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Kazumasa Aonashi* and Hisaki Eito
Meteorological Research Institute, Tsukuba, Japan
July 27, 2011
IGARSS2011
Displaced Ensemble variational Displaced Ensemble variational assimilation method assimilation method
toto incorporate microwave imager TBsincorporate microwave imager TBs into a cloud-resolving modelinto a cloud-resolving model
Satellite Observation Satellite Observation (TRMM)(TRMM)
Infrared Imager
S S T, Winds
C loud Partic les
F rozen Prec ip .
S now Aggregates
Melting Layer
Rain Drops
Radiation from RainS c attering by F rozen
Partic les
Radar
B ac k s c attering from Prec ip .
Scattering
Radiation0
Mic rowave Imager
Cloud Top Temp.
10μm
3 mm-3cm(100-10 GH z)
2cm
19 GH z 85 GHz
Cloud-Resolving Model usedCloud-Resolving Model used JMANHM ( Saito et
al,2001)• Resolution: 5 km• Grids: 400 x 400 x 38• Time interval: 15 s
Explicitly forecasts 6 species of water substances
Goal: Data assimilation of MWI TBs into CRMsGoal: Data assimilation of MWI TBs into CRMs
Hydrological ModelCloud Reslv. Model + Data Assim System
MWI TBs(PR)
Precip.
OUTLINEOUTLINE
IntroductionIntroductionMethodologyMethodology
Ensemble-based Variational Assimilation (EnVA)Ensemble-based Variational Assimilation (EnVA) Displacement error correction (DEC)Displacement error correction (DEC)
Application resultsApplication resultsCaseCase ( ( 2004/6/9) Typhoon CONSON(0404)2004/6/9) Typhoon CONSON(0404)
Assimilation ResultsAssimilation ResultsImpact on precipitation forecastsImpact on precipitation forecasts
Summary & future directionsSummary & future directions
TMI 040609.OP37437TMI 040609.OP37437
OUTLINEOUTLINE
IntroductionIntroductionMethodologyMethodology
Ensemble-based Variational Assimilation (EnVA)Ensemble-based Variational Assimilation (EnVA) Displacement error correction (DEC)Displacement error correction (DEC)
Application resultsApplication resultsCaseCase ( ( 2004/6/9) Typhoon2004042004/6/9) Typhoon200404
Assimilation ResultsAssimilation ResultsImpact on precipitation forecastsImpact on precipitation forecasts
Summary & future directionsSummary & future directions
MethodologyMethodology
Ensemble-based Variational Ensemble-based Variational Assimilation (EnVA) Assimilation (EnVA)
(Lorenc 2003, Zupanski 2005)(Lorenc 2003, Zupanski 2005)
Displacement error correction (DEC)Displacement error correction (DEC)
Data assimilation schemesData assimilation schemes
Why Why Ensemble-basedEnsemble-based method?: method?:
200km
10km
Heavy Rain Area Rain-free Area To estimate the flow-dependency of the error covariance
Ensemble forecast error corr. of PT (04/6/9/22 UTC)
Why Variational Method Why Variational Method ??
MWI TBs are non-linear function of MWI TBs are non-linear function of various CRM variables.various CRM variables.TB becomes saturated as optical thickness increases:
TB depression mainly due to frozen precipitation becomes dominant after saturation.
s
s
TTwhen
TeTBT
≈−≈− − ,)1( /2 µτε
To address the non-linearity of TBs
Presupposition of Ensemble-based assimilationPresupposition of Ensemble-based assimilation
Obs.
Analysis ensemble mean
T=t0 T=t1 T=t2
Analysis w/ errors FCST ensemble mean1
)( 111 −
≈m
Tft
ftf
t
XXP
δδ
R
Ensemble forecasts have enough spread to include (Obs. – Ens. Mean)
Displacement error betw. Observation &Displacement error betw. Observation &Ensemble forecastEnsemble forecast
Large scale displacement errors of rainy areas between the MWI observation and Ensemble forecasts
Presupposition of Ensemble assimilation is not satisfied in observed rain areas without forecasted rain.
