Updating $p$ and $q$ after reflections

Modeling of the propagating wave would be incomplete without properly updating the values of $p(s)$ and $q(s)$ after each boundary reflection. Following Bellhop the method used to update the values is given by:

$$p' = p + q r_n \hskip5mm , \hskip5mm q' = q ,$$ (5.8)

where $p$ and $q$ stand for the values before reflection, and $p'$ and $q'$ stand for the values after. The correction $r_n$ is given by the expression

$$r_n = r_m\frac { 4C_n - 2r_mC_s }{c}$$ (5.9)

where $c$ stands for the value of sound speed at the boundary,

$$r_m = {T_g}/{T_h}$$

and

$$T_g = \left( \mbox{\boldmath$\sigma$}\cdot \mbox{\boldmath$\... ...ath$\nabla$}c\cdot\mbox{\boldmath$\sigma$} \right) \hskip2mm ;$$

$\mbox{\boldmath$\tau$}_b$ corresponds to the boundary tangent, $\mbox{$\mathbf{n}$}_b$ is the boundary normal and $\mbox{\boldmath$\nabla$}c$ is the sound speed gradient; additionally, $\mbox{\boldmath$\sigma$}$ and $\mbox{\boldmath$\sigma$}_n$ stand for the slowness and normal slowness, which can be written as

$$\mbox{\boldmath$\sigma$}= \frac {1}{c} \left[ \begin{array}... ...y}{r} -\sin\theta \\ \cos\theta \end{array} \right] ;$$

all parameters should be calculated at the reflection point.