Phase and group angles and velocities

In anisotropic media the group angles and velocities do not coincide with the phase angles and velocities. The transformation from phase velocity $\widetilde{V}$to group velocity V is conventionally defined as the following Tsvankin (2001):  

$$V=\sqrt{\widetilde{V}^2+\left(\frac{d\widetilde{V}}{d\widetilde{\theta}}\right)},$$ (1)

where $\widetilde{\theta}$ is the phase propagation angle. The associated transformation from phase angles to group angles $\theta$ is defined as:  

$$\tan \theta= \frac {\tan \widetilde{\theta}+ \frac{1}{\wide... ...d \widetilde{V}}{d\widetilde{\theta}} \tan \widetilde{\theta}}.$$ (2)

Notice that throughout this paper I use the tilde symbol to distinguish between phase quantities (with a tilde) and group quantities (without a tilde).

Dellinger and Muir (1985) propose, and heuristically motivate, the following symmetric relations for the inverse transforms:  

$$\widetilde{S}=\sqrt{S^2+\left(\frac{dS}{d\theta}\right)},$$ (3)

where $\widetilde{S}$ and S are respectively the phase slowness and the group slowness, and  

$$\tan \widetilde{\theta}= \frac {\tan \theta+ \frac{1}{S}\frac{d S}{d\theta}} {1- \frac{1}{S}\frac{d S}{d\theta} \tan \theta}.$$ (4)

I use the heuristic relation in equation 4 to derive some of the analytical results presented in this paper. Furthermore, I use all the above relationships to compute the kinematic numerical results presented in this paper.

The numerical results, though not the analytical results, are also dependent on the choice of a specific approximation of the anisotropic phase-velocity function. I used the following VTI approximation for the phase velocity:  

$$\widetilde{V}^2_{\rm VTI}\left(\theta\right) = \frac { {V_V}... ... + {V_V}^2\left({V_N}^2-{V_H}^2\right) \sin^2 2 \theta } } {2},$$ (5)

where V_V, V_H, V_N, are respectively the vertical velocity, the horizontal velocity and the NMO velocity. Following Fowler (2003), the corresponding approximation for the group velocity is the following:  

$$S^2_{\rm VTI}\left(\theta\right) = \frac { {S_V}^2\cos^2 \th... ...+ {S_V}^2\left({S_N}^2-{S_H}^2\right) \sin^2 2 \theta. } } {2},$$ (6)

where S_V, S_H, S_N, are respectively the vertical slowness, the horizontal slowness and the NMO slowness.

The numerical results obtained by modeling and migrating synthetic seismic data were obtained by source-receiver depth continuation (upward for modeling and downward for migration) using the following dispersion relation:  

$$k_z= \frac{\omega}{V_V} \sqrt{\frac {\omega^2 - {V_H}^2k_x^2} {\omega^2 + \left({V_N}^2-{V_H}^2\right)k_x^2} },$$ (7)

where $\omega$ is the temporal frequency, and k_x and k_z are respectively the horizontal and vertical wavenumbers. The dispersion relation shown in equation 7 corresponds to the velocity and slowness functions in equations 5 and 6 Fowler (2003).

 

• Anisotropic parameters used for numerical tests