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Journal ArticleDOI

On seismic interferometry, the generalized optical theorem, and the scattering matrix of a point scatterer

27 Apr 2010-Geophysics (Society of Exploration Geophysicists)-Vol. 75, Iss: 3
TL;DR: In this article, a far-field approximation of the Green's function representation for seismic interferometry is analyzed, and the generalized optical theorem is derived from the nonlinear scattering matrix of a point scatterer.
Abstract: We have analyzed the far-field approximation of the Green's function representation for seismic interferometry. By writing each of the Green's functions involved in the correlation process as a superposition of a direct wave and a scattered wave, the Green's function representation is rewritten as a superposition of four terms. When the scattered waves are modeled with the Born approximation, it appears that a three-term approximation of the Green's function representation (omitting the term containing the crosscorrelation of the scattered waves) yields a nearly exact retrieval, whereas the full four-term expression leads to a significant nonphysical event. This is because the Born approximation does not conserve energy and therefore is an insufficient model to explain all aspects of seismic interferometry. We use the full four-term expression of the Green's function representation to derive the generalized optical theorem. Unlike other recent derivations, which use stationary phase analysis, our derivation uses reciprocity theory. From the generalized optical theorem, we derive the nonlinear scattering matrix of a point scatterer. This nonlinear model accounts for primary and multiple scattering at the point scatterer and conforms with well-established scattering theory of classical waves. The model is essential to explain fully the results of seismic interferometry, even when it is applied to the response of a single point scatterer. The nonlinear scattering matrix also has implications for modeling, inversion, and migration.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors extended the Optical Theorem to the case of a penetrable particle excited by a multipole source and derived the derived extinction cross-section via calculation of some specific derivatives from the scattered field at the point of the multipole location.
Abstract: Based on classic Maxwell׳s theory and the Gauss Theorem we extended the Optical Theorem to the case of a penetrable particle excited by a multipole source. We demonstrate that the derived extinction cross-section can be evaluated via calculation of some specific derivatives from the scattered field at the point of the multipole location. The obtained relation between extinction cross-section and scattering cross-section can be employed to estimate the corresponding absorption cross-section.

4 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the case of near-field cross correlation in a two-dimensional space with a single embedded scatterer that is represented by a cylindrical inclusion with small radius.
Abstract: Seismic interferometry using far-field correlation is a technique to obtain the Green’s function between two receivers using passive wave-field recordings, and often it is under the theoretical assumption that sources are in the far field when in fact they may not be. Using a heterogeneous medium enclosed by a closed boundary on which the sources are fired, we offer two views on the meaning of the far-field correlation. In the intrinsic view, the validity of the far-field correlation is based on the far-field approximation for wave scattering, and it is investigated by comparing various physical dimensions in the heterogeneous interior, regardless of the exterior. However, the extrinsic view centers on the medium properties in the exterior only without considering the interior. Previous studies showed that, with the correct scattering model and complete source coverage, no spurious arrivals should be generated if the illuminating sources are in the far field. We investigate the case of near-field cross correlation. This problem is considered in the context of two-dimensional space with a single embedded scatterer that is represented by a cylindrical inclusion with small radius. An analytical solution for the scattered wave is computed. However, if the sources are not in the far field, the cross-correlation kernel must be used to eliminate the spurious arrival, in addition to the correct scattering model and full source aperture. The kernel accounts for the near-field illumination of the region.

2 citations

Journal ArticleDOI
TL;DR: In this article, the authors derived an extended version of the optical theorem for the scattering of elastic waves by a spherical inclusion embedded in a linear elastic solid using a vector spherical harmonics representation of the waves.
Abstract: The optical theorem is an important tool for scattering analysis in acoustics, electromagnetism, and quantum mechanics. We derive an extended version of the optical theorem for the scattering of elastic waves by a spherical inclusion embedded in a linear elastic solid using a vector spherical harmonics representation of the waves. The sphere can be a rigid, empty cavity, elastic, viscoelastic, or layered material. The theorem expresses the extinction cross-section, i.e. the time-averaged power extracted from the incoming beam per its intensity, regarding the partial-wave expansion coefficients of the incident and scattered waves. We establish the optical theorem for a longitudinal spherically focused beam scattered by a sphere. Moreover, we use the optical theorem formalism to obtain the radiation force exerted on an inclusion by an incident plane wave and focused beam. Considering an iron sphere embedded in an aluminum matrix, we compute the scattering and elastic radiation force efficiencies. In addition, the elastic radiation force is obtained on a stainless steel sphere embedded in a tissue-like medium (soft solid). Remarkably, we find a relative difference of up to $98\%$ between our findings and previous lossless liquid models. Regarding some applications, the obtained results have a direct impact on ultrasound-based elastography techniques, ultrasonic nondestructive testing, as well as implantable devices activated by ultrasound.

