Inverse obstacle problem for the non-stationary wave equation with an unknown background

TL;DR

This paper presents a method to locate and estimate the position of an unknown obstacle within a domain by analyzing boundary measurements of wave propagation, applicable even with an unknown background wave speed.

Contribution

It introduces a novel approach to determine the distance to an inclusion in a wave speed, using volume reconstruction of domains of influence, applicable to both known and unknown backgrounds.

Findings

Successfully locates inclusions in known background scenarios.

Reconstructs distances to inclusions even with unknown background wave speed.

Applicable to Riemannian surfaces with simple geometry.

Abstract

We consider boundary measurements for the wave equation on a bounded domain $M \subset \R^2$ or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in $M$. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For $\tau \in C(\p M)$ the domain of influence $M(\tau)$ is the set of those points on the manifold from which the distance to some boundary point $x$ is less than $\tau(x)$.