August  2016, 36(8): 4227-4246. doi: 10.3934/dcds.2016.36.4227

Sampling, feasibility, and priors in data assimilation

1. 

Department of Mathematics, University of California, Berkeley, and Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States

2. 

College of Earth, Ocean and Atmospheric Science, Oregon State University, Corvallis, OR 97331, United States

3. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721, United States

4. 

Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States

Received  May 2015 Revised  August 2015 Published  March 2016

Importance sampling algorithms are discussed in detail, with an emphasis on implicit sampling, and applied to data assimilation via particle filters. Implicit sampling makes it possible to use the data to find high-probability samples at relatively low cost, making the assimilation more efficient. A new analysis of the feasibility of data assimilation is presented, showing in detail why feasibility depends on the Frobenius norm of the covariance matrix of the noise and not on the number of variables. A discussion of the convergence of particular particle filters follows. A major open problem in numerical data assimilation is the determination of appropriate priors; a progress report on recent work on this problem is given. The analysis highlights the need for a careful attention both to the data and to the physics in data assimilation problems.
Citation: Alexandre J. Chorin, Fei Lu, Robert N. Miller, Matthias Morzfeld, Xuemin Tu. Sampling, feasibility, and priors in data assimilation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4227-4246. doi: 10.3934/dcds.2016.36.4227
References:
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M. Ades and P. van Leeuwen, An exploration of the equivalent weights particle filter,, Quarterly Journal of the Royal Meteorological Society, 139 (2013), 820.  doi: 10.1002/qj.1995.  Google Scholar

[2]

M. Arulampalam, S. Maskell, N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,, IEEE Transaction on Signal Processing, 50 (2002), 174.  doi: 10.1109/78.978374.  Google Scholar

[3]

E. Atkins, M. Morzfeld and A. J. Chorin, Implicit particle methods and their connection with variational data assimilation,, Monthly Weather Review, 141 (2013), 1786.  doi: 10.1175/MWR-D-12-00145.1.  Google Scholar

[4]

T. Bengtsson, P. Bickel and B. Li, Curse of dimensionality revisited: The collapse of importance sampling in very large scale systems,, IMS Collections: Probability and Statistics: Essays in Honor of David A. Freedman, 2 (2008), 316.  doi: 10.1214/193940307000000518.  Google Scholar

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T. Berry and J. Harlim, Linear theory for filtering nonlinear multiscale systems with model error,, Proc. R. Soc. A, 470 (2014).  doi: 10.1098/rspa.2014.0168.  Google Scholar

[6]

P. Bickel, T. Bengtsson and J. Anderson, Sharp failure rates for the bootstrap particle filter in high dimensions,, Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 3 (2008), 318.  doi: 10.1214/074921708000000228.  Google Scholar

[7]

M. Bocquet, C. Pires and L. Wu, Beyond Gaussian statistical modeling in geophysical data assimilation,, Monthly Weather Review, 138 (2010), 2997.  doi: 10.1175/2010MWR3164.1.  Google Scholar

[8]

N. H. Chan, J. P. Kadane, R. N. Miller and W. Palma, Estimation of tropical sea level anomaly by an improved Kalman filter,, Journal of Physical Oceanography, 26 (1996), 1286.  doi: 10.1175/1520-0485(1996)026<1286:EOTSLA>2.0.CO;2.  Google Scholar

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A. J. Chorin and O. H. Hald, Stochastic Tools in Mathematics and Science,, 3rd edition, (2013).  doi: 10.1007/978-1-4614-6980-3.  Google Scholar

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A. J. Chorin and O. H. Hald, Estimating the uncertainty in underresolved nonlinear dynamics,, Mathematics and Mechanics of Solids, 19 (2014), 28.  doi: 10.1177/1081286513505465.  Google Scholar

[11]

A. J. Chorin and F. Lu, Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics,, Proceedings of the National Academy of Sciences, 112 (2015), 9804.  doi: 10.1073/pnas.1512080112.  Google Scholar

