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 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 applied to a volume of parallel lines h(x, py y + pz 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, py y + pz 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.