Finite difference and wave number migration
Journal name: Geophysical Prospecting
Issue: Vol 26, No 4, December 1978 pp. 773 - 789
Info: Article, PDF ( 1.72Mb )
Finite difference migration has been developed and popularized by J. F. Claerbout of Stanford University and is now widely used in seismic processing. For most sections finite difference migration gives results comparable to those obtained by conventional Kirchhoff migration and, where events are not dipping too much, a cleaner appearance is often apparent. However, there are two practical limitations to the method, and these occur in regions of very steep dip and where there is a large variation of the velocity in the lateral direction.
It is possible to develop successively more accurate equations to deal with the steep dip problem, but above third order these schemes become prohibitively expensive to implement. The finite difference method itself introduces errors and so imposes further limitations on the angle of dip. For the effective treatment of steeply dipping beds there appears to be no method available in the time domain which does not suffer from dispersion inaccuracies. However, by developing wavenumber migration, an exact one-way wave equation can be used, and this eliminates any error except that caused by finite sampling.
The other difficulty with wave migration is the correct migration in regions with lateral velocity variation. A number of approaches are possible of which three are discussed here. The first uses an exact theory, the second is based on the deviation from a depth stratified model, and the third uses a transformation to a depth co-ordinate system. All methods are discussed with their advantages and limitations. Finally, some examples are shown of wave migration applied to synthetic and real data.