Efficient Local Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to a Basin-Scale Ocean Turbulent Flow

Abstract In large-sized atmospheric or oceanic applications of square root or ensemble Kalman filters, it is often necessary to introduce the prior assumption that long-range correlations are negligible and force them to zero using a local parameterization, supplementing the ensemble or reduced-rank representation of the covariance. One classic algorithm to perform this operation consists of taking the Schur product of the ensemble covariance matrix by a local support correlation matrix. However, with this parameterization, the square root of the forecast error covariance matrix is no more directly available, so that any observational update algorithm requiring this square root must include an additional step to compute local square roots from the Schur product. This computation generates an additional numerical cost or produces high-rank square roots, which may deprive the observational update from its original efficiency. In this paper, it is shown how efficient local square root parameterizations can b...

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

[2]  A. Robinson,et al.  Data Assimilation via Error Subspace Statistical Estimation.Part I: Theory and Schemes , 1999, Monthly Weather Review.

[3]  M. Drévillon,et al.  The GODAE/Mercator-Ocean global ocean forecasting system: results, applications and prospects , 2008 .

[4]  Eugenia Kalnay,et al.  Weight interpolation for efficient data assimilation with the Local Ensemble Transform Kalman Filter , 2009 .

[5]  M. Verlaan,et al.  Tidal flow forecasting using reduced rank square root filters , 1997 .

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

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

[8]  G. Evensen,et al.  Assimilation of Geosat altimeter data for the Agulhas current using the ensemble Kalman filter with , 1996 .

[9]  Jean-Michel Brankart,et al.  Assimilation of sea-surface temperature and altimetric observations during 1992–1993 into an eddy permitting primitive equation model of the North Atlantic Ocean , 2003 .

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

[11]  Jean-Michel Brankart,et al.  Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals , 2010 .

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

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

[14]  Jeffrey L. Anderson.  A Local Least Squares Framework for Ensemble Filtering , 2003 .

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

[16]  Jean-Michel Brankart,et al.  Implementation of a multivariate data assimilation scheme for isopycnic coordinate ocean models: Application to a 1993–1996 hindcast of the North Atlantic Ocean circulation , 2003 .

[17]  A. Segers,et al.  Variance reduced ensemble Kalman filtering , 2001 .

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

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

[20]  Istvan Szunyogh,et al.  Assessing a local ensemble Kalman filter: perfect model experiments with the National Centers for Environmental Prediction global model , 2005 .

[21]  M. Corazza,et al.  An implementation of the Local Ensemble Kalman Filter in a quasi geostrophic model and comparison with 3D-Var , 2007 .

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

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

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

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

[26]  Pierre Brasseur,et al.  A demonstration of ensemble-based assimilation methods with a layered OGCM from the perspective of operational ocean forecasting systems , 2003 .

[27]  Jean-Michel Brankart,et al.  Implementation of a reduced rank square-root smoother for high resolution ocean data assimilation , 2010 .

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

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

[30]  Jacques Verron,et al.  A singular evolutive extended Kalman filter for data assimilation in oceanography , 1998 .

[31]  Jean-Michel Brankart,et al.  Efficient Parameterization of the Observation Error Covariance Matrix for Square Root or Ensemble Kalman Filters: Application to Ocean Altimetry , 2009 .

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