On the scattering amplitudes for elastic waves

Dassios, George; Kiriaki, Kiriakie; Polyzos, D.

doi:10.1007/bf00945822

Cited by 36 publications

(37 citation statements)

References 14 publications

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“…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.…”

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Inverse elastic scattering problems with phaseless far field data

Liu

2019

Inverse Problems

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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...

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confidence: 70%

“…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.…”

mentioning

confidence: 99%

Inverse elastic scattering problems with phaseless far field data

Liu

2019

Inverse Problems

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“…Lakhtakia in [13,14] and in his book [15] has proved reciprocity and optical theorems for a chiral scatterer in a chiral environment, extending the corresponding results for the achiral case obtained by de Hoop [8,9]. In [1], [2] and [5] there exist various scattering relations for acoustic, electromagnetic and elastic waves, respectively. Some details about optical theorems for electromagnetic scattering in achiral media can be found in [10,11].…”

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confidence: 92%

On the far field patterns for electromagnetic scattering by a chiral obstacle in a chiral environment

Athanasiadis

2005

Journal of Mathematical Analysis and Applications

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A time-harmonic plane electromagnetic wave is scattered by a chiral body in a chiral environment. The body is either a perfect conductor, or a dielectric, or a scatterer with an impedance surface. Using the Huygens's principle, we construct in closed forms both the left-circularly polarized and rightcircularly polarized electric far field patterns for such chiral media. We prove reciprocity relations and general scattering theorems for chiral materials which are a generalization of those obtained by Twersky for achiral electromagnetic scattering. In the special case when the directions of incidence and observation are the same we prove the associated forward scattering theorems.  2004 Elsevier Inc. All rights reserved.

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“…Scattering in 3D linear elasticity for an incident plane vector or a dyadic field has been studied in [11,12] and [15]- [17]. Different aspects of propagation and scattering problems in 2D linear elasticity are considered in [1]- [3], [19], [21]- [24], and [26]- [28].…”

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confidence: 99%

Scattering relations for point-generated dyadic fields in two-dimensional linear elasticity

2006

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Abstract. The problem of scattering of elastic waves by a bounded obstacle in twodimensional linear elasticity is considered. The scattering problems are presented in a dyadic form. An incident dyadic field generated by a point source is disturbed by a rigid body, a cavity, or a penetrable obstacle. General scattering theorems are proved, relating the far-field patterns due to scattering of waves from a point source set up in either of two different locations. The most general reciprocity theorem is established, and mixed scattering relations are also proved. Finally, a relation between the incident and the scattered wave which refers to the mechanism of energy transfer of the scatterer, the so-called optical theorem, is established.

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On the scattering amplitudes for elastic waves

Cited by 36 publications

References 14 publications

Inverse elastic scattering problems with phaseless far field data

Inverse elastic scattering problems with phaseless far field data

On the far field patterns for electromagnetic scattering by a chiral obstacle in a chiral environment

Scattering relations for point-generated dyadic fields in two-dimensional linear elasticity

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