Reconstruction of anisotropic stiffness tensors from partial data around one polarization

TL;DR

This paper investigates the inverse problem of reconstructing anisotropic stiffness tensors from limited data around one polarization, using algebraic geometry tools, and demonstrates conditions for unique determination in two dimensions and conjectures in three dimensions.

Contribution

It introduces a novel algebraic geometric approach to determine anisotropic stiffness tensors from partial data, revealing increased uniqueness with material asymmetry.

Findings

Unique reconstruction in 2D for generic tensors.

Partial data suffices for tensor determination in certain cases.

Counterintuitive result: more anisotropy leads to easier inverse problems.

Abstract

We study inverse problems in anisotropic elasticity using tools from algebraic geometry. The singularities of solutions to the elastic wave equation in dimension $n$ with an anisotropic stiffness tensor have propagation kinematics captured by so-called slowness surfaces, which are hypersurfaces in the cotangent bundle of $\mathbb{R}^n$ that turn out to be algebraic varieties. Leveraging the algebraic geometry of families of slowness surfaces we show that, for tensors in a dense open subset in a space of anisotropic two-dimensional stiffness tensors, a small amount of data around one polarization in an individual slowness surface uniquely determines the entire slowness surface and its stiffness tensor. In three dimensions, for generic orthorhombic and monoclinic stiffness tensors, a small number of anomalous companions give rise to the same slowness surface; nevertheless, we conjecture that in the most anisotropic setting (triclinic) the tensor is unique, as in two dimensions. The partial data needed to determine a tensor arises naturally from seismological measurements or geometrized versions of seismic inverse problems. Additionally, we explain how the reconstruction of the stiffness tensor can be carried out effectively, using Gr\"obner bases. Our uniqueness or finiteness results fail for symmetric materials (e.g., fully isotropic), evidencing the counterintuitive claim that inverse problems in elasticity can become more tractable with increasing asymmetry.