A Stochastic Collocation based. for Data Assimilation

Transcription

1 A Stochastic Collocation based Kalman Filter (SCKF) for Data Assimilation Lingzao Zeng and Dongxiao Zhang University of Southern California August 11, 2009 Los Angeles

2 Outline Introduction SCKF Algorithm Case studies Conclusions

3 Introduction Purpose: To estimate system parameters from direct or indirect measurements. Traditional approach: Gradient based methods Use observations in the entire history; Require access to the simulator source code and substantial code development and maintenance Recent method: Ensemble Kalman filter

4 Noisy observations True state Initial state with uncertainty Illustrative diagram for EnKF (Geir Evensen) Time

5 Noisy observations True state Model prediction with errors Illustrative diagram for EnKF (Geir Evensen) Time

6 Noisy observations True state Model prediction with errors Updated estimate with errors Illustrative diagram for EnKF (Geir Evensen) Time

7 Noisy observations True state New model prediction with errors Updated estimate with errors Illustrative diagram for EnKF (Geir Evensen) Time

8 Noisy observations True state Updated estimate with errors Illustrative diagram for EnKF (Geir Evensen) Time

9 Ensemble Kalman filter For two-phase flow problem. The joint state vector: T y j = ln K T, ΦT, PT, SatT, dt, j = 1, 2,K N e j The EnKF analysis: y aj = y jf + K[d obs, j Hy fj ], j = 1, 2,K N e 1 K = C y H [HC y H + R ] T T MC Realizations 2 η = 4.0, σy = 1.0 MC: 1000 realizations The cross covariance: 6.2 Head, h 1 Ne {[y fj < y f >][y jf < y f >]T }, Cy N e 1 j = x

10 Ensemble Kalman filter Remarks Easy to implement, independent of simulator. Slow convergence with ensemble size. Requirements for possible new methods Non-intrusive, no need to modify codes. Efficient to propagate uncertainty during updating.

11 PCE based Kalman filter Joint state vector in Polynomial Chaos Expansion (PCE) Q S = S Ψ j= 0 j j ( ξ ) S T T T T T j = S ln, S k, S P, S φ Sat, S d, j = 0,1, L, Q T j SC Representations 2 nd 2 Covariance represented by PCE order PCM: 28 representations, η = 4.0, σ Y = Q Q C f f f s = S jψ j S jψ j j= 1 j= 1 Q j= 1 f T 2 ( S ) = S Ψ f j j j T Head d, h x

12 PCE based Kalman filter Analysis approach 1 S a f f j = S j + K d 0δ 0 j HS j, j = 0,1, L, Q d = d 0 obs f T f T K = C H + S HCS H R Observations are treated as deterministic. Relaxation term are added to the diagonal terms of R to improve filter stability. 1

13 PCE based Kalman filter Analysis approach 2 (Square root filter) Updating mean S a f f 0 = S0 + K d HS0, Updating deviations a f f S = % j S j KHS j, j = 12 1, 2, K Q 1 f T f T K = C H + S HCS H R, 1 T 1 f T f T f T S S S ( ) ( ) K% = C H HC H + R HC H + R + R,

14 Algorithm of SCKF Solving the forwarding problem with the collocation method. Recover PCE coefficients from the representations (on the basis of collocation sets). According to observations, update each of the PCE coefficients, then get the updated representations Forward each representation to the next time when the observations are available.

15 Flowchart of SCKF Parameterize the prior uncertainty with PCE or KL expansion Forward each collocation member (representation) with time Recover PCE coefficients from collocation results, update joint coefficient vectors according to observations, update each collocation member. END

16 Remarks Use PCE for efficient uncertainty representation and propagation. Use collocation method, non-intrusive as EnKF. Continuously update PCE coefficients: The zeroth order provides mean estimation, and higher order terms give uncertainty estimation. Main computational efforts are spent on solving flow equations at collocation point sets: The number of times needed to run the simulator is the same as the number of collocation point sets.

17 Single phase 2D flow problem (, ) [ K ( ) (, ) ] (, ) = hxt s x h x t + g x t Ss t Wells: The filled and empty triangles are the pumping and injection wells, respectively. No flow boundary at top and bottom, fixed head boundary condition at the left and right. Measurements: 25 pressure head measurements at all squares with Gaussian error N(0,0.05).

