The main objective of this study is to develop a general mathematical methodology for the "inversion" of satellite measurements under varying observation geometry so as to provide estimates of the geophysical parameters of interest that influence the measurements. This inverse problem is in fact a continuum of similar inverse problems continuously indexed by the angular variables which characterize the positions of the sun and of the satellite sensor with respect to the target point on the Earth's surface. In this study, the maps to be inverted are defined on and valued in Riemannian manifolds. These differential geometrical structures arise naturally; they are among the best descriptions of the domains and ranges of the maps to be inverted. Consequently, they are core ingredients towards an improvement, in terms of accuracy and robustness, of the resolution of those inverse problems. Due to uncertainties in the measurements, as well as in the geophysical models, random objects valued in Riemannian manifolds will be considered. The investigations will yield a rigorous framework for performing the regression of a manifold valued random object on another one, and its extension to the case of a field of manifold valued random objects. Mathematical tools, based on ridge functions, will be developed to approximate, within any required accuracy, a field of regression functions between manifolds by a field of parameterized models. Dedicated algorithms to estimate the free parameters of the models will accompany those tools. This general mathematical methodology will be applied to the ocean color remote sensing problem, which consists of estimating the concentration and inherent optical properties of oceanic constituents, such as phytoplankton, sediments, and yellow substances, from top-of-atmosphere reflectance measurements. The differential-geometrical structures arising in that problem will be obtained from the analytical physical models and equations governing the radiative transfer in the ocean-atmosphere system. More precisely, the set of permitted values for the marine and top-of-atmosphere reflectance spectra, respectively, will be given the structure of a Riemannian manifold, to allow applicability of the mathematical methodology. The resulting models will be evaluated theoretically and tested on actual satellite ocean color data, from sensors such as the Sea-viewing Wide-Field-of-view Sensor (SeaWiFS) and the MODerate resolution Imaging Spectrometer (MODIS). Performance will be quantified, as well as improvements compared with other inversion schemes.
Broader Impacts: The study will contribute original results to the mathematical fields of approximation theory and statistical analysis on manifolds. In geosciences, it will provide a mostly analytical, robust and accurate inversion methodology with rigorous mathematical grounding for inversion of satellite data. The methodology will be applicable to other geophysical problems with varying observation geometry than ocean color remote sensing. The project will promote interdisciplinary and international collaboration The benefits to society will be through more accurate satellite data sets which are increasingly used in environmental applications and in the study of biogeochemistry and climate dynamics.
