Spherical versus Cartesian

To obtain the Eikonal solution using the fast marching method in Cartesian coordinates, I will use Fomel's program. For the spherical-coordinate implementation, I will use Alkhalifah and Fomel's program. As a reference solution, I will solve the spherical version of the fast marching method at a much finer grid, since finite -difference solution should converge to the exact solution as the grid size approaches zero. The errors here are exaggerated overall because the grid size used is relatively large. Specifically, we are solving the eikonal at a grid spacing of 40 m in the x-, y-, and vertical z-directions. For spherical coordinates, we use an equivalent spacing that produces results at a comparable time. The reference solution is obtained by using a much finer grid, equivalent to 20 m spacing in the all directions.

Figure  shows the errors associated with using the fast marching method in Cartesian coordinates. Specifically, we are looking at the traveltime difference between the coarse-grid Cartesian-coordinate implementation and the fine-grid spherical coordinate implementation. The traveltime errors for such a coarse-grid application are up to 80 ms.

In practice finer grid configurations are often used to solve the eikonal equation at, of coarse, a higher price. The finer grid will result in less errors (for example 8 ms instead of 80 ms). However, the distribution of the errors and the reason for their presence (the first-order nature of the solution) still applies to finer grid implementation. Such errors are inherent in the method and, as () show, when certain conditions are met.

 

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Figure 6 Vertical (left) and horizontal (right) sections of the traveltime errors map that resulted from solving the fast marching method in the Cartesian coordinates. Also displayed are the vertical and horizontal sections of the traveltime contour maps.

 

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Figure 7 Vertical (left) and horizontal (right) sections of the traveltime errors map that resulted from solving the fast marching method in the Spherical coordinates. Also displayed are the vertical and horizontal sections of the traveltime contour maps.

Figure  shows the errors associated with using the fast marching method in spherical coordinates. Again, we are looking at the traveltime difference between the coarse-grid spherical-coordinate implementation and the fine-grid spherical coordinate implementation. The traveltime errors for such a coarse-grid application are up to 60 ms, now. Unlike the Cartesian coordinate implementation, most of the errors shown here are associated with low-curvature arrivals, like head-waves. This fact is better demonstrated in Figure , where head-waves emanating from the top of the salt are clearly the source of most the errors associated with the spherical coordinate implementation. Luckily, these head-wave arrivals are of low energy, and are generally discarded when it comes to imaging applications. These head-waves, also, mask the more important direct arrival solution. Later, I will show how suggest a method to eliminate such head-wave arrivals, and thus eliminate the source of errors for the spherical coordinate implementation.