# Anisotropic 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

Since anisotropy directly influences propagation velocity, it certainly should have an impact on dip-moveout correction. Recall from principles of dip-moveout correction that 2-D dip-moveout (DMO) correction removes the dip effect on moveout velocities, and from processing of 3-D seismic data that 3-D dip-moveout correction removes both the dip and source-receiver azimuth effects on moveout velocities. Rewrite the normalized moveout velocity relation for the 2-D case  using the current notation

 ${\frac {v_{NMO(\phi )}}{v_{NMO}(0)}}={\frac {1}{\cos \phi }},$ (93)

and note that, from the standpoint of velocities, isotropic DMO correction maps the moveout velocity vNMO(ϕ) for a dipping reflector to the moveout velocity vNMO(0) = α0 for a flat reflector.

While the DMO process corrects for the dip effect on moveout velocities as described by equation (93), it also maps common-offset data, which have been moveout-corrected using the flat-event velocities vNMO(0), to zero offset. Recall from principles of dip-moveout correction that this mapping is achieved by an integral transform given by equation (5-14a) where the factor A of equation (5-5) in the current notation is given by

 $A={\sqrt {1+{\frac {h^{2}}{t_{n}^{2}}}\left[{\frac {2\sin \phi }{v_{NMO}(0)}}\right]^{2}}}.$ (94)

Here, tn is the event time after NMO correction using the flat-event velocity vNMO, h is the half-offset and ϕ is the reflector dip.

As discussed in principles of dip-moveout correction, the integral transform factor A given by equation (94) requires knowledge of the reflector dip ϕ to perform the DMO correction. To circumvent this requirement, we use the relation (Section D.1)

 $\sin \phi ={\frac {v_{NMO}(0)k_{y}}{2\omega _{0}}},$ (95)

which states that the reflector dip ϕ can be expressed in terms of midpoint wavenumber ky and frequency ω0 — the Fourier duals of midpoint y0 and zero-offset event time τ0 associated with the DMO-corrected data, respectively. Substitute equation (95) into equation (94) to get

 $A={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{t_{n}^{2}\omega _{0}^{2}}}}}.$ (96)

The frequency-wavenumber domain dip-moveout correction 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 of equation (5-14a) where A is given by equation (96).

We now examine the DMO mapping, specifically the transform factor A of equation (94) for the case of an anisotropic medium. Consider a dipping reflector situated in an anisotropic medium with its symmetry axis not coincident with the normal to the reflector. For this general case, the normal-moveout velocity is given by 

 $v_{NMO}(\phi )={\frac {\alpha (\phi )}{\cos \phi }}{\frac {\sqrt {1+{\frac {1}{\alpha (\phi )}}\left[{\frac {d^{2}\alpha (\theta )}{d\theta ^{2}}}\right]_{\theta =\phi }}}{1-{\frac {\tan \phi }{\alpha (\phi )}}\left[{\frac {d\alpha (\theta )}{d\theta }}\right]_{\theta =\phi }}},$ (97)

where θ is the phase angle measured from the vertical, ϕ is the reflector dip (Figure 11.7-2), and α is the phase velocity evaluated in the direction coincident with the reflector dip, θ = ϕ.

By using equation (84) for the phase velocity, Tsvankin  computes the derivatives in equation (97) and derives an expression for the moveout velocity associated with the case of weak anisotropy. Below, we write his equation in normalized and compact form

 $\alpha (\theta )=\alpha _{0}(1+\delta \sin ^{2}\theta \cos ^{2}\theta +\varepsilon \sin ^{4}\theta ).$ (84)

 ${\frac {v_{NMO}(\phi )}{v_{NMO}(0)}}={\frac {1}{\cos \phi }}[1+(\varepsilon -\delta )B],$ (98)

where

 $B={\frac {\sin ^{3}\phi }{1-\sin ^{2}\phi }}(4\sin ^{4}\phi -9\sin ^{2}\phi +6).$ (99)

Equation (98) states that anisotropic DMO correction maps the moveout velocity vNMO(ϕ) for a dipping reflector to the moveout velocity vNMO(0) for a flat reflector. By setting the anisotropy parameters ε = δ = 0, or making the assumption that the anisotropy is elliptical (ε − δ = 0), note that equation (98) reduces to the isotropic case given by equation (93).

