On anelliptic approximations for $qP$ velocities in transversally isotropic media

Muir approximation

Muir and Dellinger (1985) suggested representing anelliptic $qP$ phase velocities with the following approximation:

$$v_P^2(\theta) \approx e(\theta) + \frac{(q-1)\,a\,c\, \sin^2{\theta}\,\cos^2{\theta}}{e(\theta)} \;,$$ (12)

where $e(\theta)$ is the elliptical part of the velocity, defined by

$$e(\theta) = a\,\sin^2{\theta} + c\,\cos^2{\theta}\;,$$ (13)

and $q$ is the anellipticity coefficient ($q=1$ in case of elliptic velocities). Approximation (12) uses only three parameters to characterize the medium ($a$ , $c$ , and $q$ ) as opposed to the four parameters ($a$ , $c$ , $l$ , and $f$ ) in the exact expression.

There is some freedom in choosing an appropriate value for the coefficient $q$ . Assuming near-vertical wave propagation and the vertical axis of symmetry (a VTI medium) and fitting the curvature ( $d^2 v_P/d \theta^2$ ) of the exact phase velocity (4) near the vertical phase angle ( $\theta = 0$ ), leads to the definition (Dellinger et al., 1993)

$$q = \frac{l\,(c-l) + (l+f)^2} {a\,(c-l)}\;.$$ (14)

In terms of Thomsen's elastic parameters $\epsilon$ and $\delta$ (Thomsen, 1986) and the elastic parameter $\eta$ of Alkhalifah and Tsvankin (1995),

$$q = \frac{1 + 2\,\delta}{1 + 2\epsilon} = \frac{1}{1 + 2\,\eta}\;.$$ (15)

This confirms the direct relationship between $\eta$ and anellipticity. If we were to fit the phase velocity curvature near the horizontal axis $\theta=\pi/2$ (perpendicular to the axis of symmetry), the appropriate value for $q$ would be

$$\hat{q} = \frac{l\,(a-l) + (l+f)^2} {c\,(a-l)}\;.$$ (16)

Muir and Dellinger (1985) also suggested approximating the VTI group velocity with an analogous expression

$$\frac{1}{V^2_{P}(\Theta)} \approx E(\Theta) + \frac{(Q-1)\,A\,C\, \sin^2{\Theta}\,\cos^2{\Theta}}{E(\Theta)}$$ (17)

where $A=1/a$ , $C=1/c$ , $Q = 1/q$ , $\Theta$ is the group angle, and $E(\Theta)$ is the elliptical part:

$$E(\Theta) = A\,\sin^2{\Theta} + C\,\cos^2{\Theta}\;.$$ (18)

Equations (12) and (17) are consistent in the sense that both of them are exact for elliptic anisotropy ($q=Q=1$ ) and accurate to the first order in $(q-1)$ or $(Q-1)$ in the general case of transversally isotropic media.

To the same approximation order, the connection between the phase and group directions is

$$\tan{\Theta} = \tan{\theta}\,\frac{a}{c}\, \left(1 - (q-1)\,\frac... ...{\theta} - c\,\cos^2{\theta}} {a\,\sin^2{\theta} + c\,\cos^2{\theta}}\right)\;.$$ (19)

On anelliptic approximations for $qP$ velocities in transversally isotropic media