What's wrong with Snell waves?

Before the DSR was developed, I thought that the only proper way to analyze seismic data was to decompose it into Snell waves. Since a Fresnel zone seems to be about 10$^\circ$ wide, not many Snell waves should be required. The small number of required sections was important because of the limited power of computers in the 1970's. I knew that each Snell wave is analyzable by a single square-root equation, and that even multiple reflections can be handled by methods described in FGDP and in this book. Theoretically this approach was a big improvement over CDP stack, which is hardly analyzable at all. A practical problem for downgoing Snell waves, however, is that they may become complicated early if they encounter lateral velocity inhomogeneity shortly after they depart the earth's surface. I no longer believe that Snell waves are a panacea, although I am unsure what their ultimate role will be. But many waves behave a little like they are Snell waves. This motivates the development of a coordinate system that is ideal for Snell waves, and good for waves that are not far from being Snell waves.