Regularizing active set method for retrieval of the atmospheric aerosol particle size distribution function

The determination of the aerosol particle size distribution function by using the particle spectrum extinction equation is an ill-posed integral equation of the first kind [S. Twomey, J. Comput. Phys.18, 188 (1975);Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007)], since we are often faced with limited or insufficient observations in remote sensing and the observations are contaminated. To overcome the ill-posed nature of the problem, regularization techniques were developed. However, most of the literature focuses on the application of Phillips-Twomey regularization and its variants, which are unstable in several cases. As is known, the particle size distribution is always nonnegative, and we are often faced with incomplete data. Therefore, we study the active set method and propose a regularizing active set algorithm for ill-posed particle size distribution function retrieval and for enforcing nonnegativity in computation. Our numerical tests are based on synthetic data for theoretical simulations and the field data obtained with a CE 318 Sun photometer for the Po Yang lake region of Jiang Xi Province, China, and are performed to show the efficiency and feasibility of the proposed algorithms.