Multiscale data assimilation on 2D boundary fluxes of biological aerosols Yu Zou 1 Roger Ghanem 2 1...

Multiscale data assimilationon 2D boundary fluxes of biological aerosols

Yu Zou1 Roger Ghanem2

1 Department of Chemical Engineering and PACM, Princeton University2 Department of Civil Engineering, USC

WCCM VII, LAJuly 16-22, 2006

• Introduction

• Ensemble Kalman filter

• Method of extended state

• Estimation on boundary particle fluxes

• Conclusions and remarks

OutlineOutline

IntroductionIntroduction• JHU Biocomplexity Project

IntroductionIntroduction• Experimental site

IntroductionIntroduction

• Importance of multiscale data assimilation

Microscale quantities: pollen fluxes near boundary

Macroscale quantities: total pollen counts away from boundary

Potential disturbance of canopy on microscale measurements

Use macroscale measurements to calibrate microscale quantities

IntroductionIntroduction

• Sequential data assimilation methods

1. Standard Kalman filter (KF), Kalman 1960 Advantage: The updated state and error covariance can be directly computed. Disadvantage: Not valid for nonlinear systems; Not applicable to large systems.

2. Extended Kalman filter (EKF), Gelb 1974 Advantage: The updated state and error covariance can be directly computed. Disadvantage: Nonlinear models are required to be locally linearized. Not valid for strongly nonlinear systems; Not applicable to large systems.

3. Ensemble Kalman filter (EnKF), Evensen 1994 Advantage: Nonlinear models are not required to be locally linearized. Applicable to strongly nonlinear systems; Applicable to large systems.

• The EnKF is used for multiscale data assimilation due to its advantages over the other two Kalman filtering techniques.

• Explicit discrete system model

• Observation model

• Forecast Predicted state

Predicated observation Updating Updated state the Kalman gain matrix the statistical members of observation

Ensemble Kalman filterEnsemble Kalman filter

nk)L( m ,mm 1k

llkk )H( m ,z ,vmz 1k 11

,...,Ne,r)L( krr 21 ,mm k|1k

,...,Ne,r)H( kkrr 21|1 ,mh k|1k

,...,Ne,r)( kkrr

krr 21|11 ,hzKmm 1kk|1k1k

11 )

vvhhmh CCCK (k

,...,Ne,rkr

kkr 21111 ,vzz

)L( kkk 0211k ,...,,, mmmmm

• The system model may be in the form of

• Extended system model

• Extended observation model

• Microscale system model

• Macroscale system model

• Macroscale observation model

Multiscale data assimilation: Method of extended stateMultiscale data assimilation: Method of extended state

)(L kssks ,1, mm

1,11,111,1 kskssks )(H vmz

)(Fsksksksssks,ks 0,2,1,,

,...,,,,1,111 mmmmmm

)(L skskskssks 0,2,1,,,*

1, ,...,, MMMMM

1,11,1*

1,1 kskssks )(H *vMz

1,

1,11,

ks

ksks

m

mM 1,1k1,s ],[ ksM0Im

1,1ks, ][ ksMI0,m

• Extended state

Estimation on boundary particle fluxesEstimation on boundary particle fluxesUpward bridging: a 3-dimensional wind velocity field modelUpward bridging: a 3-dimensional wind velocity field model

sr.v.'Gaussian standardt independen :

0251

031251

position at the ofdeviation standard :

length Obukhov :

051

0]370[320

0.4 constant, sKarman' von :

elocityfriction v :

ydiffusiviteddy turbulent:

,

timescaleLagrangian :

31

31

2

z,i

*z

/*z

izz

/-

*

*zL

L

, z/Lu.

, z/L))L

dz((u.

Zv

L

z/L(z-d)/L, φ

, z/L-(z-d)/L..φ

kk

u

K

/φkzuKK/σt

t

yyxx DvDv ,

step time:

)-(1 ),/exp(

particle theofposition rticalcurrent ve :

series timeessdimensionl a :

)()()()(

2/12

,

1,1

Δt

tt

Z

qz

ZZtqZZv

qq

L

i

i

Zz

ziziLiiziiz

izii

i

Horizontal velocity field

Vertical velocity field Wilson’s model (Wilson et al., 1981)

Model for motion of particles

tvZZ

tvYY

tvXX

izii

yii

xii

,1

1

1

Estimation on boundary particle fluxes: Estimation on boundary particle fluxes: Model and numerical experiment set-upModel and numerical experiment set-up

Microscale quantity ms,k: particle number emitted from each cell at the top of the canopy per unit time

Macroscale quantity ms+1,k : total particle count crossing each cell at a height above the canopy top

Estimation on boundary particle fluxes: Estimation on boundary particle fluxes: Model and numerical experiment set-upModel and numerical experiment set-up

mns,k+1= mn

s,k

1, 1 1, 1 1, 1( ) ( ) ( )

1,2,3,4

z m vi

s k s k s k

j z

i j i

i

• Microscale system model

)(L kssks ,1, mm

1,11,111,1 kskssks )(H vmz)(Fsksksksssks,ks 0,2,1,,

,...,,,,1,111 mmmmmm

Macroscale measurement: Measuring particle numbers crossing four macroscale cells

Numerical upward bridging Fs+1,s

1. Nominal particle numbers mns,k+1 are

converted to actual numbers mas,k+1

2. Actual number of particles are emitted from the center of each cell 3. Particles are driven by the velocity field 4. Total particle numbers crossing cells at a height above the boundary are counted

• Macroscale system model • Macroscale observation modelInfluence of weather conditions on particle numbersemitted from a forest (Kawashima et al., 1995)

24

1 11

24

1 11

1 1

, 1 , 1

( ) /(24 ) : ( )

( ) /(24 ), : ( / )

,

m m

N

j ik kj

N

j ik kj

k k

a ns k s k

T T T N T hourly air temperature C

W W W N W hourly wind speed m s

P a T b W c

P

Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results

Otherwise 0,

11 and 11 ,/25),(

11112 mξm-mξm-smS

y-

x-

yx

• True microscale particle numbers: ms,k

true,n=60sec-1, D=0.3m/s • A priori guess for assimilation Estimate: 30sec-1

Error spectral density:

Estimates: t=0 Variances: t=0

Microscale particle numbers

31.0

30.5

30.0

29.5

29.0

120

115

110

105

10095

90

85

80

Estimates: t=10Δt Variances: t=10Δt



60

50

40

30

20

90

80

70

60

50

40

30

20




70

60

50

40

30

20

10

70

60

50

40

30

20

10




70

60

50

40

30

20

10

60

50

40

30

20

10


Covariance: t=0


80

60

40

20

0

-20

-40


Covariance: t=10Δt


60

50

40

30

0

-10

-20

20

10


Covariance: t=20Δt


50

40

30

0

-10

20

10


Covariance: t=29Δt


35

30

25

5

0

20

15

10

-5

-10

Conclusions and remarksConclusions and remarks

• A priori microscale information and a correct microscale model are needed for this approach to be implemented.

• The approach may be applied to more realistic problems and coupled with other upward bridging models for wind velocity field.

AcknowledgementsAcknowledgements

• NSF• JHU Biocomplexity research group

Multiscale data assimilation on 2D boundary fluxes of biological aerosols Yu Zou 1 Roger Ghanem 2 1...

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