The fourth statistical moment of a scalar wave propagating in a random medium is investigated. Starting with a partial differential equation obtained by various authors in a multiple-scattering approximation, we discuss the general properties of the fourth moment of an initially plane wave and then present numerical results for a two-dimensional plane wave. Results are presented for the variance and covariance of irradiance scintillations. Both results are shown to agree well with experiment. In particular, the variance exhibits the experimentally observed saturation phenomenon, and the covariance results indicate that the correlation length for irradiance scintillations is not proportional to (λz)½. in the saturation region and that the aperture-averaging effect is less than that predicted by results based on the Born or Rytov approximations.
W. P. BROWN, JR., "Fourth Moment of a Wave Propagating in a Random Medium," J. Opt. Soc. Am. 62, 966-971 (1972)