DMO-correction
This file is about DMO-correction in seismic processing and can be read in Yilmaz.
This file is about DMO-correction in seismic processing and can be read in Yilmaz.
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Principles of dip moveout correction
e objective we want to achieve with
the combination of normal-moveout and
dip-moveout correction is mapping
nonzero-offset data to the plane of zerooffset
section. Once each common-offset
section is mapped to zero-offset, the data
can then be migrated either before or
aer stack using the zero-offset theory
for migration as described in Chapter 4.
Seismic Data Analysis
Contents
Normal and dip moveout
correction
Prestack partial migration
Frequency-wavenumber
DMO correction
Log-stretch DMO
correction
See also
References
External links
Normal and dip
moveout correction
Series
Title
Author
Investigations in Geophysics
Seismic Data Analysis
Öz Yilmaz
Chapter Dip-moveout correction and
prestack migration
DOI
http://dx.doi.org/10.1190
/1.9781560801580 (http://dx.do
i.org/10.1190/1.978156080158
0)
ISBN ISBN 978-1-56080-094-1
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Figure 5.1-1a depicts the nonzero-offset
recording geometry associated with a dipping reflector. e nonzero-offset traveltime t = SRG/υ is measured along
the raypath from source S to reflection point R to receiver G, where υ is the velocity of the medium above the
dipping reflector. is arrival time is depicted on the time section in Figure 5.1-1b by point A on the trace that
coincides with midpoint y n . We want to map the amplitude at time t denoted by the sample A on the trace at
midpoint y n of the common-offset section with offset 2h to time τ 0 denoted by the sample C on the trace at
midpoint y 0 of the zero-offset section. We achieve this mapping in two steps:
1. Normal-moveout correction that maps the amplitude at time t denoted by the sample A
on the trace at midpoint y n of the common-offset section with offset 2h to time t n
denoted by the sample B on the same trace at midpoint y n of the same common-offset
section.
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2. Dip-moveout correction that maps the amplitude at time t n denoted by the sample B on
the trace at midpoint y n of the moveout-corrected common- offset section with offset 2h
to time τ 0 denoted by the sample C on the trace at midpoint y 0 of the zero-offset section.
Zero-offset migration then maps the amplitude at time τ 0 denoted by the sample C on
the trace at midpoint y 0 of the zero-offset section to the amplitude at time τ denoted by
the sample D on the trace at midpoint y m of the migrated section. Note that the
combination of NMO correction, DMO correction, and zero-offset migration achieves the
same objective as direct mapping of the amplitude at time t denoted by the sample A on
the trace at midpoint y n of the common-offset section with offset 2h to the amplitude at
time τ denoted by the sample D on the trace at midpoint y m of the migrated section. This
direct mapping procedure is the basis of algorithms for migration before stack (Section
5.3).
e important point to note is that the normal-moveout correction in step (a) is performed using the velocity of the
medium above the dipping reflector.
e NMO equation (3-8) defines the traveltime t from source location S to the reflection point R to the receiver
location G. is equation, wrien in prestack data coordinates, is
(1)
where 2h is the offset, υ is the medium velocity above the reflector, φ is the reflector dip, and t 0 is the two-way
zero-offset time at midpoint location y n .
Dip-moveout correction of step (b) is preceded by zero-dip normal-moveout correction of step (a) using the dipindependent
velocity υ:
(2)
where t n is the event time at midpoint y n aer the NMO correction. Event time t n aer the NMO correction and
event time t 0 are related as follows (Section E.2)
(3)
At first glance, equations (2) and (3) suggest a two-step approach to moveout correction:
1. Apply a dip-independent moveout correction using equation (2) to map the amplitude at
time t denoted by the sample A on the trace at midpoint y n of the common-offset section
with offset 2h to time t n denoted by the sample B on the same trace at midpoint y n of the
same common-offset section.
