A Tutorial on Bayesian Data Assimilation

doi:10.1017/9781009180412.004

3 - A Tutorial on Bayesian Data Assimilation

from Part I - Introduction

Published online by Cambridge University Press: 20 June 2023

Colin Grudzien and

Edited by

Dylan Jones and

Alik Ismail-Zadeh: Affiliation:
Karlsruhe Institute of Technology, Germany
Fabio Castelli: Affiliation:
Università degli Studi, Florence
Dylan Jones: Affiliation:
University of Toronto
Sabrina Sanchez: Affiliation:
Max Planck Institute for Solar System Research, Germany

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Summary

Abstract: This chapter provides a broad introduction to Bayesian data assimilation that will be useful to practitioners in interpreting algorithms and results, and for theoretical studies developing novel schemes with an understanding of the rich history of geophysical data assimilation and its current directions. The simple case of data assimilation in a ‘perfect’ model is primarily discussed for pedagogical purposes. Some mathematical results are derived at a high level in order to illustrate key ideas about different estimators. However, the focus of this chapter is on the intuition behind these methods, where more formal and detailed treatments of the data assimilation problem can be found in the various references. In surveying a variety of widely used data assimilation schemes, the key message of this chapter is how the Bayesian analysis provides a consistent framework for the estimation problem and how this allows one to formulate its solution in a variety of ways to exploit the operational challenges in the geosciences.

Keywords

3D-VAR 4D-VAR Adjoint Bayesian Ensemble-variational (EnVAR)Ensemble Kalman filter (EnKF)Gaussian Machine Learning (ML)Markov Maximum a posteriori (MAP)

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Type: Chapter
Information: Applications of Data Assimilation and Inverse Problems in the Earth Sciences , pp. 27 - 48

DOI: https://doi.org/10.1017/9781009180412.004 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2023

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References

Abarbanel, H. D. I., Rozdeba, P. J., and Shirman, S. (2018). Machine learning: Deepest learning as statistical data assimilation problems. Neural Computation, 30(8), 2025–55. https://doi.org/10.1162/neco_a_01094.CrossRef Google Scholar PubMed

Ait-El-Fquih, B., and Hoteit, I. (2022). Filtering with one-step-ahead smoothing for efficient data assimilation. In Park, S. K. and Xu, L. eds., Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, vol. IV. Cham: Springer, pp. 69–96. https://doi.org/10.1007/978–3–030–77722-7\_1.Google Scholar

Anderson, B. D. O., and Moore, J. B. (1979). Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar

Asch, M., Bocquet, M., and Nodet, M. (2016). Data Assimilation: Methods, Algorithms, and Applications. Philadelphia, PA: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611974546.Google Scholar

Bannister, R. N. (2017). A review of operational methods of variational and ensemble-variational data assimilation. Quarterly Journal of the Royal Meteorological Society, 143, 607–33. https://doi.org/10.1002/qj.2982.Google Scholar

Baydin, A. G., Pearlmutter, B. A., Radul, A. A., and Siskind, J. M. (2018). Automatic differentiation in machine learning: A survey. Journal of Machine Learning Research, 18, 1–43. https://doi.org/0.5555/3122009.3242010.Google Scholar

Bishop, C. H., Etherton, B. J., and Majumdar, S. J. (2001). Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Monthly Weather Review, 129, 420–36. https://doi.org/10.1175/1520–0493 Google Scholar

Bocquet, M. (2011). Ensemble Kalman filtering without the intrinsic need for inflation. Nonlinear Processes in Geophysics, 18, 735–50. https://doi.org/10.5194/npg-18–735–2011.Google Scholar

Bocquet, M., Brajard, J., Carrassi, A., and Bertino, L. (2019). Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models. Nonlinear Processes in Geophysics, 26, 143–62. https://doi.org/10.5194/npg-26-143-2019.Google Scholar

