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Kazumasa Aonashi ([email protected])Kozo Okamoto, Munehiko Yamaguchi and Seiji Origuchi (JMA/MRI)
7th IPWG workshop
Dual Scale Neighboring Ensemble Approach for the Cloud-Resolving Model
Ensemble Variational Assimilation
1
2014/1/15 GPM PI WS
GOAL: Ensemble-based Variational Assimilation System to incorporate MWR TBs into CRMs
SST, Winds
Cloud Particles
Frozen Precip.
Snow Aggregates
Melting Layer
Rain Drops
Scattering
Radiation0℃
MWR TBsMWR TBs are functionsof various atmospheric and surface variables
3 mm-3cm
Water Vapor
JMANHM(Saito et al,2001)• Resolution: 5 km• Grids: 400 x 400 x 38
Ensemble-based Variational Assim
System
Ensemble-based Variational AssimilationObs.
T=t0 T=t1 T=t2
Ensemble forecasts
Mean of Ensemble forecasts
1 1( )1
f f Tf t t
e N
X XP
RCalculation of analysis of Ensemble mean
3
Calculation of analysis of error cov. and Ensemble member.
))(())((21)()(21 11 XHYRXHYXXPXXJ fffx
Problem in EnVA: Sampling errorForecast error corr. of W (04/6/9/15z 7h fcst)
Heavy rain(170,195)
Weak rain(260,210)
Rain-free(220,150)
200 km200 km
Severe sampling error for precip-related variables
4
OUTLINE• Estimation of CRM forecast error covariance
– Ensemble forecast error analysis– Dual-scale neighboring ensemble method
• Introduction to ensemble-based variational assimilation
• Experiment results using simulated data• Summary
5
Ensemble forecastsExtra-tropical Low(Jan. 27, 2003)
Typhoon CONSON(June. 9, 2004)
C
Baiu case(June 1, 2004)
JMANHM (res. 5km, 400x400x38)100 members started with
perturbed initial dataGeostrophically-balanced perturbation plus HumidityRandom perturbation with various
horizontal and vertical scales(Mitchell et al. 2002)
6
7
Forecast error noises of precip-related variablesdue to sampling error (Typhoon case)
Vertical cross-corr. @(260,190)
V
W
Precip
RHW2
PT
U
Precip W RHW2 PT U V
Horizontal corr. of Precip@(260,190) (Z=3km)
500km
Areas with large Sampling errorTyphoon case, H~3 km
8
Averaged distance of area with horizontal correlation of precipitation over 0.5
horizontal correlation of precipitationH~3km
500 km
X
X
Horizontal correlation of ensemble forecast error (H~ 5 km)Ave over whole domain (black: U, green:RHW2, red:W)
km
9
km
Rain-free area Weak PrecipPR 1-3 mm/hr
Heavy PrecipPR >10mm/hr
TyphoonTyphoonTyphoon
LowLowLow
Dual-scale Neighboring Ensemble Method(1) Dual Scale Separation
(2) Neighboring Ensemble (NE) methodSpectrally localized forecast error correlation (Buehner 2010) :
we approximated the forecast error correlation for deviation using neighboring members of the target points (5 x 5 grids).
( 1, 2) ( 1 , 2 ) ( )sl slC x x C x s x s L s ds
10
Separation of ensemble forecast error into large-scale modes ( 65 km ave.) and local modes (deviation).
Assume that precip-related variables only have local modes.
Horizontal correlation URHW2W
Comparison of forecast error corr. between conventional and NE methods
11
ConventionalNE Method
Averaged distance of area with horizontal corr. of precip. > 0.5
Precip W RHW2 PT U V
V
W
Precip
RHW2
PT
U
Vertical cross-correlation at (260,190) Typhoon case
ConventionalNE Method
EnVA: min. cost function in the Ensemble forecast error subspace
• Minimize the cost function with non-linear Obs. term.
• Assume the analysis error belongs to the space spanned by the dual-scale NE modes:
• Horizontal pattern of S is adaptively determined.
G Gf f aNE
g g
UX X U
U
=
))(())((21)()(21 11 XHYRXHYXXPXXJ fffx
f,
( )P =
( )
tG G Gf
U NE tg g g
U U SP S
U U S
12
EnVA(OSSE)
Typhoon Conson(04/6/9/22 UTC)
JMANHM(Saito et al, 2005) 5 km res.
