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This appendix derives the expressions for
the weights to be applied to the
ADCIGs before averaging over azimuths.
These weights are based on the jacobian of the
transformation into angle domain.
The first step is therefore to
find the expressions for evaluating this jacobian.
- Biondi, B., 2001, Narrow-azimuth migration: Analysis and tests in vertically layered media: SEP-108, 105-118.
- Biondi, B., 2003, Narrow-azimuth migration of marine streamer data: SEP-113, 107-120.
- Tisserant, T., and Biondi, B., 2003, Wavefield-continuation angle-domain common-image gathers in 3-D: SEP-113, 211-220.
- Vaillant, L., and Biondi, B., 2000, Accuracy of common-azimuth migration approximations: SEP-103, 157-168.
The starting point for computing the jacobian
is the transformation into angle domain.
Tisserant and Biondi (2003) showed
that 3-D ADCIGs can be computed according
to the following mappings:
where the primes on the wavenumber indicate the rotation
of the coordinate axis by according
to the following relationships:
We need to compute the partial derivatives of
the offset wavenumbers at constant aperture angle
we start from rewriting the coplanarity condition
in equation (4)
in terms of reflections angles in the rotated coordinate system:
the aperture angle ,
the in-line dip angle ,and the cross-line dip angle .The following relationships link the wavenumber in the image
domain to these angles