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DMO-correction

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

Store

SEG Online Store (http://shop.s

eg.org/OnlineStore/ProductDeta

il/tabid/177/Default.aspx?Produ

ctId=847)

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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