Thermoacoustic tomography for an integro-differential wave equation modeling attenuation

TL;DR

This paper addresses the inverse problem in thermoacoustic tomography within attenuating media modeled by integro-differential equations, establishing uniqueness, stability, and a reconstruction method using modified time reversal.

Contribution

It introduces a novel reconstruction approach for TAT in media with fractional derivative-based attenuation, including a Neumann series formula and stability analysis.

Findings

Proved uniqueness and stability of the inverse problem.

Developed a convergent reconstruction method for variable sound speeds.

Modified time reversal yields a Neumann series reconstruction formula.

Abstract

In this article we study the inverse problem of thermoacoustic tomography (TAT) on a medium with attenuation represented by a time- convolution (or memory) term, and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters. Under the assumption of being able to measure data on the whole boundary, we prove uniqueness and stability, and propose a convergent reconstruction method for a class of smooth variable sound speeds. By a suitable modification of the time reversal technique, we obtain a Neumann series reconstruction formula.