Calculation of normals

Theoretically, in order to calculate the ray influence as accurately as possible the vector $\Delta\mbox{$\mathbf{r}$}$ (see section 5.1) is required to be perpendicular to the polarization vector $\mbox{$\mathbf{e}$}_s(s)$ (or, alternatively, parallel to $\mbox{$\mathbf{e}$}_1(s)$). For an arbitrary position of the hydrophone such condition implies traveling along the ray, until the condition $(\Delta\mbox{$\mathbf{r}$}\cdot\mbox{$\mathbf{e}$}_s) = 0$ is fullfiled. In such case the normal corresponds to

\begin{displaymath}n = \left\vert \Delta\mbox{$\mathbf{r}$} \right\vert  . \end{displaymath}

Numerical calculations indicate that there is no significant loss of accuracy (and requires much less computations) to take

0 \\
z_h - z(s)
\end{array} \right]  ,

(i.e., at each hydrophone range one interpolates ray depth, so $\Delta\mbox{$\mathbf{r}$}$ becomes a vertical vector, connecting the ray to the hydrophone) and to calculate the normal as

\begin{displaymath}n = \left\vert \Delta\mbox{$\mathbf{r}$}\cdot \mbox{$\mathbf{e}$}_1(s) \right\vert  . \end{displaymath}

Orlando Camargo Rodríguez 2012-06-21