# Appendix M: Exercises

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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 12 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store
Tyger! Tyger! burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?

—William Blake


1. In Figure M-1, discuss how the symmetric wavelet is rotated while it is framed within the envelope.

2. What is the difference between analysis of variance (Fisher, 1925[1]) and the coherence cube? [Answer: In analysis of variance, the normalized variance is computed, whereas in the coherence-cube method, the semblance is computed. What is the difference? Take the simplest possible case, the case of three traces at one time level. Suppose the first trace is at coordinates ${\displaystyle \left(x,y\right)}$, the second trace is at coordinates ${\displaystyle \left(x+\Delta x{,\ }y\right)}$, and the third trace is at coordinates ${\displaystyle \left(x{,\ }y+\Delta y\right)}$. Only one value is used on each trace — say, the value at time t. These values of the three traces are as shown in Table M-1.

In analysis of variance, we compute the average — namely, ${\displaystyle \left({1\ +\ 1\ +4}\right){/3}={6/3}={2}}$. Then we subtract this average value from each of the three values to give the table of deviations shown in Table M-2.

The sums of squares of these deviations are ${\displaystyle (1)^{2}+(1)^{2}+2^{2}=6}$. This result is the raw variance. The raw variance makes up the numerator of the expression for the normalized variance. Next, we compute the sum of squares of the original table. The result is ${\displaystyle {1}^{2}+{1}^{2}+{4}^{2}={18}}$. This result is the normalizing factor. This normalizing factor makes up the denominator of the expression for the normalized variance. Thus, the normalized variance is ${\displaystyle {6/18=}{1/3}={0.33}}$. The normalized variance represents the dissimilarity of the traces.

Figure M-1.  In each diagram, the outer curves comprise the common envelope, and the inner curve is the common wavelet with the degree of rotation indicated.

The complement of a dissimilarity value is the similarity. Thus, the similarity in this example is ${\displaystyle 1-{0.33=0.67}}$. If we wish to plot the dissimilarity of the traces, we plot the value 0.33. On the other hand, if we wish to plot the similarity of the traces, we plot the value 0.67. This example gives the essence of the analysis-of-variance method.

Next, we turn to the coherence-cube method. Again, we start with same data — that is, with the same values as those in Table M-1. First, we add together the three values to give the value of what is called the stacked trace. This result is 1 + 1 + 4 = 6. Then we square the stacked trace value to give ${\displaystyle {6}^{2}={36}}$. We next divide this result by the number of values in the original table, namely three. The result of the division is 36/3 = 12. This value makes up the numerator of the expression for the semblance. The denominator of the expression for the semblance is the same as the denominator of the expression for the normalized variance. That is, the denominator is the sum of squares from the original table, namely ${\displaystyle {1}^{2}+{1}^{2}+{4}^{2}={18}}$. Thus, the semblance is 12/18 = 2/3 = 0.67. The semblance represents the similarity of the traces. The complement of a similarity value is the dissimilarity. Thus, the dissimilarity is this example is 1 — 0.67 = 0.33. If we wish to plot the dissimilarity of the traces, we plot the value 0.33. On the other hand, if we wish to plot the similarity of the traces, we plot the value 0.67.

Table M-1. The values of three traces at the same time instant.
Coordinate x Coordinate ${\displaystyle x+\Delta x}$
Coordinate y 1 4
Coordinate ${\displaystyle y+\Delta y}$ 1 empty
Table M-2. The deviations of three traces about the mean at the same time instant.
Coordinate x Coordinate ${\displaystyle x+\Delta x}$
Coordinate y –1 2
Coordinate ${\displaystyle y+\Delta y}$ –1 empty

This example gives the essence of the coherence-cube method and the analysis of variance and shows that they lead to the same result. Differences in practice can occur because of the many optional methods available for averaging this basic result at different time levels.