mul$%scale*data*assimila$on*for* fine%resolu$on*models**50 100 150 200 −1.5 −1 −0.5 0 0.5 1...
TRANSCRIPT
Mul$-‐Scale Data Assimila$on for Fine-‐Resolu$on Models
Zhijin Li Jet Propulsion Laboratory, California Ins$tute of Technology
with James C. McWilliams (UCLA) and Kayo Ide (UMD)
Workshop on Sensi-vity Analysis and Data Assimila-on in Meteorology and Oceanography Roanoke, WV, 1-‐5 June, 2015
Copyright © All rights reserved
Atmospheric Data Assimila$on for Cloud Resolving Models
• Regional opera-onal models oGen
have a resolu-on of higher than 4
km to resolve cloud systems
• Are data assimila-on schemes
based on op-mal es-ma-on
theory are suitable for cloud
resolving models?
Oceanic Data Assimila$on for Sub-‐Mesoscale Processes
Surface Water and Ocean Topography (SWOT) Satellite
Salinity Processes in the Upper Ocean Regional Study (SPURS) (hOp://spurs.jpl.nasa.gov)
• Mesoscale 40km – 400 km • Sub-‐mesoscale 1 km -‐ 40 km
Conven$onal Data Assimila$on: Op$mal Es$ma$on
§ prescribed B § op-miza-on algorithm
Varia-onal methods (3Dvar/4Dvar):
Sequen-al methods (Kalman filter/smoother)
§ dynamically evolved B § analy-cal solu-on (matrix manipula-ons)
minxJ = 1
2(x − xb )T B−1(x − xb )+ 1
2(Hx − y)T R−1(Hx − y)
Forecast
Analysis
Observa$on
Maximum Likelihood
Error Covariance: Spreading and Filtering
2
2
2)( Dr
erc−
=
cn =D2π
exp −k2D2
2"
#$
%
&'
B = ΣCΣ
2D 2D
Correla-
on
Power Spe
ctrum
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−0.5
0
0.5
1
1.5
STAT
E DI
STRI
BUTI
ON
(a)
OBS: RMSE=0.150GUESS: RMSE=0.275
TRUEGUESSOBS
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−1
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0
0.5
1
1.5
AB−DA: RMSE=0.064MS−DA: RMSE=0.061
(b)TRUEAB−DAMS−DA
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−1
−0.5
0
0.5
1
1.5
GRID NUMBER
D=10: RMSE=0.074D=20: RMSE=0.122
(d)TRUESS−DA D=10SS−DA D=20
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−1
−0.5
0
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1
1.5
GRID NUMBER
STAT
E DI
STRI
BUTI
ON
D=5: RMSE=0.063D=35: RMSE=0.144
(c)TRUESS−DA D=5SS−DA D=35
50 100 150 200−1
−0.5
0
0.5
1
GRID NUMBER
(d)GUESSMS−DA LSMS−DA SS
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−0.5
0
0.5
1
GRID NUMBER
ANAL
YSIS
INC
REM
ENT
(c)GUESSAB−DA LSAB−DA SS
50 100 150 200−1
−0.5
0
0.5
1
(b)GUESSSS−DA D=10SS−DA D=20
50 100 150 200−1
−0.5
0
0.5
1
ANAL
YSIS
INC
REM
ENT
(a)GUESSSS−DA D=5SS−DA D=35
2
2
2)( Dr
erc−
=
Filtering Proper$es An Idealized 1D Experiment
With a large correla$on scale, DA corrects the large scale error
1D Problem with Homogeneous and Isotropic Background Error
For both global and regional domains, we have
BS ≈ FBFT =σ b2S
S = FCFT
F – Discrete Fourier Transform (DFT)
• The eigenvalues of C are its Fourier transform coefficients, namely, values of the spectral density func-on.
• The vectors that define the discrete Fourier transform are eigenvectors of C.
Filtering Proper$es
xa = xb +BHT HBHT + R( )−1y−Hxb( )
H = I
sa = Fxa
sd = F y− x f( )BS = FBF
T =σ b2S
RS = FRFT =σ o2I
sa = sb + S S +σo2
σ b2 I"
#$
%
&'
-1
sd
Filtering Proper$es: Analysis Increment Scales
Conven$onal data assimila$on is unable to update fine-‐scale informa$on
sa = sb + S S +σo2
σ b2 I!
