Local uniqueness of the density from partial boundary data for isotropic elastodynamics

Abstract

We consider an inverse problem in elastodynamics arising in seismic imaging. We prove locally uniqueness of the density of a non-homogeneous, isotropic elastic body from measurements taken on a part of the boundary. We measure the Dirichlet to Neumann map, only on a part of the boundary, corresponding to the isotropic elasticity equation of a 3-dimensional object. In earlier works it has been shown that one can determine the sheer and compressional speeds on a neighborhood of the part of the boundary (accessible part) where the measurements have been taken. In this article we show that one can determine the density of the medium as well, on a neighborhood of the accessible part of the boundary. *

Inverse problems for nonlinear hyperbolic equations

“…The singularity of U (4) is carefully analyzed in [20]. If γ xj ,ζj , j = 1, 2, 3, 4 intersect at some point q ∈…”

“…This means that the four distorted plane waves u j , j = 1, 2, 3, 4 generate a point source at q, which produce new waves propagating along L + (q). One can detect the singularities of U (4) in V and use them to recover the metric g on I + (p − ) ∩ I − (p + ). One difficulty is that the metric is unknown outside V , and we need to infer whether those distorted plane waves indeed intersect at some point from the observation.…”

“…One difficulty is that the metric is unknown outside V , and we need to infer whether those distorted plane waves indeed intersect at some point from the observation. This can be done by very careful analysis of the singularities of U (4) in [20]. In particular, one needs to distinguish the singularities generated by interactions of four waves from those generated by three waves.…”

“…Remark 1. Assume c P = 2c S except at isolated points in Ω, then the density ρ can be determined separately under the foliation condition [4] and the no conjugate points condition with extra restriction on curvatures [28].…”

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

“…For the interior determination from Λ T with T large enough, uniqueness of smooth isotropic elastic tensor and density was shown under different geometrical conditions [24,25,28,4]. We will study the interior determination of piecewise analytic parameters based on our boundary determination results,.…”