Evaluation of the dependence between multiple drivers is limited to a bivariate analysis which imposes a two-sided conditional sampling to select multivariate extremes. Compound extreme events are defined in previously published studies and selected using either annual maxima (AM) or peak over threshold (POT). Although Ward et al. (2018) performed a sensitivity analysis and compared correlations between river discharge and storm surge using both POT with two thresholds (equal to 95th and 99th percentiles) and AM sampling, here we build on that prior work to also examine the effect of the sampling method in the dependence between the four coastal flooding drivers considered. The disadvantage of the AM is that events are selected that might not be considered extreme in the dominant variable. On the other hand, the POT approach increases the number of selected extreme events but introduces two parameters in the selection process: (1) the threshold and (2) the definition of independent events established by a minimum time between peaks or below the threshold. The independence between extreme events is assured by de-clustering the events based on the duration of the storms in the study area and selecting the highest event within each storm. The criteria used to select independent events in this study comprise a storm duration of 5 d for river discharge and 3 d for precipitation, storm surge, and waves. These values were selected based on an analysis of the duration of the highest storms conditioned to each variable in the study domain and following numbers used in previous studies (Ward et al., 2018; Hendry et al., 2019; Marcos et al., 2019; Bevacqua et al., 2019). Many methodologies for an automated threshold selection have been proposed based on graphical methods combined with goodness of fit (e.g. Solari et al., 2017), but such techniques are difficult to implement in regional studies due to the different characteristics of time series of the several drivers involved in coastal flooding. Hence, we decide to apply the POT method with a threshold that guarantees three (POT3) or six (POT6) events per year to also analyse the effect of the value of the threshold.

The conditional sampling introduces another factor that could affect the definition of compound events and this is related to the selection of the concurrent value of the secondary variable to the identified extreme events of the dominant variable. Specifically, there could be a temporal lag between variables that leads to a potential coastal flooding event. This lag can be implemented after identifying both series of extremes from the two drivers (Hendry et al., 2019) or by a time window when identifying the value of the conditioned variable (Wahl et al., 2015; Ward et al., 2018; Couasnon et al., 2020). Once a time window (Δt) is established, the value of the secondary variable is selected as the maximum value within ±Δt days (d). A variety of temporal windows have been considered, from zero lag (Bevacqua et al., 2020), through ±1 d (Wahl et al., 2015) to ±3 d (Couasnon et al., 2020) and ±5 d (Ward et al., 2018; Hendry et al., 2019). Furthermore, although river discharge data have been extracted at the river mouth, not all the databases for the four coastal flooding drivers have the same spatial resolution, with the distance between grid nodes at each location of the study domain varying considerably. Therefore, here we test the sensitivity of the identification of bivariate compound events to time windows of ±10 or ±3 d, keeping the lag which provides the highest correlation coefficient between each pair of variables at each location.

Several correlation coefficient definitions (e.g. Kendall's tau, Wahl et al., 2015; or Spearman's rho, Couasnon et al., 2020) and other metrics for the characterization of the dependence between events when both drivers are extreme (e.g. joint occurrence, Hendry et al., 2019; or tail dependence, Marcos et al., 2019) have been used in previous studies. Here we analyse the extent to which characterization of compound events could be affected by the selection of these varying dependence metrics.

Correlation coefficient. We use the Kendall rank correlation coefficient tau and Spearman rank correlation coefficient rho because they are commonly used nonparametric methods of detecting associations between two variables. Significance is assessed at α=0.05 (i.e. 95 % confidence level), using corresponding p values. The Spearman rank correlation between two variables is defined as the covariance of the two variables normalized by the product of their standard deviations between the rank scores of those two variables. The Kendall tau is also a rank order correlation coefficient which quantifies the difference between the percent of concordant and discordant pairs among all possible pairwise events. The Kendall correlation is considered to be more robust than the Spearman correlation because it offers better estimates with smaller asymptotic variance and is less susceptible to outliers (Ganguli and Merz, 2019).

