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GSI-based four dimensional ensemble-variational (4DEnsVar) data assimilation: formulation and single resolution experiments with real data for NCEP GFS Ting Lei, Xuguang Wang University of Oklahoma, Norman, OK, USA Wang and Lei, MWR, 2013 Daryl Kleist (NCEP): dual resolution 4DEnsVar Acknowledgement: NCEP DA team (John Derber, Dave Parrish, Russ Treadon, Miodrag Rancic) and Jeff Whitaker (NOAA ESRL) 6 th WMO Symposium on data assimilation College Park, MD, USA 1 Oct. 7-11, 2013

Background  Over the past three years, significant efforts were conducted to develop GSI hybrid system and test with US operational Global Forecast System (GFS). The GSI hybrid DA system showed significant improvement compared to GSI 3DVAR and became operational on May 22, 2012 for GFS.  It has also been extended to a 4DEnsVar hybrid (“Tangent linear and adjoint model free”) and showed further improvements.  Efforts are being conducted to further develop and research GSI hybrid DA for operational regional forecast systems, e.g., • Xu Lu poster on GSI hybrid for Hurricane-WRF (HWRF) • Xuguang Wang Thur. talk on convective scale weather over CONUS 2 2

GSI-based hybrid ensemble-variational DA system EnKF member 1 member 1 member 1 Re-center EnKF analysis ensemble analysis 1 forecast analysis forecast EnKF to control analysis EnKF member 2 member 2 member 2 analysis 2 forecast analysis forecast Ensemble covariance EnKF member k member k member k analysis k forecast analysis forecast control control control GSI forecast analysis forecast First guess data assimilation forecast 3 Wang, Parrish, Kleist, Whitaker, MWR, 2013

Various hybrid schemes Wang and Lei, MWR, 2013 4

NCEP pre-implementation test of 3DEnsVar hybrid http://www.emc.ncep.noaa.gov/gmb/wd20rt/experiments/prd12q3s/vsdb/ • Significant improvement of operational GFS forecasts 5

GSI hybrid for GFS: GSI 3DVar vs. 3DEnsVar vs. EnKF  Hybrid was better than 3DVar due to use of flow-dependent ensemble covariance  Hybrid was better than EnKF due to the use of tangent linear normal mode balance constraint (TLNMC) Wang, Parrish, Kleist and Whitaker, MWR, 2013 6

GSI-based 4DEnsVar hybrid: motivation • Account for temporal evolution of error covariance that was ignored in 3DEnsVar hybrid. • Still need to improve computational efficiency of traditional TL/ADJ 4DVAR being developed for GSI (Rancic et al. 2012). • An alternative to TL/ADJ 4DVar, i.e., 4D-Ensemble-Variational method (4DEnsVar) is therefore developed • Conveniently avoid TL/ADJ of forecast model like earlier work and also applied localization inside variational minimization (e.g., Buehner et al. 2010) 7

GSI-based 4DEnsVar hybrid: formulation • Naturally extended from and unified with GSI-based 3DEnsVar hybrid formula (Wang 2010, MWR), which uses extended control variables to incorporate ensemble like in Lorenc 2003, Buehner 2005, Wang et al., 2007, Wang et al. 2008) Add time dimension in 4DEnsVar α ' J x , J J J 1 1 1 2 e o 1 1 1 T T 1 α α ' ' T 1 o T - 1 o x B x C ( y '-H x ) R ( y '-H x ) 1 1 static 1 2 t t t t t t t t 1 2 2 2 K ' α ' e  x x ( x ) t 1 k k t k 1 B stat 3DVAR static covariance; R observation error covariance; K ensemble size; e C correlation matrix for ensemble covariance localization; x k th ensemble perturbation; k ' ' o ' x 3DVAR increment; x total (hybrid) increment; y innovation vector; 1 H linearized observation operator; 1 weighting coefficient for static covariance; 2 weighting coefficient for ensemble covariance; α extended control variable. 8

Experiment Design • Time period : Aug. 15 2010 – Sep. 20 2010; • Model : GFS T190L64 • Observations : all operational data • Verification : global forecast and hurricane track forecasts. Experiment Description GSI3DVar The GSI 3DVar experiment 4D ensemble-variational DA experiment with hourly 4DEnsVar ensemble perturbations 4D ensemble-variational DA experiment with 2-hourly 4DEnsVar-2hr ensemble perturbations 3D ensemble-variational DA experiment 3DEnsVar Same as “4DEnsVar - 2hr” except without the use of the 4DEnsVar-nbc tangent linear normal mode balance constraint (TLNMC) 9

One obs. example for TC – 3h increment propagated by model integration 4DEnsVar 3DEnsVar t=0 t=0 t=0 * time -3h 0 3h 10

