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 and must be equal to the image sampling rates and .The other necessary conditions are that and .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 tD with respect to the midpoint coordinates of the data trace :
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.