Joint occurrence. The “joint occurrence method” (Hendry et al., 2019) consists of simply counting the number of times extreme events are identified in the two drivers analysed within the time window (Δt) considered.

Tail dependence. The dependence structure in the tails between two variables can be measured by the pair of statistics (χ,χ¯) (e.g. Coles, 2001). Both coefficients χ and χ¯ are defined as limit values which tend to 1 if both variables are asymptotically dependent over a certain threshold. The coefficient χ represents the probability of bivariate extreme events when both variables are extreme and provides a measure of the dependence strength (Marcos et al., 2019), referred to as extremal correlation (Zscheischler et al., 2021). χ¯ is the residual tail dependence coefficient and contains additional information about the association (-1<χ¯<1) between extremes of both variables when they are asymptotically independent (χ=0). We use the function taildep from the R package extRemes (Gilleland and Katz, 2016) to derive these values.

Although non-linear interactions between storm surges and waves could amplify the magnitude of the sea level, the assumption that both contributions may be linearly summed is generally adopted and has often been used as a proxy of coastal flooding driven by oceanographic variables (Rueda et al., 2016; Bevacqua et al., 2019; Marcos et al., 2019). Regarding the wave contribution to sea level, when wind-generated waves approach nearshore and break in the shallow surf zone, they induce variations in the sea level at different time and space scales, enhancing coastal flooding. The highest wave-driven contribution to the total water level, called run-up, depends on two dynamically different processes: (1) wave setup, which is a time-average sea level rise occurring over a few hours to several days, and which is determined by local wind sea and swell conditions, and (2) swash, which is a high-frequency process by which sea level fluctuates due to individual incident waves, with an additional low-frequency component generated by infragravity waves (related to the presence of groups in incident short waves). The magnitude and expanse of both components depend on the sea-state characteristics (significant wave height, period, and spectrum shape; Guza and Feddersen, 2012), as well as nearshore bathymetry and topography. Spatially, setup could extend from tens of metres in steep coastal areas to several kilometres in low-sloping coastal areas, while runup extension varies from a few metres to the order of a hundred metres in reflective and dissipative environments, respectively (Dodet et al., 2019). Runup is not usually included in the wave component of the sea level driver in coastal flooding analysis because its temporal duration is on the order of hours and requires local geological characteristics that could artificially inflate the wave contribution in global and regional studies (Aucan et al., 2019). The setup contribution is defined with different levels of sophistication. Wave setup has been approximated as the significant wave height multiplied by 0.2 (Vousdoukas et al., 2017; Bevacqua et al., 2019; Marcos et al., 2019) or by applying the Stockdon formulation (Stockdon et al., 2006) with different parameterizations (Vitousek et al., 2017; Rueda et al., 2017; Melet et al., 2018). The wave setup contribution to the total water level is very sensitive to this parameterization (Aucan et al., 2019). Following Vitousek et al., 2017, we used the Stockdon formulation for dissipative beaches (Eq. 1), which is known to provide similar results as using a beach slope of 0.02 (∼50 % of the world's beaches have slopes smaller than 0.02; Aucan et al., 2019). The two variables we use here to represent the sea level are (1) SW as the sum of S and setup given as 0.2W and (2) WL as the sum of S and the setup calculated using the Stockdon formulation. 1Setup=0.016HsL0, where Hs and L0 are the deep-water wave height and wavelength, respectively.

