ITR/AP: Optimal Nonlinear Estimation in the Geosciences

本项目开发了一种新的技术来估计随机动力系统的状态，给出了一些部分和不完全的测量信息。与传统的线性估计方法（如最小二乘变分方法或卡尔曼滤波方法）不同，这种新技术能够处理高度非线性动力学-因此是非高斯统计-以接近Kushner， Stratonovitch, Pardoux （KSP）正式精确解的最优性的方式。与KSP一样，该方法计算给定测量值的系统的条件统计量。然而，当应用于信息技术中感兴趣的问题时，如大规模地球科学或环境数据同化，新方法不会像KSP那样导致难以解决的功能随机方程。本项目采用的方法是通过变分公式来近似条件统计，该变分公式产生条件均值作为适当成本函数的最小值，协方差作为其黑森量。所提出的代价函数是通过瑞利-里兹或矩闭格式计算的，这应该使问题易于数值计算。本文的主要研究工作是开发合适的统计技术，为瑞利-里兹计算系统建模，制定有效的成本函数数值优化算法，并与现有的次优估计方案进行比较。在许多具有实际重要性的地球科学领域，将观测所得的信息与复杂的数值模型的结果结合起来，以对该系统过去或未来的状态作出最佳估计，这是至关重要的。在通过监测几个井场观察到的化学物质或放射性物质泄漏的情况下，人们希望及时追踪污染物的源头。要做到这一点，必须只掌握有关含水层和地下水流场特性的统计知识，而且测量结果本身也有误差。在数值天气预报中，目标是将卫星阵列的最新观测结果与大规模气象计算机模型的输出相结合，以预测未来的天气状况。同样，模型包含随机或混沌元素，这阻碍了基于部分和不完全信息的完美预测。本项目开发的建模方法和数值算法应提供一种实用的方案，以便在模型的随机性和观测结果的不确定性面前计算出最佳估计值。

This project develops a new technique for estimation of the state of a stochastic dynamical system, given some partial and imperfect information from measurements. Unlike traditional linear estimation methods, such as least-squares variational methods or Kalman Filter approach, this new technique is capable of handling highly nonlinear dynamics--and thus non-Gaussian statistics-- in a way that approaches the optimality of the formally exact solution by Kushner, Stratonovitch, Pardoux (KSP). Just as KSP, the new method computes the conditional statistics of the system given the measurements. However, when applied to problems of interest in information technology, such as large-scale geoscience or environmental data assimilation, the new method does not lead to functional stochastic equations that are hopelessly intractable to solve, as does KSP. The approach pursued in this project is instead to approximate the conditional statistics by means of a variational formulation, which yields the conditional mean as the minimizer of an appropriate cost function and the covariance as its Hessian. The cost function proposed is calculated by a Rayleigh-Ritz or moment-closure scheme, which should render the problem tractable to numerical computation. The main research that will be done is to develop suitable statistical techniques to model the system for the Rayleigh-Ritz calculation, to work out efficient algorithms for the numerical optimization of the cost function, and to compare with existing suboptimal estimation schemes. In many areas of the geosciences of practical importance, it is crucial to combine information from observations with the results of a sophisticated numerical model to produce the best estimate of the past or future state of the system. In the case of a chemical or radioactive spill observed by monitoring a few well sites, one wants to track the contaminant plume backward in time to its source. This must be accomplished with only statistical knowledge about the properties of the aquifer and groundwater flow field and with the measurements themselves subject to error. In numerical weather prediction, the goal is instead to combine the latest observations from satellite arrays with the output of large-scale, meteorological computer models to predict the future state of the weather. Again, the models contain stochastic or chaotic elements which prevent perfect prediction based upon partial and imperfect information. The modeling methods and numerical algorithms developed in this project should provide a practical scheme to compute the best estimate in the face of such randomness in the model and uncertainty in the observations.