Generalized Randomized Maximum Likelihood

Transcription

1 Generalized Randomized Maximum Likelihood Andreas S. Stordal & Geir Nævdal ISAPP, Delft 2015

2 Why Bayesian history matching doesn t work, cont d Previous talk:

3 Why Bayesian history matching doesn t work, cont d Previous talk: Prior model not well idenitified (missing geology and structural uncertainty)

4 Why Bayesian history matching doesn t work, cont d Previous talk: Prior model not well idenitified (missing geology and structural uncertainty) The model is wrong. Assumption of perfect model is not justified in real applications

5 Why Bayesian history matching doesn t work, cont d Previous talk: Prior model not well idenitified (missing geology and structural uncertainty) The model is wrong. Assumption of perfect model is not justified in real applications Measurement uncertainty is unknown

6 Why Bayesian history matching doesn t work, cont d Previous talk: Prior model not well idenitified (missing geology and structural uncertainty) The model is wrong. Assumption of perfect model is not justified in real applications Measurement uncertainty is unknown This talk:

7 Why Bayesian history matching doesn t work, cont d Previous talk: Prior model not well idenitified (missing geology and structural uncertainty) The model is wrong. Assumption of perfect model is not justified in real applications Measurement uncertainty is unknown This talk: We can t even find the Bayesian solution for large models

8 Why Bayesian history matching doesn t work, cont d Previous talk: Prior model not well idenitified (missing geology and structural uncertainty) The model is wrong. Assumption of perfect model is not justified in real applications Measurement uncertainty is unknown This talk: We can t even find the Bayesian solution for large models In a controlled environment we can make small improvements

9 Why Bayesian history matching doesn t work, cont d Previous talk: Prior model not well idenitified (missing geology and structural uncertainty) The model is wrong. Assumption of perfect model is not justified in real applications Measurement uncertainty is unknown This talk: We can t even find the Bayesian solution for large models In a controlled environment we can make small improvements Also, for the Norne field, the concensus is that an ensemble based Bayesian approach has given the best history matched model so far

10 Randomized Maximum Likelihood Generalization to nonlinear models of Gaussian posterior sampling via minimization [Kitanidis, 1995, Oliver et al., 1996]

11 Randomized Maximum Likelihood Generalization to nonlinear models of Gaussian posterior sampling via minimization [Kitanidis, 1995, Oliver et al., 1996] Gaussian prior Φ(X µ P), Gaussian likelihood Φ(y H(x) R).

12 Randomized Maximum Likelihood Generalization to nonlinear models of Gaussian posterior sampling via minimization [Kitanidis, 1995, Oliver et al., 1996] Gaussian prior Φ(X µ P), Gaussian likelihood Φ(y H(x) R). A sample is obtained by

13 Randomized Maximum Likelihood Generalization to nonlinear models of Gaussian posterior sampling via minimization [Kitanidis, 1995, Oliver et al., 1996] Gaussian prior Φ(X µ P), Gaussian likelihood Φ(y H(x) R). A sample is obtained by Sample ɛ Φ(0 P)

14 Randomized Maximum Likelihood Generalization to nonlinear models of Gaussian posterior sampling via minimization [Kitanidis, 1995, Oliver et al., 1996] Gaussian prior Φ(X µ P), Gaussian likelihood Φ(y H(x) R). A sample is obtained by Sample ɛ Φ(0 P) Sample η Φ(0 R)

15 Randomized Maximum Likelihood Generalization to nonlinear models of Gaussian posterior sampling via minimization [Kitanidis, 1995, Oliver et al., 1996] Gaussian prior Φ(X µ P), Gaussian likelihood Φ(y H(x) R). A sample is obtained by Sample ɛ Φ(0 P) Sample η Φ(0 R) Minimize O η,ɛ (x) = x (µ + ɛ) 2 P + y + η H(x) 2 R. Correct Bayesian sampling if H is linear. Otherwise not. Minimization using Levenberg-Marquardt.

