** Next:** artifacts caused by sparsely
** Up:** Tang: Imaging in the
** Previous:** introduction

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):

| |
(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,
*h*_{x}, *h*_{y} 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:
| |
(2) |

where is the reflection angle, *k*_{hx} is the offset wavenumber, and *k*_{z} 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
| |
(3) |

where is the reflection azimuth, is the absolute value of the offset wavenumber, *k*_{mx} and *k*_{my}
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