Do small and large floods have the same drivers of change? A regional attribution analysis in Europe

Abstract. Recent studies have shown evidence of increasing and decreasing trends in mean annual floods and flood quantiles across Europe. Studies attributing observed changes in flood peaks to their drivers have mostly focused on mean annual floods. This paper proposes a new framework for attributing flood changes to potential drivers, as a function of return period (T), in a regional context. We assume flood peaks to follow a non-stationary regional Gumbel distribution, where the median flood and the 100-year growth factor are used as parameters. They are allowed to vary in time and between catchments as a function of the drivers quantified by covariates. The elasticities of floods with respect to the drivers and the contributions of the drivers to flood changes are estimated by Bayesian inference. The prior distributions of the elasticities of flood quantiles to the drivers are estimated by hydrological reasoning and from the literature. The attribution model is applied to European flood and covariate data and aims at attributing the observed flood trend patterns to specific drivers for different return periods. We analyse flood discharge records from 2370 hydrometric stations in Europe over the period 1960–2010. Extreme precipitation, antecedent soil moisture and snowmelt are the potential drivers of flood change considered in this study. Results show that, in northwestern Europe, extreme precipitation mainly contributes to changes in both the median (q2) and 100-year flood (q100), while the contributions of antecedent soil moisture are of secondary importance. In southern Europe, both antecedent soil moisture and extreme precipitation contribute to flood changes, and their relative importance depends on the return period. Antecedent soil moisture is the main contributor to changes in q2, while the contributions of the two drivers to changes in larger floods (T > 10 years) are comparable. In eastern Europe, snowmelt drives changes in both q2 and q100.



