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Throughout the development of deconvolution theory, several alternatives have been proposed to better solve the deconvolution problem. Still, predictive deconvolution is used more than the other methods, although the minimum-phase and white reflectivity sequence assumptions have been key issues of concern.
Follow the common sequence for deconvolution of marine data in Figures 2.6-18 through 2.6-22. Note the presence of reverberations and short-period multiples in the CMP-stacked data with no deconvolution applied (Figure 2.6-18). Signature processing, in this case, was done to convert the recorded source signature to its minimum-phase equivalent (Figure 2.6-19). Therefore, aside from phase, there is no difference between the sections in Figures 2.6-18 and 2.6-19. Deconvolution before stack has helped attenuate reverberations and short-period multiples and, to some extent, compressed the basic wavelet (2.6-20). The additional step of poststack deconvolution has restored much of the high frequencies attenuated during stacking (Figure 2.6-21). Finally, time-variant spectral whitening has flattened the spectrum within the passband of the data (Figure 2.6-22) and yielded a crisp section with high resolution.
The same sequence can be followed in Figures 2.6-23 through 2.6-27. The CMP-stacked section includes reflections associated with a shallow, low-relief sedimentary strata (Figure 2.6-23). Following the signature processing (Figure 2.6-24), observe the gradual increase in the vertical resolution by prestack deconvolution (Figure 2.6-25), poststack deconvolution (Figure 2.6-26) and time-variant spectral whitening (Figure 2.6-27).
The following formal processing sequence for deconvolution theoretically should yield optimum results:
- Apply a geometric spreading compensation function to remove the amplitude loss due to wavefront divergence.
- Apply an exponential gain or minimum-phase inverse Q filter  . This compensates for frequency attenuation.
- Optionally apply signature processing to marine data. For vibroseis data, apply the filter that converts the Klauder wavelet to its minimum-phase equivalent.
- Apply predictive deconvolution to compress the basic wavelet and attenuate reverberations and short-period multiples. If required, apply surface-consistent deconvolution (Appendix B.8). This accounts for the near-surface variations effect on the wavelet because of inhomogeneities in the vicinity of sources and receivers. In step (b), the vertical variations effect on the wavelet is handled.
- Apply predictive deconvolution after stack to broaden the spectrum and attenuate short-period multiples.
- Apply time-variant spectral whitening. This provides further flattening of the spectrum within the signal bandwidth without affecting phase.
The idea is to do as much deterministic deconvolution as possible. Inverse Q filtering, signature deconvolution, and the filter that converts the Klauder wavelet to its minimum-phase equivalent are deterministic operators. Any remaining issues then are handled by statistical means. However, for most data cases, just doing the geometric spreading correction followed by predictive prestack and poststack deconvolution is adequate.
- Time-variant deconvolution
- Time-variant spectral whitening
- Frequency-domain deconvolution
- Inverse Q filtering
- The problem of nonstationarity