A Bayesian Adaptive Ensemble Kalman Filter for Sequential State and Parameter Estimation

AbstractThis paper proposes new methodology for sequential state and parameter estimation within the ensemble Kalman filter. The method is fully Bayesian and propagates the joint posterior distribution of states and parameters over time. To implement the method, the authors consider three representations of the marginal posterior distribution of the parameters: a grid-based approach, a Gaussian approximation, and a sequential importance sampling (SIR) approach with kernel resampling. In contrast to existing online parameter estimation algorithms, the new method explicitly accounts for parameter uncertainty and provides a formal way to combine information about the parameters from data at different time periods. The method is illustrated and compared to existing approaches using simulated and real data.

[1]  R. Kohn,et al.  On Gibbs sampling for state space models , 1994 .

[2]  D. Dee.  On-line Estimation of Error Covariance Parameters for Atmospheric Data Assimilation , 1995 .

[3]  Fuqing Zhang,et al.  Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation , 2016 .

[4]  Radek Hofman,et al.  Marginalized Particle Filtering Framework for Tuning of Ensemble Filters , 2011 .

[5]  L. M. Berliner,et al.  A Bayesian tutorial for data assimilation , 2007 .

[6]  P. Houtekamer,et al.  An Adaptive Ensemble Kalman Filter , 2000 .

[7]  Wolfgang Nowak,et al.  Best unbiased ensemble linearization and the quasi‐linear Kalman ensemble generator , 2009 .

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

[9]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[10]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[11]  Jean-Michel Brankart,et al.  Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals , 2010 .

[12]  Thierry Chonavel,et al.  Estimating model‐error covariances in nonlinear state‐space models using Kalman smoothing and the expectation–maximization algorithm , 2017 .

[13]  G. Kitagawa.  A self-organizing state-space model , 1998 .

[14]  G. Mason.  State and Parameter Estimation in Stochastic Dynamical Models , 2011 .

[15]  Jeffrey L. Anderson.  EXPLORING THE NEED FOR LOCALIZATION IN ENSEMBLE DATA ASSIMILATION USING A HIERARCHICAL ENSEMBLE FILTER , 2007 .

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

[17]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[18]  Michael A. West,et al.  Combined Parameter and State Estimation in Simulation-Based Filtering , 2001, Sequential Monte Carlo Methods in Practice.

[19]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .

[20]  Thomas Bengtsson,et al.  Sequential State and Variance Estimation within the Ensemble Kalman Filter , 2007 .

[21]  C. Wikle.  A kernel-based spectral model for non-Gaussian spatio-temporal processes , 2002 .

[22]  H. W. Sorenson,et al.  Kalman filtering : theory and application , 1985 .

[23]  H. Künsch,et al.  Sequential State and Observation Noise Covariance Estimation Using Combined Ensemble Kalman and Particle Filters , 2012 .

[24]  Jeffrey L. Anderson,et al.  An adaptive covariance inflation error correction algorithm for ensemble filters , 2007 .

[25]  Jeffrey L. Anderson.  An Ensemble Adjustment Kalman Filter for Data Assimilation , 2001 .

[26]  G. Ueno,et al.  Bayesian estimation of the observation‐error covariance matrix in ensemble‐based filters , 2016 .

[27]  Soroosh Sorooshian,et al.  Dual state-parameter estimation of hydrological models using ensemble Kalman filter , 2005 .

[28]  Xuguang Wang,et al.  A Comparison of Breeding and Ensemble Transform Kalman Filter Ensemble Forecast Schemes , 2003 .

[29]  Manuel Pulido,et al.  Offline parameter estimation using EnKF and maximum likelihood error covariance estimates: Application to a subgrid‐scale orography parametrization , 2013 .

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

[31]  P. Houtekamer,et al.  Data Assimilation Using an Ensemble Kalman Filter Technique , 1998 .

[32]  G. Kitagawa.  Theory and Methods , 1998 .

[33]  P. Bickel,et al.  Obstacles to High-Dimensional Particle Filtering , 2008 .

[34]  Nagatomo Nakamura,et al.  Iterative algorithm for maximum‐likelihood estimation of the observation‐error covariance matrix for ensemble‐based filters , 2014 .

[35]  Jonathan R. Stroud,et al.  Understanding the Ensemble Kalman Filter , 2016 .

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

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

[38]  Jonathan R. Stroud,et al.  An Ensemble Kalman Filter and Smoother for Satellite Data Assimilation , 2010 .

[39]  G. Storvik.  Particle filters in state space models with the presence of unknown static parameters YYYY No org found YYY , 2000 .

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

[41]  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 .

[42]  Jeffrey L. Anderson.  A Non-Gaussian Ensemble Filter Update for Data Assimilation , 2010 .

[43]  Christopher K. Wikle,et al.  Estimation of Parameterized Spatio-Temporal Dynamic Models , 2007 .

[44]  Tomoyuki Higuchi,et al.  Maximum likelihood estimation of error covariances in ensemble‐based filters and its application to a coupled atmosphere–ocean model , 2010 .