## Calculation of the a priori geometrical covariances
of travel times in a self-affine random medium

**Ludek Klimes**
### Summary

To perform the stochastic inversion of travel times,
we need the geometrical covariances of travel times, represented
by the integrals of the medium covariance function
over corresponding pairs of rays.
These covariances are also essential for estimating
the deviations of synthetic travel times calculated
in the velocity model from correct travel times.
The medium covariance function describing
the deviation of the medium from the mean model
is expressed in terms of the velocity-scaled
ellipsoidally anisotropic power-law correlation function
of a self-affine random medium.
The geometrical covariances of travel times
are semianalytically expressed in terms of the hypergeometric functions
which can efficiently be calculated.

### Keywords

Stochastic inversion of travel times, velocity model,
heterogeneous medium,
self-affine random medium, medium covariance function,
travel-time covariance.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 18,
pp. 27-70, Dep. Geophys., Charles Univ., Prague, 2008.

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