On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method

TL;DR

This paper develops a method to detect obstacles and estimate impedance properties in a medium using time-domain electromagnetic data, specifically focusing on the Leontovich boundary condition and inverse scattering problems.

Contribution

It introduces an indicator function based on electric field measurements that reveals obstacle location and impedance contrast in a time-domain Maxwell system.

Findings

Successfully determines the obstacle's distance from the source.

Differentiates whether the impedance is above or below a critical value.

Provides a new approach for inverse obstacle problems with impedance boundary conditions.

Abstract

An inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval is considered. It is assumed that the electric field $\mbox{\boldmath $E$}$ and magnetic field $\mbox{\boldmath $H$}$ which are solutions of the Maxwell system are generated only by a current density at the initial time located not far a way from an unknown obstacle. The obstacle is embedded in a medium like air which has constant electric permittivity $\epsilon$ and magnetic permeability $\mu$. It is assumed that the fields on the surface of the obstacle satisfy the impedance-or the Leontovich boundary condition $\mbox{\boldmath $\nu$}\times\mbox{\boldmath $H$} -\lambda\,\mbox{\boldmath $\nu$}\times(\mbox{\boldmath $E$}\times\mbox{\boldmath $\nu$})=\mbox{\boldmath $0$}$ with $\lambda$ an unknown positive function and $\mbox{\boldmath $\nu$}$ the unit outward normal. The observation data are given by the electric field observed at the same place as the support of the current density over a finite time interval. It is shown that an indicator function computed from the electric fields corresponding two current densities enables us to know: the distance of the center of the common spherical support of the current densities to the obstacle; whether the value of the impedance $\lambda$ is greater or less than the special value $\sqrt{\epsilon/\mu}$.