Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado Ensemble Data Assimilation...

Dusanka ZupanskiCIRA/Colorado State University

Fort Collins, Colorado

Ensemble Data Assimilation and Prediction:Applications to Environmental Science

Presentation atEWHA Department of Environmental Science and Engineering

May 26, 2005

Dusanka Zupanski, CIRA/[email protected]

Collaborators- M. Zupanski, L. Grasso, Scott Denning Group (Colorado State University)- A. Y. Hou, S. Zhang (NASA/GMAO)- M. DeMaria (NOAA/NESDIS)- Seon Ki Park (EWHA Womans University)

Research supported by NOAA Grant NA17RJ1228, NASA Grants 621-15-45-78, NAG5-12105 and NNG04GI25G, and DoD Grant DAAD19-02-2-0005.

Introduction to ensemble data assimilation methods

Maximum Likelihood Ensemble Filter (MLEF)

State augmentation approach for model error and parameter estimation

MLEF applications

Conclusions and future work


OUTLINE

Introduction: What is Ensemble Data Assimilation?

Forecast error Covariance Pf

(ensemble subspace)

DATA ASSIMILATION

Observations First guess

Optimal solution for model statex=(T,u,v,q, Te,ue,ve,qe, ,,)

ENSEMBLE FORECASTING

Analysis error Covariance Pa

(ensemble subspace)


J =12[x−xb]

T Pf-1[x−xb] +

12[H[M (x)] −yobs]

T R−1[H[M (x)] −yobs] =min

x −xb =Pf1 2 (I +C)−1 2ζ

x

- Control vector in ensemble space of dim Nensζ- Model state vector of dim Nstate >>Nens

C =ZTZ


z i =R−1 2H[M (x+ pfi )] −R−1 2H[M (x)]

- C is information matrix of dim Nens Nens

p fi =M (x+ pa

i )−M (x)

- Dynamical forecast model

- Observation operator

- zi are columns of Z

- pif and pi

a are columns of Pf and Pa

yobs - Observations vector of dim Nobs

Maximum Likelihood Ensemble Filter (MLEF)(Zupanski 2005; Zupanski and Zupanski 2005)

- Change of variable (preconditioning)

xn =Mn,n−1(xn−1)

yn =Hn(xn)

J=const.

ζ0

ζmin

x0

xmin

J=const.

Physical space (x)

Preconditioning space (ζ)

-gζ

-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

Cost function

VARIATIONAL

MLEF

H -1 =∂2 J∂x2

⎛

⎝⎜⎞

⎠⎟

-1

=Pf1/2 (I +C)−1Pf

T /2

PMLEF-1 =Pf

1/2 (I +C)−1PfT /2

xk+1 =xk −P−1gk

PVAR-1 =Pf

Milija Zupanski, CIRA/[email protected]

Ideal Hessian Preconditioning

MODE vs. MEANMODE vs. MEAN


MLEF involves an iterative minimization of functional J xmode

Minimum variance methods (EnKF) calculate ensemble mean xmean

xmode xmean

x

PDF(x)

Non-Gaussian

xmode = xmean

x

PDF(x)

Gaussian

Different results expected for non-Gaussian PDFs

Wide Range of Applications

• Data assimilation

• Ensemble forecasting

• Parameter estimation

• Model error estimation

• Targeted observations

• Information content of observations

• Advanced parallel computing

• Predictability, nonlinear dynamics

• Any phenomenon with predictive model and observations: Weather, Climate, Ocean, Ecology, Biology, Geology, Chemistry, Cosmology, …

DATA ASSIMILATION application:

TRUTH NO ASSIMILATION

Locations of min and max centers are much improved in the experiment with assimilation.

ASSIMILATION

CSU-RAMS non-hydrostatic model: Total humidity mixing ratio (level=200m, Nens=50, Nstate=54000)



- Forecast model for standard (non-augmented) model state xxn =Mn,n−1(xn−1)

MODEL ERROR AND PARAMETER ESTIMASTIONUsing state augmentation approach (Zupanski and Zupanski 2005)

bn =Gn,n−1(bn−1) - Forecast model for model error (bias) b

n = Sn,n−1(γ n−1) - Forecast model for empirical parameters

wn =Fn,n−1(wn−1)

Define augmented state vector w=(x,b, ) and use augmented forecast model F to update w from time step n-1 to n as

Minimize the following cost function J

J =12[w−wb]

T Pf-1[w−wb] +

12[H[F(w)] −yobs]

T R−1[H[F(w)] −yobs] =min

PARAMETER ESTIMATION, KdVB model(Korteweg-de Vries-Burgers model)

ESTIMATION OF DIFFUSION COEFFICIENT ν (102 , 101 )ens obs

2.00 -02E

7.00 -02E

1.20 -01E

1.70 -01E

2.20 -01E

1 11 21 31 41 51 61 71 81 91

.Cycle No

Diffusion coeficient

value

(0.07)estim value (0.07)true value (0.20)estim value

(0.20)true value

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

PDF

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

PDF

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

PDF

True parameter recovered. Improved innovation statistics.

