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 data assimilation (4DVAR) to be implemented in a computationally efficient way by replacing the minimization of the full nonlinear 4DVAR cost function with the minimization of a series of simplified cost functions. In practice, these simplified functions are usually derived from a spatial or spectral truncation of the full system being approximated. In this paper, a new method is proposed for deriving the simplified problems in incremental 4DVAR, based on model reduction techniques developed in the field of control theory. It is shown how these techniques can be combined with incremental 4DVAR to give an assimilation method that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.
SUMMARYThe Gauss-Newton (GN) method is a well-known iterative technique for solving nonlinear least-squares problems subject to dynamical system constraints. Such problems arise commonly in optimal state estimation where the systems may be stochastic. Variational data assimilation techniques for state estimation in weather, ocean and climate systems currently use approximate GN methods. The GN method solves a sequence of linear least-squares problems subject to linearized system constraints. For very large systems, low-resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new method for deriving low-order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to retain the response of the dynamical system more accurately and improve the performance of the approximate GN method.
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