Relationships among Four-Dimensional Hybrid Ensemble–Variational Data Assimilation Algorithms with Full and Approximate Ensemble Covariance Localization

Ensemble‐variational data assimilation algorithms that can incorporate the time dimension (fourdimensional or 4D) and combine static and ensemble-derived background error covariances (hybrid) are formulated in general forms based on the extended control variable and the observation-space-perturbation approaches. The properties and relationships of these algorithms and their approximated formulations are discussed. The main algorithms discussed include the following: 1) the standard ensemble 4DVar (En4DVar) algorithm incorporating ensemble-derived background error covariance through the extended control variable approach, 2) the 4DEnVar neglecting the time propagation of the extended control variable (4DEnVar-NPC), 3) the 4D ensemble‐variational algorithm based on observation space perturbation (4DEnVar), and 4) the 4DEnVar with no propagation of covariance localization (4DEnVar-NPL). Without the static background error covariance term, none of the algorithms requires the adjoint model except for En4DVar. Costly applications of the tangent linear model to localized ensemble perturbations can be avoided by making the NPC and NPL approximations. It is proven that En4DVar and 4DEnVar are mathematically equivalent, while 4DEnVar-NPC and 4DEnVar-NPL are mathematically equivalent. Such equivalences are also demonstrated by single-observation assimilation experiments with a 1D linear advection model. The effects of the non-flow-following or stationary localization approximations are also examined through the experiments. All of the above algorithms can include the static background error covariance term to establish a hybrid formulation.Whenthestatictermisincluded,allalgorithmswillrequireatangentlinearmodelandanadjoint model. The first guess at appropriate time (FGAT) approximation is proposed to avoid the tangent linear and adjoint models. Computational costs of the algorithms are also discussed.

[1]  M. Buehner.  Ensemble‐derived stationary and flow‐dependent background‐error covariances: Evaluation in a quasi‐operational NWP setting , 2005 .

[2]  Craig H. Bishop,et al.  Adaptive sampling with the ensemble transform Kalman filter , 2001 .

[3]  Takemasa Miyoshi,et al.  Ensemble-based observation impact estimates using the NCEP GFS , 2013 .

[4]  M. Fisher,et al.  Background Error Covariance Modelling , 2003 .

[5]  Qingnong Xiao,et al.  An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part III: Antarctic Applications with Advanced Research WRF Using Real Data , 2013 .

[6]  Andrew C. Lorenc,et al.  The potential of the ensemble Kalman filter for NWP—a comparison with 4D‐Var , 2003 .

[7]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory , 2007 .

[8]  Kayo Ide,et al.  An OSSE-Based Evaluation of Hybrid Variational-Ensemble Data Assimilation for the NCEP GFS. Part II: 4DEnVar and Hybrid Variants , 2015 .

[9]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. Ii: Numerical Results , 2007 .

[10]  J. Mahfouf,et al.  The ecmwf operational implementation of four‐dimensional variational assimilation. III: Experimental results and diagnostics with operational configuration , 2000 .

[11]  P. Houtekamer,et al.  Data Assimilation Using an Ensemble Kalman Filter Technique , 1998 .

[12]  P. Courtier,et al.  A strategy for operational implementation of 4D‐Var, using an incremental approach , 1994 .

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

[14]  E. Kalnay,et al.  C ○ 2007 The Authors , 2006 .

[15]  Stanley G. Benjamin,et al.  Global Ensemble Predictions of 2009’s Tropical Cyclones Initialized with an Ensemble Kalman Filter , 2011 .

[16]  Juanzhen Sun,et al.  Dynamical and Microphysical Retrieval from Doppler Radar Observations Using a Cloud Model and Its Adjoint. Part I: Model Development and Simulated Data Experiments. , 1997 .

[17]  M. Buehner,et al.  Intercomparison of Variational Data Assimilation and the Ensemble Kalman Filter for Global Deterministic NWP. Part I: Description and Single-Observation Experiments , 2010 .

[18]  G. Evensen,et al.  Asynchronous data assimilation with the EnKF , 2010 .

[19]  Craig H. Bishop,et al.  Flow‐adaptive moderation of spurious ensemble correlations and its use in ensemble‐based data assimilation , 2007 .

[20]  Fuqing Zhang,et al.  E4DVar: Coupling an Ensemble Kalman Filter with Four-Dimensional Variational Data Assimilation in a Limited-Area Weather Prediction Model , 2012 .

[21]  J. Yorke,et al.  Four-dimensional ensemble Kalman filtering , 2004 .

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

[23]  A. Lorenc,et al.  The Met Office global four‐dimensional variational data assimilation scheme , 2007 .

[24]  N. Roberts,et al.  Numerical Aspects of the Application of Recursive Filters to Variational Statistical Analysis. Part II: Spatially Inhomogeneous and Anisotropic General Covariances , 2003 .

[25]  F. Rabier,et al.  The potential of high‐density observations for numerical weather prediction: A study with simulated observations , 2003 .

[26]  Mark Buehner,et al.  Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction , 2013 .

[27]  E. Kalnay,et al.  Four-dimensional ensemble Kalman filtering , 2004 .

[28]  J. Whitaker,et al.  Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter , 2001 .

