What should an Outer Loop for ensemble data assimilation look like?

Transcription

1 What should an Outer Loop for ensemble data assimilation look like? Crai Bishop, Daniel Hodyss Naval Research Laboratory, Monterey, California Chris Snyder National Center for Atmospheric Research, Boulder, Colorado

2 Motivation When observations clearly indicate feature but none of the ensemble members has the feature, rerunnin ensemble in outer loop provides opportunity to obtain a better ensemble. Features of interest miht include clouds, precip or ribbons of chemicals/pv in the stratosphere.

3 What we ve done Alebraically derived error form or observation space form of weak and stron constraint incremental 4D VAR Based on interpretations of the meanin of the outer loop in observation space, proposed ensemble based analoues to incremental 4D VAR. Tested proposals on simple model of solitary waves.

4 Thins to consider If ensemble ives distribution of truth iven forecast then 4D ensemble covariance equals true 4D forecast error covariance matrix. In this case, (ensemble) Kalman smoother ives minimum error variance estimate (BLUE). What can an outer loop add to this? Would an outer loop turn an ensemble that is not the distribution of truth iven the forecast into one that does? Bishop and Shanley (2008, MWR) arue that some ensembles better approximate distribution of historical forecasts iven that today s ensemble mean is the truth than the prior distribution of truth. Does an ensemble outer loop make more sense from this perspective?

5 Weak constraint 4D VAR Minimize ( x) = ( x) + ( x) + ( x) J J J J b m o, where I I 1 ( ) ( f ) T 1 ( f J ) b x = x x 0, 2 0 P 0 x x T 1 J ( x) = M ( x ) x Q M ( x ) x, and m ( i 1) i i 1 i ( i 1) i i 1 i 2 i= 1 1 T 1 Jo ( x) = H( ) H( ) 2 y x R y x T

6 Linearize about uess field f x x 0 0 x 0 x 0 f f f δx= x x = x1 x1, δx = x x = x1 x1, f x2 x2 x2 x2 ( ) y' = y H x, H ( ) ( H ) ( x ) H x x = Hδ x, where = H T ( x ) M δ xi 1 = ( i 1) i i 1 ( i 1) i i x x 1, where M( i 1) i = ( ) ( ) ( M M ) i 1 i M ( x ) i 1 ( x ) ( i 1) i ( i 1) T

7 J J J b Incremental form ( δx) = ( δx) + ( δx) + ( δx) J J J J ( ) ( f ) 1 ( f δx = δx δx 0 P ) 0 0 δx δx ( ) ( ) 1 M 1 m i i i i 2 i= 1 o ( ) b m o { ( ) } Mδxi 1+ M x i 1 xi δxi T 1 ' '., where I I 1 T, { } T δx = δ + M x x x δx Q x 1 δx = δ δ 2 y H x R y H x T, and 1

8 Alebraic derivation shows that minimum of cost function occurs when ( ) ( ) ( ) 1 x x = x x + P H HP H + R y' Hδ x a f T T f, where T T T P0 P0M P0M M T T T T P = MP0 MP0M + Q MP0M M + QM, and T T T T MMP MMP M MQ MMP M M MQM Q f y ' ( f Hδ x y H x ), hence ( ) 1 H ( ) a f T T f x = x + P H HP H + R y x Because P is fixed, the outer loop only affects the analysis if chanes to the 0 uess and linearization state chane H and M.

9 Interpretation of outer loop Outer loop will consistently help analysis if it causes, and T T P P H HP H to better approximate the correspondin true forecast error covariance matrices. Improvement uaranteed if P ives distributi on of historical forecasts iven that the uess field was equal to the truth because, in this case, each improvement of the uess field causes P to better approximate a true forecast error covariance matrix. P Improvement not uaranteed if represents distribution of truth iven today's forecast because, in this case, it is not clear that centerin it closer to the analysis would improve its accuracy.

