# Principles of dip moveout correction

Series Investigations in Geophysics Seismic Data Analysis Öz Yilmaz Dip-moveout correction and prestack migration http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The objective we want to achieve with the combination of normal-moveout and dip-moveout correction is mapping nonzero-offset data to the plane of zero-offset section. Once each common-offset section is mapped to zero-offset, the data can then be migrated either before or after stack using the zero-offset theory for migration as described in Chapter 4.

## Normal and dip moveout correction

Figure 5.1-1a depicts the nonzero-offset recording geometry associated with a dipping reflector. The 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. This arrival time is depicted on the time section in Figure 5.1-1b by point A on the trace that coincides with midpoint yn. We want to map the amplitude at time t denoted by the sample A on the trace at midpoint yn of the common-offset section with offset 2h to time τ0 denoted by the sample C on the trace at midpoint y0 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 yn of the common-offset section with offset 2h to time tn denoted by the sample B on the same trace at midpoint yn of the same common-offset section.
2. Dip-moveout correction that maps the amplitude at time tn denoted by the sample B on the trace at midpoint yn of the moveout-corrected common- offset section with offset 2h to time τ0 denoted by the sample C on the trace at midpoint y0 of the zero-offset section. Zero-offset migration then maps the amplitude at time τ0 denoted by the sample C on the trace at midpoint y0 of the zero-offset section to the amplitude at time τ denoted by the sample D on the trace at midpoint ym 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 yn of the common-offset section with offset 2h to the amplitude at time τ denoted by the sample D on the trace at midpoint ym of the migrated section. This direct mapping procedure is the basis of algorithms for migration before stack (Section 5.3).
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 ynt to yntn by mapping amplitude A at time t to B at time tn on the same trace. DMO correction involves coordinate transformation from yntn to y0τ0 by mapping amplitude B at time tn on the trace at midpoint location yn of the moveout-corrected common-offset section to amplitude C at time τ0 on the trace at midpoint location y0 of the zero-offset section. Zero-offset migration involves coordinate transformation from y0 – τ0 to ymτ by mapping amplitude C at time τ0 on the trace at midpoint location y0 of the zero-offset section to amplitude D at time τ on the trace at midpoint location ym of the migrated section. Migration before stack involves direct mapping of amplitude A at time t on the trace at midpoint location yn of the common-offset section to amplitude D at time τ on the trace at midpoint location ym of the migrated section. See text for the relationships between the coordinate variables.

The 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.

The NMO equation (3-8) defines the traveltime t from source location S to the reflection point R to the receiver location G. This equation, written in prestack data coordinates, is

 ${\displaystyle t^{2}=t_{0}^{2}+{{4h^{2}{{\cos }^{2}}\phi } \over {\upsilon ^{2}}},}$ (1)

where 2h is the offset, υ is the medium velocity above the reflector, φ is the reflector dip, and t0 is the two-way zero-offset time at midpoint location yn.

Dip-moveout correction of step (b) is preceded by zero-dip normal-moveout correction of step (a) using the dip-independent velocity υ:

 ${\displaystyle t^{2}=t_{n}^{2}+{{4h^{2}} \over {\upsilon ^{2}}},}$ (2)

where tn is the event time at midpoint yn after the NMO correction. Event time tn after the NMO correction and event time t0 are related as follows (Section E.2)

 ${\displaystyle t_{n}^{2}=t_{0}^{2}-{{4h^{2}{{\sin }^{2}}\phi } \over {\upsilon ^{2}}}.}$ (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 yn of the common-offset section with offset 2h to time tn denoted by the sample B on the same trace at midpoint yn of the same common-offset section.
2. Apply a dip-dependent moveout correction using equation ('3) to map the amplitude at time tn denoted by the sample B on the trace at midpoint yn of the moveout-corrected common-offset section with offset 2h to time t0 denoted by the sample B on the same trace at midpoint yn 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 t0.

