Multimodal ensemble Kalman filtering using Gaussian mixture models

In this paper we present an extension of the ensemble Kalman filter (EnKF) specifically designed for multimodal systems. EnKF data assimilation scheme is less accurate when it is used to approximate systems with multimodal distribution such as reservoir facies models. The algorithm is based on the assumption that both prior and posterior distribution can be approximated by Gaussian mixture and it is validated by the introduction of the concept of finite ensemble representation. The effectiveness of the approach is shown with two applications. The first example is based on Lorenz model. In the second example, the proposed methodology combined with a localization technique is used to update a 2D reservoir facies models. Both applications give evidence of an improved performance of the proposed method respect to the EnKF.

[1]  J. Mandel,et al.  On the convergence of the ensemble Kalman filter , 2009, Applications of mathematics.

[2]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[3]  Ernesto Della Rossa,et al.  Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion , 2010 .

[4]  G. Evensen.  Data Assimilation: The Ensemble Kalman Filter , 2006 .

[5]  Peter S. Maybeck,et al.  Stochastic Models, Estimation And Control , 2012 .

[6]  T. Bengtsson,et al.  Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants , 2007 .

[7]  D. Grana,et al.  Petrophysical Properties Estimation In a Crosswell Study Integrated With Statistical Rock Physics , 2009 .

[8]  Geir Nævdal,et al.  Reservoir Monitoring and Continuous Model Updating Using Ensemble Kalman Filter , 2005 .

[9]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[10]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[11]  Geir Evensen,et al.  Assimilation of SLA and SST data into an OGCM for the Indian Ocean , 2002 .

[12]  Rudolph van der Merwe,et al.  Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[13]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[14]  M. Buehner,et al.  Atmospheric Data Assimilation with an Ensemble Kalman Filter: Results with Real Observations , 2005 .

[15]  Chris Snyder,et al.  Toward a nonlinear ensemble filter for high‐dimensional systems , 2003 .

[16]  Albert C. Reynolds,et al.  Assessing the Uncertainty in Reservoir Description and Performance Predictions With the Ensemble Kalman Filter , 2007 .

[17]  E. Della Rossa,et al.  Ensemble Kalman Filter for Gaussian Mixture Models , 2007 .

[18]  Dean S. Oliver,et al.  THE ENSEMBLE KALMAN FILTER IN RESERVOIR ENGINEERING-A REVIEW , 2009 .

[19]  D. Pham.  Stochastic Methods for Sequential Data Assimilation in Strongly Nonlinear Systems , 2001 .

[20]  Keston W. Smith.  Cluster ensemble Kalman filter , 2007 .

[21]  G. Evensen,et al.  Analysis Scheme in the Ensemble Kalman Filter , 1998 .

[22]  G. Evensen,et al.  An ensemble Kalman smoother for nonlinear dynamics , 2000 .

[23]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[24]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[25]  S. Cohn,et al.  Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions , 2022 .

[26]  Timothy C. Coburn,et al.  Geostatistics for Natural Resources Evaluation , 2000, Technometrics.