Some Innovative Applications and Approaches Using Nudging Four Dimensional Data Assimilation: Lili Lei and David R. Stauffer Dept. of Meteorology, Penn State University Conference on Applied Inverse Problems Data Assimilation for Geophysical Problems 23 July 2009 University of Vienna, Austria Part II. A Hybrid Nudging-EnKF for Improving Data Assimilation in the Lorenz and Shallow-Water Model Systems

UNCLASSIFIED Outline Motivation Methodology Experimental design for Lorenz model Results of Lorenz model Experimental design for shallow-water model Results of shallow-water model Conclusions Future work and acknowledgement

UNCLASSIFIED Motivation Fujita et al Mon. Wea. Rev.

UNCLASSIFIED time t obs time t obs time t obs Nudging: EnKF: Hybrid nudging-EnKF: Methodology for Hybrid Nudging-EnKF

UNCLASSIFIED Methodology for Hybrid Nudging-EnKF Ensemble state Nudging state OBS time

UNCLASSIFIED Methodology for Hybrid Nudging-EnKF The hybrid nudging coefficients: The Lorenz model equations:

UNCLASSIFIED Experimental Design Single 3000-step period experiment: From an initial condition, first 1500 time steps of integration are discarded to avoid the effects of transients, and the following time steps are analyzed. 100-sample experiment:100 initial conditions are randomly chosen, and a data assimilation cycle of 1500 time steps is executed following each initial condition. In each data assimilation cycle, the first 500 time steps of integration are discarded, and the following 1000 time steps are used for analysis. The observation error variances used to create simulated observations:

UNCLASSIFIED Experimental Design Exp. NameExp. Description TNUD Assimilate observations by traditional nudging with nudging coefficient of 10 EnKFAssimilate observations by ensemble Kalman filter (EnKF) EnKSAssimilate observations by ensemble Kalman smoother (EnKS) EnKS_lag Assimilate observations by lagged ensemble Kalman smoother (EnKS) which applies next available observation backward to previous observation time Hybrid-D Assimilate observations by hybrid nudging-EnKF with diagonal elements only HybridAssimilate observations by hybrid nudging-EnKF with full matrix

UNCLASSIFIED Experimental Design: Verification The root-mean-square (RMS) errors are computed every time step. Observation Retention (OR): the average absolute value of the RMS error difference between one time step before the observation time and that at the observation time after the data assimilation. Normalized Error and Retention (NER): sum of the average RMS error normalized by that of the EnKS and the OR normalized by that of the EnKS.

UNCLASSIFIED Comparisons of Hybrid-D and Hybrid RMSE Nudging coefficients in Hybrid-D Nudging coefficients in Hybrid

UNCLASSIFIED RMSE OR NER Average parameters in single step period experiment with ensemble size 100 and perfect model

UNCLASSIFIED RMSE OR NER Average parameters in 100-sample experiment with ensemble size 100 and perfect model

UNCLASSIFIED RMSE OR NER Average parameters in single step period experiment with ensemble size 100 and imperfect model

UNCLASSIFIED RMSE OR NER Average parameters in 100-sample experiment with ensemble size 100 and imperfect model

UNCLASSIFIED CPU Time Cost (sec) obsfreq10obsfreq25obsfreq50 TNUD455 EnKF16109 EnKS Hybrid16109

UNCLASSIFIED CPU Time Cost (sec) obsfreq10obsfreq25obsfreq50 TNUD455 EnKF16109 EnKS EnKS_lag Hybrid16109

UNCLASSIFIED Comparisons of EnKS_lag and EnKS RMSE OR

UNCLASSIFIED Summary of Lorenz Model Results A hybrid nudging-EnKF approach with potential use for NWP was explored here using the Lorenz three-variable model system. The EnKS, which is the golden standard, is more than 100 times more expensive than the EnKF and Hybrid, and it also has large data storage requirements. The EnKS_lag, which is only 4~6 times more expensive than the EnKF and Hybrid, is more practical but has somewhat larger RMS errors and Observation Retention (OR) than the EnKS. The hybrid nudging-EnKF with diagonal elements only has larger RMS error than the hybrid nudging-EnKF with full matrix. The hybrid nudging-EnKF approach produces somewhat larger / similar average RMS errors than both the EnKF in perfect / imperfect model. The hybrid nudging-EnKF has better OR than both the EnKF and the EnKS in general. The hybrid nudging-EnKF approach generally produces smaller (better) Normalized Error and Retention (NER, normalized by EnKS) than the EnKF.