2 citations

Journal ArticleDOI
TL;DR: Novel optical theorem detectors are derived that are based on the Kirchhoff-Helmholtz and Rayleigh-Sommerfeld-based formulations of diffraction, backpropagation, and boundary-value problems as well as on the canonical multipole expansion.
Abstract: We demonstrate and discuss the multitude of ways in which the extinct power of a scatterer can be measured. To tie some of the developed results to the classical statement of the optical theorem involving the imaginary part of the forward-scattering amplitude, particular attention is given to plane wave excitation. On the other hand, the general results apply to more general probing fields including near fields carrying evanescent components. Novel optical theorem detectors are derived that are based on the Kirchhoff-Helmholtz and Rayleigh-Sommerfeld-based formulations of diffraction, backpropagation, and boundary-value problems as well as on the canonical multipole expansion. The derived detectors also lead to novel expressions for the extinct power in terms of the incident and scattered fields. Applications of the derived results to scattering power sensing with near-field data are also discussed.

2 citations

Dissertation
26 Nov 2015
TL;DR: In this paper, an explicit link was found between the methods of source-receiver interferometry (SRI) and seismic imaging, a technique widely used in seismic exploration to map diffractors and reflectors in the subsurface, but also in more academic studies investigating deep crustal processes.
Abstract: Seismic or wavefield interferometry refers to a set of methods that synthesize wavefields between pairs of receivers, pairs of sources, or a source and a receiver, using wavefields propagating from and to surrounding boundaries of sources and/or receivers. Starting from cross-correlations of ambient seismic noise recordings, which provide the signal between two receivers as if one of them had been an active source, interferometric methods developed rapidly within the last decade, revolutionizing the way in which seismic, acoustic, elastic, or electromagnetic waves are used to image and monitor the interior of a medium. Only recently, an explicit link was found between the methods of source-receiver interferometry (SRI) and seismic imaging, a technique widely used in seismic exploration to map diffractors and reflectors in the subsurface, but also in more academic studies investigating, for example, deep crustal processes. This link is particularly interesting because SRI, in contrast to classical imaging schemes, does not rely on the single-scattering assumption but accounts for all multiple-scattering effects in the medium. While first non-linear imaging schemes based on SRI have been proposed, the full potential of the method remains to be explored and a number of open questions concerning, for example, the role of non-physical energy in interferometric wavefield estimates, require further investigation. The aim of this thesis is to gain more insight into the method of source-receiver interferometry in the context of wavefield construction and analysis in multiply scattering media, especially when theoretical requirements of the method (such as complete boundaries of sources and receivers, surrounding the medium of interest) are not met. First I analyse the single diffractor case using partial surface boundaries only. I find that only two out of eight terms of the SRI equation are required to construct a robust estimate of the scattered wavefield, and that one of these two terms is also used in seismic imaging. The other term provides a pseudo-physical estimate of the scattered wave; this is a new type of non-physical energy that emulates the kinematics of a physically scattered wave. I then proceed to a multiple scattering scenario, using the pseudo-physical term to predict the travel times and exact scattering paths of multiply diffracted waves. The presented algorithm is purely data-driven and fully automated and, as a by-product,

2 citations

References
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Book
01 Jan 1959
TL;DR: In this paper, the authors discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals, including interference, interferometers, and diffraction.
Abstract: The book is comprised of 15 chapters that discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals. The text covers the elements of the theories of interference, interferometers, and diffraction. The book tackles several behaviors of light, including its diffraction when exposed to ultrasonic waves.

19,815 citations

01 Oct 1999
TL;DR: In this article, the authors discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals, including interference, interferometers, and diffraction.
Abstract: The book is comprised of 15 chapters that discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals. The text covers the elements of the theories of interference, interferometers, and diffraction. The book tackles several behaviors of light, including its diffraction when exposed to ultrasonic waves.

19,503 citations

Book
01 Jan 1937

11,054 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the acoustic Green's function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points.
Abstract: The term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations. The first version of this principle was derived by Claerbout (1968), who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response. For an arbitrary 3D inhomogeneous lossless medium it follows from Rayleigh's reciprocity theorem and the principle of time-reversal invariance that the acoustic Green's function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points. The integral is along sources on an arbitrarily shaped surface enclosing these points. No assumptions are made with respect to the diffusivity of the wavefield. The Rayleigh-Betti reciprocity theorem leads to a similar representation of the elastodynamic Green's function. When a part of the enclosing surface is the earth's free surface, the integral needs only to be evaluated over the remaining part of the closed surface. In practice, not all sources are equally important: The main contributions to the reconstructed Green's function come from sources at stationary points. When the sources emit transient signals, a shaping filter can be applied to correct for the differences in source wavelets. When the sources are uncorrelated noise sources, the representation simplifies to a direct crosscorrelation of wavefield observations at two points, similar as in methods that retrieve Green's functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface.

700 citations