de Michele

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Greta Cazzaniga, Fabiola Banfi, Carlo de Michele. Statistical behavior of temporally compounding multi-peak flood events. Environmental Research Letters, 2025, 20 (5), pp.054071. 10.1088/1748-9326/add0c8. hal-05068797

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LETTER • OPEN ACCESS

Statistical behavior of temporally compounding multi-peak flood events

To cite this article: Greta Cazzaniga et al 2025 *Environ. Res. Lett.* 20 054071

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LETTER

# Statistical behavior of temporally compounding multi-peak flood events

Greta Cazzaniga¹ ID, Fabiola Banfi² ID and Carlo De Michele²,∗ ID

¹ Laboratoire des Sciences du Climat et de l'Environnement, UMR 8212 CEA-CNRS-UVSQ, Université Paris-Saclay, IPSL, Gif-sur-Yvette 91190, France

² Department of Civil and Environmental Engineering, Politecnico di Milano, Milano 20133, Italy

∗ Author to whom any correspondence should be addressed.

E-mail: carlo.demichele@polimi.it

**Keywords:** temporally compounding floods, flood frequency analysis, mixture distribution, flood design, flood peak, compound events, multi-peak floods

Supplementary material for this article is available [online](#).

## Abstract

Floods are the most frequent natural hazard globally (2003–2023) and rank second in economic losses, according to EM-DAT, the International Disaster Database. When flood events occur in close succession, they pose significant challenges for emergency management due to limited recovery time between events. This study focuses on multi-peak (MP) floods, where peaks occur within hours to days. Using discharge data from 77 hydrometric stations in Northern Italy’s Po district, we examined statistical differences between MP and single-peak (SP) flood events and analyzed their seasonal patterns and generating mechanisms across diverse river regimes. We demonstrated that SP and MP events exhibit distinct statistical behaviors. The first type of events has more skewed distributions with heavy tails, while the second displays flatter distributions with lighter tails and higher mean values. Seasonal analyses suggest that MP floods in glacial and nival-pluvial regimes are influenced by glacier and snow melting, whereas those in the Padanian regime are driven by tributary routing effects. These triggering mechanisms seem to be responsible of the lighter tails of the distribution of MP floods. By highlighting the distinct statistical behaviors and generating mechanisms of MP and SP floods, we identified recommendations for designing MP flood hydrograph, supporting flood-risk management.

## 1. Introduction

Floods are the most frequent natural hazard globally (2003–2023), ranking second in economic losses, according to the International Disaster Database (EM-DAT, www.emdat.be, Centre for research on the epidemiology of disasters 2024). The complexity of analyzing floods arises from their diverse dynamics, which sometimes classify them as compound events/objects (Zscheischler et al 2020, Bevacqua et al 2021). Following the categorization proposed by Zscheischler et al (2020), it is possible to distinguish four typologies of compound floods. (1) ‘Spatially compounding floods’ are generated by extended meteorological forcing and affect multiple catchments over a region/country/continent simultaneously (Brunner 2023), e.g. the July 2021 floods in Belgium and Germany or the May 2023 floods in

Emilia Romagna, Italy. (2) ‘Temporally compound-
ing floods’ are consecutive events in the same catch-
ment/region, amplifying impacts compared to single
events (see Hypothesis 2 of Berghuijs et al (2016)
and Zscheischler et al (2020)), as seen in California
(Bowers et al 2024). (3) ‘Multivariate floods’ are char-
acterized by the co-occurrence of multiple extreme
flood-related variables, such as extreme flood peak,
volume, and duration (Mediero 2010, Serinaldi and
Grimaldi 2011) or flood peak and sea level, e.g. the
Ravenna flood in 2015, Italy (Bevacqua et al 2017). (4)
‘Preconditioned floods’ involve the presence of pre-
conditions. Saturated soils or snow cover may act as
preconditions, while precipitation and/or snowmelt
as drivers (see Hypotheses 3 and 4 of Berghuijs et al
(2016) and Berghuijs et al (2019)), e.g. the cold rain-
on-snow event of June 2013 in the Canadian Rockies
(Pomeroy et al 2016). Note that this categorization is
not strict, and some flood events may fall into more
than one category.

