Coherencia

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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 3 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The degree of commonness for an ensemble of signals refers to the process of evaluating the information common to all of the signals in that ensemble. The process strives to find the information that is common to most or all of the signals examined, for the purpose of determining the degree of commonness among all the signals examined. The simplest procedure of this kind is to form the arithmetic average of corresponding values of a given set of signals.

What is a test for the degree of commonness? The degree of commonness among signals often is described in terms of coherence because signals drawn from a uniform or coherent signal field show a common pattern (Robinson, 1954[1]). The fact that this common pattern exists is often used as evidence that a coherent field exists. Therefore, a good test for a common pattern in a group of signals is, by inference, a good test for the existence of a coherent field from which those signals were derived.

Does the test for a common pattern reveal that pattern’s form? Many tests can be devised to reveal the existence of a common pattern, but disclosure of its exact form might have little or no bearing on a test’s usefulness. A paper by Melton and Bailey (1957)[2] gives some examples, namely:

1) Binaural location of a sound. Even though a sound is unrecognizable or transmits no particular meaning to our brain, its existence and direction are recognized.

2) Subsurface geologic structure by electrical well logging. The electrical logs are graphic plots of electrical-resistivity measurements. Even though no unique relationship exists between any single geologic bed and its electrical resistance, the graphic plot can be matched with others from nearby wells to show, on a probability basis, the shape of the geologic structure.

3) Exploration seismograph techniques. Commercial reflection and refraction seismographs record the output from many pickups on adjacent signal traces. Reflections and refractions are identified by the fact that the traces show an alignment of similar waveforms across the record. In this instance, as with electrical well logs, correlation is performed in the brain of the person viewing the records. It is difficult to say precisely what reasoning leads to the conclusion that the waveforms are similar because they are recognized as such even in cases in which they have greatly different amplitudes and are distorted by unexplained noise.

What, then, is coherence? Coherence (or coherency) is a basic word having many related meanings. The word cohere means “to stick or hold together” and refers to things that are connected logically in a united or orderly whole. Coherence is the quality or the state of a logical or orderly relationship among parts. In physics, coherence can refer to waves with a continuous relationship among phases. Coherence is characterized by a fixed phase relationship among points on a wave. A truly monochromatic wave would be perfectly coherent at all points in space. In practice, however, the region of high coherence of a seismic wave generally extends only over a finite interval.

What distinguishes a reflection event? Coherence measures the degree of similarity among more than two signals. Coherence generally is used in a qualitative sense when we are picking seismic records by eye. Quantitative measures of coherence are used in automatic picking schemes. Seismic reflection events can be linearly coherent with respect to a straight-line seismic-record dip. Seismic reflection events can be hyperbolically coherent with respect to a normal-moveout curve.

A distinguishing property of an event, and a property by which the event can be classified as a seismic reflection, is that the trace-to-trace signals making up the event are coherent. A reflection event is one for which the traces exhibit a smooth and continuous relationship among phases — that is, for an event to be a reflection, it must exhibit coherence in phase from trace to trace.

Seismic reflection events can be coherent in a nonanalytic yet systematic way with respect to geophone locations. The principal indicator for a separate seismic event is a level of coherence among members of a set of seismic traces over a short time interval (of the order of, say, 1.5 cycles of the dominant frequency), compared with less coherence observed over a longer time interval. The velocity spectrum (Taner and Koehler, 1969) and the coherence cube (Bahorich and Farmer, 1995[3]) are good examples of the use of a coherence principle in geophysics.

What is the coefficient of coherence? Coherence is a property that is definable between two wavetrains that have a well-defined phase relationship between each other. That is, coherence indicates how closely the two wave trains are “in phase” (Ursin, 1979[4]). The coefficient of coherence is a measure of the similarity of two time series. If the time series have power spectra ${\displaystyle P_{ii}}$ and ${\displaystyle P_{jj}}$ and crosspower spectra ${\displaystyle P_{ij}}$ (which might be complex), their coefficient of coherence is ${\displaystyle P_{ij}/{\left(P_{ii}P_{jj}\right)}^{\rm {l/2}}}$. The coefficient of coherence is a frequency-domain concept and is analogous to the time-domain concept of correlation. The coherence of the two time series depends on their crosscorrelation. In general, if the coefficient of coherence is unity, we say that the two time series are completely coherent. If the coefficient of coherence is zero, we say that they are completely incoherent.

What is coherent noise? Coherent noise refers to noise wavetrains that display coherence in the form of systematic phase relationships between adjacent traces. For example, source-generated seismic noise (e.g., ground roll, shallow refractions, reverberations, multiples, etc.) usually is coherent. In exploration seismology, the terms source-generated noise and signal-generated noise are the same and can be used interchangeably.

