Kirchhoff migration and modeling

The Green's functions used in Kirchhoff methods describe traveltimes between depth points in the model and survey points on the surface. One can use the described method to precompute these traveltime maps for source points that are densely spaced on the surface, and then use them when needed in the Kirchhoff summation; see for example Gray (1986), and Reshef and Kosloff (1986). If a map for a certain surface point is not available, a simple linear interpolation of traveltimes in two neighboring maps is accurate enough for most purposes.

The finite-difference method does not compute multi-valued traveltime functions, but geometry-related triplications in reflection events are still handled correctly (see Figure 10). The only problem arises when, for example, refracted waves travel faster than direct waves; this phenomenon normally occurs only in the shallow, wide-angle part of the data.

The amplitude terms in the Green's functions can be approximately estimated from the traveltime function: the geometrical spreading and obliquity terms are approximate functions of traveltime or its gradient (see also Vidale, 1989).