Data inversion methods to determine sub-3 nm aerosol size distributions using the particle size magnifier

Measuring particle size distribution accurately down to approximately 1 nm is needed for studying atmospheric new particle formation. The scanning particle size magnifier (PSM) using diethylene glycol as the working fluid has been used for measuring sub-3 nm atmospheric aerosol. A proper inversion method is required to recover the particle size distribution from PSM raw data. Similar to other aerosol spectrometers and classifiers, PSM inversion can be deduced to a problem described by the Fredholm integral equation of the first kind. We tested the performance of the step-wising method, the kernel function method (Lehtipalo et al., 2014), the H&A linear inversion method (Hagen and Alofs, 1983), and the expectation-maximization (EM) algorithm. The step-wising method and the kernel function method were used in previous studies on PSM. The H&A method and the expectation-maximization algorithm were used in data inversion for the electrical mobility spectrometers and the diffusion batteries (Maher and Laird., 1985), respectively. 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 3 nm, the step-wising method may report false sub-3 nm particle concentrations because of assuming an infinite resolution, while the kernel function method and the H&A method occasionally reports false sub-3 nm particles because of using the unstable least square 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 H&A 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 H&A method is recommended to be used for preliminary data analysis. We also gave practical suggestions on PSM operation based on the inversion analysis.