1. Sabrina Rainwater David National Research Council Postdoc at NRL with Craig Bishop and Dan Hodyss Naval Research Laboratory Multi-scale Covariance Localization 1

2. We discuss multi-scale covariance localization within the context of an EnKF. In particular, – We used a modified version of the ensemble Kalman filter described in Posselt and Bishop (2012). – It is optimal when the rank of the estimated P b is larger than the rank of R. – We modified it to accept small ensembles with a localized P b (localization increases the rank of the estimated P b ). 2 Posselt and Bishop EnKF

3. In ensemble data assimilation, Distant locations have uncorrelated background errors, But sampling error induces artificial correlations. So, we attenuate the ensemble estimated correlations with a distance function. This works well when the scale of the errors is uniform. However, … 3 Covariance Localization

4. Weather phenomena (and the associated errors) happen on a variety of scales Left: convection within a mid-latitude cyclone. Also shown: the scale of the phenomena The scale of the errors is smaller than the scale of the phenomena. 4 Our Multi-scale World

5. When the background errors are uncorrelated in space, – the background error covariance matrix P b is diagonal (zero off-diagonal correlations), – i.e only one nonzero element for each row/column of P b, – so a plot of the central row will show a spike. – similar plot if background errors are only weakly correlated, with small-scale fluctuations (red) When the background errors are correlated in space, – there are off-diagonal correlations, – so a plot of the central row of P b will be a smooth curve with a max in the center (blue). When the background errors have multi- scale correlations, – The central row of P b could look like a Prussian helmet (black), – with a smooth curve for the broad scales and a spike for the small-scales. 5 Multi-scale Covariance Construction small scales large scales Central row of P b

6. Legend: – Black: the true covariance – Blue: the estimated covariance – Magenta: the covariance localization function As mentioned previously, the ensemble estimated covariance matrix (top) is subject to sampling error. When there are multiple scales, single-scale covariance localization (bottom) compromises between – eliminating the spurious small-scale correlations, – retaining the genuine large-scale correlations. 6 Ensemble Estimate and Single-Scale Compromise Some large-scale correlations eliminated Some spurious correlations retained

7. Legend: – Black: the true covariance – Blue: the estimated covariance – Magenta: the covariance localization function Sharp localization (left) – – Pro: eliminates the spurious small- scale correlations – Con: eliminates the true large-scale correlations Broad localization (right) – – Pro: retains the large-scale correlations – Con: retains the spurious small-scale correlations Multi-scale localization (bottom) – Pro: Eliminates the spurious small- scale correlations – Pro: Retains the genuine large-scale correlations 7 Localization Functions by Scale larger r retains large-scale correlations smaller r eliminates spurious correlations controls spurious correlations without sacrificing large scale correlations

8. 8 Methodology

9. Buehner (2012) – Similar to our technique but more complex, involving wavelets. Zhang et al. (2009) – Localization scale depends on observation type Miyoshi and Kondo (2013) – Combines the analysis increments from different localization scales Bishop et al. (2007, 2009a, 2009b, 2011) – Adaptive localization scale depends on location 9 Alternate Multi-scale Localization Techniques

10. The model is a statistical two-scale 1D model (a) A multi-scale state as the sum of large-scale waves (blue) and small- scale waves (red) (b): the same as (a) except in spectral space. 10 Statistical Model Small scales Large scales Model space Spectral space

11. Lorenz Model 2 is a smoothed version of the Lorenz 40-variable model The smoothing parameter determines the scale of the waves We created a modified Model 2 with two scales 11 Modified Lorenz Models K L =32, K s = 2

12. Compared ensemble data assimilation for – No localization – Single-scale localization – Single-scale localization with cross-correlations removed (i.e. multi-scale localization with C L =C S ) – Multi-scale localization Two different models Four different ensemble sizes for each model – Localization reduces the necessary ensemble size due to a lower dimensionality locally than globally. – So for smaller ensemble sizes, localization is more important. 12 Experiments

13. 13 Results Statistical Modified M2 (c) (b) Time averaged mean squared error for various scenarios – Bar: average over 7 trials – Error bars: standard error in the mean – Asterisks: results for each trial – Purple line: theoretical minimum error (a) statistical model results (b) Modified Model 2 results Statistical Modified M2

14. Time averaged mean squared error for various scenarios – Bar: average over 7 trials – Error bars: standard error in the mean – Asterisks: results for each trial – Purple line: theoretical minimum error (a) statistical model results (b) Modified Model 2 results 14 Results Statistical Modified M2 (b) (c)

15. Multi-scale localization is always better than removed cross-correlations (green lower than sky-blue) When localization is most beneficial (small ensemble size), multi-scale localization improves upon single-scale localization. (green lower than cyan) Removing the cross-correlations does not always improve results (sky-blue sometimes higher than cyan) – Some cross-correlations could be genuine – Scale-separation techniques are imperfect 15 Results and Discussion * trial results □ average of trials -standard error ■ no localization ■ single-scale localization ■ removed cross-correlations ■ multi-scale localization Statistical Modified M2 (b) (c)

16. Operationally – Scales often treated as independent – Localization necessary, not just beneficial operationally – In those cases, multi-scale localization would be especially beneficial. 16 Results and Discussion * trial results □ average of trials -standard error ■ no localization ■ single-scale localization ■ removed cross-correlations ■ multi-scale localization Statistical Modified M2 (b) (c)

17. Weather phenomena happen on a variety of scales Single-scale localization compromises between – eliminating the spurious small-scale correlations and – retaining the genuine large-scale correlations Multi-scale localization uses a – separate localization function for each scale and – eliminates the cross-scale correlations Multi-scale localization – always better than just removing the cross-correlations – has the most benefits over single-scale localization when localization itself is most necessary 17 Summary

18. Bishop, C.H., and D. Hodyss, 2007: Flow-adaptive moderation of spurious ensemble correlations and its use in ensemble-based data assimilation. Q.J.R. Meteorol. Soc., 133, 2029-2044. Bishop, C.H., and D. Hodyss, 2009a: Ensemble covariances adaptively localized with ECO-RAP. Part 1: tests on simple error models. Tellus A, 61, 84-96. Bishop, C.H., and D. Hodyss, 2009b: Ensemble covariances adaptively localized with ECO-RAP. Part 2: a strategy for the atmosphere. Tellus A, 61, 97-111. Bishop, C.H., and D. Hodyss, 2011:Adaptive Ensemble Covariance Localization in Ensemble 4D-VAR State Estimation. Mon. Wea. Rev., 139, 1241-1255. Posselt, D.J., and C.H. Bishop, 2012: Nonlinear Parameter Estimation: Comparison of an Ensemble Kalman Smoother with a Markov Chain Monte Carlo Algorithm. Mon. Wea. Rev., 140, 1957-1974. Buehner, M., 2012: Evaluation of a Spatial/Spectral Covariance Localization Approach for Atmospheric Data Assmilation. Mon. Wea. Rev., 140, 617- 636. Miyoshi, T., and K. Kondo, 2013: A Multi-Scale Localization Approach to an Ensemble Kalman filter. SOLA, 9, 170-173, doi:10.2151/sola.2013-038. Zhang, F., Y. Weng, J.A. Sippel, Z. Meng, C.H. Bishop, 2009: Cloud-Resolving Hurricane Initialization and Prediction through Assimilation of Doppler Radar Observations with an Ensemble Kalman Filter. Mon. Wea. Rev., 137, 2105-2125. 18 References

19. Thanks to my mentor Craig Bishop. This research is supported by the Naval Research Laboratory through program element 0603207N. 19 Acknowledgments

20. 20 Questions?