# Prediction-error filters

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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
The stone unhewn and cold
Becomes a living mold,
The more the marble wastes
The more the statue grows.

—Michelangelo


Let $k=\left(k_{0}{,\ }k_{\rm {1}}{,\ }\dots {\ ,\ }k_{N-{\rm {1}}}\right)$ denote the least-squares prediction filter for prediction distance $\alpha$ The prediction-error filter results directly from the prediction filter. As we saw in equation 12 of Chapter 10, the prediction-error filter is $f={1},0,0,...,0,-k_{0},-k_{\rm {1}},-k_{N-1}$ . The prediction-error operator has $\alpha -{\rm {1}}$ zeros that lie between the leading coefficient, which is 1, and the negative prediction-filter coefficients. These $\alpha -1$ zeros constitute the gap.

Let us begin by examining a prediction-error filter, which is in fact a deconvolution filter. The associated prediction-error series is the deconvolved signal. (See Appendix K, exercise 34, at the end of this chapter for a further description of the prediction filter and the prediction-error filter.) A prediction-error filter must be causal. A successfully deconvolved signal shows improved seismic resolution and provides an estimate of the reflectivity series. Depending on a specified prediction distance $\alpha$ , we distinguish between two types of predictive deconvolution: (1) spiking deconvolution, for which the prediction distance equals one time unit, and (2) gap deconvolution, for which the prediction distance is greater than one time unit.

Let $B\left(Z\right)$ be the Z-transform of a minimum-phase wavelet b. Then $A\left(Z\right)=1/B\left(Z\right)$ is the Z-transform of the inverse wavelet $a=b^{-{\rm {1}}}$ . For prediction distance $\alpha$ , the head of b is $h=\left(b_{0}{,\ }b_{1}{,\ }b_{\alpha -1}\right)$ and the tail is $t=\left(b_{\alpha }{,\ }b_{\alpha +1}{,\ }\dots \right)$ . For both the head and tail, the first coefficient is at time 0. Thus, the wavelet is given by $b=h+{\delta }_{\alpha }*t$ , where ${\delta }_{\alpha }$ gives a pure delay of $\alpha$ time units. In terms of Z-transforms, this equation is $B=H+Z^{\alpha }T$ .

The desired output of the prediction operator is the tail. Thus, theoretical prediction operator k is defined by the equation $k*b=t$ . In terms of Z-transforms, this equation is $K\left(Z\right)B\left(Z\right)=T\left(Z\right)$ . Solving for the theoretical prediction operator, we obtain $K\left(Z\right)=A\left(Z\right)T\left(Z\right)$ , which gives $k=a*t$ .

The desired output of the prediction-error operator is the head. Thus, theoretical prediction-error operator f is defined by the equation $f*b=h$ . In terms of Z-transforms, this equation is $F\left(Z\right)B\left(Z\right)=H\left(Z\right)$ . From above, we know that $B=H+Z^{\alpha }T$ , so $H=B-Z^{\alpha }T$ .

Thus, $FB=B-Z^{\alpha }T$ , so $F=1-Z^{\alpha }TA$ . Because $TA=K$ , we see that the Z-transform of the prediction-error operator is given by $F\left(Z\right)=1-Z^{\alpha }K\left(Z\right)$ , which gives $f={\delta }_{0}-{\delta }_{\alpha }*k.$ .

Let us compare prediction operators for two cases. In the first case (Figure 1), the input wavelet b is minimum delay. In the second case (Figure 2), the input wavelet c is not minimum delay. More specifically, the nonminimum-delay wavelet is $c=p*b$ , where p is an all-pass filter.

In both figures, the prediction distance is five. In the first case the minimum-delay input is a geometrically decaying wavelet, whereas in the second case, the nonminimum-delay input is a geometrically growing wavelet. The input wavelets in both figures have the same autocorrelation. As a result, the least-squares prediction filter is the same for both wavelets. In addition, the prediction-error filter is the same for both wavelets.

The prediction errors in the two cases are quite different. The prediction in the minimum-delay case is almost perfect. (Note that the prediction indeed would be perfect if the operator were infinitely long.) The difference between desired output and prediction is almost entirely in the unreachable range, which occurs before the onset of the wavelet. In this case, the difference is essentially the advanced head of the minimum-phase wavelet, where the advance is equal to the prediction distance. The head is substantial because a minimum-phase wavelet has most of the energy up front.

The output of a prediction-error filter is the difference delayed by the prediction distance. However, the prediction in the nonminimum-delay case is poor, so the difference between desired output and prediction is spread out over the unreachable range, which occurs before the onset of the wavelet, and over the reachable range, which occurs after the onset of the wavelet. This unreachable error is the advanced head of the maximum-phase wavelet. The head is not substantial because a nonminimum-delay wavelet has its energy concentrated at the back. Hence, most of the error is in the reachable range, exactly where we do not want it to be.

Let us give an example of a case in which the input wavelet is a minimum-delay wavelet. In any deconvolution problem, the first thing to do is to compute the unit-distance prediction-error filter (i.e., the normalized spiking filter). The inverse of the normalized spiking filter gives the minimum-delay wavelet. Figure 3a (left) shows the minimum-delay input and Figure 3a (right) shows the desired output for the case of unit prediction distance. This desired output is the wavelet advanced by one time unit to the left — that is, the first nonzero coefficient of the desired output occurs at negative time n = –1. Figure 3b (left) shows the unit-distance prediction filter, and Figure 3b (right) shows the prediction. Note that the values for nonnegative times agree well with the desired output. On the other hand, values of the desired output for negative times cannot be reached by the filter. These values give the so-called unreachable prediction error, which for the case of unit prediction distance is simply a spike. Figure 3c (left) shows the unit-distance prediction-error filter, and Figure 3c (right) shows the difference between the desired output and the prediction. The output of the prediction-error filter is this difference delayed by one time unit (i.e., by the prediction distance). Figure 3.  (a) Minimum-delay input and desired output for a prediction distance of 1. (b) The prediction filter for a prediction distance of 1 and the resulting prediction. (c) The prediction-error filter for a prediction distance of 1 and the difference between desired output and prediction.

