THREE-DIMENSIONAL EXTENSION

We seek the general form of a filter to apply to a data cube $f( x,y, \tau)$that will annihilate a plane function $f(x,y,\tau )= g(\tau -p_x x -p_y y)$ where g() is an arbitrary one-dimensional function. In practice we usually need to allow the filter to vary with $(x,y,\tau )$,but first we investigate the constant dip case. For warmup we consider the case when $(p_x,p_y)=\bold 0$and later we consider nonzero values of (p_x,p_y). Clearly we can annihilate this plane with either the operator $\partial_x \equiv \partial / \partial x$or the operator $\partial_y \equiv \partial / \partial y$.The problem with the operator $\partial_x$is that it annihilates not only the vertical plane function $f(\tau )$but it annihilates other things such as functions of the form $f(\tau ,y)$,an example of which is the family of planes, $f(\tau -p_y y)$.A way to annihilate one plane and only one plane is to choose a filter with two outputs, namely the filter $(\partial_x,\partial_y)$.For this two-component output to be annihilated in a regression, both components must be annihilated. This is the lowest order filter that will do the required job. We can make a higher order filter that annihilates the plane by multiplying the column vector filter $(\partial_x,\partial_y)'$by its conjugate getting the negative of the Laplacian $\partial_{xx} +\partial_{yy}$.An apparent advantage of the Laplacian filter is that its output is a single volume. In time we will learn the practical distinction between these two monoplane annihilators. For now, I am betting on the two-output lower order filter.