Inverse source problems for the stochastic wave equations: far-field patterns

TL;DR

This paper develops a unified approach to inverse source problems for various stochastic wave equations, showing that principal symbols of covariance operators can be uniquely identified from far-field patterns using ergodicity.

Contribution

It introduces a unified framework for inverse source problems in stochastic wave equations and proves unique determination of covariance operator symbols from single realizations.

Findings

Principal symbols can be uniquely identified from far-field data.

The approach applies to multiple types of wave equations.

Single realization and frequency averaging suffice for reconstruction.

Abstract

This paper addresses the direct and inverse source problems for the stochastic acoustic, biharmonic, electromagnetic, and elastic wave equations in a unified framework. The driven source is assumed to be a centered generalized microlocally isotropic Gaussian random field, whose covariance and relation operators are classical pseudo-differential operators. Given the random source, the direct problems are shown to be well-posed in the sense of distributions and the regularity of the solutions are given. For the inverse problems, we demonstrate by ergodicity that the principal symbols of the covariance and relation operators can be uniquely determined by a single realization of the far-field pattern averaged over the frequency band with probability one.