2. Apply a dip-dependent moveout correction using equation ('3) to map the amplitude
at time t n denoted by the sample B on the trace at midpoint y n of the moveoutcorrected
common-offset section with offset 2h to time t 0 denoted by the
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sample B on the same trace at
midpoint y n of the same common-offset
section This two-step moveout
correction is equivalent to the one-step
moveout correction using equation (1)
to map event time t directly to event
time t 0 .
Our goal, however, is to map event time t not to t 0 —
the two-way zero-offset time associated with
midpoint y n between source S and receiver G, but to
τ 0 — the two-way zero-offset time at midpoint
location y 0 associated with the normal-incidence
reflection point R (Figure 5.1-1). e relationships
between (y n , t n ) coordinates of the normal-moveoutcorrected
data and (y 0 , τ 0 ) coordinates of the dipmoveout-corrected
data are given by (Section E.2):
(4)
and
(5)
where
(6)
For completeness, the relationship between event
times t n and t 0 is given by (Section E.2)
(7)
Note from equation (6) that A ≥ 1; therefore, τ 0 ≤ t n
(equation 5) and t 0 ≥ t n (equation 7).
Refer to Figure 5.1-1 and note that the normalmoveout
correction that precedes the dip-moveout
correction maps the amplitude at sample A with
Figure 5.1-1 (a) The geometry of a
nonzero-offset recording of reflections from
a dipping layer boundary; (b) a sketch of
the time section depicting the various
traveltimes. NMO correction involves
coordinate transformation from y n – t to y n
– t n by mapping amplitude A at time t to B
at time t n on the same trace. DMO
correction involves coordinate
transformation from y n – t n to y 0 – τ 0 by
mapping amplitude B at time t n on the
trace at midpoint location y n of the
moveout-corrected common-offset section
to amplitude C at time τ 0 on the trace at
midpoint location y 0 of the zero-offset
section. Zero-offset migration involves
coordinate transformation from y 0 – τ 0 to
y m – τ by mapping amplitude C at time τ 0
on the trace at midpoint location y 0 of the
zero-offset section to amplitude D at time τ
on the trace at midpoint location y m of the
migrated section. Migration before stack
involves direct mapping of amplitude A at
time t on the trace at midpoint location y n
of the common-offset section to amplitude
D at time τ on the trace at midpoint
location y m of the migrated section. See
text for the relationships between the
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coordinates (y n , t) to sample B with coordinates (y n ,
t n ). So, the midpoint coordinate is invariant under
NMO correction. e difference between the input
time t and the output time t n is defined by
coordinate variables.
(8)
which can be expressed by way of equation (2) as follows
(9)
where
(10)
Again, refer to Figure 5.1-1 and note that the dip-moveout correction maps the amplitude at sample B with
coordinates (y n ,t n ) to sample C with coordinates (y 0 ,τ 0 ). So, the midpoint coordinate is variant under DMO correction.
e lateral excursion associated with the DMO correction is given by
(11)
which can be expressed by way of equations (4) and (6) as
(12)
e difference between the input time t n and the output time t 0 is defined by
(13)
which can be expressed by way of equations (5) and (6) as
(14)
Finally, as sketched in Figure 5.1-1, the reflection point dispersal Δ = N R is defined by the distance along the
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dipping reflector between the normal-incidence points N and R associated with midpoints y n and y 0 , respectively.
By way of equations (E-18) and (11) it follows that (Section E.1)
(15)
Note from equation (15) that reflection point dispersal is nill for zero offset, and increases with the square of the
offset. Also, the larger the dip and shallower the reflector, the larger the dispersal.
A direct consequence of equation (15) is that a reflection event on a CMP gather is associated with more than one
reflection point on the reflector. Following DMO correction, reflection-point dispersal is eliminated and, hence, the
reflection event is associated with a single reflection point at normal-incidence (point R in Figure 5.1-1). While
prestack data before DMO correction can be associated with common midpoints, and thus sorted into commonmidpoint
(CMP) gathers; aer DMO correction, the data can be associated with common reflection points, and thus
can be considered in the form of common-reflection-point (CRP) gathers.
