Stable distributions for randomly positioned sound emitters in an urban environment

Abstract

Previously [D. K. Wilson, M. J. Kamrath, C. E. Haedrich, D. J. Breton, and C. R. Hart, J. Acoust. Soc. Am. 150(2), 783–800 (2021)], it was suggested that the probability distribution for sound levels in an urban environment could be usefully modeled with N randomly placed, independent sources in a circular region, with the receiver at the center. The sound decays away from the sources by a geometrical spreading law. With these assumptions, the received sound power consists of the sum of N Pareto-distributedrandom variables. Although an analytical solution for this sum is unavailable, some positively skewed, heavy-tailed distributions, such as the exponentially modified Gaussian distribution, provide reasonable approximations. The present study is motivated by the observation that, in the limit of large N, the Pareto sum must converge to a stable distribution; in particular, for spherical spreading, the limiting distribution is a Landau distribution. We have furthermore found that when N is drawn from a Poisson distribution, the Landau distribution is nearly exact for as few as eight sources. Hence, the Landau distribution provides a suitable general model for urban noise and other situations involving multiple, randomly placed sources.