PARAMETERIZATION OF ANISOTROPIC GROUP VELOCITIES

Assume that anisotropic velocities have a vertical axis of symmetry, like the transversely isotropic (TI) media described in Thomsen 1986. Although that paper describes ``weak'' anisotropy, the same equations can be applied to very strong anisotropy Tsvankin and Thomsen (1994).

Three of Thomsen's parameters, V_z, $\delta$, and $\epsilon$,are defined by the elastic constants of a general TI medium. These constants can be used to specify three different effective velocities at a single point in the model. V_z is the velocity of a wave traveling vertically along the axis of symmetry. The velocity in any horizontal direction is V_x determined by  

$$\epsilon = V_x^2 (V_z^{-2} - V_x^{-2})/2 \gt 0$$ (19)

and a ``normal moveout velocity'' (NMO) velocity V_n defined by  

$$\delta \equiv V_n^2 (V_z^{-2} - V_n^{-2})/2 < 0 .$$ (20)

Phil Anno of Conoco has shown that if the TI properties represent the equivalent medium of many isotropic layers Backus (1962); Schoenberg and Muir (1989), then the above inequalities can be expected to hold. (One additional assumption is that the V_s/V_p ratio and V_s have a positive correlation.) Notice that $V_x \approx (1+\epsilon) V_z$ and $V_n \approx (1+\delta) V_z$ , so that $V_n \leq V_z \leq V_x$.

For convenience, researchers at the Colorado School of Mines Alkhalifah and Tsvankin (1994); Tsvankin and Thomsen (1994) have also defined a constant  

$$\eta \equiv (\epsilon - \delta)/(1 + 2 \delta) = V_x^2 (V_n^{-2} - V_x^{-2})/2 \gt 0 .$$ (21)

Many combinations of three of these parameters can be used to describe a TI medium. An approximation has already dropped a fourth constant to which compressional P waves are very insensitive. The exact equations for TI phase velocity as a function of angle are rather clumsy, and no explicit form is available for group velocity. Alternative approximate equations can used which fit almost the same family of curves as the original correct equations Michelena et al. (1993). I use an approximate equation for group velocity which appears to emulate closely the exact curves for large ranges of positive $\epsilon$ and negative $\delta$.Since I aim to estimate these anisotropic velocities from noisy measurements, I expect our estimated curves to have larger errors than introduced by these approximations.

I choose approximate curves with the three velocities defined above. Let $\phi$ be the group angle of a raypath from the vertical. Then the group velocity $V(\phi)$ can be expressed as  

$$V(\phi)^{-2} = V_z^{-2} \cos^2 (\phi ) + ( V_n^{-2} - V_x^{-2} ) \cos^2 (\phi ) \sin^2 (\phi) + V_x^{-2} \sin^2 (\phi) .$$ (22)

Greg Lazear of Conoco found that a good approximation of the phase velocity $v(\theta)$as a function of the phase angle $\theta$takes a similar form, but with reciprocals of velocities:  

$$v(\theta)^{2} = V_z^{2} \cos^2 (\theta ) + ( V_n^{2} - V_x^{2} ) \cos^2 (\theta ) \sin^2 (\theta) + V_x^{2} \sin^2 (\theta) .$$ (23)

The NMO velocity also turns out to have a physical interpretation. Imagine an experiment on a homogeneous and anisotropic medium (or imagine a small scale experiment on a smooth model). Measure the traveltime t₀ between two points placed on a vertical line, separated by a vertical distance V_z t₀. Now displace the upper point a distance h along a horizontal line and measure the new traveltime t_h.

Then according to equation (22) the traveltime t_h as a function of offset h is exactly  

$$t_h^2 = t_0^2 + \left [ V_n^{-2} + (V_x^{-2} - V_n^{-2}) { h^2 \over {h^2 + V_z^2 t_0^2}} \right ] h^2 .$$ (24)

When $h \ll V_z t_0$ then the value of t_h in this ``moveout equation'' is controlled by the NMO velocity V_n rather than V_x. In the other case $h \gg V_z t_0$, the raypath is almost horizontal and V_x dominates.

I find it convenient to define a stacking velocity V_h(h) as a function of the offset h for a fixed vertical distance:  

$$V_h(h)^{-2} \equiv (t_h^2 - t_0^2)/h^2 = V_n^{-2} + (V_x^{-2} - V_n^{-2}) { h^2 \over {h^2 + V_z^2 t_0^2}} .$$ (25)

Thus, we can construct a equation which describes the best fitting hyperbola to traveltimes at zero offset and at a single finite offset h:

 

t² = t₀² + h² / V_h(h)² . (26)

Note that this stacking velocity covers the range $V_n \leq V_h(h) \leq V_x$, increasing in value as h increases. (To use two-way reflection times in 26 we need only replace the half offset h by the full offset.)

Theoretically, three measurements of traveltimes at three different offsets h should uniquely determine the three velocity constants V_z, V_x, V_n. However, the traveltimes are much more sensitive to V_n, which determines moveouts at small offsets, and to V_x, which determines moveout at larger offsets. The vertical velocity V_z affects only the rate at which the stacking velocity changes from one limit to the other. As long as V_z has roughly the correct magnitude, then we can fit all measured traveltimes very well. Remember that we expect $V_n \leq V_z \leq V_x$ for equivalent layered media.

For imaging data in time, we can set V_z = V_n and simplify our equations even further. To image surface data in depth, we can focus images very well with good values for V_x and V_n, then adjust imaged depths to tie wells with V_z, holding the other two velocities constant.

Stacking velocity analysis can be optimized with V_n because it is relatively close to V_h. If the maximum offset equals the depth, and if V_x = 1.1 V_n, then V_h = 1.02 V_n for a flat reflection, which is small enough difference for such a large anisotropy.

This anisotropy model, although certainly not the most general, describes the most important properties of transversely isotropic velocities. The very simple form allows for easy optimization and inversion. A tomographic algorithm which builds on such a model will include the necessary dependence of velocities on angle. Any refinements in the anisotropic behavior will be easy to introduce without major alterations of the computer program.