Geometric interpretation of common-azimuth downward continuation

The common-azimuth downward continuation operator derived by stationary phase has a straightforward geometric interpretation in terms of propagation directions of the rays of the continued wavefield. In Appendix A we show that expression for the stationary path of equation (7) is equivalent to the relationship  

$$\frac{p_{sy}}{p_{sz}} = \frac{p_{ry}}{p_{rz}},$$ (8)

among the ray parameters for the rays downward propagating the sources $\left( p_{sx}, p_{sy}, p_{sz}\right)$ and the ray parameters for the rays downward propagating the receivers $\left( p_{rx}, p_{ry}, p_{rz}\right)$.This relationship between the ray parameters constrains the direction of propagation of the source and receiver rays, for each possible pair of rays. In particular, the source ray and the receiver ray must lie on the same plane; with all the possible propagation planes sharing the line that connects the source and receiver location at each depth level. This geometric relationship constrains the sources and receivers at the new depth level to be aligned along the same azimuth as the source and receivers at the previous depth level, consistently with the condition that we imposed for the stationary phase derivation of the common-azimuth downward continuation operator [equation (4)]. Figure  shows a graphical representation of the geometric interpretation of common-azimuth downward continuation. The source ray and the receiver ray must lie on any of the slanted planes that share the line connecting the source and receiver locations.

In Appendix A we show that from equation (8) it is possible to derive the ray parameter equivalent of the stationary-path expression of equation (7); that is,  

$$\left(p_{ry}- p_{sy}\right) = \left(p_{ry}+ p_{sy}\right) \f... ...^2} - p_{rx}^2 }+ \sqrt{\frac{1}{v({\bf s},z)^2} - p_{sx}^2 }}.$$ (9)

In this expression the distinction between the velocity at the source location, and the velocity at the receiver location introduced in equation (3), and formally carried through the stationary phase approximation, is now physically meaningful. It implies that the source rays and receiver rays must lie on the same plane, notwithstanding different local velocities and ray bending.