We have at our disposal the bathymetry of the dike up to the deepwater point provided by the SHOM (Service Hydrographique et Océanographique de la Marine). The survey has a resolution of 3 m and spans over 6 km. As the distance from the dike to the ANEMOC (Atlas Numérique d'Etats de mer Océanique et Côtier) point (Fig. ) is greater than 6 km, we use all the available data to calculate the mean slope of the beach in front of the dike.

As there is no sensor that recorded the sea level in the immediate vicinity of the dike, which would be highly sensitive to waves anyway, we had to resort to the nearest gauge that had a large record of measures, which was located in Marseilles's harbor (Fig. ). The data of the gauge are maintained by the SHOM (https://data.shom.fr/, last access: 15 February 2023) in the REFMAR (Réseaux de rEFérence des observations MARégraphiques) database, which is part of the Global Sea Level Observing System (https://gloss-sealevel.org/, last access: 12 March 2023; GLOSS) and provides more than 100 years of hourly water elevation level. The acquisition is done using a permanent global navigation satellite system (GNSS) station. The place being located inside a port is protected from sea waves. It is located kilometers away from the actual site, but we will use these values as no on-site data are available currently. With the sea level being forced by atmospheric, mainly wind, conditions, non-linear effects are to be expected between sites, even when the tide gauges are relatively close. This underlines the importance of using on-site data whenever possible. The other issue would be the desynchronization of the data which are accounted for by our peak value selection method.

In situ data of the significant wave height provided at an hourly rate are difficult to find reliably over a long period of time (decades). This means that we have to resort to data provided by a numerical model. We use the data extracted from the ANEMOC-2 (https://candhis.cerema.fr/, last access: 10 August 2023) database currently maintained by the CEREMA (https://www.cerema.fr/fr, last access: 1 November 2022), reproducing numerically the sea conditions over a long period of time (from 1980 to 2010). The data are generated using the third generation of the spectral TOMAWAC (TELEMAC-based Operational Model Addressing Wave Action Computation), which is a part of the TELEMAC-MASCARET software suite. This model is used to interpolate sea states on many points calibrated using the GlobalWave in situ data. The significant wave height is estimated by calculating the mean value of the upper third of the recorded waves every hour. Thus, one value is given hourly at each chosen location. We have selected point 3667 (Fig. ), as it is located in front of the dike and where the water depth is high enough to be considered as offshore (≈80 m) as the continental shelf has been reached according to bathymetry maps. It should be noted that model data carry an intrinsic uncertainty compared to experimental data, which is bound to decrease the reliability of the study.

The time series data themselves are not directly exploitable, as the copula that we want to generate is based on the identification of extreme events implying a locally high value of both N and H0. We have many methods at our disposal to select extreme events, among which the peak-over-threshold is supposed to give the best results. However, selecting values over a certain peak gives a distribution bounded by the value of the threshold, which is not adapted to generate a full copula that can be used to calculate the return period of critical events (see Eqs. –). Thus, we decided to use the block maxima approach, which gives good results and asymptotically converges to the generalized extreme value (GEV) distribution that we will use to fit the resulting data.

The choice of the block size is not obvious, as a small block size decreases the accuracy of the fit to the limit distribution (GEV), leading to a bias, but a large block size reduces the size of the sample, increasing the variance. The general approach is to consider the duration of the cycles between events. The seasonality of storm cycles and the requirement of independence between events would not allow us to use a value lower than 1 year, which is the block size we chose. This limited the size of our dataset significantly, meaning that more data would probably greatly improve our analysis.

As a preliminary check, correlation coefficients have been calculated to determine the relationship between the datasets. Pearson's coefficient gave ≈0.6, indicating a strong positive relationship. However, Pearson's coefficient is only applicable for linear correlations, which is unlikely, and is inappropriate for capturing outlier influence, making it artificially high. Kendall's and Spearman's coefficients are more appropriate in this case and gave ≈0.065 (with a p value of 0.002) and ≈0.095 (with a p value of 0.003), respectively. These values indicate a weak correlation, which indicates the dependency structure might not be captured by the indices and instead calls for another approach, based on copulas.

