Filter shape considerations

Rather than using two-dimensional deconvolution for a prediction-error filter as in the LOMOPLAN case above, Ray Abma 1993 used two-dimensional deconvolution to predict and to keep events that were laterally continuous. The process is used to eliminate noise from a seismic section in a manner similar to FX-deconCanales (1984),Gulunay (1986). Two-dimensional deconvolution is more constrained than FX-decon since two-dimensional deconvolution is done in the time domain with a limited number of filter coefficients, while FX-decon is done separately within each frequency, making the number of filter coefficients fairly large.

The first filter we used for lateral prediction was simply Burg's two-dimensional filter using several columns to predict multiple linear events. This filter appeared as:  

$$\begin{array} {ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a... ...{45} \\ \cdot & a_{52} & a_{53} & a_{54} & a_{55} \end{array}.$$ (1)

Here and for the other filters displayed, the vertical axis is the time axis, the horizontal axis is the space axis. For clarity, zeros are indicated by ``''s.

The filter coefficients are calculated using a conjugate-gradient technique described in Claerbout 1992a . Generally, three coefficients in the time direction were found to be sufficient, but more may be needed if steep dips are found.

The results of the lateral prediction process using this filter was usually comparable to the results of FX-decon, but we found one case where FX-decon was producing a much better result than the two-dimensional deconvolution process. This case involved a global image provided to us by Shearer1991. To improve our result, Jon Claerbout suggested that the modified filter shown in Filter (2) would produce better results than Filter (1).  

$$\begin{array} {ccccc} \cdot & a_{12} & a_{13} & a_{14} & a_... ...{45} \\ \cdot & a_{52} & a_{53} & a_{54} & a_{55} \end{array}.$$ (2)

The difference between Filters (1) and  (2) is that the first column of Filter (2) does not contain any free coefficients. This would prevent any prediction done along the time axis in the traces. The predictions within a trace are probably so great that the lateral predictions become unimportant. For example, a noisy trace would be well predicted if the prediction along the time axis was good, even if no prediction could be done laterally. The results of applying these two filters to the problem case are compared later and may be seen in Figures 3 and  4. The calculated filters are show below in Filters (4) to  (7) and Figures 1 and  2 as a seismic plot.