Offset plane-wave downward continuation

 Offset plane wave migration reduces the computational complexity of downward continuation of common-azimuth data one step further than common-azimuth migration does. It approximates the application of the full 5-D operator expressed in equation (2) with the application of several 3-D downward continuation operators. These 3-D operators are applied to common-azimuth data after plane-wave decomposition along the offset axis. The first step of the method is thus the decomposition of the common-azimuth data into offset plane waves. Each plane wave is then independently downward continued. Full downward continuation of offset plane waves could be performed by applying following operator  

$$D_{z+\Delta z}\left(\omega,{\bf {k}_{m}},{\bf {p_{h}}}\right... ...eft(\omega,{\bf {k}_{m}},{\bf {p_{h}}}\right)e^{-ik_z\Delta z},$$ (6)

where the vertical wavenumber k_z is now function of the offset plane wave parameters $p_{x_h}=k_{x_h}/\omega$and $p_{y_h}=k_{y_h}/\omega$;that is,

$$\begin{eqnarray} k_z& = & \sqrt{ \frac{\omega^2}{v^2({{\bf s},z})} - \frac{1}{... ...{x_h}\right)^2 + \left(k_{y_m}+\omega p_{y_h}\right)^2 \right]} .\end{eqnarray}$$ (7)

Strictly speaking, only in vertically layered media can each plane wave be downward continued independently. The plane waves should be allowed to mix at each depth step when lateral velocity variations occur. Therefore, the computationally attractive feature of imaging each plane wave independently also causes limitations in accuracy. These limitations are difficult to study analytically, and thus in a following section I will analyze their effects by comparing migration results below a complex overburden (eg. a salt body).

In practice, because common-azimuth data has no cross-line offset axis, the plane wave decomposition is performed only as a function of the in-line offset ray parameter p_xh, and the cross-line offset ray parameter p_yh is assumed to be zero. This assumption introduces another approximation in the migration operator, that can be studied analytically.

When p_yh is set to zero, equation (7) becomes:

$$\begin{eqnarray} \widebar{k_z}& = & \sqrt{ \frac{\omega^2}{v^2({{\bf s},z})} - ... ...{y_m}^2}{4} - \frac{1}{4} \left(k_{x_m}+\omega p_{x_h}\right)^2} .\end{eqnarray}$$ (8)

This equation is equivalent to the offset plane wave equation presented by Mosher et al. 1997.

It is easy to verify that if we assume $v({{\bf s},z})\approx v({{\bf g},z})= v({{\bf m},z})$,the dispersion relation of equation (8) can be expressed as the cascade of a zero-offset downward continuation along the cross-line direction:  

$$k_{z_{y}}= \sqrt{ \frac{\omega^2}{v^2({{\bf m},z})} - \frac{k_{y_m}^2}{4}, }$$ (9)

and prestack downward continuation along the in-line direction:  

$$\widebar{k_z}= \sqrt{ k_{z_{y}}^2 - \frac{1}{4} \left(k_{x_... ...{z_{y}}^2 - \frac{1}{4} \left(k_{x_m}+\omega p_{x_h}\right)^2}.$$ (10)

The interpretation of this decomposition is similar to the one discussed above for common-azimuth migration. A constant velocity offset plane wave migration that uses the dispersion relation of equation (8) is equivalent to a constant-velocity cross-line zero-offset migration, followed by a constant-velocity in-line prestack migration. The order between these migrations is thus reversed with respect to the correct order. We analyze the implications of this order reversal in the following section.