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.

[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.

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