Data assimilation in highly nonlinear geophysical systems: particle filters with localization

数据同化是许多地球物理科学活动的核心，如气象学、海洋学、水文学、地震学等。在资料同化中，某一（地球物理）系统的数值模式与该系统的观测相结合。这样做的目的可以是预测、模型改进或试图更好地理解所研究的系统。首先，以天气预报为例，如果没有观测数据的持续输入，目前最先进的模型就不能很好地工作。这些模式在表示大气中的物理和化学过程方面相当好，但在作出良好的预报之前需要有关大气实际状态的信息。所有地球物理领域都是如此。在模型改进和系统理解方面，数据同化也可以发挥非常重要的作用。由于分辨率问题或对物理学知之甚少，这些模型包含了一些不能很好地描述的过程。这导致了几个（有时是数百个）鲜为人知的参数，这些参数可以通过数据同化来估计。最后，通过使用同化了观测数据的模式，可以研究真实的大气（海洋等），而不是模型表示。在大尺度地球物理系统中，已经实现了几种数据同化方法。它们都是基于某种线性化。例如集合卡尔曼滤波和四维变分方法（4D-Var）。随着模型分辨率的不断提高，模型中需要解析的过程越来越多，而且这些过程越来越趋于非线性。一个例子是云的形成和大气中的降水。数据同化社区正在努力寻找能够处理这些非线性的方法。很长一段时间以来，人们一直认为粒子过滤器很好地完成了这项工作。原则上，这些方法是完全非线性的。然而，粒子过滤器在气象学和海洋学中的应用仅限于小维度系统，因为必须使用大量的粒子。解决这个问题的一个方法是试图增加粒子集合的有效尺寸。这可以通过所谓的本地化来实现。该技术在集合卡尔曼滤波中被广泛使用，如果没有集合卡尔曼滤波，该方法将无法用于实际数值天气预报或大规模海洋模型。在局部化中，我们允许观测只对域的有限区域产生影响，只对靠近观测的区域产生影响。这导致了一个局部估计问题，并且集合成员的数量与未知的数量（仅在该区域内的那些）相比增加了相当多。如果将整个模型域划分为1000个这样的小区域，则有效集成大小将增加1000倍。我们不能在粒子滤波中直接使用定位，在这个提议中，我们在一套地球物理模型中研究了三种新的定位方法，从非常简单到接近操作。本提案的目标是：-产生一种高度非线性的大维地球物理系统的数据同化方法。由于模型和观测操作者变得越来越非线性，这一目标与NERC高度相关。所开发的方法将适用于nerc相关的所有研究领域。-使粒子滤波领域可用于地球物理学。粒子滤波是少数几种完全非线性的方法之一，在大规模系统中具有很强的应用潜力。-研究使用局部化方法进行粒子滤波，以达到上述目标。-演示本地化在气象学中的大规模应用。

Data assimilation is at the heart of many activities in geophysical sciences, being it meteorology, oceanography, hydrology, seismology etc. In data assimilation numerical models of a certain (geophysical) system are combined with observations from that system. The purpose of doing this can either be forecasting, model improvement or trying to understand the system under study better. To start with forecasting of e.g. weather, the present-day state-of-the-art models would not do a very good job without the continuous feeding of observations into them. The these models are quite good in representing the physical and chemical processes in the atmosphere, but need information on the actual state of the atmosphere before a good forecast can be made. The same is true for all geophysical fields. With regard to model improvement and system understanding data assimilation can also play a very important role. The models contain several processes that are not well described due to either resolution problems or poorly known physics. This results in several (sometimes hundreds) of poorly known parameters, which can be estimated by data assimilation. Finally, by using a model in which observations have been assimilated the real atmospheric (oceanic etc.) can be studied, instead of the model representation. Several methods to perform data assimilation have been implemented in large-scale geophysical systems. All of them are based on linearisations of some kind. Examples are the Ensemble Kalman Filter and the 4-Dimensional Variational method (4D-Var). Due to increasing model resolution more and more processes are being resolved in the models, and these processes tend to be more and more nonlinear. An example is cloud formation and precipitation in the atmosphere. The data-assimilation community is looking hard for methods that can handle these nonlinearities. It has been argued for a long time now that particle filters good do the job. In principle, these methods are fully nonlinear. However, applications of particle filters in meteorology and oceanography are limited to small dimensional systems due to the enormous number of particles that have to be used. On way to solve this problem is by trying to increase the effective size of the ensemble of particles. This can be done by so-called localization. This technique is used extensively in the Ensemble Kalman Filter, without which that method would not work on operational numerical weather prediction or large-scale ocean models. In localization one allows observations to have only influence on a limited area of the domain, only on the area close to the observation. This results in a local estimation problem, and the number of ensemble members compared to the number of unknowns (only those in that area) increases considerable. If one divides the whole model domain up into 1000 of those smaller areas, the effective ensemble size increases with a factor 1000. One cannot use localization directly in a particle filter, and in this proposal three new ways of doing it are investigated in a suite of geophysical models, running from very simple to close to operational. The objectives of this proposal are: - Generate a data-assimilation method for highly nonlinear large-dimensional geophysical systems. Since the models and observation operators are becoming more and more nonlinear, this objective is highly relevant to NERC. The methods developed will be applicable to all NERC-related research fields. - Make the field of particle filtering accessible for geophysics. Particle filters are one of the few methods that are fully nonlinear and have strong potential to be applicable in large-scale systems. - Investigate the use of localization for particle filtering to achieve the above mentioned goals. - Demonstrate the use of localization in a large-scale application in meteorology.