[12]

A. J. Chorin and M. Morzfeld, Conditions for successful data assimilation,, Journal of Geophysical Research - Atmospheres, 118 (2013), 522.  doi: 10.1002/2013JD019838.  Google Scholar

[13]

A. J. Chorin, M. Morzfeld and X. Tu, Implicit particle filters for data assimilation,, Communications in Applied Mathematics and Computational Science, 5 (2010), 221.  doi: 10.2140/camcos.2010.5.221.  Google Scholar

[14]

A. J. Chorin and X. Tu, Implicit sampling for particle filters,, Proceedings of the National Academy of Sciences, 106 (2009), 17249.  doi: 10.1073/pnas.0909196106.  Google Scholar

[15]

D. Crommelin and E. Vanden-Eijnden, Subgrid-scale parameterization with conditional Markov chains,, Journal of the Atmospheric Sciences, 65 (2008), 2661.  doi: 10.1175/2008JAS2566.1.  Google Scholar

[16]

A. Doucet, N. de Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice,, Springer, (2001).  doi: 10.1007/978-1-4757-3437-9.  Google Scholar

[17]

A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering,, Statistics and Computing, 10 (2000), 197.   Google Scholar

[18]

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

[19]

I. Fatkulin and E. Vanden-Eijnden, A computational strategy for multi-scale systems with applications to Lorenz'96 model,, Journal of Computational Physics, 200 (2004), 605.  doi: 10.1016/j.jcp.2004.04.013.  Google Scholar

[20]

G. E. Forsythe, M. A. Malcom and C. Mohler, Computer Methods for Mathematical Computations,, Prentice-Hall, (1977).   Google Scholar

[21]

A. Fournier, G. Hulot, D. Jault, W. Kuang, W. Tangborn, N. Gillet, E. Canet, J. Aubert and F. Lhuillier, An introduction to data assimilation and predictability in geomagnetism,, Space Science Review, 155 (2010), 247.   Google Scholar

[22]

J. Goodman, K. Lin and M. Morzfeld, Small-noise analysis and symmetrization of implicit Monte Carlo samplers,, Communications on Pure and Applied Mathematics, (2015).  doi: 10.1002/cpa.21592.  Google Scholar

[23]

J. Goodman and J. Weare, Ensemble samplers with affine invariance,, Communications in Applied Mathematics and Computational Sciences, 5 (2010), 65.  doi: 10.2140/camcos.2010.5.65.  Google Scholar

[24]

J. Harlim, Data assimilation with model error from unresolved scales,, , ().   Google Scholar

[25]

R. Kalman, A new approach to linear filtering and prediction theory,, Transactions of the ASME-Journal of Basic Engineering, 82 (1960), 35.   Google Scholar

[26]

M. Kalos and P. Whitlock, Monte Carlo Methods,, Second edition. Wiley-Blackwell, (2008).  doi: 10.1002/9783527626212.  Google Scholar

[27]

P. Lancaster and L. Rodman, Algebraic Riccati Equations,, Oxford University Press, (1995).   Google Scholar

[28]

J. Liu and R. Chen, Blind deconvolution via sequential imputations,, Journal of the American Statistical Association, 90 (1995), 567.  doi: 10.1080/01621459.1995.10476549.  Google Scholar

[29]

E. N. Lorenz, Deterministic non-periodic flow,, Journal of Atmospheric Science, 20 (1963), 130.   Google Scholar

[30]

E. Lorenz, Predictability: A problem partly solved,, Proc. Seminar on Predictability, (2009), 40.  doi: 10.1017/CBO9780511617652.004.  Google Scholar

[31]

A. Majda and J. Harlim, Physics constrained nonlinear regression models for time series,, Nonlinearity, 26 (2013), 201.  doi: 10.1088/0951-7715/26/1/201.  Google Scholar

[32]

R. N. Miller and L. Ehret, Application of the implicit particle filter to a model of nearshore circulation,, Journal of Geophysical Research, 119 (2014), 2363.  doi: 10.1002/2013JC009440.  Google Scholar

