Accurate estimates of the probability of extreme sea levels are pivotal for assessing risk and for designing coastal defense structures. This probability is typically estimated by modeling observed sea-level records using one of a few statistical approaches. In this study we comparatively apply the generalized-extreme-value (GEV) distribution, based on block maxima (BM) and peaks-over-threshold (POT) formulations, and the recent metastatistical extreme-value distribution (MEVD) to four long time series of sea-level observations distributed along European coastlines. A cross-validation approach, dividing available data into separate calibration and test sub-samples, is used to compare their performances in high-quantile estimation. To address the limitations posed by the length of the observational time series, we quantify the estimation uncertainty associated with different calibration sample sizes from 5 to 30 years. We study extreme values of the coastal water level – the sum of the water level setup induced by meteorological forcing and of the astronomical tide – and we find that the MEVD framework provides robust quantile estimates, especially when longer sample sizes of 10–30 years are considered. However, differences in performance among the approaches explored are subtle, and a definitive conclusion on an optimal solution independent of the return period of interest remains elusive. Finally, we investigate the influence of end-of-century projected mean sea levels on the probability of occurrence of extreme-total-water-level (the sum of the instantaneous water level and the increasing mean sea level) frequencies. The analyses show that increases in the value of total water levels corresponding to a fixed return period are highly heterogeneous across the locations explored.

The statistical analysis of extreme values of random variables is of wide conceptual and applicative importance in science and engineering

In particular, the reliable estimation of the occurrence probability of coastal flooding events of large magnitude is crucial to environmental hazard evaluation

More generally, GEV-based approaches, by construction, discard most of the observations and do not attempt to optimize the use of the available information

Here we comparatively analyze the performance of GEV-based approaches and MEVD in high-quantile estimations with application to extreme sea levels at different observation sites. The aim is to (1) identify the statistical tool affording minimal uncertainty in the estimate of extreme sea levels with assigned probability of exceedance and (2) model and understand how climate change will affect the extreme-sea-level occurrence. To achieve these objectives, we analyze selected sea-level time series along the European coastline and evaluate extreme-sea-level predictive uncertainty by adopting a cross-validation approach in which calibration and test samples are kept separate and independent. Subsequently, we use the optimized estimation method to infer possible changes in coastal flooding hazard under Intergovernmental Panel on Climate Change (IPCC) climate change scenario RCP4.5 and RCP8.5.

The structure of the paper is as follows: Sect.

The analyses were performed using daily and hourly sea-level records from four tide gauge stations (see Table

Venice sea-level data (maximum and minimum daily observations) were obtained from the “Centro Previsioni e Segnalazioni Maree” of the Venice Municipality (

All sea-level datasets span long observational periods: 148 years for Venice, 122 years for Hornbæk, 115 years for Marseille (ca. 19 missing years), and 102 years for Newlyn.

The raw data for all stations were pre-processed to eliminate (1) years with less than 6 months of water level observations and (2) days with less than 24 h of data (for the case of hourly data). This process yields four “cleaned up” time series that were subsequently used in the analyses (see Table

Information of sea-level data used in this application.

Daily maximum sea levels at different gauge stations explored after pre-processing: Venice (IT), Hornbæk (DK), Marseille (FR), and Newlyn (UK).

The sea-level sequence is highly correlated and is generated by a non-stationary process due to long-term trends in mean sea level, the deterministic tidal component, surge seasonality, and interactions between the tide and surge

The term

Two classes of methods are widely used to estimate the probability of occurrence of extreme sea levels: direct and indirect methods. Indirect methods model separately the deterministic and the stochastic components of

From a statistical point of view, this choice is justified by the fact that the random arrival of storms adds a stochastic surge contribution at unpredictable times, thereby causing

Here,

In the following discussion, we use the terms “total water level” and “coastal water level” when referring to the quantities

As highlighted by

According to the three type theorem, there are three possible non-degenerate distribution functions which can arise as limiting distributions of extremes of random samples: (i) the Gumbel distribution, or type I; (ii) the Fréchet distribution, or type II; and (iii) the reverse-Weibull distribution, or type III. The above three limiting distribution laws can be combined into a single family of three-parameter distribution known as the generalized-extreme-value (GEV) distribution given by

The second theorem of EVT defines a method to model the tail of the distribution above a threshold value

The interested reader can refer to

The typical EVT derivation starts from the premise that the maximum value among

For practical applications, the MEVD can be approximated by substituting the ensemble average in Eq. (

It is interesting to note that the POT approach, briefly described above, can be thought of as a particular case of MEVD. In fact,

