On anelliptic approximations for qP velocities in TI and orthorhombic media

Exact Expression

The phase velocity of qP waves in TI media has the following well-known explicit expression (Gassmann, 1964; Berryman, 1979):

$$v^2_{phase} = \frac{1}{2}[(c_{11}+c_{55})n^2_1 + (c_{33}+c_{55})n... ...11}-c_{55})n^2_1 - (c_{33}-c_{55})n^2_3]^2 + 4(c_{13}+c_{55})^2n^2_1n^2_3}~,\\$$ (9)

where $c_{ij}$ are density-normalized stiffness tensor coefficients in Voigt notation, $n_1 = \sin\theta$ , $n_3 = \cos\theta$ , and $\theta$ is the phase angle (measured from the vertical axis). Group velocity can be determined from phase velocity using the general expression (Cervený, 2001)

$$\mathbf{v}_{group} = v_{phase}\mathbf{n} + (\mathbf{I} - \mathbf{n}\mathbf{n}^T)\nabla_{\mathbf{n}}v_{phase}~,\\$$ (10)

where $\mathbf{I}$ denotes the identity matrix, $\mathbf{n}=\{n_1,n_3\}$ is the phase direction vector, and $\nabla_{\mathbf{n}}v_{phase} = \{\frac{\partial v_{phase}}{n_1},\frac{\partial v_{phase}}{n_3}\}^T$ is the gradient of $v_{phase}$ with respect to $\mathbf{n}$ . Using Muir-Dellinger parameters, the exact phase velocity for qP waves (equation 9) can be expressed as

$$v^2_{phase} = \frac{1}{2}\left[w_1n^2_1 + w_3n^2_3 + w_{13}\right] + \frac{1}{2}\sqrt{f}~,\\$$ (11)

where

$$f = \left[w_1n^2_1 + w_3n^2_3 - w_{13}\right]^2 + \frac{4(q_1-1)(q_3-1)(w_3-w_1)w_{13}n^2_1n^2_3}{(q_1-q_3)}~,$$

$$w_{13} = \frac{(q_1-q_3)w_1w_3}{(q_1-1)w_3-(q_3-1)w_1}~.$$

On anelliptic approximations for qP velocities in TI and orthorhombic media