The ECMWF Hybrid 4D-Var and Ensemble of Data Assimilations Lars Isaksen , Massimo Bonavita and Elias Holm Data Assimilation Section ECMWF lars.isaksen@ecmwf.int Acknowledgements to: Mike Fisher and Marta Janiskova ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 1 Assimilation at ECMWF
The ECMWF Hybrid 4D-Var and Ensemble of Data Assimilations 4D-Var Ensemble of Data Assimilations (EDA) Hybrid 4D-Var & EDA ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 2 Assimilation at ECMWF
ECMWF will continue to use deterministic 4D-Var All observations within a 12-hour period (~17,000,000) are used • simultaneously in one global (iterative) estimation problem Innovations are computed at the observation time using the high-resolution non-linear forecast model 4D-Var finds the 12-hour forecast that take account of the observations in a dynamically consistent way Based on a tangent linear and adjoint forecast models , that works in an 80,000,000 dimensional model subspace 9Z 12Z 15Z 18Z 21Z ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 3 Assimilation at ECMWF
Accuracy of Tangent-Linear and adjoint important: linearity issues Regularizations remove the most important threshold processes in physical parametrizations improving the validity of the tangent linear approximation u-wind increments Non-linear finite difference TL integration fc t+12, ~700 hPa 12 12 8 8 4 4 2 2 1 1 0.5 0.5 -0.5 -0.5 -1 -1 -2 -2 -4 -4 -8 -8 -12 -12 ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 4 Assimilation at ECMWF
The Ensemble of Data Assimilations (EDA) ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 5 Assimilation at ECMWF
The Ensemble of Data Assimilations at ECMWF (EDA) • 10 (25 from November 2013) ensemble members using 4D-Var assimilations • T399 (50km) outer loop, T95/T159 inner loops. (Deterministic 4D-Var: T1279 (16km) outer loop, T159/T255/T255 inner loops) • Observations randomly perturbed according to their estimated errors • SST perturbed with climatological perturbations • Model error represented by stochastic methods ( SPPT , Leutbecher, 2009) ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 6 Assimilation at ECMWF
The hybrid approach used at ECMWF: EDA&4D-Var Hybrid approach: Use cycled, flow-dependent background error estimates from an Ensemble of Data Assimilations in a deterministic 4D-Var analysis. This hybrid formulation has many benefits: • Introduces flow-dependent background errors into 4D-Var system • Maintain the full rank representation of B and its implicit evolution inside the assimilation window • More robust than pure EnKF for limited ensemble sizes • Ensemble perturbations are used in 4D-Var control vector space, beneficial for e.g. assimilation of radiance observations • Allows for flow-dependent Quality Control of observations ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 7 Assimilation at ECMWF
The EDA&4D-Var hybrid implementation at ECMWF EDA Cycle x b +ε i b x f + ε i f x a +ε i a Analysis Forecast y+ε i o i=1,2,…,10 (25) SST+ε i SST Variance post-process ε i Variance Variance EDA scaled f raw Recalibration Filtering variances variances 4DVar Cycle x b x a x b Analysis Forecast EDA scaled Var ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 8 Assimilation at ECMWF
Raw EDA variance estimates needs to be calibrated to become statistically consistent • We performs an online calibration (Ensemble Variance Calibration; Kolczynsky et al., 2009, 2011; Bonavita et al., 2011) • Calibration factors depend on latitude bands and parameter • Calibration factors also depend on the size of the expected error ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 9 Assimilation at ECMWF
Sampling noise due to small ensemble size is a problem Noise filtering method used until June 2012: • Use a spectral filter to disentangle noise from signal • Truncation wavenumber is determined by maximizing signal-to-noise ratio of filtered variances (Raynaud et al., 2009; Bonavita et al., 2011) A more direct strategy applied now, based on two 50-member EDAs: 1 1. Sampling noise assumed a random process P S n P S S e i j 2 2. Time average sampling noise spectrum samples 1 3. Compute raw filters and time average to smooth out n P S 1 e noise (based on Berre et al., 2010) P S raw ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 10 Assimilation at ECMWF
Introducing flow-dependent background errors in 4D-Var In the ECMWF 4D-Var, the B matrix is defined implicitly in terms of a transformation from the background departure ( x-x b ) to a control variable χ : ( x-x b ) = L χ So that the implied B=LL T . In the current wavelet formulation (Fisher, 2003), the variable transform can be written as: Σ 1 1 / 2 1 / 2 x x T C , b b j j j j T is the balance operator Σ b is the gridpoint variance of background errors C j ( λ , φ ) is the vertical covariance matrix for wavelet index j ψ j are the set of radial basis function that define the wavelet transform. ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 11 Assimilation at ECMWF
Introducing flow-dependent background errors in 4D-Var Σ 1 1 / 2 1 / 2 x x T C , b b j j j j C j ( λ , φ ) are full vertical covariance matrices, function of ( λ , φ ). They determine both the horizontal and vertical background error correlation structures ; In standard 4D-Var T and C j are computed off-line using a climatology of EDA perturbations. Σ b is computed by random sampling of the static B matrix (randomization procedure, Fisher and Courtier, 1995) How do we make this error covariance model flow-dependent ? We look for flow-dependent EDA estimates of Σ b and C j ( λ , φ ) ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 12 Assimilation at ECMWF
Improved static background-error covariance statistics based on the latest EDA (implemented June 2012) Resolution upgrades and more observations since last update resulted in sharper structure functions: reduced correlation length scales both horizontally and vertically ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 13 Assimilation at ECMWF
EDA-based flow-dependent background errors for unbalanced control variables (T u ,D u ,LNSP u ) - June 2013 EDA Variances for the Unbalanced Control Vector ( η u , ( T,p s ) u ). Var Var T M Var M Var M Var u u T T T , p N P T , p Var T , p N Var N P Var P Var T , p s u s u s u s u Explained variance Ratio for divergence and temperature ECMWF Derber, Bouttier, Fisher (1997) Similar plot for the 2013 ECMWF DA system Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 14 Assimilation at ECMWF
EDA-based flow-dependent background errors for unbalanced control variables (T u ,D u ,LNSP u ) - June 2013 Average unbalanced temperature (st.dev. in Kelvin) Previous bg error model for unbal. temp. EDA bg error for unbal. temp. Top of atmosphere Surface 90N 90S 90N 90S ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 15 Assimilation at ECMWF
Flow-dependent covariance estimation from an EDA Variance estimation needs an EDA sample size of ~10 Covariance estimation needs an EDA sample size of ~600 a) Background error covariances (JB) are computed in a post- processing step of 25 member EDA b) EDA perturbations from the past 12 days are used for a weighted running mean (Sample size: 25*12*2=600) c) Continuously updated JB is used in deterministic 4D-Var Similar activities on-going at Météo-France (Varella et al. 2011) ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 16 Assimilation at ECMWF
Flow-dependent covariance estimation from an EDA EDA-based flow-dependent variances are computed for each analysis cycle - sufficient with 10 EDA members to estimate error-of-the day. St.dev of vorticity errors and Z at 500hPA ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 17 Assimilation at ECMWF
Flow-dependent covariance estimation from an EDA Covariance estimation requires a large sample size (order 600). This is computed with a lagged 12 days running average. Correlations are representative of prevailing weather patterns, not distinct weather features! St.dev of vorticity errors and Z at 500hPA ECMWF Lars Isaksen, 6 th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 18 Assimilation at ECMWF
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