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For elliptically anisotropic media, that is, media in which the group velocity
*V* has an elliptical dependence on the angle of energy propagation
, such that
| |
(1) |

the relation between group and phase velocities and angles is
given by a closed analytical form (Levin, 1978):

| |
(2) |

| (3) |

where *v* and are the phase velocity and angle, respectively.
Equation ( 1) also represents an elliptical dispersion
relation, since and
.It turns out that elasticity theory predicts this elliptical
dependence only for *S*_{H} in a transverse isotropic symmetry, or
for all wave types in the trivial case of isotropic material where
the ellipse becomes a circle. However, the use of an elliptical
approximation for the dispersion relation can considerably simplify
the problem of estimating the elastic parameters from the recorded
wavefields.
If we consider a source and a receiver separated by a vertical
distance *z* and a horizontal distance *x*, the
traveltime *t* predicted by equation (1) is

| |
(4) |

The relation between the slownesses squared (or sloth) *M*^{x} and *M*^{z}
^{}
that best fits our data and the true vertical and horizontal sloths of
the medium *M*^{v} and *M*^{h} respectively will depend on the specific
geometry in which the data was collected. For an usual surface-seismic
geometry, equation (4) correspond to a paraxial approximation
around the vertical axis. In this case, *M*^{x} corresponds
to the estimated normal-moveout (NMO) sloth, while
is the vertical traveltime. If *z* is unknown, then *M*^{z}
is an arbitrary factor that cannot be estimated (Dellinger and Muir, 1985).
If both distances, *x* and *z*, are known
it is possible to estimate both sloth parameters by fitting the
data to equation (4). For surface seismic surveys the estimated
*M*^{z} will be close to the vertical sloth *M*^{v}, whereas for
a cross-well geometry the estimated *M*^{x} will be close to the
horizontal sloth *M*^{h}. Karrenbach (1989) described a scheme to
estimate these two sloth parameters using traveltime data from
the three wave types, for the case of a homogeneous medium.
He also derived the relations between these parameters and the
elastic constants corresponding to a transversely isotropic symmetry,
for the specific case of a cross-well geometry.

** Next:** HORIZONTALLY LAYERED INVERSION
** Up:** Cunha: anisotropic traveltime inversion
** Previous:** Introduction
Stanford Exploration Project

1/13/1998