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. Such a medium is regarded in the anisotropic world with the same importance that v(z) velocity variation has in the inhomogeneous world. 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).

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

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

Tsvankin (1995) and Alkhalifah (1997c) 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 can be considered as 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 $\eta$, 

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

Moreover, Alkhalifah and Tsvankin (1995) further show that these two parameters, $\eta$ and $V_{{\rm nmo}}(0)$, are obtainable solely from surface seismic P-wave data: specifically, from estimates of stacking velocity for reflections from interfaces having two distinct dips. These two parameters can also be resolved by examining the behavior of moveout at far offsets (Alkhalifah, 1997b). The third parameter, V_P0, is needed for time-to-depth conversion only. The two-parameter representation and inversion also holds in v(z) media (Alkhalifah, 1997a). For that situation, these two parameters are expressed in terms of the vertical time $\tau$.

Because the main assumption in the new parameterization is that the data remain in the time rather than depth domain, the post-stack migration considered here is primarily a time one.