Using analysis state to construct a forecast error covariance matrix in ensemble Kalman filter assimilation

Correctly estimating the forecast error covariance matrix is a key step in any data assimilation scheme. If it is not correctly estimated, the assimilated states could be far from the true states. A popular method to address this problem is error covariance matrix inflation. That is, to multiply the forecast error covariance matrix by an appropriate factor. In this paper, analysis states are used to construct the forecast error covariance matrix and an adaptive estimation procedure associated with the error covariance matrix inflation technique is developed.The proposed assimilation scheme was tested on the Lorenz-96 model and 2D Shallow Water Equation model, both of which are associated with spatially correlated observational systems. The experiments showed that by introducing the proposed structure of the forecast error covariance matrix and applying its adaptive estimation procedure, the assimilation results were further improved.

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

[3]  Philippe Courtier,et al.  Unified Notation for Data Assimilation : Operational, Sequential and Variational , 1997 .

[4]  T. Miyoshi.  The Gaussian Approach to Adaptive Covariance Inflation and Its Implementation with the Local Ensemble Transform Kalman Filter , 2011 .

[5]  Jeffrey L. Anderson,et al.  An adaptive covariance inflation error correction algorithm for ensemble filters , 2007 .

[6]  P. Houtekamer,et al.  A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation , 2001 .

[7]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

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

[9]  Xiangjun Tian,et al.  A POD‐based ensemble four‐dimensional variational assimilation method , 2011 .

[10]  NOTES AND CORRESPONDENCE Ensemble Square Root Filters , 2003 .

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

[12]  Emil M. Constantinescu,et al.  Ensemble-based chemical data assimilation II: Real observations , 2006 .

[13]  Q. Xiao,et al.  An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part I: Technical Formulation and Preliminary Test , 2008 .

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

[15]  J. Szpunar,et al.  Part II applications , 2003 .

[16]  Xin Li,et al.  Evolutionary Algorithm-Based Error Parameterization Methods for Data Assimilation , 2011 .

[17]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

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

[19]  Geir Evensen,et al.  The Ensemble Kalman Filter: theoretical formulation and practical implementation , 2003 .

[20]  Jeffrey L. Anderson.  Spatially and temporally varying adaptive covariance inflation for ensemble filters , 2009 .

[21]  Xiaogu Zheng.  An adaptive estimation of forecast error covariance parameters for Kalman filtering data assimilation , 2009 .

[22]  J. Butcher.  Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods , 2005 .

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

[24]  E. Kalnay,et al.  Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filter , 2009 .

[25]  Xuguang Wang,et al.  A Comparison of Breeding and Ensemble Transform Kalman Filter Ensemble Forecast Schemes , 2003 .

[26]  G. Evensen,et al.  Analysis Scheme in the Ensemble Kalman Filter , 1998 .

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

[28]  D. Dee.  On-line Estimation of Error Covariance Parameters for Atmospheric Data Assimilation , 1995 .

[29]  R. Reichle.  Data assimilation methods in the Earth sciences , 2008 .

[30]  Jeffrey L. Anderson,et al.  A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts , 1999 .

[31]  Xiao Liang,et al.  Maximum likelihood estimation of inflation factors on error covariance matrices for ensemble Kalman filter assimilation , 2012 .

[32]  Olivier Talagrand,et al.  Assimilation of Observations, an Introduction (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[33]  Emil M. Constantinescu,et al.  Ensemble‐based chemical data assimilation. I: General approach , 2007 .

[34]  J. Whitaker,et al.  Ensemble Square Root Filters , 2003, Statistical Methods for Climate Scientists.

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

[36]  G. Evensen.  Inverse methods and data assimilation in nonlinear ocean models , 1994 .

[37]  J. Whitaker,et al.  Accounting for the Error due to Unresolved Scales in Ensemble Data Assimilation: A Comparison of Different Approaches , 2005 .