Covariance regularization in data assimilation for coupled dynamical systems

Computational simulation is used in many scientific and engineering disciplines to predict the behaviour of physical systems. Applications include the prediction of coastal and river flooding, ground water flow and oil reservoir production. The ability to predict such phenomena has great impact on the economic and societal well-being of the U.K. Such systems contain tens of millions of variables and are computationally very challenging to treat in real-time. For many applications, the ability to make accurate predictions is limited by our knowledge of the current state of the physical system. Measurements of the system behaviour over time may exist, but often these measurements are sparsely distributed in space and time and they are usually noisy. The measurements must be combined with the computational simulations and other knowledge about the physical system in order to produce the best possible estimate of the current state before a forecast can be made. The technique for incorporating measurements in this way is called data assimilation.

Data assimilation aims to find the state of the system that best fits the data, while at the same time fitting the prior knowledge we have about the system. In order to do this we need to represent the uncertainty in the prior knowledge and the uncertainty in the measurements, so that we can balance these different sources of information. Previously we have proved how the assumptions we make about these uncertainties affect our ability to solve the data assimilation problem efficiently and produce an accurate solution. Recently many data assimilation practitioners have developed new 'ensemble' methods for representing the prior uncertainty in data assimilation, based on running several different predictions with slightly different conditions and quantifying the differences between them. This is expected to give a much more accurate representation of the uncertainty. However, there is currently no mathematical theory on how these uncertainties affect the properties of the hybrid assimilation techniques. In this project we will extend our previous theory to cover these new methods, developing an understanding of how they affect the efficiency of the data assimilation procedure and enabling novel methods to be derived.