Introduction
There is widespread concern that river flooding has become more frequent and severe during the last decades, and that humaninduced climate change and other drivers will further increase flood discharge and damage in many parts of the world (IPCC, 2012;Hirabayashi et al., 2013). This concern has given rise to a large number of studies investigating past changes in flood hazard, i.e. changes related to flood discharge, and flood risk, i.e. related to damage. The global pattern of increasing flood damage has been mainly attributed to increasing population, economic activities and assets in flood-prone areas (Bouwer, 2011;IPCC, 2012;Visser et al., 2014). In terms of changes in flood discharge, a variety of changes has been found (for shift in timing and trends in the magnitude of European floods, see Blöschl et al., 2017Blöschl et al., , 2019, and attempts to attribute detected changes have not resulted in a clear picture about the contribution of the underlying drivers (for a review on detecting and 25 attributing flood hazard changes in Europe see Hall et al., 2014).
The large majority of studies on past changes in flood hazard analysed the mean flood behaviour, using, for instance, the Mann-Kendall test to detect gradual changes or the Pettitt test for step changes in the mean or median annual flood (e.g. Petrow Villarini et al., 2011;Mediero et al., 2014;Mangini et al., 2018). This focus may be misleading, since changes in large floods may differ from those in the average behaviour. An illustrative example is the Mekong River, where studies 30 found negative trends in the mean flood discharge, whereas the public perception suggested that the frequency of damaging floods had increased in the past decades. Delgado et al. (2009) resolved this mismatch by analysing the temporal change in flood discharge variability. They found an upward trend in interannual variability which outweighed the decreasing mean behaviour leading to contrasting trends in the mean flood and rare floods. This change in flood variability could be attributed to changes in the Western Pacific monsoon (Delgado et al., 2012). Another recent example is the large-scale study of Bertola It has been widely acknowledged that drivers can differently affect small and large floods (e.g. Hall et al., 2014) and yet the focus has been mainly on changes in the mean flood behaviour. One reason for this may be the ability of quantifying changes in the mean more robustly than those of larger floods. However, both from theoretical and practical perspectives, 40 detection and attribution of flood changes as a function of the return period are of considerable interest for understanding how the non-linearity in the hydrological system plays out and for providing guidance for flood risk management. The shape of the flood frequency curve and its changes in time are a reflection of the interplay between atmospheric processes and catchment state (soil moisture and snow), with different characteristics depending on the region, climate and runoff generation processes (Blöschl et al., 2013). of extreme floods in 171 Mediterranean basins were likely caused by decreasing soil moisture. The relationship between the flow rate and the initial saturation state of the soil is often non-linear and the effect of antecedent soil moisture strongly depends on soil type and geology. The sensitivity of floods to initial soil moisture depends on flood magnitude, and runoff generation is more influential for smaller events. Vieux et al. (2009) analysed several watersheds in the Korean peninsula with a distributed hydrologic model and found that the sensitivity of the watershed response to the initial degree of saturation is dependent on 70 event magnitude. Zhu et al. (2018) simulated peak discharges for return periods of 2 to 500 years for several sub-watersheds in Turkey River in the Midwestern United States and found that antecedent soil moisture modulates the role of rainfall structure in simulated flood response, particularly for smaller events. Grillakis et al. (2016) analysed flash flood events in two Greek and one Austrian catchments, and found higher sensitivity of the smallest flood events to initial soil moisture, compared to larger events. These results are consistent throughout the different regions and climates, confirming that the effects of initial 75 soil moisture on flood response depend on flood magnitude.
Snow storage and melt are other important factors that modulate flood response in temperate and cold regions. Snowmelt represents the dominant flood generating process in northeastern Europe and rain-on-snow is relevant for regions in central and northwestern Europe (Berghuijs et al., 2019;Kemter et al., 2020). It was observed that in catchments where snowmelt and rain-on-snow are the dominant flood generating processes, the shape of the flood frequency curve is likely to flatten out 80 at large return periods due to the upper limit of energy available for melt (Merz and Blöschl, 2003;Merz and Blöschl, 2008).
Reduction in spring and summer snow cover extents have been detected as a result of increasing spring temperature in the Northern Hemisphere (Estilow et al., 2015). Several studies in regions dominated by snowmelt-induced peak flows reported decrease in extreme streamflow and earlier spring snowmelt peak flows, likely caused by increasing temperature (Madsen et al., 2014). The effects of changing snow storage and melt on the flood frequency curves likely depend on flood regimes and mixing 85 of different flood generating processes in the catchments. For example, in Carinthia, in the very south of Austria, the major floods tend to occur in autumn, and spring snowmelt floods represent a smaller fraction of events with small magnitude (Merz and Blöschl, 2003). Hence, changes in snow cover and melt are expected to mainly affect the smaller floods in these climates.
In contrast, in northeastern Europe where snowmelt is the dominant flood generating process of both small and large floods, the effects of decreasing snowmelt are likely important for the entire flood frequency curve.

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Overall, the contributions of different drivers to flood changes as a function of return period are currently not well understood. This is partly due to detection and attribution studies focusing generally on the mean annual flood. Several studies applied nonstationary frequency analysis to attribute past flood changes to potential drivers. These studies typically allowed the parameters of the probability distribution of floods to vary in time, using time-varying climatic covariates (e.g. Prosdocimi et al., 2014;Šraj et al., 2016;Steirou et al., 2019) and, more rarely, catchment and river covariates (e.g. López and Francés, 2013;Silva 95 et al., 2017;Bertola et al., 2019). They attempted to identify and select covariates in the non-stationary model that provide a better fit to the flood data than the alternative stationary model. However, these studies still aimed at attributing changes in the mean annual flood and did not separate the effects of drivers on floods associated with different return periods.
The aim of this paper is to address two science questions: (a) Is it possible to identify the relative contributions of different drivers to observed flood changes across Europe as a function of the return period, and if so, (b) what is the magnitude and sign 100 of these contributions across Europe? Regarding the first question, one possible outcome is for the data to provide evidence that the relative contributions differ, or alternatively, the data may contain insufficient information to separate the effects by return period. Regarding the second question, the interest resides in understanding the relative importance of potential drivers as a function of return period, provided that such information can be inferred from the data. In this study, we adopt a non-stationary flood frequency approach to attribute observed flood changes to potential drivers, used as covariates of the parameters of 105 the regional probability distribution of floods. Extreme precipitation, antecedent soil moisture and snowmelt are the potential drivers considered. The relative contribution of the different drivers to flood changes is quantified through the elasticity of flood quantiles with respect to each driver.

