Measure of coherency

Seismic Data Analysis

Series	Investigations in Geophysics
Author	Öz Yilmaz
DOI	http://dx.doi.org/10.1190/1.9781560801580
ISBN	ISBN 978-1-56080-094-1
Store	SEG Online Store

Consider the CMP gather with a single reflection sketched in Figure 3.2-11. Stacked amplitude S at two-way zero-offset time t₀ is defined as

$S=\sum \limits _{i=1}^{M}f_{i,t(i)},$

(16)

where f_i,t(i) is the amplitude value on the ith trace at two-way time t(i), and M is the number of traces in the CMP gather. Two-way time t(i) lies along the stacking hyperbola associated with a trial velocity v_stk:

$t(i)={\sqrt {{t_{0}}^{2}+{\frac {x_{i}^{2}}{v_{stk}^{2}}}}}.$

(17)

Normalized stacked amplitude is defined as

$NS={\frac {\sum \nolimits _{i=1}^{M}f_{i,t(i)}}{\sum \nolimits _{i=1}^{M}|f_{i,t(i)}|}},$

(18)

where the range of NS is 0 ≤ NS ≤ 1. As for the stacked amplitude given by equation (16), the normalized stacked amplitude given by equation (18) is defined at two-way zero-offset time.

Another quantity that is used in velocity spectrum calculations is the unnormalized crosscorrelation sum within a time gate T that follows the path corresponding to the trial stacking hyperbola across the CMP gather. The expression for the unnormalized crosscorrelation sum is given by

$CC={\frac {1}{2}}\sum \limits _{t}\left\{\left[\sum \limits _{i=1}^{M}f_{i,t(i)}\right]^{2}-\sum \limits _{i=1}^{M}f_{i,t(i)}^{2}\right\},$

(19a)

or, by way of equation (16),

$CC={\frac {1}{2}}\sum \limits _{t}\left[S_{t}^{2}-\sum \limits _{i=1}^{M}f_{i,t(i)}^{2}\right],$

(19b)

where CC can be interpreted as half the difference between the output energy of the stack and the input energy. The outer summation is over the two-way zero-offset time samples t within the correlation gate T.

A normalized form of CC is another attribute that often is used in velocity spectrum calculations and is given by

$NC=MF\sum \limits _{t}\sum \limits _{k=1}^{M-1}\sum \limits _{i=1}^{M-k}{\frac {f_{i,t(i)}f_{i+k,t(i+k)}}{{\sqrt {\sum \nolimits _{t}f_{i,t(i)}^{2}\sum \nolimits _{t}f_{i+k,t(i+k)}^{2}}},}}$

(20)

where MF = 2/[M(M − 1)].

Another coherency measure used in computing velocity spectrum is the energy-normalized crosscorrelation sum

$EC={\frac {2}{(M-1)}}{\frac {CC}{\sum \nolimits _{t}\sum \nolimits _{i=1}^{M}f_{i,t(i)}^{2}}}.$

(21)

• 

  Figure 3.2-9  Transformation of a synthetic CMP gather containing a single reflection event from offset to velocity domain. Each trace in the velocity-stack gather (b) is a stack of the traces in the CMP gather (a) using a constant-velocity NMO correction.

• 

  Figure 3.2-10  Transformation of a synthetic CMP gather containing three reflection events from offset to velocity domain. Each trace in the velocity-stack gather (b) is a stack of the traces in the CMP gather (a) using a constant-velocity NMO correction.

• 

  Figure 3.2-11  Stacked amplitude along a hyperbolic trajectory. Amplitudes f_i,t(i) along the best-fit hyperbola (equation 17) defined by optimum stacking velocity v_stk are summed to get the stacked amplitude S (equation 16).

• 

  Figure 3.2-12  (a) A CMP gather, and two ways of displaying velocity spectrum computed from this gather: (b) gated raw plot, and (c) contour plot.

The range of EC is [− 1/(M − 1)] < EC ≤ 1.

Finally, semblance, which is the normalized output-to-input energy ratio, is given by

$NE={\frac {1}{M}}{\frac {\sum \nolimits _{t}\sum \nolimits _{i=1}^{M}f_{i,t(i)}}{\sum \nolimits _{t}\sum \nolimits _{i=1}^{M}f_{i,t(i)}^{2}}}.$

(22a)

The following expression shows the relation of NE to EC:

$EC={\frac {1}{M-1}}(M\times NE-1).$

(22b)

The range of NE is 0 ≤ NE ≤ 1.

Table 3-6 shows the values of the attributes defined by equation (16) and equations (18) through (22) for the special case of a two-fold CMP gather where the second trace is a scaled version of the first as follows:

$f_{1,t}=f_{t},$

(23a)

$f_{2,t}=a,f_{t}$

(23b)

Table 3-6. Various measures of coherency as applied to the two-fold case given by equation (23).

Attribute	a = 0.5	a = −0.5
Stacked Amplitude S (equation 16)	1.5f(t)	0.5f(t)
Normalized Stacked Amplitude NS (equation 18)	1	0.333
Unnormalized Crosscorrelation Sum CC (equation 19b)	$0.5\sum \nolimits _{t}{f^{2}}(t)$	$-0.5\sum \nolimits _{t}{f^{2}}(t)$
Normalized Crosscorrelation Sum NC (equation 20)	1	1
Energy-Normalized Crosscorrelation Sum EC (equation 21)	0.8	−0.8
Semblance NE (equation 22a)	0.9	0.1

Several conclusions can be made from the results shown in Table 3-6. Note that stacked amplitude is sensitive to trace polarity. The unnormalized crosscorrelation offers a better standout of the strong reflections on the velocity spectrum, while the normalized or energy-normalized crosscorrelation brings out weak reflections on the velocity spectrum. As equation (22b) implies, semblance is a biased version of the energy-normalized crosscorrelation sum.

The velocity spectrum normally is not displayed as shown in Figures 3.2-9b or 3.2-10b. Instead, two popular types of displays are used to pick velocities in the form of a gated row plot or a contour plot as shown in Figure 3.2-12. Another quantity that helps picking is the maxima of the coherency values from each time gate displayed as a function of time next to the velocity spectrum, as shown in Figure 3.2-12. Unless otherwise indicated, the unnormalized correlation was used to construct the velocity spectrum of the synthetic CMP gather (Figure 3.2-12a) that is used in subsequent discussions.

See also

• The velocity spectrum
• Factors affecting velocity estimates
• Interactive velocity analysis
• Horizon velocity analysis
• Coherency attribute stacks
• Exercises
• Topics in moveout and statics corrections

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