16 Motivation (a) RML (b) Generalized RML Figure : RML sampling

17 Same model different behavior General statements that "RML works well for this and this model" is problematic

18 Figure : RML sampling Same model different behavior General statements that "RML works well for this and this model" is problematic Same model, same realization of reference, only change realization measurement noise. (a) RML (b) RML

19 Levenberg Marquardt Iterations At iteration k + 1, update [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 + Gk R 1 G k P 1 (µ + ɛ X k ) [ ] 1 + PGk (1 + λ k+1 )R + G k PGk (y + η H(Xk ))

20 Levenberg Marquardt Iterations At iteration k + 1, update [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 + Gk R 1 G k P 1 (µ + ɛ X k ) [ ] 1 + PGk (1 + λ k+1 )R + G k PGk (y + η H(Xk )) One model mismatch part and one data mismatch part.

21 Levenberg Marquardt Iterations At iteration k + 1, update [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 + Gk R 1 G k P 1 (µ + ɛ X k ) [ ] 1 + PGk (1 + λ k+1 )R + G k PGk (y + η H(Xk )) One model mismatch part and one data mismatch part. G k is sensitivity matrix of H(X k ) (hard to obtain without adjoint code).

22 Levenberg Marquardt Iterations At iteration k + 1, update [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 + Gk R 1 G k P 1 (µ + ɛ X k ) [ ] 1 + PGk (1 + λ k+1 )R + G k PGk (y + η H(Xk )) One model mismatch part and one data mismatch part. G k is sensitivity matrix of H(X k ) (hard to obtain without adjoint code). λ k+1 regularization parameter (step size).

23 Ensemble approximation Replace sensitivity G k with ensemble average [Chen and Oliver, 2013].

24 Ensemble approximation Replace sensitivity G k with ensemble average [Chen and Oliver, 2013]. Non standard form replacing P with P k = cov ( {X i k }N i=1) in the Hessian.

25 Ensemble approximation Replace sensitivity G k with ensemble average [Chen and Oliver, 2013]. Non standard form replacing P with P k = cov ( {Xk i=1) i }N in the Hessian. [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 k + Gk R 1 G k P 1 (µ + ɛ X k ) + P k G k [ (1 + λk+1 )R + G k P k Gk ] 1 (y + η H(Xk ))

26 Ensemble approximation Replace sensitivity G k with ensemble average [Chen and Oliver, 2013]. Non standard form replacing P with P k = cov ( {Xk i=1) i }N in the Hessian. [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 k + Gk R 1 G k P 1 (µ + ɛ X k ) + P k G k [ (1 + λk+1 )R + G k P k Gk ] 1 (y + η H(Xk )) G k = [H(X 1 k ) H k... H(X N k ) H k][x 1 k X k... X N k X k] +.

27 Ensemble approximation Replace sensitivity G k with ensemble average [Chen and Oliver, 2013]. Non standard form replacing P with P k = cov ( {Xk i=1) i }N in the Hessian. [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 k + Gk R 1 G k P 1 (µ + ɛ X k ) + P k G k [ (1 + λk+1 )R + G k P k Gk ] 1 (y + η H(Xk )) G k = [H(X 1 k ) H k... H(X N k ) H k][x 1 k X k... X N k X k] +. For the ensemble version G k P k G k and cov ( {H(X i k )}N i=1) are estimating the same quantity but without pseudo inverse (will be used later).

28 Ensemble approximation Replace sensitivity G k with ensemble average [Chen and Oliver, 2013]. Non standard form replacing P with P k = cov ( {Xk i=1) i }N in the Hessian. [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 k + Gk R 1 G k P 1 (µ + ɛ X k ) + P k G k [ (1 + λk+1 )R + G k P k Gk ] 1 (y + η H(Xk )) G k = [H(X 1 k ) H k... H(X N k ) H k][x 1 k X k... X N k X k] +. For the ensemble version G k P k G k and cov ( {H(X i k )}N i=1) are estimating the same quantity but without pseudo inverse (will be used later). For standard RML G k P k G k is only a linearization of cov ( {H(X i k )}N i=1).

