Rapid F‐K migration of zero offset marine reflection data

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Abstract
Recently developed frequency‐wave number migration algorithms have made it possible to migrate digitally recorded zero‐offset reflection data economically. The greatest contributions to the speed of these methods lie in the well‐known symmetries of the Fourier transform and the assumption of a constant velocity section. Several problems have been addressed by using a newly proposed algorithm. The problem of incorrect phase shifts caused by the assumption of a two‐dimensional medium is solved by utilizing a three‐dimensional derivation of the theory. Another problem results from the assumption of a constant migration velocity. The parts of the record section with root mean square velocities less than the migration velocity will be overmigrated and the parts where the rms velocity is greater than the migration velocity will be undermigrated. Given a velocity‐depth function, the rms velocity to a selected portion of a section is a good estimate of the velocity necessary to migrate that portion correctly. Errors in the rest of the migrated section can be reduced by applying the time‐stretching formula of Stolt (1978) prior to migration. Beyond the spatial Nyquist frequency, aliasing noise can become a problem. The amount of aliasing is dependent upon the shot spacing, the frequency content of the source, and the dip of the reflector. The shot spacings commonly in use today are sufficiently small to avoid spatial aliasing; however, shot spacings should be decreased when steeply dipping reflectors are expected and the source frequency is high. The use of automatic gain control before migration tends to reduce the signal‐to‐noise ratio of the migrated section and should be avoided if possible, especially for record sections with large areas of low reflection strength.