From elastic constants to phase velocities near the horizontal axis

P- and SV-wave phase velocities near the horizontal axis can be obtained by interchanging W₁₁ and W₃₃ and c² and s² wherever they occur in equations (3) through (8). Thus, for P-waves the result is

$$W_P (\theta) \ =\ W_{P,x} \ s^2 + W_{P,znmo} \ c^2,$$ (12)

where

$$W_{P,x} \ =\ W_{11},$$ (13)

and

$$W_{P,znmo} \ =\ W_{44} + \frac{(W_{13} + W_{44})^2}{W_{11} - W_{44}}.$$ (14)

For SV-waves, the expression for the phase velocity near the horizontal axis is

$$W_{SV} (\theta) \ =\ W_{SV,x} \ s^2 + W_{SV,znmo} \ c^2,$$ (15)

where

$$W_{SV,x} \ =\ W_{44},$$ (16)

and

$$W_{SV,znmo} \ =\ W_{33} - \frac{(W_{13} + W_{44})^2}{W_{11}- W_{44}}.$$ (17)

Near the horizontal axis the SH-wave phase velocities are

$$W_{SH} (\theta) \ =\ W_{SH,x} \ s^2 + W_{SH,znmo} \ c^2,$$ (18)

where

$$W_{SH,x} \ =\ W_{66},$$ (19)

and

$$W_{SH,znmo} \ =\ W_{SH,z} \ =\ W_{44}.$$ (20)

In the rest of the paper I refer to the elliptical parameters W_P,x, W_P,z, W_P,xnmo, W_P,znmo, W_SV,x, W_SV,z, W_SV,xnmo, W_SV,znmo, W_SH,x, W_SH,z, W_SH,xnmo, and W_SH,znmo as W_*, direct or NMO phase velocity squared for P-, SV-, and SH-waves.