Another example Temp. Height Downstream impact Upstream impact 11

Global forecasts verified against ECMWF analyses • Forecasts from 3DEnsVar are more skillful than GSI3DVar. • 4DEnsVar further improves the skill of the forecasts compared to 3DEnsVar. • The improvement of 4DEnsVar relative to 3DEnsVar is smaller than the improvement of 3DEnsVar relative to GSI3DVar. 12

6-hour forecasts verified against conv. obs. • 3DEnsVar and 4DEnsVar are more accurate than GSI3DVar at most levels. • More appreciable improvement is seen in the wind than the temperature forecasts. • Over NH and SH, 4DEnsVar shows consistent improvement relative to 3DEnsVar for wind forecasts and neutral or slightly positive impact for temperature forecast. • Over TR, 4DEnsVar shows mostly neutral impact compared to 3DEnsVar for both wind and temperature forecasts (not 13 show). K m/s

96-hour forecasts verified against conv. obs. • Temperature forecasts from 4DEnsVar show overall positive impact relative to 3DEnsVar for both NH and SH. • 4DEnsVar shows neutral impact on wind forecasts over NH and positive impact over SH. • Over TR, 4DEnsVar shows positive impact relative to 3DEnsVar only for wind forecasts at low level (not show) 14 K m/s

Verification of hurricane track forecasts: cases 16 named storms in Atlantic and Pacific basins during 2010 15

Verification of hurricane track forecasts: RMSE and percentage of better forecast • 3DEnsVar outperforms GSI3DVar. • 4DEnsVar are more accurate than 3DEnsVar after the 2- day forecast lead time. 16

Impact of number of time levels of ensemble perturbations and TLNMC • Negative impact of less time levels of ensemble perturbations. • Positive impact of TLNMC for global forecasts. 17

Impact of number of time levels of ensemble perturbations and TLNMC • Negative impact of less time levels of ensemble perturbations • Negative impact of TLNMC on TC track forecasts 18

Convergence rate and computational cost • Slightly slower convergence for the first outer loop and slightly faster convergence for the second outer loop • The cost of 4DEnsVar variational minimization is approximately 1.5 times of that of 3DEnsVar. 19

Summary and Discussion  GSI based 4DEnsVar was developed and tested for NCEP GFS.  4DEnsVar further improved upon 3DEnsVar • At short lead times, the improvement of 4DEnsVar relative to 3DEnsVar over NH was similar to that over SH. At longer forecast lead times, 4DEnsVar showed more improvement in SH than in NH. • The improvement of 4DEnsVar over TR was neutral or slightly positive when forecasts were verified against the in-situ observations. • The hurricane track forecasts initialized by 4DEnsVar were more accurate than 3DEnsVar after the 2-day forecast lead time.  Temporal localization is being developed within GSI-based 4DEnsVar. Preliminary tests showed positive impact of the temporal localization.  Further development of TLNMC.  Tests of 4DEnsVar at dual resolution mode (Daryl Kleist) 20

Cited references • Buehner, M., 2005: Ensemble-derived stationary and flow-dependent background-error covariances: evaluation in a quasi-operational NWP setting.Quart. J. Roy. Meteor. Soc., 131 ,1013-1043. • Buehner, M.,P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, 2010: Intercomparison of Variational Data Assimilation and the Ensemble Kalman Filter for Global Deterministic NWP. Part I: Description and Single-Observation Experiments. Mon. Wea. Rev., 138 , 1550-1566. • Lorenc, A. C. 2003: The potential of the ensemble Kalman filter for NWP – a comparison with 4D-VAR. Quart. J. Roy. Meteor. Soc.,129, 3183-3203. • Wang, X., C. Snyder, and T. M. Hamill, 2007: On the theoretical equivalence of differently proposed ensemble/3D-Var hybrid analysis schemes. Mon. Wea. Rev. , 135 , 222-227. • Wang, X., D. Barker, C. Snyder, T. M. Hamill, 2008: A hybrid ETKF-3DVAR data assimilation scheme for the WRF model. Part I: observing system simulation experiment. Mon. Wea. Rev. , 136 , 5116-5131. • Wang, X., 2010: Incorporating ensemble covariance in the Gridpoint Statistical Interpolation (GSI) variational minimization: a mathematical framework. Mon. Wea. Rev. , 138 ,2990-2995. • Wang, X., D. Parrish, D. Kleist and J. S. Whitaker, 2013: GSI 3DVar-based Ensemble-Variational Hybrid Data Assimilation for NCEP Global Forecast System: Single Resolution Experiments. Mon. Wea. Rev. , in press. 21