Computational Geosciences

, Volume 16, Issue 2, pp 437–453 | Cite as

Improved initial ensemble generation coupled with ensemble square root filters and inflation to estimate uncertainty

Original Paper


The performance of the Ensemble Kalman Filter method (EnKF) depends on the sample size compared to the dimension of the parameters space. In real applications insufficient sampling may result in spurious correlations which reduce the accuracy of the filter with a strong underestimation of the uncertainty. Covariance localization and inflation are common solutions to these problems. The Ensemble Square Root Filters (ESRF) is also better to estimate uncertainty with respect to the EnKF. In this work we propose a method that limits the consequences of sampling errors by means of a convenient generation of the initial ensemble. This regeneration is based on a Stationary Orthogonal-Base Representation (SOBR) obtained via a singular value decomposition of a stationary covariance matrix estimated from the ensemble. The technique is tested on a 2D single phase reservoir and compared with the other common techniques. The evaluation is based on a reference solution obtained with a very large ensemble (one million members) which remove the spurious correlations. The example gives evidence that the SOBR technique is a valid alternative to reduce the effect of sampling error. In addition, when the SOBR method is applied in combination with the ESRF and inflation, it gives the best performance in terms of uncertainty estimation and oil production forecast.


Data assimilation Ensemble Kalman filter Ensemble square root filters History matching 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aanonsen, S.I. , Nævdal, G., Oliver, D.S., Reynolds, A.C.: The ensemble Kalman filter in reservoir engineering—a review. SPE J. 14(3), 393–412 (2009)Google Scholar
  2. 2.
    Anderson, J.L., Anderson, S.L.: A Monte-Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Weather Rev. 127, 2741–2758 (1999)CrossRefGoogle Scholar
  3. 3.
    Anderson, J.L.: Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D 230, 99–111 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Anderson, J.L.: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus 59A, 210–224 (2007)Google Scholar
  5. 5.
    Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. Wiley, New York (1984 )MATHGoogle Scholar
  6. 6.
    Burgers, G., van Leeuwen, P.J., Evensen, G.: Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126, 1719–1724 (1998)CrossRefGoogle Scholar
  7. 7.
    Evensen, G.: Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99, 10143–10162 (1994)CrossRefGoogle Scholar
  8. 8.
    Evensen, G.: Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn. 54(6), 539–560 (2004)CrossRefGoogle Scholar
  9. 9.
    Furrer, R., Bengtsson, T.: Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. J. Multivar. Anal. 98, 227–255 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gaspari, G., Cohn, S.E.: Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc. 125(554), 723–757 (1999)CrossRefGoogle Scholar
  11. 11.
    Hamill, T.M., Whitaker, J.S., Snyder, C.: Distance-dependent filtering of background-error covariance estimates in an ensemble Kalman filter. Mon. Weather Rev. 129, 2776–2790 (2001)CrossRefGoogle Scholar
  12. 12.
    Houtekamer, P.L., Mitchell, H.L.: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 129, 123–137 (2001)CrossRefGoogle Scholar
  13. 13.
    Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems by Implicitely Restarted Arnoldi Methods. SIAM, Philadelphia, PA (1998)Google Scholar
  14. 14.
    Lorentzen, R.J., Nævdal, G., Vallés, B., Berg, A.M., Grimstad, A.A.: Analysis of the ensemble Kalman filter for estimation of permeability and porosity in reservoir models. In: SPE Annual Technical Conference and Exhibition, Dallas, Texas, USA, SPE 96375, 9–12 October 2005Google Scholar
  15. 15.
    Mandel, J., Cobb, L., Beezley, J.D.: On the convergence of the Ensemble Kalman Filter (2011). arXiv:0901.2951v2
  16. 16.
    Oliver, D.S., Chen, Y.: Improved initial sampling for the Ensemble Kalman Filter. Comput. Geosci. 13(1), 13–26 (2009)MATHCrossRefGoogle Scholar
  17. 17.
    Sakov, P., Oke, P.R.: Implications of the form of the ensemble trasformation in the ensemble square root filters. Mon. Weather Rev. 136, 1042–1053 (2007)CrossRefGoogle Scholar
  18. 18.
    Tippett, M.K., Anderson, J.L., Bishop, C.H., Hamill, T.M., Whitaker, J.S.: Ensemble square root filters. Mon. Weather Rev. 131, 1485–1490 (2003)CrossRefGoogle Scholar
  19. 19.
    Whitaker, J.S., Hamill, T.M.: Ensemble data assimilation without perturbed observations. Mon. Weather Rev. 130(7), 1913–1924 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Eni Exploration & ProductionSan Donato MilaneseMilanItaly

Personalised recommendations