What is the Melton coefficient, M? Suppose that ${\displaystyle x_{ct}}$ is the cth trace for the tth sample. Let C denote the number of traces. Let T denote the number of discrete amplitude values that are sampled on each trace. The Melton coefficient M (Melton and Bailey, 1957) is defined as

 {\displaystyle {\begin{aligned}M={\frac {1}{C}}{\frac {\sum \limits _{t=1}^{T}{\left|{\sum \limits _{c=1}^{C}{x_{ct}}}\right|}}{\sum \limits _{t=1}^{T}{\sum \limits _{c=1}^{C}{\left|{x_{ct}}\right|}}}}.\end{aligned}}} (2)

What is semblance? Semblance is a measure of coherence that is more familiar to exploration geophysicists (e.g., Taner and Koehler, 1969). It can be obtained from the Melton coefficient M by replacing the absolute value of a quantity by the square of that quantity. Thus, the semblance is

 {\displaystyle {\begin{aligned}F={\frac {1}{C}}{\frac {\sum \limits _{t=1}^{T}{\left({\sum \limits _{c=1}^{C}{x_{c\;t}}}\right)}^{2}}{\sum \limits _{t=1}^{T}{\sum \limits _{c=1}^{C}{x_{c\;t}^{2}}}}}.\end{aligned}}} (3)

The constant 1/C in the semblance is a normalizing factor. The sum of all the traces under consideration is called the sum trace. It is

 {\displaystyle {\begin{aligned}\sum _{c{\rm {=l}}}^{c}{x_{c\;t}}.\end{aligned}}} (4)

Semblance is the sum of squares of the sum trace divided by the sum of the squares of all the individual traces. This is equivalent to the zero-lag value of the unnormalized autocorrelation of the sum trace divided by the sum of the zero-lag values of the autocorrelations of the component traces (Table 1).

 Trace 1 0 –2 1 0 0 –2 –2 1 –3 3 Trace 2 0 1 –2 –3 3 –1 1 1 0 –2 Trace 3 3 –2 –1 2 0 2 –1 –1 –1 1 Sum 3 –3 –2 –1 3 –1 –2 1 –4 2
 Trace 1 0 4 1 0 0 4 4 1 9 9 Trace 2 0 1 4 9 9 1 1 1 0 4 Trace 3 9 4 1 4 0 4 1 1 1 1
 Trace 1 0 2 1 0 0 2 2 1 3 3 Trace 2 0 1 2 3 3 1 1 1 0 2 Trace 3 3 2 1 2 0 2 1 1 1 1

The squares of the sum trace from Table 1 are {9, 9, 4, 1, 9, 1, 4, 1, 16, 4}, which sum to 58. The squares of the individual entries are shown in Table 2 and sum to 88. Thus, the semblance is (58/88)/3, or 0.22.

In contrast, the “Melton” is the sum of the absolute values of the sum trace divided by the sum of the absolute values of all the individual traces. For example, the absolute values of the sum trace are {3, 3, 2, 1, 3, 1, 2, 1, 4, 2}, which sum to 22. The absolute values of the individual traces are shown in Table 3 and sum to 42. Thus, the Melton is (22/42)/3, or 0.17.

Where is semblance used? It is used to obtain the velocity spectrum (Taner and Koehler, 1969[5]). Schemes for velocity estimation based on the velocity spectrum are in widespread use today. The common-midpoint (CMP) gathers allow determination of root-mean-square velocities. Hyperbolic searches for semblance among the gathers form the basis on which velocities can be estimated. Measured semblances are presented as a velocity spectrum. The interpretation of this information then gives velocity estimates for both primary and multiple events.

Referencias

1. Robinson, E. A., 1954, Predictive decomposition of time series with applications to seismic exploration: Ph.D. thesis, MIT. Reprinted in Geophysics, 1967, 32, 418–484.
2. Melton, B., and L. Bailey, 1957, Multiple signal correlators: Geophysics, 22, 565–588.
3. Bahorich, M., and S. Farmer, 1995, 3-D seismic discontinuity for faults and stratigraphic features: The coherence cube: The Leading Edge, 14, no. 10, 1053–1058.
4. Ursin, B., 1979, Seismic signal detection and parameter estimation: Geophysical Prospecting, 27, 1–15.
5. Taner, M. T., and F. Koehler, 1969, Velocity spectra-digital computer derivation and applications of velocity functions: Geophysics, 34, 859–881.

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