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Slide 1
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Ensemble Data Assimilation
Massimo Bonavita
ECMWF
Acknowledgments: Lars Isaksen, Elias Holm, Mike Fisher,
Laure Raynaud
Slide 2
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Outline
•The Ensemble Data Assimilation method
•What do we do with the EDA?
•Use of EDA variances in ECMWF 4DVar
•4DVar assimilation experiments
•Conclusions and plans
Slide 2
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Outline
•The Ensemble Data Assimilation method
•What do we do with the EDA?
•Use of EDA variances in 4DVar
•4DVar assimilation experiments
•Conclusions and plans
Slide 3
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
“When simulating the error evolution of the reference system one should use the reference gain matrix K” (Berre et al. 2007)
The EDA method
KyxKHIx fa
ofa KeeKHIe
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
This is what an ensemble of 4DVar analyses with random observation and SST perturbations does!
εa/f/o = perturbations w.r.to ensemble mean
The EDA method
ofa KεεKHIε
Analysis
xb+εb
y+εo
SST+εSST (etc.)
xa+εa
Forecastxf+εf
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• After a few cycles the EDA system will have forgotten the initial background perturbations
• We are diagnosing the background error statistics of the actual analysis system
However:
• We assume that observation and SST errors are correctly specified
• We need to account for the model error component of the background error
The EDA method
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
“Stochastic Kinetic Energy Backscatter”
SKEB (Berner et al., 2009)
ΔXperturbed physics= f(KE,ΔXphysics,n)
A fraction of the dissipated energy is backscattered upscale and acts as stream function forcing on resolved-scale flow.
Spectral Markov chain: temporal and spatial correlations prescribed
Only vorticity perturbed, so only wind field directly affected
All levels are perturbed. Either randomly or partially correlated.
“Stochastically Perturbed Parameterisation Tendencies”
SPPT (Leutbecher, 2009)
ΔXperturbed physics= ( 1+μr) ΔXphysics
Random pattern r varies smoothly in space and time, with de-correlation scales 500 km and 6 hours.
Gaussian distribution with no bias and stdev 0.5 (limited to ±3stdev)Same random number r
for X=T, q, u, v
No perturbations in lowest 300 m and above 50 hPa (0≤ μ ≤1).
from: Lars Isaksen
The EDA method
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
The EDA methodComparing the impact of various stochastic perturbation methods: zonal mean
u wind ensemble background forecast spread
N S
Slide 8
Impact of SKEB on U spread Additional Impact of SPPT on U spread
from: Lars Isaksen
Slide 9
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
The EDA methodComparing the impact of various stochastic perturbation methods: zonal mean
Temperature ensemble background forecast spread
N S
Slide 9
Impact of SKEB on T spread Additional Impact of SPPT on T spread
from: Lars Isaksen
Slide 10
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
•In the EDA context model error parametrizations should increase spread where/when errors are larger => should increase spread-error correlation
•Are they doing this?
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The EDA method
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
EDA (baseline) - EDA+SKEBEDA+SPPT - EDA+SPPT+SKEB
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The EDA method
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
EDA (baseline) - EDA+SKEBEDA+SPPT - EDA+SPPT+SKEB
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The EDA method
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
•Model error parametrizations do increase spread-error correlations
•Large effect in the lower troposphere Tropics
•Small effect in the Extratropics
•Similar behaviour-performance of the two schemes and their combination
Slide 13
The EDA method
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
10 ensemble members using 4D-Var assimilations
T399 outer loop, T95/T159 inner loop (reduced number of iterations)
Observations randomly perturbed
Cloud track wind (AMV) correlations taken into account
SST perturbed with realistically scaled structures
Model error represented by stochastic methods (SPPT, Leutbecher, 2009)
All 107 conventional and satellite observations used
The EDA method
from: Lars Isaksen
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Outline
•The Ensemble Data Assimilation method
•What do we do with the EDA?
•Use of EDA variances in ECMWF 4DVar
•4DVar assimilation experiments
•Conclusions and plans
Slide 15
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• The EDA simulates the error evolution of the 4DVar analysis cycle. As such it can be applied to:
1. Compute climatology of B matrix for use at the initial time of 4DVar (Analysis-Ensemble method, Fisher 2003; see talk on B matrix modelling)
2. Provide initial conditions for ensemble forecasts (EPS)
3. Provide a flow-dependent sample of background errors at the initial time of 4DVar
What do we do with the EDA?
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Improving Ensemble Prediction System by including EDA perturbations for initial uncertainty
The Ensemble Prediction System (EPS) benefits from using EDA based perturbations. Replacing evolved singular vector perturbations by EDA based perturbations improve EPS spread, especially in the tropics.The Ensemble Mean has slightly lower error when EDA is used.
What do we do with the EDA?