Let us now give an example of a case in which the input wavelet is a mixed-delay wavelet. Figure 4a (left) show a nonminimum-delay input with the same autocorrelation as that of the minimum-delay input wavelet in Figure 3a (left). This nonminimum-delay input wavelet is obtained by passing the minimum-delay wavelet through an all-pass filter. We proceed as before. Figure 4a (right) shows the desired output for unit prediction distance. This desired output is the wavelet advanced by one time unit to the left — that is, the first nonzero coefficient of the desired output occurs at negative time $n=-1$ . Figure 4b (left) shows the unit-distance prediction filter, which is the same as that in Figure 3b (left). Figure 4b (right) shows the prediction. Now, however, we note that the values for nonnegative times no longer agree with the desired output, as was the case for the minimum-delay input. On the other hand, the filter cannot reach the values of the desired output for negative times. These values give the so-called unreachable prediction error, which for the case of unit prediction distance is simply a spike. Figure 4c (left) shows the prediction-error filter, which is the same as that in Figure 3c (left). Figure 4c (right) shows the difference between the desired output and the prediction. The output of the prediction-error filter is this difference delayed by one time unit (i.e., by the prediction distance). Figure 4.  (a) Nonminimum-delay input (i.e., mixed-delay input) and desired output for a prediction distance of 1. (b) The prediction filter for a prediction distance of 1 and the resulting prediction for the mixed-delay input. (c) The prediction-error filter for a prediction distance of 1 and the difference between desired output and prediction.

In comparing Figure 3 with Figure 4, we see that the prediction filters are identical, as are the prediction-error filters. We recall that the mixed-delay wavelet in Figure 4a (left) is obtained by passing the minimum-delay wavelet in Figure 3a (left) through an all-pass filter. It follows that the prediction in Figure 4b (right) is obtained by passing the prediction in Figure 3b (right) through the same all-pass filter. In addition, the difference in Figure 4c (right) is obtained by passing the difference in Figure 3c (right) through the same all-pass filter.

When a spiking operator is used to deconvolve a seismic trace generated by the convolution of the minimum-delay wavelet with a white reflectivity, the deconvolved trace is the reflectivity. When a spiking operator is used to deconvolve a seismic trace generated by the convolution of the mixed-delay wavelet with a white reflectivity, the deconvolved trace is not the reflectivity but instead is the convolution of the reflectivity with the all-pass filter.

We now give an example of predictive deconvolution for a prediction distance of two. Figure 5a (left) shows the minimum-delay input, and Figure 5a (right) shows the desired output for a prediction distance of two. This desired output is the wavelet advanced by two time units to the left — that is, the first nonzero coefficient of the desired output occurs at negative time $n=-2$ . Figure 5b (left) shows the two-distance prediction filter, and Figure 5b (right) shows the prediction. Note that the values for nonnegative times agree well with the desired output. On the other hand, the filter cannot reach values of the desired output for negative times. These values give the so-called unreachable prediction error, which for the case of prediction distance two is simply two spikes. Figure 5c (left) shows the unit-distance prediction-error filter, and Figure 5c (right) shows the difference between the desired output and the prediction. The output of the prediction-error filter is this difference delayed by two time units (i.e., by the prediction distance). Figure 5.  (a) Minimum-delay input and desired output for a prediction distance of 2. (b) The prediction filter for a prediction distance of 2 and the resulting prediction for the minimum-delay input. (c) The prediction-error filter for a prediction distance of 2 and the difference between desired output and prediction. Figure 6.  (a) Mixed-delay input and desired output for a prediction distance of 2. (b) The prediction filter for a prediction distance of 2 and the resulting prediction for the mixed-delay input. (c) The prediction-error filter for a prediction distance of 2 and the difference between desired output and prediction.

Figure 6a (left) show a mixed-delay wavelet with the same autocorrelation as that of the minimum-delay wavelet in Figure 5a (left). This mixed-delay wavelet is obtained from the minimum-delay wavelet by passing the minimum-delay wavelet through an all-pass filter. We proceed as before. Figure 6a (right) shows the desired output for a prediction distance of two. This desired output is the wavelet advanced by two time units to the left — that is, the first nonzero coefficient of the desired output occurs at negative time $n=-2$ . Figure 6b (left) shows the unit-distance prediction filter, which is the same as that in Figure 5b (left). Figure 6b (right) shows the prediction. Now, however, we note that the values for nonnegative times no longer agree with the desired output, as was the case for the minimum-delay input. On the other hand, the filter cannot reach the values of the desired output for negative times. These values give the so-called unreachable prediction error, which for the case of prediction distance two is simply two spikes. Figure 6c (left) shows the prediction-error filter, which is the same as that in Figure 5c (left). Figure 6c (right) shows the difference between the desired output and the prediction. The output of the prediction-error filter is this difference delayed by two time units (i.e., by the prediction distance).

In comparing Figure 5 with Figure 6, we see that the prediction filters are identical, as are the prediction-error filters. We recall that the mixed-day wavelet in Figure 6a (left) is obtained by passing the minimum-delay wavelet in Figure 5a (left) through an all-pass filter. It follows that the prediction in Figure 6b (right) is obtained by passing the prediction in Figure 5b (right) through the same all-pass filter. In addition, the difference in Figure 6c (right) is obtained by passing the difference in Figure 5c (right) through the same all-pass filter.