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

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.

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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