Dip limits of extrapolation filters
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| Series | Investigations in Geophysics |
|---|---|
| Author | Öz Yilmaz |
| DOI | http://dx.doi.org/10.1190/1.9781560801580 |
| ISBN | ISBN 978-1-56080-094-1 |
| Store | SEG Online Store |
Figure 4.4-19 shows a zero-offset section that contains a diffraction hyperbola and its migrations using the 30-, 50- and 70-degree explicit schemes. Also shown is the desired migration using the phase-shift method. Note that the low-dip algorithm causes undermigration of the diffraction hyperbola, while the steep-dip algorithm focuses the energy at the apex better than a steep-dip implicit scheme with equivalent dip accuracy (Figure 4.4-7).
Explicit schemes are designed based on a specified cutoff wavenumber kx (Section D.5), beyond which the amplitude spectrum of the filter is set to zero. Figure 4.4-20 shows the f − k spectra of the results of migration of the diffraction hyperbola as shown in Figure 4.4-19 using the three explicit extrapolators. Note that the lower the dip limit, the lower the cutoff wavenumber. The steep-dip algorithm with a 70-degree dip limit is able to replicate the amplitude response characteristics of the desired migration using the phase-shift method. This of course is at the expense of using a long, complex convolutional filter.
Figure 4.4-21 shows a zero-offset section that contains a set of dipping events and its migrations using the 30-, 50- and 70-degree explicit schemes. Also shown is the desired migration using the phase-shift method. For comparison, label the the correct position of the event with the steepest dip from the desired migration on the results of migration with different velocities. The steep-dip algorithm positions the events nearly as good as the desired migration using the phase-shift method and better than a steep-dip implicit scheme with equivalent dip accuracy (Figure 4.4-9).
Figure 4.4-22 shows the f − k spectra of the results of migration of the dipping events as shown in Figure 4.4-21 using the three explicit extrapolators. The extrapolation filter with the 30-degree dip limit has truncated the steeply dipping events at high wavenumbers kx. Except for very high wavenumbers, the extrapolation filter with the 70-degree dip limit is able to replicate the amplitude response characteristics of the desired migration using the phase-shift method. Keep in mind that we achieve stability of explicit schemes at the expense of truncating the response at wavenumbers kx above a specified cutoff value (Section D.5).
Figure 4.4-16 Tests for velocity errors in 65-degree frequency-space implicit finite-difference migration using interval velocities derived from, from top to bottom, 100, 95, 90, and 80 percent of rms velocities. The input stacked section is shown in Figure 4.2-15a, and the desired migration using the phase-shift method is shown in Figure 4.2-15b. Depth step size is 20 ms.
Figure 4.4-17 Tests for velocity errors in 65-degree frequency-space implicit finite-difference migration using interval velocities derived from, from top to bottom, 100, 105, 110, and 120 percent of rms velocities. The input stacked section is shown in Figure 4.2-15a, and the desired migration using the phase-shift method is shown in Figure 4.2-15b. Depth step size is 20 ms.
Figure 4.4-18 (a) Impulse response of a desired migration algorithm using the phase-shift method; impulse responses of (b) a 30-degree, (c) 50-degree, and (d) 70-degree frequency-space explicit scheme for migration.
A field data example of a stacked section which has been migrated using the 30-degree, 50-degree and 70-degree extrapoaltion filters is shown in Figure 4.4-23. The steep flanks of the salt diapirs are clearly better imaged by the steep-dip extrapolation filter. For comparison, Figure 4.4-24 shows the desired migration using the phase-shift method. Although the length of the extrapolation filter for a steep-dip algorithm is much longer than that for a low-dip extrapolation filter, the benefit of using the former is indisputably demonstrated by the field data example shown in Figure 4.4-23.
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