In his notes *3-D seismic imaging* ^{},
Biondo Biondi discusses partial stacking as the simplest prototype
example of an imaging operator. A simple implementation of partial
stacking can employ normal moveout (NMO) to correct for traveltime
differences in the data. Stacking irregular data after residual NMO
can be regarded then as an inverse interpolation problem, which one
can solve by optimization methods. In this section, I include a simple
example of an efficient data-space regularization for optimizing
partial stack.

Following Biondi's reproducible example, I use a potion of the Conoco North Sea dataset with offsets, windowed in the range from 400 to 600 m. The geometry of the CMP locations of the input traces is illustrated in Figure 10. The fold distribution, shown as a map view in Figure 11 and as a histogram in Figure 12 is fairly dense overall, but has noticeable holes (empty bins). The regions of zero fold make the inverse interpolation problem underconstrained and suggest an application of a regularized optimization scheme.

cmp
Midpoint geometry of the input data
Figure 10 |

fold
Fold distribution of the input data. A map view.
Figure 11 |

hist
Fold distribution of the input data. A histogram.
Figure 12 |

Figure 13 shows the result of simple binning (the adjoint of nearest neighbor interpolation), normalized by the inverse of the fold density. Obviously, the regions of zero fold don't receive any signal, which can lead to undesirable artifacts in future processing. Figure 14 is the result of non-regularized optimization, with 5 conjugate-gradient iterations at each time slice level. This approach not only fails to fill the empty holes, but also creates unbalanced output because of the poor conditioning of the inverse problem. Figure 15 shows the result of a data-space regularized inversion with a small smoothing filter as a preconditioner. The convergence is fast, and the result looks much improved. Because of the fast convergence of the data-space regularization, the inverse interpolation scheme is inexpensive to apply. One easy way of improving the result further is to change the simple nearest neighbor interpolation operator to a more accurate one. Figure 16 is the result of the regularized inverse interpolation with the Lagrange 4-point interpolator.

Figure 13

Figure 14

Figure 15

Figure 16

As demonstrated by Biondi et al. (1996), an accurate interpolation for dipping reflector events and diffractions requires the azimuth moveout operator. Another interesting, though untested, approach is to use the inverse of a prediction-error filter for preconditioning the inverse interpolation.

11/11/1997