• Baker, N. L., 2000: Observation adjoint sensitivity and the adaptive observation-targeting problem. Ph.D. dissertation, Naval Postgraduate School, 267 pp.

    • Search Google Scholar
    • Export Citation
  • Baker, N. L., and R. Daley, 2000: Observation and background adjoint sensitivity in the adaptive observation targeting problem. Quart. J. Roy. Meteor. Soc., 126 , 14311454.

    • Search Google Scholar
    • Export Citation
  • Barkmeijer, J., 1992: Local error growth in a barotropic model. Tellus, 44A , 314323.

  • Barkmeijer, J., P. Houtekamer, and X. Wang, 1993: Validation of a skill prediction method. Tellus, 45A , 424434.

  • 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 , 16951713.

    • 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 , 23332351.

    • Search Google Scholar
    • Export Citation
  • Bergot, T., 1999: Adaptive observations during FASTEX: A systematic survey of upstream flights. Quart. J. Roy. Meteor. Soc., 125 , 32713298.

    • Search Google Scholar
    • Export Citation
  • Berliner, L. M., Q. Lu, and C. Snyder, 1999: Statistical design for adaptive weather observations. J. Atmos. Sci., 56 , 25362552.

  • Bishop, C. H., and Z. Toth, 1999: Ensemble transformation and adaptive observations. J. Atmos. Sci., 56 , 17481765.

  • 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
  • Buizza, R., 1994: Localization of optimal perturbations using a projection operator. Quart. J. Roy. Meteor. Soc., 120 , 16471681.

  • Buizza, R., and T. N. Palmer, 1995: The singular vector structure of the atmospheric global circulation. J. Atmos. Sci., 52 , 14341456.

    • Search Google Scholar
    • Export Citation
  • Buizza, R., and A. Montani, 1999: Targeting observations using singular vectors. J. Atmos. Sci., 56 , 29652985.

  • Buizza, R., R. Gelaro, F. Molteni, and T. N. Palmer, 1997: The impact of increased resolution on predictability studies with singular vectors. Quart. J. Roy. Meteor. Soc., 123 , 10071033.

    • Search Google Scholar
    • Export Citation
  • Daley, R., 1991: Atmospheric Data Analysis. Cambridge University Press, 457 pp.

  • Daley, R., and E. Barker, 2000: The NAVDAS Sourcebook. Naval Research Laboratory NRL/PU/7530-00-418, 153 pp.

  • Daley, R., and E. Barker, 2001: NAVDAS: Formulation and diagnostics. Mon. Wea. Rev., 129 , 869883.

  • Davidson, E. R., 1975: The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real symmetric matrices. J. Comput. Phys., 17 , 8794.

    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., and R. M. Errico, 1995: Mesoscale predictability and the spectrum of optimal perturbations. J. Atmos. Sci., 52 , 34753500.

    • 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 , 286313.

    • Search Google Scholar
    • Export Citation
  • Errico, R. M., 2000: The dynamical balance of leading singular vectors in a primitive equation model. Quart. J. Roy. Meteor. Soc., 126 , 16011618.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., 1982: The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci., 39 , 16631686.

  • Farrell, B. F., . 1985: Transient growth of damped baroclinic waves. J. Atmos. Sci., 42 , 27182727.

  • Farrell, B. F., . 1989: Optimal excitation of baroclinic waves. J. Atmos. Sci., 46 , 11931206.

  • Fisher, M., and P. Courtier, 1995: Estimating the covariance matrices of analysis and forecast error in variational data assimilation. ECMWF Tech. Memo. 220, 35 pp.

    • Search Google Scholar
    • Export Citation
  • Gelaro, R., R. Buizza, T. N. Palmer, and E. Klinker, 1998: Sensitivity analysis of forecast errors and the construction of optimal perturbations using singular vectors. J. Atmos. Sci., 55 , 10121037.

    • Search Google Scholar
    • Export Citation
  • Gelaro, R., R. H. Langland, G. D. Rohaly, and T. E. Rosmond, 1999: An assessment of the singular-vector approach to targeted observing using the FASTEX data set. Quart. J. Roy. Meteor. Soc., 125 , 32993328.

