Adaptive Ensemble Covariance Localization in Ensemble 4D-VAR State Estimation

An adaptive ensemble covariance localization technique, previously used in ‘‘local’’ forms of the ensemble Kalman filter, is extended to a global ensemble four-dimensional variational data assimilation (4D-VAR) scheme. The purely adaptive part of the localization matrix considered is given by the element-wise square of the correlation matrix of a smoothed ensemble of streamfunction perturbations. It is found that these purely adaptive localization functions have spurious far-field correlations as large as 0.1 with a 128-member ensemble. To attenuate the spurious features of the purely adaptive localization functions, the authors multiply the adaptive localization functions with very broadscale nonadaptive localization functions. Using the Navy’s operational ensemble forecasting system, it is shown that the covariance localization functions obtained by this approach adapt to spatially anisotropic aspects of the flow, move with the flow, and are free of far-field spurious correlations. The scheme is made computationally feasible by (i) a method for inexpensively generating the square root of an adaptively localized global four-dimensional error covariance model in terms of products or modulations of smoothed ensemble perturbations with themselves and with raw ensemble perturbations, and (ii) utilizing algorithms that speed ensemble covariance localization when localization functions are separable, variable-type independent, and/or large scale. In spite of the apparently useful characteristics of adaptive localization, single analysis/forecast experiments assimilating 583 200 observations over both 6- and 12-h data assimilation windows failed to identify any significant difference in the quality of the analyses and forecasts obtained using nonadaptive localization from that obtained using adaptive localization.

[1]  Samuel Buis,et al.  Intercomparison of the primal and dual formulations of variational data assimilation , 2008 .

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

[3]  P. Houtekamer,et al.  Ensemble size, balance, and model-error representation in an ensemble Kalman filter , 2002 .

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

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

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

[7]  Craig H. Bishop,et al.  Ensemble covariances adaptively localized with ECO‐RAP. Part 2: a strategy for the atmosphere , 2009 .

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

[9]  Mark Buehner,et al.  Efficient Ensemble Covariance Localization in Variational Data Assimilation , 2011 .

[10]  Craig H. Bishop,et al.  Ensemble Transformation and Adaptive Observations , 1999 .

[11]  Craig H. Bishop,et al.  Evaluation of the Ensemble Transform Analysis Perturbation Scheme at NRL , 2008 .

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

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

[14]  Roger Daley,et al.  NAVDAS: Formulation and Diagnostics , 2001 .

[15]  Istvan Szunyogh,et al.  Extracting Envelopes of Rossby Wave Packets , 2003 .

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

[17]  Craig H. Bishop,et al.  A Local Formulation of the Ensemble Transform (ET) Analysis Perturbation Scheme , 2010 .

[18]  E. Kalnay,et al.  Ensemble Forecasting at NCEP and the Breeding Method , 1997 .

[19]  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 .

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

[21]  X. Deng,et al.  Model Error Representation in an Operational Ensemble Kalman Filter , 2009 .

[22]  T. Hogan,et al.  The Description of the Navy Operational Global Atmospheric Prediction System's Spectral Forecast Model , 1991 .

[23]  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 .

[24]  Daniel Hodyss,et al.  Ensemble covariances adaptively localized with ECO-RAP. Part 2: a strategy for the atmosphere , 2009 .

[25]  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 .

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

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

[28]  J. Kepert Covariance localisation and balance in an Ensemble Kalman Filter , 2009 .

[29]  Liang Xu,et al.  Development of NAVDAS-AR: formulation and initial tests of the linear problem , 2005 .