Inverse source problems in an inhomogeneous medium with a single far-field pattern

TL;DR

This paper establishes uniqueness results for inverse source problems in inhomogeneous media using a single far-field pattern, focusing on convex-polygonal sources and their corner properties, and proposes a data-driven inversion method.

Contribution

It introduces new uniqueness theorems for inverse source problems with convex-polygonal supports and develops a corner scattering theory-based inversion scheme.

Findings

Unique determination of source support and derivatives at corners.

Characterization of admissible source functions including harmonic ones.

Extension of radiated field across corners is impossible.

Abstract

This paper concerns time-harmonic inverse source problems with a single far-field pattern in two dimensions, where the source term is compactly supported in an a priori given inhomogeneous background medium. For convex-polygonal source terms, we prove that the source support together with the zeroth and first order derivatives of the source function at corner points can be uniquely determined. Further, we prove that an admissible set of source functions (including harmonic functions) having a convex-polygonal support can be uniquely identified by a single far-field pattern. A class of radiating sources is characterized and the extension of the radiated field across a corner point is proven impossible. The corner scattering theory leads to a data-driven inversion scheme for imaging an arbitrarily convex-polygonal source support.