On the Construction of Virtual Interior Point Source Travel Time Distances from the Hyperbolic Neumann-to-Dirichlet Map

Abstract

We introduce a new algorithm to construct travel time distances between a point in the interior of a Riemannian manifold and points on the boundary of the manifold, and describe a numerical implementation of the algorithm. It is known that the travel time distances for all interior points determine the Riemannian manifold in a stable manner. We do not assume that there are sources or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map, or its restriction, as our data. Our algorithm is a variant of the Boundary Control method, and to our knowledge, this is the first numerical implementation of the method in a geometric setting.

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

“…The construction of these sources will be recalled below in Lemma 2, but first we recall how wave caps are defined: We recall that, for all h > 0, the point x(y, s) belongs to the set cap Γ (y, s, h) and diam(cap Γ (y, s, h)) → 0 as h → 0, (see e.g. [16]). So, when h is small and φ is smooth, averaging φ over cap Γ (y, s, h) yields an approximation to φ(x(y, s)).…”

“…The difference between [16] and the present paper is that we do not use the sources f solving the control problems of the form (4) to construct boundary distance functions, instead we will use them to recover information in the interior of M . In the anisotropic case, this information is the internal data operator that gives wavefields solving (2) in semi-geodesic coordinates.…”

“…R = ∂M ) to construct boundary distance functions. In [16] we introduced a modification of [31] that allowed us to localize the problem and work with partial boundary data (i.e. R = ∂M ).…”

“…In the next two sections, we will describe techniques that allow us to do so in both the isotropic and anisotropic cases. These will be based on the control problem setup from [16] and we will recall the setup in this section.…”

“…After the N-to-D data has been generated, we use the data Λ 2T Γ ϕ i,j to discretize the connecting operator. We accomplish this using a minor modification of the procedure outlined in [16]. In particular, we discretize the connecting operator by computing a discrete approxi-mation to (6):…”