Post on 22-May-2020
A Multi-Scale Three-Dimensional Variational Data
Assimilation Scheme and Its Application to Coastal Oceans
Zhijin Li
Jet Propulsion Laboratory, California Institute of Technology
The 9th Workshop on Adjoint Model Applications in Dynamic Meteorology
Cefalu, Sicily, Italy, 10-14 October 2011
Copyright 2011 California Institute of Technology
Acknowledgements
• Dr Yi Chao and his group (JPL)
• Prof James C McWilliams and his group (UCLA)
• Prof Kayo Ide (UMD)
• We acknowledge the support from NASA Physical
Oceanography Program
3DVAR Data Assimilation and Forecast Cycle
3-day forecast
6-hour forecastxfxxx fa δ+=
Aug.100Z
Time
Aug.118Z
Aug.112Z
Aug.106Z
Initialcondition
forecast
Aug.200Z
xa
x
6-hour assimilation cycle
• Diurnal variation
• Rapid response to wind stresses
• Eddies, fronts, filaments, etc
Autonomous Ocean Sampling Network (AOSN)
Experiment August, 2003
“Bring together sophisticated new robotic vehicles with advanced ocean models to improve our ability to observe and predict the ocean”
www.mbari.org/aosn
9 km 3km 1 km
An Incremental There-Dimensional Variational Data
Assimilation (3DVAR)
f
TT
x
TfTf
x
Hxyy
yxHRyxHxBxxJ
yHxRyHxxxBxxxJ
−=
−−+=
−−+−−=
−−
−−
δ
δδδδδδδ )()(2
1
2
1)(min
)()(2
1)()(
2
1)(min
11
11
1. Real-time capability
2. Implementation with sophisticated and high resolution model
configurations
3. Flexibility to assimilate various observation simultaneously
(Li et al., 2006, MWR; Li et al., 2008, JGR)
AOSN Intensive Observations
�T/S profiles from gliders
� Ship CTD profiles
� Aircraft SSTs
� AUV sections
� HF radar velocities
0
100
200
300
400
500
600
700
800
900
213
215
217
219
221
223
225
227
229
231
233
235
237
239
241
243
245
Year Day
Num
ber
of C
asts
/Day <55
<110
<220
<440
<1100
� HF radar velocities
T/S Profile Data
Glider and AUV tracks Ships, Aircrafts, and HF radars
Comparison of Glider-Derived Currents (vertically integrated current)
Performance of ROMS3DVARAugust 2003
Black: glider Red: ROMS
(Chao and Li et al., 2009, DSR)
Southern California Coastal Ocean Observing System
(SCCOOS)
SIO Glider Tracks
Challenge: Assimilating sparse vertical profiles along with high
resolution observations for a very high resolution model
Decorrelation length scales: 15-50km
Challenges with 3DVAR
Fourier Series Expansion of Homogenous Errors
dxinxxee
inxexe
n
nn
)exp()(1
)exp()(
−=
=
∫
∑∞
∞
−∞=
dxinxxeen )exp()(2
−= ∫ ∞−π
==
≠==∗
nmc
nmee
nnm ,
,0
Wiener-Khintchine Theorem
)()()( rxexerc += Error Covariance
1
2
)exp()(
)exp()(2
1
nn
n
n
ec
dninrcrc
drinrrcc
=
=
−=
∫
∫∞
∞−
∞
∞−π
power spectral density
Error Covariance in 3DVAR: Smoothing and Spreading
2
2
2)( D
r
erc−
=
−=
2exp
2
22 DnDcn
π
A Multi-Decorrelation Length Scale Scheme for High
Resolution Models?
