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Determination of the density in the linear elastic wave equation

Jian Zhai

TL;DR

This work studies the inverse boundary value problem for the 3D isotropic elastic wave equation and proves that, with the P- and S-wave speeds known, the density $\rho$ is uniquely recoverable from boundary measurements under a strictly convex foliation condition. The authors construct high-frequency geometric optics solutions and connect the boundary data to longitudinal and transverse geodesic ray transforms of tensors that depend on $\rho$, enabling reconstruction via injectivity under the foliation hypothesis. A fourth-order elliptic equation for $\log\sqrt{\rho}$ is derived from the recovered Sym$(N)$, and unique continuation with boundary values yields interior recovery of $\rho$. This result removes the prior restriction $\lambda \neq 2\mu$ and advances seismic inversion by enabling density determination from S-wave data given fixed wave speeds.

Abstract

We study the inverse boundary value problem for the linear elastic wave equation in three-dimensional isotropic medium. We show that both the Lamé parameters and the density can be uniquely recovered from the boundary measurements under the strictly convex foliation condition.

Determination of the density in the linear elastic wave equation

TL;DR

This work studies the inverse boundary value problem for the 3D isotropic elastic wave equation and proves that, with the P- and S-wave speeds known, the density is uniquely recoverable from boundary measurements under a strictly convex foliation condition. The authors construct high-frequency geometric optics solutions and connect the boundary data to longitudinal and transverse geodesic ray transforms of tensors that depend on , enabling reconstruction via injectivity under the foliation hypothesis. A fourth-order elliptic equation for is derived from the recovered Sym, and unique continuation with boundary values yields interior recovery of . This result removes the prior restriction and advances seismic inversion by enabling density determination from S-wave data given fixed wave speeds.

Abstract

We study the inverse boundary value problem for the linear elastic wave equation in three-dimensional isotropic medium. We show that both the Lamé parameters and the density can be uniquely recovered from the boundary measurements under the strictly convex foliation condition.

Paper Structure

This paper contains 5 sections, 8 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Geometric optics solutions
  3. P-waves and related results
  4. S-waves and proof of the main theorem
  5. Proof of lemmas

Key Result

Proposition 1

(uhlmann2024invertibility2) Assume $\partial \Omega$ is strictly convex and $(\Omega,g)$ admits a smooth strictly convex function. Assume also that the second order tensor $f$ is symmetric. If for any geodesic $\gamma$ in $\Omega$ with endpoints on $\partial \Omega$, and $\eta$ a parallel vector field orthogonal to $\gamma$. Then $f=0$.

Theorems & Definitions (15)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Proposition 2
  • Corollary 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 5 more