Slide 17
EVO-SVINIEDA-SVINI
EVO-SVINIEDA-SVINI
N.-Hem. Tropics
Ensemble spread and Ensemble mean RMSE for 850hPa T
Slide 18
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• The EDA simulates the error evolution of the 4DVar analysis cycle. As such it can be applied to:
1. Compute climatology of B matrix for use at the initial time of 4DVar (Analysis-Ensemble method, see talk on B matrix modelling)
2. Provide initial conditions for ensemble forecasts (EPS)
3. Provide a flow-dependent sample of background errors at the initial time of 4DVar
What do we do with the EDA?
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• 4DVAr does not cycle error information (B), only the state estimate
• 4DVar behaves like a Kalman filter in which the covariance matrix is reset to some static matrix B every few hours (12h in ECMWF implement.)
• Two possible ways around this:
1. Run 4DVar over a long enough window so that the influence of the initial state and errors on the final state analysis is negligible
Slide 19
What do we do with the EDA?
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
T+120S.hem Lat -90.0 to -20.0 Lon -180.0 to 180.0
Root mean square error forecast500hPa Geopotential
Time series curves
15
AUGUST 200516 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
30
40
50
60
70
80
90
100
110
120
all obs
all obs
Analysis experiment started with satellite data reintroduced on 15th August 2005
Analysis experiment started with satellite data reintroduced on 15th August 2005
From Mike Fisher
Memory of the initial statedisappears after approx. 3 days
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• An analysis window of ≥3 days would allow 4DVar analysis at final time to be independent of initial state and error estimates
• However:
1. An effective model error representation must be applied to reconcile model and measurements over a long analysis time window (i.e., weak constraint 4DVar)
2. We would still lack an estimate of analysis errors
Slide 21
What do we do with the EDA?
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• Two possible ways around this:
1. Run 4DVar over a long enough window so that the influence of the initial state and errors on the final state analysis is negligible
2. Use a sequential method (EDA, EnKF) to cycle error covariance information
Slide 22
What do we do with the EDA?
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Hybrid methods
• Use of ensemble perturbations in a 3-4DVar analysis
• Cycle error information through ensemble of Data Assimilations
• Retain the implicit full rank error representation of 3-4DVar
µεσον τε και αριστονAristotle, Nic. Ethics 2.6
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Hybrid systems
• Ensemble perturbations can be used in a 3-4DVar analysis in a number of different ways:
1. Use ensemble variances for observation QC
2. Use ensemble (co)variances as starting B matrix of minimization (often in linear combination with climatological B, extra control variable)
3. Use of ensemble covariances inside 4DVar minimization (En4DVAR)
Slide 24
What do we do with the EDA?
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Outline
•The Ensemble Data Assimilation method
•What do we do with the EDA?
•Use of EDA variances in ECMWF 4DVar
•4DVar assimilation experiments
•Conclusions and plans
Slide 25
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• We want to use EDA perturbations to simulate 4DVar flow-dependent error covariance evolution
• We start with the diagonal of the Pf matrix, i.e.:
“Estimate the first guess error variances with the variance of the EDA short range forecasts”
• This has been tried before (Kucukkaraca and Fisher, 2006, Fisher 2007, Isaksen et al., 2007) but results have been inconclusive
Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Define: Veda = EDA sampled variance
V* = True error variance
Then the estimation error can be decomposed:
Veda-V* = E[Veda-V*] + (Veda-E[Veda])
systematic random
Similarly for mean square of the estimation error:
E[(Veda-V*)2] = (E[Veda-V*])2 + E[(Veda-E[Veda])2]
systematic random
We should try to minimize both!Slide 27
Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
What raw ensemble variances look like?
Spread of Vorticity FG t+9h ml=64
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• Noise level is due to sampling errors: 10 member ensemble
• EDA is a stochastic system: variance of variance estimator ~ 1/Nens
• We need a system to effectively filter out noise from first guess ensemble forecast variances: Reduce the random component of the estimation error
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
“Mallat et al.: 1998, Annals of Statistics, 26,1-47”
Define Ge(i) as the random component of the sampling
error in the estimated ensemble variance at gridpoint i:
Then the covariance of the sampling noise can be shown to be a simple function of the expectation of the ensemble-based covariance
matrix:
(1)
iiiie BEBiG
~~
2~
1
2ij
ee BEN
jGiGE
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
“Mallat et al.: 1998, Annals of Statistics, 26,1-47”
A consequence of (1) is that:
(2)
i.e., sampling noise is smaller scale than background
error. If the variance field varies on larger scales then the background error
=> we can use a spectral filter to disentangle noise
error from the sampled variance field
2
iLiL b
ee
G
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010 Slide 32
Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• There is indeed a scale separation between signal and sampling noise!
• Truncation wavenumber is determined by maximizing signal-to-noise ratio of filtered variances (details in Raynaud et al., 2009, and forthcoming Tech. Memo)
• Optimal truncation wavenumber depends on parameter and model level
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Raw Ensemble StDevVO ml64
Filtered Ensemble StDevVO ml64
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Raw Ensemble StDevSpec. Hum. ml64
Filtered Ensemble StDevSpec. Hum. ml64
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Operational StDevRandom. Method (Fisher & Courtier, 1995
Filtered Ensemble StDevVO ml64
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Is Filtering the Ensemble Variances enough to improve the analysis?
