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

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

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

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

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

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

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 Methodology

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

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

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

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 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

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)

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)

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)

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

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, Bishop, C.H., and D. Hodyss, 2009a: Ensemble covariances adaptively localized with ECO-RAP. Part 1: tests on simple error models. Tellus A, 61, Bishop, C.H., and D. Hodyss, 2009b: Ensemble covariances adaptively localized with ECO-RAP. Part 2: a strategy for the atmosphere. Tellus A, 61, Bishop, C.H., and D. Hodyss, 2011:Adaptive Ensemble Covariance Localization in Ensemble 4D-VAR State Estimation. Mon. Wea. Rev., 139, 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, Buehner, M., 2012: Evaluation of a Spatial/Spectral Covariance Localization Approach for Atmospheric Data Assmilation. Mon. Wea. Rev., 140, Miyoshi, T., and K. Kondo, 2013: A Multi-Scale Localization Approach to an Ensemble Kalman filter. SOLA, 9, , doi: /sola 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, References

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

20 Questions?