Two-sided conditional sampling has been applied to the seven pairs of drivers identified when applying the AM and the POT methods with either three or six events per year using a time window of ±3 d. Figure 2 shows the comparison of the correlation coefficients between Q and P using the three approaches when either Q or P is the dominant driver. Only this pair of variables is shown because it presents the highest correlation, and the purpose of this subsection is only to test the sampling method. Locations are divided into five regions (see Fig. S1 for the locations of these regions): northwestern Europe (NEW, 99 locations), northeastern Europe (NEE, 165 locations), southern North Atlantic coast (SNA, 60 locations), western Mediterranean Sea (WM, 99 locations), and eastern Mediterranean Sea (EM, 117 locations). Correlation is higher between the conditional pairs of extremes selected using AM while similar correlation is obtained using POT3 or POT6, with higher dispersion in the lower values of the coefficients. Correlation coefficients calculated with AM subsets of extremes are, on average, around 0.2 higher than those derived with the POT approach, and this is consistent across all regions. Scatter plots display data only for those locations where significant (p<0.05) correlation coefficients are present. Overall, conclusions about the comparisons for all other pairs of drivers are similar to those for Q–P (Fig. S2 in the Supplement).

Figure 3 shows the comparison of the number of co-occurring events between Q and P using the three approaches. The number of events is higher when using POT3 compared to AM and when using POT3 compared to POT6, as expected. The scatter plots follow roughly the 1:3 or 1:2 slope, indicating an approximate tripling or doubling in the number of events between different approaches. However, the spatial pattern is similar (correlation coefficient between the number of co-occurring events is around 0.93–0.98), which means that the three methods identify broadly equivalent areas prone to compound events with both variables being extreme. Results for all pairs of drivers are shown in Fig. S3 in the Supplement, which has a similar behaviour, albeit with an overall lower number of co-occurring events.

Figure 4 shows the comparison between the highest correlation coefficient obtained when using a time window (Δt) of ±3 d vs. using Δt=±10 d for the pairs of variables Q and P, S, W, or SW using POT3 and the joint occurrence between Q and P, S, W, or SW. Only locations with a significant correlation (p<0.05) are represented in Fig. 4. Results indicate there is no major difference in the correlation between drivers when employing the two investigated time windows. Larger differences (∼0.1 higher correlation) are obtained when Q is the dominant variable in the few locations (9 of the 540) where correlation is moderate overall (∼0.30). The joint occurrences tend to be slightly higher (10 number of co-occurring events) with a higher time window, but also fewer compound events (by half) are detected in locations with low–medium joint occurrence (<∼40) using a Δt=±3 d (lower left corner of scatter plots in Fig. 4b).

Figure 5a shows the comparisons when using the Kendall vs. Spearman rank coefficients for the pairs of variables (Q–P/P–Q, Q–S/S–Q, Q–W/W–Q, Q–SW/SW–Q, P–S/S–P, P–W/W–P, and P–SW/SW–P), using a time window of ±3 d and across the three approaches (AM, POT3, POT6) considered in the conditional sampling. There is a categorical correspondence between both correlation coefficients, with Kendall's coefficient having a tendency to be smaller than the Spearman coefficient. Therefore, to characterize compound events in terms of correlation and its spatial distribution, the information provided by both coefficients is equivalent. The number of locations with significant correlation is very similar to both correlation coefficients (see Table 1).

Regarding (χ,χ¯) statistics, the usual way to decide the threshold involves making a visual examination of the evolution of their empirical estimates for increasing threshold levels (Zscheischler et al., 2021). Here, we decided to estimate χ at a probability threshold of 0.95 after careful examination of the results for different levels. The comparison between χ and the joint occurrences divided by three events per year, and the number of years (40) for the pair of variables Q–S is shown in Fig. 5b. There is high correspondence (correlation coefficient around 0.9) between both dependence metrics mainly because both of them measure the probability of bivariate extremes when both drivers are extreme. The remaining small differences may be due to different sampling processes leading to different extreme subsets. The statistic χ is estimated using the empirical distribution of the daily time series of Q (mean daily values) and S (maximum daily values), and the extremes are selected without any clustering. On the other hand, the number of co-occurring events was calculated using a POT with a threshold that guarantees three events per year and with a storm duration of 3 or 5 d to select independent events. A similar relationship between both metrics has been found for other combinations of variable pairs (not shown here).

