Copyright © 2010 Esteban Flores and José R. León R. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We apply Rice's multidimensional formulas, in a mathematically rigorous way, to several problems which appear in random sea modeling. As a first example, the probability density function of the velocity of the specular points is obtained in one or two dimensions as well as the expectation of the number of specular points in two dimensions. We also consider, based on a multidimensional Rice formula, a curvilinear integral with respect to the level curve. It follows that its expected value allows defining the Palm distribution of the angle of the normal of the curve that defines the waves crest. Finally, we give a new proof of a general multidimensional Rice formula, valid for all levels, for a stationary and smooth enough random fields X:ℝd→ℝj(d>j).