Parameter estimation of subsurface flow models using iterative regularized ensemble Kalman filter

  • A. H. ELSheikh
  • C. C. Pain
  • F. Fang
  • J. L. M. A. Gomes
  • I. M. Navon
Original Paper


A new parameter estimation algorithm based on ensemble Kalman filter (EnKF) is developed. The developed algorithm combined with the proposed problem parametrization offers an efficient parameter estimation method that converges using very small ensembles. The inverse problem is formulated as a sequential data integration problem. Gaussian process regression is used to integrate the prior knowledge (static data). The search space is further parameterized using Karhunen–Loève expansion to build a set of basis functions that spans the search space. Optimal weights of the reduced basis functions are estimated by an iterative regularized EnKF algorithm. The filter is converted to an optimization algorithm by using a pseudo time-stepping technique such that the model output matches the time dependent data. The EnKF Kalman gain matrix is regularized using truncated SVD to filter out noisy correlations. Numerical results show that the proposed algorithm is a promising approach for parameter estimation of subsurface flow models.


ensemble Kalman filter inverse problems regularization Gaussian process regression Karhunen–Loève expansion 



The authors would like to thank the anonymous reviewers for their insightful and constructive comments which helped enhance this manuscript. Dr. A. H. Elsheikh and Dr. Jefferson L. M. A. Gomes carried this work as part of activities of the Qatar Carbonates and Carbon Storage Research Centre (QCCSRC). They gratefully acknowledge the funding of QCCSRC provided jointly by Qatar Petroleum, Shell, and the Qatar Science and Technology Park. Professor I. M. Navon acknowledges the support of NSF/CMG grant ATM-0931198.


