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

Wavefield extrapolation for a laterally varying velocity model 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 algorithm - called AGPS (adaptive Gabor phase-shift) - has a computational cost that is proportional to the complexity of the velocity model, while its accuracy is comparable to both NSPS (nonstationary phaseshift) 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. The NSPS and PSPI filters differ to the extent that the functional dependence of the velocity is either on the input coordinates (NSPS) or the output coordinates (PSPI) – in fact, they are spatial transposes of each other. This is why both methods are approximate: indeed, any accurate phase-shift operator in a v(x) medium must allow for velocity variations along the trajectory of a ray. AGPS attempts to address this problem by representing the input and output wavefields as superpositions of windowed components, each of which is approximately stationary with respect to the velocity.