Implicit Particle Methods and Their Connection with Variational Data Assimilation

AbstractThe implicit particle filter is a sequential Monte Carlo method for data assimilation that guides the particles to the high-probability regions via a sequence of steps that includes minimizations. A new and more general derivation of this approach is presented and the method is extended to particle smoothing as well as to data assimilation for perfect models. Minimizations required by implicit particle methods are shown to be similar to those that one encounters in variational data assimilation, and the connection of implicit particle methods with variational data assimilation is explored. In particular, it is argued that existing variational codes can be converted into implicit particle methods at a low additional cost, often yielding better estimates that are also equipped with quantitative measures of the uncertainty. A detailed example is presented.

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

[2]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[3]  R. P. Fedorenko.  A relaxation method for solving elliptic difference equations , 1962 .

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

[5]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[6]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[7]  F. L. Dimet,et al.  Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects , 1986 .

[8]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[9]  Goodman,et al.  Multigrid Monte Carlo method. Conceptual foundations. , 1989, Physical review. D, Particles and fields.

[10]  J. Geweke,et al.  Bayesian Inference in Econometric Models Using Monte Carlo Integration , 1989 .

[11]  L. Tierney,et al.  The validity of posterior expansions based on Laplace''s method , 1990 .

[12]  Philippe Courtier,et al.  Four‐Dimensional Assimilation In the Presence of Baroclinic Instability , 1992 .

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

[14]  Lance M. Leslie,et al.  Tropical Cyclone Prediction Using a Barotropic Model Initialized by a Generalized Inverse Method , 1993 .

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

[16]  P. Courtier,et al.  A strategy for operational implementation of 4D‐Var, using an incremental approach , 1994 .

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

[18]  Jun S. Liu,et al.  Blind Deconvolution via Sequential Imputations , 1995 .

[19]  Olivier Talagrand,et al.  Assimilation of Observations, an Introduction (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[20]  Philippe Courtier,et al.  Dual formulation of four‐dimensional variational assimilation , 1997 .

[21]  D. Zupanski.  A General Weak Constraint Applicable to Operational 4DVAR Data Assimilation Systems , 1997 .

[22]  Geir Evensen,et al.  Advanced Data Assimilation for Strongly Nonlinear Dynamics , 1997 .

[23]  Jun S. Liu,et al.  Rejection Control and Sequential Importance Sampling , 1998 .

[24]  P. Fearnhead,et al.  Improved particle filter for nonlinear problems , 1999 .

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

[26]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[27]  Robert N. Miller,et al.  Data assimilation into nonlinear stochastic models , 1999 .

[28]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[29]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[30]  Rong Chen,et al.  A Theoretical Framework for Sequential Importance Sampling with Resampling , 2001, Sequential Monte Carlo Methods in Practice.

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

[32]  Eugenia Kalnay,et al.  Atmospheric Modeling, Data Assimilation and Predictability , 2002 .

[33]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[34]  Paul Krause,et al.  Dimensional reduction for a Bayesian filter. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[35]  A. Chorin,et al.  Stochastic Tools in Mathematics and Science , 2005 .

[36]  M. Buehner.  Ensemble‐derived stationary and flow‐dependent background‐error covariances: Evaluation in a quasi‐operational NWP setting , 2005 .

[37]  Y. Trémolet.  Accounting for an imperfect model in 4D‐Var , 2006 .

[38]  Arnaud Doucet,et al.  Sequential Monte Carlo Methods , 2006, Handbook of Graphical Models.

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

[40]  E. Kalnay,et al.  C ○ 2007 The Authors , 2006 .

[41]  B. Hunt,et al.  A comparative study of 4D-VAR and a 4D Ensemble Kalman Filter: perfect model simulations with Lorenz-96 , 2007 .

[42]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory , 2007 .

[43]  Robert N. Miller,et al.  Representer‐based variational data assimilation in a nonlinear model of nearshore circulation , 2007 .

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

[45]  Eric Moulines,et al.  Adaptive methods for sequential importance sampling with application to state space models , 2008, 2008 16th European Signal Processing Conference.

[46]  Jean-Michel Marin,et al.  Adaptive importance sampling in general mixture classes , 2007, Stat. Comput..

[47]  A. Chorin,et al.  Monte Carlo without chains , 2008, 0802.1046.

[48]  P. Bickel,et al.  Sharp failure rates for the bootstrap particle filter in high dimensions , 2008, 0805.3287.

[49]  Q. Xiao,et al.  An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part I: Technical Formulation and Preliminary Test , 2008 .

[50]  A. Doucet,et al.  A note on auxiliary particle filters , 2008 .

[51]  A. Chorin,et al.  Implicit sampling for particle filters , 2009, Proceedings of the National Academy of Sciences.

[52]  Matthias Morzfeld,et al.  Implicit particle filters for data assimilation , 2010, 1005.4002.

[53]  Gauthier Hulot,et al.  An Introduction to Data Assimilation and Predictability in Geomagnetism , 2010 .

[54]  P. Leeuwen,et al.  Nonlinear data assimilation in geosciences: an extremely efficient particle filter , 2010 .

[55]  P. Leeuwen.  Efficient nonlinear data-assimilation in geophysical fluid dynamics , 2011 .

[56]  A. Chorin,et al.  Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation , 2011, 1109.3664.

[57]  Matthias Morzfeld,et al.  A random map implementation of implicit filters , 2011, J. Comput. Phys..

[58]  P. Moral,et al.  On adaptive resampling strategies for sequential Monte Carlo methods , 2012, 1203.0464.

[59]  Qingnong Xiao,et al.  An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part III: Antarctic Applications with Advanced Research WRF Using Real Data , 2013 .