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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).

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Stanford Exploration Project

1/13/1998