dusanka zupanski cira/colorado state university fort collins, colorado a general ensemble-based...
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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
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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
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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)
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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?
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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
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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
10
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1 6 11 16 21 26 31 36 41 46
Analysis cycle
ds
ds_10_ensds_20_ensds_40_ensds_80_ensds_100ens
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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.
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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
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RAMS/LES
Better timing of maxima
LWP is also assimilated
CONTROL
EXP
VERIF
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RAMS/LES
Improved timing and locations of the maxima
NO vertical structure is assimilated (LWP)
CONTROL
VERIF
EXP
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RAMS/LES
Independent observation
IFN above the inversion, as observed
IFN below inversion as cloud forms
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More results will be presented by Carrió at the AGU Fall meeting in San Francisco (Arctic clouds session)
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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
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* *, , , , , , 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
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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
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• 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-)
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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
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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]