Inverse Random Source Scattering for Elastic Waves

TL;DR

This paper addresses the inverse scattering problem for elastic waves driven by white noise sources, proposing a method to reconstruct the source's statistical properties from boundary measurements across multiple frequencies.

Contribution

It introduces a novel approach using Fredholm integral equations and the regularized Kaczmarz method to solve the inverse problem for random elastic wave sources.

Findings

Unique mild solution for the direct problem established.

Effective reconstruction of mean and variance demonstrated.

Numerical experiments validate the proposed method.

Abstract

This paper is concerned with the direct and inverse random source scattering problems for elastic waves where the source is assumed to be driven by an additive white noise. Given the source, the direct problem is to determine the displacement of the random wave field. The inverse problem is to reconstruct the mean and variance of the random source from the boundary measurement of the wave field at multiple frequencies. The direct problem is shown to have a unique mild solution by using a constructive proof. Based on the explicit mild solution, Fredholm integral equations of the first kind are deduced for the inverse problem. The regularized Kaczmarz method is presented to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.