Measure of coherency
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 t_{0} is defined as
( )
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}:
( )
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.
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