Inverse acoustic scattering for random obstacles with multi-frequency data

TL;DR

This paper presents a two-stage inversion method for reconstructing the shape and statistical properties of random obstacles in acoustic scattering using multi-frequency data, supported by theoretical analysis and numerical validation.

Contribution

It introduces a well-defined Gaussian process model for random obstacles and develops a two-stage inversion approach with theoretical justifications.

Findings

Successful reconstruction of obstacle shape and statistical features

Stable recovery demonstrated in numerical experiments

Theoretical analysis confirms model well-posedness and convergence

Abstract

We study an inverse random obstacle scattering problems in $\mathbb{R}^2$ where the scatterer is formulated by a Gaussian process defined on the angular parameter domain. Equipped with a modified covariance function which is mathematically well-defined and physically consistent, the Gaussian process admits a parameterization via Karhunen--Lo\`eve (KL) expansion. Based on observed multi-frequency data, we develop a two-stage inversion method: the first stage reconstructs the baseline shape of the random scatterer and the second stage estimates the statistical characteristics of the boundary fluctuations, including KL eigenvalues and covariance hyperparameters. We further provide theoretical justifications for the modeling and inversion pipeline, covering well-definedness of the Gaussian-process model, convergence for the two-stage procedure and a brief discussion on uniqueness. Numerical experiments demonstrate stable recovery of both geometric and statistical information for obstacles with simple and more complex shapes.