A stochastic parabolic wave equation and field-moment equations for random media having spatial variation of mean refractive index

The wave equation for propagation through random inhomogeneities is expressed in a special coordinate system which follows the ray path of the deterministic wave that would propagate in the absence of randomness. The ratio of the full wave field in the presence of random inhomogeneities to the determinstic wave field is shown to satisfy the parabolic wave equation, with acoustic wavelength depending on position along the ray. This approximation requires that the local sound speed be only slightly perturbed from its local deterministic value, although it may be very different from the sound speed at a distant point on the deterministic ray. No matter how tortuous the path of the deterministic field, this parabolic equation holds provided that the angles between the rays of the full and determinstic fields are small. The equations for moments of the field (mean field, mutual coherence function, and fourth moment) have a form very similar to the case of uniform background.