Prestack partial migration

Seismic Data Analysis

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

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 (Δy_DMO) and vertical (Δt_DMO) 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 Δy_DMO = 0 and Δt_DMO = 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 Δy_DMO and the vertical displacement Δt_DMO decrease with time t_n. 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 t_n = 0, Δy_DMO = h. This means that the largest spatial extent of the DMO operator equals the offset 2h associated with the moveout-corrected trace at t_n = 0.
4. Compare the values for Δy_DMO and Δt_DMO 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 Δy_DMO and Δt_DMO 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.

Table 5-1. Horizontal (Δy_DMO) and vertical (Δt_DMO) displacements associated with dip-moveout correction described by equations (8b) and (9b), respectively, for a given moveout-corrected time variable t_n and velocity v. The dip angle ϕ = 30 degrees, half-offset h = 1500 m. Also given are the corresponding values for the reflection point smear Δ as in equation (10).

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

Table 5-2. Horizontal (Δy_DMO) and vertical (Δt_DMO) displacements associated with dip-moveout correction described by equations (8b) and (9b), respectively, for a given moveout-corrected time variable t_n and velocity v. The dip angle ϕ = 30 degrees, half-offset h = 1500 m. Also given are the corresponding values for the reflection point smear Δ as in equation (10).

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

Table 5-3. Horizontal (Δy_DMO) and vertical (Δt_DMO) displacements associated with dip-moveout correction described by equations (8b) and (9b), respectively, for a given moveout-corrected time variable t_n and velocity v. The dip angle ϕ = 30 degrees, half-offset h = 500 m. Also given are the corresponding values for the reflection point smear Δ as in equation (10).

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

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

(5)

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

(7b)

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

(8b)

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

(9b)

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

(10)

See also

• Frequency-wavenumber DMO correction
• Log-stretch DMO correction
• Integral DMO correction
• Velocity errors
• Variable velocity
• Turning-wave migration

External links

find literature about Prestack partial migration