INNOVATION χ2 TEST (biased model)(neglect_err, 10 ens, 10 obs)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

INNOVATION χ2 TEST (biased model)(bias_estim, 10 ens, 10 obs, bias dim = 101)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

INNOVATION χ2 TEST (biased model)(bias_estim, 10 ens, 10 obs, bias dim = 10)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

INNOVATION χ2 TEST (non-biased model)(correct_model, 10 ens, 10 obs)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

NEGLECT BIAS BIAS ESTIMATION (vector size=101)

BIAS ESTIMATION (vector size=10) NON-BIASED MODEL

BIAS ESTIMATION, KdVB model

It is beneficial to reduce degrees of freedom of the model error.

BIAS ESTIMATION, KdVB model

Augmented analysis error covariance matrix is updated in each data assimilation cycle. It includes cross-covariance between the initial conditions (IC) error and model error (ME).


Impact of Lognormal observation errors: Analysis RMS errors, CSU-SWM

Height RMS error(SW model, 500 ens)

0

2

4

6

8

10

12

1 3 5 7 9 11 13 15 17 19

Cycle

RMS (m)

Obs Err

Gauss

Logn

U-wind RMS error(SW model, 500 ens)

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19

Cycle

RMS (m/s)

Obs Err

Gauss

Logn

V-wind RMS error(SW model, 500 ens)

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19

Cycle

RMS (m/s)

Obs Err

Gauss

Logn

Stddev()=1.e-3

Stddev()=1.e-2

Gaussian framework works only for small observation errors

Lognormal framework works for all error magnitudes

Height RMS error(SW model, 500 ens)

0

20

40

60

80

100

1 3 5 7 9 11 13 15 17 19

Cycle

RMS (m)

Obs Err

Gauss

Logn

U-wind RMS error(SW model, 500 ens)

0

2

4

6

8

10

1 3 5 7 9 11 13 15 17 19

Cycle

RMS (m/s)

Obs Err

Gauss

Logn

V-wind RMS error(SW model, 500 ens)

0

2

4

6

8

10

1 3 5 7 9 11 13 15 17 19

Cycle

RMS (m/s)

Obs Err

Gauss

Logn

Milija Zupanski, CIRA/[email protected]

Impact of Lognormal height observation errors: Innovation histogram

INNOVATION HISTOGRAM(SWM - 500 ens, Gauss)

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

Bin Categories

PDF

INNOVATION HISTOGRAM(SWM - 500 ens, Logn)

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

Bin Categories

PDF

Gaussian framework Lognormal framework

Stddev()=1.e-3

Stddev()=1.e-2

Height innovations )(xH−y my

−⎥⎦

⎤⎢⎣

⎡)(

lnxH

Generalized non-Gaussian framework can handle Gaussian, Lognormal, or mixed PDF errors !

INNOVATION HISTOGRAM(SWM - 500 ens, Gauss)

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

Bin Categories

PDF

INNOVATION HISTOGRAM(SWM - 500 ens, Logn)

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

Bin Categories

PDF

INFORMATION CONTENT ANALYSIS GEOS-5 column model

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

ds and

h

dsh

∑ +=+= −

i i

is trd

)1(])([

2

21

λ

λCCI

∑ +=i

ih )1ln(2

1 2λ

Degrees of freedom (DOF) for signal

Shannon information content, or entropy reduction (used for quantifying predictability)

)()( 212121212112

ffff HPRHPRHPRHPC TTT −−− ==Information matrix

New observed information

Inadequate Pf at the beginning ofdata assimilation

Ongoing work

Ensemble data assimilation and control theory (NSF)• Apply MLEF with CSU global shallow-water model, constructed on a twisted icosahedral grid• Non-Gaussian framework • Non-derivative minimization methods

Precipitation assimilation and moist processes (NASA)• Apply MLEF with NASA GEOS-5 column precipitation model• Model errors• Empirical parameter estimation

GOES-R Risk Reduction (NOAA/NESDIS)• Evaluate the impact of GOES-R measurements in applications to severe weather and tropical cyclones • Information content of GOES-R measurements

Ongoing work (continued)

NCEP Global Forecasting System model and real data (NOAA/THORPEX)

• Apply MLEF with GFS and compare with other EnKF algorithms• Compare (and combine) the conditional mean and mode PDF estimates• Develop and test double-resolution MLEF

