# Anisotropic velocity analysis

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

We shall confine the discussion on anisotropy primarily to the practical case of P-wave propagation in weakly anisotropic rocks. Consider the wavefront geometry shown in Figure 11.7-1. The P-wave phase velocity is given by [1]

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

Note that the P-wave velocity depends on the anisotropy parameters δ and ε, but not on the parameter γ. In fact, the SV-wave velocity also depends only on δ and ε, while the SH-wave velocity depends only on γ.

In the special case of vertical incidence, θ = 0, equation (84) gives the vertical P-wave velocity, α0. In the special case of horizontal incidence, θ = 90 degrees, equation (84) gives

 ${\displaystyle \alpha _{h}=\alpha _{0}(1+\varepsilon ).}$ (85)

Solving for ε, we obtain

 ${\displaystyle \varepsilon ={\frac {\alpha _{h}-\alpha _{0}}{\alpha _{0}}}.}$ (86)

This equation provides a physical meaning for the Thomsen parameter ε. Specifically, the parameter ε indicates the degree of anisotropic behavior of the rock, measured as the fractional difference between vertical P-wave velocity α0 and the horizontal P-wave velocity αh. Since for most rocks ε > 0, note that the horizontal P-wave velocity is greater than the vertical P-wave velocity.

The normal-moveout velocity vNMO(ϕ = 0), where ϕ is the dip angle, for a flat reflector in an anisotropic medium is given by Thomsen [1]

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

In the special case of δ = 0, the moveout velocity is the same as the velocity of an isotropic medium represented by α0 (normal moveout).

The P-wave traveltime equation for a flat reflector in an anisotropic medium is given by Tsvankin and Thomsen [2]

 ${\displaystyle t^{2}=t_{0}^{2}+A_{2}x^{2}+{\frac {A_{4}x^{4}}{1+Ax^{2}}},}$ (88)

where t is the two-way time from source to reflector to receiver, t0 is the two-way zero-offset time, x is the source-receiver offset, and A2 and A4 are coefficients which are written below to acknowledge their complexity

 ${\displaystyle A_{2}={\frac {1}{\alpha _{0}^{2}(1+2\delta )}},}$ (89a)

 ${\displaystyle A_{4}=-{\frac {2(\varepsilon -\delta )[1+2\delta (1-\beta _{0}^{2}/\alpha _{0}^{2})^{-1}]}{t_{0}^{2}\alpha _{0}^{4}(1+2\delta )^{4}}},}$ (89b)

and

 ${\displaystyle A={\frac {A_{4}}{\alpha _{h}^{-2}-A_{2}}}.}$ (89c)

Just for comparison, we write the traveltime equation (3-11) for a flat reflector in an isotropic medium using the current notation

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

For isotropic velocity analysis using equation (90), we only need to scan one parameter — the velocity vNMO (velocity analysis). For anisotropic velocity analysis using equation (88), note that we have to do a multiparameter scan involving α0, αh, β0, δ, and ε — an impractical and unacceptable proposal.

As a way to circumvent this insurmountable problem of a multiparameter scan in velocity analysis, Alkhalifah and Tsvankin [3] define a new effective anisotropy parameter

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

By way of equations (85), (87), (91), and (89a,89b,89c), and neglecting the effect of β0, equation (88) takes a form that is suitable for practical implementation [3]

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

where vNMO is the flat-reflector moveout velocity.

Equation (92) indicates that the traveltime for a reflector in an anisotropic medium follows a nonhyperbolic trajectory. By setting η = 0, equation (92) reduces to the isotropic case of equation (90). Note that the effect of anisotropy on reflection traveltimes is most significant at far offsets. The derivation of the anisotropic moveout equation (92) is based on a single flat layer. Nevertheless, in practice, it also is applicable to the case of a horizontally layered earth model with vertical transverse isotropy [4].

Figure 11.7-4a shows CMP raypaths associated with a flat reflector in a transversely isotropic medium with a velocity behavior as shown in Figure 11.7-2. Transverse isotropy causes fundamental departures from the CMP raypath geometry associated with an isotropic medium:

1. The zero-offset raypath is non-normal incident.
2. A single common reflection point does not exist; instead, anisotropy has given rise to reflection point dispersal even for a flat reflector.
3. The anisotropic moveout given by equation (92) in general is nonhyperbolic.
4. The NMO velocity measured from the slope of the t2x2 curve indicates that anisotropy makes the velocity offset dependent (Figure 11.7-4b).

