Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado A General Ensemble-Based Approach to Data Assimilation Model Error and Parameter.

Dusanka ZupanskiCIRA/Colorado State University

Fort Collins, Colorado

A General Ensemble-Based Approach to Data Assimilation Model Error and Parameter

Estimation

LSCE Presentation CEA/Saclay, Gif-sur-Yvette, October 19, 2005

Dusanka Zupanski, CIRA/[email protected]

Outline

State augmentation approach Issues of model error estimation

- Degrees of freedom of model error- Information content of available data

Current research projectsFuture research directions and collaborations

Major sources of forecast uncertainty

• Initial conditions• Model errors (e.g., errors in dynamical equations, errors due to

unresolved scales)• Parametric errors (errors in empirical parameters)• Forcing errors (e.g., errors in atmospheric forcing in

hydrological models)• Boundary conditions (e.g., lateral boundaries)

All sources of uncertainty should be taken into account simultaneously within a unified mathematical approach.

These uncertainties are not independent!

Verified analysis and forecast uncertainty

Dusanka Zupanski, CIRA/[email protected]

- Dynamical model for standard model state xxn =Mn,n−1(xn−1,bn−1,γn−1)

State Augmentation Approach

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

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

wn =Fn,n−1(wn−1)

Define augmented state vector w

Find optimal solution (augmented analysis) wa by minimizing J (MLEF method):

J =12[w−wb]

T Pf-1[w−wb] +

12[H[F(w)] −yobs]

T R−1[H[F(w)] −yobs] =min

wn =(xn−1,bn−1,γn−1)

And augmented dynamical model F

,

.

(Zupanski and Zupanski 2005, MWR)

Issues of Model Error (and Parameter) Estimation

Dusanka Zupanski, CIRA/[email protected]

State augmentation increases the size of the control variable

More Degrees of Freedom (DOF)!

Do we need more ensembles?

Do we need more observations?

What is the number of the effective DOF of an ensemble-based data assimilation and model error estimation system?

Dusanka Zupanski, CIRA/[email protected]

What is the number of the effective DOF?

Ensemble Data Assimilation +

State Augmentation +

Information theory

Answer can be obtained by using the following 3 components within a general framework:

∑ +=+= −

i i

is trd

)1(])([

2

21

λ

λCCI

∑ +=i

ih )1ln(2

1 2λ

Shannon information content, or entropy reduction

Degrees of freedom (DOF) for signal (Rodgers 2000; Zupanski et al. 2005)

C - information matrix in ensemble subspace

Dusanka Zupanski, CIRA/[email protected]

Information Content AnalysisGEOS-5 Single Column Model

Nstate =80; Nobs =80

ds measures effective DOF of an ensemble-based data assimilation system (e.g., MLEF). Useful for addressing DOF of the model error.

DOF for signal (ds)impact of ensemble size

0

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70

1 6 11 16 21 26 31 36 41 46

Analysis cycle

ds

ds_10_ensds_20_ensds_40_ensds_80_ensds_100ens

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.

23 2-h DA cycles: 18UTC 2 May 1998 – 00 UTC 5 May 1998(Mixed phase Arctic boundary layer cloud at Sheba site)

Experiments initialized with typical clean aerosol concentrations

May 4 was abnormal: high IFN and CCN above the inversion

x= 50m, zmax = 30m (2d domain: 50col, 40lev), t=2s, Nens=48

Sophisticated microphysics in RAMS/LES

Control variables: _il, u, v, w, N_x, R_x (8 species), IFN, CCN (dim= 22 variables x 50 columns x 40 levels = 44000)

Radar/lidar observations of IWP, LWP (LWDN, SWDN in future)

Acknowledgements Gustavo Carrió, William Cotton, and Milija Zupanski (CSU)

MLEF experiments with CSU/RAMS Large Eddy MLEF experiments with CSU/RAMS Large Eddy Simulation (LES) modelSimulation (LES) model

RAMS/LES

Better timing of maxima

LWP is also assimilated

CONTROL

EXP

VERIF

RAMS/LES

Improved timing and locations of the maxima

NO vertical structure is assimilated (LWP)

CONTROL

VERIF

EXP

RAMS/LES

Independent observation

IFN above the inversion, as observed

IFN below inversion as cloud forms

More results will be presented by Carrió at the AGU Fall meeting in San Francisco (Arctic clouds session)

