The study aims to provide a complete analysis framework applied to an earthen dike located in Camargue, France. This dike is regularly submitted to erosion on the landward slope that needs to be repaired. Improving the resilience of the dike calls for a reliable model of damage frequency. The developed system is a combination of copula theory, empirical wave propagation, and overtopping equations as well as a global sensitivity analysis in order to provide the return period of erosion damage on a set dike while also providing recommendations in order for the dike to be reinforced as well as the model to be self-improved. The global sensitivity analysis requires one to calculate a high number of return periods over random observations of the tested parameters. This gives a distribution of the return periods, providing a more general approach to the behavior of the dike. The results show a return period peak around the 2-year mark, close to reported observation. With the distribution being skewed, the mean value is higher and is thus less reliable as a measure of dike safety. The results of the global sensitivity analysis show that no particular category of dike features contributes significantly more to the uncertainty of the system. The highest contributing factors are the dike height, the critical velocity, and the coefficient of seaward slope roughness. These results underline the importance of good dike characterization in order to improve the predictability of return period estimations. The obtained return periods have been confirmed by current in situ observations, but the uncertainty increases for the most severe events due to the lack of long-term data.

The site of Salin-de-Giraud located in the Camargue area in southern France is a historically low-lying region and is thus frequently exposed to numerous storms. The latest Intergovernmental Panel on Climate Change report

The erosion problem of the dike is common in this area, and therefore assessment of erosion is necessary.
The semi-empirical approach based on hydraulic loading has been well established and traditionally used. Wave propagation from deep water to the surf zone has been well explored both analytically, numerically, and experimentally in the literature. A large overview of the theory surrounding random sea wave propagation theory was provided by

Regarding the statistical tool to predict a higher risk, the copula theory has been well accepted and used to calculate multivariate return periods of natural hazards.

The statistical study of coastal events requires relatively large, well documented, and high-quality datasets. Such historical data are not easy to find even in an area containing a dense network of coastal sensors. As a unified database of all records regarding offshore and coastal characteristics does not exist, we used data coming from different bases which contained the measures of interest with correct time synchronicity. We present the data in this section.

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.

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.

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 (

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

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

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

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

Once done, the copula can be calculated using Eq. (

The principle of the maximum-likelihood method that we use is that we try to maximize the function

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. (

The offshore significant wave height can be propagated up to the toe of the dike. Among the numerous methods, the most convenient one to use is the propagation formula written in Eq. (

This method is convenient and easy to use but can be imprecise, especially if the deepwater steepness is highly irregular and not constantly positive. The results can then be confirmed using numerical simulations, using a wave propagator such as TOMAWAC.

Once the wave reaches the toe of the dike, the wave will start interacting with the dike in what is called the overtopping phase. This phenomenon is divided into 3 steps, with the equations detailed in

The flow velocity will then decay along the crest (Eq.

According to

Main control parameters in the equation system of the framework with their reference value and their interval of variation for the GSA (global sensitivity analysis).

Defining the value of these parameters is not easy, and they may carry some amount of uncertainty that needs to be quantified. We propose a sensitivity analysis to resolve this problem.

We can now associate a terminal velocity with a set

By integrating the derivative of the copula with respect to

The showcased system is indeed able to provide return periods associated with events leading to erosion or any dangerous event defined as a criteria on flow velocity. However, added to the deep water conditions used to generate the copula are the characteristics associated with the dike, as well as many empirical parameters used to fit the laws allowing the calculations leading to the landward terminal velocity of the dike. All of these parameters carry an intrinsic amount of uncertainty which has a non-negligible impact on the results. This calls for an accurate quantification of the whole potential range of variation of each parameter. A global sensitivity analysis through the computation of global sensitivity indices will be our tool of choice. A combination of the first-order and total effect sensitivity indices (Eqs.

We estimate the value of the indices using the Saltelli estimator defined in

The first step is to define the parameters used in Eqs. (

If we provide our framework inputs that are uncertain, it should be expected that the uncertainty will be carried through the system up to the outputs. We rely on sensitivity analyses to quantify such uncertainty by comparing the influence of each parameter on the variation of the outputs relative to their respective range of variation. Since there may be a lot of interaction between parameters and we need to assess the influence of the parameters over their whole range of variation, we use a global sensitivity analysis.

