EXAMPLES IN SMOOTH MEDIA

In the following examples, velocity fields are parameterized in terms of bicubic B-splines. Any spacing can therefore be chosen for the numerical grid on which the FMM is solved. Ray paths can be obtained a posteriori by following the traveltime gradient from the receiver back through the medium to the source.

In the example below, the first-arrival wavefront is tracked through a heterogeneous medium from a source point at 30 km depth. Note that ray paths favour the faster regions as expected.

The plot below shows the convergence of traveltimes (for the above example) for four different grid sizes towards a reference solution calculated on a 50 m grid (described by a total of 1602801 nodes). The four grid sizes are 1000 m (4141 nodes), 500 m (16281 nodes), 250 m (64561 nodes) and 125 m (257121 nodes). CPU times (for a Sun Ultra 5) are also provided.

The next example demonstrates the unconditional stability of the FMM by tracking a wavefront in a medium with pathological velocity variations. Despite velocity contrasts as great as 70:1, the FMM is stable.

The convergence of traveltimes for this extreme example is demonstrated below.