10 Solitary wave example (stron constraint) h (a) t=0 t=1 true wave A (b) true P t=0 t=1 Amplitude/position Space, covariance matrices fcst wave 1 st loop P x x crest A (c) h (d) Analysis at converence of outer loop toether with truth 2 nd loop P 1 st loop P x crest Example of effect of outer loop on analysis of solitary wave. Observations are of wave amplitude at every second rid point at the final time position of the wave. x

11 Ensemble Analoue T T Approximate P, P H and HP H usin ensemble counterparts centered on uess field. In the first inner loop, the initial ensemble perturbations are added to the mean of the ensemble forecast from the previous analysis at the beinnin of the window. The covariances from this ensemble are then used to obtain uess field. In the second outer loop, the initial ensemble perturbations are added to the uess field and then propaated usin the non-linear model. If model error is present the evolvin perturbations need to be recentered around the evolvin uess field from time to time. And so on until converence.

12 Ensemble outer loop profoundly reduces analysis error Solitary wave example (stron constraint) Error Error at initial time Error at final time Errors of optimal scheme Number of outer loops

13 Ensemble outer loop for stron constraint case 1. Make an analysis usin raw ensemble 2. Create new ensemble by re centerin initial perturbations on latest analysis rather than first uess and then interatin usin non linear model. 3. Make new analysis usin new ensemble 4. Repeat steps 2 and 3 until converence. 5. (Note: At no point in the iteration should the innovation be altered).

14 Issues for weak constraint case Simple minded analoue of weak constraint 4D VAR involves continual re centerin of ensemble about uess field. Dependin on type of model error, it is not obvious that this approach would yield the best approximation to the distribution of historical forecasts iven the truth.

15 Model error case position true position Distribution of historical forecasts In presence of model error The forecasts are biased in the sense that their positions have not advanced as far as the truth. However, in eneral, such biases are unknown. In this case, the minimum error variance is obtained usin a covariance matrix of the actual errors of these biased forecasts. time

16 position Model error case true position Guess of true state after first outer loop 1 st ensemble forecast in presence of model error usin perturbed physics time Initial ensemble is not centered on the truth. It has more spread than the ensemble of historical control/mean forecasts because each model has a different set of parameters. How can the outer loop be used to enerate a covariance similar to that on the previous slide?

17 This option is the closest ensemble approximation to what happens in weak constraint 4D VAR with TLM/adjoint. Note that it does not preserve the time continuity of the actual errors. Option 1 for 2 nd ensemble (breaks) position true position Guess of true state after first outer loop 2 nd ensemble is composed of discontinuous forecasts that are periodically re-centered on the uess field. Covariances computed about ensemble mean. time

18 Option 2 for 2 nd ensemble (no breaks) position true position Guess of true state after first outer loop 2 nd ensemble is composed of continuous forecasts that were centered on the uess at the initial time Covariances computed about ensemble mean. This option preserves the continuity of the errors but the computation of covariances is still computed usin differences about the mean rather than the best available estimate of the truth. time

19 Option 3 for 2 nd ensemble (no breaks) position true position Guess of true state after first outer loop 2 nd ensemble is composed of continuous forecasts that were centered on the uess at the initial time Covariances computed about uess field. This option preserves the continuity of the errors and covariances are computed usin differences about the best available estimate of the truth (the latest uess). time

20 Test of Options t=0 t=1 t=2 Variation of 0.1 in model error corresponds to a 1 sima deviation of a parameter that affects propaation of solitary wave Green line: Analysis obtained usin covariance of historical forecasts about truth. Option 1: Black line (Weak constraint 4D-VAR analoue with breaks in ensemble) Option 2: Blue line (No breaks but differences about mean) Option 2/3: Dashed red line (as in option 2 but differences about uess after 5 th loop) There are a number of options for weak constraint form of the ensemble outer loop. The one that is closest to TLM/adjoint weak constraint 4D-VAR is not necessarily the best.

21 Concludin Remarks Outer loop for ensemble DA provides potential mechanism for dealin with problem of feature in obs but not in ensemble Experiments with simple soliton model show how rerunnin ensemble in outer loop allows the ensemble covariance to better approximate the covariance of the distribution of historical forecasts iven the truth. Ensemble outer loop offers possibility of basin covariances on differences of non linear forecasts around best uess field rather than the ensemble mean. Soliton experiments show that difference about uess can be superior to difference about mean. There is no TLM/adjoint analoue of the difference about uess framework.