Let

In order to quantify the influence of a single parameter

The total effect Sobol' index, which measures the influence of a parameter

define the input parameter space and the model output function;

generate a set of samples using Latin hypercube sampling (LHS) or another quasi-random sampling method (we used Sobol' sequence in our case);

compute the model output for each set of input parameters;

partition the output variance into components due to individual input variables and their interactions using an ANOVA-based decomposition;

calculate the first-order and total effect Sobol' indices, which measure the contribution of individual input variables and their interactions to the output variance, respectively.

Diagram highlighting the main steps of the process as well as involved methods.

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.

Cumulative distribution functions of the offshore significant wave height

We will then compute the derivative of the copula in Eq. (

The interdependence parameter can take values in the interval

Return period (in years on the contour lines) of an event composed of a couple

The contour lines of the copula in Fig.

We use the terminal velocity on the landward slope

Terminal velocity (in

Unsurprisingly, higher values of both

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

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.

Value of the first-order sensitivity (in red) and total effect (in blue) indices for each tested parameter.

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.

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.

Distribution of the return periods of an event able to provoke some amount of erosion to landward slope at the dike, with random variation of the parameters in Table

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

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

Evolution of the values of the first-order sensitivity index

Convergence has evidently been reached. It seems that we can safely use

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

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.

We have been able to build a complete automated framework allowing the user to estimate the expected return periods of events leading to erosion on the rear side of the earthen dike submitted to wave overtopping, assuming the correctly assessed ranges of variation of the parameters are provided. The framework itself needs, firstly, metocean data in order to create a reliable copula from wave and water level data, then a description of wave propagation to the toe of dike, and finally reliable laws representing wave overtopping process, run-off on the crest then on the landward slope, and bottom erosion.

The return period from which erosion on the Quenin dike located in Salin-de-Giraud starts is firstly estimated from reference parameters. This first estimate is equal to 6 years, which is significantly higher than the value of 2 years written in reports from the operating company. The framework is then able to take the parameters' uncertainty into account, which provides a generalized extreme value distribution of return periods which is right skewed with a peak around the 2 years value and a long tail in the upper range of the return periods. This result shows that a statistical study is necessary to determine a return period of damages in accordance with observed damages. Damages on a long dike are not observed on an average profile but on the weakest profile. That is why the peak of the statistical analysis is more representative than the first estimate based on average parameters. Sensitivity analysis is implemented into the framework and classifies the dike's parameters in terms of carried uncertainty. No clear trend can be observed for a specific category of parameter that would carry a significant part of the global uncertainty. However, the protocol allows us to clearly distinguish which parameters should be closely considered and which ones can be ignored. The results underline the importance of characterizing the dike using experiments and simulations in order to reduce the parameters' range of variation as much as possible. All processes contribute significantly to the uncertainty of the system, excluding the statistical treatment. This study case is indeed very specific, with a very low return period for damages and large variations of the dike crest. For any other dike, the framework is applicable by providing the appropriate input values.

Finally, the results can be provided relatively quickly without an enormous amount of computing power. They can indeed be validated using only a small set of points for the Quasi-Monte-Carlo process (around 15 000 points at most).

The data are freely available on demand from the corresponding author. REFMAR is freely accessible by creating an account at

CH – conceptualization, methodology, software, investigation, writing – original draft, data curation, visualization.AP – supervision, writing – review and editing, methodology, resources.PS – conceptualization, methodology, validation, surveillance, project administration, funding acquisition.AB – conceptualization, supervision, project administration.JJ - supervision, writing - review and editing, resources, funding acquisition, project administration.

The contact author has declared that none of the authors has any competing interests.

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

This article is part of the special issue “Hydro-meteorological extremes and hazards: vulnerability, risk, impacts, and mitigation”. It is a result of the European Geosciences Union General Assembly 2022, Vienna, Austria, 23–27 May 2022.

We hereby thank the Salins du Midi company for financially supporting the work and especially Pierre-Henri Trapy for providing useful information on the site, guidance, and access to the company's archives.

This paper was edited by Francesco Marra and reviewed by two anonymous referees.