# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 2 Seismic Anisotropy  
Seismic anisotropy is the dependence of the seismic velocity on the incident angle [53; 54; 55]. Wave propagation in anisotropic media has been discussed by various studies [53; 56; 57; 58]. To understand the seismic anisotropy of the Earth's subsurface, the relationship between the internal forces (stress) and deformation occurring in the medium (strain) should be defined. When an elastic wave propagates through the subsurface, the stress-strain relationship can be obtained according to [PERSON]'s law as shown below [59]:  
\[\sigma_{ij}=c_{ijkl}e_{kl} \tag{1}\]where \(\sigma_{ij}\) is the stress tensor, \(e_{kl}\) is the strain tensor, and \(c_{ijkl}\) is the elastic (stiffness) tensor, which is a fourth-order tensor with 81 independent components. Both the strain and stress tensors are symmetric [60], resulting in:  
\[c_{ijkl}=c_{jkl}=c_{ijlk} \tag{2}\]  
The 81 independent components are accordingly reduced to 36 due to the symmetry in the stress and strain tensors. Therefore, the elastic tensor (\(c_{ijlk}\)) is replaced with a 6 \(\times\) 6 matrix (\(c_{a\beta}\)), where [61]:  
\[\alpha=ij=ji,\beta=kl=lk \tag{3}\]  
where \(\alpha\) and \(\beta\) are the new indices of the elastic tensor. The resulting (6 \(\times\) 6) notation matrix is expressed as shown below:  
\[ \begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}.\end{bmatrix}  \tag{4}\]  
There are several types of symmetry for the independent components that determine the degree of anisotropy. The maximum anisotropy degree occurs at the lowest possible symmetry, which is called \"triclinic\" symmetry and occurs when the (\(C_{a\beta}\)) matrix has 21 different nonzero components, which is considered the maximum number of elements required to describe an anisotropic medium [58]. On the other hand, the isotropic case occurs at the highest symmetry case with a smaller amount of independent components. The most realistic symmetry case of the Earth's subsurface is polar symmetry, which has a single pole of rotational symmetry (Z-axis) that is different from the other two axes (X and Y) [61]. This anisotropy case is called transverse isotropy (TI), at which rock properties are assumed invariant about an axis of symmetry. If the symmetry axis is vertical, this means that the layers are considered horizontal and is hence called vertical transverse isotropy (VTI). Another case is called tilted transverse isotropy (TTI), which is a more realistic case because it follows the variations of the dip angles of layers. Polar symmetry can be described by a matrix of five different independent components, as shown below:  
\[ \begin{bmatrix}C_{11}&C_{11}-2C_{66}&C_{13}&0&0&0\\ C_{11}-2C_{66}&C_{11}&C_{13}&0&0&0\\ C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{66}.\end{bmatrix}  \tag{5}\]  
There are three anisotropy parameters that can define any anisotropy degree in terms of the stiffness components of a TI medium [53]. The anisotropy parameters are called Epsilon (\(\epsilon\)), Delta (\(\delta\)), and Gamma (\(\gamma\)). The weaker the anisotropy degree is, the closer these parameters are to zero. The anisotropy parameters in the weak anisotropy case have values less than 1. Epsilon is the fractional difference between the vertical and horizontal P-wave velocities. It generally has a positive sign because rocks are less compressible along bedding than across bedding. It can be obtained by the following equation:  
\[\epsilon=\frac{V_{PH}-V_{PV}}{V_{PV}}=\frac{C_{11}-C_{33}}{2C_{33}} \tag{6}\]  
where \(\epsilon\) is the anisotropy parameter Epsilon, \(V_{PH}\) is the horizontal P-wave velocity, \(V_{PV}\) is the vertical P-wave velocity, and \(C_{11}\) and \(C_{33}\) are the stiffness components of the elastic tensor.