Smoothing Problems in a Bayesian Framework and Their Linear Gaussian Solutions

Emmanuel Cosme Université Joseph Fourier/LEGI, Grenoble, France

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Jacques Verron CNRS/LEGI, Grenoble, France

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Pierre Brasseur CNRS/LEGI, Grenoble, France

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Jacques Blum Université de Nice Sophia-Antipolis/LJAD, Nice, France

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Didier Auroux Université de Nice Sophia-Antipolis/LJAD, Nice, France

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Abstract

Smoothers are increasingly used in geophysics. Several linear Gaussian algorithms exist, and the general picture may appear somewhat confusing. This paper attempts to stand back a little, in order to clarify this picture by providing a concise overview of what the different smoothers really solve, and how. The authors begin addressing this issue from a Bayesian viewpoint. The filtering problem consists in finding the probability of a system state at a given time, conditioned to some past and present observations (if the present observations are not included, it is a forecast problem). This formulation is unique: any different formulation is a smoothing problem. The two main formulations of smoothing are tackled here: the joint estimation problem (fixed lag or fixed interval), where the probability of a series of system states conditioned to observations is to be found, and the marginal estimation problem, which deals with the probability of only one system state, conditioned to past, present, and future observations. The various strategies to solve these problems in the Bayesian framework are introduced, along with their deriving linear Gaussian, Kalman filter-based algorithms. Their ensemble formulations are also presented. This results in a classification and a possible comparison of the most common smoothers used in geophysics. It should provide a good basis to help the reader find the most appropriate algorithm for his/her own smoothing problem.

Corresponding author address: Emmanuel Cosme, Université Joseph Fourier/LEGI, BP53X, 38041 Grenoble CEDEX, France. E-mail: emmanuel.cosme@hmg.inpg.fr

Abstract

Smoothers are increasingly used in geophysics. Several linear Gaussian algorithms exist, and the general picture may appear somewhat confusing. This paper attempts to stand back a little, in order to clarify this picture by providing a concise overview of what the different smoothers really solve, and how. The authors begin addressing this issue from a Bayesian viewpoint. The filtering problem consists in finding the probability of a system state at a given time, conditioned to some past and present observations (if the present observations are not included, it is a forecast problem). This formulation is unique: any different formulation is a smoothing problem. The two main formulations of smoothing are tackled here: the joint estimation problem (fixed lag or fixed interval), where the probability of a series of system states conditioned to observations is to be found, and the marginal estimation problem, which deals with the probability of only one system state, conditioned to past, present, and future observations. The various strategies to solve these problems in the Bayesian framework are introduced, along with their deriving linear Gaussian, Kalman filter-based algorithms. Their ensemble formulations are also presented. This results in a classification and a possible comparison of the most common smoothers used in geophysics. It should provide a good basis to help the reader find the most appropriate algorithm for his/her own smoothing problem.

Corresponding author address: Emmanuel Cosme, Université Joseph Fourier/LEGI, BP53X, 38041 Grenoble CEDEX, France. E-mail: emmanuel.cosme@hmg.inpg.fr
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  • Anderson, B. D. O., and J. B. Moore, 1979: Optimal Filtering. Prentice-Hall, 357 pp.

  • Anderson, J., 2010: A non-Gaussian ensemble filter update for data assimilation. Mon. Wea. Rev., 138, 41864198.

  • Barth, A., A. Alvera-Azcarate, K. Gurgel, J. Staneva, A. Port, J. Beckers, and E. V. Stanev, 2010: Ensemble perturbation smoother for optimizing tidal boundary conditions by assimilation of high-frequency radar surface currents—Application to the German Bight. Ocean Sci., 6, 161178.

    • Search Google Scholar
    • Export Citation
  • Beal, D., P. Brasseur, J. Brankart, Y. Ourmières, and J. Verron, 2010: Characterization of mixing errors in a coupled physical biogeochemical model of the North Atlantic: Implications for nonlinear estimation using Gaussian anamorphosis. Ocean Sci., 6, 247262.

    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., B. J. Etherton, and S. J. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420436.

    • Search Google Scholar
    • Export Citation
  • Bocquet, M., 2005a: Grid resolution dependence in the reconstruction of an atmospheric tracer source. Nonlinear Processes Geophys., 12, 219233.

