1B.17 ASSESSING THE IMPACT OF OBSERVATIONS AND MODEL ERRORS IN THE ENSEMBLE DATA ASSIMILATION...
1
1B.17 ASSESSING THE IMPACT OF OBSERVATIONS AND MODEL ERRORS IN THE ENSEMBLE DATA ASSIMILATION FRAMEWORK D. Zupanski 1 , A. Y. Hou 2 , S. Zhang 2 , M. Zupanski 1 , and C. D. Kummerow 1 1 Colorado State University, Fort Collins, CO 2 NASA Goddard Space Flight Center, Greenbelt, MD Introduction •A general framework that links together information theory and ensemble data assimilation is presented. •Ensemble data assimilation component provides information matrix in ensemble subspace, calculated using a flow- dependent forecast error covariance. •Information theory component provides mathematical formalism for calculation of various measures of information (e.g., degrees of freedom for signal, entropy reduction). •The general framework is examined in application to NASA GEOS-5 column precipitation model. 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) Minimize cost function J 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 Analysis error covariance Forecast error covariance • Degrees of freedom (DOF) for signal d s (Rodgers 2000) Experiments with the NASA/GEOS- 5 column model Single column version of the GOES-5 GCM GOES-5 includes a finite-volume dynamical core and full physics package The model is driven by external data (ARM observations) Model simulated “observations” with random noise 40 level model, two control variables: T and Q 10, 20, or 40 ensemble members 40 or 80 observations of T and Q 50 data assimilation cycles 6-h data assimilation interval One iteration of the minimization Impact of model error is not included in the experiments presented Increasing ensemble size generally reduces analysis errors, except in the initial cycles for Q. Acknowledgements This research is partially funded by NASA grants: 621- 15-45-78, NAG5-12105, and NNG04GI25G. 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. Rodgers, C. D., 2000: Inverse Methods for Atmospheric Sounding: Theory and Practice. World Scientific, 238 pp. Shannon, C. E., and Weaver W., 1949: The Mathematical Theory of Communication. University of Illinois Press, 144 pp. 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.Feb2005.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 ]. • Shannon information content, or entropy reduction h (Shannon and Weaver 1949; Rodgers 2000) Increased information content in the initial 1-5 cycles due to initial adjustments in P f . Increased information content in the final 10 cycles (cycles 40 – 50) due to new observed information. T obs carry more information than Q obs. Eigenvalue spectrum of C provides additional useful information. Future work Evaluate this framework in application to complex 3-d atmospheric and other geophysical models. Estimate information content of various satellite observations. ] ( [ ] ( [ 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 2 1 2 1 ) ( C I P f b x x x a Nens a a a Nens Nstate a Nstate a Nstate a Nens a a a Nens a a a Nens a a p p p p p p p p p p p p b b b . . . . . . . . . 2 1 , 2 , 1 , , 3 2 , 3 1 , 3 , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 2 1 - ) ( 2 1 2 1 C I P P f a f Nens f f f Nens Nstate f Nstate f Nstate f Nens f f f Nens f f f Nens f f p p p p p p p p p p p p b b b . . . . . . . . . 2 1 , 2 , 1 , , 3 2 , 3 1 , 3 , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 2 1 f P Columns b i f are calculated employing a non-linear forecast model M: ) ( ) ( x b x b M M a i f i Information matrix C, of dimension Nens X Nens, is a link between ensemble data assimilation and information theory: Z Z C T ) ( ) ( 2 1 2 1 x R b x R z H H f i i Columns of Z are defined as Measures of information ens N i i i s d 1 1 i -eigenvalues of C ) 1 ( 2 1 1 i N i ln h ens R M S analysis errors for T (im pactofensem ble size) 0.1 0.6 1.1 1.6 2.1 2.6 1 6 11 16 21 26 31 36 41 46 A nalysis cycle RM S T (K ) rms_analysis_40ens rms_analysis_20ens rm s_noassim R M S analysis errors for q (im pactofensem ble size) 1.00E-04 3.00E-04 5.00E-04 7.00E-04 9.00E-04 1.10E-03 1.30E-03 1.50E-03 1 6 11 16 21 26 31 36 41 46 A nalysis cycle RM S q rm s_analysis_40ens rm s_analysis_20ens rm s_noassim Innovation 2 test (im pactofensem ble size) 0 1 2 3 4 5 1 6 11 16 21 26 31 36 41 46 A nalysis cycle 2 10_ens 20_ens 40_ens Innovation statistics is satisfactory for the experiment with 40 ensemble members. D O F for signal( d s )and entropy reduction ( h ) 0 20 40 60 80 1 6 11 16 21 26 31 36 41 46 Analysiscycle d s and h ds h D O F for signal ( d s ) im pactof T and q obs 0 10 20 30 40 1 6 11 16 21 26 31 36 41 46 Analysiscycle d s ds_all_obs ds_T_obs_only ds_q_obs_only D O F for signal ( d s ) im pactofensem ble size 0 10 20 30 40 1 6 11 16 21 26 31 36 41 46 Analysiscycle ds ds_10_ens ds_20_ens ds_40_ens E igenvalues (I+C ) -1/2 0 0.2 0.4 0.6 0.8 1 1 11 21 31 E igenvalue rank m i cycle 1 cycle 2 cycle 5 E igenvalues (I+C ) -1/2 0 0.2 0.4 0.6 0.8 1 1 11 21 31 E igenvalue rank m i cycle 40 cycle 45 cycle 50
-
Upload
ramiro-bute -
Category
Documents
-
view
215 -
download
0
Transcript of 1B.17 ASSESSING THE IMPACT OF OBSERVATIONS AND MODEL ERRORS IN THE ENSEMBLE DATA ASSIMILATION...
