ELLIPTICAL ANISOTROPY

Waves propagating in a homogeneous medium can be described and extrapolated in either the time-space ``group'' or frequency-wavenumber ``phase'' domains. In the standard scalar isotropic case life is simple; there is one soundspeed characteristic of the medium, and it is the same whether we are talking about rays (group velocity) or plane waves (phase velocity). The scalar isotropic ray (or group-velocity) equation in polar coordinates is just  

$$V(\phi_{r}) = V_{\mbox{\rm\scriptsize iso}};$$ (1)

the corresponding dispersion relation (phase-velocity equation) is  

$$\omega^2 = V_{\mbox{\rm\scriptsize iso}}^2 (k_z^2 + k_x^2) .$$ (2)

With anisotropy the propagation velocity depends on the propagation direction, and the group and phase velocities and directions are no longer the same. Convenient closed-form ``group'' and ``phase'' equations like equations () and () are usually not simultaneously available. Even worse, the number of parameters required to describe the medium can be intractably large.

Scalar elliptical anisotropy does not suffer from these difficulties. There are only two velocities, horizontal and vertical. It is true that the ray equation for scalar elliptical anisotropy appears a bit complicated at first glance:  

$$V(\phi_{r}) = {1 \over \sqrt{ V_z^{-2} \cos^2(\phi_{r}) + V_x^{-2} \sin^2(\phi_{r}) }} ,$$ (3)

where $\phi_{r}$ is the group-propagation (ray) direction (with $\phi_{r} = 0$ being vertical). If we re-express equation () in terms of slowness squared M = 1/V², however, it simplifies considerably:  

$$M(\phi_{r}) = {{ M_z \cos^2(\phi_{r}) + M_x \sin^2(\phi_{r}) }} .$$ (4)

The dispersion relation for elliptical anisotropy is similarly simple:  

$$\omega^2 = W_z k_z^2 + W_x k_x^2 ,$$ (5)

where W_z is the vertical velocity squared and W_x is the horizontal velocity squared. Another form of equation () displays the fundamental nature of elliptical anisotropy more clearly. If you divide equation () through by $\vert\mbox{\bf k}\vert = k = k_z^2 + k_x^2$, you get  

$$W(\phi_{w}) = {{ W_z \cos^2(\phi_{w}) + W_x \sin^2(\phi_{w}) }} ,$$ (6)

where $W = \omega^2 / k^2$ is phase-velocity squared and $\phi_{w}$ is the wave-propagation (phase) direction, with $\phi_{w} = 0$ being vertical. The identity of form between equations () and () is not accidental. An ellipse is merely a stretched circle, and elliptical anisotropy is merely stretched isotropy. The Fourier similarity theorem tells us that a ``stretch'' (linear transformation) in the (x,z) domain corresponds to an ``inverse stretch'' (another linear transformation) in the (k_x, k_z) domain. Equations () and () look the same because they are both equations for stretched circles, i.e., ellipses.

Figure  shows how elliptical anisotropy can be used as a first-order paraxial approximation in both the group and phase domains. Although the group-velocity and phase-slowness representations of this transversely isotropic medium look very different, the first-order paraxial approximation is elliptical in both domains. (The TI medium is Greenhorn Shale from Jones and Wang (1981).)

This paraxial ellipse is also of considerable practical importance. The familiar isotropic moveout equation is  

$$T(x)^2 = T(0)^2 + (x/V_{\mbox{\rm\scriptsize NMO}})^2,$$ (7)

 

T(0)² = h² / V_z² , (8)

where T(x) is the traveltime at offset x, $V_{\mbox{\rm\scriptsize NMO}}$ is the moveout velocity, h is the vertical layer thickness, and V_z is the vertical velocity. (This is one-way traveltime.) In the isotropic case we only have one velocity, and set $V_{\mbox{\rm\scriptsize NMO}} = V_z = V_{\mbox{\rm\scriptsize iso}}$.Converting from isotropy to elliptical anisotropy with a vertical axis is as simple as replacing z with a new stretched coordinate $z^\prime \propto z$. Because the z coordinate does not occur in equation () it remains completely unchanged in the new coordinate system. The rescaled z coordinate does trivially affect equation  by scaling the numerator and denominator equally, leaving T(0) unchanged. (h and V_z scale with z, because they involve the same vertical length units.) The moveout equation for elliptical anisotropy thus has the same form as for isotropy; we merely have to let $V_{\mbox{\rm\scriptsize NMO}}$ and V_z become independent. The paraxial moveout velocity $V_{\mbox{\rm\scriptsize NMO}}$, unaffected by the stretch in z, must be the horizontal velocity V_x of the paraxial ellipse. Because h and V_z can be scaled together with no change in the recorded traveltimes, the vertical scale in elliptical anisotropy evidently can't be determined from surface kinematics alone (Dellinger and Muir, 1988).