SiB-RAMSSiB-RAMS

LPDMLPDM

meteorological fields CO2 fields and fluxes

influencefunctionsinfluencefunctions

inversiontechniquesinversiontechniques

BayesianBayesian MLEFMLEF

CO2 observations

corrected CO2 fluxes

typically run with several nested gridscovering a continental scale

run on any subdomain extracted from SiB-RAMS

MODELING FRAMEWORKMODELING FRAMEWORK

Courtesy of M. Uliasz

corrected within each inversion cycle

* *, , , , , , 0 ,

1, 1,2k R i R k i A i A k i IN k

i n i n n

C C C Cα α α= = +

= + +∑ ∑

C – observed concentration

k – index over observations (sampling times and towers)

i – index over source grid cell (both respiration & assimilation fluxes)

C*R.A – influence function integrated with respiration & assimilation

fluxes

CIN – background concentration combining effect of the flow across lateral boundaries and initial concentration at the cycle start

αR αA – corrections (biases) to be estimated

Implementation for a given inversion cycle Implementation for a given inversion cycle

Courtesy of M. Uliasz

2 ( , , ) ( , , ) ( ,( , ) ( , ,) ))O RC AF x y t R x y tx y A x yx y tα α= +CO2 fluxCO2 flux

respiration & assimilation fluxes simulated by SiB-RAMS

respiration & assimilation fluxes simulated by SiB-RAMS

time independent corrections to beestimated from concentration data

for each inversion cycle

time independent corrections to beestimated from concentration data

for each inversion cycle

ASSUMPTION: SiB-RAMS is capable to realistically reproducediurnal cycle and spatial distribution of CO2 (assimilation and respiration) fluxes. Therefore, observation data are used to correctthose fluxes for errors in atmospheric transport.

ASSUMPTION: SiB-RAMS is capable to realistically reproducediurnal cycle and spatial distribution of CO2 (assimilation and respiration) fluxes. Therefore, observation data are used to correctthose fluxes for errors in atmospheric transport.

Courtesy of M. Uliasz

• SiB-RAMS simulation: 15 days starting on July 19th , 2004 on two nested grids (10 km grid spacing on the finer grid)

• LPDM and influence function domain: 600x600km centered at WLEF tower

• Concentration pseudo-data were generated for WLEF and the ring of towers from SiB-RAMS assimilation and respiration fluxes using correction values of 1.

• Model-data mismatch error was assumed to be higher for lower towers:2 ppm for towers>100m, 3 ppm for towers > 50m, and 5 ppm for towers < 50m

• three 5 day inversion cycles were performed using Bayesian inversion technique with concentration pseudo data (initial corrections = 0.5 and their standard deviations = 0.5)

INVERSION EXPERIMENTSINVERSION EXPERIMENTS

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αR : Reduction of uncertainty (0-)

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.QuickTime™ and a

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Bayesian

MLEF

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MLEF QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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Current Research Projects

Precipitation Assimilation (NASA)• Apply MLEF to NASA GEOS-5 column precipitation model• Address model error and parameter estimation(In collaboration with A. Hou, S. Zhang - NASA/GMAO, C. Kummerow - CSU/Atmos. Sci. Dept.)

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(In collaboration with M. DeMaria - CIRA/NOAA/NESDIS, T. Vonder Haar, L. Grasso, M. Zupanski - CSU/CIRA)

Carbon Cycle Data Assimilation (NASA)• Apply MLEF to various carbon models (LPDM, SiB3, PCTM, and SiB-CASA-RAMS)• Assimilate carbon concentrations globally and locally• Address model error and parameter estimation(In collaboration with S. Denning, M. Uliasz, K. Gurney, L. Prihodko, R. Lokupitiya, I. Baker, K. Schaefer - CSU/Atmos. Sci. Dept., M. Zupanski - CSU/CIRA) Dusanka Zupanski, CIRA/CSU

[email protected]

Future Research Directions and Collaborations

Information content analysis• Quantify value added of new observations (e.g., GPM, CloudSat, GOES-R, OCO)• Determine effective DOF of an ensemble-based data assimilation system

Model bias and parameter estimation requires collaboration• Learning about model errors and uncertainties from different dynamical models• Developing diagnostic tools for new model development

Discuss issues for collaboration with NCAR/MMM • Model errors and parameter estimation for WRF model• Information content analysis - effective DOF of WRF data assimilation system

Dusanka Zupanski, CIRA/[email protected]