Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado Ensemble Ensemble Kalman Kalman Filter Filter Advanced Numerics Seminar 8 March 2006, Langen, Germany Dusanka Zupanski, CIRA/CSU [email protected]Acknowledgements: M. Zupanski, G. Carrio, S. Denning, M. Uliasz, R. Lokupitya, CSU A. Hou and S. Zhang, NASA/GMAO
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Ensemble Kalman Filter - COSMO model · Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado Ensemble Kalman Filter Advanced Numerics Seminar 8 March 2006, Langen,
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Dusanka ZupanskiCIRA/Colorado State University
Fort Collins, Colorado
Ensemble Ensemble Kalman Kalman FilterFilter
Advanced Numerics Seminar8 March 2006, Langen, Germany
Need for optimal estimate of the atmosphericstate + verifiable uncertainty of this estimate; Need for flow-dependent forecast errorcovariance matrix; and The above requirements should be applicableto most complex atmospheric models (e.g.,non-hydrostatic, cloud-resolving, LES).
Example 1: Fronts
Example 2: Hurricanes
(From Whitaker et al., THORPEX web-page)Benefits of Flow-Dependent Background Errors
Two good candidates: 4d-var method: It employs flow-dependent forecasterror covariance, but it does not propagate it in time. Kalman Filter (KF): It does propagate flow-dependent forecast error covariance in time, but it is tooexpensive for applications to complex atmosphericmodels.
EnKF is a practical alternative to KF, applicable tomost complex atmospheric models.
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A bonus benefit: EnKF does not use adjoint models!
Data assimilation should combine model and data in an optimal way.Optimal solution z can be defined in terms of optimal initial conditions xa(analysis), model error w, and empirical parameters α,β,γ.
Approach 1:Approach 1: Optimal solution (e.g., analysis xa) = Minimum varianceestimate, or conditional mean of Bayesian posterior probability densityfunction (PDF) (e.g., Kalman Kalman filterfilter; Extended Extended Kalman Kalman filterfilter; EnKFEnKF)
How can we obtain optimal solution?Two approaches are used most often:
For non-liner M or H the solution can be obtained employingExtended Extended Kalman Kalman filterfilter, or Ensemble Ensemble Kalman Kalman filterfilter.
Assuming liner M and H and independent Gaussin PDFs⇒ Kalman Kalman filterfilter solution (e.g., Jazwinski 1970)
xa is defined as mathematical expectation (i.e., mean) of the conditionalposterior p(x|y), given observations y and prior p(x).
For Gaussian PDFs and linear H and M results of all methods [KF, EnKF (with enoughensemble members), and variational] should be identical, assuming the same Pf and yare used in all methods.
Minimum variance estimate= Maximum likelihood estimate!
KF,EnKF,4d-var,
allcreatedequal?
Does this really happen?!?
TEST RESULTS EMPLOYING A LINEAR MODEL AND GAUSSIAN PDFs
Ensemble size of 500 is adequate for describing all DOFs of thisfully observed system.In later cycles more eigenvalues are approaching value 1 (noinformation).
References for further readingReferences for further reading
Anderson, J. L., 2001: An ensemble adjustment filter for data assimilation. Mon. Wea. Rev., 129,2884–2903.
Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model usingMonte Carlo methods to forecast error statistics. J. Geophys. Res., 99, (C5),. 10143-10162.
Evensen, G., 2003: The ensemble Kalman filter: theoretical formulation and practicalimplementation. Ocean Dynamics. 53, 343-367.
Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter/3D-variational analysisscheme. Mon. Wea. Rev., 128, 2905–2919.
Houtekamer, Peter L., Herschel L. Mitchell, 1998: Data Assimilation Using an Ensemble KalmanFilter Technique. Monthly Weather Review: Vol. 126, No. 3, pp. 796-811.
Houtekamer, Peter L., Herschel L. Mitchell, Gerard Pellerin, Mark Buehner, Martin Charron,Lubos Spacek, and Bjarne Hansen, 2005: Atmospheric data assimilation with an ensemble Kalman filter: Results with real observations. Monthly Weather Review: Vol.133, No. 3, pp. 604-620.
Ott, E., and Coauthors, 2004: A local ensemble Kalman filter for atmospheric data assimilation.Tellus., 56A, 415–428.
Tippett, M. K., J. L. Anderson, C. H. Bishop, T. M. Hamill, and J. S. Whitaker, 2003: Ensemblesquare root filters. Mon. Wea. Rev., 131, 1485–1490.
Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation without perturbedobservations. Mon. Wea. Rev., 130, 1913–1924.
Zupanski D. and M. Zupanski, 2006: Model error estimation employing an ensemble dataassimilation approach. Mon. Wea. Rev. (in press).
Zupanski, M., 2005: Maximum likelihood ensemble filter: Theoretical aspects. Mon. Wea. Rev.,133, 1710–1726
Time evolving model dynamics significantly reduces the noise in the initiallyprescribed perturbations!
Example: CSU SWM model of Randall et al.Solution-Truth shown
23 2-h DA cycles: 18UTC 2 May 1998 – 00 UTC 5 May 1998(Mixed phase Arctic boundary layer cloud at Sheba site)Experiments initialized with typical clean aerosol concentrationsMay 4 was abnormal: high IFN and CCN above the inversionΔx= 50m, Δzmax = 30m (2d domain: 50col, 40lev), Δt=2s, Nens=48Sophisticated microphysics in RAMS/LESControl variables: Θ_il, u, v, w, N_x, R_x (8 species), IFN, CCN (dim= 22 variables x 50 columns x 40 levels = 44000)Radar/lidar real observations of IWP, LWP are assimilated IWP and LWP are vertically integrated quantities (no informationabout the profiles of IFN, CCN is observed)
MLEF experiments with CSU/RAMS Large Eddy Simulation (LES) modelMLEF experiments with CSU/RAMS Large Eddy Simulation (LES) model
MLEF is similar to 4dvar because it seeks a maximumlikelihood solution (i.e., minimum of J). It is also similar to EnKF methods because it usesensembles to calculate forecast error covariance. MLEF uses the same definition of transformation matrixas in the ETKF (Bishop et al. 2001). It has a capability to estimate and reduce several majorsources of forecast uncertainties simultaneously: Initialconditions, model error, boundary conditions, andempirical parameters. MLEF has also a capability to take into account non-Gaussian (log-normal) PDFs (Flatcher and M. Zupanski2006)