Collaborative Research: CMG--Ensemble Data Assimilation for Nonlinear and Nondifferentiable Problems in Geosciences

数据同化是试图观测和预测大气状态的重要组成部分，它被定义为特定位置的温度、压力、湿度、风速和其他变量的值。数据同化系统通常有两个组成部分：1)一组不完美的、在空间和时间上分布不均的观测值，这些观测值以复杂的方式与状态相关(例如，卫星感知辐射，而辐射与温度和湿度间接相关，雷达回波与降水间接相关)；以及2)复杂和不完美的预报模型，它提供对大气状态的“初步猜测”。数据同化的目标是以最佳方式将模式的初步猜测和观测结合起来，以产生对状态的最佳表示，同时估计由于观测和预报模式的限制而造成的状态不确定性。集合数据同化(EnsDA)是一种数据同化方法，在每个同化周期中使用预报集合，以便预报集合成员之间的差异提供了一种表达模式产生的第一次猜测的概率性质的手段。例如，单一预报可以预测给定位置的降雨或不降雨，而一系列预报可以估计降雨的概率。由于大气变化的复杂性和观测与状态相关的间接方式，EnsDA方法通常需要简化假设才能实用。常见的简化假设有1)观测值可以通过简单的线性函数与状态相关；2)大气平稳演化，因此大气状态可以被视为在空间和时间上以平滑、可微的方式变化。虽然方便，但这些假设在物理上是不合理的，本提案中的研究是试图找到不依赖于这些假设的新的EnsDA方法。这项工作首先使用“成本函数”来量化估计大气状态中的误差，该“成本函数”被最小化以产生同化状态。大气状态演化和状态观测关系中的非线性和不可微性导致代价函数中的非线性和不可微性。这项研究通过1)评估适用于EnsDA的不可微代价函数最小化方法；2)检验混合集合数据同化方法在非线性和不可微应用中的价值；以及3)开发和评估一种非线性和不可微EnsDA方法，旨在量化现实高维地球科学应用中的不确定性。本研究旨在找到更好的方法来使用现有的数据和模型来理解和预测大气行为。这些努力最终将导致对恶劣天气的更好预报，这将使社会受益。此外，为大气开发的EnsDA技术将适用于海洋和用于预测气候变化的大气-海洋耦合模式。这笔赠款还将通过资助研究生的教育和培训来帮助培训下一代科学家。

Data assimilation is an essential component of attempts to observe and predict the state of the atmosphere, defined as the values of temperature, pressure, humidity, wind speed, and other variables at specific locations. A data assimilation system typically has two components:1) a set of observations which are imperfect, unevenly distributed in space and time, and related to the state in complex ways (e.g. satellites sense radiation which is indirectly related to temperature and moisture, radar returns are indirectly related to precipitation); and 2) a complex and imperfect forecast model, which provides a "first guess" of the atmospheric state. The goal of data assimilation is to optimally combine the model first guess and the observations to produce the best possible representation of the state, accompanied by an estimate of the state uncertainty caused by the limitations of the observations and the forecast model. Ensemble data assimilation (EnsDA) is a data assimilation method in which an ensemble of forecasts is used in each assimilation cycle, so that differences among the forecast ensemble members provide a means of expressing the probabilistic nature of the model-generated first guess. For example, a single forecast will predict either rain or no rain at a given location, whereas an ensemble of forecasts can estimate the probability of rainfall.Due to the complexity of atmospheric variability and the indirect ways in which observations are related to the state, EnsDA methods usually require simplifying assumptions in order to be practically useful. Among the common simplifying assumptions are 1) that the observations can be related to the state through simple linear functions; and 2) that the atmosphere evolves smoothly, so that the atmospheric state can be treated as varying in space and time in a smooth, differentiable way. While convenient, these assumptions are not physically justifiable, and the research in this proposal is an attempt to find new EnsDA methods which do not rely on these assumptions. The work begins by quantifying the error in the estimated atmospheric state using a "cost function", which is minimized to produce the assimilated state. Nonlinearity and nondifferentiability in the evolution of the atmospheric state and in state-observation relationships leads to nonlinearity and nondifferentiability in the cost function. This research addresses the lack of smoothness in the cost function by 1) evaluating nondifferentiable cost function minimization methods suitable for EnsDA; 2) examining the value of hybrid ensemble data assimilation methods for nonlinear and nondifferentiable applications; and 3) developing and evaluating a nonlinear and nondifferentiable EnsDA method designed to quantify uncertainty in realistic high-dimensional geosciences applications.The research is intended to find better ways to use existing data and models to understand and predict the behavior of the atmosphere. These efforts will ultimately lead to better forecasts of severe weather which will benefit society. In addition, EnsDA techniques developed for the atmosphere will be applicable to the ocean and to coupled atmosphere-ocean models used to anticipate climate change. The grant will also contribute to the training of the next generation of scientists, by funding the education and training of a graduate student.