Transcript Multi-scale Covariance Localization - PSU WRF/EnKF Real-time

Multi-scale
Covariance Localization
Sabrina Rainwater David
National Research Council
Postdoc at NRL
with
Craig Bishop and Dan Hodyss
Naval Research Laboratory
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Posselt and Bishop EnKF
• 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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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, …
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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.
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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.
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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
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Localization Functions by Scale
• Legend:
– Black: the true covariance
– Blue: the estimated covariance
– Magenta: the covariance localization
function
• Sharp localization (left) –
– Pro: eliminates the spurious smallscale 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 smallscale correlations
– Pro: Retains the genuine large-scale
correlations
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Methodology
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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, 2009a, 2009b, 2011)
– Adaptive localization scale depends on location
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Statistical Model
Model space
Spectral space
Large
scales
Small
scales
• The model is a
statistical two-scale
1D 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.
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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
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Experiments
• 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.
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Results
• Time averaged mean
squared error for various
scenarios
Statistical
Statistical
– Bar: average over 7 trials
– Error bars: standard error in
the mean
– Asterisks: results for each
trial
– Purple line: theoretical
minimum error
Modified M2
Modified M2
(b)
(c)
• (a) statistical model results
• (b) Modified Model 2 results
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Results
• Time averaged mean
squared error for various
scenarios
Statistical
Modified M2
(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
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Results and Discussion
•
Statistical
•
•
Modified M2
(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
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Results and Discussion
• Operationally
Statistical
Modified M2
(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
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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
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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, 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, 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.
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Acknowledgments
• Thanks to my mentor Craig Bishop.
• This research is supported by the Naval
Research Laboratory through program
element 0603207N.
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Questions?
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