# Seismic anisotropy

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

A medium is anisotropic if its intrinsic elastic properties, measured at the same location, change with direction, and it is isotropic if its properties do not change with direction [1]. Most of seismic data analysis is based on the assumption that the subsurface behaves seismically isotropic. A case of anisotropy that is of interest in exploration seismology is the change in velocity with direction.

A medium is transversely isotropic if its elastic properties do not change in any direction perpendicular to an axis of symmetry. The usual meaning of seismic anisotropy is variation of seismic velocity, which itself depends on the elastic properties of the medium, with the direction in which it is measured [2]. Anisotropic variation of seismic velocity must not be confused with the source-receiver azimuthal variation of moveout velocity for a dipping reflector in an isotropic medium (equation 7-3).

There are two cases of seismic anisotropy that we shall review in this section, both of which are special cases of transverse isotropy. For convenience, consider a horizontally layered earth model. First, is the vertical transverse isotropy, otherwise simply referred to as transverse isotropy, for which velocities do not vary from one lateral direction to another, but vary from one direction to another on a vertical plane that coincides with a given lateral direction. Horizontal bedding and fracturing parallel to the bedding produce transverse isotropy. Second, is the horizontal transverse isotropy, otherwise known as azimuthal anisotropy, for which velocities vary from one lateral direction to another. Fracturing in a direction other than the bedding direction gives rise to azimuthal anisotropy.

Transverse isotropy stems from the fact that, as a result of a depositional process, velocities in a layer are different in the direction of bedding and the direction perpendicular to the bedding. Specifically, fine layering of isotropic beds that constitute the depositional unit give rise to an anisotropy with its axis of symmetry perpendicular to the bedding plane. Azimuthal anisotropy can stem from the fact that, as a result of a tectonic process, rock material associated with a layer may have different rigidity in different azimuthal directions. A specific example is secondary fracturing in rocks whereby the velocity in the fracture direction is higher than the velocity in the orthogonal direction, again, giving rise to an anisotropy with its axis of symmetry parallel to the bedding plane.

Refer to Figure 11.7-1 to review the physical aspects of wave propagation in an anisotropic medium. The directional change of the velocity is illustrated by the skewed ellipse in Figure 11.7-1a, with the fast velocity in the direction of the major axis and the slow velocity in the direction of the minor axis. Now consider Huygens’ secondary sources situated along the wavefront A, say at time t. This wavefront actually coincides with an exploding reflector dipping at an angle ϕ. Because of the anisotropic behavior of the propagation medium, the Huygens’ sources do not emanate semicircular wavefronts; instead, the wavefronts are skewed in the direction of the fast velocity. These skewed wavefronts will form the plane wavefront B at a later time t + Δt. While the energy was transmitted along the raypath SP at group velocity, the wavefront that represents a constant phase actually traveled from position A to B along TP normal to the wavefront at phase velocity. Because the group velocity is associated with the raypath, it is sometimes referred to as the ray velocity. Similarly, because the phase velocity is associated with the wavefront, it is sometimes referred to as the wavefront velocity. Note that the wavefront angle θ associated with the phase velocity is different from the ray angle ϕ associated with the group velocity. Only if the medium were isotropic, Huygens’ secondary sources would produce semicircular wavefronts and the phase angle θ would coincide with the ray angle ϕ.

Note from Figure 11.7-1 that the zero-offset raypath SP does not make a right angle with the reflector; thus, in the case of anisotropy, the zero-offset ray is not a normal-incident ray as would be in the case of isotropy. This behavior can be better explained by sketching the isotropic and anisotropic wavefronts as shown in Figure 11.7-1b. Specifically, in the case of isotropic medium, the wavefront emanating from a point P is circular and the zero-offset ray is normal-incident to the reflector. Whereas, in the case of an anisotropic medium, the wavefront is skewed and the zero-offset ray impinges on the reflector at a non-normal incidence angle.

An example of how velocity changes with direction in an anisotropic medium is shown in Figure 11.7-2 [3]. Based on velocity characteristics of the Green River shale [4], this polar plot of group velocities shows that, as for most rock types, the P-wave velocity is nonelliptical, whereas the SH-wave velocity behaves elliptically anisotropic. Also note that the horizontal P-wave velocity is greater than the vertical P-wave velocity.

We shall associate seismic anisotropy primarily with seismic velocities. Aside from anisotropic velocity analysis, this means that we will need to review those processes that are intimately influenced by velocity anisotropy, such as dip-moveout correction, migration and AVO analysis.

