CMG: Understanding Ensemble Approaches to Environmental Data Assimilation

本项目的重点是环境数据同化--广义上定义为利用所有相关信息描述环境系统状况的过程。一类被称为集合方法的序列数据同化技术为处理环境应用中重要的非线性、高维和不确定性问题提供了一种很有前途的方法。系综方法从相对较少的随机重复(由随机初始条件和模型输入驱动的模型模拟)有效地构造复杂非线性系统的低维近似。集合数据同化方法所使用的近似可以看作原始状态空间在连续变化的随机子空间上的投影。这些子空间的结构反映了物理约束的影响。但在近似过程中也有很强的偶然性。这种随机性似乎提供了确定性近似方法所不能轻易实现的稳健性和灵活性，特别是对于大型非线性问题。尽管集合数据同化正越来越频繁地应用于环境问题，但没有一套理论解释它何时或为什么起作用。特别是，研究人员不理解为什么在随机线性子空间上的投影可以为高度非线性问题产生改进的近似。该项目将加深对集合估计的理解，并为集合方法的应用和改进提供良好的理论基础。它将以分阶段的方式将理论分析和计算实验结合起来，逐步从简单的问题转移到更复杂的问题。该项目将依靠随机矩阵理论的最新数学进展和特别适合于集合应用的并行计算资源的可用性。地球科学正处于一场重要的变革之中，这在很大程度上是由于可用信息的数量和质量的急剧增加。遥感已经对气象学、水文学和海洋学产生了重大影响。吸收所有这些新信息是一项具有挑战性的任务。然而，潜在的好处是巨大的，特别是在气候变化、人口对自然资源的压力以及全球要素循环的重大变化日益引起关注的时候。该项目将研究吸收和合并大量不同卫星数据的非常有效但知之甚少的方法。要调查的数学问题涉及从手机网络到海洋学的各种应用。该项目将支持两项主要的教育/推广活动：1)让本科生研究人员参与数学教学实验室的开发和地球科学研究项目，2)与波士顿科学博物馆现有的科学与技术中心(CS&amp；T)建立公共推广伙伴关系(包括展览、多媒体演示和实践讲习班)。所有这些节目都将把研究成果带给广泛和多样化的受众。

The focus in this project is on environmental data assimilation -- broadly defined as the process of characterizing the state of an environmental system, using all relevant information. A class of sequential data assimilation techniques known as ensemble methods provides a promising way to deal with the issues of nonlinearity, high dimensionality, and uncertainty, which are all important in environmental applications. Ensemble methods effectively construct low dimensional approximations of complex nonlinear systems from a relatively small number of random replicates (model simulations driven by random initial conditions and model inputs). The approximations used by ensemble data assimilation methods can be viewed as projections of the original state space on continually changing random subspaces. These subspaces have structure that reflects the influence of physical constraints. But there is also a strong element of chance in the approximation process. This randomness seems to provide robustness and flexibility that cannot be so readily achieved with deterministic approximation methods, especially for large nonlinear problems. Although ensemble data assimilation is being applied to environmental problems with increasing frequency there is no body of theory that explains when or why it works. In particular, researchers do not understand why projections onto random linear subspaces can yield improved approximations for highly nonlinear problems. This project will advance understanding of ensemble estimation and provide a sound theoretical basis for the application and improvement of ensemble methods. It will combine theoretical analysis and computational experiments in a staged approach that gradually moves from simple to more complex problems. The project will rely on recent mathematical advances in random matrix theory and on the availability of parallel computing resources that are especially well-suited to ensemble applications.The earth sciences are in the midst of an important transformation, due in large part to a dramatic increase in the quantity and quality of available information. Remote sensing has already had a significant impact on meteorology, hydrology and oceanography. Assimilation of all this new information is a challenging task. Nevertheless, the potential benefits are substantial, particularly at a time when climate change, population pressures on natural resources, and major modifications in global element cycles are attracting increasing attention. This project will investigate very efficient but poorly understood methods for assimilating and merging large amounts of diverse satellite data. The mathematical issues to be investigated are relevant to applications ranging from cell phone networks to oceanography. This project will support two major education/outreach activities: 1) involvement of undergraduate researchers in development of mathematical teaching laboratories and in earth science research projects, 2) a public outreach partnership with the Current Science & Technology Center (CS&T) at the Boston Museum of Science (including exhibits, multimedia presentations, and hands-on workshops). All of these programs will bring research results to a broad and diverse audience.