Next generation Numerical Weather Prediction: 4DVar ensembles and Particle Filters

资料同化是一种将数值模式与观测相结合的方法。它被用于所有的环境科学，并且对于能够模拟真实世界而不是与现实无关的纯粹模型世界至关重要。随着地球物理模型分辨率的提高，模型的大小和非线性都在增加。此外，观测的数量增加，将模式变量与观测联系起来的观测操作者也变得越来越复杂和非线性，就像气象预报中的新卫星观测和雷达观测一样。显然，数据同化方法必须充分考虑这些非线性。目前数值天气预报的实现都是基于线性化的。例如，（Ensemble） Kalman Filter假设线性更新，4DVar等变分方法通过线性迭代解决弱非线性问题。变分方法的另一个问题是很难得到误差估计，而且对于高度非线性的问题是不准确的。一些天气预报中心已经开始试验使用4DVar系统。这有可能解决非线性问题，同时提供误差估计。最近，欧洲中期天气预报中心（ECMWF）开始试验4DVar解决方案的集合，通过扰动观测产生，结果非常有希望。从卡尔曼滤波（或者更确切地说是更平滑）理论可知，当这个集合经过几个数据同化循环时，它的传播将近似于系统的误差协方差。在这种情况下，集合是来自正确分布的样本。然而，对于一个强非线性系统，卡尔曼滤波理论并不成立，并且不清楚集合意味着什么，并且有强烈的科学和操作需要了解这些集合意味着什么，以及我们如何改进它们。另一方面，众所周知，在所谓的粒子滤波器中，我们可以通过一组粒子，即一组模型状态来表示底层分布。粒子滤波在模型演化和分析过程中都是完全非线性的。一个基本问题，即所谓的“维度诅咒”阻碍了它们在地球科学应用中的应用。最近，PI发现了一种解决方案，有可能彻底改变高度非线性地球物理系统的数据同化（Van Leeuwen, 2010a; Van Leeuwen, 2010b）。后一篇论文描述了相对简单（高达1000维）的高度非线性系统的应用，这些系统以前需要数百到数千个模型集成，而现在只需要20个模型集成。本研究计划探讨将4DVar集合与粒子滤波思想相结合的可能性，用于下一代数值天气预报。一个简单而令人兴奋的想法是使用4DVar解决方案作为粒子过滤器中的粒子，这是我们将要研究的方向之一。但我们也将研究在非线性系统中产生有意义的4DVar集成的其他方法。一个优点是，我们可以直接访问运行ECMWF系统，使我们能够在相对简单的学术模型和实际世界之间有效地进行操作。

Data assimilation is a method to combine numerical models with observations. It is used in all environmental sciences and essential to be able to simulate the real world, instead of a pure model world which has little to do with reality. With the increasing resolution of geophysical models both the size and the nonlinearity of these models increase. Also the number of observations increases and the observation operators, which connect the model variables to the observations, become more and more complex and nonlinear, like new satellite observations and radar observations in weather forecasting. Obviously, the data-assimilation methods have to fully allow for these nonlinearities. Present-day implementations in numerical weather prediction are all based in linearisations. For example, the (Ensemble) Kalman Filter assumes linear updates, and variational methods like 4DVar solve a weakly nonlinear problem through linear iterations. A further problem with variational methods is that error estimates are hard to obtain, and for highly nonlinear problems inaccurate.A few operational weather prediction centres have started experimenting with ensembles of 4DVar's. This has the potential of solving the nonlinearity problem, and at the same time provides an error estimate. Recently, the European Centre for Medium Range Weather Forecasts (ECMWF) started experimenting with ensembles of 4DVar solutions, generated by perturbing the observations, with very promising results. It is known from Kalman Filter (or rather Smoother) theory that when this ensemble is cycled through several data-assimilation cycles its spread will approximate the error covariance of the system. In that case, the ensemble is a sample from the correct distribution. However, for a strongly nonlinear system the Kalman filter theory does not hold, and it is unclear what the ensemble means, and there is a strong scientific and operational need to understand what these ensembles mean, and how we can improve them. On the other hand, it is well-known that we can represent the underlying distributions by a set of particles, i.e. a set of model states, in a so-called particle filter. Particle filters are fully nonlinear both in model evolution and analysis step. A fundamental problem, the so-called 'curse of dimensionality' has hampered their use in geoscience applications. Very recently a solution has been found by the PI that has the potential to revolutionize data assimilation in highly nonlinear geophysical systems (Van Leeuwen, 2010a; Van Leeuwen, 2010b). The latter paper describes applications to relatively simple (up to 1000-dimensional) highly nonlinear systems that previously needed hundreds to thousands of model integrations, and now only of the order of 20 model integrations. This research proposal explores the possibilities of combining 4DVar ensembles with ideas from Particle Filtering for the next generation numerical weather prediction. A simple and exciting idea is to use 4DVar solutions as particles in the Particle Filter, and this is one of the directions we will investigate. But we will also investigate other ways to generate 4DVar ensembles that are meaningful in nonlinear systems. A strong point is that we will have direct access to the operational ECMWF system, allowing us to efficiently operate between relatively simple academic models and the operational world.