COMPARISON%20OF%20ENSEMBLE-BASED%20AND%20VARIATIONAL-BASED%20DATA%20ASSIMILATION%20SCHEMES%20IN%20A%20QUASI-GEOSTROPHIC%20MODEL

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ENSEMBLE KALMAN FILTER IN THE PRESENCE OF MODEL
ERRORS
Hong Li Eugenia Kalnay

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If we assume a perfect model, we can grossly
underestimate the errors
imperfect model (obs from NCEP- NCAR Reanalysis
NNR)
Perfect model
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We compare several methods to handle model errors
imperfect model (obs from NCEP- NCAR Reanalysis
NNR)
perfect model
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• SPEEDY MODEL (Molteni 2003)
• T30L7 global spectral model
• total 96x48 grid points on each level
• State variables u,v,T,Ps,q
• Data Assimilation LETKF
• Methods to handle model errors
• Multiplicative/additive inflation
• Dee daSilva (1998) (DdS)
• Low-dimensional method (LDM, Danforth et al, MWR,
  2007)

Dense Observations
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Control run 100 inflation Dee da
Silva Low-order
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Model error estimation schemes (1)

• 1a. Covariance inflation (multiplicative)

(Ideal KF)
(EnKF)
1b. Covariance inflation (additive)
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2. Dee and daSilva bias estimation scheme (1998)
Model error estimation schemes (DdS)
Do data assimilation twice first for model error
then for model state (expensive)
and need to be tuned
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Model error estimation schemes (LDM)
3. Low-dim method (Danforth et al, 2007
Estimating and correcting global weather model
error. Mon. Wea. Rev)

• Generate a long time series of model forecast
  minus reanalysis from the training period

model
NNR
NNR
NNR
NNR
t0
NNR
t6hr
We collect a large number of estimated errors
and estimate bias, etc.
Time-mean model bias
Diurnal model error
Forecast error due to error in IC
State dependent model error
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Further explore the Low-dimensional method
Include Bias, Diurnal and State-Dependent model
errors
Time-mean model bias
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BIAS
one month
climatological
debiased
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Diurnal model errors
Leading EOFs for 925 mb TEMP

• Generate the leading EoFs from the forecast error
  anomalies fields for temperature.

pc1 pc2

• Lack of diurnal forcing generates wavenumber 1
  structure

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925hPa Temperature
Black line
Blue line
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State-dependent model errors
the local state anomalies (Contour) and the
forecast error anomalies (Color)
SVD2
SVD1
SVD4
SVD3
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Correct state-dependent model errors
500hPa Uwind
500hPa Height
Black line
Blue line
Univariate SVD (not account for the relations
between different variables)
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Impact of model error, and different approaches
to handle it
Perfect model
imperfect model (obs from Reanalysis)
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Simultaneous estimation of inflation and
observation errors

• Hong Li
• Eugenia Kalnay

University of Maryland
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Motivation

• Any data assimilation scheme requires accurate
  statistics for the observation and background
  errors. Unfortunately those statistics are not
  known and are usually tuned or adjusted by gut
  feeling.
• Ensemble Kalman filters need inflation (additive
  or multiplicative) of the background error
  covariance, but
• 1) Tuning the inflation parameter is expensive
  especially if it is regionally dependent, and it
  may depend on time
• 2) Miyoshi and Kalnay 2005 (MK) proposed a
  technique to objectively estimate the covariance
  inflation parameter.
• 3) This method works, but only if the
  observation errors are known.
• Here we introduce a method to simultaneously
  estimate observation errors and inflation.

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MK method to estimate the inflation parameter
(Miyoshi 2005, MiyoshiKalnay 2005)
obs. increment in obs. space
Should be satisfied if R, Pb and are correct
(they are not!)
So, at any given analysis time, and computing the
inner product

• Assumption R is known

(1)
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Diagnosis of observation error statistics
(Desroziers et al, 2006, Navascues et al, 2006)
Desroziers et al, 2006, introduced two new
statistical relationships
if the R and B statistics are correct and errors
are uncorrelated
Writing their inner products we obtain two
equations which we can use to observe R and

(2)
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Simultaneous estimation of inflation and
observation errors
(1)
(2)

• Model Lorenz-96 model / SPEEDY model
• Perfect model scenario
• Data assimilation scheme Local ensemble
  transform Kalman filter (LETKF, Hunt et al. 2006)
• We estimate both and R online at each
  analysis time

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Tests within LETKF with Lorenz-96 model
40 observations with true Rt1, 10 ensemble
member. Optimally tuned rms0.20
Perfect R, estimate inflation using (1) it
works
Wrong R, estimate inflation using (1) it fails
(1)
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Tests within LETKF with L96 model
Now we estimate observation error and optimal
inflation simultaneously using (1) and (2) it
works!
Estimated R
Estimated
Rinit
 
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Tests within LETKF with SPEEDY

• SPEEDY MODEL (Molteni 2003)
• Primitive equations, T30L7 global spectral model
• total 96x46 grid points on each level
• State variables u,v,T,Ps,q

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Tests within LETKF with SPEEDY

• OBSERVATIONS
• Generated from the truth plus random errors
  with error standard deviations of 1 m/s (u), 1
  m/s(v), 1K(T), 10-4 kg/kg (q) and 100Pa(Ps).
• Dense observation network 1 every 2 grid points
  in x and y direction

• EXPERIMENTAL SETUP
• Run SPEEDY with LETKF for two months ( January
  and February 1982) , starting from wrong
  (doubled) observational errors of 2 m/s (u), 2
  m/s(v), 2K(T), 210-4 kg/kg (q) and 200Pa(Ps).
• Estimate and correct the observational errors
  and inflation adaptively.

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online estimated observational errors
The original wrongly specified R converges to the
correct R quickly (in about 5-10 days)
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Estimation of the inflation
Estimated Inflation
Using an initially wrong R and but
estimating them adaptively
Using a perfect R and estimating adaptively
After R converges, they give similar inflation
factors (time dependent)
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Global averaged analysis RMS
500hPa Height
500hPa Temperature
Using an initially wrong R and but
estimating them adaptively
Using a perfect R and estimating adaptively
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Summary

• The online (adaptive) estimation of inflation
  parameter alone does not work without estimating
  the observational errors.
• Estimating both of the observational errors and
  the inflation parameter simultaneously our
  approach works well on both the Lorenz-96 and the
  SPEEDY global model. It can also be applied to
  other ensemble based Kalman filters.
• SPEEDY experiments show our approach can
  simultaneously estimate observational errors for
  different instruments.
• Current work shows our method also works in the
  presence of random model errors.

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A few more slides

• Junjie Liu Adaptive observations
• Junjie Liu Estimation of the impact of
  observations
• Shu-Chih Yang Comparison of EnKF, simple hybrid
  (3D-Var Bred Vectors) and 4D-Var
• Shu-Chih Yang 4D-Var and initial and final SVs,
  EnKF and initial and final BVs
• No cost smoother for reanalysis

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Adaptive sampling with the LETKF-based ensemble
spreadJunjie Liu

• Purpose
• Sample 10 adaptive DWL wind observations to get
  90 improvement of full coverage
• Compare ensemble spread method with other
  sampling strategies
• How the results are sensitive to the data
  assimilation schemes (3D-Var and LETKF)
• Note
• same adaptive observations from ensemble spread
  method are assimilated by both 3D-Var and LETKF

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500hPa zonal wind RMS error
Rawinsonde climatology uniform random
ensemble spread ideal 100
3D-Var
LETKF
4.04 2.36 0.92 0.74 0.43 0.36 0.30
1.18 0.38 0.36 0.33 0.32 0.29 0.23
RMSE

• With 10 adaptive observations, the analysis
  accuracy is significantly improved for both
  3D-Var and LETKF.
• 3D-Var is more sensitive to adaptive strategies
  than LETKF. Ensemble spread strategy gets best
  result among operational possible strategies

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500hPa zonal wind RMS error (2 adaptive obs)
Rawinsonde climatology uniform random
ensemble spread ideal 100
3D-Var
LETKF

• With fewer (2) adaptive observations, ensemble
  spread sampling strategy outperforms the other
  methods in LETKF
• For 3D-Var 2 adaptive observations are clearly
  not enough

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Analysis sensitivity study within LETKF

• The self sensitivity is the trace of the matrix
  S.
• It can show the analysis sensitivity with respect
  to
• different types of observations (e.g.,
  rawinsonde, satellite, adaptive observation and
  routine observations)
• the observations in different area (e.g., SH, NH)

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Analysis sensitivity of adaptive observation (one
obs. selected from ensemble spread method over
ocean) and routine observations (every grid point
over land) in Lorenz-40 variable model
10-day forecast RMS error
Analysis sensitivity

• About 17 information of the analysis comes from
  observations over land.
• About 85 information comes from observation for
  the adaptive observation (a single observation
  over ocean).
• The single adaptive observation is more
  important than single observation over land.

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Comparison of ensemble-based and
variational-based data assimilation schemes in a
Quasi-Geostrophic model.Shu-Chih Yang et al.
3D-Var
Hybrid (3DVar20 BVs)
12-hour 4D-Var
LETKF (40 ensemble)
24-hour 4D-Var
3D-Var HYBD 4D-Var 4D-Var LETKF LETKF LETKF LETKF
3D-Var HYBD 12hr 24 hr l3 l5 l7 l9
RMS error (?10-2) 1.44 0.70 0.56 0.35 0.67 0.48 0.44 0.44
Time (minutes) 0.15 1.5 1.8 2.5 0.3 1.0 1.9 2.4
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Analysis increment (color shaded) vs.
dynamically fast growing errors (contours)
12Z Day 24
00Z Day 25
Initial increment (smoother) vs. BV
analysis increment vs. BV
LETKF
analysis increment vs. Final SV
Initial increment vs. Initial SV
12-hour 4DVAR
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Analysis increment (color shaded) vs.
dynamically fast growing errors (contours)
00Z Day 24
00Z Day 25
analysis increment vs. Final SV
Initial increment vs. Initial SV
24-hour 4DVAR
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3D-LETKF
time
to
t1
4D-LETKF
No-cost LETKF smoother (cross) apply at t0 the
same weights found optimal at t1, works for 3D-
or 4D-LETKF
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No-cost LETKF smoother
LETKF analysis at time i
LETKF Analysis
Smoother reanalysis
Smoother analysis at time i-1
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LETKF minimizes the errors of the day and thus
provides an excellent first guess to the 3D-Var
analysis
3DVar
3DVar with the background of the first 50 days
provided from LETKF
3DVar with the background provided from LETKF
(forecast mean)
LETKF
We conclude from this experiment that the errors
of the day (and not just ensemble averaging) are
important in LETKF and 3D-Var.