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，即提供一类实际可扩展的数据同化（或过滤）方案，为地球物理流体动力学的状态估计提供坚实的理论基础。这个建议是一个副产品的PI最近成功的努力，设计准确，减少过滤方法与廉价的随机模型作为替代昂贵的模型。提出了四个项目：1.设计计算速度更快的随机滤波器，以便在热带地区存在多种云类型的情况下同化大气红外探测器（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。该项目涉及设计一种新的AR模型参数化方案，避免使用长时间序列的经典回归策略，但尊重最佳AR滤波的充分条件，建立在PI的最近的工作。3.研究了当从经典平均理论得到的简化模型用于过滤具有适度尺度分离的湍流信号模式之间的多尺度相互作用时，奇异摄动展开的高阶项的作用。该研究涉及形式渐近展开和严格的误差估计。PI将表明，高阶项对于避免存在模型误差时的协方差低估非常重要。4.发展一个快速的过滤框架来同化多尺度动力系统和“超参数化”，一个快速的数值方案来解决云尺度动力学和大尺度热带对流大气的相互作用。新算法将包括一个在线小规模估计方案，该方案在大规模和小规模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各种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一篇文章所述，这一长期存在的问题阻碍了全球天气模型预报技能从每周一次提高到每月一次。从这个建议的结果将改变未来的设计计算方法的各种预测相关的问题，在模式误差的存在下，特别是数值天气预报。该提案支持研究生的跨学科研究培训环境，涉及数学分析，统计建模和科学计算。PI被共同任命为PSU数学和气象系的教师，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.