Approximation of the power distribution from multiple sound sources in the atmosphere using sums of gamma random variables

A sum of gamma random variables is a mathematically tractable approach to approximate multiple sources when the power (amplitude squared) of a single source is nearly gamma distributed because the sum can be expressed analytically for a wide variety of cases. Previous work indicates that the gamma distribution is a good two-parameter empirical approximation of received power from a single, elevated sound source in a turbulent atmosphere. The gamma distribution can be conceptualized as a sum of k independent and identically distributed exponential random variables, each with mean m. Here, k and m are called the shape and scale parameters, respectively. Thus, the summation of multiple gamma random variables with the same scale parameter still yields a gamma distribution. When multiple sources can be approximated well using a gamma distribution, then those sources could be grouped together for simplicity and conceived as a single source in a turbulent atmosphere. In addition, multiple authors have derived analytic expressions for the sum of N independent gamma random variables with distinct parameters in terms of the confluent hypergeometric functions. If the gamma random variables are correlated, then the result is still analytic using recurrence relations.