P1.8 QUANTIFYING AND REDUCING UNCERTAINTY BY EMPLOYING MODEL ERROR ESTIMATION METHODS Dusanka...
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P1.8 QUANTIFYING AND REDUCING UNCERTAINTY BY EMPLOYING MODEL ERROR ESTIMATION METHODS Dusanka Zupanski Cooperative Institute for Research in the Atmosphere Colorado State University, Fort Collins, CO Introduction Data assimilation methods can be effectively used to estimate errors in dynamical models of the atmosphere and ocean. Variational methods have been successfully used for model error estimation in recent years. New generation data assimilation techniques, referred as Ensemble Kalman fileter-like methods, are also capable of estimating and correcting model errors of dynamical models in terms of biases and empirical parameters. These new methods do not require prescribed background error covariances, since the background error covariances are calculated via ensemble forecasting. Thus, ensemble based data assimilation offers a potential for quantifying and reducing uncertainty of our knowledge about atmospheric and oceanic processes. Methodology Maximum Likelihood Ensemble Filter (MLEF, Zupanski 2005; Zupanski and Zupanski 2005) Developed using ideas from Variational data assimilation (3DVAR, 4DVAR) Iterated Kalman Filters Ensemble Transform Kalman Filter (ETKF, Bishop et al. 2001) Parameter estimation KdVB model min ] ( [ ] ( [ 2 1 ] [ ] [ 2 1 1 1 obs T obs b - f T b H H J y x R y x x x x x ) ) P Minimize cost function J ζ C I P f b 2 1 2 1 ) ( x x ) ( ) ( 2 1 2 1 2 1 2 1 2 1 1 2 f f f f HP R HP R HP R H P C T T T x ζ Change of variable - augmented control variable of dim Nstate >>Nens (includes initial conditions, model error, empirical parameters) - control variable in ensemble space of dim Nens a Nens a a f a b b b C I P P . ) ( 2 1 2 1 - 2 1 2 1 ) ( ) ( 2 1 2 1 2 1 2 1 x H R b x H R HP R f i f Analysis error covariance ) ( ) ( x M b x M b a i f i f Nens f f f b b b P . 2 1 2 1 Forecast error covariance Experiments Estimate empirical parameters and model errors of a one-dimensional Korteweg-de Vries-Burgers (KdVB) model Estimate model errors of a 3- dimensional non-hydrostatic model (CSU-RAMS) Calculate uncertainties of the estimated model errors in terms of analysis error covariance matrix P a “True” diffusion parameters are successfully estimated. Correct innovation statistic (expected to be Gaussian for this experiment) indicates that the estimated P a is reliable. Cross-covariance between model state and model error is complex for both models. It would be hard to correctly prescribe this cross- covariance, even for simple dynamical models, such as KdVB. Ensemble based approaches, such as MLEF, can produce complex time evolving covariances for both model state and model error. Acknowledgements This research was partially funded by DoD grant DAAD19- 02-2-0005 and NOAA grant NA17RJ1228 . References Bishop, C. H., B. J. Etherton, and S. Majumjar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part 1: Theoretical aspects. Mon. Wea. Rev., 129, 420–436. Zupanski, D., and M. Zupanski, 2005: Model error estimation employing ensemble data assimilation approach. Submitted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/Zupanski/manuscripts/MLEF_mo del_err.revised2.pdf] Zupanski, M., 2005: Maximum likelihood ensemble filter: Theoretical aspects. Accepted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/milija/papers/MLEF_MWR.pdf]. M - a non-linear dynamical model ES T IM AT IO N O F D IFFU SIO N C O EFFICIEN T (102 ens,101 obs) 2.00E -02 4.00E -02 6.00E -02 8.00E -02 1.00E -01 1.20E -01 1.40E -01 1.60E -01 1.80E -01 2.00E -01 2.20E -01 2.40E -01 2.60E -01 1 11 21 31 41 51 61 71 81 91 C ycle N o. D iffusion coeficientvalue estim value (0.07) true value (0.07) estim value (0.20) true value (0.20) Inn ovatio n h isto gram (inco rrectdiffu sio n ) (neg lect_err, 10 ens,101 obs) 0.00E +00 1.00E -01 2.00E -01 3.00E -01 4.00E -01 5.00E -01 -5 -4 -3 -2 -1 0 1 2 3 4 5 C ategory bins PDF In novation histog ram (incorrectd iffusio n) (param _estim , 10 ens,101 obs) 0.00E +00 1.00E -01 2.00E -01 3.00E -01 4.00E -01 5.00E -01 -5 -4 -3 -2 -1 0 1 2 3 4 5 C ategory bins PDF Innovation histogram (correctdiffusion) (correct_m odel, 10 ens,101 obs) 0.00E +00 1.00E -01 2.00E -01 3.00E -01 4.00E -01 5.00E -01 -5 -4 -3 -2 -1 0 1 2 3 4 5 C ateg ory bin s PDF Bias estimation of KdVB model Estimated components of the augmented analysis error covariance matrix Bias estimation of CSU-RAMS model Estimated components of the augmented analysis error covariance matrix AMS Ed Lorenz Symposium - experiments with simulated observations - model error created by using erroneous Coriolis force in the model’s equations
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Transcript of P1.8 QUANTIFYING AND REDUCING UNCERTAINTY BY EMPLOYING MODEL ERROR ESTIMATION METHODS Dusanka...
