• Andersson, E., , M. Fisher, , R. Munro, , and A. McNally, 2000: Diagnosis of background errors for radiances and other observable quantities in a variational data assimilation scheme, and the explanation of a case of poor convergence. Quart. J. Roy. Meteor. Soc., 126, 14551472.

    • Search Google Scholar
    • Export Citation
  • Bannister, R. N., 2008a: A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances. Quart. J. Roy. Meteor. Soc., 134, 19511970.

    • Search Google Scholar
    • Export Citation
  • Bannister, R. N., 2008b: A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics. Quart. J. Roy. Meteor. Soc., 134, 19711996.

    • Search Google Scholar
    • Export Citation
  • Berre, L., , and G. Desroziers, 2010: Filtering of background error variances and correlations by local spatial averaging: A review. Mon. Wea. Rev., 138, 36933720.

    • Search Google Scholar
    • Export Citation
  • Bowler, N. E., , A. Arribas, , K. R. Myle, , K. B. Robertson, , and S. E. Beare, 2008: The MOGREPS short-range ensemble prediction system. Quart. J. Roy. Meteor. Soc., 134, 703722.

    • Search Google Scholar
    • Export Citation
  • Bowler, N. E., , A. Arribas, , S. E. Beare, , K. R. Myle, , and G. J. Shutts, 2009: The local ETKF and SKEB: Upgrades to the MOGREPS short-range ensemble prediction system. Quart. J. Roy. Meteor. Soc., 135, 767776.

    • Search Google Scholar
    • Export Citation
  • Buehner, M., , P. L. Houtekamer, , C. Charette, , H. L. Mitchel, , and B. He, 2010: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part II: One-month experiments with real observations. Mon. Wea. Rev., 138, 15671586.

    • Search Google Scholar
    • Export Citation
  • Carrassi, A., , and S. Vannitsem, 2010: Accounting for model error in variational data assimilation: A deterministic formulation. Mon. Wea. Rev., 138, 33693386.

    • Search Google Scholar
    • Export Citation
  • Courtier, P., , J.-N. Thépaut, , and A. Hollingsworth, 1994: Accounting for an imperfect model in 4D-Var. Quart. J. Roy. Meteor. Soc., 120, 13671387.

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

    • Search Google Scholar
    • Export Citation
  • Fisher, M., 2003: Background error covariance modelling. Extended Abstracts, Seminar on Recent Developments in Data Assimilation, Reading, United Kingdom, ECMWF, 45–64.

    • Search Google Scholar
    • Export Citation
  • Fisher, M., , and P. Courtier, 1995: Estimating the covariance matrices of analysis and the forecast error in variational data assimilation. ECMWF Tech. Memo. 220, 1–28. [Available from ECMWF, Shinfield Park, Reading, Berkshire RG2 9AX, United Kingdom.]

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , and C. Snyder, 2000: A hybrid ensemble Kalman filter–3D variational analysis scheme. Mon. Wea. Rev., 128, 29052919.

  • 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
  • Hollingsworth, A., , and P. Lönnberg, 1986: The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field. Tellus, 38A, 111136.

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

  • Kalnay, E., , H. Li, , T. Miyoshi, , S.-H. Yang, , and J. Ballabrera-Poy, 2007: 4-D-Var or ensemble Kalman filter? Tellus, 59A, 758773.

  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112, 11771194.

  • Lorenc, A. C., 2003a: Modelling of error covariances by 4D-Var assimilation. Quart. J. Roy. Meteor. Soc., 129, 31673182.

  • Lorenc, A. C., 2003b: The potential of the ensemble Kalman filter for NWP–A comparison with 4D-Var. Quart. J. Roy. Meteor. Soc., 129, 31833203.

    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., , and F. Rawlins, 2005: Why does 4D-Var beat 3D-Var? Quart. J. Roy. Meteor. Soc., 131, 32473257.

  • Lorenc, A. C., and Coauthors, 2000: The Met Office global three-dimensional variational data assimilation scheme. Quart. J. Roy. Meteor. Soc., 126, 29913012.

    • Search Google Scholar
    • Export Citation
  • Magnusson, L., , J. Nycander, , and E. Källén, 2009: Flow-dependent versus flow-independent initial perturbations for ensemble prediction. Tellus, 61A, 194209.

    • Search Google Scholar
    • Export Citation
  • Ménard, R., , and R. Daley, 1996: The application of Kalman smoother theory to the estimation of 4D-Var error statistics. Tellus, 48, 221237.

    • Search Google Scholar
    • Export Citation
  • Mitchell, H., , P. Houtekamer, , and G. Pellerin, 2002: Ensemble size, balance, and model-error representation in an ensemble Kalman filter. Mon. Wea. Rev., 130, 27912808.

