Uniqueness of the density in an inverse problem for isotropic elastodynamics

Abstract

Abstract. We consider the unique determination of the density of a nonhomogeneous, isotropic elastic object from measurements made at the surface. We model the behavior of the bounded, 3-dimensional object by the linear, hyperbolic system of operators for isotropic elastodynamics. The material properties of the object (its density and elastic properties) correspond to the smooth coefficients of these differential operators. The data for this inverse problem, in the form of the correspondence between applied surface tractions and resulting surface displacements, is modeled by the dynamic Dirichlet-toNeumann map on a finite time interval. In an earlier paper we show that the speeds c p/s of (compressional and sheer) wave propagation through the object are uniquely determined by the Dirichlet-to-Neumann map. Here we extend that result by showing that the density is also determined in the interior by the Dirichlet-to-Neumann map in the case, for example, that cp = 2cs at only isolated points in the object. We use techniques from microlocal analysis and integral geometry to solve this fully three-dimensional problem.

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

“…(See also Pestov [27] for weaker conditions needed to invert the ray transform for tensor fields.) THEOREM 3 (Uniqueness for isotropic elastodynamics in Euclidean space (Rachele [29][30][31]…”

“…It follows that uniqueness results for classes of elastic media may be extended partially to any anisotropic elastic media in the orbits of those classes. We conclude, for example, in Section 2.1 that Rachele's uniqueness results [29][30][31][32] for isotropic elastodynamics (with or without residual stress) extend to certain anisotropic elastic media. Therefore, our next aim is to describe the orbits of density functions &(x) and elasticity tensors C(x) at a given but arbitrary point x 2 , under the action by pull-back of diffeomorphisms that fix the boundary to first order.…”

“…It follows from the proof of covariance for the nonlinear case that 0 0 is a motion of the transformed elastic body (31) and the reasoning in (45) and (47). It follows that A 0 ¼ *A since A depends tensorially on C, F, and the metric.…”

On uniqueness in the inverse problem for transversely isotropic elastic media with a disjoint wave mode

“…In particular, they show that the travel time of the wave through the object is uniquely determined by the associated DN map. Uniqueness also holds for isotropic elastodynamics [12], [13], [14], and, up to the natural obstruction, for isotropic media with residual stress [15], [16]. Hansen and Uhlmann [16] use a microlocal analytic approach to the study of reflection of singularities to prove that the scattering relation at the surface is determined by the Dirichlet-to-Neumann map, which extends Rachele's result [15] to the case that caustics are present.…”

“…To solve the inverse problem we assume that the material parameters are determined to infinite order at the boundary of the elastic object, and we proceed as in [11] and also [13], [14], [15], to extend the parameters to all of R 3 in a smooth fashion, here in a way that preserves strong ellipticity, the property of having a disjoint mode, and, for the corollary, the GWP property for the disjoint mode. To make this smooth extension in the case of general anisotropy we assume that R 3 Ω is diffeomorphic to the exterior of a ball in R 3 , but in the case of transverse isotropy we do not need this assumption.…”