Estimation of the local Snell parameter

Figure  shows a descending (incident) wavefield crossing an ascending (reflected) wavefield at a given time step of the backward propagation part of the scheme. The crossing point defines the point of the interface that was imaged at that time, and the angles $\alpha$ and $\beta$ are measured, respectively, from the tangent and from the normal to the interface at that point.

 

reflect Figure 10 The points where the ascending and descending wavefronts overlap define the location of the reflector. The reflection angle can be determined by the gradients of the two wavefields at the reflection point, at the time when the reflection occurred.

From the figure we get the following relation,

$$\cos (2\alpha) = {\bf \hat{i} \cdot \hat{r}},$$

where the unit vectors ${\bf \hat{i}}$ and ${\bf \hat{r}}$ represent, respectively, the directions of incidence and reflection. I define the local Snell parameter $\tilde{p}$ as the slowness component parallel to the interface at the reflection point  

$$\tilde{p} \; \; = \; \; {\sin (\beta) \over v_p} \; \; = \; \; \sqrt{{\bf \hat{i} \cdot \hat{r}} + 1 \over 2 v_p^2},$$ (13)

where v_p is the P wave group velocity at that particular location. This definition is restricted to an isotropic assumption, but a more general definition can be formulated that includes the anisotropic extension of Snell's law.