Methods
2.1 Regional driver-informed model 110 In this study, we use non-stationary flood frequency analysis to attribute observed flood changes across Europe (see e.g., Blöschl et al., 2019;Bertola et al., 2020) to potential drivers, used as time-varying covariates. In the spirit of Bertola et al. (2020), we formulate the flood model as a regional Gumbel model. The Gumbel distribution has two parameters (i.e. the location µ and scale σ parameters) and its cumulative distribution function is: The two Gumbel parameters can be inferred from knowledge of two flood quantiles, e.g., the 2-year and the 100-year flood. We adopt here the same alternative parameters as in Bertola et al. (2020), i.e. the 2-year flood q 2 and the 100-year growth factor x 100 . The T-year flood can be obtained with the following relationship: where a T = (y T − y 2 )/(y 100 − y 2 ), with y being the Gumbel reduced variate, which is related to the return period by: https://doi.org/10.5194/hess-2020-396 Preprint. Discussion started: 18 August 2020 c Author(s) 2020. CC BY 4.0 License.
We adopt the following regional change model accounting for catchment area: ln q 2 = ln α 20 + γ 20 ln S + α 21 ln X 1 + α 22 ln X 2 + α 23 ln X 3 + ε (4a) Where X 1 , X 2 and X 3 are three covariates (i.e. time series of the potential drivers of flood change), S is catchment area and the Greek symbols represent the parameters of the model to be estimated. The ε term, here assumed normally distributed, accounts for additional local variability (i.e. not explained by catchment area and the covariates) of q 2 .
The elasticity of the generic flood quantile q T with respect to the covariate X i is defined as: It represents the percentage change in q T , due to a 1% change in X i , i.e., how sensitive flood peaks are to changes in the drivers. However, the elasticity alone does not tell how much the flood quantiles have actually changed (in time) due to observed changes of the drivers. Hence, we define the contribution of X i to the changes in q T as: It represents the percentage change in q T , due the actual change in X i . The total change in q T due to the changes in the drivers,

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assuming that the contributions are additive, is: A measure of relative contribution of X i to the change in q T is expressed here by: In the change model, the flood and covariate data are pooled and used simultaneously to attribute any observed changes in floods to their drivers. This pooling increases the robustness of the estimates (see e.g., Viglione et al., 2016) but requires an assumption of homogeneity. Specifically, we assume here that for a given return period and catchment scale, the elasticities of the flood discharges to their drivers are uniform within the region. We do allow the drivers to vary between catchments.
We frame the estimation problem in Bayesian terms through a Markov chain Monte Carlo (MCMC) approach, using the Spatial correlation of floods is not directly accounted for in the proposed regional change model of Sect. 2.1 and it may result in underestimated sample uncertainties (see e.g., Stedinger, 1983;Castellarin et al., 2008;Sun et al., 2014). Here, we adopt an approach proposed by Ribatet et al. (2012) and based on the work of Smith (1990), consisting in a magnitude adjustment to the likelihood function in a Bayesian framework, which accounts for the overall dependence in space and allows to obtain reliable credible intervals. The adjusted likelihood is defined as: where L is the likelihood under the assumption of spatial independence, θ is the vector of unknown parameters and k is the magnitude adjustment factor to be estimated, such as 0 < k ≤ 1 (see Appendix A). The magnitude adjustment factor k represents the overall reduction of hydrological information in the data caused by the presence of spatial correlation and results in an inflated posterior variance of the parameters. If floods at different sites are spatially independent, k is 1; on the contrary, 160 if floods are strongly cross-correlated, k assumes values close to 0. In this latter case, the sample uncertainty resulting from the adjusted likelihood will be larger, compared to the model where spatial cross-correlation is not accounted for. For further details on the adjustment to the likelihood and its application to hydrological data see Smith (1990), Ribatet et al. (2012) and Sharkey and Winter (2019).  3.2. It consists of nine cells, corresponding to 600 km × 600 km, whose central cell is black shaded. Three regions analysed in Sect. 3.3, respectively located in northwestern, southern and eastern Europe, are shown with coloured circles and the shaded regions represent their central cells.

