Multi-Model data assimilation techniques for flood forecasting

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.