Filtering Partially Observed Multiscale Systems with Heterogeneous Multiscale Methods–Based Reduced Climate Models

Emily L. Kang Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina

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John Harlim Department of Mathematics, North Carolina State University, Raleigh, North Carolina

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Abstract

This paper presents a fast reduced filtering strategy for assimilating multiscale systems in the presence of observations of only the macroscopic (or large scale) variables. This reduced filtering strategy introduces model errors in estimating the prior forecast statistics through the (heterogeneous multiscale methods) HMM-based reduced climate model as an alternative to the standard expensive (direct numerical simulation) DNS-based fully resolved model. More importantly, this approach is not restricted to any analysis (or Bayesian updating) step from various ensemble-based filters. In a regime where there is a distinctive separation of scales, high filtering skill is obtained through applying the HMM alone with any desirable analysis step from ensemble Kalman filters. When separation of scales is not apparent as typically observed in geophysical turbulent systems, an additional procedure is proposed to reinitialize the microscopic variables to statistically reflect pseudo-observations that are constructed based on the unbiased estimates of the macroscopic variables. Specifically, these pseudo-observations are constructed offline from the conditional distributions of the microscopic forcing to the macroscopic dynamics given the macroscopic variables with the method-of-moments estimator. This HMM-based filter is comparable to the more expensive standard DNS-based filter on a stringent test bed, the two-layer Lorenz’96 model, in various regimes of scale gap, including the not so apparent one. This high filtering skill is robust in the presence of additional model errors through inconsistent pseudo-observations and even when macroscopic observations are spatially incomplete.

Current affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio.

Corresponding author address: Emily L. Kang, Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH 45221-0025. E-mail: kangel@ucmail.uc.edu

Abstract

This paper presents a fast reduced filtering strategy for assimilating multiscale systems in the presence of observations of only the macroscopic (or large scale) variables. This reduced filtering strategy introduces model errors in estimating the prior forecast statistics through the (heterogeneous multiscale methods) HMM-based reduced climate model as an alternative to the standard expensive (direct numerical simulation) DNS-based fully resolved model. More importantly, this approach is not restricted to any analysis (or Bayesian updating) step from various ensemble-based filters. In a regime where there is a distinctive separation of scales, high filtering skill is obtained through applying the HMM alone with any desirable analysis step from ensemble Kalman filters. When separation of scales is not apparent as typically observed in geophysical turbulent systems, an additional procedure is proposed to reinitialize the microscopic variables to statistically reflect pseudo-observations that are constructed based on the unbiased estimates of the macroscopic variables. Specifically, these pseudo-observations are constructed offline from the conditional distributions of the microscopic forcing to the macroscopic dynamics given the macroscopic variables with the method-of-moments estimator. This HMM-based filter is comparable to the more expensive standard DNS-based filter on a stringent test bed, the two-layer Lorenz’96 model, in various regimes of scale gap, including the not so apparent one. This high filtering skill is robust in the presence of additional model errors through inconsistent pseudo-observations and even when macroscopic observations are spatially incomplete.

Current affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio.

Corresponding author address: Emily L. Kang, Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH 45221-0025. E-mail: kangel@ucmail.uc.edu
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  • Anderson, J., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903.

  • Anderson, J., 2003: A local least squares framework for ensemble filtering. Mon. Wea. Rev., 131, 634642.

  • 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
  • Casella, G., and R. L. Berger, 2002: Statistical Inference. 2nd ed. Duxbury Press, 7000 pp.

  • Chorin, A., and P. Krause, 2004: Dimensional reduction for a Bayesian filter. Proc. Natl. Acad. Sci. USA, 101 (42), 15 01315 017.

  • Crommelin, D., and E. Vanden-Eijnden, 2008: Subgrid-scale parameterization with conditional Markov chains. J. Atmos. Sci., 65, 26612675.

    • Search Google Scholar
    • Export Citation
  • Dimet, F.-X. L., and O. Talagrand, 1986: Variational algorithm for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus, 38A, 97110.

    • Search Google Scholar
    • Export Citation
  • E, W., and B. Engquist, 2003: The heterogeneous multiscale methods. Commun. Math. Sci., 1, 87132.

  • E, W., B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden, 2007: The heterogeneous multiscale method: A review. Commun. Comput. Phys., 2, 367450.

    • Search Google Scholar
    • Export Citation
  • Engl, H., and C. Groetsch, 1987: Optimal parameter choice for ordinary and iterated Tikhonov regularization. Inverse and Ill-Posed Problems, H. Engl and C. Groetsch, Eds., Academic Press, 97–125.

