# Calculation of traveltimes

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

A direct method to compute traveltimes is ray tracing through the specified velocity-depth model. A bundle of rays emerging from a source location at the surface can be sprayed down into the earth and traced through the subsurface while accounting for ray bending caused by changes in velocity gradient and refraction at layer boundaries with velocity contrast. Reflection points along each of the raypaths are identified as the intersection points of the rays with the layer boundaries. The traveltime from the source location at the surface and a reflection point at the subsurface is then calculated by integrating the elements of distance along the raypath divided by the velocity associated with that element. By applying reciprocity, the traveltime from a receiver location at the surface to a reflection point in the subsurface can be computed in the same manner. Finally, for a given source-receiver pair at the surface and a reflection point in the subsurface, the total traveltime is computed by adding the traveltime from the source to the reflection point to the traveltime from the reflection point to the receiver.

Figure 8.5-3  Traveltime contours derived from ray tracing through a velocity-depth model that includes a salt sill represented by the white zone [1]. Compare with Figure 8.5-4 and 8.5-6.

The two-point ray tracing described above is conceptually simple, but is computationally intensive. Efficient ray tracing through complex velocity-depth models is not a trivial task. Alternatives to two-point ray tracing, however, have been developed and implemented with sufficient accuracy. Examples include paraxial ray tracing [2] [3] and Gaussian beam ray tracing [4].

Given the circumstances described above, one can identify potential problems with the ray tracing. There will not always be a raypath combination associated with a source-receiver pair and a reflection point. Depending on the recording geometry and the complexity of the velocity-depth model, there may be zones through which some rays may be missed. This complication is compounded by the computational load involved in ray tracing itself. As such, direct ray tracing is rarely used for traveltime computations required for prestack depth migration.

Figure 8.5-3 shows the traveltime contours through a velocity-depth model that includes a salt sill with velocity higher than the surrounding sediments. The discontinuities along the traveltime contours correspond to locations where ray bending is implied by ray tracing. Note, however, that the locations where ray bending takes place do not coincide with the salt sill boundary where the largest velocity contrast exists. Problems with ray tracing through a complex model as outlined above also cause physically implausable rapid variations in the traveltime contours.

An alternative to two-paint ray tracing is wavefront construction [5], which involves tracing not just one ray but a fan of rays together. As such, the medium represented by the velocity-depth model used in ray tracing is covered adequately by controlling the ray density along wavefronts [6]. In areas with low ray density, additional ray bundles may be created by paraxial rays.

## References

1. Nichols, 1996, Nichols, D. E., 1996, Maximum-energy traveltimes calculated in the seismic frequency band: Geophysics, 61, 253–263.
2. Keho and Beydoun, 1988, Keho, T. H. and Beydoun, W. B., 1988, Paraxial ray Kirchhoff migration: Geophysics, 53, 1540–1546.
3. Červený et al., 1982, Červený, V., Popov, M. M., and Pšenčík, I., 1982, Computation of wavefields in inhomogeneous media — Gaussian beam approach: Geophys. J. Roy. Astr. Soc., 70, 109–128.
4. Červený et al., 1984, Červený, V., Klimes, L., and Pšenčík, I., 1984, Paraxial ray approximation in the computation of seismic wavefields in inhomogeneous media: Geophys. J. Roy. Astr. Soc., 79, 89–104.
5. Vinje et al., 1993, Vinje, V., Iversen, E., and Gjoystdal, H., 1993, Traveltime and amplitude estimation using wavefront construction: Geophysics, 58, 1157–1166.
6. Lecomte, 1999, Lecomte, I., 1999, Local and controlled prestack depth migration in complex areas: Geophys. Prosp., 47, 799–818.