Sabrina Rainwater DavidNational Research Council

Postdoc at NRLwith

Craig Bishop and Dan HodyssNaval Research Laboratory

Multi-scale Covariance Localization

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• 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 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).

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Posselt and Bishop EnKFPosselt 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, …

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Covariance LocalizationCovariance 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.

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Our Multi-scale WorldOur Multi-scale World

• When the background errors are uncorrelated in space,– 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 multi-

scale correlations,– 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

Multi-scale Covariance ConstructionMulti-scale Covariance Construction

small scales

large scales

Central row of Pb

• 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.

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Ensemble Estimate andEnsemble Estimate andSingle-Scale CompromiseSingle-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

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Localization Functions by Scale Localization Functions by Scale

larger r retains

large-scale correlationssmaller r eliminates

spurious correlations

controls

spurious correlations

without sacrificing

large scale correlations

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MethodologyMethodology

• 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

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Alternate Alternate Multi-scale Localization TechniquesMulti-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.

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Statistical ModelStatistical 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

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Modified Lorenz ModelsModified Lorenz Models

KL=32, Ks= 2

• Compared ensemble data assimilation for– No localization– Single-scale 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

ExperimentsExperiments

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ResultsResults

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

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ResultsResults

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

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Results and DiscussionResults 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.

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Results and DiscussionResults 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

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SummarySummary

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

ReferencesReferences

• Thanks to my mentor Craig Bishop.• This research is supported by the Naval

Research Laboratory through program element 0603207N.

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AcknowledgmentsAcknowledgments

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Questions?Questions?