Lateral invariance

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 $P(t) \, \times \, \hbox{const} (x)$.Snell waves for any particular nonzero p-value are also laterally invariant. That is, with

$$\begin{eqnarray} t' \ \ \ &=&\ \ \ t\ -\ p\,x \\ x' \ \ \ &=&\ \ \ x\end{eqnarray}$$ (27) (28)

lateral invariance is given by the statement  

$$P(x,t) \eq P' ( t' ) \ \times \ \hbox{const} ( x' )$$ (29)

Obviously, when an apparently two-dimensional problem can be reduced to one dimension, great conceptual advantages result, to say nothing of sampling and computational advantages. Before proceeding, study equation (29) until you realize why the wavefield can vary with x but be a constant function of x' when (28) says x = x'.

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.