## 3 Extreme Value Analysis Considering Seasonality
### Simplified Approach for Return Period Correlated Values
\[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]2 Comparison of exact and approximated (Equations (11) and (12)) 95% confidence intervals, considering October (upper row; (**a**) and (**b**) for non-linear and linear cases, respectively) and February (lower row; (**c**) and (**d**) for non-linear and linear cases, respectively). papers preserve roughly similar trends than without considering the \(T_{\_{3\prime\prime}}\) values and follow an approximately normal distribution. Figure 9a,b is analogous to Figure 6a,b and they are comparable, as can be observed in these figures, with slightly less wider confidence intervals in Figure 9; as in Figure 6 for October, \(\delta\) = 2.3 is also used in \[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]3 For illustration purposes, the expression of Pierson-Moskowitz (PM), the one from the International Towing Tank Conference (ITTC) recommendations and a fitted equation for West Shetland at a North Atlantic location, are shown in Figure 9a, and the equations recommended by the NATO-STANAG 4194 and the 2013 Interim Guidelines proposed in MEPC 232 (65) are shown in Figure 9b; all these equations are retrieved from [26], where details and proper citation can be found. Some of these equations are given as a function of \(V_{\_}w\) instead of \(H\)s [26]; in such cases, equations are solved for \(V_{\_}w\) as a function of \(H\)s. The PM, ITTC and West Shetland expressions have the functional form of Equation (8) with values of \(b\) equal to 6.376, 4.636 and 5.31, respectively, and values of \(\alpha\) equal to 0.5, 0.709 and 0.603, respectively. Likewise, for the NATO-STANAG and 2013 Interim Guidelines, the corresponding functional form is given by Equation (10) with parameters \(b\) equal to 0 and 6.9 and \(\alpha\) equal to 3.2 and 2.2, respectively. For the previous expressions \(H\)s is given in \(m\) and \(V_{\_}w\) in \(m\)s. Note that the NATO-STANAG and 2013 Interim Guidelines equations are developed for coastal waters but included in Figure 9b for comparison. Figure 9 shows that the ITTC and the 2013 Interim Guidelines equations are similar in shape to those developed in this study, but under- or overestimating the predicted values for October the former and the latter, respectively. The other equations lead to not so dissimilar results as those reported here, and they are roughly in between the confidence intervals for future values. To inspect how the proposed approach varies when considering seasonality, and how results compare with respect to the equations reported in [26] for other seasons, results like those shown in Figure 7 are also shown in Figures 10 and 11, but for the months of February and August, when the highest predicted \(T_{  
It can be observed from Figures 7, 10 and 11 that expected \(V_{w}\) values as function of \(H\)s depend on the season and month considered, that the uncertainty also varies for each case and that the regression is relatively irrespective of the selected scheme (linear or logarithmic space) for October and February, but it does vary when August is considered. Figure 11 for August corresponds to the largest estimated \(T_{\_{yr}}\) values for \(H\)s (hurricane season) and, as shown in Figure 11, the regression line predicts better estimates when the regression is performed in the logarithmic space; in fact, when the \(T_{\_{yr}}\) values are included in the regression (solid line), there is not significant deviation from the regression with the observed data only (dashed-dotted line); conversely, an important change in the slope of the regression line is observed for August. \[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]4 Regression lines and confidence intervals including (solid and dashed lines, respectively) and excluding (dashed–dotted and dotted lines, respectively) \(T_{\_{yr}}\) values for February; (**a**) and (**b**) for logarithmic space and linear space, respectively. Filled circles represent \(T_{\_{yr}}\) values with embedded perpendicular lines indicating confidence intervals from GEV models.