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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
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
Microscale particle numbers
Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results
60
50
40
30
20
90
80
70
60
50
40
30
20
Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results
Estimates: t=20Δt Variances: t=20Δt
Microscale particle numbers
70
60
50
40
30
20
10
70
60
50
40
30
20
10
Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results
Estimates: t=29Δt Variances: t=29Δt
Microscale particle numbers
70
60
50
40
30
20
10
60
50
40
30
20
10
Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results
Covariance: t=0
Microscale particle numbers
80
60
40
20
0
-20
-40
Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results
Covariance: t=10Δt
Microscale particle numbers
60
50
40
30
0
-10
-20
20
10
Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results
Covariance: t=20Δt
Microscale particle numbers
50
40
30
0
-10
20
10
Estimation on boundary particle fluxes: Numerical resultsEstimation on boundary particle fluxes: Numerical results
Covariance: t=29Δt
Microscale particle numbers
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