Trivariate copula to design coastal structures

Some coastal structures must be redesigned in the future due to rising sea levels caused by global warming. The 5 design of structures subjected to the actions of waves requires an accurate estimate of the long return period of such parameters as wave height, wave period, storm surge and more specifically their joint exceedance probabilities. The Defra method that is currently used makes it possible to directly connect the joint exceedance probabilities to the product of the univariate probabilities by means of a simple factor. These schematic correlations do not, however, represent all the complexity of the reality and may lead to damaging errors in coastal structure design. 10 The aim of this paper is therefore to remedy the lack of accuracy of these current approaches. To this end, we use copula theory with a copula function that aggregates joint distribution function to its univariate margins. We select a bivariate copula that is adapted to our application by the likelihood method with a copula parameter that is obtained by the error method. In order to integrate extreme events, we also resort to the notion of tail dependence. We can select the copulas with the same tail dependence as data. In the event of an opposite tail dependence structure, we resort to the survival copula. The tail 15 dependence parameter makes it possible to estimate the optimal copula parameter. The most accurate copulas for our practical case with applications in Saint-Malo and Le Havre (France), are the Clayton normal copula and the Gumbel survival copula. The originality of this paper is the creation of a new and accurate trivariate copula. Firstly, we select the fittest bivariate copula with its parameter for the two most correlated univariate margins. Secondly, we build a trivariate function. For this purpose, we aggregate the bivariate function with the remaining univariate margin with its parameter. We show that this trivariate 20 function satisfies the mathematical properties of the copula. We can finally represent joint trivariate exceedance probabilities for a return period of 10, 100 and 1000 years.


Introduction
The design of coastal structures requires the multiplicity of variables and their degree of correlation to be taken into account.
We must therefore address the lack of accuracy of the dependencies between the different variables characterizing the sea state 25 (Sergent et al., 2014;Hawkes, 2005) such as wave height H, wave period T and storm surge S. The design of coastal structures is based in particular on the return periods of wave overtopping or of armour damage. The aim of this paper is to improve the accuracy of estimating them in order to avoid costly and inappropriate decisions (Li et al., 2008). To this end, we provide accurate estimates of the correlations between the variables H, T and S and obtain reliable return periods. Currently, in reference manuals such as the Rock Manual (Ciria, 2009), it is recommended that a factor be applied to the return periods of the variables 30 taken separately to determine the joint period.
Copulas are relatively innovative mathematical tools for modelling the dependence structure of several random variables. The theory of copulas was developed by the mathematician Sklar (1959). The copula is a written form of the joint distribution function that provides all the information on the dependency structure. The recent interest in copulas started in financial risk management and insurance. Its use in environmental science especially concerns hydrology. In order to estimate the probability 35 of failure of coastal or offshore structures caused in particular by the critical appearance of the combinations of parameters during a storm, Salvadori et al. (2007) use a copula in order to link the intensity of storm surge to its duration.
Using the copula theory, Hawkes (2005) obtains, for example, all the pairs of variables wave height H and surge S for a given return period. The bivariate return period can be generalized to the multivariate case. In this paper we propose the use of https://doi.org/10.5194/nhess-2020-80 Preprint. Discussion started: 19 March 2020 c Author(s) 2020. CC BY 4.0 License. copulas to take into account the dependence between three variables H, T and S. Copulas generally only allow two parameters. 40 The purpose of this article is the creation of a new trivariate copula and the evaluation of its accuracy even though this is a complicated exercise according to Nelsen (1985). In the literature the Chakak and Koehler (1995) method is found but it has a compatibility problem. Corbella and Strech (2013) nevertheless use the Chakak and Koehler procedure and also the conditional mixture to create the trivariate copula from the bivariate copulas.
In a first part, we define the theory by presenting the marginal distribution, the recommended method of the Rock Manual, the 45 normal copula, the bivariate copula, the tail dependence, the survival copula, the trivariate copula and iso-values for different return periods. We obtain a bivariate copula and the copula parameter by the method of maximum likelihood and the method of the error. We show that the trivariate function that is obtained satisfies the mathematical properties of a copula. In a second part, we present the isovalues for applications at the ports of Le Havre and Saint-Malo (France) with bivariate copulas corresponding to different return periods. We show that the Clayton and Gumbel copulas are the most accurate copulas for our 50 practical applications of coastal engineering. Finally, in a third part, we apply trivariate copulas on the same sites.

