Approximate Gauss-Newton methods for optimal state estimation using reduced-order models

The 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. Copyright (C) 2007 John Wiley & Sons, Ltd.