Prepared by
Dusanka Zupanski and ……
Maximum Likelihood Ensemble Filter:application to carbon problems
Maximum Likelihood Ensemble Filter (MLEF)(Zupanski 2005; Zupanski and Zupanski 2005)
Developed using ideas from:Variational data assimilation (3DVAR, 4DVAR)Iterated Kalman FiltersEnsemble Transform Kalman Filter (ETKF, Bishop et al. 2001)
Characteristics of the MLEF
Calculates optimal estimates of:- model state variables (e.g., carbon fluxes, sources, sinks)- empirical parameters (e.g., light response, allocation, drought stress)- model error (bias)- boundary conditions error (lateral, top, bottom boundaries)
Calculates uncertainty of all estimates
Fully non-linear approach. Adjoint models are not needed.
Provides more information about PDF (higher order moments could be calculated from ensemble perturbations)
Non-derivative minimization (first variation instead of first derivative is used).Dusanka Zupanski, CIRA/[email protected]
min]([]([21][][
21 11
obsT
obsb-f
Tb HHJ yxRyxxxxx ))P
MLEF APPROACH
ζCIPfb2121 )( xx
Change of variable (preconditioning)
x
- control vector in ensemble space of dim Nens
Minimize cost function J
ζ
- model state vector of dim Nstate >>Nens
)()( 212121212112
ffff HPRHPRHPRHPC TTT
Dusanka Zupanski, CIRA/[email protected]
)()( 21212121 xHRbxHRHPR fif
C - information matrix of dim Nens Nens
MLEF APPROACH (continued)
21-)(2121 CIPP fa
Analysis error covariance
aNens
aa
aNensNstate
aNstate
aNstate
aNens
aa
aNens
aa
aNens
aa
bbb
ppp
ppppppppp
.
.....
.
.
.
21
,2,1,
,32,31,3
,22,21,2
,12,11,1
Dusanka Zupanski, CIRA/[email protected]
Forecast error covariance
fNens
ff
fNensNstate
fNstate
fNstate
fNens
ff
fNens
ff
fNens
ff
bbb
ppp
ppppppppp
.
.....
.
.
.
21
,2,1,
,32,31,3
,22,21,2
,12,11,1
21
fP
)()( xMbxMb ai
fi
Forecast model M essential for propagating in time (updating) columns of Pf.
J=const.
0
min
x0
xmin
J=const.
Physical space (x)
Preconditioning space ()
-g
-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
Cos
t fun
ctio
n
Ideal Hessian Preconditioning
VARIATIONAL
MLEF
2/12/11
21 )( T
ff
-- J PAIPH
x
2
2/12/11 )( Tff
-MLEF PAIP P
kkk gP 11
xx
f-
VAR P1P
Milija Zupanski, CIRA/[email protected]
STATE AUGMENTATION APPROACH as a part of the MLEF
Example: parameter estimation
11,1
1, 0
0
nnn
n
1-nnn
n
nn F
Mz
xI
xz
- augmented state variablenz
1, nnF - augmented forecast model
Assumption: parameter remains constant, or changes slowly with time
1nn
SAME FRAMEWORK IS USED FOR MODEL BIAS ESTIMATION(use bias instead of a parameter to augment state variable)
Parameters are randomly perturbed only in the first cycle. In later cycles, the MLEF updates ensemble perturbations.
TRANSCOM- ….-….(Ravi, perhaps you can include a couple of bullets for Transcom)
SiB Parameter estimation- Estimate control parameters on the fluxes- MLEF calculates uncertainties of all parameters (in terms of Pa and Pf)
LPDM- Estimate monthly mean carbon fluxes, empirical parameters- Estimate uncertainties of the mean fluxes and empirical parameters
SiB-CASA-RAMS- Use various observations of weather, eddy-covariance fluxes, CO2- Estimate carbon fluxes, empirical parameters (e.g., light response,
allocation, drought stress, phonological triggers)- Time evolution of state variables, provided by the coupled model, is critical for updating Pf
Applications of the MLEF to carbon studies
Dusanka Zupanski, CIRA/[email protected]
TRANSCOM
Dusanka Zupanski, CIRA/[email protected]
Ravi, you might want to add more detail about TRANSCOM
Preliminary results using RAMS
Dusanka Zupanski, CIRA/[email protected]
RMS analysis error(analysis-truth)
0.00E+00
2.00E-01
4.00E-01
6.00E-01
8.00E-01
1 11 21 31Cycle No.
RM
S (m
/s)
rms_urms_u_noobs
RMS analysis error(analysis-truth)
0.00E+00
2.00E-04
4.00E-04
6.00E-04
8.00E-04
1 11 21 31Cycle No.
RM
S
rms_r_totalrms_r_total_noobs
Hurricane Lili case35 1-h DA cycles: 13UTC 1 Oct 2002 – 00 UTC 3 Oct30x20x21 grid points, 15 km grid distance (in the Gulf of Mexico)Control variable: u,v,w,theta,Exner, r_total (dim=54000)Model simulated observations with random noise (7200 obs per DA cycle)Nens=50Iterative minimization of J (1 iteration only)
RMS errors of the analysis (control experiment without assimilation)
Hurricane entered the model domain. Impact of assimilation more pronounced.
Example: Total humidity mixing ratio, level=200 m, cycle 31
Locations of min and max centers are much improved in the experiment with assimilation.
TRUTH NO ASSIMILATION
ASSIMILATION
SUMMARY
Dusanka Zupanski, CIRA/[email protected]
The MLEF is currently being evaluated in various atmospheric science applications, showing encouraging results.
The MLEF is suitable for assimilation of numerous new carbon observations, employing complex non-linear coupled models.
Work in carbon applications has just started. Results will be presented in the future.
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