UNCLASSIFIED Methodology for Hybrid Nudging-EnKF The shallow water model equations: L = 500 km, D = 300 km

UNCLASSIFIED Experimental Design: Initial Conditions Case I - WaveCase II - Vortex

UNCLASSIFIED Case ICase II D300 km L500 km f10 -4 s -1 g0.5 ms ms m 2 s -1 dx / dy10 km dt30 sec B.C. Periodic B.C in west-east Free-slip rigid wall B.C in south- north Tendencies of height and wind components = 0.0 in south-north Inflation factor1.1 Localization scale500 km100 km Half-period nudging time window 1 h

UNCLASSIFIED Experimental Design: Observations Simulated 3-hourly observations are generated by finer-scale model simulations. The fine domain has grid spacing of 1 km. The observation error variances used to create simulated observations: Observation networks: Case I: Case II: OBSN I: 1 OBS OBSN II: 19 OBS in X direction OBSN III: 11 OBS in Y direction OBSN IV: OBSN II + OBSN III

UNCLASSIFIED Experimental Design Exp. NameExp. Description TNUD Assimilate observations by traditional nudging with nudging coefficient of s -1 EnKFAssimilate observations by ensemble Kalman filter (EnKF) EnKS Assimilate observations by lagged ensemble Kalman smoother (EnKS) which applies next available observation backward to previous observation time every 30 minutes HybridAssimilate observations by hybrid nudging-EnKF with full matrix

UNCLASSIFIED Experimental Design: Verification The verification data is based on the 1-km model simulation and available on every grid point of the 10-km coarse domain. The verification data is the average value of surrounding 10*10 1-km grid points from the 1-km “ truth ” domain. The root-mean-square (RMS) errors of height and wind are computed separately every minute. Normalized RMS error: the RMS error computed against the “ truth ” divided by the RMS error of the “ truth ” computed against its domain- average value. Observation Retention (OR): the average absolute value of the RMS error difference between one time step before the observation time and that at the observation time after the data assimilation. Normalized Error and Retention (NER): sum of the average RMS error normalized by that of the EnKS and the OR normalized by that of the EnKS.

UNCLASSIFIED Normalized RMS Error of Case I with OBSN II Height Wind

UNCLASSIFIED RMSE RMSE – 30min OR NER NER – 30min Average parameters of height field with different observation frequencies (in hours) in OBSN II

UNCLASSIFIED RMSE RMSE – 30min OR NER NER – 30min Average parameters of wind field with different observation frequencies in OBSN II

UNCLASSIFIED RMSE OR NER Average parameters of height field with three-hourly observations in different observation networks

UNCLASSIFIED RMSE OR NER Average parameters of wind field with three-hourly observations in different observation networks

UNCLASSIFIED Normalized RMS Error of Case II with OBSN II Height Wind

UNCLASSIFIED RMSE RMSE – 30min OR NER NER – 30min Average parameters of height field with different observation frequencies in OBSN II

UNCLASSIFIED RMSE RMSE – 30min OR NER NER – 30min Average parameters of wind field with different observation frequencies in OBSN II

UNCLASSIFIED RMSE OR NER Average parameters of height field with three-hourly observations in different observation networks

UNCLASSIFIED RMSE OR NER Average parameters of wind field with three-hourly observations in different observation networks

UNCLASSIFIED A hybrid nudging-EnKF data assimilation approach is further investigated using a shallow-water model. A quasi-stationary wave (Case I) and a moving vortex (Case II) are used to test the hybrid nudging-EnKF scheme. Three kinds of observation frequencies and four observation networks are applied in the 24-h data assimilation experiments for each case. The hybrid EnKF reduces the RMS errors compared to those of the traditional nudging and EnKF applied separately. The hybrid EnKF also has the ability to reduce the RMS error as well as or even better than the “ gold standard ” EnKS, and also to produce better observation retention than the EnKS at a reduced computational cost more similar to that of the EnKF. Summary of Shallow-Water Model Results

UNCLASSIFIED General Conclusions A hybrid nudging-EnKF data assimilation approach is investigated using the Lorenz model and a shallow-water model. The hybrid nudging-EnKF retains the spatial (flow-dependent) error correlation weighting function from the EnKF and the gradual corrections of the continuous nudging approach (digital filter unnecessary) to avoid the strong corrections and discontinuities (error spikes) at the analysis steps. In the hybrid nudging-EnKF, the model equations assist in the data assimilation process.

UNCLASSIFIED Future Work Test the hybrid EnKF in strongly forced / unstable conditions Test the hybrid EnKF in forecasting … Transition hybrid EnkF to WRF

UNCLASSIFIED ACKNOWLEDGEMENTS This research is supported by DTRA contract no. HDTRA1-07-C-0076 under the supervision of John Hannan of DTRA. The authors would like to thank Aijun Deng, Sue Ellen Haupt, George S. Young and Fuqing Zhang for helpful discussions and comments.