Bocquet, M., Brajard, J., Carrassi, A., and Bertino, L. (2020). Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization. Foundations of Data Science, 2, 55–80. https://doi.org/10.3934/fods.2020004.Google Scholar

Bocquet, M., Raanes, P. N., and Hannart, A. (2015). Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation. Nonlinear Processes in Geophysics, 22, 645–62. https://doi.org/10.5194/npg22–645-2015.Google Scholar

Bocquet, M., and Sakov, P. (2012). Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems. Nonlinear Processes in Geophysics, 19, 383–99. https://doi.org/10.5194/npg-19-383-2012.CrossRef Google Scholar

Bocquet, M., and Sakov, P. (2013). Joint state and parameter estimation with an iterative ensemble Kalman smoother. Nonlinear Processes in Geophysics, 20, 803–18. https://doi.org/10.5194/npg-20-803-2013.Google Scholar

Bocquet, M., and Sakov, P. (2014). An iterative ensemble Kalman smoother. Quarterly Journal of the Royal Meteorological Society, 140, 1521–35. https://doi.org/10.1002/qj.2236.Google Scholar

Bonavita, M., Isaksen, L., and Hólm, E. (2012). On the use of EDA background error variances in the ECMWF 4D-VAR. Quarterly Journal of the Royal Meteorological Society, 138, 1540–59. https://doi.org/10.1002/qj.1899.Google Scholar

Brajard, J., Carrassi, A., Bocquet, M., and Bertino, L. (2020). Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: A case study with the Lorenz 96 model. Journal of Computational Science, 44, 101171. https://doi.org/10.1016/j.jocs.2020.101171.Google Scholar

Burgers, G., van Leeuwen, P. J., and Evensen, G. (1998). Analysis scheme in the ensemble Kalman filter. Monthly Weather Review, 126, 1719–24. https://doi.org/10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2.Google Scholar

Carrassi, A., Bocquet, M., Bertino, L., and Evensen, G. (2018). Data assimilation in the geosciences: An overview on methods, issues, and perspectives. WIREs Climate Change, 9, e535. https://onlinelibrary.wiley.com/doi/abs/10.1002/wcc.535.Google Scholar

Carrassi, A., Bocquet, M., Demaeyer, J., Gruzien, C., Raanes, P. N. and Vannitsem, S. (2022). Data assimilation for chaotic dynamics. In Park, S. K. and Xu, L. eds., Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, vol. IV. Cham: Springer, pp. 1–42. https://doi.org/10.1007/978-3–030–77722–7\_1.Google Scholar

Carrassi, A., Bocquet, M., Hannart, A., and Ghil, M. (2017). Estimating model evidence using data assimilation. Quarterly Journal of the Royal Meteorological Society, 143, 866–80. https://doi.org/10.1002/qj.2972.Google Scholar

Chen, Y., and Oliver, D. S. (2012). Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Mathematical Geosciences, 44, 1–26. https://doi.org/10.1007/s11004-011-9376-z.Google Scholar

Cohn, S. E., Sivakumaran, N. S., and Todling, R. (1994). A fixed-lag Kalman smoother for retrospective data assimilation. Monthly Weather Review, 122, 2838–67. https://doi.org/10.1175/1520–0493(1994)122<2838:AFLKSF>2.0.CO;2.Google Scholar

Corazza, M., Kalnay, E., Patil, D. J. et al. (2003). Use of the breeding technique to estimate the structure of the analysis errors of the day. Nonlinear Processes in Geophysics, 10, 233–43. https://doi.org/10.5194/npg-10-233-2003.CrossRef Google Scholar

Cosme, E., Verron, J., Brasseur, P., Blum, J., and Auroux, D. (2012). Smoothing problems in a Bayesian framework and their linear Gaussian solutions. Monthly Weather Review, 140, 683–95. https://doi.org/10.1175/MWR-D-10-05025.1.Google Scholar

