The enclosure method for inverse obstacle scattering over a finite time interval: IV. Extraction from a single point on the graph of the response operator

TL;DR

This paper introduces a simplified enclosure method for inverse obstacle problems governed by wave equations, using minimal boundary data to determine the distance from a point outside the domain to an unknown obstacle.

Contribution

It presents a new formulation of the enclosure method that requires only a single boundary measurement to extract obstacle distance information.

Findings

Enables obstacle distance extraction from a single boundary response

Uses only Neumann and Dirichlet data on the boundary

Applicable to wave equation governed inverse problems

Abstract

Now a final and maybe simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a {\it spacial domain with an outer boundary} over a finite time interval is fixed. The method employs only a single pair of a quite natural Neumann data prescribed on the outer boundary and the corresponding Dirichlet data on the same boundary of the solution of the governing equation over a finite time interval, that is a single point on the graph of the so-called {\it response operator}. It is shown that the methods enables us to extract the distance of a given point outside the domain to an embedded unknown obstacle, that is the maximum sphere centered at the point whose exterior encloses the unknown obstacle. To make the explanation of the idea clear only an inverse obstacle problem governed by the wave equation is considered.