Theoretical approach
After recalling the notion of a bivariate distribution function, currently used in applications, we present the notion of copula with the selection and the construction of trivariate copulas.

Data used 55
To characterize oceanic forcing, we introduce three random variables, wave height H, wave period T and storm surge S. 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 probability density functions (PDF) fH, fT and fS are the result of calibrations of statistic exponential laws of data recorded at high tide and collected by the Candhis wave buoy network for waves and by tide gauge measurements recorded in the ports 60 for storm surge.
As the study focuses on the integration of tidal range in the macrotidal environment in the calculation of the probability of joint occurrence of waves and water levels, the data used are those of waves and surges taken at high tide. For low and moderate values the density functions are the empirical density functions. For the strongest and extreme values, the density functions result from an adjustment of the exponential law. 65

Bivariate cumulative distribution function
We denote by the cumulative distribution function (CDF) of a random variable defined by : where P is the probability.
We must model the correlation between, for example, wave heights H and storm surges S by proposing a relation defining the joint cumulative distribution function from the univariate cumulative distribution functions. We thus seek to obtain a function C which links the bivariate cumulative distribution frequency ( , ) to the univariate cumulative distribution frequencies ( ) and ( ) by integrating a correlation parameter.

Current practice
The Defra method [2005] currently used makes it possible to directly connect the joint probability density function to the product of the univariate probability density functions and through a dependence factor denoted FD : 85 The dependence factor FD depends on the correlation coefficient ρ obtained from the Gaussian copula (see definition in § 2.4).
The variables X and Y for the bivariate analysis are generally wave height H and storm surge S. The dependence factor is site specific and results from the analysis of the local correlation between wave heights and storm surges.
The correspondence table between the correlation coefficient ρ and the dependence factor FD is given by Kergadallan [2013]. 90 This table recommends, for example, for the North Sea, Channel and Atlantic coast the use of a minimum dependence factor FD of 25.

Copulas
The copula is a statistical tool to characterize the dependence between several random variables where linear correlations are generally not able to represent them accurately (Charpentier, 2014). According to the latter, copulas have become an important 95 tool for modelling a multivariate law that "couples" univariate cumulative distribution functions, hence the Latin name "copula" name chosen by Sklar [1959].
If C is the copula associated with a random variable vector (X, Y) then the copula C couples the univariate cumulative distribution functions ( ) and ( ) using (4).

Name Copula Generator
Inverse generator

Elliptic copulas
Elliptic copulas are Gaussian and Student's copulas : 125 The Gaussian copula is written as follows : is a distribution function of , with X = ( 1 , 2 , … , ) a Gaussian random vector (X~ (0, ∑)), where ∑ is a covariance matrix. 130 Student's copula is written as follows : is a distribution function of the univariate Student distribution law with ν degrees of freedom.
They are symmetrical copulas. They are widely used in finance. They are implicit and therefore do not have an explicit 135 analytical form.

Archimax copulas
Archimax copulas include a large number of copulas, including Archimedean copulas.
We will adopt the following notation (0) = lim u−>0 (t) −1 (s) = 0, for s ≥ (0) For more information, refer to reference books such as Joe [1997] and Nelsen [1999]. The reader may also refer to Clayton [1978]. 150 2.5 Selection of the best bivariate copula by two methods

The error method
We illustrate the method for the random variables wave height H and storm surge S. This method consists in determining the mean error e between the calculated joint cumulative distribution function (ℎ, , ) with the copula C and its parameter θ and the observed joint cumulative distribution function (ℎ, ). 155 with n the number of pairs of values (ℎ , ).
For each copula, we first determine the parameter θ that minimizes the error e. We then select the copula with the lowest minimum error.

The maximum likelihood method 160
Let us call X the sample of measures (x1, x2, ... , xn) with bivariate xi = (hi, si), i = 1, .., n. The likelihood function L is defined by equation (16) : where is the probability density function of the bivariate cumulative distribution function . θ is the parameter of the copula.
The maximum likelihood method consists in finding the parameter θ, which maximizes the probability of obtaining the sample 165 (Tassi, 2004). Since likelihood is a product of density we take its log-likelihood in order to facilitate calculations. We can thus work with the sum and derive it with respect to θ.
The best copula is the copula with the largest likelihood.

Construction of a trivariate copula
For more than two variables, C is not generally a copula (impossibility theorem of Genest [1995]). According to Nelsen [2006], 170 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.

