# Dictionary:Transverse isotropy

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Also called polar or azimuthal anisotropy; see Figure T-13. It involves elastic properties that are the same in any direction perpendicular to a symmetry axis and it has five independent elastic constants. This symmetry is like a crystal having hexagonal symmetry; see Figure S-29.

Layering and parallel fracturing tend to produce transverse isotropy. A sequence of isotropic layers (such as sedimentary bedding) produces thin-layer (also called periodic thin-layer PTL) anisotropy, although the layering need not be periodic) for wavelengths that are appreciably larger than the layer thicknesses. The axis of symmetry is perpendicular to the bedding with the velocities of P- and SH-waves parallel to the bedding being larger than that perpendicular to the bedding. The velocity parallel to the bedding is greater because the higher-velocity members carry the first energy whereas in measurements perpendicular to the bedding all members contribute proportionally to the time taken to traverse their thicknesses. Parallel isotropic layering where there are more than eight or so layers per wavelength behaves as a transversely isotropic medium.

Near-vertical jointing/fracturing/microcracks (azimuthal anisotropy, sometimes called extensive dilatancy anisotropy, EDA) tends to have a horizontal axis of symmetry perpendicular to the fracturing and the velocity of waves that are polarized parallel to the fracturing is larger than for those perpendicular to the fracturing. The symmetry axis may be tilted. This situation is involved in shear-wave splitting (q.v.) or birefringence.

Vertically fractured horizontal bedding may produce orthorhombic symmetry (the symmetry of a brick) where velocity is different along the three orthogonal symmetry axes. This situation involves nine independent elastic constants and leads to different S-wave splitting in the three directions.

With a vertical symmetry axis, pure S- and P-waves may exist only in certain directions. SH-wavefronts are ellipsoidal in shape (elliptical anisotropy, q.v.; see Figure A-10c) and SV- and P-modes of propagation are coupled with wavefronts that in general are not orthogonal to the directions of wave propagation. Phase velocity is velocity perpendicular to a surface of constant phase (wavefront) and ray velocity (in the direction of energy transport, also called group velocity) is generally not in the same direction as phase velocity, phase and ray velocities being different (see Figure A-14a). The reciprocal of phase velocity, also a vector quantity, is called slowness. The surfaces for SV-wavefronts may have cusps.

## Transverse isotropy in geophysics

In geophysics, a common assumption is that the rock formations of the crust are locally polar anisotropic (transversely isotropic); this is the simplest case of geophysical interest. Backus upscaling[1] is often used to determine the effective transversely isotropic elastic constants of layered media for long wavelength seismic waves.

Assumptions that are made in the Backus approximation are:

• All materials are linearly elastic
• No sources of intrinsic energy dissipation (e.g. friction)
• Valid in the infinite wavelength limit, hence good results only if layer thickness is much smaller than wavelength
• The statistics of distribution of layer elastic properties are stationary, i.e., there is no correlated trend in these properties.

For shorter wavelengths, the behavior of seismic waves is described using the superposition of plane waves. Transversely isotropic media support three types of elastic plane waves:

• a quasi-P wave (polarization direction almost equal to propagation direction)
• a quasi-S wave
• a S-wave (polarized orthogonal to the quasi-S wave, to the symmetry axis, and to the direction of propagation).

Solutions to wave propagation problems in such media may be constructed from these plane waves, using Fourier synthesis.

### Backus upscaling (Long wavelength approximation)

A layered model of homogeneous and isotropic material, can be up-scaled to a transverse isotropic medium, proposed by Backus.[1]

Backus presented an equivalent medium theory, a heterogeneous medium can be replaced by a homogeneous one which will predict the wave propagation in the actual medium.[2] Backus showed that layering on a scale much finer than the wavelength has an impact and that a number of isotropic layers can be replaced by a homogeneous transversely isotropic medium that behaves exactly in the same manner as the actual medium under static load in the infinite wavelength limit.

If each layer ${\displaystyle i}$ is described by 5 transversely isotropic parameters ${\displaystyle (a_{i},b_{i},c_{i},d_{i},e_{i})}$, specifying the matrix

${\displaystyle {\underline {\underline {{\mathsf {C}}_{i}}}}={\begin{bmatrix}a_{i}&a_{i}-2e_{i}&b_{i}&0&0&0\\a_{i}-2e_{i}&a_{i}&b_{i}&0&0&0\\b_{i}&b_{i}&c_{i}&0&0&0\\0&0&0&d_{i}&0&0\\0&0&0&0&d_{i}&0\\0&0&0&0&0&e_{i}\\\end{bmatrix}}}$

The elastic moduli for the effective medium will be

${\displaystyle {\underline {\underline {{\mathsf {C}}_{\mathrm {eff} }}}}={\begin{bmatrix}A&A-2E&B&0&0&0\\A-2E&A&B&0&0&0\\B&B&C&0&0&0\\0&0&0&D&0&0\\0&0&0&0&D&0\\0&0&0&0&0&E\end{bmatrix}}}$

where

{\displaystyle {\begin{aligned}A&=\langle a-b^{2}c^{-1}\rangle +\langle c^{-1}\rangle ^{-1}\langle bc^{-1}\rangle ^{2}\\B&=\langle c^{-1}\rangle ^{-1}\langle bc^{-1}\rangle \\C&=\langle c^{-1}\rangle ^{-1}\\D&=\langle d^{-1}\rangle ^{-1}\\E&=\langle e\rangle \\\end{aligned}}}

${\displaystyle \langle \cdot \rangle }$ denotes the volume weighted average over all layers.

