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In cases such as cross-borehole tomography where true vertical and
horizontal scale measurements are available, elliptical anisotropy can
be useful as a ``first step'' away from isotropy.
If only surface kinematic data are available, though,
equation () is enough for doing NMO and
assuming elliptical anisotropy accomplishes nothing more
than making explicit the depth ambiguity that was always there.
To add another velocity parameter to this isotropic (really
elliptically anisotropic) case, we need to add some sort of an
*an*elliptic parameter.
Note that
equations (), (), and ()
share a common polynomial form:

| |
(9) |

We add an *an*elliptic parameter to equation ()
by generalizing it to
| |
(10) |

Varying the cross term *F* lets us vary the behavior away
from perfect ellipticity in between the coordinate axes.
If the numerator becomes a perfect square and
equation () reverts to the original elliptic form in
equation () for all propagation directions.
Why this particular generalization?
This anelliptic form needs only one more parameter
beyond elliptical anisotropy (or isotropy for the surface survey case),
retains the properties that make elliptical anisotropy so convenient
(in approximation),
and makes a good approximation to exact transverse isotropy,
especially in the phase-slowness domain. (See Figure .)

Following this template,
equation () (the ray equation) becomes

| |
(11) |

where
,,and, as before, *M* indicates slowness squared
(*M*_{x} = 1/ *V*_{x}^{2},
,and *M*_{z} = 1/ *V*_{z}^{2}).
Note this equation explicity separates the
true horizontal velocity *V*_{x} from the paraxial NMO velocity
.
We can similarly extend equation ()
(the dispersion relation) to

| |
(12) |

where
,,and as before *W* indicates velocity squared
(*W*_{x} = *V*_{x}^{2} = 1 / *M*_{x},
,and *W*_{z} = *V*_{z}^{2} = 1 / *M*_{z}).
Note the symmetry between
equations and ;
as for elliptical anisotropy, converting from the representation in
one domain to the other is as simple as replacing each of the velocity
parameters with its reciprocal.

** Next:** Consistency
** Up:** Dellinger, Muir, & Karrenbach:
** Previous:** ELLIPTICAL ANISOTROPY
Stanford Exploration Project

11/17/1997