March  2014, 34(3): 1061-1078. doi: 10.3934/dcds.2014.34.1061

Analysis of the 3DVAR filter for the partially observed Lorenz'63 model

1. 

Warwick Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom, United Kingdom, United Kingdom

Received  December 2012 Revised  April 2013 Published  August 2013

The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.
Citation: Kody Law, Abhishek Shukla, Andrew Stuart. Analysis of the 3DVAR filter for the partially observed Lorenz'63 model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1061-1078. doi: 10.3934/dcds.2014.34.1061
References:
[1]

A. Apte, C. K. R. T. Jones, A. M. Stuart and J. Voss, Data assimilation: Mathematical and statistical perspectives,, International Journal for Numerical Methods in Fluids, 56 (2008), 1033. doi: 10.1002/fld.1698. Google Scholar

[2]

A. Bennett, "Inverse Modeling of the ocean and Atmosphere,", Cambridge, (2002). doi: 10.1017/CBO9780511535895. Google Scholar

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

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D. Blömker, K. Law, A. M. Stuart and K. C. Zygalakis, Accuracy and stability of the continuous-time 3dvar filter for the navier-stokes equation,, Preprint, (2012). Google Scholar

[5]

C. E. A. Brett, K. F. Lam, K. J. H. Law, D. S. McCormick, M. R. Scott and A. M. Stuart, Accuracy and stability of filters for dissipative pdes,, Physica D: Nonlinear Phenomena, 245 (2013), 34. doi: 10.1016/j.physd.2012.11.005. Google Scholar

[6]

A. Doucet, N. De Freitas and N. Gordon, "Sequential Monte Carlo Methods in Practice,", Springer Verlag, (2001). Google Scholar

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G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation,, Ocean Dynamics, 53 (2003), 343. doi: 10.1007/s10236-003-0036-9. Google Scholar

[8]

G. Evensen, "Data Assimilation: The Ensemble Kalman Filter,", Springer Verlag, (2009). doi: 10.1007/978-3-642-03711-5. Google Scholar

[9]

C. Foias, M. S. Jolly, I. Kukavica and E. S. Titi, The lorenz equationas a metaphor for the navier-stokes equation, discrete and continous dynamical systems,, Discrete and Continous Dynamical Systems, 7 (2001), 403. doi: 10.3934/dcds.2001.7.403. Google Scholar

[10]

C. Foias and G. Prodi, Sur le comportement global des solutions nonstationnaires des équations de navier-stokes en dimension 2,, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1. Google Scholar

[11]

A. C. Harvey, "Forecasting, Structural Time Series Models and the Kalman Filter,", Cambridge Univ Pr, (1991). Google Scholar

[12]

K. Hayden, E. Olson and E. S. Titi, Discrete data assimilation in the lorenz and 2d navier-stokes equations,, Physica D: Nonlinear Phenomena, 240 (2011), 1416. doi: 10.1016/j.physd.2011.04.021. Google Scholar

[13]

A. H. Jazwinski, "Stochastic Processes and Filtering Theory,", Academic Pr, (1970). Google Scholar

[14]

D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the navier-stokes equations,, Indiana University Mathematics Journal, 42 (1993), 875. doi: 10.1512/iumj.1993.42.42039. Google Scholar

[15]

E. Kalnay, "Atmospheric Modeling, Data Assimilation, and Predictability,", Cambridge Univ. Pr., (2002). doi: 10.1017/CBO9780511802270. Google Scholar

[16]

E. Kalnay, H. Li, T. Miyoshi, S.-C. Yang and J. Ballabrera-Poy, 4DVAR or ensemble Kalman filter?,, Tellus A, 59 (2008), 758. Google Scholar

[17]

K. J. H. Law and A. M. Stuart, Evaluating data assimilation algorithms,, Monthly Weather Review, 140 (2012), 3757. doi: 10.1175/MWR-D-11-00257.1. Google Scholar

[18]

A. C. Lorenc, Analysis methods for numerical weather prediction,, Quart. J. R. Met. Soc., 112 (2000), 1177. doi: 10.1002/qj.49711247414. Google Scholar

[19]

