Next: Real Data
Up: Sava and Biondi: Amplitudepreserved
Previous: Jacobian for Commonazimuth migration
The transformation Jacobians, given by Equation (23)
for reflectionangle gathers and by Equation (28)
for offset rayparameter gathers, are strictly speaking
valid only for constant velocity media.
In the case of variable velocity, downward continuation is
implemented in the mixt and domain. The dispersion
relation is approximated using an expansion of the DSR equation.
In one of the most usual forms, the dispersion relation can be
written as
 

 
 (39) 
Equations (39) give the effective dispersion
relation for which we need to implement the Jacobian weighting.
Appendix A outlines the derivation for the expresion of the
Jacobian in variable velocity media. For the case of migration
with output in offset, the Jacobian expression is:
 
(40) 
where and are respectively the slowness perturbations
at the source and receiver. In constant velocity media,
and , Equation (40) obviously
takes the same form as Equation (23) we derived earlier
for the case of constant velocity.
For the expressions of all Jacobians, we need to
compute the quantity which may lead
to numerical instability when k_{z} approaches zero.
A simple way so stabilize the Jacobian, which we have used for
the current examples, is to add a small positive quantity to
the denominator, and compute .
Another approach, which we have not tested yet,
but which appears to have the potential to be more accurate, is
to write
 

 (41) 
For small values of we can
compute using the following Taylor series expansion:
 
(42) 
This expansion is very similar to the transformation used by
Huang et al. (1999) for the extended Local Born Fourier migration
method, or by Sava and Biondi (2000) in waveequation MVA.
This interesting possibility, however, awaits future research.
Next: Real Data
Up: Sava and Biondi: Amplitudepreserved
Previous: Jacobian for Commonazimuth migration
Stanford Exploration Project
4/30/2001