Using model reduction techniques within the incremental 4D-Var method

This thesis is devoted to the development and study of new numerical methods for data assimilation of large dimensional problems. Incremental four-dimensional variational data assimilation is the method of choice in many operational atmosphere and ocean data assimilation systems. It allows the four dimensional variational technique (4D-Var) to be implemented in a computationally efficient way by replacing the minimization of the full nonlinear 4D-Var cost function with the minimization of a series of simplified linear cost functions. In practice these simplified functions are usually derived from a spatial or spectral truncation of the full system being approximated. This thesis proposes a new method for deriving the simplified problems in incremental 4D-Var based on model reduction techniques developed in the field of control theory. Such a procedure ensures that no essential information is lost. It supplies a better accuracy of the forecast compared to commonly used approximation techniques. A main contribution of this thesis is the derivation of a new interpretation of the data assimilation problem as a control theoretic problem incorporating all statistical information. This new approach makes the use of model reduction techniques within data assimilation methods possible. Moreover, it is important to take into account that numerical weather prediction systems usually lead to unstable control systems. It is shown how a proper choice of the model reduction method is able to cope with this additional difficulty.Various numerical experiments using shallow water test models underline that the combination of model reduction techniques with incremental 4D-Var gives an assimilation method that retains more of the dynamical information of the full system. Numerical tests with varying observing networks illustrate the superior performance of the model reduction approach compared to standard truncation techniques. Additionally, the numerical experiments confirm how the incorporation of all statistical information in the model reduction procedure improve the accuracy of the approximation