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