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*, *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.

11/12/1997