SPECULATIONS ON THE USE OF VARIOUS FILTER SHAPES

Here we propose a number of other possible filter shapes and their corresponding applications.

Claerbout's LOMOPLAN filter might be replaced with an a₁₁ constrained version of the filter as shown in Filter (10) if no prediction is needed along the time axis.  

$$\begin{array} {cc} \cdot & a_{12} \\ \cdot & a_{22} \\ 1 & a_{32} \\ \cdot & a_{42} \\ \cdot & a_{52} \end{array}$$ (10)

The action of the process is unlikely to change much if a time deconvolution is done first, but the number of coefficients to be calculated would be reduced somewhat.

Where Claerbout's steep-dip deconvolution does both the traditional single trace deconvolution in time plus the removal of steeply dipping events with a single process, I propose a separate removal of the undesired dipping events. A prediction process might use a filter such as shown here  

$$\begin{array} {ccccc} \cdot & \cdot & \cdot & \cdot & a_{15... ...& a_{45} \\ \cdot & \cdot & \cdot & \cdot & a_{55} \end{array}$$ (11)

to predict events within a desired range of dips.

Another approach, that of predicting and removing undesired events might be to apply a filter shaped like that of Filter (12).  

$$\begin{array} {ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a... ...\cdot \\ \cdot & a_{92} & a_{93} & a_{94} & a_{95} \end{array}$$ (12)

This filter shape is likely to produce results similar to the steep-dip deconvolution technique, since the lower four rows of the filter could be understood to eliminate events predicted by the upper-right portion of the steep-dip deconvolution filter.

Another method of modifying the prediction quality is to introduce a gap between the output point of the filter and the free coefficients. For instance, Filter (11) might be modified to appear as  

$$\begin{array} {ccccccc} \cdot & \cdot & \cdot & \cdot & \cd... ...t & \cdot & \cdot & \cdot & \cdot & \cdot & a_{55} \end{array}.$$ (13)

This gap would prevent the nearby samples from contributing to the output, forcing the prediction to be done with events that were coherent over a longer distance. While I (Abma) have tried a few tests using small gaps in the filter in the lateral prediction problem, I found only small effects on the prediction.

To remove events with a limited velocity range, filters such as

$\begin{array} {ccccc} \cdot & \cdot & \cdot & \cdot & a_{15} \\ \cdot & \cdo... ... & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & a_{95} \end{array}$,$\begin{array} {ccccc} \cdot & \cdot & \cdot & \cdot & a_{15} \\ \cdot & \cdo... ...& \cdot & \cdot & a_{85} \\ \cdot & \cdot & \cdot & \cdot & a_{95} \end{array}$,$\begin{array} {ccccc} \cdot & \cdot & \cdot & \cdot & a_{15} \\ \cdot & \cdo... ... \cdot & a_{74} & a_{85} \\ \cdot & \cdot & \cdot & \cdot & a_{95} \end{array}$, or $\begin{array} {ccccc} \cdot & \cdot & \cdot & \cdot & a_{15} \\ \cdot & \cdo... ...& \cdot & a_{84} & \cdot \\ \cdot & \cdot & \cdot & \cdot & a_{95} \end{array}$might be used. The form of the filter would depend on the known velocity range of the undesired events and how well the position of the coefficients fit the events. If the a15 and a95 coefficients fit the velocity of the undesired events perfectly, only these two coefficients would be needed. In general, since the grid on which the coefficients sit would not fit a given velocity, the second filter with a15, a25, a85, and a95 free would be needed. If more than one event within the velocity range exists, or if events dipping both directions with the same velocity exist within a window, more columns with free coefficients will be needed. The last of the previous filters may be needed in a practical case.

For the lateral prediction problem, the optimum filter shape is unclear. Ideally, a small filter with few coefficients is desired. As discussed above, a tall narrow filter such as Filter (8) would predict a single linear event. Adding more columns, such as in Filter (1) would allow more linear vents to be predicted. The number of time coefficients might be determined by the maximum dip expected. The safest filter, one with a large number of columns and rows, might be too expensive to calculate practically.

One might assume that a filter which is symmetric about the time and space axes would not require two passes of prediction. However, this is not the case. First, the calculation of Filter (14) shown below should be symmetric about the central coefficient, so calculating the other coefficients might be considered a waste of time.  

$$\begin{array} {ccccccccc} a_{11} & \cdot & \cdot & \cdot & ... ...t & \cdot & \cdot & \cdot & \cdot & \cdot & a_{59} \end{array}$$ (14)

A second, more important issue is that Filter (14) looks like an interpolation-error filter. In Claerbout's1992a discussion of one-dimensional interpolation-error filters, the output is shown to have spectrum that is the inverse of the input. This is generally undesirable. Therefore, the filters we generate will be one-sided in either time or in space.

As seen in the examples above in Filters (6) and  (7), a filter with no free coefficients in the output column appears to have a forward prediction filter that is simply the reverse of the reverse prediction. This is not true with the filter with free coefficients in the output column.

While calculating small filters might be efficient, calculating larger filters involves n³ operations, where n is the number of coefficients to be calculated. Breaking large filters into smaller ones might reduce the computation significantly.