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.