Partial Uniqueness and Obstruction to Uniqueness in Inverse Problems for Anisotropic Elastic Media

Abstract

We consider the inverse problem of identifying the density and elastic moduli for threedimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface. These surface measurements are modelled by the dynamic Dirichlet-to-Neumann map on a finite time interval. For linear or nonlinear anisotropic hyperelastic bodies we show that the displacement-to-traction surface measurements do not change when the density and elasticity tensor in the interior are transformed tensorially by a change of coordinates fixing the surface of the body to first order. Our main tool, a new approach in inverse problems for elastic media, is the representation of the equations of motion in a covariant form (following Marsden and Hughes, 1983) that preserves the underlying physics.In the case of classical linear elastodynamics we then investigate how the type of anisotropy changes under coordinate transformations. That is, we analyze the orbits of general linear, anisotropic elasticity tensors under the action by pull-back of diffeomorphisms that fix the surface of the elastic body to first order, and derive a pointwise characterization of parts of the orbits under this action. For example, we show that the orbit of isotropic elastic media, at any point in the body, consists of some transversely isotropic and some orthotropic elastic media. We then derive the first uniqueness result in the dynamic inverse problem for anisotropic media using surface displacement-traction data: uniqueness of three elastic moduli for tensors in the orbit of isotropic elasticity tensors. (2000): 35R30, 74B05, 70G45, 74B20. Mathematics Subject Classifications

Particle relabelling transformations in elastodynamics

“…The above results generalize and extend those of Mazzucato & Rachele (2006) for finite hyperelasticity. Our approach, however, uses only elementary methods, and does not require lengthy calculations in local coordinates.…”

“…Pendry et al 2006;Rahm et al 2008). Of particular relevance to geophysics is the work of Mazzucato & Rachele (2006), who establish non-uniqueness results for both finite and linearized elasticity. Though the situation considered by these authors (the so-called Dirichlet-to-Neumann map) is not directly relevant to surface observations in seismology, it is clear that their methods could be extended in a suitable manner.…”

“…At face value, such results imply that there exists an infinite-fold non-uniqueness for seismic inverse problems even with error-free data everywhere on the Earth's surface, and this clearly has significant and worrying implications for seismic tomography. The work of Mazzucato & Rachele (2006) is based on a covariant formulation of elasticity due to Marsden & Hughes (1983), but involves some lengthy calculations in local coordinates. In this paper we re-derive and extend their results using elementary methods.…”

A geometric theory of thermal stresses

“…Geometric linearization of elasticity was first introduced by Marsden and Hughes [29] and was further developed by Yavari and Ozakin [43]. See also [33] for similar discussions. Here, we start with a temperature-dependent material manifold and its motion in an ambient space.…”

On Transversely Isotropic Elastic Media with Ellipsoidal Slowness Surfaces

“…Proof of Equation (17). It follows, for example, from the proof of [25,Lemma 2.7], that the determinant of the principal symbol of P ρ,C is given by (−ρτ 2 + α)…”