Inverse random potential scattering for elastic waves

TL;DR

This paper investigates the inverse scattering problem for a three-dimensional elastic wave in a random potential modeled as a Gaussian random field, demonstrating unique recovery of the potential's microlocal strength from high-frequency backscattered data.

Contribution

It establishes the unique determination of the random potential's microlocal strength using a single high-frequency backscattered wave measurement, advancing inverse scattering theory for random media.

Findings

Unique recovery of the microlocal strength of the random potential.

Well-posedness of the direct scattering problem in the distribution sense.

Application of microlocal analysis and ergodicity in inverse scattering.

Abstract

This paper is concerned with the inverse elastic scattering problem for a random potential in three dimensions. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic Gaussian random field whose covariance operator is a classical pseudo-differential operator. Given the potential, the direct scattering problem is shown to be well-posed in the distribution sense by studying the equivalent Lippmann--Schwinger integral equation. For the inverse scattering problem, we demonstrate that the microlocal strength of the random potential can be uniquely determined with probability one by a single realization of the high frequency limit of the averaged compressional or shear backscattered far-field pattern of the scattered wave. The analysis employs the integral operator theory, the Born approximation in the high frequency regime, the microlocal analysis for the Fourier integral operators, and the ergodicity of the wave field.