Remarks

We consider expression (A-1) the lowest order finite difference filter that destroys exclusively planes. The expression (A-1) and the proof above can be reformulated so that it includes a normalized vector ${\bf p}$ and only two instead of three components.

A zero output of neither a single component equation (A-1) nor a single 3-D filter suffice to conclude that the function g is a volume of parallel planes. For example, the first component $(p_y \partial_z - p_z \partial_y)$ applied to a volume of parallel lines h(x, p_y y + p_z z) would yield zero, but h is not a volume of parallel planes. In contrast, a 3-D volume filter removes events other than parallel planes. Similarly, a 3-D filter removes the volume of parallel lines h(x, p_y y + p_z z).

Nevertheless, our filter formulation might be more restrictive than necessary: essentially, we do not care how the filter operates on events that it does not encounter in seismic subsurface images.