Editorial Type:
Article
Article Type:
Research Article

Ensemble Filtering and Low-Resolution Model Error: Covariance Inflation, Stochastic Parameterization, and Model Numerics

I. Grooms Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

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Y. Lee Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

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A. J. Majda Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

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Print Publication:
01 Oct 2015
DOI:
https://doi.org/10.1175/MWR-D-15-0032.1
Page(s):
3912–3924
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Abstract

The use of under-resolved models in ensemble data assimilation schemes leads to two kinds of model errors: truncation errors associated with discretization of the large-scale dynamics and errors associated with interactions with subgrid scales. Multiplicative and additive covariance inflation can be used to account for model errors in ensemble Kalman filters, but they do not reduce the model error. Truncation errors can be reduced by increasing the accuracy of the numerical discretization of the large-scale dynamics, and subgrid-scale parameterizations can reduce errors associated with subgrid-scale interactions. Stochastic subgrid-scale parameterizations both reduce the model error and inflate the ensemble spread, so their effectiveness in ensemble assimilation schemes can be gauged by comparing with covariance inflation techniques. The effects of covariance inflation, stochastic parameterizations, and model numerics in two-layer periodic quasigeostrophic turbulence are compared on an f plane and on a β plane. The stochastic backscatter schemes used here model backscatter in the inverse cascade regime of quasigeostrophic turbulence, as appropriate to eddy-permitting ocean models. Covariance inflation improves the performance of a benchmark model with no parameterizations and second-order numerics. Fourth-order spatial discretization and the stochastic parameterizations, alone and in combination, are superior to covariance inflation. In these experiments fourth-order numerics and stochastic parameterizations lead to similar levels of improvement in filter performance even though the climatology of models without stochastic parameterizations is poor.

Corresponding author address: Ian Grooms, Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012. E-mail: grooms@cims.nyu.edu

Abstract

The use of under-resolved models in ensemble data assimilation schemes leads to two kinds of model errors: truncation errors associated with discretization of the large-scale dynamics and errors associated with interactions with subgrid scales. Multiplicative and additive covariance inflation can be used to account for model errors in ensemble Kalman filters, but they do not reduce the model error. Truncation errors can be reduced by increasing the accuracy of the numerical discretization of the large-scale dynamics, and subgrid-scale parameterizations can reduce errors associated with subgrid-scale interactions. Stochastic subgrid-scale parameterizations both reduce the model error and inflate the ensemble spread, so their effectiveness in ensemble assimilation schemes can be gauged by comparing with covariance inflation techniques. The effects of covariance inflation, stochastic parameterizations, and model numerics in two-layer periodic quasigeostrophic turbulence are compared on an f plane and on a β plane. The stochastic backscatter schemes used here model backscatter in the inverse cascade regime of quasigeostrophic turbulence, as appropriate to eddy-permitting ocean models. Covariance inflation improves the performance of a benchmark model with no parameterizations and second-order numerics. Fourth-order spatial discretization and the stochastic parameterizations, alone and in combination, are superior to covariance inflation. In these experiments fourth-order numerics and stochastic parameterizations lead to similar levels of improvement in filter performance even though the climatology of models without stochastic parameterizations is poor.

Corresponding author address: Ian Grooms, Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012. E-mail: grooms@cims.nyu.edu
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  • Anderson, J., 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

  • Anderson, J., 2007: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus, 59A, 210–224, doi:10.1111/j.1600-0870.2006.00216.x.

    • Search Google Scholar
    • Export Citation

  • Anderson, J., 2009: Spatially and temporally varying adaptive covariance inflation for ensemble filters. Tellus, 61A, 72–83, doi:10.1111/j.1600-0870.2008.00361.x.

    • Search Google Scholar
    • Export Citation

  • Anderson, J., and S. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 2741–2758, doi:10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Arakawa, A., 1966: Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I. J. Comput. Phys., 1, 119–143, doi:10.1016/0021-9991(66)90015-5.

