Transcript Ensemble Kalman Filter Methods

Ensemble Kalman Filter Methods
Dusanka Zupanski
CIRA/Colorado State University
Fort Collins, Colorado
NOAA/NESDIS Cooperative Research Program (CoRP)
Third Annual Science Symposium
15-16 August 2006, Hilton Fort Collins, CO
Collaborators:
M. Zupanski, L. Grasso, M. DeMaria, S. Denning, M. Uliasz, R. Lokupityia, C.
Kummerow, G. Carrio, T. Vonder Haar, D. Randall, CSU
A. 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
Dusanka Zupanski, CIRA/CSU
[email protected]
OUTLINE
 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 directions
Dusanka Zupanski, CIRA/CSU
[email protected]
Typical KF
Forecast error
Covariance Pf
(full-rank space)
Observations
First guess
DATA ASSIMILATION
Analysis error
Covariance Pa
(full-rank space)
Optimal solution for model state
x=(T,u,v,w, q, …)
LINEARISED FORECAST MODEL
Dusanka Zupanski, CIRA/CSU
[email protected]
Typical EnKF
Forecast error
Covariance Pf
(reduced-rank ensemble subspace)
Observations
First guess
DATA ASSIMILATION
Analysis error
Covariance Pa
(reduced-rank ensemble subspace)
Optimal solution for model state
x=(T,u,v,w, q, …)
NON-LINEAR ENSEMBLE OF FORECAST MODELS
Dusanka Zupanski, CIRA/CSU
[email protected]
Typical variational method
Prescribed Forecast error
Covariance Pf
(full-rank space)
Observations
First guess
DATA ASSIMILATION
Analysis error
Covariance Pa
(full-rank space)
Optimal solution for model state
x=(T,u,v,w, q, …)
NON-LINEAR FORECAST MODEL
Dusanka Zupanski, CIRA/CSU
[email protected]
Maximum Likelihood Ensemble Filter (MLEF)
(Zupanski 2005; Zupanski and Zupanski 2006)
 Linear full-rank MLEF = KF (Full-rank means Nens=Nstate)
x  xb +  Pf HT (HPfT HT  R)1[ y  H(xb )] ; for =1
MLEF= KF valid under Gaussian error assumption.
For Non-Gaussian case, ask M. Zupanski, S. Fletcher and
collaborators.
 Non-linear full-rank MLEF, without updating of Pf = 3d-var
1
1
J(x)  [x  xb ]T Pf 1[x  xb ]  [ y  H (x)]T R 1[ y  H (x)]
2
2

Comparisons of KF and 3d-var within the same algorithm.
Dusanka Zupanski, CIRA/CSU
[email protected]
Information measures in ensemble subspace
(Bishop et al. 2001; Wei et al. 2005; Zupanski et al. 2006, subm. to JAS)
C  ZT Z
C
- information matrix in ensemble subspace of dim Nens x Nens
for linear H and M
i
zi  R1 2 H[M(x  pai )]  R1 2 H[M(x)]  R1 2 HPf 1 2 z - are columns of Z
x  xb  Pf1 2 (I  C)1 2

x
- control vector in ensemble space of dim Nens
- model state vector of dim Nstate >>Nens
Degrees of freedom (DOF) for signal (Rodgers 2000):
i2
d s  tr [( I  C ) C ]  
2
i (1  i )
1
i2
- eigenvalues of C
Shannon information content,
or entropy reduction
h
1
ln(1  i2 )

2 i
Errors are assumed Gaussian in these measures.
Dusanka Zupanski, CIRA/CSU
[email protected]
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/CSU
[email protected]
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)
Inadequate Pf
Large Pf
DOF for signal (ds), 40 obs
30
25
ds_10ens
ds_80ens_KF
ds_80ens_3dv
ds
20
15
10
5
0
1
11
21
31
41
51
61
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).
Data assimilation cycles
q true (g kg-1)
Small ensemble
size (10 ens),
even though not
perfect, captures
main data
signals.
Vertical levels
T true (K)
Data assimilation cycles
Dusanka Zupanski, CIRA/CSU
[email protected]
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)
Inadequate Pf (ensemble members far from the truth):
T_brightness, Background
T_brightness, Observations
T_brightness, Analysis
DOF=49.39, end
ineffective use of the
observations (the
analysis is close to the
background).
Dusanka Zupanski, CIRA/CSU
[email protected]
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)
Adequate Pf (ensemble members close to the truth):
T_brightness, Background
T_brightness, Observations
T_brightness, Analysis
DOF=14.73, and effective
use of the observations
(the analysis is close to
the truth).
Dusanka Zupanski, CIRA/CSU
[email protected]
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/CSU
[email protected]
Thank you.
Dusanka Zupanski, CIRA/CSU
[email protected]
Is the increased amount of information a simple consequence
of a large magnitude of Pf?
Trace P f (T -component)
2400
KF1_40obs
KF2_40obs
3dv1_40obs
3dv2_40obs
1600
2
Trace (K )
2000
Large Pf
1200
800
400
0
-400
1
11
21
31
41
51
61
Cycles
Inadequate Pf
Trace P f (q -component)
1.20E-03
KF1_40obs
KF2_40obs
3dv1_40obs
3dv2_40obs
Trace (kg2/kg2)
1.00E-03
8.00E-04
Large Pf
6.00E-04
4.00E-04
2.00E-04
0.00E+00
1
11
21
31
Cycles
41
51
61
Dusanka Zupanski, CIRA/CSU
[email protected]
The GEOS-5 results indicated the following impact of Pf
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)
Dusanka Zupanski, CIRA/CSU
[email protected]
Benefits of Flow-Dependent Background Errors
(From Whitaker et al., THORPEX web-page)
Example 1: Fronts
Example 2: Hurricanes