1B.17 ASSESSING THE IMPACT OF OBSERVATIONS AND MODEL ERRORS IN THE
ENSEMBLE DATA ASSIMILATION FRAMEWORKD. Zupanski1, A. Y. Hou2, S. Zhang2, M. Zupanski1, and C. D. Kummerow1
1Colorado State University, Fort Collins, CO2NASA Goddard Space Flight Center, Greenbelt, MD
Introduction
•A general framework that links together information theory and ensemble data assimilation is presented.•Ensemble data assimilation component provides information matrix in ensemble subspace, calculated using a flow-dependent forecast error covariance.•Information theory component provides mathematical formalism for calculation of various measures of information (e.g., degrees of freedom for signal, entropy reduction).•The general framework is examined in application to NASA GEOS-5 column precipitation model.
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)
Minimize cost function J
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
Analysis error covariance
Forecast error covariance
• Degrees of freedom (DOF) for signal ds
(Rodgers 2000)
Experiments with the NASA/GEOS-5 column modelSingle column version of the GOES-5 GCMGOES-5 includes a finite-volume dynamical core and full physics package The model is driven by external data (ARM observations)Model simulated “observations” with random noise40 level model, two control variables: T and Q10, 20, or 40 ensemble members40 or 80 observations of T and Q50 data assimilation cycles6-h data assimilation intervalOne iteration of the minimizationImpact of model error is not included in the experiments presented
Increasing ensemble size generally reduces analysis errors, except in the initial cycles for Q.
AcknowledgementsThis research is partially funded by NASA grants: 621-15-45-78, NAG5-12105, and NNG04GI25G.
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.Rodgers, C. D., 2000: Inverse Methods for Atmospheric Sounding: Theory and Practice. World Scientific, 238 pp.Shannon, C. E., and Weaver W., 1949: The Mathematical Theory of Communication. University of Illinois Press, 144 pp. 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.Feb2005.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].
• Shannon information content, or entropy reduction h(Shannon and Weaver 1949; Rodgers 2000)
Increased information content in the initial 1-5 cycles due to initial adjustments in Pf.Increased information content in the final 10 cycles (cycles 40 – 50) due to new observed information.T obs carry more information than Q obs.Eigenvalue spectrum of C provides additional useful information.
Future workEvaluate this framework in application to complex 3-d atmospheric and other geophysical models.Estimate information content of various satellite observations.
]([]([2
1][][
2
1 11obs
Tobsb
-f
Tb HHJ yxRyxxxxx ))P
2121 )( CIPfbxx
x
aNens
aa
aNensNstate
aNstate
aNstate
aNens
aa
aNens
aa
aNens
aa
ppp
ppp
ppp
ppp
bbb .
.
....
.
.
.
21
,2,1,
,32,31,3
,22,21,2
,12,11,1
21-)(2121 CIPP fa
fNens
ff
fNensNstate
fNstate
fNstate
fNens
ff
fNens
ff
fNens
ff
ppp
ppp
ppp
ppp
bbb .
.
....
.
.
.
21
,2,1,
,32,31,3
,22,21,2
,12,11,1
21
fP
Columns bif are calculated employing a non-linear
forecast model M:
)()( xbxb MM ai
fi
Information matrix C, of dimension Nens X Nens, is a link between ensemble data assimilation and information theory:
ZZC T
)()( 2121 xRbxRz HH fii
Columns of Z are defined as
Measures of information
ensN
i i
isd
1 1
i -eigenvalues of C
)1( 2
1
1i
N
i
lnhens
RMS analysis errors for T(impact of ensemble size)
0.1
0.6
1.1
1.6
2.1
2.6
1 6 11 16 21 26 31 36 41 46
Analysis cycle
RM
S T
(K
)
rms_analysis_40ensrms_analysis_20ensrms_noassim
RM S analysis errors for q(impact of ensemble size)
1.00E-04
3.00E-04
5.00E-04
7.00E-04
9.00E-04
1.10E-03
1.30E-03
1.50E-03
1 6 11 16 21 26 31 36 41 46
Analysis cycle
RM
S q
rms_analysis_40ensrms_analysis_20ensrms_noassim
Innovation 2 test (impact of ensemble size)
0
1
2
3
4
5
1 6 11 16 21 26 31 36 41 46
Analysis cycle
2
10_ens20_ens40_ens
Innovation statistics is satisfactory for the experiment with 40 ensemble members.
DOF for signal (d s ) and entropy reduction (h )
0
20
40
60
80
1 6 11 16 21 26 31 36 41 46
Analysis cycle
d s
and
h
dsh
DOF for signal (d s )
impact of T and q obs
0
10
20
30
40
1 6 11 16 21 26 31 36 41 46
Analysis cycle
d s
ds_all_obsds_T_obs_onlyds_q_obs_only
DOF for signal (d s )
impact of ensemble size
0
10
20
30
40
1 6 11 16 21 26 31 36 41 46
Analysis cycle
d s
ds_10_ensds_20_ensds_40_ens
Eigenvalues (I+C)-1/2
0
0.2
0.4
0.6
0.8
1
1 11 21 31
Eigenvalue rank
mi
cycle 1cycle 2cycle 5
Eigenvalues (I+C)-1/2
0
0.2
0.4
0.6
0.8
1
1 11 21 31
Eigenvalue rankm
i
cycle 40cycle 45cycle 50
![J T P-1 H M )] T R 1 H M )] 1 Carbon data assimilation ... · Carbon data assimilation using Maximum Likelihood Ensemble filter (MLEF) Dusanka Zupanski1, A. Scott Denning2, ... Zupanski](https://static.fdocuments.net/doc/165x107/5b4b7f377f8b9a691e8cdb8e/j-t-p-1-h-m-t-r-1-h-m-1-carbon-data-assimilation-carbon-data-assimilation.jpg)
