Discussion

The analytical band-limit of equation (5) is the necessary criteria to appropriately delimit non-aliased wavenumbers for either wavefield continuation migration method. The amount of compression of a data axis in the Fourier domain by removing samples from the space domain dictates the areal extent of the source function or the wavenumber limit for the limited energy imaging strategy. This problem will manifest itself primarily with shot-profile migration strategies. This is due to the fact that two conflicting sampling schemes are simultaneously available. Choosing the finer sampling for migration, results in an aliased image. Choosing the coarser sampling for the migration throws away valuable information. We have shown two methods thus far to eliminate aliased contributions to the image based on equation (24). We feel that it is more appropriate to use selective energy imaging conditions rather than a fat source function or band-limited continuation steps while propagating on the finest available data grid.

In practice, lateral velocity variation will cause individual wavenumber energy constituents to move about the fk-plane. At some continuation step, energy could move back and forth across the prescribed image space band-limit, and yet still be appropriate for that individual profile propagation. To allow for any beneficence from this movement, it would be unwise to either: a) propagate a band-limited source function, or b) to eliminate energy from the source and receiver wavefields during propagation steps (by either coarse resampling, or band-limiting the propagation wavefields). Therefore we recommend migrating individual shots on the fine grid, and accounting for operator aliasing in the imaging step at a modest cost increase. The output resolution will still be at least as good as a source-geophone migration.