# Prestack deconvolution

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Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Figure 2.5-1 shows a CMP gather that contains five prominent reflections at around 1.1, 1.35, 1.85, 2.15, and 3.05 s. The gather also contains strong reverberations associated with these reflections. The examination of the deconvolution parameters will begin with an analysis of the time gate to estimate the autocorrelation function. A first gate selected may be the entire length (6 s) of the record as seen in panel (a). The solid lines on the CMP gathers refer to the gate start and end times. The autocorrelogram of the record is shown at the bottom of each panel. A second choice might be to exclude the deeper part of the record where ambient noise dominates. The start of the gate is chosen as the first arrival path as shown in panel (b). A third choice may be to exclude not only the deeper portion, but also the early part of the record that contains energy corresponding to the guided waves as shown in panel (c). These waves travel within the water layer and are not part of the signal reflected from the substrata.

By comparing the autocorrelograms from these different windows, note that the third choice best represents the reverberatory character of the data as shown in panel (c) over most of the offsets. All of the traces in the autocorrelogram within approximately the first 150 ms have a common appearance. This early portion of the autocorrelogram characterizes the basic seismic wavelet contained in the data.

In general, the autocorrelation window should include part of the record that contains useful reflection signal, and should exclude coherent or incoherent noise. An autocorrelation function contaminated by noise is undesirable since the deconvolution process is most effective on noise-free data (assumption 4).

Another aspect of the autocorrelation window is length. Panel (d) of Figure 2.5-1 shows the autocorrelogram estimated from a narrow window. The autocorrelogram estimated from the narrower part of the time gate (the right side of the record) in some data cases may lack the characteristics of the reverberations, and even those of the basic seismic wavelet.

In general, any autocorrelation function is biased; that is, the first lag value is computed from, say, n nonzero samples, the second lag value is computed from n – 1 nonzero samples, and so on. If n is not large enough, then there can be an undesirable biasing effect. How large should the data window be to avoid such biasing? If the largest autocorrelation lag used in designing the deconvolution operator were m, an accepted rule of thumb is that the number of data samples should be no less than 8m.

Now that the autocorrelation window is determined, we examine operator length. In Figure 2.5-2, prediction lag (4 ms, the same as the sampling rate) and percent prewhitening (0.1%) are fixed. The autocorrelograms (at the bottom of each gather) are displayed for diagnostic purposes. From the analyses of the single spike, sparse spike, and reflectivity models (predictive deconvolution in practice), recall that the short (40-ms) operator leaves some residual energy that corresponds to the basic wavelet and reverberating wavetrain in the record. For a spiking deconvolution with a 160-ms-long operator, no remnant of the energy is associated with the basic wavelet and reverberations. Any operator longer than 160 ms does not change the result, significantly. From the output of the 160-ms operator, note that the prominent reflections (at 1.1, 1.35, 1.85, and 2.15 s at the near-offset) have been uncovered, the seismic wavelet has been compressed, and the reverberations have been significantly suppressed.

The effect of prediction lag now is examined. In Figure 2.5-3, the 160-ms operator length and 0.1% prewhitening are fixed, while prediction lag is varied. If prediction lag were increased, the deconvolution process would be increasingly less effective in broadening the spectrum, and the autocorrelograms would contain increasingly more energy at nonzero lags. In the extreme, the deconvolution process is ineffective with a 128-ms prediction lag. In practice, common values for the prediction lag are unity (spiking deconvolution) or the first or second zero crossing of the autocorrelation function (predictive deconvolution).

Finally, the percent of prewhitening is varied, while the 4-ms prediction lag and 160-ms operator length are fixed. These tests are shown in Figure 2.5-4. By increasing the percent prewhitening, the deconvolution process becomes less effective. The high end of the spectrum is not flattened as much as the rest of the spectrum (Figure 2.4-24). Note that the autocorrelograms contain increasingly more energy at nonzero lags with increasing percent prewhitening. In practice, it is not advisable to assign a large percent of prewhitening. Typically, a value between 0.1 and 1 percent is sufficient to ensure stability in designing the deconvolution operator (equation 32).

 ${\displaystyle {\begin{pmatrix}\beta r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&\beta r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&\beta r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &\beta r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}1\\0\\0\\\vdots \\0\end{pmatrix}},}$ (32)

We now examine the effect of operator length and prediction lag on amplitude spectrum. Figure 2.5-5a shows a common-shot gather with its autocorrelogram and average amplitude spectrum. The field record is prepared for deconvolution by first muting the guided waves (Figure 2.5-5b) and applying t2-scaling (Figure 2.5-5c). Figure 2.5-5d shows the same record after spiking deconvolution. Note the flattening of the spectrum within the passband of the recorded data and attenuation of the energy at nonzero lags of the autocorrelogram. With prediction lag greater than unity (Figure 2.5-6), for the same operator length, we note insufficient flattening at the high-frequency end of the spectrum. At larger prediction lags, note the insufficient flattening at the low-frequency end of the spectrum. A very large prediction lag causes the amplitude spectrum of the deconvolved data remain similar to that of the input data (compare Figures 2.5-6c with 2.5-5c).

The data sometimes must be preconditioned for deconvolution. If the data were too noisy, then a wide band-pass filter could be necessary before deconvolution. If there is significant coherent noise in the data, dip filtering (frequency-wavenumber filtering and the slant-stack transform) can be applied before deconvolution so that coherent noise is not included in the autocorrelation estimate. Alternatively, time-variant spectral whitening (the problem of nonstationarity) can be applied to balance the spectrum before deconvolution.