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Statistical analysis

Procedure

Return periods. There is no unique definition of CF return periods, with none of them being a priori superior. Each definition provides different information, and one should be chosen on the basis of the context and aim of the study (37). Here, the AND return period is chosen, which measures the probability of both individual hazards, sea level and precipitation, exceeding a chosen threshold. This definition allows for disentangling and understanding the dynamics of the potential flooding caused only by the concurrence of high sea level and precipitation values (1-year return levels). Specifically, we defined the bivariate CF return periods (26) as the mean waiting time between events where sea level and precipitation simultaneously exceed the individual 1-year return levels (i.e., the ~99.7th percentiles s99.7 and p99.7). To allow for a robust estimation, we applied a parametric copula-based bivariate probability distribution. Applying a parametric model for the full range of values, one would run the risk of biasing the representation of the extreme tail by the bulk of the bivariate distribution where most data occur. Therefore, we applied the model only to pairs of high values. We selected pairs where, simultaneously, sea level and precipitation values exceed the individual 95th percentiles (ssel and psel, respectively). In a very few locations, one might end up with selecting few pairs only. Here, we reduced the selection threshold 0.95 to ensure that at least 20 pairs of values were selected (never below 0.9). Clusters of selected event pairs separated by less than 3 days were replaced by a unique event, which assumed the maximum sea level S and precipitation P observed in the cluster (fig. S9).

The bivariate return period is thus given as(1)where μ is the average time elapsing between the selected pairs, uS99.7 = FS(s99.7), FS is the marginal cumulative distribution of the excesses over the selection threshold (accordingly for precipitation), and CSP is the copula modeling the dependence between the selected pairs.

The marginal distributions of sea level and precipitation beyond the selection thresholds were modeled by a generalized Pareto distribution. Copulas were fitted to (uS, uP) [obtained via empirical marginal cumulative distribution function (CDF) (26)] and selected via Akaike information criterion from the families: Gaussian, t, Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7, and BB8. Marginal distributions and copulas were fitted through a maximum likelihood estimator [via the ismev (38) and VineCopula (39) R-packages]. Goodness of fit of marginals and copulas was tested on the basis of the Cramer-von-Mises criterion (one-tailed; Nboot = 100 for copulas) [via the eva (40) and VineCopula (39) R packages, respectively]. The projected change (%) of the return period T was estimated for the individual CMIP5 models as ΔT(%) = 100 ⋅ (T2070−2099T1970−2004)/T1970−2004 (Fig. 3A and fig. S10).

Sampling uncertainty of ERA-Interim–based CF return periods. To obtain the 95% sampling uncertainty range of the ERA-Interim–based CF return periods in Fig. 3C, we applied a resampling procedure. The uncertainty was computed in the 11 representative locations whose return periods are shown in black in Fig. 3C. We based our estimate of sampling uncertainty on the previously generated 240 bivariate sea level/precipitation time series (where surge and precipitation are identical; only the astronomical tides were resampled). Each of these 240 bivariate time series were used for a further resampling procedure by combining bootstrapped numerator and denominator values of the return period expression (Eq. 1). The numerator-bootstrapped values of μ were obtained via resampling of the observed times elapsing between the selected pairs (si, pi) used for fitting the parametric probability density function (pdf); the denominator-bootstrapped values were obtained via resampling of the observed pairs (si, pi) used for the fit of the pdf. The final return period sampling uncertainty range was defined as the 2.5th to 97.5th percentile interval of the 240 ⋅ 240 return period estimates. This procedure is preferred to a classic resampling of all the pairs, which, here, would bias the obtained median return period due to the serial correlation of the sea level time series. On the basis of a large sample of data without any serial correlation, we estimated that the 95% sampling uncertainty range is overestimated by 30% from our procedure (with respect to a classic resampling procedure). Thus, conclusions about the detection of a climate change signal in the future (Fig. 3C) are conservative.

Delta change approach. We computed CF return periods for the future via the delta change approach (27), i.e., multiplying the ERA-Interim–based historical return period $TEra1980−2004$ by the individual CMIP5 model i variation of the CF return periods . The present-day reference period is the intersection of the ERA-Interim and the historical CMIP5 data, for which sea level simulations are available. See fig. S5 for comparing return periods based on ERA-Interim and individual CMIP5 models.

Attribution of return period variation. We carried out three experiments (3) to assess how the CF probability would change in future when only considering variation—with respect to the present—of (a) the dependence between sea level and precipitation, (b) the sea level overall marginal distributions (i.e., the distribution of the sea level without reference to precipitation), and (c) the precipitation overall marginal distribution. We estimated the relative change of the probability that would have occurred for experiment (i) as (Fig. 4), where Tpres is the return period for the present period and is computed as follows. Experiment (a): Given the variables (Sfut, Pfut), we got the associated empirical cumulative distribution (USfut, UPfut). From the variables Spres and Ppres, we defined the empirical CDFs FSpres and FPpres, through which we defined $Sa=FSpres−1(USfut)$ and $Pa=FPpres−1(UPfut)$. The variables (Sa, Pa) have the same Spearman correlation and tail dependence (3) as (Sfut, Pfut), but marginal distributions as in the present period. We computed the return period based on (Sa, Pa). Experiment (b): Given the variable Spres, we got the associated empirical cumulative distribution USpres. From the variable Sfut, we defined the empirical CDFs FSfut, through which we defined $Sb=FSfut−1(USpres)$. The variables (Sb, Ppres) have the same Spearman correlation and tail dependence as during the present, but the marginal distribution of Sb is that of the future. We computed the return period based on (Sb, Ppres). Experiment (c): as experiment (b), exchanging precipitation and sea level variables.

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