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## Anelliptic NMO for TI media

Equations () and () relate TI elastic constants and NMO velocities, but to perform and interpret anelliptic moveout we will also need equations relating TI elastic constants and the anelliptic parameter FW. For the SH wavetype ,because SH waves in TI media are exactly elliptically anisotropic. For the qP and qSV wavetypes we have to begin by deriving a general form. We proceed by constructing a power series for T(p) in the same form as equation (). That done, we can just match coefficients on like powers of p.

Start with the equation for the traveltime through a layer,
 (30)
where is the traveltime through a layer of thickness h, is the group velocity, and is the group propagation direction. After some algebra to convert from group to phase variables equation () becomes
 (31)
where T is the group traveltime through a layer of thickness h, S is the square of the sine of the phase propagation direction, W(S) gives the squared phase velocity as a function of S, and is the first derivative of W. (See equation () for W(S) in the TI case.)

The ray parameter p can also be expressed as a simple function of S and W(S):
 (32)
Expand this equation and equation () to get power series for p2(S) and T(S), respectively.

Finally, we revert p2(S) and composite the resulting series S(p2) with the series T(S), obtaining
 (33)
We have used the identity (refer back to equation ()) to emphasize the similarity between equations () and (). The term is also invariant under changes of vertical scale (as we knew it must be from the symmetry of the problem).

Equating the p4 terms in equations () and () we obtain the desired equation for the paraxial value of FW:
 (34)

Although we could substitute equation () into equation () directly, the resulting hash of elastic constants is best avoided. The answer is much simpler in terms of ,,and .Using these variables, for the TI qP mode we obtain
 (35)

and for the qSV mode
 (36)

These equations and equation () allow us to find the layer FW for a given set of TI elastic constants. (Note these equations are not weak-anisotropy approximations.) Unfortunately, given a layer FW (perhaps found by anelliptic three-term velocity'' analysis) these equations do not appear to put simple constraints on the corresponding layer elastic constants.

Next: Conclusions Up: NMO Previous: Dix's equation
Stanford Exploration Project
11/17/1997