Residual of three two-dimensional prediction-error filters

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   where g is the original wavefield, A_x, A_y and A_z are the individual filter operators, and is the prediction-error vector.

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.