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For reflectionangle gathers, the dispersion relation used to
downward continue the wavefield is given by the DSR equation:
 

 (22) 
where s is the local slowness, and are respectively the vertical wavenumber for the source and receiver
components, and and are respectively the midpoint and
offset wavenumbers.
ghmig
Figure 10 Reflectionangle gather implemented using
Equation (20). We do not use the correct weighting of
the transformation Jacobian, therefore the amplitudes are distorted.
gpmig
Figure 11 Offset rayparameter gather implemented using
Equation (21). We do not use the correct weighting of
the transformation Jacobian, therefore the amplitudes are distorted.
The Jacobian for this transformation is thus the common
prestack Stolt migration Jacobian:
 

 (23) 
As mentioned in the preceding sections, in the case of variable
velocity media, those quantities are evaluated at the reflector location.
For an arbitrary 2D reflector geometry (Figure 1),
we can rewrite Equation (23) as
 
(24) 
where is the structural dip angle, and is the reflection
angle.
For flat reflectors, defined by and ,
the Jacobian takes the simple form
 
(25) 
chmig
Figure 12 Reflectionangle gather computed in the
image space. The weighting factors restore correct amplitudes.
Compare with the theoretical response in Figure 10.
cpmig
Figure 13 Offset rayparameter gather computed in the
data space. The weighting factors restore correct amplitudes.
Next: Rayparameter gathers
Up: Transformation Jacobians
Previous: Transformation Jacobians
Stanford Exploration Project
4/30/2001