Courtier, P., Thépaut, J.-N., and Hollingsworth, A. (1994). A strategy for operational implementation of 4D-VAR, using an incremental approach. Quarterly Journal of the Royal Meteorological Society, 120, 1367–87. https://doi.org/10.1002/qj.49712051912.Google Scholar

Daley, R. (1991). Atmospheric Data Analysis. New York: Cambridge University Press.Google Scholar

Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B, 1–38. https://doi.org/10.1111/j.2517–6161.1977.tb01600.x.Google Scholar

Desbouvries, F., Petetin, Y., and Ait-El-Fquih, B. (2011). Direct, prediction- and smoothing-based Kalman and particle filter algorithms. Signal Processing, 91, 2064–77. https://doi.org/10.1016/j.sigpro.2011.03.013.Google Scholar

Desroziers, G., Camino, J.-T., and Berre, L. (2014). 4DEnVar: link with 4D state formulation of variational assimilation and different possible implementations. Quarterly Journal of the Royal Meteorological Society, 140, 2097–110. https://doi.org/10.1002/qj.2325.Google Scholar

Evensen, G. (1994). Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. Journal of Geophysical Research, 99, 10143–62. https://doi.org/10.1029/94JC00572.Google Scholar

Evensen, G. (2003). The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dynamics, 53, 343–67. https://doi.org/10.1007/s10236-003–0036–9.Google Scholar

Evensen, G., and van Leeuwen, P. J. (2000). An ensemble Kalman smoother for nonlinear dynamics. Monthly Weather Review, 128, 1852–67. https://doi.org/10.1175/1520–0493(2000)128<1852:AEKSFN>2.0.CO;2.Google Scholar

Fablet, R., Chapron, B., Drumetz, L. et al. (2021). Learning variational data assimilation models and solvers. Journal of Advances in Modeling Earth Systems, 13, e2021MS002572. https://doi.org/10.1029/2021MS002572.Google Scholar

Farchi, A., and Bocquet, M. (2019). On the efficiency of covariance localisation of the ensemble Kalman filter using augmented ensembles. Frontiers in Applied Mathematics and Statistics, 5, 3. https://doi.org/10.3389/fams.2019.00003.Google Scholar

Farchi, A., Laloyaux, P., Bonavita, M., and Bocquet, M. (2021). Using machine learning to correct model error in data assimilation and forecast applications. Quarterly Journal of the Royal Meteorological Society, 147(739), 3067–84. https://doi.org/10.1002/qj.4116.Google Scholar

Fillion, A., Bocquet, M., Gratton, S., Gürol, S. and Sakov, P. (2020). An iterative ensemble Kalman smoother in presence of additive model error. SIAM/ASA Journal of Uncertainty Quantification, 8, 198–228. https://doi.org/10.1137/19M1244147.Google Scholar

Ghahramani, Z., and Roweis, S. T. (1999). Learning nonlinear dynamical systems using an EM algorithm. In Kearns, M., Solla, S., and Cohn, D., eds., Advances in Neural Information Processing Systems, vol. 11. Cambridge, MA: MIT Press, pp. 431–7. https://doi.org/0.5555/340534.340696.Google Scholar

Grewal, M. S., and Andrews, A. P. (2014). Kalman Filtering: Theory and Practice with MATLAB. Hoboken, NJ; John Wiley & Sons. https://doi.org/10.1002/9781118984987.Google Scholar

Griewank, A. (1989). On automatic differentiation. Mathematical Programming: Recent Developments and Applications, 6, 83–107.Google Scholar

Griewank, A., and Walther, A. (2003). Introduction to automatic differentiation. Proceedings in Applied Mathematics and Mechanics, 2(1), 45–9. https://doi.org/10.1002/pamm.200310012.Google Scholar

Grudzien, C., and Bocquet, M. (2022). A fast, single-iteration ensemble Kalman smoother for sequential data assimilation. Geoscientific Model Development, 15, 7641–7681. https://doi.org/10.5194/gmd--15-7641-2022.Google Scholar

