Implementations of a square-root ensemble analysis and a hybrid localisation into the POD-based ensemble 4DVar

ABSTRACT The main purpose of this study is to propose a hybrid localisation technique for the proper orthogonal decomposition (POD)-based ensemble 4-D variational assimilation (PODEn4DVar) method with the aim of facilitating its implementation. This hybrid localisation scheme takes full advantages of both the implicit and explicit localisations and is relatively easy to be achieved by parallel programming. Besides, to strengthen its performance, we also incorporate a square root analysis scheme into the PODEn4DVar instead of its original one. The feasibility and effectiveness of the modified PODEn4DVar are demonstrated in a 2-D shallow-water equation model with simulated observations. It is found that the PODEn4DVar performs robust and is capable of outperforming the local ensemble transform Kalman filter (LETKF), its 4-D extension 4D-LETKF and another En4DVar method under both perfect- and imperfect-model scenarios.

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

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

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

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

[5]  Adrian Sandu,et al.  A hybrid approach to estimating error covariances in variational data assimilation , 2010 .

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

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

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

[9]  T. Matsuno.  Numerical Integrations of the Primitive Equations by a Simulated Backward Difference Method , 1966 .

[10]  Yunfeng Wang,et al.  An ensemble-based three-dimensional variational data assimilation scheme , 2011, 2011 International Conference on Multimedia Technology.

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

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

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

[14]  H. Tran,et al.  Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor , 2002 .

[15]  Qin Xu.  Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part I: Basic Issues and Heuristic Examples , 1996 .

[16]  H. Tran,et al.  Modeling and control of physical processes using proper orthogonal decomposition , 2001 .

[17]  Chris Snyder,et al.  A Comparison between the 4DVAR and the Ensemble Kalman Filter Techniques for Radar Data Assimilation , 2005 .

[18]  Christian L. Keppenne,et al.  Data Assimilation into a Primitive-Equation Model with a Parallel Ensemble Kalman Filter , 2000 .

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

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

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

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

[23]  J. M. Lewis,et al.  The use of adjoint equations to solve a variational adjustment problem with advective constraints , 1985 .

[24]  B. Hunt,et al.  Four-dimensional local ensemble transform Kalman filter: numerical experiments with a global circulation model , 2007 .

[25]  G. Evensen.  Sampling strategies and square root analysis schemes for the EnKF , 2004 .

[26]  Liang Xu,et al.  Development of NAVDAS-AR: non-linear formulation and outer loop tests , 2006 .

[27]  Xiaolei Zou,et al.  Toward Developing an Objective 4DVAR BDA Scheme for Hurricane Initialization Based on TPC Observed Parameters , 2004 .

[28]  Jean-Noël Thépaut,et al.  Impact of MODIS polar winds in ECMWF's 4DVAR data assimilation system , 2004 .

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

[30]  Li Wei,et al.  Fitting model fields to observations by using singular value decomposition: An ensemble‐based 4DVar approach , 2007 .

[31]  Jidong Gao,et al.  Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part V: Coarse-Grain Adjoint and Problems in Gradient Check , 1998 .

[32]  Peter Bauer,et al.  Implementation of 1D+4D‐Var assimilation of precipitation‐affected microwave radiances at ECMWF. II: 4D‐Var , 2006 .

[33]  Istvan Szunyogh,et al.  A Local Ensemble Kalman Filter for Atmospheric Data Assimilation , 2002 .

[34]  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 .

[35]  Juanzhen Sun,et al.  Impacts of Initial Estimate and Observation Availability on Convective-Scale Data Assimilation with an Ensemble Kalman Filter , 2004 .

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