J T P-1 H M )] T R 1 H M )] 1 Carbon data assimilation ... · Carbon data assimilation using...
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Carbon data assimilation using Maximum Likelihood Ensemble filter (MLEF) Dusanka Zupanski 1 , A. Scott Denning 2 , Marek Uliasz 2 , Andrew Schuh 2 , and Milija Zupanski 1 1 Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO 80523 2 Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523 Corresponding author E-mail: [email protected] Goals • Develop an innovative general approach for carbon flux inversion • Estimate and reduce carbon flux uncertainty • Estimate and reduce biases in the carbon fluxes Methodology Maximum Likelihood Ensemble Filter (MLEF) (M. Zupanski 2005, MWR; D. Zupanski and M. Zupanski 2006, MWR) J = 1 2 [ x ! x b ] T P f -1 [ x ! x b ] + 1 2 [ H[ M (x)] ! y obs ] T R !1 [ H[ M (x)] ! y obs ] = min x ! x b = P f 12 ( I + C) !12 " x - Control vector in ensemble space of dim Nens ! - Model state vector of dim Nstate >>Nens C = Z T Z z i = R !12 H[ M ( x + p f i )] ! R !12 H[ M ( x)] - C is information matrix of dim Nens × Nens p f i = M ( x + p a i ) ! M ( x) - Dynamical forecast model - Observation operator - z i are columns of Z - p i f and p i a are columns of P f (forecast error cov) and P a (analysis error cov) y obs - Observations vector of dim Nobs - Change of variable (preconditioning) x n = M n, n!1 (x n!1 ) y n = H n (x n ) For N ens = N state (full-rank problem) and linear M and H : MLEF = classical Kalman filter For N ens = N state , non-linear M and H, constant P f : MLEF = 3d-var This research was funded by NASA Grant NNG05GD15G Data assimilation experiments with an off-line Lagrangian Particle Dispersion Model (LPDM) Bias estimation problem: Integrate SiB-RAMS to pre-compute CO 2 fluxes (photosynthesis and respiration) Employ MLEF to estimate spatially dependent SiB-RAMS biases in carbon fluxes: β GPP (photosynthesis bias) and β Resp (respiration bias) Employ LPDM as “observation operator” for CO 2 concentration observations (e.g., observations from ring of towers in northern Wisconsin) Experiments are performed for Single tower (WLEF) and Ring of Towers: Simulated observations from WLEF tower and Ring of towers Real SiB-RAMS influence functions Data assimilation interval: 10 days Simulation over 70-day period starting 1 June 2004 600km x 600km domain, Δx=20km N obs =1200 (WLEF only) and N obs =2640 (Ring of Towers) Obs_err varies between 1ppm and 45 ppm N state =1800 (900 grid-points x 2 variables) Background values β GPP = β Resp =0.75 Background error for β GPP and β Resp assumed equal 0.2 in all 1800 grid-points Kalman filter diagonal start Cycle 1 Cycle 7 Cycle 3 Kalman flter smooth start True β Resp Using smooth initial background error covariance (smooth start) works better than using diagonal initial error covariance (diag start)! MLEF 40 Ens smooth start MLEF 100 Ens smooth start Cycle 1 Cycle 7 Cycle 3 Reduced-rank MLEF results as good as full-rank Kalman filter results! True β Resp NEE( x, y, t ) = ! RESP ( x, y) RESP( x, y, t ) " ! GPP ( x, y)GPP( x, y, t ) WLEF tower Ring of Towers Uncertainties of the estimated biases decrease with time: the MLEF system has learning capability. Conclusions MLEF approach is promising for carbon flux inversion problems: it can be used to estimate carbon fluxes, their biases, and the associated uncertainties. Using more data (from more towers) has a clearly beneficial impact. We need more towers! A relatively small ensemble size (100 ens) can produce results comparable to Kalman filter results (small ensemble size is a practical advantage over Kalman filter). No need to use adjont models (practical advantage over variational methods) Future work Apply the same approach to the entire North American region. Use real observations. Employ a fully coupled atmospheric and carbon transport model (computationally demanding but could improve the results even further). References Zupanski D. and M. Zupanski, 2006: Model error estimation employing ensemble data assimilation approach. Mon. Wea. Rev., 134, 1337-1354. Zupanski, D., A. S. Denning, and M. Uliasz, 2005: Ensemble data assimilation applications to atmospheric and carbon cycle science. Extended abstracts of the AMS 10th Symposium on Integrated Observing and Assimilation Systems for the Atmosphere, Oceans, and Land Surface, 29 January – 2 February 2006, Atlanta, GA, 4 pp. Zupanski, M., 2005: Maximum Likelihood Ensemble Filter: Theoretical Aspects. Mon. Wea. Rev., 133, 1710–1726.
