# Dictionary:Anisotropy (seismic)

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Variation of seismic velocity depending on either the direction of travel (for P- or S- waves) or the direction of polarization (for S-waves). Velocity anisotropy (or coefficient of anisotropy) is sometimes taken as the fractional difference between the maximum and minimum velocities in different directions (within a given dataset), ${\displaystyle {\frac {V_{max}-V_{min}}{V_{max}}}}$, often expressed as a percentage, sometimes as the ratio of maximum and minimum velocities, ${\displaystyle {\frac {V_{max}}{V_{min}}}}$; the numerical value usually makes clear which is meant. Thus defined, the value depends on the survey (e.g. the angular aperture), and may not represent the medium itself.

P-wave anisotropy is usually meant (in exploration geophysics), unless S-wave anisotropy is specified. Of course, anisotropy of P-waves usually implies anisotropy for S-waves, and vice-versa. Detection of P-wave anisotropy requires more than one source-receiver pair (i.e. more than one propagation angle), whereas "shear-wave splitting" may be detected at a single station.

Anisotropy may be analyzed in terms of the symmetry of the medium:

• The general elasticity tensor (stiffness or its inverse compliance) is a 3X3X3X3 tensor relating stress and strain. Among its 81 components, it contains up to 21 independent constants, the number depending on the symmetry (see symmetry systems). Because of symmetries, this tensor may be written as a 6X6 matrix. In the general case, for each direction of plane-wave propagation there is one "quasi-longitudinal" wave and two "quasi-shear" waves (with two characteristic angles of polarization, and two different velocities, leading to "shear-wave splitting".[1]
• In isotropic media (the simplest case) there are only two independent constants among 12 nonzero elements of this matrix. These may be taken as the longitudinal modulus M and the shear modulus ${\displaystyle \mu }$. A non-zero linear combination, ${\displaystyle \lambda =M-(2*\mu )}$ also appears in the matrix. Other elastic parameters, such as the bulk modulus ${\displaystyle K=M-(4/3)\mu }$ may also be computed from these, see E-5. In the isotropic case, the longitudinal wave is exactly longitudinal (polarization vector exactly parallel to wave vector), and the two shear waves degenerate to one (with a velocity independent of polarization direction, which may lie anywhere within the plane exactly perpendicular to the wave vector, hence no shear-wave splitting).
Anisotropy. (a) Application of Huygens’ principle to anisotropic velocity illustrates why phase and ray velocities may differ in both direction and magnitude. (b) The application of Fermat’s principle to anisotropic velocity illustrates why the angle of incidence for a reflection for a coincident source and receiver may not make a right angle with the reflector. (c) SH-wavefronts in transversely isotropic media are elliptical but P- and SV-wavefronts are not.
• Polar anisotropy (transverse isotropy) is the simplest geophysical case of anisotropy. It has elastic properties that are independent of the azimuth about a polar axis (hence the name) of symmetry, which is usually vertical. It is associated with, for example, unfractured shales or thinly-bedded sequences (see Physical Causes of Anisotropy, below). An example is given in Figure A-14. Polar anisotropy has five independent constants among 12 nonzero elements of the stiffness (or compliance) matrix. However, the anisotropic seismic behaviour is governed directly by certain combinations of these five stiffnesses; for body waves they may be taken as the Thomsen anisotropic parameters[2][3][4].
• Azimuthal anisotropy is a general term describing all lower symmetries, which all have azimuthal variation of elastic properties. In exploration geophysics, it is usually caused by aligned fractures.
• The simplest plausible case of azimuthal anisotropy, in exploration geophysics, is that of Orthorhombic anisotropy (more properly: "orthotropic"). It has the symmetry of a brick, with nine independent constants among 12 nonzero elements of the stiffness (or compliance) matrix. In exploration geophysics, it is usually caused by a single set of aligned fractures in an otherwise polar anisotropic medium, or perhaps two such sets, orthogonal to each other. Despite the complexity, it is commonly feasible to analyze modern wide-azimuth (WAZ) datasets in terms of orthorhombic anisotropy [5].
• The most realistic case of anisotropy is that of monoclinic anisotropy. It has twelve independent constants among 18 nonzero elements of the stiffness (or compliance) matrix. However, this is also more difficult to analyze than are the simpler symmetries.