The theory of wavefield extrapolation by phase-shift and its extension from the Fourier domain to the space-frequency domain are reviewed. The extension of the phase-shift method to strong lateral velocity variations is also reviewed. It is shown that the space-frequency formulation developed from constant-velocity phase-shift can be trivially modified to accommodate any of three phase-shift expressions: nonstationary phase-shift, phase-shift plus interpolation, and Weyl-form extrapolation. These space-frequency equivalents to the Fourier methods are shown to be Kirchhoff-style summation operators that are applied by spatial convolution. Formulae for the 3-D and 2-D Kirchhoff equivalents of each of the Fourier methods are given. An extended 2-D numerical investigation shows that the Kirchhoff approach produces virtually identical results to the Fourier techniques. The importance of including the near-field term is demonstrated. An examination of algorithmic costs shows that the Kirchhoff approach can be dramatically faster than the Fourier method for strongly heterogeneous media.
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