Inverse scattering for the biharmonic wave equation with a random potential

TL;DR

This paper investigates the inverse scattering problem for the biharmonic wave equation with a Gaussian random potential, establishing uniqueness results and the ability to determine the potential's correlation from high-frequency data.

Contribution

It proves unique continuation and well-posedness for the biharmonic wave equation with rough potentials, and shows the correlation strength can be uniquely identified from high-frequency scattering data.

Findings

Unique continuation principle established for rough potentials.

Correlation strength of the random potential is uniquely determined by high-frequency data.

Single realization data suffices for uniqueness in lossless media.

Abstract

We consider the inverse random potential scattering problem for the two- and three-dimensional biharmonic wave equation in lossy media. The potential is assumed to be a microlocally isotropic Gaussian rough field. The main contributions of the work are twofold. First, the unique continuation principle is proved for the fourth order biharmonic wave equation with rough potentials and the well-posedness of the direct scattering problem is established in the distribution sense. Second, the correlation strength of the random potential is shown to be uniquely determined by the high frequency limit of the second moment of the scattered field averaged over the frequency band. Moreover, we demonstrate that the expectation in the data can be removed and the data of a single realization is sufficient for the uniqueness of the inverse problem with probability one when the medium is lossless.