Conclusions

The main conclusion of this paper is simple: plane-wave destructors with an improved finite-difference design can be a valuable tool in processing multidimensional seismic data. On several examples, I showed their good performance in such problems as fault detection, missing data interpolation, and noise attenuation. Further experiments will be necessary to gain more experience with plane-wave destructors and to improve the practical aspects of their usage.

It might be useful to summarize the similarities and differences between plane-wave destructors and T-X prediction-error filters.

Similarities:

• Both types of filters operate in the original time-and-space domain of recorded data.
• Both filters aim at predicting local plane-wave events in the data.
• In most problems, one filter type can be replaced by the other, and certain techniques, such as Claerbout's trace interpolation method, are common for both approaches.

Differences:

• The design of plane-wave destructors is purely deterministic and follows the plane-wave differential equation. The design of T-X PEF has statistical roots in the framework of the maximum-entropy spectral analysis Burg (1975). In principle, T-X PEF can characterize more complex signals than local plane waves.
• In the case of PEF, we estimate filter coefficients. In the case of plane-wave destructors, the estimated quantity is the local plane slope. Several important distinctions follow from that difference:
  • The filter estimation problem is linear. The slope estimation problem, in the case of the improved filter design, is non-linear, but can be iteratively linearized. In general, non-linearity is an undesirable feature because of local minima and the dependence on initial conditions. However, we can sometimes use it creatively. For example, it helped me avoid aliased dips in the trace interpolation example.
  • Non-stationarity is handled gracefully in the local slope estimation. It is a much more difficult issue for PEFs because of the largely underdetermined problem.
  • Local slope has a clearly interpretable physical meaning, which allows for an easy quality control of the results. The coefficients of T-X PEFs are much more difficult to interpret.
• Plane-wave destructors are stable filters by construction. Stability is not guaranteed in the traditional PEF estimation and often can be a serious practical problem.
• The efficiency of the two approaches is difficult to compare. Plane-wave destructors are generally more efficient to apply because of the optimally small number of filter coefficients. However, they may require more computation at the estimation stage because of the already mentioned non-linearity problem.