Hierarchical model for the scale-dependent velocity of waves in random media

Elastic waves of short wavelength propagating through the upper layer of the Earth appear to move faster at large separations of source and receiver than at short separations. Existing perturbation theories predict a linear increase of the velocity shift with increasing separation and cannot describe the saturation of the velocity shift at large separations that is seen in computer simulations. We point out that this nonperturbative problem can be solved using a model developed originally for the study of directed polymers. The saturation velocity is found to scale with the four-thirds power of the root-mean-square amplitude of the velocity fluctuations, in good agreement with the computer simulations.