[29]  J. Derber,et al.  Introduction of the GSI into the NCEP Global Data Assimilation System , 2009 .

[30]  Stanley G. Benjamin,et al.  Predictions of 2010’s Tropical Cyclones Using the GFS and Ensemble-Based Data Assimilation Methods , 2011 .

[31]  Jimy Dudhia,et al.  Four-Dimensional Variational Data Assimilation for WRF : Formulation and Preliminary Results , 2009 .

[32]  Monique Tanguay,et al.  Extension of 3DVAR to 4DVAR: Implementation of 4DVAR at the Meteorological Service of Canada , 2007 .

[33]  D. Dee,et al.  Variational bias correction of satellite radiance data in the ERA‐Interim reanalysis , 2009 .

[34]  M. Buehner,et al.  Intercomparison of Variational Data Assimilation and the Ensemble Kalman Filter for Global Deterministic NWP. Part II: One-Month Experiments with Real Observations , 2010 .

[35]  Craig H. Bishop,et al.  Ensemble covariances adaptively localized with ECO-RAP. Part 1: tests on simple error models , 2009 .

[36]  M. Tanguay,et al.  Four-Dimensional Variational Data Assimilation for the Canadian Regional Deterministic Prediction System , 2012 .

[37]  Kayo Ide,et al.  An OSSE-Based Evaluation of Hybrid Variational–Ensemble Data Assimilation for the NCEP GFS. Part I: System Description and 3D-Hybrid Results , 2015 .

[38]  Nils Gustafsson,et al.  Discussion on ‘4D-Var or EnKF?’ , 2007 .

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

[40]  J. Whitaker,et al.  Ensemble Data Assimilation without Perturbed Observations , 2002 .

[41]  Thomas M. Hamill,et al.  Ensemble Data Assimilation with the NCEP Global Forecast System , 2008 .

[42]  Ian Roulstone,et al.  A comparison of 4DVar with ensemble data assimilation methods , 2014 .

[43]  Craig H. Bishop,et al.  Comparison of Hybrid Ensemble/4DVar and 4DVar within the NAVDAS-AR Data Assimilation Framework , 2013 .

[44]  Joaquim Ballabrera-Poy,et al.  Response to the discussion on “4-D-Var or EnKF?” by Nils Gustafsson , 2007 .

[45]  Craig H. Bishop,et al.  Adaptive Ensemble Covariance Localization in Ensemble 4D-VAR State Estimation , 2011 .

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

[47]  Istvan Szunyogh,et al.  Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter , 2005, physics/0511236.

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

[49]  A. Lorenc,et al.  Operational implementation of a hybrid ensemble/4D‐Var global data assimilation system at the Met Office , 2013 .

[50]  Neill E. Bowler,et al.  Comparison of Hybrid-4DEnVar and Hybrid-4DVar Data Assimilation Methods for Global NWP , 2015 .

[51]  N. B. Ingleby,et al.  The Met. Office global three‐dimensional variational data assimilation scheme , 2000 .

[52]  Jean-Thomas Camino,et al.  4DEnVar: link with 4D state formulation of variational assimilation and different possible implementations , 2014 .

[53]  G. Evensen.  Using the Extended Kalman Filter with a Multilayer Quasi-Geostrophic Ocean Model , 1992 .

[54]  David D. Parrish,et al.  GSI 3DVar-Based Ensemble-Variational Hybrid Data Assimilation for NCEP Global Forecast System: Single-Resolution Experiments , 2013 .

[55]  Jeffrey L. Anderson.  An Ensemble Adjustment Kalman Filter for Data Assimilation , 2001 .

[56]  Olivier Pannekoucke,et al.  On the Merits of Using a 3D-FGAT Assimilation Scheme with an Outer Loop for Atmospheric Situations Governed by Transport , 2010 .

[57]  R. Purser,et al.  Three-Dimensional Variational Analysis with Spatially Inhomogeneous Covariances , 2002 .

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

[59]  John Derber,et al.  The National Meteorological Center's spectral-statistical interpolation analysis system , 1992 .

[60]  P. L. Houtekamer,et al.  Ensemble Kalman filtering , 2005 .

[61]  Ming Hu,et al.  A GSI-Based Coupled EnSRF-En3DVar Hybrid Data Assimilation System for the Operational Rapid Refresh Model: Tests at a Reduced Resolution , 2014 .

[62]  Jean-Noël Thépaut,et al.  Impact of the Digital Filter as a Weak Constraint in the Preoperational 4DVAR Assimilation System of Météo-France , 2001 .

[63]  Ming Xue,et al.  A four‐dimensional asynchronous ensemble square‐root filter (4DEnSRF) algorithm and tests with simulated radar data , 2013 .

[64]  Q. Xiao,et al.  An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part II: Observing System Simulation Experiments with Advanced Research WRF (ARW) , 2009 .

[65]  Craig H. Bishop,et al.  Vertical Covariance Localization for Satellite Radiances in Ensemble Kalman Filters , 2010 .

[66]  T. Hamill,et al.  On the Theoretical Equivalence of Differently Proposed Ensemble 3DVAR Hybrid Analysis Schemes , 2007 .

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