EXAMPLES

In the first example I calculate traveltimes for a constant-velocity medium with a velocity of 2.5 km/s. The grid is evenly sampled in depth and laterally, with a sample interval of 10 m. Figure  shows the traveltime function for a source on the surface at the middle of the model, and Figure  displays the difference with the analytical solution.

Errors accumulate as depth increases, and are largest on the outer sides of the model, where rays travel at angles closer to the horizontal. Since the finite-difference scheme is designed for downward traveling rays, this behavior is expected. The maximum error is about .6 ms, which is one order of magnitude smaller than the standard time sampling interval of 4 ms (see Figure ). Finite-difference traveltime calculations of one shot on a $100\times 100$ grid take about .1 s in CPU time on the Convex C-1.

Figure  shows the difference between Vidale's scheme and the analytical solution. I have only implemented the plane wave extrapolation method, and errors can probably be reduced if a combination of plane and circular wave extrapolation is used. The errors are largest away from the vertical, diagonal and horizontal direction, where the plane wave approximation breaks down. The errors at the bottom of the model are of the same order of magnitude as the errors in the method described here (see again Figure ). However, Vidale's scheme does not vectorize, and on the Convex C-1 the finite-difference calculations are about 5-10 times faster than his method.

The next example illustrates the calculations for a more complicated model. The model is shown in Figure ; it consists of 3 layers and a wedge intrusion. The velocity in the top layer is 2 km/s, the middle layer has a velocity of 1.75 km/s, and the bottom velocity is 2.5 km/s. The velocity in the wedge that intrudes the middle layer from the right is 2.75 km/s. Figure  shows the result of tracing rays from the surface downwards; because of the velocity contrasts, rays cross and shadow zones are apparent.

As is obvious from Figure , interpolating traveltimes from the rays onto the grid is not easy for this model. However, the finite-difference calculation correctly fills in the problem areas as can be seen in Figure : the contour lines in the plot reveal the correct curvature of the wave fronts in the high- and low-velocity regions. Figure  provides a more quantitative check; it compares interpolated traveltimes of the rays at the bottom of the model with the finite-difference traveltimes. Since the rays are calculated for a spline model that is fitted to the grid model (Van Trier, 1988), some discrepancies may be expected, especially if one realizes that a slight change in velocity contrast can drastically change the direction of the ray. Except for the region near the shadow zone, the traveltime curves match reasonably well.