[33]

R. N. Miller, M. Ghil and F. Gauthiez, Advanced data assimilation in strongly nonlinear systems,, Journal of Atmospheric Science, 51 (1994), 1037.  doi: 10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2.  Google Scholar

[34]

M. Morzfeld and H. Godinez, Estimation and prediction for an orbital propagation model using data assimilation,, Advances in the Astronautical Sciences Spaceflight Mechanics, 152 (2015), 2003.   Google Scholar

[35]

M. Morzfeld and D. Hodyss, Analysis of the ensemble Kalman filter for marginal and joint posteriors,, submitted for publication., ().   Google Scholar

[36]

M. Morzfeld, X. Tu, E. Atkins and A. J. Chorin, A random map implementation of implicit filters,, Journal of Computational Physics, 231 (2012), 2049.  doi: 10.1016/j.jcp.2011.11.022.  Google Scholar

[37]

M. Morzfeld, X. Tu, J. Wilkening and A. J. Chorin, Parameter estimation by implicit sampling,, Communications in Applied Mathematics and Computational Science, 10 (2015), 205.  doi: 10.2140/camcos.2015.10.205.  Google Scholar

[38]

T. Moselhy and Y. Marzouk, Bayesian inference with optimal maps,, Journal of Computational Physics, 231 (2012), 7815.  doi: 10.1016/j.jcp.2012.07.022.  Google Scholar

[39]

C. Nicolis, Probabilistic aspects of error growth in atmospheric dynamics,, Quarterly Jourmal of the Royal Meteorological Society, 118 (1992), 553.   Google Scholar

[40]

A. B. Owen, Monte Carlo Theory, Methods and Examples, 2013., URL , ().   Google Scholar

[41]

Y. Pokern, A. Stuart and P. Wiberg, Parameter estimation for partially observed hypoelliptic diffusions,, Journal of the Royal Statistical Society: Series B, 71 (2009), 49.  doi: 10.1111/j.1467-9868.2008.00689.x.  Google Scholar

[42]

C. Rasmussen and C. Williams, Gaussian Processes for Machine Learning,, MIT Press, (2006).   Google Scholar

[43]

N. Recca, A New Methodology for Importance Sampling,, Masters Thesis, (2011).   Google Scholar

[44]

A. Samson and M. Thieullen, A contrast estimator for completely or partially observed hypoelliptic diffusion,, Stochastic Processes and their Applications, 122 (2012), 2521.  doi: 10.1016/j.spa.2012.04.006.  Google Scholar

[45]

C. Snyder, Particle filters, the "optimal'' proposal and high-dimensional systems,, Proceedings of the ECMWF Seminar on Data Assimilation for Atmosphere and Ocean, (2011).   Google Scholar

[46]

C. Snyder, T. Bengtsson, P. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering,, Monthly Weather Review, 136 (2008), 4629.   Google Scholar

[47]

C. Snyder, T. Bengtsson and M. Morzfeld, Performance bounds for particle filters using the optimal proposal,, Monthly Weather Review, 143 (2015), 4750.   Google Scholar

[48]

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

[49]

M. Tippet, J. Anderson, C. Bishop, T. Hamil and J. Whitaker, Ensemble square root filters,, Monthly Weather Review, 131 (2003), 1485.   Google Scholar

[50]

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

[51]

P. van Leeuwen, Nonlinear data assimilation in geosciences: an extremely efficient particle filter,, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1991.   Google Scholar

[52]

E. Vanden-Eijnden and J. Weare, Rare event simulation and small noise diffusions,, Communications on Pure and Applied Mathematics, 65 (2012), 1770.  doi: 10.1002/cpa.21428.  Google Scholar

[53]

E. Vanden-Eijnden and J. Weare, Data assimilation in the low noise, accurate observation regime with application to the Kuroshio current,, Monthly Weather Review, 141 (2013), 1822.   Google Scholar

