Editorial Type:
Article
Article Type:
Research Article

Ensemble Forecasts and the Properties of Flow-Dependent Analysis-Error Covariance Singular Vectors

Thomas M. Hamill University of Colorado, and NOAA–CIRES Climate Diagnostics Center, Boulder, Colorado

Search for other papers by Thomas M. Hamill in
Current site
Google Scholar
PubMed Close
,
Chris Snyder National Center for Atmospheric Research, Boulder, Colorado*

Search for other papers by Chris Snyder in
Current site
Google Scholar
PubMed Close
, and
Jeffrey S. Whitaker NOAA–CIRES Climate Diagnostics Center, Boulder, Colorado

Search for other papers by Jeffrey S. Whitaker in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Aug 2003
DOI:
https://doi.org/10.1175//2559.1
Page(s):
1741–1758
Restricted access

Abstract

Approximations to flow-dependent analysis-error covariance singular vectors (AEC SVs) were calculated in a dry, T31 L15 primitive-equation global model. Sets of 400-member ensembles of analyses were generated by an ensemble-based data assimilation system. A sparse network of simulated rawinsonde observations were assimilated, and a perfect model was assumed. Ensembles of 48-h forecasts were also generated from these analyses. The structure of evolved singular vectors was determined by finding the linear combination of the forecast ensemble members that resulted in the largest forecast-error variance, here measured in a total-energy norm north of 20°N latitude. The same linear combination of analyses specifies the initial-time structure that should evolve to the forecast singular vector under assumptions of linearity of error growth.

The structures of these AEC SVs are important because they represent the analysis-error structures associated with the largest forecast errors. If singular vectors using other initial norms have very different structures, this indicates that these structures may be statistically unlikely to occur. The European Centre for Medium-Range Weather Forecasts currently uses singular vectors using an initial total-energy norm [“total-energy singular vectors” or (TE SVs)] to generate perturbations to initialize their ensemble forecasts. Approximate TE SVs were also calculated by drawing an initial random ensemble with perturbations that were white in total energy and applying the same approach as for AEC SVs. Comparing AEC SVs and approximate TE SVs, the AEC SVs had maximum amplitude in midlatitudes near the tropopause, both at the initial and evolved times. The AEC SVs were synoptic in scale, deep, and did not appear to be geographically localized nor tilted dramatically upshear. This contrasts with TE SVs, which started off relatively smaller in scale, were tilted upshear, and had amplitudes typically largest in the lower to midtroposphere.

The difference between AEC SVs and TE SVs suggests that operational ensemble forecasts based on TE SVs could be improved by changing the type of singular vector used to generate initial perturbations. This is particularly true for short-range ensemble forecasts, where the structure of the forecast ensemble is more closely tied to the analysis ensemble.

Corresponding author address: Dr. Thomas M. Hamill, NOAA–CIRES CDC, R/CDC 1, 325 Broadway, Boulder, CO 80305-3328. Email: tom.hamill@noaa.gov

Abstract

Approximations to flow-dependent analysis-error covariance singular vectors (AEC SVs) were calculated in a dry, T31 L15 primitive-equation global model. Sets of 400-member ensembles of analyses were generated by an ensemble-based data assimilation system. A sparse network of simulated rawinsonde observations were assimilated, and a perfect model was assumed. Ensembles of 48-h forecasts were also generated from these analyses. The structure of evolved singular vectors was determined by finding the linear combination of the forecast ensemble members that resulted in the largest forecast-error variance, here measured in a total-energy norm north of 20°N latitude. The same linear combination of analyses specifies the initial-time structure that should evolve to the forecast singular vector under assumptions of linearity of error growth.

The structures of these AEC SVs are important because they represent the analysis-error structures associated with the largest forecast errors. If singular vectors using other initial norms have very different structures, this indicates that these structures may be statistically unlikely to occur. The European Centre for Medium-Range Weather Forecasts currently uses singular vectors using an initial total-energy norm [“total-energy singular vectors” or (TE SVs)] to generate perturbations to initialize their ensemble forecasts. Approximate TE SVs were also calculated by drawing an initial random ensemble with perturbations that were white in total energy and applying the same approach as for AEC SVs. Comparing AEC SVs and approximate TE SVs, the AEC SVs had maximum amplitude in midlatitudes near the tropopause, both at the initial and evolved times. The AEC SVs were synoptic in scale, deep, and did not appear to be geographically localized nor tilted dramatically upshear. This contrasts with TE SVs, which started off relatively smaller in scale, were tilted upshear, and had amplitudes typically largest in the lower to midtroposphere.

The difference between AEC SVs and TE SVs suggests that operational ensemble forecasts based on TE SVs could be improved by changing the type of singular vector used to generate initial perturbations. This is particularly true for short-range ensemble forecasts, where the structure of the forecast ensemble is more closely tied to the analysis ensemble.

Corresponding author address: Dr. Thomas M. Hamill, NOAA–CIRES CDC, R/CDC 1, 325 Broadway, Boulder, CO 80305-3328. Email: tom.hamill@noaa.gov

Save
Cover Monthly Weather Review
Monthly Weather Review

  • Anderson, J. L., 1996: A method for producing and evaluating probabilistic forecasts from ensemble model integrations. J. Climate, 9 , 1518–1530.