The effect of the definition of sea level (oceanographic flooding drivers) in the characterization of compound events is analysed here by comparing the Kendall correlation coefficients obtained between the pair of variables Q or P when using the two varying sea level definitions (SW and WL) used in this study (Fig. 6). Differences in the obtained correlation coefficients are typically small (mainly around 0.05), except along the southern Atlantic coast where the differences are slightly higher than 0.15. The southern Atlantic coast region presents the highest correlations between pluvial or fluvial sources and oceanographic drivers (correlation coefficients around 0.6–0.7) within the entire study domain, meaning that the identification of this region as an area with significance dependence is still preserved. The effect of the sea level definition on the correlation when Q or P is the dominant variable in the conditional sampling is much smaller (around 0.05, not shown here).

Here we consider the results obtained using the POT3 method and a time window of ±3 d to characterize the multivariate compound events along the North Atlantic, Mediterranean, and Black Sea coasts. The analysis of the dependence between S–W is performed at an hourly temporal resolution and using a smaller Δt (1 d) because it is considered that both drivers contribute simultaneously to the definition of sea level. The dependence between S–W as calculated using daily data is compared with the results using hourly data (Fig. S4 in the Supplement). A reduction of the correlation coefficients in the whole study domain, especially along the Atlantic coasts of Spain and France and along the Baltic Sea coast is evident while the number of joint occurrences increases in almost all locations, especially along the Atlantic coast of the Iberian Peninsula up to north of Africa and in the Mediterranean Sea.

Kendall's correlation coefficient and the joint occurrences between the four pairs of variables: Q–P, Q–S, Q–W, and Q–SW are represented in Fig. 7. The variables Q and P (Fig. 7a) present correlation coefficients of around 0.6–0.7 along the most southern coasts of the Atlantic study region and also in some locations in the Mediterranean Sea (coast of Gibraltar Strait, Algeria, southern Italy, east coast of Turkey and Levante region in the eastern Mediterranean), while correlation coefficients of around 0.1–0.2 are more predominant along northern European coasts (except the west coast of Jutland and west coast of the UK). Similar spatial correlations are found whether Q (red scale) or P (blue scale) is used as the dominant variable in the conditional sampling, except in locations along the French coast of the English Channel, the eastern coast of the UK, and the coasts of Tunisia and Libya, where higher correlations are obtained when the compound events are conditioned to P. The joint occurrence (circle size) presents a similar spatial pattern to the correlation coefficient between these two variables, with the maximum number of co-occurring events close to 100.

For Q and S (Fig. 7b), the highest correlations are of around 0.3–0.4 when both drivers are dominant, mainly along the southern Atlantic coasts of the Iberian Peninsula, north of Africa and the Gibraltar Strait. The dependency is slightly higher when Q is the main variable in the identification of compound events, even the only correlation along most of the northern coasts of the Mediterranean Sea. Higher joint occurrences are detected in locations with higher correlation, with around 50–60 of co-occurring events. However, similar numbers of joint occurrences are found in locations along the eastern coast of Italy and the eastern north Mediterranean coasts with lower correlation.

For Q and W (Fig. 7c), the spatial distribution is quite similar to Q–S, with slightly smaller correlation along the most southern Atlantic coast and higher along the west coast of the Iberian Peninsula. Correlation only when Q is the dominant variable is even more pronounced in locations along the Mediterranean Sea (e.g. eastern coast of Spain). These spatial patterns are also reflected in the distribution of the correlation coefficient between Q and SW as a combination of both (Fig. 7d).

