In this study, we use non-stationary flood frequency analysis to attribute observed flood changes across Europe see, for example, to potential drivers, used as time-varying covariates. In the spirit of , 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 1FX(x)=p=e-e-x-ξσ. The two Gumbel parameters can be inferred from knowledge of two flood quantiles, for example, the 2-year and the 100-year flood. Flood quantiles q, associated with fixed annual exceedance probabilities 1-p, are expressed here in terms of return periods T, through p=1-1/T. We adopt here the same alternative parameters as in , i.e. the 2-year flood q2 and the 100-year growth factor x100′. The relationships linking q2 and x100′ to the Gumbel parameters are 2q2=ξ+σy2x100′=σ(y100-y2)/(ξ+σy2), where y2=-ln⁡(-ln⁡(0.5)) and (y100-y2)=-ln⁡(-ln⁡(0.99))+ln⁡(-ln⁡(0.5)).

The T-year flood can be obtained with the following relationship: 3qT=q21+aTx100′, where aT=(yT-y2)/(y100-y2), with y being the Gumbel reduced variate, which is related to the return period by 4yT=-ln⁡-ln⁡1-1T=-ln⁡-ln⁡p.

We adopt the following regional change model accounting for catchment area (S): 5ln⁡q2=ln⁡α20+γ20ln⁡S+α21ln⁡X1+α22ln⁡X2+α23ln⁡X3+εln⁡x100′=ln⁡αg0+γg0ln⁡S+αg1ln⁡X1+αg2ln⁡X2+αg3ln⁡X3,ε∼N(0,σ) where X1, X2 and X3 are three covariates (i.e. time series of the potential drivers of flood change), and the α and γ terms represent regional model parameters to be estimated. The ε term, here assumed normally distributed, is a station-specific error term that accounts for additional local variability (i.e. not explained by catchment area and the covariates) of q2. Similar to the index flood method of and , we assume here that the growth curve x100′ is the same across all sites within the region (i.e. it depends on catchment area and the covariates only), while the median flood q2 (the index flood) is allowed to vary between sites, through the error term ε.

The elasticity of the generic flood quantile qT with respect to the covariate Xi is defined as 6ST,Xi=XiqT∂qT∂Xi=α2i+αgi1-11+aTx100′. It represents the percentage change in qT, due to a 1 % change in Xi, i.e. how sensitive flood peaks are to changes in the drivers. However, the elasticity alone does not tell us how much the flood quantiles have actually changed (in time) due to observed changes of the drivers. Hence, we define the contribution of Xi to the changes in qT as 7CT,Xi=XiqT∂qT∂Xi⋅1XidXidt. It represents the percentage change in qT due to the actual change in Xi. The total change in qT due to the changes in the drivers, assuming that the contributions are additive, is 81qTdqTdt=∑iCT,Xi=∑iXiqT∂qT∂Xi⋅1XidXidt. A measure of relative contribution of Xi to the change in qT is expressed here by 9RT,Xi=abs(CT,Xi)∑iabs(CT,Xi), where ∑iRT,Xi=1.

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, for example, 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 R package rStan , which makes use of a Hamiltonian Monte Carlo algorithm to sample the posterior distribution . For each inference, we generate four chains of 10 000 simulations each, with different initial values, and we check for their convergence. We use prior information on the model parameters to constrain their estimation to hydrologically plausible values (see Sect. ).

Spatial correlation of floods is not directly accounted for in the proposed regional change model of Sect. , and it may result in underestimated sample uncertainties see, for example,. Here, we adopt an approach proposed by and based on the work of , consisting in a magnitude adjustment to the likelihood function in a Bayesian framework, which accounts for the overall dependence in space and allows reliable credible intervals to be obtained. The adjusted likelihood is defined as 10L*θ,y=Lθ,yk, 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 ). 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, 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 in which spatial cross-correlation is not accounted for. For further details on the adjustment to the likelihood and its application to hydrological data, see , and .

Consistent with and , we analyse long series of annual maximum discharges between 1960 and 2010, from 2370 hydrometric stations in 33 European countries (https://github.com/tuwhydro/europe_floods, last access: 9 April 2020). Stations affected by strong artificial alterations (such as large reservoirs in the proximity of the gauges) are not included in this database . The location of the stations is shown in Fig. . Their contributing catchment areas range from 5 to 100 000 km2, and the median record length is 51 years. The density of stations in the database is highest in central Europe and lowest in eastern and southern Europe, where time series are generally shorter (Fig. ). The catchment boundaries relative to each hydrometric station are derived from the CCM River and Catchment Database . Daily gridded precipitation and mean surface temperature are obtained from the E-OBS dataset (version 18.0e, resolution 0.1∘; ). It covers the area 25–71.5∘ N × 25∘ W–45∘ E for the period 1950–2018.

