ENSEMBLE KALMAN FILTER IN THE PRESENCE OF MODEL ERRORS Hong Li Eugenia Kalnay

If we assume a perfect model, we can grossly underestimate the errors ~Perfect model imperfect model (obs from NCEP- NCAR Reanalysis NNR)

We compare several methods to handle model errors ~perfect model imperfect model (obs from NCEP- NCAR Reanalysis NNR)

SPEEDY MODEL (Molteni 2003) T30L7 global spectral model total 96x48 grid points on each level State variables u,v,T,Ps,q Data Assimilation: LETKF Methods to handle model errors 1)Multiplicative/additive inflation 2)Dee & daSilva (1998) (DdS) 3)Low-dimensional method (LDM, Danforth et al, MWR, 2007) Dense Observations

Control run 100% inflation Dee & da Silva Low-order

1a. Covariance inflation (multiplicative) Model error estimation schemes (1) (Ideal KF) (EnKF) 1b. Covariance inflation (additive)

2. Dee and daSilva bias estimation scheme (1998) and need to be tuned Model error estimation schemes (DdS) Do data assimilation twice: first for model error then for model state (expensive)

Generate a long time series of model forecast minus reanalysis from the training period 3. Low-dim method ( Danforth et al, 2007: Estimating and correcting global weather model error. Mon. Wea. Rev) t=0 t=6hr model Model error estimation schemes (LDM) We collect a large number of estimated errors and estimate bias, etc. NNR Time-mean model bias Diurnal model error State dependent model error Forecast error due to error in IC NNR

Further explore the Low-dimensional method Include Bias, Diurnal and State-Dependent model errors: Time-mean model bias

BIAS climatological debiased one month

Diurnal model errors Generate the leading EoFs from the forecast error anomalies fields for temperature. Leading EOFs for 925 mb TEMP  Lack of diurnal forcing generates wavenumber 1 structure pc1 pc2

Black line: Blue line: 925hPa Temperature

the local state anomalies (Contour) and the forecast error anomalies (Color) SVD1 SVD2 SVD3 SVD4 State-dependent model errors

Correct state-dependent model errors Black line: Blue line: 500hPa Uwind500hPa Height Univariate SVD (not account for the relations between different variables)

Impact of model error, and different approaches to handle it Perfect model imperfect model (obs from Reanalysis)

Simultaneous estimation of inflation and observation errors Hong Li Eugenia Kalnay University of Maryland

Motivation  Any data assimilation scheme requires accurate statistics for the observation and background errors. Unfortunately those statistics are not known and are usually tuned or adjusted by gut feeling.  Ensemble Kalman filters need inflation (additive or multiplicative) of the background error covariance, but 1) Tuning the inflation parameter is expensive especially if it is regionally dependent, and it may depend on time 2) Miyoshi and Kalnay 2005 (MK) proposed a technique to objectively estimate the covariance inflation parameter. 3) This method works, but only if the observation errors are known.  Here we introduce a method to simultaneously estimate observation errors and inflation.

MK method to estimate the inflation parameter (Miyoshi 2005, Miyoshi&Kalnay 2005) Assumption: R is known Should be satisfied if R, P b and are correct (they are not!) obs. increment in obs. space So, at any given analysis time, and computing the inner product (1)

Diagnosis of observation error statistics (Desroziers et al, 2006, Navascues et al, 2006) if the R and B statistics are correct and errors are uncorrelated Desroziers et al, 2006, introduced two new statistical relationships: Writing their inner products we obtain two equations which we can use to “observe” R and : (2)

Simultaneous estimation of inflation and observation errors  Model : Lorenz-96 model / SPEEDY model  Perfect model scenario  Data assimilation scheme: Local ensemble transform Kalman filter (LETKF, Hunt et al. 2006)  We estimate both and R online at each analysis time (2) (1)

Tests within LETKF with Lorenz-96 model Wrong R, estimate inflation using (1) : it fails 40 observations with true Rt=1, 10 ensemble member. Optimally tuned rms=0.20 Perfect R, estimate inflation using (1) : it works method rms (1) method rms (1)

Now we estimate observation error and optimal inflation simultaneously using (1) and (2): it works! R init Estimated R Estimated Tests within LETKF with L96 model R method method rms (2) (1)

SPEEDY MODEL (Molteni 2003) Primitive equations, T30L7 global spectral model total 96x46 grid points on each level State variables u,v,T,Ps,q Tests within LETKF with SPEEDY

OBSERVATIONS Generated from the ‘truth’ plus “random errors” with error standard deviations of 1 m/s (u), 1 m/s(v), 1K(T), kg/kg (q) and 100Pa(Ps). Dense observation network: 1 every 2 grid points in x and y direction Tests within LETKF with SPEEDY EXPERIMENTAL SETUP Run SPEEDY with LETKF for two months ( January and February 1982), starting from wrong (doubled) observational errors of 2 m/s (u), 2 m/s(v), 2K(T), 2*10 -4 kg/kg (q) and 200Pa(Ps). Estimate and correct the observational errors and inflation adaptively.