    • Search Google Scholar
    • Export Citation
  • Gelaro, R., C. A. Reynolds, R. H. Langland, and G. D. Rohaly, 2000: A predictability study using geostationary satellite wind observations during NORPEX. Mon. Wea. Rev., 128 , 37893807.

    • Search Google Scholar
    • Export Citation
  • Gelaro, R., C. A. Reynolds, and R. M. Errico, 2002: Transient and asymptotic perturbation growth in a simple model. Quart. J. Roy. Meteor. Soc., 128 , 205227.

    • 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 , 35253539.

    • Search Google Scholar
    • Export Citation
  • Hogan, T., and L. Brody, 1993: Sensitivity studies of the navy's global forecast model parameterizations and evaluation of improvements to NOGAPS. Mon. Wea. Rev., 121 , 23732395.

    • Search Google Scholar
    • Export Citation
  • Klinker, E., F. Rabier, and R. Gelaro, 1998: Estimation of key analysis errors using the adjoint technique. Quart. J. Roy. Meteor. Soc., 124 , 19091933.

    • Search Google Scholar
    • Export Citation
  • Langland, R. H., and G. D. Rohaly, 1996: Adjoint-based targeting of observations for FASTEX cyclones. Preprints, Seventh Conference on Mesoscale Processes, Reading, United Kingdom, Amer. Meteor. Soc., 369–371.

    • Search Google Scholar
    • Export Citation
  • Langland, R. H., and Coauthors. 1999: The North Pacific Experiment (NORPEX-98): Targeted observations for improved North American weather forecasts. Bull. Amer. Meteor. Soc., 80 , 13631384.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1965: A study of the predictability of a 28-variable atmospheric model. Tellus, 17 , 321333.

  • Marshall, J., and F. Molteni, 1993: Toward a dynamical understanding of planetary-scale flow regimes. J. Atmos. Sci., 50 , 17921818.

  • Molteni, F., and T. N. Palmer, 1993: Predictability and finite-time instability of the northern winter circulation. Quart. J. Roy. Meteor. Soc., 119 , 269298.

    • 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 , 73119.

    • Search Google Scholar
    • Export Citation
  • Montani, A., A. J. Thorpe, R. Buizza, and P. Unden, 1999: Forecast skill of the ECMWF model using targeted observations during FASTEX. Quart. J. Roy. Meteor. Soc., 125 , 32193240.

    • Search Google Scholar
    • Export Citation
  • Mureau, R., F. Molteni, and T. N. Palmer, 1993: Ensemble prediction using dynamically conditioned perturbations. Quart. J. Roy. Meteor. Soc., 119 , 299323.

    • Search Google Scholar
    • Export Citation
  • Palmer, T. N., R. Gelaro, J. Barkmeijer, and R. Buizza, 1998: Singular vectors, metrics, and adaptive observations. J. Atmos. Sci., 55 , 633653.

    • Search Google Scholar
    • Export Citation
  • Penrose, R., 1955: A generalized inverse for matrices. Proc. Cambridge Philos. Soc., 51 , 406413.

  • Pu, Z., and E. Kalnay, 1999: Targeting observations with the quasi-inverse linear and adjoint NCEP global models: Performance during FASTEX. Quart. J. Roy. Meteor. Soc., 125 , 33293338.

    • Search Google Scholar
    • Export Citation
  • Reynolds, C., and T. N. Palmer, 1998: Decaying singular vectors and their impact on analysis and forecast correction. J. Atmos. Sci., 55 , 30053023.

    • Search Google Scholar
    • Export Citation
  • Reynolds, C., and R. M. Errico, 1999: Convergence of singular vectors toward Lyapunov vectors. Mon. Wea. Rev., 127 , 23092323.

  • Reynolds, C., R. Gelaro, and T. N. Palmer, 2000: An examination of targeting methods in a simplified setting. Tellus, 52A , 391411.

