Measuring particle size distribution accurately down to approximately 1nm is needed for studying atmospheric new particle formation. The scanning particle size magnifier (PSM) using diethylene glycol as a working fluid has been used for measuring sub-3nm atmospheric aerosol. A proper inversion method is required to recover the particle size distribution from PSM raw data. Similarly to other aerosol spectrometers and classifiers, PSM inversion can be deduced from a problem described by the Fredholm integral equation of the first kind. We tested the performance of the stepwise method, the kernel function method (Lehtipalo et al., 2014), the HA linear inversion method (Hagen and Alofs, 1983), and the expectation–maximization (EM) algorithm. The stepwise method and the kernel function method were used in previous studies on PSM. The HA method and the expectation–maximization algorithm were used in data inversion for the electrical mobility spectrometers and the diffusion batteries, respectively (Maher and Laird, 1985). In addition, Monte Carlo simulation and laboratory experiments were used to test the accuracy and precision of the particle size distributions recovered using four inversion methods. When all of the detected particles are larger than 3nm, the stepwise method may report false sub-3nm particle concentrations because an infinite resolution is assumed while the kernel function method and the HA method occasionally report false sub-3nm particles because of the unstable least squares method. The accuracy and precision of the recovered particle size distribution using the EM algorithm are the best among the tested four inversion methods. Compared to the kernel function method, the HA method reduces the uncertainty while keeping a similar computational expense. The measuring uncertainties in the present scanning mode may contribute to the uncertainties of the recovered particle size distributions. We suggest using the EM algorithm to retrieve the particle size distributions using the particle number concentrations recorded by the PSM. Considering the relatively high computation expenses of the EM algorithm, the HA method is recommended for preliminary data analysis. We also gave practical suggestions on PSM operation based on the inversion analysis.
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