Stability for Time-domain Elastic Wave Equations

TL;DR

This paper develops a method for reconstructing density in elastic wave equations using boundary control and complex geometric optics, establishing stability through Carleman estimates and connect operators.

Contribution

It introduces an explicit density reconstruction formula and proves stability for the inverse problem in elastic wave equations.

Findings

Explicit reconstruction formula for density

Stable observability via Carleman estimate

Density stability established through connect operator

Abstract

This paper is concerned with the inverse scattering problem involving the time-domain elastic wave equations in a bounded $d$-dimensional domain. First, an explicit reconstruction formula for the density is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator.