# Gap deconvolution

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

In gap deconvolution, one more step must be added to spiking deconvolution. This step amounts to reaveraging the prediction errors (namely, the reflectivity series). We now describe the head filter and the tail filter and explain their relationship to gap deconvolution.

The first step in understanding gap deconvolution requires splitting the minimum-phase wavelet b into two parts. One part is the head,

 {\displaystyle {\begin{aligned}h=(b_{0},\;b_{1},\;b_{2},\;\ldots ,b_{\alpha -1}).\end{aligned}}} (29)

The head consists of the first ${\displaystyle \alpha }$ coefficients, with ${\displaystyle b_{0}}$ in time spot 0. The second part is the tail,

 {\displaystyle {\begin{aligned}t=(b_{\alpha },\;b_{\alpha +1},\;b_{\alpha +2},\ldots ).\end{aligned}}} (30)

The tail consists of the remaining coefficients, advanced in time so that ${\displaystyle b_{\alpha }}$ occurs in time spot 0. Hence, the tail must be delayed in the reconstruction of the wavelet given by

 {\displaystyle {\begin{aligned}b=\left(b_{0}{,\ }b_{1}{,\ }b_{2}{,\dots ,\ }b_{\alpha -1}\right)+\left({0,0,0,\ }\dots {\ ,\ 0,\ }b_{\alpha }{\ ,\ }b_{\alpha +1}{\ ,\ }b_{\alpha {+2}}{,\ }\dots \right)=h+{\delta }_{\alpha }*t.\end{aligned}}} (31)

Given the minimum-phase wavelet (which is obtained as the inverse of the spiking-deconvolution filter), we have three ways to compute the gap-deconvolution operator. They are the head-filtering method, the tail-shaping method, and the head-shaping method.

Let the minimum-phase wavelet be the input to a prediction filter with prediction distance ${\displaystyle \alpha }$. At each time index, the filter tries to predict the value of the input, ${\displaystyle \alpha }$ time units ahead. Thus, at time instant 0, the filter tries to predict ${\displaystyle b_{\alpha }}$. At time instant 1, the filter tries to predict ${\displaystyle b_{\alpha +1}}$. At time instant 2, the filter tries to predict ${\displaystyle b_{\alpha {+2}}}$, and so on. However, these values make up the tail. Thus, the desired output of the prediction filter is the tail of the minimum-phase wavelet. We therefore can find the prediction filter by merely convolving the spiking filter with the tail; that is,

 {\displaystyle {\begin{aligned}k=a*t\mathrm {\;\;\;or\;\;\;\;} k_{i}=\sum _{s={0}}^{\infty }{b_{\alpha +}},a_{i-s},\mathrm {where} \;\;\;\;i={0},1,2,\ldots \end{aligned}}} (32)

For further discussion, see Robinson (1954[1]), in which his equation 5.332 is the same as our equation 32 here. The corresponding prediction-error filter given by equation 13 in the present case becomes

 {\displaystyle {\begin{aligned}f={\delta }_{0}-{\delta }_{\alpha }*a*t.\end{aligned}}} (33)

The head of the wavelet gives us what we might call the unreachable prediction error for the prediction-error filter.

Let us now find another expression for the prediction-error filter. Because ${\displaystyle {\delta }_{0}=a*b}$, the prediction-error filter as given by equation 33 can be written as

 {\displaystyle {\begin{aligned}f=a*b-{\delta }_{\alpha }*a*t=a*\left(b-{\delta }_{\alpha }*t\right).\end{aligned}}} (34)

However, equation 31 gives

 {\displaystyle {\begin{aligned}h=b-{\delta }_{\alpha }*t.\end{aligned}}} (35)

Thus, the prediction-error filter, as given by equation 34, becomes

 {\displaystyle {\begin{aligned}f=a*h.\end{aligned}}} (36)

## References

1. Robinson, E. A., 1954, Predictive decomposition of time series with applications to seismic exploration: Ph.D. thesis, Massachusetts Institute of Technology. (Reprinted in Geophysics, 32, 418-484, 1967.)