    • Search Google Scholar
    • Export Citation

  • Anderson, J. L., 1997: The impact of dynamical constraints on the selection of initial conditions for ensemble predictions: Low-order perfect model results. Mon. Wea. Rev., 125 , 2969–2983.

    • Search Google Scholar
    • Export Citation

  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129 , 2884–2903.

  • Anderson, J. L., and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127 , 2741–2758.

    • Search Google Scholar
    • Export Citation

  • Badger, J., and B. J. Hoskins, 2001: Simple initial-value problems and mechanisms for baroclinic growth. J. Atmos. Sci., 58 , 38–49.

  • Barkmeijer, J., M. van Gijzen, and F. Bouttier, 1998: Singular vectors and estimates of the analysis error covariance metric. Quart. J. Roy. Meteor. Soc., 124 , 1695–1713.

    • Search Google Scholar
    • Export Citation

  • Barkmeijer, J., R. Buizza, and T. N. Palmer, 1999: 3D-Var Hessian singular vectors and their potential use in the ECMWF ensemble prediction system. Quart. J. Roy. Meteor. Soc., 125 , 2333–2351.

    • Search Google Scholar
    • Export Citation

  • Bénard, P., A. Marki, P. N. Neytchev, and M. T. Prtenjak, 2000: Stabilization of nonlinear vertical diffusion schemes in the context of NWP models. Mon. Wea. Rev., 128 , 1937–1948.

    • Search Google Scholar
    • Export Citation

  • Boer, G. J., and T. G. Shepherd, 1983: Large-scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci., 40 , 164–184.

  • Buizza, R., 1994: Sensitivity of optimal unstable structures. Quart. J. Roy. Meteor. Soc., 120 , 429–451.

  • Buizza, R., and T. N. Palmer, 1995: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52 , 1434–1456.

    • Search Google Scholar
    • Export Citation

  • Buizza, R., J. J. Tribbia, F. Molteni, and T. N. Palmer, 1993: Computation of optimal unstable structures for a numerical weather prediction model. Tellus, 45A , 388–407.

    • Search Google Scholar
    • Export Citation

  • Burgers, G., P. J. van Leeuwen, and G. Evensen, 1998: Analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126 , 1719–1724.

  • Ehrendorfer, M., 1994: The Liouville equation and its potential usefulness for the prediction of forecast skill. Part I: Theory. Mon. Wea. Rev., 122 , 703–713.

    • Search Google Scholar
    • Export Citation

  • Ehrendorfer, M., and R. M. Errico, 1995: Mesoscale predictability and the spectrum of optimal perturbations. J. Atmos. Sci., 52 , 3475–3500.

    • Search Google Scholar
    • Export Citation

  • Ehrendorfer, M., and J. J. Tribbia, 1997: Optimal prediction of forecast error covariances through singular vectors. J. Atmos. Sci., 54 , 286–313.

    • Search Google Scholar
    • Export Citation

  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasigeostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99 (C5) 10143–10162.

    • Search Google Scholar
    • Export Citation

  • Evensen, G., and P. J. van Leeuwen, 1996: Assimilation of Geosat altimeter data for the Agulhas current using the ensemble Kalman filter with a quasigeostrophic model. Mon. Wea. Rev., 124 , 85–96.

    • Search Google Scholar
    • Export Citation

  • Farrell, B. F., 1989: Optimal excitation of baroclinic waves. J. Atmos. Sci., 46 , 1193–1206.

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

    • Search Google Scholar
    • Export Citation

  • Gelaro, R., T. E. Rosmond, and R. Daley, 2002: Singular vector calculations with an analysis error variance metric. Mon. Wea. Rev., 130 , 1166–1186.

    • Search Google Scholar
    • Export Citation

  • Gilmour, I., L. A. Smith, and R. Buizza, 2001: Linear regime duration: Is 24 hours a long time in synoptic weather forecasting? J. Atmos. Sci., 58 , 3525–3539.

    • Search Google Scholar
    • Export Citation

  • Hamill, T. M., 2001: Interpretation of rank histograms for verifying ensemble forecasts. Mon. Wea. Rev., 129 , 550–560.

  • Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter–3D variational analysis scheme. Mon. Wea. Rev., 128 , 2905–2919.

  • Hamill, T. M., and C. Snyder, 2002: Using improved background error covariances from an ensemble Kalman filter for adaptive observations. Mon. Wea. Rev., 130 , 1552–1572.

    • Search Google Scholar
    • Export Citation

  • Hamill, T. M., C. Snyder, and R. E. Morss, 2000: A comparison of probabilistic forecasts from bred, singular vector, and perturbed observation ensembles. Mon. Wea. Rev., 128 , 1835–1851.

    • 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.

    • Search Google Scholar
    • Export Citation

  • Hamill, T. M., J. S. Whitaker, and R. E. Morss, 2002a: Analysis-error statistics of a quasigeostrophic model using three-dimensional variational assimilation. Mon. Wea. Rev., 130 , 2777–2790.