Drivers of flood change
Because stations with substantial artificial alterations are not included in the database, in this study we consider three potential 175 climatic drivers of flood change: (i) extreme precipitation, (ii) antecedent soil moisture and (iii) snowmelt. For each driver we obtain catchment-averaged time series, as described in detail in the following paragraphs, which are used as covariates in the regional model of Sect. 2.1. Unlike Viglione et al. (2016), scale dependence is here accounted for by the data, as we use local (i.e. catchment-averaged) covariates, and not directly into the model.
Daily series of catchment-averaged precipitation between 1960 and 2010 are calculated for each hydrometric station from the daily gridded E-OBS precipitation and the catchment boundaries. For each station we identify a window around the average date of occurrence of floodsD, in which extreme precipitation is considered to be typically relevant for the generation of the annual peaks. The width of the window w is set between 90 and 360 days and it is taken proportional to 1 − R, with R being the concentration of the date of occurrence around the average date, through the following equation: D and R are obtained with circular statistics (see Appendix B). The window of dates is centred aroundD, in a way that two thirds of the window occur before the average date of occurrence of floods (as shown in Fig. 2 for an example series in one example year). For each year in the period of interest, we calculate the 7-day maximum precipitation within the identified window (which varies between catchments but is fixed between years).

Antecedent soil moisture index
An index of antecedent soil moisture is obtained from daily catchment-averaged precipitation. For each year and each station, we calculate the 30-day precipitation preceding the 7-day window identified for extreme precipitation above. We use this index (for brevity, hereinafter referred to as 'antecedent soil moisture') based on precipitation instead of modelled soil moisture, as in Blöschl et al. (2019), in order to more strongly rely on observational data.

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Snowmelt Similar to precipitation, daily series of catchment-averaged temperature between 1960 and 2010 are obtained for each hydrometric station. We calculate daily series of catchment-averaged snowmelt according to a simple degree-day model (Parajka and Blöschl, 2008) as a function of mean daily air temperature T A and precipitation P : Where M and P s are the daily snowmelt depth and snow water equivalent storage, DDF is the degree day factor and T m , T s and T R are the temperature thresholds that control the occurrence of melt, snow and rainfall, respectively. Here we assume T m = T s = 0 • C, T R = 2.5 • C and DDF = 2.5 mm day −1 • C −1 (Parajka and Blöschl, 2008;He et al., 2014). For each station, the time series of 7-day maximum snowmelt is obtained from daily snowmelt, using the same procedure illustrated above for 205 the case of extreme precipitation.
As in Bertola et al. (2019), this study aims at attributing flood changes to the long-term evolution of the covariates rather than their year-to-year variability. For this reason, we smooth the annual series of the drivers with the locally weighted polynomial regression LOESS (Cleveland, 1979) using the R function loess. The subset of data over which the local polynomial regression is performed is 10 years (i.e. 10 data-points of the series) and the degree of the local polynomials is set equal to 0, which is 210 equivalent to a weighted 10-year moving average.

