The approach is applied in two ports in northern France, Saint-Malo and Le Havre, which are presented in Fig. 1.

To characterize oceanic forcing, we introduce three random variables: wave height H, wave period T and storm surge S. The wave height is the significant wave height that is noted H in order to simplify the notation. By convention, the random variables are written in capital letters, and the realizations of these random variables are written in lowercase (h, t, s). The wave data are derived from the Anemoc numerical database, which reconstructs sea conditions over a period of nearly 25 years by hindcast. Sea levels come from the tide gauges of various ports. The storm surges are deduced from these observations by removing from the water level the value of the astronomical tide obtained using the Shom Predit software. This software also allows the density of high tides to be obtained.

As the study focuses on the integration of tidal range in a macrotidal environment in the calculation of the probability of joint occurrence of waves and water levels, the used data are those of waves and surges taken at high tide. The sample is made of 706 events per year using the same definition as in the Rock Manual. In coastal engineering, it is customary to calculate the probability of occurrence of extreme tidal sea levels by making the product of convolution of high tide densities with the survival function of storm surges. As 706 annual tides occur, the annual probability of exceeding a given level is chosen in reference to this figure of 1/706. In addition, because coastal structures are located at shallow depths, the conditions that require the structures in terms of their stability or wave overtopping correspond to waves occurring with high water levels. That is why the privileged situations are high tides. The analysed samples are therefore pairs of storm surge associated with wave height at high tide. However, for safety, the wave height that is chosen is the maximum value over the 3 h time interval on either side of high tide. Kergadallan (2015) recommends selecting the maximum H value within a time window centred on the time of high water. Using the same data as him, this recommendation is followed. Statistical samples therefore contain tuples of wave and storm surge values at high tides at the rate of 706 annual pairs. Separated by about 12 h, the values may not fully meet the independent and identically distributed (i.i.d) assumption. However, this aspect is not considered here as it is often accepted in practice (see for example Hawkes, 2005).

Another approximation is the assumption of the presence of a unique wave population. This assumption is also not completely valid when we consider the wave direction of extreme events. The topic has already been discussed by Hawkes et al. (2002), Mazas (2019), and Mazas and Hamm (2017), among others. The treatment of wave direction can also be considered to be a fourth random variable of the oceanic forcing but has not been included in this work.

For low and moderate values the cumulative distribution functions (CDFs) FH, FT and FS are the empirical functions. For the strongest and extreme values, these cumulative distribution functions result from an adjustment of the exponential law. Survival functions, whether for storm surge or wave height, are therefore adjusted by piece by exponential-type analytical functions as close as possible to empirical frequencies, whereas extrapolations for extreme values use exponential laws.

For more than two variables, C is not generally a copula (impossibility theorem of Genest, 1995). According to Nelsen (2006), it is difficult to construct n-order copulas from n-1 copulas. We present two methods for the construction of trivariate copulas. In the first method, a trivariate copula generalizes the bivariate copula with three random variables and one parameter. In the second method, a trivariate copula associates two bivariate copulas with their two respective parameters.

The determination of the contour of equal joint exceedance probability P(H>h,T>t,S>s) consists of obtaining all the variables (H, T, S) associated with different return periods: T10 (10-year event), T100 (100-year event) and T1000 (1000-year event).

It is necessary to treat the extreme events that are characterized by a very low occurrence. The difficulty of taking them into account is of a statistical nature: the scarcity of observations. In order to take the extreme events into account, we introduce the concept of tail dependence. For a bivariate copula, the tail dependence measures the probability of simultaneous extreme realizations (Clauss, 2008). It is a highly relevant tool for the study of extreme values. We distinguish lower- and upper-tail dependences. They are characterized by their lower- and upper-tail dependence coefficients that are deduced from the following conditional probabilities, whose value is given by Eqs. (35) and (36) that are given by Clauss (2008) for cumulative distribution functions U1 and U2: 35P(U1≤u1|U2≤u2)=P(U1≤u1,U2≤u2)P(U2≤u2)=C(u1,u2)u236P(U1>u1|U2>u2)=P(U1>u1,U2>u2)P(U2>u2)=1+C(u1,u2)-u1-u21-u2.