The copula method is a popular approach for estimating return periods of extreme events in hydrology or finance and is commonly used as a tool of risk management, as univariate statistical analysis might not be enough to provide reliable probabilities with correlated variables as stated by . One key advantage of the copula method over other methods is that it allows more flexible modeling of the dependence structure between variables. Other methods, such as the traditional joint probability method or the design storm method, assume a specific type of dependence structure (e.g., independence or fixed correlation), which may not accurately reflect the true relationship between the variables. In addition, the copula method can provide more accurate estimates of extreme events and their return periods, especially for events with very low probabilities of occurrence. This is because the copula method allows for a more precise estimation of the tail dependence between variables, which is important for accurately estimating extreme events. As we are looking for rather rare events in this study, this represents a considerable advantage. It however appears that the practitioner has a large selection of copulas to choose from depending on the nature of the data. The choice of which copula to choose varies from the type of data as well as the physics of the setup and even so we are left with a rather large selection. Merging multiple copulas in order to combine their properties has also been explored by , complicating the decision process further. suggested that the Gumbel–Hougaard copula was particularly adapted when combining water level and wave intensity, although they used the time integral of the wave height over a threshold instead of the significant wave height, and the region of interest was the North Sea. also recommended using the Gumbel–Hougaard or Clayton copulas for coastal waves on the Atlantic shores of France. An application of the Gumbel–Hougaard copula has also been explored on UK shores, aiming to study extreme coastal waves by . This motivates us to directly use the Gumbel–Hougaard copula as the most adapted choice (Eq. ). 1F(u,v|θ)=exp⁡-(-log⁡(u))θ+(-log⁡(v))θ1/θ, where u and v are the cumulative distribution functions of the histograms originating from the datasets. The copula parameter θ represents the interdependency of the data.

The value of this copula parameter is important and can be calculated using a panel of different methods, i.e., the error method (see Appendix B for the equation as written by ) and the maximum-likelihood method.

Once done, the copula can be calculated using Eq. (), attributing a probability of occurrence of any event E with one of the variables having a value smaller or equal to the defined ones, written as E(H≤h||N≤n). The logical inverse E(H>h,N>n) can then be obtained by calculating the survival copula Cθ-1, defined as 2Cθ-1=Cθ+u+v-1. Finally, we can associate each value of Cθ-1 to a return period using the formula provided in : 3RP=μCθ-1, where μ is the average interarrival time between two events of interest, i.e., the storms. The offshore conditions have been determined by a couple (N,H), the water level, and the significant wave height, respectively, with an associated return period. This gives us the properties of an offshore wave.

The principle of the maximum-likelihood method that we use is that we try to maximize the function L in Eq. (), yielding the likelihood of generating the observed data for a set value of θ. It essentially means that, given a set of data, a high value of L indicates that the function is highly likely to have been able to generate the data sample. 4L(θ)=∑i=0ncθ(u(i),v(i)), where cθ is the copula density, which can be obtained by calculating the derivative of the copula function with respect to its cumulative density functions in Eq. (): 5cθ(u,v)=∂2Cθ(u,v)∂u∂v.

We are able to link a deep water state to a return period. However, this does not give us any information on the probability of the occurrence of an event that would provoke erosion. Hence, we need to assess what kind of event provokes erosion using Eqs. ()–() to calculate the terminal velocity of the flow on the landward slope.

We can now associate a terminal velocity with a set Svt={(N,H0),f(N,H0)=vt} that is the set of couples (N,H0) through the function f to a terminal velocity vt.

By integrating the derivative of the copula with respect to H0 along the isoline Svt, we can obtain the return period of event Evt={vt*>vt}, which is any event implying a terminal velocity equal or higher than vt (see Eqs.  to ). 12P(vt*>vt)=∬C∂2CN,H0∂N∂H0dNdH0,13P(vt*>vt)=∫0∞∂CN,H0∂H0S(H0)∞dH0,14P(vt*>vt)=-∫0∞∂CN,H0∂H0(S(H0),H0)dH0, where C is the surface of integration, which is the area above the velocity curve, and S(H0) the velocity curve. This means that we can calculate the return period associated with a certain terminal velocity threshold for a defined dike by fixing the parameters in Table . We give reference values to these parameters. They are obtained either experimentally from in situ data or extracted from the literature when observations are unavailable. The details of the values are explained in Sect. .