This includes isotropic layers, as the layer is isotropic if ${\displaystyle b_{i}=a_{i}-2e_{i}}$, ${\displaystyle a_{i}=c_{i}}$ and ${\displaystyle d_{i}=e_{i}}$.

### Short and medium wavelength approximation

Solutions to wave propagation problems in linear elastic transversely isotropic media can be constructed by superposing solutions for the quasi-P wave, the quasi S-wave, and a S-wave polarized orthogonal to the quasi S-wave. However, the equations for the angular variation of velocity are algebraically complex and the plane-wave velocities are functions of the propagation angle ${\displaystyle \theta }$ are.[3] The direction dependent wave speeds for elastic waves through the material can be found by using the Christoffel equation and are given by[4]

{\displaystyle {\begin{aligned}V_{qP}(\theta )&={\sqrt {\frac {C_{11}\sin ^{2}(\theta )+C_{33}\cos ^{2}(\theta )+C_{44}+{\sqrt {M(\theta )}}}{2\rho }}}\\V_{qS}(\theta )&={\sqrt {\frac {C_{11}\sin ^{2}(\theta )+C_{33}\cos ^{2}(\theta )+C_{44}-{\sqrt {M(\theta )}}}{2\rho }}}\\V_{S}&={\sqrt {\frac {C_{66}\sin ^{2}(\theta )+C_{44}\cos ^{2}(\theta )}{\rho }}}\\M(\theta )&=\left[\left(C_{11}-C_{44}\right)\sin ^{2}(\theta )-\left(C_{33}-C_{44}\right)\cos ^{2}(\theta )\right]^{2}+\left(C_{13}+C_{44}\right)^{2}\sin ^{2}(2\theta )\\\end{aligned}}}

where {\displaystyle {\begin{aligned}\theta \end{aligned}}} is the angle between the axis of symmetry and the wave propagation direction, ${\displaystyle \rho }$ is mass density and the ${\displaystyle C_{ij}}$ are elements of the elastic stiffness matrix. The Thomsen parameters are used to simplify these expressions and make them easier to understand.

#### Thomsen parameters

Thomsen parameters[5] are dimensionless combinations of elastic moduli which characterize transversely isotropic materials, that are encountered, for example, in geophysics. In terms of the components of the elastic stiffness matrix, these parameters are defined as:

{\displaystyle {\begin{aligned}\epsilon &={\frac {C_{11}-C_{33}}{2C_{33}}}\\\delta &={\frac {(C_{13}+C_{44})^{2}-(C_{33}-C_{44})^{2}}{2C_{33}(C_{33}-C_{44})}}\\\gamma &={\frac {C_{66}-C_{44}}{2C_{44}}}\end{aligned}}}

where index 3 indicates the axis of symmetry (${\displaystyle \mathbf {e} _{3}}$) . These parameters, in conjunction with the associated P wave and S wave velocities, can be used to characterize wave propagation through weakly anisotropic, layered media. It is found empirically that, for most layered rock formations the Thomsen parameters are usually much less than 1.

The name refers to Leon Thomsen, professor of geophysics at the University of Houston, who proposed these parameters in his 1986 paper "Weak Elastic Anisotropy".

#### Simplified expressions for wave velocities

In geophysics the anisotropy in elastic properties is usually weak, in which case ${\displaystyle \delta ,\gamma ,\epsilon \ll 1}$. When the exact expressions for the wave velocities above are linearized in these small quantities, they simplify to

{\displaystyle {\begin{aligned}V_{qP}(\theta )&\approx V_{P0}(1+\delta \sin ^{2}\theta \cos ^{2}\theta +\epsilon \sin ^{4}\theta )\\V_{qS}(\theta )&\approx V_{S0}\left[1+\left({\frac {V_{P0}}{V_{S0}}}\right)^{2}(\epsilon -\delta )\sin ^{2}\theta \cos ^{2}\theta \right]\\V_{S}(\theta )&\approx V_{S0}(1+\gamma \sin ^{2}\theta )\end{aligned}}}

where

${\displaystyle V_{P0}={\sqrt {C_{33}/\rho }}~;~~V_{S0}={\sqrt {C_{44}/\rho }}}$

are the P and S wave velocities in the direction of the axis of symmetry (${\displaystyle \mathbf {e} _{3}}$) (in geophysics, this is usually, but not always, the vertical direction). Note that ${\displaystyle \delta }$ may be further linearized, but this does not lead to further simplification.

The approximate expressions for the wave velocities are simple enough to be physically interpreted, and sufficiently accurate for most geophysical applications. These expressions are also useful in some contexts where the anisotropy is not weak.

## References

1. Backus, G. E. (1962), Long-Wave Elastic Anisotropy Produced by Horizontal Layering, J. Geophys. Res., 67(11), 4427–4440
2. Ikelle, Luc T. and Amundsen, Lasse (2005),Introduction to petroleum seismology, SEG Investigations in Geophysics No. 12
3. Nye, J. F. (2000). Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press.
4. G. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4
5. Thomsen, Leon (1986). "Weak Elastic Anisotropy". Geophysics 51 (10): 1954–1966. doi:10.1190/1.1442051.