TWO-D DECONVOLUTION FILTERING

Two-dimensional deconvolution creates a filter that minimizes the energy in a window using a least squares criteria. The filters used in the examples in this article have a form similar to the one below:  

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

The vertical axis is the time axis, and the horizontal axis is the space axis. For clarity, ``''s are used to indicate zeros.

While the size of this filter may be modified to match the desired output, generally I've found three coefficients in the time direction are sufficient unless steep dips are found. The number of coefficients in the space direction may depend on how many dips occur within the window and how much prediction is desired. The filter coefficients are calculated using a conjugate-gradient technique described in Claerbout1992a.

One advantage of the two-dimensional deconvolution approach to lateral prediction is that the shape of the filter can be modified to match the range of dips desired. As an example, the filter below might be used to predict a range of low-angle dips:  

$$\begin{array} {ccccc} \cdot & \cdot & \cdot & \cdot & a_{15... ...& a_{45} \\ \cdot & \cdot & \cdot & \cdot & a_{55} \end{array}$$ (3)

To keep the calculation and application of the filter symmetrical, the two-dimensional deconvolution examples shown in this paper have two filters calculated and applied for each window: one forward in space and one reversed. In the program used here, the filter is not extended beyond times where a full filter response is available, so the two-dimensional deconvolution results shown have the original data at the top and bottom of the section.

Two-dimensional deconvolution uses only two filters within each window, compared to FX-decon, which uses a separate filter for each frequency. While the two-dimensional deconvolution filter will have more coefficients than each filter used in the FX-decon, the FX-decon will have a different filter for each frequency. Thus, the two-dimensional deconvolution filter is much more constrained in predicting events than FX-decon. The computer time needed to apply these two processes are comparable, since the many filters calculated by FX-decon are very short and the one filter calculated by the two-dimensional deconvolution is longer. Since the time to calculate a filter is proportional to n², where n is the number of filter coefficients, the calculation times used by the two approaches are about equal.

The similarity of the results of two-dimensional deconvolution and FX-decon may be explained by the similarity of the two filters in the time domain. If the FX-decon filters are collected and transformed into the space-time domain, they will transform into a filter that appears similar to Filter (3), but having more rows. The number of rows will be the number of time samples in the window, giving the FX-decon a long effective length in time. This extended length in time allows the FX-decon to create false lineups if the time-domain representation of the FX-decon operator has coefficients of significant strength far from the output point. For a more complete comparison of the two filters see Abma and Claerbout (1993) in this volume.