[54]

B. Weir, R. N. Miller and Y. Spitz, Implicit estimation of ecological model parameters,, Bulletin of Mathematical Biology, 75 (2013), 223.  doi: 10.1007/s11538-012-9801-6.  Google Scholar

[55]

D. Wilks, Effects of stochastic parameterizations in the Lorenz '96 model,, Quarterly Journal of the Royal Meteorological Society, 131 (2005), 389.   Google Scholar

[56]

V. Zaritskii and L. Shimelevich, Monte Carlo technique in problems of optimal data processing,, Automation and Remote Control, 36 (1975), 2015.   Google Scholar

show all references

References:
[1]

M. Ades and P. van Leeuwen, An exploration of the equivalent weights particle filter,, Quarterly Journal of the Royal Meteorological Society, 139 (2013), 820.  doi: 10.1002/qj.1995.  Google Scholar

[2]

M. Arulampalam, S. Maskell, N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,, IEEE Transaction on Signal Processing, 50 (2002), 174.  doi: 10.1109/78.978374.  Google Scholar

[3]

E. Atkins, M. Morzfeld and A. J. Chorin, Implicit particle methods and their connection with variational data assimilation,, Monthly Weather Review, 141 (2013), 1786.  doi: 10.1175/MWR-D-12-00145.1.  Google Scholar

[4]

T. Bengtsson, P. Bickel and B. Li, Curse of dimensionality revisited: The collapse of importance sampling in very large scale systems,, IMS Collections: Probability and Statistics: Essays in Honor of David A. Freedman, 2 (2008), 316.  doi: 10.1214/193940307000000518.  Google Scholar

[5]

T. Berry and J. Harlim, Linear theory for filtering nonlinear multiscale systems with model error,, Proc. R. Soc. A, 470 (2014).  doi: 10.1098/rspa.2014.0168.  Google Scholar

[6]

P. Bickel, T. Bengtsson and J. Anderson, Sharp failure rates for the bootstrap particle filter in high dimensions,, Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 3 (2008), 318.  doi: 10.1214/074921708000000228.  Google Scholar

[7]

M. Bocquet, C. Pires and L. Wu, Beyond Gaussian statistical modeling in geophysical data assimilation,, Monthly Weather Review, 138 (2010), 2997.  doi: 10.1175/2010MWR3164.1.  Google Scholar

[8]

N. H. Chan, J. P. Kadane, R. N. Miller and W. Palma, Estimation of tropical sea level anomaly by an improved Kalman filter,, Journal of Physical Oceanography, 26 (1996), 1286.  doi: 10.1175/1520-0485(1996)026<1286:EOTSLA>2.0.CO;2.  Google Scholar

[9]

A. J. Chorin and O. H. Hald, Stochastic Tools in Mathematics and Science,, 3rd edition, (2013).  doi: 10.1007/978-1-4614-6980-3.  Google Scholar

[10]

A. J. Chorin and O. H. Hald, Estimating the uncertainty in underresolved nonlinear dynamics,, Mathematics and Mechanics of Solids, 19 (2014), 28.  doi: 10.1177/1081286513505465.  Google Scholar

[11]

A. J. Chorin and F. Lu, Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics,, Proceedings of the National Academy of Sciences, 112 (2015), 9804.  doi: 10.1073/pnas.1512080112.  Google Scholar

[12]

A. J. Chorin and M. Morzfeld, Conditions for successful data assimilation,, Journal of Geophysical Research - Atmospheres, 118 (2013), 522.  doi: 10.1002/2013JD019838.  Google Scholar

[13]

A. J. Chorin, M. Morzfeld and X. Tu, Implicit particle filters for data assimilation,, Communications in Applied Mathematics and Computational Science, 5 (2010), 221.  doi: 10.2140/camcos.2010.5.221.  Google Scholar

[14]

A. J. Chorin and X. Tu, Implicit sampling for particle filters,, Proceedings of the National Academy of Sciences, 106 (2009), 17249.  doi: 10.1073/pnas.0909196106.  Google Scholar

