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 kinds of anisotropies can exist (i.e., orthrohombic anisotropy), the large amount of shales present in the subsurface implies that the TI model has the most influence on P-wave data Banik (1984).

In homogeneous transversely isotropic media with a vertical symmetry axis (VTI media), P- and SV-waves can be described by the vertical velocities V_P0 and V_S0 of P- and S-waves, respectively, and two dimensionless parameters $\epsilon$ and $\delta$ Thomsen (1986).

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

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

Alkhalifah (1997a) demonstrated that P-wave velocity and traveltime are practically independent of V_S0, even for strong anisotropy. This implies that, for practical purposes, P-wave kinematic signatures is a function of just three parameters: V_P0, $\delta$, 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 correction, and prestack and post-stack time migration. These two parameters are the normal-moveout velocity for a horizontal reflector  

$$V_{{\rm nmo}}(0)=V_{p0} \sqrt{1+2 \delta} \, ,$$ (1)

and the anisotropy coefficient $\eta$, 

$$\eta \equiv \frac{1}{2}(\frac{V_h^2}{V_{{\rm nmo}}^2(0)}-1)=\frac{\epsilon-\delta}{1+2 \delta} \, ,$$ (2)

where V_h is the horizontal velocity. Instead of $V_{\rm nmo}$, I will use v to represent the interval NMO velocity in both isotropic and TI media. The midpoint-offset traveltime equations, like any other time-domain equations, are expected to be dependent on these to parameters as well.