SL xxx +=
Background Error
SL eee +=
SL
TSL
BBB
ee
+=
= 0
3DVAR with a Background Error Covariance of
Multi-Decorrelation Length Scales
( ) )()(2
1
2
1)(min 11 yxHRyxHxBBxxJ T
SLT
xδδδδδδδ −−++= −−
( )
( ) )()(2
1
2
1)(min
)()(2
1
2
1)(min
11
11
yxHRHHByxHxBxxJ
yxHRHHByxHxBxxJ
ST
LT
SSST
SSx
LT
ST
LLLT
LLx
S
L
δδδδδδδ
δδδδδδδ
δ
δ
−+−+=
−+−+=
−−
−−
(Li et al., 2011, QJRMS, in revision))|(
)|(
yxp
yxp
S
L(Lorenc, 1986)
Multi-Scale Representativeness Errors
to xHye −= δδ
TS HHBR +Observational Error
Covariance for Large Scale
( ) ( ) ( )orS
orom
tS
bS
ttt
tL
oL
eee
xxHyHxyy
xHye
++=
−−−−−=
−= δδ
Measurement error + Representativeness error + Multi-scale representativeness error
Multi-Scale Data Assimilation
hS
hL
h yyy +=
High resolution Observation
( )
( ) )()(2
1
2
1min
)()(2
1
2
1min
11
11
hSSS
ThSSSS
TSS
x
hLLL
ThLLLL
TLL
x
yxHRyxHxBxxJ
yxHRyxHxBxxJ
S
L
δδδδδδδ
δδδδδδδ
δ
δ
−−+=
−−+=
−−
−−
Multi-scale DA
3DVAR Formulations
)()(2
1)()(
2
1min 11 yHxRyHxxxBxxJ TfTf
x−−+−−= −−
( ) )()(11
min11 yxHRHHByxHxBxJ TTT δδδδδδ −+−+=
−−
3DVAR
AB-3DVAR
( )
( ) )()(2
1
2
1min
)()(2
1
2
1min
11
11
yxHRHHByxHxBxJ
yxHRHHByxHxBxJ
ST
LT
SSST
Sx
LT
ST
LLLT
Lx
S
L
δδδδδδ
δδδδδδ
δ
δ
−+−+=
−+−+=
−−
−−
)()(2
1
2
1min
)()(2
1
2
1min
11
11
hSSS
ThSSSS
TS
x
hLLL
ThLLLL
TL
x
yxHRyxHxBxJ
yxHRyxHxBxJ
S
L
δδδδδδ
δδδδδδ
δ
δ
−−+=
−−+=
−−
−−
MS-3DVAR
Experiments with Idealized Problems
cos1
0 N
nkaSx t
k
K
k
tk
tn
+= ∑=
φπ
tkb
tk
K
k
bk
bn
aa
N
nkaSx
β
φπ
=
+= ∑=
cos1
0
True StateBackground/First Guess
5/,200
)1,1(,
1
NKN
ka
N
kktk
tk
k
==
−∈=
=
=
απαφ
γtk
kbk aa β=
ooe
tn
o eaxy +=Observations
Difference between SD-3DVAR, MD-3DVAR,
and MS-3DVAR Solutions
Patchy Observation
( )
−−=
2
22
2exp
D
jibB eij
Analysis Errors
MS-3DVAR Work Flow
SL yy δδ ,
yδObservation innovationForecast fx
fS
fL xx ,
Increment
SS-3DVARLS-3DVAR
IncrementaSxδ
aS
faL
a xxx δ+=aL
ffaL xxx δ+=
aLxδ
SL BB ,
( )
( )
)()(2
1
)()(2
1
2
1min
)()(2
1
)()(2
1
2
1min
1
11
1
11
hSSS
ThSS
ST
LT
SSST
Sx
hLLL
ThLL
LT
ST
LLLT
Lx
yxHRyxH
yxHRHHByxHxBxJ
yxHRyxH
yxHRHHByxHxBxJ
S
L
δδδδ
δδδδδδ
δδδδ
δδδδδδ
δ
δ
−−+
−+−+=
−−+
−+−+=
−
−−
−
−−
Kronecker Product Formulation of
3D Error Correlations
130
100
80
60
Cx(z
a, z
a)
DIS
TA
NC
E F
RO
M S
HO
RE
0.2
0.4
0.6
0.8
1
100
150
200
250D
IST
AN
CE
FR
OM
S. B
OU
ND
AR
Y
Ch(z
a, z
a)
37.50.8
1
ΣΣ= CB
“NMC” Method: 48h-24h Forecast
( ) ( )( )( )T
TT
GGGG
GGGGC
CCC
ηξκηξκ
ηηξκξκξηκ
ηξκξηκ
⊗⊗=
⊗=
⊗=
130 100 80 60 40 20
40
20
DISTANCE FROM SHORE (km)
DIS
TA
NC
E F
RO
M S
HO
RE
0
0.2
50100150200250
50
100
DISTANCE FROM S. BOUNDARY (km)
DIS
TA
NC
E F
RO
M S
. BO
UN
DA
RY
−123.5 −123 −122.5 −122 −121.535
35.5
36
36.5
37
LAT
ITU
DE
(° N
)
LONGITUDE (° W)
−0.2
0
0.2
0.4
0.6
−123.5 −123 −122.5 −122 −121.5LONGITUDE (° W)
Improved Performance with SCCOOS
MS-3DVAR Performance
1989 Exxon Valdez Supertanker Oil Spill
in the Prince William Sound
Struggling sea lion during the tragedy days
Prince William Sound
Blind sea lion present day
Field Experiment 2009 Prediction of
Drifter Trajectories in the Prince William Sound
L0 10km
L1 3.6km
L2 1.2km
Oil Spill: 1989 Exxon Tanker Wreck ,
Prince William Sound, Alaska
Surface CurrentsHF radar observed (Red), ROMS (Black)
Effective Assimilation of High Frequency Radar High
Resolution Velocities during Field Experiment 2009
(Schoch and Chao, 2010, EOS)
Summary
• A multi-scale 3DVAR scheme with partitioned cost functions was
developed
• MS-3dVAR used multi-decorrelation length scales to construct
background error covariancebackground error covariance
• Effectiveness of the assimilation of both sparse and high resolution
observations was improved
� Observation oriented covariance
� Reduced representativeness errors