First guess: Ensemble forecast (04/6/9/15 ) :100 members started with perturbed initial dataObservation: Conventional + TMI TB
13
Surface Precipitation (mm/h) and TB19v (K)for 04/6/9/22 UTC
14
First guess(Mean of Ens. FCST)
Truth TBo
Analysis ofEns. mean
TB19vRainyareas
SurfPrecip>0.1 mm/h
Ensemble Forecast for Hourly PrecipitationNoDA (Init: 04/6/9/22 UTC, 100 member)
started with the first guess
16
FT=1 FT=2 FT=3
FT=4 FT=5 FT=6
Ensemble Forecast for Hourly PrecipitationDA(Init: 04/6/9/22 UTC, 100 member)
started with the EnVA analyses
17
FT=1 FT=2 FT=3
FT=4 FT=5 FT=6
Areas Classified with Precip(1mm/hr)Green (Forecast O, Truth O); Blue (Forecast X, Truth O);
Orange (Forecast O, Truth X)NoDA (Init: 04/6/9/22 UTC) FT=1-6h
18
FT=1 FT=2 FT=3
FT=4 FT=5 FT=6
Areas Classified with Precip(1mm/hr)Green (Forecast O, Truth O); Blue (Forecast X, Truth O);
Orange (Forecast O, Truth X)DA (Init: 04/6/9/22 UTC) FT=1-6h
19
FT=1 FT=2 FT=3
FT=4 FT=5 FT=6
POD for NoDA(dash) and DA(solid)(Init: 04/6/9/22 UTC) FT=1-6h
Blue: Th=1 mm/hrRed: Th=3 mm/hr
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6
POD
POD1ENS
POD1ETKF
POD3ENS
POD3ETKF
Summary
• Estimation of CRM forecast error covariance– Ensemble forecast error analysis– Dual-scale neighboring ensemble method
• Introduction to ensemble-based variational assimilation
• Experiment results using simulated data• Future directions (real data, FA cycle)
21
Sample error-damping methods of previous studies
• Spatial Localization (Lorenc, 2003)
• Spectral Localization (Buehner and Charron, 2007)
– When transformed into spatial domain
• Variable Localization (Kang, 2011)
ˆ ˆ ˆ( 1, 2) ( 1, 2) ( 1, 2)sl ENS slC k k C k k L k k
( 1, 2) ( 1 , 2 ) ( )sl ENS slC x x C x s x s L s ds
1,2( 1, 2) ( 1, 2) ( )sp ENSC x x C x x S
( 1, 2) ( 1, 2) ( 1, 2)v ENSC v v C v v v v
23
Power spectral of horizontal ensemble forecast error (H~5000m) : Typhoon
W U
1) Precip and W are diagonal, other had significant amplitudesfor low-frequency, off-diagonal modes.2) The presumption of the spectral localization “Correlations in spectral space decreases as the difference in wave number increases” is valid.
24
1)Cross correlation between precipitation-related variables and other variables increases with precipitation rate.2) Variables can be classified in terms of precipitation rate.
Cross correlation of CRM variables in the vertical Ave. whole Domain (Typhoon)
Weak rain areas
Rain-free areas
Heavy rain areas
25
Neighboring ensemble• Hypothesis of Buehner and Charron (2007)
Correlations in spectral space decreases as the difference in wave number increases.
• Spectral Localization
• When transformed into spatial domain
Spectral-Localized correlation is a weighted, spatially-shifted average of correlation over the neighboring points.
• we approximated the forecast error correlation using neighboring ensemble (NE) members of the target points (5 x 5 grids).
ˆ ˆ ˆ( 1, 2) ( 1, 2) ( 1, 2)sl slC k k C k k L k k
( 1, 2) ( 1 , 2 ) ( )sl slC x x C x s x s L s ds
Power spectral of horizontal ensemble forecast error (H~5000m) : Typhoon
W U
26
Horizontal correlation cal. from the SVD modes
27
Precip
U
No separation Separation
To reduce degree of freedom, we use SVD modes of vertical cross correlation of NE forecast error
Separation of NE into large-scale modes ( 65 km ave.) and local modes (derivation).
500km
500km
500km
500km
Averaged distance of area with horizontal correlation of precipitation over 0.5
(Typhoon case, H~3 km)
28
Conventional NE Method
29
Vertical cross-correlation at a large sampling error point (260,190 Typhoon)
Conventional NE MethodPrecip W RHW2 PT U
V
W
Precip
RHW2
PT
U
V Precip W RHW2 PT U V
Space spanned by NE vertical SVD modes
• NE member (~2500) =Ensemble (100 mem.) x Grid box (5x5)
• To reduce degree of freedom, we use SVD modes of vertical cross correlation of NE forecast error
• Space spanned by the NE vertical SVD modes:
30
1
1
1
1
(1) (1)( ) ( )
(1) (1)( ) ( )
G G N
G h G N hfNE
g g N
g h g N h
u uu K M u K M
Uu u
u K M u K M
EnVADetermination of cost function• Cost function in the NE vertical SVD space
31
11 1( ) ( ) { ( ) } { ( ) }2 2
a f t f a f a t aU UJ U U P U U H U Y R H U Y
,
( )( )
G Ga f aNE
g g
tG G Gf f
U U NE tg g g
UU U
U
U U SP P S
U U S
1 1
1 1 1
: 1 2 { } 1 2{ ( ) } { ( ) }
1 2 { } 1 2 { } 1 2{ ( , ) } { ( , ) }
at a a t a
t t tG G G g g g G g G g
J trace S H Y R H Y
trace S trace S H Y R H Y
EnVADetermination of cost function• Since vertical SVD modes are independent ,
cost function results in that of horizontal components:
32
1 1
1
1
: 1 2 { } 1 2{ ( ) } { ( ) }
1 2 { } 1 2{ }
1 2{ ( , ) } { ( , ) }
h
at a a t ah h h h h
t tGh Gh Gh g g
tGh g Gh g
J J
trace S H Y R H Y
trace S
H Y R H Y
EnVADiagonalization of background
term• Block-Diagonalize
• Approximation of (Adaptive Localization)
33
GhS1
1
: 1 2 { } 1 2{ }
1 2{ ( , ) } { ( , ) }kb
at a tkb kb kb g g
a t akb g kb g
J trace S
H Y R H Y
1/ 2kbS
2 2 1/ 2 1/ 2| | , fkb center center kb kbS Corr S C
1 2 akb kb kbS
Adaptive Localization:
200km
10km
Heavy Rain Area Rain-free AreaTo estimate the flow-dependency
of the error correlation
Ensemble forecast error corr. of PT (04/6/9/22 UTC) 34
EnVAMinimization of cost function
• Minimize the cost function
• To solve the non-linear min. problem, we follow Aonashi and Eito (2011).
• Approximate the gradient of the observation with the finite differences about the forecast error.
35
10 0
1 11 2{ } [ ( , ) ] [ ( , ) ]2 2
t t tkb kb g g kb g kb g
kbJ H d R H d