While research has been conducted on spatially
(Najibi et al 2023, Fang et al 2024), and multivari-
ate typologies of compound floods (Wahl et al 2015,
Bevacqua et al 2017, 2019, 2020), preconditioned
and temporally compounding floods are less fre-
quently studied, despite their significant implications
for flood management. For example, temporally-
clustered flood events are highly dangerous for
hydraulic structures such as dams. The first wave(s)
can fill the storage, and the consecutive wave can
lead to a flood due to the previously filled storage
that otherwise may not have happened (Klein et al
2010). Similarly, in retention basins, the efficiency of
the reservoir is generally evaluated considering only a
single-peak (SP) event. It means that the structure is
assumed to be empty and all the storage fully avail-
able. This may result in an overestimation of flood
attenuation, if a temporally-clustered event occur.

From a statistical perspective, clustering beha-
vior introduces serial correlation into the flood time
series, invalidating the assumption of independent
and identically distributed (i.i.d.) flood events, that
is commonly used in flood frequency analysis (Merz
et al 2016). The neglect of temporal clustering can
lead to inaccurate flood quantile estimates for flood
design, increasing uncertainty in the estimation pro-
cess (Koutsoyiannis 2005). Flood frequency analysis is
typically based on the i.i.d. assumption, but temporal
clustering in floods may introduce serial correlation,
challenging this assumption.

Temporally-clustered behavior may be driven
by various meteorological, geomorphological, and
hydrological factors (Brunner *et al* 2017, Jiang *et al* 2024), including catchment characteristics, rain-
fall patterns, and other weather-related phenomena.
Based on the timescale involved, two types of flood
clusters can be defined: low-frequency and high-
frequency (Barton *et al* 2016). Low-frequency clusters
occur over months to years and are often related
to seasonal flooding. Most of the research on flood
clusters has focused on the low-frequency type. It was
shown that in Europe or China, many catchments
exhibit temporal clustering for moderate floods, at
timescales of 1 year to a few years (Gu *et al* 2016, Merz
*et al* 2016, Lun *et al* 2020). This type of flood clus-
tering appears to be influenced by large-scale climate
indices (Villarini *et al* 2013, Gu *et al* 2016, Liu and
Zhang 2017). Nonetheless, as shown by Tarasova *et al*
(2023), flood generation processes also play a signific-
ant role in modulating the anomalies observed during
flood-rich and flood-poor periods. High-frequency
clusters are flood events characterized by multiple
peaks and involve hourly and daily temporal scales.
Hereafter we will refer to high-frequency clusters as
multi-peak (MP) events. When looking at MP flood
events, we can assume that the triggering mechan-
isms are no longer slowly evolving climate factors,

but are predominantly meteorological, geomorpho-
logical, and hydrological factors (Brunner *et al* 2017,
Jiang *et al* 2024) (e.g. catchment size, altitude, flow
regime, type of soil) and weather-related phenom-
ena, which operate over shorter temporal scales (a few
hours or days). For example, in large-size catchments,
spatially compound rainfall events (Zscheischler *et al*
2020) may cause many simultaneous flood events
over various tributaries (Chen *et al* 2012) that are
routed towards the mainstream with different lag
times. Moreover, the temporal clustering of rainfall
events, even in small-size catchments, may not allow
the complete depletion of each triggered flood event,
thus generating hydrographs with more than one
peak (Barton *et al* 2016, Bevacqua *et al* 2021, Banfi
and De Michele 2022). MP events are very frequent
among all streamflow events (not only floods). In
Tarasova *et al* (2018) they are assumed to have stat-
istically different properties.

