Variation of seismic velocity depending on either the direction of travel for P- or Swaves  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, (Vmax-Vmin)/Vmax, often expressed as a percentage, sometimes as the ratio of maximum and minimum velocities, Vmax/Vmin; the numerical value usually makes clear which is meant. P-wave anisotropy is usually meant unless S-wave anisotropy is specified, but anisotropy of P-waves usually implies anisotropy for S-waves and vice-versa.

a) The general elasticity tensor stiffness or its inverse compliance, relating stress and strain contains up to 21 independent constants, the number depending on the symmetry see symmetry systems. Because of symmetries, this 3X3X3X3 tensor may be written as a 6X6 matrix (Fig. 1). In isotropic media there are only two independent constants among 12 nonzero elements of this matrix.

Fig. 1 Elastic constants for isotropic media expressed in terms of each other and P- and S-wave velocities (Vp and Vs,) and density p.

b) Polar anisotropy (transverse isotropy) involves elastic properties that are the same in any direction perpendicular to a symmetry axis but different parallel to the axis. Layering is the most common cause of this situation; see Fig. 2 and polar anisotropy. Polar anisotropy involves five independent elastic constants. This symmetry is similar to that of a crystal having hexagonal symmetry.

Fig. 2. 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.

c) Azimuthal asymmetry involving orthorhombic symmetry the symmetry of a brick gives a different P-wave velocity along the three orthogonal symmetry axes and different shear-wave splitting (see d below) in the three directions. Vertically fractured horizontal layering may produce this situation. Orthorhombic asymmetry involves nine independent elastic constants.

d) In an arbitrary polar or orthorhombic anisotropic medium, for each travel direction only two orthogonal polarizations of plane shear-waves are allowed although they are not necessarily transverse to the propagation direction; they may travel with different velocities. An S-wave of arbitrary polarization entering such a region in a direction other than along the symmetry axis splits into two S-waves; this is called shearwave splitting, birefringence, S-wave splitting, or double refraction.

e) Monoclinic anisotropy is similar to orthorhombic anisotropy except that one of the three axes is not orthogonal to the other axes. It involves eleven independent elastic constants.


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