# 3-D dip-moveout correction

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

Equation (2) indicates that each trace in a common-cell gather would have to be moveout-corrected using a different moveout velocity (equation 3) which accounts for the source-receiver azimuth associated with that trace.

 $t^{2}=t_{0}^{2}+{\frac {4h^{2}(1-\sin ^{2}\phi \cos ^{2}\theta )}{v^{2}}}$ (2)

 $v_{NMO}={\frac {v}{\sqrt {1-\sin ^{2}\phi \cos ^{2}\theta }}}$ (3)

In dip-moveout correction and prestack migration, we learned that the dip-moveout (DMO) process corrects the moveout velocity of a dipping reflector for the true dip angle and yields a dip-independent moveout velocity at the unmigrated position of the dipping reflector. Rewrite equations (2) and (3) for the 2-D case by setting the azimuth angle θ = 0 to get

 $t^{2}=t_{0}^{2}+{\frac {4h^{2}\cos \phi }{v^{2}}}$ (4)

and

 $v_{NMO}={\frac {v}{\cos \phi }}.$ (5)

So, the 2-D DMO correction maps velocity vNMO to velocity v. A single velocity function at a given CMP location can then be used to stack events with conflicting dips that otherwise would have different stacking velocities.

Again, refer to equation (3) and this time define an apparent dip angle ϕ′:

 $\sin \phi ^{\prime }=\sin \phi \sin \theta .$ (6)

Using this definition, the NMO velocity given by equation (3) is rewritten as

 $v_{NMO}={\frac {v}{\cos \phi ^{\prime }}}.$ (7)

This equation is identical to the 2-D equation (5), except that equation (7) refers to apparent dip, while equation (5) refers to true dip angle.

The implication of equation (7) is that the 3-D DMO process corrects for the effect of apparent dip on stacking velocities. Note from equation (6) that the apparent dip accounts for the combined effect of the true dip as measured along the dip-line direction and the source-receiver azimuth. The apparent dip is equal to true dip when the source-receiver azimuth coincides with the direction of the dip line. We conclude that 3-D DMO not only corrects for the true dip but also for the source-receiver azimuth effects on moveout velocity.

Application of 3-D DMO correction followed by 2-D inverse DMO correction have the effect of correcting for stacking velocities only for the azimuth angle θ in equation (3). This is called azimuth-moveout correction (AMO). The resulting CMP gathers can be treated as if they belong to a 2-D seismic line with shots and receivers placed along a straight traverse.

In dip-moveout correction and prestack migration, we learned that the 2-D DMO process removes the relection-point smear Δ defined as (Section E.1)

 $\Delta ={\frac {h^{2}}{vt_{0}}}\sin 2\phi .$ (8)

Equation (8), by way of equation (5-6), is the same as equation (5-10) of dip-moveout correction and prestack migration with offset 2h. The 3-D equivalent of equation (8) is

 $\Delta ={\frac {2h^{2}}{vt_{0}}}\sin \phi \cos \theta {\sqrt {1-\sin ^{2}\phi \cos ^{2}\theta }},$ (9)

where, again, θ is the azimuth angle between the structural dip direction and the direction of the profile line (Figure 7.2-8).

Old marine 3-D surveys used to be conducted with single-cable or dual-cable geometry which did not incur large source-receiver azimuthal variations. In such cases, source-receiver azimuths are confined to a feathering angle, generally less than 7 degrees. (For two opposite shooting directions, it is twice the feathering angle.) As a result, the 3-D aspect of the DMO process is not significant for old marine 3-D surveys. In contrast, modern marine 3-D surveys are conducted using multicable recording geometries that incur large source-receiver azimuthal variations. Therefore, the 3-D aspect of the DMO process becomes important. Similarly, land 3-D surveys are usually carried out using swath shooting. This type of recording also gives rise to large source-receiver azimuthal variations; the DMO process, therefore, has to be applied in a 3-D sense.

The 2-D DMO operator (principles of dip-moveout correction and E.4) maps the moveout-corrected amplitudes on a common-offset section along truncated elliptical trajectories that narrow at increasingly late times . The 3-D DMO operator maps the amplitudes on the plane of source-receiver azimuths over an ellipsoid of revolution (Figure 7.2-11) . The fidelity of 3-D DMO correction, therefore, depends on source-receiver azimuthal coverage, in particular, how well the ellipsoid is defined.

