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Exact Expression

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

$\displaystyle 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)

$\displaystyle \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

$\displaystyle 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
$\displaystyle f$ $\displaystyle =$ $\displaystyle \left[w_1n^2_1 + w_3n^2_3 - w_{13}\right]^2 + \frac{4(q_1-1)(q_3-1)(w_3-w_1)w_{13}n^2_1n^2_3}{(q_1-q_3)}~,$  
$\displaystyle w_{13}$ $\displaystyle =$ $\displaystyle \frac{(q_1-q_3)w_1w_3}{(q_1-1)w_3-(q_3-1)w_1}~.$  


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Next: Muir and Dellinger Approximations Up: Transversely isotropic media Previous: Transversely isotropic media

2017-04-14