Multidimensional autoregression |

The first stage is that of the previous section,
finding the optimal PEF while
carefully avoiding using any regression equations
that involve boundaries or missing data.
For the second stage, we take the PEF as known and
find values for the empty bins so that
the power out of the prediction-error filter is minimized.
To do this we find missing data with
module `mis2()` .

This two-stage method avoids the nonlinear problem we would otherwise face if we included the fitting equations containing both free data values and free filter values. Presumably, after two stages of linear least squares we are close enough to the final solution that we could switch over to the full nonlinear setup described near the end of this chapter.

The synthetic data in Figure 15 is a superposition of two plane waves of different directions, each with a random (but low-passed) waveform. After punching a hole in the data, we find that the lost data is pleasingly restored, though a bit weak near the side boundary. This imperfection could result from the side-boundary behavior of the operator or from an insufficient number of missing-data iterations.

hole
Original data (left), with a zeroed hole, restored,
residual selector (right).
Figure 15. |
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The residual selector in Figure 15 shows where the filter output has valid inputs. From it you can deduce the size and shape of the filter, namely that it matches up with Figure 14. The ellipsoidal hole in the residual selector is larger than that in the data because we lose regression equations not only at the hole, but where any part of the filter overlaps the hole.

The results in Figure 15 are essentially perfect representing the fact that that synthetic example fits the conceptual model perfectly. Before we look at the many examples in Figures 16-19 we will examine another gap-filling strategy.

- Adding noise (Geostat)
- Inversions with geostat
- Infill of 3-D seismic data from a quarry blast
- Imposing prior knowledge of symmetry
- Hexagonal coordinates

Multidimensional autoregression |

2013-07-26