  1. Allen M, Frame D, Kettleborough J, Stainforth D (2006) Model error in weather and climate forecasting. In: Palmer T, Hagedorn R (eds) Predictability of weather and climate. Cambridge University Press, CambridgeGoogle Scholar
  2. Anderson JL (2001) An ensemble adjustment Kalman filter for data assimilation. Mon Weather Rev 129(12):2884–2903CrossRefGoogle Scholar
  3. Anderson JL (2003) A local least squares framework for ensemble filtering. Mon Weather Rev 131(4):634–642CrossRefGoogle Scholar
  4. Anderson JL, Anderson SL (1999) A monte carlo implementation of the nonlinear filtering problem to produce ensemble assimilation and forecasts. Mon Weather Rev 127:2741–2758CrossRefGoogle Scholar
  5. Anderson BDO, Moore JB (1979) Optimal filtering. Information and system sciences series. Prentice-Hall, Inc., Englewood Cliffs, NJGoogle Scholar
  6. Bengtsson T, Snyder C, Nychka D (2003) Toward a nonlinear ensemble filter for high-dimensional systems. J Geophys Res 108(D24):8775–8785Google Scholar
  7. Blum J, Le Dimet FX, Navon IM (2008) Data assimilation for geophysical fluids. In: Ciarlet PG, Temam R, Tribbia J (eds) Computational methods for the atmosphere and the oceans. Handbook of numerical analysis, vol 14. Elsevier, Amsterdam, pp 385–442Google Scholar
  8. Bulygina N, Gupta H (2010) How bayesian data assimilation can be used to estimate the mathematical structure of a model. Stoch Environ Res Risk Assess 24(6):925–937CrossRefGoogle Scholar
  9. Carrera J, Alcolea A, Medina A, Hidalgo J, Slooten LJ (2005) Inverse problem in hydrogeology. Hydrogeol J 13(1):206–222CrossRefGoogle Scholar
  10. Chen Z (2007) Reservoir simulation: mathematical techniques in oil recovery. Society for Industrial and Applied Mathematics, Philadelphia, PACrossRefGoogle Scholar
  11. Chen Y, Oliver D (2012) Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Math Geosci 44(1):1–26CrossRefGoogle Scholar
  12. Chilès JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New YorkCrossRefGoogle Scholar
  13. Christie M, Blunt M (1995) Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv Eval Eng 4:308–317Google Scholar
  14. Cohn SE (1997) An introduction to estimation theory. J Meteorol Soc Jpn Ser II 75(1B):257–288Google Scholar
  15. Dostert P, Efendiev Y, Mohanty B (2009) Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models. Adv Water Resour 32(3):329–339CrossRefGoogle Scholar
  16. Dovera L, Della Rossa E (2011) Improved initial ensemble generation coupled with ensemble square root filters and inflation to estimate uncertainty. Comput Geosci 1–17Google Scholar
  17. Efendiev Y, Datta-Gupta A, Ginting V, Ma X, Mallick B (2005) An efficient two-stage markov chain monte carlo method for dynamic data integration. Water Resour Res 41(12)Google Scholar
  18. ELsheikh AH, Jackson MD, Laforce TC (2012) Bayesian reservoir history matching considering model and parameter uncertainties. Math Geosci 44(5):515–543CrossRefGoogle Scholar
  19. Evensen G (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. J Geophys Res 99(C5):10,143–10,162CrossRefGoogle Scholar
  20. Evensen G, van Leeuwen PJ (2000) An ensemble Kalman smoother for nonlinear dynamics. Mon Weather Rev 128(6):1852–1867CrossRefGoogle Scholar
  21. Fang H, Gong G, Qian M (1997) Annealing of iterative stochastic schemes. SIAM J Control Optim 35(6):1886–1907CrossRefGoogle Scholar
  22. Fang F, Pain CC, Navon IM, Piggott MD, Gorman GJ, Allison PA, Goddard AJH (2009) Reduced-order modelling of an adaptive mesh ocean model. Int J Numer Methods Fluids 59(8):827–851CrossRefGoogle Scholar
  23. Fu J, Gomez-Hernandez J (2009) A blocking markov chain monte carlo method for inverse stochastic hydrogeological modeling. Math Geosci 41(2):105–128CrossRefGoogle Scholar
  24. Gaspari G, Cohn SE (1999) Construction of correlation functions in two and three dimensions. Q J R Meteorol Soc 125(554):723–757CrossRefGoogle Scholar
  25. Gelfand S, Mitter S (1991) Simulated annealing type algorithms for multivariate optimization. Algorithmica 6(1):419–436CrossRefGoogle Scholar
  26. Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer-Verlag New York, Inc., New YorkCrossRefGoogle Scholar
  27. Golub G, Van Loan C (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, BaltimoreGoogle Scholar
  28. Gu Y, Oliver DS (2007) An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE J 12(4):438–446Google Scholar
  29. Hansen C (1998) Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, PhiladelphiaGoogle Scholar
  30. Hillery A, Chin R (1991) Iterative wiener filters for image restoration. IEEE Trans Signal Process 39(8):1892–1899CrossRefGoogle Scholar
  31. Houtekamer PL, Mitchell HL (2001) A sequential ensemble Kalman filter for atmospheric data assimilation. Mon Weather Rev 129(1):123–137CrossRefGoogle Scholar
  32. Houtekamer PL, Mitchell HL (2005) Ensemble Kalman filtering. Q J R Meteorol Soc 131(613):3269–3289CrossRefGoogle Scholar
  33. Jazwinski AH (1970) Stochastic processes and filtering theory. Academic Press, New YorkGoogle Scholar
  34. Kac M, Siegert AJF (1947) An explicit representation of a stationary Gaussian process. Ann Math Stat 18:438–442CrossRefGoogle Scholar
  35. Karhunen K (1947) Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann Acad Sci Fenn Ser A I Math Phys 1947(37):79Google Scholar
  36. Krymskaya MV, Hanea RG, Verlaan M (2009) An iterative ensemble Kalman filter for reservoir engineering applications. Comput Geosci 13(2):235–244CrossRefGoogle Scholar
  37. Kushner HJ (1987) Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: global minimization via monte carlo. SIAM J Appl Math 47(1):169–185CrossRefGoogle Scholar