Prestack partial migration
While conventional normal-moveout correction involves only a time shi given by equation (9), dip-moveout
correction involves mapping both in time and space given by equations (12) and (14), respectively. is 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 zerooffset
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 (12) and (14). Tables 1, 2, and 3 show horizontal (Δ yDMO )
and vertical (Δ tDMO ) displacements associated with dip-moveout correction described by equations (12) and (14),
respectively. Combined with equations (12) and (14), we make the following observations:
1. Set φ = 0 in equations (12) and (15), and note that Δ yDMO = 0 and Δ tDMO = 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 1 that the horizontal displacement Δ yDMO and the vertical displacement
Δ tDMO 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 (6) into equation (12) and note that, in the limit t n = 0, Δ yDMO = 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 Δ yDMO and Δ tDMO in Tables 1 and 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.
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5. For a specific reflector dip φ, compare the values for Δ yDMO and Δ tDMO in Tables 2 and 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 1, 2 and 3 that the reflection point smear Δ given by equation
(15) decreases in time and for small offsets.
Frequency-wavenumber DMO correction
Refer to Figure 5.1-1 and recall that our objective with DMO correction is to transform the normal-moveoutcorrected
prestack data P n (y n , t n ; h) from y n – t n coordinates to y 0 – τ 0 coordinates so as to obtain the dip-moveoutcorrected
zero-offset data P 0 (y 0 , τ 0 ; h). Note, however, the transformation equations (4) and (5) require knowledge
of the reflector dip φ to perform the DMO correction. To circumvent this requirement, Hale),. [1] developed a
method for DMO correction in the frequency-wavenumber domain. First, we use the relation from Section D.1
(16)
which states that the reflector dip φ can be expressed in terms of wavenumber k y and frequency ω 0 , which are the
Fourier duals of midpoint y 0 and event time τ 0 , respectively. By way of equation (16), the transformation equations
(4) and (5) are recast explicitly independent of reflector dip as
(17)
and
(18)
where A of equation (6) now is of the form
(19)
e frequency-wavenumber domain dip-moveout correction process that transforms the normal-moveoutcorrected
prestack data with a specific offset 2h from y n – t n domain to y 0 – τ 0 domain is achieved by the integral
(20)
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Derivation of the integral transform of equation (20) is given in Section E.2.
Once dip-moveout correction is applied, the data are inverse Fourier transformed
(21)
e amplitude scaling (2A 2 – 1)/A 3 in equation (20) is by Black et al.) [2] and is represented by A –1 in the original
derivation by Hale) [1] e difference is due to the fact that Hale) [1] defined the output time variable for DMO
correction as to of equation (7), whereas Black et al.) [2] correctly defined the output time variable as τ 0 of equation
(5). Fortunately, the phase term exp(–iω 0 t n A) as in equation (20) is identical in the case of both derivations. ere is
one other variation of the amplitude term by Liner (1989) and Bleistein) [3] given by (2A 2 – 1)/A. Nevertheless,
within the context of a conventional processing sequence which includes geometric spreading correction prior to
DMO correction, the amplitude scaling (2A 2 – 1)/A 3 described here preserves relative amplitudes.
We now outline the steps in dip-moveout correction in the frequency-wavenumber domain:
1. Start with prestack data in midpoint-offset y – h coordinates, P(y, h, t) and apply normal
moveout correction using a dip-independent velocity υ.
2. Sort the data from moveout-corrected CMP gathers P n (y n , h, t n ) to common-offset
sections P n (y n , t n ; h).
3. Perform Fourier transform of each common-offset section in midpoint y n direction, P n (k y ,
t n ; h).
4. For each output frequency ω 0 , apply the phase-shift exp(–iω 0 t n A), scale by (2A 2 – 1)/A 3 ,
and sum the resulting output over input time t n as described by equation (20).
5. Finally, perform 2-D inverse Fourier transform to obtain the dip-moveout corrected
common-offset section P 0 (y 0 , τ 0 ; h) (equation 21). A flowchart of the dip-moveout
correction described above is presented in Figure 5.1-2.