The generalized form of Hooke’s law, which is the foundation of linear elastic theory, states that each stress component can be expressed as a linear combination of all the strain components [5]. Hooke’s law is based on the assumption that elastic deformations in solids are infinitesimally small. The stiffness matrix, otherwise known as the elastic modulus matrix [4], that relates the stress components to the strain components is (Section L.1)

 ${\displaystyle \{c_{ij}\}={\begin{pmatrix}c_{11}&c_{12}&c_{13}&c_{14}&c_{15}&c_{16}\\c_{21}&c_{22}&c_{23}&c_{24}&c_{25}&c_{26}\\c_{31}&c_{32}&c_{33}&c_{34}&c_{35}&c_{36}\\c_{41}&c_{42}&c_{43}&c_{44}&c_{45}&c_{46}\\c_{51}&c_{52}&c_{53}&c_{54}&c_{55}&c_{56}\\c_{61}&c_{62}&c_{63}&c_{64}&c_{65}&c_{66}\\\end{pmatrix}}.}$ (82a)

The elements of the stiffness matrix are the elastic constants of an elastic solid. Since this matrix is symmetric, cij = cji, there are 21 independent constants for an elastic medium.

For an isotropic solid, the elastic behavior of which is independent of direction at a point within the solid, the number of independent elastic constants is only two, known as Lamé’s constants, λ and μ. Consequently, the stiffness matrix given by equation (82a) reduces to the special form

 ${\displaystyle \{c_{ij}\}={\begin{pmatrix}\lambda +2\mu &\lambda &\lambda &0&0&0\\\lambda &\lambda +2\mu &\lambda &0&0&0\\\lambda &\lambda &\lambda +2\mu &0&0&0\\0&0&0&2\mu &0&0\\0&0&0&0&2\mu &0\\0&0&0&0&0&2\mu \\\end{pmatrix}}.}$ (82b)

To describe the P- and S-wave propagation in isotropic solids, only two elastic parameters are needed — Lamé’s constants, λ and μ. Other elastic parameters — Young’s modulus E, Poisson’s ratio σ and bulk modulus κ, and the P- and S-wave velocities can all be expressed in terms of λ and μ (Figure 11.0-1 and Appendix L.1).

For a transversely isotropic solid, the elastic behavior of which is the same in two orthogonal directions but different in the third direction, the number of independent constants is five [5] [4] [2]. For the more specific case of vertically transverse isotropy (VTI), which has a vertical symmetry axis, the five independent elastic constants are c11, c13, c33, c44, and c66 [4].

To explicitly describe the effect of anisotropy in wave propagation, Thomsen [4] has elegantly redefined the five elastic constants for the VTI media — the vertical P- and S-wave velocities, α0 and β0, in the vertical direction,

 ${\displaystyle \alpha _{0}={\sqrt {\frac {c_{33}}{\rho }}},}$ (83a)

 ${\displaystyle \beta _{0}={\sqrt {\frac {c_{44}}{\rho }}},}$ (83b)

and three constants that describe the degree of anisotropy, ε, γ, and δ, in terms of the five constants c11, c13, c33, c44, and c66

 ${\displaystyle \varepsilon ={\frac {c_{11}-c_{33}}{2c_{33}}},}$ (83c)

 ${\displaystyle \delta ={\frac {(c_{13}+c_{44})^{2}-(c_{33}-c_{44})^{2}}{2c_{33}(c_{33}-c_{44})}},}$ (83d)

and

 ${\displaystyle \gamma ={\frac {c_{66}-c_{44}}{2c_{44}}}.}$ (83e)

For most sedimentary rocks, the parameters ε, δ, and γ are of the same order of magnitude and usually much less than 0.2 [4]. In fact, the Thomsen parameters given by equations (83c,83d,83e) relate to the case of weak anisotropy described by small values (≪ 1) of ε, γ, and δ. While applications of anisotropy in seismic data analysis are primarily based on the assumption of weak anisotropy, these parameters still are useful to describe the general case of transverse isotropy.

The special case of δ = ε is known as elliptical anisotropy [7]. The ellipticity is associated with the shape of the wavefront expanding from a point source. Albeit its underlying theory is simpler than the general theory of anisotropy, elliptical anisotropy occurs in nature only rarely. Figure 11.7-3 shows a crossplot of the two Thomsen parameters, ε and δ, for various types of sedimentary and crystalline rocks based on field and laboratory studies [4]. Note that a few of the rock samples closely satisfy the ellipticity condition, ε = δ. Also, for most rock types, the Thomsen parameters, ε and δ, are positive constants less than 0.2; thus, the supporting evidence for weak anisotropy theory.

## References

1. Winterstein, 1990, Winterstein, D. F., 1990, Velocity anisotropy terminology for geophysicists: Geophysics, 55, 1070–1088.
2. Sheriff, 1991, Sheriff, R. E., 1991, Encyclopedic dictionary of exploration geophysics: Soc. Expl. Geophys.
3. 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.
4. Thomsen, 1986, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.
5. Officer, 1958, Officer, C. B., 1958, Introduction to the theory of sound transmission with application to the ocean: McGraw-Hill Book Co.
6. 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.
7. Daley and Hron, 1979, Daley, P. F. and Hron, F., 1979, Reflection and transmission coefficients for seismic waves in ellipsoidally anisotropic media: Geophysics, 44, 27–38.