Parabolic and white-noise approximations for elastic waves in random media

In this paper we carry out an asymptotic analysis of the elastic wave equations in random media in the parabolic white-noise regime. In this regime, the propagation distance is much larger than the initial beam width, which is itself much larger than the typical wavelength; moreover, the correlation length of the random medium is of the same order as the initial beam width, and the amplitude of the random fluctuations is small. In this distinguished limit we show that wave propagation is governed by a system of random paraxial wave equations. The equations for the shear waves and pressure waves have the form of Schrödinger equations driven by two correlated Brownian fields. The diffraction operators can be expressed in terms of transverse Laplacians. The covariance structure of the Brownian fields is determined by the two-point statistics of the density and Lamé parameters of the random medium.