There is no question that two-way finite difference methods offer the potential for the most accurate images any migration algorithm can produce. If we know the exact subsurface wavefield at the last recorded time, the output from this technique provides an exact answer. It images all events from which reflections occur at precisely the correct reflectivity. Consequently, it is the ultimate goal of all migration algorithm developers. Its only algorithmic problems are its extreme computational requirements and that it can produce unusual artifacts that are difficult to explain when incorrect Earth models are used.
However, full two-way reverse time migration has no velocity sensitivities, it has no dip limitations, and, in the prestack sense, it is the only approach that accurately handles all amplitude issues.
Phase-shift-plus-interpolation (PSPI) represents a very simple extension to the pure phase-shift approach. It uses the phase shift algorithm with multiple constant velocities, and then interpolates as needed to achieve the proper image at each image point on the current depth or time slice. It was very likely the first FK-style method that was able to at least partially remove both the v ( z ) assumption and still retain a reasonable dip response. The quality of PSPI algorithms is still a function of the accuracy of the implementation, but, nevertheless, most such algorithms are quite good. Like its phase-shift counterpart, it can be extended to include some full two-way propagation. Because the interpolation step in PSPI can be somewhat difficult, and because each application of the phase shift method adds to the overall cost, alternatives to the method have been sought.
Split-step methods attempt to avoid the interpolation step of PSPI methods through a different approximation to the underlying wave-equation. In effect, this approach was one of the first to use a dual domain approach. The "shift" is accomplished in the FK domain, while the modification of the interpolation step is accomplished in the FX domain. When this method was published, it was thought to provide an improved approach to PSPI, but this did not prove to be the case.
Generalized phase screens are really split-step algorithms with additional terms to increase the overall dip-response and improve accuracy. The key difference between a split-step algorithm and a phase screen method is that the phase screen methods have additional correction terms in ( F;X ) space, and, as a result, should produce a more accurate, less sensitive algorithm when properly implemented. Again, like the original phase shift method, generalized phase screens can be modified to include some forms of two-way propagation.
