Outline Introduction Sigma-point Kalman filters Introduction - PDF document

8 February 2008 Advanced Data Assimilation in strongly nonlinear systems Youmin Tang and Jaison Thomas Ambadan Environmental Science & Engineering Program Natural Resources & Environmental Science Data assimilation workshop at Banff, Feb 3-8 8 February 2008 1 Outline Introduction Sigma-point Kalman filters • Introduction Simulations with Lorenz model • Sigma-point Kalman filters A Reduced SPKF • Simulations with Lorenz model Summary & Conclusion • A Reduced SPKF • Summary & Conclusion. 8 February 2008 2 1

8 February 2008 Introduction Introduction Data assimilation – a dynamical Sigma-point Kalman filters state space estimate problem Simulations with Lorenz model A Reduced SPKF = State equation x f ( x , u ); (1) Summary & Conclusion − k k 1 k = Observatio n equation y h ( x , v ) (2) k k k x where is state vector at time k , f state transition k u function, and process noise with known k y distribution; is observations at time instant k , h k v observation function, and observation noise with k known distribution. 8 February 2008 3 Statistical estimation Introduction KF & EKF Sigma-point Kalman filters − − = + − ˆ ˆ ( ( ( ˆ , ))), x x K y E h x v Simulations with k k k k k k Analysis step Lorenz model − = − P ( I K H ) P A Reduced SPKF x k k x k k = − + − − Summary & Conclusion T T 1 [ ] K P H R H P H k x k k k x k k − = ˆ [ ( ˆ , )], x E f x u The forecast − − k k 1 k 1 step − = + T P L P L Q − − − x k 1 x k 1 k 1 − k k 1 = − − T Q E [( u u )( u u ) ], k k k k k = − − T R E [( v v )( v v ) ] k k k k k 8 February 2008 4 2

8 February 2008 Statistical estimation Introduction EnKF--- Popular approach Sigma-point Kalman filters − − = + − Simulations with x ˆ x ˆ K y E h x ˆ v ( ( ( , ))), Analysis step Lorenz model k k k k k k − − − = + T T 1 A Reduced SPKF K P H [ R H P H ] k x k k k x k k Summary & Conclusion − = ˆ ˆ x E [ f ( x , u )], The forecast − − k k k 1 1 step − − − = − − T ˆ ˆ P E [( x x )( x x ) ] x k k k Linear assumption for measurement function. 8 February 2008 5 Statistical estimation Introduction EnKF--- classic approach Sigma-point Kalman filters = − + − − Simulations with ˆ ˆ ˆ x x K ( y E ( h ( x , v ))), Analysis step Lorenz model k k k k k k − = − T P P K P K A Reduced SPKF x x k y k k k k − = Summary & Conclusion 1 K P P k x y y k k k − = x ˆ E [ f ( x ˆ , u )], − − k k 1 k 1 The forecast = − − − − T P E [( x ˆ x )( y E ( h ( x ˆ , v ))) ] step x y k k k k k = − − − − T ˆ ˆ P E [( y E ( h ( x , v )))( y E ( h ( x , v ))) ] y k k k k k No linear assumption for measurement function. Ref: (Gelb, 1974) 8 February 2008 6 3

8 February 2008 Sigma-point Kalman filter Introduction Sigma-point Kalman • The SPKF makes use of a reformulated Kalman gain K filters and “chooses” the ensemble deterministically in such a way that it can capture the statistical moments of the Simulations with Lorenz model nonlinear model accurately; in other words, the forecast A Reduced SPKF error covariance equation is computed using deterministically chosen samples, called “sigma-points”. Summary & Conclusion The SPKF algorithm has been successfully implemented in • many areas like robotics, artificial intelligence, natural language processing, and global positioning systems navigation. Ref: (Julier et al. 1995; Nørgad Magnus et al. 2000; Ito and Xiong 2000; Lefebvre et al. 2002;Wan and Van Der Merve 2000; Haykin 2001; Van der Merwe 2004, Van der Merwe and Wan4 April 2001,M;). 8 February 2008 7 SP-Unscented Kalman filter Introduction Unscented transformation ( Julier et al. 1995; Julier 1998; Wan and Van Der Merve 2000; Julier 2002 ). Sigma-point Kalman filters Simulations with Consider the propagation pf a L-dimensional random Lorenz model variable x through an arbitrary nonlinear function: A Reduced SPKF ϕ = g ( x ); Summary & Conclusion x has mean x and covariance P x = χ S { w , ); k = = i i i w i 0 + 0 χ = L k x ; 0 1 = = χ = + + ,..., w i 1 2 L x ( ( L k ) P ) , + i i x i 2 ( L k ) χ = − + x ( ( L k ) P ) k is a scaling parameter i x i ψ = g χ ( ); i i the dimension of the state-space L increases, the radius of the sphere that bounds all the sigma-points increases as well. 8 February 2008 8 4

