We implemented TFMES in both 2-D and 3-D cases. The 3-D constant velocity medium is used as a benchmark test to verify the accuracy of the new algorithm. We use the same sampling interval in both the Cartesian and the tetragonal coordinates to make a fair comparison. As shown in Figure 7, the Cartesian implementation tends to over-estimate the traveltime in the diagonal direction, while the tetragonal result matches the analytical result very accurately.

Figure 7

More complex Marmousi and SEG/EAGE saltdome models are used to test its
stability when handling more complex models.
Figure 8 is the test of 2-D Marmousi model. The source
is located on the surface at coordinates (*x*=4100*m*, *z*=0*m*). In most areas,
the two results match each other. When passing through the complex
structure in the middle, they begin to deviate from each other.
The trigonal result is not as smooth as the Cartesian result. This is
because the trigonal result has six neighboring points instead of four
points in the Cartesian coordinates, which makes it more capable of
simulating complex wavefront. The Cartesian implementation over-estimates
the traveltime compared with the trigonal result, which is similar to the
conclusion reached by Alkhalifah and Fomel
1997.

Figure 8

Figure 9 and 10
show three traveltime slices from the SEG/EAGE saltdome model. The source is
located at coordinates (*x*=7200*m*, *y*=7320*m*, *z*=1680*m*).
Figure 9 was obtained with a constant depth *z*=1560*m*,
while Figure 10 were extracted in the diagonal and
anti-diagonal directions of *x-y* plane.

Figure 9

Figure 10

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