1. A comparison of 4D-Var with 4D-En-Var D. Fairbairn. and S. R. Pring
A comparison of 4D-Var with 4D-En-Var D. Fairbairn* and S.R. Pring * University of Surrey
1. Aim
Develop a toy model test bed;
Compare the deterministic analyses of 4D-Var, 4D-En-Var and 4D-Var-Ben (4D-Var with flow-dependent B matrix);
Understand the importance of a flow-dependent B matrix;
Link the toy model results to NWP.
4. a) Experimental setup – The toy model
Experiments are performed on Lorenz 2005 model 2. The model describes the propagation of waves of some unspecified atmospheric quantity across a latitude circle. It was chosen because the nonlinearity and spatial continuity can be controlled.
Model error introduced from having the truth at higher resolution than DA.
5. Results (b) 4D-En-Var vs 4D-Var-Ben
Analysis errors taken at the end of the window
Figure 2 shows the RMS analysis errors of the 4D-En-Var and 4D-Var-Ben methods against the ensemble size for the perfect model. For observations only at the beginning of the window they are identical. For evenly spaced observations 4D-Var-Ben performs significantly better than 4D-En-Var when the ensemble size is small. For a small ensemble the time correlation of the observations in 4D-En-Var is damaged by the severe localization that is required, since the localization matrix and the nonlinear model do not commute:
Figure 3 is equivalent to Figure 2 but for the imperfect model. The results more or less agree with the results for the perfect model.
2. Motivation
Operational centres are currently interested in the value of the ‘errors of the day’ in the background error covariance matrix;
Relatively new DA methods such as 4D-En-Var and 4D-Var-Ben take advantages from both variational and ensemble DA methods. However, they have not been tested extensively using toy models;
Operational experiments by Buehner et al (2010) have shown than 4D-Var does less well than flow-dependent DA methods in the Southern Hemisphere (SH) but the results are similar in the Northern Hemisphere (NH). Some possible reasons for their results were:
SH has sparser surface observations than NH;
Localization issues with radiance observations in SH;
Balance values of climatological B.
Buehner et al (2010) also found their 4D-Var-Ben performed slightly better than their 4D-En-Var;
These experiments aim to compare similar methods to Buehner et al (2010), but for a toy model system.
4. b) Experimental setup – The DA
DA methods use timesteps of 0.1 time units and an assimilation window length of 0.5 time units (corresponds to about 6 hours in the real atmosphere);
Range of observation densities tested for two scenarios:
a) Observations only at the beginning of the window;
b) Observations evenly spaced in time.
Observations are random in space and R = N(0,0.1);
DEnKF analysis update of 4D-En-Var to maintain ensemble spread:
Fixed covariance inflation and Gaspari-Cohn localization (since ensemble size << model dimension) applied to 4D-En-Var;
Perfect climatological B tuned for 4D-Var.
3. DA methods
Notation:
6. Conclusion
In our toy model, 4D-Var cannot compete with the 4D-En-Var or 4D-Var-Ben methods since it does not measure the ‘errors of the day’ in the background error covariance matrix;
The results from these toy model experiments cannot directly be related to operational results. The toy model has relatively few degrees of freedom, which can be quite well sampled by the ensemble. An NWP has millions of degrees of freedom, even with clever localization techniques, it is unlikely that a relatively small ensemble can sample them all, while the B used in 4D-Var can;
4D-En-Var produces analyses that are similar to 4D-Var-Ben when the ensemble size is large. For a small ensemble, 4D-En-Var suffers from localization issues, which damages the time correlation of the observations;
The results would suggest that for the same ensemble size, 4D-Var-Ben would perform better than 4D-En-Var in operational practice, which agrees with the operational experiments by Buehner et al (2010).
If the extra computational cost of implementing 4D-Var-Ben were spent on a larger ensemble for 4D-En-Var, then perhaps 4D-En-Var would produce a more accurate deterministic analysis than 4D-Var-Ben – but this requires research on operational systems!
5. Results (a) 4D-Var vs 4D-En-Var
Analysis errors taken at the end of the window and perfect model used;
Figure 1: 4D-En-Var vs 4D-Var. 4D-En-Var beats 4D-Var when the ensemble size is large enough, since 4D-En-Var estimates the ‘errors of the day’ in the background error covariance matrix.
Acknowledgements
David Fairbairn’s supervisor Andrew Lorenc has been very helpful. This work was funded by EPSRC and the Met Office.
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