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![]() | Test case for PEF estimation with sparse data II | ![]() |
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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 ![]() ![]() |
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The problem is illustrated in Figure 3. Given a plane wave with
dip , we must set the filter coefficients
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, 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.
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interp-comp
Figure 4. Interpolation schemes compared. Deconvolution of random image with labeled steering filters. |
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![]() | Test case for PEF estimation with sparse data II | ![]() |
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