An Exact Redatuming Procedure for the Inverse Boundary Value Problem for the Wave Equation

TL;DR

This paper presents an exact method for transforming wave measurement data from one boundary configuration to another on a manifold, using unique continuation and stability analysis, with computational validation.

Contribution

It introduces a two-step exact redatuming procedure for the wave equation on manifolds, leveraging known boundary metrics and stability results.

Findings

The first step of the procedure has conditional Hölder stability.

The method successfully reconstructs data in computational experiments.

The approach applies to wave equations on bounded domains and manifolds.

Abstract

Redatuming is a data processing technique to transform measurements recorded in one acquisition geometry to an analogous data set corresponding to another acquisition geometry, for which there are no recorded measurements. We consider a redatuming problem for a wave equation on a bounded domain, or on a manifold with boundary, and model data acquisition by a restriction of the associated Neumann-to-Dirichlet map. This map models measurements with sources and receivers on an open subset $\Gamma$ contained in the boundary of the manifold. We model the wavespeed by a Riemannian metric, and suppose that the metric is known in some coordinates in a neighborhood of $\Gamma$. Our goal is to move sources and receivers into this known near boundary region. We formulate redatuming as a collection of unique continuation problems, and provide a two step procedure to solve the redatuming problem. We investigate the stability of the first step in this procedure, showing that it enjoys conditional H\"{o}lder stability under suitable geometric hypotheses. In addition, we provide computational experiments that demonstrate our redatuming procedure.