parameterization in Anisotropic media

In this paper, I consider the simplest and probably most practical anisotropic model, that is a transversely isotropic (TI) medium with a vertical symmetry axis. Such a medium is given the same importance in the anisotropic world that v(z) velocity variation has in the inhomogeneous world. Although more complicated anisotropies can exist (such as orthrohombic anisotropy), the large amount of shales Banik (1984) present in the subsurface makes the TI model the most influential for P-wave data.

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 the two dimensionless parameters $\epsilon$ and $\delta$ Thomsen (1986), as follows:

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

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

Tsvankin and Thomsen (1994), and more generally Alkhalifah (1997) have demonstrated that P-wave velocity and traveltime are practically independent of V_S0, even for strong anisotropy. This finding implies that, for practical purposes, P-wave kinematic signatures can be considered 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 nonhyperbolic 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  

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

and the anisotropy coefficient  

$$\eta \equiv 0.5(\frac{V_h^2}{V_{{\rm nmo}}^2(0)}-1)=\frac{\epsilon-\delta}{1+2 \delta} \, ,$$ (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.

However, if depth is of concern, as in this paper, the vertical P-wave velocity (v_v or V_P0) is also needed to characterize the medium. Again, the vertical shear wave velocity has little influence on P-wave propagation in such media, and as a result, it is set to zero with little loss in the accuracy of the VTI equations.