Measure of coherency
|Series||Investigations in Geophysics|
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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
where fi,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 vstk:
Normalized stacked amplitude is defined as
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
or, by way of equation (16),
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
where MF = 2/[M(M − 1)].
Another coherency measure used in computing velocity spectrum is the energy-normalized crosscorrelation sum
The range of EC is [− 1/(M − 1)] < EC ≤ 1.
Finally, semblance, which is the normalized output-to-input energy ratio, is given by
The following expression shows the relation of NE to EC:
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:
|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)|
|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.
- The velocity spectrum
- Factors affecting velocity estimates
- Interactive velocity analysis
- Horizon velocity analysis
- Coherency attribute stacks
- Topics in moveout and statics corrections