# # # # An Application of Maximum Likelihood Ensemble Filter (MLEF) to Carbon Problems Ravindra...
1
# # # # An Application of Maximum Likelihood Ensemble Filter (MLEF) to Carbon Problems Ravindra Lokupitiya 1 , Scott Denning 1 , Dusanka Zupanski 2 , Kevin Gurney 3 , and Milija Zupanski 2 1 Department of Atmospheric Science, 2 Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO 3 Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN Estimation of surface-atmosphere exchanges from observations of CO 2 by inversion of atmospheric transport requires a transport model and an optimization scheme. Batch Bayesian synthesis inversion is efficient for sparse observations, and provides self-consistent estimates of uncertainty, but fails when the number of observations becomes very large (e.g., for hourly tower data or daily satellite retrievals). Variational techniques (e.g., 4d-var) are computationally efficient for very large data sets, but require a model adjoint and uncertainty estimation is problematic. Ensemble data assimilation (EnsDA) can also handle extremely large data sets, requires no adjoint, and estimates error covariance, but may be more computationally expensive than 4d- var. We have applied a new EnsDA method, the Maximum Likelihood Ensemble Filter (MLEF) to the source estimation problem. We show results of two experiments using synthetic data: (1) a very large problem, with observations defined in every model grid cell, and (2) a more realistic problem, with a network of 85 weekly observations. Results are encouraging, though the under-determined problem requires strong covariance localization and smoothing. Introduction Methods ledgements: We would like to thank Wouter Peters for his valuable comments and David Baker for providing the low resolution PCTM model. This research is supported by grants from National Aeronautics and Space Administration (NNG0GD15G & NNG0541G) Experiment 1: Even-constrained problem -- one observation in every surface grid cell A methodological test, to evaluate model performance and test our ability to invert a system with a very large observation vectors. Imagine a satellite which could observe PBL CO2 as a weekly mean everywhere. MLEF-PCTM solves this very large problem just as fast as the under-constrained problem. Summary and Conclusions •The MLEF-PCTM can efficiently process very large observation vectors, and is thus suited for flux estimation using continuous-sampling towers and global satellite retrievals. •Severely under-constrained nature of the flux estimation problem with the current observing system requires aggressive covariance localization and smoothing to obtain reasonable results •In the under-constrained problem, ocean fluxes and under-represented land regions are poorly recovered References: Peters, W., J.B. Miller, J. Whitaker, A.S. Denning, A. Hirsch, M.C. Krol, D. Zupanski, L. Bruhwiler, and P.P. Tans, 2005: An ensemble data assimilation system to estimate CO2 surface fluxes from atmospheric trace gas observations. J. Geophys. Res, 110 (D24) Zupanski, M.,2005:Maximum likelihood ensemble filter: Theoretical aspects, Monthly Weather Review, 133, 1710-1726 Experiment 2: Current weekly flask sampling network Number of unknowns much greater than number of observation sites. • We reduced the degrees of freedom by considering solving for fluxes map of much lower spatial resolution (10° × 6°). • An eight-week moving window was considered and assimilation of observations was done at the end of the window. • Ensembles created using covariance localization based on Prior / Posterior 60 th Percentile of the sigma ratio distribution • Covariance smoothing: decorrelation length scale of 900 km on land and 2000 km over oceans Flask Sites Optimizati on Observed CO 2 concentrations Modeled CO 2 concentrations Global transport model (PCTM) Land fluxes (Bios + FF) Ocean fluxes Real-World Problem M: Offline transport model based on NASA GSFC fvGCM GEOS 4 winds, convective mass fluxes, turbulence (PBL) 2.5° lon x 2° lat x 25 levels, t = 15 min Pseudo-data: • SiB3 fluxes over land • Takahashi ocean fluxes • 4-year integration in PCTM Ensemble data Assimilation: Discrete stochastic-dynamic model Discrete stochastic observation model M: x k = M (x k−1 )+ G (x k−1 ) w k−1 x k-1 – model state w k-1 – model error (stochastic forcing) M – non-linear dynamic (NWP) model G – operator reflecting the state dependence of model error k k k x H y ε + = ) ( : D ε k – measurement + representativeness error H – non-linear observation operator ( M M ) x ζ min ] ( [ ] ( [ 2 1 ] [ ] [ 2 1 1 1 = − − + − − = − ob T ob b - f T b H H J y x R y x x x x x ) ) P x −x b = P f 12 ( I + C ) −1 2 ζ Change of variable - control vector in ensemble space of dim Nens Minimize cost function J - model state vector of dim Nstate >>Nens ) ( ) ( 2 1 2 1 2 1 2 1 2 1 1 2 f f f f HP R HP R HP R H P C T T T − − − = = 2 1 - ) ( 2 1 2 1 C I P P f a + = Analysis error covariance [ ] a Nens a a a Nens Nstate a Nstate a Nstate a Nens a a a Nens a a a Nens a a b b b p p p p p p p p p p p p . . . . . . . . . 