Carbon Cycle data assimilation (NASA)• Apply MLEF with fully coupled SiB-CASA-RAMS atmospheric-carbon-biomass model• Assimilate carbon concentration globally and locally• Empirical parameter estimation in carbon models

Microscale ensemble data assimilation (CGAR-DoD)• Apply MLEF with Univ. Purdue non-hydrostatic model • Assimilation of real data• Collaboration with the Army Research Laboratory

Future Research Directions

Development of a fully non-Gaussian algorithm• Allow for non-Gaussian state variable components• Generalized algorithm with a list of PDFs

Model bias and parameter estimation• Improve prediction models by learning about its errors and uncertainties• Develop as a probabilistic tool for new model development

Information content analysis• Value added of new observations (e.g., GPM, CloudSat, GOES-R, OCO)

Cross-over the existing scientific boundaries• Apply probabilistic assimilation/prediction to different science disciplines (e.g., atmospheric, oceanic, ecological, hydrological sciences)• General, adaptive, algorithmically simple algorithm opens new possibilities

Anderson, J. L. 2003: A local least squares framework for ensemble filtering. Mon. Wea. Rev., 131, 634-642.

Bishop, B., Etherton, J. and Majmudar, S. J. 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420-436.

Cohn, S.E., 1997: An introduction to estimation theory. J. Meteor. Soc. Japan, 75, 257-288.

Evensen, G. 2003: The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynamics, 53, 343-367.

Fletcher, S. J., and M. Zupanski, 2005: A framework for data assimilation which allows for non-Gaussian errors. Submitted to Proc. Royal Soc. of London A. [ftp://ftp.cira.colostate.edu/milija/papers/Steven_nongauss.pdf]

Hamill, T.M., and C. Snyder, 2000: A hybrid ensemble Kalman filter-3D variational analysis scheme. Mon. Wea. Rev., 128, 2905-2919.

Houtekamer, P.L., H.L. Mitchell, G. Pellerin, M. Buehner, M. Charron, L. Spacek, and B. Hansen, 2005: Atmospheric data assimilation with an Ensemble Kalman Filter: Results with real observations. Mon. Wea. Rev., 133, 604-620.

Jazwinski, A.H., 1970: Stochastic processes and filtering theory. Academic Press, New York, 376 pp.

Ott, E., Hunt, B. R., Szunyogh, I., Zimin, A. V., Kostelich, E. J., Corazza, M., Kalnay, E., Patil, D. J. and Yorke, J. A. 2004: A Local Ensemble Kalman Filter for Atmospheric Data Assimilation. Tellus, 56A, No. 4, 273-277.

Reichle, R. H., McLaughlin, D. B. and Entekhabi, D. 2002a: Hydrologic data assimilation with the Ensemble Kalman Filter. Mon. Wea. Rev., 130, 103-114.

Snyder, C., and Zhang, F. 2003: Assimilation of simulated Doppler radar observations with an ensemble Kalman filter. Mon. Wea. Rev., 131, 1663-1677.

Tippett, M., J.L. Anderson, C.H. Bishop, T.M. Hamill, and J.S. Whitaker, 2003: Ensemble square-root filters. Mon. Wea. Rev., 131, 1485-1490.

Whitaker, J. S., and Hamill, T. M. 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130,1913-1924.

Zupanski, M. 2005: Maximum Likelihood Ensemble Filter: Theoretical Aspects. Mon.Wea.Rev., in print. [ftp://ftp.cira.colostate.edu/milija/papers/MLEF_MWR.pdf]

Zupanski, D. and M. Zupanski, 2005: Model error estimation employing ensemble data assimilation approach. Submitted to Mon. Wea. Rev. [ftp://ftp.cira.colostate.edu/Zupanski/manuscriptss/MLEF_model_err.Feb2005.pdf]

Literature

Review paper

General

Review paper

Review paper

Summary of basic characteristics of the MLEF

Developed using ideas from-Variational data assimilation (3DVAR, 4DVAR)-Iterated Kalman Filters-Ensemble Transform Kalman Filter (ETKF, Bishop et al. 2001)

Calculates optimal estimates of:- model state variables (e.g., air temperature, soil moisture, tracer fluxes)- empirical parameters (e.g., diffusion coefficient, drought stress)- model error (bias)- boundary conditions error (lateral, top, bottom boundaries)

Calculates uncertainty of all estimates (in terms of Pa and Pf) Calculates information measures (e.g., degrees of freedom for signal, entropy reduction)


Fully non-linear approach. Adjoint models are not needed.Non-derivative minimization (finite difference is used instead of first derivative).It can use Gaussan and non-Gaussian errors

DATA ASSIMILATION application: CSU-RAMS atmospheric non-hydrostatic model


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.

RMS (m/s)

rms_u

rms_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.

RMS rms_r_total

rms_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.

Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado Ensemble Data Assimilation...

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