Low-dimensional representation of error covariance

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensionalapproximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time-independent systems are used to identify factors that cause thesteady-state analysis error covariance to admit a low-dimensional representation. A usefulmeasure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-stateanalysis error covariance matrix eigenvalues are derived from the bound matrix. The estimatesgeneralize to time-dependent systems. If much of the steady-state analysis error variance is dueto a few dominant modes, the leading eigenvectors of the bound matrix approximate those ofthe steady-state analysis error covariance matrix. The analytical results are illustrated in twonumerical examples where the Kalman filter is carried to steady state. The first example usesthe dynamics of a generalized advection equation exhibiting non-modal transient growth.Failure to observe growing modes leads to increased steady-state analysis error variances.Leading eigenvectors of the steady-state analysis error covariance matrix are well approximatedby leading eigenvectors of the bound matrix. The second example uses the dynamics of adamped baroclinic wave model. The leading eigenvectors of a lowest-order approximation ofthe bound matrix are shown to approximate well the leading eigenvectors of the steady-stateanalysis error covariance matrix.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

[3]  B. Farrell.  Transient Growth of Damped Baroclinic Waves , 1985 .

[4]  Brian F. Farrell,et al.  Optimal Excitation of Baroclinic Waves , 1989 .

[5]  B. Goswami,et al.  Predictability of a coupled ocean-atmosphere model , 1991 .

[6]  J. Barkmeijer,et al.  Singular vectors and estimates of the analysis‐error covariance metric , 1998 .

[7]  S. Cohn,et al.  Assessing the Effects of Data Selection with the DAO Physical-Space Statistical Analysis System* , 1998 .

[8]  Michael Ghil,et al.  Tracking Atmospheric Instabilities with the Kalman Filter. Part II: Two-Layer Results , 1996 .

[9]  Brian F. Farrell,et al.  Stochastic Dynamics of Baroclinic Waves , 1993 .

[10]  S. Cohn,et al.  Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions , 2022 .

[11]  Dick Dee,et al.  Maximum-Likelihood Estimation of Forecast and Observation Error Covariance Parameters. Part I: Methodology , 1999 .

[12]  Stephen E. Cohn,et al.  An Introduction to Estimation Theory (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[13]  D. Blake,et al.  The compatibility between aircraft and ground-based air quality measurements , 1994 .

[14]  S. Cohn,et al.  An Introduction to Estimation Theory , 1997 .

[15]  Gene H. Golub,et al.  Matrix computations , 1983 .

[16]  Dick Dee,et al.  Maximum-Likelihood Estimation of Forecast and Observation Error Covariance Parameters. Part II: Applications , 1999 .

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

[18]  P. Courtier,et al.  The ECMWF implementation of three‐dimensional variational assimilation (3D‐Var). I: Formulation , 1998 .

[19]  Lippincott Williams Wilkins,et al.  Part I , 1997, Neurology.

[20]  Robert Vautard,et al.  Four-dimensional variational assimilation and predictability in a quasi-geostrophic model , 1998 .

[21]  P. Courtier,et al.  The ECMWF implementation of three‐dimensional variational assimilation (3D‐Var). II: Structure functions , 1998 .

[22]  Eugenia Kalnay,et al.  Ensemble Forecasting at NMC: The Generation of Perturbations , 1993 .

[23]  Istvan Szunyogh,et al.  A comparison of Lyapunov and optimal vectors in a low-resolution GCM , 1997 .

[24]  Michael Ghil,et al.  Tracking Atmospheric Instabilities with the Kalman Filter. Part 1: Methodology and One-Layer Resultst , 1994 .

[25]  Dick Dee,et al.  Observability of discretized partial differential equations , 1988 .

[26]  Jeffrey S. Whitaker,et al.  A Linear Theory of Extratropical Synoptic Eddy Statistics , 1998 .

[27]  Michael K. Tippett,et al.  Upper bounds for the solution of the discrete algebraic Lyapunov equation , 1999, Autom..

[28]  Petros,et al.  Generalized Stability Theory . Part I : Autonomous Operators , 2022 .

[29]  R. Todling,et al.  Approximate Data Assimilation Schemes for Stable and Unstable Dynamics , 1996 .

[30]  Takehiro Mori,et al.  Upper and lower bounds for the solution to the discrete Lyapunov matrix equation , 1982 .

[31]  F. Molteni,et al.  The ECMWF Ensemble Prediction System: Methodology and validation , 1996 .

[32]  Roberto Buizza,et al.  The impact of increased resolution on predictability studies with singular vectors , 1997 .

[33]  Michael K. Tippett,et al.  Bounds for the solution of the discrete algebraic Lyapunov equation , 1998, Autom..

[34]  P. L. Houtekamer,et al.  A System Simulation Approach to Ensemble Prediction , 1996 .

[35]  M. Cane,et al.  On the prediction of ENSO: a study with a low-order Markov model , 1994 .

[36]  Herschel L. Mitchell,et al.  Discretization error and signal/error correlation in atmospheric data assimilation , 1997 .

[37]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[38]  Michael K. Tippett,et al.  Conditioning of the Stable, Discrete-Time Lyapunov Operator , 2000, SIAM J. Matrix Anal. Appl..

[39]  Michael K. Tippett,et al.  Bounds for solutions of the discrete algebraic Lyapunov equation , 1999, IEEE Trans. Autom. Control..

[40]  Roger Daley,et al.  Spectral Characteristics of Kalman Filter Systems for Atmospheric Data Assimilation , 1993 .

[41]  Roberto Buizza,et al.  The Singular-Vector Structure of the Atmospheric Global Circulation , 1995 .

[42]  Brian F. Farrell,et al.  Generalized Stability Theory. Part II: Nonautonomous Operators , 1996 .

[43]  Dick Dee,et al.  Ooce Note Series on Global Modeling and Data Assimilation Maximum-likelihood Estimation of Forecast and Observation Error Covariance Parameters , 2022 .

[44]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

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

[46]  C. V. L. Charlier,et al.  Periodic Orbits , 1898, Nature.

[47]  F. Smithies.  A HILBERT SPACE PROBLEM BOOK , 1968 .

[48]  M. Verlaan,et al.  Tidal flow forecasting using reduced rank square root filters , 1997 .

[49]  Arlindo da Silva,et al.  Data assimilation in the presence of forecast bias , 1998 .

[50]  Roberto Buizza,et al.  Sensitivity Analysis of Forecast Errors and the Construction of Optimal Perturbations Using Singular Vectors , 1998 .

[51]  Herschel L. Mitchell,et al.  Discretization error and signal/error correlation in atmospheric data assimilation : (I). All scales resolved , 1997 .

[52]  R. Daley.  Forecast-error statistics for homogeneous and inhomogeneous observation networks , 1992 .

[53]  Anna Trevisan,et al.  Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System , 1998 .