Since we fix the lower-tail dependence coefficient λL and upper-tail dependence coefficient λU by equations 37λL=limu→0P(U1≤u|U2≤u),38λU=limu→1P(U1>u|U2>u), we deduce the definitions of tail dependence coefficients.

Definition: The lower-tail dependence coefficient is defined by 39λL=limu→0C(u,u)u.

The copula C has a lower-tail dependence if λL exists with λL∈]0,1].

If λL=0 then the copula does not have a lower-tail dependence.

Definition: The upper-tail dependence coefficient is defined by 40λU=limu→11+C(u,u)-2u1-u.

The copula C has an upper-tail dependence if λU exists with λU∈]0,1].

If λU=0 then the copula does not have an upper-tail dependence.

In the following section, we use survival copula C‾ and survival function u‾. The lower-tail dependence corresponds then to high wave heights and water levels.

The tail dependences of the different copulas are determined in Nelsen (2006) and Roncalli (2002) from their tail dependence coefficients. They are expressed in Table 1. The Ali–Mikhail–Haq copula and Gaussian copula are noted AMH and Gauss, respectively, in what follows.

We find that some copulas do not have lower- and upper-tail dependence coefficients. They are inappropriate in case of extreme dependence. Some copulas have a lower-tail dependence; others have an upper-tail dependence. The tail dependence of the sample is firstly checked. For this we graphically represent the evolution of C‾u‾,u‾u‾ and determine its limit when u‾ tends to 0. Since u‾ is used, we note the tail dependence coefficients λ‾L and λ‾U. We can therefore decide whether the sample does or does not have a lower- or upper-tail dependence. In choosing the copula, it is essential to satisfy the class of tail dependence of the sample. If the sample does not have a tail dependence, then the use of the Gaussian copula or another copula with the same class or tail dependence is recommended. If the sample has a lower-tail dependence, the use of a copula with a lower-tail dependence or the survival copula with an upper-tail dependence is recommended. If the sample has an upper-tail dependence, the use of a copula with an upper-tail dependence or the survival copula with a lower-tail dependence is recommended. We can also deduce the parameter of the copula from the tail dependence coefficient given by the sample. The method that is proposed here for assessing the sample dependence refers to lower-tail dependence. Other methods exist such as the chi-plot proposed by Fisher and Switzer (1985, 2001) and used in coastal analyses by Mazas and Hamm (2017) for instance.

The representation of the contours requires knowledge of the statistical laws of adjustment of the different parameters. We therefore present these laws. For the two sites of Saint-Malo and Le Havre, we have used data files that provide the values for wave height, wave period and storm surge at high tide over a time period of about 20 years. The file for the Le Havre site includes, for example, around 15 000 values. The wave data are extracted from the Anemoc numerical database. Sea levels at high tide are extracted from tide gauge measurements. The astronomical tide is obtained from the Shom Predit software.

Adjustments of the statistical laws are made according to the peaks-over-threshold (POT) method on the basis of the exponential law.

The copula parameters are calibrated from samples where wave height values less than 1 m are excluded (see Fig. 2), thus reducing the sample size to about 3000 values. The copulas are fitted to all pairs or triplets of observations where the wave height exceeds 1 m. This threshold of 1 m that is used for filtering wave height excludes the swells and leaves only a very homogeneous population of pure wind waves. This treatment removes long wave periods and increases the dependence between wave height and wave period.

The use of the simplified Defra method in Ciria et al. (2007) is common among European coastal engineers for the study of wave overtopping or armour damages in coastal structures. It refers to the complete Defra method presented for example by Hawkes (2005) that is based on the Gauss copula. The complete Defra method is close to the method that is used in this paper. The main difference is the choice of the Gauss copula that does not present tail dependence. The simplified Defra method refers to univariate survival functions F‾H and F‾S rather than cumulative distribution functions of wave height and storm surge as coastal engineers usually work with exceedance probability rather than with non-exceedance probability. In this simplified method, the bivariate survival function is related to univariate survival functions by the expression 41F‾HS=FDF‾HF‾S.

In France, the order of magnitude for the dependence factor FD coefficient is about 20. Kergadallan (2013) recommends however a minimum value of 25 for safety reasons. This factor corresponds to a weak dependence. For a very strong dependence, FD is between 500 and 1000.

