Inverse elastic scattering problems with phaseless far field data

Abstract

This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for inverse elastic scattering problems with phaseless far field data. The phaseless far field data is closely related to the outward energy flux, which is easily measured in practice. Systematically, we study two basic models, i.e., inverse scattering of plane waves by rigid bodies and inverse scattering of sources with compact support. For both models, we show that the phaseless far field data is invariant under translation of the underlying scattering objects, which implies that the location of the objects can not be uniquely recovered by the data. To solve this problem, we consider simultaneously the incident point sources with one fixed source point and at most three scattering strengths. With this technique, we establish some uniqueness results for source scattering problem with multi-frequency phaseless far field data. Furthermore, a fast and stable phase retrieval approach is proposed based on a simple geometric result which provides a stable reconstruction of a point in the plane from three distances to given points. Difficulties arise for inverse scattering by rigid bodies due to the additional unknown far field pattern of the point sources. To overcome this difficulty, we introduce an artificial rigid body into the system and show that the underlying rigid bodies can be uniquely determined by the corresponding phaseless far field data at a fixed frequency. Noting that the far field pattern of the scattered field corresponding to point sources is very small if the source point is far away from the scatterers, we propose an appropriate phase retrieval method for obstacle scattering problems, without using the artificial rigid body. Finally, we propose several sampling methods for shape reconstruction with phaseless far field data directly. For inverse obstacle scattering problems, two different direct sampling methods are proposed with data at a fixed frequency. For inverse source scattering problems, we introduce two direct sampling methods for source supports with sparse multi-frequency data. The phase retrieval techniques are also combined with the classical sampling methods for the shape reconstructions. Extended numerical examples in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval techniques and sampling methods.the readers interested in a more comprehensive treatment of the direct and inverse elastic scattering problems, we suggest consulting [9,14,36,37] on this subject.The two well known difficulties of the inverse scattering problems are nonlinearity and ill-posedness. Actually, in many cases of practical interest, the third difficulty is incomplete data, i.e., only partial information can be measured directly. There are two cases of incomplete data. The first one is the limited-aperture data, where the measurements are only available in limited directions or positions. Limited-aperture data can present a severe ch...

An inverse acoustic-elastic interaction problem with phased or phaseless far-field data

“…In [17], a phase retrieval technique combined with the direct sampling method was proposed to reconstruct the location and shape of an obstacle from phaseless far-field data. The method was extended to the phaseless inverse elastic scattering problem and phaseless IAEIP [16]. We refer to [37,38,41,43,44] for the uniqueness results on the inverse scattering problems by using phaseless data.…”

Unique determinations in inverse scattering problems with phaseless near-field measurements

“…The uniqueness of a coefficient inverse scattering problem with phaseless near-field data has been established in [30]. By using the superposition of incident point sources and supplementing a reference ball of impedance type, the uniqueness for an inverse interior scattering problem with phaseless data was recently established in [60]. We also refer to [29,44,45] for some recovery algorithms for the inverse medium scattering problems with phaseless near-field data.…”

Inverse Electromagnetic Source Scattering Problems with MultiFrequency Sparse Phased and Phaseless Far Field Data

“…Actually, we will show in the next section that phaseless electric far field pattern of the scattered electromagnetic wave is invariant under the translation of the source, thus the source cannot be uniquely determined by the phaseless far field data, even multi-frequency data are considered. The same problem also appears in the acoustic and elastic source scattering problems [28,31]. Other than developing methods using phaseless data directly, we take an alternative approach to first recover the phase information.…”

“…We first give a uniqueness result following the ideas for the inverse acoustic and elastic scattering problems [1,28].…”