Uncertainty dynamics and predictability in chaotic systems

Quarterly Journal of the Royal Meteorological Society

Volumn 125, Issue 560, 1999, Pages 2855-2886

  Smith, L A ^a   Ziehmann, C ^b   Fraedrich, K ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b NONE

Author keywords

Ensemble prediction; Error growth; Lyapunov exponents; Nonlinear systems; Perfect model; Predictability; Singular vectors; Uncertainty

Indexed keywords

AERODYNAMICS; MODEL; WEATHER FORECASTING;

EID: 0033427662     PISSN: 00359009     EISSN: None     Source Type: Journal    
DOI: 10.1256/smsqj.56004     Document Type: Article

References (71)

1. Abarbanel, H. D. I., Brown, R. and Kennel, M. B. 1991 Variation of Lyapunov exponents on a strange attractor. J. Nonlinear Sci. 1, 175-199
2. Anderson, J. L. and van den Dool, H. M. 1994 Skill and return of skill in dynamic extended-range forecasts. Mon. Weather Rev., 122, 507-516
3. Barkmeijer, J. 1996 Constructing fast-growing perturbations for the nonlinear regime. J. Atmos. Sci., 53, 2838-2851
4. Benzi, R. and Carnevale, G. F. 1989 A possible measure of local predictability. J. Atmos. Sci., 46, 3595-3598
5. Broomhead, D. S. and Lowe, D. 1988 Multivariable functional interpolation and adaptive networks. J. Complex Systems, 2, 321-355
6. Buizza, R. 1996 Optimal perturbation time evolution and sensitivity of ensemble prediction to perturbation amplitude. Q. J. R. Meteorol. Soc., 122, 1705-1738
7. Buizza, R. and Palmer, T. N. 1995 The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52, 1434-1455
8. Casdagli, M. 1989 Nonlinear prediction of chaotic time series. Physica D, 35, 335-356
9. Doerner, R., Hübinger, B., Martiensen, W., Grossmann, S. and Thomae, S. 1991 Predictability portraits for chaotic motions. Chaos, Solitons and Fractals, 1, 553-571
10. Eckhardt, B. and Yao, D. 1993 Local Lyapunov exponents in chaotic systems. Physica D, 65, 100-108
11. Farmer, J. D. and Sidorowich, J. 1987 Predicting chaotic time series. Phys. Rev. Lett., 59, 845-848
12. Farrell, B. F. 1990 Small error dynamics and the predictability of atmospheric flows. J. Atmos. Sci., 47, 2409-2416 1994 Evolution and revolution in cyclogenesis theory. Pp. 101-110 in The life cycles of extratropical cyclones, Vol. 3
13. Farrell, B. F. 1990 Small error dynamics and the predictability of atmospheric flows. J. Atmos. Sci., 47, 2409-2416 1994 Evolution and revolution in cyclogenesis theory. Pp. 101-110 in The life cycles of extratropical cyclones, Vol. 3
14. Gilmour, I. 1998 'Nonlinear model evaluation: t-shadowing, probabilistic prediction and weather forecasting'. D. Phil. thesis, University of Oxford
15. Greene, J. and Kim, J.-S. 1987 The calculation of Lyapunov Spectra. Physica D, 24, 213-225
16. Haubs, G. and Haken, H. 1985 Quantities describing local properties of chaotic attractors. Zeitschrift für Physik B, 59, 459-468
17. Houtekamer, P. L. and Derome, J. 1994 Prediction experiments with two-member ensembles. Mon. Weather Rev., 122, 2179-2191
18. Ikeda, K. 1979 Multiple-valued stationary state and instability of the transmitted light by a ring cavity system. Opt. Commun., 30, 257
19. Jetschke, G. 1989 Mathematik der Selbstorganisation. Friedrich Vieweg unt Sohn, Wiesbaden, Germany
20. Kostelich, E. and Yorke, J. 1990 Noise reduction: Finding the simplest dynamical system consistent with the data. Physica D, 41, 183-196
21. Kurths, J. and Herzel, H. 1987 An attractor in a solar time series. Physica D, 25, 165-172
22. Lacarra, J.-F. and Talagrand, O. 1988 Short range evolution of small perturbations in a barotropic model. Tellus, 40 A, 81-95
23. Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130-141 1965 A study of the predictability of a 28-variable atmospheric model. Tellus, 17, 321-333
24. Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130-141 1965 A study of the predictability of a 28-variable atmospheric model. Tellus, 17, 321-333
25. May, R. 1996 Necessity and chance. Bull. Am. Math. Soc., 32(3), 291-308
26. Mees, A. I. 1993 Parsimonious dynamical reconstructions. Int. J. Bif. and Chaos, 3, 669-675
27. Molteni, F. and Plamer, T. N. 1993 Predictability and finite-time instability of the northern winter circulation. Q. J. R. Meteorol. Soc., 119, 269-298
28. Moore, D. W. and Spiegel, E. A. 1966 A thermally excited nonlinear oscillator. Astrophys. J., 143(3), 871-887
29. Mukougawa, H., Kimoto, M. and Yoden, S. 1991 A relationship between local error growth and quasi-stationary states. J. Atmos. Sci., 48, 1231-1237
30. Nese, J. M. 1989 Quantifying local predictability in phase space. Physica D, 35, 237-250
31. Nicolis, C. 1992 Probabilistic aspects of error growth in atmospheric dynamics. Q. J. R. Meteorol. Soc., 118, 553-568
