Bayesian analysis for inversion of aerosol size distribution data

Obtaining the continuous aerosol size distribution from a set of discrete measurements is an ill-posed problem. We use a new methodology based on a synthesis of Bayesian probability analysis and Monte Carlo simulations to estimate the parameters of a bimodal lognormal size distribution, i.e. the mass median diameters and geometric standard deviations of the two modes, and the fraction of the total mass in the first mode. In the Bayesian view, a measurement process serves to refine previous knowledge of physical parameters by narrowing their probability distributions. The approach avoids the direct solution of the inverse problem; instead the “forward” problem is solved repeatedly for many sets of input size distribution parameters. From a statistical analysis of the model outputs, the input size distribution parameters are inferred. The method makes explicit use of information available prior to making any measurements and the nature of the measurement errors. Once the updated probability distributions of the size distribution parameters are determined, a number of possible solutions can be obtained. The choice could either be the maximum likelihood solution, or a minimum variance Bayes estimate that minimizes the difference between the actual measurements and those calculated using the solution. Good retrievals were obtained for numerically generated data as well as experimental data. Bayes-Monte Carlo gives reconstructions that are better than those obtained using a weighted least-squares method, even though they may sometimes come with higher computation costs. The method is not limited to determining the parameters of a certain functional form of the size distribution; it is flexible enough to allow estimation of size distributions without predetermined functional forms.