    • Search Google Scholar
    • Export Citation
  • Bocquet, M., 2005b: Reconstruction of an atmospheric tracer source using the principle of maximum entropy. I: Theory. Quart. J. Roy. Meteor. Soc., 131, 21912208.

    • Search Google Scholar
    • Export Citation
  • Bocquet, M., 2005c: Reconstruction of an atmospheric tracer source using the principle of maximum entropy. II: Applications. Quart. J. Roy. Meteor. Soc., 131, 22092223.

    • Search Google Scholar
    • Export Citation
  • Bocquet, M., C. A. Pires, and L. Wu, 2010: Beyond Gaussian statistical modeling in geophysical data assimilation. Mon. Wea. Rev., 138, 29973023.

    • Search Google Scholar
    • Export Citation
  • Brankart, J., C. Ubelmann, C. Testut, E. Cosme, P. Brasseur, and J. Verron, 2009: Efficient parameterization of the observation error covariance matrix for square root or ensemble Kalman filters: Application to ocean altimetry. Mon. Wea. Rev., 137, 19081927.

    • Search Google Scholar
    • Export Citation
  • Brankart, J., E. Cosme, C. Testut, P. Brasseur, and J. Verron, 2010: Efficient adaptive error parameterizations for square root or ensemble Kalman filters: Application to the control of ocean mesoscale signals. Mon. Wea. Rev., 138, 932950.

    • Search Google Scholar
    • Export Citation
  • Brankart, J., E. Cosme, C. Testut, P. Brasseur, and J. Verron, 2011: Efficient local error parameterizations for square root or ensemble Kalman filters: application to a basin-scale ocean turbulent flow. Mon. Wea. Rev., 139, 474493.

    • Search Google Scholar
    • Export Citation
  • Brusdal, K., J. Brankart, G. Halberstadt, G. Evensen, P. Brasseur, P. van Leeuwen, E. Dombrowsky, and J. Verron, 2003: A demonstration of ensemble-based assimilation methods with a layered OGCM from the perspective of operational ocean forecasting systems. J. Mar. Syst., 40–41, 253289.

    • Search Google Scholar
    • Export Citation
  • Buehner, M., P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, 2010: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part I: Description and single-observation experiments. Mon. Wea. Rev., 138, 15501566.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. E., 1997: An introduction to estimation theory. J. Meteor. Soc. Japan, 75 (1B), 257288.

  • Cohn, S. E., N. S. Sivakumaran, and R. Todling, 1994: A fixed-lag Kalman smoother for retrospective data assimilation. Mon. Wea. Rev., 122, 28382867.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. E., A. da Silva, J. Guo, M. Sienkiewicz, and D. Lamich, 1998: Assessing the effects of data selection with the DAO physical-space statistical analysis system. Mon. Wea. Rev., 126, 29132926.

    • Search Google Scholar
    • Export Citation
  • Cosme, E., F. Hourdin, C. Genthon, and P. Martinerie, 2005: Origin of dimethylsulfide, non-sea-salt sulfate, and methanesulfonic acid in eastern Antarctica. J. Geophys. Res., 110, D03302, doi:10.1029/2004JD004881.

    • Search Google Scholar
    • Export Citation
  • Cosme, E., J. Brankart, J. Verron, P. Brasseur, and M. Krysta, 2010: Implementation of a reduced-rank, square-root smoother for ocean data assimilation. Ocean Modell., 33, 87100.

    • Search Google Scholar
    • Export Citation
  • Dee, D., 1995: On-line estimation error covariance parameters for atmospheric data assimilation. Mon. Wea. Rev., 123, 11281145.

  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasigeostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99 (C5), 10 14310 162.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., 2003: The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dyn., 53, 343367.

  • Evensen, G., and P. J. van Leeuwen, 2000: An ensemble Kalman smoother for nonlinear dynamics. Mon. Wea. Rev., 128, 18521867.

  • Fearnhead, P., D. Wyncoll, and J. Tawn, 2010: A sequential smoothing algorithm with linear computational cost. Biometrika, 97, 447464.

    • Search Google Scholar
    • Export Citation
  • Fieguth, P. W., W. C. Karl, A. S. Willsky, and C. Wunsch, 1995: Multiresolution optimal interpolation and statistical analysis of TOPEX/POSEIDON satellite altimetry. IEEE Trans. Geosci. Remote Sens., 33, 280292.