We start by compiling the selected storm surge events into a histogram, giving the univariate probability densities of both datasets. However, since we only work with about 20 years of hourly data, we need to fit the cumulative histogram in order to create a cumulative distribution function that allows us to extrapolate to rarer events. We use the generalized extreme value distribution, which is used for the estimation of tail risks and is currently applied in hydrology for rainfalls and river discharges in the context of extreme events as in .

This means that the events can then be sorted into a histogram for us to observe their respective univariate distributions. In this case, the sample limits us to events that can happen up to once every 20 years, since we have no data covering a larger period. Thus, we can obtain information about more extreme events by extrapolating the data using a fitted distribution on each individual sample. Using the block maxima event selection, the Fisher–Tippett–Gnedenko Theorem indicates that extreme values selected this way asymptotically follow the generalized extreme value distribution (GEV) (Eq. ), which we use to fit the data. The distribution is described here: 19F(x)=exp⁡(t(x))witht=1+ξx-μσ-1/ξif ξ≠0exp⁡-x-μσif ξ=0, with (μ,σ,ξ) representing the location, scale, and shape factor, respectively. The results are displayed in Fig. . The laws are fitted using the maximum-likelihood method. The fit gives a R2 score higher than 0.999 for both datasets. Hence, we consider the fit almost perfect, and it will not be accounted for during future calculations of uncertainty, as it is considered insignificant compared to other sources.

We will then compute the derivative of the copula in Eq. () and maximize the value of L(θ) from Eq. ().

The interdependence parameter can take values in the interval 1,+∞, where 1 is the independent copula, and +∞ means absolute correlation. A value of 1.23 means that there is a moderate correlation between the two distributions. This can be seen visually in Fig. , as the contour lines form an L shape, indicating that the values are strongly linked. Hence, we can generate the copula using Eq. (). The cumulative distribution function yields the probability of a value laying under a threshold. Hence, we use Eq. () to inverse the copula and obtain the survival copula (Fig. ). This allows us to evaluate the return period of any event E so that E(N≤x||H0≤y) using Eqs. (–).

The contour lines of the copula in Fig.  show that the data are coupled to some degree. Indeed, since the data are correlated, a high value of the water level N and the significant wave height H should be more probable than if the data were uncorrelated, thus decreasing the return period and driving the contour lines toward the smaller values.

We use the terminal velocity on the landward slope vt as a criterion of erosion. This means that damage starts to occur when vt>vc, where vc is the critical velocity which has to be determined using the literature. Using Eqs. ()–(), we can calculate it from any couple (N, H0) of offshore water level and significant wave height, given that the mean slope of the bathymetry is known. The results are shown in Fig. .

Unsurprisingly, higher values of both N or H0 induce higher values of terminal velocities. All values below the “0.0” line (Fig. ) failed to produce overtopping and thus generate a null value while in fact there is no water flowing on the slope.

Typically, we observe that the Quenin dike's landward slope is covered by rubble mounds which have an average diameter of 20 cm. Applying Peterka's formula (Eq. ), which is used by the U.S. Bureau of Reclamation, we can obtain the critical velocity of erosion on the dike. 20v*=d50/0.043, where v* is the critical erosion velocity, and d50 is the median block parameter. For blocks with a diameter of 20 cm, we obtain a critical erosion velocity of approximately 2 ms-1. Our calculations estimate that such flow velocity will occur on average once every 5.86 years. This gives a higher value than what is reported by the Salins du Midi company, currently exploiting the dike. The company reports significant damage that needs to be repaired approximately once every 2 years. This has been confirmed by its archives. This gap can be caused by uncertainty on the parameters which will be further estimated via sensitivity analysis.

After generating a sample of parameter values, each set is computed through the framework, giving an associated return period from which we calculate the global sensitivity indices of both first order and total effect (Fig. ).