Trivariate copula with one parameter : a multi-level archimedian trivariate
Since we are looking for the correlation between three variables, the first idea is to generalize the bivariate copula ( 1 , 2 ) 180 to obtain ( 1 , 2 , 3 ). We must check that ( 1 , 2 , 3 ) is a copula, which is difficult. However Archimedean copulas like Gumbel and Clayton can be extended to an order greater than 2 using the property of equation (10).
For a Clayton copula of order n, this gives : For Clayton copula of order 3, it gives : For Gumbel copula of order n, it gives : 185 For Gumbel copula of order 3, it gives : By taking a single copula parameter for the three variables, we do not differentiate the two-to-two correlations of the variables even though some variables may be more correlated than others.

Trivariate copula with two parameters : a fully nested hierarchical copula
To better take into account the correlations of variables two by two, one option is to build trivariate functions from bivariate 190 copulas as a fully nested hierarchical copula: Corbella [2013] tests a fully nested hierarchical copula but he uses a unique bivariate copula and does not distinguish the two bivariate copulas 1 and 2 . 1 is a bivariate copula with 1 as copula parameter. 2 is a bivariate copula with 2 as copula parameter. We must check that this function (24) is a copula and satisfies the properties of equations (18). We first aggregate the two most correlated variables with the copula 2 and its copula parameter. We then add the third random variable with the 195 copula 1 and its copula parameter. We will show later that this order provides the most accurate copula.

Validity of copula properties for 2.6.3
We do not know any general methods to build high order copulas from low order copulas (Durrleman, 2010). Generally satisfies the three properties of equation (18) with d = 3, which is difficult. However Charpentier [2014] points out that C is a 200 copula if it satisfies i) or ii). i) 1 and 2 are both Clayton or Gumbel copulas with parameters 1 for 1 and 2 for 2 positive and growing.
ii) 1 and 2 are both Archimedean copulas of respective generator 1 , 2 with 2 o 1 −1 being the inverse of a Laplace transform.
For Gumbel and Clayton copulas 1 and 2 that are Archimedean copulas we check the condition (ii) that there is a function 205 f for which the inverse Laplace transform −1 satisfies : For 1 and 2 Clayton copulas we have as the generator of 2 and as the inverse generator of 1 : This gives : We can find that : 210 With Γ( , ) the incomplete Gamma function set by for a complex with real part(a) > 0 : We can conclude that there is a function f such that ϕ 2 oϕ 1 −1 = −1 [ ] : For 1 and 2 Gumbel copulas we have as generator of 2 and as the inverse generator of 1 : This gives : We can find that : 215 With Γ Gamma function, defined by : We can conclude that there is a function f such that 2 o 1 −1 = −1 [ ] :

Determination of the contour of equal joint exceedance probability 220
The determination of the contour of equal joint exceedance probability consists in 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).

Bivariate probability without tide
We deal with a set of pairs of values (h, s) that satisfy : ̅ is the selected bivariate survival copula. ̅ , ̅ are survival functions associated with the variables. The values f10, f100 or f1000 225 are the frequencies corresponding to the ten-year, hundred-year and thousand-year periods.

Bivariate probability with tide
The bivariate probability with tide requires the development of the copula connecting wave height and storm surge. We can then define the joint survival function of the wave height and the storm surge. The chosen calculation method favors high tide.
The sea levels considered are therefore the sums of the astronomical high tide (generated by the attraction of the moon and the 230 sun without weather disturbance) and the storm surges raised at the time of these astronomical high tides. This method is of course valid only for macrotidal seas. The equation (37) established by Simon [1994] gives the probability that the sea level at high tide N exceeds a given value n : z is the height of the high tide, between the minimum and maximum values and respectively at high tide. 235 ( ) is the probability that the high tide is between z and z + dz.
̅ (s) is the probability of observing a storm surge S larger than s, thus ̅ (s) = P(S > s).
The bivariate survival function for wave height H and sea level N is therefore written as follows : This can be written by introducing the survival copula ̅ : The set of pairs (h, n) corresponding to the different return periods, the ten-year, hundred-year and thousand-year periods, 240 satisfies : It is thus possible to represent the contour of equal joint exceedance probability associated with the variables wave height and sea level .

Trivariate joint exceedance probability with tide
The trivariate survival function for wave height H, wave period T and sea level N is written as follows: This can be written by introducing the selected survival copula ̅ 2 : 260 This expression can be written by introducing the survival copula ̅ 1 connecting F ̅ and ̅ .