29 GRML part one - weighting the prior O γ η,ɛ(x) = γ x (µ + ɛ) 2 P + y + η H(x) 2 R

30 GRML part one - weighting the prior O γ η,ɛ(x) = γ x (µ + ɛ) 2 P + y + η H(x) 2 R (L-M) GRML update equation: [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 + γ 1 Gk R 1 G k P 1 (µ + ɛ X k ) [ ] 1 + PGk (1 + λ k+1 )γr + G k PGk (y + η H(Xk ))

31 Non standard ensemble versions GEnRML [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 k + γ 1 Gk T R 1 G k P 1 (µ + ɛ X k ) [ ] 1 + P k Gk (1 + λ k+1 )γr + G k P k Gk T (y + η H(Xk ))

32 Non standard ensemble versions GEnRML [ ] 1 X k+1 = X k + (1 + λ k+1 )P 1 k + γ 1 Gk T R 1 G k P 1 (µ + ɛ X k ) [ ] 1 + P k Gk (1 + λ k+1 )γr + G k P k Gk T (y + η H(Xk )) GEnRML-approx X k+1 = X k + P k G k [ ] 1 (1 + λ k+1 )γr + G k P k Gk (y + η H(Xk ))

33 Difficult γ For the correct posterior π (necessary condition) E π [ d dx O(X) ] = 0, O(x) = x µ 2 P + y H(x) 2 R.

34 Difficult γ For the correct posterior π (necessary condition) E π [ d dx O(X) ] = 0, O(x) = x µ 2 P + y H(x) 2 R. For RML E RML [ d dx O(X) ] = 2E [ ] G X R 1 η = 0 iff H(x) = Hx.

35 Difficult γ For the correct posterior π (necessary condition) E π [ d dx O(X) ] = 0, O(x) = x µ 2 P + y H(x) 2 R. For RML E RML [ d dx O(X) ] = 2E [ ] G X R 1 η = 0 iff H(x) = Hx. For GRML E GRML [ d dx O(X) ] = 2E [ ] [ ] G X R 1 η + 2(1 γ)e P 1 (X µ).

36 Difficult γ For the correct posterior π (necessary condition) E π [ d dx O(X) ] = 0, O(x) = x µ 2 P + y H(x) 2 R. For RML E RML [ d dx O(X) ] = 2E [ ] G X R 1 η = 0 iff H(x) = Hx. For GRML E GRML [ d dx O(X) ] = 2E [ ] [ ] G X R 1 η + 2(1 γ)e P 1 (X µ). In theory: minimize absolute value with respect to γ.

37 (a) RML (b) γ = 0.2 (c) RML (d) γ = 3 (e) RML (f) γ = 1.9

38 Practical Problems (Chen and Oliver [2013]) "Unfortunately, the method (EnRML) generally stopped reducing the data mismatch at a level that was still unacceptably high." (Psuedo inverse)

39 Practical Problems (Chen and Oliver [2013]) "Unfortunately, the method (EnRML) generally stopped reducing the data mismatch at a level that was still unacceptably high." (Psuedo inverse) Measure: Drop model mismatch term in gradient (EnRML-approx).

40 Practical Problems (Chen and Oliver [2013]) "Unfortunately, the method (EnRML) generally stopped reducing the data mismatch at a level that was still unacceptably high." (Psuedo inverse) Measure: Drop model mismatch term in gradient (EnRML-approx). "In general, the total objective function is difficult to compute, so in most practical applications stopping is based on the magnitude of the data mismatch."

