## 3 Applications and Results
### Overtopping Reliability of a Breakwater  
In this section, we consider for the coastal engineering application the example reported in [10], where certain conditions are assumed and where the reader is referred to for further details and used references. It is a breakwater with deterministic slope, \(tan\)\(\tau=1/1.5\), and freeboard, \(F_{b}=10\) m. When the water runs up the breakwater, overtopping could occur (i.e., the water surpasses the freeboard), which is considered as a failure. This is defined by the LSF given by  
\[g_{bkw}=F_{c}-A_{u}H\left(1-e^{B_{u}1.25T\frac{tan.\tau}{vH}}\right) \tag{25}\]  
where \(A_{u}\) and \(B_{u}\) are coefficients characterized as independent normally distributed random variables, with mean values equal to \(1.05\) and \(-0.67\), respectively, and coefficients of variation both equal to \(0.2\)[10]; \(H\) denotes de wave height and \(T\) represents wave period. \(H\) and \(T\) are random variables probabilistically characterized by the joint Longuet-Higgins distribution [20] with parameter \(\
u=0.25\). The joint PDF of the Longuet-Higgins distribution is given by  
\[f_{H_{n}T_{n}}(H_{n},T_{n})=L(\
u)\frac{2}{\pi^{1/2}\
u}\left(\frac{H_{n}^{2}} {T_{n}^{2}}\right)e^{-H_{n}^{2}}\left[1+\left(1-\frac{1}{T_{n}}\right)^{2}/ \
u^{2}\right] \tag{26}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline \(n_{sim}\) & \(\beta\) & \(x_{B}/(\alpha_{B})\) & \(x_{fy}/(\alpha_{fy})\) & \(x_{Fd}/(\alpha_{fc})\) & \(x_{D}/(\alpha_{D})\) & \(x_{Vl}/(\alpha_{V})\) & \(x_{b}/(\alpha_{b})\) & \(x_{d}/(\alpha_{d})\) \\ \hline \(1\times 10^{4}\) & — & — & — & — & — & — & — & — \\ \(5\times 10^{4}\) & — & — & — & — & — & — & — & — \\ \(1\times 10^{5}\) & 3.0798 & 0.9429 & 455.528 & 29.802 & 420.037 & 534.156 & 301.950 & 959.454 \\  & & (\(-0.3521\)) & (\(-0.245\)) & (\(-0.1248\)) & (\(0.1170\)) & (\(0.8520\)) & (\(-0.0276\)) & (\(-0.2453\)) \\ \(2\times 10^{5}\) & 3.0747 & 0.9354 & 453.722 & 30.099 & 420.169 & 539.456 & 302.109 & 956.271 \\  & & (\(-0.3732\)) & (\(-0.2575\)) & (\(-0.0992\)) & (\(0.1134\)) & (\(0.8395\)) & (\(-0.0223\)) & (\(-0.2580\)) \\ \(1\times 10^{6}\) & 3.0842 & 0.9350 & 453.914 & 30.042 & 419.88 & 539.275 & 301.844 & 955.342 \\  & & (\(-0.3747\)) & (\(-0.2546\)) & (\(-0.1028\)) & (\(0.1090\)) & (\(0.8375\)) & (\(-0.0289\)) & (\(-0.2648\)) \\
**FORM** & 3.130 & 0.9424 & 455.332 & 30.393 & 419.026 & 702.302 & 302.151 & 958.785 \\  & & (\(-0.3562\)) & (\(-0.2489\)) & (\(-0.0841\)) & (\(0.1019\)) & (\(0.8543\)) & (\(-0.0224\)) & (\(-0.2518\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Design point, \(\beta\) and \(\alpha_{i}\) by using multi-linear regression and FORM for the RCB. where \(H_{n}=H/H_{s}\) and \(T_{n}=T/T_{z}\) are normalized wave heights and periods by considering \(H_{s}=5\) m and \(T_{z}=10\) s, which are the significant wave height and the zero up-crossing mean period, respectively, which define the sea state [10]; \(L(\
u)\) is a normalization factor implying only positive values of \(T_{n}\) and defined by  
\[L(\
u)=\left(\frac{1}{2}[1+(1+v^{2})^{-1/2}]\right)^{-1} \tag{27}\]  
First, the FORM is applied to Equation (25). Salient points of performing the FORM to this overtopping LSF are briefly described in the following. First, it is noted that since \(H_{n}\) and \(T_{n}\) are not independent, the Rosenblatt transformation is performed for the joint distribution to map the equivalent distribution parameters into the normal space by using [17]  
\[\left\{ \begin{array}{c}z_{1}=\Phi^{-1}(F_{H_{n}}(H_{n}))\\ z_{2}=\Phi^{-1}\Big{(}F_{T_{n}|H_{n}}(T_{n}|H_{n})\Big{)}\end{array} \right.