Anisotropic media parameterization

Here, I consider the simplest and probably most practical anisotropic model, that is a transversely isotropic (TI) medium with a vertical symmetry axis. Although, more complicated anisotropies can exist (i.e. orthorhombic anisotropy), the large amount of shales present in the subsurface makes the TI model the most influential on P-wave data Banik (1984).

In homogeneous transversely isotropic media with vertical symmetry axis (VTI media), P- and SV-waves (I omit the qualifiers in ``quasi-P-wave" and ``quasi-SV-wave" for brevity) can be described by the vertical velocities v_v and v_sv of P- and S-waves, respectively, and two dimensionless parameters $\epsilon$ and $\delta$ Thomsen (1986).

$$\epsilon \equiv \frac{c_{11} - c_{33}}{2 c_{33}},$$

$$\delta \equiv \frac{(c_{13}+c_{44})^2 - (c_{33} - c_{44})^2}{ 2 c_{33} (c_{33} - c_{44})} \, .$$

Tsvankin (1996) and Alkhalifah (1997b) demonstrated that P-wave velocity and traveltime are practically independent of v_sv, even for strong anisotropy. This implies that, for practical purposes, P-wave kinematic signatures can be considered a function of just three parameters: v_v, $\delta$, and $\epsilon$.

Alkhalifah and Tsvankin (1995) further demonstrated that a new representation in terms of just two parameters is sufficient for performing all time-related processing, such as normal moveout correction (including non-hyperbolic moveout correction, if necessary), dip-moveout removal, and prestack and post-stack time migration. These two parameters are the normal-moveout velocity for a horizontal reflector  

$$v=v_v \sqrt{1+2 \delta} \, ,$$ (2)

and the anisotropy coefficient $\eta$, 

$$\eta \equiv 0.5(\frac{v_h^2}{v^2}-1)=\frac{\epsilon-\delta}{1+2 \delta} \, ,$$ (3)

where v_h is the horizontal velocity. Since we are dealing here with time processing only two parameters will be used in this paper, v and $\eta$.