scholarly journals Tutorial on seismic interferometry: Part 1 — Basic principles and applications

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. 75A195-75A209 ◽  
Author(s):  
Kees Wapenaar ◽  
Deyan Draganov ◽  
Roel Snieder ◽  
Xander Campman ◽  
Arie Verdel
Keyword(s):  
Surface Wave ◽  
Plane Wave ◽  
Green’S Function ◽  
Green's Function ◽  
Ambient Noise ◽  
Source Function ◽  
Stationary Points ◽  
Seismic Interferometry ◽  
Direct Wave ◽  
Reflected Wave

Seismic interferometry involves the crosscorrelation of responses at different receivers to obtain the Green’s function between these receivers. For the simple situation of an impulsive plane wave propagating along the [Formula: see text]-axis, the crosscorrelation of the responses at two receivers along the [Formula: see text]-axis gives the Green’s function of the direct wave between these receivers. When the source function of the plane wave is a transient (as in exploration seismology) or a noise signal (as in passive seismology), then the crosscorrelation gives the Green’s function, convolved with the autocorrelation of the source function. Direct-wave interferometry also holds for 2D and 3D situations, assuming the receivers are surrounded by a uniform distribution of sources. In this case, the main contributions to the retrieved direct wave between the receivers come from sources in Fresnel zones around stationary points. The main application of direct-wave interferometry is theretrieval of seismic surface-wave responses from ambient noise and the subsequent tomographic determination of the surface-wave velocity distribution of the subsurface. Seismic interferometry is not restricted to retrieving direct waves between receivers. In a classic paper, Claerbout shows that the autocorrelation of the transmission response of a layered medium gives the plane-wave reflection response of that medium. This is essentially 1D reflected-wave interferometry. Similarly, the crosscorrelation of the transmission responses, observed at two receivers, of an arbitrary inhomogeneous medium gives the 3D reflection response of that medium. One of the main applications of reflected-wave interferometry is retrieving the seismic reflection response from ambient noise and imaging of the reflectors in the subsurface. A common aspect of direct- and reflected-wave interferometry is that virtual sources are created at positions where there are only receivers without requiring knowledge of the subsurface medium parameters or of the positions of the actual sources.

scholarly journals Three-station interferometry and tomography: coda versus direct waves

2020 ◽  
Vol 221 (1) ◽  
pp. 521-541
Author(s):  
Shane Zhang ◽  
Lili Feng ◽  
Michael H Ritzwoller
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Ambient Noise ◽  
Signal To Noise Ratio ◽  
Wave Dispersion ◽  
Correction Method ◽  
Coda Wave ◽  
Surface Wave Dispersion ◽  
Direct Wave ◽  
Noise Interferometry

SUMMARY Traditional two-station ambient noise interferometry estimates the Green’s function between a pair of synchronously deployed seismic stations. Three-station interferometry considers records observed three stations at a time, where two of the stations are considered receiver–stations and the third is a source–station. Cross-correlations between records at the source–station with each of the receiver–stations are correlated or convolved again to estimate the Green’s function between the receiver–stations, which may be deployed asynchronously. We use data from the EarthScope USArray in the western United States to compare Rayleigh wave dispersion obtained from two-station and three-station interferometry. Three three-station interferometric methods are distinguished by the data segment utilized (coda-wave or direct-wave) and whether the source–stations are constrained to lie in stationary phase zones approximately inline with the receiver–stations. The primary finding is that the three-station direct wave methods perform considerably better than the three-station coda-wave method and two-station ambient noise interferometry for obtaining surface wave dispersion measurements in terms of signal-to-noise ratio, bandwidth, and the number of measurements obtained, but possess small biases relative to two-station interferometry. We present a ray-theoretic correction method that largely removes the bias below 40 s period and reduces it at longer periods. Three-station direct-wave interferometry provides substantial value for imaging the crust and uppermost mantle, and its ability to bridge asynchronously deployed stations may impact the design of seismic networks in the future.

scholarly journals Seismic Interferometry from Correlated Noise Sources

2021 ◽  
Vol 13 (14) ◽  
pp. 2703
Author(s):  
Daniella Ayala-Garcia ◽  
Andrew Curtis ◽  
Michal Branicki
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Ambient Noise ◽  
Ocean Waves ◽  
Seismic Interferometry ◽  
Correlated Noise ◽  
Highway Traffic ◽  
Noise Sources ◽  
Estimation Errors ◽  
Band Limited

