INTRODUCTION
Seismic anisotropy is the variation of velocity as a function of the signal propagation direction. Winterstein^{[1]} restates the definition of anisotropy as variation of one or more properties of a material with direction. The important features of wave propagation in anisotropic solids are i) the variation of wave velocities with direction ii) the threedimensional (3D) displacement of the particle which leads to shearwave splitting and iii) the propagation of energy deviated both in velocity and direction from phase propagation.
The variation of properties for purely elastic solids, such as crystals, solid containing aligned cracks or one made up of periodic thinlayers, can be simulated by anisotropic elastic constants and fully described by fourorder tensors of anisotropic constants. There are eight anisotropic systems or crystalline symmetry (including isotropic symmetry) and two subsystems, which can be specified by patterns of elastic constants.
Dealing with wave phenomena in anisotropic media, one must distinguish between group velocity and phase velocity. Group velocity is the speed at which wave energy travels radially outward from a point source in a homogeneous elastic anisotropic medium^{[1]}. Phase velocity is the velocity in the direction of the phase propagation vector, normal to the surface of constant phase^{[2]}. Field measurements of traveltime and distance often yield group velocity, which could be performed in laboratory setting^{[3,4]}. In anisotropic media, group and phase velocities can coincide along particular trajectories. For instance, for vertical and horizontal propagation in transversely isotropic material with a vertical symmetry axis (TIV), group velocity equals phase velocity.
Measurements of seismic velocity anisotropy from traveltimes of P, Vsand SHwaves^{[57]} have shown that many sedimentary rocks are anisotropic. Seismic anisotropy can provide important quantitative information about structure and Lithology of the sedimentary rocks^{[8,9]} and provide more geological information and better understanding of the earth^{[10]}.
Since seismic particle motion is vector polarization, the potential value of shearwave propagation lies in the fact that each shearwave component carries threedimensional information about the symmetry structure along the raypath and contains much more information about the nature of the raypath than is possible with the polarizations of Pwaves. Different shear waves have different behavior at interface and at internal structures along the raypath, which splits the shear wave into several arrivals with different polarizations and different velocities.
The simplest anisotropy case of widespread geophysical applicability is called traverse isotropy or hexagonal symmetry.
Anisotropy and shearwave splitting: The presence of anisotropy in the
earth, which manifests itself most diagnostically in terms of shearwave splitting
in multicomponent seismic data, can lead to substantial complication in the
processing and interpretation of both surface seismic and VSP shearwave data.

Fig. 1: 
Differential particle displacement in a deformed medium 
However, analysis of anisotropic wavepropagation phenomena, such as shearwave
splitting, could lead to more information, such as strike direction and density
of vertical particle within reservoir^{[11,12]} which helps in the understanding
of the earth. Transverse isotropy with vertical symmetry axis (TIV) serves as
a good introduction to anisotropy for geophysicist and helps to define the basic
terminology and methodology for anisotropy studies^{[1]}.
Particle displacement and strain field: When the particles of a medium
are displaced from their equilibrium positions internal restoring forces arise
which lead to oscillatory motion of the medium. Each particle is assigned an
equilibrium position vector and
the displacement position vector
The displacement of the particle located at
in the equilibrium state is defined by Fig. 1.
However, since the particle displacement field
is nonzero for rigid motions, it does not itself provide a satisfactory measure
of material deformation. A more convenient quantity is:
which measures the difference between the distance of two neighboring particles
in the equilibrium state and deformed state. In Fig. 1 two
displaced particle positions at fixed time are shown for two neighboring particles
separated
by in the equilibrium state.
The deformation measure
is calculated from
by using relations:
where,
for continuous medium, is the 3x3 matrix made from the derivatives of
with respect to
. Thus
where, the matrix elements
are termed as components of the strain field.
The strain field determines the deformation
in terms of the particle displacement
and reduces it to zero for all rigid motions.
