Statistical inversion of aerosol size measurement data

We consider the determination of the particle size distribution function fr om Poisson distributed observations arising in aerosol size distribution me asurements with the differential mobility particle sizer (DMPS), The DMPS m easurement data consists of counts of aerosol particles classified into dif ferent 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 th is inverse problem by treating both the observations and the unknown parame ters as random variables. We construct a realistic posterior model for the aerosol size distribution function by using the Bayes' theorem. In the cons truction of this model we assume that the measurements obey Poisson statist ics and that the solution is a smooth non-negative function. We discuss the computation of two point estimates from the posterior density. These are t he maximum a posteriori estimate, which is computationally an optimisation problem, and the conditional mean which is computationally an integration p roblem. The former is solved by using an exterior point algorithm and the l atter with a Markov chain Monte Carlo (MCMC) method. The virtue of using MC MC methods for drawing samples from the posterior distribution is not limit ed to computing the conditional mean only - they can also be used for the c omputation of other moments and confidence intervals. The point estimates a s well as some marginal distributions and confidence intervals are investig ated using artificially generated data. The estimates are also compared to those obtained by using Gaussian statistical assumptions.