The variational Kalman filter and an efficient implementation using limited memory BFGS

In the field of state space estimation and data assimilation, the Kalman filter (KF) and the extended Kalman filter (EKF) are among the most reliable methods used. However, KF and EKF require the storage of, and operations with, matrices of size n x n, where n is the size of the state space. Furthermore, both methods include inversion operations for m x m matrices, where m is the size of the observation space. Thus, KF methods become impractical as the dimension of the system increases. In this paper, we introduce a variational Kalman filter (VKF) method to provide a low storage, and computationally efficient, approximation of the KF and EKF methods. Furthermore, we introduce a variational Kalman smoother (VKS) method to approximate the fixed-lag Kalman smoother (FLKS) method. Instead of using the KF formulae, we solve the underlying maximum a posteriori optimization problem using the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method. Moreover, the LBFGS optimization method is used to obtain a low storage approximation of state estimate covariances and prediction error covariances. A detailed description of the VKF and VKS methods with LBFGS is given. The methodology is tested on linear and nonlinear test examples. The simulated results of the VKF method are presented and compared with KF and EKF, respectively. The convergence of BFGS/LBFGS methods is tested and demonstrated numerically.

[1]  Fabrice Veersé,et al.  Variable-storage Quasi-Newton Operators as Inverse Forecast/Analysis Error Covariance Matrices in Variational Data Assimilation , 1999 .

[2]  I. Yu. Gejadze,et al.  On Analysis Error Covariances in Variational Data Assimilation , 2008, SIAM J. Sci. Comput..

[3]  Didier Auroux,et al.  Limited-Memory BFGS Diagonal Preconditioners for a Data Assimilation Problem in Meteorology , 2000 .

[4]  K. Emanuel,et al.  Optimal Sites for Supplementary Weather Observations: Simulation with a Small Model , 1998 .

[5]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[6]  T. Hamill,et al.  A Hybrid Ensemble Kalman Filter-3D Variational Analysis Scheme , 2000 .

[7]  D. Dee.  Simplification of the Kalman filter for meteorological data assimilation , 1991 .

[8]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[9]  A. Simmons,et al.  The ECMWF operational implementation of four‐dimensional variational assimilation. I: Experimental results with simplified physics , 2007 .

[10]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[11]  Jorge Nocedal,et al.  Representations of quasi-Newton matrices and their use in limited memory methods , 1994, Math. Program..

[12]  J. Greenstadt.  Variations on variable-metric methods. (With discussion) , 1970 .

[13]  J. Greenstadt.  Variations on Variable-Metric Methods , 1970 .

[14]  Alexey Kaplan,et al.  Mapping tropical Pacific sea level : Data assimilation via a reduced state space Kalman filter , 1996 .

[15]  E. Lorenz.  Predictability of Weather and Climate: Predictability – a problem partly solved , 2006 .

[16]  Xiangjun Tian,et al.  An ensemble-based explicit four-dimensional variational assimilation method , 2008 .

[17]  Heikki Haario,et al.  LARGE-SCALE KALMAN FILTERING USING THE LIMITED MEMORY BFGS METHOD , 2009 .

[18]  M. Zupanski.  Maximum Likelihood Ensemble Filter: Theoretical Aspects , 2005 .

[19]  J. Derber.  A Variational Continuous Assimilation Technique , 1989 .

[20]  Philippe Courtier,et al.  A New Hessian Preconditioning Method Applied to Variational Data Assimilation Experiments Using NASA General Circulation Models , 1996 .

[21]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[22]  Heikki Haario,et al.  VARIATIONAL KALMAN FILTERING ON PARALLEL COMPUTERS , 2007 .

[23]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[24]  Arto Voutilainen,et al.  A filtering approach for estimating lake water quality from remote sensing data , 2007, Int. J. Appl. Earth Obs. Geoinformation.

[25]  F. L. Dimet,et al.  Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects , 1986 .

[26]  Martin Leutbecher.  A data assimilation tutorial based on the Lorenz-95 system , 2010 .

[27]  Fuqing Zhang,et al.  Coupling ensemble Kalman filter with four-dimensional variational data assimilation , 2009 .

[28]  Heikki Haario,et al.  OPTIMAL APPROXIMATION OF KALMAN FILTERING WITH TEMPORALLY LOCAL 4D-VAR IN OPERATIONAL WEATHER FORECASTING , 2005 .