Stability of Explicit Depth Extrapolation Through Laterally-varying Media

Explicit table-driven frequency-space depth extrapolation methods are the heart of accurate 2-D and 3-D depth migration (Blacquiere, et (Soubaras, 1991). These methods and their cousin, the phase-shift plus interpolation (pspi) method (Gazdag and Squazzero, 1984) are based on using the phase-shift method (either in the wavenumber domain or using a space-domain equivalent) for one-way extrapolation. Since the phase-shift method (Gazdag, 1980) (without any additional amplitude treatment) is unconditionally stable, it was assumed that a composed extrapolation, one where the dispersion relation used to produce a given output point was allowed to vary from point to point would also be stable.