Link between operator anti-aliasing and image anti-aliasing

The constraints on the data frequency to avoid operator aliasing and image aliasing are different in general; and both sets of constraints must be respected to assure a high-quality image. However, the two sets of constraints are equivalent in many practical situations. The analysis is simple for zero-offset or constant-offset data, where we can assume that the summation surfaces $t_{D}=t_{D}({\bf \vec\xi},x_m,y_m)$are only functions of the image point coordinates ${\bf \vec\xi}$ and of the midpoint coordinates of the data trace $\left(x_m,y_m\right)$.

The first condition for linking image aliasing to operator aliasing is that the data are not spatially aliased, and thus the operator anti-aliasing constraints are the ones expressed in equation (3), and not the constraints expressed in equation (5). Comparing the constraints for operator anti-aliasing [equation (3)] with the constraints for image anti-aliasing [equation (2)] we can easily notice that a necessary condition for them being uniformly equivalent is that the data sampling rates $\Delta x_{D}$ and $\Delta y_{D}$must be equal to the image sampling rates $\Delta x_\xi$ and $\Delta y_\xi$.The other necessary conditions are that $p^{{\rm op}}_x= p^{{\rm \xi}}_x {dt_{D}}/{d\tau_\xi}$and $p^{{\rm op}}_y= p^{{\rm \xi}}_y {dt_{D}}/{d\tau_\xi}$.These conditions are fulfilled in the important case of spatially invariant imaging operators, as it can be shown by applying the chain rule to the derivative of the summation surfaces t_D with respect to the midpoint coordinates of the data trace $\left(x_m,y_m\right)$:

$$\begin{eqnarray} p^{{\rm op}}_x=\frac{dt_{D}}{dx_m} & = & \frac{dt_{D}}{dz_\xi}\... ...y_\xi} \frac{dy_\xi}{dy_m} = \frac{dt_{D}}{dz_\xi}p^{{\rm \xi}}_y.\end{eqnarray}$$ (6)

The equalities in (6) rely on the horizontal invariance of the imaging operator, by requiring that the derivatives of the horizontal coordinates of the image point with respect to the horizontal coordinates of the data trace to be equal to one. For migration, these conditions are strictly fulfilled only in horizontally layered media, though they are approximately fulfilled when migration velocity varies smoothly. Because equalities in (6) do not require any other assumptions on the shape of the summation surfaces; the same link between operator aliasing and image aliasing exists for all spatially-invariant integral operators, such as DMO and AMO.

The distinction between operator aliasing and image aliasing can thus be safely ignored when time migrating well sampled zero-offset data Bevc and Claerbout (1992); Claerbout (1995); Lumley et al. (1994), but it ought be respected when depth migrating irregularly sampled prestack data. This distinction is also important when a priori assumptions on the dips in the data permit setting less stringent operator anti-aliasing constraints, and thus the reflectors can be imaged with high-resolution and without operator-aliasing artifacts.

An open, and more subtle, question remains regarding the operator aliasing of prestack data, that in general do not constitute a minimal data set Padhi and Holley (1997). However, also in this case the constraints to avoid image aliasing [equation (2)] must be respected to produce high-quality and interpretable images.