REFERENCES

Biondi, B., 2001, Narrow-azimuth migration: Analysis and tests in vertically layered media: SEP-108, 105-118.

Biondi, B., 2003, Narrow-azimuth migration of marine streamer data: SEP-113, 107-120.

Tisserant, T., and Biondi, B., 2003, Wavefield-continuation angle-domain common-image gathers in 3-D: SEP-113, 211-220.

Vaillant, L., and Biondi, B., 2000, Accuracy of common-azimuth migration approximations: SEP-103, 157-168.

A This appendix derives the expressions for the weights to be applied to the ADCIGs before averaging over azimuths. These weights are based on the jacobian of the transformation into angle domain. The first step is therefore to find the expressions for evaluating this jacobian.

The starting point for computing the jacobian is the transformation into angle domain. Tisserant and Biondi (2003) showed that 3-D ADCIGs can be computed according to the following mappings:

$$\begin{eqnarray} k_{x_h}' &=& -\tan \gamma\sqrt{k_{y_m}'^2 + k_z^2}, \\ k_{y_h}' &=& -\frac{k_{y_m}'k_{x_m}'k_{x_h}'}{k_z^2 + k_{y_m}'^2};\end{eqnarray}$$ (3) (4)

where the primes on the wavenumber indicate the rotation of the coordinate axis by $\phi$ according to the following relationships:

$$\begin{eqnarray} k_{x_m}' &=& \cos \phi k_{x_m}- \sin \phi k_{y_m}, \\ k_{y_m}' &=& \sin \phi k_{x_m}+ \cos \phi k_{y_m},\end{eqnarray}$$ (6) (7)

and similarly

$$\begin{eqnarray} k_{x_h}' &=& \cos \phi k_{x_h}- \sin \phi k_{y_h}, \\ k_{y_h}' &=& \sin \phi k_{x_h}+ \cos \phi k_{y_h}.\end{eqnarray}$$ (8) (9)

We need to compute the partial derivatives of the offset wavenumbers at constant aperture angle $\gamma$.Therefore, we start from rewriting the coplanarity condition in equation (4) in terms of reflections angles in the rotated coordinate system: the aperture angle $\gamma'$ , the in-line dip angle $\alpha'_x$,and the cross-line dip angle $\alpha'_y$.The following relationships link the wavenumber in the image domain to these angles  

$$\tan \alpha'_x= \frac{k_{x_m}'}{k_z},$$ (10)

 

$$\tan \alpha'_y= \frac{k_{y_m}'}{k_z},$$ (11)

and  

$$\tan \gamma'= - \cos \alpha'_y\frac{k_{x_h}'}{k_z}.$$ (12)

Then equation (4) becomes:  

$$k_{y_h}'=k_z\tan \gamma'\tan \alpha'_x\sin \alpha'_y,$$ (13)

and equation (3) becomes:  

$$k_{x_h}'= - k_z\frac{\tan \gamma'}{\cos \alpha'_y}.$$ (14)

 

• Evaluation of ${\partial k_{x_h}'}/{\partial \phi}$and ${\partial k_{y_h}'}/{\partial \phi}$
• Evaluation of ${\partial k_{x_h}}/{\partial \phi}$and ${\partial k_{y_h}}/{\partial \phi}$
• Computation of scaling factor $W_{\gamma,\phi}$