STATISTICS OF VARIATIONAL DATA ASSIMILATION IN CONTINUOUS TIME

Many natural phenomena manifest themselves as dynamical processes, that is processes evolving in time. Mathematical descriptions of such phenomena usually consist of equations describing temporal evolution, such as differential or difference equations; we will refer to them as dynamical models. Dynamical models have been conceived for the atmosphere, the sun, human crowd behaviour, traffic, and the stock market, just to name a few. A dynamical model allows to forecast the future behaviour of the real world dynamical process through numerical simulations (usually on a computer).

However, in order to forecast the future behaviour of the dynamical process, its current state has to be known. Data assimilation, which is the main theme of this project, means to gather past and present observations of the dynamical process and estimate its current state or even whole trajectories in a dynamically consistent fashion. (E.g. data assimilation will not only result in a series of snapshots of the global wind fields; the evolution of these snapshots will be consistent with the physics describing air motion.) For this reason, data assimilation is a core step in forecasting with dynamical models.

Data assimilation is carried out already in a wide range of applications, for example in weather forecasting. To some extent though, data assimilation rests on an ad--hoc methodology with only part of it being completely understood. A thorough understanding of data assimilation though is vital, as the performance and thus the value of every forecast depends crucially on the data assimilation.

This project aims at providing data assimilation with further mathematical foundations. In particular, the following points will be investigated:

* Dynamical models are often formulated in continuous time, and data assimilation is much nicer to analyse in continuous time than in discrete time, for mathematical reasons. In practice though, weather observations are sampled at discrete points in time, which seems to necessitate a discrete time framework for data assimilation. How do we take care of this important practical problem while at the same time rescuing the elegance and power of a continuous time approach?

* Many data assimilation approaches provide solutions that appear reasonable, but the precise properties are not properly understood. A particularly pressing problem is the uncertainty associated with the estimated trajectories. Suppose the observations contain measurement error, this error will clearly feed through the entire data assimilation machinery onto the estimated trajectories. How do we take care of this?

* The practitioneer needs a formalism to perform some quality control of her or his data assimilation results. The problem here is this: simply comparing the output of data assimilation with the observations is dangerous, since the observations have already been used to find the underlying trajectory, so this approach might give overly optimistic results. In statistics, this is known as ``in sample evaluation'', and several methods have been conceived to avoid them. In data assimilation, something similar is needed, although the problem is more complex as the observations are usually heavily dependent; a series of wind observations cannot be treated like a series of patients in medical trials. But building on previous work, a formalism will be developed allowing for more realistic performance assessment of data assimilation.

Dynamical models are employed to describe a vast range of processes in fields ranging from physics and engineering to economics and the social sciences. Further, dynamical models are a topic of intensive research in pure mathematics, not least because of their practical importance. In particular, a model allows to forecast the future behaviour of the dynamical process through numerical simulations (usually on a computer).

However, in order to forecast the future behaviour of the dynamical process, its current state has to be known. Data assimilation (which is a collective term used in the geosciences) means to gather past and present observations of the dynamical process and estimate its current state in a dynamically consistent fashion. Hence, data assimilation is a core step in forecasting with dynamical models, and thus this project bears, at least indirectly, on any research area or application where dynamical models are employed for forecasting.

Data assimilation is carried out already in a wide range of applications, but very often using an ad--hoc methodology with only part of it being thoroughly understood. A proper understanding of data assimilation though is vital, as the performance and thus the value of every forecast depends crucially on the data assimilation. The main aim of this project is to provide data assimilation with proper mathematical foundations. Thus, in the widest sense, the project contributes to improving all kinds of forecasts which rely on data assimilation. In the short term, direct nonacademic beneficiaries will be whoever carries out data assimilation as part of their business, ranging from meteorological services such as the MetOffice or the European Centre for Medium Range Weather Forecasts, to companies maintaining oil rigs or offshore wind farms.

The project, would help meteorological services offering better products, or in more general terms, increase the effectiveness of public services. Thus, any weather dependent industries such as energy providers or environmental engineering will benefit indirectly. This could, for example lead to more efficient use of resources such as fossile energy through better prediction of demand and availability of green energy sources. Beneficiaries here would be energy providers such as Scottish Southern Electricity or EDF, but also the customer through lower energy costs, and the whole society through reduction in pollution.

In addition, forecasts might become relevant for yet further applications for which there is currently no market as the skill of the forecast is not sufficiently high. Every improvement in skill might open these markets to business. One example would be personalised ultra short term and micro area forecasts for public and private, e.g. for event management and recreational outdoor activities; this could become reality in one form or another in a few years.

Further long term (on the scale of decades) benefits of this project could be better mitigation of environmental hazards, both globally and nationally, and long term planning and living with environmental changes. On an academic level, this proposal would also contribute to a number of very important research disciplines nationally and internationally. Data assimilation is an important application of statistics, probability theory, and dynamical systems; this project might thus stimulate further research in these fields.

On the other hand, data assimilation is a fundamental tool in the analysis of our past and present climate. So called reanalysis experiments are regularly undertaken which essentially consist of assimilating observations of several decades worth of observations into current climate models. These experiments provide invaluable insights into our present climate, and improving either the understanding of data assimilation, its performance, and its limitations will increase the value of conclusions gained from reanalysis experiments.