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

Maarten V De Hoop,Paul Kepley,Lauri Oksanen

doi:10.1137/17m1151481

Abstract

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semigeodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.

Highlights

We study the inverse boundary value problem for the wave equation from a reconstruction point of view through the design of an algorithm
We propose an approach to reducing the anisotropic inverse boundary value problem to a problem with data in the interior of M
To elliptic inverse problems with internal data [1], this hyperbolic internal data problem may be of independent interest, and we show a Lipschitz stability result for the problem under a geometric assumption

Summary

Introduction

We study the inverse boundary value problem for the wave equation from a reconstruction point of view through the design of an algorithm. One of the main components used to calculate these averages is a family of sources that solve blind control problems with target functions of the form \phi = 1B, where B is a set known as a wave cap. The following lemma is an amalgamation of results from [18] and shows that there is a family of sources \psi h,\alpha which produce approximately constant wave fields u\psi h,\alpha (T, \cdot ) on wave caps, and that these sources can be constructed from the boundary data \Lambda 2\GammaT ,\scrR. On the other hand, it is possible to construct the map \Phi g(y, s), and, the wave speed was determined in Belishev's original paper [7] by first showing that the internal data uf (t, x) can be recovered in Cartesian coordinates, and using the identity. \| \Phi g\~ - \Phi g\| C1(\Sigma ) \leq C \| Hc\~ - Hc\| C1(M)\rightarC1(\Sigma )

This together with

We use the notation

Note that

Note also