A Vine-Copula Model for Time Series of Significant Wave Heights and Mean Zero-Crossing Periods in the North Sea

Stochastic description and simulation of oceanographic variables are essential for coastal and marine engineering applications. In the past decade, copula-based approaches have become increasingly popular to estimate the multivariate distribution of some variables at the peak of a storm along with its duration. The modeling of the storm shape, which contributes to its impact, is often simplified. This article proposes a vine-copula approach to characterize hourly significant wave heights and corresponding mean zero-crossing periods as a random process in time. The model is applied to a data set in the North Sea and time series with the duration of an oceanographic winter are simulated. The synthetic wave scenarios emulate storms as well as daily conditions. The results are for example useful as input for coastal risk analyses and for planning offshore operations. Nonetheless, selecting a vine structure, finding appropriate copula families and estimating parameters is not straightforward. The validity of the model, as well as the conclusions that can be drawn from it, are sensitive to these choices. A valuable by-product of the proposed vine-copula approach is the bivariate distribution of significant wave heights and corresponding mean zero-crossing periods at the given location. Its dependence structure is approximated by the flexible skew-t copula family and preserves the limiting wave steepness condition. INTRODUCTION Wind-induced seawaves affect coastlines, marine structures and offshore operations. Unfavorable conditions can cause significant morphological change, damages or downtime. Intuitively, the higher a wave, the more energy it carries and the more destructive 1 Jäger, May 4, 2017 This is an Accepted Manuscript of an article published by ASCE library ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering Vol. 3, Issue 4 (December 2017) See more at: https://doi.org/10.1061/AJRUA6.0000917 it is. At beaches and dunes, which act as defenses to coastal developments, high waves with long periods lead to higher run-up and intensify erosion rates (van Gent et al. 2008). Wave overtopping is also problematic at non-sandy coasts. Moreover, extreme wave heights and periods can contribute to structural failure of marine structures. In particular offshore, wave periods close to the resonant heave period of a vessel or a structure pose an additional threat (e.g., Faltinsen 1993). Wave heights and periods are strongly interrelated, usually arising from a common meteorological system. In addition, wave heights are limited by their associated periods in terms of a so-called maximum steepness condition, which postulates that too steep waves break and reduce in height. Understanding the joint behavior of wave heights and periods and being able to estimate possible extreme conditions is important, for instance, to determine design criteria and for risk analyses of marine structures and coastal environments (Hawkes et al. 2002; Salvadori et al. 2014; Gouldby et al. 2014) or for scheduling and budgeting offshore operations (Leontaris et al. 2016). Instead of considering individual waves, one generally uses statistics that describe the sea state under stationary conditions. For example, in this article, we concentrate on the significant wave height, Hm0, and the mean zero-crossing period, Tm02, which are computed from the zerothand second-order moments of the variance density spectrum of a wave record (Holthuijsen 2010). Multiple studies have been dedicated tomodeling the dependencies between oceanographic variables. Most common are analyses of maxima or peak over threshold values to model storms, with variables of interest being, for instance, significant wave heights, peak wave periods and water levels. A popular approach is based on copulas, which isolate the marginal properties from the dependence structure of random variables (e.g., Genest and Favre 2007 for an introduction). A combination of any copula with any marginal distribution leads to a valid specification of a joint distribution, enabling representations of a wide range of complex multivariate behaviors. In the bivariate case, many different copula families have proven to be useful (e.g., Salvadori et al. 2014; Masina et al. 2015, and references therein). For more than two oceanographic variables, nested (also called hierarchical) Archimedean copulas (Wahl et al. 2012; Corbella and Stretch 2013; Lin-Ye et al. 2016) and elliptical copulas, such as Gaussian or t, (Li et al. 2014a; Wahl et al. 2016; Rueda et al. 2016) have been implemented and found valuable, but also dependence trees (Poelhekke et al. 2016) and vines (De Michele et al. 2007; Montes-Iturrizaga and Heredia-Zavoni 2016), which are a generalization thereof, have been proposed. Callaghan et al. (2008) and Serafin and Ruggiero (2014) adopt two other approaches to model dependencies. They use a bivariate logistics model (Tawn 1988) and they specify parameters for a conditional distribution of one variable based on the value of the conditioning variable. Not only the dependencies between variables are important for impact assessment, but also their temporal evolution; impacts amass during long-lasting or recurring extreme conditions (e.g., Karunarathna et al. 2014, and references therein for impacts on a sandy beach). Storm sequences (i.e., time series of storm events) have been modeled as different types of renewal processes (De Michele et al. 2007; Callaghan et al. 2008; Corbella and Stretch 2013; Li et al. 2014b; Wahl et al. 2016) and storm shapes are 2 Jäger, May 4, 2017