LIMITATIONS

The main limitation of the method was noted before: it computes only the first-arrival field, as that is the unique viscosity (thus stable) solution to the eikonal equation. Another limitation follows from the ``thermodynamical'' nature of viscosity: it is one-directional in time, as it tends to suppress high-frequency components. Even in the viscosity limit defining the solution class of interest to us, the computed solution depends on the direction of ``time'', which is actually a spatial direction varying from place to place.

It is necessary to assume that the least-time field is the admissible viscosity weak solution of the eikonal equation with respect to the ``time'' variable implicitly defined by the schedule of expanding circular or rectangular fronts -- and this assumption might or might not be correct! Apparently the algorithm always computes some piece of a branch of the traveltime, but it may well stop before filling the computational domain. This feature is shared with Vidale's code. Both codes rely on an a-priori prescription of a version of ``time'', through the family of computational fronts along which the solution is marched. If the time field to be computed does not have an outward-pointing gradient at each front -- i.e. if the time gradient becomes parallel to the computational front (a turned ray) -- then the calculation stops. It is easy to construct examples that cause any a-priori prescribed family of computational fronts to fail to have this essential outflow property. Therefore neither this code nor Vidale's can be relied upon completely -- both succeed only when the user has general knowledge of the shape of the time field. This limitation is essential -- it comes from the necessary role of ``thermodynamical'' considerations in picking out the ``correct'' weak (nonclassical) solution.