## 3 Extreme Value Analysis Considering Seasonality
### Simplified Approach for Return Period Correlated Values  
The proposal to estimate the simultaneous occurrence of extreme metocean variables is based on a modified version of the classical linear regression technique. This consists in including estimated values from the time-dependent GEV models in the previous section--together with the simultaneously observed monthly data for maximum \(H\)s and companion \(V_{w}\)--in the regression analysis, to assess how adequate is the use of the return period value of the explanatory variable (in the regression analysis sense) to predict the return period value of the other variable and its uncertainty. This is to be carried out for each month of the year to estimate seasonal extreme (and companion) values associated with given return periods, so that they can be considered as a demand which is applied simultaneously to a maritime system (or other system). From this standpoint, significant wave heights and wind velocities can be considered as concurrent extreme metocean hazards accounting for seasonality, since they can be translated into load effects affecting a given maritime structure at the same time, at any desired season of the year. Moreover, such demands are usually related to given return periods for design purposes; consequently, it is convenient to establish them in terms of return periods. This is also the case for using contour maps in ocean engineering [20]. Figure 2: Monthly companion and 30-yr return period values of \(V_{w}\) using different time ranges. Before proceeding to introduce the proposal, the classical regression analysis using _Hs_ as the explanatory variable and \(V_{w}\) as the variable to be predicted is described as follows by using the observed data only. Buoy 42,001 is used to illustrate the procedure, but this can be extended to any buoy with simultaneous recorded metocean data. _Hs_ is selected as the explanatory variable because it was used in a previous study [28] to investigate maxima. As noted before, \(V_{w}\) is not necessarily the maximum, but the wind velocity observed simultaneously with the monthly maximum _Hs_. Naturally, it could be considered otherwise, e.g., maximum \(V_{w}\) and associated simultaneously recorded companion _Hs_, but also other pairs of concurrent values of _Hs_ and \(V_{w}\) could be selected. This aspect is discussed later. From a designer perspective, it can be thought of as selecting an _Hs_ value and using it to predict a \(V_{w}\) value that is expected to occur simultaneously; then, both can be used to estimate the imposed (adding) effects of both metocean variables acting over a system, which capacity is to be designed to withstand such demand. Since both metocean variables are random, this can be expressed in mathematical terms [32; 33] as:  
\[E[V_{w}|Hs=hs]=b+\alpha\,hs \tag{7}\]  
where the expected value of \(V_{w}\), given that _Hs_ = _hs_, is a linear function of _hs_; \(b\) and \(\alpha\) are constants to be determined by linear regression analysis. However, guideline expressions that relate wind velocities and wave heights, as well as fitted functions found in the literature [26], commonly have the following mathematical form:  
\[V_{w}=bH_{s}^{\alpha} \tag{8}\]  
which can be linearized by taking the natural logarithm, leading to:  
\[\ln V_{w}=\beta+\alpha\ln H_{s} \tag{9}\]  
where \(\beta\) = ln \(b\); the notation in terms of the expected value is skipped for simplicity. Equations (7) and (9) are used in the linear regression analysis. For using Equation (9), the wind velocities and significant wave heights are first transformed into logarithmic space [33]. Figure 3a,b shows the linear regression analysis by using the nonlinear model (i.e., Equation (9)) and the linear model (i.e., Equation (7)), respectively, for October (around hurricane season); the residuals versus the fitted values of _ln_\(V_{w}\) (Figure 3c) and \(V_{w}\) (Figure 3d) are also shown.