    • Search Google Scholar
    • Export Citation
  • Fraser, D. C., and J. E. Potter, 1969: The optimum linear smoother as a combination of two optimum linear filters. IEEE Trans. Automat. Contrib., 14, 387390.

    • Search Google Scholar
    • Export Citation
  • Fukumori, I., 2002: A partitioned Kalman filter and smoother. Mon. Wea. Rev., 130, 13701383.

  • Fukumori, I., 2006: What is data assimilation really solving, and how is the calculation actually done? Ocean Weather Forecasting: An Integrated View of Oceanography, E. Chassignet and J. Verron, Eds., Springer, 271–316.

    • Search Google Scholar
    • Export Citation
  • Gazeaux, J., D. Batista, C. Ammann, P. Naveau, C. Jegat, and C. Gao, 2011: Extracting common pulse-like signals from multiple ice core time series. Comput. Stat. Data Anal., in press.

    • Search Google Scholar
    • Export Citation
  • Grewal, M. S., and A. P. Andrews, 2001: Kalman Filtering: Theory and Practice Using MATLAB. 2nd ed. Wiley-Interscience, 416 pp.

  • Greybush, S. J., E. Kalnay, T. Miyoshi, K. Ide, and B. R. Hunt, 2011: Balance and ensemble Kalman filter localization techniques. Mon. Wea. Rev., 139, 511522.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., J. S. Whitaker, and C. Snyder, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 27762790.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796811.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 2001: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129, 123137.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., H. L. Mitchell, and X. Deng, 2009: Model error representation in an operational ensemble Kalman filter. Mon. Wea. Rev., 137, 21262143.

    • Search Google Scholar
    • Export Citation
  • Hunt, B. R., and Coauthors, 2004: Four-dimensional ensemble Kalman filtering. Tellus, 56A, 273277.

  • Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112126.

    • Search Google Scholar
    • Export Citation
  • Ihler, A. T., S. Kirshner, M. Ghil, A. W. Robertson, and P. Smyth, 2007: Graphical models for statistical inference and data assimilation. Physica D, 230, 7287.

    • Search Google Scholar
    • Export Citation
  • Kalman, R. E., 1960: A new approach to linear filter and prediction problems. J. Basic Eng., 82, 3545.

  • Khare, S. P., J. L. Anderson, T. J. Hoar, and D. Nychka, 2008: An investigation into the application of an ensemble Kalman smoother to high-dimensional geophysical systems. Tellus, 60A, 97112.

    • Search Google Scholar
    • Export Citation
  • Lermusiaux, P. F. J., 2006: Uncertainty estimation and prediction for interdisciplinary ocean dynamics. J. Comput. Phys., 217, 176199.

    • Search Google Scholar
    • Export Citation
  • Lermusiaux, P. F. J., and A. R. Robinson, 1999: Data assimilation via error subspace statistical estimation. Part I: Theory and schemes. Mon. Wea. Rev., 127, 13851407.

    • Search Google Scholar
    • Export Citation
  • Lermusiaux, P. F. J., A. R. Robinson, P. J. H. Haley, and W. G. Leslie, 2002: Advanced interdisciplinary data assimilation: Filtering and smoothing via Error Subspace Statistical Estimation. Proc. The OCEANS 2002 MTS/IEEE Conf., Biloxi, MS, IEEE, 795–802.

    • Search Google Scholar
    • Export Citation
  • Losa, S., G. Kivman, J. Schroter, and M. Wenzel, 2003: Sequential weak constraint parameter estimation in an ecosystem model. J. Mar. Syst., 43, 3149.

    • Search Google Scholar
    • Export Citation
  • Ménard, R., S. E. Cohn, L. Chang, and P. M. Lyster, 1996: The application of Kalman smoother theory to the estimation of 4DVAR error statistics. Tellus, 48A, 221237.

    • Search Google Scholar
    • Export Citation
  • Mitchell, H. L., and P. L. Houtekamer, 2000: An adaptive ensemble Kalman filter. Mon. Wea. Rev., 128, 416433.

