An Inverse Problem in Elastodynamics: Uniqueness of the Wave Speeds in the Interior

Abstract

We consider the unique determination of internal properties of a nonhomogeneous, isotropic elastic object from measurements made at the surface. The 3-dimensional object is modelled by solutions of the linear hyperbolic system of equations for elastodynamics, whose (leading) coefficients correspond to the internal properties of the object (its density and elasticity). We model surface measurements by the Dirichlet-to-Neumann map on a finite time interval. In a previous paper the author has shown that the density and elastic properties of the surface of the object are uniquely determined by the Dirichlet-to-Neumann map. Here we apply that result to conclude that certain properties of the interior of the object (the wave speeds) are also determined. We then observe that the elastodynamic polarization is determined outside the object by the Dirichlet-to-Neumann map. We conclude, in the case that the wave speeds are constant, that the polarization data do not determine the density in the interior. This problem and techniques used in its study are closely related to those in, for example, seismology and medical imaging. The techniques used here, though (from geometric optics, integral geometry, and microlocal analysis) lead to the solution of this fully three-dimensional problem. Academic Press

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.…”

Carleman estimates for the non-stationary Lamé system and the application to an inverse problem

“…For inverse problems for the non-stationary Lamé equation by infinitely many boundary observations (i.e., Dirichlet-toNeumann map), we refer to Rachele [42], for example. A monograph of Yahkno [48] is concerned with the inverse problems for the Lamé system.…”

An inverse problem for the dynamical Lamé system with two sets of boundary data

“…For the case of many boundary measurements (the Dirichlet-to-Neumann map), we refer the reader to Rachele [20] and Yakhno [24].…”

“…In spite of the importance of the inverse problem for the Lamé system, there are only a few papers concerning its mathematical analysis. For the case of many boundary measurements (the Dirichlet-to-Neumann map), we refer the reader to Rachele [20] and Yakhno [24].…”