Implementation of the LEKF on the NCEP GFS:

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Implementation of the LEKF on the NCEP GFS
Perfect Model Experiments

• Chaos-Weather Team
• University of Maryland College Park

Camp Springs, MD, February 27, 2004
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UMCP Chaos-Weather Team

• Founders E. Kalnay and J. Yorke
• Theory B. Hunt and E. Ott et al.
• 4D Extension B. Hunt, T. Sauer, and J. Yorke et
  al.
• GFS Implementation I. Szunyogh, E. Kostelich and
  G. Gyarmati et al.

An interdisciplinary team of experts in dynamical
systems theory, meteorology, mathematics, and
scientific computing
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Outline

• The LEKF Concept
• 4-D Extension
• Implementation on the NCEP GFS
• Conclusions
• Future Plans

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Data Assimilation
Observation pdf
Analysis pdf For the Grid
Initial Condition Analysis Mean
t3
Forecast pdf For the Grid
t1
t2
Kalman Filter The analysis uncertainty is
evolved using the model dynamics to obtain the
forecast uncertainty
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Ensemble Kalman Filtering
Ensemble of Initial conditions
Background Ensemble
t1
t2

• Ensemble Kalman Filters (EnKF) (i) Multiple
  analyses are prepared, and then (ii) the most
  likely state is determined by the mean of the
  analysis ensemble
• Ensemble Square-root Filters (EnSF) (ii) The
  most likely state is determined first, and then
  (ii) the mean state is perturbed to obtain the
  analysis ensemble

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Examples

• EnKF Evensen (1994), Houtekamer and Mitchell
  (1998, 2001), Hamill and Snyder (2000), Anderson
  and Anderson (1999), Keppene and Rienecker (2002)
• EnSF Anderson (2001), Bishop et al. (2001),
  Whitaker and Hamill (2002), Tippett et al.
  (2002), LEKF (Ott et al., 2003a, b)
• What is the difference between our scheme and
  other square-root filters? The LEKF is not a
  sequential data assimilation scheme.

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Sequential Schemes

• The observations are assimilated sequentially
• A local region around the grid point is defined
  by the Gaussian filter of Gaspari and Cohn (1998)
• The state estimate is updated at all grid-points
  within the local region

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Local Ensemble Kalman Filter

• The state estimate is updated concurrently at the
  different grid-points
• A local region is defined (the shape is
  arbitrary, no forced distance dependent reduction
  of the correlation)
• All observations within the local region are
  assimilated

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Potential Advantages of the LEKF

• Allows for significant reduction of the
  dimension, based on estimating the complexity of
  the dynamics (efficient filtering of redundant
  information from the observations)
• Efficient for observations with correlated errors
• The computation can be performed concurrently for
  each grid point
• We believe that this formulation is advantageous,
  when many observations, especially those with
  correlated errors, are assimilated

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Potential Disadvantages of the LEKF

• When the observations are sparse (when the local
  regions must be large) the Gaussian filter may be
  a better tool to localize the covariance
  information (more efficient filtering of spurious
  long distance correlations)
• When the number of observations is significantly
  lower than the number of grid points, the LEKF is
  probably more expensive than a sequential scheme
• We expect that sequential schemes are more
  suitable than the LEKF, when relatively few
  observations are assimilated (e.g. Whitaker et
  al. 2003 Reanalysis without radio-sondes )

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Schematic of the LEKF
Analysis ensemble at t-1
Evolve model from t-1 to t
Global analysis ensemble at t
Background ensemble at t
Obtain Global Analysis Ensemble

Form local vectors
Local Analysis Ensembles
Local background vectors
Do local analysis
Local analysis and Analysis Error covariance
matrix
Obtain Local Analysis Ensembles
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Illustration by the Lorenz-96 model
Unstable non-linear waves with a typical
wavelength of 5-6 grid points (the wavelength is
independent of the M)
xMx1
x2
xM-1
x3
Eastward group velocity
For M40 the Lyapunov dimension is 27.1
Prototype of a spatio-temporally chaotic system
with a finite correlation length
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LEKF vs. Global EKF
Global

• The global scheme requires an increasing number
  of ensemble members as the size of the system
  increases
• The number of ensemble members needed in the LEKF
  is much lower than in the global scheme and
  independent of the system size

M40
Lyapunov dimension
M80
M120
Local
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4D Ensemble Kalman Filter
Hunt et. al 2004a,b
Ensemble Kalman Filters allow for assimilating
observations at the correct time
yh(x)
model state
observation
observational operator
xbEbe
background mean
background ensemble
vector of linear weights
H does not exist for a single background
y-Hxoy-HEoey-Hxb
innovation at obs. time
ensemble at obs. time
modified H
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4D Ensemble Kalman Filter
Illustration by the 40-variable Lorenz model
Approach used in 3D-Var Schemes
Only data at analysis Times are used
4D Ens. KF
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NCEP GFS

• A time series of true states is generated by a
  long integration of the model started from the
  operational NCEP analysis at January 1, 2000.
• The observations are created by adding Gaussian
  random noise to the true state (the errors 1 K
  for temperature, 1.1 m/s for wind vector
  components, and 1 hPa for surface pressure)
• The lower boundary condition of an analysis
  ensemble member is copied from the associated
  background ensemble member
• The simulated observations are assimilated and
  the result is compared to the true state
• This experimental design allows for testing our
  fundamental hypothesis, that the local
  dimensionality of the model is low (whether we
  can stay close to the true state by estimating
  the state in local low-dimensional spaces)
• This is a crucial step, which is needed for us to
  be able to distinguish between problems in our
  formulation (and implementation) and the effects
  of model errors

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Experiments

• Red (Base Line) 80-member, the surface
  pressure, temperature and wind are observed at
  each grid point, ozone and humidity copied from
  truth
• Green Same as Red, except 40-members and 90 of
  the observations are removed
• Dark Blue Same as Red, except 40-members are
  removed, and ozone and humidity are copied from
  background
• Light Blue (Realistic) Same as Red, except 40
  members and 90 of the observations are removed,
  ozone and humidity are copied from background

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Similarly good convergence was Also found for the
other variables
All implementations are Close to the Base line
Observational error
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Conclusions

• While some modest improvement is expected from
  further tuning parameters in the Perfect Model
  environment, the LEKF is ready for testing with
  real observations
• Based on the Perfect Model experiments it is
  expected that the LEKF analyses will have the
  highest quality in the extra-tropic (where a
  modest size ensemble, and a modest number of
  observations are sufficient), while analyses in
  the Tropics will improve with increased
  observational density. Some modest improvement
  can be expected when the ensemble size is
  increased

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Near Future Work

• Implementation of the 4D extension (in progress)
• Assimilation of radiosonde observations (in
  progress)
• Implementation on the NCEP RSM (in progress)
• Surface Analysis (your help is wanted!)
• Implementation on the NASA model (starts soon)
• Handling of model errors (in the theoretical
  phase)
• Direct minimization of the cost function (in the
  theoretical phase)