3DVAR%20on%20the%20JMA-NHM%20and%20EnKF%20on%20the%20SPEEDY%20model

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3DVAR on the JMA-NHMandEnKF on the SPEEDY model
6/21/05 EMC/NCEP seminar

• Takemasa Miyoshi
• Department of Meteorology, University of Maryland
• and
• Numerical Prediction Division, Japan
  Meteorological Agency

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Two topics

• 3DVAR on the JMA-NHM
• Before coming to UMD, I developed the 3DVAR for
  nonhydrostatic model (JMA-NHM) at NPD/JMA (Apr.
  2002- Jun. 2003)
• Currently, 4DVAR has been developed by Mr. Honda
  (NPD) and several scientists at MRI/JMA
• EnKF on the SPEEDY primitive equations model
• Doctoral dissertation (advisor Prof. Eugenia
  Kalnay)
• Defended on June 7
• Purpose is to develop a path towards operational
  implementation

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JMA-Nonhydrostatic Model (NHM)

• In operation since Sep. 2004
• Prognostic Variables
• U,V,W,Prs,?,qv,qc,qr,qs,qci,qg
• Terrain-Following Vertical Coordinates (z)
• Staggered Grid Structure
• Arakawa C type (Horizontal)
• Lorenz type (Vertical)

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Design of control variables

• Gaussian horizontal correlation
• Recursive filter, Cholesky decomposition
• Full vertical correlation
• Inter-variable correlation

• Geostrophic Balance (Synoptic Scale)
• Mass Conservation

PB Hydrostatic Pressure
Control Variable Unbalanced Stream Func.
Unbalanced Vertical Wind
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Control variables
Control Variables Definition
Unbalanced Stream Function
Unbalanced Velocity Potential
Unbalanced Vertical Wind
Unbalanced Pressure
Potential Temperature the bottom level pressure
Water Vapor Quantity
Different variables are assumed to be uncorrelated
Regression coefficients r1-r4 are determined
statistically
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Background error statistics

• NMC Method
• Error 12hrs fcst - 6hrs fcst (at the same valid
  time)
• Model Setting
• 62cases Statistics in 4/24/2002 - 5/28/2002
• Grid 12212245
• Horizontal Resolution 5km
• Region

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A case study

• Aircraft overran at Narita Intl Airport at
  around 2150

Synoptic chart 2100JST
Rapid growth of the Low
No clear front around NRT in synoptic scale
analysis
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Local front around NRT airport

• Very Shallow Structure
• Low Temperature Side
• Weak Northern Wind
• High Temperature Side
• Strong South Wind

Cross Section of Local Front
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7-hr fcsts by several initial conditions
Valid 2200JST
Initial 1500JST
Hourly 3DVAR performs as well as the routine
MSM4DVAR!
Still, NRT is in the warmer side in all cases.
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Keep updating
Newer analyses by Hourly 3DVAR
04Z
05
06
07
08
09
10

3DVAR
3DVAR
3DVAR
3DVAR
The latest analysis available without hourly
analysis

• Observations assimilated
• Synoptic conventional observations
• Wind profiler radar
• Aircraft observations

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Forecasts valid at the same time (22JST)
Hourly New Initial Values
Different initial conditions
?
?
FT6
FT1
?
?
FT5
FT2
?
?
FT3
FT4
Initial 1900JST
Initial 1800JST
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Analysis field by hourly 3DVAR
Valid 2200JST
?
Strong shear line at the height of 2-300m
?
Low Temperature zone and the local front is well
analyzed
NRT is located in the low temperature side of the
front
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Summary

• I developed 3DVAR on JMA-NHM during 2002-2003.
  (Currently, 4DVAR has been developed by Mr. Honda
  and others.)
• 3DVAR shows reasonable analysis performance.
• The hourly updating cycle using 3DVAR shows an
  advantage in a case study.

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Ensemble Kalman filter experiments with a
Primitive-Equation global model

• Dissertation Committee
• Prof. Eugenia Kalnay (chair/advisor)
• Prof. James Carton
• Prof. Owen Thompson
• Prof. Brian Hunt
• Dr. Joaquim Ballabrera

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Overview
Introduction
Data assimilation
A way to keep the model state close to the nature
using observations
Ensemble Kalman filtering (EnKF)
A sophisticated method of data assimilation
Several implementations have been proposed but
not compared
Model imperfections
No model can simulate the Mother Nature perfectly
Sophisticated methods are generally more
sensitive to model imperfections
Experiments
Perfect model experiments
Compare EnKF methods to 3DVAR
Imperfect model experiments
Investigate the effect of model imperfections
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Purpose
The ultimate goal
To develop a path towards an operational ensemble
forecast-analysis system.
Realized by ensemble Kalman filtering (EnKF)
Relevant scientific questions

1. What are the relative advantages and
   disadvantages among different implementations of
   EnKF?
2. How do they compare with 3DVAR?
3. What is the effect of model imperfections in data
   assimilation?

