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| | |
2|c|Dip Filters for Monochromatic Data ( Const) | |
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2|c| | |
Low Pass | High Pass |
To apply these filters in the space domain it is necessary only to interpret k^{2} 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) |
Turn your attention from narrow-band data to data with a somewhat broader spectrum and consider
2|c| | |
2|c|Dip Filters for Moderate Bandwidth Data () | |
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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.