Editorial Type:
Article
Article Type:
Research Article

Nonglobal Parameter Estimation Using Local Ensemble Kalman Filtering

Thomas Bellsky School of Mathematics and Statistical Sciences, Arizona State University, Tempe, Arizona

Search for other papers by Thomas Bellsky in
Current site
Google Scholar
PubMed Close
,
Jesse Berwald Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota

Search for other papers by Jesse Berwald in
Current site
Google Scholar
PubMed Close
, and
Lewis Mitchell Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont

Search for other papers by Lewis Mitchell in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Jun 2014
DOI:
https://doi.org/10.1175/MWR-D-13-00200.1
Page(s):
2150–2164
Restricted access

Abstract

The authors study parameter estimation for nonglobal parameters in a low-dimensional chaotic model using the local ensemble transform Kalman filter (LETKF). By modifying existing techniques for using observational data to estimate global parameters, they present a methodology whereby spatially varying parameters can be estimated using observations only within a localized region of space. Taking a low-dimensional nonlinear chaotic conceptual model for atmospheric dynamics as a numerical test bed, the authors show that this parameter estimation methodology accurately estimates parameters that vary in both space and time, as well as parameters representing physics absent from the model.

Corresponding author address: Thomas Bellsky, School of Mathematics and Statistical Sciences, Arizona State University, Physical Sciences, A-Wing, P.O. Box 871804, Tempe, AZ 85287-1804. E-mail: bellskyt@asu.edu

Abstract

The authors study parameter estimation for nonglobal parameters in a low-dimensional chaotic model using the local ensemble transform Kalman filter (LETKF). By modifying existing techniques for using observational data to estimate global parameters, they present a methodology whereby spatially varying parameters can be estimated using observations only within a localized region of space. Taking a low-dimensional nonlinear chaotic conceptual model for atmospheric dynamics as a numerical test bed, the authors show that this parameter estimation methodology accurately estimates parameters that vary in both space and time, as well as parameters representing physics absent from the model.

Corresponding author address: Thomas Bellsky, School of Mathematics and Statistical Sciences, Arizona State University, Physical Sciences, A-Wing, P.O. Box 871804, Tempe, AZ 85287-1804. E-mail: bellskyt@asu.edu
Save
Cover Monthly Weather Review
Monthly Weather Review

  • Aksoy, A., F. Zhang, and J. W. Nielsen-Gammon, 2006: Ensemble-based simultaneous state and parameter estimation in a two-dimensional sea-breeze model. Mon. Wea. Rev., 134, 2951–2970, doi:10.1175/MWR3224.1.

    • Search Google Scholar
    • Export Citation

  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 2884–2903, doi:10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Annan, J. D., 2005: Parameter estimation using chaotic time series. Tellus, 57A, 709–714, doi:10.1111/j.1600-0870.2005.00143.x.

  • Annan, J. D., and J. C. Hargreaves, 2007: Efficient estimation and ensemble generation in climate modelling. Philos. Trans. Roy. Soc. A, 365, 2077–2088, doi:10.1098/rsta.2007.2067.

    • Search Google Scholar
    • Export Citation

  • Annan, J. D., J. C. Hargreaves, N. R. Edwards, and R. Marsh, 2005: Parameter estimation in an intermediate complexity earth system model using an ensemble Kalman filter. Ocean Modell., 8, 135–154, doi:10.1016/j.ocemod.2003.12.004.

    • Search Google Scholar
    • Export Citation

  • Baek, S.-J., B. Hunt, E. Kalnay, E. Ott, and I. Szunyogh, 2006: Local ensemble Kalman filtering in the presence of model bias. Tellus, 58A, 293–306, doi:10.1111/j.1600-0870.2006.00178.x.

    • Search Google Scholar
    • Export Citation

  • Bishop, C. H., and D. Hodyss, 2007: Flow-adaptive moderation of spurious ensemble correlations and its use in ensemble-based data assimilation. Quart. J. Roy. Meteor. Soc., 133, 2029–2044, doi:10.1002/qj.169.

    • 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, 420–436, doi:10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Buizza, R., and T. N. Palmer, 1995: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52, 1434–1456, doi:10.1175/1520-0469(1995)052<1434:TSVSOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Carrassi, A., and S. Vannitsem, 2011: Treatment of the error due to unresolved scales in sequential data assimilation. Int. J. Bifurcation Chaos, 21, 3619–3626, doi:10.1142/S0218127411030775.

    • Search Google Scholar
    • Export Citation

  • Corazza, M., E. Kalnay, and S.-C. Yang, 2007: An implementation of the local ensemble Kalman filter in a quasigeostrophic model and comparison with 3D-Var. Nonlinear Processes Geophys., 14, 89–101, doi:10.5194/npg-14-89-2007.