Our goal, however, is to map event time t not to t0 — the two-way zero-offset time associated with midpoint yn between source S and receiver G, but to τ0 — the two-way zero-offset time at midpoint location y0 associated with the normal-incidence reflection point R (Figure 5.1-1). The relationships between (yn, tn) coordinates of the normal-moveout-corrected data and (y0, τ0) coordinates of the dip-moveout-corrected data are given by (Section E.2):

 ${\displaystyle y_{0}=y_{n}-{{h^{2}} \over {t_{n}A}}\left({{2\sin \phi } \over \upsilon }\right),}$ (4)

and

 ${\displaystyle \tau _{0}={{t_{n}} \over A},\ }$ (5)

where

 ${\displaystyle A={\sqrt {1+{{h^{2}} \over {t_{n}^{2}}}{{\left({{2\sin \phi } \over \upsilon }\right)}^{2}}.}}}$ (6)

For completeness, the relationship between event times tn and t0 is given by (Section E.2)

 ${\displaystyle t_{0}=t_{n}A.\ }$ (7)

Note from equation (6) that A ≥ 1; therefore, τ0tn (equation 5) and t0tn (equation 7).

Refer to Figure 5.1-1 and note that the normal-moveout correction that precedes the dip-moveout correction maps the amplitude at sample A with coordinates (yn , t) to sample B with coordinates (yn , tn). So, the midpoint coordinate is invariant under NMO correction. The difference between the input time t and the output time tn is defined by

 ${\displaystyle \Delta {t_{NMO}}=t-t_{n},\ }$ (8)

which can be expressed by way of equation (2) as follows

 ${\displaystyle \Delta {t_{NMO}}=t(A_{n}-1),}$ (9)

where

 ${\displaystyle A_{n}={\sqrt {1+{{h^{2}} \over {t_{n}^{2}}}{{\left({2 \over \upsilon }\right)}^{2}}.}}}$ (10)

Again, refer to Figure 5.1-1 and note that the dip-moveout correction maps the amplitude at sample B with coordinates (yn,tn) to sample C with coordinates (y00). So, the midpoint coordinate is variant under DMO correction. The lateral excursion associated with the DMO correction is given by

 ${\displaystyle {\Delta {y_{DMO}}}=\left|{y_{n}-y_{0}}\right|,}$ (11)

which can be expressed by way of equations (4) and (6) as

 ${\displaystyle {\Delta {y_{DMO}}}={{h^{2}} \over {t_{n}A}}\left({{2\sin \phi } \over \upsilon }\right).}$ (12)

The difference between the input time tn and the output time t0 is defined by

 ${\displaystyle \Delta {t_{DMO}}=t_{n}-\tau _{0},}$ (13)

which can be expressed by way of equations (5) and (6) as

 ${\displaystyle \Delta {t_{DMO}}=t_{n}\left({1-{1 \over A}}\right).}$ (14)

Finally, as sketched in Figure 5.1-1, the reflection point dispersal Δ = N R is defined by the distance along the dipping reflector between the normal-incidence points N and R associated with midpoints yn and y0, respectively. By way of equations (E-18) and (11) it follows that (Section E.1)

 ${\displaystyle \Delta ={{h^{2}} \over {t_{n}A}}\left({{\sin 2\phi } \over \upsilon }\right).}$ (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 common-midpoint (CMP) gathers; after 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 shift given by equation (9), dip-moveout correction involves mapping both in time and space given by equations (12) and (14), 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 (12) and (14). Tables 1, 2, and 3 show horizontal (ΔyDMO) and vertical (ΔtDMO) 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 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 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 (6) into equation (12) 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 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.
5. For a specific reflector dip φ, compare the values for ΔyDMO and ΔtDMO 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-moveout-corrected prestack data Pn(yn, tn; h) from yn – tn coordinates to y0 – τ0 coordinates so as to obtain the dip-moveout-corrected zero-offset data P0(y0, τ0; 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

 ${\displaystyle \sin \phi ={{\upsilon k_{y}} \over {2\omega _{0}}},}$ (16)

which states that the reflector dip φ can be expressed in terms of wavenumber ky and frequency ω0, which are the Fourier duals of midpoint y0 and event time τ0, respectively. By way of equation (16), the transformation equations (4) and (5) are recast explicitly independent of reflector dip as

 ${\displaystyle y_{0}=y_{n}-{{h^{2}k_{y}} \over {t_{n}A\omega _{0}}},}$ (17)

and

 ${\displaystyle \tau _{0}={{t_{n}} \over A},\ }$ (18)

where A of equation (6) now is of the form

 ${\displaystyle A={\sqrt {1+{{h^{2}k_{y}^{2}} \over {t_{n}^{2}\omega _{0}^{2}}}.}}}$ (19)

The frequency-wavenumber domain dip-moveout correction process that transforms the normal-moveout-corrected prestack data with a specific offset 2h from yn – tn domain to y0 – τ0 domain is achieved by the integral

