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Kalman Filtering for Coarse Time-Stepper Based Multiscale Data Assimilation

Kalman Filtering for Coarse Time-Stepper Based Multiscale Data Assimilation. WCCM VI – APCOM’04. Roger Ghanem Yu Zou. Department of Civil Engineering Johns Hopkins University. September 2004. Introduction. Importance of multiscale data assimilation

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Kalman Filtering for Coarse Time-Stepper Based Multiscale Data Assimilation

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  1. Kalman Filtering for Coarse Time-Stepper Based Multiscale Data Assimilation WCCM VI – APCOM’04 Roger Ghanem Yu Zou Department of Civil Engineering Johns Hopkins University September 2004

  2. Introduction • Importance of multiscale data assimilation Multiscale phenomena are common in the nature. Due to inaccurate information on initial conditions in fine scales of a system, experimental observations are required to updated predictions in different scales. Coarse-scale measurements are in need to update fine-scale states since fine-scale measurements are usually not available. WCCM VI – APCOM’04

  3. 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. Disadvantage: The updated state and error covariance are calculated based on ensemble members in the evolution. Introduction WCCM VI – APCOM’04

  4. The EnKF is expected to be used as the engine for multiscale data assimilation due to its advantages over the other two Kalman filtering techniques. Application of the ensemble Kalman filter has been limited to one-scale problems so far. Forecasts of the Lorenz model are updated by simulated measurements of one component of the forecast vector. Evensen 1996 Measured L-band brightness temperature are used to updated model predictions for soil moisture and temperature. Reichle 2002 Introduction WCCM VI – APCOM’04

  5. Explicit discrete system model mk+1=L(mk), m2Rn Observation model zk+1=H(mk+1)+vk+1, z2Rl v2Rl Forecast Predicted statemrk+1|k=L(mrk), r =1,…, Ne Predicated observation hrk+1|k=H(mrk+1|k), r=1,…, Ne Updating Updated state mrk+1= mrk+1|k+ Kk+1(zrk+1 - hrk+1|k), where the Kalman gain matrix Kk+1 = Cmh(Chh+Cvv)-1, the statistical members of observation zrk+1 = zk+1 + vrk+1 Ensemble Kalman Filter WCCM VI – APCOM’04

  6. Ensemble Kalman Filter WCCM VI – APCOM’04

  7. The coarse time-stepper is used to describe the interplay between states in different scales. Gear, Kevrekidis et al 2002 Identification of coarse and fine states Lifting: ψ: Ma× Ω → Mi, Ma: the set of coarse-scale states;Ω: sampling space; Mi: the set of fine-scale states Restriction: φ: Mi → Ma Coarse Time-Stepper Coarse State 1 Coarse State 2 Lifting Restriction Fine State 1 Fine State 2 Microscale evolution WCCM VI – APCOM’04

  8. X Xj 1 0 Uj U Components of the coarse time-stepper for 1-dimensional particle systems • Coarse-scale state: Inverse CDF of particle positions • Fine-scale state: particle positions • Lifting • Generating particle positions • in terms of the ICDF • Xj=IF(Uj), Uj~U(0,1) • Microscale evolution • Evolving particle positions • via microscale simulators • Restriction • Sorting particle positions and • generating the numerical ICDF X 1 0 U WCCM VI – APCOM’04

  9. Fine-scale evolution mri,k+1=Li(mri,k), r=1,…,Ne Restriction mra,k+1|k=φ(mri,k), r=1,…,Ne Ensemble Kalman filter Obtain updated coarse-scale state mra,k+1 frommra,k+1|k Lifting mr,si,k+1= ψ(mra,k+1, ωs), r=1,…,Ne , ωs 2 Ω, s=1,…, Nr The 1st- and 2nd-order error statistics for components of mi,k+1 are calculated by wheremi,k+1,αandmi,k+1,βare the αth and βth components ofmi,k+1. A new ensemble of Ne fine-scale states are generated based on the error statistics calculated above. Coarse time-stepper based multiscale data assimilation WCCM VI – APCOM’04

  10. Coarse time-stepper based multiscale data assimilation Illustration Ne Nr Note :By the law of large numbers, as NeNr → ∞, the calculated moments approach the true moments. Since Ne→∞ implies NeNr → ∞, the value of Nr does not impact significantly the updated fine-scale error statistics as Ne is sufficiently large. WCCM VI – APCOM’04

  11. Reporting regions 0 Numerical example: Estimation on positions of diffusive particles Positions of non-interactive identical Brownian particles are estimated viameasurements on number of particles in the reporting regions. -4RD -3RD -2RD -RD RD 2RD 3RD 4RD • Fine-scale state: [X1,X2,…,XN]T, Xj is the position of the jth particle, j=1,2,…,N, N=2000. • Fine-scale system model: dXj=DdWj, j=1,2,…,N. D is the diffusive coefficient, D = 5cm/s1/2 and Wj are independent Wiener processes. • Coarse-scale state: [ΔF1, ΔF2, ΔF3, ΔF4, ΔF5, ΔF6, F(d1), F(d2), …, F(d101)]T where d1 and d100 are minimum and maximum component values in the WCCM VI – APCOM’04

  12. ensemble of vectors of particle positions. dm=d1+(m-1)(d101-d1)/100, m=1,2,…,101. F(d) is the CDF of particle positions at the location d. ΔF1= F(-3RD)-F(-4RD), ΔF2= F(-2RD)-F(-3RD), ΔF3= F(-RD)-F(-2RD), ΔF4= F(2RD)-F(RD), ΔF5= F(3RD)-F(2RD), ΔF6= F(4RD)-F(3RD), where RD=D/2√t . Coarse-scale observation model: ND,p/N= ΔFp+ v,p=1,2,…,6 where ND,p are the number of particles located within the reporting regions and v is the observation error, v = 0.01 The true initial condition for fine-scale particle positions: Xj,0= 4(j-(N+1)/2)/N (cm),j=1,2,…,N The a priori information on the initial condition for fine-scale particle positions: Estimate E(Xj,0) = 10(j-(N+1)/2)/N (cm),j=1,2,…,N Error variance σ2(Xj,0) = 16.0 (cm2) Numerical example: Estimation on positions of diffusive particles WCCM VI – APCOM’04

  13. Numerical example: Estimation on positions of diffusive particles t=10Δt Ne=200, Nr=10, Δt=0.01sec, red lines: true statistics; blue lines: estimated statistics t=30Δt t=60Δt t=100Δt WCCM VI – APCOM’04

  14. Numerical example: Estimation on positions of diffusive particles t=10Δt Ne=200, Nr=100, Δt=0.01sec, red lines: true statistics; blue lines: estimated statistics t=30Δt t=60Δt t=100Δt WCCM VI – APCOM’04

  15. Conclusions • A multiscale data assimilation technique is constructed by combining coarse time-steppers and the ensemble Kalman filter. • Coarse-scale measurements can be used to effectively estimate fine-scale states through this technique. • As the ensemble size, Ne, of the initial fine-scale state is sufficiently large, variation of the ensemble size, Nr, of the fine-scale state lifted from each coarse-scale ensemble member does not significantly impact the estimated statistics. WCCM VI – APCOM’04

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