Accounting for Atmospheric motion Vector Error Correlations in the ECMWF 4D-Var and Ensembles of Data Assimilations

Transcription

1 Accounting for Atmospheric motion Vector Error Correlations in the 4D-Var and Ensembles of Data Assimilations Lars Isaksen and Gábor Radnóti, Reading, UK Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

2 Reasons for the investigations Presently in 4DVAR the observation error correlation matrix R is diagonal observations are assumed to be uncorrelated We know that this assumption is not fully correct: Bormann et al. (2003), MWR, showed that SATOB winds are correlated due to e.g. height assignment errors To compensate we at the moment apply excessive thinning of data and use inflated observation error A proper account for error correlations would allow more correct use of observations and use of more observations Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

3 Proposed error correlation method Emphasis is on horizontal correlations, but time and vertical correlations are also accounted for in a simple way A block diagonal approximation of R for groups of intercorrelated data (same instrument, method, channel) Approximation is full-rank for each inter-correlated block The implementation is based on a truncated Eigen-vector representation in order to save computational cost Based on an algorithm proposed by Mike Fisher, (RD Memorandum, R48.3/MF/05106, 2005) Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

4 The method explained J o = z T zeff z normalized departure; z eff the effective norm. departures N 1 RETAIN 1 1 z = R z = z + γ v γ 1 i eff i i i i, k, α i k= 1 λi, k α i k i = v T i, k i, k z i Departure coordinate in eigen-space We can see from this formulation: 1. We don t need the matrix R explicitly available (only its leading N RETAIN Eigen-vectors) 2. Two scans through the obs. set are needed: 1. to compute gamma (and of course Jo contrib. of uncorrelated data) 2. to compute the effective departures and thus the Jo contrib. of correlated data 3. Eigen-vectors are stored in observation space Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb Eigen-pairs Parameter accounting for truncation

5 The correlation model Horizontal correlations introduced as a convolution multiplication with a weight-function in spectral space Going to observation space by interpolation (horizontally, vertically and in time) Then the square-root correlation model looks like: U = TS D G 1 1/2 s T 1 * 1 T R = UU = TS GS T T: interpolator S: spectral transform (typically T159) D: spectral inner product weight matrix G: spectral weight function of the convolution (Hankel transform of the correlation function) Implemented method: Lánczos algorithm to compute the truncated eigen-system of R, R as sequence of linear operators Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

6 Eigen vectors in observation space 46 N 24 N 2 N GOES VIS1 850hPa 30 minutes ~500 points EV10 20 S 179 W W W W 99.8 W 8 Correlation length scale L=200km EV1 EV100 Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

7 Idealized experiments with perfectly known correlated errors added to radiosonde data These experiments show the method works. The algorithm works well both in terms of degraded analysis fit to observations and improved forecast performance - when error correlations are large enough and perfectly known Eigenspectrum of correlation matrix is relatively flat: sufficiently large number of leading Eigen-pairs are needed (typically, when 2000 measurements are correlated, N RETAIN =50) If errors are uncorrelated and we incorrectly assume a correlated Jo the forecast performance is degraded Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

8 Eigen-spectrum of SATOB error correlation matrix Eigen-spectrum of SATOB error correlation matrix L=400km WhN=5% Dim=7750 Jo expressed with the normalized Departures in the Eigen-space of the error correlation matrix: T 1 corr.: Jo = z Λ z uncorr : J o = T z z Spectra of over-penalizing departures with respect to Ideal, full eigen-representation Uncorrelated representation Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

9 Real experimentation with SATOB winds A month of 4D-Var, T511/L91. exp:f74o /DA (black) v. f74n/da (12) SATOB-Uwind N.Midlat used U 100 STD.DEV exp -ref nobsexp 2 background departure o-b(ref) background departure o-b analysis departure o-a(ref) analysis departure o-a BIAS 100 Three assimilations performed: REF (all obs. assumed uncorrelated) Obs corr (assume AMV correlated) NoAMV (REF without AMV) Assumed correlation length scale: L=200km Based on Bormann et al., 2003: R( r) = R (1 + r / L) e 0 0 r / L R = ; L = km Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb Pressure (hpa) exp:f74o /DA (black) v. f74n/da (12) SATOB-Uwind Tropics used U Pressure (hpa) STD.DEV Ref. exp. is red exp -ref nobsexp BIAS background departure o-b(ref) background departure o-b analysis departure o-a(ref) analysis departure o-a Obs.corr. exp. is black

10 R M S E 5.5 M/S Verified against operational analysis FORECAST VERIFICATION 500 hpa VECTOR WIND ROOT MEAN SQUARE ERROR FORECAST AREA=TROPICS TIME=00 MEAN OVER 8 CASES DATE1= /... DATE2= /... Corr Ref FORECAST VERIFICATION 500 hpa VECTOR WIND MEAN ERROR FORECAST AREA=TROPICS TIME=00 MEAN OVER 8 CASES DATE1= /... DATE2= /... Corr Ref MAGICS 6.12 midgard - dag Mon Mar 2 14:52: Verify SCOCOM Forecast Day Improved RMSE and bias in the tropics where Bormann et al. (2003) found the largest correlation distances, neutral (N.Hem.); slightly negative (S.Hem.) for 8-10 day forecasts Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb MAGICS 6.12 midgard - dag Mon Mar 2 14:52: Verify SCOCOM * 1 ERROR(S) FOUND * FORECAST VERIFICATION 100 hpa VECTOR WIND Forecast Day MEAN ERROR FORECAST AREA=TROPICS TIME=00 MEAN OVER 8 CASES DATE1= /... DATE2= / MAGICS 6.12 midgard - dag Mon Mar 2 14:52: Verify SCOCOM * 1 ERROR(S) FOUND * Forecast Day Corr Ref B I A S

11 Verified against operational analysis 100 % 98 N.Hemisphere 500hPa geopotential FORECAST VERIFICATION 500 hpa GEOPOTENTIAL ANOMALY CORRELATION FORECAST AREA=N.HEM TIME=00 MEAN OVER 30 CASES DATE1= /... DATE2= /... DATE3= /... Corr REF NoAMV Forecast Day Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

12 Verification against radiosonde observations 4.6 M/S 4.4 FORECAST VERIFICATION 700 hpa VECTOR WIND ROOT MEAN SQUARE ERROR FORECAST AREA=TROPICS TIME=00 MEAN OVER 30 CASES DATE1= /... DATE2= /... DATE3= /... Corr REF NoAMV Forecast Day Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb

13 Summary A new method that accounts for horizontally correlated observation errors has been developed at The truncated Eigen-value decomposition method will be affordable to use operationally The results at T511/L91 shows slightly positive results for SATOB winds in the tropics and N.Hemisphere. The results are slightly negative in the S.Hemisphere Recent results at T1279/L91 show neutral results The same method has been tested for ATOVS data with neutral results Further investigation and experimentation required before operational implementation can be envisaged Isaksen The 10 th International Winds Workshop, Tokyo, Japan Feb