The scope of this study is twofold. We first invest-
igate the statistical properties of MP floods, and
make a systematic comparison with SP. Specifically,
we examine statistical differences, comparing the
distributions of flood peak, volume, and duration,
between MP and SP events. Investigating the beha-
vior of these three types of flood-related variable is
fundamental, as these variables interact to determ-
ine flood severity (De Michele et al 2005, Rahimi
et al 2021). Not accounting for volume and dura-
tion, would in fact provide a limited description of
a flood event (Mediero 2010), leading to an inef-
fective flood damage prevention and the design of
hydraulic structures. The second aim is establish-
ing links between the statistical features of MP and
SP floods and their respective triggering mechan-
isms. Through a seasonal analysis and a screening of
the literature we developed hypotheses on the flood-
triggering mechanisms. Eventually, we provide sug-
gestions on how to possibly integrate the concept of
MP flood events into flood frequency analysis, flood
design, and flood-risk management. Integrating the
concept of MP events into flood hydrograph design is
crucial, as it enhances flood-risk management by cap-
turing the varying impacts of different flood behavi-
ors on inundation areas (Sikorska et al 2015, Brunner
et al 2017).

## 2. Case study and data

The study relies on 77 observed discharge time
series aggregated at an hourly time scale, from rivers
belonging to the Po district, in northern Italy. The
Po district covers the Po Basin, which has an area
of around 71 060 km², and several small-sized basins
flowing into the Adriatic Sea, near the Po Delta. These
latter pertain to Reno, Fissero, Tartaro, Canalbianco,
Marecchia-Conca rivers, and all the small-size basins
located in Romagna. Figure 1 illustrates the Po district
<figure><img src="image_6.png" /></figure>

boundaries, as well as the selected water catchment boundaries with their corresponding outlet sections/hydrometric stations, where discharge data are available. In the study area, there exists a broad variety of basin sizes as well as diverse altitudes (from 0 to 4800 m a.s.l.). This ensures a broad representation of different flow regimes. The Po district undergoes a natural hydrological regime transformation as it flows from upstream to downstream. In the area are present four flow regimes (AdbPo 2016): *glacial*, *nival-pluvial*, *pluvial*, and *padanian*. In figure 1, the hydrometric stations are grouped by flow regime and area of the corresponding catchments. In the supplementary material further information on the catchments and the flow regimes are available.

## 3. Methods

We developed a methodology to assess whether SP and MP flood events have different statistical characteristics and how to include this information in flood design. The methodology is articulated in the following steps: (1) Sampling flood events; (2) classifying events as SP or MP, (3) Making statistical analysis of flood event samples, (4) fitting the random variables with suitable probability density functions (PDFs), (5) investigating the upper tail behavior of flood peaks. The following subsections describe each step of the methodology.

### 3.1. Step 1: sampling flood events

To sample flood events from the hourly streamflow time series we used a peak-over-threshold (POT) analysis (figure 2(a)), which is one of the most common approaches in literature (Salvadori et al 2007, Serinaldi and Kilsby 2013, Brunner et al 2019, Rahimi et al 2021). The starting/ending of an event is defined as the time where the discharge exceeds/drops below the threshold. Typically, the independence of events

is ensured by defining a minimum number of days between two flood events. In the case where two sampled events occur too close in time, just the one with the highest peak is considered (Rahimi et al 2021). We adapted this approach by grouping individual events occurring within a maximum time lag of 6, 24, 48, and 72 hours for basins with areas on the order of magnitude of 10, 10², 10³, and 10⁴ km², respectively. These events are then assumed as temporally dependent. These time lags were identified as representative values that could reasonably fit the different groups of basin areas, based on the relationship between catchment area and the concentration time in Ferro (2006). Considering two dependent flood events as one, the starting and ending times of the event are the starting point of the first one and the ending point of the second one, respectively (see e.g. event 1 in figure 2(a)). We finally implemented two additional rules to discard some of the sampled events (refer to the supplementary material).

The event volume, $V$, is calculated as the area underlying the hydrograph between the starting and the ending times. The event duration, $D$, is defined as the time elapsed between starting and ending points. The flood peak, $Q$, is the maximum discharge value. These three variables are represented in figure 2(a) for two illustrative events.

