Assessing a Local Ensemble
Kalman Filter
Istvan Szunyogh
“Chaos-Weather Team”
University of Maryland College Park
IPAM DA Workshop, UCLA, February 22-25, 2005
System Components
• Data assimilation scheme: Local Ensemble
(Transform) Kalman Filter (Ott et al. 2002, 2004;
Hunt 2005) implemented by Eric Kostelich (ASU)
and I. Sz.
• Model: Operational Global Forecast System
(GFS) of the National Centers for Environmental
Prediction/National Weather Service
• Model Resolution: T62 (~150 km) in the
horizontal directions and 28 vertical level
dimension of the state vector:1,137,024;
dimension of the grid space (analysis space):
2,544,768]
WARNING!!!!!
All results shown in this
presentation were obtained for
the perfect model scenario
Why a Perfect Model?
• Easier to find bugs in the code
• To expose weaknesses of the scheme
(model errors cannot be blamed for
unexpected bad results)
• To establish a reference needed to assess the
effects of model errors
• To learn more about the dynamics of the
model (predictability, dimensionality, etc.)
Local Ensemble Kalman Filter
Illustration on a two dimensional grid
• The state estimate is
updated at the center grid
point
• The background state is
considered only from a
local region (yellow dots)
• All observations are
considered from the local
region (purple diamonds)
Base Experiment
• Number of ensemble members: 40
• Local regions: 7x7xV grid point cubes; V=1, 3, 5,
7
• Variance Inflation: Multiplicative, uniform 4%
(needed to compensate for the loss of variance due
to nonlinearities and sampling errors)
• Observations: 2000 vertical sounding of wind,
temperature, and surface pressure
Depth of Local Cubes
Lower stratosphere
Upper troposphere
Mid-troposphere
Lower troposphere
Dimension of Local
State Vector ~1,700
Time evolution of errors
surface pressure
Observational error
Rms
analysis
error
analysis cycle (time)
The error settles at a similarly rapid speed for all variables
15-days (60 cycles) is a safe upper bound estimate for the transient
Zonal-Mean Analysis Error (45-day mean)
The analysis errors are much smaller than the observational
errors
Temperature
u-wind
The “largest” errors: deep convection (maximum CAPE), polar regions
Time-Mean Analysis Error
45-day average
Temperature 60 kPa
Euro-Asia
Africa
u-wind 30 kPa
NH Extratropics
N-America
Tropics
Australia S-America
SH Extratropics
The figures confirm the conclusions drawn based on zonal means
E-dimension
A local measure of complexity
Illustration in 2D model grid space
A spatio-temporally
changing scalar value
is assigned to each
grid point
Based on the
eigenvalues i
of the ensemble
based estimate of
the local covariance

matrix:
 
