The challenge the researchers set out to solve is fundamentally statistical. Coastal flooding simulations typically rely on time series of offshore conditions—water levels, waves and atmospheric surge—that force hydrodynamic models at their seaward boundaries. When scientists want to explore rare, catastrophic scenarios, they need to generate synthetic time series that are more extreme than anything actually observed, yet statistically consistent with the record. The standard toolkit for this task is extreme value theory, which has been extended in recent years to functional data: entire curves or time series treated as single observations in an infinite-dimensional space. But this functional framework rests on two demanding assumptions. The time series must be independent and identically distributed, and they must exhibit so-called regular variation, which implies that their marginal distributions have heavy, Pareto-type tails.

Real meteoceanic data, it turns out, satisfy neither condition. Working with a hindcast database covering the years 1979 to 2016 at the Gâvres site in southern Brittany, the team extracted 26,652 surge time series, each spanning a half tidal cycle—the six-hour window within plus or minus three hours of high tide, sampled at ten-minute intervals to give 37 points per cycle. Autocorrelation analysis revealed that successive tidal cycles are correlated, with correlation levels around 0.10 persisting even at a lag of 50 cycles, and that cross-correlations link different time points across neighboring cycles. The data also show clear seasonality, with the largest surges concentrated in the winter months of September through March. More troubling still, diagnostics of the tail behavior told a decisive story: the Hill estimator, which converges only under heavy-tailed assumptions, showed no stable region, and both the moments and maximum-likelihood estimators confirmed that the shape parameter of the generalized Pareto distribution was not positive. The surge series are short-tailed, violating the regular variation hypothesis outright.

Rather than discarding the functional extreme value framework, Gorse and his colleagues devised a two-stage preprocessing strategy to bring the data back into its realm of validity. The first step, which they call whitening, tackles temporal dependence. After detrending the series, they fit autoregressive models that capture the dependence of each cycle on its predecessors, retaining only one observation out of every three cycles to reduce residual correlation. The residuals of these models form, by construction, a sequence of independent random variables, while preserving the shape of the surge evolution within each cycle. The second step addresses the tail problem through a marginal transformation: at each of the 37 time points, the empirical distribution of the residuals is approximated by a semiparametric mixture, using the empirical cumulative distribution function in the non-extreme region and a generalized Pareto distribution above a threshold set at the 90th percentile. Applying the inverse of the logarithmic transformation then yields residuals with unit Fréchet marginals—distributions with exactly the heavy, Pareto-type tails that regular variation demands.

With the preprocessed data now satisfying the framework assumptions, the team constructed a probabilistic model using a polar coordinate representation, a technique borrowed from recent theoretical work on regular variation in Hilbert spaces. Each extreme time series is decomposed into a radial component, given by its L2 norm, and an angular component that captures its shape. Asymptotically, these two components become independent, and the radial part follows a Pareto distribution whose simulation is straightforward. The angular component is the hard part. The researchers reduced its dimensionality using principal component analysis, finding that the first three eigenvectors explain 83 percent of the variance of the extreme observations. They then modeled the joint distribution of the three principal component scores using vine copulas—hierarchical constructions that assemble bivariate copulas into a tree structure capable of capturing complex multivariate dependence without imposing a rigid parametric form on the whole vector. A Student t copula was selected for most pairs, though the team switched to a rotated Tawn copula where diagnostics revealed asymptotic independence between the first two coordinates.

Simulation then proceeds in reverse. New angular components are drawn from the fitted copula model, combined with freshly sampled Pareto radii, and pushed back through the inverse marginal transformation to recover extreme residuals on the original scale. Finally, the autoregressive model is inverted: each simulated residual is added to a scaled version of an initial reference time series, producing a complete synthetic surge curve. Crucially, this inversion step gives the method a remarkable flexibility. Because the output depends on the chosen reference series, the level of extremeness can be tuned at will. Selecting a non-extreme reference produces data-like simulations that follow the same statistical law as the observed extremes, while choosing an extreme reference—such as the series recorded during Storm Johanna itself—generates sequences of consecutive extreme events, a scenario of obvious relevance for flood risk assessment.

Validating a generator of synthetic extremes is a subtle business, and the authors attacked it from several angles simultaneously. Comparisons of empirical percentiles showed that the simulated levels fall within bootstrap confidence bands constructed from the observed extremes. A Kolmogorov–Smirnov test on the first principal component of the normalized series found no significant difference between simulations and observations, though a difference emerged in the second dimension, hinting at residual imperfections in the shape modeling. An extremogram analysis, which quantifies the dependence between extreme values at different time points within a cycle, placed the simulated dependence structure squarely within the confidence band of the observations. Return level plots, expressed in terms of return periods assuming roughly seven extreme observations per year, showed that the highest simulated values constitute a plausible extrapolation of the observed data toward larger, unobserved magnitudes.

Perhaps the most striking validation comes from machine learning. The team framed the problem as a two-sample classification test: if the simulated surge curves are truly indistinguishable from real extreme observations, then classifiers trained to tell them apart should perform no better than chance. Support-vector machines, generalized linear models and random forests were each trained on hundreds of labeled series, with the procedure repeated a hundred times to quantify uncertainty. When given the raw time series as input, all three classifiers hovered near 50 percent accuracy—the signature of statistical indistinguishability. Only when fed the angular component alone did accuracy rise to around 60 percent, suggesting that the simulator reproduces the magnitude of extremes more faithfully than some finer aspects of their shape. The team also showed that this success depends on their conditional sampling of the reference series; an unconditional approach allowed the classifiers to detect the simulations, confirming that the dependence between neighboring cycles must be handled carefully.

The implications extend well beyond one town on the Breton coast. Atmospheric storm surge is, in many coastal environments, a larger contributor to the storm tide than wave set-up, making it the natural first target for this methodology. The approach is implemented in R with publicly available code and the surge dataset deposited on Zenodo, lowering the barrier for other coastal engineering groups to adopt it. The authors are candid about the limitations and the road ahead: the current framework handles a single variable, whereas real forcing conditions are multivariate, involving wave height, period and direction alongside surge and wind. Extending the method to the multivariate case, or replacing the copula-based angular model with Gaussian mixtures or generative deep learning approaches, are flagged as promising directions. For now, the study delivers something coastal flood modelers have long needed—a principled way to conjure statistically faithful extreme scenarios from stubbornly non-ideal data, and with them, a sharper picture of the worst the sea may one day deliver.

Subject of Research: Statistical simulation of extreme storm surge time series over tidal cycles for coastal flooding analysis

Article Title: Simulation of extreme functionals in meteoceanic data: application to surge evolution over tidal cycles

Article References: Gorse, N., Roustant, O., Rohmer, J., & Idier, D. (2026). Simulation of extreme functionals in meteoceanic data: application to surge evolution over tidal cycles. Advances in Statistical Climatology, Meteorology and Oceanography, 12(1), 123-148. https://doi.org/10.5194/ascmo-12-123-2026

Image Credits: AI Generated

DOI: 10.5194/ascmo-12-123-2026

Keywords: storm surge, coastal flooding, extreme value theory, functional data analysis, copulas, autoregressive models, tidal cycles, Gâvres, France, hydrodynamic modeling, principal component analysis, machine learning validation