Spatial distribution of the dependence between P and S, W, or SW (Fig. S5 in the Supplement) is relatively similar to the spatial distribution between Q and the three oceanographic variables. The highest correlation between P–S or P–W (with values around 0.3–0.4) is concentrated on the southern coast of the North Atlantic Ocean. In the case of the Mediterranean Sea, similar areas with high dependence are detected including the western coast of the Black Sea and excluding many locations along the Greek coast. Correlation is higher when the conditional sampling is conditioned to oceanographic variables. S and W (Fig. S5d) are the drivers with the highest correlation, with coefficients of around 0.6–0.7 in the north Atlantic Ocean to minimum values of 0.2 in some regions in the Mediterranean Sea.

Here we define an index, based on driver severity, to be included in the characterization of the spatial patterns of compound flooding potential. The driver severity is calculated as the sum of normalized thresholds of each driver, applied in the conditional sampling (multiplied by 0.2 in the case of W). Q thresholds, which cover a wide range of values, have been categorized into 10 intervals [0–10–25–50–100–250–500–750–1000–5000 → 25 000 m3s-1] to avoid skewing the driver severity due to very high discharge magnitude in several locations. Driver severity is divided into 11 scores from 0 to 1. Figure 8d shows the spatial distribution of the severity index (SI). Areas with the highest SI are concentrated in the North Sea, the northwest of the Iberian Peninsula, the eastern coast of the Adriatic Sea, the eastern coast of the Black Sea, and a few locations that represent large rivers. Coastal areas with the lowest SI are mainly concentrated along the southern coast of the Mediterranean Sea and the most southern coast of the Atlantic Ocean of our study domain. The SI spatial distribution indicates that an identical SI ranking can be determined by different combinations of driver extremes. To facilitate this analysis, we classify the thresholds of the four drivers (shown in Fig. S6 in the Supplement) into 10 clusters to define the main combinations of driver extremeness (Fig. 8a). The probability of occurrence of each cluster (number of locations of the study domain represented by each cluster) associated with each SI rank (Fig. 8b) provides which combinations of the four driver thresholds have an equal SI rank. For example, locations with SI equal to 0 are associated with only one cluster (represented in light green), which is defined by a combination of the lowest thresholds of Q, P, and S and low W severity. On the other hand, locations with SI equal to 1 are associated with clusters 1 and 2 (in yellow and orange, respectively) which are characterized mainly by the severity of one driver (Q or S, respectively) but also associated with clusters 3, 7, and 8 (in red and dark and light blue, respectively) with high severity of two or three drivers (Q, P, and W; or Q and P; or Q and S, respectively). The spatial distribution of these clusters (Fig. 8c) allows identification of the representative combination of the four driver thresholds for each location.

The characterization of compound flooding potential can be summarized using the combination of two metrics: the Kendall correlation (τ) and the joint occurrence (JO) for the pairs Q–P, Q–SW, and P–SW and the number of co-occurring events when all three variables are extreme JO(Q–P–SW). The two-step cascade classification method is applied to the 11-dimensional array Xi=[τ1(Q–P)i, τ2(P–Q)i, JO(Q–P)i, τ1(Q–SW)i, τ2(SW–Q)i, JO(Q–SW)i, τ1(P–SW)i, τ2(SW–P)i, JO(P–SW)i, JO(Q–P–SW)i, SIi], where the subscript represents the ith grid point. Each parameter is normalized to avoid assigning different weights in the classification process. We first use the SOM algorithm to obtain a large collection of centroids (20×20=400) projected onto a 2D organized lattice that helps to analyse the dependence between the 11 parameters. The hexagonal SOM of 20×20 size of the compound flooding potential derived from the 11 metrics outlined above for the study sites is shown in Fig. 9. Results are shown in individual panels (Fig. 9a–k) over the same 2D lattice for the different metrics defining the SOM centroids (the hexagons in a certain position correspond to the same map unit in each figure). Note that each figure has a different scale. For example, we can observe that the three parameters related with the pair Q–P (τ1, τ2, JO; see Fig. 9a–c) present a similar distribution in the lattice, which means that there is a high dependence between them. Locations with the highest correlation between Q–P(P–Q) (Fig. 9a and b) also have the highest number of joint occurrences (Fig. 9c). Centroids with the highest dependence parameters between Q–SW and P–SW are concentrated in the upper right corner of the lattice (Fig. 9d–j). They are also characterized by the highest number of joint occurrences between Q–P (Fig. 9c) and high–medium dependence between Q–P (Fig. 9a and b). Accordingly, the highest joint occurrences between all three variables are also found in the upper right corner (Fig. 9i). Other centroids located in the upper left side of the lattice represent locations with high dependence between P–SW (Fig. 9g–h) but not between Q–SW and relatively smaller dependence between Q–P.