online estimated observational errors The original wrongly specified R converges to the correct R quickly (in about 5-10 days)

Estimation of the inflation Using a perfect R and estimating adaptively Using an initially wrong R and but estimating them adaptively Estimated Inflation After R converges, they give similar inflation factors (time dependent)

Global averaged analysis RMS 500hPa Height Using a perfect R and estimating adaptively Using an initially wrong R and but estimating them adaptively 500hPa Temperature

Summary  The online (adaptive) estimation of inflation parameter alone does not work without estimating the observational errors.  Estimating both of the observational errors and the inflation parameter simultaneously our approach works well on both the Lorenz-96 and the SPEEDY global model. It can also be applied to other ensemble based Kalman filters.  SPEEDY experiments show our approach can simultaneously estimate observational errors for different instruments.  Current work shows our method also works in the presence of random model errors.

A few more slides Junjie Liu: Adaptive observations Junjie Liu: Estimation of the impact of observations Shu-Chih Yang: Comparison of EnKF, simple hybrid (3D-Var + Bred Vectors) and 4D-Var Shu-Chih Yang: 4D-Var and initial and final SVs, EnKF and initial and final BVs No cost smoother for reanalysis

Adaptive sampling with the LETKF- based ensemble spread Junjie Liu Purpose –Sample 10% adaptive DWL wind observations to get 90% improvement of full coverage –Compare ensemble spread method with other sampling strategies –How the results are sensitive to the data assimilation schemes (3D-Var and LETKF) Note –same adaptive observations from ensemble spread method are assimilated by both 3D-Var and LETKF

500hPa zonal wind RMS error D-VarLETKF RMSE Rawinsonde; climatology; uniform; random; ensemble spread; “ideal”; 100%  With 10% adaptive observations, the analysis accuracy is significantly improved for both 3D-Var and LETKF.  3D-Var is more sensitive to adaptive strategies than LETKF. Ensemble spread strategy gets best result among operational possible strategies

500hPa zonal wind RMS error (2% adaptive obs) 3D-VarLETKF  With fewer (2%) adaptive observations, ensemble spread sampling strategy outperforms the other methods in LETKF  For 3D-Var 2% adaptive observations are clearly not enough Rawinsonde; climatology; uniform; random; ensemble spread; “ideal”; 100%

Analysis sensitivity study within LETKF The self sensitivity is the trace of the matrix S. It can show the analysis sensitivity with respect to: a)different types of observations (e.g., rawinsonde, satellite, adaptive observation and routine observations) b)the observations in different area (e.g., SH, NH)

Analysis sensitivity of adaptive observation (one obs. selected from ensemble spread method over ocean) and routine observations (every grid point over land) in Lorenz-40 variable model 10-day forecast RMS errorAnalysis sensitivity About 17% information of the analysis comes from observations over land. About 85% information comes from observation for the adaptive observation (a single observation over ocean). The single adaptive observation is more important than single observation over land.

Comparison of ensemble-based and variational-based data assimilation schemes in a Quasi-Geostrophic model. Shu-Chih Yang et al. 3D-Var Hybrid (3DVar+20 BVs) 12-hour 4D-Var LETKF (40 ensemble) 24-hour 4D-Var 3D-VarHYBD 4D-VarLETKF 12hr24 hrl=3l=5l=7l=9 RMS error (  ) Time (minutes)

Analysis increment (color shaded) vs. dynamically fast growing errors (contours) Initial increment (smoother) vs. BV analysis increment vs. Final SV 12Z Day 2400Z Day 25 Initial increment vs. Initial SV LETKF 12-hour 4DVAR analysis increment vs. BV

Analysis increment (color shaded) vs. dynamically fast growing errors (contours) analysis increment vs. Final SVInitial increment vs. Initial SV 24-hour 4DVAR 00Z Day 2400Z Day 25

time 4D-LETKF 3D-LETKF toto t1t1 No-cost LETKF smoother (cross): apply at t 0 the same weights found optimal at t 1, works for 3D- or 4D-LETKF

No-cost LETKF smoother “Smoother” reanalysis LETKF Analysis LETKF analysis at time i Smoother analysis at time i-1

LETKF minimizes the errors of the day and thus provides an excellent first guess to the 3D-Var analysis 3DVar with the background provided from LETKF (forecast mean) 3DVar 3DVar with the background of the first 50 days provided from LETKF LETKF We conclude from this experiment that the errors of the day (and not just ensemble averaging) are important in LETKF and 3D-Var.