### 3.2. Step 2: classifying events as SP or MP

This part of the algorithm aims at detecting flood peaks within a flood event and to classify such event as either SP or MP. Figure 2(b) illustrates both event types. For each flood event, we identify local minima and maxima (green and light blue points in figure 2(b)). If the first stationary point (in the mathematical calculus sense, i.e. having the temporal derivative of discharge equal to zero) is a local maximum, the preceding local minimum is identified
<figure><img src="image_7.png" /><figcaption>Figure 2. Methodology to sample independent flood events and classify them into SP and MP events. Q, V, and D are the flood peak, volume, and duration, respectively. Δ is the difference in absolute value between the successive local minimum and local maximum points, or vice versa.</figcaption></figure>

before the starting time of the event. The discharge differences, Δ, between each local minimum and its neighboring maxima are then calculated. Any local maximum with a discharge difference below a set threshold is excluded to avoid capturing minor fluctuations, such as those caused by intermittent rainfall. This threshold is event-specific and set to 5% of the maximum discharge observed during the event. The number of peaks is determined by counting the remaining local maxima. If more than one peak is detected, the event is classified as MP (Event 1 in figure 2(b)); otherwise, it is classified as SP (Event 2 in figure 2(b)).

### 3.3. Step 3: making statistical analysis of flood event samples

For each random variable, Q, V, and D, we applied two statistical tests, the non-parametric Kolmogorov-Smirnov two-sample test (Lehmann et al 1986, Kottegoda and Rosso 2008) and the Cramér-von Mises two-sample test (Anderson 1962), to compare the empirical distributions of SP and MP samples. The distributions of the two statistics are calculated under the null hypothesis that the samples are drawn from the same distribution.

### 3.4. Step 4: fitting a mixture model to flood peak, volume, and duration

If step 3 confirms that Q, V, and D follow distinct distributions for SP and MP events, a mixture distribution is typically needed to account for the presence of multiple underlying processes within the

overall population. This approach may ensure more robust estimates (McLachlan et al 2019). The mixture model is a weighted summation of N (cumulative) distributions (that can also belong to different families) defined by their own set of parameters {$F_1(x; \Theta_1), \dots, F_N(x; \Theta_N)$}, where the weights (also called mixing probabilities) {$w_1, \dots, w_N$} sum to one (McLachlan et al 2019)

$$F(x; \Theta_1, \dots, \Theta_N) = \sum_{n=1}^{N} w_n F_n(x; \Theta_n). \quad (1)$$

We present here the case of flood peak Q. For this, we propose using a mixture distribution based on two Generalized Pareto distributions (GPDs). The mixture cumulative distribution is hence formulated as the weighted sum of the two underlying GPDs, $F_{sp}$ and $F_{mp}$:

$$\begin{aligned} F(x; \sigma_{sp}, \xi_{sp}, \mu_{sp}, \sigma_{mp}, \xi_{mp}, \mu_{mp}) \\ = w_{sp} \cdot F_{sp}(x; \sigma_{sp}, \xi_{sp}, \mu_{sp}) + w_{mp} \cdot F_{mp} \\ \times (x; \sigma_{mp}, \xi_{mp}, \mu_{mp}) \end{aligned} \quad (2)$$

holding that $w_{sp} + w_{mp} = 1$, where $w_{sp}$ and $w_{mp}$ are respectively the contributions of SP, sp, and MP, mp components to the overall distribution. The cumulative distribution function, $F_{sp}$ can be written as:

$$F_{sp}(x; \sigma_{sp}, \xi_{sp}, \mu_{sp}) = \left[ 1 - \left( 1 + \xi_{sp} \frac{x - \mu_{sp}}{\sigma_{sp}} \right)^{-\frac{1}{\xi_{sp}}} \right] \quad (3)$$
where $\sigma_{sp}$, $\xi_{sp}$, and $\mu_{sp}$ are respectively the scale, shape, and location parameters of the GPD for the SP component. Similarly, for the cumulative distribution function of the MP component, with the parameters $\sigma_{mp}$, $\xi_{mp}$, and $\mu_{mp}$. As we can properly label each observation with respect to SP and MP categories, the parameters can be estimated separately for the two distributions leveraging traditional estimation methods, such as the probability-weighted moment method, or the maximum likelihood estimation method. While we can use the relative frequency

