Initial and boundary conditions

We have presented the numerical scheme in polar coordinates for the sole purpose of solving the point-source problem. In polar coordinates the initial condition at the source is easily specified as $u(0,\th) = 0$. However, the same problem may be solved in Cartesian coordinates if calculations are done on a sequence of expanding rectangular fronts. Although slightly more complicated, this approach saves the cost of mapping the slowness and traveltime fields to and from polar coordinates. The rectangular computational fronts are also used by Vidale (1988).

When one assumes outgoing rays at the boundaries, one-sided finite-differences can be used at the left and right sides 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 $\t_\r$ over $\r$ using a simple trapezoidal rule, where $\t_\r$ is found from equation (5).