Improved initial sampling for the ensemble Kalman filter

In this paper, we discuss several possible approaches to improving the performance of the ensemble Kalman filter (EnKF) through improved sampling of the initial ensemble. Each of the approaches addresses a different limitation of the standard method. All methods, however, attempt to make the results from a small ensemble as reliable as possible. The validity and usefulness of each method for creating the initial ensemble is based on three criteria: (1) does the sampling result in unbiased Monte Carlo estimates for nonlinear flow problems, (2) does the sampling reduce the variability of estimates compared to ensembles of realizations from the prior, and (3) does the sampling improve the performance of the EnKF? In general, we conclude that the use of dominant eigenvectors ensures the orthogonality of the generated realizations, but results in biased forecasts of the fractional flow of water. We show that the addition of high frequencies from remaining eigenvectors can be used to remove the bias without affecting the orthogonality of the realizations, but the method did not perform significantly better than standard Monte Carlo sampling. It was possible to identify an appropriate importance weighting to reduce the variance in estimates of the fractional flow of water, but it does not appear to be possible to use the importance weighted realizations in standard EnKF when the data relationship is nonlinear. The biggest improvement came from use of the pseudo-data with corrections to the variance of the actual observations.

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

[2]  Man-Suk Oh,et al.  Integration of Multimodal Functions by Monte Carlo Importance Sampling , 1993 .

[3]  L. Mark Berliner,et al.  Monte Carlo Based Ensemble Forecasting , 2001, Stat. Comput..

[4]  Alan E. Gelfand,et al.  Bayesian statistics without tears: A sampling-resampling perspective , 1992 .

[5]  Tor Arne Johansen,et al.  Incorporating 4D Seismic Data in Reservoir Simulation Models Using Ensemble Kalman Filter , 2007 .

[6]  Jens Schröter,et al.  A comparison of error subspace Kalman filters , 2005 .

[7]  James A. Hansen.  Accounting for Model Error in Ensemble-Based State Estimation and Forecasting , 2002 .

[8]  Xian-Huan Wen,et al.  Some Practical Issues on Real-Time Reservoir Model Updating Using Ensemble Kalman Filter , 2007 .

[9]  Dynamically Stratified Monte Carlo Forecasting , 1992 .

[10]  Dean S. Oliver,et al.  Evaluation of Monte Carlo Methods for Assessing Uncertainty , 2003 .

[11]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[12]  Dean S. Oliver,et al.  An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation , 2007 .

[13]  Dongxiao Zhang,et al.  On importance sampling Monte Carlo approach to uncertainty analysis for flow and transport in porous media , 2003 .

[14]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[15]  D. Rubin.  Using the SIR algorithm to simulate posterior distributions , 1988 .

[16]  Dan Crisan,et al.  Particle Filters - A Theoretical Perspective , 2001, Sequential Monte Carlo Methods in Practice.

[17]  Peter R. Oke,et al.  Ensemble member generation for sequential data assimilation , 2008 .

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

[19]  Dinh-Tuan Pham,et al.  A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific , 2002 .

[20]  Luca Capriotti,et al.  Least-squares Importance Sampling for Monte Carlo security pricing , 2007, physics/0703181.

[21]  P. J. van Leeuwen,et al.  A variance-minimizing filter for large-scale applications , 2003 .

[22]  Dean S. Oliver,et al.  Conditioning Permeability Fields to Pressure Data , 1996 .

[23]  L. M. Berliner,et al.  Approximate importance sampling Monte Carlo for data assimilation , 2007 .