ITR/AP: Optimal Nonlinear Estimation in the Geosciences

本计画发展一种新的技术，用来估计一个随机动态系统的状态，并从量测中得到部分或不完全的资讯。 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态-因此非高斯统计-以接近Kushner，Stratonovitch，Pardoux（KSP）的形式精确解的最优性的方式。 正如KSP一样，新方法计算给定测量值的系统的条件统计量。 然而，当应用于信息技术的利益，如大规模的地球科学或环境数据同化的问题，新方法不会导致功能随机方程是无可救药地难以解决，KSP。 在这个项目中所追求的方法，而不是近似的条件统计的变分公式，它产生的条件平均值作为一个适当的成本函数的最小值和协方差作为其海森。 提出的成本函数计算的瑞利-里兹或时刻封闭计划，这应该使问题易于数值计算。 主要的研究，将做的是开发合适的统计技术来模拟系统的瑞利-里兹计算，制定有效的算法的数值优化的成本函数，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，将观测到的联合收割机信息与复杂的数值模型的结果结合起来，对系统的过去或未来状态作出最佳估计是至关重要的。在通过监测几个井场观察到化学品或放射性泄漏的情况下，人们希望及时追踪污染物羽流到其源头。 只有在掌握有关含水层和地下水流场特性的统计知识以及测量本身可能出现误差的情况下，才能做到这一点。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，这些模型包含随机或混沌元素，这些元素阻止了基于部分和不完美信息的完美预测。 在这个项目中开发的建模方法和数值算法应该提供一个实用的计划，以计算在面对这种随机性的模型和观测的不确定性的最佳估计。

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.