Tail dependence of the sample
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, it measures the probability of simultaneous extreme realizations (Clauss, 2009). It describes the dependences of distribution tails for the simultaneous occurrence of extreme values. It is a highly 270 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 equations (49) and (50) that, in tum, are given by (Clauss, 2009) : P( 1 > 1 | 2 > 2 ) = ( 1 > 1 , 2 > 2 ) ( 1 > 1 ) = 1 + ( 1 , 2 ) − 1 − 2 1 − 2 Since we fix the lower tail dependence coefficient and upper tail dependence coefficient by equations (51) and (52) : 275 We deduce the definitions of tail dependence coefficients.

Definition:
The lower tail dependence coefficient is defined by : The copula C has a lower tail dependence if exists with Є ]0,1]. 280 If = 0 then it does not have a lower tail dependence.

Definition :
The upper tail dependence coefficient is defined by : The copula C has an upper tail dependence if exists with Є ]0,1]. The tail dependences of the different copulas are determined in (Nelsen, 2006) and (Roncalli, 2002) from their tail 285 dependence coefficients. They are expressed in Table 2. The tail dependence is firstly checked. For this we graphically represent the evolution of C(u, u)/u and determine its limit when 290 u tends to 0. We can therefore decide whether the sample has or does not have a lower or upper tail dependence.
In choosing the copula, it is essential to satisfy the tail dependence of the sample.
If the sample does not have a tail dependence, then the use of Gaussian copula or Student copula or other copula with the same tail dependence characteristics is recommended.
If the sample has a lower tail dependence, the use of a copula with a lower tail dependence or the survival copula of a copula 295 with an upper tail dependence is recommended.
We can also deduce the parameter of the copula from the tail dependence coefficient given by the sample.

Results for bivariate copulas
We select the most appropriate copulas at both the Le Havre and Saint-Malo (France) sites using two methods. We analyze the tail dependence of the two samples. We represent the contour of equal joint exceedance probability with the selected 300 copulas for three return periods in order to assess the relevance of the copulas.

State of the art : Defra method
Until now the simplified Defra method has been quite popular among coastal engineers. The method refers to univariate survival functions ̅ and ̅ of wave height and storm surge. The reason is that coastal engineers usually work with 310 exceedance probability rather than with non exceedance probability. In this simplified method, the bivariate survival function is related to univariate survival functions by expression (55). In France, the order of magnitude for the FD coefficient is about 20. In order to improve the results we now introduce the copula theory.

Analysis of the tail dependence
The sample is analyzed in order to determine its tail dependence. This will affect the choice of copula. Since the sample has a 320 tail dependence, it should be known whether it has a lower tail dependence or an upper tail dependence. Indeed, the result will condition the choice of the copula depending on whether the sample has the same tail dependence as the copula or not. To simplify the notation, we will choose the survival copula ̅ of equation (40) as copula C. We determine its limit for u tending to 0. https://doi.org/10.5194/nhess-2020-80 Preprint. Discussion started: 19 March 2020 c Author(s) 2020. CC BY 4.0 License.
Note : as the Gumbel copula has an upper tail dependence, the use of its survival copula is recommended. This analysis of the sample makes it possible to understand why the Gumbel survival copula gives a minimum of error much close to the minimum error of the Clayton copula. We can therefore expect Gumbel survival copula results that are close to the results obtained by Clayton copula.  For the set of copulas we determine their maximum likelihood with their parameter. We will select the copula that has the same tail dependence as the sample with the largest likelihood. 340

The Log-likelihood method
For the Saint-Malo sample, we chose the Clayton copula, which has the same tail dependence as the sample, with a loglikelihood of 291 in table 3. For the Le Havre sample, we also chose the Clayton copula, which has the same tail dependence as the sample, with a log-likelihood of 387.
The Clayton copula parameters obtained by the tail dependence coefficients come close to those obtained by the Log likelihood method for the Le Havre sample (3040 values) and the Saint-Malo sample (5888 values). 345 For Le Havre, we obtain 0.74 as the parameter of the Clayton copula using the method of maximum likelihood and 0.76 with the tail dependence coefficient.
For Saint-Malo, we obtain as 0.38 the parameter of the Clayton copula using the method of maximum likelihood and 0.43 with the tail dependence coefficient.
The value of the log likelihood of the Gumbel survival copula is large. It comes close to the Clayton copula. In addition, the 350 Gumbel survival copula has the same tail dependence as the Clayton. It is therefore as suitable as the Clayton copula.
The Gauss, Student and especially the AMH copula have a relatively large likelihood. However, they do not have a correct tail dependence. They cannot therefore correctly represent the tail dependence. We will come back later to the AMH copula which has a special property.

The error method for the Clayton, Gumbel and survival Gumbel Copula 355
In order to select the most relevant copula, we represent the mean error e between the calculated survival function (ℎ, , ) with the copula C and its parameter and the measured (ℎ, ).  Figure 4 for the ports of Saint-Malo and Le Havre shows that the error that is obtained with the Gumbel survival copula is very close to that obtained with the Clayton copula. The curve of the error obtained by the Gumbel copula survival however has a 360 very acute minimum. Obtaining the parameter of this copula will therefore be very sensitive to the value of its minimum error.
It will therefore be necessary to determine it very precisely.
We note Emin the minimum of the error e and Error rate = exp (Emin) -1.  Table 4 : Emin, error rate and copula parameter for the Clayton, Gumbel and Gumbel survival copula in the ports of Le Havre and Saint Malo. 370 Table 4 is used to verify that Clayton copula is the most accurate copula. It also appears that Gumbel survival copula is also an appropriate option.
We have therefore shown by two methods that the Clayton copula is the most relevant for the Saint-Malo and Le Havre sites.
The parameters obtained by the different copulas using the two methods are shown below.  The parameters of the copula obtained by the error method are close to those obtained by the method of maximum likelihood for the Clayton copula.

Comparison of observed and calculated joint frequencies 380
In order to assess the accuracy of the copulas, we show the observed and calculated joint frequencies for the Le Havre sample (3040 pairs of values). The copula represents reality more closely as the points approach the bisector y = x.
The 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 points obtained by the Defra method are very far from the bisector.
The Clayton copula provides a good representation of the reality of joint frequencies for wave height and storm surge. The 385 points obtained by the Clayton copula come close to the bisector.
In contrast, the Gumbel copula does not give a good representation of the reality of the joint frequencies for wave height and storm surge. The points obtained by the Gumbel copula move away from the bisector. The explanation is therefore in the analysis of the sample carried out in § 3.2 : we showed that the sample had a lower tail dependence whereas the Gumbel copula has an upper tail dependence. 390 The Gumbel survival copula provides a good representation of the reality of joint frequencies for wave height and storm surge.
The points obtained by the Gumbel survival copula come close to the bisector. The explanation lies in the fact of introducing the survival copula. The tail dependence of the Gumbel survival copula is opposite to the tail dependence of the Gumbel copula. We therefore reestablish a right tail dependence which gives correct results.
The results obtained by the AMH are surprisingly correct. Kumar [2010] shows that the AMH copula does not have tail 395 dependence except if the copula parameter is equal to 1. In our case, the copula parameter is close to 1. The copula seems therefore to behave like a copula with a lower tail dependence.
We show 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 in copula selection of the tail dependence analysis of the sample. If the sample has a tail 400 dependence it is necessary to select a copula with the same tail dependence. The Clayton copula that has the same tail https://doi.org/10.5194/nhess-2020-80 Preprint. Discussion started: 19 March 2020 c Author(s) 2020. CC BY 4.0 License. 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 the bisector but is slightly less accurate than the Clayton copula. It 405 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.  As can be seen in Figure 6, contours of equal joint exceedance probabilities obtained by Clayton are very far from those obtained by Gumbel and the Defra method. On the contrary, the joint exceedance curves obtained using the Gumbel survival copula are very similar to those obtained with Clayton. Results are therefore very sensitive to the choice of copula. A poor choice may lead to undersizing and may have economic consequences. 420  With tide the effect of storm surge on the sea level is small. The tidal range, which has an amplitude much larger than the 425 storm surge especially for the port of Saint Malo, mitigates the variations due to the storm surge. In particular, for the port of Saint Malo, it can be seen that sea level is less sensitive to variations in the return periods than storm surge (cf. Figure 7).

Conclusion on selecting of the best bivariate copula
We selected the Clayton copula for the ports of Le Havre and Saint-Malo using three methods. In order to validate the Clayton 430 copula, we analyzed 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 2). The sample always has a lower tail dependence (see Appendix 1). We can therefore conclude that the Clayton copula is the most appropriate copula for our application. For this purpose, the following  Even though in 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 behavior. 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 accuracy falls. Since the accuracy depends on the copula parameter and on the site, it cannot be recommended for a general use. reasons. He therefore recommends the nested hierarchical construction with Archimedean copulas. We have tested hierarchical construction,.using a fully nested hierarchical archimedean copula. In this type of construction, we build a bivariate copula between two parameters, then we create a trivariate copula with the previous copula and the third parameter. Unlike Corbella we introduce two parameters.

Construction of the best trivariate copula for the port of Le Havre 450
We first determine the most appropriate copula for two parameters: (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.

Bivariate copula for the three random variables
To determine the best bivariate copula we assess the maximum likelihood between (FH, FS), (FT, FS) Table 7 : Log-likelihood and copula parameter for the different bivariate copulas between the parameters H and S, T and S 460 then H and T.

Determination of the best trivariate copula
We determine the maximum likelihood between FH,T and FS with the different copulas In conclusion, we have thus aggregated the most correlated H and T parameters with the best performing Clayton copula. We also used Clayton copula to aggregate , and . The aggregation requires two different parameters. 470

Statistical law for adjusting wave heights, surges and periods
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 475 wave height, wave period and storm surge at high tide over a time period of about twenty years. The file for the Le Havre site includes, for example, around 15.000 values. The wave data are extracted from the ANEMOC digital 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 POT method on the basis of the exponential law.
The copula parameters were calibrated from samples where wave height values less than one meter were excluded, thus 480 reducing the sample size to about 3.000 values.

Contours of equal joint exceedance probability with a trivariate copula
We represent trivariate joint exceedance probability for return periods of 10, 100 and 1.000 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 485 FHT and FS.
In order to better visualize the incidence of return periods on trivariate joint exceedance probability, cross-sections along (H,     In Figure 8, a constant wave period is fixed corresponding to an annual return period. We show the joint exceedance probability 490 of wave height and storm surge for three return periods of 10, 100 and 1.000 years.
In Figure 9, a constant wave height is fixed corresponding to an annual return period. We show the joint exceedance probability of the storm surge and the wave period for three return periods of 10, 100 and 1.000 years.
In Figure 10, a constant storm surge is fixed corresponding to an annual return period. We show the joint exceedance probability of the wave height and the wave period for three return periods of 10, 100 and 1.000 years. 495 In the three latter figures we recognize the usual pattern and the characteristics of a strong correlation for (H, T).
In Figure 11, a relationship between H and T is obtained with a trivariate copula with (H,S) satisfying a joint exceedance probability of 1.000 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.

Error rate and goodness of fit for trivariate copulas
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 equation (15) and the definition of the error rate given by exp(e) -1.

Copula
Clayton Gumbel ( ( , ), )  Table 9 : Error rate of the different trivariate copulas for the port of Le Havre. 505 The results obtained by the trivariate copula constructed by two bivariate copulas and two parameters are generally good.
However, by aggregating the most correlated variables first, accuracy improves. 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  The best results are obtained with two parameters. With one parameter Archimedian copula and fully nested hierarchical 515 copula are exactly the same copula as shown in Table 10.
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.

Conclusion 520
Wave structure designers must accurately estimate return periods of parameters such as storm surge, wave height and wave period, and more specifically, their joint probabilities of exceedance. In present practice, this joint probability of exceedance is related to the product of univariate probabilities by means of a simple factor. This method can cause damaging design errors.
After highlighting the limit of the current method, the theory of copula is introduced. Copulas make it possible to couple the marginal laws in order to obtain a multivariate law. 525 Analysis of the tail dependence of the sample is used to make an initial selection of the copulas. This is because if the sample has lower tail dependence (upper tail dependence, respectively), the copula with the same tail dependence or an inverse tail dependence is chosen by taking the survival copula. The correlation between the storm surge and wave height is modelled using the Clayton copula and the survival Gumbel copula.
In order to take into account the three variables (wave height, wave period, and storm surge), we show that a fully nested 530 Archimedian trivariate copula with two parameters is the best construction technique. This function satisfies the mathematical properties of the copulas. The error rate of 3.8 % is lower than the trivariate copula obtained by generalizing the Clayton copula with a single parameter (error rate of 8.8 %). We confirm that the best results are obtained by first aggregating the most correlated variables that are here wave height and wave period. Nevertheless, the choice of method of aggregation is much less important than the choice of the copula. 535