 ${\displaystyle P_{0}(k_{y},\omega _{0};h)=\int {{2A^{2}-1} \over {A^{3}}}\times P_{n}(k_{y},t_{n};h)\exp(-i\omega _{0}t_{n}A)dt_{n}.}$ (20)

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

 ${\displaystyle P_{0}(y_{0},\tau _{0};h)=\int \int P_{0}(k_{y},\omega _{0};h)\times \exp(-ik_{y}y_{0}+i\omega _{0}\tau _{0})dk_{y}d\omega _{0}.}$ (21)

The amplitude scaling (2A2 – 1)/A3 in equation (20) is by Black et al.)[2] and is represented by A–1 in the original derivation by Hale)[1] The 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 τ0 of equation (5). Fortunately, the phase term exp(–iω0tnA) as in equation (20) is identical in the case of both derivations. There is one other variation of the amplitude term by Liner (1989) and Bleistein)[3] given by (2A2 – 1)/A. Nevertheless, within the context of a conventional processing sequence which includes geometric spreading correction prior to DMO correction, the amplitude scaling (2A2 – 1)/A3 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 Pn(yn, h, tn) to common-offset sections Pn(yn, tn; h).
3. Perform Fourier transform of each common-offset section in midpoint yn direction, Pn(ky, tn; h).
4. For each output frequency ω0, apply the phase-shift exp(–iω0tnA), scale by (2A2 – 1)/A3, and sum the resulting output over input time tn as described by equation (20).
5. Finally, perform 2-D inverse Fourier transform to obtain the dip-moveout corrected common-offset section P0(y0, τ0; 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 scatterers and dipping events. Figure 5.1-3 depicts six point scatterers buried in a constant-velocity medium. A synthetic data set that comprises 32 common-offset sections, each with 63 midpoints, was created. The 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. The offset range used in stacking is 50 – 1550 m. At the apex of the traveltime trajectory for each point scatterer, the event dip is zero. Therefore, 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. The 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. The 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 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.

The 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.
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 zero-offset 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). The 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. The 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.

#### Log-stretch DMO correction

The frequency-wavenumber DMO correction described in this section is computationally intensive.[1][2] Specifically, for each output frequency ω0, one has to apply the phase-shift exp(–iω0tnA), scale by (2A2 – 1)/A3, and sum the resulting output over input time tn as described by equation (20). A computationally more efficient DMO correction can be formulated in the logarithmic time domain.[4][5][6][7][8] The 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-shift operator in the Fourier transform domain.

Define the following logarithmic variables that correspond to the time variables τ0 and tn of equation (18):

 ${\displaystyle T_{0}=\ln \tau _{0},\ }$ (22)

and

 ${\displaystyle T_{n}=\ln t_{n},\ }$ (23)

where, for convenience, a constant scalar with its unit in time is omitted. Hence, the inverse relationships are given by

 ${\displaystyle \tau _{0}={e^{T_{0}}},\ }$ (24)

and

 ${\displaystyle t_{n}={e^{T_{n}}}.\ }$ (25)

Our goal is to derive equations for DMO correction in the log-stretch coordinates (y0, T0). The transform relation between the input log-stretch time variable Tn and the output log-stretch time variable T0 is given by

 ${\displaystyle T_{0}=T_{n}-\ln A_{e},\ }$ (26)

and the expression for the midpoint variable y0 in the log-stretch domain is given by

 ${\displaystyle y_{0}=y_{n}-{{h^{2}k_{y}} \over {A_{e}\Omega _{0}}},}$ (27)

where

 ${\displaystyle A_{e}={\sqrt {1+{{h^{2}k_{y}^{2}} \over \Omega _{0}^{2}}.}}}$ (28)

The variable Ω0 is the Fourier transform dual of the variable T0 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 left to Section E.3.

The log-stretch dip-moveout correction process is achieved by the following relationship (Section E.3):

 ${\displaystyle P_{0}(k_{y},\Omega _{0};h)=\exp \left({-i{{h^{2}k_{y}^{2}} \over {A_{e}\Omega _{0}}}+i\Omega _{0}\ln A_{e}}\right)P_{n}(k_{y},\Omega _{0};h).}$ (29)

Note that the relationship of input Pn(ky, Ω0; h) to output P0(ky, Ω0; h) given by equation (29) computationally is much simpler than that of equation (20). The log-stretch domain implementation of DMO correction involves application of a phase-shift given by the exponential in equation (29) to the input data; whereas, the frequency-wavenumber implementation involves an integral transform given by equation (20).