Covariance regularization in data assimilation for coupled dynamical systems

计算模拟在许多科学和工程学科中被用来预测物理系统的行为。应用包括对沿海和河流洪水、地下水流量和油藏产量的预测。预测这种现象的能力对英国的经济和社会福祉有很大影响。这种系统包含数千万个变量，在计算上非常困难，难以实时处理。对于许多应用程序，做出准确预测的能力受到我们对物理系统当前状态的了解的限制。系统行为随时间的测量可能存在，但通常这些测量在空间和时间上是稀疏分布的，并且它们通常是噪声的。在做出预测之前，必须将测量结果与计算模拟和有关物理系统的其他知识相结合，以便对当前状态做出可能的最佳估计。以这种方式合并测量数据的技术称为数据同化。数据同化的目的是找到最符合数据的系统状态，同时符合我们已有的关于系统的先验知识。为了做到这一点，我们需要表示先验知识中的不确定性和测量中的不确定性，以便我们能够平衡这些不同的信息源。以前我们已经证明了我们对这些不确定性的假设如何影响我们有效地解决数据同化问题并产生准确解的能力。最近，许多数据同化实践者开发了新的“集成”方法来表示数据同化中的先验不确定性，这些方法基于几个条件略有不同的不同预测并量化它们之间的差异。预计这将更准确地反映不确定性。然而，目前还没有关于这些不确定性如何影响混合同化技术特性的数学理论。在这个项目中，我们将扩展我们以前的理论，以涵盖这些新方法，发展对它们如何影响数据同化过程的效率的理解，并使新方法得以产生。

Computational simulation is used in many scientific and engineering disciplines to predict the behaviour of physical systems. Applications include the prediction of coastal and river flooding, ground water flow and oil reservoir production. The ability to predict such phenomena has great impact on the economic and societal well-being of the U.K. Such systems contain tens of millions of variables and are computationally very challenging to treat in real-time. For many applications, the ability to make accurate predictions is limited by our knowledge of the current state of the physical system. Measurements of the system behaviour over time may exist, but often these measurements are sparsely distributed in space and time and they are usually noisy. The measurements must be combined with the computational simulations and other knowledge about the physical system in order to produce the best possible estimate of the current state before a forecast can be made. The technique for incorporating measurements in this way is called data assimilation.Data assimilation aims to find the state of the system that best fits the data, while at the same time fitting the prior knowledge we have about the system. In order to do this we need to represent the uncertainty in the prior knowledge and the uncertainty in the measurements, so that we can balance these different sources of information. Previously we have proved how the assumptions we make about these uncertainties affect our ability to solve the data assimilation problem efficiently and produce an accurate solution. Recently many data assimilation practitioners have developed new 'ensemble' methods for representing the prior uncertainty in data assimilation, based on running several different predictions with slightly different conditions and quantifying the differences between them. This is expected to give a much more accurate representation of the uncertainty. However, there is currently no mathematical theory on how these uncertainties affect the properties of the hybrid assimilation techniques. In this project we will extend our previous theory to cover these new methods, developing an understanding of how they affect the efficiency of the data assimilation procedure and enabling novel methods to be derived.