A Spectral Target Signature for Thin Surfaces with Higher Order Jump Conditions

TL;DR

This paper develops a method to uniquely identify and analyze thin, anisotropic, dissipative inhomogeneities in 2D and 3D from scattering data by using a spectral approach based on eigenvalues, without requiring detailed governing equations.

Contribution

It introduces a novel spectral target signature and eigenvalue-based method for inverse scattering of thin surfaces with complex jump conditions, enabling coefficient identification from far field data.

Findings

Unique determination of surface coefficients from far field patterns.

Eigenvalues can be extracted from scattering data to identify surface properties.

Preliminary numerical results support the effectiveness of the proposed inversion method.

Abstract

In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in ${\mathbb R}^m$, $m=2,3$ from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open $m-1$ dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in \cite{Screens}. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach.