Reconstruction of anisotropic stiffness tensors from partial data around one polarization

TL;DR

This work links microlocal elastic wave propagation to algebraic geometry to address the inverse problem of reconstructing anisotropic stiffness tensors from partial travel-time data. By treating the slowness polynomial $P_{\mathbf{a}}(\mathbf{p})=\det(\Gamma(\mathbf{p})-I_n)$ as the central object, the authors show that, generically, a small open patch of a single slowness sheet determines the entire slowness surface and, in 2D, the full stiffness tensor; in 3D, generic orthorhombic/monoclinic tensors admit only a finite set of anomalous companions with the same slowness polynomial. The analysis hinges on the irreducibility of slowness polynomials across a base of stiffness tensors, using Grothendieck’s generic geometric integrality and, practically, Gröbner-bases computations to reconstruct tensors from polynomial data. They also develop a two-layer model to study layered media, proving that outer-layer data can determine the outer stiffness uniquely and that the inner layer can be recovered under generic conditions. The results reveal that increasing anisotropy can improve identifiability in elasticity inverse problems, with positivity constraints characterized by Cayley cubic geometry, and they provide constructive algorithms for stiffness reconstruction from slowness data, including explicit examples and computational pathways.

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 Gröbner 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.

Figures (3)

• Figure 1: A 2-dimensional slowness curve $S_x$ with $x\in\mathbb{R}^2$ fixed and parameters $b_{11}=10$, $b_{22}=12$, $b_{33}=20$, $b_{12}=2$, $b_{23}=5$, $b_{13}=3$; cf. §\ref{['sec:alg-principles']}. The slowness curve is on the left and the group velocity curve is on the right. Here, the multiplicity of each eigenvalue in the Christoffel matrix is one for all $\mathbf{p}\neq0$, so different eigenvectors (polarizations) correspond to different eigenvalues $\lambda_i$. The slowness curve $S_x$ is the set where at least one of the $\lambda_i$'s equals $1$. The bigger eigenvalue corresponds to quasi-pressure polarization (qP) and the smaller one to quasi-shear (qS). The gradients of the eigenvalue functions on the left correspond to points on the right and vice versa, via Legendre duality. The blue curve on the right is the unit sphere of the Finsler geometry whose geodesics the wave packets follow. Polarization vectors are not indicated in these figures. For a description of how the slowness curve arises in geophysics and microlocal analysis see, e.g., Yedling Yedling, who provides a geometric analysis of the wave front in a homogeneous anisotropic material.
• Figure 2: A cartoon of the proof of Theorem \ref{['thm:2-layers']}. In the first step, we use short geodesics near the boundary, depicted as the blue line segments to the right. In the second step, we vary this family of line segments until they hit $\partial\omega_i$ and no longer exist, depicted as the dashed parts of the blue line segments. In the third step we take two points on $\partial\omega_i$ and find their qP distance, depicted as the solid red line, hitting them with all possible rays through the now known mantle, depicted as the dashed red lines.
• Figure 3: The Cayley cubic surface $1 + 2xyz - x^2 - y^2 - z^2 = 0$ in $\mathbb{R}^3$. Points in the roughly tetrahedral shape in the middle correspond to anomalous companions of an orthorhombic stiffness tensor that are physically realizable, while points in the four cone-shaped regions extending to infinity do not.

Theorems & Definitions (41)

• Definition 1
• Remark 4
• Remark 5
• Theorem A: Uniqueness of slowness surfaces and stiffness tensor from partial data
• Theorem B: Two-layer model
• Remark 7
• Theorem C: Generic irreducibility
• Theorem D: Generic Reconstruction
• Conjecture E
• Remark 8