  38. Li G, Reynolds AC (2009) Iterative ensemble Kalman filters for data assimilation. SPE J 14(3):496–505Google Scholar
  39. Loève M (1948) Fonctions aléatoires de second order. In: Levy P (ed) Processus Stochastiques et Movement Brownien. Hermann, ParisGoogle Scholar
  40. Lorentzen RJ, Naevdal G (2011) An iterative ensemble Kalman filter. IEEE Trans Autom Control 56(8):1990–1995CrossRefGoogle Scholar
  41. Ma X, Al-Harbi M, Datta-Gupta A, Efendiev Y (2008) An efficient two-stage sampling method for uncertainty quantification in history matching geological models. SPE J 13(1):77–87Google Scholar
  42. MacKay DJC (1999) Comparison of approximate methods for handling hyperparameters. Neural Comput 11(5):1035–1068CrossRefGoogle Scholar
  43. McLaughlin D, Townley LR (1996) A reassessment of the groundwater inverse problem. Water Resour Res 32(5):1131–1161CrossRefGoogle Scholar
  44. Moradkhani H, Sorooshian S, Gupta HV, Houser PR (2005) Dual state–parameter estimation of hydrological models using ensemble Kalman filter. Adv Water Resour 28(2):135–147CrossRefGoogle Scholar
  45. Naevdal G, Johnsen L, Aanonsen S, Vefring E (2005) Reservoir monitoring and continuous model updating using ensemble Kalman filter. SPE J 10(1):66–74Google Scholar
  46. Navon IM (1998) Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography. Dyn Atmos Oceans 27(1–4):55–79CrossRefGoogle Scholar
  47. Nocedal J, Wright SJ (2006) Numerical Optimization, 2nd edn. Springer VerlagGoogle Scholar
  48. Oliver D, Cunha L, Reynolds A (1997) Markov chain monte carlo methods for conditioning a permeability field to pressure data. Math Geol 29(1):61–91CrossRefGoogle Scholar
  49. Ott E, Hunt BR, Szunyogh I, Zimin AV, Kostelich EJ, Corazza M, Kalnay E, Patil DJ, Yorke JA (2004) A local ensemble Kalman filter for atmospheric data assimilation. Tellus A 56(5):415–428CrossRefGoogle Scholar
  50. Pham DT, Verron J, Roubaud MC (1998) A singular evolutive extended Kalman filter for data assimilation in oceanography. J Mar Syst 16(3–4):323–340CrossRefGoogle Scholar
  51. Rasmussen CE, Williams CKI (2005) Gaussian processes for machine learning (adaptive computation and machine learning). MIT Press, CambridgeGoogle Scholar
  52. Sadegh P, Spall J (1998) Optimal random perturbations for stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans Autom Control 43(10):1480–1484CrossRefGoogle Scholar
  53. Sætrom J, Omre H (2011) Ensemble Kalman filtering with shrinkage regression techniques. Comput Geosci 15(2):271–292CrossRefGoogle Scholar
  54. Sakov P, Oliver DS, Bertino L (2012) An iterative enkf for strongly nonlinear systems. Mon Weather Rev 140(6):1988–2004CrossRefGoogle Scholar
  55. Simon E, Bertino L (2009) Application of the gaussian anamorphosis to assimilation in a 3-D coupled physical-ecosystem model of the north atlantic with the ENKF: a twin experiment. Ocean Sci 5(4):495–510CrossRefGoogle Scholar
  56. Smith KW (2007) Cluster ensemble Kalman filter. Tellus A 59(5):749–757CrossRefGoogle Scholar
  57. Sørensen JVT, Madsen H (2004) Data assimilation in hydrodynamic modelling: on the treatment of non-linearity and bias. Stoch Environ Res Risk Assess 18(4):228–244CrossRefGoogle Scholar
  58. Spall J (2003) Introduction to stochastic search and optimization: estimation, simulation, and control. Wiley-Interscience series in discrete mathematics and optimization. Wiley-Interscience, HobokenGoogle Scholar
  59. Sun AY, Morris A, Mohanty S (2009) Comparison of deterministic ensemble Kalman filters for assimilating hydrogeological data. Adv Water Resour 32(2):280–292CrossRefGoogle Scholar
  60. Thacker WC (1989) The role of the hessian matrix in fitting models to measurements. J Geophys Res 94(C5):6177–6196CrossRefGoogle Scholar
  61. Tippett MK, Anderson JL, Bishop CH, Hamill TM, Whitaker JS (2003) Ensemble square root filters. Mon Weather Rev 131(7):1485–1490CrossRefGoogle Scholar
  62. Tong J, Hu B, Yang J (2010) Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data. Stoch Environ Res Risk Assess 24(8):1211–1223CrossRefGoogle Scholar
  63. Tong J, Hu B, Yang J (2012) Assimilating transient groundwater flow data via a localized ensemble Kalman filter to calibrate a heterogeneous conductivity field. Stoch Environ Res Risk Assess 26(3):467–478CrossRefGoogle Scholar
  64. Wan E, Van Der Merwe R (2000) The unscented Kalman filter for nonlinear estimation. In: Adaptive systems for signal processing, communications, and control symposium 2000. AS-SPCC. The IEEE 2000, Lake Louise, pp 153 –158Google Scholar
  65. Zhang D, Lu Z, Chen Y (2007) Dynamic reservoir data assimilation with an efficient, dimension-reduced Kalman filter. SPE J 12(1):108–117Google Scholar
  66. Zhou E, Fu M, Marcus S (2008) A particle filtering framework for randomized optimization algorithms. In: Simulation conference, 2008. WSC 2008, Winter, pp 647–654Google Scholar
  67. Zhou H, Gomez-Hernandez JJ, Franssen HJH, Li L (2011) An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering. Adv Water Resour 34(7):844–864CrossRefGoogle Scholar
  68. Zupanski M (2005) Maximum likelihood ensemble filter: theoretical aspects. Mon Weather Rev 133(6):1710–1726CrossRefGoogle Scholar
  69. Zupanski M, Navon IM, Zupanski D (2008) The maximum likelihood ensemble filter as a non-differentiable minimization algorithm. Q J R Meteorol Soc 134(633):1039–1050CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • A. H. ELSheikh
    • 1
  • C. C. Pain
    • 1
  • F. Fang
    • 1
  • J. L. M. A. Gomes
    • 1
  • I. M. Navon
    • 2
  1. 1.Department of Earth Science and EngineeringImperial College LondonLondonUK
  2. 2.Department of Scientific ComputingFlorida State UniversityFLUSA

Personalised recommendations