8 February 2008 SP-UKF Algorithm Introduction = − + − − ˆ ˆ ˆ x x K ( y h ( x )), Analysis step Sigma-point Kalman k k k k k filters − = − T P P K P K Simulations with x x k y k Lorenz model k k k − = 1 A Reduced SPKF K P P k x y y k k k Measurement Summary & Conclusion Forecast step step = χ i i Y h ( ), k k ψ = χ i i f ( ), 2 L ∑ k k − = i i ˆ y w Y 2 L ∑ k k − = ψ i i x ˆ w = i 0 k k = i 0 2 L ∑ = − − − − i i i T ˆ ˆ P w ( Y y )( Y y ) 2 L ∑ − = ψ − − ψ − − y k k k k i i i T P w ( x ˆ )( x ˆ ) k = x k k k k i 0 k = i 0 2 L ∑ = ψ − − − − i i i T ˆ ˆ P w ( x )( Y y ) x y k k k k k k = i 0 8 February 2008 9 SP-Central Difference KF Introduction Sigma-point Kalman • In SP-CDKF the analytical derivatives in EKF filters Simulations with are replaced by numerically evaluated central Lorenz model divided differences, based on Sterling’s A Reduced SPKF polynomial interpolation. Summary & Conclusion ( Ito and Xiong 2000; Nørga°d Magnus et al. 2000; Lefebvre et al. 2002). 8 February 2008 10 5

8 February 2008 Lorenz model Introduction Sigma-point Kalman filters Simulations with Lorenz model A Reduced SPKF Summary & Conclusion True value: integrate the model over 4000 time steps using prescribed parameters and initial conditions. Observation: true value plus normal distribute noise; Experimental conditions are the same as those used by Miller (Miller, 1994) and Evensen (Evensen 1997) 8 February 2008 11 Lorenz model: state estimation Introduction Observation and initial conditions: their true values Sigma-point Kalman plus normal distributed noise . The assimilation N ( 0 , 2 ) filters interval is 25 time steps. Simulations with Lorenz model A Reduced SPKF Summary & Conclusion 8 February 2008 12 6

8 February 2008 1 = − true 2 Error ( x x ) k k N Introduction Sigma-point Kalman filters Simulations with Lorenz model A Reduced SPKF Summary & Conclusion 8 February 2008 13 Introduction EnKF with 19 members Sigma-point Kalman filters Simulations with Lorenz model A Reduced SPKF Summary & Conclusion 8 February 2008 14 7

8 February 2008 Stronger noise (10 times) Introduction Sigma-point Kalman filters Simulations with Lorenz model A Reduced SPKF Summary & Conclusion 8 February 2008 15 Strong noise + Less observations Introduction Sigma-point Kalman filters Simulations with Lorenz model A Reduced SPKF Summary & Conclusion 8 February 2008 16 8

8 February 2008 Parameter estimation Introduction • If the measurement function is nonlinear, it Sigma-point Kalman has to be linearized in the EnKF. filters Simulations with Lorenz model Λ = Λ + Constraints & issues q − − k k 1 k 1 Summary & Conclusion = Λ + y f ( x , ) r k k k k 8 February 2008 17 Parameter estimation Introduction Initial parameters: true parameters plus normal distributed noise of covariance 100. Two parameters are estimated: Sigma-point Kalman filters Simulations with Lorenz model ρ β and A Reduced SPKF Summary & Conclusion Kivman, G.A. 2003: Sequential parameter estimation for stochastic system, Nonlinear. Process. in Geophysics, 10, 253-259. 8 February 2008 18 9

8 February 2008 Joint estimation (state +parameters) Introduction Sigma-point Kalman filters Simulations with Lorenz model A Reduced SPKF Summary & Conclusion 8 February 2008 19 A “big” drawback of SPKF Introduction Sigma-point Kalman filters • For an L-dimensional system, the number of sigma-points Simulations with required to estimate the true statistics is 2L+1 Lorenz model A Reduced SPKF Summary & Conclusion • 2L+1 sigma-point integration is impossible, when the dimension system is of the order of tens of millions as it happens often in GCMs 8 February 2008 20 10