From elastic constants to phase velocities near the vertical axis

Expanding equation (1a) around $\theta = 0$ and neglecting terms in $\sin^4 \theta$, results in:

$$\begin{eqnarray} 2 W_{P,SV} (\theta) & = & (W_{33} + W_{44}) \cos^2 \theta + (W_... ...} + W_{44})^2}{W_{33} - W_{44}} \sin^2 \theta \right). \nonumber\end{eqnarray}$$ (2)

Choosing the positive root yields the P-wave phase velocity near the vertical axis, as follows:

$$W_{P} (\theta) \ =\ W_{P,z} \ c^2 + W_{P,xnmo} \ s^2 ,$$ (3)

where $c \ =\ \cos \theta$, $s \ =\ \sin \theta$,

$$W_{P,z} \ =\ W_{33},$$ (4)

and

$$W_{P,xnmo} \ =\ W_{44} + \frac{(W_{13} + W_{44})^2}{W_{33} - W_{44}}.$$ (5)

W_P,z is the vertical P-wave phase velocity squared and W_P,xnmo is the horizontal normal moveout (NMO) phase velocity squared.

Choosing the negative root in equation (2) yields SV-wave phase velocities near the vertical axis, as follows:

$$W_{SV} (\theta) \ =\ W_{SV,z} \ c^2 + W_{SV,xnmo} \ s^2,$$ (6)

where

$$W_{SV,z} \ =\ W_{44},$$ (7)

and

$$W_{SV,xnmo} \ =\ W_{11} - \frac{(W_{13} + W_{44})^2}{W_{33}- W_{44}}.$$ (8)

The previous expressions for the NMO velocities agree with the results of Thomsen (1986) and Vernik and Nur (1992).

The expression for SH-wave phase velocities near the vertical axis is

$$W_{SH} (\theta) \ =\ W_{SH,z} \ c^2 + W_{SH,xnmo} \ s^2,$$ (9)

where

$$W_{SH,z} \ =\ W_{44},$$ (10)

and

$$W_{SH,xnmo} \ =\ W_{SH,x} \ =\ W_{66}.$$ (11)