Estimation of background error statistics in ARPEGE 4D-var

1. Estimation of background error statistics in ARPEGE 4D-var Margarida Belo Pereira (Instituto de Meteorologia, Lisboa) Loïk Berre (Météo-France, Toulouse)

2. Importance of background error estimative - The analysis field results from a combination of observations and background (short range forecast) - The weights given to the observations and to the background depend on error statistics - The background errors statistics determines the way as the information from observations is spread spatially - How to estimate the background error statistics?

3. Obs Obs Obs Obs NMC method (operational in ARPEGE 4D-VAR) Forecast error Analysis error

4. Experiments Ensemble with five 4D-VAR cycles of the non-stretched version of ARPEGE model with T299 and 41 levels Period 1 of February to 24 of March of 2002 Alternative to NMC method? Ensemble Analysis Method Random numbers (5) + observations Perturbed observations (5) Background differences->B Perturbed background (5) Data assimilation Perturbed analysis (5) 6hforecast

5. Standard deviation (normalized) of vorticity background error Ensemble Method Level 21 (500hPa) Truncation T42 Level 32 (850hPa)

6. Impact of geographical variation of standard deviation of background errors Geopotential (anomaly correlation) Forecast against ECMWF analysis Forecast against observations

7. Impact of geographical variation of standard deviation of background errors Wind speed (anomaly correlation) Forecast against ECMWF analysis Forecast against observations

8. Ensemble Method versus NMC Method Spectral Space

9. Spectra of vorticity background error Ensemble Method:the errors for the wind field have a bigger contribution from the mesoscale and subsynoptic scalesthan with the NMC method Ensemble method This result is valid also for the other variables, except for divergence

10. How to estimate the length scale of autocorrelation function? Which assumptions are made? Background error covariances are assumed to be - stationary - spatially homogeneous - isotropic (horizontal length scale doesn’t depend on direction) autocorrelation function Definition of length scale (L) of the autocorrelation function (Daley, 1991) for the one-dimensional case L is a measure of the inverse curvature of the autocorrelation function at the origin For a sharp autocorrelation function, the curvature is large, so L is small. So, L gives an idea about how the autocorrelation function decays with distance, from its initial value. In pratice, L is a measure of the influence radius of one observation.

11. Autocorrelation of background errors The autocorrelation tends to zero faster in Ensemble method than in NMC method Surface pressure Length scale of autocorrelation Length scale of background errors are smaller in Ensemble method than in NMC method The length scale of vorticity is smaller than the one of temperature, this difference is smaller in Ensemble method than in NMC method

12. Ensemble Method NMC Method NMC Method Ensemble Method North-South variation of vertical correlation of background error Temperature Vorticity

13. NMC x 0.9 Model levels Vorticity Standard deviation Vertical profile of standard deviation of background error On operationalARPEGE 4D-VAR, the vertical profiles of total standard deviation of the background errors are rescaled by a factor of 0.9 to account for mismatch between the magnitudes of the 12/36-hours forecast differences and the 6-hour forecast errors To use the statistics from Ensemble Method it is need to rescale the vertical profile? what is the best factor? 1.5 (green curve)

14. Impact of background covariances estimated by Ensemble method (against NMC method) Forecast against ECMWF analysis Geopotential Wind

15. N Ly Lx E Autocorrelation function in gridpoint space Isotropic: Lx= Ly Lx < Ly Lx > Ly o OR

16. Length scale of autocorrelation autocorrelation Covariance of stream function => standard deviation of background error Helmholtz’s theorem => Rotational component of meridional wind Covariance of v between 2 points => Zonal length scale of autocorrelation => Meridional length scale of autocorrelation =>

17. Ensemble Method versus NMC Method Gridpoint Space Length scale of autocorrelation

18. Horizontal length scale of vorticity Lx Ly Ensemble LAND NMC SEA Ensemble SEA Lx Ly Ensemble EURATL NMC SEA Ly is larger than Lx, this difference is larger in EURATL region Ensemble GLOBAL NMC EURATL

19. Lx is smaller over land than over sea, mainly in Ensemble method Horizontal length scale of temperature Lx Ly Both Lx and Ly in Ensemble method are smaller than in NMC method Horizontal length scale of geopotential Lx Ly Ly is larger than Lx, also for temperature and geopotential

20. LAND SEA Horizontal length scale of wind (Zonal and meridional length scale of zonal and meridional wind) Lx (u) > Ly(u), except in PBL Lx (v) u is more anisotropic over sea than over land, except near surface Ensemble Method Ly (v) Lx (u) EURATL u is more isotropic in EURATL region Ly (v) > Lx(v), mainly in EURATL region => v is more anisotropic in this region

21. North-South variation of horizontal length scale of wind background error Lx(u) and Lx (v) in Ensemble method are smaller than in NMC method and both are larger in the tropics Lx(u) is larger than Lx (v) in both method

22. Conclusions • In Ensemble method the length scale of autocorrelation is shorter than in NMC method • This difference has a positive impact on forecasts • The meridional length scale is larger than the zonal length scale for all variables, except zonal wind • The meridional length scale is more homogeneous than the zonal length scale • The zonal length scale is larger over the tropics