Recovery of a source term or a speed with one measurement and applications

TL;DR

This paper addresses the inverse problem of recovering source terms and sound speed in wave equations using a single measurement, providing uniqueness and stability conditions with applications to thermoacoustic tomography.

Contribution

It introduces sharp conditions for uniqueness and stability in recovering source terms and sound speed from minimal data, combining Carleman estimates with geometric analysis.

Findings

Established uniqueness conditions for source recovery.

Derived stability criteria for sound speed reconstruction.

Applied results to thermoacoustic tomography.

Abstract

We study the problem of recovery the source $a(t,x)F(x)$ in the wave equation in anisotropic medium with $a$ known so that $a(0,x)\not=0$ with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery the sound speed in the equation $u_{tt} -c^2(x)\Delta u =0$ with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented.