Gap interpolation

Irregular gaps occur in the recorded data for many different reasons, and prediction-error filters are known as a powerful method for interpolating them. Interpolating irregularly spaced data also reduces to gap interpolation after binning.

Figure 10 shows a simple synthetic example of gap interpolation from Claerbout (1999). The input data has a large elliptic gap cut out from a two plane-wave model. I estimate both dip components from the input data by using the method of equations (16-18). 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 (two middle plots in Figure 10.) Although the estimation program did not make any assumption about dip being constant, it correctly estimated nearly constant values with the help of regularization equations (17-18). The rightmost plot in Figure 10 shows the result of gap interpolation with a two-plane local plane-wave destructor. The result is nearly perfect and compares favorably with the analogous result of the T-X PEF technique Claerbout (1999).

 

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Figure 10 Synthetic gap interpolation example. From left to right: original data, input data, first estimated dip, second estimated dip, interpolation output.

Figure 11 is another benchmark gap interpolation example from Claerbout (1999). The data are ocean depth measurements from one day SeaBeam acquisition. The data after normalized binning are shown in the left plot of Figure 11. From the known part of the data, we can partially see a certain elongated and faulted structure on the ocean floor. 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 11. 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 fully agreeable with the data. Clapp (2000b) shows on the same data example how to obtain multiple statistically equivalent realizations of the interpolated data.

 

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Figure 11 Depth of the ocean from SeaBeam measurements. Left plot: after binning. Right plot: after binning and gap interpolation.

A 3-D interpolation example is shown in Figure 12. 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.

 

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Figure 12 3-D gap interpolation in passive seismic data. The left 12 panels are slices of the input data. The right 12 panels are the corresponding slices in the interpolation output.

We can conclude that plane-wave destructors provide an effective method of gap filling and missing data interpolation.