Kazumasa Aonashi*, Hisaki Eito, and Seiji Origuchi Meteorological Research Institute, Tsukuba,...
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Kazumasa Aonashi*, Hisaki Eito, and Seiji Origuchi Meteorological Research Institute, Tsukuba, Japan [email protected] Jan. 18 th , 2012 JCSDA Seminar Ensemble-Based Variational Assimilation Method Ensemble-Based Variational Assimilation Method to Incorporate Microwave Imager Data to Incorporate Microwave Imager Data into a Cloud-Resolving Model into a Cloud-Resolving Model 1
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Kazumasa Aonashi*, Hisaki Eito, and Seiji Origuchi Meteorological Research Institute, Tsukuba, Japan [email protected]. Jan. 18 th , 2012 JCSDA Seminar. Ensemble-Based Variational Assimilation Method to Incorporate Microwave Imager Data into a Cloud-Resolving Model. - PowerPoint PPT Presentation
Transcript of Kazumasa Aonashi*, Hisaki Eito, and Seiji Origuchi Meteorological Research Institute, Tsukuba,...
Kazumasa Aonashi*, Hisaki Eito, and Seiji Origuchi
Meteorological Research Institute, Tsukuba, Japan [email protected]
Jan. 18th, 2012
JCSDA Seminar
Ensemble-Based Variational Assimilation Method Ensemble-Based Variational Assimilation Method to Incorporate Microwave Imager Data to Incorporate Microwave Imager Data
into a Cloud-Resolving Modelinto a Cloud-Resolving Model
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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）
2cm１９ GH ｚ ８５ GHz
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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
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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.
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OUTLINEOUTLINEIntroductionIntroductionEnsemble-based Variational Assimilation (EnVA)Ensemble-based Variational Assimilation (EnVA)MethodologyMethodologyProblems in EnVA for CRMProblems in EnVA for CRMDisplacement error correction (DEC)+EnVADisplacement error correction (DEC)+EnVA MethodologyMethodology Application results for Typhoon CONSON (T0404)Application results for Typhoon CONSON (T0404)
Sampling error damping method for CRM EnVASampling error damping method for CRM EnVASample error-damping methods of previous studiesSample error-damping methods of previous studiesCheck the validity of presumptions of these methodsCheck the validity of presumptions of these methods
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Ensemble-based Variational Assimilation Ensemble-based Variational Assimilation (EnVA)(EnVA)
• MethodologyMethodology (Lorenc 2003, Zupanski 2005)(Lorenc 2003, Zupanski 2005)• Problems in EnVA for CRMProblems in EnVA for CRM
– Displacement errorDisplacement error– Sampling errorSampling error
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EnVA: min. cost function in the EnVA: min. cost function in the Ensemble forecast error Ensemble forecast error
subspace subspace Minimize the cost function with non-linear Obs. term.
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
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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)8
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
TTwhenTeTBT
,)1( /2
To address the non-linearity of TBs
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Detection of the optimum analysis Detection of the optimum analysis Detection of the optimum by minimizing
where is diagonalized with U eigenvectors of S ：
Approximate the gradient of the observation with the finite differences about the forecast error:
To solve non-linear min. problem, we performed iterations.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
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Problem in EnVA (1): Problem in EnVA (1): Displacement error Displacement error
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 TB18v (2003/1/27/04z)
Mean of Ensemble Forecast(2003/1/26/21 UTC FT=7h ）
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Presupposition of Ensemble-based assimilationPresupposition of Ensemble-based assimilation
Obs.
Analysis ensemble mean
T=t0 T=t1 T=t2
Analysis w/ errors FCST ensemble mean
1 1( )1
f f Tf t t
e N
X XP
R
Ensemble forecasts have enough spread to include (Obs. – Ens. Mean)
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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!13
Problem in EnVA (2): Problem in EnVA (2): Sampling errorSampling errorForecast 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
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Displacement error correction (DEC)Displacement error correction (DEC)+EnVA+EnVA
MethodologyMethodology Application results for Typhoon CONSON Application results for Typhoon CONSON (T0404)(T0404)CaseCaseAssimilation ResultsAssimilation ResultsImpact on precipitation forecastsImpact on precipitation forecasts
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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
arg max ( , | , )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
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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
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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 ( ( ( ))) ( ( ( )))} 2
2f t f
d dJ Y H X d R Y H X d d
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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
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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
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Case Case （（ 2004/6/9/22 UTC) TY 2004/6/9/22 UTC) TY CONSONCONSON
TMI TB19 ｖ RAM (mm/hr)Assimilate TMI TBsAssimilate TMI TBs (10v, 19v, 21v(10v, 19v, 21v )) at at 22UTC22UTC
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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
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Ensemble Forecasts & RTM codeEnsemble Forecasts & RTM code
Ensemble forecasts100 members started with perturbed initial data at 04/6/9/15 UTC (FG)Geostrophically-balanced perturbation (Mitchell et al. 2002) plus Humidity
RTM: Guosheng Liu (2004)One-dimensional model (Plane-parallel)Mie Scattering (Sphere)4 stream approximation
Ensemble mean (FG)Rain mix. ratio
TB19v cal. from FG
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TB19v from TMI and CRM TB19v from TMI and CRM outputsoutputs
FG ：First guess
DE ：After DEC
ND:NoDE+EnVA
CN ：DE+EnVA
TMI
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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
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RAM and Rain mix. ratio analysis RAM and Rain mix. ratio analysis (z=930m)(z=930m)
FG DE
RAM
ND CN
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RH(contours) and W(shades) along N-SRH(contours) and W(shades) along N-S
FG DE
CNNDＳ
Ｎ
Ｓ Ｎ Ｓ Ｎ
M
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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
220210
QR9 (g/ kg)
1.0.8.6.4.20.0
TBc1
9v (K
)
270
260
250
240
230
220210
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
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Hourly Precip. forecastsHourly Precip. forecasts (FT=0-1 h) 22-23Z 9th(FT=0-1 h) 22-23Z 9th
RAM
DE
CNND
FG
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Hourly Precip. Forecasts (FT=3-4 h) 01-02Z 10th Hourly Precip. Forecasts (FT=3-4 h) 01-02Z 10th
RAM
DE
CNND
FG
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SummarySummaryEnsemble-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.
