134
Views
9
CrossRef citations to date
0
Altmetric
Original Articles

Low-dimensional representation of error covariance

, , &
Pages 533-553 | Received 04 May 1998, Accepted 24 Jan 2000, Published online: 15 Dec 2016

References

  • Barkmeijer, J., Gijzen, M. V. and Bouttier, F. 1998. Singular vectors and estimates of the analysis-error covariance matrix. Q. J. R. Meteorol. Soc. 124, 1695–1713.
  • Blumenthal, M. B. 1991. Predictability of a coupled ocean—atmosphere model. J. Clim. 4, 766–784.
  • Buizza, R. and Palmer, T. N. 1995. The singular vector structure of the atmospheric global circulation. J. Atmos. Sc i. 52, 1434–1472.
  • Buizza, R., Gelaro, R., Molteni, F. and Palmer, T. N. 1997. The impact of increased resolution on predict-ability studies with singular vectors. Q. J. R. Meteorol. Soc. 123, 1007–1033.
  • Cane, M. A., Kaplan, A., Miller, R., Tang, B., Hackert, E. and Busalacchi, A. J. 1996. Mapping tropical Pacific sea level: Data assimilation via a reduced state space Kalman filter. J. Geophys. Res. 101, 22, 599–22, 617.
  • Cohn, S. E. 1997. An introduction to estimation theory. J. Meteor. Soc. Japan 75, 257–288.
  • Cohn, S. E. and Dee, D. P. 1988. Observability of discret-ized partial differential equations. SIAM J. Numerical Analysis 25, 586–617.
  • Cohn, S. E. and Todling, R. 1996. Approximate data assimilation schemes for stable and unstable dynamics. J. Meteor. Soc. Japan 74, 63–75.
  • Cohn, S. E., Da Silva, A. M., Guo, J., Sienkiewicz, M. and Lamich, D. 1998. Assessing the effects of data selection with the DAO physical-space statistical ana-lysis system. Mon. Wea. Re v. 126, 2913–2926.
  • Courtier, P. and Talagrand, O. 1987. Variational assimilation of meteorological observations with the adjoint vorticity equation (II). Numerical results. Quart. J. Roy. Meteor. Soc. 113, 1329-1347.
  • Courtier, P., Andersson, E., Heckley, W. J. P., Vasiljevic, D., Hamrud, M., Hollingsworth, A., Rabier, E. and Fisher, M. 1998. The ECMWF implementation of three-dimensional variational assimilation (3D-Var) (I). Formulation. Q. J. R. Meteorol. Soc. 124, 1783–1807.
  • Daley, R. 1992. Forecast-error statistics for homogeneous and inhomogeneous observation networks. Mon. Wea. Re v. 120, 627–643.
  • Daley, R. and Ménard, R. 1993. Spectral characteristics of Kalman filter systems for atmospheric data assim-ilation. Mon. Wea. Re v. 121, 1554–1565.
  • Dee, D. P. and Da Silva, A. M. 1998. Data assimilation in the presence of forecast bias. Quart. J. Roy. Meteor. Soc. 124, 269–297.
  • Dee, D. P. Da Silva, A. M. 1999. Maximum-likelihood estimation of forecast and observation error covari-ance parameters. Part I: Methodology. Mon. Wea. Re v. 127, 1822–1834.
  • Dee, D. P.„ Gaspari, G., Redder, C., Rukhovets, L. and Da Silva, A. M. 1999. Maximum-likelihood estimation of forecast and observation error covariance par-ameters. Part II: Aplications. Mon. Wea. Rev. 127, 1835–1849.
  • Evensen, G. 1994. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99, 1043–1062.
  • Farrell, B. F. 1985. Transient growth of damped baro-clinic waves. J. Atmos. Sc i. 42, 2718–2727.
  • Farrell, B. F. 1989. Optimal excitation of baroclinic waves. J. Atmos. Sc i. 46, 1193–1206.
  • Farrell, B. F. and Ioannou, P. J. 1993. Stochastic dynamics of baroclinic waves. J. Atmos. Sc i. 50, 4044–4057.
  • Farrell, B. F. and Ioannou, P. J. 1996. Generalized stability theory (I). Autonomous operators. J. Atmos. Sc i. 53, 2025–2040.
  • Gajié, Z. and Qureshi, M. 1995. Lyapunov matrix equation in system stability and control. Academic Press, San Diego. 255 pp.
  • Gaspari, G. and Cohn, S. E. 1999. Construction of cor-relation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc. 125, 723–757.
