Modelling Sea Surface Dynamics Using Crossing Distributions

Abstract: The thesis deals mainly with modelling sea surface dynamics. We consider two different scales. The short-term scale,known as sea state, in which the sea surface over a restricted time period and space can be modelled as a stationary random field, and the long-term scale in which we study the evolution of wave characteristics like the significant wave height $H_s$, over long periods of time and at great geographic regions. The main statistical tools are crossing distributions, which are given by a generalisation of Rice's formula which is valid under mild conditions. In the short-term scale, we consider the sea surface as a Gaussian stationary random field. Study of the motion of such a surface should include the notion of velocity. Different velocities, that capture different aspects of the sea dynamics, are defined and their statistical distributions are obtained. Also of interest is the effect the wave kinematics have on the distribution of global maximum. It is observed that taking into account time dynamics of spatial characteristics results in distributions different than those obtained for the static case. Satellites orbiting around the earth provide with global spatial coverage of the ocean surfaces. The logarithmic values of $H_s$ are modelled as a locally stationary Gaussian random field. The mean value varies seasonally and geographically and the covariance structure is modelled as a sum of two independent sources, one in a coarser and one in a finer scale. To capture the temporal variability velocities, that enter the covariance structure as parameters, are used. Wave climate of $H_s$ is of importance for different applications, like for example estimation of the fatigue accumulated by a vessel sailing a certain route.