Both authors are grateful to the referees for numerous suggestions that have improved the presentation of this material. In particular, we thank Chris Snyder. KJHL is grateful to the EPSRC for funding. AMS is grateful to EPSRC, ERC, and ONR for funding.
Anderson, J., 1996: A method for producing and evaluating probabilistic forecasts from ensemble model integrations. J. Climate, 9, 1518–1530.
Apte, A., M. Hairer, A. Stuart, and J. Voss, 2007: Sampling the posterior: An approach to non-Gaussian data assimilation. Physica D, 230, 50–64.
Apte, A., C. Jones, A. Stuart, and J. Voss, 2008a: Data assimilation: Mathematical and statistical perspectives. Int. J. Numer. Methods Fluids, 56, 1033–1046.
Arulampalam, M., S. Maskell, N. Gordon, and T. Clapp, 2002: A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process., 50, 174–188.
Auvinen, H., J. Bardsley, H. Haario, and T. Kauranne, 2009: Large-scale Kalman filtering using the limited memory BFGS method. Electron. Trans. Numer. Anal., 35, 217–233.
Bain, A., and D. Crişan, 2008: Fundamentals of Stochastic Filtering. Springer Verlag, 390 pp.
Banks, H., 1992: Computational issues in parameter estimation and feedback control problems for partial differential equation systems. Physica D, 60, 226–238.
Banks, H., and K. Kunisch, 1989: Estimation Techniques for Distributed Parameter Systems. Birkhauser, 315 pp.
Bengtsson, T., C. Snyder, and D. Nychka, 2003: Toward a nonlinear ensemble filter for high-dimensional systems. J. Geophys. Res., 108, 8775, doi:10.1029/2002JD002900.
Bennett, A., 2002: Inverse Modeling of the Ocean and Atmosphere. Cambridge University Press, 234 pp.
Brett, C., A. Lam, K. Law, D. McCormick, M. Scott, and A. Stuart, 2012: Accuracy and stability of filters for dissipative PDEs. Physica D, in press.
Brooks, S., and A. Gelman, 1998: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat., 7, 434–455.
Bryson, A., and M. Frazier, 1963: Smoothing for linear and nonlinear dynamic systems. U.S. Air Force Tech. Rep. AFB-TDR-63-119, Wright-Patterson Air Force Base, OH, Aeronautical Systems Division, 353–364.
Carrassi, A., M. Ghil, A. Trevisan, and F. Uboldi, 2008: Data assimilation as a nonlinear dynamical systems problem: Stability and convergence of the prediction-assimilation system. Chaos, 18, 023112, doi:10.1063/1.2909862.
Chorin, A., M. Morzfeld, and X. Tu, 2010: Implicit particle filters for data assimilation. Commun. Appl. Math. Comput. Sci., 5, 221–240.
Cotter, S., M. Dashti, J. Robinson, and A. Stuart, 2009: Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Probl., 25, 115008, doi:10.1088/0266-5611/25/11/115008.
Cotter, S., M. Dashti, and A. Stuart, 2011: Variational data assimilation using targetted random walks. Int. J. Numer. Methods Fluids, 68, 403–421.
Courtier, P., and O. Talagrand, 1987: Variational assimilation of meteorological observations with the adjoint vorticity equation. II: Numerical results. Quart. J. Roy. Meteor. Soc., 113, 1329–1347.
Cox, H., 1964: On the estimation of state variables and parameters for noisy dynamic systems. IEEE Trans. Autom. Control, 9, 5–12.
Doucet, A., N. De Freitas, and N. Gordon, 2001: Sequential Monte Carlo Methods in Practice. Springer Verlag, 581 pp.
Evensen, G., 2009: Data Assimilation: The Ensemble Kalman Filter. Springer Verlag, 307 pp.
Evensen, G., and Coauthors, 1994: Assimilation of Geosat altimeter data for the Agulhas Current using the ensemble Kalman filter with a quasigeostrophic model. Mon. Wea. Rev., 124, 85–96.
Fisher, M., M. Leutbecher, and G. Kelly, 2005: On the equivalence between Kalman smoothing and weak-constraint four-dimensional variational data assimilation. Quart. J. Roy. Meteor. Soc., 131, 3235–3246.
Hamill, T., C. Snyder, and R. Morss, 2000: A comparison of probabilistic forecasts from bred, singular-vector, and perturbed observation ensembles. Mon. Wea. Rev., 128, 1835–1851.
Harlim, J., and A. Majda, 2008: Filtering nonlinear dynamical systems with linear stochastic models. Nonlinearity, 21, 1281, doi:10.1088/0951-7715/21/6/008.
Harvey, A., 1991: Forecasting, Structural Time Series Models, and the Kalman Filter. Cambridge University Press, 554 pp.
Hesthaven, J., S. Gottlieb, and D. Gottlieb, 2007: Spectral Methods for Time-Dependent Problems. Cambridge University Press, 273 pp.