    • Search Google Scholar
    • Export Citation

  • Hamill, T. M., J. S. Whitaker, and C. Snyder, 2002b: Generating initial conditions for ensemble forecasts: Monte-Carlo vs. dynamic methods. Preprints, Symp. on Observations, Data Assimilation, and Probabilistic Prediction, Orlando, FL, Amer. Meteor. Soc., J1–J6.

    • Search Google Scholar
    • Export Citation

  • Heemink, A. W., M. Verlaan, and A. J. Segers, 2001: Variance-reduced ensemble Kalman filtering. Mon. Wea. Rev., 129 , 1718–1728.

  • Held, I. M., and M. J. Suarez, 1994: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteor. Soc., 75 , 1825–1830.

    • 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 , 111–136.

    • Search Google Scholar
    • Export Citation

  • Hoskins, B. J., R. Buizza, and J. Badger, 2000: The nature of singular vector growth and structure. Quart. J. Roy. Meteor. Soc., 126 , 1565–1580.

    • Search Google Scholar
    • Export Citation

  • Houtekamer, P. L., and J. Derome, 1995: Methods for ensemble prediction. Mon. Wea. Rev., 123 , 2181–2196.

  • Houtekamer, P. L., and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126 , 796–811.

    • 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 , 123–137.

    • Search Google Scholar
    • Export Citation

  • Houtekamer, P. L., L. Lafaivre, and J. Derome, 1996: The RPN ensemble prediction system. Proc. ECMWF Seminar on Predictability, Vol. II, Reading, United Kingdom, ECMWF, 121–146. [Available from ECMWF, Shinfield Park, Reading, Berkshire RG2 9AX, United Kingdom.].

    • Search Google Scholar
    • Export Citation

  • Ide, K., P. Courtier, M. Ghil, and A. C. Lorenc, 1997: Unified notation for data assimilation: Operational, sequential, and variational. J. Meteor. Soc. Japan, 25 , 181–189.

    • Search Google Scholar
    • Export Citation

  • Keppenne, C. L., 2000: Data assimilation into a primitive equation model with a parallel ensemble Kalman filter. Mon. Wea. Rev., 128 , 1971–1981.

    • Search Google Scholar
    • Export Citation

  • Legras, B., and R. Vautard, 1996: A guide to Lyapunov vectors. Proc., ECMWF Seminar on Predictability, Vol. I, Reading, United Kingdom, ECMWF, 143–156. [Available from ECMWF, Shinfield Park, Reading, Berkshire RG2 9AX, United Kingdom.].

    • 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 , 1385–1407.

    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112 , 1177–1194.

  • Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20 , 130–141.

  • Mitchell, H. L., and P. L. Houtekamer, 2000: An adaptive ensemble Kalman filter. Mon. Wea. Rev., 128 , 416–433.

  • Mitchell, H. L., P. L. Houtekamer, and G. Pellerin, 2002: Ensemble size, balance, and model-error representation in an ensemble Kalman filter. Mon. Wea. Rev., 130 , 2791–2808.

    • Search Google Scholar
    • Export Citation

  • Molteni, F., R. Buizza, T. N. Palmer, and T. Petroliagis, 1996: The ECMWF ensemble prediction system: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122 , 73–119.

    • Search Google Scholar
    • Export Citation

  • Morgan, M. C., 2001: A potential vorticity and wave activity diagnosis of optimal perturbation evolution. J. Atmos. Sci., 58 , 2518–2544.

    • Search Google Scholar
    • Export Citation

  • Mukogawa, H., and T. Ikeda, 1994: Optimal excitation of baroclinic waves in the Eady model. J. Meteor. Soc. Japan, 72 , 499–513.

  • Parrish, D. F., and J. C. Derber, 1992: The National Meteorological Center's spectral statistical-interpolation analysis system. Mon. Wea. Rev., 120 , 1747–1763.

    • Search Google Scholar
    • Export Citation

  • Pham, D. T., 2001: Stochastic methods for sequential data assimilation in strongly nonlinear systems. Mon. Wea. Rev., 129 , 1194–1207.

  • Reichle, R. H., D. B. McLaughlin, and D. Entekhabi, 2002: Hydrologic data assimilation with the ensemble Kalman filter. Mon. Wea. Rev., 130 , 103–114.

    • Search Google Scholar
    • Export Citation

  • Reynolds, C. A., R. Gelaro, and J. D. Doyle, 2001: Relationship between singular vectors and transient features in the background flow. Quart. J. Roy. Meteor. Soc., 127 , 1731–1760.

    • Search Google Scholar
    • Export Citation

  • Snyder, C., T. M. Hamill, and S. J. Trier, 2003: Linear evolution of error covariances in a quasigeostrophic model. Mon. Wea. Rev., 131 , 189–205.

    • 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.

  • Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125 , 3297–3319.

  • van Leeuwen, P. J., 1999: Comment on “Data assimilation using an ensemble Kalman filter technique.”. Mon. Wea. Rev., 127 , 1374–1377.

    • Search Google Scholar
    • Export Citation

  • Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130 , 1913–1924.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 145 54 6
PDF Downloads 102 43 3