E. N. Lorenz, Deterministic nonperiodic flow,, Atmos. J. Sci., 20 (1963), 130. Google Scholar

[20]

E. N. Lorenz, Predictability: A problem partly solved,, in, 1 (1996), 1. doi: 10.1017/CBO9780511617652.004. Google Scholar

[21]

A. J. Majda and J. Harlim, "Filtering Complex Turbulent Systems,", Cambridge University Press, (2012). doi: 10.1017/CBO9781139061308. Google Scholar

[22]

A. J. Majda, J. Harlim and B. Gershgorin, Mathematical strategies for filtering turbulent dynamical systems,, Discrete and Continuous Dynamical Systems - Series A, 27 (2010), 441. doi: 10.3934/dcds.2010.27.441. Google Scholar

[23]

X. Mao, "Stochastic Differential Equations And Applications,", Horwood, (1997). doi: 10.1533/9780857099402. Google Scholar

[24]

D. S. Oliver, A. C. Reynolds and N. Liu, "Inverse Theory for Petroleum Reservoir Characterization and History Matching,", Cambridge University Press, (2008). doi: 10.1017/CBO9780511535642. Google Scholar

[25]

E. Olson and E. S. Titi, Determining modes for continuous data assimilation in 2D turbulence,, Journal of Statistical Physics, 113 (2003), 799. doi: 10.1023/A:1027312703252. Google Scholar

[26]

D. F. Parrish and J. C. Derber, The National Meteorological Center's spectral statistical-interpolation analysis system,, Monthly Weather Review, 120 (1992), 1747. doi: 10.1175/1520-0493(1992)120<1747:TNMCSS>2.0.CO;2. Google Scholar

[27]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Cambridge Texts in Applied Mathematics. Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[28]

C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors,", Springer, (1982). Google Scholar

[29]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numer., 19 (2010), 451. doi: 10.1017/S0962492910000061. Google Scholar

[30]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 68 of Applied Mathematical Sciences. Springer-Verlag, 68 (1997). Google Scholar

[31]

W. Tucker, A rigorous ode solver and smale's 14th problem,, Journal of Foundations of Computational Mathematics, 2 (2002), 53. Google Scholar

[32]

P. J. Van Leeuwen, Particle filtering in geophysical systems,, Monthly Weather Review, 137 (2009), 4089. Google Scholar

show all references

References:
[1]

A. Apte, C. K. R. T. Jones, A. M. Stuart and J. Voss, Data assimilation: Mathematical and statistical perspectives,, International Journal for Numerical Methods in Fluids, 56 (2008), 1033. doi: 10.1002/fld.1698. Google Scholar

[2]

A. Bennett, "Inverse Modeling of the ocean and Atmosphere,", Cambridge, (2002). doi: 10.1017/CBO9780511535895. Google Scholar

[3]

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

[4]

D. Blömker, K. Law, A. M. Stuart and K. C. Zygalakis, Accuracy and stability of the continuous-time 3dvar filter for the navier-stokes equation,, Preprint, (2012). Google Scholar

[5]

C. E. A. Brett, K. F. Lam, K. J. H. Law, D. S. McCormick, M. R. Scott and A. M. Stuart, Accuracy and stability of filters for dissipative pdes,, Physica D: Nonlinear Phenomena, 245 (2013), 34. doi: 10.1016/j.physd.2012.11.005. Google Scholar

[6]

A. Doucet, N. De Freitas and N. Gordon, "Sequential Monte Carlo Methods in Practice,", Springer Verlag, (2001). Google Scholar

[7]

G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation,, Ocean Dynamics, 53 (2003), 343. doi: 10.1007/s10236-003-0036-9. Google Scholar

[8]

G. Evensen, "Data Assimilation: The Ensemble Kalman Filter,", Springer Verlag, (2009). doi: 10.1007/978-3-642-03711-5. Google Scholar

[9]

C. Foias, M. S. Jolly, I. Kukavica and E. S. Titi, The lorenz equationas a metaphor for the navier-stokes equation, discrete and continous dynamical systems,, Discrete and Continous Dynamical Systems, 7 (2001), 403. doi: 10.3934/dcds.2001.7.403. Google Scholar

