Xuguang Wang NOAA/ESRL/PSD, Boulder, CO Thomas M. Hamill Jeffrey S. Whitaker

1. A comparison of hybrid ensemble transform Kalman filter(ETKF)-3DVAR and ensemble square root filter (EnSRF) analysis schemes Xuguang Wang NOAA/ESRL/PSD, Boulder, CO Thomas M. Hamill Jeffrey S. Whitaker NOAA/ESRL/PSD, Boulder, CO Craig H. Bishop NRL, Monterey, CA

2. Why hybrid ETKF-3DVAR ? • Hybrid ETKF-3DVAR is expected to be less expensive than EnSRF, and still can benefit from ensemble-estimated error statistics. • The hybrid may be more robust for small ensemble size, since can adjust the amount of static vs. ensemble covariance used. • The hybrid can be conveniently adapted to the existing variational framework.

3. perturbation 1 perturbation 2 updated perturbation 3 updated perturbation 1 updated perturbation 2 perturbation 3 ensemble mean updated mean Hybrid ETKF-3DVAR member 1 forecast member 1 analysis member 1 forecast ETKF update perturbations member 2 forecast member 2 analysis member 2 forecast member 3 forecast member 3 analysis member 3 forecast ETKF-3DVAR update mean data assimilation forecast

4. Hybrid ETKF-3DVAR updates mean • Background error covariance is approximated by a linear combination of the sample covariance matrix of the ETKF forecast ensemble and the static covariance matrix. • Can be conveniently adapted into the operational 3DVAR through augmentation of control variables (Lorenc 2003; Buehner 2005; Wang et al. 2006).

5. ETKF updates perturbations • ETKF transforms forecast perturbations into analysis perturbations by where is chosen by trying to solve the Kalman filter error covariance update equation, with forecast error covariance approximated by ensemble covariance. • Latest formula for (Wang et al. 2004;2006, MWR) • Computationally inexpensive for ensemble size of o(100), since transformation fully in perturbation subspace.

6. Experiment design • Numerical model • Dry 2-layer spectral PE model run at T31; • Model state consists of vorticity, divergence and layer thickness of Exner function • Error doubling time is 3.78 days at T31 • Perfect model assumption (Hamill and Whitaker MWR, 2005) • Observations • 362 Interface and surface Exner functions taken at equally spaced locations • Observation values are T31 truth plus random noise drawn from normal distribution • Assimilated every 24h

7. First estimate BHTand HBHTis constructed from 250 24h fcst. errors with covariance localization (iteratively) Run a huge number of DA cycles (7000 > number of model dimension) with BHTand HBHT Construct BHTand HBHT with this huge sample of 24h forecast errors Experiment design • Update of mean (OI) and formulation of B • In this experiment, the hybrid updates the mean using whose solution is equivalent to that if solved variationally under our experiment design. • The static error covariancemodel is constructed iteratively from a large sample of 24h fcst. errors.

8. RMS analysis errors: 50 member • Improved accuracy of EnSRF over 3DVAR can be mostly achieved by the hybrid. • Covariance localization applied on the ETKF ensemble when updating the mean (but not applied when updating the perturbations) improved the analyses of the hybrid.

9. RMS analysis errors: 20 member • Both 20mem hybrid and EnSRF worse than 50mem, but still better than 3DVAR • Hybrid nearly as accurate (KE, 2 norms) or even better (surface  norm) compared to EnSRF

10. RMS analysis errors: 5 member EnSRF filter divergence • EnSRF experienced filter divergence for all localization scales tried. • Hybrid was still more accurate than the 3DVAR. • Hybrid is more robust in the presence of small ensemble size.

11. Comparison of flow dependent background error covariance models (Hybrid 50 mem. result) (EnSRF 50 mem. result)

12. Initial-condition balance • Analysis is more imbalanced with more severe localization. • Analyses of the hybrid with the smallest rms error are more balanced than those of the EnSRF, especially for small ensemble size (not shown). (50 mem. result)

13. Spread-skill relationships • Overall average of the spread is approximately equal to overall average of rms error for both EnSRF and Hybrid. • Abilities to distinguish analyses of different error variances are similar for EnSRF and Hybrid. (20 mem. result)

14. Summary • The hybrid analyses achieved similar improved accuracy of the EnSRF over 3DVAR. • The hybrid was more robust when ensemble size was small. • The hybrid analyses were more balanced than the EnSRF analyses, especially when ensemble size was small. • The ETKF ensemble variance was as skillful as the EnSRF. • The hybrid can be conveniently adapted into the existing operational 3DVAR framework. • The hybrid is expected to be less expensive than the EnSRF in operational settings.

15. Preliminary results with resolution model error • T127 run as truth; imperfect model run at T31 • 200-member ensembles • Additive model error parameterization • comparable rms errors for hybrid and EnSRF

16. BUFRyo OBS PROC yo Xa1 …. XaK mean Xb VAR Xa sum WRF Xf1 …. XfK δXf1 …. δXfK gen_be Pb ETKF H(Xf1) …. H(XfK) VAR da_ntmax=0 Hybrid ETKF-3DVAR for WRF(collaborated with Dale Barker and Chris Snyder) • α-control variable method (Lorenc 2003) to incorporate ensemble in WRF-VAR

17. Statistics of iteratively constructed B balance rms analysis errors

18. Ensemble square-root filter(Whitaker and Hamill,MWR,2002) • Background error covariance is estimated from ensemble with covariance localization. • Mean state is corrected to new observations, weighted by the Kalman gain. • Reduced Kalman gain is calculated to update ensemble perturbations. • Observations are serially processed. So cost scales with the number of observations. obs1 obs2 obs3 EnSRF EnSRF EnSRF • Covariance localization can produce imbalanced initial conditions.

19. Maximal perturbation growth • Find linear combination coefficients b to maximize • Maximal growth in the ETKF ensemble perturbation subspace is faster than that in the EnSRF ensemble perturbation subspace.