NDLI: Ensemble filter methods with perturbed observations applied to nonlinear problems

Content Provider	Springer Nature Link
Author	Zhang, Yanfen Liu, Ning Oliver, Dean S.
Copyright Year	2009
Abstract	The ensemble Kalman filter (EnKF) has been shown repeatedly to be an effective method for data assimilation in large-scale problems, including those in petroleum engineering. Data assimilation for multiphase flow in porous media is particularly difficult, however, because the relationships between model variables (e.g., permeability and porosity) and observations (e.g., water cut and gas–oil ratio) are highly nonlinear. Because of the linear approximation in the update step and the use of a limited number of realizations in an ensemble, the EnKF has a tendency to systematically underestimate the variance of the model variables. Various approaches have been suggested to reduce the magnitude of this problem, including the application of ensemble filter methods that do not require perturbations to the observed data. On the other hand, iterative least-squares data assimilation methods with perturbations of the observations have been shown to be fairly robust to nonlinearity in the data relationship. In this paper, we present EnKF with perturbed observations as a square root filter in an enlarged state space. By imposing second-order-exact sampling of the observation errors and independence constraints to eliminate the cross-covariance with predicted observation perturbations, we show that it is possible in linear problems to obtain results from EnKF with observation perturbations that are equivalent to ensemble square-root filter results. Results from a standard EnKF, EnKF with second-order-exact sampling of measurement errors that satisfy independence constraints (EnKF (SIC)), and an ensemble square-root filter (ETKF) are compared on various test problems with varying degrees of nonlinearity and dimensions. The first test problem is a simple one-variable quadratic model in which the nonlinearity of the observation operator is varied over a wide range by adjusting the magnitude of the coefficient of the quadratic term. The second problem has increased observation and model dimensions to test the EnKF (SIC) algorithm. The third test problem is a two-dimensional, two-phase reservoir flow problem in which permeability and porosity of every grid cell (5,000 model parameters) are unknown. The EnKF (SIC) and the mean-preserving ETKF (SRF) give similar results when applied to linear problems, and both are better than the standard EnKF. Although the ensemble methods are expected to handle the forecast step well in nonlinear problems, the estimates of the mean and the variance from the analysis step for all variants of ensemble filters are also surprisingly good, with little difference between ensemble methods when applied to nonlinear problems.
Starting Page	249
Ending Page	261
Page Count	13
File Format	PDF
ISSN	14200597
Journal	Computational Geosciences
Volume Number	14
Issue Number	2
e-ISSN	15731499
Language	English
Publisher	Springer Netherlands
Publisher Date	2009-07-22
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Ensemble Kalman filter Second-order-exact sampling Square-root filter Nonlinearity Sequential data assimilation Mathematical Modeling and Industrial Mathematics Soil Science & Conservation Hydrogeology Geotechnical Engineering
Content Type	Text
Resource Type	Article
Subject	Computers in Earth Sciences Computational Theory and Mathematics Computer Science Applications Computational Mathematics

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