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For the case of offset ray-parameter gathers, we can rewrite the
DSR equation (22) as
| |
|
| (26) |
In this case, the imaging Jacobian becomes
| |
|
| (27) |
which can be re-arranged as:
| |
(28) |
For an arbitrary 2-D reflection geometry (Figure 1),
we can write Equation (28) as
| |
(29) |
For flat reflectors, defined by and ,
the Jacobian takes the simple form
| |
(30) |
which is equivalent to the weighting factor introduced by
Wapenaar et al. (1999).
For the case of flat reflectors, we also have
| |
(31) |
which explains the opposite behavior of the uncorrected migration
amplitudes for reflection-angle gathers (Figure 10)
and offset ray-parameter gathers (Figure 11).
After we apply the Jacobian weights, we obtain the corrected
angle-gathers shown in Figures 12 and 13.
As expected, the amplitudes are constant for the entire usable
angular range.
Next: Jacobian for Common-azimuth migration
Up: Transformation Jacobians
Previous: Angle-gathers
Stanford Exploration Project
4/16/2001