This direction switching is forced by the appearance of discontinuities (shocks) in the flow variables. For the eikonal equation, the gradient components of traveltime can be discontinuous: rays, whose direction is determined by the gradient components, ``break'' at interfaces in the model where velocity contrasts occur. Reshef and Kosloff (1986) observe stability problems in a centered finite-difference scheme at these interfaces. Upwind finite-difference methods, on the other hand, are stable because they mimic the underlying physics of the problem in two important aspects. First, they add numerical viscosity to the equations, thus finding a smooth viscosity solution. Second, they copy the behavior of continuum flow by taking their information from upstream.

Careful inspection of Vidale's ordering of gridpoints reveals that
he applies the basic ``box'' (four-point) difference formula only
in the upwind sense (see Vidale, 1988, Figure 5), having used an
alternative extrapolation at
the few points at which the flow (i.e. ray field)
changes direction along the
computational front. Those points are local traveltime minima. His scheme
is thus an upwind finite-difference method, although not
presented as such. It is an *implicit* method, as
it connects more than one value on the grid level being updated.
The ordering of points is required to achieve a closed-form
solution of the difference formulas, as opposed to an iterative
approximate solution (as is often chosen with implicit schemes).
Our formulas are *explicit*, in contrast, and do not
require any special ordering of points.

1/13/1998