Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

The Green's function of waves that propagate between two receivers can be found by cross-correlating multiply scattered waves recorded at these receivers. This technique obviates the need for a source at one of these locations, and is therefore called "passive imaging." This principle has been explained by assuming that the normal modes of the system are uncorrelated and that all carry the same amount of energy (equipartitioning). Here I present an alternative derivation of passive imaging of the ballistic wave that is not based on normal modes. The derivation is valid for scalar waves in three dimensions, and for elastic surface waves. Passive imaging of the ballistic wave is based on the destructive interference of waves radiated from scatterers away from the receiver line, and the constructive interference of waves radiated from secondary sources near the receiver line. The derivation presented here shows that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.

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Extracting the Green's function from the correlation of coda waves: A derivation based on stationary phase

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.

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Green's function representations for seismic interferometry

[1] It has been demonstrated experimentally and theoretically that an estimate of the Green's function between two seismic stations can be obtained from the time-derivative of the long-time average cross correlation of ambient noise between these two stations. This TDGF estimate from just the noise field includes all tensor components of the Green's function and these Green's function estimates can be used to infer Earth structure. We have computed cross correlations using 1 to 30 continuous days of ambient noise recorded by over 150 broadband seismic stations located in Southern California. The data processing yielded thousands of cross-correlation pairs, for receiver separations from 4–500 km, which clearly exhibit coherent broadband propagating dispersive wavetrains across frequency band 0.1–2 Hz.

/pdf/extracting-time-domain-green-s-function-estimates-from-23y2zfok4r.pdf

Extracting time-domain Green's function estimates from ambient seismic noise

A multi-decadal view of seismic methods for detecting precursors of magma movement and eruption

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 x-axis, the crosscorrelation of the responses at two receivers along the x-axis gives the Green’s function of the direct wave between these receivers. When the source function of the plane wave is a transientas in exploration seismology or a noise signalas 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 the retrieval of seismic surface-wave responses from ambient noise and the subsequent tomographic determination of the surfacewave 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.

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

One application of seismic interferometry is to retrieve the impulse response (Green's function) from crosscorrelation of ambient seismic noise. Various researchers show results for retrieving the surface-wave part of the Green's function. However, reflection retrieval has proven more challenging. We crosscorrelate ambient seismic noise, recorded along eight parallel lines in the Sirte basin east of Ajdabeya, Libya, to obtain shot gathers that contain reflections. We take advantage of geophone groups to suppress part of the undesired surface-wave noise and apply frequency-wavenumber filtering before crosscorrelation to suppress surface waves further. After comparing the retrieved results with data from an active seismic exploration survey along the same lines, we use the retrieved reflection data to obtain a migrated reflection image of the subsurface.

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Reflection images from ambient seismic noise

The retrieval of the earth's reflection response from cross?correlations of seismic noise recordings can provide valuable information, which may otherwise not be available due to limited spatial distribution of seismic sources. We cross?correlated ten hours of seismic background?noise data acquired in a desert area. The cross?correlation results show several coherent events, which align very well with reflections from an active survey at the same location. Therefore, we interpret these coherent events as reflections. Retrieving seismic reflections from background?noise measurements has a wide range of applications in regional seismology, frontier exploration and long?term monitoring of processes in the earth's subsurface.

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Retrieval of reflections from seismic background?noise measurements

SUMMARY Seismic interferometry, also known as Green’s function retrieval by crosscorrelation, has a wide range of applications, ranging from surface wave tomography using ambient noise, to creating virtual sources for improved reflection seismology. Despite its successful applications, the crosscorrelation approach also has its limitations. The main underlying assumptions are that the medium is lossless and that the wave field is equipartitioned. These assumptions are in practice often violated: the medium of interest is often illuminated from one side only, the sources may be irregularly distributed, and, particularly for EM applications, losses may be significant. These limitations may partly be overcome by reformulating seismic interferometry as a multidimensional deconvolution (MDD) process. We present a systematic analysis of seismic interferometry by crosscorrelation and by MDD. We show that for the non-ideal situations mentioned above, the correlation function is proportional to a Green’s function with a blurred source. The source blurring is quantified by a so-called point-spread function which, like the correlation function, can be derived from the observed data (i.e., without the need to know the sources and the medium). The source of the Green’s function obtained by the correlation method can be deblurred by deconvolving the correlation function for the point-spread function. This is the essence of seismic interferometry by MDD. We illustrate the crosscorrelation and MDD methods for controlled-source and passive data applications with numerical examples and discuss the advantages and limitations of both methods.

/pdf/seismic-interferometry-by-crosscorrelation-and-by-1qjvmc533a.pdf

Deyan Draganov

Papers

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

Reflection images from ambient seismic noise

Retrieval of reflections from seismic background?noise measurements

Seismic interferometry by crosscorrelation and by multidimensional deconvolution: a systematic comparison

Seismic interferometry by crosscorrelation and by multi-dimensional deconvolution: a systematic comparison