Grudzien, C., Carrassi, A., and Bocquet, M. (2018). Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error. Nonlinear Processes in Geophysics, 25, 633–48. https://doi.org/10.5194/npg-25-633-2018.Google Scholar

Gu, Y., and Oliver, D. S. (2007). An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE Journal, 12, 438–46. https://doi.org/10.2118/108438-PA.CrossRef Google Scholar

Hamill, T. M., and Snyder, C. (2000). A hybrid ensemble Kalman filter-3D variational analysis scheme. Monthly Weather Review, 128, 2905–19. https://doi.org/10.1175/1520-0493(2000)128<2905:AHEKFV>2.0.CO;2.Google Scholar

Härdle, W. K., Okhrin, O., and Okhrin, Y. (2017). Basic Elements of Computational Statistics. Cham: Springer. https://link.springer.com/book/10.1007/978-3-319-55336-8.Google Scholar

Hascoët, L. (2014). An introduction to inverse modelling and parameter estimation for atmosphere and ocean sciences. In Blayo, É, Bocquet, M., Cosme, E., and Cugliandolo, L. F., eds., Advanced Data Assimilation for Geosciences. Oxford: Oxford University Press, pp. 349–69.Google Scholar

Hsieh, W. W., and Tang, B. (1998). Applying neural network models to prediction and data analysis in meteorology and oceanography. Bulletin of the American Meteorological Society, 79, 1855–70. https://doi.org/10.1175/1520-0477(1998)079<1855: ANNMTP>2.0.CO;2.Google Scholar

Hunt, B. R., Kostelich, E. J., and Szunyogh, I. (2007). Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112–26. https://doi.org/10.1016/j.physd.2006.11.008.CrossRef Google Scholar

Iglesias, M. A., Law, K. J. H., and Stuart, A. M. (2013). Ensemble Kalman methods for inverse problems. Inverse Problems, 29, 045001. https://doi.org/10.1088/0266-5611/29/4/045001.Google Scholar

Janjić, T., Bormann, N., Bocquet, M. et al. (2018). On the representation error in data assimilation. Quarterly Journal of the Royal Meteorological Society, 144, 1257–78. https://doi.org/10.1002/qj.3130.Google Scholar

Kalnay, E. (2003). Atmospheric Modeling, Data Assimilation and Predictability. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511802270.Google Scholar

Kalnay, E., and Yang, S.-C. (2010). Accelerating the spin-up of ensemble Kalman filtering. Quarterly Journal of the Royal Meteorological Society, 136, 1644–51. https://doi.org/10.1002/qj.652.Google Scholar

Kolmogorov, A. N. (2018). Foundations of the Theory of Probability, 2nd English ed. Mineola, NY: Dover Publications.Google Scholar

Kovachki, N. B., and Stuart, A. M. (2019). Ensemble Kalman inversion: a derivative-free technique for machine learning tasks. Inverse Problems, 35, 095005. https://doi.org/10.1088/1361-6420/ab1c3a.Google Scholar

Laloyaux, P., Bonavita, M., Dahoui, M., et al. (2020). Towards an unbiased stratospheric analysis. Quarterly Journal of the Royal Meteorological Society, 146, 2392–409. https://doi.org/10.1002/qj.3798.Google Scholar

Lawson, W. G., and Hansen, J. A. (2004). Implications of stochastic and deterministic filters as ensemble-based data assimilation methods in varying regimes of error growth. Monthly Weather Review, 132, 1966–81. https://doi.org/10.1175/1520-0493(2004)132<1966:IOSADF>2.0.CO;2.Google Scholar

Le Dimet, F.-X., and Talagrand, O. (1986). Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus A, 38, 97–110. https://doi.org/10.3402/tellusa.v38i2.11706.√Google Scholar

Liu, C., Xiao, Q. and Wang, B. (2008). An ensemble-based four-dimensional variational data assimilation scheme. Part I: Technical formulation and preliminary test. Monthly Weather Review, 136, 3363–73. https://doi.org/10.1175/2008MWR2312.1.√CrossRef Google Scholar

