Traveltime maps

To implement prestack Kirchhoff migration, we need first to construct traveltime maps (the Greens function) used to define the migration summation trajectories. Ray tracing is used for this task, and in particular ($x-\tau$)-domain ray tracing. However, ray tracing only provides traveltime information along the ray paths. As a result, cubic interpolation is used to place the traveltime information on a regular grid.

 

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Figure 3 Rays (solid lines) and wavefront (dashed lines) calculated using the $(x-\tau)$-domain ray tracing for waves emanating from a source at surface location 4000 m. In the background the smooth version of the Marmousi velocity model used for the traveltime calculation is displayed.

Figure 3 shows rays emanating from a source at a surface location 4000 m. Traveltime along these rays are used to create a traveltime map with time values given by the dashed contour curves. These dashed curves, thus represent the wavefronts of the propagating waves. In ($x-\tau$)-domain, the wavefronts do not necessarily have to be perpendicular to the rays, even for isotropic media. Faster wavefronts are in agreement with high velocity zones, displayed in background. Clearly, from Figure 3 we can see regions of minimal ray coverage, which makes the task of interpolating of the traveltime information to these regions a little more ambiguous. However, such regions also have low energy and thus contribute little to the image. These areas of low ray coverage exist mainly in areas surrounded by cusps. For the purpose of migration, traveltime information in regions of large lateral distance from the source, compared to depth (or vertical time in this case), are often discarded. This typically includes the area directly under surface location 6000 m, which seemingly has suspect traveltime contours.

Traveltimes can also be calculated directly from finite difference solution of the eikonal equation. Specifically, I use the fast marching method in polar coordinates Alkhalifah and Fomel (1997) to generate traveltimes in the (x-z)-domain. To compare these traveltimes with the ones obtained from ($x-\tau$)-domain raytracing, I convert the raytracing traveltime maps from time to depth.

 

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Figure 4 Contours of traveltime plotted in depth calculated using $(x-\tau)$-domain ray tracing followed by a time-to-depth conversion (solid lines), and the eikonal solver (dashed curve), again for waves emanating from a source at surface location 4000 m. The smoothed Marmousi model is again displayed in the background.

Figure 4 shows contour lines from both traveltime maps, with the velocity field displayed in the background. Clearly, the traveltime contours obtained using ($x-\tau$)-domain raytracing agree well with those obtained via the eikonal solver in areas around the source. Differences occur in areas dominated by traveltime triplications, as can be deduced from Figure 3. The finite difference solution of the eikonal equation provides only the fastest energy solution, not necessarily the most useful (energetic), while the raytracing equations are capable of producing all ray-theoretical solutions, and thus one can choose the most energetic solution.

 

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Figure 5 The absolute difference between the two traveltime maps shown in Figure 4. This image is clipped at a maximum error of 10 ms.

The absolute difference between the two traveltime maps is shown in Figure 5. Errors of 10 ms and beyond are given the color black. The areas of clear differences between the two traveltime solutions coincide with areas of multi-arrival traveltimes (triplications), as can be seen in Figure 3. The eikonal solution provides the fastest arrival; ray tracing via interpolation, provides the most energetic solution.