Retrieval of the aerosol particle size distribution function by incorporating a priori information

The determination of the aerosol particle size distribution function using the particle spectrum extinction equation is an ill-posed integral equation of the first kind. To overcome the ill-posedness, regularization techniques such as Phillips–Twomey's regularization, Tikhonov's smooth regularization as well as some iterative methods were developed. However, most of the literature focuses on improving the solvability of the problem without introducing some useful a priori   information, which are essential to ill-posed inverse problems, since it is known that, we are often faced with limited/insufficient observations in remote sensing. Therefore, we restudy this problem. We first formulate the problem in l1l1 space and solve a nonnegative constrained problem to obtain a useful a priori   solution, then we use this solution to reformulate the problem in l2l2 space and solve a regularized problem. Numerical results are based on the remotely sensed data by Sun-photometer CE 318 for Poyang lake region of Jiangxi Province and are performed to show the efficiency and feasibility of the proposed algorithms.