Green’s function representations for seismic interferometry

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.

Download Full-text

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

Geophysics ◽

10.1190/1.3457445 ◽

2010 ◽

Vol 75 (5) ◽

pp. 75A195-75A209 ◽

Cited By ~ 149

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.

Download Full-text

Seismic Interferometry from Correlated Noise Sources

Remote Sensing ◽

10.3390/rs13142703 ◽

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.

Download Full-text

Directional balancing for seismic and general wavefield interferometry

Geophysics ◽

10.1190/1.3298736 ◽

2010 ◽

Vol 75 (1) ◽

pp. SA1-SA14 ◽

Cited By ~ 43

Author(s):

Andrew Curtis ◽

David Halliday

Keyword(s):

Green’S Function ◽

Green's Function ◽

Acoustic Scattering ◽

Data Driven ◽

Seismic Interferometry ◽

Noise Sources ◽

Naturally Occurring ◽

Passive Seismic ◽

Incoming Energy ◽

Solid Bodies

In passive seismic interferometry using naturally occurring, heterogeneous noise sources and in active-source seismic interferometry where sources can usually only be distributed densely on the exterior of solid bodies, bias can be introduced in Green’s function estimates when amplitudes of energy have directional variations. We have developed an algorithm to remove bias in Green’s function estimates constructed using seismic interferometry when amplitudes of energy used have uncontrollable directional variations. The new algorithm consists of two parts: (1) a method to measure and adjust the amplitudes of physical, incoming energy using an array of receivers and (2) a method to predict and remove nonphysical energy that remains (and can be accentuated) in interferometrically derived Green’s functions after applying the method in step 1. To accomplish step 2, we have created two data-driven methods to predict the nonphysical energy using direct computation or move-out-based methods, and a way to suppress such energy using (in this case) helical least-squares filters. Two-dimensional acoustic scattering examples confirm the algorithm’s effectiveness.

Download Full-text

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

Geophysics ◽

10.1190/1.3463440 ◽

2010 ◽

Vol 75 (5) ◽

pp. 75A211-75A227 ◽

Cited By ~ 101

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.

Download Full-text