## 3 Applications and Results
### Overtopping Reliability of a Breakwater
\tag{28}\]  
where \(\Phi^{-1}(\bullet)\) denotes the inverse of the CDF of a standard normal variable, \(F_{H_{n}}(H_{n})\) and \(F_{T_{n}|H_{n}}(T_{n}|H_{n})\) are the marginal distribution of \(H_{n}\) and the conditional distribution of \(T_{n}\) given \(H_{n}\) for Equation(26), respectively, and defined by  
\[F_{H_{n}}(H_{n})=H_{n}L(\
u)e^{-H_{n}^{2}}[1+erf(R_{n}/\
u)] \tag{29}\]  
\[F_{T_{n}|H_{n}}(T_{n}|H_{n})=2(\pi^{2}\
u[1+erf(H_{n}/\
u)])^{-1}\Bigg{(}\frac {H_{n}}{T_{n}^{2}}\Bigg{)}e^{-H_{n}^{2}(1-\
u/\
u_{x})^{2}/\
u^{2}} \tag{30}\]  
where the error function is given by  
\[erf(H_{n}/\
u)=\frac{2}{\sqrt{\pi}}\int\limits_{0}^{H_{n}/\
u}e^{-t^{2}}dt \tag{31}\]  
In Equation (29) the equivalent version reported in [25], rather than the original version in [10], is considered. This is so, simply because the error function used in [25] is more readily available in current software. Then, to derive the conditional probability distribution, we divided Equation (26) by Equation (29) yielding Equation (30) given above. Since the CDFs of Equation (29) and Equation (30) are also required to obtain the equivalent parameters mapped in the standardized normal space, other point to highlight is that they were obtained numerically at the design point, unlike for the normal distributed random variables, where simple analytical expressions can be used (which is also possible for other common PDFs). Additionally, it is noted that as part of the procedure to obtain the reliability index in each iteration of the FORM, usually a vector obtained by multiplying each partial derivative of the LSF (i.e., Equation (25)), evaluated at the design point, by the equivalent second moment in the normal space (for the corresponding random variable) is enough. However, this approach is not possible for the joint random variables in this example. Therefore, the Jacobian (and its inverse) is required [17]; once the inverse of the Jacobian is computed, it is multiplied by the vector of partial derivatives evaluated at the design point mentioned above, and the reliability index can then be obtained in each iteration in the regular way for the FORM (i.e., as when the variables are independent). This approach is followed in the present study. Note that for a set of jointly distributed random variables \(x_{i}\) (\(z_{i}\) in the normalized space), the inverse of the Jacobian is a lower-triangular matrix determined (often numerically) as [17]  
\[J_{ij}^{-1}=\frac{\partial z_{i}}{\partial x_{i}}=\left\{ \begin{array}{cc}0, &i<j\\ &\frac{f(x_{i}|y_{1-\
u},x_{i-1})}{\phi(z_{i})}\\ &\frac{x_{i}}{x_{i}}(x_{i}|y_{1-\
u},x_{i-1})\\ &\frac{x_{i}}{x_{i}}(x_{i}|z_{1-\
u},x_{i-1})\\ &\phi(z_{i}),&i>j\end{array} \right.,\qquad i=j \tag{32}\]where \(\Phi\) (\(z_{i}\)) is the PDF of a standard normal random variable, with the argument \(z_{i}\) obtained in an analogous way to Equation (28); \(f_{i}\) and \(F_{i}\) refers to the PDF and CDF for the variable with subscript \(i\), respectively. It was noticed that for the present example, disregarding the elements outside the Jacobian diagonal does not impact very significantly the computed reliability indices. A few final important aspects regarding the FORM worth to mention, include that the order of the variables in defining Equation (28) does matter, although similar results may be expected [17]. For instance, in [10] the marginal distribution of \(T_{n}\) and the conditional distribution of \(H_{n}\) given \(T_{n}\) are used to define Equation (28) (i.e., the order of the variables is inverted as compared with this study), which results in a reliability index, \(\beta\), equal to 2.01 for the problem in question, whereas \(\beta\) = 2.10 is obtained in this study with the FORM formulation described earlier, and adopted in the following; \(\beta\) = 2.10 is also closer to the exact value to be discussed later.