CMG: Theory and Modeling of Differential-Geometrical Structures in Ocean-Atmosphere Optics for Inversion of Satellite Data

摘要：0417748CMG：海洋-大气光学中用于卫星数据反演的微分几何结构理论和建模本研究的主要目的是发展一种通用的数学方法，用于在不同观测几何下“反演”卫星测量，从而提供对影响测量的感兴趣的地球物理参数的估计。这个反问题实际上是一系列类似的反问题的连续体，这些反问题连续地以角变量为索引，这些角变量表示太阳和卫星传感器相对于地球表面目标点的位置。在本研究中，要反转的映射被定义在黎曼流形上并在黎曼流形上赋值。这些不同的几何结构是自然产生的；它们是对待反转地图的域和范围的最好描述之一。因此，它们是改进这些逆问题的精度和鲁棒性的核心成分。由于测量和地球物理模型的不确定性，将考虑黎曼流形中的随机对象。研究将产生一个严格的框架来执行一个流形值随机对象对另一个的回归，并将其扩展到流形值随机对象的域的情况。将开发以脊函数为基础的数学工具，以便在任何要求的精度范围内，通过参数化模型来近似流形之间的回归函数场。用于估计模型自由参数的专用算法将伴随这些工具。这种一般的数学方法将应用于海洋颜色遥感问题，该问题包括通过大气顶部反射率测量估计海洋成分（如浮游植物、沉积物和黄色物质）的浓度和固有光学特性。该问题中产生的微分几何结构将从控制海洋-大气系统辐射传输的解析物理模型和方程中得到。更准确地说，海洋和大气顶反射光谱的容许值将分别被赋予黎曼流形的结构，以允许数学方法的适用性。由此产生的模型将在理论上进行评估，并在实际卫星海洋颜色数据上进行测试，这些数据来自于海景广域传感器（SeaWiFS）和中分辨率成像光谱仪（MODIS）等传感器。性能将被量化，以及与其他反演方案相比的改进。广泛影响：本研究将为流形的近似理论和统计分析等数学领域贡献原创成果。在地球科学方面，它将为卫星数据的反演提供一种具有严格数学基础的主要分析、稳健和准确的反演方法。该方法将适用于海洋色遥感以外具有不同观测几何形状的其他地球物理问题。该项目将促进跨学科和国际合作，通过越来越多地用于环境应用和生物地球化学和气候动力学研究的更准确的卫星数据集，将为社会带来好处。

Abstract: 0417748CMG: Theory and Modeling of Differential-Geometrical Structures in Ocean-Atmosphere Optics for Inversion of Satellite Data 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.