a multi-scale three-dimensional variational data assimilation scheme
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A multi-scale three-dimensional variational data assimilation scheme
Zhijin Li, , Yi Chao (JPL) James C. McWilliams (UCLA), Kayo Ide (UMD)
The 8th International Workshop on Adjoint Model Applications in Dynamic Meteorology
May 18-22, 2009, Tannersville, PA
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Outline
1. Motivations
2. A multi-scale three-dimensional variational data assimilation (MS-3DVAR) scheme
3. Applications and evaluations
4. Summary
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Data Assimilation and Forecast Cycle
3-day forecast
Aug.100Z
Time
Aug.118Z
Aug.112Z
Aug.106Z
Initialcondition
6-hour forecast
Aug.200Z
xa
xf
6-hour assimilation cycle
xxx fa
Time scales comparable with those of the atmosphere
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Observations and Data assimilation
0
100
200
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700
800
900
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Year Day
Nu
mb
er o
f C
asts
/Day
<55
<110
<220
<440
<1100
T/S profiles from gliders Ship CTD profiles Aircraft SSTs AUV sections HF radar velocities
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Comparison of Glider-Derived Currents (vertically integrated current)
Black: SIO glider; Red: ROMS
SALT(PSU)
Performance of ROMS3DVAR:AOSN-II, August 2003
(Chao et al., 2009, DSR)
TEMP(C)
Glider temperature/salinity profiles
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Challenge: An Example with SCCOOS
Model domain, the resolution of 1km
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Southern California Coastal Ocean Observing System (SCCOOS)
SIO Glider Tracks
Challenge: How to assimilate sparse vertical profiles along with high resolution observations for a very high resolution model
Aug 2008, simulation without DA
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Data Assimilation Formulation
)()(21
)()(21
min 11 yHxRyHxxxBxxJ TfTf
x
prescribed B optimization algorithm
Variational methods (3Dvar/4Dvar):
Sequential methods (Kalman filter/smoother)
dynamically evolved B analytical solution
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Error Covariance and correlations
CB
C: Correlation matrix : RMSE diagonal matrix
Correlations are the vehicle for spreading out information of observations
Correlation scales are about
15-30km
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Forecast Error Covariance B:Single Observation Experiment
One single observation of SSH
afa xxx (increment)
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Southern California Coastal Ocean Observing System (SCCOOS)
SIO Glider Tracks
Motivation: assimilating sparse vertical profiles along with high resolution observations for a very high resolution model
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Multi-Scale Data Assimilation: Concept
HL
HL
eee
xxx
)()(2
1)()(
2
1min
)()(2
1)()(
2
1min
11
11
HHHHT
HHHfHHH
TfHH
x
LLLLT
LLLfLLL
TfLL
x
yxHRyxHxxBxxJ
yxHRyxHxxBxxJ
S
L
HL
HL yyy
Hxy
Background
Observation
Prerequisites
Multi-scale DA
0
0
THL
THLee
SCCOOS Glider Tracks
(Boer, 1983, MWR)
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Multi-Scale Data Assimilation: Scheme
Low Resolution (LR)
)()(2
1)()(
2
1min 11 yHxRyHxxxBxxJ Taf
LaTaf
Lx
)()(2
1)()(
2
1min 11
LLLLT
LLLfLLL
TfLL
xyxHRyxHxxBxxJ
L
aL
fL
aL xxx
aL
fafL xxx
High Resolution (HR)
Sparse Obs
High Resolution Obs
SCCOOS Glider Tracks
High resolution HF radar
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Work Flowchart of MS-3DVAR
Satellite SST/SSH HF Radar
LR-3DVAR
fLx
Forecast
Smoothed
fx
Start
HR-3DVAR
Increment aLx
afa xxx
aL
fafL xxx
End
Glider/Argo/MooringSmoothed
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Glider Observations vs Analyses
Aug 2008
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CALCOFI Observations
Aug 14-30, 2008 (not assimilated)
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Aug, 2008
Monthly Means
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Analysis vs HF Radar Observations
Taylor diagram (Taylor, 2001, JGR)
19 Surface Currents
Forecasts vs Analyses
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Summary
A multi-scale 3DVAR (MS-3DVAR) scheme has been formulated and developed.
It has been implemented in support of SCCOOS and AOOS-PWS.
The scheme has demonstrated the capability of assimilating sparse and high resolutions observations simultaneously, effectively and reliably.
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3 nested levels: L0 / L1 / L2.
Resolution : 10km / 3.6km / 1.2 km
(L2 : Prince William Sound)
Alaska Ocean Observing System -Prince William Sound:AOOS-PWS
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From USGS
Oil Spill: 1989 Exxon Tanker Wreck Prince William Sound, Alaska
Today
March 24, 1989
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Decomposition of Large and Small Scalesand Estimation of Error Covariance
Generate perturbations: difference between 24h and 48h forecasts, valid at the same time.
Decompose perturbations
Large scales: smoothed fields, with weight of
where L=25km, which is the decorrelation length scale from Russ’s estimation Small scales: the residuals
)2/exp( 22 Lr
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Observational Errors: Representativeness Errors
• Large scale observations: T/S vertical profiles from SIO, mooring and Argo
• Observational errors at the depth of z:
• Tentative values: briefly and empirically
estimated from the RMS profile of the small
scale components
2,0~ zoz Ne
oz
Lz
Lz
Lz zzNe )]/exp(1[,,0~ 2
022
mz 30,6.1 0 mz 50,6.1 0
For temperature
For salinity
CALCOFI Section
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Future Coastal Observing System
(National Research Council, 2003)
Buoy and glider profiles are sparse, with distances between profiles larger than decorrelation scales
Radar and satellite measurements may have very high resolutions, as high as the model resolution
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Glider Temperature and Salinity Profiles
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Surface Tidal Current Comparison (M2)
Corr. RMS Mean Sept. 0.43 3.6 6.6 Oct. 0.44 3.8 6.6 Nov. 0.51 3.7 6.3
Length of Major Axis (cm/s)
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A There-Dimensional Variational Data Assimilation (3DVAR)
• Real-time capability• Implementation with sophisticated and high resolution
model configurations• Flexibility to assimilate various observation
simultaneously
• Development for more advanced scheme
(Li et al., 2006, MWR; Li et al., 2008, JGR, Li et al., 2008, JAOT)
),,,,(
)()(2
1)()(
2
1min 11
vuSTx
yHxRyHxxxBxxJ TfTf
x
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