Rice formulae and Gaussian waves

TL;DR

This paper applies Rice formulae to compute moments of level functionals related to oceanography and optics, such as specular points and wave dislocations, aiding statistical inference and model testing.

Contribution

It introduces new computations of moments for level functionals in Gaussian waves, enabling statistical inference and central limit theorems in related fields.

Findings

Computed expectations and second moments of level functionals.

Derived central limit theorems for certain functionals.

Provided tools for statistical inference in wave models.

Abstract

We use Rice formulae in order to compute the moments of some level functionals which are linked to problems in oceanography and optics: the number of specular points in one and two dimensions, the distribution of the normal angle of level curves and the number of dislocations in random wavefronts. We compute expectations and, in some cases, also second moments of such functionals. Moments of order greater than one are more involved, but one needs them whenever one wants to perform statistical inference on some parameters in the model or to test the model itself. In some cases, we are able to use these computations to obtain a central limit theorem.