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Dusanka Zupanski CIRA/Colorado State University
Fort Collins, Colorado
Ensemble Kalman FilterEnsemble Kalman Filter
Guest Lecture at AT 753: Atmospheric Water Cycle 21 April 2006, CSU/ATS Dept., Fort Collins, CO
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Acknowledgements: M. Zupanski, C. Kummerow, S. Denning, and M. Uliasz, CSUA. Hou and S. Zhang, NASA/GMAO
Why Ensemble Data Assimilation?
Kalman filter and Ensemble Kalman filter Maximum likelihood ensemble filter (MLEF)
Examples of MLEF applications
Future research directions
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
OUTLINE
Why Ensemble Data Assimilation?
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Three main reasons :
Need for optimal estimate of the atmospheric state + verifiable uncertainty of this estimate; Need for flow-dependent forecast error covariance matrix; and The above requirements should be applicable to 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
Are there alternatives?
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Two good candidates: 4d-var method: It employs flow-dependent forecast error 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 too expensive for applications to complex atmospheric models.
EnKF is a practical alternative to KF, applicable to most complex atmospheric models.
A bonus benefit: EnKF does not use adjoint models!
Typical EnKF
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Forecast error Covariance Pf
(ensemble subspace)
DATA ASSIMILATION
Observations First guess
Optimal solution for model statex=(T,u,v,f, )
ENSEMBLE FORECASTING
Analysis error Covariance Pa
(ensemble subspace)
INFORMATION CONTENT ANALYSIS
Tb,ub,vb,fb, ,,
Hessian preconditioning
Non-Gaussian PDFs
Maximum Likelihood Ensemble Filter
x
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Data Assimilation Equations
xn =Mn,n−1(xn−1) +G(xn−1)wn−1
Equations in model space:
Pf =E x−E x( )⎡⎣ ⎤⎦ x−E x( )⎡⎣ ⎤⎦T
{ }
Prior (forecast) error covariance of x (assumed known):
- Dynamical model for model state evolution (e.g., NWP model)M
- Model state vector of dim Nstate ; w - Model error vector of dim Nstate
G - Dynamical model for state dependent model error
Model error covariance (assumed known):
Q =E wwT( )
E - Mathematical expectation;
GOAL: Combine Model and Data to obtain optimal estimate of dynamical state x
n - Time step index
y - Observations vector of dim Nobs ;
yn =Hn(xn) + εn
Observation error covariance, includes also representatives error (assumed known):
H
ε
R =E εε T( )
- Observation operator
Equations in data space:
- Observation error
Data Assimilation Equations
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
n - Time step index (denoting observation times)
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 ,,.
Approach 1:Approach 1: Optimal solution (e.g., analysis xa) = Minimum variance estimate, or conditional mean of Bayesian posterior probability density function (PDF) (e.g., Kalman filterKalman filter; Extended Kalman filterExtended Kalman filter; EnKFEnKF)
xa =E x y( ) = xp(x y)∫ dx= xp(y x)p(x)
p(y)∫ dx p - PDF
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
How can we obtain optimal solution?Two approaches are used most often:
For non-liner M or H the solution can be obtained employing Extended Kalman filterExtended Kalman filter, or Ensemble Kalman filterEnsemble Kalman filter.
Assuming liner M and H and independent Gaussin PDFs Kalman filterKalman filter solution (e.g., Jazwinski 1970)
xa is defined as mathematical expectation (i.e., mean) of the conditional posterior p(x|y), given observations y and prior p(x).
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Approach 2:Approach 2: Optimal solution (e.g., analysis xa) = Maximum likelihood estimate, or conditional mode of Bayesian posterior p(x|y)(e.g., variationalvariational methods; MLEFMLEF)
J =12[x−xb]
T Pf-1[x−xb] +
12[H[M (x)] −y]T R−1[H[M (x)] −y]
For independent Gaussian PDFs, this is equivalent to minimizing cost function J:
xk+1 =xk −H −1∇J k
Solution can be obtained (with ideal preconditioning) in one iteration for liner H and M. Iterative solution for non-linear H and M:
H−1
- Preconditioning matrix = inverse Hessian of J
xa= Maximum of posterior p(x|y), given observations and prior p(x).
xa =max p x y( )⎡⎣ ⎤⎦=maxp(y x)p(x)
p(y)=min −log p x y( )⎡⎣ ⎤⎦{ }
J=const.
ζ0
ζmin
x0
xmin
J=const.