of SP and MP events as $w_{sp}$ and $w_{mp}$. Another possibility may be using the expectation maximization algorithm (Ghojogh et al 2019), which allows the optimization of all the parameters at once (including the mixing probabilities: $w_{sp}$ and $w_{mp}$). When the parameters are optimized, the probability that an observation belongs to a specific component can be calculated using Bayes' theorem. This probability is conditional and is called responsibility, $\gamma$. For example, the responsibility of the MP component, $\gamma_{mp}$, given an observation $x$ is:

$$ \gamma_{mp} = P(\text{mp} | x) = \frac{P(\text{mp}) \cdot P(x | \text{mp})}{P(x)} = \frac{w_{mp}^* \cdot F_{\text{mp}}(x, \sigma_{\text{mp}}^*, \xi_{\text{mp}}^*, \mu_{\text{mp}}^*)}{w_{mp}^* \cdot F_{\text{mp}}(x, \sigma_{\text{mp}}^*, \xi_{\text{mp}}^*, \mu_{\text{mp}}^*) + w_{sp}^* \cdot F_{\text{sp}}(x, \sigma_{\text{sp}}^*, \xi_{\text{sp}}^*, \mu_{\text{sp}}^*)}, \quad (4) $$

where $\sigma^*$, $\xi^*$, $\mu^*$, and $w^*$ are the optimized GPD parameters and mixing probabilities of the two populations.

The estimation of this responsibility can be effectively leveraged in flood design. Depending on the values of these responsibility, they can guide the decision on whether to use an SP or MP hydrograph. We propose a straightforward approach: select an MP hydrograph when the responsibility associated with the MP component for the design quantile(s) is greater than 0.5, and opt for an SP hydrograph otherwise. Please note that if the three random variables ($Q$, $V$, and $D$) are modeled using a copula to account for their dependency relationship, a single responsibility value can be assigned to the design quantile triplet ($q_T$, $v_T$, and $d_T$).

### 3.5. Step 5: investigating the upper tail behavior of flood peaks

The behavior of the upper tail of flood peaks is an extremely important aspect to investigate, as it affects the reliability of flood design and risk management, which are based on the probability of extremes. As an indicator of tail behavior, we used the shape parameter (Coles et al 2001), $\xi$ of the GPDs fitted on sampled events. The shape parameter shows indeed a relatively low sensitivity to the sample size and reasonable uncertainty compared to other indicators Wietzke et al (2020). Thus, it enables us to perform a comparison across samples with different lengths. The shape parameter has been estimated through the use of the probability-weighted moment method for the hydrometric stations where the Cramér-von Mises test applied to $Q$ resulted significant. $\xi$ parameter has been estimated on the overall sample (SP + MP), but also separately for the two samples, SP and MP. Please note that we used the version of the GPD as reported in equation (3), which implies a heavy-tailed (Pareto tail) distribution for $\xi > 0$,

a light-tailed (shifted exponential) distribution for $\xi = 0$, and an upper-bounded tail for $\xi < 0$. To estimate the uncertainty in the shape parameter, we generated 1000 bootstrap resamples by randomly sampling the data with replacement. For each resample, we computed the shape parameter. The uncertainty was then determined as the standard deviation of the shape parameter values obtained from the bootstrap distribution.

## 4. Results and discussion

### 4.1. Statistical tests on flood peak, volume, and duration

For the sampling of flood events, we selected the 85th percentile as the threshold for the POT analysis. The supplementary material includes a sensitivity analysis evaluating different thresholds and justifying this choice. It also provides the number of detected events for each water catchment. For each sampled flood event, we computed $Q$, $V$, and $D$ as described in section 3.1. Next, we applied the Cramér-von Mises and Kolmogorov-Smirnov tests to compare the distributions of SP and MP samples for each random variable, to determine whether they follow the same distribution or not. The tests were conducted separately for each hydrometric station, with a 5% significance level. Overall, it emerges that the two components originate from distinct distributions. Results from the Cramér-von Mises test are illustrated in figure 3 for $Q$, $V$, and $D$ and each station. The tests consistently have statistically significant results, with the exception of 4 stations for flood peak analysis (2 stations belonging to the padanian regime, 1 to the glacial, and 1 to the nival-pluvial) and 1 station for flood volume. Similar results are obtained with the Kolmogorov-Smirnov test (see the supplementary material). We can further investigate the nature of these differences. The empirical density
<figure><img src="image_8.png" /></figure>