    • Search Google Scholar
    • Export Citation

  • DelSole, T., and X. Yang, 2010: State and parameter estimation in stochastic dynamic models. Physica D, 239, 1781–1788, doi:10.1016/j.physd.2010.06.001.

    • Search Google Scholar
    • Export Citation

  • Done, J., C. A. Davis, and M. L. Weisman, 2004: The next generation of NWP: Explicit forecasts of convection using the Weather Research and Forecasting (WRF) Model. Atmos. Sci. Lett., 5, 110–117, doi:10.1002/asl.72.

    • Search Google Scholar
    • Export Citation

  • Engelen, R. J., S. Serrar, and F. Chevallier, 2009: Four-dimensional data assimilation of atmospheric CO2 using AIRS observations. J. Geophys. Res., 114, D03303, doi:10.1029/2008JD010739.

    • Search Google Scholar
    • Export Citation

  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte-Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 143–10 162, doi:10.1029/94JC00572.

    • Search Google Scholar
    • Export Citation

  • Frederiksen, J. S., T. J. O’Kane, and M. J. Zidikheri, 2012: Stochastic subgrid parameterizations for atmospheric and oceanic flows. Phys. Scr.,85, 068202, doi:10.1088/0031-8949/85/06/068202.

  • Ghil, M., and P. Malanotte-Rizzoli, 1991: Data assimilation in meteorology and oceanography. Advances in Geophysics, Vol. 33, Academic Press, 141–266, doi:10.1016/S0065-2687(08)60442-2.

    • Search Google Scholar
    • Export Citation

  • Grabowski, W. W., 2001: Coupling cloud processes with the large-scale dynamics using the Cloud-Resolving Convection Parameterization (CRCP). J. Atmos. Sci., 58, 978–997, doi:10.1175/1520-0469(2001)058<0978:CCPWTL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Hacker, J. P., and W. M. Angevine, 2013: Ensemble data assimilation to characterize surface-layer errors in numerical weather prediction models. Mon. Wea. Rev., 141, 1804–1821, doi:10.1175/MWR-D-12-00280.1.

    • Search Google Scholar
    • Export Citation

  • Hakkarainen, J., A. Solonen, A. Ilin, J. Susiluoto, M. Laine, H. Haario, and H. Järvinen, 2013: A dilemma of the uniqueness of weather and climate model closure parameters. Tellus, 65A, 20147, doi:10.3402/tellusa.v65i0.20147.

    • 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, 2776–2790, doi:10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Higham, D. J., X. Mao, and A. M. Stuart, 2003: Strong convergence of Euler-type methods for nonlinear stochastic differential equation. SIAM J. Numer. Anal., 40, 1041–1063, doi:10.1137/S0036142901389530.

    • Search Google Scholar
    • Export Citation

  • Houtekamer, P., and H. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796–811, doi:10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Hunt, B., E. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112–126, doi:10.1016/j.physd.2006.11.008.

    • Search Google Scholar
    • Export Citation

  • Kalman, R., 1960: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng., 82, 35–45, doi:10.1115/1.3662552.

    • Search Google Scholar
    • Export Citation

  • Kalnay, E., 2002: Atmospheric Modeling, Data Assimilation, and Predictability. Cambridge University Press, 364 pp.

  • Kang, J. S., E. Kalnay, J. Liu, and I. Fung, 2011: “Variable localization” in an ensemble Kalman filter: Application to the carbon cycle data assimilation. J. Geophys. Res., 116, D09110, doi:10.1029/2010JD014673.

    • Search Google Scholar
    • Export Citation

  • Koyama, H., and M. Watanabe, 2010: Reducing forecast errors due to model imperfections using ensemble Kalman filtering. Mon. Wea. Rev., 138, 3316–3332, doi:10.1175/2010MWR3067.1.

    • Search Google Scholar
    • Export Citation

  • Lieb-Lappen, R. M., and C. M. Danforth, 2012: Aggressive shadowing of a low-dimensional model of atmospheric dynamics. Physica D, 241, 637–648, doi:10.1016/j.physd.2011.12.001.

    • Search Google Scholar
    • Export Citation

  • Lorenz, E. N., 1996: Predictability: A problem partly solved. Proc. Seminar on Predictability, Vol. I, Reading, United Kingdom, ECMWF, 1–19.

  • Majda, A. J., I. Timofeyev, and E. Vanden-Eijnden, 2001: A mathematical framework for stochastic climate models. Commun. Pure Appl. Math., 54, 891–974, doi:10.1002/cpa.1014.