Compare equations (93) and (98) and note that, for the case of ε − δ > 0, the isotropic DMO correction factor is smaller than the anisotropic DMO correction factor. This means that an anisotropic DMO correction would have a larger aperture than the isotropic DMO correction. However, depending on the magnitude of the anisotropy parameters, ε and δ, and their signs, the opposite could also be true. In fact, setting ε = δ in equation (98), the case of elliptical anisotropy, the effect of anisotropy on dip-moveout cancels out altogether.

We are now ready to redefine the transform factor A of equation (94) for anisotropic DMO correction . To use the anisotropic moveout velocity of equation (98), first, rewrite equation (94) as

 $A={\sqrt {1+{\frac {4h^{2}}{t_{n}^{2}}}\left[{\frac {1-\cos ^{2}\phi }{v_{NMO}^{2}(0)}}\right]}}.$ (100a)

Then substitute equation (93) to get the desired expression

 $A={\sqrt {1+{\frac {4h^{2}}{t_{n}^{2}}}\left[{\frac {1}{v_{NMO}^{2}(0)}}-{\frac {1}{v_{NMO}^{2}(\phi )}}\right]}}.$ (100b)

By substituting equations (98) and (95) into equation (100b), and making the weak anisotropy assumption, we obtain the transform factor given by Anderson and Tsvankin , which we write below in compact form,

 $A={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{t_{n}^{2}\omega _{0}^{2}}}{\Big [}1+(\varepsilon -\delta )C{\Big ]}}},$ (101)

where

 $C=8\sin ^{4}\phi -18\sin ^{2}\phi +12,$ (102)

and sin ϕ is given by equation (95) in terms of vNMO(0), ky, and ω0.

To implement the frequency-wavenumber domain anisotropic dip-moveout correction, again, use the integral of equation (5-14a) where A is given by equation (101).

By setting the anisotropy parameters ε = δ = 0, or making the assumption that the anisotropy is elliptical (ε − δ = 0), note that equation (101) reduces to the isotropic case given by equation (96). The implementation of anisotropic DMO correction requires a simple modification to the isotropic implementation, given by the factor [1 + (ε − δ)C]. Note that both for the moveout velocity given by equation (98) and the DMO transform factor given by equation (101), it is the difference ε − δ, and not the individual anisotropy parameters, that dictates the anisotropic effect. What remains to be determined is the difference ε − δ. Combine equations (87) and (91) to get the relation

 $v_{NMO}(0)=\alpha _{0}{\sqrt {1+2\delta }}.$ (87)

 $\eta ={\frac {\varepsilon -\delta }{1+2\delta }}.$ (91)

 $\varepsilon -\delta =\left[{\frac {v_{NMO}(0)}{\alpha _{0}}}\right]^{2}\eta .$ (103)

The effective anisotropy parameter η and the flat-reflector moveout velocity vNMO(0) are estimated by anisotropic velocity analysis described earlier in this section (equation 92). The scaling velocity α0 is associated with a vertically incident P-wave in an isotropic, horizontally layered earth. Finally, note that isotropic DMO correction (equation 96) is velocity-independent, whereas anisotropic DMO correction (equation 101) is velocity-dependent by way of equation (102).

The anisotropic DMO impulse response can vary in shape and depart from the familiar elliptical shape associated with the isotropic DMO impulse response. Figure 11.7-13 shows some of the anisotropic impulse responses created by Anderson and Tsvankin  using a frequency-wavenumber domain DMO correction based on the above formulation. For ε − δ = 0, the case of elliptical anisotropy, the impulse response kinematically is identical to the isotropic impulse response. For ε − δ > 0, the aperture of the anisotropic impulse response broadens while maintaining the upward curvature, and for ε − δ < 0, the curvature is reversed downward.

