• Anderson, J. L., 2001: An ensemble adjustment filter for data assimilation. Mon. Wea. Rev., 129 , 28842903.

  • Arnold, C., , and C. Dey, 1986: Observation system simulation experiments: Past, present, and future. Bull. Amer. Meteor. Soc., 67 , 687695.

    • 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., , M. van Gijzen, , and F. Bouttier, 1998: Singular vectors and estimates of the analysis error covariance metric. Quart. J. Roy. Meteor. Soc., 126 , 14311454.

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

  • Bishop, C. H., 1993: On the behaviour of baroclinic waves undergoing horizontal deformation. Part 2: Error bound amplification and Rossby wave diagnostics. Quart. J. Roy. Meteor. Soc., 119 , 241269.

    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., , and A. J. Thorpe, 1994: Frontal stability during moist deformation frontogenesis. Part II: The suppression of nonlinear wave development. J. Atmos. Sci., 51 , 852873.

    • 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 , 420436.

    • 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 , 14341456.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. E., 1997: An introduction to estimation theory. J. Meteor. Soc. Japan, 75 , 257288.

  • Cohn, S. E., , and D. P. Dee, 1988: Observability of discretized partial differential equations. SIAM. J. Numer. Anal., 25 , 586617.

  • Dabberdt, W. F., and Coauthors. 1996: Research opportunities from emerging atmospheric observing and modeling capabilities. Bull. Amer. Meteor. Soc., 77 , 305323.

    • Search Google Scholar
    • Export Citation
  • Daley, R., 1991: Atmospheric Data Analysis. Vol. 2. Cambridge University Press, 457 pp.

  • Dritschel, D. G., , P. H. Haynes, , M. N. Juckes, , and T. G. Shepherd, 1991: The stability of a two-dimensional vortices filament under uniform strain. J. Fluid Mech., 230 , 647665.

    • Search Google Scholar
    • Export Citation
  • Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1 , 1731.

  • Emanuel, K., and Coauthors. 1995: Report of the First Prospectus Development Team of the U.S. Weather Research Program to NOAA and the NSF. Bull. Amer. Meteor. Soc., 76 , 11941208.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K., and Coauthors. 1997: Observations in aid of weather prediction for North America: Report of Prospectus Development Team Seven. Bull. Amer. Meteor. Soc, 78 , 28592868.

    • Search Google Scholar
    • Export Citation
  • Evans, M. N., , A. Kaplan, , and M. A. Cane, 1998: Optimal sites for coral-based reconstruction of global sea surface temperature. Paleoceanography, 13 , 502516.

    • Search Google Scholar
    • Export Citation
  • Farrell, B., 1988: Optimal excitation of neutral Rossby waves. J. Atmos. Sci., 45 , 163172.

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

  • Farrell, B., , and P. J. Ioannou, 2001a: Accurate low-dimensional approximation of the linear dynamics of fluid flows. J. Atmos. Sci., 58 , 27712789.

    • Search Google Scholar
    • Export Citation
  • Farrell, B., , and P. J. Ioannou, 2001b: State estimation using a reduced-order Kalman filter. J. Atmos. Sci., 58 , 36663680.

  • Gajic, Z., , and M. Qureshi, 1995: Lyapunov Matrix Equation in System Stability and Control. Academic Press, 255 pp.

  • Golub, G. H., , and C. F. van Loan, 1996: Matrix Computations. 3d ed. The Johns Hopkins University Press, 694 pp.

  • Haarsma, R. J., , J. D. Opsteegh, , F. M. Selten, , and X. Wang, 2001: Rapid transitions and ultra-low frequency behaviour in a 40 kyr integration with a coupled climate model of intermediate complexity. Climate Dyn., 17 , 559570.

    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., , and P. J. Valdes, 1990: On the existence of storm-tracks. J. Atmos. Sci., 47 , 18541864.

  • Hoskins, B. J., , M. E. McIntyre, , and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111 , 877946.

    • 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 , 123137.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., , L. Lefaivre, , J. Derome, , H. Ritchie, , and H. L. Mitchell, 1996: A system simulation approach to ensemble prediction. Mon. Wea. Rev., 124 , 12251242.

    • Search Google Scholar
    • Export Citation
  • Ide, K., , P. Coutier, , M. Ghil, , and A. C. Lorenc, 1997: Unified notation for data assimilation: Operational, sequential and variational. J. Meteor. Soc. Japan, 75 , 181189.

    • Search Google Scholar
    • Export Citation
  • Jazwinski, A., 1970: Stochastic Processes and Filtering Theory. Academic Press, 376 pp.

  • Lehoucq, R. B., , D. C. Sorenson, , and C. Yang, 1998: ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Society for Industrial and Applied Mathematics, 160 pp.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., , and B. Farrell, 1980: A simple approximate result for the maximum growth rate of baroclinic instabilities. J. Atmos. Sci., 37 , 16481654.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., , and K. A. Emanuel, 1998: Optimal sites for supplementary observation sites: Simulation with a small model. J. Atmos. Sci., 55 , 399414.

