Two Methods of Localization  Model Covariance Matrix Localization (B Localization)  Accomplished by taking a Schur product between the model covariance matrix and a matrix whose elements are dependent upon the distance between the corresponding grid points. (Hamill et al., 2001)  Model grid points that are far apart have zero error covariance.  Observation Covariance Matrix Localization (R Localization)  Observations that are far away from a grid point have infinite error. EnKF Localization Techniques and Balance 1 Steven J. Greybush, 1 Eugenia Kalnay, 2 Takemasa Miyoshi 1 University of Maryland, College Park, MD, U.S.A. 2 Numerical Prediction Division, Japan Meteorological Agency, Tokyo, Japan Localization  A modification of the covariance matrices in the Kalman gain formula that reduces the influence of distant regions. (Houtekamer and Mitchell, 2001)  Removes spurious long distance correlations due to sampling error of the model covariance from finite ensemble size. (Anderson, 2007)  Takes advantage of the ensemble's lower dimensionality in local regions. (Hunt et. al., 2007)  Ultimately creates a more accurate analysis (reduces RMSE). Experimental Setup  Model  The shallow water equations in a rotating, inviscid fluid.  Variation only along the x-axis.  The variables of interest are h and v.  Linearize the equations, and apply a harmonic form to the solution.  Substituting into the governing equations, and assuming geostrophic balance, yields the following solutions for h and v:  Ensemble  Initially geostrophic waveform for truth and 2 ensemble members.  101 Grid Points every 50 km along domain. Results from Assimilation of Gridded Observations Assimilate 20 observations of h and v (randomly perturbed from the truth) at regular intervals along the domain; observe accuracy and balance of the analysis. Results from Monte Carlo Simulations  Three scales in the problem: W = wavelength of solutions L = localization distance D = distance between observations  Explore the phase space of the three scales.  Obtain robust results by repeating each scenario 100x with random observation errors. Future Work  Complement this study by comparing balance for B localization and R localization EnSQRT data assimilation on the SPEEDY GCM (Molteni, 2003), using realistic observation densities and locations.  Further investigate the mathematical properties of the two localization methods. WCRP/THORPEX WORKSHOP on 4D-VAR and ENSEMBLE KALMAN FILTER COMPARISONS Buenos Aires, Argentina; November 10-13, 2008 Acknowledgements Thanks to Kayo Ide and Jeff Anderson for their helpful comments and critiques of this project. Balance  Lorenc (2003) and Kepert (2006) argue that localization reduces the balance information encoded in the model covariance matrix.  Houtekamer and Mitchell (2005) noted balance issues when applying a localized EnKF to the Canadian GCM.  Imbalanced analyses project information onto inertial-gravity waves, which are filtered out (geostrophic adjustment, digital filtering, etc.), resulting in a loss of information and a suboptimal analysis. Research Questions  How does localization introduce imbalance into an analysis? Can it be avoided?  How do the analyses produced by B-localization and R-localization EnKF compare in terms of accuracy (RMSE) and (geostrophic) balance? R loc = R * exp(+(d) 2 / 2L 2 ) B loc = B * exp(-(r i -r j ) 2 / 2L 2 ) Conclusions  Both types of localization do introduce imbalance into analysis increments, especially for short localization distances.  R localization is significantly more balanced than B localization, but is slightly less accurate. L Use ageostrophic wind as measure of imbalance. v ageo = v – g/f dh/dx Localize the waveforms with L=1500 km. |-dh/dx| increases while |v| decreases, disrupting the balance between the wind field and the mass field. (Lorenc 2003) Localization Distance L=1000 km; Distance between Observations D = 500 km; Wavelength W = 2000 km Analysis RMS Error: B Localization < R Localization < No Localization Analysis Imbalance: B Localization >> R Localization > No Localization K = BH T (HBH T + R) -1 Initial Waveforms: Analysis Increments: Constant D=500 km, W=2000 km, vary L: Constant D=250 km, vary L and W: (warmer colors show greater imbalance) Initial Ensemble and Truth Localized Ensemble Imbalance (Ageostrophic Wind)