A few observations can be made about the results. It appears that there is some correlation between the first-order Sobol' index and the total effect index. This is not surprising, as the total effect index encapsulates all orders, including the first order. However, many parameters showing a value close to zero at first order had much higher values on the total effect index. It is safe to assume that the other parameters presenting a high total effect value likely show some uncertainty when interacting with the three parameters and thus that the first-order index is clearly not sufficient. Then, the considered parameters contribute very differently to the uncertainty of the system. It is essential to remind one that high contribution to uncertainty can mean either that the parameter is very influential or that it is very uncertain (or both). We will focus on the three main parameters showing both high first-order and total effect values.

γf. This parameter intervenes at the beginning of the overtopping process and thus should present a high degree of interactivity with other parameters, which explains the very high total effect index value. The parameter is undoubtedly very influential, since it is a direct coefficient of the overtopping wave height. However, this parameter is very difficult to assess experimentally, and we had to rely on reference values for the variation interval. This means that a part of the uncertainty can be explained by a large range of variation defined in the bound of the global sensitivity analysis. This result is corroborated by , where they proceed similarly and find that the majority of the uncertainty is held by the overtopping process.

Launching such a high number of calculations allows us to compile the return periods into a histogram to evaluate the probability of the return periods taking into account uncertainties. The results are compiled in Fig. .

The distribution of values appears to form a cluster close the 1-year mark, with many outliers showing very high values. Note that for clarity purpose the histogram has been truncated, but values range from 0 up to 3000 years. With the mean value being biased because of the very high values of the return periods, the median is a more appropriate metric, which gave a value of ≈1.1 years. This value is very close the in situ observation at the Quenin dike relayed by the operator and comforts us in the reliability of our current analysis.

In order to make sure that the estimation of the sensitivity indices is accurate, we need to ensure that the convergence of the estimator has been reached. We will do this by plotting the values of the indices and incrementally increasing the number of points generated by the Sobol' sequence; this is called a validation curve. Note that the number of plotting points is limited because the Sobol' sequence, being a non independent sample, is only valid for 2n points. The results are displayed in Fig. .

Convergence has evidently been reached. It seems that we can safely use ≈40000 points which in our case is still fairly low as the computation of the terminal velocity is pretty fast. However, should the computation time increase by changing the methods of calculation, this could become a problem which would require more intensive optimizations.

Results from the global sensitivity analysis give indications on how the dike could be reinforced in order to increase the most the return periods. The recommendation would be to act upon the most significant parameters of the analysis, meaning the ones which yield the highest values of Sobol' indices. This indicates that the geometrical features of the dike, the crest height as well as the slopes, should be acted upon first whenever possible. Elevating the dike or decreasing its seaward steepness should bring good results, while altering the erosion properties of the landward slope does not look so promising. This focus on the geometrical features of the dike is supported by . Generally, the recommendations of the USACE (United States Army Corps of Engineering) seem to focus mainly on geometrical features and secondly on erosion resistance when considering the design of levees. Approaching the problem using Sobol' indices in this particular use case had not been done before and seems to provide similar results, confirming the value of the method. The recommendations stated here do not include, however, an analysis of cost-effectiveness, which should be one of the next milestones of the work that is presented here.

The framework provides a rather complete approach but obviously suffers some limitations. Some of them are inherent to the system itself, while others call for future improvements. Our main focus was to obtain an assessment of the risk of erosion on the landward slope of the dike. Coastal protection is nonetheless submitted to many other damages such as erosion in other locations like the crest of the seaward slope. A more general criterion of security such as “any damage to the dike” would require one to broaden the calculations to take all possible damages into account. We have also limited our criteria of interest as a condition of whether or not the critical velocity has been overreached on the landward slope. The possibility of a breach or the actual amount of eroded material is therefore not quantified. For practical reasons, we calculated return periods on an averaged profile of the dike, which as stated by the global sensitivity analysis can lead to a return period different from the local profile. A location-wise study could bring reduced uncertainty and bring more relevant results. The data sources used to make our analysis are incomplete, located in far away places, or generated using numerical models. It is necessary to use in situ data extracted from experimental devices in order to improve the reliability of the study. Moreover, the study uses long-term data, but the impact of climate change implies an elevation of the water level as well as changes in the characteristics of storm surges such as intensity, duration, or frequency. Integrating these effects should be done in the future in order to improve long-term studies. This problem is especially important for more resilient dikes dealing with higher return periods.