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| Theory of 3-D angle gathers in wave-equation seismic imaging | |
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Common-azimuth migration (Biondi and Palacharla, 1996) is a downward continuation
imaging method tailored for narrow-azimuth streamer surveys that can be
transformed to a single common azimuth with the help of azimuth moveout
(Biondi et al., 1998) Employing the common-azimuth approximation, one
assumes the reflection plane stays confined in the acquisition azimuth.
Although this assumption is strictly valid only in the case of constant
velocity (Vaillant and Biondi, 2000), the modest azimuth variation in
realistic situations justifies the use of the method (Biondi, 2003).
To restrict equations of the previous section to the common-azimuth
approximation, it is sufficient to set the cross-line offset
to zero
assuming the
coordinate is oriented along the acquisition azimuth. In
particular, from equations (8-9), we obtain
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(19) |
With the help of equations (6),
(7), and (10), equation (21)
transforms to the form
suggested by Biondi and Palacharla (1996). Combining equations (6),
(7), (10), and (20) and transforming to the
frequency-wavenumber domain, we obtain the common-azimuth dispersion
relationship
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(23) |
which shows that, under the common-azimuth approximation and in a laterally
homogeneous medium, 3-D seismic migration amounts to a cascade of a
2-D prestack migrations in the in-line direction and a 2-D zero-offset
migration in the cross-line direction (Canning and Gardner, 1996).
Under the common-azimuth approximation, the angle-dependent
relationship (13) takes the form
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(24) |
which is identical to the 2-D equation (14). This proves that
under this approximation, one can perform the structural correction
independently for each cross-line wavenumber.
The post-imaging equation (16) transforms to the equation
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(25) |
obtained previously by Biondi et al. (2003). In the absence of cross-line
structural dips (
), it is equivalent to the 2-D
equation (18).
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| Theory of 3-D angle gathers in wave-equation seismic imaging | |
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Next: Algorithm I: Angle gathers
Up: Fomel: 3-D angle gathers
Previous: Traveltime derivatives and dispersion
2013-07-26