Missing-data interpolation was introduced in Chapter as a simple case of data regularization when the input data are already binned to regular grid locations but with remaining uncovered gaps.
Figure shows a simple synthetic example of gap interpolation from Claerbout (1999). The input data has a large elliptic gap cut out in a two plane-wave model. I estimate both dip components from the input data by using the method of equations (-). The initial values for the two local dips were 1 and 0, and the estimated values are close to the true dips of 2 and -1 (the third and fourth plots in Figure .) Although the estimation program does not make any assumption about dip being constant, it correctly estimates nearly constant values with the help of regularization equations (-). The rightmost plot in Figure shows the result of gap interpolation with a two-plane local plane-wave destructor. The result is nearly ideal and compares favorably with the analogous result of the T-X PEF technique Claerbout (1999).
Figure is another benchmark gap interpolation example from Claerbout (1999), already featured in Chapter (Figures -). The data are ocean-depth measurements from one day SeaBeam acquisition. The data after normalized binning are shown in the left plot of Figure . From the known part of the data, we can partially see a certain elongated and faulted structure on the ocean floor created by fractures around an ocean ridge. Estimating a smoothed dominant dip in the data and interpolating with the plane-wave destructor filters produces the image in the right plot of Figure . The V-shaped acquisition pattern is somewhat visible in the interpolation result, which might indicate the presence of a fault. Otherwise, the result is both visually pleasing and in full agreement with the input data. Clapp (2000b) uses the same data example to obtain multiple statistically equivalent realizations of the interpolated data.
A 3-D interpolation example is shown in Figure . The input data resulted from a passive seismic experiment Cole (1995) and originally contained many gaps because of instrument failure. I interpolated the 3-D gaps with a pair of two orthogonal plane-wave destructors in the manner proposed by Schwab and Claerbout (1995) for T-X prediction filters. The interpolation result shows a visually pleasing continuation of locally plane events through the gaps. It compares favorably with an analogous result of a stationary T-X PEF.
We can conclude that plane-wave destructors provide an effective method of gap filling and missing-data interpolation.