Solids differ widely in their deformabilities. For rigid materials, the displacement
gradient must be kept below unity if permanent deformation is to be avoided.
For displacement derivatives much smaller than unity, the quadratic terms in
equation (5) are negligible and this allows using the linearized
strain field:
Since the strain field is symmetric, one subscript rather than two can specify each component. Following Voigt notation, we can obtain:
The strain field may also be written as a sixelement column matrix rather than a nineelement square matrix, such as follows:
and it can also be decomposed in more simple form by introducing the operator:
Traction forces and stress fieldequation of motion: A method of exciting vibrations in a material body is to apply external forces at its surface. In this case the applied excitation does not act directly on particles within the body but is transmitted to them by means of the Hook forces acting between neighboring particles. To specify these forces, three force components are required for each face of the particle; the traction force acting on the area element facing i direction is vector:
The components
of these forces are called stress components;
is the ith component of force acting on the +j face of an infinitesimal volume
element at position .
The abbreviated subscript notation introduced for the strain field can also be used to describe the stress components:
In this case the convention is to omit the factor ½ that appeared in
equation (7) and the stress can now be written as a sixelement
column matrix:
The forces associated with the vibration of material particle are the traction forces applied to its surface by the neighboring particles:
where,
is the normal vector to the surface.
Green’s formula applied on the integrated surface acting on the particle gives:
with
being the divergence of the stress matrix
The equation of motion is obtained using Newton’s law:
ρ is the density of the medium and
is the particle velocity.
Hooke’s law: For small deformations it is experimentally observed that the strain in a deformed body is linearly proportional to the applied stress, i.e.
where, c are the elastic stiffness constants.
In the full index notation c is a tensor of order 4 and there are 3^{4}
= 81 elastic stiffness constants. These are not all independent, however, since
(because S and T are symmetric matrices), the number of independent constants
is reduced to (3+2+1)^{2} = 36. Some remarks about energy give
, thus only 6+5+4+3+2+1 = 21 independent constants left. This is the maximum
number of elastic constants for any medium. For an isotropic medium, there are
only two independent constants
called the Lame’s parameters.
Christoffel equation
By equation (16), we have
and using equations (15), (17) and (9),
the following can be obtained
Equation (19) is a wave equation for general homogeneous
media for which a plane wave analysis can be performed. A uniform plane wave
propagating along the direction
proportional to,
where,
and k is the wavenumber;
is the velocity of the advance of wavefront and is called the phase velocity.
Operators and act on a plane wave like
After replacing the relevant parts in equation (19) with
and equation (20), the following dispersion relation can
be derived
where, Γ is a 3x3 matrix called the Christoffel matrix, whose elements
are functions only of the plane wave propagation direction
and of the stiffness constants c_{KL} of the medium.
The dispersion relation (21) is an eigenvalues problem (are
the eigenvalues). It has the unique solution
(i.e. no propagation in the medium!), if the determinant of the system is nonzero;
since this is not physically attainable, the determinant should be zero.
There are 3 possible solutions for the phase velocity
in equation (22), only waves with one of these phase velocities
can propagate in the medium.
Associated with each eigenvalue, there is an eigenvector corresponding to the polarization of the wave propagating with the phase velocity. Mathematically the three eigenvectors are mutually orthogonal, which suggests that physically the three polarizations are in the direction of propagation: one of these is the quasilongitudinal wave and is simply denoted as P. Another eigenvector is orthogonal to the first one but not to the direction of propagation: this is the quasitraverse shear wave and is denoted SV. The last eigenvector is orthogonal to the direction of propagation and also to the other eigenvectors: this is the exactly traverse shear wave and is denoted SH.
Hexagonal symmetry and vertical axis: Up to now, the hexagonal system
has been the anisotropic symmetry most frequently used. This system is of rotational
symmetry, which means that the tensor c_{ijkl} does not change with
rotation around the axis of symmetry. In other words, in the plane perpendicular
to this axis, the tensor behaves isotropically. Therefore, the symmetry is also
sometimes called transverse isotropy, especially in case when the axis of rotational
symmetry coincides with x_{3}the axis of the coordinate system.