AMSRE TB19v (2003/1/27/04z)
Mean of Ensemble Forecast(2003/1/26/21 UTC FT=7h )
Ensemble-based assimilation for observed rain Ensemble-based assimilation for observed rain areas without forecasted rainareas without forecasted rain
Obs.
Analysis ensemble mean
T=t0 T=t1 T=t2
Analysis w/ errors FCST ensemble mean
R
Assimilation can give erroneous analysis when the presupposition is not satisfied.
Signals from rain can bemisinterpreted as thosefrom other variables
Displacement error correction is needed!
Displaced Ensemble variational Displaced Ensemble variational assimilation methodassimilation method
In addition to , we introduced to assimilation.The optimal analysis value maximizes :
Assimilation results in the following 2 steps: 1) DEC scheme to derive from 2)EnVA scheme using the DEC Ensembles to
derive from
Xv
dv
arg max ( , | , )fP X d Y Xvv
( , | , ) ( | , ) ( | , , )f f fP X d Y X P d Y X P X d Y X=v v vv v
( | , )fP d Y Xv
( | , , )a fP X d Y Xvv
adv
aXv
Fig. 1:CRM Ensemble
Forecasts
Displacement ErrorCorrection
Ensemble-basedVariational Assimilation
( , ,
( ))
f fe
c f
X P
TB X
Y
d%
MWI TBs
( ( ), ( ),
( ( )),
( ( )))
f fe
c f
c fi
X d P d
TB X d
TB X d
% %
%
%( , ,
)
a ae
ai
X P
X
%v
Assimilation methodAssimilation method
DEC scheme: min. cos t func t ion for dDEC scheme: min. cos t func t ion for d
Bayes’ Theorem
can be expressed as the cond. Prob. of Y given :
We assume Gaussian dist. of : where is the empirically determined scale of the
displacement error.We derived the large-scale pattern of by minimizing
(Hoffman and Grassotti ,1996) :
( | , ) ( , | ) ( ) / ( , )f f fP d Y X P Y X d P d P Y X=v v v
( , | )fP Y X dv
( )fX dv
1( , | ) exp 1 2( ( ( )) ( ( ( ))f f t fP Y X d Y H X d R Y H X d−= − − −v v v
( )P dv
2 2( ) exp ( / 2 )dP d d σ= −v v
dσ
d%
dJ21 21
( ( ( ))) ( ( ( ))) 22
f t fd dJ Y H X d R Y H X d d σ−= − − +
v v v
Detection of the large-scale Detection of the large-scale pattern of optimum displacementpattern of optimum displacement
We derived the large-scale pattern of from , following Hoffman and Grassotti (1996) :
We transformed into the control variable in wave space, using the double Fourier expansion.
We used the quasi-Newton scheme (Press et al. 1996) to minimize the cost function in wave space.
we transformed the optimum into the large-scale pattern of by the double Fourier inversion.
21 21( ( ( ))) ( ( ( ))) 2
2f t f
d dJ Y H X d R Y H X d d σ−= − − +v v v
dv
dv
dJd%
rv
rv
Fig. 1:CRM Ensemble
Forecasts
Displacement ErrorCorrection
Ensemble-basedVariational Assimilation
( , ,
( ))
f fe
c f
X P
TB X
Y
d%
MWI TBs
( ( ), ( ),
( ( )),
( ( )))
f fe
c f
c fi
X d P d
TB X d
TB X d
% %
%
%( , ,
)
a ae
ai
X P
X
%v
Assimilation methodAssimilation method
EnVA: min. cost function in the EnVA: min. cost function in the Ensemble forecast error Ensemble forecast error
subspace subspace Minimize the cost function
Assume the analysis error belongs to the Ensemble forecast error subspace ( Lorenc, 2003):
Forecast error covariance is determined by localization
Cost function in the Ensemble forecast error subspace :
f/2e
fX X− Ωv
o=P/ 2
1 2[ , , , , , ]f f f f f f fe NP X X X X X X= − − −
v v v1 2 N=[w , w ,, , , , w ]Ω v v v
))(())((21)()(21 11 XHYRXHYXXPXXJ fffx
vvvv−−+−−= −−
f feP =P So
1 1( ) 1 2 1 2 ( ( )) ( ( )) t tJ trace S H X Y R H X Y− −Ω = Ω Ω + Ω − Ω −v v
Calculation of the optimum analysis Calculation of the optimum analysis
Detection of the optimum by minimizing • Transform of using eigenvectors of S :
• Minimize the diagonalized cost functionApproximate the gradient of the observation with the finite differences about the forecast error:
Following Zupanski (2005), we calculated the analysis of each Ensemble members, from the Ensemble analysis error covariance.
