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

Let us now discuss a model consisting of a surface interface overlying two buried interfaces, with the three interfaces separated by arbitrary two-way layer traveltimes. Let the reflection coefficients be given by a, b, c. Let the two-way traveltime between the surface and the first buried interface be S, and let the two-way traveltime between the first buried interface and the second buried interface be T. In other words, the surface reflection coefficient is , the reflection coefficient for the first buried interface is , and the reflection coefficient for the second buried interface is . The Z-transform of the reflectivity is


(30)

The right-hand side of the autocorrelation of the reflectivity is


(31)

If we replace by 1, we obtain the feedback loop


(32)

There are three reverberations. The reverberation between the surface and the first buried interface contributes , the reverberation between the second and third buried interfaces contributes , and the reverberation between the surface and third buried interface contributes . The synthetic trace now is given by the feedforward-feedback filter (Figure 11):

Figure 11.  (a) The reflectivity for the case a = 0.8, b = –0.4, and c = 0.7. (b) The resulting impulsive synthetic trace.


(33)

Suppose that . Then the synthetic trace is given by


(34)

which is


(35)

The shot occurs at time 0, the first primary b occurs at time 2, the first-surface- first-interface multiple occurs at time 4, the second-surface–first-interface multiple occurs at time 6, the second primary occurs at time 7, the third-surface–first-interface multiple occurs at time 8, the peg-leg multiple occurs at time 9, and so on. Figure 11a shows the reflectivity for the case . The leading portion of the corresponding impulsive synthetic trace is shown in Figure 11b.


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