Inverse random potential scattering for the polyharmonic wave equation using far-field patterns

TL;DR

This paper investigates the inverse scattering problem for a random potential in the polyharmonic wave equation, establishing uniqueness of recovering the potential's statistical properties from high-frequency far-field data.

Contribution

It introduces a novel approach to uniquely recover the covariance and relation operators of a complex Gaussian random potential using only a single high-frequency far-field measurement.

Findings

Well-posedness of the direct problem at high wavenumbers

Uniqueness in recovering statistical properties of the potential

Recovery achieved with a single realization of far-field data

Abstract

This paper addresses the inverse scattering problem of a random potential associated with the polyharmonic wave equation in two and three dimensions. The random potential is represented as a centered complex-valued generalized microlocally isotropic Gaussian random field, where its covariance and relation operators are characterized as conventional pseudo-differential operators. Regarding the direct scattering problem, the well-posedness is established in the distribution sense for sufficiently large wavenumbers through analysis of the corresponding Lippmann--Schwinger integral equation. Furthermore, in the context of the inverse scattering problem, the uniqueness is attained in recovering the microlocal strengths of both the covariance and relation operators of the random potential. Notably, this is accomplished with only a single realization of the backscattering far-field patterns averaged over the high-frequency band.