extracting angle-domain common-image gathers

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):  

$$I(x, y, h_x, h_y) = \sum_\omega D^*(x-h_x, y-h_y, \omega)U(x+h_x, y+h_y, \omega),$$ (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 $\omega$ 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:  

$$\tan \gamma = -\frac{k_{h_x}}{k_z} = -\frac{\partial z}{\partial h_x},$$ (2)

where $\gamma$ 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  

$$\tan \gamma = - \frac{\left\vert{\bf k_h}\right\vert}{k_z}\f... ...m_x}}{k_z}\sin\beta + \frac{k_{m_y}}{k_z} \cos\beta\right)^2}},$$ (3)

where $\beta$ is the reflection azimuth, $\left\vert{\bf k_h}\right\vert$ is the absolute value of the offset wavenumber, k_mx and k_my are the components of the midpoint wavenumber.