The effect of random isotropic inhomogeneities on the phase velocity of seismic waves

In this paper we study, theoretically and numerically, the influence of 2-D and 3-D random isotropic stationary inhomogeneities on the phase velocities of the transmitted compressional wavefield of an initially plane (or spherical) wave. Due to scattering by the inhomogeneities the wavefield becomes distorted as the wave propagates through the medium. The traveltimes fluctuate when considering different wavefield registrations acquired at the points of surfaces that are parallel to the wavefront of the initial wave. It is usually observed that the slowness obtained from the averaged traveltime differs from the averaged slowness of the medium. In the geophysical literature this effect has been termed the 'velocity shift'. Using the Rytov approximation we establish formulas for the frequency- and travel-distance-dependent phase velocity of the transmitted wavefield in 2-D and 3-D randomly inhomogeneous media. We also compare our analytical results with finite-difference simulations. Good agreement between numerical simulations and theory is observed. The low-frequency limit of our analytical results coincides with the known effective-medium limit of the phase velocity in statistically isotropic inhomogeneous fluids with constant densities. In the high-frequency limit our results coincide with the results previously obtained by the ray-perturbation theory. However, in contrast to the ray theory, our description is not restricted to media with differentiable correlation functions of fluctuations. Moreover, our results quantify the frequency dependence of the velocity shift in the intermediate-frequency range. This frequency dependence is of major importance for estimating this effect in realistic situations.