Reflected pulses from a refractive random medium at grazing incidence

A problem of acoustic pulse reflection by a one-dimensional refractive random medium is considered in the case of grazing angle incidence. The material parameters of the medium are assumed to vary with a random microscale and a deterministic macroscale. A system of stochastic equations for random scattering variables is derived based upon the random modelling of three separate scales of variations. The statistical properties of the reflected pulses are characterized by an asymptotic diffusion limit theorem of stochastic differential equations with multiple scales. The transport equations governing the limiting stochastic distributions of the random reflection coefficient are obtained in the propagating regime, which leads to the power spectral densities of the reflected pressure and particle velocity fields.