A priori on model parameters
In the attribution analysis we use informative priors on the parameters controlling the relationship between flood and covariate changes (see Bertola et al., 2019). This is done because we do not want to use the time patterns of the covariates X i only to discriminate between drivers, which may lead to spurious correlations, but to hydrologically 'inform' the attribution analysis.
covariates considered in this study, the elasticities of flood quantiles to the drivers (defined in Eq. 5) are expected to be positive (i.e. we expect the changes in X i and q T to have the same sign). For T=2 and 100 years, this translates respectively into: Eq. 12a represents the lower limit for the elasticity parameters of q 2 . The lower limit for α gi is obtained from Eq. 12b and depends on α 2i and on the growth factor: For simplicity, we assume q 100 = 2q 2 as a reasonable approximation valid for Europe (Blöschl et al., 2013;Alfieri et al., 2015), and we simplify Eq. 13 to: The prior distributions of α 2i and on α gi are modelled as normal distributions N (0, 2) with truncated lower tail, as summarised in Tab. 1. For the remaining parameters we set an improper uniform prior distribution. Elasticity of x 100 to X 3 −2α 23 Truncated normal Table 1. A priori on model elasticity parameters 2.6 Regional analyses  Fig. 1 and each of the considered regions is composed of nine adjacent cells, (e.g. the black bordered region in Fig. 1). In each region, we estimate the elasticity of q 2 and q 100 to the drivers X i and the contribution of each driver to flood changes, obtained by multiplying the elasticity by the average driver trend in the region (Eq. 6). In regions where the average 7-day maximum snowmelt is less than km cell of the region (e.g. the shaded cell in the black bordered region in Fig. 1). The rationale of the homogeneity assumption is that the spatial windows, given their size, have rather homogeneous climatic conditions (and hence flood generation processes and processes driving flood changes) relative to the overall variability within Europe. The results of this analysis are shown 240 in Sect. 3.2. In Sect. 3.3, the elasticities of flood quantiles to the drivers and their contributions to flood change are further analysed as a function of the return period, for three regions located respectively in northwestern, southern and eastern Europe (see Fig. 1). are observed in southern countries and in few spots in central Europe (Fig. 3d). Similar spatial patterns appear for antecedent soil moisture ( Fig. 3b and 3e), but the negative changes tend to be more widespread and with stronger (negative) magnitude.

Drivers of flood change
Mean snowmelt is largest in northeastern Europe and in the Alpine region (Fig. 3c). Its changes are mostly negative across all Europe, with the exception of the very North and few isolated spots (Fig. 3f).

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The obtained time series of catchment-averaged extreme precipitation, antecedent soil moisture and snowmelt are used as covariates in the regional driver-informed model. Figure 4 shows maps of the elasticity of the 2-year flood q 2 and the 100-year flood q 100 to each of the three drivers, as defined in Eq. 5, resulting from fitting the regional model to the pooled flood and covariate data in moving windows across Europe. The value of the posterior median of the elasticities is shown together with the 90% credible bounds. The elasticity of q 2 to extreme precipitation (Fig. 4a) is large (0.6 to 1.5) in western, central and southern