The bivariate survival functions F‾HS of Table 4.15 of Rock Manual (Ciria et al., 2007) are determined with Eq. (41). Figure 3 shows the differences between observed bivariate survival functions and calculated bivariate survival functions using the simplified Defra method. The points of calculations in blue lie far from the first bisector in black in the figure. This shows that the use of the Defra simplified method is inappropriate. This is due mostly to the use of the simplified Defra method of Eq. (41), but the complete Defra method with the Gauss copula would not also perfectly represent the extreme events because the Gauss copula has no tail dependence, as we see later.

In order to improve the results, we now introduce the copula theory.

The sample is analysed in order to determine its tail dependence. Indeed, the result will condition the choice of the copula depending on whether the sample has the same class of tail dependence as the copula or not. In Eqs. (22), (26), (30) and (34), the survival copula C‾ is used with survival functions. In the following section, we use survival copula C‾ and survival function u‾. Upper-tail dependence and lower-tail dependence will be inverted. We are interested in the extreme events with high wave heights and water levels. For survival copula C‾, we determine below its limit for survival function u‾ tending to 0. The lower-tail dependence corresponds to these high wave heights and water levels.

For the Saint-Malo sample, C‾u‾,u‾u‾ tends to around 0.2 when u‾ tends to 0.

For the Le Havre sample, C‾u‾,u‾u‾ tends to around 0.4 when u‾ tends to 0.

Using the survival function u‾, these two samples have a lower-tail dependence, which justifies the use of the Clayton copula. We determine the Clayton copula parameter from the lower-tail dependence coefficient of the sample. With the Clayton copula, we can determine the value of its copula parameter in Saint-Malo and Le Havre with the equation 42θ=-ln⁡2ln⁡λ‾L.

This copula parameter is 0.43 and 0.76, respectively.

Note: as the Gumbel copula has an upper-tail dependence it is not recommended. In contrast, its survival copula with a lower-tail dependence is appropriate. This analysis of the sample makes it possible to understand why the survival Gumbel copula gives a minimum error close to the minimum error of the Clayton copula. We can therefore expect survival Gumbel copula results to be close to the results obtained by the Clayton copula.

In order to assess the robustness of the copulas, we show in Fig. 6 the observed and calculated joint frequencies for the Le Havre sample (3 040 pairs of values). The copula represents reality more closely when the points approach the bisector y=x. The simplified Defra method currently in use does not give a good representation of the reality of the joint frequencies for wave height and storm surge. The Clayton copula provides a good representation. In contrast, the Gumbel copula gives a bad representation. The explanation is in the analysis of the sample carried out in Sect. 3.3: we show that the sample had a lower-tail dependence, whereas the Gumbel copula has an upper-tail dependence. In contrast, the survival Gumbel copula provides a good representation of the reality of joint frequencies for wave height and storm surge. The explanation lies in introducing the survival copula. The tail dependence of the survival Gumbel copula is opposite to the tail dependence of the Gumbel copula. The survival Gumbel copula therefore has the right class of tail dependence. The results obtained by the AMH copula are surprisingly correct. Kumar (2010) shows that the AMH copula does not have tail dependence except if the copula parameter is equal to 1. In our case, the copula parameter is close to 1. The copula therefore seems to behave like a copula with a lower-tail dependence. We show here the utility of the Clayton copula in comparison with the Gumbel copula and the Defra method that is currently in use.

The results highlight the importance of the class of tail dependence of the sample in copula selection. If the sample has a tail dependence it is necessary to select a copula with the same tail dependence. The Clayton copula that has the same class of tail dependence as the sample gives a calculated joint frequency close to the observed joint frequency. Conversely the Gumbel copula does not correctly represent the observed joint frequency: it moves away from the bisector for the extreme points. This is because the sample has a tail dependence opposite to that of the Gumbel copula. In order to restore the proper tail dependence, we resort to the survival copula. The latter comes close to the bisector but is slightly less robust than the Clayton copula. It should be noted that calibration is performed on the entire sample. By truncating the sample for joint frequency values below 0.01, we would have obtained a much larger parameter for the Gumbel copula with results that are closer to measurements.

