Corrigendum to "Inverse Problems for abstract evolution equations II: higher order differentiability for viscoelasticity

TL;DR

This paper corrects an error in previous work on the mathematical analysis of full waveform inversion in viscoelastic media, specifically regarding higher order differentiability of the parameter-to-solution map.

Contribution

It provides corrected analytic expressions for derivatives of the parameter-to-solution map in viscoelastic FWI, enhancing the mathematical foundation of the inverse problem.

Findings

Corrected the first and second order Fréchet derivatives expressions.

Clarified the impact of the correction on previous results.

Strengthened the theoretical basis for FWI in viscoelasticity.

Abstract

In our paper [SIAM J.\ Appl.~Math.\ 79-6 (2019), https://doi.org/10.1137/19M1269403] we considered full waveform inversion (FWI) in the viscoelastic regime. FWI entails the nonlinear inverse problem of recovering parameter functions of the viscoelastic wave equation from partial measurements of reflected wave fields. We have obtained explicit analytic expressions for the first and second order Fr\'echet derivatives and their adjoints (adjoint wave equations) of the underlying parameter-to-solution map. In the present manuscript we correct an error which occurred in a basic result and thus affected several other statements, which we have also corrected.