APPLICATION IN KIRCHHOFF METHODS

My main use for the method has been in Kirchhoff migration and modeling. Although ray tracing in itself may be faster than finite-difference calculations, the interpolation of traveltimes along rays onto the grid can be quite cumbersome and expensive: special care is needed in handling crossing rays and shadow zones. The finite-difference method does not handle multi-valued traveltime functions, however, and ray tracing is still necessary if one wants to correctly include triplications in the migration.

The Green's functions that are used in Kirchhoff methods describe traveltimes between depth points in the model and survey points on the surface. One can precompute these traveltime maps with the described method for densely spaced surface points, store them on disk (if enough disk space is available), and then read them when needed in the Kirchhoff summation. If a map for a certain surface point is not on disk, a simple linear interpolation of traveltimes in two neighboring maps is accurate enough for most purposes.

The amplitudes 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. Ray propagation angles are readily available from the gradient vector of the traveltime field (see previous section), and can be used accordingly in amplitude calculations.