Seismic Reciprocity

Knopoff, L.; Gangi, Anthony F.

doi:10.1190/1.1438647

Cited by 134 publications

(49 citation statements)

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“…an infinite plate, with one face rigid and the other free), we thus obtain the reciprocal property the tractions vanish on S ; div w(r, r") = div" w(r", r) (9) in which each divergence is calculated in receiver co-ordinates. Equation (9) is a special case of the very general reciprocity relations established for mechanical systems by Rayleigh about a century ago: other important examples of reciprocity in elastic media have been pointed out by Knopoff & Gangi (1959), Burridge & Knopoff (1964), Payton (1964), and (for linear, viscoelastic media) by De Hoop (1966).…”

Section: J Jmentioning

confidence: 97%

An Elasticity Theorem for Heterogeneous Media, with an Example of Body Wave Dispersion in the Earth

Richards

1971

Geophysical Journal International

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An explicit surface integral expression is derived, for the divergence (div u) of elastic displacement in an inhomogeneous anisotropic medium. In isotropic media, simple ray theory methods permit approximate evaluation of the integral; and (to the same order) an approximate relation is found between div u and the irrotational component of u. The resulting formula for P-wave displacement is expected to find application in the study of seismic sources: it is used here to evaluate the frequencydependent interference properties of P-waves, near the shadow boundary of the Earth's core.

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Section: J Jmentioning

confidence: 97%

An Elasticity Theorem for Heterogeneous Media, with an Example of Body Wave Dispersion in the Earth

Richards

1971

Geophysical Journal International

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“…In the following, we consider two independent elastodynamic states ͑i.e., sources and wavefields͒ distinguished by subscripts A and B. For an arbitrary spatial domain D enclosed by boundary ‫ץ‬D with outward-pointing normal n ‫ס‬ ͑n 1 ,n 2 ,n 3 ͒, the Rayleigh-Betti reciprocity theorem that relates these two states is given by Gangi, 1959;de Hoop, 1966;Aki and Richards, 1980͒. This theorem is also known as a reciprocity theorem of the convolution type because all products in the frequency domain, such as v i,A ij,B , correspond to convolutions in the time domain.…”

Section: Elastodynamic Representationmentioning

confidence: 99%

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

et al. 2010

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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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“…Graffi derived the first convolutional reciprocity theorem for an isotropic, homogeneous, elastic solid. Its extension to inhomogeneous elastic anisotropic media was achieved by Knopoff andGangi (1959). De Hoop (1966) showed that reciprocity holds in inhomogeneous, anelastic (viscoelastic), and anisotropic media.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Elimination of the overburden response from multicomponent source and receiver seismic data, with source designature and decomposition into PP-, PS-, SP-, and SS-wave responses

Holvik

Amundsen

2005

GEOPHYSICS

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This paper shows that Betti's reciprocity theorem gives an integral equation procedure to eliminate from the physical multicomponent-source, multicomponent-receiver seismic measurements the effect of the physical source radiation pattern and the response of the physical overburden (that is, the medium above the receiver plane). The physical multicomponent sources are assumed to be orthogonally aligned anywhere above the multicomponentreceiver depth level. Other than the position of the sources, no source characteristics are required. The method, denoted the Betti designature/elastic demultiple, has the following additional characteristics: it preserves primary amplitudes while eliminating all waves scattered from the overburden; it requires no knowledge of the medium below the receiver level; it requires no knowledge of the medium above the receiver level; it requires information only of the local density and elastic wave propagation velocities at the receiver level to decompose the physical seismic measurements into upgoing and downgoing waves.Following the Betti designature/elastic demultiple step is an elastic wavefield decomposition step that decomposes the data into PP-, PS-, SP-, and SS-wave responses that would be recorded from idealized compressional-wave and shear-wave sources and receivers. The combined elastic wavefield decomposition on the source and receiver side gives data equivalent to data from a hypothetical survey with overburden absent, with single-component compressional and shear-wave sources, and single-component compressional and shear-wave receivers.When the medium is horizontally layered, the Betti designature/elastic demultiple scheme followed by the elastic source-receiver decomposition scheme greatly simplifies and is conveniently implemented as deterministic multidimensional deconvolution and elastic source-receiver wavefield decomposition of common-source gathers (or common-receiver gathers when source array variations are negligible).Betti designature/elastic demultiple followed by sourcereceiver wavefield decomposition applies to three different seismic experiments: a 9-component (9C) land seismic experiment, a 12-component (12C) ocean-bottom seismic experiment, and an 18-component (18C) borehole seismic experiment. For the land and ocean-bottom seismic experiments, an additional geophone should be deployed below the zero-offset geophone to predict the source-induced vertical traction vector at the source location.A numerical example for the 12C ocean-bottom seismic experiment over a horizontally layered medium validates the Betti designature/elastic demultiple scheme.

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Seismic Reciprocity

Cited by 134 publications

References 0 publications

An Elasticity Theorem for Heterogeneous Media, with an Example of Body Wave Dispersion in the Earth

An Elasticity Theorem for Heterogeneous Media, with an Example of Body Wave Dispersion in the Earth

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

Elimination of the overburden response from multicomponent source and receiver seismic data, with source designature and decomposition into PP-, PS-, SP-, and SS-wave responses

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