COMMON-AZIMUTH MIGRATION

Common-azimuth migration is the second component of the common-azimuth imaging procedure. Common-azimuth migration Biondi and Palacharla (1996) is based on a downward-continuation operator derived from the full 3-D prestack downward-continuation operator. Common-azimuth data sets are recursively evaluated at increasing depth levels, starting from the common-azimuth data set recorded at the surface. Common-azimuth data has zero cross-line offset, and thus it is only four-dimensional. Consequently, the common-azimuth operator is also only four dimensional, whereas the full 3-D prestack downward continuation operator is five dimensional. This reduction in dimensionality of the continuation operator results in a substantial reduction of computational and storage requirements, though at the cost of some potential loss in accuracy.

The full prestack downward continuation operator is expressed in the frequency-wavenumber domain by the Double Square Root (DSR) dispersion relation

$$\begin{eqnarray} k_z& = & \sqrt{ \frac{\omega^2}{v({{\bf \vec s},z)}^2} - \fra... ..._{{x}}+k_{hx}\right)^2 + \left(k_{{y}}+k_{hy}\right)^2 \right]} ,\end{eqnarray}$$ (2)

where $\omega$ is the temporal frequency, k_x and k_y are the midpoint wavenumbers, and k_hx and k_hy are the offset wavenumbers; $v({{\bf \vec s},z)}$ and $v({{\bf \vec g},z)}$ are respectively the propagation velocities at the source and receiver location.

A stationary-phase approximation of the full DSR  yields the common-azimuth dispersion relation. This new dispersion relation can be expressed as the cascade of two 2-D relations. The first relation is that for 2-D prestack downward-continuation along the in-line direction,  

$$k_{z_{x}}= \sqrt{ \frac{\omega^2}{v({{\bf \vec s},z)}^2} - ... ...2} - \frac{1}{4} \left[\left(k_{x_m}+k_{x_h}\right)^2 \right]}$$ (3)

and the second relation is the one for 2-D zero-offset downward continuation along the cross-line axis,  

$$\widehat{k_z}= \sqrt{k_{z_{x}}^2 - k_{{y}}^2} .$$ (4)

The common-azimuth downward continuation operator is not as a general operator as the full DSR, and consequently it introduces some approximations. However, the components of the wavefield that contribute significantly to the final image are correctly extrapolated Biondi and Palacharla (1996).