1. Korea Institute of Atmospheric Prediction Systems (KIAPS) ( 재 ) 한국형수치예보모델개발사업단 Identical Twin Experiments for the Representer Method with a Spectral Element Shallow Water System Byoung-Joo Jung, Sangil Kim, Youngsoon Jo, Hyeon-Jin Shin Youngsoon Jo, Hyeon-Jin Shin KMS Spring Meeting April 17-19, 2013, Seoul, Korea

2. Outline Numerical model Representer method Background error covariance Identical Twin Experiment - design Result Summary 2

3. HOMME SW 3 NE=8, NP=4 Higher-Order Multiscale Modeling Environment Cubed sphere Continuous Galerkin (CG) or Spectral Element (e.g., CAM-SE) Gauss-Legendre-Lobatto (GLL) quadrature points Shallow water system

4. The method of Representer Bennett (1992,2002) 4 Representer function: time-space (four-dimensional) covariance function between a given k th observation and model space Representer coefficient Representer matrix: (KxK) time-space covariance matrix between given observations

5. 5 Read obs. data Specify initial guess state NL_model Get the innovation Get the Representer matrix & Sanity test Calculate the Rep. coeff. Construct analysis NL_model Single impulse forcing ADJ_model Covariance model TL_model Sampling of observations The method of Representer For this, the TL & ADJ model is developed and tested for the shallow water system.  KIAPS W. Session IV (Dr. Sangil Kim), Poster 대기역학 -10 (Ms. Youngsoon Jo)

6. Background Error Covariance In this study, The pseudo-heat equation is used for the spatial correlation. The variable correlation is not considered. 6 (Uni-variate) Spatial Correlation SQRT(Uni-variate error variance) Variable Correlation Covariance modeling and evaluation including variable transform  Poster 기상기술 -13

7. Identical Twin Experiment How to ? 1.Make a reference solution as a true state (or nature run) 2.(Estimate variance information) 3.Make simulated observation including errors 4.Data assimilation experiment 5.Evaluation 7

8. Nature Run, guess, and simulated obs. 8 Day 05 14 15 13 Good guess Bad guess Nature run (as a true state) Shallow water standard test cast (Williamson et al. 1992) Case 5: Isolated mountain case Simulated observation (Network-A) Make Observing System for Twin Exp. 6 hour window, every 1 hour except starting and ending time (i.e., 1, 2, 3, 4, 5 hour time) U-wind observation on (2,2) GLL grid for the randomly selected elements, with probability 0.5

9. Exp. design and evaluation 9 EXPObs. H. Correlation Length scale background EXP01Network-A5%mediumgood (13Day) EXP02A20%mediumgood EXP03A1%mediumgood EXP04A5%Nogood EXP05A5%smallgood EXP06A5%largegood EXP07A5%mediumbad (5Day) * Evaluation a) observational cost functional (Jo) from innovation (O-B) or residual (O-A) statistics (NOT independent observation) b) RMSE for u, v wind components and geopotential height verified with true state (reference solution) c) Representer function at initial time for obs #1 d) Analysis increment

10. Result 10 EXPObs. H. Correlation Length scale background EXP01Network-A5%mediumgood (13Day) EXP02A20%mediumgood EXP03A1%mediumgood EXP04A5%Nogood EXP05A5%smallgood EXP06A5%largegood EXP07A5%mediumbad (5Day) -16.1%-4.8%-43.6%-55.2%-34.2%-11.3%-25.4%

11. Result 11 EXPObs. H. Correlation Length scale background EXP01Network-A5%mediumgood (13Day) EXP02A20%mediumgood EXP03A1%mediumgood EXP04A5%Nogood EXP05A5%smallgood EXP06A5%largegood EXP07A5%mediumbad (5Day)

12. Result 12 EXP H. Correlation Length scale background EXP01mediumgood (13Day) EXP04Nogood EXP05smallgood EXP06largegood EXP07mediumbad (5Day)

13. Result 13 EXP H. Correlation Length scale EXP015%medium EXP0220%medium EXP031%medium EXP045%No EXP055%small EXP065%large

14. Observing System for Twin Exp. Network-B Observe U-wind on 20% station of radiosonde network (Bi-linear interpolation) Good or Bad guess 14

15. Result 15

16. Result 16

17. Summary A series of identical twin experiments are performed to evaluate the characteristics of the representer method, a variational method, for spectral element shallow water system. To do this, the tangent linear and adjoint model for SW system is developed and verified, also for observation operator. For background error covariance, the spatial correlation module is developed by solving the pseudo-heat equation. Identical Twin Experiment – Ratio between background & observational error covariance – Quality of background field – Spatial correlation length in background error covariance – Verifying measures: Jo term (misfit from observations), RMSE (w.r.t. reference solution), structure of initial representer function, analysis increment field Development of the representer method continues for a 3D hydrostatic model (HOMME/KIAPS). 17

19. Result 19 EXPObs. H. Correlation Length scale background EXP01Network-A5%mediumgood (13Day) EXP02A20%mediumgood EXP03A1%mediumgood EXP04A5%Nogood EXP05A5%smallgood EXP06A5%largegood EXP07A5%mediumbad (5Day)

20. Result 20 EXP H. Correlation Length scale background EXP015%mediumgood (13Day) EXP0220%mediumgood EXP031%mediumgood EXP045%Nogood EXP055%smallgood EXP065%largegood EXP075%mediumbad (5Day)