# Prediction distance

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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

Many seismic-processing systems contain no guidelines for choosing the prediction distance. Should the prediction distance be set equal to the first or second zero crossing of the trace autocorrelation? Are there ways to automatically pick zero crossings? Does one have to examine the autocorrelations by eye? In fact, can a seismic processing system automatically determine what prediction distance is best to use? Automation would help greatly here.

Several questions arise in the use of deconvolution. Often, deconvolution parameters are chosen by educated guesses after a few autocorrelations have been examined. Should the first or the second zero crossing of an input autocorrelation be used for the optimal prediction distance? Does use of the first zero-crossing criterion produce a relatively sharp pulse, and does the second zero-crossing criterion produce a broader wavelet that has both positive and negative side lobes?

Predictive deconvolution is based on the convolutional model, which states that the trace within a given gate is the convolution of two components — a minimum-phase wavelet and a white-reflectivity series. Spike deconvolution is a method of finding the components of the convolutional model, and it always should be performed as the first step of predictive deconvolution. The second step involves application of a postdeconvolution filter to yield the final deconvolved trace. Let $\alpha$ denote the prediction distance. The head is defined as the first $\alpha$ coefficients of the minimum-phase wavelet. In the case in which the parameter $\alpha$ is equal to one, the head is a spike. A prediction-error filter with prediction distance $\alpha$ equal to one is called a spike (or spiking) deconvolution filter (Chapter 10). The spike-deconvolution filter is equivalent (within a constant scale factor) to the shaping filter that shapes the minimum-phase wavelet into its head. Convolution of the spike-deconvolution filter with the trace yields the spike-deconvolved trace.

A gap-deconvolution filter is a least-squares prediction-error filter with a prediction distance $\alpha$ greater than one (Chapter 10). The gap is equal to $\alpha -1$ . For the spike filter, the gap is zero because $\alpha -1=1-1={0}$ . The gap-deconvolution filter is equivalent (within a constant scale factor) to the shaping filter that shapes the minimum-phase wavelet into its head. The gap-deconvolution filter is equal to the convolution of the spike-deconvolution filter with the head. Convolution of the gap-deconvolution filter with the trace yields the gap-deconvolved trace. The gap-deconvolved trace equals the convolution of the spike-deconvolved trace with the head.

It often is stated that gap deconvolution is a more generalized approach than spike deconvolution is. In fact, the opposite is often true. Let us explain. Some claim that the gap-deconvolution technique allows one to control the length of the desired output wavelet and hence to control the desired degree of resolution. In gap deconvolution, the desired output wavelet is the head of the wavelet. Thus, in gap deconvolution, the length of the desired output can be specified, but the shape cannot. Because often the head is not minimum phase, in such cases, gap deconvolution destroys the minimum-phase character of the seismic trace, and subsequent filtering operations, such as wavelet processing, are more difficult or even impossible to perform.

Gap deconvolution should be used only when proper safeguards are taken. For example, gap deconvolution can be used to attenuate multiple reflections, and as such, it often is referred to as predictive deconvolution. One such procedure is given in the next section.

Instead of gap deconvolution, spike deconvolution should be used on a routine basis, followed by carefully designed postdeconvolution filters. A postdeconvolution filter controls not only the length of the desired output but also its shape. This filter can be either a minimum-phase filter if further deconvolution operations are to be applied or it can be a zero-phase filter if wavelet processing is desired.