  • Parrish, D. F., and S. E. Cohn, 1985: A Kalman filter for a two-dimensional shallow-water model: Formulation and preliminary experiments. Tech. Rep., Office Note 304, National Meteorological Center, Washington, DC, 64 pp.

    • Search Google Scholar
    • Export Citation
  • Pham, D. T., J. Verron, and M. C. Roubaud, 1998: A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Mar. Syst., 16, 323340.

    • Search Google Scholar
    • Export Citation
  • Rauch, H. E., F. Tung, and C. T. Striebel, 1965: Maximum likelihood estimates of linear dynamic systems. AIAA J., 3, 14451450.

  • Ravela, S., and D. McLaughlin, 2007: Fast ensemble smoothing. Ocean Dyn., 57, 123134.

  • Sakov, P., and L. Bertino, 2010: Relation between two common localisation methods for the EnKF. Comput. Geosci., 15, 225237.

  • Sakov, P., G. Evenson, and L. Bertino, 2010: Asynchronous data assimilation with the EnKF. Tellus, 62A, 2429.

  • Simon, D., 2006: Optimal State Estimation. Wiley & Sons, 530 pp.

  • Simon, E., and L. Bertino, 2009: Application of the Gaussian anamorphosis to assimilation in a 3-D coupled physical-ecosystem model of the North Atlantic with the EnKF: A twin experiment. Ocean Sci., 5, 495510.

    • Search Google Scholar
    • Export Citation
  • Skandrani, C., J. Brankart, N. Ferry, J. Verron, P. Brasseur, and B. Barnier, 2009: Controlling atmospheric forcing parameters of global ocean models: Sequential assimilation of sea surface mercator-ocean reanalysis data. Ocean Sci., 5, 403419.

    • Search Google Scholar
    • Export Citation
  • Tippett, M. K., J. L. Anderson, C. H. Bishop, T. M. Hamill, and J. S. Whitaker, 2003: Ensemble square root filters. Mon. Wea. Rev., 131, 14851490.

    • Search Google Scholar
    • Export Citation
  • van Leeuwen, P. J., 1999: The time mean circulation in the Agulhas region determined with the ensemble smoother. J. Geophys. Res., 104, 13931404.

    • Search Google Scholar
    • Export Citation
  • van Leeuwen, P. J., 2001: An ensemble smoother with error estimates. Mon. Wea. Rev., 129, 709728.

  • van Leeuwen, P. J., 2009: Particle filtering in geophysical systems. Mon. Wea. Rev., 137, 40894114.

  • van Leeuwen, P. J., and G. Evensen, 1996: Data assimilation and inverse methods in terms of a probabilistic formulation. Mon. Wea. Rev., 124, 28982913.

    • Search Google Scholar
    • Export Citation
  • Verlaan, M., and A. W. Heemink, 1997: Tidal flow forecasting using reduced-rank square root filter. Stochastic Hydrol. Hydraul., 11, 349368.

    • Search Google Scholar
    • Export Citation
  • Wikle, C. K., and L. M. Berliner, 2007: A Bayesian tutorial for data assimilation. Physica D, 230, 116.

  • Willsky, A. S., 2002: Multiresolution Markov models for signal and image processing. Proc. IEEE, 90, 13961458.

  • Zheng, F., and Y. Zhu, 2008: Balanced multivariate model errors of an intermediate coupled model for ensemble Kalman filter data assimilation. J. Geophys. Res., 113, C12006, doi:10.1029/2008JC005065.

    • Search Google Scholar
    • Export Citation
  • Zhou, Y., D. McLaughlin, D. Entekhabi, and G. Crystal Ng, 2008: An ensemble multiscale filter for large nonlinear data assimilation problems. Mon. Wea. Rev., 136, 678698.

    • Search Google Scholar
    • Export Citation
  • Zhu, Y., R. Todling, and S. E. Cohn, 1999: Technical remarks on smoother algorithms. Tech. Rep., NASA/GSFC/Data Assimilation Office, Office Note 99-02, 45 pp.

    • Search Google Scholar
    • Export Citation
  • Zhu, Y., R. Todling, J. Guo, S. E. Cohn, I. M. Navon, and Y. Yang, 2003: The GEOS-3 retrospective data assimilation system: The 6-hour lag case. Mon. Wea. Rev., 131, 21292150.

    • Search Google Scholar
    • Export Citation
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