We shall now test the frequency-wavenumber DMO correction using modeled data for point scaerers and dipping
events. Figure 5.1-3 depicts six point scaerers buried in a constant-velocity medium. A synthetic data set that
comprises 32 common-offset sections, each with 63 midpoints, was created. e offsets range is from 0 to 1550 m
with an increment of 50 m.
Figure 5.1-4 shows two constant-velocity stacks (CVS) of the CMP gathers from the synthetic data set associated
with the velocity-depth model depicted in Figure 5.1-3. e offset range used in stacking is 50 – 1550 m. At the
apex of the traveltime trajectory for each point scaerer, the event dip is zero. erefore, stack response is best
with moveout velocity equal to the medium velocity (3000 m/s). Along the flanks of the traveltime trajectories,
optimum stack response varies as the event dip changes. e steeper the dip, the higher the moveout (or stacking)
velocity.
Selected common-offset sections associated with the subsurface model in Figure 5.1-3 are shown in Figure 5.1-5a.
e well-known nonhyperbolic table-top trajectories are apparent at large offsets. Selected CMP gathers from the
model of Figure 5.1-3 are shown in Figure 5.1-5b. Only selected gathers that span the right side of the center
midpoint are displayed, since the common-offset sections are symmetric with respect to the center midpoint (CMP
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32). Note that the traveltimes at the center midpoint are perfectly hyperbolic, while the traveltimes at CMP gathers
away from the center are increasingly nonhyperbolic.
Figure 5.1-2 A
flowchart for
frequencywavenumber
dipmoveout
correction
algorithm. The scalar
A is given by
equation (19) and B
= (2A 2 – 1)/A 3 as in
equation (20).
Figure 5.1-3 Depth
model of six point
scatterers buried in a
constant-velocity
medium. The
asterisks indicate the
positions of the point
scatterers.
Figure 5.1-4 Stack
response of six point
scatterers buried in a
constant-velocity
earth model (3000
m/s) as depicted in
Figure 5.1-3: (a) zerooffset
section, (b)
stack with NMO
velocity of 3000 m/s,
(c) stack with NMO
velocity of 3600 m/s.
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Figure
5.1-5 Intermediate
results from DMO
processing the
nonzero-offset
synthetic data
derived from the
depth model in Figure
5.1-3: (a) commonoffset
sections with
offset range from 50
to 1550 m and an
increment of 300 m;
(b) CMP gathers
sorted from the
common-offset
sections as in (a) at
midpoint locations
from 32 to 63 as
denoted in Figure
5.1-3 with an
increment of 3; (c)
the CMP gathers as in
(b) after NMO
correction and
muting.
e following DMO processing is applied to the data as in Figure 5.1-5a:
1. Figure 5.1-5c shows the NMO-corrected gathers, with stretch muting applied. The medium
velocity (3000 m/s) was used for NMO correction (equation 2), an essential requirement
for subsequent DMO correction. As a result, the events at and in the vicinity of the center
midpoint (CMP 32) are flat after NMO correction, while the events at midpoints away from
the center midpoint are increasingly overcorrected.
2. The stacked section derived from these gathers (Figure 5.1-5c) is shown in Figure 5.1-4b.
Because medium velocity was used for NMO correction, the stack response is best for
zero dip. Note the poor stack response along the steeply dipping flanks. The desired
section is the zero-offset section in Figure 5.1-4a.
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3. We sort the NMO-corrected gathers (Figure 5.1-5c) into common-offset sections for DMO
processing. These are shown in Figure 5.1-6a.
4. Each common-offset section is individually corrected for dip moveout. The impulse
responses of the dip-moveout operator for the corresponding offsets are shown in Figure
5.1-6b, and the resulting common-offset sections are shown in Figure 5.1-6c. Note the
following effects of DMO:
1. DMO is a partial migration process. The flanks of the nonhyperbolic trajectories have
been moved updip just enough to make them look like zero offset trajectories, which
are hyperbolic. As a result, each common-offset section after NMO and DMO
corrections is approximately equivalent to the zero-offset section (Figure 5.1-4a).
