Abstract
The classical ensemble Kalman filter (EnKF) is known to underestimate the prediction uncertainty. This can potentially lead to low forecast precision and an ensemble collapsing into a single realisation. In this paper, we present alternative EnKF updating schemes based on shrinkage methods known from multivariate linear regression. These methods reduce the effects caused by collinear ensemble members and have the same computational properties as the fastest EnKF algorithms previously suggested. In addition, the importance of model selection and validation for prediction purposes is investigated, and a model selection scheme based on cross-validation is introduced. The classical EnKF scheme is compared with the suggested procedures on two-toy examples and one synthetic reservoir case study. Significant improvements are seen, both in terms of forecast precision and prediction uncertainty estimates.
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Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Vallès, B.: Ensemble Kalman filter in reservoir engineering—a review. SPE J. 14(3), 393–412 (2009)
Adabir, K.M., Magnus, J.R.: Matrix Algebra. Cambridge University Press, New York (2005)
Anderson, J.L.: A local least squares framework for ensemble filtering. Mon. Weather Rev. 131(4), 634–642 (2003)
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York (2003)
Barker, M., Rayens, W.: Partial least squares for discrimination. J. Chemom. 17, 166–173 (2003)
Cook, D.R.: Fisher lecture: dimension reduction in regression. Stat. Sci. 22(1), 1–26 (2007)
Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10(3), 197–208 (2000)
Efron, B.: The estimation of prediction error: covariance penalties and cross-validation. J. Am. Stat. Assoc. 99(467), 619–632 (2004)
Evensen, G.: Sequential data assimilation with nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99, 10,143–10,162 (1994)
Evensen, G.: The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dyn. 53(4), 343–367 (2003)
Evensen, G.: Data Assimilation. The Ensemble Kalman Filter. Springer, New York (2007)
Farrer, D.E., Glauber, R.R.: Multicollinearity in regression analysis: the problem revisited. Rev. Econ. Stat. 49(1), 92–107 (1967)
Furrer, R., Bengtsson, T.: Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. J. Multivar. Anal. 98(2), 227–255 (2007)
GeoQuest: ECLIPSE reference manual 2009.1. Schlumberger, GeoQuest (2009)
Hadi, A.S., Ling, R.F.: Some cautionary notes on the use of principal components regression. Am. Stat. 52(1), 15–19 (1998)
Hastie, T., Tibshirani, R.: Efficient quadratic regularization for expression arrays. Biostatistics 5(3), 329–340 (2004)
Hastie, T., Tibshirani, R., Freidman, J.: The Elements of Statistical Learning; Data Mining, Inference, and Prediction, 2nd edn. Springer, New York (2009)
Hegstad, B.K., Omre, H.: Uncertainty in production forecasts based on well observations, seismic data and production history. Soc. Pet. Eng. J. 6(4), 409–425 (2001)
Helland, I.S.: Some theoretical aspects of partial least squares regression. Chemometr. Intell. Lab. Syst. 58(2), 97–107 (2001)
Höskuldsson, A.: PLS regression methods. J. Chemom. 2, 211–228 (1988)
Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24(6), 417–441 (1933)
Houtekamer, P.L., Mitchell, H.L.: Data assimilation using an ensemble Kalman filter technique. Mon. Weather Rev. 126, 796–811 (1998)
Houtekamer, P.L., Mitchell, H.L.: Reply. Mon. Weather Rev. 127(6), 1378–1379 (1999)
Jolliffe, I.: A note on the use of principal components in regression. Appl. Statist. 31(3), 300–303 (1982)
Jolliffe, I.T.: Principal Component Analysis, 2nd edn. Springer, Berlin (2002)
Kalivas, J.H.: Cyclic subspace regression with analysis of the hat matrix. Chemometr. Intell. Lab. Syst. 45(1–2), 215–224 (1999)
Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82(Series D), 35–45 (1960)
Kaspar, M.H., Ray, W.H.: Partial Least Squares modelling as successive Singular Value Decomposition. Comput. Chem. Eng. 17(10), 985–989 (1993)
Ledoit, O., Wolf, M.: A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal. 88(2), 365–411 (2004)
van Leeuwen, P.J.: Comments on Data assimilation using an ensemble Kalman filter technique. Mon. Weather Rev. 127, 1374–1377 (1999)
Mardia, K.V., Kent, J.T., Bibby, J.M.: Multivariate Analysis. Academic, London (1979)
Myrseth, I.B., Omre, H.: Large-scale Inverse Problems and Quantification of Uncertainty, chap. The Ensemble Kalman Filter and Related Filters. Wiley, New York (2010)
Ränner, S., Lindgren, F., Gelandi, P., Wold, S.: A PLS kernel algorithm for data sets with many variables and fewer objects. Part 1: theory and algorithm. J. Chemom. 8, 111–125 (1994)
Ripley, B.D.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (1996)
Rosipal, R., Krämer, N.: Subspace, latent structure and feature selection techniques. In: Lect. Notes Comput. Sci. Chap. Overview and Recent Advances in Partial Least Squares, vol. 2940, pp. 34–51. Springer, New York (2006)
Sacher, W., Bartello, P.: Sampling errors in ensemble Kalman filtering. Part I: theory. Mon. Weather Rev. 136(8), 3035–3049 (2008)
Seber, G.A.F., Lee, A.J.: Linear Regression Analysis. Wiley, New York (2003)
Skjervheim, J.A., Evensen, G., Aanonsen, S., Ruud, B.O., Johansen, T.A.: Incorporating 4D seismic data in reservoir simulation models using ensemble Kalman filter. SPE J. 12(3), 282–292 (2007)
Strang, G.: Linear Algebra and Its Applications. Thomson Learning, London (1988)
Tikhonov, A.N., Arsenin, V.A.: Solution of Ill-Posed Problems. Winston, Washington (1977)
Wold, H.: Quantitative sociology: international perspectives on mathematical and statistical model building, chap. Path models with latent variables: the NiPALS approach, pp. 307–357. Academic, New York (1975)
Zeng, X.Q., Wang, M.W., Nie, J.Y.: Text classification based on partial least square analysis. In: The 22nd Annual ACM Symposium on Applied Computing, Special Track on Information Access and Retrieval (2007)
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Sætrom, J., Omre, H. Ensemble Kalman filtering with shrinkage regression techniques. Comput Geosci 15, 271–292 (2011). https://doi.org/10.1007/s10596-010-9196-0
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DOI: https://doi.org/10.1007/s10596-010-9196-0