Lorenc, A. C. (1986). Analysis methods for numerical weather prediction. Quarterly Journal of the Royal Meteorological Society, 112, 1177–94. https://doi.org/10.1002/qj.49711247414.√CrossRef Google Scholar

Lorenc, A. C. (2003). The potential of the ensemble Kalman filter for NWP: A comparison with 4D-Var. Quarterly Journal of the Royal Meteorological Society, 129, 3183–203. https://doi.org/10.1256/qj.02.132.√Google Scholar

Mahalanobis, P. C. (1936). On the generalized distance in statistics. Proceedings of the National Institute of Science of India, 2, 49–55.Google Scholar

Ménétrier, B., and Auligné, T. (2015). An overlooked issue of variational data assimilation. Monthly Weather Review, 143, 3925–30. https://doi.org/10.1175/MWRD-14–00404.1.Google Scholar

Meyer, C. D. (2000). Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics.Google Scholar

Miller, R. N., Ghil, M., and Gauthiez, F. (1994). Advanced data assimilation in strongly nonlinear dynamical systems. Journal of Atmospheric Sciences, 51, 1037–56. https://doi.org/10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2.√2.0.CO;2>CrossRef Google Scholar

Nguyen, V. D., Ouala, S., Drumetz, L., and Fablet, R. (2019). EM-like learning chaotic dynamics from noisy and partial observations. arXiv:1903.10335. https://arxiv.org/abs/1903.10335.Google Scholar

Nocedal, J., and Wright, S. (2006). Numerical Optimization. New York: Springer. https://doi.org/10.1007/978-0-387-40065-5.Google Scholar

Pawitan, Y. (2001). In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford: Oxford University Press.Google Scholar

Penny, S. G. (2017). Mathematical foundations of hybrid data assimilation from a synchronization perspective. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27, 126801. https://doi.org/10.1063/1.5001819.√Google Scholar

Peyron, M., Fillion, A., Gürol, S., et al. (2021). Latent space data assimilation by using deep learning. Quarterly Journal of the Royal Meteorological Society, 147, 3759–77. https://doi.org/10.1002/qj.4153.√Google Scholar

Raanes, P. N. (2016). On the ensemble Rauch–Tung–Striebel smoother and its equivalence to the ensemble Kalman smoother. Quarterly Journal of the Royal Meteorological Society, 142, 1259–64. https://doi.org/10.1002/qj.2728.√Google Scholar

Raanes, P. N., Bocquet, M., and Carrassi, A. (2019a). Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures. Quarterly Journal of the Royal Meteorological Society, 145, 53–75. https://doi.org/10.1002/qj.3386.Google Scholar

Raanes, P. N., Stordal, A. S., and Evensen, G. (2019b). Revising the stochastic iterative ensemble smoother. Nonlinear Processes in Geophysics, 26, 325–38. https://doi.org/10.5194/npg-26-325-2019.Google Scholar

Reich, S., and Cotter, C. (2015). Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9781107706804.Google Scholar

Rojas, R. (2013). Neural Networks: A Systematic Introduction. New York: Springer. https://doi.org/10.1007/978-3-642-61068-4.Google Scholar

Ross, S. M. (2014). Introduction to Probability Models, 11th edition. San Diego, CA: Academic Press. https://doi.org/10.1016/C2012-0-03564-8.Google Scholar

Sakov, P., and Bertino, L. (2011). Relation between two common localisation methods for the EnKF. Computational Geosciences, 15, 225–37. https://doi.org/10.1007/s10596-010–9202–6.CrossRef Google Scholar

Sakov, P., and Bocquet, M. (2018). Asynchronous data assimilation with the EnKF in presence of additive model error. Tellus A, 70, 1414545. https://doi.org/10.1080/16000870.2017.1414545.Google Scholar

