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| Theory of 3-D angle gathers in wave-equation seismic imaging | |
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This algorithm follows from equation (13). It consists of the
following steps, applied at each propagation depth
:
- Generate local offset gathers and transform them to the wavenumber
domain. In the double-square-root migration, the local offset wavenumbers
are immediately available. In the shot gather migration, local offsets are
generated by cross-correlation of the source and receiver wavefields
(Rickett and Sava, 2002).
- For each frequency
, transform the local offset wavenumbers
into the angle coordinates
according
to equation (13). The angle coordinates depend on velocity but
do not depend on the local structural dip. In the 2-D case, each frequency
slice is simply the
plane, and each angle coordinate
corresponds to a circle in that plane centered at the origin and described
by equation (14). Figure 3 shows an example of
a 2-D frequency slice transformed to angles.
- Accumulate contributions from all frequencies to
apply the imaging condition in time.
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angle1
Figure 3. Constant-depth constant-frequency
slice mapped to reflection angles according to the 2-D version of
Algorithm I. Zero offset wavenumber maps to zero (normal incidence)
angle. The top right corner is the evanescent region.
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This algorithm is applicable for targets localized in depth. The local
offset gathers need to be computed for all lateral locations, but
there is no need to store them in memory, because conversion to angles
happens on the fly. The algorithm outputs not angles directly, but
velocity-dependent parameters
. Alkhalifah and Fomel (2009) extend
this algorithm to transversally-isotropic media.
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| Theory of 3-D angle gathers in wave-equation seismic imaging | |
|
Next: Algorithm II: Post-migration angle
Up: Fomel: 3-D angle gathers
Previous: Common-azimuth approximation
2013-07-26