Direct and inverse scattering problem by an unbounded rough interface with buried obstacles

TL;DR

This paper investigates the scattering of time-harmonic waves by an unbounded rough interface with a buried obstacle, establishing well-posedness, approximation properties, and a uniqueness result for the inverse problem using near-field data.

Contribution

It introduces a novel reciprocity relation and proves the unique determination of the rough surface and buried obstacle from near-field measurements.

Findings

Well-posedness of the direct scattering problem established.

Scattered fields due to HSPSWs can be approximated by PSWs.

Unique determination of the rough surface and obstacle from near-field data.

Abstract

In this paper, we consider the direct and inverse problem of scattering of time-harmonic waves by an unbounded rough interface with a buried impenetrable obstacle. We first study the well-posedness of the direct problem with a local source by the variational method; the well-posedness result is then extended to scattering problems induced by point source waves (PSWs) and hyper-singular point source waves (HSPSWs). For PSW or HSPSW incident waves, the induced total field admits a uniformly bounded estimate in any compact subset far from the source position. Moreover, we show that the scattered field due to HSPSWs can be approximated by the scattered fields due to PSWs. With these properties and a novel reciprocity relation of the total field, we prove that the rough surface and the buried obstacle can be uniquely determined by the scattered near-field data measured only on a line segment above the rough surface. The proof substantially relies upon constructing a well-posed interior transmission problem for the Helmholtz equation.