Inversion methods to determine two-dimensional aerosol mass-mobility distributions II: Existing and novel Bayesian methods

Abstract

Moving towards two-dimensional distributions of particle properties is important to the study of aerosol formation, aerosol climate impacts, and aerosols in material science. This paper builds on existing work to examine Bayesian or statistical approaches to inverting tandem particle mass analyzer (PMA) and differential mobility analyzer (DMA) data to retrieve the two-dimensional mass-mobility distribution. We first consider the Bayesian representation of derivative-based Tikhonov regularization, focusing on the first-order case. We demonstrate a new Bayesian model selection scheme to choose the regularization parameter, which generally outperforms the L-curve approach for derivative-based Tikhonov regularization. We also perform a Bayesian-based uncertainty analysis to evaluate the quality of the reconstructions, noting that uncertainties are lowest in regions close to device setpoints. We then present a new exponential distance prior, a variant of generalized Tikhonov regularization that provides a natural approach to regularizing the two-dimensional aerosol size distribution problem by allowing smoothing preferentially along the length of the distribution. The exponential distance approach is observed to reduce errors in the reconstructions by up to 60%, with the benefit to using the exponential distance prior increasing as the distributions become increasingly narrow, i.e. more highly correlated. Finally, Bayesian model selection is shown to also be a good candidate to optimize the regularization parameters in the exponential distance prior.