# Gap deconvolution of a mixed-delay wavelet

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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 11 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Much work has been done on the problem of deconvolution in the case of a mixed-delay (or nonminimum-delay) wavelet. See, for example, Eisner and Hampson (1990)[1]; Ford (1978)[2]; Lazear (1984)[3]; Lines and Ulrych (1977)[4]; Porsani and Ursin (1998)[5]; Sacchi and Ulrych (2000)[6]; Sengbush et al. (1987)[7]; Tygel et al. (1991)[8]; Ulrych et al. (1995)[9]; and Yilmaz (1987)[10]. Here, we give another approach.

Suppose various signature-deconvolution procedures are used to remove the source signature, absorption, and instrument response from a seismic trace. However, the process is not perfect, so a short causal nonminimum-delay residual wavelet u remains in their place. Thus, the resulting trace is ${\displaystyle x=u*m*\varepsilon }$, where m is the minimum-delay multiple and ${\displaystyle \varepsilon }$ is the white reflectivity. Suppose the multiple consists only of the water-layer reverberation given above — that is, ${\displaystyle m=b}$. For example, suppose the reverberation is

 {\displaystyle {\begin{aligned}m=b=\left({1,\ 0,\ }-{0.5,0,0.25}{,0,\ }-{0.125,0,\ }\dots \right).\end{aligned}}} (5)

The seismic wavelet w is defined as the convolution of the residual wavelet and the minimum-delay reverberation train. Thus, the seismic trace is ${\displaystyle x=w*\varepsilon }$. If the residual wavelet is the maximum-delay wavelet ${\displaystyle u={1},2}$, then the seismic wavelet is the mixed-delay wavelet

 {\displaystyle {\begin{aligned}w=u*b=\left({1,\ 2,\ }-{0.5,\ }-{1,0.25,0.5,\ }-{0.125,\ }-{0.25,\ }\dots \right).\end{aligned}}} (6)

By inspection, the prediction operator (for a prediction distance of two) for the seismic wavelet is ${\displaystyle k=\left(-{0.5,0,0,0,\ }\dots \right)}$. We can verify that this indeed is the prediction operator by carrying out the convolution

 {\displaystyle {\begin{aligned}k*w=\left(-{0.5,\ -1,\ 0.25,\ 0.5,\ }-{0.125,\ }-{0.25,\ }\dots \right),\end{aligned}}} (7)

which is the causal part of the advanced seismic wavelet. Thus, the prediction-error operator is ${\displaystyle f=\left({1\ ,\ 0,0.5,0,0,0,0,\ ..\ .}\right)}$ which, in fact, is the dereverberation operator. We carry out the convolution

 {\displaystyle {\begin{aligned}f*w=(1,2,0,0,0,\ldots )\end{aligned}}} (8)

which is the residual wavelet u. If we deconvolve the seismic trace with this prediction-error operator, we obtain

 {\displaystyle {\begin{aligned}f*x=f*w*\varepsilon =u*\varepsilon .\end{aligned}}} (9)

Thus, the deconvolved trace is the reflectivity smoothed by the residual wavelet.

By wavelet processing, this residual wavelet can be shaped into a desirable interpreter wavelet. We note that the deconvolution operator f is equal to the dereverberation ${\displaystyle a=b^{-{\rm {1}}}}$. In effect, the nonminimum-delay residual wavelet falls thought the cracks in the gap-deconvolution process. This example illustrates that in the case of a nonminimum-delay seismic wavelet, gap deconvolution can be used judiciously in the correct circumstances to remove the reverberations.

Let us now give another example of gap deconvolution. We shall illustrate the method of predictive deconvolution for the case of a water layer, which produces a reverberation train. Suppose the water depth is 100 ft (30 m). Let the water-surface reflection coefficient be unity, and let the water-bottom reflection coefficient be 0.5. Let the velocity of the water be 5000 ft (830 m)/s. Then the two-way normal-incidence traveltime in the water layer is 2(100)/5000 = 0.040 s, which is 40 ms. Assume that the sampling increment is 4 ms, so that the reverberation cycle time in discrete units is 40/4 or 10.

Figure 9.  (a) The components of the convolutional model of a reverberating trace. (b) The prediction-error operator, the deconvolved trace, the dereverberation operator, and the smoothed reflectivity for the trace shown in Figure 7a.

Figure 9a shows the components of the convolutional model ${\displaystyle x=b*u*\varepsilon }$. Figure 9b depicts the prediction-error operator (PEO) f for a prediction distance equal to the cycle time, the deconvolved trace ${\displaystyle y=f*x}$, the theoretical dereverberation operator a, and the smoothed reflectivity ${\displaystyle u*\varepsilon }$. How well the least-squares deconvolution worked can be determined as follows. First compare the prediction-error operator (PEO) with the dereverberation operator. The closer they are, the better the deconvolution is. Second, compare the deconvolved trace with the smoothed reflectivity. Again, the closer they are, the better the deconvolution is. Although there are differences in the details, the agreement is quite good.

Deconvolution removes the reverberation but leaves the nonminimum-phase residual wavelet in place (if the length of the residual wavelet is less than the reverberation cycle time). The important point is that the method of gap deconvolution works even if the residual wavelet is not minimum delay but is instead mixed delay. It is necessary to use the correct value of the prediction distance (namely, the reverberation cycle time) in the design of the prediction-error operator.

How does one determine the prediction distance? One way is to compute the autocorrelation of the seismic trace and see if it exhibits periodicity. If it does, the cycle time can be set equal to that period. A set of prediction distances clustering around such an estimate of cycle time should be used to determine empirically the “best-looking” dereverberation operator. Within statistical accuracy, such a method of predictive deconvolution can be described as follows: The prediction-error filter operating on the trace wipes out the reverberations so that the reverberations do not appear in the deconvolved trace. The residual wavelet (because its length is less than the cycle time) falls through the cracks of the prediction-error operator. Hence, the source wavelet does appear in the deconvolved trace. This deconvolved trace is equal to the reflectivity smoothed by the residual wavelet.