Statistical inversion of aerosol size measurement data

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Abstract
We consider the determination of the particle size distribution function from Poisson distributed observations arising in aerosol size distribution measurements with the differential mobility particle sizer (DMPS). The DMPS measurement data consists of counts of aerosol particles classified into different size ranges and the goal is to compute an estimate for the particle size distribution function on the basis of this data. This leads to an ill-posed inverse problem. The approach we take in this paper is to consider this inverse problem by treating both the observations and the unknown parameters as random variables. We construct a realistic posterior model for the aerosol size distribution function by using the Bayes' theorem. In the construction of this model we assume that the measurements obey Poisson statistics and that the solution is a smooth non-negative function. We discuss the computation of two point estimates from the posterior density. These are the maximum a posteriori estimate, which is computationally an optimisation problem, and the conditional mean which is computationally an integration problem. The former is solved by using an exterior point algorithm and the latter with a Markov chain Monte Carlo (MCMC) method. The virtue of using MCMC methods for drawing samples from the posterior distribution is not limited to computing the conditional mean only – they can also be used for the computation of other moments and confidence intervals. The point estimates as well as some marginal distributions and confidence intervals are investigated using artificially generated data. The estimates are also compared to those obtained by using Gaussian statistical assumptions.