MEVD has been applied in several earth-science contexts. In rainfall extreme estimates, the ordinary value distribution is assumed to be Weibull when applied to point daily rainfall

In the present context, we define as ordinary values any coastal water elevation (i.e., the maximum water level reached during a storm event) greater than a site-specific threshold value. This threshold is chosen to be as small as possible (differently from the POT approach) to retain as much of the observational information as possible and will be dependent on the magnitude of the local tidal range (sea-level difference between high and low water level over a tidal cycle) and of storm contributions. Additionally, the threshold is set to be large enough to filter out coastal-water-level peaks that are likely fully determined by tidal fluctuation in the absence of any storm contribution. Given the above constraints, we also choose the threshold value that minimizes the estimation error under the MEVD framework.

As suggested by several rainfall applications, ordinary distribution parameters are here estimated using the probability-weighted moments (PWMs) method in non-overlapping estimation windows of 5 years. In the present application, the optimal estimation window length was set to 5 years to obtain a more robust parameter estimation, especially when few values in each year are available. PWM estimation, introduced by

The GEV-based approaches are fit on either annual peak maxima (GEV–BM) or on a few water level peaks over a high threshold (POT–GPD), which can be assumed to be realizations of independent stochastic variables. The MEVD requires that all ordinary values (coastal-water-level peaks in this case) within one block may be assumed to be realizations from independent random variables. This hypothesis, in turn, requires that observed peaks are filtered to only retain events that may be considered to be independent, through a declustering process

Statistical modeling aims to use sample information to infer the probability distribution of the population from which the data are extracted. This inference is uncertain due to imperfect parameter estimates and to the possible inability of the chosen distribution to capture the statistical properties of the underlying population. Although these sources of uncertainty are inherent in any statistical model, their impact can be minimized by a careful choice of the model and by an effective use of all sources of information

The procedure can be summarized as follows: (a) we randomly reshuffle the observational years on record while keeping all the independent water level peaks in their original year to (1) preserve both the ordinary value frequency distribution in each year and the distribution of the number of events per year and (2) remove possible non-stationarity and correlation in the time series; (b) we divide the observational sample into two independent sub-samples obtained by randomly selecting

Future increases in the frequency of extreme total water levels (i.e., the variable previously referred as

It is very likely that sea-level rise will continue to accelerate over time, thereby increasing the frequency of extreme-sea-level events, leading to severe flooding in many low-lying coastal cities and small islands

Based on these elements, here we estimate the probability of future total water levels along European coastlines by assuming that changes in the tidal and storm-surge components are negligible with respect to changes in mean sea level, an assumption common to previous approaches

To assess how the exceedance probabilities of extreme total water levels might change in the future, the projections of sea-level rise through 2100 from the IPCC’s Fifth Assessment Report (AR5) are used. In particular, we explore an intermediate (RCP4.5) and an extreme scenario (RCP8.5), using CMIP5 model outputs from the “Integrated Climate Data Center” (ICDC) database (University of Hamburg:

For each tide gauge, our approach can be summarized as follows: (1) we infer the probability distribution of extreme coastal water levels (annual maxima) from observed independent events whose intensity (maximum coastal water level attained,

One of the main objectives of frequency analysis is to calculate the average recurrence interval or return period. It is a widely used concept in hydrological and geophysical risk analysis. If a process is stationary, the return period (

We start by computing mean sea level on a yearly basis and by subtracting it from observed total water level. The first question that we want to explore is the presence of long-term trends, unrelated to sea-level rise and associated with other factors (e.g., human-induced factors, morphological variations), in the “cleaned up” signal, i.e., the observed measurements without mean sea level. To answer this question, in this work we focus on the deviation of yearly maxima from yearly mean sea level and test for the presence of a trend by the two-tail Mann–Kendall test ^{Venice} ^{Hornbæk} ^{Marseille} ^{Newlyn}

Deviation of yearly maxima from yearly mean sea level (blue line) and 19-year running mean (black line) calculated for Venice (IT), Hornbæk (DK), Marseille (FR), and Newlyn (UK).

The MEVD formulation requires the choice of an optimal distribution of ordinary values that can represent the characteristics of the natural phenomenon under analysis. Different candidate distributions for the

As mentioned above (Sect.

Considering the above threshold values, the observed and estimated distributions of coastal water level are compared by plotting their quantiles against each other. By comparing measures of in-sample and out-of-sample test predictive accuracy, the results are presented by means of quantile–quantile (QQ) plots. The reader can refer to Fig.

Total number of independent events and average number of events per year for all the gauge stations explored.