An interesting theoretical conjecture can be drawn from the behavior of isotropic and anisotropic DMO impulse responses shown in Figures 11.7-13a,b. While anisotropy causes the DMO aperture to be broadened, vertically increasing velocity causes the DMO aperture to be narrowed (principles of dip-moveout correction). As a result of these two counteracting effects, we may conclude that anisotropic DMO correction that accounts for vertically varying velocity may be equivalent to constant-velocity isotropic DMO correction. Does this mean that, for most field data cases where velocities vary vertically, it suffices to perform constant-velocity isotropic DMO correction even in the presence of anisotropy? This would implicitly account for any anisotropic behavior that might be present in the data. In fact, Levin  conducted numerical studies of reflection times from a dipping plane in a transversely isotropic media using elastic parameters associated with four different sandstone, shale, and limestone samples, and concluded that, when the symmetry axis of transverse anisotropy is perpendicular to the reflector, the isotropic moveout velocity given by equation (93) is largely valid. Larner  extended this work to the case of a medium with vertically varying velocity and reached a similar conclusion that, for most cases of weak anisotropy, isotropic DMO correction is largely valid.

We refer to the field data example shown in Figure 11.7-14. Following isotropic DMO correction, the CMP gather as in Figure 11.7-5 exhibits the hockey-stick behavior of the anisotropic moveout more distinctly. Whatever the velocity pick, the isotropic moveout correction using equation (90) fails to flatten the events within the time gate 1.5-2 s (Figure 11.7-15). By first applying isotropic moveout correction using the near-offset traces in the gather, we derive an estimate of the normal-moveout velocity vNMO(0) as shown in Figure 11.7-16. This then is followed by the analysis of the anisotropy parameter η and applying the fourth-order moveout correction using equation (92) (Figure 11.7-17). Note that, despite the DMO correction that did not account for anisotropy, the subsequent anisotropic moveout correction has been successful in flattening the events almost completely.

 $t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{NMO}^{2}}}.$ (90)

Scan the anisotropic velocity analyses shown in Figures 11.7-18 through 11.7-23 to examine the extent of anisotropy manifested by these DMO-corrected gathers and how well the anisotropy has been accounted for by the post-DMO velocity analysis. The failure in flattening the events completely in some gathers may stem from various sources of error associated with the assumptions made about anisotropy, and it may even be caused by not correcting for anisotropy during DMO correction that preceded the velocity analysis.

Figures 11.7-24 and 11.7-25 show portions of a stacked section with isotropic DMO correction which was followed by velocity analysis with and without anisotropy accounted for. Compare with the corresponding panels in Figures 11.7-9 and 11.7-10 and note that the steep fault-plane reflections and diffraction flanks have been preserved by DMO correction. In fact, much of the task of preserving steep dips with conflicting dips already has been achieved by isotropic dip-moveout processing as seen in panel (a) in both Figures 11.7-24 and 11.7-25 that shows the full-fold stack based on isotropic velocity analysis and moveout correction using the first two terms of equation (92). For comparison, panel (b) in both Figures 11.7-24 and 11.7-25 shows the near-offset stack based on isotropic processing. Panel (c) of both Figures 11.7-24 and 11.7-25 shows the full-fold DMO stack derived from post-DMO anisotropic velocity analysis and moveout correction using both the second-order and fourth-order terms in equation (92). Compared with panel (a) in the same figures, the difference between the isotropic and anisotropic processing may be viewed at best as marginal. Again, this inconclusive result is attributable to the fact that DMO correction was done without accounting for anisotropy. The marginal differences are also evident on the migrated sections shown in Figures 11.7-26 and 11.7-27. By accounting for anisotropy in DMO correction, however, the subsequent imaging of the steeply dipping fault-plane reflections can be improved .

 $t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{NMO}^{2}}}-{\frac {2\eta x^{4}}{v_{NMO}^{2}\left[t_{0}^{2}v_{NMO}^{2}+(1+2\eta )x^{2}\right]}},$ (92)

Throughout the various stages in data analysis, we encounter different moveout types:

1. Normal moveout associated with flat events,
2. Dip moveout associated with dipping events,
3. Azimuthal moveout associated with 3-D recording geometry of varying source-receiver directions, and
4. Anisotropic moveout caused by directional changes in velocity.

In practice, the moveout associated with an event observed in a CMP gather actually is a combination of all the contributions of the moveout effects listed above. It makes sense in practice to attempt to break up the total moveout into individual components and correct for them one at a time. A pragmatic workflow for moveout correction is NMO correction (moveout type (a)), followed by DMO correction (combined moveout types (b) and (c)), and anisotropic moveout correction (moveout type (d)). The 2-D field data example shown in Figures 11.7-26 and 11.7-27 is based on this moveout correction sequence with the exclusion of moveout type (c) related to 3-D data.