Conclusions

Several choices exist in selecting the regularization operator for iterative data regularization.

Splines in tension represent an approach to data regularization suitable for smooth data. The constraint is embedded in a user-specified tension parameter. The two boundary values of tension correspond to cubic and linear interpolation. By applying the method of spectral factorization on a helix, I have been able to define a family of two-dimensional minimum-phase filters, which correspond to the spline interpolation problem with different values of tension. These filters contribute to the collection of useful helical filters. I have used them for preconditioning in data-regularization problems with smooth surfaces. In general, they are applicable for preconditioning various estimation problems with smooth models.

I demonstrate that adaptive local plane-wave destructors with an improved finite-difference design can be a valuable tool in processing multidimensional seismic data. In several examples, I have shown a good performance of plane-wave destructors in application to data regularization. It may be useful to summarize here 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 to predict 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 allows for an iterative linearization. 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 under-determined problem.
  • Local slope has a clearly interpretable physical meaning, which allows for easy quality control of the results. The coefficients of T-X PEFs are much more difficult to interpret.
• 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.

I have shown that a 3-D plane-wave destruction filter can be designed from a pair of two-dimensional filters by using helix transform and the Wilson-Burg spectral factorization algorithm. A special approach to designing plane-wave destructor filters follows from the general B-spline regularization method.

Differential offset continuation provides a valuable tool for regularization of reflection seismic data. Starting from analytical frequency-domain solutions of the offset continuation differential equation, I have designed accurate finite-difference filters for implementing offset continuation as a local convolutional operator. A similar technique works for shot continuation across different shot gathers.

Differential offset continuation serves as a bridge between integral and convolutional approaches to data interpolation. It shares the theoretical grounds with the integral approach but is applied in a manner similar to that of prediction-error filters in the convolutional approach.

Tests with synthetic and real data demonstrate that the offset-continuation regularization can succeed in complex structural situations where more simplistic methods fail.