Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

Barreira, L. and Pesin, Y.: Lyapunov Exponents and Smooth Ergodic Theory, Student Mathematical Library, American Mathematical Society, 38–40, 2002.

Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica, 15, 9–20, 1980.

Bocquet, M.: Ensemble Kalman filtering without the intrinsic need for inflation, Nonlin. Processes Geophys., 18, 735–750, https://doi.org/10.5194/npg-18-735-2011, 2011.

Bocquet, M. and Carrassi, A.: Four-dimensional ensemble variational data assimilation and the unstable subspace, Tellus A, 69, 1304 504, 2017.

Bocquet, M. and Sakov, P.: Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems, Nonlin. Processes Geophys., 19, 383–399, https://doi.org/10.5194/npg-19-383-2012, 2012.

Bocquet, M., Raanes, P. N., and Hannart, A.: Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation, Nonlin. Processes Geophys., 22, 645–662, https://doi.org/10.5194/npg-22-645-2015, 2015.

Bocquet, M., Gurumoorthy, K., Apte, A., Carrassi, A., Grudzien, C., and Jones, C.: Degenerate Kalman Filter Error Covariances and Their Convergence onto the Unstable Subspace, SIAM/ASA J. Uncert. Quant., 5, 304–333, 2017.

Carrassi, A., Trevisan, A., and Uboldi, F.: Adaptive observations and assimilation in the unstable subspace by breeding on the data-assimilation system, Tellus A, 59, 101–113, 2007.

Carrassi, A., Trevisan, A., Descamps, L., Talagrand, O., and Uboldi, F.: Controlling instabilities along a 3DVar analysis cycle by assimilating in the unstable subspace: a comparison with the EnKF, Nonlin. Processes Geophys., 15, 503–521, https://doi.org/10.5194/npg-15-503-2008, 2008.

Carrassi, A., Vannitsem, S., Zupanski, D., and Zupanski, M.: The maximum likelihood ensemble filter performances in chaotic systems, Tellus A, 61, 587–600, 2009.

Carrassi, A., Bocquet, M., Bertino, L., and Evensen, G.: Data assimilation in the geosciences: An overview of methods, issues, and perspectives, Wiley Interdisciplinary Reviews: Climate Change, 9, p. e535, https://doi.org/10.1002/wcc.535, 2018.

Chandrasekar, J., Kim, I., Bernstein, D., and Ridley, A.: Cholesky-based reduced-rank square-root Kalman filtering, in: American Control Conference, 2008, 3987–3992, IEEE, 2008.

Corazza, M., Kalnay, E., and Yang, S.: An implementation of the Local Ensemble Kalman Filter in a quasi geostrophic model and comparison with 3D-Var, Nonlin. Processes Geophys., 14, 89–101, https://doi.org/10.5194/npg-14-89-2007, 2007.

de Leeuw, B., Dubinkina, S., Frank, J., Steyer, A., Tu, X., and Van Vleck, E.: Projected Shadowing-based Data Assimilation, arXiv preprint arXiv:1707.09264, 2017.

Ershov, S. and Potapov, A.: On the concept of stationary Lyapunov basis, Physica D, 118, 167–198, 1998.

Frank, J. and Zhuk, S.: A detectability criterion and data assimilation for non-linear differential equations, arXiv preprint arXiv:1711.05039, 2017.

Froyland, G., Hüls, T., Morriss, G., and Watson, T.: Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Physica D, 247, 18–39, 2013.

González-Tokman, C. and Hunt, B.: Ensemble data assimilation for hyperbolic systems, Physica D, 243, 128–142, 2013.

Grudzien, C., Carrassi, A., and Bocquet, M.: Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error, SIAM/ASA J. Uncert. Quant., 2018.

Grudzien, C.: cgrudz/role_of_covariance_inflation_ supplementary_materials: First release (Version v1.0), Zenodo, http://doi.org/10.5281/zenodo.1404189, last access: 27 August 2018.

Gurumoorthy, K., Grudzien, C., Apte, A., Carrassi, A., and Jones, C.: Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution, SIAM J. Control Optim., 55, 741–759, 2017.

Hamill, T. and Snyder, C.: A hybrid ensemble Kalman filter–3D variational analysis scheme, Mon. Weather Rev., 128, 2905–2919, 2000.

