Inverse obstacle scattering problem for elastic waves with phased or phaseless far-field data

TL;DR

This paper develops a novel method for reconstructing the shape and location of rigid obstacles in elastic wave scattering using phased or phaseless far-field data, employing Helmholtz decomposition and iterative algorithms.

Contribution

It introduces a new approach combining Helmholtz decomposition, a Nyström discretization, and iterative methods with a reference ball technique for phaseless data in elastic inverse scattering.

Findings

The method accurately reconstructs obstacle shape and position.

The approach effectively handles phaseless data by breaking translation invariance.

Numerical experiments confirm robustness and efficiency.

Abstract

This paper concerns an inverse elastic scattering problem which is to determine the location and the shape of a rigid obstacle from the phased or phaseless far-field data for a single incident plane wave. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The relation is established between the compressional or shear far-field pattern for the elastic wave equation and the corresponding far-field pattern for the coupled Helmholtz equations. An efficient and accurate Nystr\"{o}m type discretization for the boundary integral equation is developed to solve the coupled system. The translation invariance of the phaseless compressional and shear far-field patterns are proved. A system of nonlinear integral equations is proposed and two iterative reconstruction methods are developed for the inverse problem. In particular, for the phaseless data, a reference ball technique is introduced to the scattering system in order to break the translation invariance. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.