Refraction correction

Besides the integration of the dynamic equations, and the update of $p(s)$ and $q(s)$ after reflections, TRACEO borrows from Bellhop an undocumented correction of refraction, which can be written as

$$p' = p + q \; \tilde{r}_n ,$$

where $'$ stands for the corrected value,

$$\tilde{r}_n = - \tilde{r}_m\frac { ( 2C_{nj} - \tilde{r}_mC_{sj} )}{ c } ,$$

$$\tilde{r}_m = \sigma_r(s)/\sigma_z(s) ,$$

$$C_{nj} = \delta\mbox{\boldmath$\nabla$}c \cdot \mbox{\boldma... ...\boldmath$\nabla$}c \cdot \mbox{\boldmath$\sigma$}\hskip5mm ,$$

$c$ represents the sound speed at the arrival position, and $\delta\mbox{\boldmath$\nabla$}c$ represents the ``jump'' of the gradient, i.e., the variation of $\mbox{\boldmath$\nabla$}c$ between the initial and final positions.