Efficient wave-equation migrations of common-azimuth data

The full downward continuation of 3-D prestack data can be expressed in the frequency-wavenumber domain by the following phase-shift operator  

$$D_{z+\Delta z}\left(\omega,{\bf {k}_{m}},{\bf {k}_{h}}\right... ...eft(\omega,{\bf {k}_{m}},{\bf {k}_{h}}\right)e^{-ik_z\Delta z},$$ (1)

where $\omega$ is the temporal frequency, ${\bf {k}_{m}}$ is the midpoint-wavenumber vector, ${\bf {k}_{h}}$ is the offset-wavenumber vector, and $v({{\bf s},z})$ and $v({{\bf g},z})$ are respectively the velocity at the source and receivers locations. The vertical wavenumber k_z is given by the Double Square Root (DSR),

$$\begin{eqnarray} k_z& = & \sqrt{ \frac{\omega^2}{v^2({{\bf s},z})} - \frac{1}{... ...x_m}+k_{x_h}\right)^2 + \left(k_{y_m}+k_{y_h}\right)^2 \right]} .\end{eqnarray}$$ (2)

This operator is a function of the cross-line component of the offset wavenumber k_yh, while common-azimuth data are independent of k_yh because they are different from zero only at y_h=0. Therefore, the exact full downward continuation is performed by applying a 5-D operator on a data set that is only 4-D. While accurate, this procedure is tremendously wasteful of computational efforts, because only a small subset of the 5-D wavefield contributes to the final image. The final image is formed by extracting the zero-offset cube from the downward-continued wavefield. This data extraction is equivalent to the summation of the wavefield along both offset-wavenumber axes. Most of the wavefield components destructively interfere in the imaging step. In fact, only a 4-D slice of the 5-D wavefield contributes to the image when no multipathing occurs, such as in constant velocity or in a vertically layered media. Even when multipathing occurs, most of the wavefield components destructively interfere in the imaging step.

It is therefore natural to limit the computational cost by reducing the dimensionality of the downward continuation operator from 5-D to 4-D. Both common-azimuth migration and offset plane wave migration achieve this goal, though in different ways.