functions (see the supplementary material) of MP samples, for Q, V, and D, display a significant rightward shift in the mean compared to the SP component. We observed this behavior in 96% of the stations for Q, and across all stations for V and D. The analysis also revealed differences in the skewness of SP and MP distributions. The PDFs of the SP component mostly showed a strong positive skewness with a high spike at lower values, while the MP distributions were more flattened across the entire range of values. This behavior was observed in the 87%, 76%, and 69% of the cases for Q, V, and D, respectively.

Our findings indicate the tendency of MP events to feature longer durations and to involve greater peaks and volume of waters than SP events. Establishing the statistically significant difference in the distributions of V and D random variables based on the compoundness level (namely, the categorization into SP and MP) of flood events holds evident relevance from a design perspective. However, such an outcome, although seldom stressed, may be considered trivial. For example, we can imagine that a temporal cluster of rainfall events leads to a temporal cluster of flood events and this inevitably involves a longer time window and a higher volume, with respect to a single event. A much less straightforward outcome lies in the results of the statistical analysis on flood peaks, Q, proofing that MP flood events, on average, exhibit higher peaks compared to SP ones.

## 4.2. Upper tail behavior of flood peaks

In figure 4, the values of ξ for the three cases (SP + MP, SP, and MP) are presented. The dots and the error bars represent, respectively, the estimates and the uncertainties (from the bootstrap procedure) of the shape parameter for the 73 hydrometric stations where the Cramér-rvon Mises test applied to Q resulted significant. At first glance, general differences

are observable among the three cases. In the dataset including all observations (SP + MP), positive shape parameters are estimated for the vast majority of stations (69 out of 73). This means that the GPDs typically exhibit a heavy tail, as confirmed by scientific works investigating the causes of heavy tails of flood distributions (Macdonald *et al* 2022, Merz *et al* 2022) and the flood study report in north-western Italy (Progetto Speciale per la Valutazione delle Piene in Italia, VAPI report) (De Michele and Rosso 1999). In the latter, the parameters of the GEV distribution were derived to reproduce the growing factor of flood annual maxima across four hydrological homogeneous regions in the Po district. In all four regions identified, heavy-tailed GEV distributions were found, in agreement with the heavy-tailed GPDs found here. The four stations with negative shape parameters belong to the nival-pluvial (1 station) and the padanian regime (3 stations). This is again in accordance with literature that identifies less pronounced heavy-tailed distribution for large catchments (Rosbjerg *et al* 2013). When analyzing only SP events, 72 out of 73 GPDs still exhibit a heavy tail. Only one station has a negative shape parameter and is the one associated to the padanian regime. Different outcomes are found in MP samples. There is a significant rise in the prevalence of GPD distributions that either have an upper bound or display a light tail (16 out of 73). These results show that the flood peaks of SP events have in general heavier tails compared to MP events. A heavy tail behavior is characterized by a higher probability of the occurrence of extreme events compared to distributions with bounded or light tails. This means that in general it is more likely to experience an unprecedented and surprisingly SP large flood rather than a MP one. In terms of design, light-tailed distributions result in lower design quantiles, as extreme events are
<figure><img src="image_9.png" /><figcaption>**Figure 4.** Estimates of GPD shape parameter for MP, SP, and SP + MP samples, grouped by flow regime and colored by the altitude of the outlet. The bars represent the uncertainty of the parameter estimate, calculated as ± the standard deviation of the parameter bootstrap distribution.</figcaption></figure>

less likely and the tail of the distribution decays more rapidly.

