Reconstruction of interfaces from the elastic farfield measurements using CGO solutions

TL;DR

This paper develops a method to reconstruct interfaces in elastic media from farfield measurements using complex geometrical optics solutions, enabling the estimation of both convex and non-convex parts of the interface with limited data.

Contribution

It introduces a novel approach to inverse elastic scattering that employs CGO solutions with explicit decay properties to recover interfaces from minimal farfield data.

Findings

Successfully estimates convex hulls of interfaces

Reconstructs non-convex interface parts using single wave measurements

Applicable to both penetrable and impenetrable obstacles

Abstract

In this work, we are concerned with the inverse scattering by interfaces for the linearized and isotropic elastic model at a fixed frequency. First, we derive complex geometrical optic solutions with linear or spherical phases having a computable dominant part and an $H^\alpha$-decaying remainder term with $\alpha <3$, where $H^{\alpha}$ is the classical Sobolev space. Second, based on these properties, we estimate the convex hull as well as non convex parts of the interface using the farfields of only one of the two reflected body waves (pressure waves or shear waves) as measurements. The results are given for both the impenetrable obstacles, with traction boundary conditions, and the penetrable obstacles. In the analysis, we require the surfaces of the obstacles to be Lipschitz regular and, for the penetrable obstacles, the Lam\'e coefficients to be measurable and bounded with the usual jump conditions across the interface.