Kazumasa Aonashi* and Hisaki Eito Meteorological Research Institute, Tsukuba, Japan [email protected]July 27, 2011 IGARSS2011 Displaced Ensemble variational Displaced Ensemble variational assimilation method assimilation method to to incorporate microwave imager TB incorporate microwave imager TB into a cloud-resolving model into a cloud-resolving model
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Kazumasa Aonashi* and Hisaki Eito Meteorological Research Institute, Tsukuba, Japan [email protected] July 27, 2011 IGARSS2011 Displaced Ensemble variational.
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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)
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
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.
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