CAM stationary path

In the cartesian coordinate system, the components of the source and receiver slowness vectors along a ray are

$$\begin{eqnarray} p_{sx} &=& \frac{\sin \theta_s \cos \phi_s}{v(z)} \qquad \qquad... ...\; \qquad \qquad \qquad \qquad p_{rz} = \frac{\cos \theta_r}{v(z)}\end{eqnarray}$$ (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:  

$$\hat{k}_{hy} = k_{my} \frac{\sqrt{1-v(z)^2 p_{rx}^2} - \sqrt... ...{sx}^2}} {\sqrt{1-v(z)^2 p_{rx}^2} + \sqrt{1-v(z)^2 p_{sx}^2}}$$ (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 $60^\circ$ and oriented at $45^\circ$ 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 phy (thick solid line) and of the approximated phy (dashed curve) calculated with CAM stationary-phase approximation, in the case of a reflector dipping at about $60^\circ$ and oriented at $45^\circ$ with respect to the in-line direction.