Stable determination of an impedance obstacle by a single far-field measurement

TL;DR

This paper proves the first stability estimates for uniquely determining an impedance obstacle of polygonal shape using only a single far-field measurement, advancing inverse scattering theory.

Contribution

It introduces new stability estimates for impedance obstacle determination from a single measurement, including explicit geometric relationships and handling corner singularities.

Findings

First stability result for impedance obstacle with one measurement

Stability independent of boundary impedance for polygonal obstacles

Explicit geometric and wave field relationship at corners

Abstract

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in $\mathbb{R}^2$. The obstacle is of general polygonal shape and the impedance parameter can be variable. We establish the stability results by using a single far-field pattern, which constitutes a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. If the obstacle is of a generally polygonal shape, the stability in determining the obstacle is established in terms of a modified Hausdorff distance and is independent of the boundary impedance parameter. If the obstacle is further known to be convex, the stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. There are several technical novelties and development in the mathematical strategy developed for establishing the aforementioned stability results. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships among the geometric configurations of the obstacle and the vanishing order of the wave field at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner as well as to tackle the impedance boundary condition.