  • Gelaro, R., Buizza, R., Palmer, T. N. and Klinker, E. 1998. Sensitivity analysis of forecast errors and the construction of optimal perturbations using singular vectors. J. Atmos. Sc i. 55, 1012–1037.
  • Gohberg, I. C. and Krein, M. G. 1969. Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, RI.
  • Golub, G. H. and Van Loan, C. F. 1996. Matrix computations, 3rd edition. The Johns Hopkins University Press, Baltimore. 694 pp.
  • Halmos, P. R. 1967. A Hilbert space problem book. Van Nostrand-Reinhold, New York. 365 pp.
  • Horn, R. A. and Johnson, C. R. 1985. Matrix analyis. Cambridge University Press, New York. 561 pp.
  • Houtekamer, P. L., Lefaivre, L., Derome, J., Ritchie, H. and Mitchel, H. L. 1996. A system simulation approach to ensemble prediction. Mon. Wea. Re v. 124, 1225–1242.
  • Kalman, R. 1960. A new approach to linear filtering and prediction problems. Trans. ASME, Ser. D, J. Basic Eng. 82, 35–45.
  • Legras, B. and Vautard, R. 1996. A guide to Lyapunov vectors. In: Proceedings of a Seminar on Predictability, Vol. 1, European Centre for Medium-Range Forecasts, Reading, England, pp. 143-156.
  • Mitchell, H. L. and Daley, R. 1997. Discretization error and signal/error correlation in atmospheric data assimilation (I). All scales resolved. Tellus 49A, 32-53. Molteni, F., Buizza, R., Palmer, T. N. and Petroliagis, T. 1996. The ECMWF ensemble prediction system: Methodology and validation. Q. J. R. Meteorol. Soc. 122, 73–119.
  • Mori, T., Fukuma, N. and Kuwahara, M. 1982. Upper and lower bounds for the solution to the discrete Lyapunov matrix equation. International Journal of Control 36, 889–892.
  • Rabier, F., McNally, A., Andersson, E., Courtier, P., Unden, P., Eyre, J., Hollingsworth, A. and Bouttier, F. 1998. The ECMWF implementation of three-dimen-sional variational assimilation (3D-Var) (II). Structure functions. Q. J. R. Meteorol. Soc. 124, 1809–1829.
  • Swanson, K. L., Vautard, R. and Pires, C. 1998. Four-dimensional variational assimilation and predictability in a quasi-geostrophic model. Tellus 50A, 369–390.
  • Szunyogh, I., Kalnay, E. and Toth, Z. 1997. A com-parison of Lyapunov and optimal vectors in a low-resolution GCM. Tellus 49A, 200–227.
  • Tippett, M. K. and Marchesin, D. 1999a. Bounds for solutions of the discrete algebraic Lyapunov equation. IEEE Trans. Automat. Contr. 44, 214–218.
  • Tippett, M. K. and Marchesin, D. 1999b. Upper bounds for the solution of the discrete algebraic Lyapunov equation. Automatica 35, 1485–1489.
  • Tippett, M. K., Cohn, S., Todling, R. and Marchesin, D. 2000. Conditioning of the stable, discrete-time Lyapu-nov operator. SIAM J. Matrix Anal. Appl., in press.
  • Todling, R. and Ghil, M. 1996. Tracking atmospheric instabilities with the Kalman filter. Part I: Method-ology and one-layer results. Mon. Wea. Re v. 122, 183–204.
  • Toth, Z. and Kalnay, E. 1993. Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc. 74, 2317–2330.
  • Trefethen, L. N., Trefethen, A. E. and Reddy, S. C. 1993. Hydrodynamic stability without eigenvalues. Science 261, 578–584.
  • Trevisan, A. and Pancotti, F. 1998. Periodic orbits, Lya-punov vectors and singular vectors in the Lorenz system. J. Atmos. Sc i. 55, 390–298.
  • Verlaan, M. and Heemink, A. W. 1997. Tidal flow forecasting using reduced rank square filters. Stoch-astic Hydrology and Hydraulics 11, 349–368.
  • Whitaker, J. S. and Sardeshmukh, P. D. 1998. A linear theory of extratropical synoptic eddy statistics. J. Atmos. Sc i. 55, 237–258.
  • Xue, Y., Cane, M. A., Zebiak, S. E. and Blumenthal, M. B. 1994. On the prediction of ENSO: A study with a low-order Markov model. Tellus 46A, 512–528.