Not quite…
N.Hem. Z 500hPA AC S.Hem. Z 500hPA AC
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Should we also do something about the systematic component of the estimation
error?
E[(Veda-V*)2] = (E[Veda-V*])2 + E[(Veda-[Veda])2]
systematic random
A statistically reliable ensemble satisfies:
Slide 38
Use of EDA variances in 4DVar
Mean_Errord_Ens_Square_VarianceEns11
1111
ensens NN
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Vorticity ml 30 (~50hPa) Ensemble Error Ensemble Spread
Spread - Error
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Vorticity ml 78 (~850hPa) Ensemble Error Ensemble Spread
Spread - Error
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Is the ensemble fg statistically calibrated?
• Calibration factors needs to be model level, latitude and parameter dependent
• Calibration factors seems also to be flow-dependent, i.e. depend on the size of the expected error
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010 Slide 42
Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• Calibration factors need to be flow-dependent, too!
• Do they also change in time?
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010 Slide 44
Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010 Slide 45
Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• There is not a large day-to-day variability but seasonal variability is important
• General solution: slowly varying adaptive calibration coefficients
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Use of EDA variances in 4DVar
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Variance post-process
xa+εia
Analysis ForecastSST+εi
SST
y+εio
xb+εib
xf+εif
i=1,2,…,10
EDA Cycle
εif raw
variances
Variance Recalibration
Variance Filtering
EDA scaled variances
4DVar Cycle
xa
Analysis ForecastEDA scaled Varxb xf
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010 Slide 48
Slide 49
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Outline
•The Ensemble Data Assimilation method
•What do we do with the EDA?
•Use of EDA variances in ECMWF 4DVar
•4DVar assimilation experiments
•Conclusions and plans
Slide 49
Slide 50
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
Deterministic DA experiments with EDA variances
1. CY35r3_esuite, T799L91, 7/01 – 16/02 2009
2. Control f8a4
3. Experiment fb4k with ensemble DA variances:
a) Calibration step: adaptive, flow-dependent, regionally varying, for each parameter and model level
b) Filtering step: “Optimal” spectral filtering
c) EDA variances are used both in observation QC and start of 4DVar minimization
Slide 50
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
N.HEM Z acJan-Feb
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
S.HEM Z acJan-Feb
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
TROP. VW RMSEJan-Feb
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
Deterministic DA experiments with EDA variances
1. CY36r2_esuite, T1279L91, 7/08 – 16/09 2008
2. Control far9
3. Experiment fb9x with ensemble DA variances:
a) Calibration step: adaptive, flow-dependent, regionally varying, for each parameter and model level
b) Filtering step: “Optimal” spectral filtering
c) EDA variances are used both in observation QC and start of 4DVar minimization
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
N.HEM Z acAug-Sep
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
S.HEM Z acAug-Sep
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
TROP VW RMSEAug-Sep
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
Deterministic DA experiments with EDA variances
1. Results are generally positive in the Extra-Tropics (more so in the NH)
2. Small positive impact in the Tropics
Slide 59
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
4DVar assimilation with EDA variances
Slide 60
• Observation departures statistics support the idea that (slightly) different observation usage is the result of smarter OBS QC decisions
Slide 61
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Outline
•The Ensemble Data Assimilation method
•What do we do with the EDA?
•Use of EDA variances in ECMWF 4DVar
•4DVar assimilation experiments
•Conclusions and plans
Slide 61
Slide 62
ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
• First step towards an error cycling DA
• Use of flow dependent EDA variances has the potential to improve the deterministic scores
• A careful post-processing step of the raw ensemble first guess forecast is necessary to:
a)Filter sampling noise
b)Adaptively calibrate the ensemble
Conclusions and plans
Slide 62
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
•Better OBS QC decisions seems to be playing a part
•Further improvements in model error parameterizations will directly benefit the system
•Increase in ensemble size will benefit the system
Conclusions and plans
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
WHERE NEXT
•Operational implementation and testing
•Further tuning of system at full operational resolution (T1279L91)
Conclusions and plans
Slide 64
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
•Generalize the use of EDA variances to unbalanced components of control vector
•Relax the assumption: analysis=“truth” in the calibration step
Conclusions and plans
Slide 65
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Medium term:
•Refine the representation of initial uncertainties (correlated perturbations, surface fields uncertainties) in stochastic EDA
•Evaluate EnKF covariances
•Further develop the hybridization of 4DVar with EDA (investigate the use of EDA covariances)
Conclusions and plans
Slide 66
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ECMWF Data Assimilation Training Course – May 2010 ECMWF Data Assimilation Training Course – May 2010
Thanks for your attention!
I welcome your questions/comments…
Slide 67
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