Importantly, the second panel does not show the aliasing problems present in the second panel of Figure 3 despite the same level of shot decimation. In this experiment, the subsampling of the shot-axis is partially mapped into both of the two new coordinate axes before migration. The coordinate transform of equation (1) thus distributes the axis' lower Nyquist limit equally to the new coordinates prior to migration. Since the output image coordinates inherit this same sampling, the resampled data naturally adhere to the band-limiting criterion of equation (5).

The image location and subsurface offset variables of these migrations have the same meaning as those discussed in the shot-profile section previously Biondi (2003). There have been two important modifications however due to the initial coordinate transform. First, notice the range of wavenumbers included in the second image space is drastically limited from the panel to the left showing the migration of all available shots. The resorting has effectively band-limited the image space honoring the Nyquist requirement appropriate for the image given the shot axis subsampling. Thus, the algebraic combination of source and receiver coordinates in the numerator makes this an inherently band-limited propagation method. Second, the division by two of both new axes stretches their Fourier dual domains. Notice that the alias replications in the right panel of Figure 4 appear at a wavenumber of 20 rather than the 10 seen in the shot-profile migration example. Despite the fact that the same number and sampling interval for the shot axis was used in both experiments, the division associated with the coordinate mapping has decreased the sampling interval in the space domain and stretched energy along the Fourier domain. This has happened independently of the three modes of aliasing described above and needs undoing separately as well.

Figure 4

5/23/2004