Sakov, P., Haussaire, J.-M., and Bocquet, M. (2018). An iterative ensemble Kalman filter in presence of additive model error. Quarterly Journal of the Royal Meteorological Society, 144, 1297–309. https://doi.org/10.1002/qj.3213.Google Scholar

Sakov, P., and Oke, P. R. (2008). Implications of the form of the ensemble transformation in the ensemble square root filters. Monthly Weather Review, 136, 1042–53. https://doi.org/10.1175/2007MWR2021.1.Google Scholar

Sakov, P., Oliver, D. S., and Bertino, L. (2012). An iterative EnKF for strongly nonlinear systems. Monthly Weather Review, 140, 1988–2004. https://doi.org/10.1175/MWR-D-11–00176.1.Google Scholar

Schillings, C., and Stuart, A. M. (2018). Convergence analysis of ensemble Kalman inversion: The linear, noisy case. Applicable Analysis, 97, 107–23. https://doi.org/10.1080/00036811.2017.1386784.Google Scholar

Tabeart, J. M., Dance, S. L., Haben, S. A., et al. (2018). The conditioning of least-squares problems in variational data assimilation. Numerical Linear Algebra with Applications, 25, e2165. https://doi.org/10.1002/nla.2165.Google Scholar

Talagrand, O., and Courtier, P. (1987). Variational assimilation of meteorological observation with the adjoint vorticity equation. I: Theory. Quarterly Journal of the Royal Meteorological Society, 113, 1311–28. https://doi.org/10.1002/qj.49711347812.Google Scholar

Tandeo, P., Ailliot, P., Bocquet, M., et al. (2020). A review of innovation-based approaches to jointly estimate model and observation error covariance matrices in ensemble data assimilation. Monthly Weather Review, 148, 3973–94. https://doi.org/10.1175/MWRD-19–0240.1.Google Scholar

Taylor, M. E. (1996). Partial Differential Equations. Vol. 1: Basic Theory. New York: Springer. https://doi.org/10.1007/978-1-4419-7055-8.Google Scholar

Tippett, M. K., Anderson, J. L., Bishop, C. H., Hamill, T. M., and Whitaker, J. S. (2003). Ensemble square root filters. Monthly Weather Review, 131, 1485–90. https://doi.org/10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2.2.0.CO;2>CrossRef Google Scholar

Tong, Y. L. (2012). The Multivariate Normal Distribution. New York: Springer. https://doi.org/10.1007/978-1-4613-9655-0.Google Scholar

Trémolet, Y. (2006). Accounting for an imperfect model in 4D-Var. Quarterly Journal of the Royal Meteorological Society, 132, 2483–504. https://doi.org/10.1256/qj.05.224.Google Scholar

Trevisan, A., and Uboldi, F. (2004). Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle. Journal of the Atmospheric Sciences, 61, 103–13. https://doi.org/10.1175/1520-0469(2004)061<0103:AOSATO>2.0.CO;2.Google Scholar

Whitaker, J. S., and Loughe, A. F. (1998). The relationship between ensemble spread and ensemble mean skill. Monthly Weather Review, 126, 3292–302. https://doi.org/10.1175/1520-0493(1998)126<3292:TRBESA>2.0.CO;2.Google Scholar

Yang, S.-C., Lin, K.-J., Miyoshi, T., and Kalnay, E. (2013). Improving the spinup of regional EnKF for typhoon assimilation and forecasting with Typhoon Sinlaku (2008). Tellus A, 65, 20804. https://doi.org/10.3402/tellusa.v65i0.20804.Google Scholar

Zupanski, M. (2005). Maximum likelihood ensemble filter: Theoretical aspects. Monthly Weather Review, 133, 1710–26. https://doi.org/10.1175/MWR2946.1.Google Scholar

Zupanski, M., Navon, I. M., and Zupanski, D. (2008). The maximum likelihood ensemble filter as a non-differentiable minimization algorithm. Quarterly Journal of the Royal Meteorological Society, 134, 1039–50. https://doi.org/10.1002/qj.251.Google Scholar

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