Statistical properties of reflection traveltimes in 3-D randomly inhomogeneous and anisomeric media

We study the statistical properties of seismic reflection traveltimes in the presence of the double-passage effect (DPE) in order to characterize the inhomogeneities of the reflector overburden. A detailed analysis of the traveltime covariance function is presented for more general conditions than in the pioneering works by Touati, Iooss and Gaerets et al., namely 3-D geometry, quasi-homogeneous fluctuations of the medium parameters, anisomeric (i.e. statistically anisotropic fluctuations), longitudinal and transversal positions of receivers, and curved rays. The behaviour of the traveltime covariance function is elucidated by some numerical examples. It is shown, in particular, that at large offsets the effective longitudinal correlation scale associated with the ray trajectory is much larger than the transverse correlation scale and their product equals the product of the correlation lengths of the fluctuations of the medium