A homogeneous isotropic example

Figure 2 shows three examples of finite-difference traveltimes in homogeneous media. For the moment limit your attention to the top plot, which shows an isotropic medium. The wavefronts in this example are nearly circular, but they are not as accurate as the example in Figure 3 of Van Trier and Symes (page 49 of this report).

 

homo
Figure 2 Finite-difference traveltime calculations for three different homogeneous media. The ``*'' shows the position of the source, the thin lines show equally spaced traveltime contours, and the thick dashed curve shows a theoretical contour for comparison.

There are several reasons for this. We used extrapolation in z instead of r, which is not as accurate for near-circular wavefronts. It especially has trouble at the top of the model, where the extrapolation is (almost) at right angles to the wave propagation direction. Calculating the initial boundary conditions is harder when extrapolating in z, too. Our model shows a ``bump'' directly under the source. This is due to the initial discontinuity in $u = \partial \tau / \partial x$ at the source. The initial error at z=0 is propagated along when we integrate $F(u) = \partial \tau / \partial z$ over z to get the final result $\tau(x,z)$, causing the vertical line of ``bumps'' under the source.

 

hetero
Figure 3 A heterogeneous example. The top and left part of the model are isotropic. The block on the right is very anisotropic.