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Data assimilation (filtering)
Data assimilation is an analysis technique in
which the observed information is accumulated
into the model state by taking advantage of
consistency constraints with laws of time
evolution and physical properties. Bouttier and
Courtier (1999)
Accumulate past observations
Obs
Obs
Obs
Analysis
Time
Assimilate
Analysis
Analysis
Model integration
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Kalman filtering (KF)
Kalman filtering (KF) is an optimal weighted mean
between forecast and observation.
Here, the optimal weight (Kalman gain) is given as
Error covariance is estimated by the forecast
KF (estimation of P) is optimal when the model is
linear and perfect.
Error covariance forecast is usually
underestimated partly because of model
nonlinearity. Thus, covariance inflation is
required.
Inflation parameter
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Ensemble Kalman filtering (EnKF)
Ensemble formulation
Assumption Limited number of ensembles can
estimate P
To avoid sampling errors, we localize covariance.
By the ensemble formulation, we can forecast
error covariance from ensemble forecasting
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An advantage of Kalman filtering
In Kalman filtering, we forecast P
We use constant P in 3DVAR
An example of using the flow-dependent P There is
a cold front in our area
Kalman filtering can consider the flow-dependent
error structure.
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EnKF approaches
We do not investigate this
Perturbed observation (PO) method
EnKF
Square root filtering (SRF) method
Serial EnSRF (Whitaker and Hamill 2002)
Effective in serial treatment of observations
ETKF (Bishop et al. 2001)
EAKF (Anderson 2001)
LEKF (Ott et al. 2002 2004)
Simultaneous treatment of observations
We implement and compare Serial EnSRF and LEKF
Weighting function
Serial EnSRF localizes covariance P around
observation locations.
Local patch
LEKF treats local patches independently.
N local patches cover the entire globe.
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Results on the Lorenz-96 model
Lorenz-96 model (Lorenz 1996 Lorenz and Emanuel
1998)
Observe 20 points out of 40 total grid points,
with error stdev of 1.0
3DVAR RMSE 1.15
LEKF with obs. localization
Serial EnSRF
Covariance inflation factor
0.33
0.34
Localization length scale
Localization length scale
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The SPEEDY model (Molteni 2003)
Prognostic variables
Level Sigma (p/ps) Pressure (hPa)
1 0.950 925
2 0.835 850
3 0.685 700
4 0.510 500
5 0.340 300
6 0.200 200
7 0.080 100
u, v, T, q, Ps
Resolution
T30L7 ( 96 x 48 x 7 )
Primitive-equation dynamic core
Simplified physics
Convection (a simplified mass-flux scheme)
Large-scale condensation, Clouds
Short-wave radiation (two spectral bands)
Long wave radiation (four spectral bands)
Surface fluxes of momentum and energy (bulk
aerodynamic formula)
Vertical diffusion
No diurnal forcing
Computational time
2 seconds for 6-hour forecast on a 2.7GHz Celeron
PC
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Data assimilation experiments
OSSEs (Observation Systems Simulation Experiments)
Nature run
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Experimental setup
Perfect model experiments
Nature run is generated by the SPEEDY
Data assimilation systems
3DVAR
Serial EnSRF
LEKF
Observational network
Uniform (mainly used)
Realistic
Variables Obs. Err. Stdev.
u m/s 1.0
v m/s 1.0
T K 1.0
q kg/kg 0.0001
Ps Pa 100.0
Cov. inflation
4 fixed, multiplicative
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Results 500hPa height RMSE
Average skills in 500hPa height RMSE meters
3DVAR 31m
Serial EnSRF
D filter divergence
Localization scale 1 2 3 4 5 6
NBV10 12 25 40 D D D
NBV20 11 11 11 D 22 D
NBV30 15 5 5 8 10 17
LEKF
Localization scale 1 2 3 4 5 6
NBV10 60 76 D D D D
NBV20 35 20 17 31 D D
NBV30 18 17 9 8 13 15
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Results time sequence
30 ensemble members
3DVAR (31m)
LEKF (8m)
Serial EnSRF (5m)
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Results zonal mean RMSE
Zonal mean RMSE of u-wind
3DVAR shows clearer striped pattern corresponding
to observational locations.
3DVAR
Serial EnSRF
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Analysis increments
EnSRF analysis increment is better capturing
error structures.
Lattice-like pattern in the 3DVAR background
error field.
3DVAR
EnSRF
Shades background error, Contour analysis
increment
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Error growth
3DVAR
SPREAD
RMSE
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Online estimation of inflation parameter
Concepts of the method
Kalnay suggested that covariance inflation
parameter can be estimated using the statistics
At each analysis step, we can have an estimate of
Assuming this estimate as observation, estimate
online using KF.
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Online estimation of inflation parameter
Results for EnSRF with 10 members
We confirm this online estimation provides stable
performance in a simple system (the Lorenz-96
model).
On the SPEEDY, the relative amplitude of analysis
RMSE and ensemble spread is adjusted, but
analysis RMSE is not decreased.
Inflation Analysis RMSE Ensemble spread
4 multiplicative 2.3 4.9
Online estimation 2.4 2.1
Energy norm
With this method, we can avoid tuning the
parameter.
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Observational error covariance localization
Concepts of the method
Hunt suggested covariance localization by
localizing the observational error covariance.
Weight the observational variance by the inverse
of the Gaussian
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Observational error covariance localization
Results for LEKF with 10 members
Local patch parameter
With the localization, LEKF is improved.
With this method, we can fix the patch size as
large and tune the localization scale.
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Summary of the perfect model experiments

