A comparison of hybrid ensemble transform Kalman filterETKF3DVAR and ensemble square root filter EnS

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A comparison of hybrid ensemble transform Kalman
filter(ETKF)-3DVAR and ensemble square root
filter (EnSRF) analysis schemes

• Xuguang Wang
• NOAA/ESRL/PSD, Boulder, CO
• Thomas M. Hamill
• Jeffrey S. Whitaker
• NOAA/ESRL/PSD, Boulder, CO
• Craig H. Bishop
• NRL, Monterey, CA

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Why hybrid ETKF-3DVAR ?

• Hybrid ETKF-3DVAR is expected to be less
  expensive than EnSRF, and still can benefit from
  ensemble-estimated error statistics.
• The hybrid may be more robust for small ensemble
  size, since can adjust the amount of static vs.
  ensemble covariance used.
• The hybrid can be conveniently adapted to the
  existing variational framework.

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Hybrid ETKF-3DVAR
member 1 forecast
member 1 analysis
member 1 forecast
ETKF update perturbations
member 2 forecast
member 2 analysis
member 2 forecast
member 3 forecast
member 3 analysis
member 3 forecast
ETKF-3DVAR update mean
data assimilation
forecast
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Hybrid ETKF-3DVAR updates mean

• Background error covariance is approximated by a
  linear combination of the sample covariance
  matrix of the ETKF forecast ensemble and the
  static covariance matrix.

• Can be conveniently adapted into the operational
  3DVAR through augmentation of control variables
  (Lorenc 2003 Buehner 2005 Wang et al. 2006).

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ETKF updates perturbations

• ETKF transforms forecast perturbations into
  analysis perturbations by
• where is chosen by trying to solve the
  Kalman filter error covariance update equation,
  with forecast error covariance approximated by
  ensemble covariance.
• Latest formula for (Wang et al. 20042006,
  MWR)
• Computationally inexpensive for ensemble size of
  o(100), since transformation fully in
  perturbation subspace.

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Experiment design

• Numerical model
• Dry 2-layer spectral PE model run at T31
• Model state consists of vorticity, divergence
  and layer thickness of Exner function
• Error doubling time is 3.78 days at T31
• Perfect model assumption

(Hamill and Whitaker MWR, 2005)

• Observations
• 362 Interface and surface Exner functions taken
  at equally spaced locations
• Observation values are T31 truth plus random
  noise drawn from normal distribution
• Assimilated every 24h

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Experiment design

• Update of mean (OI) and formulation of B

• In this experiment, the hybrid updates the mean
  using

whose solution is equivalent to that if solved
variationally under our experiment design.

• The static error covariance model is constructed
  iteratively from a large sample of 24h fcst.
  errors.

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RMS analysis errors 50 member

• Improved accuracy of EnSRF over 3DVAR can be
  mostly achieved by the hybrid.
• Covariance localization applied on the ETKF
  ensemble when updating the mean (but not applied
  when updating the perturbations) improved the
  analyses of the hybrid.

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RMS analysis errors 20 member

• Both 20mem hybrid and EnSRF worse than 50mem,
  but still better than 3DVAR
• Hybrid nearly as accurate (KE, ??2 norms) or
  even better (surface ? norm) compared to EnSRF

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RMS analysis errors 5 member
EnSRF filter divergence

• EnSRF experienced filter divergence for all
  localization scales tried.
• Hybrid was still more accurate than the 3DVAR.
• Hybrid is more robust in the presence of small
  ensemble size.

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Comparison of flow dependent background error
covariance models
(Hybrid 50 mem. result)
(EnSRF 50 mem. result)
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Initial-condition balance

• Analysis is more imbalanced with more severe
  localization.
• Analyses of the hybrid with the smallest rms
  error are more balanced than those of the EnSRF,
  especially for small ensemble size (not shown).

(50 mem. result)
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Spread-skill relationships

• Overall average of the spread is approximately
  equal to overall average of rms error for both
  EnSRF and Hybrid.
• Abilities to distinguish analyses of different
  error variances are similar for EnSRF and Hybrid.

(20 mem. result)
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Summary

• The hybrid analyses achieved similar improved
  accuracy of the EnSRF over 3DVAR.
• The hybrid was more robust when ensemble size was
  small.
• The hybrid analyses were more balanced than the
  EnSRF analyses, especially when ensemble size was
  small.
• The ETKF ensemble variance was as skillful as the
  EnSRF.
• The hybrid can be conveniently adapted into the
  existing operational 3DVAR framework.
• The hybrid is expected to be less expensive than
  the EnSRF in operational settings.

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Preliminary results with resolution model error

• T127 run as truth imperfect model run at T31
• 200-member ensembles
• Additive model error parameterization
• comparable rms errors for hybrid and EnSRF

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Hybrid ETKF-3DVAR for WRF(collaborated with Dale
Barker and Chris Snyder)

• a-control variable method (Lorenc 2003) to
  incorporate ensemble in WRF-VAR

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Statistics of iteratively constructed B
balance
rms analysis errors
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Ensemble square-root filter(Whitaker and
Hamill,MWR,2002)

• Background error covariance is estimated from
  ensemble with covariance localization.

• Mean state is corrected to new observations,
  weighted by the Kalman gain.

• Reduced Kalman gain is calculated to update
  ensemble perturbations.

• Observations are serially processed. So cost
  scales with the number of observations.

obs1
obs2
obs3
EnSRF
EnSRF
EnSRF

• Covariance localization can produce imbalanced
  initial conditions.

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Maximal perturbation growth

• Find linear combination coefficients b to
  maximize
• Maximal growth in the ETKF ensemble perturbation
  subspace is faster than that in the EnSRF
  ensemble perturbation subspace.