Because stations with substantial artificial alterations are not included in the database, in this study we consider three potential 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. . As the interest of this study resides in attributing flood changes to the long-term evolution of the drivers, we use average flood seasonality and its variability to identify time windows in which drivers are typically relevant for the generation of the annual peaks, rather than pairing floods with the corresponding event precipitation (which would be instead relevant for event attribution). Unlike in , scale dependence is here accounted for by the data, as we use local (i.e. catchment-averaged) covariates.

As in , 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 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 equivalent to a weighted 10-year moving average.

In the attribution analysis we use informative priors of the parameters controlling the relationship between flood and covariate changes (i.e. the elasticities; see Eq. ). This is done because we do not want to use the time patterns of the covariates Xi only to discriminate between drivers, which may lead to spurious correlations, but to “inform” the attribution analysis based on hydrological knowledge. Therefore, we set a priori constraints on the model parameters, based on qualitative reasoning and on prior literature. The changes in flood quantiles are expected to be caused by changes of the same sign in the drivers, given the covariates considered in this study (i.e. extreme precipitation, antecedent soil moisture and snowmelt). For example, increasing floods can be attributable to increasing extreme precipitation, while decreasing precipitation cannot reasonably cause increasing floods (which would correspond to a negative elasticity). In other words, we expect the elasticity of qT to Xi to be positive. For T=2 and 100 years, this translates respectively into 13aα2i>013bα2i+αgi1-11+x100′>0. Eq. represents the lower limit for the elasticity parameters of q2. The lower limit for αgi is obtained from Eq. () and depends on α2i and on the growth factor: 14αgi>-α2i1-q2q100. For simplicity, we assume q100=2q2 as a reasonable approximation valid for Europe , and we simplify Eq. () to 15αgi>-2α2i. The prior distributions of α2i and on αgi are modelled as normal distributions N0,2 with a truncated lower tail, as summarised in Table . For the remaining parameters, we set an improper uniform prior distribution.

Following the spatial moving window approach of , we identify several regions of size 600 km × 600 km across Europe, which overlap by 200 km in both directions. We fit the regional flood change model of Sect.  to pooled flood and covariate data of sites within each region. The resulting 200 km × 200 km grid cells are shown in Fig. , and each of the considered regions is composed of nine adjacent cells, (e.g. the black bordered region in Fig. ). All station-years contribute to the likelihood, and the likelihood is corrected using the magnitude adjustment to account for spatial cross-correlation between sites. The rationale behind the homogeneity assumption is that the spatial windows, given their size, are characterised by rather homogeneous climatic conditions relative to the overall variability within Europe.

In each region, we estimate the elasticity of q2 and q100 to the drivers Xi and the contribution of each driver to flood changes, obtained by multiplying the elasticity by the average driver trend in the region (Eq. ). In regions where the average 7 d maximum snowmelt is less than 2 mm per day, only extreme precipitation and antecedent soil moisture are considered as potential drivers (i.e. Eq.  is modified by removing the contribution of X3). The resulting elasticity and contribution are plotted in the central 200 km × 200 km cell of the region (e.g. the shaded cell in the black bordered region in Fig. ). The results are shown for a hypothetical catchment area S = 1000 km2, corresponding to a medium-sized catchment. This is because it is of interest to show average driver contributions to changes in flood quantiles within each region, rather than model results corresponding to an existing single catchment in the region. 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. The results of this analysis are shown in Sect. . In Sect. , 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. ).

Time series of catchment-averaged (i) extreme precipitation, (ii) antecedent soil moisture and (iii) snowmelt are obtained for each hydrometric station for the period 1960–2010, as described in Sect. . Figure  shows maps of the mean value and the change of these drivers for each station in the period of interest. Extreme precipitation (Fig. a) exhibits its largest mean values in central and western Europe, particularly in the Alpine region and on the western Atlantic coast. Positive changes of extreme precipitation are observed in the Alpine region, northwestern and central Europe, Scandinavia and Poland; negative changes are observed in southern countries and in few spots in central Europe (Fig. d). Similar spatial patterns appear for antecedent soil moisture (Fig. b and e), but the negative changes tend to be more widespread and with stronger (negative) magnitude. Mean snowmelt is largest in northeastern Europe and in the Alpine region (Fig. c). Its changes are mostly negative across all Europe, with the exception of the very north and a few isolated spots (Fig. f).

