## 3 Extreme Value Analysis Considering Seasonality
### Simplified Approach for Return Period Correlated Values
\[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]5 Exact and approximated 95% confidence intervals for October considering the \(T_{\_{yr}}\) values: regression (**a**) in logarithmic space and (**b**) in linear space. Legends indicate expressions referred in [28]. Figure 11b, implying better estimates of correlated values for given \(T_{\_}yr\) if the regression is carried out in the logarithmic space. Figures 9, 12 and 13 show that a single equation cannot capture the seasonality of correlated metocean variables, because it can overestimate or underestimate the correlated values when comparing against the equations compiled in [26], at least for the case of \(V_{\_}w\) and \(H\)s and buoy 42001, but it is possibly the case for other metocean variables and buoys, in the Gulf of Mexico or anywhere else. Therefore, suitable sets of region-specific and site-specific equations should be developed to take into account seasonal effects. \[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]6 Regression lines and confidence intervals including (solid and dashed lines, respectively) and excluding (dashed–dotted and dotted lines, respectively) \(T_{\_}yr\) values for August; (**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. \[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]7 Exact and approximated 95% confidence intervals for February considering the \(T_{\_}yr\) values: regression (**a**) in logarithmic space and (**b**) in in linear space. Legends indicate expressions referred in [28]. Nonetheless, it is interesting to note that the equation developed in [26] for West Shetland is practically coincident with the regression line in Figure 13a at hurricane season. For values associated with \(T_{\text{-}yr}\) levels in Figure 11a (or in other figures), the uncertainty from the GEV models for each \(T_{\text{-}yr}\) (i.e., the embedded perpendicular lines in the black filled circles) are a sort of envelope roughly covered by the confidence intervals for mean values in Figure 11a. Furthermore, such an envelope can be covered by using Equation (12) and a smaller value of \(\delta\) (e.g., \(\delta=1\)) than that used for the confidence intervals of future values. Therefore, Equations (8) and (10) can be considered as possible design aids--simple expressions for design--like those contained in guidelines [26], but also accounting for the uncertainty in a simplified way (Equations (11) and (12)), either based on simplified confidence intervals for future values or confidence intervals within which the GEV uncertainty is confined. Since sets of such equations can be obtained for any desired season, the proposal here represents a simpler alternative than the more complex models reported in the introduction; consequently, they could be more attractive for designers since they are consistent with expressions recommended in guidelines. Results for other months (not shown before) are reported in Figures A1-A9 in Appendix B. By observing Figures 7, 10 and 11 and Figures A1-A9, it can be concluded that the regression in the logarithmic or linear space could be both alternatives, with better results depending on the month considered. An exception is shown by Figure A8 for November, where neither a good fit is obtained for the logarithmic case (Figure A8a) nor for the linear one (Figure A8b) (also see Table 1). This is attributed to atypical simultaneous occurrence of larger \(H\!s\) and smaller \(V_{w}\) values, as can be observed in Figure A8, but also to a large dispersion of the data along the whole range of \(H\!s\) values, leading to a regression line somewhat horizontal (and roughly around 12 m/s), implying no correlation of \(H\!s\) and \(V_{w}\) for November. It would be interesting to investigate in the future why this behavior is found specifically for November at buoy 42,001 in the Gulf of Mexico.