Multi-Model data assimilation techniques for flood forecasting

由于气候变化和社会经济增长的共同作用，预计到2080年，欧洲每年因洪水造成的损失将增加到1000亿欧元。欧洲洪水预警系统（European Flood Awareness System，EFAS）是一个监测和预测欧洲洪水的实用系统。欧洲洪水预报系统的预报增加了防洪准备措施的准备时间，特别是对大型河流流域。这个博士项目的目标是开发新的数学方法，通过将观测数据与计算模型输出相结合，使用称为数据同化的复杂技术来改善洪水预报。EFAS由一个区域水文模型组成，预测欧洲大江大河的水流（称为径流预测）。这个计算模式是由数值天气预报得出的降雨预报集合驱动的。在有河规数据的地方，所得到的径流预测会经过统计校准，以匹配当地的观测结果。目前，大约有800个地点以这种方式进行校准。然而，校正的价值是有限的，因为它们没有考虑到河流网络中上游和下游点之间自然存在的空间关系。此外，目前的校准系统只能处理流量计观测，尽管其他观测类型，如水位信息的可用性。相比之下，数据同化提供了一个数学框架的状态估计和校准，允许异质观测类型的最佳组合，以及非本地更新，其中观测的影响在空间上延伸，根据位置之间的动态关系。该项目的目的是开发一个用于EFAS预报校准的数据同化系统。虽然集合数据同化技术在数值天气预报应用中已经很成熟，但其在水文应用中的应用仍处于起步阶段。因此，EFAS同化系统的开发将需要开发新的数学来解决新应用的挑战：1。沿着河网的权矩阵正则化。在数据同化中，将观测数据和模式数据结合起来，同时考虑到它们的相对不确定性。不同空间位置的不确定性之间的关系用加权矩阵表示，称为误差协方差矩阵。预报误差协方差矩阵可以直接从EFAS集合输出估计。然而，由于有限的系综大小，估计可能是有噪声的。为了保证资料同化结果不受噪声污染，有必要对加权矩阵进行正则化（或重新调整）。在数学金融学、不确定性量化和数值线性代数文献中存在许多正则化方法。学生将学习和调整这些技术的洪水预报，其中的距离动态一致的概念是沿着河流网络的几何形状，而不是“乌鸦飞”。2.多模式集合资料同化。现有的实际集合数据同化技术假设每个集合预报是针对同一域（地理区域）和相同的基本气候进行的。然而，EFAS系统是由多模式集成的降雨预报，因此，EFAS计算运行在不同的域。学生将开发新的数学理论多模式数据同化，基于高斯混合分布统计。他们将研究新方法的实际数值实现，首先通过理想化建模，然后使用EFAS数据。

The annual cost due to flood damage in Europe is expected to rise to 100 billion EUR by the year 2080, due to a combination of climate change and socio-economic growth. The EU-funded European Flood Awareness System (EFAS) is an operational system that monitors and forecasts floods across Europe. EFAS forecasts increase the lead time available for flood-preparedness measures, particularly for large river basins. The goal of this PhD project is to develop new mathematical methods to improve such flood forecasts by combining observational data with computational model output, using the sophisticated technique known as data assimilation. EFAS consists of a regional hydrological model, forecasting the flow of water in large rivers across Europe (known as streamflow forecasts). This computational model is driven by an ensemble of rainfall forecasts derived from numerical weather prediction. Where river gauge data is available, the resulting streamflow forecasts are statistically calibrated to match the observations locally. Currently, there are approximately 800 locations that are calibrated in this way. Nevertheless, the value of the corrections is limited, as they do not take into account the spatial relationships that naturally exist between points up- and downstream in the river network. Furthermore, the current calibration system is only able to deal with flow gauge observations, despite the availability of other observation-types such as water height information. In contrast, data assimilation provides a mathematical framework for state estimation and calibration that allows for both optimal combination of heterogeneous observation types, as well as non-local updates, where observation influence extends spatially, according to dynamical relationships between locations. The aim of the project is to develop a data assimilation system for EFAS forecast calibration. While ensemble data assimilation techniques are well established in numerical weather prediction applications, their use for hydrological applications is in its infancy. Thus, the development of an assimilation system for EFAS will require the development of new mathematics to address the challenges of a new application:1. Weighting matrix regularization along a river network. In data assimilation, observations and model data are combined, taking account their relative uncertainties. The relationships between uncertainties at different spatial locations are expressed in a weighting matrix, known as an error covariance matrix. The forecast error covariance matrix can be estimated directly from the EFAS ensemble output. However, the estimate is likely to be noisy due to the limited ensemble size. To ensure that the data assimilation results are not contaminated by noise, it is necessary to regularize (or recondition) the weighting matrix. A number of regularization methods exist in the mathematical finance, uncertainty quantification and numerical linear algebra literature. The student will study and adapt these techniques to the geometry of flood forecasting, where the dynamically consistent notion of distance is along the river network, rather than "as the crow flies". 2. Multi-model ensemble data assimilation. Existing practical ensemble data assimilation techniques assume that each ensemble forecast is carried out for the same domain (geographical area) and with the same underlying climatology. However, the EFAS system is driven by a multi-model ensemble of rainfall forecasts, and consequently, EFAS computations are run on different domains. The student will develop new mathematical theory for multi-model data assimilation, based on Gaussian mixture distributions from statistics. They will investigate the practical numerical implementation of the new methods, first through idealized modelling, and then using EFAS data.