On anelliptic approximations for qP velocities in TI and orthorhombic media

Exact Expressions

As the analog of equation 9, qP waves have the following well-known explicit exact expression for phase velocity in orthorhombic media (Tsvankin, 2012; Schoenberg and Helbig, 1997; Tsvankin, 1997):

$$v^2_{phase} = 2\sqrt{\frac{-d}{3}}\cos(\frac{\nu}{3})-\frac{a}{3} ~,\\$$ (40)

where

$$\nu = \arccos \left(\frac{-q}{2\sqrt{(-d/3)^3}}\right)~,$$

$$q = 2\left(\frac{a}{3}\right)^3 - \frac{ab}{3}+c~,~~~d~~=~~-\frac{a^2}{3} + b~,$$

$$a = -(G_{11}+G_{22}+G_{33})~,$$

$$b = G_{11}G_{22}+G_{11}G_{33}+G_{22}G_{33}-G^2_{12}-G^2_{13}-G^2_{23}~,$$

$$c = G_{11}G^2_{23}+G_{22}G^2_{13}+G_{33}G^2_{12}-G_{11}G_{22}G_{33}-2G_{12}G_{13}G_{23}~,$$

and

$$G_{11} = c_{11}n^2_1 + c_{66}n^2_2 + c_{55}n^2_3~,$$

$$G_{22} = c_{66}n^2_1 + c_{22}n^2_2 + c_{44}n^2_3~,$$

$$G_{33} = c_{55}n^2_1 + c_{44}n^2_2 + c_{33}n^2_3~,$$

$$G_{12} = (c_{12}+c_{66})n_1n_2~,$$

$$G_{13} = (c_{13}+c_{55})n_1n_3~.$$

Here, $c_{ij}$ are density-normalized stiffness tensor coefficients, $n_1 = \sin\theta \cos\phi$ , $n_2 = \sin\theta \sin\phi$ , $n_3 = \cos\theta$ , $\theta$ is zenith phase angle (measured from $n_3$ ), and $\phi$ is azimuthal phase angle (measured from $n_1$ ) in the local orthorhombic frame of reference where the axes $n_1$ , $n_2$ , and $n_3$ are intersections of the corresponding planes of symmetry. The corresponding group-velocity expression can be determined from equation 10 extended to 3D.

On anelliptic approximations for qP velocities in TI and orthorhombic media