False start

Curiously I find that people do not like the idea of a filter needing to have two outputs. A false start (for me too) is to guess an operator like

$$G_1 \quad=\quad p_y {\partial\ \over\partial x} + p_x {\partial\ \over\partial y} +2 p_x p_y {\partial\ \over\partial t}$$ (2)

to kill the plane $f(\tau -p_x x - p_y y)$.While it is true that G₁ kills this plane, it also kills the two planes $f_1(\tau -2p_y y)$ and f₂(p_x x - p_y y) so it does not do the job we need, a low order filter that rejects one and only one plane.

Thus, the way to annihilate any signal on any dipping plane $f(\tau -p_x x - p_y y)$without annihilating signals on any other planes is to annihilate the two volume outputs of the vector filter $(\partial_x + p_x \partial_\tau, \, \partial_y + p_y \partial_\tau)$.Where these two volumes are already zero the migrated data fits our notions of what a migrated data set should be, so there is nothing more to be done--no parameters can be fitted to embellish the model. Where either of the two volumes is nonzero, there more than one planar event occurs on the input (or the (p_x,p_y) field was incorrectly estimated).