We have tested the newly derived rotated McClellan filter for depth extrapolating a 3-D wavefield in a poststack migration program. We migrated a band-limited (10-100 Hz) impulse delayed .48 seconds assuming a constant propagation velocity of 2,250 meters per second. The spatial sampling was 5 meters in both in-line and cross-line directions and the depth step was constant for all frequencies and equal to 5 meters. The coefficients of the depth-extrapolation filters were derived by windowing the ideal impulse response of the operator with a Gaussian window Nautiyal et al. (1993). To test the accuracy of the proposed method at high dips we used a fairly long extrapolator with half-length equal to 80 samples. As explained in the previous section, during the downward propagation we alternated between depth steps the application of and as transformation filters. Further, we applied a spatial lowpass filter to the input data to attenuate waves with wavenumbers outside the circle defined by .These waves would be not correctly propagated in any case because the McClellan transforms cause the replication of the extrapolation operator along the radial directions in the wavenumber domain.
Figure and Figure show respectively the depth sections obtained by migrating the impulse using the 17-point McClellan filter and the proposed averaged filter. These sections were obtained by cutting the migrated cube along one of the vertical diagonals; that is, along the direction with largest error for both migration schemes. Both sections show the typical semi-circular migration impulse response; however the reflector computed using the 17-point filter is frequency dispersed above a depth of about 300 meters, equivalent to a dip of about 55 degrees. On the contrary, very little dispersion is present in the result computed using the proposed method. The reflector is correctly migrated up to the maximum dip propagated by the extrapolator operator used for the test; that is, a dip of about 78 degrees.
Figure and Figure show a depth slice of the migrated results, respectively computed using the 17-point McClellan filter and the proposed filter. The depth slices were cut at a depth of 200 meters, equivalent to a dip of about 68 degrees. The results of migration using the 17-point McClellan filter are highly dispersed. On the contrary, the reflector migrated using the proposed method appears to be perfectly circular, and very little dispersion of the wavefield is present.