American Institute of Mathematical Sciences

Advanced
January  2014, 1(1): 177-189. doi: 10.3934/jcd.2014.1.177

On the consistency of ensemble transform filter formulations

Sebastian Reich 1, and  Seoleun Shin 2,

1. 

Universität Potsdam, Institut für Mathematik, Am Neuen Palais 10, D-14469 Potsdam, Germany
2. 

Korea Institute of Atmospheric Prediction Systems, 4F Korea Computer Bldg., 35 Boramae-ro 5-gil, Dongjak-gu, Seoul 156-849, South Korea

Received  October 2011 Revised  July 2012 Published  April 2014

In this paper, we consider the data assimilation problem for perfect differential equation models without model error and for either continuous or intermittent observational data. The focus will be on the popular class of ensemble Kalman filters which rely on a Gaussian approximation in the data assimilation step. We discuss the impact of this approximation on the temporal evolution of the ensemble mean and covariance matrix. We also discuss options for reducing arising inconsistencies, which are found to be more severe for the intermittent data assimilation problem. Inconsistencies can, however, not be completely eliminated due to the classic moment closure problem. It is also found for the Lorenz-63 model that the proposed corrections only improve the filter performance for relatively large ensemble sizes.
Keywords: ensemble Kalman filter, Data assimilation, particle filter..
Mathematics Subject Classification: Primary: 93E11, 60G35, 65C05; Secondary: 62M20, 62F15, 65C3.
Citation: Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177
References:
[1]

J. Amezcua, E. Kalnay, K. Ide and S. Reich, Ensemble transform Kalman-Bucy filters, Q. J. Royal Meteorological Soc., published online 25. July 2013. doi: 10.1002/qj.2186.

[2]

J. Anderson and S. Anderson, A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Mon. Wea. Rev., 127 (1999), 2741-2758. doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

[3]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, vol. 60 of Stochastic modelling and applied probability, Springer-Verlag, New-York, 2009.

[4]

K. Bergemann, G. Gottwald and S. Reich, Ensemble propagation and continuous matrix factorization algorithms, Q. J. R. Meteorological Soc., 135 (2009), 1560-1572. doi: 10.1002/qj.457.

[5]

K. Bergemann and S. Reich, A localization technique for ensemble Kalman filters, Q. J. R. Meteorological Soc., 136 (2010), 701-707. doi: 10.1002/qj.591.

[6]

K. Bergemann and S. Reich, A mollified ensemble Kalman filter, Q. J. R. Meteorological Soc., 136 (2010), 1636-1643. doi: 10.1002/qj.672.

[7]

K. Bergemann and S. Reich, An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorolog. Zeitschrift, 21 (2012), 213-219. doi: 10.1127/0941-2948/2012/0307.

[8]

D. Crisan and J. Xiong, Approximate McKean-Vlasov representation for a class of SPDEs, Stochastics, 82 (2010), 53-68. doi: 10.1080/17442500902723575.

[9]

D. Crisan and J. Xiong, Numerical solution for a class of SPDEs over bounded domains, Stochastics, published online 02 September 2013. doi: 10.1051/proc:071916.

[10]

G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, New York, 2006. doi: 10.1007/978-3-642-03711-5.

[11]

A. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, New York, 1970.

[12]

J. Lei and P. Bickel, A moment matching ensemble filter for nonlinear and non-Gaussian data assimilation, Mon. Weath. Rev., 139 (2011), 3964-3973. doi: 10.1175/2011MWR3553.1.

[13]

E. Lorenz, Deterministic non-periodic flows, J. Atmos. Sci., 20 (1963), 130-141.

[14]

S. Reich, A dynamical systems framework for intermittent data assimilation, BIT Numer Math, 51 (2011), 235-249. doi: 10.1007/s10543-010-0302-4.

[15]

S. Reich, A Gaussian mixture ensemble transform filter, Q. J. R. Meterolog. Soc., 138 (2012), 222-233. doi: 10.1002/qj.898.

[16]

M. Tippett, J. Anderson, G. Bishop, T. Hamill and J. Whitaker, Ensemble square root filters, Mon. Wea. Rev., 131 (2003), 1485-1490. doi: 10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2.

