# Seismic anisotropy

A medium is seismically anisotropic if its intrinsic elastic properties, measured at a point, 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 isotropically. However, recent experience has shown that a more realistic assumption is that the subsurface is often anisotropic. (Much of the following material is taken from Seismic_Data_Analysis).

A medium is polar anisotropic ("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 with propagation angle, although it also includes the variation of (shear) velocity with polarization angle. 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 are reviewed in Seismic_Data_Analysis, both of which are special cases of transverse isotropy. First is the vertical transverse isotropy ("VTI"), 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. Shaliness, thin horizontal bedding, and fracturing parallel to the bedding all produce transverse isotropy.

Second is the horizontal transverse isotropy ("HTI"), which uses the same equations as VTI, but with the symmetry axis horizontal. This makes a special case of azimuthal anisotropy. It could, theoretically, be caused by a single set of aligned vertical, penny-shaped cracks, embedded in an otherwise isotropic medium, although this situation is vanishingly rare in geophysics.

These cases are the simplest; other cases, with azimuthal variation (e.g. orthorhombic and monoclinic) may be more realistic.

Vertical transverse isotropy stems from the fact that, as a result of depositional processes, velocities in a layer are different in the direction of bedding and the direction perpendicular to the bedding. Specifically, fine layering of isotropic (or polar anisotropic) 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 [2]. Based on velocity characteristics of the Green River shale [3], this polar plot of group velocities shows that, as for most rock types, the P-wave velocity is nonelliptical, whereas the SH-wave velocity is elliptic. 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 [4]. 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 [3], that relates the stress components to the strain components is (Section L.1)

 ${\displaystyle \{c_{\alpha \beta }\}={\begin{pmatrix}c_{11}&c_{12}&c_{13}&c_{14}&c_{15}&c_{16}\\c_{12}&c_{22}&c_{23}&c_{24}&c_{25}&c_{26}\\c_{13}&c_{23}&c_{33}&c_{34}&c_{35}&c_{36}\\c_{14}&c_{24}&c_{34}&c_{44}&c_{45}&c_{46}\\c_{15}&c_{25}&c_{35}&c_{45}&c_{55}&c_{56}\\c_{16}&c_{26}&c_{36}&c_{46}&c_{56}&c_{66}\\\end{pmatrix}}.}$ (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 2, known as Lamé’s constants, λ and μ. Consequently, the stiffness matrix given by equation (82a) reduces to the special form

 ${\displaystyle \{c_{ij}^{iso}\}={\begin{pmatrix}\lambda +2\mu &\lambda &\lambda &0&0&0\\\lambda &\lambda +2\mu &\lambda &0&0&0\\\lambda &\lambda &\lambda +2\mu &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \\\end{pmatrix}}.}$ (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 polar anisotropic ("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 5 [4] [3] . 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 [3], with a corresponding stiffness matrix

 ${\displaystyle \{c_{\alpha \beta }^{VTI}\}={\begin{pmatrix}c_{11}&c_{11}-2c_{66}&c_{13}&0&0&0\\c_{11}-2c_{66}&c_{11}&c_{13}&0&0&0\\c_{13}&c_{13}&c_{33}&0&0&0\\0&0&0&c_{44}&0&0\\0&0&0&0&c_{44}&0\\0&0&0&0&0&c_{66}\\\end{pmatrix}}.}$ (82b)

The corresponding velocities are complicated functions of these five elastic moduli:

quasi-P:  ${\displaystyle V_{P}^{2}(\theta )={\frac {1}{2\rho }}\left[c_{33}+c_{44}+(c_{11}-c_{33})\sin ^{2}(\theta )+D\right]}$

quasi-S:  ${\displaystyle V_{SV}^{2}(\theta )={\frac {1}{2\rho }}\left[c_{33}+c_{44}+(c_{11}-c_{33})\sin ^{2}(\theta )-D\right]}$

pure-S:   ${\displaystyle V_{SH}^{2}(\theta )={\frac {1}{\rho }}\left[c_{44}\cos ^{2}(\theta )+c_{66}\sin ^{2}(\theta )\right]}$


where the main complications are contained within the term

${\displaystyle D=\left\{(c_{33}-c_{44})^{2}+2[2(c_{13}+c_{44})^{2}-(c_{33}-c_{44})(c_{11}+c_{33}-2c_{44})]\sin ^{2}(\theta )+[(c_{11}+c_{33}-2c_{44})-4(c_{13}+c_{44})]\sin ^{4}(\theta )\right\}^{\frac {1}{2}}}$


Note that the only difference between ${\displaystyle V_{P}}$ and ${\displaystyle V_{SV}}$ (the velocity of a quasi-S wave whose polarization vector contains a Vertical component) is this complicated term ${\displaystyle D}$, so ${\displaystyle D}$ is the difference between a P-wave and an SV-wave! Since there are two shear-modes, with different velocities, this leads to shear-wave splitting, except at vertical incidence, where ${\displaystyle V_{SV}=V_{SH}}$.

These expressions are too complicated to be useful in geophysics. However, Thomsen [3] has elegantly re-parametrized the five elastic constants for VTI media as: 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 [3]. In fact, in the case of weak anisotropy (described by small values (≪ 1) of ε, γ, and δ), the equations for the elastic velocities simplify considerably. 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 arbitrarily strong transverse isotropy.