### 4.3. The processes causing MP and SP upper tail behavior

In section 4.2, we found out that on average SP floods exhibit heavier tails than MP floods. For some hydrological regimes, namely glacial, nival-pluvial, and padanian the upper tails turn even bounded. Here, we try to identify in the meteorological and hydrological processes the mechanisms behind the generation of such distinct types of tails. To do this we look at the seasonality in the incidence of SP and MP flood events, as reported in figure 5. For each basin, we identified the season with the highest number of flood events. In the figure, the height of each bar represents the number of basins (grouped by flow regime) where that specific season had the peak occurrence of flood events. Glacial regimes are usually characterized by a peak shifted towards summer (Sikorska et al 2015). This is in line with figure 5, where the majority of events, both SP and MP, occur in summer for most of the glacial catchments. However, the divergences in the shape parameter estimates (see figure 4) suggest that the processes generating them differ between the two components, even if they occur in the same season. Sikorska et al (2015) classified flood events in different basins in Switzerland, observing that floods in glacerized basins may also be associated to long-lasting rainfall and rain-on-snow events. An hypothesis of the observed differences may be that the heavy-tail distributions, rather than with glacier or

snow melting, are associated with rainfall events. Many works have indeed revealed that floods caused by heavy, often convective, rainfall tend to have heavier rainfall tails (Bernardara et al 2008, Smith et al 2019). Conversely, the upper-bounded or light-tails of MP events, given the constraint imposed by available energy on meltwater release, could be attributed to processes such as snow melting or glacial ablation (as confirmed in Merz and Blöschl (2003), Thorarinsdottir et al (2018)). However further and more specific studies should be conducted to understand the different tail behaviors. Concerning the nival-pluvial regime, we observe a temporal shift in the peak of MP events, typically occurring in spring, while SP events predominantly unfold during summer. Nival-pluvial catchments feature a core flow resulting from the late spring melting of the snowpack, occurring earlier than in glacial regimes. The timing of MP events, observed in catchments associated with this flow regime (above all in Lombardia region), once again suggests that MP events in this case are influenced by processes related to snowmelt. Rain-on-snow events occur mainly at the change between cold and warm season and snowmelt flood in spring to summer (Merz and Blöschl 2003). Simultaneously, the peak occurrence of SP events in summer and the corresponding heavy-tailed distribution may be related to short-rainfall events. Indeed Sikorska et al (2015) observed an increasing importance of this mechanism with decreasing average catchment altitude in Switzerland. In figure 5, we note that also in pluvial regimes there is a shift between the peak
<figure><img src="image_10.png" /></figure>

of occurrences of SP and MP events. This time MP events are delayed in time and occur in winter, while SP events take place in autumn. For both the components the distributions are found to be heavy-tailed, however we observe a damping of the heaviness (lower values of the shape parameter) from SP to MP events. Moving to padanian regimes, we encounter another distinct behavior. The discharge along the Po river is influenced by the amount of water entering from the tributaries. Given the area of the catchment, the inflows may have peaks simultaneously or with a time lag, or only some of them may observe high discharge values, depending on the driving mechanisms or the size of the precipitation event. In addition, the routing effect of individual flood events play a crucial role in attenuating the final flood peaks at the outlet section. Here MP shows light or upper-bounded tails and we observe low values of the shape parameter also for SP events. In larger catchments, the spatial aggregation of precipitation and runoff generation may contribute to less pronounced heavy-tailed flood distributions (Merz and Blöschl 2009, Rosbjerg et al 2013).

This study emphasizes the significance of recognizing temporally-clustered flood events, particularly MP floods, in flood risk analysis. MP floods are defined as events with multiple peaks occurring within hours or a few days. This specific type of event can be particularly dangerous because the affected sectors (e.g. populations, ecosystems, infrastructures) and the emergency response systems, may not have sufficient time to recover between consecutive peaks. This lack of recovery time can lead to significant damage, even when the individual peaks are not extreme. Based on an analysis of 77 discharge time series from the Po River district in Italy, we found that SP and MP events exhibit distinct statistical behaviors in terms of flood peak, volume, and duration. Specifically, the

