Method for improving forecast statistics in large-scale variational data assimilation problems

在过去的二十年里，数据同化方法已经成为分析和预测大气和海洋中复杂现象的重要工具。也许，这些方法最广为人知的应用是中短期天气预报，它已经成为指导我们日常生活的一项常规服务。更广泛的前景揭示了此类预测的巨大经济和安全价值。最新的例子可能包括火山灰云在大气中的扩散和海洋中的石油泄漏跟踪。精确监测和预测这类不幸事件的重要性几乎不需要任何进一步的宣传。数据同化方法本质上是一种数学方法，它允许将大气和海洋流动的数学模型、观测数据(由气象站或卫星测量)和一些反映我们在几十年的观察和分析中积累的经验的额外信息结合在一起。气象学和海洋学中的数据同化问题的一个显著特点是它们的大尺度，这意味着流动模型包含大量的变量，例如$\sim 10^9$或更多。这就是为什么，尽管计算能力有了非常显著的提高，研究人员和实践者仍然只有非常有限的选择来解决这些问题，合适的数学工具，以及随后的算法工具。在几种可行的解决这些问题的方法中，变分资料同化方法是一些主要业务中心实施的首选方法，如英国气象局、欧洲气象局、法国气象局等。虽然这种方法产生的预报通常是相当好的，但目前在4D-Var框架下的预报协方差矩阵的评估问题仍然没有得到解决。需要这个协方差矩阵来评估预测的好坏和可用性；如果没有这样的评估，预测本身可能没有什么价值。当观测到的现象是高度非线性的(大多数情况下就是这种情况)时，目前采用的技术可能不能提供合理质量的预测协方差矩阵。让我们再次强调，数学和统计学中有一些方法原则上能够解决这个问题；正是问题的维度使它们中的大多数无法使用。因此，这项研究的主要目的是开发一种计算上可行的技术，以评估预测协方差矩阵在4D-Var框架中的准确逼近。为实现这一目标所建议的方法将以申请人最近在高影响计算期刊上发表的最新结果为基础。它代表了用于解决数据同化问题的确定性和统计学(贝叶斯)方法之间的微妙平衡，并有可能最终在实际应用中使用。这一潜力已得到已发表论文的审查者和该领域一些主要专家的私下交流的证实。然而，在目前的状态下，该方法需要理论和算法的升级，更重要的是，需要用实际的大规模模型进行更实质性的验证，例如正在迅速流行的尼莫海洋模型。所有这些步骤都将在拟议的研究中进行。

Over the past two decades, methods of data assimilation have become vital tools for analysis and prediction of complex phenomena in the atmosphere and the ocean. Probably, the most well known application of these methods is the short- and medium-range weather prediction which has become a routine service guiding our everyday life. A wider outlook reveals the great economic and security value of such forecasts. The latest examples may include the volcano ash cloud dispersion in the atmosphere and oil spill tracking in the ocean. The importance of precise monitoring and prediction of such unfortunate events hardly requires any further promotion. The data assimilation methods are, essentially, mathematical methods which allow the combination of mathematical models of the atmospheric and oceanic flows, observation data (measured by weather stations or by satellites) and some additional information which reflects our experience with these phenomena, accumulated during decades of observations and analysis. A distinguishing feature of the data assimilation problems in meteorology and oceanography is their large-scale, which means that flow models contain a huge number of variables, say $\sim 10^9$ and more. That is why, despite a very significant increase in computational power, the researchers and practitioners still have a very limited choice of suitable mathematical and, subsequently, algorithmic tools for solving these problems. Among few methods feasible for solving these problems the variational data assimilation method called '4D-Var' is the preferred method implemented at some major operational centers, such as the UK Met Office, ECMWF, Meteo France, etc. While the forecasts produced by this method are usually reasonably good, the problem of evaluating the forecast covariance matrix in the 4D-Var framework currently remains unsolved. This covariance matrix is needed to assess how good and usable the forecast is; without such assessment the forecast itself may be of littlevalue. The technique currently adopted may not provide the forecast covariance matrix of a reasonable quality when the observed phenomena are highly nonlinear (which is the case in most circumstances). Let us stress again that there are methods in mathematics and statistics capable of solving this problem in principle; it is the problem's dimensionality that makes most of them unusable. Therefore, the main objective of this research is to develop a computationally feasible technique for evaluating an accurate approximation of the forecast covariance matrix in the 4D-Var framework. The suggested methodology to achieve this objective will be based on the latest results recently published by the applicant in high impact computational journals. It represents a fine balance between deterministic and statistical (Bayesian) methods used for solving data assimilation problems and has a potential for the eventual operational use in real-life applications. This potential has been confirmed both by the reviewers of the published papers and in private communications involving some leading experts in the field. However, in its current state the method needs a theoretical and algorithmic upgrade and, importantly, more substantial verification with a realistic large-scale models, such as the NEMO ocean model, which is rapidly becoming popular. All these steps will be conducted in the proposed research.