Kalnay, E.: Atmospheric modeling, data assimilation and predictability, Cambridge University Press, 212–215, 2003.

Kuptsov, P. and Parlitz, U.: Theory and Computation of Covariant Lyapunov Vectors, J. Nonlinear Sci., 22, 727–762, 2012.

Legras, B. and Vautard, R.: A guide to Lyapunov vectors, in: ECMWF Workshop on Predictability, 135–146, ECMWF, Reading, United Kingdom, 1996.

Lorenz, E. and Emanuel, K.: Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci., 55, 399–414, 1998.

Mitchell, L. and Carrassi, A.: Accounting for model error due to unresolved scales within ensemble Kalman filtering, Q. J. Roy. Meteorol. Soc., 141, 1417–1428, 2015.

Nerger, L., Hiller, W., and Schröter, J.: A comparison of error subspace Kalman filters, Tellus A, 57, 715–735, 2005.

Ng, G., McLaughlin, D., Entekhabi, D., and Ahanin, A.: The role of model dynamics in ensemble Kalman filter performance for chaotic systems, Tellus A, 63, 958–977, 2011.

Palatella, L. and Grasso, F.: The EKF-AUS-NL algorithm implemented without the linear tangent model and in presence of parametric model error, SoftwareX, 7, 28–33, 2018.

Palatella, L. and Trevisan, A.: Interaction of Lyapunov vectors in the formulation of the nonlinear extension of the Kalman filter, Physical Rev. E, 91, 042 905, 2015.

Palatella, L., Carrassi, A., and Trevisan, A.: Lyapunov vectors and assimilation in the unstable subspace: theory and applications, J. Phys. A, 46, 254 020, 2013.

Penny, S.: Mathematical foundations of hybrid data assimilation from a synchronization perspective, Chaos: An Interdisciplinary J. Nonlinear Sci., 27, 126 801, 2017.

Pham, D., Verron, J., and Roubaud, M.: A singular evolutive extended Kalman filter for data assimilation in oceanography, J. Marine Syst., 16, 323–340, 1998.

Pires, C., Vautard, R., and Talagrand, O.: On extending the limits of variational assimilation in nonlinear chaotic systems, Tellus A, 48, 96–121, 1996.

Raanes, P., Carrassi, A., and Bertino, L.: Extending the square root method to account for additive forecast noise in ensemble methods, Mon. Weather Rev., 143, 3857–3873, 2015.

Raanes, P., Bocquet, M., and Carrassi, A.: Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures, arXiv preprint arXiv:1801.08474, 2018.

Sakov, P., Haussaire, J., and Bocquet, M.: An iterative ensemble Kalman filter in presence of additive model error, Q. J. Roy. Meteorol. Soc., https://doi.org/10.1002/qj.3213, 2018.

Shimada, I. and Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems, Prog. Theor. Phys., 61, 1605–1616, 1979.

Toth, Z. and Kalnay, E.: Ensemble forecasting at NCEP and the breeding method, Mon. Weather Rev., 125, 3297–3319, 1997.

Trevisan, A. and Palatella, L.: Chaos and weather forecasting: the role of the unstable subspace in predictability and state estimation problems, Int. J. Bifurcat. Chaos, 21, 3389–3415, 2011a.

Trevisan, A. and Palatella, L.: On the Kalman Filter error covariance collapse into the unstable subspace, Nonlin. Processes Geophys., 18, 243-250, https://doi.org/10.5194/npg-18-243-2011, 2011b.

Trevisan, A. and Uboldi, F.: Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle, J. Atmos. Sci., 61, 103–113, 2004.

Trevisan, A., D'Isidoro, M., and Talagrand, O.: Four-dimensional variational assimilation in the unstable subspace and the optimal subspace dimension, Q. J. Roy. Meteorol. Soc., 136, 487–496, 2010.

Vannitsem, S.: Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics, Chaos: An Interdisciplinary J. Nonlinear Sci., 27, 032 101, 2017.

Whitaker, J. and Hamill, T.: Evaluating Methods to Account for System Errors in Ensemble Data Assimilation, Mon. Weather Rev., 140, 3078–3089, 2012.