It is a well-established principle that cross-correlating seismic observations at different receiver locations can yield estimates of band-limited inter-receiver Green’s functions. This principle, known as Green’s function retrieval or seismic interferometry, is a powerful technique that can transform noise into signals which enable remote interrogation and imaging of the Earth’s subsurface. In practice it is often necessary and even desirable to rely on noise already present in the environment. Theory that underpins many applications of ambient noise interferometry assumes that the sources of noise are uncorrelated in time. However, many real-world noise sources such as trains, highway traffic and ocean waves are inherently correlated in space and time, in direct contradiction to the these theoretical foundations. Applying standard interferometric techniques to recordings from correlated energy sources makes the Green’s function liable to estimation errors that so far have not been fully accounted for theoretically nor in practice. We show that these errors are significant for common noise sources, always perturbing or entirely obscuring the phase one wishes to retrieve. Our analysis explains why stacking may reduce the phase errors, but also shows that in commonly encountered circumstances stacking will not remediate the problem. This analytical insight allowed us to develop a novel workflow that significantly mitigates effects arising from the use of correlated noise sources. Our methodology can be used in conjunction with already existing approaches, and improves results from both correlated and uncorrelated ambient noise. Hence, we expect it to be widely applicable in ambient noise studies.

scholarly journals Green’s function representations for seismic interferometry

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. SI33-SI46 ◽  
Author(s):  
Kees Wapenaar ◽  
Jacob Fokkema
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Layered Medium ◽  
Closed Surface ◽  
Reciprocity Theorem ◽  
Stationary Points ◽  
Seismic Interferometry ◽  
Noise Sources ◽  
Shaping Filter ◽  
Reversal Invariance

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.

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

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. 75A211-75A227 ◽  
Author(s):  
Kees Wapenaar ◽  
Evert Slob ◽  
Roel Snieder ◽  
Andrew Curtis
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Seismic Interferometry ◽  
Virtual Source ◽  
Bending Waves ◽  
Complex Source ◽  
Recent Developments ◽  
Diffusion Phenomena ◽  
Quantum Mechanical Scattering ◽  
Moving Media

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 differ-ent 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.

Fast plane-wave reverse time migration

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. S549-S556 ◽  
Author(s):  
Xiongwen Wang ◽  
Xu Ji ◽  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Time Delay ◽  
Plane Wave ◽  
Green’S Function ◽  
Green's Function ◽  
Time Domain ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Wave Source ◽  
Computation Cost ◽  
Time Migration

Plane-wave reverse time migration (RTM) could potentially provide quick subsurface images by migrating fewer plane-wave gathers than shot gathers. However, the time delay between the first and the last excitation sources in the plane-wave source largely increases the computation cost and decreases the practical value of this method. Although the time delay problem is easily overcome by periodical phase shifting in the frequency domain for one-way wave-equation migration, it remains a challenge for time-domain RTM. We have developed a novel method, referred as to fast plane-wave RTM (FP-RTM), to eliminate unnecessary computation burden and significantly reduce the computational cost. In the proposed FP-RTM, we assume that the Green’s function has finite-length support; thus, the plane-wave source function and its responding data can be wrapped periodically in the time domain. The wrapping length is the assumed total duration length of Green’s function. We also determine that only two period plane-wave source and data after the wrapping process are required for generating the outcome with adequate accuracy. Although the computation time for one plane-wave gather is twice as long as a normal shot gather migration, a large amount of computation cost is saved because the total number of plane-wave gathers to be migrated is usually much less than the total number of shot gathers. Our FP-RTM can be used to rapidly generate RTM images and plane-wave domain common-image gathers for velocity model building. The synthetic and field data examples are evaluated to validate the efficiency and accuracy of our method.

Measurement of interstation phase and group velocities and Q using Wiener filtering

1982 ◽  
Vol 72 (1) ◽  
pp. 73-91
Author(s):  
Steven R. Taylor ◽  
M. Nafi Toksöz
Keyword(s):  
Surface Wave ◽  
Green’S Function ◽  
Green's Function ◽  
Amplitude Spectrum ◽  
Wiener Filter ◽  
Wave Path ◽  
Phase And Group Velocities ◽  
Filtering Technique ◽  
Surface Wave Data ◽  
Group Velocities

abstract A method for calculating interstation phase and group velocities and attenuation coefficients using a Wiener (least-squares) filtering technique is presented. The interstation Green's (or transfer) function is estimated from surface wave data from two stations laying along the same great circle path. The estimate is obtained from a Wiener filter which is constructed to estimate the signal recorded at the station further from the source from the signal recorded at the nearer station. The interstation group velocity is obtained by applying the multiple-filtering technique to the Green's function, and the interstation phase velocity from the phase spectrum of the Green's function. The amplitude spectrum of the Green's function is used to calculate average attenuation between the two stations. Using synthetic seismograms contaminated by noise, it is shown that the Q values calculated from the Green's function are significantly more stable and accurate than those obtained by taking spectral ratios. The method is particularly useful for paths involving short station separations and is applied to a surface wave path crossing the Iranian Plateau.

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