Figure 11.7-5 shows a CMP gather and its velocity spectrum computed using the isotropic traveltime equation (90). Note that there are some overcorrected events between the two-way zero-offset times of 1.5-2 s and some undercorrected events above the two-way zero-offset time of 1.5 s. The undercorrected events are clearly multiples which produce distinct semblance peaks on the velocity spectrum. The overcorrected events result from one of the three possibilities — that the overcorrection is caused by the dip effect on moveout velocities, anisotropy, or a combination of the two phenomena. In Figure 11.7-5, a close-up display of the zone of interest between 1.5-2 s shows the overcorrected events with a distinct moveout behavior that is not quite the same as what a typical overcorrected event with hyperbolic moveout looks like. Specifically, we see that the events are nearly flat with no moveout within the near-offset range, while the moveout is like the end of a hockey-stick within the far-offset range.

Try flattening the events by experimenting with various velocity picks as shown in Figure 11.7-6. Note that, whatever the pick you make on the velocity spectrum, the hockey-stick events do not quite flatten; there is always some curvature left along the moveout-corrected event.

Equation (92) indicates that, for anisotropic velocity analysis, we need to scan two parameters, vNMO and η. Note that the parameter η is only present in the fourth-order moveout term that is significant at far offsets. This suggests a two-stage parameter scan as described below.

1. Perform hyperbolic velocity analysis using only the first two terms on the right-hand side of equation (92) and the near offsets where there is no hockey-stick effect. Figure 11.7-7 shows the same CMP gather as in Figure 11.7-5, but with far-offset traces muted, and the associated velocity spectrum. This first-stage analysis would give an estimate of the moveout velocity vNMO as in equation (92).
2. Next, insert the estimated vNMO function into equation (92) and compute the η spectrum as shown in Figure 11.7-8. After picking an η function in time, apply the fourth-order moveout correction given by equation 92 to the CMP gather.

Figures 11.7-9 and 11.7-10 show portions of a CMP stacked section with and without accounting for anisotropy in velocity analysis and moveout correction. The portion shown in Figure 11.7-9 is abundantly rich in diffractions and the portion shown in Figure 11.7-10 contains steeply dipping fault-plane reflections conflicting with gently dipping reflections. Panel (a) in both figures shows the full-fold stack and panel (b) shows the near-offset stack based on isotropic velocity analysis and moveout correction using the first two terms of equation (92). Note that the full-fold stack has been adversely affected by the hockey-stick effect on moveout at far offsets, while the near-offset stack has better preserved the diffractions and dipping events. Panel (c) of both Figures 11.7-9 and 11.7-10 shows the full-fold stack derived from anisotropic velocity analysis and moveout correction using both the second-order and fourth-order terms in equation (92). Note that the full-fold stacking based on anisotropic moveout correction has better preserved the diffractions and dipping events compared to the full-fold stacking based on isotropic moveout correction. These observations are more evident on the migrated sections shown in Figures 11.7-11 and 11.7-12.

## References

1. Thomsen, 1986, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.
2. Tsvankin and Thomsen (1994), Tsvankin, I. and Thomsen, L., 1994, Nonhyperbolic reflection moveout in anisotropic media: Geophysics, 59, 1290–1304.
3. Alkhalifah and Tsvankin (1995), Alkhalifah, T. and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550–1566.
4. Grechka and Tsvankin, 1997, Grechka, V. and Tsvankin, I., 1997, Inversion of nonhyperbolic moveout in transversely isotropic media: 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1685–1688.
5. Uren et al., 1990a, Uren, N. F., Gardner, G. H. F. and McDonald, J. A., 1990a, The migrator’s equation for anisotropic media: Geophysics, 55, 1429–1434.
6. Sheriff, 1991, Sheriff, R. E., 1991, Encyclopedic dictionary of exploration geophysics: Soc. Expl. Geophys.
7. Uren et al., 1990b, Uren, N. F., Gardner, G. H. F. and McDonald, J. A., 1990b, Dip moveout in anisotropic media: Geophysics, 55, 863–867.
8. Uren et al., 1990c, Uren, N. F., Gardner, G. H. F. and McDonald, J. A., 1990c, Normal moveout in anisotropic media: Geophysics, 55, 1634–1636.