J Ω,a awΩ
( ) 1 ( )ti m im d U mχ = Ω
( ) / ~ ( ) ( )/fiH X H X p H Xαδ α∂ ∂Ω + −
v v v
Ω
aiXv
Application results
Case ( 2004/6/9) Typhoon CONSON
(0404)
Assimilation Results
Impact on precipitation forecasts
Case Case (( 2004/6/9/22 UTC) TY 2004/6/9/22 UTC) TY CONSONCONSON
1) Assimilate TMI TBs1) Assimilate TMI TBs
(10v, 19v, 21v(10v, 19v, 21v ))
2) 100 member Ensemble2) 100 member Ensemble(init. 04/6/9/15 UTC(init. 04/6/9/15 UTC :: FG)FG)TMI TB19 v RAM (mm/hr)
TB19v from TMI and CRM output sTB19v from TMI and CRM output s
FG :First guess
DE :After DEC
ND:NoDE+EnVA
CN :DE+EnVA
TMI
RAM and Rain mix . ra t io analys i s RAM and Rain mix . ra t io analys i s (z=930m)(z=930m)
FG DE
RAM
ND CN
RH(contours) and W(shades) along N-SRH(contours) and W(shades) along N-S
FG DE
CNNDS
N
S N S N
M
RTW17 (%)
120100806040
TBc2
1v (K)
280
270
260
250
240
RTW17 (%)
120100806040
TBc2
1v (K)
280
270
260
250
240
QR9 (g/ kg)
1 .0.8.6.4.20.0
TBc1
9v (K)
270
260
250
240
230
220
210
QR9 (g/ kg)
1 .0.8.6.4.20.0
TBc1
9v (K)
270
260
250
240
230
220
210
CRM Variables vs. TBc at Point MCRM Variables vs. TBc at Point M
FG
FG
DE
DE
Qr(930m) vs.TB19v
RTWRTW(3880m)(3880m) vs.TB21vvs.TB21v
Hourly Precip. forecasts (FT=0-1 h) 22-23Z 9thHourly Precip. forecasts (FT=0-1 h) 22-23Z 9th
RAM
DE
CNND
FG
Hourly Precip. Forecasts (FT=3-4 h) 01-02Z 10th Hourly Precip. Forecasts (FT=3-4 h) 01-02Z 10th
RAM
DE
CNND
FG
SummarySummary
Ensemble-based data assimilation can give erroneous analysis, particularly for observed rain areas without forecasted rain. In order to solve this problem, we developed the Ensemble-based assimilation method that uses Ensemble forecast error covariance with displacement error correction. This method consisted of a displacement error correction scheme and an Ensemble-based variational assimilation scheme.
SummarySummaryWe applied this method to assimilate TMI TBs (10, 19, and 21 GHz with vertical polarization) for a Typhoon case (9th June 2004). The results showed that the assimilation of TMI TBs alleviated the large-scale displacement errors and improved precip forecasts. The DEC scheme also avoided misinterpretation of TB increments due to precip displacements as those from other variables.
Forecast error corr. of W (04/6/9/15z 7h fcst)Forecast 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
Thank you for your attention.Thank you for your attention.
