Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | 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]}

**(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, *α*_{0}. In the special case of horizontal incidence, *θ* = 90 degrees, equation (**84**) gives

**(85)**

Solving for *ε*, we obtain

**(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 *α*_{0} and the horizontal *P*-wave velocity *α _{h}*. 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 *v _{NMO}*(

*ϕ*= 0), where

*ϕ*is the dip angle, for a flat reflector in an anisotropic medium is given by Thomsen

^{[1]}

**(87)**

In the special case of *δ* = 0, the moveout velocity is the same as the velocity of an isotropic medium represented by *α*_{0} (normal moveout).

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

**(88)**

where *t* is the two-way time from source to reflector to receiver, *t*_{0} is the two-way zero-offset time, *x* is the source-receiver offset, and *A*_{2} and *A*_{4} are coefficients which are written below to acknowledge their complexity

**(89a)**

**(89b)**

and

**(89c)**

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

**(90)**

For isotropic velocity analysis using equation (**90**), we only need to scan one parameter — the velocity *v _{NMO}* (velocity analysis). For anisotropic velocity analysis using equation (

**88**), note that we have to do a multiparameter scan involving

*α*

_{0},

*α*,

_{h}*β*

_{0},

*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

**(91)**

By way of equations (**85**), (**87**), (**91**), and (**89a**,**89b**,**89c**), and neglecting the effect of *β*_{0}, equation (**88**) takes a form that is suitable for practical implementation ^{[3]}

**(92)**

where *v _{NMO}* 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:

- The zero-offset raypath is non-normal incident.
- A single common reflection point does not exist; instead, anisotropy has given rise to reflection point dispersal even for a flat reflector.
- The anisotropic moveout given by equation (
**92**) in general is nonhyperbolic. - The NMO velocity measured from the slope of the
*t*^{2}−*x*^{2}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.

**Figure 11.7-1**(a) Application of Huygens’ principle to anisotropic plane-wave propagation from an exploding reflector after^{[5]}. (b) The isotropic and anisotropic wavefront associated with an expanding*P*-wave after^{[6]}. See text for details.**Figure 11.7-2**Group velocity functions for*P*- and*SH*-waves in Green River shale (After^{[7]}^{[1]}.**Figure 11.7-3**Crossplot of*P*-wave anisotropy parameters,*ε*and*δ*, associated with various rock samples after^{[1]}.**Figure 11.7-4**(a) CMP raypaths associated with a flat reflector in a transversely isotropic medium; (b) the*t*^{2}−*x*^{2}plot that shows the offset-dependent behavior of the NMO velocity (After^{[8]}.**Figure 11.7-5**(a) An NMO-corrected CMP gather and (b) its velocity spectrum based on isotropic moveout equation (**90**); (c) and (d) are enlargements of (a) and (b), respectively.**Figure 11.7-6**(a) The CMP gather and (b) its velocity spectrum as in Figure (11.7-5c,d), except that the NMO correction has been applied using a different velocity function.**Figure 11.7-7**(a) The NMO-corrected CMP gather as in Figure 11.7-5a but with a different mute, and (b) its velocity spectrum; (c) and (d) are enlargements of (a) and (b), respectively.**Figure 11.7-8**(a) The NMO-corrected CMP gather as in Figure 11.7-5c and (b) its*η*-spectrum computed from the anisotropic moveout equation (**92**); (c) the same gather as in (a) after anisotropic moveout correction using the*η*function picked from the spectrum in (d).

Equation (**92**) indicates that, for anisotropic velocity analysis, we need to scan two parameters, *v _{NMO}* 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.

- 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*v*as in equation (_{NMO}**92**). - Next, insert the estimated
*v*function into equation (_{NMO}**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.

**Figure 11.7-9**Portions of (a) full-fold isotropic stack, (b) near-offset isotropic stack and (c) full-fold anisotropic stack.**Figure 11.7-10**Portions of (a) full-fold isotropic stack, (b) near-offset isotropic stack and (c) full-fold anisotropic stack.**Figure 11.7-11**Portions of (a) full-fold isotropic stack, (b) near-offset isotropic stack and (c) full-fold anisotropic stack as in Figure 11.7-9 after poststack time migration.**Figure 11.7-12**Portions of (a) full-fold isotropic stack, (b) near-offset isotropic stack and (c) full-fold anisotropic stack as in Figure 11.7-10 after poststack time migration.

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Thomsen, 1986, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966. - ↑ Tsvankin and Thomsen (1994), Tsvankin, I. and Thomsen, L., 1994, Nonhyperbolic reflection moveout in anisotropic media: Geophysics, 59, 1290–1304.
- ↑
^{3.0}^{3.1}Alkhalifah and Tsvankin (1995), Alkhalifah, T. and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550–1566. - ↑ 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.
- ↑ 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.
- ↑ Sheriff, 1991, Sheriff, R. E., 1991, Encyclopedic dictionary of exploration geophysics: Soc. Expl. Geophys.
- ↑ 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.
- ↑ 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.

## See also

- Seismic anisotropy
- Anisotropic dip-moveout correction
- Anisotropic migration
- Effect of anisotropy on AVO
- Shear-wave splitting in anisotropic media