The obtained time series of catchment-averaged extreme precipitation, antecedent soil moisture and snowmelt are used as covariates in the regional driver-informed model of Sect. . Figure  shows maps of the elasticity of the 2-year flood q2 and the 100-year flood q100 to each of the three drivers, as defined in Eq. (), resulting from fitting the regional model to the pooled flood and covariate data in moving windows across Europe. The elasticities are measured here in percent per percent (%/%) and represent the percentage change in qT, due to a 1 % change in Xi, i.e. how sensitive flood peaks are to changes in the drivers. The value of the posterior median of the elasticities is shown together with the 90 % credible bounds, which represent a measure of the uncertainty associated with the estimate and take into account the different density of stations across Europe (i.e. larger uncertainties are typically observed in data-scarce regions).

The elasticity of q2 to extreme precipitation (Fig. a) is large (0.6 to 1.5) in western, central and southern Europe (indicating that the 2-year flood increases by 0.6 % to 1.5 % if extreme precipitation increases by 1 %), and lower values (0 to 0.25) are observed in northeastern Europe (i.e. the 2-year flood increases by 0 % to 0.25 % following a 1 % increase in extreme precipitation). Similar values of elasticity to extreme precipitation are observed for the 100-year flood across Europe (Fig. b), 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 (i.e. q2 increases more than q100 if soil moisture increases by 1 %), and it is largest in southern Europe (0.25 to 0.6; Fig. b and e). Overall, the elasticities of q2 and q100 to soil moisture are smaller than those to extreme precipitation. The elasticity of floods to snowmelt is largest in northeastern Europe (Fig. c and d), where values above 1 are observed (i.e. a change of 1 % in snowmelt translates into a change in flood quantiles larger than 1 %). In northeastern Europe the elasticities of q2 and q100 to snowmelt are similar, while in central Europe and the Balkans they decrease with the return period.

Figure  shows maps of the contributions of each of the three drivers to changes in q2 and q100, as defined in Eq. (). They are obtained by multiplying the elasticities of flood quantiles to the drivers by the average changes (in percent per decade) in the drivers in each region over the period 1960–2010 (Eq. ). They represent the change in flood quantiles, in percent per decade, caused by the change in a specific driver. Extreme precipitation (Fig. a and d) contributes to positive flood changes in northwestern and central Europe and to negative flood changes in southern and eastern Europe. The absolute value of the contributions of extreme precipitation appears to slightly decrease when moving from q2 to q100. Antecedent soil moisture contributes mostly to negative flood changes in southern Europe, and the magnitude of this contribution is smaller in absolute values for large floods than for the median floods (Fig. b and e). Snowmelt (Fig. c and f) contributes to marked negative changes in q2 and q100 in eastern Europe and to positive flood changes in northern Europe. We overall observe smaller contributions in absolute values to changes in q100 than q2. In data-scarce regions the credible bounds tend to be larger; i.e. the attribution results have larger uncertainties. Overall the uncertainties associated with the contribution of the drivers to changes in q100 do not seem to increase much compared to q2.

In order to further investigate the differences in terms of (absolute) contributions of the drivers to changes in large (i.e. q100) versus small floods (i.e. q2), we compute for each driver the ratio between these two quantities (Fig. ). In the case of extreme precipitation (Fig. a), the ratio between its contributions to changes in q100 and q2 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 q100 is smaller, in absolute value, compared to changes in q2. 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 in q100 compared to q2 (i.e. the ratio is <1; Fig. b and c). Large uncertainties in the ratio of elasticities are observed in northeastern Europe, in the case of extreme precipitation and antecedent soil moisture (Fig. a and b), and in southern Europe, in the case of snowmelt (Fig. c). They result from values of the contribution of the drivers to q2 that are close to zero (see Fig. ), indicating that, in these regions, flood changes are not explained by changes in extreme precipitation or antecedent soil moisture.

Finally, for each region we obtain the relative contribution of the three drivers to changes in q2 and q100, as defined in Eq. () (Fig. ). They represent the fraction of the regional trend in flood quantiles qT that is explained by changes in one driver. The relative contribution of extreme precipitation is the largest of all the drivers in most of western and central Europe for both q2 and q100 (Fig. a and d). The relative contribution is slightly smaller for large floods than for the median flood in northwestern Europe, while the opposite is the case in the south. In southern Europe antecedent soil moisture gives the largest relative contribution to changes in q2 (Fig. b), and its relative importance tends to decrease for more extreme floods (Fig. e). The relative contribution of snowmelt to flood changes clearly prevails over the other drivers in eastern Europe, with slightly decreasing strength for the higher return period.

In this section we select three example regions among those analysed in Sect. , located respectively in northwestern, southern and eastern Europe (see Fig. ). For these three regions we further show in Fig.  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. c). Additionally, in Fig. , flood and driver time series are shown for the stations in each of the three regions, as well as their average changes in time within the regions.

