Representation theorems and Green’s function retrieval for scattering in acoustic media

Vasconcelos, Ivan; Snieder, Roel; Douma, Huub

doi:10.1103/physreve.80.036605

Cited by 54 publications

(71 citation statements)

References 39 publications

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“…According to equation 29, it is proportional to the energy scattered by the point scatterer ͑Carney et al., 2004;Vasconcelos et al, 2009;Curtis and Halliday, 2010͒. By adding the fourth term of equation 10, L͑G A s ,G B s ͒, resulting from the crosscorrelation of the scattered Green's functions, the residual event at t B ‫מ‬ t A ‫ס‬ 0.033 s is canceled, as shown in Figure 12.…”

Section: Derivation Of Scattering Matrix Of a Point Scatterer From Opmentioning

confidence: 94%

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

2010

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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 waves͒ 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 fourterm 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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Section: Derivation Of Scattering Matrix Of a Point Scatterer From Opmentioning

confidence: 94%

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

2010

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“…with ⌬ ͑x,͒ ‫ס‬ ͑x,͒ ‫מ‬ *͑ x,͒ ͑Vasconcelos and Snieder, 2008a;Douma, 2009;Vasconcelos et al, 2009͒. Here, the quantities with/without a bar refer to the reference/monitor state ͑for simplicity, we assume that time-lapse changes occur only in the compressibility͒. The equivalent theory for source-receiver interferometry is given in Halliday and Curtis ͑2010͒.…”

Section: Time-lapse Seismic Interferometrymentioning

confidence: 99%

Tutorial on seismic interferometry: Part 2 — Underlying theory and new advances

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In the 1990s, the method of time-reversed acoustics was developed. This method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal. When ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers, they focus at the position of the original source, even when the medium is very complex. In seismic interferometry the time-reversed responses are not physically sent into the earth, but they are convolved with other measured responses. The effect is essentially the same: The time-reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers. A mathematical derivation, based on reciprocity theory, formalizes this principle: The crosscorrelation of responses at two receivers, integrated over different sources, gives the Green's function emitted by a virtual source at the position of one of the receivers and observed by the other receiver. This Green's function representation for seismic interferometry is based on the assumption that the medium is lossless and nonmoving. Recent developments, circumventing these assumptions, include interferometric representations for attenuating and/or moving media, as well as unified representations for waves and diffusion phenomena, bending waves, quantum mechanical scattering, potential fields, elastodynamic, electromagnetic, poroelastic, and electroseismic waves. Significant improvements in the quality of the retrieved Green's functions have been obtained with interferometry by deconvolution. A trace-by-trace deconvolution process compensates for complex source functions and the attenuation of the medium. Interferometry by multidimensional deconvolution also compensates for the effects of one-sided and/or irregular illumination.

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“…This is done using the scattering reciprocity theorems of Vasconcelos et al ͑2009a͒. We then show that it is possible to create an explicit link between this form of seismic interferometry and inverse-scattering seismic imaging theory ͑also see Vasconcelos et al, 2009b͒.…”

Section: Introductionmentioning

confidence: 98%

An interferometric theory of source-receiver scattering and imaging

Halliday

Curtis

2010

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It is known that there is a link between the theory of seismic interferometry and theories of seismic imaging and inversion. However, although this has been discussed in several studies, there are few where any explicit links have been derived. We use reciprocity theorems for scattering media to derive a new form of seismic interferometry that describes the scattered wavefield between a source and a receiver in an acoustic medium, using both sources and receivers on two enclosing boundaries. This form of seismic interferometry is equivalent to a generalized imaging condition ͑IC͒ that combines the full wavefield inside any finite-sized subregion of the medium of interest. By using the Born ͑single-scattering͒ approximation, this generalized IC reduces to the method of imaging by double-focusing originally derived by Michael Oristaglio in 1989. Thus an explicit link is made between seismic interferometry, new generalized full-wavefield ICs, and existing single-scattering imaging methods.

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Representation theorems and Green’s function retrieval for scattering in acoustic media

Cited by 54 publications

References 39 publications

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

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

Tutorial on seismic interferometry: Part 2 — Underlying theory and new advances

An interferometric theory of source-receiver scattering and imaging

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