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Given the eikonal equation, there is an abundance of
raytracing methods in literature.
Finite-difference travel time schemes are one class of methods
which calculate
the first arrival of the wave-front from the eikonal
equation. The finite-difference scheme introduced by Vidale
(1988) has an explicit minimization condition (i.e., the time along
a new computation front is calculated using the minimum
time along the previous computation front), important for
keeping the scheme stable. The Van Trier and Symes (1989; 1990) algorithm
has a similar minimization condition which is applied
locally to three points of the four point finite-difference
stencil.
I analyze their first-order upwind finite-difference scheme
and show the implicit
minimization condition and the errors introduced in favor
of greater computational speed.
I extend the algorithm to a 3-D medium and show results of traveltime
fields for constant and slowly varying velocity media.
The 3-D algorithm is much more susceptible to
numerical instability when rapid variations in velocity are present.

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Stanford Exploration Project

12/18/1997