In the region in northwestern Europe, both extreme precipitation and antecedent soil moisture contribute to positive flood changes, with extreme precipitation representing the most important driver. Its contribution to flood trends decreases with increasing return period, while the contribution stays almost constant in the case of antecedent soil moisture (Fig. d and g). In the region in southern Europe extreme precipitation and antecedent soil moisture represent both important drivers. The elasticity of floods to extreme precipitation is larger than that to antecedent soil moisture (Fig. b). However, antecedent soil moisture contributes to a larger extent to negative flood changes for small return periods (i.e. T=2–10 years) due to larger (negative) changes in antecedent soil moisture (Fig. b and e). Its contribution decreases in absolute values with increasing return period (Fig. e). For more extreme events (T>10 years) the relative contribution of extreme precipitation increases and becomes comparable to that of antecedent soil moisture (Fig. h). The contribution of snowmelt to flood changes clearly dominates in the region located in eastern Europe at all return periods (Fig. c, f and i).

In this latter case, we observe that the posterior distribution of the elasticity of qT to antecedent soil moisture (representing the change in flood quantiles due to a 1 % change in the driver) is concentrated and flattened around zero. This results from the adopted informative priors of the elasticity parameters, which set their lower bound to zero, in order to exclude hydrologically implausible values (i.e. a negative elasticity would imply that decreasing floods are attributed to increasing antecedent soil moisture, or vice versa; see Sect. ). In fact, antecedent soil moisture slightly increases over time in this region (Fig. f), while flood magnitude decreases for both T=2 and 100 years (Fig. i). As a consequence, the elasticity would tend to be negative (in case non-informative priors on the elasticity parameters are adopted), but it is constrained by the lower bound of the priors.

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 q2 and q100 for the most part, and the contribution of antecedent soil moisture is of secondary importance. Similarly, in 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 it is indeed possible to identify the relative contributions to changes in q2 and q100 with the presented approach.

Extreme precipitation contributes to positive flood changes in northwestern Europe (about 3.3 % to 2.8 % per decade in Fig. ), and its effect decreases slightly with return period in the region analysed in Sect. . 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), depending on the return period. The contribution of antecedent soil moisture is negative in southern Europe and decreases in absolute value (from -3.8 % to -2.3 % per decade) with the return period in the analysed region. Finally, in eastern Europe snowmelt strongly contributes to negative flood changes in a similar way at all return periods (about -3 % per decade for the region in Sect. ). This study more generally suggests that the changes in flood quantiles potentially caused by the three considered drivers are overall compatible, in terms of patterns and magnitude, with the flood changes observed in previous studies . Some discrepancies are nevertheless observed, for instance, in Scandinavia, where the contributions of the drivers are all positive or close to zero, while mostly moderate negative flood trends were observed in previous studies. This discrepancy points to other drivers not accounted for in the presented model, such as river regulation effects , or non-linear relationships between the drivers not captured by the model.

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. This selection is motivated by recent studies pointing out potential correlations between timing and magnitude of floods and the considered drivers across Europe . 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 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.

In this study, we directly model the changes in flood quantiles because, in a Bayesian framework, it is typically easier for experts to formulate prior beliefs in terms of flood quantiles associated with large return periods, which they are familiar with, rather than in terms of distribution parameters see, for example, the causal information expansion based on expert judgement in. 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 q2 and q100 to the drivers is 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 q100 is reflected in a lower bounded prior distribution of the elasticity of the growth factor x100′, which depends on the ratio between q100 and q2 (Sect. ). For simplicity, we assume this ratio to be equal to 2. This assumption is reasonably valid for humid catchments see, for example, and is in overall agreement with flood maps of the mean annual flood and q100 in Europe presented by . However, in arid regions, larger values of this ratio e.g. 4; see would be more appropriate (corresponding to stricter priors on the elasticity of the growth factor) because the flood frequency curves tend to be steeper.

The change model of Sect.  is fitted to the pooled flood and covariate data of several regions across Europe, where elasticities of flood quantiles to their drivers are assumed homogeneous. The rationale behind the homogeneity assumption is that the spatial windows, given their size, are characterised by rather homogeneous climatic conditions and presumably processes driving flood changes, relative to the overall variability within Europe. 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. Even though we have not assessed the statistical homogeneity of the regions in terms of the flood change model used here, we expect the effect of heterogeneity on the average regional behaviour to be less relevant than for the local behaviour. As already stated, the presented results should be interpreted at the European scale. Average driver contributions to changes in flood quantiles over the 5 analysed decades are presented, and this study does not aim at estimating driver contributions locally, in ungauged basins.

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 in which floods are considered spatially independent. Overall the obtained uncertainties associated with the contribution of the drivers to changes in q100 do not seem to increase much compared to q2, while a relevant increase would be reasonably expected. These results are valid under the assumption of the adopted model (i.e. Gumbel distribution), which may be too stringent. The model assumptions could be relaxed (e.g. adopting a generalised extreme value distribution) in order to allow for larger model flexibility.