Generalized nonhyperbolic moveout approximation

Appendix E: HOMOGENEOUS VTI MODEL

According to the acoustic approximation of Alkhalifah (1998), one can use the following parametric equations to define the traveltime-offset relationship in a homogeneous VTI model:

$$x(p) = \frac{2\,H}{v_z}\,\frac{p\,v^2}{(1-2\,\eta\,p^2\,v^2)^2\,\sqrt{1-\frac{p^2\,v^2}{1-2\,\eta\,p^2\,v^2}}}\;,$$ (78)

$$t(p) = \frac{2\,H}{v_z}\,\frac{(1-2\,\eta\,p^2\,v^2)^2 + 2\,\eta\,p^4\,v^4} {(1-2\,\eta\,p^2\,v^2)^2\,\sqrt{1-\frac{p^2\,v^2}{1-2\,\eta\,p^2\,v^2}}}\;,$$ (79)

where $p$ is the ray parameter, $H$ is the depth of the reflector, $v_z$ is the vertical velocity, $v$ is the NMO velocity, and $\eta$ is the dimensionless parameter introduced by Alkhalifah and Tsvankin (1995).

At small offsets, the homogeneous VTI traveltime behaves as

$$t^2(x) \approx t_0^2 + \frac{x^2}{v^2} - \frac{2\,\eta\,x^4}{t_0^2\,v^4}\;,$$ (80)

which allows us to define $A = - 4\,\eta$ according to equation 18.

At large offsets, the homogeneous VTI traveltime behaves as

$$t^2(x) \approx t_0^2\,(1+2\,\eta) + \frac{x^2}{v^2\,(1+2\,\eta)}\;.$$ (81)

Comparing with equation 24, we note that $T_{\infty} = t_0\,\sqrt{1+2\,\eta}$ and $P_{\infty} = 1/(v\,\sqrt{1+2\,\eta})$. Substituting into equations 25-26, we derive the coefficients $B$ and $C$ to be

$$B = \frac{1 + 8\,\eta + 8\,\eta^2}{1 + 2\,\eta}\;,$$ (82)

$$C = \frac{1}{(1 + 2\,\eta)^2}\;.$$ (83)

Generalized nonhyperbolic moveout approximation