adaptive localization and adaptive inflation methods with the letkf
DESCRIPTION
5/20/2010, 5 th International EnKF Workshop, Bergen, Norway. Adaptive localization and adaptive inflation methods with the LETKF. Takemasa Miyoshi University of Maryland, College Park [email protected]. With many thanks to E. Kalnay, B. Hunt, K. Ide, J.-S. Kang, H. Li, - PowerPoint PPT PresentationTRANSCRIPT
Adaptive localization and adaptive inflation methods with the LETKF
Takemasa MiyoshiUniversity of Maryland, College Park
With many thanks toE. Kalnay, B. Hunt, K. Ide, J.-S. Kang, H. Li,
UMD Chaos-Weather group
5/20/2010, 5th International EnKF Workshop, Bergen, Norway
Adaptive covariance localization
Covariance localization (e.g., Houtekamer and Mitchell 1998)
fPfP
Empirical treatment for…• reducing sampling noise• increasing the rank
Localized
Difficulties of localizationDifficulties include…• depending on (x, y, z, t)• reducing flow-dependence
95.0 51.0 08.0
Adaptive localization
• Hierarchical filter by Anderson (2007)– Cross-validation by groups of ensembles
• ECO-RAP by Bishop and Hodyss (2009)– Smooth the sample correlations raised to a power
• High sample correlation = more reliable = more weight
• Spatial smoothing to reduce noisiness of the sample correlation
1m
TyXCross-
validationLocalization
weights1m
TyX
1m
TyX1m
TyX
LETKF algorithm (Hunt et al. 2007)
Each grid point is treated independently.
Multiple observations are treated simultaneously.
Local Ensemble Transform Kalman Filter
Localization in LETKF
),,(1 iif
iifim
fi
ai dRYTx1xx
Analysis of the i-th variable:
1. Selecting a subset of global obs for the i-th variable
iif
i dRY ,,
2. Obs error std. is weighted by the localization factor
are composed of only selected local obs.
iii RR 1~ so that far-away obs have large error.
)( mN )( mN )( mm
Two steps of localization:
R-localization, Hunt et al. (2007)
Adaptive localization with LETKFThe localization factor for R-localization is given by two adaptive components:
A. Cross-validation (Anderson 2007)as
a
ccw
2
||1 21
B. Use the sample covariance (Bishop and Hodyss 2009)High sample correlation = more reliable = more weight
bsb cw ||
),( yX Cor
),( yX Cor
c1
c2
m/2-member correl.
m-member correl.baadaptive ww ~
Advantages and disadvantages
• Advantages:– Minimal additional computations– Minimal changes to the existing LETKF code– Automatic inter-variable 4-D localization
• Disadvantages– Sampling error issue remains.
Not the best, but we can effectively avoid optimization.
Results with the Lorenz-96 model
system dim. 40
obs. network 13 regularly
ensemble size 10
inflation 10% multiplicative
adaptivefixedcombined ~~~ noisynot-noisy
Preliminary results with an AGCM
Adaptive covariance inflation
Covariance inflation (e.g., Houtekamer and Mitchell 1998)
Empirical treatment for…• covariance underestimation
Error covariance is underestimated due to various sources of imperfections:
• limited ensemble size• nonlinearity• model errors
inflation
No inflation 5% inflation
Difficulties of inflationFixed covariance inflation has difficulties such as…• ensemble spread exaggerates observing density pattern• should depend on (x, y, z, t); tuning is very difficult
Temperature spread of JMA’s global LETKF w/ fixed multiplicative inflation
~50 hPa~500 hPa
adapted from Miyoshi et al. (2010)
Additive inflation• introduces new directions to span the error space• solves the problem of the spread pattern• has difficulties in obtaining reasonable additive noise• random noise does not grow initially
Temperature spread of JMA’s global LETKF w/ additive inflation
adapted from Miyoshi et al. (2010)
~50 hPa~500 hPa
Adaptive inflation
Li et al. (2009) used the statistical relationship derived by Desroziers et al. (2005):
TfTob
ab
Tob
ob
TfTfTob
ab
ob
abaab
HHPdd
ddRHHPHHPdd
HKdHdxHxHxd
1)(
RHHPdd TfTob
ob using
In the EnKF, )( ff PP TfTo
bab HPHdd )( i.e.,
Tf
Tob
ab
HPHtr
ddtr
)( We estimate the inflation parameter:
Anderson (2007; 2009) developed an adaptive inflation algorithm using a hierarchical Bayesian approach.
Adaptive inflation in the LETKF
),,(1 iif
iifim
fi
ai dRYTx1xx
Analysis of the i-th variable:
)( mN )( mN )( mm
When computing Ti, we compute the following statistics simultaneously:
11
11
)~)((~
)~(
ii
Tfi
fii
ii
Tob
ab
itr
ddtr
RYY
R
iii RR 1~ R-localization, Hunt et al. (2007)
Normalization factor
3-dimensional field of inflation
Time smoothingDue to the limited sample size in the local region at a single time step, it is essential to apply time smoothing to include more samples in time.
We use the Kalman filter approach.
a tuning parameter that determines the strength of time smoothing002.0b
po
1
22
21
2
bo
obtot
For example,
p denotes the number of observations (i.e., sample size)
i.e., more samples, more reliable.
For example, when p = 100, 1.0o
More considerations
Localization function:is a measure of reliability
~
1
o
1~0
Considering the sample size weighted by the reliability:
Results with the Lorenz 96 model
5% inflation adaptive inflation
Observed No obs Observed No obs
6-month average after 6-month spin-up
0.31
3.07
0.31
2.71
Results with the SPEEDY model
Estimated adaptive inflation
Time series of adaptive inflation
bOvershooting
002.0b gives stable performance
4-mo. exp.
still spinning up
~20% reduction of RMSE
4% fixed inflation adaptive inflation
m/s
Too large spread Improved spread
Averages of the final 1 mo. of 4-mo. experiments
m/s
Regular obs network
With regular obs network
4% fixed inflation adaptive inflation
~20% reduction of RMSE
Improved spread
m/s m/s
Expt. with model errors (Ji-Sun Kang, 2010)
CO2 data assimilation with the SPEEDY-C model by Ji-Sun Kang
(u, v, T, q, ps, C, CF)Analyzed variables:
All variables are prognosticusing the SPEEDY-VEGAS model with dynamical vegetation
prognostic
Nature run:
like a parameterwith fixed vegetation
Results with model errors
m/s m/s K 0.1g/kg hPa ppmv 10-8kg/m2/s
by Ji-Sun Kang (2010)
RMSE and Spread
Fixed inflation
Globally constantadaptive inflation
This adaptive inflation
This adaptive inflationwith model bias correction
by Ji-Sun Kang (2010)
CF estimates by Ji-Sun Kang (2010)
RMSE and Spread of CF estimates
Fixed inflation
Globally constantadaptive inflation
This adaptive inflation
by Ji-Sun Kang (2010)
Using both adaptive localization and adaptive inflation simultaneously
Preliminary results with Lorenz-96
Adaptive localization and inflation
Adaptive inflation significantly stabilizes the filter,giving nearly optimal performance.
Future challenges
• Application to other systems– Realistic NWP models– Ocean models– Martian atmosphere models
• Assimilation of real observations– Model errors– Temporally varying observing network
• (e.g., aircraft, satellites)
The LETKF code is available at:
http://code.google.com/p/miyoshi/