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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.
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Sampling error damping method for CRM Sampling error damping method for CRM EnVAEnVA
•Sample error-damping methods of previous studies•Check the validity of presumptions of these methods
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Sample error-damping methods of Sample error-damping methods of previous studiesprevious 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
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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
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Ensemble ForecastsEnsemble ForecastsExtra-tropical Low(Jan. 27, 2003)
Typhoon CONSON(June. 9, 2004)
C
Baiu case(June 1, 2004)
100 members started with perturbed initial dataGeostrophically-balanced perturbation plus HumidityRandom perturbation with various horizontal and vertical scales (Mitchell et al. 2002)
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Horizontal correlation of ensemble forecast Horizontal correlation of ensemble forecast error (H~ 5000 m) : Typhoon error (H~ 5000 m) : Typhoon （（ black: U, green:RHW2, redblack: U, green:RHW2, red ：： WW ））
Rain-free area Weak Precip PR 1-3 mm/hr
Heavy PrecipPR >10mm/hr
km km km
1) (precip, W) had narrow correlation scales (~ 15 km).2) Horizontal correlation scales of (U, V, PT, RH) decreased (160 km -> 40 km) with precipitation rate. 37
Simple spatial localization is not usable.
Power spectral of horizontal ensemble Power spectral of horizontal ensemble forecast errorforecast error (H~5000m)(H~5000m) : Typhoon : Typhoon
W U
1) Precip and W are diagonal, other had significant amplitudes for 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. 38
Spectral localization may be applicable.
Cross correlation of CRM variables in the Cross correlation of CRM variables in the vertical (Typhoon): Rain-free areasvertical (Typhoon): Rain-free areas
Precip
W RHW2 PT U V
Precip
WR
HW
2P
TU
V
PS
Cross correlation of CRM variables in the Cross correlation of CRM variables in the vertical (Typhoon): Weak rain vertical (Typhoon): Weak rain ((1-3 mm/hr)
Precip
W RHW2 PT U V
Precip
WR
HW
2P
TU
V
PS
Cross correlation of CRM variables in the Cross correlation of CRM variables in the vertical (Typhoon):Heavy rain vertical (Typhoon):Heavy rain ((>10mm/hr)
Precip
W RHW2 PT U V
Precip
WR
HW
2P
TU
V
PS
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 Cross correlation of CRM variables in the vertical (Typhoon)vertical (Typhoon)
Weak rain areas
Rain-free areas
Heavy rain areas
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Variable localization needs classification in terms of precipitation.
Introducing sampling error Introducing sampling error damping ideas to EnVAdamping ideas to EnVA
Spetctal Localization >Use of ensemble forecasts at neighboring grid points
Heterogeneity of forecast covariance >
Classification of CRM variables and assumption of zero cross correlation between different classes.
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SummarySummary
We checked the validity of presumptions of the sampling error damping methods..
Simple spatial localization is not usable.Spectral localization may be applicable.Variable localization needs classification in terms of precipitation.We should consider heterogeneity of the forecast covariance (Michel et al, 2011).
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SummarySummaryEnsemble-based Variational Assimilation (EnVA)Ensemble-based Variational Assimilation (EnVA)MethodologyMethodologyProblems in EnVA for CRMProblems in EnVA for CRMDisplacement error correction (DEC)+EnVADisplacement error correction (DEC)+EnVA MethodologyMethodology Application results for Typhoon CONSON (T0404)Application results for Typhoon CONSON (T0404)
Sampling error damping method for CRM EnVASample error-damping methods of previous studiesCheck the validity of presumptions of these methods
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Thank you for your attention.Thank you for your attention.
End
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