# Small and white reflection coefficients

Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 8 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Finally, let us consider the case in which the reflection coefficients are not only small but also are white. In such a case, the autocorrelation coefficients ${\displaystyle g_{1},g_{2},g_{3},.\dots }$ are each approximately zero. As a result, the denominator polynomial ${\displaystyle 1+g_{1}Z+g_{2}Z^{2}+\ldots +g_{N}Z^{N}}$ reduces to 1, and thus ${\displaystyle M\left(Z\right)={1}}$. This equation says that in effect, the multiples cancel each other out. Thus, when the reflection coefficients are small and white, the trace becomes ${\displaystyle x=s*\varepsilon }$, which is the multiple-free trace given by equation 3, the synthetic seismogram (without multiples). In other words, in the case of small and white reflection coefficients, the dynamic convolutional model (equation 1) reduces to the time-honored synthetic seismogram (without multiples), which is by far the most used type of synthetic.

The synthetic seismogram (without multiples) corresponds to a sequence of wavelets each weighted by a reflection coefficient. It is interesting that the amplitude of each reflection coefficient is pristine because the primary reflection appears to suffer no transmission losses on the trip down to the respective interface and then back up. In other words, under the small-and-white hypothesis, the effects of all the multiples are only good. That is, the multiples jack up the primary reflections on the seismogram to their proper strengths, and yet the multiples are otherwise invisible.

In all science, there might not be any better example of randomness causing a beautiful result. This result has had practical value because it made possible much of the oil discovered by seismic means in the period from 1930 to 1960. During that period, the seismic prospects that could be interpreted with good results either approximately fitted this model or else benefited from some other fortuitous circumstance. Many of these prospects occurred at shallow depths.

The vast majority of oil-bearing regions, both in extent and in depth, including virtually all offshore regions, could not be explored for a long time because interfering multiples and reverberations made the seismic record uninterpretable. To explore those regions, the oil industry turned to the digital computer, and in so doing, it was the first industry to do so. The digital computer and associated instrumentation are responsible for most of the oil discovered after 1965. If the digital computer were to disappear, no additional significant oil discoveries would be made. No further space exploration would be done. Promising avenues in science and medicine would be abandoned. Business and finance would be crippled. The Internet would disappear.

In the words of Dawson (2008[1], p. 117), the seismic “industry’s tools and methods have become increasingly more accurate; however, target objectives have become much more obscure. Consequently, the future will require considerably more thought and effort.” The computer should not be used as an excuse for not learning the multiplications table. Instead, the computer is the reason that the most advanced methods of geophysics, mathematics, physics, and electrical engineering must be mastered.

Let us now give an example of the small-and-white hypothesis. This example is made purposely simple to illustrate what the mathematics is doing. We have a water layer with the surface-reflection coefficient ${\displaystyle {\varepsilon }_{0}={0.9}}$ and the bottom-reflection coefficient ${\displaystyle {\varepsilon }_{1}=-{0.7}}$. The reflection coefficients within the time gate from time 2 to time 9 are

 {\displaystyle {\begin{aligned}\left({\varepsilon }_{2},{\varepsilon }_{2}{,\ }\dots ,{\varepsilon }_{9}\right)={(}-{0.014},0.056,0.017,0.016,-{0.005,}-{0.0056},0.041,0.043),\end{aligned}}} (36)

which are small and white. The model for the small-reflection-coefficient hypothesis is

 {\displaystyle {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+\dots {.}+{\varepsilon }_{9}Z^{9}}{{1}+g_{l}Z+g_{2}Z^{2}+...+g_{9}Z^{9}}}.\end{aligned}}} (37)

Because the reflection coefficients are small and white within the gate, we can make the approximations

 {\displaystyle {\begin{aligned}g_{1}={\varepsilon }_{0}{\varepsilon }_{1}{,\ }g_{2}=g_{3}=\dots =g_{9}=0.\end{aligned}}} (38)

Thus, we have the approximation

 {\displaystyle {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+\dots +{\varepsilon }_{9}Z^{9}}{{1}+g_{1}Z}}.\end{aligned}}} (39)

This approximation is not dynamic, but it is stationary. In the equation ${\displaystyle h=m*\varepsilon }$, the multiple wavelet m has the Z-transform

 {\displaystyle {\begin{aligned}M\left(Z\right)={\frac {1}{{1}+g_{1}Z}}\end{aligned}}} (40)
Figure 12.  The trace (dark bars of each pair) within the gate for the small-reflection-coefficient model. The trace (the light bars of each pair) within the gate for the small-and white-reflection-coefficient model. The two bars of each pair are almost the same. This result demonstrates that the approximation is good.
Figure 13.  The reflection coefficients (dark bars) within the gate. The trace (light bars) within the gate for the small-reflection-coefficient model. The two bars of each pair are far different. This result demonstrates that the recorded trace fails to show the reflection coefficients.
Figure 14.  The reflection coefficients (dark bars) within the gate. The deconvolved trace (light bars) within the gate for the small-reflection-coefficient model. The two bars of each pair are almost the same. This result demonstrates that the deconvolved trace shows the reflection coefficients. Notice the vertical scale change from Figure 13 to Figure 14.

which, as we know, is a reverberation. We illustrate this model by Figures 12, 13, and 14. Each figure is made up of two touching bars. The gate is from time 2 to time 9. As we see in the figures, deconvolution gates should be chosen between major reflecting horizons, where the reflection coefficients are more likely to be small and white.

## References

1. Dawson, L. D., 2008, Sixty years with SEG: The Leading Edge, 27, no. 1, 117.