A note on the enclosure method for an inverse obstacle scattering problem with a single point source

TL;DR

This paper explores the enclosure method for inverse obstacle scattering, demonstrating unique determination of convex hulls of sound-hard obstacles from a single wave source, with applications to various obstacle types.

Contribution

It introduces a formula to extract the support function of obstacles from wave data, extending the enclosure method to new scenarios and obstacle configurations.

Findings

Unique determination of convex hulls from single wave data

A formula for the support function of obstacles

Applications to thin obstacles and layered media

Abstract

This paper gives a note on an application of the enclosure method to an inverse obstacle scattering problem governed by the Helmholtz equation in two dimensions. It is shown that one can uniquely determine the convex hull of an unknown sound-hard polygonal obstacle from the trace of the total wave that was exerted by a single point source onto a known circle surrounding the obstacle provided the source is sufficiently far from the obstacle. The result contains a formula that extracts the value of the support function of the obstacle at a generic direction. Some other applications to thin obstacles, obstacles in a layered medium and the far-field equation in the linear sampling method are also included.