Estimating the Variance in Solutions to the Aerosol Data Inversion Problem

Abstract

Regularization has been successfully used for solving a wide variety of ill-posed problems such as the inversion of aerosol size distribution data. The solutions are well characterized and converge nicely to the true distribution as the number of data increases. If there are few data, then there can be many reasonable distributions that are consistent with the measurements. Here, in addition to knowing an optimal solution, one should also have an estimate of the variance of the solution or a characterization of the size of the solution set. We set up the necessary machinery to allow one to estimate the variance of linear functionals of the size distribution, e.g., the concentration of particles in a given size interval. This estimate depends on the form of the weighted average, the variance in one's a priori estimate of the size distribution, the data, and the uncertainty in the data. There are many applications. We demonstrate, for example, how to determine which of two instruments will better allow one to estimate the concentration of particles in a given size interval. Also, we determine the number of measurements necessary to ensure the variance of the estimated concentration is less than a specified value.