# Prediction lag

So far, we have learned that predictive deconvolution has two uses: (a) spiking deconvolution — the case of unit prediction lag, and (b) predicting the input seismogram at a future time defined by the prediction lag. Case (b) is used to predict and attenuate multiples.

Now, the effect of the prediction lag parameter is examined from an interpretive point of view. Consider the single, isolated minimum-phase wavelet in Figure 2.4-14. Here, operator length and percent prewhitening are kept constant, while prediction lag is varied. When prediction lag is equal to the sampling rate, then the result is equivalent to spiking deconvolution. Predictive deconvolution using a prediction lag greater than unity yields a wavelet of finite duration instead of a spike. Given an input wavelet of *α* + *n* samples, predictive deconvolution using prediction filter with length *n* and prediction lag *α* converts this wavelet into another wavelet that is *α* samples long. The first *α* lags of the autocorrelation are preserved, while the next *n* lags are zeroed out. Additionally, the amplitude spectrum of the output increasingly resembles that of the input wavelet as prediction lag is increased (Figure 2.4-14). At a 94-ms prediction lag, predictive deconvolution does nothing to the input wavelet because almost all the lags of its autocorrelation have been left untouched. This experiment has an important practical implication: Under the ideal, noise-free conditions, resolution on the output from predictive deconvolution can be controlled by adjusting the prediction lag. Unit prediction lag implies the highest resolution, while a larger prediction lag implies less than full resolution. However, in reality, these assessments are dictated by the signal-to-noise ratio. The deconvolved output using a unit prediction lag contains high frequencies; nevertheless, resolution may be degraded if the high-frequency energy is mostly noise, not signal.

In Figure 2.4-14, prediction lags of 8 and 22 ms correspond to the first and second zero crossings on autocorrelation of the input wavelet, respectively. The first zero crossing produces a spike with some width, while the second zero crossing lag produces a wavelet with a positive and negative lobe.

The relationship between prediction lag and whitening also holds for the sparse-spike series in Figure 2.4-15 and when the input wavelet is unknown (Figure 2.4-16).

The effect of prediction lag on the output from predictive deconvolution of a synthetic seismogram, which was obtained from the sonic log (Figure 2.1-1a), is demonstrated in Figures 2.4-17 and 2.4-18. As the prediction lag is increased, the output spectrum becomes increasingly less broadband. Predictive deconvolution of seismograms constructed from the mixed-phase wavelet again demonstrates that output resolution can be controlled by adjusting prediction lag (Figures 2.4-19 through 2.4-23).

**Figure 2.1-1**(a) A segment of a measured sonic log, (b) the reflection coefficient series derived from (a), (c) the series in (b) after converting the depth axis to two-way time axis, (d) the impulse response that includes the primaries (c) and multiples, (e) the synthetic seismogram derived from (d) convolved with the source wavelet in Figure 2.1-4. One-dimensional seismic modeling means getting (e) from (a). Deconvolution yields (d) from (e), while 1-D inversion means getting (a) from (d). Identify the event on (a) and (b) that corresponds to the big spike at 0.5 s in (c). Impulse response (d) is a composite of the primaries (c) and all types of multiples.

If prediction lag is increased, then the output amplitude spectrum becomes increasingly band-limited. The output also can be band-limited by applying a band-pass filter on the spiking deconvolution output. Are these two ways of band-limiting equivalent? Refer to the results from both the minimum- and mixed-phase wavelets in Figures 2.4-14 and 2.4-19, respectively. Note that the output of the 22-ms prediction lag has an amplitude spectrum that is band-limited to approximately 0 to 100 Hz. However, the spectral shape within this bandwidth is not a boxcar, but rather similar to that of the input wavelet. The boxcar shape would be the case if a band-pass filter (0 to 100 Hz) were applied to the output of the spiking deconvolution (2-ms prediction lag). Hence, *spiking deconvolution followed by band-pass filtering is not equivalent to predictive deconvolution with a prediction lag greater than unity*.

In conclusion, if prediction lag is increased, the output from predictive deconvolution becomes less spiky. This effect can be used to our advantage, since it allows the bandwidth of deconvolved output to be controlled by adjusting the prediction lag. The application of spiking deconvolution to field data is not always desirable, since it boosts high-frequency noise in the data. The most prominent effect of the nonunity prediction lag is suppression of the high-frequency end of the spectrum and preservation of the overall spectral shape of the input data. This effect is seen in Figures 2.4-18 and 2.4-23, which correspond to the minimum-and mixed-phase seismic wavelets. If prediction lag is increased further, then the low-frequency end of the spectrum is affected as well, making the output more band-limited.