Propagation of singularities and inverse problems for the viscoacoustic wave equation

TL;DR

This paper investigates an inverse problem for the viscoacoustic wave equation, demonstrating unique recovery of medium parameters and the memory kernel using propagation of singularities and geometric analysis.

Contribution

It introduces a novel approach to analyze the inverse problem for viscoacoustic media without restrictions on sound speed, linking it to lens rigidity and geodesic transforms.

Findings

Unique determination of sound speed and memory kernel derivatives from exterior data.

Construction of solutions concentrating near geodesics for singularity analysis.

Extension of results to the extended Maxwell model.

Abstract

We study an inverse problem for the viscoacoustic wave equation, an integro-differential model describing wave propagation in viscoacoustic media with memory in the leading order term. The medium is characterized by a spatially varying sound speed and a space-time dependent memory kernel. Assuming that waves are generated by sources supported outside the region of interest, we consider exterior measurements encoded by the source-to-solution map. To study this inverse problem, we construct solutions concentrating near fixed geodesics and establish a corresponding propagation of singularities result for the semiclassical wave front set. These results are valid without any restriction on the underlying sound speed. Then, under certain geometric conditions, we prove that the exterior data uniquely determine not just the sound speed inside the domain but also all time derivatives at zero of the memory kernel. This involves a reduction to the lens rigidity and geodesic ray transform inverse problems. As an application, we establish uniqueness for the recovery of variable parameters in the extended Maxwell model.