Transcript COMPARISON OF ENSEMBLE-BASED AND VARIATIONAL …

ENSEMBLE KALMAN FILTER IN THE PRESENCE OF
MODEL ERRORS
Hong Li
Eugenia Kalnay
If we assume a perfect model, we can
grossly underestimate the errors
~Perfect model
imperfect model
(obs from NCEP- NCAR
Reanalysis NNR)
We compare several methods to handle
model errors
~perfect model
imperfect model
(obs from NCEP- NCAR
Reanalysis NNR)
SPEEDY MODEL (Molteni 2003)
•
T30L7 global spectral model
•
total 96x48 grid points on each level
•
State variables u,v,T,Ps,q
Dense
Observations
Data Assimilation: LETKF
Methods to handle model errors
1) Multiplicative/additive inflation
2) Dee & daSilva (1998) (DdS)
3) Low-dimensional method (LDM, Danforth et al, MWR, 2007)
Control run
100% inflation
Dee & da Silva
Low-order
Model error estimation schemes (1)
1a. Covariance inflation
(multiplicative)
Pi f  M xa Pia1M xa
i 1
i 1
T
Q
1 K f
Pi 
( xi  x f )(xif  x f )T

k  1 i 1
f
(Ideal KF)
(EnKF)
~
Pi f  (1 ) * Pi f  Pi f  Pi f
Q
1b. Covariance inflation
(additive)
xia '  xia  ia ; ia  N(0,Q)
Model error estimation schemes (DdS)
2. Dee and daSilva bias estimation scheme (1998)
bt f  bta1
Do data assimilation twice:
b a  b f  L[ y o  ( Hx f  Hb f )]
first for model error
L  P bias H T ( HPbias H T  HP f H T  R) 1
~
x f  x f  ba
then for model state (expensive)
xa  ~
x f  K [ y o  H~
xf]
K  P f H T ( HP f H T  R) 1
P
bias
 *P
f
0  ,  1
1 K f
P 
( xi  x f )(xif  x f )T

k  1 i 1
f
and need to be tuned
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
xf
x6ehr
model
NNR
NNR
NNR
NNR
x truth
NNR
t=0
t=6hr
xf
We collect a large number of estimated errors and estimate bias, etc.
L
M
l 1
m 1
 nf1  x nf 1  xtn1  M (x an )  M (xtn )  b    n,l el    n,mf m
Forecast error
due to error in IC
Time-mean
model bias
State dependent
Diurnal
model error
model error
Further explore the Low-dimensional method
Include Bias, Diurnal and State-Dependent model errors:
10
10
l 1
m1
mod el _ error  b   n,l el    n,m fm
Time-mean
model bias
BIAS
one month
climatological
debiased
Leading EOFs for 925 mb TEMP
Diurnal model errors
• Generate the leading EoFs
from the forecast error
anomalies fields for
temperature.
e '
6 hr ( i )
x
 x6ehr  x6ehr
e '
6 hr ( i )
EOF[ x
]
pc1
pc2
Lack of diurnal forcing generates
wavenumber 1 structure
'
EOF[ x6ehr (i ) ]  EOF[ x6ehr  x6ehr ]
925hPa Temperature
Black line:
~
x f  xf b
Blue line:
10
~
x f  x f  (b    n,l el )
l 1
State-dependent model errors
the local state anomalies (Contour) and the forecast error anomalies (Color)
SVD1
SVD2
SVD3
SVD4
Correct state-dependent model errors
500hPa Uwind
500hPa Height
L
M
l 1
m 1
~
x f  x f  [b    n,l el    n,mf m ]
~
x f  xf b
Black line:
~
x f  xf b
Blue line:
~
x  x  [b    n,mf m ]
10
f
f
m 1
Univariate SVD (not account for the relations between different variables)
Impact of model error, and different
approaches to handle it
Perfect model
imperfect model
(obs from Reanalysis)
Simultaneous estimation of
inflation and observation
errors
Hong Li
Eugenia Kalnay
University of Maryland
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.
MK method to estimate the inflation parameter
(Miyoshi 2005, Miyoshi&Kalnay 2005)
dob  yo  H(xb )
 dobd
T
ob
obs. increment in obs. space
Should be satisfied if R, Pb and  are
correct (they are not!)
 (1  )HP H  R
b
T
So, at any given analysis time, and computing the inner product
dTobdob  (1 o )Tr(HPb HT )  Tr(R)
T
~ d ob d ob  Tr (R)