"#
$
%&
-1
sd
σ o
σ b =12
0 50 100 150 200 250 300 3500
0.2
0.4
0.6
0.8
1
DISTANCE r (KM)
CORR
ELAT
ION
BACKGROUND ERROR
exp(−r2/2D2)
D=50KMD=10KM
200 100 50 40 30 25 200
0.2
0.4
0.6
0.8
1
WAVELENGTH L (KM)
STAN
DARI
ZED
POW
ER S
PECT
RUM
ANALYSIS INCREMENT
INC., 50KMINC., 10KM
2D 2D
Meso-‐ and Small Scale Component in Background Error Covariance
1. Meso-‐ and small-‐ scale
systems are intensive, but
are localized and
intermiOently occur.
2. The forecast/background
error covariance is primarily
determined by large scale
systems
3. The correla$on scale is inevitable to be large scale
Ques$ons and Challenges
• Do we need to update fine-‐scale informa-on in data assimila-on?
maybe not ?
• If yes, how can we update fine scale informa-on?
We here suggest a scheme to update fine scale informa$on
Mul$-‐Scale Data Assimila$on: Data Assimila$on Separately for Dis$nct Scales
SL
SL
eeexxx
+=
+=
( )
( ) )()(21
21min
)()(21
21min
11
11
yxHRHHByxHxBxJ
yxHRHHByxHxBxJ
ST
LT
SSSTSx
LT
ST
LLLTLx
S
L
δδδδδδ
δδδδδδ
δ
δ
−+−+=
−+−+=
−−
−−
Scale decomposi$on
Mul$-‐scale DA
SL
TSL
BBB
ee
+=
= 0
bxxx −=δ (Li et al., 2015, MWR)
Proper$es of Mul$-‐Scale Data Assimila$on
• The decomposed cost func-on can be derived by maximizing the condi-onal
probability
• Separate es-mate of the state for dis-nct scales using decomposed cost func-ons
• Explicit incorpora-on of mul-ple decorrela-on scales, thus, Mul$-‐Scale Data
Assimila$on
p(xL | y)p(xS | y)
Mul$-‐Scale Representa$veness Errors and Aliasing
( )
( ) )()(21
21min
)()(21
21min
11
11
yxHRHHByxHxBxJ
yxHRHHByxHxBxJ
ST
LT
SSSTSx
LT
ST
LLLTLx
S
L
δδδδδδ
δδδδδδ
δ
δ
−+−+=
−+−+=
−−
−−
sa = sb + S S +σo2
σ b2 I!
"#
$
%&
-1
sd
Scales untangled
Scales tangled Scale aliasing/contamina$on ?
Mul$-‐scale representa$veness error
Assimila$on of Decomposed Observa$ons
Mul$-‐scale representa$veness error
δy = δyL +δyS
minδxL
J = 12δxL
TBL−1δxL +
12(HδxL −δyL )
T HBSHT + RL( )
−1(HδxL −δyL )
minδxS
J = 12δxS
TBS−1δxS +
12(HδxS −δyS )
T HBLHT + RS( )
−1(HδxS −δyS )
NEXRAD Reflec$vity: Meso-‐Scale Connec$ve System (MCS): June 14, 2007
UTC 00:00 UTC 03:00
UTC 06:00 UTC 09:00
Improved Reflec$vity: UTC 06, 14 June, 2007
NO DA
DA
Simulated 900 hPa Reflec$vity
(Li et al., 2015, JGR)
Summary
• Due to the filtering proper-es, conven-onal data assimila-on can not
effec-vely update fine-‐scale informa-on.
• To update fine-‐scale informa-on, it is suggested that the fine scale should be
treated separately from larger scales.
• The cost func-on is mathema-cally decomposed for formula-ng a MS-‐DA
scheme .
• The decomposed cost func-on allows for the background error covariance to
explicitly incorporate mul-ple decorrela-on scales.
• Experiments show promising performance of the MS-‐DA scheme in 3Dvar for
both oceanic and atmospheric applica-ons