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Carbon data assimilation using Maximum Likelihood Ensemble filter (MLEF)Dusanka Zupanski1, A. Scott Denning2, Marek Uliasz2, Andrew Schuh2, and Milija Zupanski1
1Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO 805232Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523
Corresponding author E-mail: [email protected]
Goals• Develop an innovative general approach for carbonflux inversion• Estimate and reduce carbon flux uncertainty• Estimate and reduce biases in the carbon fluxes
MethodologyMaximum Likelihood Ensemble Filter (MLEF)
(M. Zupanski 2005, MWR; D. Zupanski and M. Zupanski 2006, MWR)
J =1
2[x ! xb ]
TPf-1[x ! xb ]+
1
2[H[M (x)]! yobs ]
TR
!1[H[M (x)]! yobs ] = min
x ! xb = Pf1 2(I + C)
!1 2"
x
- Control vector in ensemble space of dim Nens!
- Model state vector of dim Nstate >>Nens
C = ZT
Z
zi= R
!1 2H[M (x + pf
i)]! R
!1 2H[M (x)]
- C is information matrix of dim Nens × Nens
pfi= M (x + pa
i) ! M (x)
- Dynamical forecast model
- Observation operator
- zi are columns of Z
- pif and pi
a are columns of Pf (forecast errorcov) and Pa (analysis error cov)
yobs - Observations vector of dim Nobs
- Change of variable (preconditioning)
xn= M
n,n!1(x
n!1)
yn= H
n(x
n)
For Nens = Nstate (full-rank problem) and linear M and H :MLEF = classical Kalman filter
For Nens = Nstate, non-linear M and H, constant Pf :MLEF = 3d-var
This research was funded by NASA Grant NNG05GD15G
Data assimilation experiments with an off-lineLagrangian Particle Dispersion Model (LPDM)
Bias estimation problem: Integrate SiB-RAMS to pre-compute CO2 fluxes(photosynthesis and respiration) Employ MLEF to estimate spatially dependent SiB-RAMSbiases in carbon fluxes: βGPP (photosynthesis bias) and βResp(respiration bias)
Employ LPDM as “observation operator” for CO2concentration observations (e.g., observations from ring oftowers in northern Wisconsin)
Experiments are performed for Single tower (WLEF) andRing of Towers:
Simulated observations from WLEF tower and Ring oftowers Real SiB-RAMS influence functions Data assimilation interval: 10 days Simulation over 70-day period starting 1 June 2004 600km x 600km domain, Δx=20km Nobs=1200 (WLEF only) and Nobs=2640 (Ring of Towers) Obs_err varies between 1ppm and 45 ppm Nstate=1800 (900 grid-points x 2 variables) Background values βGPP= βResp =0.75 Background error for βGPP and βResp assumed equal 0.2in all 1800 grid-points
Kalman filterdiagonal start
Cycle 1 Cycle 7Cycle 3
Kalman fltersmooth start
True βResp
Using smooth initial background errorcovariance (smooth start) works better thanusing diagonal initial error covariance (diagstart)!
MLEF 40 Enssmooth start
MLEF 100 Enssmooth start
Cycle 1 Cycle 7Cycle 3
Reduced-rank MLEF results as good asfull-rank Kalman filter results!
True βResp
NEE(x, y,t) = !RESP (x, y)RESP(x, y,t) " !GPP (x, y)GPP(x, y,t)
WLEF tower
Ring of Towers
Uncertainties of the estimated biasesdecrease with time: the MLEF systemhas learning capability.
Conclusions MLEF approach is promising for carbon flux inversion problems: itcan be used to estimate carbon fluxes, their biases, and theassociated uncertainties. Using more data (from more towers) has a clearly beneficialimpact. We need more towers! A relatively small ensemble size (100 ens) can produce resultscomparable to Kalman filter results (small ensemble size is apractical advantage over Kalman filter). No need to use adjont models (practical advantage overvariational methods)
Future work Apply the same approach to the entire North American region. Use real observations. Employ a fully coupled atmospheric and carbon transport model(computationally demanding but could improve the results evenfurther).
ReferencesZupanski D. and M. Zupanski, 2006: Model error estimation employing ensemble dataassimilation approach. Mon. Wea. Rev., 134, 1337-1354.Zupanski, D., A. S. Denning, and M. Uliasz, 2005: Ensemble data assimilation applicationsto atmospheric and carbon cycle science. Extended abstracts of the AMS 10th Symposiumon Integrated Observing and Assimilation Systems for the Atmosphere, Oceans, and LandSurface, 29 January – 2 February 2006, Atlanta, GA, 4 pp.Zupanski, M., 2005: Maximum Likelihood Ensemble Filter: Theoretical Aspects. Mon. Wea.Rev., 133, 1710–1726.
