Unique Recovery of Piecewise Analytic Density and Stiffness Tensor from the Elastic-Wave Dirichlet-To-Neumann Map

Abstract

We study the recovery of piecewise analytic density and stiffness tensor of a threedimensional domain from the local dynamical Dirichlet-to-Neumann map. We give global uniqueness results if the medium is transversely isotropic with known axis of symmetry or orthorhombic with known symmetry planes on each subdomain. We also obtain uniqueness of a fully anisotropic stiffness tensor, assuming that it is piecewise constant and that the interfaces which separate the subdomains have curved portions. The domain partition need not to be known. Precisely, we show that a domain partition consisting of subanalytic sets is simultaneously uniquely determined.

Uniqueness for the inverse boundary value problem of piecewise homogeneous anisotropic elasticity in the time domain

“…The second main result is in the case when the geometry of this pieces is unknown. In Section 2 we briefly review a boundary determination result that has been obtained previously in [11], [13]. In Section 3 we discuss interior determination results for both Dirichlet-to-Neumann and Neumann-to-Dirichlet maps.…”

“…In this section we follow [13] to show that the DN map Λ T,Σ ρ,C , or the ND map Φ T,Σ ρ,C , determines the values of ρ| Σ and C| Σ . More precisely we will show the following.…”

“…That is for the hexagonal elasticity equation with ellipsoidal slowness surfaces (see [19]). In [13] a uniqueness result is proved for the piecewise constant elastic coefficients in the generally anisotropic case (or piecewise analytic, but with greater symmetry assumptions).…”

“…The argument showing the uniqueness in [13], as is typical for the above types of measurement data setups, consists of two steps. The first step is a "boundary determination" result.…”

“…We will consider the inverse boundary value problem for the piecewise homogeneous, generally anisotropic, elasticity equation in the space-time domain and prove the above type of uniqueness result. We will use the boundary determination result of [13]. In Section 2 we sketch the connection between the time domain equation and the elliptic one via the finite time Laplace transform and then quote the relevant boundary uniqueness result from [11], [13], [20].…”