Improved Uncertainty Quantification in the Ensemble Kalman Filter Using Statistical Model-Selection Techniques
- Jon Sætrom (Norwegian University of Science and Technology) | Joakim Hove (Statoil Research Centre) | Jan-Arild Skjervheim (Statoil Research Centre) | Jon Gustav Vabø (Statoil Research Centre)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2012
- Document Type
- Journal Paper
- 152 - 162
- 2012. Society of Petroleum Engineers
- 5.6.9 Production Forecasting
- Reservoir Characterisation, Cross-Validation, Regression Model Overfitting, Dimension Reduction
- 0 in the last 30 days
- 436 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 10.00|
|SPE Non-Member Price:||USD 30.00|
The ensemble Kalman filter (EnKF) is a sequential Monte Carlo method for solving nonlinear spatiotemporal inverse problems, such as petroleum-reservoir evaluation, in high dimensions. Although the EnKF has seen successful applications in numerous areas, the classical EnKF algorithm can severely underestimate the prediction uncertainty. This can lead to biased production forecasts and an ensemble collapsing into a single realization.
In this paper, we combine a previously suggested EnKF scheme based on dimension reduction in the data space, with an automatic cross-validation (CV) scheme to select the subspace dimension. The properties of both the dimension reduction and the CV scheme are well known in the statistical literature. In an EnKF setting, the former can reduce the effects caused by collinear ensemble members, while the latter can guard against model overfitting by evaluating the predictive capabilities of the EnKF scheme. The model-selection criterion traditionally used for determining the subspace dimension, on the other hand, does not take the predictive power of the EnKF scheme into account, and can potentially lead to severe problems of model overfitting. A reservoir case study is used to demonstrate that the CV scheme can substantially improve the reservoir predictions with associated uncertainty estimates.
|File Size||14 MB||Number of Pages||11|
Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., and Vallès, B.2009. The Ensemble Kalman Filter In Reservoir Engineering--a Review. SPEJ. 14 (3): 393-412. SPE-117274-PA. http://dx.doi.org/10.2118/117274-PA.
Anderson, J.L. 2003. A local least squares framework for ensemble filtering.Monthly Weather Review 131 (4): 634-642. http://dx.doi.org/10.1175/1520-0493(2003)131<0634:ALLSFF>2.0.CO;2
Burgers, G., van Leeuwen, P.J., and Evensen, G. 1998. Analysis Scheme in theEnsemble Kalman Filter. Monthly Weather Review 126 (6):1719-1724. http://dx.doi.org/10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2
Doucet, A., Godsill, S., and Andrieu, C. 2000. On sequential Monte Carlosampling methods for Bayesian filtering. Statistics and Computing 10 (3): 197-208. http://dx.doi.org/10.1023/A:1008935410038.
Evensen, G. 1994. Sequential data assimilation with a nonlinearquasi-geostrophic model using Monte Carlo methods to forecast error statistics.J. of Geophysical Research 99 (C5): 10143-10162.
Evensen, G. 2004. Sampling strategies and square root analysis schemes forthe EnKF. Ocean Dynamics 54 (6): 539-560. http://dx.doi.org/10.1007/s10236-004-0099-2.
Evensen, G. 2007. Data Assimilation: The Ensemble Kalman Filter.Berlin: Springer Verlag.
Farrer, D.E. and Glauber, R.R. 1967. Multicollinearity in RegressionAnalysis: The Problem Revisited. The Review of Economics and Statistics 49 (1): 92-107. http://dx.doi.org/10.2307/1937887
Hastie, T. and Tibshirani, R. 2004. Efficient quadratic regularization forexpression arrays. Biostatistics 5 (3): 329-340. http://dx.doi.org/10.1093/biostatistics/kxh010
Hastie, T., Tibshirani, R., and Friedman, J. 2009.Elements of StatisticalLearning: Data Mining, Inference, and Prediction, second edition. New York:Series in Statistics, Springer.
Hoerl, A.E. and Kennard, R.W. 1970. Ridge Regression: Biased Estimation forNonorthogonal Problems. Technometrics 12 (3): 55-67. http://dx.doi.org/10.2307/1267351
Hotelling, H. 1933. Analysis of a Complex of Statistical Variables intoPrincipal Components. Journal of Educational Psychology 24(6): 417-441.
Houtekamer, P.L. and Mitchell, H.L. 1998. Data Assimilation Using anEnsemble Kalman Filter Technique. Monthly Weather Review 126 (3): 796-811. http://dx.doi.org/10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2
Kalman, R.E. 1960. A new approach to linear filtering and predictionproblems. Transactions of the ASME--Journal of Basic Engineering 82 (Series D): 35-45.
Ledoit, O. and Wolf, M. 2004. A well-conditioned estimator forlarge-dimensional covariance matrices. Journal of Multivariate Analysis 88 (2): 365-411. http://dx.doi.org/10.1016/S0047-259X(03)00096-4
Mardia, K.V., Kent, J.T. and Bibby, J.M. 1979. Multivariate Analysis.London: Academic Press.
Picard, R.R. and Cook, D.R. 1984. Cross-Validation of Regression Models.Journal of the American Statistical Association 79 (387):575-583. http://dx.doi.org/10.2307/2288403.
Sætrom, J. and Omre, H. 2010. Ensemble Kalman filtering using shrinkageregression techniques. Computational Geosciences 15 (2):271-292. http://dx.doi.org/10.1007/s10596-010-9196-0
Seber, G.A.F. and Lee, A.J. 2003. Linear Regression Analysis, secondedition. Hoboken, New Jersey: Wiley Series in Probability and Statistics, JohnWiley & Sons.
Seiler, A., Rivenæs, J.C., Aanonsen, S.I., and Evensen, G. 2009. StructuralUncertainty Modelling and Updating by Production Data Integration. Paper SPE125352 presented at the SPE/EAGE Reservoir Characterization and SimulationConference, Abu Dhabi, UAE, 19-21 October. http://dx.doi.org/10.2118/125352-MS.
Skjervheim, J.-A., Evensen, G., Aanonsen, S.I., Ruud, B.O., and Johansen,T.A. 2007. Incorporating 4D Seismic Data in Reservoir Simulation Model UsingEnsemble Kalman Filter. SPE J. 12 (3): 282-292.SPE-95789-PA. http://dx.doi.org/10.2118/95789-PA.
Strang, G. 1988. Linear Algebra and its Applications, third edition.New York: Harcourt Brace Jovanovich.