Physical space (x)
Preconditioning space (ζ)
-gζ
-gx
IMPACT OF MATRIX CIN HESSIAN PRECONDITIONING
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1 6 11 16 21 26 31 36 41 46 51
Number of iterations
Cost function
VARIATIONAL
MLEF
H -1 =∂2 J∂x2
⎛
⎝⎜⎞
⎠⎟
-1
=Pf1/2 (I +C)−1Pf
T /2
PMLEF-1 =Pf
1/2 (I +C)−1PfT /2
xk+1 =xk −P−1gk
PVAR-1 =Pf
Milija Zupanski, CIRA/CSUZupanskiM@CIRA.colostate.edu
Ideal Hessian Preconditioning
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
xmode xmean
x
p(x)
Non-Gaussian
xmode = xmean
x
p(x)
Gaussian
MEAN vs. MODE
For Gaussian PDFs and linear H and M results of all methods [KF, EnKF (with enough ensemble members), and variational] should be identical, assuming the same Pf , R, and y are used in all methods.
Minimum variance estimate= Maximum likelihood estimate!
KF,EnKF,4d-var,
all created equal?
Does this really happen?!?
TEST RESULTS EMPLOYING A LINEAR MODEL AND GAUSSIAN PDFs
(M.Uliasz)
(D. Zupanski)
xa
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
- Optimal estimate of x (analysis)
Kalman filter solutionKalman filter solution
xa =xb + Pf H
T (HPf HT + R)−1 y − H (xb )[ ]
Analysis step:
xb - Background (prior) estimate of x
Pa =[I − Pf HT (HPf HT + R)−1 H ]Pf = I − KH( ) Pf
Pa - Analysis (posterior) error covariance matrix (Nstate x Nstate)
Forecast step:
x0 =xa ;
Pf =MPaM
T + GQGT - Update of forecast error covariance
K - Kalman gain matrix (Nstate x Nobs)
xn =Mn,n−1(xn−1) +G(xn−1)wn−1 Often neglected
Ensemble Kalman Filter (EnKF) solutionEnKF as first introduced by Evensen (1994) as a Monte Carlo filter.
Analysis solution defined for each ensemble member i:
xa
i =xbi + Pf
eH T (HPfe HT + Re )−1( y i − H (xb
i ))
Mean analysis solution:
xa =xb + Pf
eH T (HPfe HT + Re )−1( y − H (xb ))
Analysis error covariance in ensemble subspace:
Analysis step:
bia = xa
i - xa
Analysis ensemble perturbations:
=
p1,1a p1,2
a . p1,Nensa
p2,1a p2,2
a . p2,Nensa
p3,1a p3,2
a . p3,Nensa
. . . .
pNstate,1f pNstate,2
f . pNstate,Nensf
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
= b1a b2
a . bNensa⎡⎣ ⎤⎦
Pae( )
1 2
Pa
e =1
Nens -1Pa
e( )1 2
Pae( )
1 2⎡⎣
⎤⎦
T
Sample analysis covariance
Equations given here following Evensen (2003)
Ensemble Kalman Filter (EnKF)
Forecast step:
xnj =Mn,n−1(xn−1
j ) +G(xn−1j )wn−1
j
Forecast error covariance calculated using ensemble perturbations:
Ensemble forecasts employing a non-linear model M
=
p1,1f p1,2
f . p1,Nensf
p2,1f p2,2
f . p2,Nensf
p3,1f p3,2
f . p3,Nensf
. . . .
pNstate,1f pNstate,2
f . pNstate,Nensf
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
= b1f b2
f . bNensf⎡⎣ ⎤⎦
bif =M (xa
i ) −M (xa)
Pf
e( )1 2
Pf
e =1
Nens -1Pf
e( )1 2
Pfe( )
1 2⎡⎣
⎤⎦
T
;
Sample forecast covarianceNon-linear forecast perturbations
There are many different versions of EnKF
Monte Carlo EnKF (Evensen 1994; 2003)
EnKF (Houtekamer et al. 1995; 2005; First operational version)
Hybrid EnKF (Hamill and Snyder 2000) EAKF (Anderson 2001)
ETKF (Bishop et al. 2001)
EnSRF (Whitaker and Hamill 2002)
LEKF (Ott et al. 2004)
MLEF (Zupanski 2005; Zupanski and Zupanski 2006)
Minimum variance solution
Maximum likelihood solution
Why maximum likelihood solution? It is more adequate for employing non-Gaussian PDFs (e.g., Fletcher and Zupanski 2006).
Current status of EnKF applications
EnKF is operational in Canada, since January 2005 (Houtekamer et al.). Results comparable to 4d-var.
EnKF is better than 3d-var (experiments with NCEP T62 GFS) - Whitaker et al., THORPEX presentation ).
Very encouraging results of EnKF in application to non-hydrostatic, cloud resolving models (Zhang et al., Xue et al.).
Very encouraging results of EnKF for ocean (Evensen et al.), climate (Anderson et al.), and soil hydrology models (Reichle et al.).
Theoretical advantages of ensemble-based DA methods are getting confirmed in an increasing number of practical applications.
