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ADCIGs can be extracted either before applying an imaging condition Mosher and Foster (2000); Prucha and Symes (1999); Soubaras (2003); Xie and Wu (2002)
or afterward Biondi and Symes (2004); Sava and Fomel (2003). The advantage of extracting
the angle gathers after the imaging step is that it is a model-space processing, which offers more versatility and generally more efficiency. The
same transformation can be used for images produced by source-receiver migration Sava and Fomel (2003),
shot-profile migration Rickett and Sava (2002)
and reverse time migration Biondi and Shan (2002).
There are basically two steps to extract the angle gathers after imaging: First, compute the SODCIGs. Second, transform the SODCIGs
into ADCIGs. For source-receiver migration, the SODCIGs are immediately available after downward continuation of the wavefields; for
shot-profile migration, a multi-offset imaging condition should be applied to get the SODCIGs Rickett and Sava (2002):
| ![\begin{displaymath}
I(x, y, h_x, h_y) = \sum_\omega D^*(x-h_x, y-h_y, \omega)U(x+h_x, y+h_y, \omega),\end{displaymath}](img1.gif) |
(1) |
where I is the image in the subsurface-offset domain, D is the source wavefield, * means the complex conjugate,
U is the receiver wavefield, x, y are the components of midpoint,
hx, hy are the components of subsurface half offset, and
is frequency.
Sava and Fomel (2003) derived the following radial-trace transformation in the Fourier domain to transform the SODCIGs into ADCIGs in 2-D:
| ![\begin{displaymath}
\tan \gamma = -\frac{k_{h_x}}{k_z} = -\frac{\partial z}{\partial h_x},\end{displaymath}](img3.gif) |
(2) |
where
is the reflection angle, khx is the offset wavenumber, and kz is the depth wavenumber.
The transformation is independent of geological dip in 2-D, but the 3-D formulation must be corrected for a crossline dip component.
Tisserant and Biondi (2003) show that we can make this 3-D correction by re-writing the angle-gather transformation as
| ![\begin{displaymath}
\tan \gamma = - \frac{\left\vert{\bf k_h}\right\vert}{k_z}\f...
...m_x}}{k_z}\sin\beta + \frac{k_{m_y}}{k_z} \cos\beta\right)^2}},\end{displaymath}](img5.gif) |
(3) |
where
is the reflection azimuth,
is the absolute value of the offset wavenumber, kmx and kmy
are the components of the midpoint wavenumber.
Next: artifacts caused by sparsely
Up: Tang: Imaging in the
Previous: introduction
Stanford Exploration Project
5/6/2007