Inversion techniques for personal cascade impactor data

Abstract

We examined two inversion procedures for solving the Fredholm integral equation of the first kind to obtain aerosol particle size distributions from a set of measured masses collected on the various stages of a personal cascade impactor. The problem is essentially ill-conditioned, in that many solutions satisfy exactly an integral equation slightly perturbed from the original due to measurement error. The two methods, although derived from different families of inversion techniques, fit into the general framework of Tikhonov regularization. Both try to optimize the a posteriori degree of matching of the solution to the measured data and the a priori judgments about the likelihood of a solution in terms of its smoothness. The first method uses a weighted least squares optimization and zeroth-order regularization to fit a priori bi-modal log-normal distribution functions, using an intermediate step to define an appropriate starting point for the optimization routine. The second involved “blind” inversion of the impactor data to express the second derivative of the particle size distribution function as a linear combination of orthogonal basis functions, chosen so that the resulting solution is smooth and positive. The orthogonal functions are constructed from the eigenvectors and eigenvalues of a kernel covariance matrix. The personal inhalable dust spectrometer (PIDS), used to illustrate the application of these methods, is an eight-stage cascade impactor which selects the inhalable fraction of the aerosol by means of a specially designed inlet. Both inversion methods explicitly include consideration of the aerosol that is collected in the sampler entry between the inlet and the first impactor stage, something that applies to all cascade impactors but which has not usually been taken into account in the past. An important parameter in inversions, the expected value of measurement error for each stage, was estimated from a series of wind-tunnel experiments. Both methods work well for simulated PIDS data as well as for experimental wind-tunnel data for a wide range of sets of aerosol size distribution parameters.