In the cartesian coordinate system, the components of the source and receiver slowness vectors along a ray are
(6) | ||
(7) | ||
(8) |
In this context, we can reformulate the expression for the stationary path in CAM theory Biondi and Palacharla (1996), which gives the cross-line offset ray parameter as a function of velocity and ray parameters:
(9) |
Moreover, since wave propagation can be handled completely analytically in constant gradient velocity, we can calculate the theoretically ``exact'' cross-line offset wavenumber and compare it to the values given by the stationary-phase approximation (Equation (9)), as shown in Figure 6. Here, in the case of a reflection on a plane dipping at and oriented at with respect to the in-line direction, the stationary path given by Equation (9) is a seriously biased approximation.
phy_cam
Figure 6 Comparison of the exact cross-line offset ray parameter p_{hy} (thick solid line) and of the approximated p_{hy} (dashed curve) calculated with CAM stationary-phase approximation, in the case of a reflector dipping at about and oriented at with respect to the in-line direction. |