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Propagation of singularities and inverse problems for the viscoacoustic wave equation

Giovanni Covi, Maarten de Hoop, Mikko Salo

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.

Propagation of singularities and inverse problems for the viscoacoustic wave equation

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.

Paper Structure

This paper contains 9 sections, 11 theorems, 154 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Motivation and connection to the literature
  4. Organization of the article
  5. Existence of approximate solutions and propagation of singularities
  6. Well-posedness of the direct problem and source-to-solution map
  7. Inverse problem
  8. Applications
  9. Derivation of the viscoacoustic equation

Key Result

Theorem 1.1

Let $\Omega \Subset\Omega'\subset\mathbb R^n$, $n\geq 2$, be open bounded sets, and assume $T>0$. Define the spacetime cylinders $M:= [0,T]\times\overline\Omega$ and $Z:=[0, T]\times\overline\Omega'$. Let $c \in C^\infty(\overline{\Omega}')$ be a positive function and consider a bicharacteristic $\g such that in $Z$, and $r_N \in L^2((0,T),L^2(\Omega'))$ satisfies The functions $\varphi, a'_l, a

Figures (3)

  • Figure 1: Represented in red is a propagating singularity. An example of a stationary singularity is given in orange.
  • Figure :
  • Figure :

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • proof : Proof of Theorem \ref{['GO-construction-general']}
  • Lemma 2.2
  • proof : Proof of Theorem \ref{['propagation']}
  • Remark 2.3
  • ...and 11 more