Decision metrics for automated target recognition and classification rely upon statistical distributions for the signal of interest and the background noise. This paper describes distribution models for situations in which an acoustic, RF, optical, or seismic signal is randomly scattered by the environment and received at one or two sensors, with the scattering strength varying randomly in space and time. A new distribution, called the compound variance gamma, is introduced, which applies to partially correlated data between two sensors with random scattering strength. This distribution reduces to several previously known scattering distributions as special cases. Calculation of receiver operating characteristic (ROC) curves using the new distribution is also discussed. A second new distribution, involving a product of modified Bessel functions, is also introduced to describe the magnitude of the cross product between a pair of sensors as needed to calculate the ROC curves. It is shown that the randomized scattering strength and correlation between the two sensors significantly impact signal detection.

Gamma distributions for signal intensity (or, equivalently, Nakagami distributions for signal amplitude) are often used in acoustics and electromagnetics to describe the single-point statistics of randomly scattered signals and noise in urban and other complex environments. An extension of the gamma distribution, called the compound gamma (CG), has been furthermore shown [D. K. Wilson, M. J. Kamrath, C. E. Haedrich, D. J. Breton, and C. R. Hart, “Urban noise distributions and the influence of geometric spreading on skewness,” J. Acoust. Soc. Am. 150(2), 783–800 (2021)] to usefully generalize the gamma to scenarios in which the distribution becomes skewed due to random variations in the scattering strength or loudness of the noise sources. Here we present the compound variance gamma (CVG), which further generalizes the compound gamma to two-point statistics and is, therefore, useful for representing statistics on arrays of sensors or networks. The CVG distribution, which is expressed with a Gauss hypergeometric function, is shown to be in excellent agreement with realistic simulations of sound scattering in the atmosphere.

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