• Results using the Lorenz-96 model show that with
  the observation localization, LEKF outperforms
  EnSRF
• EnKF strongly outperforms 3DVAR on the SPEEDY
  model
• 2-day forecast errors of the day are similar to
  analysis errors, but not similar among different
  methods. In ensemble low-dimensional regions, the
  errors show similar structures
• Serial EnSRF generally outperforms LEKF, but the
  difference is substantially reduced when we apply
  observational error covariance localization or
  increase the ensemble size

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Imperfect model experiments
Use NCEP/NCAR reanalysis (NNR) as the nature run
Time evolving model of NNR is different from the
SPEEDY model.
(the SPEEDY model error) (6-hour forecast)
(NNR)
Initial condition is the NNR
The statistics gives model bias (time average)
and model error EOFs
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Imperfect model experiments
Data assimilation experiments

• No treatment of the model error (use the same DA
  system as perfect model cases)

• Dee and da Silva (1998) model bias estimation
  scheme

• Low order bias estimation scheme (Kalnay and
  Danforth)

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Model bias fields
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Model error EOFs
EOF1
EOF2
Since the SPEEDY model has no diurnal forcing,
first 2 EOFs show diurnal signals with phase
differences.
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Results without bias correction
The advantage of EnKF is not as large in the
presence of model imperfections. (31m 12m in
the perfect model case)
3DVAR is more sensitive to observational networks.
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Dee and da Silvas bias correction
Concepts of the method (Dee and da Silva 1998)
Persistence for the model bias evolution
Observational increment
Observation of the model error bias
Apply EnKF for bias estimation
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Dee and da Silvas bias correction
Results with 1010 members
With the bias correction, the filter diverges.
Possibly because low dimensional assumption of
EnKF is not good due to the dynamics of the model
bias (we assumed identity (persistence) as
time-evolving law of the model bias)
Dee and da Silva bias estimation
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Low order bias estimation
The same equations are applied to estimate the
low dimensional bias variable
Nbias
1
Bias amplitude to be estimated
1
N

Nbias
Each column is a model field with the shape of
the bias
Mean bias field
EOFs
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Results with low-order bias correction
Bias correction works better in EnKF.
500hPa height RMSE
Nbias 0 1 2 3 4 5 6
RMSE m 66.6 42.8 40.6 39.2 47.6 43.9 42.5
Mean bias
Mean bias EOFs
No correction
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Summary of imperfect model experiments

• Without the bias correction, the advantage of
  EnKF over 3DVAR is reduced
• Bias correction improved EnKF substantially
• Full dimensional bias estimation by Dee and da
  Silva (1998) did not work with 10 members and is
  computationally expensive
• We obtained substantial improvement by low order
  bias correction, which is less expensive

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Towards an operational EnKF
This research addresses

• EnKF has the potential to outperform 3DVAR in
  operations.

• LEKF is efficient with a large number of
  observations and parallel computers, it may be
  the only possible choice in operations.

• With observational error covariance
  localization, the disadvantage of LEKF is reduced.

• Online estimation of covariance inflation
  parameter can be used instead of tuning.

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Towards an operational EnKF
Future

• Implement more efficient LETKF

• Comparison with 4DVAR

• Application of 4D-EnKF (Hunt et al. 2004) to
  treat asynoptic observations

I will work on the development of 4D-LETKF in JMA
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Acknowledgements

• I express my deep gratitude to Prof. Kalnay and
  the Chaos group, especially Profs. Hunt, Ott and
  Szunyogh, and to Dr. Ballabrera.
• The Japanese Government financially supported my
  graduate study.
• Drs. Franco Molteni and Fred Kucharski kindly
  provided the SPEEDY model.
• Many thanks to the committee professors and other
  faculty and students for insightful discussions.
• Many many thanks to all for help and support.