  i  
i 
Complexity:
2
2
i
i


Introduced by
Patil, Hunt et al. (2001)
Studied in details by
Oczkowski et al (2005)
E-dimension
1
Number of Ensemble
The more unevenly distributed the variance in Members-1
the ensemble space, the lower the E-dimension
Explained (Background) Error
Variance
Illustration for a rank-2 covariance matrix
(3-member ensemble)
True state
Eigenvector 1
Explained Variance:  be2/  b2
b
Background
mean
be
Projection on the plane of the
eigenvectors
Eigenvector 2
A perfect explained variance of 1 implies that the space of uncertainties is
correctly captured by the ensemble, but it does not guarantee that the
distribution of the variance within that space is correctly represented by the
ensemble
E-dimension, Explained
Variance, Analysis Error
• Background ensemble perturbations span the space, where
corrections to the estimate of the state can be made => Edimension characterizes the distribution of variance
between distinct state space directions, EV measures the
potential for a correction
• A correction is realized, when the difference between the
observation and the background mean has a projection on
the subspace that contributes to the explained variance
• Low explained variance or low E-dimension would be a
problem if the error in the resulting state estimate was
large
E-Dimension and Explained
Variance
E-dimension
Explained BackgroundVariance
The E-dimension and Explained Background Variance seem to be strongly
anti-correlated. The E-dimensions is the highest, the explained variance is the lowest,
in the Tropics.
Szunyogh et al. (2005)
E-Dimension vs. Explained Variance
Low E-dimension always indicates high explained variance
No Averaging
E-dimension
Zonal Mean
Correlation:-0.91
Explained Variance
Correlation:-0.9
Explained Variance
High E-dimension always indicates low explained variance
Szunyogh et al. (2005)
Sensitivity to the Size of the
Local Region: Part I
3x3xV
5x5xV
7x7xV
9x9xV
11x11xV
best
worst
mid-troposphere
Temperature, Global Error
The performance is only
modestly sensitive to the
local region size
Observational
error
rms error
Szunyogh et al. (2005)
Sensitivity to the Size of the
Local Region: Part II
u-wind, NH extra-tropics
Best: 5x5xV
Worst:11x11xV
Observational error
u-wind, Tropics
The tropical wind
is the most sensitive
analysis variable
Szunyogh et al. (2005)
Sensitivity to the Size of the
Local Region and the Ensemble
Size
u-wind, Tropics, 80-member ensemble
the5x5xV localization
improves only a little with
increasing the ensemble size
Observational error
The 7x7xV localization
breaks even with the
5x5xV localization
Szunyogh et al. (2005)
Sensitivity to the Number of
Observations
Global Temperature Error
Wind Error in Tropics
500 soundings
1000 soundings
2000 soundings
18048 soundings
(All locations)
Szunyogh et al. (2005)
E-dimension and Explained Variance
(fully observed atmosphere)
E-dimension
Explained Variance
The largest E-dimension did not change
The smallest explained variance
Error reduction in the tropics is
was reduced by about 0.05
about 46%
(about 12%)
Szunyogh et al. (2005)
Evolution of the Forecast Errors
As forecast time
increases the
extratropical
storm track
regions become
the regions of
largest error
45-day mean
D. Kuhl et al.
Evolution of the E-dimension
The E-dimension rapidly
Decreases in the storm
Track regions
The error growth and
the decrease of the
E-dimension is closely
related
D. Kuhl et al.
Evolution of the Explained Variance
The explained variance is
the largest in the storm track
regions and it increases with
time
Large error growth,
low E-dimension, and
large explained variance
are closely related
There seems to exist a ‘local
analogue’ to the unstable
subspace
D. Kuhl et al.
E-dimension
The scatter plots confirm
the increasingly close
correspondence between
low E-dimensionality and
high explained variance
(improving
ensemble performance)
Explained Variance
D. Kuhl et al.
Time Mean Evolution of the
Forecast Errors
Average Forecast Error
4
Extra Trop. NH
Extra Trop. SH (exponential
Tropics
3.5
3
growth)
(linear growth)
2.5
2
1.5
Curves fitted for
First 72 hours
1
0.5
0
0
12
24
36
48
60
72
84
96
108 120 132
Forecast (Hour)
The error doubling time in the extratropics is about
35-37 hours
D. Kuhl et al.
The effect of Local Patch Size on the
Error Growth in the NH Extratropics…
is negligible
Average Forecast Error
10
Patch Size 9x9
Patch Size 7x7
Patch Size 5x5
1
0.1
0
12
24
36
48
60
72
84
Forecast (Hour)
Forecast hour
D. Kuhl et al.
The Effect of the Ensemble Size on the
Forecast Errors in the NH Extratropics…
is negligible
Average Forecast Error
10
40 Mem. Ensemble
80 Mem. Ensemble
1
0.1
0
12
24
36
48
Forecast (Hour)
Forecast hour
60
72
84
D. Kuhl et al.
The Effect of the Number of Observations on
the Forecast Errors in the NH Extratropics
Average Forecast Error
10
500 Observations
1,000 Observations
2000 Observations
17,848 Observations
1
0.1
0
12
24
36
48
60
72
84
Forecast (Hour)
Forecast hour
The slightly larger growth rate for the initially smaller errors
indicates the presence of saturation processes
D. Kuhl et al.
Conclusions and Challenges
• The state-of-the art model shows local low-dimensional
behavior. Is it reasonable to assume that the real
atmosphere shows a similar behavior? (My guess: yes)
• Local low dimensionality helps obtain more accurate
estimate of the initial state and more accurate prediction of
the forecast uncertainties.
• Localization in the physical space seems to be a practical
way to apply low dimensional concepts to a very high
dimensional system. Is it possible to develop a rigorous
theoretical framework to support this phenomenological
result? (I have no guess)
• On the practical side, the LETKF assimilates an operational
observation file (excluding satellite radiances) in 5 minutes
References
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Kuhl, D., I. Szunyogh, E. J. Kostelich, G. Gyarmati, D.J. Patil, M.
Oczkowski, B. Hunt, E. Kalnay, E. Ott, J. A. Yorke, 2005: Assessing
predictability with a Local Ensemble Kalman Filter (to be submitted)
Szunyogh, I, E. J. Kostelich, G. Gyarmati, D. J. Patil, B. R. Hunt, E.
Kalnay, E. Ott, and J. A. Yorke, 2005: Assessing a local ensemble
Kalman filter: Perfect model experiments with the NCEP global model.
Tellus 57A. [in print]
Oczkowski, M., I. Szunyogh, and D. J. Patil, 2005: Mechanisms for the
development of locally low dimensional atmospheric dynamics. J.
Atmos. Sci. [in print].
Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M.
Corazza, E. Kalnay, D. J. Patil, J. A. Yorke, 2004: A local ensemble
Kalman Filter for atmospheric data assimilation.Tellus 56A , 415-428.
Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M.
Corazza, E. Kalnay, D. J. Patil, and J. A. Yorke, 2004: Estimating the
state of large spatio-temporally chaotic systems. Phys. Lett. A., 330,
365-370.
Patil, D. J., B. R. Hunt, E. Kalnay, J. A. Yorke, and E. Ott, 2001: Local
low dimensionality of atmospheric dynamics, Phys. Rev. Let., 86, 58785881.
Reprints and preprints of papers by our group are available at
http://keck2.umd.edu/weather/weather_publications.htm