1. Multiscale data assimilation on 2D boundary fluxes of biological aerosols Yu Zou 1 Roger Ghanem 2 1 Department of Chemical Engineering and PACM, Princeton University 2 Department of Civil Engineering, USC WCCM VII, LA July 16-22, 2006

2. Introduction Ensemble Kalman filter Method of extended state Estimation on boundary particle fluxes Conclusions and remarks Outline

3. Introduction JHU Biocomplexity Project

4. Introduction Experimental site

5. Introduction 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

6. Introduction 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.

7. 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 filter The system model may be in the form of

8. Extended system model Extended observation model Microscale system model Macroscale system model Macroscale observation model Multiscale data assimilation: Method of extended state Extended state

9. Estimation on boundary particle fluxes Upward bridging: a 3-dimensional wind velocity field model Horizontal velocity field Vertical velocity field Wilson’s model (Wilson et al., 1981) Model for motion of particles

10. Estimation on boundary particle fluxes: Model and numerical experiment set-up Microscale quantity m s,k : particle number emitted from each cell at the top of the canopy per unit time Macroscale quantity m s+1,k : total particle count crossing each cell at a height above the canopy top

11. Estimation on boundary particle fluxes: Model and numerical experiment set-up m n s,k+1 = m n s,k Microscale system model Macroscale measurement: Measuring particle numbers crossing four macroscale cells Numerical upward bridging F s+1,s 1. Nominal particle numbers m n s,k+1 are converted to actual numbers m a s,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 model Influence of weather conditions on particle numbers emitted from a forest (Kawashima et al., 1995)

12. Estimation on boundary particle fluxes: Numerical results True microscale particle numbers: m s,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 29.0 120 115 110 105 100 95 90 85 80

13. Estimates: t=10ΔtVariances: t=10Δt Microscale particle numbers Estimation on boundary particle fluxes: Numerical results 60 50 40 30 20 90 80 70 60 50 40 30 20

14. Estimation on boundary particle fluxes: Numerical results Estimates: t=20ΔtVariances: t=20Δt Microscale particle numbers 70 60 50 40 30 20 10 70 60 50 40 30 20 10

15. Estimation on boundary particle fluxes: Numerical results Estimates: t=29ΔtVariances: t=29Δt Microscale particle numbers 70 60 50 40 30 20 10 60 50 40 30 20 10

16. Estimation on boundary particle fluxes: Numerical results Covariance: t=0 Microscale particle numbers 80 60 40 20 0 -20 -40

17. Estimation on boundary particle fluxes: Numerical results Covariance: t=10Δt Microscale particle numbers 60 50 40 30 0 -10 -20 20 10

18. Estimation on boundary particle fluxes: Numerical results Covariance: t=20Δt Microscale particle numbers 50 40 30 0 -10 20 10

19. Estimation on boundary particle fluxes: Numerical results Covariance: t=29Δt Microscale particle numbers 35 30 25 5 0 20 15 10 -5 -10

20. Conclusions 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. Acknowledgements NSF JHU Biocomplexity research group