    • Search Google Scholar
    • Export Citation

  • Berner, J., G. Shutts, M. Leutbecher, and T. Palmer, 2009: A spectral stochastic kinetic energy backscatter scheme and its impact on flow-dependent predictability in the ECMWF ensemble prediction system. J. Atmos. Sci., 66, 603–626, doi:10.1175/2008JAS2677.1.

    • Search Google Scholar
    • Export Citation

  • Berner, J., T. Jung, and T. Palmer, 2012: Systematic model error: The impact of increased horizontal resolution versus improved stochastic and deterministic parameterizations. J. Climate, 25, 4946–4962, doi:10.1175/JCLI-D-11-00297.1.

    • Search Google Scholar
    • Export Citation

  • Branicki, M., and A. Majda, 2012: Quantifying uncertainty for predictions with model error in non-Gaussian systems with intermittency. Nonlinearity, 25, 2543, doi:10.1088/0951-7715/25/9/2543.

    • Search Google Scholar
    • Export Citation

  • Burgers, G., P. van Leeuwen, and G. Evensen, 1998: Analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126, 1719–1724, doi:10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2.

    • 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

  • Evensen, G., 2009: Data Assimilation: The Ensemble Kalman Filter. 2nd ed. Springer, 307 pp.

  • Frenkel, Y., A. Majda, and B. Khouider, 2012: Using the stochastic multicloud model to improve tropical convective parameterization: A paradigm example. J. Atmos. Sci., 69, 1080–1105, doi:10.1175/JAS-D-11-0148.1.

    • Search Google Scholar
    • Export Citation

  • Gaspari, G., and S. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723–757, doi:10.1002/qj.49712555417.

    • Search Google Scholar
    • Export Citation

  • Grooms, I., and A. Majda, 2013: Efficient stochastic superparameterization for geophysical turbulence. Proc. Natl. Acad. Sci. USA, 110, 4464–4469, doi:10.1073/pnas.1302548110.

    • Search Google Scholar
    • Export Citation

  • Grooms, I., and A. Majda, 2014: Stochastic superparameterization in quasigeostrophic turbulence. J. Comput. Phys., 271, 78–98, doi:10.1016/j.jcp.2013.09.020.

    • Search Google Scholar
    • Export Citation

  • Grooms, I., and Y. Lee, 2015: A framework for variational data assimilation with superparameterization. Nonlinear Processes Geophys. Discuss., 2, 513–536, doi:10.5194/npgd-2-513-2015.

  • Grooms, I., Y. Lee, and A. Majda, 2014: Ensemble Kalman filters for dynamical systems with unresolved turbulence. J. Comput. Phys., 273, 435–452, doi:10.1016/j.jcp.2014.05.037.

    • Search Google Scholar
    • Export Citation

  • Grooms, I., Y. Lee, and A. Majda, 2015: Numerical schemes for stochastic backscatter in the inverse cascade regime of quasigeostrophic turbulence. Multiscale Model. Simul., in press.

    • Search Google Scholar
    • Export Citation

  • Ha, S., J. Berner, and C. Snyder, 2015: A comparison of model error representations in mesoscale ensemble data assimilation. Mon. Wea. Rev., doi:10.1175/MWR-D-14-00395.1, in press.

  • 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

  • Houtekamer, P., H. Mitchell, G. Pellerin, M. Buehner, M. Charron, L. Spacek, and B. Hansen, 2005: Atmospheric data assimilation with an ensemble Kalman filter: Results with real observations. Mon. Wea. Rev., 133, 604–620, doi:10.1175/MWR-2864.1.

    • Search Google Scholar
    • Export Citation

  • Houtekamer, P., H. Mitchell, and X. Deng, 2009: Model error representation in an operational ensemble Kalman filter. Mon. Wea. Rev., 137, 2126–2143, doi:10.1175/2008MWR2737.1.

