Comparison of 3D-Var and LETKF in an Atmospheric GCM: SPEEDY

Transcription

1 Comparison of 3D-Var and LEKF in an Atmospheric GCM: SPEEDY Catherine Sabol Kayo Ide Eugenia Kalnay, akemasa Miyoshi Weather Chaos, UMD 9 April 2012

2 Outline SPEEDY Formulation Single Observation Eperiments Observation Network Eperiments Long-term Instabilities and Biases Summary and Future Work 2

3 SPEEDY (Molteni 2003) Model Description Simplified Parameterizations, primitive-equation DYnamics Global atmospheric general circulation model of intermediate compleity 30 spectral resolution grid points 7 vertical levels (sigma coordinates) Eperimental Setup Output every 6 hours (Miyoshi 2005) Eperiments begin on January 1 st, 1982 after 1 year of spin-up Observation ype u v q p s Observation Error 1 m/s 1 m/s 1 K 10-4 kg/kg 100 Pa 3

4 Observation Networks 4

5 . 3D-Var Formulation with Preconditioning he analysis is obtained by finding the minimum (δv a ) of the cost function: J 1 1 ob 1 ob v v v d HU v R d HU v a analysis (R N ) b background (R N ) d ob innovation (R L ) H linear observation operator (R L N ) R observation error covariance matri (R L L ) δv preconditioned control vector (R N ) : 2 U v 2 a b U v a B U U U S F S background error standard deviation (R N N ) F spatial correlations (R N N ) (recursive filter, Purser 2003a) 5

6 LEKF Formulation Local Ensemble ransform Kalman Filter (Hunt et al 2007) Analysis is computed locally for each grid point Xˆ 1,..., a m a Xˆ m ensemble (R N M ) ensemble mean (R N ) M a ensemble spread (R N M ) 20 members Adaptive Inflation Horizontal Localization: 500 km Vertical Localization: 0.1 ln p where w a b b Xˆ w ~ a b ob P 1 1 Yˆ R d where Xˆ a W a ˆ b a X W ~ a P 1/ 2 (R M ) (R M M ) ~ a ~ 1 P b 1 b b P Yˆ R Yˆ ~ b P 1 k 1 I (R M M ) (R M M ) 6

7 Geostrophic Constraint he horizontal velocity, V, can be broken down into its balanced and unbalanced components: V rv g V u where V g geostrophic wind fk V g Rln p r linear regression coefficient for V and V g E r E g εε g g ε ε ε difference in 18 and 24 hr forecasts of V verifying at the same time (NMC method, Parrish and Derber 1992) r is computed so that V u and V g are statistically uncorrelated s

8 3D-Var, Geostrophic Constraint 8 o apply to 3D-Var, we transform the increment: where G linearized geostrophic transformation U z background error for the δz rather than δ he cost function becomes: p s q v u s u u p q v u z z GU z z HGU d R z HGU d z z z z ob z ob J

9 LEKF, Geostrophic Constraint 9 o apply to LEKF, we transform the ensemble to include the unbalanced wind: he analysis is performed on z rather than : Variable localization Removes correlation between V u and (, p s ) Employed through choice of observations used in local calculations m g z m s m m m m m m p q v u s m m m u m u m m p q v u z w Z z z b b a ˆ a b a W Z Z ˆ ˆ

10 Single Observation - (sig=0.51) u 3D-Var, Without Constraint Constant = 0 3D-Var, With Constraint 10

11 Single Observation - (sig=0.51) u LEKF, Without Constraint LEKF, With Constraint 11

12 Comparison: 3D-Var 3D-Var No Constraint Dense Sparse Realistic Free Run 3D-Var Constraint 12

13 Comparison: 3D-Var vs. LEKF 2-month 3D-Var and LEKF 15-year Dense No Constraint 3D-Var, No Geo 3D-Var, Geo LEKF, No Geo LEKF, Geo 13

14 Comparison: 3D-Var vs. LEKF 3D-Var, No Geo 3D-Var, Geo LEKF, No Geo LEKF, Geo Dense Sparse Realistic 14

15 Analysis Bias, (sig=0.51), 02/01-03/01 Dense Network No Constraint Constraint 3D-Var LEKF 15

16 Effect of Observation Networks Randomizing Observation Locations for the 3D-Var, Dense, Geostrophic Constraint Case Regularly Spaced Irregularly Spaced 16

17 Effect of Observation Networks 3D-Var, Sparse Network, Geostrophic Constraint Analysis Analysis Bias 17

18 Bias: Spatial Pattern Stationary waves Wave length: 4 grid points Observation locations occur between the crests and the troughs 18

19 K K K Bias: Evolution here is a significant warm temperature bias, highest in the upper troposphere. Analysis Step Global Analysis Bias for the (sig=0.34) Forecast Step 19

20 Why Assimilation Cannot Correct What the assimilation sees Innovation Analysis Increment 20

21 What Can We Do? Irregular observation locations Bias correction Change the length scale Reduce noise: Digital filter Spatial emporal not effective at removing biases stationary in time Smoothing the background error etends the assimilation Not assimilating q observations above the 4 th model level he value of q is equal or less than the observation error 21

22 What Can We Do? une the Background Error oo small Observations are not taken seriously enough, no convergence oo large Observations are taken too serious sharp, large increments increase noise 22

23 Summary 3D-Var and LEKF with and without the geostrophic constraint are implemented in the SPEEDY model. For each observational network and constraint option, LEKF outperforms 3D- Var. Biases and stability issues were encountered for the regularly spaced network cases when using the geostrophic constraint. Currently under investigation. Can be resolved by not using regularly spaced observations. Future work: Hybrid 3D-Var/LEKF for SPEEDY Evaluate for usefulness in the creation of a new reanalysis data set. 23

24 hank You 24