P1.8 QUANTIFYING AND REDUCING UNCERTAINTY BY EMPLOYING MODEL ERROR ESTIMATION METHODS
Dusanka Zupanski
Cooperative Institute for Research in the AtmosphereColorado State University, Fort Collins, CO
Introduction
Data assimilation methods can be effectively used to estimate errors in dynamical models of the atmosphere and ocean. Variational methods have been successfully used for model error estimation in recent years. New generation data assimilation techniques, referred as Ensemble Kalman fileter-like methods, are also capable of estimating and correcting model errors of dynamical models in terms of biases and empirical parameters. These new methods do not require prescribed background error covariances, since the background error covariances are calculated via ensemble forecasting. Thus, ensemble based data assimilation offers a potential for quantifying and reducing uncertainty of our knowledge about atmospheric and oceanic processes.
Methodology
Maximum Likelihood Ensemble Filter (MLEF, Zupanski 2005; Zupanski and Zupanski 2005)
Developed using ideas fromVariational data assimilation (3DVAR, 4DVAR)Iterated Kalman FiltersEnsemble Transform Kalman Filter (ETKF, Bishop et al. 2001)
Parameter estimation KdVB model
min]([]([2
1][][
2
1 11 obs
Tobsb
-f
Tb HHJ yxRyxxxxx ))P
Minimize cost function J
ζCIPfb2121 )( xx
)()( 212121212112
ffff HPRHPRHPRHPC TTT
x
ζ
Change of variable
-augmented control variable of dim Nstate >>Nens
(includes initial conditions, model error, empirical
parameters)
-control variable in ensemble space of dim Nens
aNens
aafa bbbCIPP .)( 21
21-2121
)()( 21212121 xHRbxHRHPR fif
Analysis error covariance
)()( xMbxMb ai
fi
fNens
fff bbbP .21
21
Forecast error covariance
Experiments
Estimate empirical parameters and model errors of a one-dimensional Korteweg-de Vries-Burgers (KdVB) model
Estimate model errors of a 3-dimensional non-hydrostatic model (CSU-RAMS)
Calculate uncertainties of the estimated model errors in terms of analysis error covariance matrix Pa
“True” diffusion parameters are successfully estimated. Correct innovation statistic (expected to be Gaussian for this experiment) indicates that the estimated Pa is reliable.
Cross-covariance between model state and model error is complex for both models. It would be hard to correctly prescribe this cross-covariance, even for simple dynamical models, such as KdVB. Ensemble based approaches, such as MLEF, can produce complex time evolving covariances for both model state and model error.
AcknowledgementsThis research was partially funded by DoD grant DAAD19-02-2-0005 and NOAA grant NA17RJ1228 .
ReferencesBishop, C. H., B. J. Etherton, and S. Majumjar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part 1: Theoretical aspects. Mon. Wea. Rev., 129, 420–436.Zupanski, D., and M. Zupanski, 2005: Model error estimation employing ensemble data assimilation approach. Submitted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/Zupanski/manuscripts/MLEF_model_err.revised2.pdf]Zupanski, M., 2005: Maximum likelihood ensemble filter: Theoretical aspects. Accepted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/milija/papers/MLEF_MWR.pdf].
M - a non-linear dynamical model
ESTIMATION OF DIFFUSION COEFFICIENT (102 ens, 101 obs)
2.00E-02
4.00E-02
6.00E-02
8.00E-02
1.00E-01
1.20E-01
1.40E-01
1.60E-01
1.80E-01
2.00E-01
2.20E-01
2.40E-01
2.60E-01
1 11 21 31 41 51 61 71 81 91
Cycle No.
Dif
fusi
on
co
efic
ien
t va
lue
estim value (0.07)
true value (0.07)
estim value (0.20)
true value (0.20)
Innovation histogram (incorrect diffusion)(neglect_err, 10 ens, 101 obs)
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
-5 -4 -3 -2 -1 0 1 2 3 4 5
Category bins
PD
F
Innovation histogram (incorrect diffusion)(param_estim, 10 ens, 101 obs)
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
-5 -4 -3 -2 -1 0 1 2 3 4 5
Category bins
PD
F
Innovation histogram (correct diffusion)(correct_model, 10 ens, 101 obs)
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
-5 -4 -3 -2 -1 0 1 2 3 4 5
Category bins
PD
F
Bias estimation of KdVB model
Estimated components of the augmented analysis error covariance matrix
Bias estimation of CSU-RAMS model
Estimated components of the augmented analysis error covariance matrix
AMS Ed Lorenz Symposium
- experiments with simulated observations- model error created by using erroneous Coriolis force in the model’s equations