41 Practical Problems (Chen and Oliver [2013]) "Unfortunately, the method (EnRML) generally stopped reducing the data mismatch at a level that was still unacceptably high." (Psuedo inverse) Measure: Drop model mismatch term in gradient (EnRML-approx). "In general, the total objective function is difficult to compute, so in most practical applications stopping is based on the magnitude of the data mismatch." Measure: Drop model mismatch term in objective function.

42 Practical Problems (Chen and Oliver [2013]) "Unfortunately, the method (EnRML) generally stopped reducing the data mismatch at a level that was still unacceptably high." (Psuedo inverse) Measure: Drop model mismatch term in gradient (EnRML-approx). "In general, the total objective function is difficult to compute, so in most practical applications stopping is based on the magnitude of the data mismatch." Measure: Drop model mismatch term in objective function. Claim: First is good for ensemble versions, second is potentially bad.

43 Practical Problems (Chen and Oliver [2013]) "Unfortunately, the method (EnRML) generally stopped reducing the data mismatch at a level that was still unacceptably high." (Psuedo inverse) Measure: Drop model mismatch term in gradient (EnRML-approx). "In general, the total objective function is difficult to compute, so in most practical applications stopping is based on the magnitude of the data mismatch." Measure: Drop model mismatch term in objective function. Claim: First is good for ensemble versions, second is potentially bad. Second part leads to Bayesian algorithms that are stopping criteria dependent and thus are formulated only in terms of the method used for minimization.

44 GEnRML part two (pgenrml) If H is linear and invertible x (µ + ɛ) 2 P = Hx H(µ + ɛ) 2 HPH

45 GEnRML part two (pgenrml) If H is linear and invertible x (µ + ɛ) 2 P = Hx H(µ + ɛ) 2 HPH For nonlinear model, use (a number) of measurements for prior term and define O γ η,ɛ(x) = γ Hx H(µ + ɛ) 2 HPH + y + η H(x) 2 R,

46 GEnRML part two (pgenrml) If H is linear and invertible x (µ + ɛ) 2 P = Hx H(µ + ɛ) 2 HPH For nonlinear model, use (a number) of measurements for prior term and define O γ η,ɛ(x) = γ Hx H(µ + ɛ) 2 HPH + y + η H(x) 2 R, with notation HPH = cov (H(X))

47 GEnRML part two (pgenrml) If H is linear and invertible x (µ + ɛ) 2 P = Hx H(µ + ɛ) 2 HPH For nonlinear model, use (a number) of measurements for prior term and define O γ η,ɛ(x) = γ Hx H(µ + ɛ) 2 HPH + y + η H(x) 2 R, with notation HPH = cov (H(X)) HPH has to be estimated, ideally choose num meas < N.

48 GEnRML part two (pgenrml) If H is linear and invertible x (µ + ɛ) 2 P = Hx H(µ + ɛ) 2 HPH For nonlinear model, use (a number) of measurements for prior term and define O γ η,ɛ(x) = γ Hx H(µ + ɛ) 2 HPH + y + η H(x) 2 R, with notation HPH = cov (H(X)) HPH has to be estimated, ideally choose num meas < N. Prior term easy to evaluate and much better estimated.

49 GEnRML part two (pgenrml) If H is linear and invertible x (µ + ɛ) 2 P = Hx H(µ + ɛ) 2 HPH For nonlinear model, use (a number) of measurements for prior term and define O γ η,ɛ(x) = γ Hx H(µ + ɛ) 2 HPH + y + η H(x) 2 R, with notation HPH = cov (H(X)) HPH has to be estimated, ideally choose num meas < N. Prior term easy to evaluate and much better estimated. Ideally other functionals should be used for non-gaussian models.

