Predictive deconvolution in practice

It now is appropriate to review the implications of the assumptions stated in the convolutional model and inverse filtering that underlie the process of deconvolution within the context of predictive deconvolution.

Assumptions 1, 2, and 3 are the basis for the convolutional model of the recorded seismogram (the convolutional model). In practice, deconvolution often yields good results in areas where these three assumptions are not strictly valid.
Assumption 3 can be relaxed in practice by considering a time-variant deconvolution (the problem of nonstationarity). In this technique, a seismogram is divided into a number of time gates, typically three or more. Deconvolution operators then are designed from each gate and convolved with data within that gate. Alternatively, time-variant spectral whitening can be used to account for nonstationarity (the problem of nonstationarity).
Not much can be done about assumption 4. However, noise can be minimized in the recording process. Deconvolution operators can be designed using time gates and frequency bands with low noise levels. Poststack deconvolution can be used in an effort to take advantage of the noise reduction inherent in the stacking process.
If the source wavelet were minimum-phase and known (assumption 5), then a perfect result could be obtained from deconvolution in the noise-free case as in trace (c) of Figures 2.4-1 and 2.4-2.
If assumption 6 were violated and if the source waveform were not known, then you would have problems as in trace (d) of Figures 2.4-1 and 2.4-2.
The quality of the output from spiking deconvolution is degraded further when the source wavelet is not minimum-phase as in Figures 2.4-3 and 2.4-4; that is, when assumption 7 is violated.
Finally, in addition to violating assumptions 5 and 7, if there were noise in the data, that is, when assumption 4 is violated, then the result of the deconvolution would be unacceptable as in Figure 2.4-5.

Figures 2.4-1 through 2.4-5 test our confidence in the usefulness of predictive deconvolution. In reality, deconvolution has been applied to billions of seismic traces; most of the time it has yielded satisfactory results. Figures 2.4-1 through 2.4-5 emphasize the critical assumptions that underlie predictive deconvolution. When deconvolution does not work on some data, the most probable reason is that one or more of the above assumptions has been violated. In the remaining part of this section, a series of numerical experiments will be performed to examine the validity of these assumptions. The purpose of these experiments is to gain a basic understanding of deconvolution from a practical point of view.

Figure 2.4-1 (a) Impulse response, (b) seismogram, (c) spiking deconvolution using known, minimum-phase wavelet, (d) deconvolution assuming an unknown, minimum-phase source wavelet. Impulse response (a) is a sparse-spike series. For an unknown source wavelet (in violation of assumption 4), spiking deconvolution yields a less than perfect result (compare (c) and (d)).
Figure 2.4-2 (a) Impulse response, (b) seismogram, (c) spiking deconvolution using known, minimum-phase source wavelet, (d) deconvolution assuming an unknown, minimum-phase source wavelet. Impulse response (a) is based on a sonic log (Figure 2.1-1a). For the unknown source wavelet (in violation of assumption 4), spiking deconvolution yields a less than perfect result. (Compare (c) and (d).
Figure 2.4-3 (a) Impulse response, (b) seismogram, (c) deconvolution using a known, mixed-phase source wavelet, (d) deconvolution assuming an unknown, mixed-phase source wavelet. Impulse response (a) is a sparse-spike series. For a mixed-phase source wavelet (in violation of assumption 5), spiking deconvolution yields a degraded output (d), even when the wavelet is known (c).
Figure 2.4-4 (a) Impulse response, (b) seismogram, (c) deconvolution using a known, mixed-phase source wavelet, (d) deconvolution assuming an unknown, mixed-phase source. Impulse response (a) is based on the sonic log of Figure 2.1-1a. For the mixed-phase source wavelet (in violation of assumption 5), spiking deconvolution yields a degraded output (d) even when the wavelet is known (c).
Figure 2.4-5 (a) Impulse response, (b) seismogram with noise, (c) deconvolution assuming an unknown, mixed-phase source wavelet. Impulse response (a) is based on the sonic log of Figure 2.1-1a. In the presence of random noise (in violation of assumption 3), spiking deconvolution can produce a result with nelation to the earth’s reflectivity (compare (a) to (c)).