[10]

C. Foias and G. Prodi, Sur le comportement global des solutions nonstationnaires des équations de navier-stokes en dimension 2,, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1. Google Scholar

[11]

A. C. Harvey, "Forecasting, Structural Time Series Models and the Kalman Filter,", Cambridge Univ Pr, (1991). Google Scholar

[12]

K. Hayden, E. Olson and E. S. Titi, Discrete data assimilation in the lorenz and 2d navier-stokes equations,, Physica D: Nonlinear Phenomena, 240 (2011), 1416. doi: 10.1016/j.physd.2011.04.021. Google Scholar

[13]

A. H. Jazwinski, "Stochastic Processes and Filtering Theory,", Academic Pr, (1970). Google Scholar

[14]

D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the navier-stokes equations,, Indiana University Mathematics Journal, 42 (1993), 875. doi: 10.1512/iumj.1993.42.42039. Google Scholar

[15]

E. Kalnay, "Atmospheric Modeling, Data Assimilation, and Predictability,", Cambridge Univ. Pr., (2002). doi: 10.1017/CBO9780511802270. Google Scholar

[16]

E. Kalnay, H. Li, T. Miyoshi, S.-C. Yang and J. Ballabrera-Poy, 4DVAR or ensemble Kalman filter?,, Tellus A, 59 (2008), 758. Google Scholar

[17]

K. J. H. Law and A. M. Stuart, Evaluating data assimilation algorithms,, Monthly Weather Review, 140 (2012), 3757. doi: 10.1175/MWR-D-11-00257.1. Google Scholar

[18]

A. C. Lorenc, Analysis methods for numerical weather prediction,, Quart. J. R. Met. Soc., 112 (2000), 1177. doi: 10.1002/qj.49711247414. Google Scholar

[19]

E. N. Lorenz, Deterministic nonperiodic flow,, Atmos. J. Sci., 20 (1963), 130. Google Scholar

[20]

E. N. Lorenz, Predictability: A problem partly solved,, in, 1 (1996), 1. doi: 10.1017/CBO9780511617652.004. Google Scholar

[21]

A. J. Majda and J. Harlim, "Filtering Complex Turbulent Systems,", Cambridge University Press, (2012). doi: 10.1017/CBO9781139061308. Google Scholar

[22]

A. J. Majda, J. Harlim and B. Gershgorin, Mathematical strategies for filtering turbulent dynamical systems,, Discrete and Continuous Dynamical Systems - Series A, 27 (2010), 441. doi: 10.3934/dcds.2010.27.441. Google Scholar

[23]

X. Mao, "Stochastic Differential Equations And Applications,", Horwood, (1997). doi: 10.1533/9780857099402. Google Scholar

[24]

D. S. Oliver, A. C. Reynolds and N. Liu, "Inverse Theory for Petroleum Reservoir Characterization and History Matching,", Cambridge University Press, (2008). doi: 10.1017/CBO9780511535642. Google Scholar

[25]

E. Olson and E. S. Titi, Determining modes for continuous data assimilation in 2D turbulence,, Journal of Statistical Physics, 113 (2003), 799. doi: 10.1023/A:1027312703252. Google Scholar

[26]

D. F. Parrish and J. C. Derber, The National Meteorological Center's spectral statistical-interpolation analysis system,, Monthly Weather Review, 120 (1992), 1747. doi: 10.1175/1520-0493(1992)120<1747:TNMCSS>2.0.CO;2. Google Scholar

[27]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Cambridge Texts in Applied Mathematics. Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[28]

C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors,", Springer, (1982). Google Scholar

[29]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numer., 19 (2010), 451. doi: 10.1017/S0962492910000061. Google Scholar

[30]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 68 of Applied Mathematical Sciences. Springer-Verlag, 68 (1997). Google Scholar

[31]

W. Tucker, A rigorous ode solver and smale's 14th problem,, Journal of Foundations of Computational Mathematics, 2 (2002), 53. Google Scholar

[32]

P. J. Van Leeuwen, Particle filtering in geophysical systems,, Monthly Weather Review, 137 (2009), 4089. Google Scholar

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