We develop a model for the probability density function (pdf) of the irradiance fluctuations of an optical wave propagating through a turbulent medium. The model is a two-parameter distribution that is based on a doubly stochastic theory of scintillation that assumes that small-scale irradiance fluctuations are modulated by large-scale irradi- ance fluctuations of the propagating wave, both governed by indepen- dent gamma distributions. The resulting irradiance pdf takes the form of a generalized K distribution that we term the gamma-gamma distribution. The two parameters of the gamma-gamma pdf are determined using a recently published theory of scintillation, using only values of the refractive-index structure parameter C n (or Rytov variance) and inner scale l 0 provided with the simulation data. This enables us to directly calculate various log-irradiance moments that are necessary in the scaled plots. We make a number of comparisons with published plane wave and spherical wave simulation data over a wide range of turbu- lence conditions (weak to strong) that includes inner scale effects. The gamma-gamma pdf is found to generally provide a good fit to the simu- lation data in nearly all cases tested. © 2001 Society of Photo-Optical Instrumen-

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Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media

Publisher Summary This chapter discusses the effects of atmospheric turbulence in optical astronomy, summarizes the present state of the theory, reviews the experimental checks that have been made, and discusses the implications in the domain of astronomical observations. Diffraction limited resolution has been obtained with large telescopes up to magnitude 13 by means of speckle interferometry. A Michelson interferometer is already working in the visible on a 20 m baseline up to magnitude 4, and many projects are nearing completion. However, such technological progress can be useful only if accurate quantitative measurements prove to be feasible through atmospheric turbulence. In addition, much work remains to be done before achieving a full understanding of atmospheric effects. The relationship between isoplanicity or speckle boiling and the structure of the atmosphere is still not clear.

V The Effects of Atmospheric Turbulence in Optical Astronomy

Current speckle filters attempt to restore the radar reflectivity using only the multiplicative speckle noise assumption. The best known filters, namely the Frost, Lee or Kuan niters are adaptive filters based on the local statistics, computed in a fixed square window. In this way, the speckle is reduced as a function of the heterogeneity measured by the local coefficient of variation. When the radar reflectivity undergoes significant variations due to the presence of strong scatterers or structural features (edges or lines) in the processing window, such speckle filtering is less effective. In this paper it is shown that the filtering process can be controlled both by the coefficient of variation and by various geometrical ratio detectors. Through shape adaptive windowing, these detectors allow the use of large window sizes for better speckle reduction while preserving spatial resolution and structural features. The backscattered intensity is modelled as K-distributed within speckled targets and...

Structure detection and statistical adaptive speckle filtering in SAR images

A new class of distributions, G distributions, arising from the multiplicative model is presented, along with their main properties and relations. Their densities are derived for complex and multilook intensity and amplitude data. Classical distributions, such as K, are particular cases of this new class. A special case of this class called G/sup 0/, that has as many parameters as K distributions, is shown able to model extremely heterogeneous clutter, such as that of urban areas, that cannot be properly modeled with K distributions. One of the parameters of this special case is related to the degree of homogeneity, and a limiting case is that of a scaled speckle. The advantage of the G/sup 0/ distribution becomes evident through the analysis of a variety of areas (urban, primary forest and deforested) from two sensors.

A model for extremely heterogeneous clutter

Optical wireless, also known as free-space optics, has received much attention in recent years as a cost-effective, license-free and wide-bandwidth access technique for high data rates applications. The performance of free-space optical (FSO) communication, however, severely suffers from turbulence-induced fading caused by atmospheric conditions. Multiple laser transmitters and/or receivers can be placed at both ends to mitigate the turbulence fading and exploit the advantages of spatial diversity. Spatial diversity is particularly crucial for strong turbulence channels in which single-input single-output (SISO) link performs extremely poor. Atmospheric-induced strong turbulence fading in outdoor FSO systems can be modeled as a multiplicative random process which follows the K distribution. In this paper, we investigate the error rate performance of FSO systems for K-distributed atmospheric turbulence channels and discuss potential advantages of spatial diversity deployments at the transmitter and/or receiver. We further present efficient approximated closed-form expressions for the average bit-error rate (BER) of single-input multiple-output (SIMO) FSO systems. These analytical tools are reliable alternatives to time-consuming Monte Carlo simulation of FSO systems where BER targets as low as 10-9 are typically aimed to achieve.

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Optical wireless links with spatial diversity over strong atmospheric turbulence channels

A mathematical model for non-Rayleigh microwave sea echo is developed which describes explicitly the dependence of statistical properties of the radar cross section on the area of sea surface illuminated by the radar. In addition to the first probability distribution of the scattered radiation, its temporal and spatial correlation functions are also considered. It is shown that, in general, these correlation functions decay on at least two scales, the second, non-Rayleigh, contributions being strongly dependent on the properties of a "single scatterer." Predictions of the model are found to be in qualitative agreement with existing experimental data. A new class of probability distributions, the " K -distributions," is introduced, which may prove useful for fitting such data.

P. Pusey

Papers

A model for non-Rayleigh sea echo