We selected the Clayton copula for the ports of Le Havre and Saint-Malo using three methods. In order to validate the Clayton copula, we analysed samples from 19 sites of the French coast along the Atlantic and English Channel with the maximum-likelihood method. We always obtained the greatest maximum likelihood with the Clayton copula or the AMH copula (see Appendix C). The sample always has lower-tail dependence (see Appendix B). Even though at some sites the AMH copula provides a larger likelihood than the Clayton copula, it should not be chosen because it has a particular kind of behaviour. It has a lower-tail dependence if the copula parameter is 1 (or close to 1 in practice). If the parameter is not 1, the AMH copula does not have tail dependence, and its interests disappear. Since the robustness depends on the copula parameter and on the site, it cannot be recommended for general use. We can therefore conclude that the Clayton copula is the most appropriate copula for our application. For this purpose, Table 4 gives the parameters of the different sites.

We show in Fig. 9 that there is a coastal area with a maximal dependence from Concarneau to Port-Bloc (in grey in the figure). This area is the most exposed to wind that comes mainly from the west direction along the French Atlantic coast.

We have tested hierarchical construction using a fully nested hierarchical Archimedean copula. In this type of construction, we build a bivariate copula between two variables; then we create a trivariate copula with the previous copula and the third variable using another bivariate copula. Unlike Corbella and Stretch (2013), who uses the same copula parameter for the two bivariate copulas, we introduce two different copula parameters.

With the fully nested hierarchical copula method, we first determine the most appropriate bivariate copula for the three variables: (T,S), (H,T) and then (H,S). We construct the bivariate distribution function using the selected copula for the two most correlated variables. We determine the most relevant copula between the function obtained with the two most correlated variables and the third variable.

We represent in Fig. 10 trivariate joint exceedance probability for return periods of 10, 100 and 1000 years. The trivariate copula used is therefore constructed from a Clayton copula parameter 2.37 connecting H and T and a copula parameter 0.56 connecting FHT and FS.

In order to better visualize the incidence of return periods on trivariate joint exceedance probability, cross sections along (H,T), (H,S) and (T,S) are shown for T=T1, H=H1 and S=S1 in Fig. 10a–c, respectively.

In Fig. 10a–c, a constant wave period, a constant wave height and a constant storm surge, respectively, are fixed corresponding to an annual return period. We show the joint exceedance probability of wave height and storm surge, of storm surge and wave period, and of wave height and wave period, respectively, for three return periods of 10, 100 and 1000 years. In the three figures we recognize the usual pattern and the characteristics of a strong correlation for (H,T). In Fig. 10c we recognize indeed the classic pattern of contours for very dependent variables. Wave height and wave period are the most correlated variables. This result is not surprising all the more since we deal with pure wind waves after we have removed the swell. In Fig. 10d, a relationship between H and T is obtained with a trivariate copula with (H,S) satisfying a joint exceedance probability of 1000 years and with T, which maximizes the trivariate joint probability density function. This relationship enables us to obtain the wave period from the wave height and the storm surge.

In order to show the utility of the constructed trivariate copula, we determine the error rate of the different copulas in the Le Havre area using the formula of the error given by Eq. (1) and the definition of the error rate given by exp⁡(e)-1 (see Table 7).

The results obtained by the trivariate copula constructed by two bivariate copulas and two parameters are the best of the four fitted trivariate copulas. However, by aggregating the most correlated variables first, the robustness improves, as stated by Charpentier (2014).

As expected, with one parameter the Archimedean copula is less robust than the fully nested hierarchical copula with two parameters.

It can also be seen that by associating the most correlated variables (H,T), the Clayton copula gives better results than the Gumbel copula. For a single parameter the trivariate copula constructed with the Clayton copula is significantly more accurate than the Gumbel copula.

Finally, Table 7 shows that the choice of the copula is much more important than the choice of the trivariate method of construction. This result validates our choice of a simple method of construction that can even lead to the most robust results according to Corbella and Stretch (2013).

Table 8 presents the goodness of fit of the trivariate copulas for the port of Le Havre through the chi-squared test (CHI-2) and the Kolmogorv–Smirnov test (KS). The best results are obtained with two parameters. As expected, the fit of the single-parameter Archimedean copula and of the fully nested hierarchical copula is exactly the same copula as shown in Table 8. The results highlight the contribution of trivariate copulas constructed as a fully nested hierarchical copula with the help of two Clayton bivariate copulas and two parameters by first aggregating the two most correlated parameters.