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Horizontal and Vertical Regularity of Elastic Wave Geometry

Joonas Ilmavirta, Pieti Kirkkopelto, Antti Kykkänen

TL;DR

This work develops a rigorous link between stiffness-tensor regularity and the Finsler geometry that models elastic-wave propagation, focusing on the qP-waves. It introduces anisotropic function spaces and shows that, under global separation of the qP slowness surface, the elastic Finsler function is $C^k$ on the manifold and smooth on fibers, with Legendre transforms preserving this regularity. The paper provides complete 2D vertical-regularity criteria and a near-isotropic stability result in higher dimensions, then leverages these to prove injectivity of the geodesic X-ray transform and a travel-time rigidity result for inverse problems. It also develops a detailed algebraic framework distinguishing varieties and schemes to analyze slowness-surface singularities, offering broad implications for seismic imaging under low-regularity elasticity and Finslerian modeling.

Abstract

The elastic properties of a material are encoded in a stiffness tensor field and the propagation of elastic waves is modeled by the elastic wave equation. We characterize analytic and algebraic properties a general anisotropic stiffness tensor field has to satisfy in order for Finsler-geometric methods to be applicable in studying inverse problems related to imaging with elastic waves.

Horizontal and Vertical Regularity of Elastic Wave Geometry

TL;DR

This work develops a rigorous link between stiffness-tensor regularity and the Finsler geometry that models elastic-wave propagation, focusing on the qP-waves. It introduces anisotropic function spaces and shows that, under global separation of the qP slowness surface, the elastic Finsler function is on the manifold and smooth on fibers, with Legendre transforms preserving this regularity. The paper provides complete 2D vertical-regularity criteria and a near-isotropic stability result in higher dimensions, then leverages these to prove injectivity of the geodesic X-ray transform and a travel-time rigidity result for inverse problems. It also develops a detailed algebraic framework distinguishing varieties and schemes to analyze slowness-surface singularities, offering broad implications for seismic imaging under low-regularity elasticity and Finslerian modeling.

Abstract

The elastic properties of a material are encoded in a stiffness tensor field and the propagation of elastic waves is modeled by the elastic wave equation. We characterize analytic and algebraic properties a general anisotropic stiffness tensor field has to satisfy in order for Finsler-geometric methods to be applicable in studying inverse problems related to imaging with elastic waves.

Paper Structure

This paper contains 43 sections, 26 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Horizontal Regularity
  4. Vertical Regularity
  5. Applications to Inverse Problems
  6. Algebraic Notions of Singularities and Implications for Analysis
  7. Motivation, Context and Related Results
  8. Organization of the Article
  9. Acknowledgments
  10. Geometric preliminaries
  11. Elasticity
  12. Finsler Geometry
  13. Metric Structure
  14. Geodesics
  15. Horizontal Regularity
  16. ...and 28 more sections

Key Result

Theorem 1

Let $M\subset\mathbb{R}^n$ be a smooth domain with or without boundary and let $c\in C^k(M)$ be a stiffness tensor field such that the qP-branch of the slowness surface associated to $c$ is globally separate. Then the associated Finsler function $F^c_{qP}$ is of class $C^k$ along the manifold and sm

Theorems & Definitions (61)

  • Theorem 1: Proven in Section \ref{['sec:thm-1-proof']}
  • Theorem 2: Proven in Section \ref{['sec:singularity-2D']}
  • Theorem 3: Proven in Section \ref{['sec:singularity-3D']}
  • Remark 4
  • Corollary 5: Proven in Section \ref{['sec:xrt']}
  • Corollary 6: Proven in Section \ref{['sec:travel-time']}
  • Definition 7
  • Proposition 8: Comparison of concepts of singularity for real slowness surfaces; proven in Section \ref{['sec:interpretation-sing-pt']}
  • Proposition 9: Comparison of concepts of singularity for complex slowness surfaces; proven in Section \ref{['sec:interpretation-sing-pt']}
  • Remark 10
  • ...and 51 more