The 3-D DMO process can be conceptualized as the extension of the 2-D DMO process (Figure 7.2-11a). Take a moveout-corrected trace from a common-cell gather, which for the swath shooting coincides with a common-midpoint gather, and map the amplitude at time sample A to neighboring cells that are coincident with the source-receiver azimuthal direction associated with that input trace, along the elliptical trajectory of the 2-D DMO operator. The DMO ellipse is described by equation (5-21) and the amplitude mapping is described by equation (5-24). The excursion from the center trace depends upon the source-receiver separation, and the NMO-corrected time at the center trace. The amplitude at an output cell location along the mapping trajectory is the amplitude at the center trace scaled by the excursion from the center trace and the source-receiver separation. Notice that mapping is done over the plane of NMO-corrected time versus the source-receiver azimuthal direction.

Continue the process for all the traces from the same common-cell gather and map the amplitudes in the same manner. The output of the DMO process at any one cell location is simply a superposition of the contributions from the mapping of all the input traces.

Consider the hypothetical recording geometry with a common-cell gather coincident with a common-midpoint gather. Also, consider the traces in the gather covering a 360-degree source-receiver azimuthal range, but having the same source-receiver separation. As a result of the DMO correction described above, the amplitude at some specific input time sample A of each input trace is mapped along an elliptical trajectory. When combined, the elliptical trajectories associated with all the traces constitute an ellipsoid of revolution as shown in Figure 7.2-11a. This ellipsoid represents the kinematic aspect of the impulse response of the 3-D DMO operator in a constant-velocity medium. As for the 2-D DMO correction, vertical velocity variations may be accounted for in the design of a 3-D DMO operator .

In reality, there never exists a situation that corresponds to the recording geometry of Figure 7.2-11a. Instead, 3-D recording geometries give rise to a nonuniform source-receiver azimuthal coverage with each trace in a common-cell gather associated with a different source-receiver separation as illustrated in Figure 7.2-11b. Again, the amplitude at some specific input time sample A of each input trace is mapped along an elliptical trajectory. When combined, however, the elliptical trajectories associated with all the traces constitute only a sparse representation of the surface of the ideal DMO ellipsoid.

Aside from nonuniform source-receiver azimuthal and offset coverage, in the case of marine 3-D geometry, the fact that a common-cell gather does not coincide with a common-midpoint gather causes nonuniform spatial sampling of the data output from 3-D DMO correction. Figure 7.2-11b shows the surface projections of the output samples represented by the solid circles along each of the DMO ellipses. A similar distribution of the output samples is also shown in a map view in Figure 7.2-12 for the general case of irregular spatial sampling caused by the arbitrary source-receiver azimuthal and offset coverage with midpoint scattering over the survey area. As illustrated by the source-receiver pair S1R1 at the top portion of the cell grid in Figure 7.2-12, if there were no cable feathering, the output samples (again, denoted by the solid circles) from DMO correction applied to the trace at midpoint location M1 could be placed at the centers of the cells along the receiver cable. The output samples from DMO correction applied to a trace with cable feathering, however, do not necessarily coincide with the cell centers along the source-receiver direction. This realistic situation is illustrated by the source-receiver pair S2R2 in the middle portion of the cell grid in Figure 7.2-12.

Now imagine the superposition of the distribution of the output samples from 3-D DMO correction applied to all input traces in a common-cell gather. Just as the midpoint distribution in a common-cell gather input to 3-D DMO correction can be nonuniform because of cable feathering (Figure 7.1-5), the distribution and population of the output samples that fall within a cell from 3-D DMO correction also are nonuniform. The resulting fold of coverage from 3-D DMO correction may exhibit significant variations over the survey area. Moreover, the fold of coverage is time-dependent since the parameters of DMO ellipse are time dependent. As for conventional CMP stacking, the amplitudes after DMO correction need to be compensated for the variations in fold of coverage in a time-dependent manner  . Coverage and binning issues are reviewed by Goodway and Ragan  for land, and by Brink and Ronen  for marine 3-D surveys.

Note that, ideally, the output samples, denoted by the solid circles in Figure 7.2-12, from DMO correction should be placed equidistant away from the midpoint location M2 along the line that connects the source S2 and the receiver R2 locations. In practice, however, the output samples are assigned to the cell centers denoted by the open circles. The unfortunate consequence of this output sampling is the vulnerability of the DMO-corrected data to spatial aliasing.

A way to correct for the irregular spatial sampling and undersampling of the 3-D DMO ellipse is to spread the output amplitudes away from the source-receiver azimuthal direction. . Specifically, as illustrated in Figure 7.2-12, amplitudes not only are mapped onto the cell centers, denoted by the open circles, nearest the line that joins the source S2 and receiver R2 locations, but also to the neighboring cell centers denoted by × using a set of scalars that are inversely proportional to the distance of the cell from the source-receiver line. For instance, the output amplitude at A is not only scaled and placed at the cell center B but is distributed also to cell centers denoted by × using the appropriate scaling factors.