Location and size estimation of small rigid bodies using elastic far-fields

TL;DR

This paper develops methods for detecting and estimating the size of small rigid obstacles in 3D elastic media using various wave components, based on the Foldy-Lax approximation, with proven effectiveness under certain conditions.

Contribution

It demonstrates that either pressure or shear waves suffice for inverse detection and size estimation of small elastic scatterers, extending the applicability of elastic scattering analysis.

Findings

Detection and size estimation are possible with either P or S waves.

Shear-horizontal and shear-vertical waves are also sufficient for localization.

Finer size estimates are derived under convexity assumptions.

Abstract

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by (possibly many) small rigid obstacles of arbitrary Lipschitz regular shapes in 3D. Based on the Foldy-Lax approximation, valid under a sufficient condition on the number of the obstacles, the size and the minimum distance between them, we show that any of the two body waves, namely the pressure waves P or the shear waves S, is enough for solving the inverse problem of detecting these scatterers and estimating their sizes. Further, it is also shown that the shear-horizontal part SH or the shear vertical part SV of the shear waves S are also enough for the location detection and the size estimation. Under some extra assumption on the scatterers, as the convexity assumption, we derive finer size estimates as the radius of the largest ball contained in each scatterer and the one of the smallest ball containing it. The two estimates measure, respectively, the thickness and length of each obstacle.