Post on 07-Feb-2016
description
Comparative study on the error covariance matrices
for KF and EnKF using the barotropic S-model
H.L. Tanaka and K. KondoUniversity of Tsukuba
Introduction
KF (Kalman Filter) (Kalman 1960) implementation needs to calculate inverse of a matrix with the dimension of model variables.
We can’t directly implement KF in recent numerical models.
EnKF (Ensemble Kalman Filter) approximates KF(Evensen 1994).
Dimension of the barotropic S-model is low (Tanaka 2003).
We can directly implement EKF (Extended Kalman Filter) in the barotropic S-model.
EKF vs EnKF
Introduction
Methods
)0(,...,3,2,1
mi
fwwriwid
dwikj
jkijkii
i
Barotropic S-ModelBarotropic S-Model
EnKFEnKF Local Ensemble Transform Kalman Filter: L
ETKF (Hunt 2005) Ensemble member: 51
Kalman Filter Ensemble Kalman Filter
ai
fi 1Mxx
QMMPP Ta
ifi 1
1 RHHPHPK Tfi
Tfii
)( 0 fiii
fi
ai HxyKxx
),..,1(1
),..,1( mai
mfi M xx
Tfifi
fi EEP
111 RHEHERHEIEK TT
fii
ai PHKIP
)( 0 fiii
fi
ai xHyKxx
fii
ai PHKIP
Methods EKF and EnKF are implemented in perfe
ct model experiments with the barotropic S-model.
Observation data ＝ Ture data + noise EKF and EnKF assimilated observation d
ata to forecast data in every 6 hours.
Results
Initial 1990/01/01/00z
EKF
EnKF Observational Error
EKF vs EnKF
Pf norm of EnKF Pa norm of EnKF Pf norm of EKF Pa norm of EKF
1 3.5 25
1day(24hr)
Pf of EKF Pa of EKF
260
1day(24hr)
Pf of EnKF Pa of EnKF Pf of EKF Pa of EKF
Pf of EKF Pa of EKF
110
3.5day(84hr)
Pf of EnKF Pa of EnKF Pf of EKF Pa of EKF
3.5day(84hr)
Pf of EKF Pa of EKF
50
25day(600hr)
Pf of EnKF Pa of EnKF Pf of EKF Pa of EKF
25day(600hr)
1day(24hr)
1day(24hr)
3.5day(84hr)
3.5day(84hr)
25day(600hr)
25day(600hr)
Conclusions Performance of EnKF is as good as that of EK
F. The assimilation cycle leads to degenerate di
mensions of the error covariance matrices. Eigenvalue of the error covariance matrices of
EKF is about 50% of that of EnKF. Eigenvector of the error covariance matrices of
EnKF is similar to that of EKF.