LOMOPLAN DEFINITION IN 2-D

First we review two-dimensional deconvolution filters. The dip and deconvolution character of such filters is described in considerable detail in chapter 8 of my recent book Claerbout (1992a). The coefficients of deconvolution filters are chosen to produce minimum power out of a given input. The coefficients in a 2-D LOMOPLAN filter fit the pattern:  

$$\begin{array} {cc} c &d \\ b &e \\ a &f \\ 1 &g \\ \cdot &h \\ \cdot &i \\ \cdot &j \end{array}$$ (1)

The vertical axis is time. The horizontal axis is space. The ``''s are zeros. The filter can be lengthened in time but not in space. The choice of exactly two columns is a choice to have an analytic form that can exactly destroy a single plane, but cannot destroy two. With two signals that are statistically independent, the filter reduces to the well-known prediction-error filter in the left column and zeros in the right column. If the filter coefficients were extended in both directions on t and to the right on x, it would flatten the two-dimensional spectrum of the input 1992a. Imagine all the coefficients in (1) vanished but d=-1 and the given 1. Such a filter would annihilate an appropriately sloping plane wave.

The coefficients (a,b,c) in the filter (1) have an interesting but not fully understood role. To begin with, their presence is suggested by the representation of 2-D spectra by the inverse of their autoregression filters 1992a. The presence of these coefficients will cause the output to be white on the time axis. However, if we were to omit these coefficients, or constrain them to zero, the (d,e,f,g,h,i,j) coefficients would probably attempt to adjust themselves toward making the output temporally white (because the earth tends to have flat dip). Thus, it seems that our goals for studying the spatial aspects of the data become bound together with the temporal aspects. Since data is always oversampled in time, and we are not interested in whitening it in time, I needed to design something like a gapped filter, or do postfiltering to give the output a more normal color. Typically, I included the (a,b,c) coefficients and then applied a triangular smoothing filter on the time axis after LOMOPLAN.