[17]

C. Villani, Topics in Optimal Transportation, American Mathematical Society, Providence, Rhode Island, NY, 2003. doi: 10.1007/b12016.

[18]

X. Xiong, I. Navon and B. Uzungoglu, A note on the particle filter with posterior Gaussian resampling, Tellus, 58 (2006), 456-460. doi: 10.1111/j.1600-0870.2006.00185.x.

[19]

T. Yang, P. Mehta and S. Meyn, Feedback particle filter, IEEE Trans. on Automatic Control, 58 (2013), 2465-2480. doi: 10.1109/TAC.2013.2258825.

show all references

References:
[1]

J. Amezcua, E. Kalnay, K. Ide and S. Reich, Ensemble transform Kalman-Bucy filters, Q. J. Royal Meteorological Soc., published online 25. July 2013. doi: 10.1002/qj.2186.

[2]

J. Anderson and S. Anderson, A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Mon. Wea. Rev., 127 (1999), 2741-2758. doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

[3]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, vol. 60 of Stochastic modelling and applied probability, Springer-Verlag, New-York, 2009.

[4]

K. Bergemann, G. Gottwald and S. Reich, Ensemble propagation and continuous matrix factorization algorithms, Q. J. R. Meteorological Soc., 135 (2009), 1560-1572. doi: 10.1002/qj.457.

[5]

K. Bergemann and S. Reich, A localization technique for ensemble Kalman filters, Q. J. R. Meteorological Soc., 136 (2010), 701-707. doi: 10.1002/qj.591.

[6]

K. Bergemann and S. Reich, A mollified ensemble Kalman filter, Q. J. R. Meteorological Soc., 136 (2010), 1636-1643. doi: 10.1002/qj.672.

[7]

K. Bergemann and S. Reich, An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorolog. Zeitschrift, 21 (2012), 213-219. doi: 10.1127/0941-2948/2012/0307.

[8]

D. Crisan and J. Xiong, Approximate McKean-Vlasov representation for a class of SPDEs, Stochastics, 82 (2010), 53-68. doi: 10.1080/17442500902723575.

[9]

D. Crisan and J. Xiong, Numerical solution for a class of SPDEs over bounded domains, Stochastics, published online 02 September 2013. doi: 10.1051/proc:071916.

[10]

G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, New York, 2006. doi: 10.1007/978-3-642-03711-5.

[11]

A. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, New York, 1970.

[12]

J. Lei and P. Bickel, A moment matching ensemble filter for nonlinear and non-Gaussian data assimilation, Mon. Weath. Rev., 139 (2011), 3964-3973. doi: 10.1175/2011MWR3553.1.

[13]

E. Lorenz, Deterministic non-periodic flows, J. Atmos. Sci., 20 (1963), 130-141.

[14]

S. Reich, A dynamical systems framework for intermittent data assimilation, BIT Numer Math, 51 (2011), 235-249. doi: 10.1007/s10543-010-0302-4.

[15]

S. Reich, A Gaussian mixture ensemble transform filter, Q. J. R. Meterolog. Soc., 138 (2012), 222-233. doi: 10.1002/qj.898.

[16]

M. Tippett, J. Anderson, G. Bishop, T. Hamill and J. Whitaker, Ensemble square root filters, Mon. Wea. Rev., 131 (2003), 1485-1490. doi: 10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2.

[17]

C. Villani, Topics in Optimal Transportation, American Mathematical Society, Providence, Rhode Island, NY, 2003. doi: 10.1007/b12016.

[18]

X. Xiong, I. Navon and B. Uzungoglu, A note on the particle filter with posterior Gaussian resampling, Tellus, 58 (2006), 456-460. doi: 10.1111/j.1600-0870.2006.00185.x.

[19]

T. Yang, P. Mehta and S. Meyn, Feedback particle filter, IEEE Trans. on Automatic Control, 58 (2013), 2465-2480. doi: 10.1109/TAC.2013.2258825.

