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Identification of sound-soft 3D obstacles from phaseless data. (English) Zbl 1220.35194

Summary: 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 [O. Ivanyshyn, ibid. 1, No. 4, 609–622 (2007; Zbl 1194.35502)] to the three-dimensional case. It is known, see [R. Kress and W. Rundell, in: H. W. Engl (ed.) et al., Inverse Problems in medical imaging and nondestructive testing. Wien: Springer. 75–92 (1997; Zbl 0880.65105)], 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.

MSC:

35R30 Inverse problems for PDEs
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
76Q05 Hydro- and aero-acoustics
65Z05 Applications to the sciences
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