AUTHOR=Dubovik Oleg , Fuertes David , Litvinov Pavel , Lopatin Anton , Lapyonok Tatyana , Doubovik Ivan , Xu Feng , Ducos Fabrice , Chen Cheng , Torres Benjamin , Derimian Yevgeny , Li Lei , Herreras-Giralda Marcos , Herrera Milagros , Karol Yana , Matar Christian , Schuster Gregory L. , Espinosa Reed , Puthukkudy Anin , Li Zhengqiang , Fischer Juergen , Preusker Rene , Cuesta Juan , Kreuter Axel , Cede Alexander , Aspetsberger Michael , Marth Daniel , Bindreiter Lukas , Hangler Andreas , Lanzinger Verena , Holter Christoph , Federspiel Christian 

TITLE=A Comprehensive Description of Multi-Term LSM for Applying Multiple a Priori Constraints in Problems of Atmospheric Remote Sensing: GRASP Algorithm, Concept, and Applications

JOURNAL=Frontiers in Remote Sensing

VOLUME=Volume 2 - 2021

YEAR=2021

URL=https://www.frontiersin.org/journals/remote-sensing/articles/10.3389/frsen.2021.706851

DOI=10.3389/frsen.2021.706851

ISSN=2673-6187

ABSTRACT=We describe an approach called the Multi-term Least Square Method (LSM) that has been used to develop complex aerosol inversion algorithms for a number of years and applied to retrievals of laboratory and ground-based measurements. Theoretically, it was shown how to unite the advantages of a variety of approaches and to provide transparency and flexibility in development of practically efficient retrievals.  From a practical viewpoint, this approach provides a methodology for using multiple a priori constraints to atmospheric problems where rather different groups of parameters should be retrieved simultaneously.  For example, Dubovik and King (2000) used multi-term LSM for designing the algorithm that retrieves aerosol size distribution and spectrally dependent complex index of refraction from Sun/sky-radiometer ground-based observations. Furthermore, the significant potential of the multi-term LSM approach was demonstrated with the development of the GRASP (Generalized Retrieval of Aerosol and Surface Properties) algorithm. The GRASP algorithm is based on several generalization principles with the idea to develop a scientifically rigorous and versatile algorithm.  It has significantly extended capabilities and areas of applicability and can be applied to diverse remote sensing observations. This paper also illustrates the practical applicability of GRASP and, therefore the multi-term LSM, in diverse situations.
GRASP has two main independent modules. The first module is a numerical inversion that includes general mathematical operations not related to a particular physical nature of the inverted data. Numerical inversion is implemented as a statistically optimized fitting of observations following the multi-term LSM strategy. The presentation of the GRASP numerical inversion provides a profound description of the main methodological aspects used for establishing a multi-term LSM approach continuum that is aimed at applying multiple a priori constraints in the retrieval. The foundation of this approach uses the fundamental frameworks of the Method of Maximum Likelihood (MML) and LSM statistical estimation concepts. We discuss the asymptotical optimal properties of MML and LSM estimates in order to emphasize the importance of statistical estimation methods in remote sensing. 
We also compare the multi-term LSM with other established statistical optimization approaches  such as Bayesian concepts and the Optimal Estimation approach [Rodgers 2000].