## Automatic selection of reference velocities for recursive depth migration

### Hugh D. Geiger, Gary F. Margrave

Wave equation depth migration methods such as phase-shift plus interpolation, extended split-step Fourier, or Fourier finite difference plus interpolation require a limited set of reference velocities for efficient wavefield extrapolation through laterally inhomogeneous velocity models. In basic implementations, reference velocities are selected as either a linear or a geometric progression spanning the range of model velocities. However, it is unlikely that the model velocities are distributed linearly or geometrically. If the reference velocities can be distributed statistically to more closely approximate the actual distribution, the accuracy of the extrapolation step can be improved. In this paper, we present a modification to a previously published algorithm for statistical selection of reference velocities (Bagaini et al. 1995).

Key features of our automatic reference velocity selection algorithm are 1) division of the velocity distribution into clusters 2) entropy based statistical control to determine the minimal number of reference velocities required within a cluster, 3) a novel 'greedy search' that selects reference velocities within each cluster at or near peaks in the probability distribution, and 4) calculation of a single reference velocity if the velocity range within each cluster is less than a minimum threshold. We show that our method is superior to Bagaini et al.'s method, which includes only step (2) above, and - in place of step (3) - selection of reference velocities by piecewise constant interpolation of the probability distribution of the underlying velocities. Our automatic velocity selection algorithm can be run using a single parameter - the approximate desired velocity step expressed as a percentage - although the maximum step and minimum threshold can be specified if the defaults are not suitable. The automatic velocity selection algorithm is implemented as part of a prestack PSPI depth migration of the 2-D Marmousi model. The resulting images are clearly superior to images created using a linear or geometrical distribution of a similar number of reference velocities. The algorithm is suitable for 3-D data, where the tradeoff between accuracy and efficiency is more pronounced.