Inverse Problem and the Pseudo-Empirical Orthogonal Function Method of Solution. Part 1. Theory. Part 2. Application

摘要

A library of mathematical functions or set of observations is used to form a set of orthonormal basis functions. It is assumed that any unknown solution can be constructed from a linear sum of these basis functions. A solution with a smoothing constraint and/or positivity constraints is developed. An analysis of the information contained in the measurements about the unknown solution is given. The amount of information (types of solutions, accuracy, and moments of the solution) about tropospheric rural aerosol size distribution that can in theory be obtained from backscattered measurements, without using any additional information about the anticipated assumptions (constraints) must be used to solve for aerosol size distribution, The inferred solution reflects assumptions and is therefore non objective. The quality of the solution depends on the applicability of the constraints to the given problem. Solutions for the inverse problem using the pseudo-empirical orthogonal method of solution were obtained using two types of constraints: positivity and positivity combined with smoothing. Results of the method, when backscattered radiation is used as measurements, are presented. Discussion on the limitations of the method and the effects upon the solution of the different assumptions that are used is given. Some possible uses of the solution are considered.