Kazumasa Aonashi* and Hisaki Eito

Meteorological Research Institute, Tsukuba, Japan

aonashi@mri-jma.go.jp

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

SST, Winds

Cloud Particles

Frozen Precip.

Snow Aggregates

Melting Layer

Rain Drops

Radiation from RainScattering by Frozen

Particles

Radar

Back scattering from Precip.

Scattering

Radiation0℃

Microwave 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

IntroductionIntroduction

MethodologyMethodology

Ensemble-based Variational Assimilation (EnVA)Ensemble-based Variational Assimilation (EnVA)

Displacement error correction (DEC)Displacement error correction (DEC)

Application resultsApplication results

CaseCase 　（　（ 2004/6/9) Typhoon CONSON(0404)2004/6/9) Typhoon CONSON(0404)

Assimilation ResultsAssimilation Results

Impact on precipitation forecastsImpact on precipitation forecasts

Summary & future directionsSummary & future directions

TMI 040609.OP37437TMI 040609.OP37437

OUTLINEOUTLINE

IntroductionIntroduction

MethodologyMethodology

Ensemble-based Variational Assimilation (EnVA)Ensemble-based Variational Assimilation (EnVA)

Displacement error correction (DEC)Displacement error correction (DEC)

Application resultsApplication results

CaseCase 　（　（ 2004/6/9) Typhoon2004042004/6/9) Typhoon200404

Assimilation ResultsAssimilation Results

Impact 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 MethodWhy 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

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

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

X

d

argmax ( , | , )fP X d Y X

( , | , ) ( | , ) ( | , , )f f fP X d Y X P d Y X P X d Y X

( | , )fP d Y X

( | , , )a fP X d Y X

ad

aX

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

Assimilation methodAssimilation method

DEC scheme: min. cost DEC scheme: min. cost function for dfunction 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

( , | )fP Y X d

( )fX d

1( , | ) exp{ 1 2( ( ( )) ( ( ( ))}f f t fP Y X d Y H X d R Y H X d

( )P d

2 2( ) exp{ ( / 2 )}dP d d

d

d

dJ21 21

( ( ( ))) ( ( ( )))} 22

f t fd dJ Y H X d R Y H X d d

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( ( ( ))) ( ( ( )))} 22

f t fd dJ Y H X d R Y H X d d

d

d

dJd

r

r

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

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

＝Ｐ/ 2

1 2[ , , , , , ]f f f f f f fe NP X X X X X X

1 2 N=[w ,w ,, , , ,w ]

))(())((21)()(21 11 XHYRXHYXXPXXJ fffx

f feP =P S

1 1( ) 1 2 { } 1 2{ ( ( )) } { ( ( )) }t tJ trace S H X Y R H X Y

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 ,a aw

( ) 1 { } ( )ti m im d U m

( ) / ~ { ( ) ( )}/fiH X H X p H X

aiX

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 TB19v from TMI and CRM outputsoutputs

FG ：First guess

DE ：After DEC

ND:NoDE+EnVA

CN ：DE+EnVA

TMI

RAM and Rain mix. ratio analysis RAM and Rain mix. ratio analysis (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

CNND

Ｓ

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. forecastsHourly Precip. forecasts (FT=0-1 h) 22-23Z 9th(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

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 primitivePrecipitation scheme:

Moist convective adjustment + Arakawa-Schubert + Large scale condensation

Resolution: 20 kmGrids: 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)

Rain Mixing Ratio (H ~ 2 km)

.0016

.0014

.0012

.0010

.0008

.0006

.0004

.0002

0.0000

- .0002

Emis

sion

Sig

nal a

t 18

GHz

4.6

4.4

4.2

4.0

3.8

3.6

3.4

3.2

RWC

6543210- 1

Emis

sion

Sin

gal a

t 18

GHz

4.6

4.4

4.2

4.0

3.8

3.6

3.4

3.2

Snow Mixing Ratio at H~ 4 km

.005.004.003.002.0010.000- .001

Scat

terin

g Si

ngal

at

89 G

Hz

.1

0.0

- .1

- .2

- .3

IWC

121086420-2

Scat

terin

g Si

gnal

at

89 G

Hz

.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

Assimilation methodAssimilation method

Post-fit residualsPost-fit residuals

FG ： DE ：

ND:CN ：

LN:DE+EnVA.1st

))(())((21)()(21 11 XHYRXHYXXPXXJ fffx

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