## Local determination of weak anisotropy parameters
from qP-wave slowness and particle motion measurements

**Xuyao Zheng** **&**
**Ivan Psencik**
### Summary

We propose an algorithm for local evaluation of weak anisotropy (WA)
parameters from measurements of slowness vector components and/or of
particle motions of *qP* waves at individual receivers in a borehole
in a multi-azimuthal multiple-source offset VSP experiment. As a
byproduct the algorithm yields approximate angular variation of *qP*-wave
phase velocity. The formulae are derived under assumption of weak
but arbitrary anisotropy and lateral inhomogeneity of the medium. The
algorithm is thus independent of structural complexities between the
source and the receiver. If complete slowness vector is determinable
from observed data, then the information about polarization can be used
as an independent additional constraint. If only the component of the
slowness along the borehole can be determined from observations (which
is mostly the case), the inversion without information about polarization
is impossible. We present several systems of equations, which can be
used when different numbers of components of the slowness vector are
available. The SVD algorithm is used
to solve an overdetermined system of linear equations for WA parameters
for two test examples of synthetic multi-azimuthal multiple-source offset
VSP data. The system of equations results from approximate first-order
perturbation equations for the slowness and polarization vectors of
the *qP* wave. Analysis of singular values and of variances of WA
parameters is used for the estimation of chances to recover the sought
parameters. Effects of varying number of profiles with sources and of
noise added to "observed" data are illustrated. An important observation
is that although, due to insuficient data, we often cannot recover
all individual WA parameters with sufficient accuracy, angular phase
velocity variation can be recovered rather well.

### Whole paper

The paper is available in
PostScript (703 kB)
and GZIPped PostScript (147 kB).

In: Seismic Waves in Complex 3-D Structures, Report 11,
pp. 15-41, Dep. Geophys., Charles Univ., Prague, 2001.

To appear in *Pure and Applied Geophysics*, **159** (2002), 1881-1905.

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