Derivatives of the radiation field and their application to the solution of inverse problems

Abstract

Spectral distribution of the solar radiation traveling through the Earth’s atmosphere contain an important information about numerous atmospheric and surface parameters. This information can be gained from the measured spectra employing so-called inverse theory and the problem to be solved thereby is usually referred to as the inverse problem. The first step to be done in the solution of any inverse problem is to formulate a model usually referred to as the forward model which will allow us to simulate the measured quantity assuming all relevant atmospheric and surface parameters to be known. Generally, in the case of the scattered, reflected, or transmitted solar radiance measured in the ultraviolet, visible, or near-infrared spectral range by means of satellite, airborne, or ground-based instruments, the corresponding forward model is nonlinear, i.e., there is no linear relationship between measured values of intensity and atmospheric parameters. However, the theoretical basis of the inverse problem solution is well investigated in the case of linear inverse problems only [19]. Thus, to make use of the existing numerical methods the forward model has to be linearized, i.e., a linear relationship between intensity of radiation and the atmospheric parameters has to be obtained. This can formally be done considering the intensity as a function or functional of the corresponding parameters and expanding it in the Taylor series with respect to the variations of the desired parameters.