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Many different distributions are used to model statistics of waves that have been randomly scattered in atmospheric and terrain environments. These distributions have varying analytical advantages and ranges of physical applicability. This report reviews several basic distributions and discusses how they can be extended to include spatial and temporal variability in the scattering process. For this purpose, a compound probability density function (pdf) can be introduced in which a basic pdf describing the underlying scattering process is modulated by a second pdf describing parametric uncertainties in the scattering. We describe some useful new formulations based on the compound pdf, including strong and Rytov (lognormal) scattering processes modulated by the environment. These new formulations lead to relatively simple marginalized signal power distributions (Lomax and lognormal, respectively). Furthermore, we show how the conditional scattered signal pdf may be viewed as a likelihood function in which the modulating pdf is the Bayesian conjugate prior. The parameters of the modulating process can thus be refined by simple sequential Bayesian updating. Finally, the impact of the parametric uncertainties on signal detection and receiver operating characteristic curves is discussed and shown to be a very important consideration in practical applications. DISCLAIMER: The contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. All product names and trademarks cited are the property of their respective owners. The findings of this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. DESTROY THIS REPORT WHEN NO LONGER NEEDED. DO NOT RETURN IT TO THE ORIGINATOR.

Impact of Parametric Uncertainties on Scattered Signal Distributions and Receiver Operating Characteristics