# Summation strategies

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

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^{[1]}^{[2]}, - 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^{[3]}, - 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^{[4]}, 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 ^{[5]} ^{[6]}.

## References

- ↑ 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.

## See also

- 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

## External links