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

Elastic waves of short wavelength propagating through the upper layer of th e Earth appear to move faster at large separations of source and receiver t han at short separations. Existing perturbation theories predict a linear i ncrease of the velocity shift with increasing separation and cannot describ e the saturation of the velocity shift at large separations that is seen in computer simulations. We point out that this nonperturbative problem can b e solved using a model developed originally for the study of directed polym ers. The saturation velocity is found to scale with the four-thirds power o f the root-mean-square amplitude of the velocity fluctuations, in good agre ement with the computer simulations.