When the number of steps in a random walk varies, the distribution of the resultant vector components in the limit of large mean step number may be non-Gaussian The statistics and temporal correlation properties of one class of such non-Gaussian limit distributions are derived and some of its potential applications are reviewed briefly

https://iopscience.iop.org/article/10.1088/0305-4470/13/1/006/pdf

On the statistics of K-distributed noise

This paper addresses problems associated with the development of widely applicable non-Gaussian noise models, particularly with reference to the statistical properties of scattered waves. A combination of phenomenological arguments and exact solutions of specific scattering problems are used to elucidate the significance of one model—K-distributed noise—which has several attractive features and has already found many applications. A full statistical-mechanical formulation is developed for non-Gaussian compound Markov processes, with this model as a special case. The implications for numerical simulation of correlated non-Gaussian noise are explored and comparisons made with experimental data. A brief review of current applications of the K-distribution model is given.

Non-Gaussian models for the statistics of scattered waves

The distribution of intensity scattered by a plane mirror placed in the Fresnel region behind a smoothly varying deep random phase screen is calculated for arbitrary angles of incidence of the illuminating beam. It is found that enhancement occurs in the backward direction because of the coherent addition of doubly scattered waves, as in particle-scattering systems, and also as a result of statistical geometrical-optics effects. These effects are largest when reflection takes place near the focusing plane of the phase screen and lead to an enhancement factor proportional to the logarithm of the phase variance introduced by the screen. The coherence and higher-order statistical properties of the scattered wave are briefly examined.

Enhanced backscattering through a deep random phase screen

Consideration of the density of rays emanating from an infinite, corrugated Gaussian surface with fractal slope reveals a hitherto unexplored shortwave scattering regime that is uncomplicated by the presence of caustics in the scattered wave field. The statistical and coherence properties of the ray-density fluctuations in this regime are calculated as a function of fractal dimension D, and it is shown that in the Brownian case (D = 1.5) the problem can be solved exactly. The properties of the intensity pattern in a coherent scattering configuration are also investigated. The contrast of the pattern is computed as a function of propagation distance, and the asymptotic behavior in the strong scattering limit is again found to be exactly solvable when D = 1.5. It is shown that the intensity fluctuations are K distributed in this case. The effects of a finite outer-scale size are evaluated and discussed.

Fresnel scattering by a corrugated random surface with fractal slope

The critical exponent of the intensity moment is computed from the Thorn-Arnol'd classification of caustics as catastrophes The caustics are studied on the torus whose coordinates are 16 N random phases theta 1 theta N for members of the ensemble describing the phase screen or inhomogeneous medium responsible for the disorder of the wave The results indicate that when N to infinity (Gaussian random medium) the exponents depend only on whether the waves propagate in two or three space dimensions

/pdf/focusing-and-twinkling-critical-exponents-from-catastrophes-3dnvw799eh.pdf

Focusing and twinkling: critical exponents from catastrophes in non-Gaussian random short waves

The role of the phase autocorrelation function in the generation of speckle, focusing and other scintillation effects produced when radiation is scattered by a deep random phase screen is investigated. New analytical formulae are obtained which reveal the mathematical origin of the observed physical phenomena, and these are supported by a range of numerically computed results.

https://iopscience.iop.org/article/10.1088/0305-4470/10/9/016/pdf

E Jakeman

Papers

Correlation function dependence of the scintillation behind a deep random phase screen