Wavefield extrapolation by nonstationary phase shift

Abstract

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 an analytic form for the popular phase shift plus interpolation (PSPI) 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. In the (x, ω) domain, these processes are the transpose of each other; however, only NSPS has the physical interpretation of forming the scaled, linear superposition of laterally‐variable impulse responses (i.e.,Huygen...