IPI
The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain
Masaru Ikehata
Inverse Problems & Imaging 2016, 10(1): 131-163 doi: 10.3934/ipi.2016.10.131
In this paper, a time domain enclosure method for an inverse obstacle scattering problem of electromagnetic wave is introduced. The wave as a solution of Maxwell's equations is generated by an applied volumetric current having an orientation and supported outside an unknown obstacle and observed on the same support over a finite time interval. It is assumed that the obstacle is a perfect conductor. Two types of analytical formulae which employ a single observed wave and explicitly contain information about the geometry of the obstacle are given. In particular, an effect of the orientation of the current is catched in one of two formulae. Two corollaries concerning with the detection of the points on the surface of the obstacle nearest to the centre of the current support and curvatures at the points are also given.
keywords: Enclosure method inverse obstacle scattering obstacle electromagnetic wave reflection mean value theorem. Maxwell`s equations
IPI
On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method
Masaru Ikehata
Inverse Problems & Imaging 2017, 11(1): 99-123 doi: 10.3934/ipi.2017006

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 $\boldsymbol{E}$ and magnetic field $\boldsymbol{ H}$ 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 $ε$ and magnetic permeability $μ$. It is assumed that the fields on the surface of the obstacle satisfy the Leontovich boundary condition $\boldsymbol{ ν}×\boldsymbol{H}-λ\,\boldsymbol{ ν}×(\boldsymbol{ E}×\boldsymbol{ ν})=\boldsymbol{ 0}$ with admittance $λ$ an unknown positive function and $\boldsymbol{ ν}$ 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 admittance $λ$ is greater or less than the special value $\sqrt{ε/μ}$.

keywords: Enclosure method inverse obstacle scattering electromagnetic wave Maxwell’s equations obstacle Leontovich boundary condition
IPI
An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method
Masaru Ikehata Mishio Kawashita
Inverse Problems & Imaging 2014, 8(4): 1073-1116 doi: 10.3934/ipi.2014.8.1073
This paper studies a prototype of inverse initial boundary value problems whose governing equation is the heat equation in three dimensions. An unknown discontinuity embedded in a three-dimensional heat conductive body is considered. A single set of the temperature and heat flux on the lateral boundary for a fixed observation time is given as an observation datum. It is shown that this datum yields the minimum length of broken paths that start at a given point outside the body, go to a point on the boundary of the unknown discontinuity and return to a point on the boundary of the body under some conditions on the input heat flux, the unknown discontinuity and the body. This is new information obtained by using enclosure method.
keywords: corrosion thermal imaging cavity heat equation enclosure method. Inverse boundary value problem
IPI
On finding a buried obstacle in a layered medium via the time domain enclosure method
Masaru Ikehata Mishio Kawashita
Inverse Problems & Imaging 2018, 12(5): 1173-1198 doi: 10.3934/ipi.2018049

An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.

keywords: Enclosure method inverse obstacle scattering problem buried obstacle wave equation subsurface radar ground probing radar
IPI
Inverse obstacle scattering with limited-aperture data
Masaru Ikehata Esa Niemi Samuli Siltanen
Inverse Problems & Imaging 2012, 6(1): 77-94 doi: 10.3934/ipi.2012.6.77
Inverse obstacle scattering aims to extract information about distant and unknown targets using wave propagation. This study concentrates on a two-dimensional setting using time-harmonic acoustic plane waves as incident fields and taking the obstacles to be sound-hard with smooth or polygonal boundary. Measurement data is simulated by sending one incident wave towards the area of interest and computing the far field pattern (1) on the whole circle of observation directions, (2) only in directions close to backscattering, and (3) only in directions close to forward-scattering. A variant of the enclosure method is introduced, based on applying the far field operator to an explicitly constructed density, yielding information about the convex hull of the obstacle. The numerical evidence presented suggests that the convex hull of obstacles can be approximately recovered from noisy limited-aperture far field data.
keywords: Inverse problem inverse obstacle scattering. enclosure method

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