Joint modelling of wave spectral parameters for extreme sea states

Characterising the dependence between extremes of wave spectral parameters such as significant wave height (H_S) and spectral peak period (T_P) is important in understanding extreme ocean environments and in the design and assessment of marine structures. For example, it is known that mean values of wave periods tend to increase with increasing storm intensity. Here we seek to characterise joint dependence in a straightforward manner, accessible to the ocean engineering community, using a statistically sound approach.Many methods of multivariate extreme value analyses are based on models which assume implicitly that in some joint tail region each parameter is either independent of or asymptotically dependent on other parameters; yet in reality the dependence structure in general is neither of these. The underpinning assumption of multivariate regular variation restricts these methods to estimation of joint regions in which all parameters are extreme; but regions where only a subset of parameters are extreme can be equally important for design. The conditional approach of Heffernan and Tawn (2004), similar in spirit to that of Haver (1985) but with better theoretical foundation, overcomes these difficulties.We use the conditional approach to characterise the dependence structure of H_S and T_P. The key elements of the procedure are: (1) marginal modelling for all parameters, (2) transformation of data to a common standard Gumbel marginal form, (3) modelling dependence between data for extremes of pairs of parameters using a form of regression, (4) simulation of long return periods to estimate joint extremes. We demonstrate the approach in application to measured and hindcast data from the Northern North Sea, the Gulf of Mexico and the North West Shelf of Australia. We also illustrate the use of data re-sampling techniques such as bootstrapping to estimate the uncertainty in marginal and dependence models and accommodate this uncertainty in extreme quantile estimation.We discuss the current approach in the context of other approaches to multivariate extreme value estimation popular in the ocean engineering community.