[1]

Andreas Bock, Colin J. Cotter. Learning landmark geodesics using the ensemble Kalman filter. Foundations of Data Science, 2021, 3 (4) : 701-727. doi: 10.3934/fods.2021020
[2]

Junyoung Jang, Kihoon Jang, Hee-Dae Kwon, Jeehyun Lee. Feedback control of an HBV model based on ensemble kalman filter and differential evolution. Mathematical Biosciences & Engineering, 2018, 15 (3) : 667-691. doi: 10.3934/mbe.2018030
[3]

Alexander Bibov, Heikki Haario, Antti Solonen. Stabilized BFGS approximate Kalman filter. Inverse Problems and Imaging, 2015, 9 (4) : 1003-1024. doi: 10.3934/ipi.2015.9.1003
[4]

Russell Johnson, Carmen Núñez. The Kalman-Bucy filter revisited. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4139-4153. doi: 10.3934/dcds.2014.34.4139
[5]

Jin-Won Kim, Amirhossein Taghvaei, Yongxin Chen, Prashant G. Mehta. Feedback particle filter for collective inference. Foundations of Data Science, 2021, 3 (3) : 543-561. doi: 10.3934/fods.2021018
[6]

Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337
[7]

Andrea Arnold, Daniela Calvetti, Erkki Somersalo. Vectorized and parallel particle filter SMC parameter estimation for stiff ODEs. Conference Publications, 2015, 2015 (special) : 75-84. doi: 10.3934/proc.2015.0075
[8]

Sahani Pathiraja, Wilhelm Stannat. Analysis of the feedback particle filter with diffusion map based approximation of the gain. Foundations of Data Science, 2021, 3 (3) : 615-645. doi: 10.3934/fods.2021023
[9]

Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 727-746. doi: 10.3934/dcdss.2021097
[10]

Qifeng Cheng, Xue Han, Tingting Zhao, V S Sarma Yadavalli. Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter. Journal of Industrial and Management Optimization, 2019, 15 (1) : 177-198. doi: 10.3934/jimo.2018038
[11]

Jiangqi Wu, Linjie Wen, Jinglai Li. Resampled ensemble Kalman inversion for Bayesian parameter estimation with sequential data. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 837-850. doi: 10.3934/dcdss.2021045
[12]

Mojtaba F. Fathi, Ahmadreza Baghaie, Ali Bakhshinejad, Raphael H. Sacho, Roshan M. D'Souza. Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother. Journal of Computational Dynamics, 2020, 7 (2) : 469-487. doi: 10.3934/jcd.2020019
[13]

Wawan Hafid Syaifudin, Endah R. M. Putri. The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3433-3443. doi: 10.3934/jimo.2021119
[14]

Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri, Diego P. Ruiz. A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system. Mathematical Biosciences & Engineering, 2014, 11 (3) : 573-597. doi: 10.3934/mbe.2014.11.573
[15]

Geir Evensen, Javier Amezcua, Marc Bocquet, Alberto Carrassi, Alban Farchi, Alison Fowler, Pieter L. Houtekamer, Christopher K. Jones, Rafael J. de Moraes, Manuel Pulido, Christian Sampson, Femke C. Vossepoel. An international initiative of predicting the SARS-CoV-2 pandemic using ensemble data assimilation. Foundations of Data Science, 2021, 3 (3) : 413-477. doi: 10.3934/fods.2021001
[16]

Neil K. Chada, Claudia Schillings, Simon Weissmann. On the incorporation of box-constraints for ensemble Kalman inversion. Foundations of Data Science, 2019, 1 (4) : 433-456. doi: 10.3934/fods.2019018
[17]

Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems and Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036
[18]

Hai Huyen Dam, Kok Lay Teo. Variable fractional delay filter design with discrete coefficients. Journal of Industrial and Management Optimization, 2016, 12 (3) : 819-831. doi: 10.3934/jimo.2016.12.819
[19]

Xueling Zhou, Bingo Wing-Kuen Ling, Hai Huyen Dam, Kok-Lay Teo. Optimal design of window functions for filter window bank. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1119-1145. doi: 10.3934/jimo.2020014
[20]

Anugu Sumith Reddy, Amit Apte. Stability of non-linear filter for deterministic dynamics. Foundations of Data Science, 2021, 3 (3) : 647-675. doi: 10.3934/fods.2021025

 Impact Factor: 

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy