## 3 Applications and Results
### Overtopping Reliability of a Breakwater
Another slight difference between [10] and this study when applying the FORM, is that in the present work, when assuming initial design points, one is determined by setting \(g_{blox}\) = 0, to ensure that the design point is on the failure boundary (e.g., [26]). To inspect the variation of \(\beta\) for different \(F_{c}\) values, the FORM is performed by varying the freeboard between 9 m and 12 m and the resulting reliability index is shown in Figure 5a with a black dashed line. As expected, it can be observed that \(\beta\) increases for increasing freeboard; if the slope of the breakwater is increased to \(tan\) \(\tau\) = 1/2 and the FORM is carried out for the same range of \(F_{c}\), it further increases reliability levels, as shown by the dashed grey line in Figure 5a. These results are used as reference and for comparison purposes, with respect to the results from the normality polynomial and multi-linear regression approaches revisited in this study. The simulations for this coastal engineering application, used as the basis of the revisited methods, are much more computationally intensive than for the classical and structural examples, because of the dependency between the wave height and period and inclusion of the Longuet-Higgins distribution, which imposes numerical computing for the probability levels (e.g., values from CDFs) and a different  
Figure 5: Reliability of breakwater and error estimation in the reliability index. (**a**) Reliability index as a function of \(F_{c}\); (**b**) Reliability index as a function of \(n_{sim}\); (**c**) Computed error for the reliability index; (**d**) Fitted mean and standard deviation for the error in the reliability index estimation. method for the sampling. This latter aspect, i.e., the generation of jointly distributed random numbers when a set of \(x_{i}\) variables are dependent, is based on expressing the joint PDF as [22]  
\[f_{X}(x)=f_{X1}(x_{1})f_{X2}(x_{2}|x_{1})\dots f_{Xn}(x_{n}|x_{1},\dots,x_{n-1}) \tag{33}\]  
with the corresponding CDF given by  
\[F_{X}(x)=F_{X1}(x_{1})F_{X2}(x_{2}|x_{1})\dots F_{Xn}(x_{n}|x_{1},\dots,x_{n-1}) \tag{34}\]  
Using the previous concepts, and considering a set of values \(\mathbf{U}\) generated from \(n\) independent standard uniformly distributed random variables, the set of dependent random variables can be determined as  
\[\left\{ \begin{array}{c}x_{1}=F_{x_{1}}^{-1}(u_{1})\\ x_{2}=F_{x_{2}}^{-1}(u_{2}|x_{1})\\ \cdot\\ x_{n}=F_{x_{n}}^{-1}(u_{n}|x_{1},\dots,x_{n-1})\end{array} \right. \tag{35}\]  
where \(F^{-1}(\bullet)\) denotes the inverse of the CDF. The obtaining of this inverse of the CDF can be relatively straightforward for some common probability distributions, where an analytical expression can be used for Equation (35). This is not the case for the Longuet-Higgins distribution. In this case, the jointly distributed random wave height and period must be determined numerically. Figure 6 shows samples of jointly generated random values of wave height and period in the normalized space (for \(n_{sim}=1\times 10^{3}\), \(5\times 10^{3}\); \(1\times 10^{4}\) and \(5\times 10^{4}\)). A few contours of the theoretical Longuet-Higgins distribution (i.e., Equation (26)) are also shown in Figure 6; it can be observed that they are in good agreement. The values in the non-normalized space can be obtained simply by recognizing that \(H_{n}=H\)/\(H_{s}\) and \(T_{n}=T\)/\(T_{z}\). As mentioned before, the sampling procedure is significantly more time-consuming than for the LSFs in previous sections. Therefore, MCS are sampled only up to \(1\times 10^{6}\) simulations for all the variables of the LSF represented by Equation (25), and the reliability index is determined by employing Equation (17) and Equation (7), only for the case reported in [10] (i.e., \(F_{c}=10\) m and \(tan\)\(\tau=\)1/1.5). Nonetheless, results depicted in Figure 5b indicate that the reliability index stabilizes,from approximately \(n_{sim}=1\times 10^{5}\) and over, to a reliability index practically equal to 2.2 (gray solid line).