  • Riishojgaard, L., 2000: A method of estimating the analysis error variance in a physical space data assimilation system. Quart. J. Roy. Meteor. Soc., 126 , 13671386.

    • Search Google Scholar
    • Export Citation
  • Rosmond, T. E., 1997: A technical description of the NRL Adjoint Modeling System. NRL/MR/7532/97/7230, Naval Research Laboratory, 62 pp.

    • Search Google Scholar
    • Export Citation
  • Strang, G., 1976: Linear Algebra and Its Applications. Academic Press, 374 pp.

  • Strang, G., . 1986: Introduction to Applied Mathematics. Wellesley-Cambridge Press, 758 pp.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 0 0 0
PDF Downloads 0 0 0

Singular Vector Calculations with an Analysis Error Variance Metric

View More View Less
  • 1 Naval Research Laboratory, Monterey, California
Restricted access

Abstract

Singular vectors of the navy's global forecast model are calculated using an initial norm consistent with an estimate of analysis error variance provided by the Naval Research Laboratory's (NRL) Atmospheric Variational Data Assimilation System (NAVDAS). The variance estimate is based on a decomposition of the block diagonal preconditioner for the conjugate-gradient descent algorithm used in NAVDAS. Because the inverse square root of the operator that defines the variance norm is readily computed, the leading singular vectors are obtained using a standard Lanczos algorithm, as with diagonal norms such as total energy.

The resulting singular vectors are consistent with the expected distribution of analysis errors. Compared with singular vectors based on a total energy norm, the variance singular vectors at initial time have less amplitude over well-observed areas, as well as greater amplitude in the middle and upper troposphere. The variance singular vectors are in some ways similar to the full covariance (Hessian) singular vectors developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). However, unlike the Hessian singular vectors, the variance singular vectors exhibit only minor difference in structure and growth rate compared with total energy singular vectors. This is because the variance singular vectors exclude covariance information used in NAVDAS that significantly penalizes smaller scales.

The 20 leading analysis error variance singular vectors explain approximately the same fraction of forecast error variance as the total energy singular vectors in a linear context, but less in a nonlinear context. Deficiencies in the current experimental configuration are among the reasons suspected for this. Implications for targeted observing are also examined. The results show that the variance norm can have a significant impact on determining the locations for supplemental observations.

Current affiliation: Data Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, Maryland

Corresponding author address: Ronald Gelaro, Data Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, MD 20771. Email: gelaro@dao.gsfc.nasa.gov

Abstract

Singular vectors of the navy's global forecast model are calculated using an initial norm consistent with an estimate of analysis error variance provided by the Naval Research Laboratory's (NRL) Atmospheric Variational Data Assimilation System (NAVDAS). The variance estimate is based on a decomposition of the block diagonal preconditioner for the conjugate-gradient descent algorithm used in NAVDAS. Because the inverse square root of the operator that defines the variance norm is readily computed, the leading singular vectors are obtained using a standard Lanczos algorithm, as with diagonal norms such as total energy.

The resulting singular vectors are consistent with the expected distribution of analysis errors. Compared with singular vectors based on a total energy norm, the variance singular vectors at initial time have less amplitude over well-observed areas, as well as greater amplitude in the middle and upper troposphere. The variance singular vectors are in some ways similar to the full covariance (Hessian) singular vectors developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). However, unlike the Hessian singular vectors, the variance singular vectors exhibit only minor difference in structure and growth rate compared with total energy singular vectors. This is because the variance singular vectors exclude covariance information used in NAVDAS that significantly penalizes smaller scales.

The 20 leading analysis error variance singular vectors explain approximately the same fraction of forecast error variance as the total energy singular vectors in a linear context, but less in a nonlinear context. Deficiencies in the current experimental configuration are among the reasons suspected for this. Implications for targeted observing are also examined. The results show that the variance norm can have a significant impact on determining the locations for supplemental observations.

Current affiliation: Data Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, Maryland

Corresponding author address: Ronald Gelaro, Data Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, MD 20771. Email: gelaro@dao.gsfc.nasa.gov

Save