A common-midpoint gather holding data with only one velocity should stack OK without need for antialiasing. It is nice when antialiasing is not required because then high temporal frequencies need not be filtered away simply to avoid aliased spatial frequencies. When several velocities are simultaneously present on a CDP gather, we will find crossing waves. These waves will be curved, but aliasing concepts drawn from plane waves are still applicable. We designed the antialiasing of migration by expecting hyperbola flanks to be orthogonal to horizontal beds or dipping beds of some chosen dip. With a CDP gather we chose not a dip, but a slowness s0. The slope of a wave of slowness s on a CDP gather is xs2/t. The greater the contrast in dips, the more need for antialiasing. The slope of a wave with slowness s0 is xs02/t. The difference between this slope and that of another wave is xs2/t - xs02/t or (s2-s02)x/t which in the program is the slope for the purpose of antialiasing. The choice of s0 has yet to be determined according to the application. For illustration, I prepared a figure with three velocities, a very slow surface wave, a water wave, and a fast sediment wave. I chose s0 to match the water wave. In practice s0 might be the earth's slowness as a function of traveltime depth.
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Figure 16 The air wave and fast wave are broadened increasingly with offset, but the water wave does not. This broadening enables crossing events to maintain their orthogonality. |  |