Examples of MLEF applications
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
- Dynamical model for standard model state xxn =Mn,n−1(xn−1,bn−1,n−1)
Maximum Likelihood Ensemble Filter
bn =Gn,n−1(bn−1) - Dynamical model for model error (bias) b
n = Sn,n−1(γ n−1) - Dynamical model for empirical parameters
zn =Fn,n−1(zn−1)
Define augmented state vector z
Find optimal solution (augmented analysis) za by minimizing J
(MLEF method):
J =12[z−zb]
T Pf-1[z−zb] +
12[H[F(z)] −yobs]
T R−1[H[F(z)] −yobs] =min
zn =(xn−1,bn−1,n−1)
And augmented dynamical model F
,
.
(Zupanski 2005; Zupanski and Zupanski 2006)
Both the magnitude and the spatial patterns of the true bias are successfully captured by the MLEF.
40 Ens
100 Ens
True R
Cycle 1 Cycle 3 Cycle 7
Bias estimation: Respiration bias R, using LPDM carbon transport model (Nstate=1800, Nobs=1200, DA interv=10 days)
Domain with larger bias (typically land)
Domain with smaller bias (typically ocean)
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 MWR)
λi2
- eigenvalues of C
for linear H and M
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
GEOS-5 Single Column Model: DOF for signal(Nstate=80; Nobs=80, 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 approach does not capture this variability.
DOF for signal (ds), 80
0
5
10
15
20
25
30
35
40
45
1 11 21 31 41 51 61
Cycles
ds
ds_10ensds_20ensds_40ensds_80ensds_80ens_KFds_80ens_3dv
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Small ensemble size (10 ens), even though not perfect, captures main data signals.
T obs (K) q obs (g kg-1)
Data assimilation cycles
Ver
tical
leve
ls
RMS Analysis errors for T, q:------------------------------------ 10ens ~ 0.45K; 0.377g/kg 20ens ~ 0.28K; 0.265g/kg40ens ~ 0.23K; 0.226g/kg80ens ~ 0.21K; 0.204g/kg-------------------------------------No_obs ~ 0.82K; 0.656g/kg
ds =λi
2
(1+ λi2 )i
∑
Non-Gaussian (lognormal) MLEF framework: CSU SWM Non-Gaussian (lognormal) MLEF framework: CSU SWM (Randall et al.)(Randall et al.)
( ) ( )i
N
iS
T
ff
Tfobs y
my
my
xJ ∑=
−−⎥⎦
⎤⎢⎣
⎡+⎟⎟⎠
⎞⎜⎜⎝
⎛−⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎥
⎦
⎤⎢⎣
⎡+−−=
1
11
)(ln
)(ln
)(ln
2
1
2
1)(
xxR
xxxPxx
HHH
Beneficial impact of correct PDF assumption – practical advantagesDusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Cost function derived from posterior PDF ( x-Gaussian, y-lognormal):
Lognormaladditional nonlinear term
Normal(Gaussian)
Courtesy of M. Zupanski
Height RMS Error(SW model, 520 ensembles, Height obs 3_1)
0
2
4
6
8
10
12
14
16
1 6 11 16 21 26 31 36
Cycles
Height Error(m)
GaussianMLEF
LognormalMLEF
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
ba refxx
x exp)(H
Future Research Directions
Covariance inflation and localization need further investigations: Are these techniques necessary?
Model error and parameter estimation need further attention: Do we have sufficient information in the observations to estimate complex model errors?
Information content analysis might shed some light on DOF of model error and also on the necessary ensemble size.
Non-Gaussian PDFs have to be included into DA (especially for cloud variables).
Characterize error covariances for cloud variables.
Account for representativeness error.
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
References for further readingReferences for further reading
Anderson, J. L., 2001: An ensemble adjustment filter for data assimilation. Mon. Wea. Rev., 129, 2884–2903.Fletcher, S.J., and M. Zupanski, 2006: A data assimilation method for lognormally distributed observational
errors. Q. J. Roy. Meteor. Soc. (in press).Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo
methods to forecast error statistics. J. Geophys. Res., 99, (C5),. 10143-10162.Evensen, G., 2003: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean
Dynamics. 53, 343-367.Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter/3D-variational analysis scheme. Mon. Wea. Rev., 128, 2905–2919.Houtekamer, Peter L., Herschel L. Mitchell, 1998: Data Assimilation Using an Ensemble Kalman Filter
Technique. Mon. Wea. Rev., 126, 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. Mon. Wea. Rev., 133, 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: Ensemble square root
filters. Mon. Wea. Rev., 131, 1485–1490.Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea.
Rev., 130, 1913–1924.Zupanski D. and M. Zupanski, 2006: Model error estimation employing an ensemble data assimilation approach. Mon. Wea. Rev. 134, 1337-1354.Zupanski, M., 2005: Maximum likelihood ensemble filter: Theoretical aspects. Mon. Wea. Rev., 133, 1710–1726
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Thank you.
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