    • Search Google Scholar
    • Export Citation
  • Farrell, B., and P. Ioannou, 2001: State estimation using a reduced-order Kalman filter. J. Atmos. Sci., 58, 36663680.

  • Fatkulin, I., and E. Vanden-Eijnden, 2004: A computational strategy for multiscale systems with applications to Lorenz 96 model. J. Comput. Phys., 200, 605638.

    • Search Google Scholar
    • Export Citation
  • Gershgorin, B., and A. Majda, 2008: A nonlinear test model for filtering slow-fast systems. Commun. Math. Sci., 6 (3), 611649.

  • Gershgorin, B., and A. Majda, 2010: Filtering a nonlinear slow-fast system with strong fast forcing. Commun. Math. Sci., 8 (1), 6792.

    • Search Google Scholar
    • Export Citation
  • Harlim, J., 2011: Numerical strategies for filtering partially observed stiff stochastic differential equations. J. Comput. Phys., 230 (3), 744762.

    • Search Google Scholar
    • Export Citation
  • Harlim, J., and B. Hunt, 2007: Four-dimensional local ensemble transform Kalman filter: Numerical experiments with a global circulation model. Tellus, 59A, 731748.

    • Search Google Scholar
    • Export Citation
  • Harlim, J., and A. Majda, 2008: Mathematical strategies for filtering complex systems: Regularly spaced sparse observations. J. Comput. Phys., 227 (10), 53045341.

    • Search Google Scholar
    • Export Citation
  • Harlim, J., and A. Majda, 2010: Filtering turbulent sparsely observed geophysical flows. Mon. Wea. Rev., 138, 10501083.

  • Hunt, B., E. 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
  • Kurtz, T., 1975: Semigroups of conditional shifts and approximations of Markov processes. Ann. Probab., 3, 618642.

  • Lorenz, E. N., 1995: Predictability—A problem partly solved. Proc. 1995 ECMWF Seminar on Predictability, Reading, United Kingdom, ECMWF, 1–18.

    • Search Google Scholar
    • Export Citation
  • Majda, A., and M. Grote, 2009: Mathematical test models for superparametrization in anisotropic turbulence. Proc. Natl. Acad. Sci. USA, 2106 (14), 54705474.

    • Search Google Scholar
    • Export Citation
  • Majda, A., I. Timofeyev, and E. Vanden-Eijnden, 2006: Stochastic models for selected slow variables in large deterministic systems. Nonlinearity, 19, 769794.

    • Search Google Scholar
    • Export Citation
  • Mellor, G., and P. Durbin, 1975: The structure and dynamics of the ocean surface mixed layer. J. Phys. Oceanogr., 5, 718728.

  • Ott, E., B. Hunt, I. Szunyogh, A. Zimin, E. Kostelich, M. Corrazza, E. Kalnay, and J. Yorke, 2004: A local ensemble Kalman filter for atmospheric data assimilation. Tellus, 56A, 415428.

    • Search Google Scholar
    • Export Citation
  • Papanicolaou, G., 1976: Some probabilistic problems and methods in singular perturbations. Rocky Mountain J. Math., 6, 653673.

  • Pavliotis, G., and A. Stuart, 2007: Parameter estimation for multiscale diffusions. J. Stat. Phys., 127 (4), 741781.

  • Skamarock, W., J. Klemp, J. Dudhia, D. Gill, D. Barker, W. Wang, and J. Poweras, 2005: A description of the Advanced Research WRF version 2. Tech. Rep. NCAR/TN-468+STR, National Center for Atmospheric Research, 88 pp.

    • Search Google Scholar
    • Export Citation
  • Tibshirani, R., 1996: Regression shrinkage and selection via the Lasso. J. Roy. Stat. Soc., 58B (1), 267288.

  • Tikhonov, A., 1963: Regularization of incorrectly posed problems. Sov. Math. Dokl., 4, 16241627.

  • Wilks, D. S., 2005: Effects of stochastic parametrizations in the Lorenz ‘96 system. Quart. J. Roy. Meteor. Soc., 131, 389407.

  • Xing, Y., A. J. Majda, and W. W. Grabowski, 2009: New efficient sparse space–time algorithms for superparameterization on mesoscales. Mon. Wea. Rev., 137, 43074324.

    • Search Google Scholar
    • Export Citation
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