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