18 Single phase 2D flow problem Random fields: log permeability 0.20 Initial statistics: Gaussian random field with separable exponential covariance function n 2 x1 x2 y1 y 2 Cx ( 1, y1; x2, y2) = σ exp, (a) n=1 10 λx λy λ = 200 m, λ = 100 m, σ = 1.0 x y Y λ n x (b) n=4 10 x Use Karhunen-Loeve expansion to parameterize the Gaussian random field with independent Gaussian random variables. Use collocation points based on Stroud-2 rule, up to 1st order of PCE (Xiu, 2008) x (c) 10 n=10 x x 1 N = n= x (d) 10 n=20 x x 1 Y (, xω) ξ ( ω) λ f () x n n n

19 Single phase 2D flow problem Computational efforts to solve the governing equation EnKF: number of ensemble size SCKF: number of collocation sets, (M+1) for M-dimensional problem with Stroud-2 rule. Performance measure Mean estimation Uncertainty estimation N 1 t 2 RMSE= EY ( ( xi)) Y ( xi), N i = 1 N 1 SPREAD = VAR( xi ). N i = 1 Match between RMSE and SPREAD, involved with 1st two moments estimation 2 1 N t t { } { } N RMSE SPREAD = E[ Y ( x )] E[ Y x )] Y x Y x E[ Y x ] + i ( i ( i) ( i) ( i) N i= 1 N i= 1

20 Performance comparison EnKF with 100 realizations EnKF with 100 realizations EnKF with 200 realizations EnKF with 200 realizations

21 Performance comparison EnKF with 1000 realizations SCKF with different modes SCKF with different modes

22 Mean estimation Reference EnKF 1000 realizations SCKF with 100 modes SCKF with 200 modes

23 Variance estimation EnKF 1000 realizations SCKF with 100 modes SCKF with 200 modes

24 Pressure prediction Head prediction at day 20 0 solid line are reference dashed lines Head prediction at day 20.0, solid line are reference, dashed lines are computed from mean estimation of SCKF with 100 modes

25 Larger Variance a σ 2 = 2.0 (a) RMSE for EnKF with 200 realizations (b) SPREAD for EnKF with 200 realizations (c) RMSE and SPREAD for SCKF with 200 modes.

26 Shorter correlation eato length λ x = 80 m, λ y = 80m (a) RMSE for EnKF with 200 realizations (b) SPREAD for EnKF with 200 realizations (c) RMSE and SPREAD for SCKF with 200 modes.

27 Fewer e observations o s at 9 filled squares (a) RMSE for EnKF with 200 realizations (b) SPREAD for EnKF with 200 realizations (c) RMSE and SPREAD for SCKF with 200 modes.

28 Discussion Under certain conditions, SCKF performs better than EnKF with similar computational efforts The superiority is decreased with increase of variance, decrease of correlation ratio, or reduction of observations. SCKF suffers from curse of dimensionality. ( M + d )! Q + 1 =. M! d!

29 Two phase flow problem: history matching Water-oil two-phase system: subject to the constraint S o + S = 1 w Stronger nonlinearity, up to 2 nd order PCE are used; Probabilistic collocation method (Li & Zhang 2007) is used to obtain PCE coefficients from collocation representations.

30 Case study 2D two phase problem Prior statistics of log K: mean 5.0, variance 1.0, correlation length 600 in both directions. Updating step is at every 100 days, from day 0 to day The PCE terms are up to 2nd order. The KL terms are truncated up to the first 20 terms, 80% energy. Similar Computational efforts: The ensemble size and the number of collocation point sets are the same as 231.

31 Performance comparison RMSE SPREAD Blue solid lines are for SCKF, red dashed lines are for EnKF. Both filters are of similar computational efforts.

32 Match to production data SCKF EnKF EnKF Oil production rate of production well 2

33 Match to production data SCKF EnKF EnKF Oil production rate of production well 4

34 Match to production data SCKF EnKF EnKF Water cut of production well 4

35 Match to production data SCKF EnKF EnKF Bottom hole pressure of injection well

36 Non-Gaussianality caused by conditioning Although the unconditional field is assumed to be a Gaussian random field (KL expansion), higher PCE coefficients of log permeability will be produced d during the updating (non-gaussian conditional field represented by PCE) At one node, PCE coefficients before and after the first analysis.

37 Conclusions EnKF or SCKF? It depends EnKF: Slow convergence with ensemble size, realization dependent before convergence, but independent of random dimensionality. SCKF: Efficient than EnKF under certain conditions, but impractical when the random dimensionality is huge. Both are suboptimal for nonlinear problems Ways for Improvement: iteration, reparameterization, ation etc.

38 Thanks