When the phrase complex media is used, the Marmousi model often comes to mind. The huge amount of folding and faulting induced in this model have created a rather interesting distribution of velocity anomalies and discontinuities. As a result, the Marmousi model has served as a performance measure used to evaluate various migration and traveltime calculation algorithms. In this section, we use the unsmoothed version of the Marmousi model to test the fast marching methods accuracy and stability. The lateral grid spacing was reduced to obtain a more efficient finite-difference solution of the acoustic wave equation, for comparison. Because the second-order scheme van Trier and Symes (1991), even with the adaptive measures, is not stable in such a complex model, we use a version of the fast marching eikonal solver, implemented on a very fine grid in polar coordinates to reduce the first-order traveltime derivate errors. The solution of this fine-grid implementation will serve as a reference used to compare the accuracies of Cartesian versus polar coordinate implementation of the method on a more practical grid spacing.
Figure 9 shows contours of traveltime using the fast marching method in Cartesian coordinates (dashed curves), and in polar coordinates (gray curves). The black curves show traveltimes for the more accurate reference solution based on a fine-grid implementation. For vertical and horizontal wave propagation, the various contours practically overlap. At a near 45-degree angular wave propagation, the dashed curves tend to predict faster times than the actual solution. In some regions, the polar coordinate implementation may give the worse results, for example, at distance 3000 m and depth 2000 m, but, overall, fast marching in polar coordinates gives better results than the Cartesian-coordinate implementation as Figure 10 shows. Both methods were executed on a 200MHz Pentuim processor and took less than two seconds of computer time.
Figure 11 shows the one-second contour curves superimposed on a snapshot of the finite difference solution of the acoustic wave equation at time one second caused by a source at time zero. The source is a ricker wavelet that starts at zero time with the peak value slightly delayed. As a result, the curves act as an envelope for the energy propagation, which in this case, matches most of the more energetic waves. (Later in this section, we show examples in which the most energetic waves are not necessarily the fastest predicted by the eikonal solver, and as a result the eikonal solution does not provide the desired solution.)
Figure 12 shows contours of traveltime using the fast marching method in Cartesian coordinates (dashed curves) and in polar coordinates (gray curves). The black curves show traveltimes for the more accurate reference solution. The source is placed above a complicated region of the Marmousi, which causes multi-arrival traveltimes. For vertical and horizontal wave propagation, the various contour curves practically coincide. At near 45-degree angular wave propagation, the dashed curves, which are the result of the Cartesian coordinate implementation, provide quite inaccurate traveltimes (see Figure 13). A closer look, given by Figure 14, reveals how much the Cartesian coordinate implementation hampered the results. The errors are as large as three percent in this area, which is clearly unacceptable. Finer grid coverage in the Cartesian-coordinate implementation will reduce such errors, but at a higher computational cost. At the same computational cost, the polar coordinate implementation of the fast marching method provides far better results even in complex models.
Figure 15 shows the 0.5 second contour curves superimposed on a snapshot of the finite difference solution of the acoustic wave equation at time 0.5 seconds caused by a source at time zero. The source is above a complicated area of the Marmousi model and some evidence of the departure of the eikonal solution from the most energetic solutions appears, especially for waves traveling vertically. A snapshot at later time, one second, (Figure 16) shows how much the eikonal solution departed from the most energetic solution. This departure results in less than desirable traveltimes when using the eikonal solution for a process like migration. However, this is the price we pay for such a highly efficient method of calculating traveltimes.
The above examples demonstrate that for an algorithm of the same cost, the accuracy of the polar (or spherical) implementation of the fast marching eikonal solver is far superior to the Cartesian version, even in complex models. Both methods are unconditionally stable with no limitations imposed on the direction of the wavefront propagation. Also, while the eikonal solver may provide reasonable solutions in some areas of the Marmousi models, the complexity of other areas results in less than optimal results.