Explicit form of the random field spectral representation and some applications

TL;DR

This paper derives an explicit spectral representation for stationary random fields, enabling easier calculation of statistics, geometric interpretation, and straightforward numerical generation, with practical applications demonstrated.

Contribution

It provides an explicit form of the spectral measure for stationary random fields, improving analysis and simulation methods.

Findings

Explicit spectral measure expression simplifies higher order statistics calculation.

The convergence mechanism differs from Fourier transform, with a new formalism as a limit case.

Facilitates numerical generation and geometric interpretation of random fields.

Abstract

We present here an explicit form of the random spectral measure element, what allows us to express a stationary random field as a stochastic integral explicitly depending on its power spectrum and a spectral tensor if the field is a vector one. It has been shown here that convergence mechanism of such integral is significantly different from the one of the Fourier transform and that the traditional formalism is a partial limiting case of the one presented here. The fact that there is an explicit expression of a random field makes calculation of higher order statistics of it much more straightforward (see for example Chepurnov et al. 2020). For a vector field such expression contains a projection of an isotropically distributed random vector by a spectral tensor, what makes geometrical interpretation of harmonics behavior possible, simplifying its analysis (see Sect. 2). This spectral representation also makes straightforward numerical generation of a random field, what is extensively used by Chepurnov et al. 2020. We also present here some practical applications of this formalism.