The severity index (Fig. 9k) reveals that locations with the highest driver severity do not present dependence between any pair of drivers (lower right corner of each individual panel in Fig. 9). These locations are the ones represented principally by clusters with the severity determined by only one driver (clusters 1 and 2 in Fig. 8a). The probability of many SOM centroids is null (Fig. 9l) because a large lattice size (20×20) is required to guarantee that SOM centroids cover the multidimensional data space defined by the 11 parameters.

In the second step, we apply the KMA to find a reduced number of clusters applied to the SOM centroids. We obtain an optimal number of eight clusters by applying the Davis–Bouldin and gap criteria. Figure 10b shows the KMA classification in eight groups over the SOM lattice and highlights the similarity between KMA clusters because neighbouring centroids in the 2D lattice have similar values of the 11 parameters (see black contours of Fig. 9 which delimitate the SOM centroids belonging to each KMA cluster). Figure 10a shows the mean value of the 11 parameters associated with each cluster and the variability within each group (25th, 75th, 5th, and 95th percentiles). The number of locations represented by each cluster is shown in Fig. 10c.

The KMA classification reveals two clusters (red and pink groups) that represent locations where more compound flood events can occur. The pink group is characterized by the highest dependence (both correlation and joint occurrences) between Q–SW (SW–Q) and P–SW (SW–Q) and high dependence between Q–P (P–Q), which is reflected in the highest number of joint occurrences between the three drivers. The SI centroid presents a medium–high severity with a wide variability. The red group represents locations with the highest dependence between Q–P (P–Q) and medium correlation, only when extremes are selected conditioned to oceanographic drivers, and high joint occurrence between Q–SW and Q–P. It is characterized by a low–medium severity index because most of the locations present mild meteocean conditions (represented by cluster 4 in Fig. S8a in the Supplement). The purple cluster follows the other two in terms of compound flooding potential. It is characterized by a high joint occurrence between Q–SW and P–SW but lower dependence between Q–P and represents locations with a high severity index. Green and blue clusters stand out for the high numbers of compound events resulting mainly from the combination of P–SW. The green cluster is also characterized by significant dependence between Q–P and Q–SW at locations with low-severity meteocean climates. In contrast, the blue cluster represents locations without compound events generated by the combination of Q and SW. The dark green cluster represents locations where only compound events from the combination of Q–P can occur. The two remaining clusters, which represent 43 % of all study locations, are characterized by negligible compound flooding potential and are distinguished by the severity of the drivers, with the yellow cluster representing locations with high severity index and the light blue cluster representing locations with low severity index.

The geographical distribution of the eight KMA clusters represents the compound flooding potential patterns across the study domain (see Fig. 11). For example, the pink cluster which characterizes the pattern where the most compound events occur from the combination of the four drivers is distributed along the southern coasts of the North Atlantic Ocean, the eastern coast of France, and scattered locations along the northern coast and off the eastern coast of the Mediterranean Sea. On the other hand, the red cluster is mainly localized along the most southern coast of the North Atlantic Ocean and isolated locations in the Mediterranean and Black Sea; recall that this cluster represents locations with low driver severity. Other locations identified with significant compound flooding potential, including along the western coast of France and the UK or the northeastern coast of the Mediterranean Sea, are part of the purple cluster.