Parameterization in Anisotropic media

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). Tsvankin and Thomsen (1994) and Alkhalifah (1997a) demonstrated that P-wave velocity and traveltime are practically independent of V_S0, even for strong anisotropy. Thus, 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, assuming that the velocity varies only vertically. 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  

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

where V_h is the horizontal velocity. Instead of $V_{\rm nmo}$, we use v to denote the interval NMO velocity in both isotropic and TI media.