Determination of an anisotropic perturbation in elastic inverse scattering

Abstract: We consider a linearized inverse scattering problem for elastic waves. We prove that a fully anisotropic perturbation of the elastic parameters around an isotropic and homogeneous reference can be uniquely determined by (single-)scattered waves. We also give a quantitative stability estimate for an isotropic perturbation, and as a consequence a rigidity result is established.

• The paper proves that the linearized forward map is injective for reconstructing all 22 parameters from compactly supported perturbations.
• It establishes a Lipschitz-type stability estimate for isotropic perturbations, ensuring local rigidity even with single incident direction data.
• The analysis employs microlocal techniques and progressive wave expansions to connect measured scattered fields with underlying tensor and density perturbations.

Determination of Fully Anisotropic Perturbations in Elastic Inverse Scattering

Problem Statement and Motivations

This paper addresses a longstanding question in the inverse theory of elastic wave equations: the unique determination of fully anisotropic perturbations of elastic parameters, together with the density, from measurements of scattered waves originating from known incident elastic plane waves. The linearized (Born approximation) setting considered focuses on perturbations around a homogeneous and isotropic reference medium. The authors also consider the rigidity and stability properties for the nonlinear map in the isotropic case.

The ability to reconstruct the complete set of 21 independent parameters of the anisotropic elastic tensor plus the density by using only scattered field measurements is of major theoretical and practical significance for areas such as seismology, nondestructive testing, and geophysical imaging. Previously, unique recovery for isotropic or partially anisotropic settings was established, but the fully anisotropic scenario was largely unresolved.

Main Results

Uniqueness for Fully Anisotropic Perturbations

At the core of the contribution is the proof that the linearized forward map at the isotropic background, denoted as $\mathcal{A}'[\mathbf{C}_0,1]$, is injective when acting on compactly supported perturbations of both the fourth-order elasticity tensor $\mathbf{C}$ (allowing full anisotropy) and the density $\rho$. That is, knowledge of the scattered time-domain wavefields measured outside the defect region for a discrete set of incident plane waves suffices to uniquely determine all 22 parameters (21 in the tensor, 1 in density) of the localized perturbation.

Importantly, the analysis demonstrates that no nontrivial gauge freedoms are present in this inverse problem in contrast to known results for anisotropic acoustic or Maxwell systems, despite the apparent high-dimensionality of the unknowns.

Stability and Rigidity for Isotropic Perturbations

In the case of isotropic perturbations, the authors establish a quantitative Lipschitz-type stability estimate for the nonlinear forward operator in a neighborhood of the isotropic reference. This result implies local rigidity: if two sets of isotropic parameters (Lamé coefficients and density) produce identical scattering data (with a single incident direction and both $P$ and $S$ polarizations), then they must coincide, provided the parameters are sufficiently close.

The stability is explicitly expressed in Sobolev norms, providing a rigorous quantification of the stability of inversion from a practical measurement setting involving only a single incident direction, as opposed to the full Dirichlet-to-Neumann (DtN) map which would require data for all angles.

Analytical Approach

The analysis hinges on a careful microlocal study of the linearized elastic wave equations and their responses to different types of incident fields (plane $P$-waves and $S$-waves) and their resulting scattered components. The authors:

• Utilize progressive wave expansions for the scattered field, tracking various singularity structures (Dirac delta distributions and their derivatives) corresponding to the primary, converted, and multiple-scattered elastic waves.
• Employ a unique continuation principle to propagate information from the exterior region (accessible by measurement) into the support of the perturbation.
• Systematically analyze the divergence and curl of the scattered field for different combinations of incident and measured wave types ($P$-$P$, $S$-$\mathbf{C}$0, $\mathbf{C}$1-$\mathbf{C}$2, $\mathbf{C}$3-$\mathbf{C}$4), leading to a sequence of transport and elliptic equations for the unknown tensor components.
• Use combinations of these analytic identities, together with the compact support property, to bootstrap the uniqueness result to all tensor and density components.
• For the isotropic case, establish stability via quantitative estimates relating Sobolev norms of the perturbations to the boundary data, leveraging the structure of the linearized forward map and the abstract framework of Stefanov-Uhlmann interpolation inequalities.

Notable Technical Points

• Dimensional Closure: Despite measurements involving only finitely many incident directions and polarizations, the analysis exploits the tensor symmetries and structure to ensure the system is determined for all 21 tensor parameters plus density.
• Absence of Gauge Freedoms: The injectivity result stands in contrast with the presence of nontrivial gauge equivalences in the analogous anisotropic acoustic and electromagnetic problems, and a discussion is given in the context of recent literature.
• Single-direction data: The Lipschitz stability for the isotropic case using only a single incident direction with both polarizations is a strong assertion, relevant for real-world experimental settings.
• Highly constructive identities: The derivation of explicit identities relating singular parts of the scattered field to derivatives of the elastic parameters provides a blueprint for potential algorithmic approaches.

Implications and Future Directions

The established uniqueness and stability results ensure that, in the linearized regime, the inverse elastic scattering problem for fully anisotropic media is, in principle, solvable from exterior measurements with a small set of experiments. This has direct consequences for methodologies in geophysical imaging and materials science where anisotropy is prevalent.

The techniques developed also serve as a foundation for addressing the nonlinear inverse problem, where full (not linearized) determinability remains open and significantly more difficult. The absence of gauge invariance, shown here in the linear regime, suggests the possibility of analogous results in the nonlinear setting, although substantial challenges remain.

The stability results point toward the feasibility of robust inversion in the presence of noise and modelling error for small deviations from isotropy—a scenario common in practice.

Potential future research avenues include:

• Extension of similar uniqueness and stability arguments to the frequency domain and more general background media.
• Investigation of the nonlinear (beyond Born) problem for fully anisotropic tensors, possibly through iterative and regularization methods informed by the present analysis.
• Development of inversion algorithms leveraging the progressive wave expansion and transport identities derived herein.

Conclusion

This work answers an outstanding question in the inverse elasticity literature by demonstrating that a fully anisotropic, compactly supported perturbation of the elastic tensor and density can be uniquely and stably determined from time-domain scattered wave measurements in the linearized regime. The analytical methodology is comprehensive and sets a new standard for the conceptual and technical treatment of inverse problems in elastodynamics. The results have both direct theoretical significance and practical impact for imaging anisotropic media using elastic-wave-based modalities.