From elastic constants to phase velocities

The phase velocity expression for P- and SV-waves in TI media is (Auld, 1990)

$$\begin{eqnarray} 2 W_{P,SV} (\theta) & = & (W_{33} + W_{44}) \cos^2 \theta + (W_... ...^2 + 4 (W_{13} + W_{44})^2 \sin^2 \theta \cos^2 \theta}, \nonumber\end{eqnarray}$$ (1)

where $W (\theta)$ is the phase velocity squared and $\theta$ is the phase angle from the vertical. W_ij is the (ij)^th elastic modulus divided by density, with units of velocity squared; I refer to the quantity W_ij as an ``elastic constant'' in the remainder of this paper. The ``+'' sign in front of the square root corresponds to P-waves and the ``-'' to SV-waves. For SH-waves, the expression for the phase velocity is (Auld, 1990)

$$W_{SH} (\theta) \ =\ W_{44} \cos^2 \theta + W_{66} \sin^2 \theta.$$ (2)

Near the vertical and horizontal axes equation (1a) is elliptical. The velocities that describe the corresponding ellipses are called elliptical velocities. In the next two subsections, I rederive equations contained in previous papers (Levin, 1979; Levin, 1980; Muir, 1990b) that are needed to calculate these elliptical velocities from the elastic constants. The expressions that result are used later in the paper to solve the inverse mapping when the data have narrow aperture.