An inverse kinematic problem with internal sources

Abstract

Abstract. Given a bounded domain M in R n with a conformally Euclidean metric g = ρ dx 2 , in this paper we consider the inverse problem of recovering a semigeodesic neighborhood of a domain Γ ⊂ ∂M and the conformal factor ρ in the neighborhood from the travel time data (defined below) and the Cartesian coordinates of Γ. We develop an explicit reconstruction procedure for this problem. The key ingredient is the relation between the reconstruction and a Cauchy problem of the conformal Killing equation.

Reconstruction of a compact manifold from the scattering data of internal sources

“…A generic property. We will now formulate a generic property in Met(N ) that is related to the complete scattering data (21).…”

“…and the above steps prove that the map Ψ is a diffeomorphism. Also we will explicitly construct the smooth structure for R E ∂M (M ) only using the data (21). In the steps below, we will often consider only one manifold and do not use the sub-indexes, M 1 and M 2 , when ever it is not necessary.…”

“…Then K(p) is not empty and K(p) is determined by r and data (21). More precisely if (N i , g i ) and M i are as in Theorem 1.1 such that (15) and (23) hold, then for every p ∈ M 1 the set K(p) = ∅ and DΦ(K(p)) = K(Ψ(p)).…”

“…In the next Lemma we will show that we can find the set of conjugate directions with respect to p from data (21) and there exist lots of (q, η) ∈ r := R E ∂M (p) such that η is not a conjugate direction with respect to p. If (q, η) ∈ K(p) is not a conjugate direction with respect to p, we will later construct coordinates for p such that (n − 1)-coordinates are given by η. (45) Then K G (p) is not empty, π(K G (p)) ⊂ ∂M is open.…”

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

“…to determine the metric g up to boundary fixing isometries). While several methods to recover the geometry from the boundary distance functions have been proposed [14,21,22,35], these have not been implemented computationally to our knowledge. It appears to us that, at least in the isotropic case, it is better to recover the wave speed directly without first recovering the boundary distance functions.…”