The general method of B-spline regularization, outlined in Chapter , is easily applicable for the case of local plane-wave destruction. The continuous regularization operator D in this case comes from the theoretical plane-wave differential equation (). We simply need to construct the auto-correlation filter dj according to formula () and factorize it with the Wilson-Burg method. Figure shows three plane waves constructed from three distant spikes by application of inverse recursive filtering with two different B-spline regularizers. The left plot was obtained with first-order B-splines (equivalent to linear interpolation). This type of regularizer is identical to Clapp's steering filters Clapp et al. (1997) and suffers from numerical dispersion effects. The right plot was obtained with third-order splines. Most of the dispersion is suppressed by using a more accurate interpolation.
Equipped with the powerful B-spline plane-wave construction, we can now approach the main goal of this work: three-dimensional seismic data regularization. For an illustrative test, I chose the North Sea dataset, which was previously used for testing azimuth moveout Biondi et al. (1998) and common-azimuth migration Biondi (1996). Figure in the introduction showed the highly irregular midpoint geometry for a selected in-line and cross-line offset bin in the data. The data irregularity is also evident in the bin fold map, shown in Figure . The goal of data regularization is to create a regular data cube at the specified bins from the irregular input data, which have been preprocessed by normal moveout.
Figure 19 Map of the fold distribution for the 3-D data test.
The data cube after normalized binning is shown in Figure . Binning works reasonably well in the areas of large fold but fails to fill the zero fold gaps and has an overall limited accuracy.
For efficiency, I perform regularization on individual time slices. Figure shows the result of regularization using bi-linear interpolation and smoothing preconditioning with the minimum-phase Laplacian filter. The empty bins are filled in a consistent manner but the data quality is distorted because simple smoothing fails to characterize the complicated data structure. Instead of continuous events, we see smoothed blobs in the time slices. The events in the in-line and cross-line sections are also not clearly pronounced.
We can use the smoothing regularization result to estimate the local dips in the data, design invertible local plane-wave destruction filters, and repeat the regularization process. Inverse interpolation using bi-linear interpolations with plane-wave preconditioning is shown in Figure . The regularization result is improved: the continuous reflection events become clearly visible in the time slices. As expected, a higher quality result is achieved with cubic B-spline (Figure ). Regularization works again in constant time slices, using recursive filter preconditioning with plane-wave destructor filters analogous to those in Figure . Despite the irregularities in the input data, the regularization result preserves both flat reflection events and steeply-dipping diffractions. Preserving diffractions is important for correct imaging of sharp edges in the subsurface structure Biondi and Palacharla (1996b).
For simplicity, I assumed only a single local dip component in the data. This assumption degrades the result in the areas of multiple conflicting dips, such as the intersections of plane reflections and hyperbolic diffractions in Figure . One could improve the regularization result by considering multiple local dips. In the next section of this chapter, I describe an alternative offset-continuation approach, which uses a physical connection between neighboring offsets instead of assuming local continuity in the midpoint domain.
The 3-D results of this subsection were obtained with an efficient 2-D regularization in time slices. This approach is computationally attractive because of its easy parallelization: different slices can be interpolated independently and in parallel. Figure shows the interpolation result for four selected time slices. Local plane waves, barely identifiable after binning (left plots in Figure ), appear clear and continuous in the interpolation result (right plots in Figure ). Different time slices are assembled together to form the 3-D cube shown in Figure .
A more powerful, although less convenient, approach to 3-D data regularization, is the full 3-D plane-wave destruction. I discuss it in the next subsection.