Wavefield extrapolation by nonstationary phase shift
Gary F. Margrave, Robert James Ferguson
The phase shift method of wavefield extrapolation applies a phase shift in the Fourier domain to deduce a scalar wavefield at one depth level given its value at another. The phase shift operator varies with frequency and wavenumber and assumes constant velocity across the extrapolation step. We use nonstationary filter theory to generalize this method to nonstationary phase shift (NSPS) which allows the phase shift to vary laterally depending upon the local propagation velocity. For comparison, we derive the popular PSPI (phase shift plus interpolation) method in the limit of an exhaustive set of reference velocities. NSPS and this limiting form of PSPI can be written as generalized Fourier integrals which reduce to ordinary phase shift in the constant velocity limit. However, only NSPS has the physical interpretation of a laterally varying phase shift which forms the scaled, linear superposition of impulse responses (i.e. Huygen's wavelets).
The difference between NSPS and PSPI is clear when they are compared in the caseof a piecewise constant velocity variation. Define a set of windows such that the jth window is unity when the propagation velocity is the jth distinct velocity and is zero otherwise. NSPS can be computed by applying the window set to the input data to create a set of windowed wavefields, each of which is phase shift extrapolated with thecorresponding constant velocity, and the extrapolated set is superimposed. PSPI proceeds by phase shift extrapolating the input data for each distinct velocity, applying the jth window to the jth extrapolation, and superimposing. PSPI has the unphysical limit that discontinuities in the lateral velocity variation cause discontinuities in the wavefield while NSPS shows the expected wavefront "healing".
We then formulate a finite aperture compensation for NSPS which has the practicalresult of absorbing lateral boundaries for all incidence angles. Wavefield extrapolation can be regarded as the crosscorrelation of the wavefield with the expected response of a point diffractor at the new depth level. Aperture compensation simply applies a laterally varying window to the infinite, theoretical diffraction response. The cross correlation becomes spatially variant, even for constant velocity, and hence is a nonstationary filter. The nonstationary effects of aperture compensation can be simultaneously appliedwith the NSPS extrapolation through a laterally variable velocity field.