Hinze, M., R. Pinnau, M. Ulbrich, and S. Ulbrich, 2009: Optimization with PDE Constraints. Springer, 270 pp.
Jazwinski, A., 1970: Stochastic Processes and Filtering Theory. Academic Press, 376 pp.
Kaipio, J., and E. Somersalo, 2005: Statistical and Computational Inverse Problems. Springer, 339 pp.
Kalnay, E., 2003: Atmospheric Modeling, Data Assimilation, and Predictability. Cambridge University Press, 341 pp.
Kelley, C., 2003: Solving Nonlinear Equations with Newton’s Method. Vol. 1, Fundamentals of Algorithms, Society for Industrial Mathematics, 104 pp.
Lawless, A., N. Nichols, and S. Ballard, 2003: A comparison of two methods for developing the linearization of a shallow-water model. Quart. J. Roy. Meteor. Soc., 129, 1237–1254.
Lei, J., P. Bickel, and C. Snyder, 2010: Comparison of ensemble Kalman filters under non-Gaussianity. Mon. Wea. Rev., 138, 1293–1306.
Leutbecher, M., 2003: Adaptive observations, the Hessian metric and singular vectors. Proc. ECMWF Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, Reading, United Kingdom, ECMWF, 8–12.
Lorenz, E., 1996: Predictability: A problem partly solved. Proc. Seminar on Predictability, Reading, United Kingdom, ECMWF, 1–18.
Majda, A., J. Harlim, and B. Gershgorin, 2010: Mathematical strategies for filtering turbulent dynamical systems. Dyn. Syst., 27, 441–486.
Meng, Z., and F. Zhang, 2008: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part IV: Comparison with 3DVAR in a month-long experiment. Mon. Wea. Rev., 136, 3671–3682.
Miller, R., M. Ghil, and F. Gauthiez, 1994: Advanced data assimilation in strongly nonlinear dynamical systems. J. Atmos. Sci., 51, 1037–1056.
Nocedal, J., and S. Wright, 1999: Numerical Optimization. Springer Verlag, 636 pp.
Palmer, T., R. Gelaro, J. Barkmeijer, and R. Buizza, 1998: Singular vectors, metrics, and adaptive observations. J. Atmos. Sci., 55, 633–653.
Quinn, J., and H. Abarbanel, 2010: State and parameter estimation using Monte Carlo evaluation of path integrals. Quart. J. Roy. Meteor. Soc., 136, 1855–1867.
Saad, Y., 1996: Iterative Methods for Sparse Linear Systems. 1st ed. PWS Publishing, 447 pp.
Snyder, T., T. Bengtsson, P. Bickel, and J. Anderson, 2008: Obstacles to high-dimensional particle filtering. Mon. Wea. Rev., 136, 4629–4640.
Talagrand, O., and P. Courtier, 1987: Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. Quart. J. Roy. Meteor. Soc., 113, 1311–1328.
Tarantola, A., 2005: Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial Mathematics, 342 pp.
Temam, R., 2001: Navier–Stokes Equations: Theory and Numerical Analysis. American Mathematical Society, 408 pp.
Tippett, M., J. Anderson, C. Bishop, T. Hamill, and J. Whitaker, 2003: Ensemble square root filters. Mon. Wea. Rev., 131, 1485–1490.
Trefethen, L., and D. Bau, 1997: Numerical Linear Algebra. Society for Industrial Mathematics, 361 pp.
van Leeuwen, P., 2010: Nonlinear data assimilation in geosciences: An extremely efficient particle filter. Quart. J. Roy. Meteor. Soc., 136, 1991–1999.
Vogel, C., 2002: Computational Methods for Inverse Problems. Society for Industrial Mathematics, 183 pp.
Vogel, C., and J. Wade, 1995: Analysis of costate discretizations in parameter estimation for linear evolution equations. SIAM J. Control Optim., 33, 227–254.
Zhang, M., and F. Zhang, 2012: E4DVAR: Coupling an ensemble Kalman filter with four-dimensional variational data assimilation in a limited-area weather prediction model. Mon. Wea. Rev., 140, 587–600.
Zhang, M., F. Zhang, X. Huang, and X. Zhang, 2010: Intercomparison of an ensemble Kalman filter with three- and four-dimensional variational data assimilation methods in a limited-area model over the month of June 2003. Mon. Wea. Rev., 139, 566–572.
Zupanski, D., 1997: A general weak constraint applicable to operational 4DVAR data assimilation systems. Mon. Wea. Rev., 125, 2274–2292.
Note that our observations include data at time t = 0. Because the prior is Gaussian and the observational noise is Gaussian we could alternatively redefine the prior to incorporate this data point, which can be done in closed form; the observations would then start at time t = h.
Here “w.p.” denotes “with probability.”
As an aside, we note that a more sophisticated improved version we have not seen yet in the literature would include the higher-order drift term involving the Hessian. Although adding significant expense there could be scenarios in which it would be worthwhile to attempt this.