American Institute of Mathematical Sciences

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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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

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