[15]

D. Crommelin and E. Vanden-Eijnden, Subgrid-scale parameterization with conditional Markov chains,, Journal of the Atmospheric Sciences, 65 (2008), 2661.  doi: 10.1175/2008JAS2566.1.  Google Scholar

[16]

A. Doucet, N. de Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice,, Springer, (2001).  doi: 10.1007/978-1-4757-3437-9.  Google Scholar

[17]

A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering,, Statistics and Computing, 10 (2000), 197.   Google Scholar

[18]

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

[19]

I. Fatkulin and E. Vanden-Eijnden, A computational strategy for multi-scale systems with applications to Lorenz'96 model,, Journal of Computational Physics, 200 (2004), 605.  doi: 10.1016/j.jcp.2004.04.013.  Google Scholar

[20]

G. E. Forsythe, M. A. Malcom and C. Mohler, Computer Methods for Mathematical Computations,, Prentice-Hall, (1977).   Google Scholar

[21]

A. Fournier, G. Hulot, D. Jault, W. Kuang, W. Tangborn, N. Gillet, E. Canet, J. Aubert and F. Lhuillier, An introduction to data assimilation and predictability in geomagnetism,, Space Science Review, 155 (2010), 247.   Google Scholar

[22]

J. Goodman, K. Lin and M. Morzfeld, Small-noise analysis and symmetrization of implicit Monte Carlo samplers,, Communications on Pure and Applied Mathematics, (2015).  doi: 10.1002/cpa.21592.  Google Scholar

[23]

J. Goodman and J. Weare, Ensemble samplers with affine invariance,, Communications in Applied Mathematics and Computational Sciences, 5 (2010), 65.  doi: 10.2140/camcos.2010.5.65.  Google Scholar

[24]

J. Harlim, Data assimilation with model error from unresolved scales,, , ().   Google Scholar

[25]

R. Kalman, A new approach to linear filtering and prediction theory,, Transactions of the ASME-Journal of Basic Engineering, 82 (1960), 35.   Google Scholar

[26]

M. Kalos and P. Whitlock, Monte Carlo Methods,, Second edition. Wiley-Blackwell, (2008).  doi: 10.1002/9783527626212.  Google Scholar

[27]

P. Lancaster and L. Rodman, Algebraic Riccati Equations,, Oxford University Press, (1995).   Google Scholar

[28]

J. Liu and R. Chen, Blind deconvolution via sequential imputations,, Journal of the American Statistical Association, 90 (1995), 567.  doi: 10.1080/01621459.1995.10476549.  Google Scholar

[29]

E. N. Lorenz, Deterministic non-periodic flow,, Journal of Atmospheric Science, 20 (1963), 130.   Google Scholar

[30]

E. Lorenz, Predictability: A problem partly solved,, Proc. Seminar on Predictability, (2009), 40.  doi: 10.1017/CBO9780511617652.004.  Google Scholar

[31]

A. Majda and J. Harlim, Physics constrained nonlinear regression models for time series,, Nonlinearity, 26 (2013), 201.  doi: 10.1088/0951-7715/26/1/201.  Google Scholar

[32]

R. N. Miller and L. Ehret, Application of the implicit particle filter to a model of nearshore circulation,, Journal of Geophysical Research, 119 (2014), 2363.  doi: 10.1002/2013JC009440.  Google Scholar

[33]

R. N. Miller, M. Ghil and F. Gauthiez, Advanced data assimilation in strongly nonlinear systems,, Journal of Atmospheric Science, 51 (1994), 1037.  doi: 10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2.  Google Scholar

[34]

M. Morzfeld and H. Godinez, Estimation and prediction for an orbital propagation model using data assimilation,, Advances in the Astronautical Sciences Spaceflight Mechanics, 152 (2015), 2003.   Google Scholar

[35]