1
e
Tr (HP H)
(1)
Assumption: R is
known
Diagnosis of observation error statistics
(Desroziers et al, 2006, Navascues et al, 2006)
Desroziers et al, 2006, introduced two new statistical relationships:
 doadTob  R
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  :
p
 o2  dToa dob / p   (yoj  yaj )(yoj  ybj ) / p
j 1
(2)
Simultaneous estimation of inflation and
observation errors
dTobd ob  Tr (R)

1
e
Tr (HP H)
(1)
p
 o2  dToa dob / p   (yoj  yaj )(yoj  ybj ) / p
(2)
j 1
 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
Tests within LETKF with Lorenz-96 model
40 observations with true Rt=1, 10 ensemble member.
Optimally tuned rms=0.20
Perfect R, estimate inflation using (1) : it works
 method
Rs

rms
(1)
1
0.044
0.202
Wrong R, estimate inflation using (1) : it fails
 method
(1)
Rs
4.0

rms
0.027
1.632
Tests within LETKF with L96 model
Now we estimate observation error and optimal inflation
simultaneously using (1) and (2): it works!
R method
(2)

method
(1)
Rinit
Estimated Estimated
R

rms
0.25
1.001
0.042
0.202
4.0
1.008
0.040
0.204
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
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), 2*10-4 kg/kg (q) and 200Pa(Ps).
• Estimate and correct the observational errors and inflation
adaptively.•
online estimated observational
errors
The original wrongly specified R converges to the
correct R quickly (in about 5-10 days)
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)
Global averaged analysis RMS
500hPa Temperature
500hPa Height
Using an initially wrong R and  but estimating them adaptively
Using a perfect R and estimating  adaptively
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.
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
Adaptive sampling with the LETKFbased ensemble spread
Junjie 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
500hPa zonal wind RMS error
Rawinsonde; climatology; uniform; random; ensemble spread; “ideal”; 100%
3D-Var
RMSE
4.04
2.36
0.92
0.74
0.43
LETKF
0.36
0.30
1.18
0.38
0.36
0.33
0.32
0.29
0.23
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
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
Analysis sensitivity study within LETKF
Hx a
S
 R 1HPa HT
y
The self sensitivity is the trace of the matrix S.
It can show the analysis sensitivity with respect to:
a) different types of observations (e.g., rawinsonde,
satellite, adaptive observation and routine
observations)
b) the observations in different area (e.g., SH, NH)
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.
Comparison of ensemble-based and variational-based
data assimilation schemes in a Quasi-Geostrophic model.
Shu-Chih Yang et al.
3D-Var
Hybrid (3DVar+20 BVs)
12-hour 4D-Var
LETKF (40 ensemble)
24-hour 4D-Var
4D-Var
3D-Var
HYBD
RMS error (10-2)
1.44
Time (minutes)
0.15
LETKF
12hr
24 hr
l=3
l=5
l=7
l=9
0.70
0.56
0.35
0.67
0.48
0.44
0.44
1.5
1.8
2.5
0.3
1.0
1.9
2.4
Analysis increment (color shaded)
vs. dynamically fast growing errors (contours)
12Z Day 24
Initial increment (smoother) vs. BV
00Z Day 25
analysis increment vs. BV
LETKF
Initial increment vs. Initial SV
12-hour 4DVAR
analysis increment vs. Final SV
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
No-cost LETKF smoother
LETKF analysis
at time i
Smoother analysis
at time i-1
xai  xif  Eif Wi
x
i 1
a
x
i1
a
i 1
a
LETKF Analysis
E W
i
“Smoother” reanalysis
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.