Initial and boundary conditions

Equation (8) is used to extrapolate $\u$ downward in z. Initial values for $\u$ 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 $\u$.

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 $\u$ has been computed, traveltimes are calculated by integrating $\u$ over x using Simpson's rule.