The nice thing about a vertically incident source of plane waves p=0 in a horizontally stratified medium is that the ensuing wavefield is laterally invariant. In other words, an observation or a theory for a wavefield would in this case be of the form .Snell waves for any particular nonzero p-value are also laterally invariant. That is, with
The coordinate system (27) and (28) is a retarded coordinate system, not a moving coordinate system. Moving coordinate systems work out badly in solid-earth geophysics. The velocity function is never time-variable in the earth, but it becomes time-variable in a moving coordinate system. This adds a whole dimension to computational complexity.
The goal is to create images from data using a model velocity that is a function of all space dimensions. But the coordinate system used will have a reference velocity that is a function of depth only.