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Definition of a recursive dip filter

Let P denote raw data and Q denote filtered data. When seismic data is quasimonochromatic, dip filtering can be achieved with spatial frequency filters. The table below shows filters with an adjustable cutoff parameter .

 2|c|Dip Filters for Monochromatic Data ( Const) 2|c| 2|c| Low Pass High Pass

To apply these filters in the space domain it is necessary only to interpret k2 as the tridiagonal matrix T with (-1,2,-1) on the main diagonal. Specifically, for the low-pass filter it is necessary to solve a tridiagonal set of simultaneous equations like
 (21)
in which q and p are column vectors whose elements denote different places on the x-axis. Previously, this was done while solving the heat-flow equation. To make the filter space-variable, the parameter can be taken to depend on x so that is replaced by an arbitrary diagonal matrix. It doesn't matter whether p and q are represented in the -domain or the t-domain.

Turn your attention from narrow-band data to data with a somewhat broader spectrum and consider

 2|c|Dip Filters for Moderate Bandwidth Data () 2|c| 2|c| Low Pass High Pass

Naturally these filters can be applied to data of any bandwidth. However the filters are appropriately termed dip filters'' only over a modest bandwidth.

To understand these filters look in the -plane at contours of constant i.e. .Such contours, examples of which are shown in Figure 3, are curves of constant attenuation and constant phase shift.

 dipfil Figure 3 Constant-attenuation contours of dip filters. Over the seismic frequency band these parabolas may be satisfactory approximations to the dashed straight line. Pass/reject zones are indicated for the low-pass filter. (Hale)

An interesting feature of these dip filters is that the low-pass and the high-pass filters constitute a pair of filters which sum to unity. So nothing is lost if a dataset is partitioned by them in two. The high-passed part could be added to the low-passed part to recover the original dataset. Alternately, once the low-pass output is computed, it is much easier to compute the high-pass output, because it is just the input minus the low-pass. The low-pass filter has no phase shift in the pass zone, but there is time differentiation in the attenuation zone. This is apparent from the defining equation. The high-pass filter has no phase shift in the flat pass zone, but there is time integration in the attenuating zone.

Next: Recursive-dip-filter implementation Up: RECURSIVE DIP FILTERS Previous: RECURSIVE DIP FILTERS
Stanford Exploration Project
10/31/1997