Practical Filtering Methods with Model Errors

本提案中的项目是PI长期职业目标的一部分，该目标是为地球物理流体动力学的状态估计提供一类具有坚实理论基础的实际可扩展的数据同化（或过滤）方案。这一建议是PI最近成功设计精确、简化过滤方法的成果，该方法使用廉价的随机模型来替代昂贵的模型。提出了四个项目：1.；设计计算速度更快的随机滤波器，以便在热带多种云型存在的情况下吸收大气红外探测器（AIRS）。2. 为非线性、弱混沌动力系统开发稳定的线性自回归（AR）滤波器。该项目涉及为AR模型设计一种新的参数化方案，该方案避免使用经典回归策略中的长时间序列，但尊重PI最近工作中建立的最佳AR滤波的充分条件。3. 研究了用经典平均理论的简化模型滤波具有中等尺度分离的湍流信号模态间多尺度相互作用时，高阶项在奇异摄动展开中的作用。本研究涉及形式渐近展开和严格误差估计。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很重要。4. 利用“超参数化”发展同化多尺度动力系统的快速滤波框架，解决云尺度动力学与大尺度热带对流大气相互作用的快速数值方案。新算法将包括一个在线小规模估计方案，该方案在大变量和小变量之间施加统计一致性。实时天气预报的基本问题是模型误差。这个问题是由于对物理的不完全理解和我们缺乏计算资源来解决不同时间和长度尺度的物理过程。即使在解析了100亿个变量之后，现代业务天气模式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式预报技术从每周提高到每月。该建议的结果将改变未来设计的计算方法，以解决存在模式误差的各种预测相关问题，特别是数值天气预报。本提案为研究生提供跨学科的研究训练环境，包括数学分析、统计建模和科学计算。作为PSU数学系和气象系的联合教员，PI将开发一门跨学科的研究生课程，重点是PDE和大气和海洋建模中的波浪。

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.