    • Search Google Scholar
    • Export Citation
  • Lust, K., , D. Roose, , A. Spence, , and A. R. Champneys, 1998: An adaptive Newton–Picard algorithm with subspace iteration for computing periodic solutions. SIAM J. Sci. Comput., 19 , 11881209.

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

  • Morss, R. E., , K. A. Emanuel, , and C. Snyder, 2001: Idealized adaptive observation strategies for improving numerical weather prediction. J. Atmos. Sci., 58 , 210232.

    • Search Google Scholar
    • Export Citation
  • Orrell, D., , L. Smith, , J. Barkmeijer, , and T. N. Palmer, 2001: Model error in weather forecasting. Nonlinear Processes Geophys., 8 , 357371.

    • 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
  • Petterssen, S., , and S. J. Smebye, 1971: On the development of extratropical cyclones. Quart. J. Roy. Meteor. Soc., 97 , 457482.

  • Simmons, A. J., , and B. J. Hoskins, 1979: The downstream and upstream development of unstable baroclinic waves. J. Atmos. Sci., 36 , 12391254.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., , and A. Hollingsworth, 2002: Some aspects of the improvement in skill of numerical weather prediction. Quart. J. Roy. Meteor. Soc., 128 , 647678.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., , R. Mureau, , and T. Petroliagis, 1995: Error growth and estimates of predictability from the ECMWF forecasting system. Quart. J. Roy. Meteor. Soc., 121 , 17391771.

    • Search Google Scholar
    • Export Citation
  • Tippett, M. K., , S. E. Cohn, , R. Todling, , and D. Marchesin, 2000: Low-dimensional representation of error covariance. Tellus, 52A , 533553.

    • Search Google Scholar
    • Export Citation
  • Toth, Z., , and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125 , 32973319.

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

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 12 12 2
PDF Downloads 4 4 0

Optimization of the Fixed Global Observing Network in a Simple Model

View More View Less
  • 1 Naval Research Laboratory, Monterey, California
  • 2 International Research Institute for Climate Prediction, Palisades, New York
© Get Permissions
Restricted access

Abstract

An exact closed form expression for the infinite time analysis and forecast error covariances of a Kalman filter is used to investigate how the locations of fixed observing platforms such as radiosonde stations affect global distributions of analysis and forecast error variance. The solution pertains to a system with no model error, time-independent nondefective unstable dynamics, time-independent observation operator, and time-independent observation error covariance. As far as the authors are aware, the solutions are new. It is shown that only nondecaying normal modes (eigenvectors of the dynamics operator) are required to represent the infinite time error covariance matrices. Consequently, once a complete set of nondecaying eigenvectors has been obtained, the solution allows for the rapid assessment of the error-reducing potential of any observational network that bounds error variance.

Atmospherically relevant time-independent basic states and their corresponding tangent linear propagators are obtained with the help of a (T21L3) quasigeostrophic global model. The closed form solution allows for an examination of the sensitivity of the error variances to many different observing configurations. It is also feasible to determine the optimal location of one additional observation given a fixed observing network, which, through repetition, can be used to build effective observing networks.

Effective observing networks result in error variances several times smaller than other types of networks with the same number of column observations, such as equally spaced or land-based networks. The impact of the observing network configuration on global error variance is greater when the observing network is less dense. The impact of observations at different pressure levels is also examined. It is found that upper-level observations are more effective at reducing globally averaged error variance, but midlevel observations are more effective at reducing forecast error variance at and downstream of the baroclinic regions associated with midlatitude jets.

University Corporation for Atmospheric Research Visiting Scientist

Corresponding author address: Craig H. Bishop, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Monterey, CA 93943-5502. Email: bishop@nrlmry.navy.mil

Abstract

An exact closed form expression for the infinite time analysis and forecast error covariances of a Kalman filter is used to investigate how the locations of fixed observing platforms such as radiosonde stations affect global distributions of analysis and forecast error variance. The solution pertains to a system with no model error, time-independent nondefective unstable dynamics, time-independent observation operator, and time-independent observation error covariance. As far as the authors are aware, the solutions are new. It is shown that only nondecaying normal modes (eigenvectors of the dynamics operator) are required to represent the infinite time error covariance matrices. Consequently, once a complete set of nondecaying eigenvectors has been obtained, the solution allows for the rapid assessment of the error-reducing potential of any observational network that bounds error variance.

Atmospherically relevant time-independent basic states and their corresponding tangent linear propagators are obtained with the help of a (T21L3) quasigeostrophic global model. The closed form solution allows for an examination of the sensitivity of the error variances to many different observing configurations. It is also feasible to determine the optimal location of one additional observation given a fixed observing network, which, through repetition, can be used to build effective observing networks.

Effective observing networks result in error variances several times smaller than other types of networks with the same number of column observations, such as equally spaced or land-based networks. The impact of the observing network configuration on global error variance is greater when the observing network is less dense. The impact of observations at different pressure levels is also examined. It is found that upper-level observations are more effective at reducing globally averaged error variance, but midlevel observations are more effective at reducing forecast error variance at and downstream of the baroclinic regions associated with midlatitude jets.

University Corporation for Atmospheric Research Visiting Scientist

Corresponding author address: Craig H. Bishop, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Monterey, CA 93943-5502. Email: bishop@nrlmry.navy.mil

Save