Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T15:19:29.670Z Has data issue: false hasContentIssue false

3 - A Tutorial on Bayesian Data Assimilation

from Part I - Introduction

Published online by Cambridge University Press:  20 June 2023

By
Colin Grudzien  and
Marc Bocquet
Edited by
Alik Ismail-Zadeh ,
Fabio Castelli ,
Dylan Jones  and
Sabrina Sanchez
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
Get access

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-VAR4D-VARAdjointBayesianEnsemble-variational (EnVAR)Ensemble Kalman filter (EnKF)GaussianMachine Learning (ML)MarkovMaximum a posteriori (MAP)
Type
Chapter
Information
Applications of Data Assimilation and Inverse Problems in the Earth Sciences , pp. 27 - 48
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.CrossRefGoogle ScholarPubMed
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/10.1175/1520–0493Google 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.√CrossRefGoogle 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.√CrossRefGoogle 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>CrossRefGoogle 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.CrossRefGoogle 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>CrossRefGoogle 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

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×