M. Morzfeld and D. Hodyss, Analysis of the ensemble Kalman filter for marginal and joint posteriors,, submitted for publication., ().   Google Scholar

[36]

M. Morzfeld, X. Tu, E. Atkins and A. J. Chorin, A random map implementation of implicit filters,, Journal of Computational Physics, 231 (2012), 2049.  doi: 10.1016/j.jcp.2011.11.022.  Google Scholar

[37]

M. Morzfeld, X. Tu, J. Wilkening and A. J. Chorin, Parameter estimation by implicit sampling,, Communications in Applied Mathematics and Computational Science, 10 (2015), 205.  doi: 10.2140/camcos.2015.10.205.  Google Scholar

[38]

T. Moselhy and Y. Marzouk, Bayesian inference with optimal maps,, Journal of Computational Physics, 231 (2012), 7815.  doi: 10.1016/j.jcp.2012.07.022.  Google Scholar

[39]

C. Nicolis, Probabilistic aspects of error growth in atmospheric dynamics,, Quarterly Jourmal of the Royal Meteorological Society, 118 (1992), 553.   Google Scholar

[40]

A. B. Owen, Monte Carlo Theory, Methods and Examples, 2013., URL , ().   Google Scholar

[41]

Y. Pokern, A. Stuart and P. Wiberg, Parameter estimation for partially observed hypoelliptic diffusions,, Journal of the Royal Statistical Society: Series B, 71 (2009), 49.  doi: 10.1111/j.1467-9868.2008.00689.x.  Google Scholar

[42]

C. Rasmussen and C. Williams, Gaussian Processes for Machine Learning,, MIT Press, (2006).   Google Scholar

[43]

N. Recca, A New Methodology for Importance Sampling,, Masters Thesis, (2011).   Google Scholar

[44]

A. Samson and M. Thieullen, A contrast estimator for completely or partially observed hypoelliptic diffusion,, Stochastic Processes and their Applications, 122 (2012), 2521.  doi: 10.1016/j.spa.2012.04.006.  Google Scholar

[45]

C. Snyder, Particle filters, the "optimal'' proposal and high-dimensional systems,, Proceedings of the ECMWF Seminar on Data Assimilation for Atmosphere and Ocean, (2011).   Google Scholar

[46]

C. Snyder, T. Bengtsson, P. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering,, Monthly Weather Review, 136 (2008), 4629.   Google Scholar

[47]

C. Snyder, T. Bengtsson and M. Morzfeld, Performance bounds for particle filters using the optimal proposal,, Monthly Weather Review, 143 (2015), 4750.   Google Scholar

[48]

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

[49]

M. Tippet, J. Anderson, C. Bishop, T. Hamil and J. Whitaker, Ensemble square root filters,, Monthly Weather Review, 131 (2003), 1485.   Google Scholar

[50]

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

[51]

P. van Leeuwen, Nonlinear data assimilation in geosciences: an extremely efficient particle filter,, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1991.   Google Scholar

[52]

E. Vanden-Eijnden and J. Weare, Rare event simulation and small noise diffusions,, Communications on Pure and Applied Mathematics, 65 (2012), 1770.  doi: 10.1002/cpa.21428.  Google Scholar

[53]

E. Vanden-Eijnden and J. Weare, Data assimilation in the low noise, accurate observation regime with application to the Kuroshio current,, Monthly Weather Review, 141 (2013), 1822.   Google Scholar

[54]

B. Weir, R. N. Miller and Y. Spitz, Implicit estimation of ecological model parameters,, Bulletin of Mathematical Biology, 75 (2013), 223.  doi: 10.1007/s11538-012-9801-6.  Google Scholar

[55]

D. Wilks, Effects of stochastic parameterizations in the Lorenz '96 model,, Quarterly Journal of the Royal Meteorological Society, 131 (2005), 389.   Google Scholar

[56]

V. Zaritskii and L. Shimelevich, Monte Carlo technique in problems of optimal data processing,, Automation and Remote Control, 36 (1975), 2015.   Google Scholar

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