2. This partial migration is subtly different from conventional migration in one respect.
Unlike conventional migration, note from the impulse responses in Figure 5.1-6b that
the dip-moveout correction becomes greater at increasingly shallow depths.
3. While it does nothing to the zero-offset section, dip-moveout correction also is greater
at increasingly large offsets (Figure 5.1-6c).
4. Finally, as with conventional migration, the steeper the event, the greater partial
migration takes place, with flat events remaining unaltered (Figure 5.1- 6c).
5. Following the DMO correction, the data are sorted back to CMP gathers (Figure 5.1-6d).
Compare the gathers in Figure 5.1-6d to the CMP gathers without DMO correction (Figure
5.1-5b). The DMO correction has left the zero-dip events unchanged (at and in the vicinity
of CMP 32), while it substantially corrected steeply dipping events on the CMP gathers
away from the center midpoint (CMP 32). The events on the CMP gathers now are
flattened (Figure 5.1-6d). Also, since DMO correction is a migration-like process, it causes
the energy to move from one CMP gather to neighboring gathers in the updip direction.
Energy depletion at the CMP gathers in Figure 5.1-6d farther from the center midpoint
occurred because there was no other CMP gather to contribute energy beyond CMP 63.
6. Stacking the NMO- and DMO-corrected gathers (Figure 5.1-6d) yields a section (Figure
5.1-7c) that more closely represents the zero-offset section (Figure 5.1-7a) than the
stacked section without the DMO correction (Figure 5.1-7b). Note the enhanced stack
response along the steeply dipping flanks in Figure 5.1-7c. (The sections all have the
same display gain.)
We now examine results of DMO processing of a modeled data set for dipping events. Figure 5.1-8a shows a zerooffset
section that consists of events with dips from 0 to 45 degrees with a 5-degree increment. Medium velocity is
constant (3500 m/s). Several velocity analyses were performed along the line; an example is shown in Figure 5.1-9a.
Note the dip-dependent semblance peaks. Selected CMP gathers are shown in Figure 5.1-10a. By using the
optimum stacking velocities picked from the densely spaced velocity analyses, we apply NMO correction to the
CMP gathers (Figure 5.1-10b), then stack them (Figure 5.1-8b). Aside from the conflicting dips at location A, stack
response is close to the zero-offset section (Figure 5.1-8a). e DMO processing requires NMO correction using
medium velocity (Figure 5.1-10c). Stack response using the medium velocity (Figure 5.1-8c) clearly degrades at
steep dips. By applying DMO correction (Figure 5.1-10d) to the NMO-corrected gathers (Figure 5.1-10b), we get the
improved stacked section in Figure 5.1-8d. e DMO stack is closest to the zero-offset section (Figure 5.1-8a).
DMO correction also yields dip-corrected velocity functions that can be used in subsequent migration. Refer to the
velocity analysis in Figure 5.1-9b and note that all events have semblance peaks at 3500 m/s, which is the medium
velocity for this model data set.
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Figure
5.1-6 Intermediate
results from DMO
processing the
nonzero-offset
synthetic data
derived from the
depth model in Figure
5.1-3: (a) commonoffset
sections with
offset range from 50
to 1550 m and an
increment of 300 m
sorted from the NMOcorrected
gathers as
in Figure 5.1-4c; (b)
impulse responses of
the DMO operators
applied to the
common-offset
gathers; (c) commonoffset
sections as in
(a) after DMO
correction; (d) CMP
gathers sorted from
the common-offset
sections as in (c) at
midpoint locations
from 32 to 63 as
denoted in Figure
5.1-3 with an
increment of 3.
Figure 5.1-7 (a)
Zero-offset section
associated with the
depth model in Figure
5.1-3, (b) stack
derived from the CMP
gathers as in Figure
5.1-5c, (c) DMO stack
derived from the CMP
gathers as in Figure
5.1-6d.