50 Ensemble updates pgenrml-approx Hessian approximation ( G k ( (1 + λk )(HP k H ) 1 + γ 1 R 1) G k ) 1.

51 Ensemble updates pgenrml-approx Hessian approximation ( G k ( (1 + λk )(HP k H ) 1 + γ 1 R 1) G k ) 1. However, in the ensemble framework HP k H = G k P k Gk so ( ( G k (1 + λk )(HP k H ) 1 + γ 1 R 1) ) 1 G k = ( ) 1 = (1 + λ k )P 1 k + γ 1 Gk R 1 G k

52 Ensemble updates pgenrml-approx Hessian approximation ( G k ( (1 + λk )(HP k H ) 1 + γ 1 R 1) G k ) 1. However, in the ensemble framework HP k H = G k P k Gk so ( ( G k (1 + λk )(HP k H ) 1 + γ 1 R 1) ) 1 G k = ( ) 1 = (1 + λ k )P 1 k + γ 1 Gk R 1 G k The approx update (pgenrml-approx) is X k = X k 1 + P k G ( (1 + λ)g k P k G k + γr ) 1 (di H(X k )).

53 Ensemble updates pgenrml-approx Hessian approximation ( G k ( (1 + λk )(HP k H ) 1 + γ 1 R 1) G k ) 1. However, in the ensemble framework HP k H = G k P k Gk so ( ( G k (1 + λk )(HP k H ) 1 + γ 1 R 1) ) 1 G k = ( ) 1 = (1 + λ k )P 1 k + γ 1 Gk R 1 G k The approx update (pgenrml-approx) is X k = X k 1 + P k G ( (1 + λ)g k P k G k + γr ) 1 (di H(X k )). Only stopping criteria and γ is different from EnRML-approx!

54 Same model as in Chen and Oliver [2013] (a) RML (b) RML approx (c) p γ = 1 (d) p γ = 3.5 Figure : Estimates with R=1

55 (a) RML (b) RML approx (c) p γ = 1 (d) p γ = 4 Figure : Estimates with R=16

56 MCMC on a 2D model grid with unknown permeability values. Gaussian prior.

57 MCMC on a 2D model grid with unknown permeability values. Gaussian prior. Model and the simulator (SimSim) used to solve the flow equations are developed at TU Delft Jansen [2011] (slightly compressible two phase flow).

58 MCMC on a 2D model grid with unknown permeability values. Gaussian prior. Model and the simulator (SimSim) used to solve the flow equations are developed at TU Delft Jansen [2011] (slightly compressible two phase flow). 40 measurements (bhp in injector, oil rates in 4 producers, 8 time steps).

59 MCMC on a 2D model grid with unknown permeability values. Gaussian prior. Model and the simulator (SimSim) used to solve the flow equations are developed at TU Delft Jansen [2011] (slightly compressible two phase flow). 40 measurements (bhp in injector, oil rates in 4 producers, 8 time steps). Two independent MCMC chains of runs.

60 MCMC on a 2D model grid with unknown permeability values. Gaussian prior. Model and the simulator (SimSim) used to solve the flow equations are developed at TU Delft Jansen [2011] (slightly compressible two phase flow). 40 measurements (bhp in injector, oil rates in 4 producers, 8 time steps). Two independent MCMC chains of runs. After burn-in and de-correlation 1960 (independent) samples are chosen.

61 MCMC on a 2D model grid with unknown permeability values. Gaussian prior. Model and the simulator (SimSim) used to solve the flow equations are developed at TU Delft Jansen [2011] (slightly compressible two phase flow). 40 measurements (bhp in injector, oil rates in 4 producers, 8 time steps). Two independent MCMC chains of runs. After burn-in and de-correlation 1960 (independent) samples are chosen. Run different EnRML versions with 100 ensemble members.

62 MCMC on a 2D model grid with unknown permeability values. Gaussian prior. Model and the simulator (SimSim) used to solve the flow equations are developed at TU Delft Jansen [2011] (slightly compressible two phase flow). 40 measurements (bhp in injector, oil rates in 4 producers, 8 time steps). Two independent MCMC chains of runs. After burn-in and de-correlation 1960 (independent) samples are chosen. Run different EnRML versions with 100 ensemble members. Note: EnRML and GEnRML use full covariance matrix while pgenrml-approx use an estimated.

