Estimating Observation Impact in a Hybrid Data Assimilation System: Experiments with a Simple Model

Transcription

1 Estimating Observation Impact in a Hybrid Data Assimilation System: Experiments with a Simple Model NOAA / NCEP / EMC College Park, MD 20740, USA 10 February 2014

2 Overview Goal Sensitivity Theory Adjoint Sensitivity Theory Ensemble Sensitivity Theory Observation Impact Hybrid Data Assimilation Experiments Summary

3 Goal Evaluate observation impact in a hybrid data assimilation system using ensemble statistics and validate them with adjoint methods

4 W A R N I N G

5 Sensitivity Theory Definitions state vector, perturbation, ensemble matrix: x, δx, δx forecast metric, perturbation, ensemble vector: J, δj, δj analysis time, verification time: t 0, t TLM: M t,t0 = M t,t δt M ti +δt,t i M t0+δt,t 0 ADJ: M T t,t 0 = M T t 0+δt,t 0 M T t i +δt,t i M T t,t δt

6 Adjoint Sensitivity Theory Linearized dynamics and metric response: δx t M t,t0 δx t0 δj J T δx t x t

7 Adjoint Sensitivity Theory Linearized dynamics and metric response: δx t M t,t0 δx t0 δj J T δx t x t δj J T [ M t,t0 δx t0 = x t M T t,t 0 ] T J δx x J t0 t x t0 T δx t0

8 Adjoint Sensitivity Theory Linearized dynamics and metric response: δx t M t,t0 δx t0 δj J T δx t x t δj J T [ M t,t0 δx t0 = x t M T t,t 0 ] T J δx x J t0 t x t0 T δx t0 J = M T J t,t x 0 = M T t t0 x 0+δt,t 0 M T t i +δt,t i M T t,t δt t J x t

9 Ensemble Sensitivity Theory δj and δx are random variables, assume Gaussian distributed: Let { } denote expectation. P = { δx t0 δx T t 0 } cov (Xt0, X t0 ) denote error covariance matrix Ancell and Hakim 2007

10 Ensemble Sensitivity Theory δj and δx are random variables, assume Gaussian distributed: Let { } denote expectation. P = { δx t0 δx T t 0 } cov (Xt0, X t0 ) denote error covariance matrix δj = J T δx t0 x t0 Ancell and Hakim 2007

11 Ensemble Sensitivity Theory δj and δx are random variables, assume Gaussian distributed: Let { } denote expectation. P = { δx t0 δx T t 0 } cov (Xt0, X t0 ) denote error covariance matrix { δjδx T t 0 δj = J T δx t0 x t0 } = J T δx t0 δx T t x 0 t0 Ancell and Hakim 2007

12 Ensemble Sensitivity Theory δj and δx are random variables, assume Gaussian distributed: Let { } denote expectation. P = { δx t0 δx T t 0 } cov (Xt0, X t0 ) denote error covariance matrix { δjδx T t 0 δj = J T δx t0 x t0 } = J T δx t0 δx T t x 0 t0 { } δjδx T J T { t0 = δxt0 δx T x t0} t0 Ancell and Hakim 2007

13 Ensemble Sensitivity Theory δj and δx are random variables, assume Gaussian distributed: Let { } denote expectation. P = { δx t0 δx T t 0 } cov (Xt0, X t0 ) denote error covariance matrix { δjδx T t 0 δj = J T δx t0 x t0 } = J T δx t0 δx T t x 0 t0 { } δjδx T J T { t0 = δxt0 δx T x t0} t0 cov (J, X t0 ) = J T P x t0 Ancell and Hakim 2007

14 Adjoint v/s Ensemble Sensitivity Theory Ensemble method recovers adjoint sensitivity J x t0 = P 1 cov (X t0, J) = M T J t,t 0 x t Simultaneous multivariate regression

15 Adjoint v/s Ensemble Sensitivity Theory Ensemble method recovers adjoint sensitivity J x t0 = P 1 cov (X t0, J) = M T J t,t 0 x t Simultaneous multivariate regression Ensemble Sensitivity - Pros and Cons Pros No adjoint model is required. No assumptions about on/off or moist processes. Rapidly evaluate many J (cf. new adjoint run for each J). Can apply statistical significance testing.

