Figure 8.5-10 shows the zero-offset wavefield responses of a point diffractor buried in media with varying degrees of complexity. The traveltime trajectories associated with the point diffractors buried in a constant-velocity medium, beneath an overburden with mild to moderate lateral velocity variations, and beneath an overburden with strong lateral velocity variations, all are single-valued. Therefore, ray tracing through such models would produce unambiguous traveltimes for Kirchhoff summation.
The zero-offset traveltime trajectory associated with the point diffractor buried beneath an overburden with severe lateral velocity variations, however, is multivalued (Figures 8.5-10d). One would have to decide as to what summation path to choose:
- The travelpath that corresponds to the first arrivals; that is, minimum-time summation trajectory  ,
- The travelpath that corresponds to the bowties that contain the most significant portion of the energy associated with the zero-offset wavefield response; that is, maximum-energy summation trajectory ,
- The travelpath that corresponds to the shortest distance between the source or receiver point at the surface and the reflection point at the subsurface; that is, minimum-distance summation trajectory , or
- The entire multivalued travelpath.
From the zero-offset wavefields associated with a diffractor buried in a medium with varying complexity shown in Figure 8.5-10, it can be inferred that the minimum-time strategy may be suitable for cases of moderate to strong lateral velocity variations (Figure 8.5-10b,c), whereas the maximum-energy strategy may be imperative for a case of a complex overburden with severe lateral velocity variations (Figure 8.5-10d). Ideally, it would be desirable not to exclude any portion of the traveltime trajectory and use a multivalued summation path. This, however, can be formidably costly and often is not needed in practice. Efficient traveltime calculation and choice of a summation path are important considerations for the 3-D prestack depth migration of large volumes of seismic data  .
- Vidale, 1988, Vidale, J., 1988, Finite-difference calculation of traveltimes: Bull. Seis. Soc. Am., 78, 2026–2076.
- Podvin and Lecomte, 1991, Podvin, P. and Lecomte, I., 1991, Finite-difference computation of traveltimes for velocity-depth models with strong velocity contrast across layer boundaries — a massively parallel approach: Geophys. J. Int., 105, 271–284.
- Nichols, 1996, Nichols, D. E., 1996, Maximum-energy traveltimes calculated in the seismic frequency band: Geophysics, 61, 253–263.
- Moser, 1991, Moser, T. J., 1991, Shortest-path calculation of seismic rays: Geophysics, 56, 59–67.
- Meshbey et al., 1993, Meshbey, V., Kosloff, D., Ragoza, Y., Meshbey, O., Egozy, U., and Cozzens, J., 1993, A method for computing traveltimes for an arbitrary velocity model: Presented at the 45th Ann. EAGE Mtg.
- Sethian and Popovici, 1999, Sethian, J. A. and Popovici, A. M., 1999, 3-D traveltime computation using the fast marching method: Geophysics, 64, 516–523.
- 3-D prestack depth migration
- Kirchhoff summation
- Calculation of traveltimes
- The eikonal equation
- Fermat’s principle
- Migration aperture
- Operator antialiasing
- 3-D common-offset depth migration