## 3 Extreme Value Analysis Considering Seasonality
### Simplified Approach for Return Period Correlated Values
As shown in Figures 3 and 5, the data are approximately contained within the confidence intervals for future values in all cases. It is noticed that the confidence intervals for future values can be simplified by using Equation (11a,b), for the lower and upper interval, respectively. \[V_{wlow} =b+\alpha Hs-\delta\sigma_{e} \tag{11a}\] \[V_{wupp} =b+\alpha Hs+\delta\sigma_{e} \tag{11b}\]  
where \(\sigma_{e}=\sqrt{\sigma_{e}^{2}}\) is the so-called RMSE which--advantageously--is part of the information that can be obtained when a regression analysis is performed; \(\delta\) is a constant value to be discussed. If Equation (8) is to be considered to derive simplified expressions for the confidence intervals, expressions like Equation (11) can be written by including \(\pm\delta\) times the RMSE in Equation (9), except that the RMSE corresponds to the regression in logarithmic space. If the equation in the original space is to be used to compute the expected value (i.e., Equation (8)), it can be shown with some algebraic manipulations that the 95% confidence intervals for future values can be approximated by using Equation (12a,b):  
\[V_{wlow}=bH_{s}^{\alpha}\Big{(}\frac{1}{e^{\delta\sigma_{e}}}\Big{)} \tag{12a}\]  
\[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]0 Regression and residual analysis by using (**a**) nonlinear (Equation (9)) and (**b**) linear (Equation (7)) models for February. Residuals using (**c**) Equation (9) and (**d**) Equation (7). \[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]1  
The value of \(\delta\) is adequate to represent the 95% confidence intervals (denoted simply as confidence intervals from now on) for future values by adopting constant values approximately between 1.9 and 2.3, depending on the season considered and for both sets of equations (i.e., Equations (11) or (12)). Exact confidence intervals (dashed lines) and approximated confidence intervals (Equations (11) or (12); dotted lines) are shown in Figure 6a-d for the four cases depicted in Figure 3a,b and Figure 5a,b, respectively; the original instead of the logarithmic scale is used for the cases corresponding to Equations (8) and (12). A value of \(\delta\) equal to 2.3 is used for October (Figure 6a,b), while \(\delta\) = 2.2 is employed for February (Figure 6c,d). It can be observed that Equations (11) and (12) are reasonable alternatives to represent the uncertainty in a simplified (practical) way. Even though simple, as far as the authors know, Equations (11) and (12) are not reported in the literature for extreme values accounting for seasonality; therefore, they are an important contribution of this study that include the uncertainty in correlated metocean variables for design purposes. The proposed approach to estimate concurrent metocean variables (\(H\)s and \(V_{w}\) in this case), is based on using the time-dependent GEV model for \(H\)s obtained in the previous section, to compute a \(H\)s value associated with a selected return period in years, \(T_{\text{-}yr}\) (the subscript is used to denote the selected return period, e.g., \(T_{\text{-}30}\) corresponds to a 30-yr return period); the \(H\)s for a given \(T_{\text{-}yr}\) can then be used as input for Equations (8) and (10)-(12), to determine the corresponding expected companion \(V_{w}\) for the same \(T_{\text{-}yr}\), including an uncertainty measure (i.e., the confidence intervals). To investigate whether the previous proposal is consistent with both time-dependent GEV models in the previous section, estimated values of \(H\)s and \(V_{w}\) associated with \(T_{\text{-}20}\), \(T_{\text{-}30}\), \(T_{\text{-}50}\), \(T_{\text{-}75}\) and \(T_{\text{-}100}\) are included in the scattergrams and regression analyses as can be observed in Figure 7, where the \(T_{\text{-}yr}\) values for October are indicated by black filled circles with embedded perpendicular lines (lines to be explained below). Selection of these return periods is somewhat arbitrary but based on the usage of such return periods for design.