    • Search Google Scholar
    • Export Citation

  • Mitchell, L., and G. A. Gottwald, 2013: Controlling model error of underdamped forecast models in sparse observational networks using a variance-limiting Kalman filter. Quart. J. Roy. Meteor. Soc., 139, 212–225, doi:10.1002/qj.1968.

    • Search Google Scholar
    • Export Citation

  • Molteni, F., 2003: Atmospheric simulations using a GCM with simplified physical parametrizations. I: Model climatology and variability in multi-decadal experiments. Climate Dyn., 20, 175–191, doi:10.1007/s00382-002-0268-2.

    • Search Google Scholar
    • Export Citation

  • Orrell, D., 2003: Model error and predictability over different timescales in the Lorenz 96 systems. J. Atmos. Sci., 60, 2219–2228, doi:10.1175/1520-0469(2003)060<2219:MEAPOD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ott, E., and Coauthors, 2004: A local ensemble Kalman filter for atmospheric data assimilation. Tellus, 56A, 415–428, doi:10.1111/j.1600-0870.2004.00076.x.

    • Search Google Scholar
    • Export Citation

  • Palmer, T. N., 1999: A nonlinear dynamical perspective on climate prediction. J. Climate, 12, 575–591, doi:10.1175/1520-0442(1999)012<0575:ANDPOC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Palmer, T. N., 2001: A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models. Quart. J. Roy. Meteor. Soc., 127, 279–304, doi:10.1256/smsqj.57201.

    • Search Google Scholar
    • Export Citation

  • Patil, D., B. R. Hunt, E. Kalnay, J. Yorke, and E. Ott, 2001: Local low dimensionality of atmospheric dynamics. Phys. Rev. Lett., 86, 5878–5881, doi:10.1103/PhysRevLett.86.5878.

    • Search Google Scholar
    • Export Citation

  • Peacock, S., 2012: Projected twenty-first-century changes in temperature, precipitation, and snow cover over North America in CCSM4. J. Climate, 25, 4405–4429, doi:10.1175/JCLI-D-11-00214.1.

    • Search Google Scholar
    • Export Citation

  • Randall, D., M. Khairoutdinov, A. Arakawa, and W. W. Grabowski, 2003: Breaking the cloud parameterization deadlock. Bull. Amer. Meteor. Soc., 84, 1547–1564, doi:10.1175/BAMS-84-11-1547.

    • Search Google Scholar
    • Export Citation

  • Rodwell, M. J., and T. N. Palmer, 2007: Using numerical weather prediction to assess climate models. Quart. J. Roy. Meteor. Soc., 133, 129–146, doi:10.1002/qj.23.

    • Search Google Scholar
    • Export Citation

  • Ruiz, J. J., M. Pulido, and T. Miyoshi, 2013: Estimating model parameters with ensemble-based data assimilation: A review. J. Meteor. Soc. Japan, 91, 79–99, doi:10.2151/jmsj.2013-201.

    • Search Google Scholar
    • Export Citation

  • Solonen, A., P. Ollinaho, M. Laine, H. Haario, J. Tamminen, and H. Järvinen, 2012: Efficient MCMC for climate model parameter estimation: Parallel adaptive chains and early rejection. Bayesian Anal., 7, 715–736, doi:10.1214/12-BA724.

    • Search Google Scholar
    • Export Citation

  • Stainforth, D. A., and Coauthors, 2005: Uncertainty in predictions of the climate response to rising levels of greenhouse gases. Nature, 433, 403–406, doi:10.1038/nature03301.

    • Search Google Scholar
    • Export Citation

  • Toth, Z., and E. Kalnay, 1993: Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc., 74, 2317–2330, doi:10.1175/1520-0477(1993)074<2317:EFANTG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Urban, N. M., and K. Keller, 2010: Probabilistic hindcasts and projections of the coupled climate, carbon cycle and Atlantic meridional overturning circulation system: A Bayesian fusion of century-scale observations with a simple model. Tellus, 62A, 737–750, doi:10.1111/j.1600-0870.2010.00471.x.

    • Search Google Scholar
    • Export Citation

  • Xu, K. M., and D. A. Randall, 2001: Updraft and downdraft statistics of simulated tropical and midlatitude cumulus convection. J. Atmos. Sci., 58, 1630–1649, doi:10.1175/1520-0469(2001)058<1630:UADSOS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Yang, X., and T. DelSole, 2009: Using the ensemble Kalman filter to estimate multiplicative model parameters. Tellus, 61A, 601–609, doi:10.1111/j.1600-0870.2009.00407.x.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 149 57 1
PDF Downloads 85 32 1