## Inversion of qP wave travel times: synthetic study

**JingSong Liu** **&**
**Ivan Psencik**
### Summary

An iterative procedure to determine lateral variation of 21 elastic
parameters from *qP* wave travel times is proposed. The procedure consists
of two steps. In the first step, an isotropic starting model is
updated using formulae for weak anisotropy. In this way, 15 *qP* wave weak
anisotropy (WA) parameters are determined. The WA parameters serve as
a basis for an estimate of all 21 elastic parameters, which are iteratively
updated until the difference between observed and calculated travel times
stops to vary. No a priori assumptions about heterogeneity and anisotropy
of a model are made except the variation of elastic parameters with respect
to spatial coordinates in the model box is assumed to be trilinear.

The procedure is tested on a synthetic multi-azimuthal multiple-source
offset VSP experiment. Travel times were picked from noiseless *qP* wave
synthetics generated by sources distributed along 6 profiles itersecting
at the mouth of a borehole with receivers. The model considered
is vertically inhomogeneous of hexagonal symmetry, with nearly horizontal
axis of symmetry. The model may thus simmulate a cracked medium with a
slightly tilted system of vertical parallel cracks. The SVD technique is
used for solving the system of linear equations resulting from the
condition of minimization of the object function. Due to insufficient
illumination of the studied structure, the inversion process is surprisingly
slow. At least for *qP* waves, however, it converges to elastic parameters
of the true model. In case of *qS* waves, the resulting phase velocities
roughly show a proper angular variation but the phase velocity surfaces
based on inverted data have a nearly constant offset from the true
surfaces. The experiment shows that a single iteration is insufficient for
accurate recovery of the structure.

### Whole paper

The paper is available in
PostScript (957 kB, colour figures)
and GZIPped PostScript (238 kB, colour figures).

In: Seismic Waves in Complex 3-D Structures, Report 8,
pp. 27-46, Dep. Geophys., Charles Univ., Prague, 1999.

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