# Frequency-wavenumber DMO correction

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

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 y0τ0 coordinates so as to obtain the dip-moveout-corrected zero-offset data P0(y0, τ0; h). Note, however, the transformation equations (4a) and (4b) 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

 ${\displaystyle {\rm {sin}}\phi ={\frac {vk_{y}}{2{\omega _{0}}}},}$ (11)

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 (11), the transformation equations (4a) and (4b) are recast explicitly independent of reflector dip as

 ${\displaystyle y_{0}=y_{n}-{\frac {h^{2}k_{y}}{t_{n}A\omega _{0}}},}$ (12a)

and

 ${\displaystyle \tau _{0}={\frac {t_{n}}{A}},}$ (12b)

where A of equation (5) now is of the form

 ${\displaystyle A={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{t_{n}^{2}\omega _{0}^{2}}}}}.}$ (13)

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 y0τ0 domain is achieved by the integral

 ${\displaystyle P_{0}\left(k_{y},\omega _{0};h\right)=\int {\frac {2A^{2}-1}{A^{3}}}\times P_{n}\left(k_{y},t_{n};h\right){\rm {exp}}\left(-i\omega _{0}t_{n}A\right)dt_{n}.}$ (14a)

Derivation of the integral transform of equation (14a) is given in Section E.2.

Once dip-moveout correction is applied, the data are inverse Fourier transformed

 ${\displaystyle P_{0}\left(y_{0},\tau _{0};h\right)=\int \int P_{0}\left(k_{y},\omega _{0};h\right)\times {\rm {exp}}\left(-ik_{y}y_{0}+i\omega _{0}\tau _{0}\right)dk_{y}d\omega _{0}.}$ (14b)

The amplitude scaling (2A2 − 1)/A3 in equation (14a) is by Black [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 t0 of equation (6), whereas Black [2] correctly defined the output time variable as τ0 of equation (4b). Fortunately, the phase term exp(−0tnA) as in equation (14a) 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 v.
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(−0tnA), scale by (2A2 − 1)/A3, and sum the resulting output over input time tn as described by equation (14a).
5. Finally, perform 2-D inverse Fourier transform to obtain the dip-moveout corrected common-offset section P0(y0, τ0; h) (equation 14b).

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.

Figure 5.1-2  A flowchart for frequency-wavenumber dip-moveout correction algorithm. The scalar A is given by equation (13) and B = (2A2 − 1)/A3 as in equation (14a).

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 travel-times 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:
• 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).
• 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.
• While it does nothing to the zero-offset section, dip-moveout correction also is greater at increasingly large offsets (Figure 5.1-6c).
• Finally, as with conventional migration, the steeper the event, the greater partial migration takes place, with flat events remaining unaltered (Figure 5.1-6c).
1. 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.
2. 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).

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.

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.

## References

1. Hale (1984), Hale, D., 1984, Dip moveout by Fourier transform: Geophysics, 49, 741–757.
2. Black et al. (1993), Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58, 47–66.
3. Bleistein (1990), Bleistein, N., 1990, Born DMO revisited: 60th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1366–1369.