# Polar anisotropy

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Polar anisotropy has a pole of rotational symmetry (hence the name). This corresponds to undeformed (and unfractured) shales, and to sequences of thin (compared to the seismic wavelength) beds of isotropic and/or polar-ansiotropic symmetry. It is also called "Transverse Isotropy" because all directions normal to the pole have the same velocities. In seismics, the corresponding elastic stiffness matrix (symmetric) has five independent components (in Voigt notation):

${\displaystyle \{c_{\alpha \beta }\}={\begin{pmatrix}c_{11}&c_{12}&c_{13}&0&0&0\\c_{12}&c_{11}&c_{13}&0&0&0\\c_{13}&c_{13}&c_{33}&0&0&0\\0&0&0&c_{44}&0&0\\0&0&0&0&c_{44}&0\\0&0&0&0&0&c_{66}\\\end{pmatrix}}}$

with ${\displaystyle c_{12}=c_{11}-2c_{66}}$. These indices refer to coordinate directions in the natural coordinate system of the medium; the 3-direction (normally vertical) is the pole of symmetry.

The resulting expressions for the seismic velocities at any incidence angle may be found with conventional algebraic techniques; they are quite complicated. [1] [2]. The assumption of weak[3] polar anisotropy makes it feasible to analyze real data for polar anisotropy.

## References

1. Rudzki,M. P., 1915. Über die Theorie der Erdbebenwellen: Die Naturwissenschaften,3, 201–204.
2. Thomsen, L., 2014. Seismic Anisotropy in Exploration and Exploitation, the SEG/EAGE Distinguished Instructor Short Course #5 Lecture Notes, 2nd Edition, Soc. Expl. Geoph., Tulsa
3. Thomsen, L., 1986. Weak Elastic Anisotropy, Geophysics, 51(10), pp. 1954-1966.