The three two-dimensional one-column prediction-error filters shown in Figure 39 predict and remove a plane-wave volume (stacked pancake model). Any other image then a plane-wave volume leads to nonzero prediction error in at least one of the three output images. In contrast to a three-dimensional vector, the three filters will not predict and remove a volume of random constant amplitude lines (spaghetti model). In that sense the three filters are a restriction of the general three-dimensional filter. Symbolically, the three fold prediction-error filtering amounts to

(15) |

The combination of the three filters
generalizes the finite difference implementation of the
cross-product operator of section *Plane wave misfit*.
When trained at an identical plane-wave volume,
the prediction-error filters and
the finite difference approximations of the crossproduct operator
are essentially identical.
When encountering two competing plane-wave components
separated by a fault,
the cross-product operator estimates an average dip
that leads to a large residual of the crossproduct everywhere.
Similarly,
the three prediction-error filters adjust their coefficients to reject
both plane waves. Since the plane waves are not superposed, however,
the prediction error is significant everywhere.

3/8/1999