The special case of δ = ε is known as elliptical anisotropy [6]. 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 [3]. 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.

Physical Causes of Anisotropy

Anisotropy is always the large-scale expression of small-scale structure with preferred orientation. This means that the measurement of seismic anisotropy will vary with the wavelength used to measure it.

• At the scale of a single crystal (giga-Hertz frequencies), the small-scale structure is the arrangement of atomic cells.
• At the scale of a core (mega-Hertz frequencies), the small-scale structure is the micro-geometry of the arrangement of grains and pores. In some rocks, the crystals are randomly oriented, so that although each is intrinsically anisotropic, the random averaging means that the rock itself is isotropic (this might describe some sandstones). In other rocks (e.g. shales), there is a preferred orientation of some crystals (e.g. clays, which often have the shapes of platelets, and commonly lie horizontally), usually established by the direction of gravity during sedimentation and lithification. (The complementary pore space, occupied by fluid or kerogen, is then also flat-lying.) By itself, this usually leads to polar anisotropy.

Also at the core scale, there may be other physical causes in special circumstances. For example, if a salt body has flowed into place (rather than precipitated into place), the flow process can include re-crystallization, with preferential orientation of salt grains, related to the flow. (Tectonic flow in the upper mantle may include re-crystallization of olivine and other minerals, with preferred orientations related to that flow.)

At this core scale, there may also be micro-fractures, related to the state of stress, or a previous state of stress. It is sometimes thought that the stress itself (if the 3 principle stresses are not equal) makes for anisotropy . However, this direct stress-anisotropy is purely elastic, and so disappears once the stress is removed. By contrast, if the finite stress changes the micro-structure of the rock, e.g. by creating micro-fractures, this is called indirect stress-anisotropy, and may not be reversible, with the removal of the stress. (This could happen, for example, if the fractures were open for a long time, and minerals precipitated out of the fluid, on the crack-faces, so that the cracks do not close when the stress is removed.) In the lab, it is possible to distinguish these cases; normally one finds that the direct effect of stress is much smaller than the indirect effect.

If stress-aligned micro-fractures are present, they may be preferentially aligned with flat faces perpendicular to the direction of least compressive stress. There may be a second set, preferentially aligned with flat faces perpendicular to the direction of intermediate compressive stress. Rarely, there is a third set, preferentially aligned with flat faces perpendicular to the direction of most compressive stress. If (as is common in sedimentary basins), the maximum stress is oriented vertically, then the first two sets mentioned here are vertical, and orientated in two orthogonal horizontal directions.

• At the logging scale (kilo-Hertz frequencies), the small-scale structure may also include the layering which results from the sedimentary process, with layer-thicknesses small compared to the sonic wavelengths. By itself, this usually leads to polar anisotropy. The thin-layering need not not be periodic, although if the statistics of the layering are not stationary, then the sequence is effectively inhomogeneous, as well as anisotropic[7]. There may also be stress-induced anisotropy (both direct and indirect), caused by the stress concentrations near the borehole, which accompany the creation of the borehole wall. This will cause lower symmetry of anisotropy.
• At the reservoir scale, (seismic frequencies), the small-scale structure may also include the layering which results from the sedimentary process, with layer-thicknesses small compared to the seismic wavelength. By itself, this usually leads to polar anisotropy. (In the sub-crustal lithosphere, there may be azimuthal anisotropy at this scale, caused by sheeted-dike emplacement at spreading ridges.)

There may also be both direct and indirect (or crack-related) stress-induced anisotropy; in field data, it may not be possible to distinguish between these causes. Hence, the observed anisotropy orientations might indicate directions of either the current stress-state, or a previous stress-state, or both. If the fractures yield monoclinic anisotropy, this is an indication of a complex geologic history, which places its complex character on the current anisotropy.

If aligned cracks are present, their shapes may be limited, top and bottom, by fracture-resistant beds. Since there is no corresponding limitation in the horizontal direction, they may be "ribbon-shaped" joints, rather than "penny-shaped" micro-cracks. Such joints usually dominate the hydraulic anisotropy, although they may or may not dominate the seismic anisotropy.

## References

1. Winterstein, D. F., 1990, Velocity anisotropy terminology for geophysicists: Geophysics, 55, 1070–1088.
2. Uren, N. F., Gardner, G. H. F. and McDonald, J. A., 1990a, The migrator’s equation for anisotropic media: Geophysics, 55, 1429–1434.
3. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.
4. Officer, C. B., 1958, Introduction to the theory of sound transmission with application to the ocean: McGraw-Hill Book Co.
5. Uren, N. F., Gardner, G. H. F. and McDonald, J. A., 1990b, Dip moveout in anisotropic media: Geophysics, 55, 863–867.
6. Daley, P. F. and Hron, F., 1979, Reflection and transmission coefficients for seismic waves in ellipsoidally anisotropic media: Geophysics, 44, 27–38.
7. Backus, George E. (1962). "Long-wave elastic anisotropy produced by horizontal layering". Journal of GeophysicalResearch 67 (11): 4427–4440. doi:10.1029/JZ067i011p04427.