variables characterizing SP events have skewed distributions with heavy tails, which are responsible for the occurrence of unprecedented or surprisingly large floods. These extreme events are particularly hazardous, especially from a preparedness perspective (Merz et al 2022). In contrast, MP events display flatter distributions with lighter tails but higher mean values. This implies that while the probability of experiencing a large flood (in terms of peak, volume, and duration) is lower for MP events compared to SP events, MP events generally have higher average values for flood-related variables. The combination of these factors underscores the importance of considering both SP and MP events in comprehensive flood risk assessments. Seasonal analyses show that MP and SP floods have different driving mechanisms that affect the different tail behaviors (Ashley and Ashley 2008, Kochanek et al 2012, Tarasova et al 2019). The analysis suggests that MP floods in glacial and nival-pluvial regimes are mainly influenced by glacier or snow melting, whereas those in the Padanian regime are driven by tributary routing effects. These triggering mechanisms are responsible for the lighter tails of the flood peaks statistical distributions.

Finally, we suggest the use of mixture distributions to model the behavior of the three considered variables describing the flood event (discharge, duration, volume). The value of the responsibility (that in this case is the probability that a quantile refers to a MP or SP event) may guide the decision on whether to use an SP or MP flood hydrograph. Even though a mixture distribution could better reproduce the different behaviors of MP and SP events, it should be taken into account that small sample sizes may eventually result in more uncertainty. It is then important to balance between the increased uncertainty and the added information gained using a mixture distribution.
The results of this work could possibly sup-
port further studies focused on investigating how
to transfer the concept of MP events into risk man-
agement and prevention. According to the data
retrieved in the post-flood event reports in Emilia
Romagna (https://www.arpae.it/it/temi-ambientali/
meteo/report-meteo/rapporti-post-evento), during
the period 2010-2023, clustered flood events rep-
resent the 41% of floods that occurred. Therefore,
incorporating clustered flood events into flood fre-
quency analysis could improve the robustness of
quantile estimations. In addition, it highlights the
need of a more realistic flood hydrograph, ensuring
that the infrastructure can withstand clustered flood
scenarios. Hydraulic engineers and planners should
consider adopting this type of investigation in flood
risk management and infrastructure design to better
account for the risks posed by temporally clustered
floods, leading to more resilient flood protection sys-
tems. Incorporating the concept of clustered flood
events into flood risk management policies and reg-
ulations would instead improve the ability of author-
ities to evaluate and plan for potential flood impacts
more comprehensively. Future studies should there-
fore translate the present results into the development
of MP design hydrographs. They would then allow to
compare the differences between SP (from traditional
methods) and MP flood hydrographs for infrastruc-
tures design and verification, also considering the
uncertainty associated with the estimation of para-
meters for the related distributions.

The results presented here are valid for the specific watershed considered as case study, other catchments in other parts of the world may exhibit different behaviors (e.g. due to different floods’ drivers). Future studies should therefore assess how to extend the results in other geographical areas, including other databases or regionalization techniques, and determine if SP and MP flood events consistently exhibit heavy and lighter tails, respectively. In addition, stationarity was assumed here, while under future climate conditions the proposed methodology and conclusions should be further assessed.

Data availability statement

The data that support the findings of this study are
openly available at the following URL/DOI: https://
www.arpae.it/it; https://www.arpalombardia.it/dati-
e-indicatori; https://www.arpa.piemonte.it; https://
www.arpa.vda.it.

Acknowledgment

This study was carried out within the RETURN
project (Multi-risk science for resilient com-
munities under a changing climate, https://www.
fondazionereturn.it/) with Extended Partnership,
which received funding from the European Union

Next-GenerationEU (National Recovery and
Resilience Plan-NRRP, Mission 4, Component 2,
Investment 1.3–D.D. 1243 2/8/2022, PE0000005).

ORCID iDs

Greta Cazzaniga ● https://orcid.org/0000-0002-6290-7762

Fabiola Banfi ● https://orcid.org/0000-0002-9774-9071

Carlo De Michele ● https://orcid.org/0000-0002-7098-4725

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