Figure 5.1-8 DMO
processing of dipping
events: (a) zerooffset
section with
the medium velocity
of 3500 m/s; (b) stack
using optimum
velocity picks from
velocity spectra
along the line, such
as that shown in
Figure 5.1-12a; (c)
stack using the
medium velocity of
3500 m/s; (d) DMO
stack using velocity
picks from velocity
spectra along the
line, such as that
shown in Figure
5.1-12b. Location A
refers to an example
of events with
conflicting dips.
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Figure
5.1-9 Velocity
analysis (a) without
and (b) with DMO
correction at
analysis. The stacked
sections without and
with DMO correction
are shown in Figures
5.1-8b and d.
Log-stretch DMO correction
e frequency-wavenumber DMO correction described in this section is computationally intensive. [1][2]
Specifically, for each output frequency ω 0 , one has to apply the phase-shi exp(–iω 0 t n A), scale by (2A 2 – 1)/A 3 , and
sum the resulting output over input time t n as described by equation (20). A computationally more efficient DMO
correction can be formulated in the logarithmic time domain. [4][5][6][7][8] e log-stretch time variable enables
linearization of the coordinate transform equation (18), and as a result, the DMO correction is achieved by a simple
multiplication of the input data with a phase-shi operator in the Fourier transform domain.
Define the following logarithmic variables that correspond to the time variables τ 0 and t n of equation (18):
(22)
and
(23)
where, for convenience, a constant scalar with its unit in time is omied. Hence, the inverse relationships are given
by
(24)
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and
(25)
Our goal is to derive equations for DMO correction in the log-stretch coordinates (y 0 , T 0 ). e transform relation
between the input log-stretch time variable T n and the output log-stretch time variable T 0 is given by
(26)
and the expression for the midpoint variable y 0 in the log-stretch domain is given by
(27)
where
(28)
e variable Ω 0 is the Fourier transform dual of the variable T 0 in the log-stretch domain. Equations (26,27) and
(28) correspond to equations (17,18) and (19) in the log-stretch domain. Mathematical details of the derivation of
equations (26,27) are le to Section E.3.
e log-stretch dip-moveout correction process is achieved by the following relationship (Section E.3):
(29)
Note that the relationship of input P n (k y , Ω 0 ; h) to output P 0 (k y , Ω 0 ; h) given by equation (29) computationally is
much simpler than that of equation (20). e log-stretch domain implementation of DMO correction involves
application of a phase-shi given by the exponential in equation (29) to the input data; whereas, the frequencywavenumber
implementation involves an integral transform given by equation (20).
See also
Dip-moveout correction and prestack migration
Principles of dip-moveout correction
Prestack partial migration
Frequency-wavenumber DMO correction
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Log-stretch DMO correction
Integral DMO correction
Velocity errors
Variable velocity
Turning wave migration
References
1. Hale, D., 1984, Dip moveout by Fourier transform: Geophysics, 49, 741 – 757.
2. Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout:
Geophysics, 58, 47 – 66.
3. Bleistein, N., 1990, Born DMO revisited: 60th Ann. Internat. Mtg., Soc. Expl. Geophys.,
Expanded Abstracts, 1366 – 1369.
4. Bolondi, G., Loinger, E. and Rocca, F., 1982, Offset continuation of seismic sections:
Geophys. Prosp., 30, 813 – 828.
5. Bale, R. and Jacubowitz, H., 1987, Poststack prestack migration: 57th Ann. Internat. Mtg.,
Soc. Expl. Geophys., Expanded Abstracts, 714 – 717.
6. Notfors, C. D. and Godfrey, R. J., 1987, Dip-moveout in the frequency-wavenumber
domain: Geophysics, 52, 1718 – 1721.
7. Liner, C. L., 1990, General theory and comparative anatomy of dip-moveout: Geophysics,
55, 595 – 607.
8. Zhou, B, Mason, I. M., and Greenalgh, S., A., 1996, Accurate and efficient shot-gather dipmoveout
processing in the log-stretch domain: Geophys. Prosp., 43, 963 – 978.
External links
find literature about
Principles of dip moveout
correction
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