16 Adjoint v/s Ensemble Sensitivity Theory Ensemble method recovers adjoint sensitivity J x t0 = P 1 cov (X t0, J) = M T J t,t 0 x t Simultaneous multivariate regression Ensemble Sensitivity - Pros and Cons Pros No adjoint model is required. No assumptions about on/off or moist processes. Rapidly evaluate many J (cf. new adjoint run for each J). Can apply statistical significance testing. Cons Sampling error. Computing P 1 is impractical for high-dimensional problems.

17 Observation Impact obs b g f t 0 a t δe g f = ( (y Hx b ), K T Jg + J ) f x b x a Langland and Baker 2004

18 Observation Impact Adjoint Framework K T J g x b = K T M T t,t 0 J g x g K T J f x a = K T M T t,t 0 J f x f

19 Observation Impact Adjoint Framework K T J g x b = K T M T t,t 0 J g x g K T J f x a = K T M T t,t 0 J f x f K = BH T [ HBH T + R ] 1 = AH T R 1

20 Observation Impact Adjoint Framework K T J g x b = K T M T t,t 0 J g x g K T J f x a = K T M T t,t 0 J f x f K = BH T [ HBH T + R ] 1 = AH T R 1 Ensemble Framework K T J g x b = K T B 1 cov (X b, J g ) = [ HBH T + R ] 1 cov (HXb, J g ) K T J f x a = K T A 1 cov (X a, J f ) = R 1 cov (HX a, J f )

21 Hybrid Data Assimilation System EnKF EnKF EnKF x a x a x a Obs! Obs! Obs! B e B e B e J(x) = 1 2 [x x b] T B 1 h [x x b ] [Hx b y] T R 1 [Hx b y] B h = (1 β) B s + βb e C

22 Configuration Lorenz 1996 X i t = (X i+1 X i 2 ) X i 1 X i + F I = 40 F = 8.0 (perfect model), 8.4 Data Assimilation Three-D Var B s derived from very long integration with EnKF EnKF, Ne = 20 Inflation = 2% Localization = 4 points β = 0.75

23 Tuning the hybrid data assimilation system β e β e 1.2 RMSE - 3DVar RMSE RMSE - EnKF RMSE

24 Experiments Varying Observation Location H = 1 : All 40 observed H = 2 : Alternate 20 observed H = 3 : Random 20 observed Varying Observation Quality R = 1 : Uniform ob. quality R = 2 : Alternate good/bad ob. R = 3 : Random ob. quality (δj e δj a )/δj a 3??? R 2??? 1??? H

25 Experiment H = 1, R = 1 All 40 obs., Same quality assimilate obs. at all locations all obs. have same obs. error Adjoint-based mean ( δj 10 a ) δj = δj a + δj b Ensemble-based mean ( δj 10 e ) δj mean δj e : / mean δj a : / Assimilation Step

26 Experiment H = 3, R = 3 Random 20 obs., Random quality assimilate obs. at all locations all obs. have same obs. error Adjoint-based mean ( δj 10 a ) δj mean δj e : / mean δj a : / δj = δj a + δj b Assimilation Step Ensemble-based mean ( δj 10 e )

27 Experiments - Result Summary (δj e δj a )/δj a % -2.93% % R % -2.16% -6.86% % -2.16% -0.29% H 15

28 Summary and Future Work Summary Ensemble-method recovers adjoint sensitivity Applied ensemble sensitivity and validated against adjoint method based observation impact in Lorenz Future Work Apply ensemble technique in a full NWP system. Extend observation impact to 4DEnsVar. Your thoughts?

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30 Why Ne = 20? 1.8 RMSE RMSE Ensemble Size