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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 presence of internal multiples in seismic data can lead to artifacts in subsurface images obtained by conventional migration algorithms, which can be ameliorated by removing mult...
Abstract: The presence of internal multiples in seismic data can lead to artifacts in subsurface images obtained by conventional migration algorithms. This problem can be ameliorated by removing mult...

1 citations

Journal ArticleDOI
Haruo Sato1
TL;DR: In this article, the authors studied the condition for the Green's function retrieval in relation to the energy conservation for a single obstacle of arbitrary shape, and explicitly derived the constraint for the spherical Hankel function expansion coefficients as the sufficient condition.
Abstract: SUMMARY For imaging the earth structure, the cross-correlation function (CCF) of random waves as ambient noise or coda waves has been widely used for the estimation of the Green’s function. We precisely study the condition for the Green’s function retrieval in relation to the energy conservation for a single obstacle of arbitrary shape. When an obstacle is placed in a 2-D homogeneous medium, the Green’s function is written by a double series expansion using Hankel functions of the first kind which represent outgoing waves. When two receivers and the scattering obstacle are illuminated by uncorrelated noise sources randomly and uniformly distributed on a closed circle of a large radius surrounding them, the lag-time derivative of the CCF of random waves at the two receivers can be written by a convolution of the antisymmetrized Green’s function and the autocorrelation function of the noise source time function. We explicitly derive the constraint for the Hankel function expansion coefficients as the sufficient condition for the Green’s function retrieval. We show that the constraint is equal to the generalized optical theorem derived from the energy conservation principle. Physical meaningofthegeneralizedopticaltheorembecomesclearwhentheHankelfunctionexpansion coefficients are transformed into scattering amplitudes in the framework of the conventional scattering theory. In the 3-D case, the Green’s function is written by a double series expansion using spherical Hankel functions of the first kind and spherical harmonic functions. When two receivers and the scattering obstacle are illuminated by noise sources randomly and uniformly distributed on a closed spherical shell of a large radius surrounding them, we explicitly derive the constraint for the spherical Hankel function expansion coefficients for the Green’s function retrieval and the energy conservation. We note that the derivation of the constraint does not assume that two receivers are in the far field of the scattering obstacle.

1 citations

Journal ArticleDOI
TL;DR: In this article, the authors extended the Optical Theorem to the case of a penetrable particle excited by a local source deposited near a plane interface and derived the extinction cross-section involving the total point source radiating crosssection and some definite integrals responsible for the scattering.
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 local source deposited near a plane interface. We demonstrate that the derived Extinction Cross-Section involves the total point source radiating cross-section and some definite integrals responsible for the scattering by the interface. The derived extinction cross-section can be employed to estimate the quantum yield and the optical antenna efficiency without computation of the absorption cross-section.

1 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