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To estimate *W*_{11}, *W*_{33}, *W*_{44}, and *W*_{13} directly from
equation (1a), we need at least four
phase velocities at four different angles between 0 and 90 degrees. *W*_{ij}
is the solution of a system of nonlinear equations where the independent term
is formed by these phase velocities. Along the axes, the system of equations
is almost diagonal and the estimation of three elastic constants
(*W*_{33}, *W*_{11}, and *W*_{44})
is straightforward:

| |
(1) |

| (2) |

| (3) |

The elastic constants are estimated directly
from phase velocities along the axes.
*W*_{13} can be estimated from the previous elastic constants and
one phase velocity at an
oblique angle, typically 45 degrees.
This approach, although simple in theory, is not applicable in
many practical situations
because wide aperture data are required
(the angles of the observations must
include 0, 90 degrees, and one intermediate measurement) in order to
simplify the system of equations.
This is not the case for most single-geometry data sets (either surface,
or cross-well or VSP) where no rays travel along (at least) one of the
axes and therefore, phase velocities along both axes cannot be estimated
without having to assume a velocity symmetry (e.g.,
isotropic, elliptical,
TI).

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Stanford Exploration Project

11/17/1997