    • Search Google Scholar
    • Export Citation
  • Parrish, D., , and J. Derber, 1992: The National Meteorological Center’s Spectral Statistical-Interpolation Analysis System. Mon. Wea. Rev., 120, 17471763.

    • Search Google Scholar
    • Export Citation
  • Rawlins, F., , S. Ballard, , K. Bovis, , A. Clayton, , D. Li, , G. Inverarity, , A. C. Lorenc, , and T. Payne, 2007: The Met Office Global Four-Dimensional Variational Data Assimilation Scheme. Quart. J. Roy. Meteor. Soc., 133, 347362.

    • Search Google Scholar
    • Export Citation
  • Trémolet, Y., 2006: Accounting for an imperfect model in 4D-Var. Quart. J. Roy. Meteor. Soc., 132, 24832504.

  • Wang, X., , and C. H. Bishop, 2003: A comparison of breeding and ensemble transform Kalman filter ensemble forecast schemes. J. Atmos. Sci., 60, 11401158.

    • Search Google Scholar
    • Export Citation
  • Wang, X., , D. M. Barker, , C. Snyder, , and T. M. Hamill, 2008: A hybrid ETKF–3DVAR data assimilation scheme for the WRF Model. Part II: Real observation experiments. Mon. Wea. Rev., 136, 51325147.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924.

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Growth of Forecast Errors from Covariances Modeled by 4DVAR and ETKF Methods

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  • 1 Met Office, Exeter, United Kingdom
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Abstract

Numerical weather forecasting errors grow with time. Error growth results from the amplification of small perturbations due to atmospheric instability or from model deficiencies during model integration. In current NWP systems, the dimension of the forecast error covariance matrices is far too large for these matrices to be represented explicitly. They must be approximated.

This paper focuses on comparing the growth of forecast error from covariances modeled by the Met Office operational four-dimensional variational data assimilation (4DVAR) and ensemble transform Kalman filter (ETKF) methods over a period of 24 h. The growth of forecast errors implied by 4DVAR is estimated by drawing a random sample of initial conditions from a Gaussian distribution with the standard deviations given by the background error covariance matrix and then evolving the sample forward in time using linearized dynamics. The growth of the forecast error modeled by the ETKF is estimated by propagating the full nonlinear model in time starting from initial conditions generated by an ETKF. This method includes model errors in two ways: by using an inflation factor and by adding model perturbations through a stochastic physics scheme. Finally, these results are compared with a benchmark of the climatological error.

The forecast error predicted by the implicit evolution of 4DVAR does not grow, regardless of the dataset used to generate the static background error covariance statistics. The forecast error predicted by the ETKF grows more rapidly because the ETKF selects balanced initial perturbations, which project onto rapidly growing modes. Finally, in both cases it is not possible to disentangle the contribution of the initial condition error from the model error.

Corresponding author address: Chiara Piccolo, Met Office, FitzRoy Road, Exeter EX1 3PB, United Kingdom. E-mail: chiara.piccolo@metoffice.gov.uk

This article is included in the Intercomparisons of 4D-Variational Assimilation and the Ensemble Kalman Filter special collection.

Abstract

Numerical weather forecasting errors grow with time. Error growth results from the amplification of small perturbations due to atmospheric instability or from model deficiencies during model integration. In current NWP systems, the dimension of the forecast error covariance matrices is far too large for these matrices to be represented explicitly. They must be approximated.

This paper focuses on comparing the growth of forecast error from covariances modeled by the Met Office operational four-dimensional variational data assimilation (4DVAR) and ensemble transform Kalman filter (ETKF) methods over a period of 24 h. The growth of forecast errors implied by 4DVAR is estimated by drawing a random sample of initial conditions from a Gaussian distribution with the standard deviations given by the background error covariance matrix and then evolving the sample forward in time using linearized dynamics. The growth of the forecast error modeled by the ETKF is estimated by propagating the full nonlinear model in time starting from initial conditions generated by an ETKF. This method includes model errors in two ways: by using an inflation factor and by adding model perturbations through a stochastic physics scheme. Finally, these results are compared with a benchmark of the climatological error.

The forecast error predicted by the implicit evolution of 4DVAR does not grow, regardless of the dataset used to generate the static background error covariance statistics. The forecast error predicted by the ETKF grows more rapidly because the ETKF selects balanced initial perturbations, which project onto rapidly growing modes. Finally, in both cases it is not possible to disentangle the contribution of the initial condition error from the model error.

Corresponding author address: Chiara Piccolo, Met Office, FitzRoy Road, Exeter EX1 3PB, United Kingdom. E-mail: chiara.piccolo@metoffice.gov.uk

This article is included in the Intercomparisons of 4D-Variational Assimilation and the Ensemble Kalman Filter special collection.

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