Practical Filtering Methods with Model Errors

该方案中的项目是PI长期职业目标的一部分，目的是提供一类具有坚实的地球物理流体动力学状态估计理论基础的实用可伸缩数据同化(或过滤)方案。这一建议是PI最近成功地设计了准确的、简化的过滤方法的结果，该方法使用廉价的随机模型作为昂贵模型的替代方案。提出了四个方案：1.设计计算速度更快的随机滤波器，以同化热带多种云型下的大气红外探测仪(AIRS)。2.为非线性弱混沌动力系统设计稳定的线性自回归(AR)滤波器。这个项目涉及到设计一种新的AR模型的参数化方案，该方案避免了使用经典回归策略中的长时间序列，但尊重PI最近的工作中建立的最优AR滤波的充分条件。3.研究了将经典平均理论的简化模型用于滤除具有适度尺度间隔的湍流信号模式间的多尺度相互作用时，奇异摄动展开的高阶项所起的作用。该研究涉及形式渐近展开和严格的误差估计。PI将表明，在存在模型误差的情况下，高阶项对于避免协方差低估是重要的。4.发展了一个同化多尺度动力系统的“超参数”快速滤波框架，这是一种解决云-尺度动力与大尺度热带对流大气相互作用的快速数值方案。新的算法将包括一个在线小规模估计方案，该方案强制大小尺度变量之间的统计一致性。实时天气预报的基本问题是模式误差。这个问题归因于对物理学的不完全理解，以及我们缺乏计算资源来解决不同时间和长度尺度上的物理过程。即使在分解了100亿个变量之后，现代可操作的天气模型也很难再现热带观测记录。正如世界气象组织公报最近的一篇文章所报道的那样，这个长期存在的问题阻碍了全球天气模型预报技能从每周到每月的提高。这一建议的结果将改变未来对各种预报相关问题的计算方法的设计，在存在模式误差的情况下，特别是数值天气预报。这项建议支持为研究生提供跨学科的研究培训环境，包括数学分析、统计建模和科学计算。PI被联合任命为巴黎州立大学数学和气象系的教员，将开发一门跨学科的研究生课程，重点是PDE和用于大气和海洋建模的波。

The projects in this proposal are part of the PI's long-term career goal to deliver a class of practically scalable data assimilation (or filtering) schemes with solid theoretical foundations for state estimation of geophysical fluid dynamics. This proposal is an outgrowth of the PI's recent successful effort in designing accurate, reduced filtering methods with cheap stochastic models as alternatives to expensive models. Four projects are proposed: 1. Design computationally faster stochastic filters to assimilate atmospheric infrared sounder (AIRS) in the presence of multiple cloud types in the tropics. 2. Develop stable linear autoregressive (AR) filters for nonlinear, weakly chaotic dynamical systems. This project involves designing a novel parameterization scheme for AR models that avoids utilizing a long time series as in classical regression strategy, yet respects the sufficient conditions for optimal AR filtering, established in the PI's recent work. 3. Study the role of higher order terms of the singular perturbation expansion when a reduced model from classical averaging theory is used in filtering multi-scale interaction between modes of turbulent signals with moderate separation of scales. This study involves formal asymptotic expansion and rigorous error estimation. The PI will show that the higher order terms are important to avoid covariance underestimation in the presence of model errors. 4. Develop a fast filtering framework to assimilate multi-scale dynamical systems with "superparameterization", a fast numerical scheme to resolve interaction of cloud-scale dynamics and large-scale tropical convecting atmosphere. The new algorithm will include an online small-scale estimation scheme that imposes statistical consistency between the large and small-scale variables. Fundamental issues in real-time weather prediction are model errors. This problem is attributed to incomplete understanding of the physics and our lack of computational resources to resolve physical processes in various time and length scales. Modern operational weather models poorly reproduce the tropical observational records even after resolving 10 billion variables. This long-standing issue prevents the global weather model forecasting skill to improve from weekly to monthly, as reported in a recent article in the World Meteorological Organization bulletin. The results from this proposal will transform the future design of computational methods for various prediction related problems in the presence of model errors, in particular numerical weather prediction. This proposal supports an interdisciplinary research training environment for a graduate student, involving mathematical analysis, statistical modeling, and scientific computing. The PI, who is jointly appointed as a faculty in the mathematics and meteorology departments at PSU, will develop an interdisciplinary graduate course with emphasis on PDE and waves for atmospheric and ocean modeling.