International Journal of Stochastic Analysis
Volume 2010 (2010), Article ID 621038, 22 pages
Research Article

Level Sets of Random Fields and Applications: Specular Points and Wave Crests

1Departamento de Matemáticas, Facultad Experimental de Ciencias y Tecnología, Universidad de Carabobo, Valencia 2001, Venezuela
2Escuela de Matemática, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47197 Los Chaguaramos, Caracas 1041-A, Venezuela

Received 23 September 2009; Revised 22 February 2010; Accepted 22 February 2010

Academic Editor: Deli Li

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.


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:dj(d>j).