End
Ensemble-based Variational Ensemble-based Variational Assimilation MethodAssimilation Method
Why Why Ensemble-basedEnsemble-based Assimilation Assimilation
method?:method?:
To address the flow-dependency of the error covariance
Why Variational Assimilation MethodWhy Variational Assimilation Method ? ? :: To address the non-linearity of TBs
Why Why Ensemble-basedEnsemble-based method?: method?: Ensemble forecast corr. of PT (04/6/9/22 UTC)Ensemble forecast corr. of PT (04/6/9/22 UTC)
200km
10km
1000 km
Heavy Rain Area Rain-free Area
To address the flow-dependency of the error covariance
Presupposition of Ensemble-based assimilationPresupposition of Ensemble-based assimilation
Ensemble forecasts have enough spread to include (Obs. – Ens. Mean)
Assimilation can give erroneous analysis when the presupposition is not satisfied.
Cloud-Resolving Model usedCloud-Resolving Model used JMANHM ( Saito et
al,2001)• Resolution: 5 km• Grids: 400 x 400 x 38• Time interval: 15 s
Initial and boundary dataJMA’s operational regional model
• Basic equations : Hydrostatic primitive
• Precipitation scheme: Moist convective adjustment + Arakawa-Schubert + Large scale condensation• Resolution: 20 km• Grids: 257 x 217 x 36
Explicit cloud microphysics scheme based on bulk method ( Lin et al.,1983; Murakami,
1990; Ikawa and Saito, 1991 )
The water substances are categorized into 6 water species (water vapor, cloud water, rain, cloud ice, snow and graupel)
Explicitly predicting the mixing ratios and the number concentrations of frozen particles
Cloud Microphysical SchemeCloud Microphysical Scheme
Why EnVA?Why EnVA?Emission & Scattering signals vs. hydrometersEmission & Scattering signals vs. hydrometers
Convective rain (Jan. 27, 2003)Convective rain (Jan. 27, 2003)
Ra in Mix ing Ra t io (H ~ 2 km)
.0016
.0014
.0012
.0010
.0008
.0006
.0004
.0002
0.0000
- .0002
Em
issi
on S
igna
l at
18G
Hz
4.6
4.4
4.2
4.0
3.8
3.6
3.4
3.2
RWC
6543210- 1
Em
issi
on S
inga
l at
18 G
Hz
4.6
4.4
4.2
4.0
3.8
3.6
3.4
3.2
Snow Mix ing Ra t io a t H~ 4 km
.005.004.003.002.0010.000- .001
Sca
tter
ing
Sin
gal a
t 89
GHz
.1
0.0
- .1
- .2
- .3
IWC
121086420- 2
Sca
tter
ing
Sig
nal a
t 89
GHz
.1
0.0
- .1
- .2
- .3
EmissionSingalsAt 18 GHzτ ∝ LN(Ts-TB)Ts-TB=Ts(1-εs)exp(-2τ)
ScatteringSingalsAt 89 GHzτ ∝ LN(TB/TBflh)TB=TBflh Exp(-τ)
Fig. 1:CRM Ensemble
Forecasts
Displacement ErrorCorrection
Ensemble-basedVariational Assimilation
( , ,
( ))
f fe
c f
X P
TB X
Y
d%
MWI TBs
( ( ), ( ),
( ( )),
( ( )))
f fe
c f
c fi
X d P d
TB X d
TB X d
% %
%
%
( , ,
)
a ae
ai
X P
X
%v
Assimilation methodAssimilation method
Post - f i t re s idual sPos t - f i t re s idual s
FG : DE :
ND:CN :
LN:DE+EnVA.1st
))(())((21)()(21 11 XHYRXHYXXPXXJ fffx
vvvv−−+−−= −−
Jx=24316.6Jb=0Jo=24316.6
Jx= 9435.2Jb=0Jo= 9435.2
Jx= 4105.4Jb= 834.5Jo= 3270.9
Jx= 2431.9Jb= 290.9Jo= 2141.0
Jx= 6883.0Jb= 14.5Jo= 6868.4