# Coherence cube (C3)

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 12 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

A fault is an extended break in a body of rock and is marked by relative displacement and discontinuity of strata on either side of a particular surface. In other words, a fault is a displacement of rocks along a shear surface, which is called the fault plane (and which can be a curved surface and not a plane at all). How can faults be delineated?

Seismic reflections come from subsurface interfaces (the horizons). The geologic images provided by seismic processing (e.g., the migrated records) show these interfaces in their correct spatial positions. Seismic reflections commonly do not arise directly from faulting, so the images generally do not show faults as such. As a result, in seismic interpretation, the spatial positions of faults generally are inferred from breaks (or discontinuities) in the horizons on a seismic image.

Coherency is a measure of the similarity existing among two or more signals. At a fault plane, coherency suffers a disruption because the strata are displaced along the fault plane. A coherency matrix, described in Robinson (1954, equation 5.672, p. 102), determines the spectra of all possible linear combinations of the traces in question.

Let us give a simple example of the use of coherency to determine the location of a fault (Figure 6). We use a layer-cake model cut vertically by a fault. In other words, the fault plane is vertical. The right side of the layer-cake model slips downward a distance (measured in time units) of $\alpha ={4}$ with respect to the left side. A subsection of traces between two time points is shown in Figure 6a. We consider two adjacent traces at a time. More specifically, we consider traces 1 and 2, then traces 2 and 3, then traces 3 and 4, and so on. The coefficient of coherency (in the frequency domain) for two traces is given by Robinson (1954, equation 5.696, p. 104), which for our simple case reduces to Figure 6.  A subsection of traces from a layer-cake model that has been cut by a vertical fault. (b) The corresponding subsection of the computed coherency, which shows the location and the slip of the fault.

 {\begin{aligned}{\rm {coh}}\,{\rm {(}}\omega {\rm {)}}\,{\rm {=}}\,\rho e^{-i\omega \alpha },\end{aligned}} (7)

where $\rho$ is a constant of magnitude less that one. Thus, the Z-transform of the coefficient of coherency is obtained by the substitution $e^{-i\omega }=Z$ . The Z-transform is

 {\begin{aligned}{\rm {coh}}\,{\rm {(}}Z{\rm {)}}\,{\rm {=}}\,\rho Z^{\alpha }.\end{aligned}} (8)

Thus, in the time domain, the coefficient of coherency is

 {\begin{aligned}{\rm {co}}{\rm {h}}_{t}=\rho {\delta }_{t-\alpha }.\end{aligned}} (9)

For all pairs of adjacent traces, the value of $\alpha$ is 0 (indicating no slippage) except for pair (4, 5), for which the value of $\alpha$ is 4 (indicating a slippage of 4 time units). In practice, we compute equation 9 for each pair of consecutive traces and then plot the coherency subsection as shown in Figure 6b. The coherency subsection clearly indicates that the location of the fault is between traces 4 and 5 and that the slippage is 4 time units. In seismology, we use this procedure systematically on consecutive subsections of the entire seismic prospect to map the locations of faults.

What is the coherence-cube method? Bahorich and Farmer (1995, 1996, 1998) introduced the coherence cube (or C3). Here, the term cube refers to any 3D array of data. The coherence cube can reveal fault surfaces within a 3D volume for which no fault reflections have been recorded. The method involves processing a given geologic image for the purpose of accentuating geologic discontinuities instead of accentuating the seismic reflections, as usually is done. Discontinuities include faults and some stratigraphic features. An algorithm is used to compare the similarity of nearby regions of the given 3D image. A trace segment that is similar to its neighbors is assigned a low discontinuity value, whereas a trace segment that is not similar to its neighbors is assigned a high discontinuity value. The final result is a discontinuity cube with fault surfaces enhanced but with noise and coherent stratigraphic features attenuated.

How do the seismic imprints of horizons and faults differ? A reflected event is an echo from a horizon. A reflected event can be thought of as a “barking dog” — that is, the horizon makes its presence known by giving off reflections. On the other hand, generally speaking, few reflections arise from faults. A fault does not represent a barking dog. To find a fault, we must find the dog that does not bark. Because of the importance of faulting in geologic interpretation, the problem of finding that dog must be solved. Traditionally, the use of attributes was devoted almost exclusively to detecting reflections hidden by noise. In contrast, discontinuity attributes (or dissimilarity attributes) are required to detect faults.

A coherence cube uses 3D seismic discontinuities to detect faults and stratigraphic features. As the input to the coherence-cube method, a 3D volume (cube) of data is selected. Generally, this selection is a geologic image resulting from the seismic-processing stage, such as the migrated record. Discontinuity processing systematically cuts through the data volume trace by trace without regard for the geologic horizons. An attribute is computed that gives a measure of the data dissimilarity from trace to trace. The output is a new 3D volume of data (the enhanced image made up of dissimilarity attributes). This dissimilarity image reveals faults and other subtle stratigraphic changes that stand out as prominent anomalies in otherwise homogeneous data. Discontinuity processing gives a new way to view seismic data by revealing the degree of dissimilarity from trace to trace. Interpretation can be based on the image of dissimilarity as well as on the original geologic image. Use of both types of image makes interpretation easier and more reliable. Figure 7.  Graphic representation of the coherence-cube method.

Seismic data usually are acquired and processed for imaging reflections. Here, we have described a method of processing seismic data for imaging discontinuities (e.g., faults and stratigraphic features). One application of this nontraditional process is a 3D volume, or cube, of coherence coefficients within which faults are revealed as numerically separated surfaces. Figure 7 compares a traditional 3D reflection amplitude time slice with the results of the new method.