Topological Sensitivity Based Far-Field Detection of Elastic Inclusions

TL;DR

This paper develops and analyzes topological sensitivity algorithms for detecting small elastic inclusions using far-field scattering data, addressing mode-conversion issues to improve detection stability and resolution.

Contribution

It introduces a weighted location indicator that effectively handles mode-conversion, enhancing the stability and resolution of elastic inclusion detection.

Findings

The unweighted indicator has low resolution with mode-conversion.

The weighted indicator achieves stable detection with Rayleigh resolution.

The algorithms are rigorously analyzed for stability and performance.

Abstract

The aim of this article is to present and rigorously analyze topological sensitivity based algorithms for detection of diametrically small inclusions in an isotropic homogeneous elastic formation using single and multiple measurements of the far-field scattering amplitudes. A $L^2-$cost functional is considered and a location indicator is constructed from its topological derivative. The performance of the indicator is analyzed in terms of the topological sensitivity for location detection and stability with respect to measurement and medium noises. It is established that the location indicator does not guarantee inclusion detection and achieves only a low resolution when there is mode-conversion in an elastic formation. Accordingly, a weighted location indicator is designed to tackle the mode-conversion phenomenon. It is substantiated that the weighted function renders the location of an inclusion stably with resolution as per Rayleigh criterion.