The representation of a velocity field is crucial to the accuracy and efficiency of traveltime and amplitude calculations. Because the traveltime and amplitude calculations are more accurate in a continuous velocity model than in a discontinuous velocity model, I map a given velocity field onto a grid formed by regular triangular cells. Within each triangular cell, velocity varies linearly, as follows:
v(x,z) = v_{o}+v_{x}(x-x_{o})+v_{z}(z-z_{o}), | (1) |
(2) |
The reason for using linearly varying velocities within triangular cells is that seismic waves propagate along circular ray paths in a linearly varying velocity medium. Therefore, local ray-tracing in these triangular cells can be done analytically. An alternative is to use linearly varying sloth (slowness squared) within each triangular cell. The ray paths in such a medium are hyperbolas.
In a manner similar to the finite difference method developed by Vidale (1988), two schemes need to be built for the local ray-tracing method. In the following sections, I first develop the scheme used to propagate a local wavefront through a triangular cell. I then describe the scheme that addresses the order in which the local computations proceed.