Prepared by Dusanka Zupanski and …… Maximum Likelihood Ensemble Filter: application to carbon...

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


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


TRANSCOM


Ravi, you might want to add more detail about TRANSCOM

Preliminary results using RAMS


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


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.

Prepared by Dusanka Zupanski and …… Maximum Likelihood Ensemble Filter: application to carbon...

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