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 2 1 f f HP R HP R C T − − = ) ( ) ( 2 1 2 1 2 1 2 1 x H R b x H R HP R f i f − − − − + = ) ( ) ( x M b x M b a i f i − + = Forecast error covariance [ ] f Nens f f f Nens Nstate f Nstate f Nstate f Nens f f f Nens f f f Nens f f b b b p p p p p p p p p p p p . . . . . . . . . 2 1 , 2 , 1 , , 3 2 , 3 1 , 3 , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 2 1 f P N ens = 500 for exp 1 N ens = 200 for exp 2 Covariance Localization Prior estimates: • CASA + Taka + noise • model-data = 1 ppm North A m erica Tem perate -2.50E-08 -2.00E-08 -1.50E-08 -1.00E-08 -5.00E-09 0.00E+00 5.00E-09 1.00E-08 1.50E-08 2.00E-08 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Time(weeks) Analysis True Europe -5.00E-08 -4.00E-08 -3.00E-08 -2.00E-08 -1.00E-08 0.00E+00 1.00E-08 2.00E-08 3.00E-08 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Time(weeks) Analysis True Eurasian Boreal -5.00E-08 -4.00E-08 -3.00E-08 -2.00E-08 -1.00E-08 0.00E+00 1.00E-08 2.00E-08 3.00E-08 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Time(weeks) Analysis True Southern O cean -6.00E-10 -5.00E-10 -4.00E-10 -3.00E-10 -2.00E-10 -1.00E-10 0.00E+00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Tim e(weeks) Analysis True North A m erica Tem perate -4.00E-08 -3.00E-08 -2.00E-08 -1.00E-08 0.00E+00 1.00E-08 2.00E-08 3.00E-08 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Time(weeks) Analysis True Southern O cean -2.00E-09 -1.50E-09 -1.00E-09 -5.00E-10 0.00E+00 5.00E-10 1.00E-09 1.50E-09 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Tim e(weeks) Analysis True Experiment 1: Even-constrained problem -- one observation in every surface grid cell A methodological test, to evaluate model performance and test our ability to invert a system with a very large observation vectors. Imagine a satellite which could observe PBL CO2 as a weekly mean everywhere. MLEF-PCTM solves this very large problem just as fast as the under-constrained problem. Europe -6.00E-08 -5.00E-08 -4.00E-08 -3.00E-08 -2.00E-08 -1.00E-08 0.00E+00 1.00E-08 2.00E-08 3.00E-08 4.00E-08 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Time(weeks) Analysis True Eurasian Boreal -6.00E-08 -5.00E-08 -4.00E-08 -3.00E-08 -2.00E-08 -1.00E-08 0.00E+00 1.00E-08 2.00E-08 3.00E-08 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Time(weeks) Analysis True
-
Upload
ami-french -
Category
Documents
-
view
233 -
download
2
Transcript of # # # # An Application of Maximum Likelihood Ensemble Filter (MLEF) to Carbon Problems Ravindra...
#
#
#
#
An Application of Maximum Likelihood Ensemble Filter (MLEF) to Carbon Problems
Ravindra Lokupitiya1, Scott Denning1 , Dusanka Zupanski2 , Kevin Gurney3, and Milija Zupanski2
1 Department of Atmospheric Science, 2 Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO
3Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN
Estimation of surface-atmosphere exchanges from observations of CO2 by inversion of atmospheric transport requires a transport model and an optimization scheme. Batch Bayesian synthesis inversion is efficient for sparse observations, and provides self-consistent estimates of uncertainty, but fails when the number of observations becomes very large (e.g., for hourly tower data or daily satellite retrievals). Variational techniques (e.g., 4d-var) are computationally efficient for very large data sets, but require a model adjoint and uncertainty estimation is problematic. Ensemble data assimilation (EnsDA) can also handle extremely large data sets, requires no adjoint, and estimates error covariance, but may be more computationally expensive than 4d-var.
We have applied a new EnsDA method, the Maximum Likelihood Ensemble Filter (MLEF) to the source estimation problem. We show results of two experiments using synthetic data: (1) a very large problem, with observations defined in every model grid cell, and (2) a more realistic problem, with a network of 85 weekly observations. Results are encouraging, though the under-determined problem requires strong covariance localization and smoothing.
Introduction
Methods
Acknowledgements: We would like to thank Wouter Peters for his valuable comments and David Baker for providing the low resolution PCTM model. This research is supported by grants from National Aeronautics and Space Administration (NNG0GD15G & NNG0541G)
North America Temperate
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Southern Ocean
-2.00E-09
-1.50E-09
-1.00E-09
-5.00E-10
0.00E+00
5.00E-10
1.00E-09
1.50E-09
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Experiment 1: Even-constrained problem -- one observation in every surface grid cell
A methodological test, to evaluate model performance and test our ability to invert a system with a verylarge observation vectors.
