Multiscale Covariance Localization Sabrina Rainwater David National Research
Multi-scale Covariance Localization Sabrina Rainwater David National Research Council Postdoc at NRL with Craig Bishop and Dan Hodyss Naval Research Laboratory 1
Posselt and Bishop En. KF • We discuss multi-scale covariance localization within the context of an En. KF. • 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 Pb is larger than the rank of R. – We modified it to accept small ensembles with a localized Pb (localization increases the rank of the estimated Pb). 2
Covariance Localization • 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
Our Multi-scale World • 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
Multi-scale Covariance Construction • When the background errors are uncorrelated in space, Central row of Pb – the background error covariance matrix Pb is diagonal (zero off-diagonal correlations), – i. e only one nonzero element for each row/column of Pb , – 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 Pb will be a smooth curve with a max in the center (blue). • When the background errors have multiscale correlations, small scales large scales – The central row of Pb could look like a Prussian helmet (black), – with a smooth curve for the broad scales and a spike for the small-scales. 5
Ensemble Estimate and Single-Scale Compromise • 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. Some spurious correlations retained Some large-scale correlations eliminated 6
Localization Functions by Scale • Legend: – Black: the true covariance – Blue: the estimated covariance – Magenta: the covariance localization function ates n i elim lations r r – Pro: eliminates the spurious smalle ll e sma us corr scale correlations rio – Con: eliminates the true large-scale spu • Sharp localization (left) – correlations ins a t e ns r r o i r t e a l larg e corre scal e g lar • Broad localization (right) – – Pro: retains the large-scale correlations – Con: retains the spurious small-scale correlations • Multi-scale localization (bottom) – Pro: Eliminates the spurious smallscale correlations – Pro: Retains the genuine large-scale correlations ls ons o r t con rrelati s co rificing s u o i spur out sac relation with ale cor sc e g r la 7
Methodology • 8
Alternate Multi-scale Localization Techniques • 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, 2009 a, 2009 b, 2011) – Adaptive localization scale depends on location 9
Statistical Model space Spectral space Large scales Small scales • The model is a statistical two-scale 1 D model • (a) A multi-scale state as the sum of large-scale waves (blue) and smallscale waves (red) • (b): the same as (a) except in spectral space. 10
Modified Lorenz Models • 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 KL=32, Ks= 2 11
Experiments • Compared ensemble data assimilation for – No localization – Single-scale localization with cross-correlations removed (i. e. multi-scale localization with CL=CS) – 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
Results • Time averaged mean squared error for various scenarios Statistical – Bar: average over 7 trials – Error bars: standard error in the mean – Asterisks: results for each trial – Purple line: theoretical minimum error Modified M 2 (b) (c) • (a) statistical model results • (b) Modified Model 2 results 13
Results • Time averaged mean squared error for various scenarios Statistical Modified M 2 (b) (c) – 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 and Discussion • Statistical • • Modified M 2 (b) 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 (c) * □ ■ ■ trial results average of trials standard error no localization single-scale localization removed cross-correlations multi-scale localization 15
Results and Discussion • Operationally Statistical Modified M 2 (b) (c) – Scales often treated as independent – Localization necessary, not just beneficial operationally – In those cases, multi-scale localization would be especially beneficial. * □ ■ ■ trial results average of trials standard error no localization single-scale localization removed cross-correlations multi-scale localization 16
Summary • 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
References 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, 2009 a: 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, 2009 b: 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 4 D-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, 617636. 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
Acknowledgments • Thanks to my mentor Craig Bishop. • This research is supported by the Naval Research Laboratory through program element 0603207 N. 19
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