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** Previous:** Dix's equation

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 *F*_{W}.
For the SH wavetype ,because SH waves in TI media are exactly elliptically anisotropic.
For the *q*P and *q*SV 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
*p*^{2}(*S*) and *T*(*S*), respectively.
Finally, we revert *p*^{2}(*S*) and composite the resulting series *S*(*p*^{2})
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 *p*^{4} terms
in equations () and () we obtain
the desired equation for the paraxial value of *F*_{W}:

| |
(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 *q*P mode we obtain

| |
(35) |

and for the *q*SV mode
| |
(36) |

These equations and equation () allow us to find the layer *F*_{W}
for a given set of TI elastic constants. (Note these equations are
not weak-anisotropy approximations.)
Unfortunately, given a layer *F*_{W} (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