Digression: Accuracy of Dip Filters

Given a pure plane wave section, i.e., a wavefield where all events have linear moveout, designing a discrete multichannel filter to annihilate events with a given dip seems a trivial task. In fact, it is quite a tricky task; an exercise in interpolation. For many applications, accuracy considerations make the choice of interpolation algorithm critical. Accuracy here means localization of the filter's dip spectrum -- ideally the filter should annihilate only the desired dip, or a narrow range of dips.

 

steering Figure 3 Steering filter schematic. Given a plane wave with dip p, choose the ai to best annihilate the plane wave.

The problem is illustrated in Figure 3. Given a plane wave with dip p, we must set the filter coefficients a_i to best annihilate the plane wave. Achieving good dip spectrum localization implies a filter with many coefficients, by the uncertainly principle Bracewell (1986). If computational cost was not an issue, the best choice would be a sinc function with as many coefficients as time samples. Realistically, however, a compromise must be found between pure sinc and simple linear interpolation. The reader is referred to Fomel's 2000 paper, which discusses these issues much more thoroughly. The model of Figure 1 was computed using an 8-point tapered sinc function. Figure 4 compares the result of using, for the same task, dip filters computed via four different interpolation schemes: 8-point tapered sinc, 6-point local Lagrange, 4-point cubic convolution, and simple 2-point linear interpolation. As expected, we see that the more accurate interpolation schemes lead to increased spatial coherency in the model panel. Clapp (2000) has been successful in using as few as 3 coefficients in steering filters for regularizating tomography problems, so we see that the needed amount of steering filter accuracy is a problem-dependent parameter.

 

interp-comp
Figure 4 Interpolation schemes compared. Deconvolution of random image with labeled steering filters.