    • Search Google Scholar
    • Export Citation

  • Janjic, Z., T. Janjic, and R. Vasic, 2011: A class of conservative fourth-order advection schemes and impact of enhanced formal accuracy on extended-range forecasts. Mon. Wea. Rev., 139, 1556–1568, doi:10.1175/2010MWR3448.1.

    • Search Google Scholar
    • Export Citation

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

  • Keating, S., A. Majda, and K. Smith, 2012: New methods for estimating ocean eddy heat transport using satellite altimetry. Mon. Wea. Rev., 140, 1703–1722, doi:10.1175/MWR-D-11-00145.1.

    • Search Google Scholar
    • Export Citation

  • Li, H., E. Kalnay, and T. Miyoshi, 2009: Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filter. Quart. J. Roy. Meteor. Soc., 135, 523–533, doi:10.1002/qj.371.

    • Search Google Scholar
    • Export Citation

  • Majda, A., and J. Harlim, 2012: Filtering Complex Turbulent Systems. Cambridge University Press, 368 pp.

  • Majda, A., and I. Grooms, 2014: New perspectives on superparameterization for geophysical turbulence. J. Comput. Phys., 271, 60–77, doi:10.1016/j.jcp.2013.09.014.

    • Search Google Scholar
    • Export Citation

  • Mitchell, H., and P. Houtekamer, 2000: An adaptive ensemble Kalman filter. Mon. Wea. Rev., 128, 416–433, doi:10.1175/1520-0493(2000)128<0416:AAEKF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Miyoshi, T., 2011: The Gaussian approach to adaptive covariance inflation and its implementation with the local ensemble transform Kalman filter. Mon. Wea. Rev., 139, 1519–1535, doi:10.1175/2010MWR3570.1.

    • Search Google Scholar
    • Export Citation

  • Randall, D., M. Branson, M. Wang, S. Ghan, C. Craig, A. Gettelman, and J. Edwards, 2013: A Community Atmosphere Model with superparameterized clouds. Eos, Trans. Amer. Geophys. Union, 94, 221–222, doi:10.1002/2013EO250001.

    • Search Google Scholar
    • Export Citation

  • Shutts, G., 2005: A kinetic energy backscatter algorithm for use in ensemble prediction systems. Quart. J. Roy. Meteor. Soc., 131, 3079–3102, doi:10.1256/qj.04.106.

    • Search Google Scholar
    • Export Citation

  • Shutts, G., 2013: Coarse graining the vorticity equation in the ECMWF integrated forecasting system: The search for kinetic energy backscatter. J. Atmos. Sci., 70, 1233–1241, doi:10.1175/JAS-D-12-0216.1.

    • Search Google Scholar
    • Export Citation

  • Stammer, D., 1997: Global characteristics of ocean variability estimated from regional TOPEX/POSEIDON altimeter measurements. J. Phys. Oceanogr., 27, 1743–1769, doi:10.1175/1520-0485(1997)027<1743:GCOOVE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Szunyogh, I., E. Kostelich, G. Gyarmati, E. Kalnay, B. Hunt, E. Ott, E. Satterfield, and J. Yorke, 2008: A local ensemble transform Kalman filter data assimilation system for the NCEP global model. Tellus, 60A, 113–130, doi:10.1111/j.1600-0870.2007.00274.x.

    • Search Google Scholar
    • Export Citation

  • Whitaker, J., and T. Hamill, 2012: Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev., 140, 3078–3089, doi:10.1175/MWR-D-11-00276.1.

    • Search Google Scholar
    • Export Citation

  • Whitaker, J., T. Hamill, X. Wei, Y. Song, and Z. Toth, 2008: Ensemble data assimilation with the NCEP global forecast system. Mon. Wea. Rev., 136, 463–482, doi:10.1175/2007MWR2018.1.

    • Search Google Scholar
    • Export Citation

  • Zupanski, D., and M. Zupanski, 2006: Model error estimation employing an ensemble data assimilation approach. Mon. Wea. Rev., 134, 1337–1354, doi:10.1175/MWR3125.1.

    • Search Google Scholar
    • Export Citation
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