A multivariate model of sea storms using copulas

Sea storms are customarily characterized in terms of significant wave height, peak period, and storm direction. Much less investigated is the storm duration, which is also very important when dealing with coastal dynamics and durability of large structures at sea. Moreover, these random variables are often treated using univariate frequency analysis, even if they are generally non-independent. A thorough statistical description requires the joint use of their marginal laws to obtain a multivariate distribution. In this paper we focus the attention on the multidimensional frequency analysis of sea storm significant wave height (H), storm duration (D), storm direction (A), and storm interarrival time (I) (i.e. the calm period separating two successive storms). In particular, we explain how copulas can be used for this purpose, for they constitute a novel and efficient mathematical tool for the analysis of multivariate events. An accurate model of these four random variables would undoubtedly be useful in aiding decision making processes with regard to marine operations, and this work could provide a guide to the oceanographic community not already familiar with copulas. Several important issues are considered, and the corresponding practical problems are solved using new copula-based techniques. (1) The construction of a bivariate model for the pair (H, D): in turn, this yields the statistics of the sea storm magnitude M. (2) The calculation of the return period of multivariate events: this gives the possibility to calculate the probability of occurrence of super-critical events, and yields an estimation of the minimum energetic content of sea storms having assigned (multivariate) return period. (3) The construction of a trivariate model for the triple (H, D, A): this provides useful indications about the relation between sea storm magnitude and direction. (4) The extension to storm interarrival duration I: this yields a trivariate model for the triple (D, I, A), that casts new light on the relation between sea storm timing and direction. (5) The construction of a global model for the vector (H, D, I, A): the overall structure is that of a reward alternating renewal process, whose dynamics develops along a random direction. In turn, this gives the possibility to simulate a sequence of sea storm events, accounting for all the variables of interest and their mutual relations. A case study is presented, based on real data collected by the Italian Sea Wave Measurement network. These statistical analyses are very important when dealing with coastal dynamics, marine structures reliability, or the planning of operations at sea. A proper evaluation of the main sea state parameters is crucial for a correct structural design, for reliable prognosis of the evolution of coastal environments, and for proper decision making in marine operations, due to the nonlinearities and the multidimensional structure of most of the design or failure function formulations.