Abstract
The inverse problem for time-harmonic acoustic wave scattering to recover a sound-soft obstacle from a given incident field and the far field pattern of the scattered field is considered. We split this problem into two subproblems; first to reconstruct the shape from the modulus of the data and this is followed by employing the full far field pattern in a few measurement points to find the location of the obstacle. We extend a nonlinear integral equation approach for shape reconstruction from the modulus of the far field data [6] to the three-dimensional case. It is known, see [13], that the location of the obstacle cannot be reconstructed from only the modulus of the far field pattern since it is invariant under translations. However, employing the underlying invariance relation and using only few far field measurements in the backscattering direction we propose a novel approach for the localization of the obstacle. The efficient implementation of the method is described and the feasibility of the approach is illustrated by numerical examples.
Highlights
In practical applications such as nondestructive testing, radar, sonar or medical imaging, the inverse obstacle scattering problem for acoustic waves occurs for frequencies in the resonance region, that is, for scatterers and wave numbers k such that the wavelength 2π/k is of a comparable size to the diameter of the scatterer
Given an obstacle D, i.e., a bounded domain D ⊂ IR3 with C2 boundary Γ such that Γ can be bijectively mapped onto a sphere, consider the scattering of a plane wave ui(x) = eikx·dwith wave number k > 0 and a unit vector ddescribing the direction of propagation
In this paper we investigate the following three inverse problems: Inverse Problem (IP1): Given the far field pattern u∞ on Ω for one incident wave ui, determine the shape and the location of the boundary surface Γ of the scatterer D
Summary
In practical applications such as nondestructive testing, radar, sonar or medical imaging, the inverse obstacle scattering problem for acoustic waves occurs for frequencies in the resonance region, that is, for scatterers and wave numbers k such that the wavelength 2π/k is of a comparable size to the diameter of the scatterer. Inverse Problem (IP2): Given the modulus of far field pattern |u∞| on Ω for one incident plane wave ui, determine the shape of the boundary surface Γ of the scatterer D.
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