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1. Introduction The original ensemble Kalman filter (EnKF; Evensen 1994 ) has been developed with the aim to enable the application of sequential data assimilation algorithms based on the Kalman filter with large-scale numerical models. Burgers et al. (1998) and Houtekamer and Mitchell (1998) clarified that the EnKF requires an ensemble of perturbed observations for statistical consistency. The EnKF represents the state estimate by the mean of an ensemble of model state realizations, while
1. Introduction The original ensemble Kalman filter (EnKF; Evensen 1994 ) has been developed with the aim to enable the application of sequential data assimilation algorithms based on the Kalman filter with large-scale numerical models. Burgers et al. (1998) and Houtekamer and Mitchell (1998) clarified that the EnKF requires an ensemble of perturbed observations for statistical consistency. The EnKF represents the state estimate by the mean of an ensemble of model state realizations, while
1. Introduction The ensemble Kalman filter (EnKF; Evensen 1994 , 2003 , 2007 ) has become a popular tool for data assimilation because of its computational efficiency and flexibility ( Anderson 2001 ; Whitaker and Hamill 2002 ; Ott et al. 2004 ; Bengtsson et al. 2003 ; Evensen 2007 ). In various versions of EnKFs, one major difference is how to get the updated ensemble after obtaining the updated mean and variance. Stochastic methods ( Houtekamer and Mitchell 1998
1. Introduction The ensemble Kalman filter (EnKF; Evensen 1994 , 2003 , 2007 ) has become a popular tool for data assimilation because of its computational efficiency and flexibility ( Anderson 2001 ; Whitaker and Hamill 2002 ; Ott et al. 2004 ; Bengtsson et al. 2003 ; Evensen 2007 ). In various versions of EnKFs, one major difference is how to get the updated ensemble after obtaining the updated mean and variance. Stochastic methods ( Houtekamer and Mitchell 1998
1. Introduction Ensemble square root Kalman filters are an efficient deterministic variant of the original ensemble Kalman filter (EnKF; Evensen 1994 ; Burgers et al. 1998 ). Common members of this class of filters are the ensemble transform Kalman filter (ETKF; Bishop et al. 2001 ), the ensemble adjustment Kalman filter (EAKF; Anderson 2001 , 2003 ), and the ensemble square root Kalman filter with serial processing of observations (EnSRF; Whitaker and Hamill 2002 ). Recently, also the
1. Introduction Ensemble square root Kalman filters are an efficient deterministic variant of the original ensemble Kalman filter (EnKF; Evensen 1994 ; Burgers et al. 1998 ). Common members of this class of filters are the ensemble transform Kalman filter (ETKF; Bishop et al. 2001 ), the ensemble adjustment Kalman filter (EAKF; Anderson 2001 , 2003 ), and the ensemble square root Kalman filter with serial processing of observations (EnSRF; Whitaker and Hamill 2002 ). Recently, also the
1. Introduction One of the main goals of data assimilation is to improve the forecast performance of a dynamical model. Many data assimilation algorithms are based on Kalman filtering, where the state of a system is estimated by combining all the information that is available about the system in accord with their statistical uncertainty. The main computational issue in Kalman filter–based data assimilation is the propagation of the forecast error covariance matrix. A number of methods have been
1. Introduction One of the main goals of data assimilation is to improve the forecast performance of a dynamical model. Many data assimilation algorithms are based on Kalman filtering, where the state of a system is estimated by combining all the information that is available about the system in accord with their statistical uncertainty. The main computational issue in Kalman filter–based data assimilation is the propagation of the forecast error covariance matrix. A number of methods have been
1. Introduction Motivated by the work of Evensen (1994) and Burgers et al. (1998) , various ensemble Kalman filter (EnKF) approaches have been proposed for Monte Carlo–based data assimilation in high-dimensional atmospheric systems. Examples include serial assimilation algorithms ( Houtekamer and Mitchell 2001 ; Anderson 2001 ), covariance stabilization techniques ( Hamill et al. 2001 ; Anderson 2007 ), and square root filters ( Bishop et al. 2001 ; Tippett et al. 2003 ). These EnKF
1. Introduction Motivated by the work of Evensen (1994) and Burgers et al. (1998) , various ensemble Kalman filter (EnKF) approaches have been proposed for Monte Carlo–based data assimilation in high-dimensional atmospheric systems. Examples include serial assimilation algorithms ( Houtekamer and Mitchell 2001 ; Anderson 2001 ), covariance stabilization techniques ( Hamill et al. 2001 ; Anderson 2007 ), and square root filters ( Bishop et al. 2001 ; Tippett et al. 2003 ). These EnKF