Imagine a satellite which could observe PBL CO2 as a weekly mean everywhere. MLEF-PCTM solves this very large problem just as fast as the under-constrained problem.
Summary and Conclusions
•The MLEF-PCTM can efficiently process very large observation vectors, and is thus suited for flux estimation using continuous-sampling towers and global satellite retrievals.
•Severely under-constrained nature of the flux estimation problem with the current observing system requires aggressive covariance localization and smoothing to obtain reasonable results
•In the under-constrained problem, ocean fluxes and under-represented land regions are poorly recovered
•Current localization is limited to the land points due to the strong land signal. New localization scheme, which separates land
and ocean points is under development.
References:Peters, W., J.B. Miller, J. Whitaker, A.S. Denning, A. Hirsch, M.C. Krol, D. Zupanski, L. Bruhwiler, and P.P. Tans, 2005: An ensemble data assimilation system to estimate CO2 surface fluxes from atmospheric trace gas observations. J. Geophys. Res, 110 (D24)
Zupanski, M.,2005:Maximum likelihood ensemble filter: Theoretical aspects, Monthly Weather Review, 133, 1710-1726
Experiment 2: Current weekly flask sampling network
Number of unknowns much greater than number of observation sites.
• We reduced the degrees of freedom by considering solving for fluxes map of much lower spatial resolution (10° × 6°).
• An eight-week moving window was considered and assimilation of observations was done at the end of the window.
• Ensembles created using covariance localization based on Prior/ Posterior 60th Percentile of the sigma ratio distribution
• Covariance smoothing: decorrelation length scale of 900 km on land and 2000 km over oceans
Flask Sites
Optimization
Observed CO2 concentrations
Modeled CO2 concentrations
Global transport model (PCTM)
Land fluxes (Bios + FF)Ocean fluxes
Real-World Problem
PCTM: • Offline transport model based on NASA GSFC fvGCM• GEOS 4 winds, convective mass fluxes, turbulence (PBL)• 2.5° lon x 2° lat x 25 levels, t = 15 min
Pseudo-data:• SiB3 fluxes over land• Takahashi ocean fluxes• 4-year integration in PCTM
Ensemble data Assimilation:
Discrete stochastic-dynamic model
Discrete stochastic observation model
M: xk =M (xk−1) +G(xk−1)wk−1
x k-1 – model state
w k-1 – model error (stochastic forcing)
M – non-linear dynamic (NWP) model
G – operator reflecting the state dependence
of model error
kkk xHy ε+= )( :D
ε k – measurement + representativeness error
H – non-linear observation operator ( )
x
ζ
min]([]([2
1][][
2
1 11 =−−+−−= −obs
Tobsb
-f
Tb HHJ yxRyxxxxx ))P
x−xb = Pf
1 2 (I + C)−1 2ζ
Change of variable
- control vector in ensemble space of dim Nens
Minimize cost function J
- model state vector of dim Nstate >>Nens
)()( 212121212112
ffff HPRHPRHPRHPC TTT −−− ==
21-)(2121 CIPP fa +=
Analysis error covariance
)()( 21212121
ff HPRHPRC T −−=
)()( 21212121 xHRbxHRHPR fif
−−− −+=
)()( xMbxMb ai
fi −+=
Forecast error covariance
21fP
Nens = 500 for exp 1Nens = 200 for exp 2
Covariance Localization
Prior estimates:• CASA + Taka + noise• model-data = 1 ppm
North America Temperate
-2.50E-08
-2.00E-08
-1.50E-08
-1.00E-08
-5.00E-09
0.00E+00
5.00E-09
1.00E-08
1.50E-08
2.00E-08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Europe
-5.00E-08
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Eurasian Boreal
-5.00E-08
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Southern Ocean
-6.00E-10
-5.00E-10
-4.00E-10
-3.00E-10
-2.00E-10
-1.00E-10
0.00E+00
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/sec)
Analysis
True
North America Temperate
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Southern Ocean
-2.00E-09
-1.50E-09
-1.00E-09
-5.00E-10
0.00E+00
5.00E-10
1.00E-09
1.50E-09
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Experiment 1: Even-constrained problem -- one observation in every surface grid cell
A methodological test, to evaluate model performance and test our ability to invert a system with a verylarge observation vectors.
Imagine a satellite which could observe PBL CO2 as a weekly mean everywhere. MLEF-PCTM solves this very large problem just as fast as the under-constrained problem.
Europe
-6.00E-08
-5.00E-08
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
4.00E-08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
Eurasian Boreal
-6.00E-08
-5.00E-08
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
Time(weeks)
Flux(KgC/m2/Sec)
Analysis
True
