Numerical Inversion Methods in Geoscience and Quantitative Remote Sensing

To estimate structural parameters and spectral component signatures of Earth surface cover type, quantitative remote sensing seems to be an appropriate way to deal with these problems. Since the real physical system that couples the atmosphere with the land surface is very complicated and should be continuous, sometimes it requires comprehensive parameters to describe such a system, so any practical physical model can only be approximated by a model which includes only a limited number of the most important parameters that capture the major variation of the real system. The pivot problem for quantitative remote sensing is the inversion. Inverse problems are typically ill-posed. The ill-posed nature is characterized by (C1) the solution may not exist; (C2) the dimension of the solution space may be infinite; (C3) the solution is not continuous with the variation of the observed signals. These issues exist for all quantitative remote sensing inverse problems. For example, when sampling is poor, i.e., there are very few observations, or directions are poorly located, the inversion process would be underdetermined, which leads to the large condition number of the normalized systems and the significant noise propagation. Hence (C2) and (C3) would be the chief difficulties for quantitative remote sensing inversion. This chapter will address the theory and methods from the viewpoint that the quantitative remote sensing inverse problems can be represented by kernel-based operator equations.