QQ plots of extreme-coastal-water-level quantiles, computed with the MEVD framework, for the

We now focus on evaluating the performance of the three approaches (GEV–BM, POT–GPD, and MEVD) in high-quantile estimation. We explore the predictive performance of the MEVD and GEV distribution as a function of the NDE (Sect.

Distribution of the nondimensional error (NDE) for maximum sample return period (

Kernel density estimates for the nondimensional error (NDE) distributions obtained with a calibration sample size (

Results for the Marseille site show a peculiar behavior that requires a specific discussion. In this case, the application of the traditional extreme-value theory is advantageous when compared with the MEVD (Fig.

We also provide a comparative analysis between the three methods to evaluate if the tested extreme-value distributions are representative of the entire range of return times of interest. To achieve this purpose, we evaluate method performance also for intermediate

Results of the evaluation metric obtained for all the gauge stations and for calibration sample sizes (

Median of the nondimensional error (NDE) for return period greater than the calibration sample size in test sub-sample for the GEV–BM, POT–GPD, and MEVD approaches (magenta, blue, and green dots, respectively). The results are obtained for the Venice (IT), Hornbæk (DK), and Newlyn (UK) sites and by estimating the distribution parameters on 30-year calibration sub-samples.

Future total water level projections, with respect to the current mean sea level, in Venice (IT; panels

We next explore how sea-level rise may influence the frequency of extreme total water levels across the sites analyzed. As described in Sect.

We use site-specific sea-level projections from IPCC’s AR5

Changes in sea-level extremes can also be studied by focusing on changes in the return period of a fixed value of the total water level. To this end, one can define a sensitivity measure (SM) as

Results of the percentage changes in total water level (

The comparative examination of extreme-value distributions applied to observed sea levels at several sites along European coasts provides insights into the predictive performance of traditional and new approaches. Our analyses confirm some practical and conceptual advantages of the MEVD with respect to traditional methods. A cross-validation scheme (with 1000 realizations for each site) was used to compare model performance in high-quantile estimation. The use of two independent sub-samples (calibration and test sample) allows the quantification of actual predictive uncertainty.

We find that the MEVD approach provides reliable estimates of high quantiles for almost all the gauge stations explored, particularly when sufficiently long calibration sample sizes are considered. Differences in performance between the MEVD framework and GEV-based approaches are not large, and a definitive conclusion on an optimal solution independent of the return period of interest remains elusive. However, small differences in the estimation accuracy are relevant for engineering applications when dealing with rare extreme events. If we focus on high-return-period quantile estimation, our analyses show that the MEVD approach provides reliable estimates for almost all the gauge stations explored. Data from the Marseille gauge station exhibit a behavior that deviates from those from other sites, showing an inferior predictive performance of the MEVD with respect to GEV-based approaches. We interpret this fact to be linked to the small average number of sea-level peaks every year: the small sample of yearly ordinary events available prevents the MEVD from adding significant information with respect to GEV–BM and POT–GPD. Conversely, when we evaluate method performance for intermediate return period values, GEV–BM displays an overall greater robustness, and MEVD exhibits a greater absolute value of the estimation error.

Unfortunately, the size of the available datasets does not allow us to explore model performance for greater values of the return period. Future work could investigate if the estimation error can be reduced, with respect to what was found here, by using different approaches, e.g., by assuming “time-invariant” parameters in the ordinary distribution, whose estimation would thus be performed on the entire calibration dataset rather than on relatively short sliding windows. Synthetic water level time series may be produced by one of the several existing numerical models to extend analyses to arbitrarily long return periods.

Finally, we explored projections of the frequency of extreme total water levels driven by changes in mean sea level. The sensitivity of extreme-water-level frequency to sea-level rise is location-dependent, and we find that, at a given site and for a set value of the total water level extreme, the relative change in return time grows linearly with return time itself.

All data used are publicly available from sources cited in the main text. Venice sea-level data were obtained from the “Centro Previsioni e Segnalazioni Maree” of the Venice Municipality (

The supplement related to this article is available online at:

MM designed and coordinated the research. MFC performed the research and analyzed the data. Both authors contributed to the writing and editing of the manuscript.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Maria Francesca Caruso acknowledges the PhD School in Civil Sciences and Environmental Engineering at the University of Padova for funding her PhD. Marco Marani acknowledges the support of the Venezia2021 research program.

Scientific activity was performed as part of the research program Venezia2021, coordinated by CORILA, with the contribution of the Provveditorato for the Public Works of Veneto, Trentino Alto Adige, and Friuli Venezia Giulia, provided through the concessionary of State Consorzio Venezia Nuova.

This paper was edited by Philip Ward and reviewed by two anonymous referees.