Uniqueness and stability in determining the wave equation from a single passive boundary measurement

TL;DR

This paper proves that both the wave speed and initial source in a multidimensional wave equation can be uniquely and stably determined from a single boundary measurement, broadening applicability to practical imaging scenarios.

Contribution

It establishes new uniqueness and stability results for recovering wave speed and source from minimal boundary data, relaxing previous restrictive conditions.

Findings

Proves uniqueness and Hölder stability in determining wave parameters

Extends results to general piecewise constant sound speeds

Does not require decay in time of solutions

Abstract

This article addresses the inverse problem of simultaneously recovering both the wave speed coefficient and an unknown initial condition (acting as the source) for the multidimensional wave equation from a single passive boundary measurement. Specifically, we establish uniqueness and H\"{o}lder stability estimates for determining these parameters in the wave equation on $\mathbb{R}^3$, where only a single boundary measurement of the solution--generated by the unknown source--is available. Our work connects to thermoacoustic and photoacoustic tomography (TAT/PAT) for the physically relevant case of piecewise constant sound speeds. We significantly relax the stringent conditions previously required for resolving this problem, extending results to general classes of piecewise constant sound speeds over inclusions with unknown locations. Moreover, we do not require decay properties in time of solutions to the wave equation, which enables our study to accommodate a much broader class of unknown sources. The approach combines low frequency-domain solution representations with distinctive properties of elliptic and hyperbolic equations.