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Europe and lower values (0 to 0.25) are observed in northeastern Europe. Similar values of elasticity to extreme precipitation are observed for the 100-year flood across Europe (Fig 4b), with small differences in northeastern Europe. This means that the elasticity of flood quantiles to extreme precipitation does not vary much with return period. In contrast, the elasticity of flood quantiles to soil moisture decreases with return period (Fig. 4b and 4e) and it is largest in southern Europe (0.25 to 0.6).
Overall, the elasticities of q 2 and q 100 to soil moisture are smaller than those to extreme precipitation. The elasticity of floods 265 to snowmelt is largest in northeastern Europe ( Fig. 4c and 4d), where values above 1 are observed (i.e. a change of 1% in 12 https://doi.org/10.5194/hess-2020-396 Preprint. Discussion started: 18 August 2020 c Author(s) 2020. CC BY 4.0 License.
snowmelt translates into a change in flood quantiles larger than 1%). In northeastern Europe the elasticities of q 2 and q 100 to snowmelt are similar, while in central Europe and the Balkans they decrease with the return period. period. The contributions of snowmelt to changes in q 2 and q 100 are predominantly negative and marked in Eastern Europe, with small differences towards smaller contributions in absolute values with return period (Fig. 5c and 5f). In contrast, snowmelt contributes to positive flood changes in Scandinavia, and to a lesser extent for q 100 than for q 2 . Overall the uncertainties associated with the contribution of the drivers to changes in q 100 do not seem to increase much compared to q 2 . In order to further investigate the differences in terms of (absolute) contributions of the drivers to changes in large (i.e. q 100 ) 280 versus small floods (i.e. q 2 ), we compute for each driver the ratio between these two quantities (Fig. 6). In the case of extreme precipitation (Fig. 6a), the ratio between its contributions to changes in q 100 and q 2 is between 0 and 1 in the Atlantic region, Spain, Italy, the Balkans, southern Germany, Austria and Finland, i.e., in these regions the contribution of extreme precipitation to changes in q 100 is smaller, in absolute value, compared to changes in q 2 . In southern France, eastern Europe and Turkey the opposite is observed (i.e. the ratio is larger than 1). Antecedent soil moisture and snowmelt generally contribute less to changes 285 14 https://doi.org/10.5194/hess-2020-396 Preprint. Discussion started: 18 August 2020 c Author(s) 2020. CC BY 4.0 License.
in q 100 compared to q 2 ( Fig. 6b and 6c). Large uncertainties in the ratio of elasticities are observed in northeastern Europe, in the case of extreme precipitation and antecedent soil moisture ( Fig. 6a and 6b), and in southern Europe, in the case of snowmelt (Fig. 6c), and they result from values of the contribution of the drivers to q 2 that are close to zero in these regions (see Fig. 5). Figure 6. Same as Fig. 4, but for the ratios of the contributions of extreme precipitation (a), antecedent soil moisture (b) and snow melt (c) to changes in q100 relative to q2. Values below 1 (red colour) indicate that the contribution of the driver to q100 is smaller than the contribution to q2; values above 1 (blue colour) indicate that the contribution of the driver to q100 is larger than the contribution to q2.
Finally, for each region we obtain the relative contribution of the three drivers to changes in q 2 and q 100 , as defined in Eq. 8 (Fig. 7). The relative contribution of extreme precipitation is the largest of all the drivers in most of western and central Europe 290 for both q 2 and q 100 ( Fig. 7a and 7d). The relative contribution decreases somewhat with return period in northwestern Europe, while the opposite is the case in the South. In southern Europe antecedent soil moisture has the largest relative contribution to changes in q 2 (Fig. 7b) and its relative importance tends to decrease for more extreme floods (Fig. 7e). The relative contribution of snowmelt to flood changes clearly prevails in eastern Europe, with slightly decreasing strength for the higher return period.

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In this section we select three example regions among those analysed in Sect. 3.2, located respectively in northwestern, southern and eastern Europe (see Fig. 1). For these three regions we further show in Fig. 8 the elasticities of floods to the drivers (first row), the contributions (second row) and relative contributions (third row) of the drivers to flood change, as a function of the return period. In the regions located in northwestern and southern Europe, snowmelt is excluded from the potential drivers as it does not represent a relevant process for most of the catchments in these regions (see Fig. 3c). In the region in northwestern 300 Europe extreme precipitation and antecedent soil moisture contribute positively to flood change, with extreme precipitation representing the most important driver. Its contribution to flood trends decreases with return period, while the contribution stays almost constant in the case of antecedent soil moisture ( Fig. 8d and 8g). In the region in southern Europe extreme precipitation and antecedent soil moisture represent both important drivers. The elasticity of floods to extreme precipitation 16 https://doi.org/10.5194/hess-2020-396 Preprint. Discussion started: 18 August 2020 c Author(s) 2020. CC BY 4.0 License.
is larger than that to antecedent soil moisture (Fig. 8b). However, antecedent soil moisture contributes (negatively) to a larger 305 extent to flood changes for small return periods (i.e. T=2-10 years). Its contribution decreases in absolute values with increasing return period (Fig. 8e). For more extreme events (T>10 years) the relative contribution of extreme precipitation increases and becomes comparable to that of antecedent soil moisture (Fig. 8h). In the region in eastern Europe snowmelt is clearly the dominant driver at all return periods (Fig. 8c, 8f and 8i).
In this study, we attribute the changes in flood discharges that have occurred in Europe during the period 1960-2010 (Blöschl et al., 2019;Bertola et al., 2020) to potential drivers as a function of the return period, while previous detection and attribution studies have generally focused on the mean flood behaviour. In particular, we compare the relative contribution of extreme precipitation, antecedent soil moisture and snow melt to changes in the median and the 100-year flood. The attribution study is framed in terms of a non-stationary flood frequency analysis and the parameters of the distribution are estimated in a regional 315 context with Bayesian inference.

4.1
Is it possible to identify the relative contributions of different drivers to q 100 changes as compared to q 2 changes?
Our results suggest that in northwestern and eastern Europe, changes in small and large floods are driven mainly by one single driver, which dominates at all return periods. In northwestern Europe, extreme precipitation contributes to changes in both q 2 and q 100 for the most part and the contribution of antecedent soil moisture is of secondary importance. Similarly, in 320 eastern Europe, snowmelt clearly drives flood changes at all return periods. In southern Europe both antecedent soil moisture and extreme precipitation significantly contribute to flood changes and their relative importance depends on the return period.
Antecedent soil moisture contributes the most to changes in small floods (i.e. T=2-10 years), while the two drivers contribute with comparable magnitude to changes in more extreme events (T>10 years). Given the relative driver contributions and their credible bounds obtained in the analysis, the findings suggest that is indeed possible to identify the relative contributions to q 2 325 and q 100 clearly.
4.2 What is the nature (sign and magnitude) of these contributions?
The contribution of extreme precipitation is positive in northwestern Europe (about 3.3 to 2.8% per decade in Fig. 8) and decreases slightly with return period in the region analysed in Sect. 3.3. In contrast, in the region selected in southern Europe extreme precipitation contributes to 37 to 45% of the negative flood changes (corresponding to -2.2 to -1.8% per decade), previous studies (Blöschl et al., 2019;Bertola et al., 2020). This discrepancy points to other drivers not accounted for in the presented model, such as river regulation effects (Arheimer and Lindström, 2019), or non linear relationships between the drivers not captured by the model.

Model assumptions
Prior information on the elasticities is used in order to 'inform' the attribution analysis, based on hydrological reasoning and the literature. Specifically, the prior distribution of the elasticities of q 2 and q 100 to the drivers are assumed positive. This is because any changes in the considered covariates are expected to translate into flood changes with the same sign. In practice, the prior distribution of the elasticity of q 100 is reflected in a lower bounded prior distribution on the elasticity of the growth factor x 100 , which depends on the ratio between q 100 and q 2 (Sect. 2.5). For simplicity, we assume this ratio approximately 345 equal to 2. This assumption is reasonably valid for humid catchments (see e.g., Blöschl et al., 2013) and is in overall agreement with flood maps of the mean annual flood and q 100 in Europe presented by Alfieri et al. (2015). However, in arid regions, larger values of this ratio (e.g. 4, see Blöschl et al., 2013) would be more appropriate (corresponding to stricter priors on the elasticity of the growth factor) because the flood frequency curves tend to be steeper.
We fitted the change model of Sect. 2.1 to the pooled flood and covariate data of several regions across Europe, where 350 elasticities of flood quantiles to their drivers are assumed homogeneous. This assumption is reasonable because of the spatial proximity of the catchments that is reflected into similar climatic conditions, flood generation processes and processes driving flood changes. The attribution analysis is thereby performed at the regional scale, where average regional contributions of the decadal changes in the drivers to average regional trends in flood quantiles are estimated. Figure 8 shows the contributions of the drivers to flood changes as a function of the return period for three regions selected respectively in northwestern, southern 355 and eastern Europe. Similar results would be obtained by fitting the model to larger regions over northwestern, southern and eastern Europe, that present comparatively homogeneous flood regime changes and processes driving flood changes (e.g. the three macro-regions in Blöschl et al., 2019;Bertola et al., 2020).
Overall the obtained uncertainties associated with the contribution of the drivers to changes in q 100 do not seem to increase much compared to q 2 , while a relevant increase would be reasonably expected. These results are valid under the assumption 360 of the adopted model (i.e. Gumbel distribution) which may be too stringent. The model assumptions could be relaxed (e.g. adopting a Generalized Extreme Value distribution) in order to allow for larger model flexibility.
Spatial cross-correlation of floods at different sites is taken into account through an approach based on a magnitude adjustment to the likelihood. This results in larger uncertainties of the posterior distribution of the estimated parameters, compared to the case where floods are considered spatially independent.

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As already noted, one of the main assumptions in our analysis is that the three drivers, i.e., extreme precipitation, soil moisture and snowmelt, are the only candidates for explaining river flood changes. The effects of other drivers not accounted for in this study, such as land cover change or river regulation, are probably not very large at the scale of Europe as we are focusing on catchments with minimum alteration. However, in contexts where anthropogenic alterations are important it will be useful to extend the analysis for such effects. This attribution analysis may be repeated with catchment (e.g. land-use or 370 land-cover changes) and river drivers (e.g. construction of reservoirs in the catchment) in addition to atmospheric covariates, if detailed information about changes in land-use/land-cover and river structures were available for European catchments and flood data of affected stations were collected.

Conclusions
This study complements recent research on past changes in European floods by formally attributing the detected trends to 375 potential drivers (i.e., extreme precipitation, antecedent soil moisture and snowmelt) as function of return period. The proposed method allows to identify the relative contribution of different drivers to changes in flood quantiles and to estimate the sign and magnitude of these contributions. The results show that in northwestern and eastern Europe changes in both the 2-year and the 100-year flood are driven by a single driver only (i.e. respectively extreme precipitation and snowmelt), while in southern Europe two drivers contribute to flood changes (i.e. soil moisture and extreme precipitation), with different relative 380 contributions depending on the return period. Even though this study focuses on observed flood changes, the understanding of past processes is a fundamental step for the prediction of flood changes in future climate scenarios.
Data availability. The flood discharge data used in this paper can be obtained from the Supplement of Blöschl et al. (2019)

Appendix A: Adjustment to the likelihood
Under the assumption of spatial independence of the data, the asymptotic distribution of the maximum likelihood estimatorθ of the independence likelihood is:θ ∼ N θ 0 , n −1 H −1 V H −1 , where θ 0 is the true value of θ and H −1 V H −1 is the modified 390 covariance matrix, whereH = −E∇ 2 l θ 0 , y and V = Cov∇l θ 0 , y . If the assumption of spatial independence is correct, we have that H = V . In Sect 2.2 we described an approach, proposed by Ribatet et al. (2012), that enables to account for spatial cross-correlation in spatial datasets and consists in an overall adjustment to the likelihood. In this analysis we adopted a magnitude adjustment, through a factor k (Eq. 9). Ribatet et al. (2012) proposed to estimate k by setting: where p is the number of parameters in the independence likelihood and λ i are the eigenvalues of the matrix H −1 V . The matrix H is approximated by the observed information matrix ∇ 2 l θ , y and V is estimated by decomposing the likelihood into independent yearly contributions.
the average date are obtained with circular statistics, by conversion of the date of occurrence of a flood in the year i into an angular value D i : Where n is the number of peaks registered at that station, m i is the number of days in the year i andm is the average number of days per year. When floods occur equally throughout the year R = 0, while R = 1 when floods always occur on the same 410 date.
Author contributions. All co-authors designed the overall study. MB performed the analysis and prepared the paper. All co-authors contributed to the interpretation of the results and writing of the paper.