1. Introduction Reduced-state filters ( Lawson and Hansen 2004 ) have been extensively studied in the past few years as an alternative to the extended Kalman filter (e.g., Evensen 1992 ), which has shown limitations when used with strongly nonlinear dynamics ( Evensen 1992 ; Gauthier et al. 1993 ; Bouttier 1994 ). Various versions using Monte Carlo methods, many of them reviewed by Evensen (2003) , have been the subject of studies in meteorological, oceanographic, and hydrological contexts
1. Introduction Reduced-state filters ( Lawson and Hansen 2004 ) have been extensively studied in the past few years as an alternative to the extended Kalman filter (e.g., Evensen 1992 ), which has shown limitations when used with strongly nonlinear dynamics ( Evensen 1992 ; Gauthier et al. 1993 ; Bouttier 1994 ). Various versions using Monte Carlo methods, many of them reviewed by Evensen (2003) , have been the subject of studies in meteorological, oceanographic, and hydrological contexts
1. Introduction The Kalman filter (KF) ( Kalman 1960 ) is a sequential data assimilation algorithm. For linear stochastic systems, it can be shown that the KF is an optimal linear estimator that minimizes the variance of the estimation error ( Simon 2006 , chapter 5). Because of its relative simplicity in implementation, the KF is suitable for many data assimilation problems. However, for high-dimensional systems such as weather forecasting models, direct application of the KF is prohibitively
1. Introduction The Kalman filter (KF) ( Kalman 1960 ) is a sequential data assimilation algorithm. For linear stochastic systems, it can be shown that the KF is an optimal linear estimator that minimizes the variance of the estimation error ( Simon 2006 , chapter 5). Because of its relative simplicity in implementation, the KF is suitable for many data assimilation problems. However, for high-dimensional systems such as weather forecasting models, direct application of the KF is prohibitively
applied variational methods to analyze a displacement field and align coherent structures. More recently, Lawson and Hansen (2005) proposed a new error model that includes a non-Gaussian alignment error and a Gaussian additive error, and they corrected two types of errors separately with an ensemble Kalman filter (EnKF). These existing analysis techniques are meant to “move” the vortex a distance that is significant compared to the horizontal scale of the vortex. When the error of the forecast
applied variational methods to analyze a displacement field and align coherent structures. More recently, Lawson and Hansen (2005) proposed a new error model that includes a non-Gaussian alignment error and a Gaussian additive error, and they corrected two types of errors separately with an ensemble Kalman filter (EnKF). These existing analysis techniques are meant to “move” the vortex a distance that is significant compared to the horizontal scale of the vortex. When the error of the forecast
-alone ensemble Kalman filter (EnKF) and a stand-alone variational (Var) system, operational NWP centers worldwide have implemented or are implementing various forms of hybrid EnVar operationally for both global and regional NWP (e.g., Buehner et al. 2010a , b ; Wang 2010 ; Bonavita et al. 2012 ; Clayton et al. 2013 ; Wang et al. 2013 ; Kuhl et al. 2013 ; Wang and Lei 2014 ; Wang and Wang 2017 ; Lu et al. 2016 , 2017 ; Duda et al. 2019 ). One challenge associated with the estimation of multiscale
-alone ensemble Kalman filter (EnKF) and a stand-alone variational (Var) system, operational NWP centers worldwide have implemented or are implementing various forms of hybrid EnVar operationally for both global and regional NWP (e.g., Buehner et al. 2010a , b ; Wang 2010 ; Bonavita et al. 2012 ; Clayton et al. 2013 ; Wang et al. 2013 ; Kuhl et al. 2013 ; Wang and Lei 2014 ; Wang and Wang 2017 ; Lu et al. 2016 , 2017 ; Duda et al. 2019 ). One challenge associated with the estimation of multiscale
most of the concepts described in this paper apply to the conservation of other quantities, such as angular momentum and energy, we focus on mass conservation to make the discussion more specific and to illustrate ideas with simple examples. When ensemble Kalman filters are used for data assimilation two distinct mass conservation issues arise. Mass should be conserved in each member (replicate) of the forecast ensemble produced by propagating states over time with a forecast model, and it should
most of the concepts described in this paper apply to the conservation of other quantities, such as angular momentum and energy, we focus on mass conservation to make the discussion more specific and to illustrate ideas with simple examples. When ensemble Kalman filters are used for data assimilation two distinct mass conservation issues arise. Mass should be conserved in each member (replicate) of the forecast ensemble produced by propagating states over time with a forecast model, and it should
