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  ,
\end{displaymath} (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}
\end{displaymath} (5.9)

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

\begin{displaymath}r_m = {T_g}/{T_h} \end{displaymath}


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}
-\sin\theta \\
\right]  ;

all parameters should be calculated at the reflection point.

Orlando Camargo Rodríguez 2012-06-21