Post on 21-Dec-2015
Dusanka Zupanski CIRA/Colorado State University
Fort Collins, Colorado
Ensemble Kalman Filter MethodsEnsemble Kalman Filter Methods
NOAA/NESDIS Cooperative Research Program (CoRP)Third Annual Science Symposium
15-16 August 2006, Hilton Fort Collins, CO
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Collaborators: M. Zupanski, L. Grasso, M. DeMaria, S. Denning, M. Uliasz, R. Lokupityia, C. Kummerow, G. Carrio, T. Vonder Haar, D. Randall, CSUA. Hou and S. Zhang, NASA/GMAO
Grant support:
NASA Grant NNG05GD15G, NASA NNG04GI25G, NOAA Grant NA17RJ1228, and DoD Grant DAAD19-02-2-0005 P00007
Computational support from NASA Halem and Columbia super-computers, CIRA and Atmospheric Science Dept. Linux clusters
Kalman filter, ensemble Kalman filter and variational methods
Maximum Likelihood Ensemble Filter (MLEF) KF vs. 3d-var, as special cases of the MLEF
Information content analysis of data (e.g., TRMM, GPM, GOES-R)
• NASA/GEOS-5 single column model (complex, 1-d model)• CSU/RAMS non-hydrostatic model (complex, 3-d model)
Conclusions and future research directionsDusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
OUTLINE
Typical KF
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Forecast error Covariance Pf
(full-rank space)
DATA ASSIMILATION
Observations First guess
Optimal solution for model statex=(T,u,v,w, q, …)
LINEARISED FORECAST MODEL
Analysis error Covariance Pa
(full-rank space)
Typical EnKF
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Forecast error Covariance Pf
(reduced-rank ensemble subspace)
DATA ASSIMILATION
Observations First guess
Optimal solution for model statex=(T,u,v,w, q, …)
NON-LINEAR ENSEMBLE OF FORECAST MODELS
Analysis error Covariance Pa
(reduced-rank ensemble subspace)
Typical variational method
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Prescribed Forecast error Covariance Pf
(full-rank space)
DATA ASSIMILATION
Observations First guess
Optimal solution for model statex=(T,u,v,w, q, …)
NON-LINEAR FORECAST MODEL
Analysis error Covariance Pa
(full-rank space)
Maximum Likelihood Ensemble Filter (MLEF) (Zupanski 2005; Zupanski and Zupanski 2006)
Linear full-rank MLEF = KF (Full-rank means Nens=Nstate)
Non-linear full-rank MLEF, without updating of Pf = 3d-var
x =xb + α Pf H T (HPf
T H T + R)−1[ y − H (xb )]
J(x) =
12[x−xb]
T Pf−1[x−xb] +
12[ y−H(x)]T R−1[ y−H(x)]
; for =1
Comparisons of KF and 3d-var within the same algorithm.Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
MLEF= KF valid under Gaussian error assumption. For Non-Gaussian case, ask M. Zupanski, S. Fletcher and collaborators.
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Information measures in ensemble subspace
∑ +=+= −
i i
is trd
)1(])([
2
21
λ
λCCI
∑ +=i
ih )1ln(2
1 2λ
Shannon information content, or entropy reduction
Degrees of freedom (DOF) for signal (Rodgers 2000):
- information matrix in ensemble subspace of dim Nens x Nens
x −xb =Pf1 2 (I +C)−1 2ζ
C =ZTZ
z i =R−1 2H[M (x+ pai )] −R−1 2H[M (x)] ≈R−1 2HPf
1 2 - are columns of Z
- control vector in ensemble space of dim Nensζ
z i
C
x - model state vector of dim Nstate >>Nens
Errors are assumed Gaussian in these measures.
(Bishop et al. 2001; Wei et al. 2005; Zupanski et al. 2006, subm. to JAS)
λi2
- eigenvalues of C
for linear H and M
KF vs. 3d-var: GEOS-5 Single Column Model (Nstate=80; Nobs=40, Nens=80, seventy 6-h DA cycles,
assimilation of simulated T,q observations)
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
GEOS-5 Single Column Model: DOF for signal(Nstate=80; Nobs=40, Nens=80 or Nens=10, seventy 6-h DA cycles,
assimilation of simulated T,q observations)
DOF for signal varies from one analysis cycle to another due to changes in atmospheric conditions. 3d-var does not capture this variability (straight line).
T true (K)
DOF for signal (d s), 40 obs
0
5
10
15
20
25
30
1 11 21 31 41 51 61
Data assimilation cycles
ds
ds_10ensds_80ens_KFds_80ens_3dv
q true (g kg-1)
Data assimilation cycles
Ver
tica
l le
vels
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Small ensemble size (10 ens), even though not perfect, captures main data signals.
Large PfInadequate Pf
Is this applicable to CSU/RAMS? (Nstate=2138400; Nobs=5940, Nens=50,
assimilation of simulated GOES-R 10.35 brightness temperature observations, hurricane Lili case)
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Inadequate Pf (ensemble members far from the truth):
DOF=49.39, end ineffective use of the observations (the analysis is close to the background).
T_brightness, Background T_brightness, Analysis
T_brightness, Observations
Is this applicable to CSU/RAMS? (Nstate=2138400; Nobs=5940, Nens=50,
assimilation of simulated GOES-R 10.35 brightness temperature observations, hurricane Lili case)
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Adequate Pf (ensemble members close to the truth):
DOF=14.73, and effective use of the observations (the analysis is close to the truth).
T_brightness, Background T_brightness, Analysis
T_brightness, Observations
Conclusions and Future Research Directions
Conclusions Flow-dependent forecast error covariance is of fundamental importance for both analysis and information measures.
Ensemble-based data assimilation methods employ flow-dependent forecast error covariance.
Information matrix defined in ensemble subspace is practical to calculate in many applications due to small ensemble size.
Future work
Evaluate DOF in the presence of model error.
Apply the information content analysis to WRF model and real satellite observations.
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Thank you.
Is the increased amount of information a simple consequence of a large magnitude of Pf?
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Trace P f (T -component)
-400
0
400
800
1200
1600
2000
2400
1 11 21 31 41 51 61
Cycles
Trace (K
2)
KF1_40obsKF2_40obs3dv1_40obs3dv2_40obs
Trace P f (q -component)
0.00E+00
2.00E-04
4.00E-04
6.00E-04
8.00E-04
1.00E-03
1.20E-03
1 11 21 31 41 51 61
Cycles
Trace (kg
2/kg
2)
KF1_40obsKF2_40obs3dv1_40obs3dv2_40obs
Large Pf
Large Pf
Inadequate Pf
The GEOS-5 results indicated the following impact of Pf
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Inadequate Pf Increased information content of data, but poor analysis quality (ineffective use of observed information)
Adequate Pf Reduced information content of data, but good analysis quality (effective use of observed information)
Example 1: Fronts
Example 2: Hurricanes
(From Whitaker et al., THORPEX web-page)
Benefits of Flow-Dependent Background Errors