An algorithm is described for inverting individual particle properties from statistics of ensemble observations, thereby dispelling the notion that coincident particles create inherently erroneous data in particle probes. The algorithm assumes that the observed property obeys superposition, that the particles are independently randomly distributed in space, and that the particle distribution is stationary over the accumulation distance. The fundamental principle of the algorithm is based on a derived analytical relationship between ensemble and individual particle statistics with fully defined derivatives. This enables rapid convergence of forward inversions. Furthermore, this relationship has no dependence on the particular instrument realization, so the accuracy of the relationship is not fundamentally constrained by the accuracy to which a measurement system can be characterized or modeled. This algorithm is presented in terms of a single observed property, but the derivation is valid for correlated multiparameter retrievals. Because data are collected in histograms, this technique would require relatively little storage and network bandwidth on an aircraft data system. This statistical analysis is derived here for measuring particle geometric extinction cross sections, but it could also be applied to other particle properties, such as scattering cross-section and phase matrix elements. In this example application, a simulated beam passes through a sampled environment onto a single detector to periodically measure beam extinction. This measured extinction may be the result of one or more particles, but it is shown that the probability distribution function of the ensemble (multiparticle) extinction measurement can be used to obtain the distribution of individual particle extinction cross sections (used here as a proxy for particle size distribution).
