Inverse obstacle scattering with limited-aperture data

Masaru Ikehata,Samuli Siltanen,Esa Niemi

doi:10.3934/ipi.2012.6.77

Masaru Ikehata, Samuli Siltanen + Show 1 more

Open Access

https://doi.org/10.3934/ipi.2012.6.77

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Journal: Inverse Problems and Imaging	Publication Date: Jan 1, 2012
Citations: 47	License type: cc-by

Affiliation: Gunma University

Abstract

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.

Highlights

Inverse obstacle scattering aims to extract information about distant and unknown targets using wave propagation
The argument for the derivation employs an idea in [18] reducing the integral equation (6) to another integral equation via the Vekua transform [36, 37] which maps harmonic functions into solutions of the Helmholtz equation
Our method for computing the far field pattern data is based on layer potential representation and boundary integral equations, for details see for example [5]

Summary

Introduction

Inverse obstacle scattering aims to extract information about distant and unknown targets using wave propagation. In this work we concentrate on the limited-aperture case of sending one incident wave towards the area of interest and measuring the scattered field in various directions. Consider the scattering of the plane wave eikx·d with incident direction d ∈ S1 and wave number k > 0. We present in Subsection 2.1 an argument for the derivation of the density gN given by (7) with coefficients satisfying (8). The argument for the derivation employs an idea in [18] reducing the integral equation (6) to another integral equation via the Vekua transform [36, 37] which maps harmonic functions into solutions of the Helmholtz equation.

Methods

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Conclusion