LIMITATIONS

The flux-conservative form of the Eikonal equation is essentially a one-way equation. This means that rays can only be propagated in one direction, and that only single-valued traveltime curves are computed. So far, I have been discussing downward traveling rays, but if $\u=\t_z$ instead of $\u = \t_x$ is used in equation (2), one can calculate traveltimes of rays that travel sideways. Initial conditions then have to be specified along a vertical line.

In the downward-propagation finite-difference scheme, the numerical errors get large if $F\Delta z\ll \Delta x$, which occurs when rays travel almost horizontally. This numerical noise can be reduced by decreasing the stepsize in z when rays have large propagation angles at the current depth level, and increasing it when the propagation angles get smaller. If greater accuracy is still required, an other alternative is to use higher-order finite difference schemes.

Since the method does not calculate ray paths, it is only of limited use in tomographic methods. It is possible, however, to calculate ray directions on the grid from the traveltime gradient. One component of the slowness vector, $\t_x$, is already available as $\u$, the other one, $\t_z$, is calculated using $\t_z= \sqrt{s^2 - \u^2}$.