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

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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$. 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.