## Fast wavefield extrapolation by phase-shift in the nonuniform Gabor domain

### Jeff P. Grossman, Gary F. Margrave, Michael P. Lamoureux

Wavefield extrapolation for a laterally varying velocity can be achieved by applying a nonstationary phase-shift filter to an adaptive, nonuniform Gabor transform over the lateral coordinate. A family of adaptive Gabor frames can be constructed from a molecular decomposition of unity, each molecule of the latter being built by conjoining neighbouring atoms from a uniform partition of unity - consisting of translates of a single atom along the lateral coordinate - according to a local stationarity criterion derived from the velocity model.

The resulting extrapolation scheme - called AGPS (adaptive Gabor phase-shift) - has a computational cost that is proportional to the complexity of the velocity model, v(x), while its accuracy is comparable to both NSPS (nonstationary phase-shift) and generalized PSPI (phase-shift plus interpolation). AGPS includes NSPS and PSPI as complementary limiting cases, yet the cost of AGPS ranges from an order of magnitude less to about the same order. This range is based on two extremes: a simple step between two constant velocities, and a velocity that varies randomly at each offset.