With regard to hexagonally symmetric material with a vertical axis of symmetry,
matrix
can be expressed as:
The hexagonal symmetry is described by five independent elastic parameters.
The coordinate system in which the matrix
is given should be described as
. In the above case, the axis x_{3} coincides with the axis of rotational
symmetry, while x_{1}, x_{2} are both located in the isotropic
plane.
In the case of the hexagonal symmetry, the Christoffel equation
(22) can be expressed as:
Thus, there are only five independent elastic constants for describing a crystal with hexagonal symmetry: c_{11}, c_{13}, c_{33}, c_{44}, c_{66}.
The propagation vector
could be written as:
where, θ is the angle between the propagation direction and the vertical axis.
Solving the zerodeterminant equation (22) yields the solutions
satisfying
Thomsen’s notations: As introduced by Thomsen^{[13]}, it
is useful to recast equations (25) using notations involving
only two elastic moduli (e.g. vertical P and S velocities) plus three measures
of anisotropy. In order to simplify equations (25), these
three anisotropy coefficients should be nondimensional and efficient combinations
of elastic moduli (c_{11},…….,c_{66}). Furthermore,
they should reduce to zero in case of isotropy. One suitable combinations the
author derived is:
Since the vertical P and S velocities are:
equation (25) can be rewritten as:
where, D(θ) is given by:
with
ε, δ and γ are called Thomsen’s parameters and are convenient variables to support calculus.
Phase velocity and group velocity: In a general homogeneous elastic medium, where the velocity is constant in any given direction, it is obvious that a particle moves along a straight line; the energy propagates along this line with group velocity, while group angle φ is formed between the direction of propagation and the vertical axis.
However, due to anisotropy, the wavefront is non spherical. The wave vector
is locally perpendicular to the wavefront and the phase angle θ with the
vertical axis. The phase velocity that suggests the speed of advance of the
wavefront along the direction is given by the Thomsen’s formula (28).
Figure 2 shows that a point on the wavefront can be reached either by energy traveling with group velocity V(φ) and group angle φ in the direction of energy propagation or by phase traveling with phase velocity V(θ) and phase angle θ in the direction perpendicular to the wavefront. Note that the phase velocity direction does not start from the source point.
Thus, in a homogeneous medium, wavefronts are defined by:
From
We derive:
which suggests that the traveltime along the ray is a linear function of the
sourcewavefrontdistance. Also, along a ray of energy propagation, the direction
of vector normal to the wavefront is constant . Thus, θ is irrelevant to time and is constant when φ is fixed.
The phase velocity is defined as the projection of the group velocity on the vector normal to the wavefront (Fig. 2). Thus, the group velocity is given in terms of the phase velocity as:

Fig. 2: 
Phase (wavefront) angle θ at two consecutive times and
group (ray) angle φ 
and with the general form of equation (28) we have
The general relation between group angle φ and phase angle θ is (Fig. 2).
In anisotropic media, wavefronts traveling outward from a point source are not, in general, spherical as a result of dependence of velocity upon direction of propagation. Shown in Fig. 2 are two wave fronts in space separated by unit time. The group velocity, V(φ), denotes the velocity with which energy travels from the source, while the phase velocity, v(φ), is the velocity with which a wavefront propagates at a local point. Here, the group angle φ specifies the direction of the ray from the source point to the point of interest, while the phase angleφ (also called wavefrontnormal angle) specifies the direction of the vector that is normal to the wavefront. In general, they are different at any point of propagation, except at certain singular points.
CONCLUSIONS
This study has presented anisotropy in detail. In particular, Transverse isotropy (TIV) is described by five elastic parameters. This has been achieved by using simplifications of notations introduced by Thomsen, 1986. Furthermore, the group velocity is derived as a function of the group angle from the phase velocity that varies with the phase angle.