Prestack partial migration

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

While conventional normal-moveout correction involves only a time shift given by equation (7b), dip-moveout correction involves mapping both in time and space given by equations (8b) and (9b), respectively. This means that dip-moveout correction, strictly speaking, is not a moveout correction in conventional terms; rather, it is a process of partial migration before stack applied to common-offset data. We therefore may speak of a dip-moveout operator with a specific impulse response as for the migration process itself. Following this partial migration to map nonzero-offset data to the plane of zero-offset, each common-offset section is then fully migrated using a zero-offset migration operator.

A dip-moveout operator maps amplitudes on a moveout-corrected trace of a common-offset section along its impulse response trajectory. Before we derive the expression for its impulse response, we shall first make some inferences about the DMO process based on equations (8b) and (9b). Tables 5-1, 5-2, and 5-3 show horizontal (ΔyDMO) and vertical (ΔtDMO) displacements associated with dip-moveout correction described by equations (8b) and (9b), respectively. Combined with equations (8b) and (9b), we make the following observations:

1. Set ϕ = 0 in equations (8b) and (10b), and note that ΔyDMO = 0 and ΔtDMO = 0. Hence, the DMO operator has no effect on a flat reflector, irrespective of the offset. The steeper the dip, the larger the DMO correction.
2. Note from Table 5-1 that the horizontal displacement ΔyDMO and the vertical displacement ΔtDMO decrease with time tn. This means that the spatial aperture of the dip-moveout operator, in contrast with a migration operator, actually decreases with event time.
3. Substitute equation (5) into equation (8b) and note that, in the limit tn = 0, ΔyDMO = h. This means that the largest spatial extent of the DMO operator equals the offset 2h associated with the moveout-corrected trace at tn = 0.
4. Compare the values for ΔyDMO and ΔtDMO in Tables 5-1 and 5-2, and note that the lower the velocity, the larger the DMO correction. This also implies that the shallower the event, the more significant the DMO term, since lower velocities generally are found in shallow parts of the seismic data.
5. For a specific reflector dip ϕ, compare the values for ΔyDMO and ΔtDMO in Tables 5-2 and 5-3, and note that the larger the offset 2h, the more the DMO correction. Whatever the reflector dip, DMO correction has no effect on zero-offset data with h = 0.
6. Finally, note from Tables 5-1, 5-2 and 5-3 that the reflection point smear Δ given by equation (10) decreases in time and for small offsets.
 tn (s) v (m/s) ΔyDMO (m) ΔtDMO (s) Δ (m) 0.5 2400 1,170 0.188 1,013 1.0 2400 793 0.152 687 1.5 2400 575 0.115 497 2.0 2400 446 0.090 386 2.5 2400 363 0.075 314 3.0 2400 305 0.063 264 4.0 2400 230 0.008 199
 tn (s) v (m/s) ΔyDMO (m) ΔtDMO (s) Δ (m) 0.5 1800 1,284 0.243 1,111 1.0 2000 900 0.200 779 1.5 2200 620 0.135 536 2.0 2400 446 0.090 386 2.5 2700 324 0.060 280 3.0 3000 246 0.036 213 4.0 4000 140 0.016 121
 tn (s) v (m/s) ΔyDMO (m) ΔtDMO (s) Δ (m) 0.5 1800 241 0.064 209 1.0 2000 121 0.030 105 1.5 2200 74 0.015 64 2.0 2400 51 0.010 44 2.5 2700 37 0.008 32 3.0 3000 27 0.004 23 4.0 4000 15 0.000 13

Equations

 ${\displaystyle A={\sqrt {1+{\frac {h^{2}}{t_{n}^{2}}}\left({\frac {2{\rm {sin}}\phi }{v}}\right)^{2}}}.}$ (5)

 ${\displaystyle \Delta t_{NMO}=t_{n}\left(A_{n}-1\right),}$ (7b)

 ${\displaystyle \Delta y_{DMO}={\frac {h^{2}}{t_{n}A}}\left({\frac {2{\rm {sin}}\phi }{v}}\right).}$ (8b)

 ${\displaystyle \Delta t_{DMO}=t_{n}\left(1-{\frac {1}{A}}\right).}$ (9b)

 ${\displaystyle \Delta ={\frac {h^{2}}{t_{n}A}}\left({\frac {{\rm {sin2}}\phi }{v}}\right).}$ (10)