# Measure of coherency

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

 ${\displaystyle S=\sum \limits _{i=1}^{M}f_{i,t(i)},}$ (16)

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:

 ${\displaystyle t(i)={\sqrt {{t_{0}}^{2}+{\frac {x_{i}^{2}}{v_{stk}^{2}}}}}.}$ (17)

Normalized stacked amplitude is defined as

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

 ${\displaystyle CC={\frac {1}{2}}\sum \limits _{t}\left\{\left[\sum \limits _{i=1}^{M}f_{i,t(i)}\right]^{2}-\sum \limits _{i=1}^{M}f_{i,t(i)}^{2}\right\},}$ (19a)

or, by way of equation (16),

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

 ${\displaystyle NC=MF\sum \limits _{t}\sum \limits _{k=1}^{M-1}\sum \limits _{i=1}^{M-k}{\frac {f_{i,t(i)}f_{i+k,t(i+k)}}{{\sqrt {\sum \nolimits _{t}f_{i,t(i)}^{2}\sum \nolimits _{t}f_{i+k,t(i+k)}^{2}}},}}}$ (20)

where MF = 2/[M(M − 1)].

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

 ${\displaystyle EC={\frac {2}{(M-1)}}{\frac {CC}{\sum \nolimits _{t}\sum \nolimits _{i=1}^{M}f_{i,t(i)}^{2}}}.}$ (21)

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

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

 ${\displaystyle NE={\frac {1}{M}}{\frac {\sum \nolimits _{t}\sum \nolimits _{i=1}^{M}f_{i,t(i)}}{\sum \nolimits _{t}\sum \nolimits _{i=1}^{M}f_{i,t(i)}^{2}}}.}$ (22a)

The following expression shows the relation of NE to EC:

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

 ${\displaystyle f_{1,t}=f_{t},}$ (23a)

 ${\displaystyle f_{2,t}=a,f_{t}}$ (23b)
 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) ${\displaystyle 0.5\sum \nolimits _{t}{f^{2}}(t)}$ ${\displaystyle -0.5\sum \nolimits _{t}{f^{2}}(t)}$ 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.