PE FILTERS VERSUS DIFFERENTIAL OPERATORS

We could implement the filters in the cross product expression (4) as finite difference operators. Finite difference operators are susceptible to frequency dispersion and cannot deal with aliased data. Alternatively, we could implement Prediction Error (PE) filters that are related to these finite difference filters, but that overcome aliasing and frequency dispersionClaerbout (1994).

A planar or 2-D PE filter,  

$$p_z \partial_x - p_x \partial_z \sim \left[ \begin{array} ... ...\\ g &b \\ 1 &c \\ \cdot &d \\ \cdot &e \end{array}\right]$$ (6)

corresponds to a finite-difference operator $n_z \partial_x - n_x \partial_z$.The vertical axis is z, the horizontal axis is x and the ``''s are zeros. Imagine all the coefficients but d=-1 and the given 1 have vanished. Such a filter would annihilate the appropriately sloping plane wave. Slopes that are not exact integers are also approximately extinguishable because the adjustable filter coefficients can interpolate in time.

Consequently, an alternative (but related) method for analyzing the dip of an image volume is to replace the finite difference operators implied in equation (1) by PE filters. We then find the PE filter coefficients that minimize the output of the filters convolved with the image volume. The optimal PE filter coefficients implicitly contain the dip information of the image volume, as we can easily show, by displaying the filter's amplitude spectrum.

The shape of a 2-D PE filter is constrained by its objective: the filter's height can be changed, but its width has to remain two columns wide. Two columns can destroy a single plane, which corresponds to our model of a single plane reflector dip within a sub-cube. Any additional column could destroy an additional plane and, consequently, could remove more energy than the single plane reflector model would justify. The length of the 2-D PE filter has to be larger than the maximal step-out (dip) of the input data. The coefficients in the first column of the PE filter (6) accomplish a deconvolution that is not implied in the original cross product formulation. However, these coefficients are necessary to ensure that the filter output is white Claerbout (1994). If the image does not resemble the plane reflector model, but, for example, the superposition of two plane layered volumes, the filter will probably try to remove both partially, rather than exclusively removing the dominant contribution.

By forcing a 1-D, two-term filter to predict both backward and forward, we avoid perfect predictability on growing 1-D exponentialsClaerbout (1976). To prevent predictions of exponentially growing planes, we require the 2-D PE filter to extinguish two copies of the data: the original copy and a second copy with both x and z reversed. Dips on the reversed copy are the same as on the original, but amplitude growth on one is decay on the other.