1. Initial trials of 4DEnVAR
Neill Bowler, EMS
Andrew Lorenc, Peter Jermey, David Fairbairn, Eunjoo Lee and Stephen Pring

2. 4DEnVar (Ensemble-Var)
Current operational scheme is hybrid-4DVar (2003/2011)
44 member ensemble (ETKF) (~33km resolution 800x600)
Motivation for change?
Maintainability of 4DVar is a challenge (particularly given we expect to change dynamical core)
Scalability of 4DVar is a future issue
Why 4DEnVar?
No use of linear models (tangent linear/adjoint models)
Cheaper minimisation algorithm (similar to hybrid-3DVAR)
Pre-calculated ensemble members define 4D ensemble covariance

3. 4D-Var Hybrid MOGREPS-G 4D-Var Linear model Linear model 21Z 0Z 3Z 6Z
With a 4D-Var hybrid the ensemble forecast runs to the start of the next DA window, then provides information to the assimilation. 4D-Var uses a linear model to propagate this information through the time window. The resulting analysis is used to inform the starting conditions of the next ensemble forecast, which in turn provides uncertainty information to the assimilation as before. Note that it is the use of a linear model which greatly increases the cost of 4D-Var above other data-assimilation methods.
Linear model
Linear model
21Z 0Z 3Z 6Z 9Z
12Z
15Z

4. 4D-Ens-Var MOGREPS-G 4D-Ens-Var 21Z 0Z 3Z 6Z 9Z 12Z 15Z
4D-Ens-Var uses the fact that the ensemble forecast normally runs through the entire time of the next data assimilation window. Rather than providing uncertainty information only at the start of the DA window, information is provided throughout the entire window. This means that a linear model is no longer required to propagate the ensemble information through the window, making the assimilation much cheaper without great loss of accuracy. As before, once the analysis is produced this is used by the ensemble in producing the next set of forecasts, which are once again fed to the data assimilation.
This is an approach which has been tested with the Canadian ensemble, with promising results, and is being developed for use in America.
21Z 0Z 3Z 6Z 9Z
12Z
15Z

5. Trial configurations
Analysis increments calculated at ~60km (432x325 grid)
44 ensemble members at ~60km resolution (ETKF)
Deterministic forecast resolution ~25km (1024x769)
Trial period – 9/10/ /11/2011
Hybrid-3DVar /50 = Climatological B / Ensemble B
4DVar /0 = Climatological B / Ensemble B
Hybrid-4DVar /50 = Climatological B / Ensemble B
4DEnVar /50 = Climatological B / Ensemble B

6. IAU-like interface with forecast model
4D-Var control variables gives initial δx, implicitly defining δx.
δx is initialised by Jc term.
Natural to add δx at beginning of forecast.
4DEnVar δx is defined for all window.
There is no internal initialisation.
Nudge in δx during forecast, as part of an IAU-like initialisation. (Bloom et al. 1996)
© Crown copyright Met Office

7. Results ±2% RMSE thresholds (Better/neutral/worse)
Forecast period: 9th October 2011 – 9th November 2011

8. 4DEnVar44 v hybrid-3DVar44

9. 4DEnVar44 v 4DVar

10. 4DEnVar44 v hybrid-4DVar44

11. Hybridisation/Localisation/Balance
Climatological B is now only 3D at all times in the window
Ensemble must represent the correct flow-dependent errors
Localisation
Correlations can move outside localisation radius
The localisation function used is exp(-z2/2L2) L=1200km
Covariances scaled by 0.61 at z=1200km
Covariances scaled by 0.14 at z=2400km
Balance of analysis increments
Is the IAU-like initialisation optimal?

12. Hybridisation 100:0
4DEnVar
L=1200km
100% Ens
0% Clim
4DVar

13. Localisation 100:0; L=500km
4DEnVar
L=500km
100% Ens
0% Clim
4DVar

14. Hybridisation 50:50
4DEnVar
L=1200km
50% Ens
50% Clim
4DVar

15. Hybridisation Ens/Clim=0/100
4DEnVar
L=1200km
0% Ens
100% Clim
4DVar
u=+10m/s
500hPa

16. Single obs with/without Jc
4DVar – no Jc
L=1200km
0% Ens
100% Clim
4DVar – Jc
u=+10m/s
500hPa

17. 4DEnVar176 vs 4DEnVar44

18. Conclusions 4DEnVar has some attractions Our results indicate
Computational cost and scalability
No linear model to develop / maintain
Our results indicate
Hybrid 4DVar performs better than hybrid 4DEnVar
Largest differences: strong use of static covariance
[Localisation and balance are also issues]
176m ETKF better than 44m ETKF
Need to understand weightings

19. Future work Improve the ensemble Localisation
An ensemble of 4DEnVARs (currently being set up)
Inflation – additive inflation, relaxation to prior spread
Perturbed observations?
Localisation
Flow adaptive and scale-dependent localisation
Tests with large ensemble sizes
Can we give more weight to the ensemble covariance?
Localisation in spectral space (wavebands) – coded and being tested
Optimisation of the code