63 Highlighted results - Distance to MCMC mean Method grad obj func γ mean #iter EnRML approx D D GEnRML D and M D and M EnRML D and M D and M GEnRML approx D D and M pgenrml-approx D D and DM Table : Summary statistics from different EnRML approaches

64 Results with γ = 1 Method mean #iter EnRML approx penrml approx Table : Summary statistics from different EnRML approaches

65 Conclusions RML (EnRML) is a very promising tool for Bayesian history matching.

66 Conclusions RML (EnRML) is a very promising tool for Bayesian history matching. The performance of RML (and EnRML) can be improved by weighting the prior term of the objective function

67 Conclusions RML (EnRML) is a very promising tool for Bayesian history matching. The performance of RML (and EnRML) can be improved by weighting the prior term of the objective function For EnRML the model mismatch term in the gradient typically leads to premature convergence and should be dropped

68 Conclusions RML (EnRML) is a very promising tool for Bayesian history matching. The performance of RML (and EnRML) can be improved by weighting the prior term of the objective function For EnRML the model mismatch term in the gradient typically leads to premature convergence and should be dropped If this term is dropped for the objective function, the result is typically overfitting

69 Conclusions RML (EnRML) is a very promising tool for Bayesian history matching. The performance of RML (and EnRML) can be improved by weighting the prior term of the objective function For EnRML the model mismatch term in the gradient typically leads to premature convergence and should be dropped If this term is dropped for the objective function, the result is typically overfitting If the model mismatch term is problematic, a prior term can be evaluated in (part of) the measuerement space

70 Conclusions RML (EnRML) is a very promising tool for Bayesian history matching. The performance of RML (and EnRML) can be improved by weighting the prior term of the objective function For EnRML the model mismatch term in the gradient typically leads to premature convergence and should be dropped If this term is dropped for the objective function, the result is typically overfitting If the model mismatch term is problematic, a prior term can be evaluated in (part of) the measuerement space The gradient without model mismatch is equal to that of EnRML-approx

71 Conclusions RML (EnRML) is a very promising tool for Bayesian history matching. The performance of RML (and EnRML) can be improved by weighting the prior term of the objective function For EnRML the model mismatch term in the gradient typically leads to premature convergence and should be dropped If this term is dropped for the objective function, the result is typically overfitting If the model mismatch term is problematic, a prior term can be evaluated in (part of) the measuerement space The gradient without model mismatch is equal to that of EnRML-approx Results better than EnRML, much better than EnRML-approx and as good as modified EnRML for a 2D model

72 Acknowledgements The first author acknowledges the Uni Research CIPR/IRIS cooperative research project "4D Seismic History Matching" which is funded by its industry partners, Total, Eni and Petrobras, and the Research Council of Norway (PETROMAKS2). The second author acknowledges the national IOR centre of Norway and its participants, The Research Council of Norway, ConocoPhillips, BP, Det Norske Oljeselskap, Eni, Maersk, DONG Energy, Statoil, GDF Suez, Lundin, Halliburton, Schlumberger and Wintershall.

73 References Y. Chen and D. S. Oliver. Levenberg Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Computational Geosciences, pages 1 15, H. W. Engl, M. Hanke, and A. Neubauer. Regularization of inverse problems, volume 375. Springer Science & Business Media, J. D. Jansen. SimSim: A simple reservoir simulator. TU Delft, Departement of Geotechnology, P. K. Kitanidis. Quasi-linear geostatistical theory for inversing. Water resources research, 31(10): , D. S. Oliver, N. He, and A. C. Reynolds. Conditioning permeability fields to pressure data. In European Conference for the Mathematics of Oil Recovery, V, pages 1 11, 1996.