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的理论和建模本研究的主要目的是发展一种在不同观测几何条件下卫星测量“反演”的一般数学方法，以便提供影响测量的感兴趣的地球物理参数的估计。该逆问题实际上是由角变量连续索引的类似逆问题的连续体，所述角变量表征太阳和卫星传感器相对于地球表面上的目标点的位置。在这项研究中，映射被反演定义在黎曼流形上，并在黎曼流形中取值。这些微分几何结构是自然产生的，它们是待反演映射的域和范围的最佳描述。因此，它们是提高这些逆问题的精度和鲁棒性的核心要素。由于测量的不确定性，以及在地球物理模型中，随机对象值黎曼流形将被考虑。调查将产生一个严格的框架进行回归的一个流形值的随机对象上的另一个，其扩展的情况下，一个领域的流形值的随机对象。数学工具，脊函数的基础上，将开发近似，在任何所需的精度，一个领域的回归函数之间的流形参数化模型的字段。这些工具将附带用于估计模型自由参数的专用算法。这种一般的数学方法将适用于海洋颜色遥感问题，其中包括估计浓度和固有的光学特性的海洋成分，如浮游植物，沉积物和黄色物质，从大气层反射率测量。该问题中出现的微分几何结构将从控制海洋-大气系统中辐射传输的分析物理模型和方程中获得。更确切地说，海洋和大气层顶反射光谱的允许值集将分别给出黎曼流形的结构，以允许数学方法的适用性。将对由此产生的模型进行理论评估，并根据来自海景宽视场传感器和中分辨率成像光谱仪等传感器的实际卫星海洋颜色数据进行测试。性能将被量化，以及与其他反演方案相比的改进。更广泛的影响：这项研究将为流形上的近似理论和统计分析的数学领域贡献原创成果。在地球科学方面，它将提供一种主要是分析性的、可靠的和准确的反演方法，为卫星数据的反演提供严格的数学基础。该方法将适用于其他地球物理问题与不同的观测几何比海洋颜色遥感。 该项目将促进跨学科和国际合作。社会将受益于更准确的卫星数据集，这些数据集越来越多地用于环境应用以及地球化学和气候动力学研究。

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.