THREE‐DIMENSIONAL KINEMATIC MIGRATION IN VARIABLE VELOCITY MEDIA*

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Abstract
A three‐dimensional (3‐D) kinematic migration algorithm for media in which migration velocity varies linearly with depth is developed, implemented and tested. The algorithm is based on the concept that a single reflection or diffraction in a (zero‐ or finite‐offset) trace may have originated at any point on a constant traveltime surface within the Earth defined by the observed two‐way traveltime. The envelope of all such constant time surfaces, for all observed reflections and diffractions produced by one reflector, is the desired migrated 3‐D image. The optimal envelope position in depth is determined, beneath each point on a regular grid, by a statistical imaging condition; an incremental function of depth containing the number of constant time surfaces passing through that depth increment is cross‐correlated with a Gaussian function whose width is chosen to correspond to the vertical scale of the features of interest.The numerical procedures are based on the observation that, in a medium in which velocity varies linearly with depth, ray segments are circular so traveltimes can be computed analytically. Also, traveltimes are independent of azimuth so the 3‐D problem can be collapsed into an equivalent 2‐D problem.The algorithm is illustrated and tested by application to synthetic data and to scale‐model data from the Seismic Acoustics Laboratory at the University of Houston.