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Equation (8) is used to extrapolate downward in *z*.
Initial values for have to be specified at the surface. In many cases
the velocity model is constant at the surface, and the initial
conditions can be calculated analytically for a given source
position. For models with non-constant surface velocities, traveltimes
can be computed along the surface with a simple 1-D finite-difference
approximation. The derivative of the traveltime function with respect
to *x* then gives .
When one assumes outgoing rays at the boundaries, one-sided finite
differences can be used at the left and right side of the model.
These boundary conditions are of the same order as the scheme
inside the model, and generally do not cause any problems.
After has been computed, traveltimes are calculated by
integrating over *x* using Simpson's rule.

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Stanford Exploration Project

1/13/1998