32. Nicolis, C., Vannitsem, S. and Royer, J.-F. 1995 Short-range predictability of the atmosphere: mechanisms for superexponential error growth. Q. J. R. Meteorol. Soc., 121, 705-722
33. Nicolis, J. S., Meyer-Kress, G. and Haubs, G. 1983 Non-uniform chaotic dynamics with implications for information processing. Zeitschrift für Naturforschung, 38a, 1157-1169
34. Orr, W. 1907 Stability of instability of the steady motions of a perfect fluid. Proc. R. Irish. Acad., A27, 9-138
35. Oseledec, V. I. 1968 A multiplicative ergodic theorem. Trans. Moscow Math. Soc., 19, 197-231
36. Ott, E. 1994 Chaos in Dynamical Systems. Cambridge University Press, Cambridge, UK
37. Palmer, T. N. 1988 Medium and extended range predictability and stability of the Pacific/North American mode. Q. J. R. Meteorol. Soc., 114, 691-713 1993a Extended-range atmospheric prediction and the Lorenz model. Bull. Am. Meteorol. Soc., 74, 49-64 1993b A nonlinear dynamical perspective on climate change. Weather, 48, 314-326
38. Palmer, T. N. 1988 Medium and extended range predictability and stability of the Pacific/North American mode. Q. J. R. Meteorol. Soc., 114, 691-713 1993a Extended-range atmospheric prediction and the Lorenz model. Bull. Am. Meteorol. Soc., 74, 49-64 1993b A nonlinear dynamical perspective on climate change. Weather, 48, 314-326
39. Palmer, T. N. 1988 Medium and extended range predictability and stability of the Pacific/North American mode. Q. J. R. Meteorol. Soc., 114, 691-713 1993a Extended-range atmospheric prediction and the Lorenz model. Bull. Am. Meteorol. Soc., 74, 49-64 1993b A nonlinear dynamical perspective on climate change. Weather, 48, 314-326
40. Palmer, T. N., Molteni, F., Mureau, R., Buizza, R., Chapelet, P. and Tribbia, J. 1992 'Ensemble prediction'. Tech. Memo. No. 188. Available from ECMWF, Shinfield Park, Reading, UK
41. Papoulis, A. 1991 Probability, random variables, and stochastic processes. McGraw-Hill International Editions, New York, USA
42. Parker, T. S. and Chua, L. O. 1989 Practical numerical algorithms for chaotic systems. Springer-Verlag, New York, USA
43. Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. 1987 Numerical recipes. Cambridge University Press, Cambridge, UK
44. Pires, C., Vautard, R. and Talagrand, O. 1996 On extending the limits of variational assimilation in non-linear chaotic systems. Tellus, 48A, 96-121
45. Read, P., Bell, M. J., Johnson, D. W. and Small, R. M. 1992 Quasi-periodic and chaotic flow regimes in a thermally driven, rotating fluid annulus. J. Fluid Mech., 238, 599-632
46. Rössler, O. E. 1976 An equation for continuous chaos. Phys. Lett. A, 57A(5), 397-398
47. Smith, L. A. 1992 Identification and prediction of low-dimensional dynamics. Physica D, 58, 50-76 1994a Visualising predictability with chaotic ensembles. Pp. 293-304 in Advanced signal processing: Algorithms, architectures and implementations. Ed. F. T. Luk. SPIE, Bellingham, USA 1994b Local optimal prediction. Philos. Trans. R. Soc. London, A384, 371-381 1996 'Accountability and error in ensemble prediction in baroclinic flows'. Pp. 351-368 in Proceedings of seminar of predictability (vol. 1). European Centre for Medium-Range Weather Forecasts, September 1995. Available from ECMWF, Shinfield, Reading, UK 1997 'The maintenance of uncertainty'. Pp. 177-246 in Proceedings of the International School of Physics 'Enrico Fermi', Course CXXXIII. Italian Physical Society, Bologna, Italy
48. Smith, L. A. 1992 Identification and prediction of low-dimensional dynamics. Physica D, 58, 50-76 1994a Visualising predictability with chaotic ensembles. Pp. 293-304 in Advanced signal processing: Algorithms, architectures and implementations. Ed. F. T. Luk. SPIE, Bellingham, USA 1994b Local optimal prediction. Philos. Trans. R. Soc. London, A384, 371-381 1996 'Accountability and error in ensemble prediction in baroclinic flows'. Pp. 351-368 in Proceedings of seminar of predictability (vol. 1). European Centre for Medium-Range Weather Forecasts, September 1995. Available from ECMWF, Shinfield, Reading, UK 1997 'The maintenance of uncertainty'. Pp. 177-246 in Proceedings of the International School of Physics 'Enrico Fermi', Course CXXXIII. Italian Physical Society, Bologna, Italy
49. Smith, L. A. 1992 Identification and prediction of low-dimensional dynamics. Physica D, 58, 50-76 1994a Visualising predictability with chaotic ensembles. Pp. 293-304 in Advanced signal processing: Algorithms, architectures and implementations. Ed. F. T. Luk. SPIE, Bellingham, USA 1994b Local optimal prediction. Philos. Trans. R. Soc. London, A384, 371-381 1996 'Accountability and error in ensemble prediction in baroclinic flows'. Pp. 351-368 in Proceedings of seminar of predictability (vol. 1). European Centre for Medium-Range Weather Forecasts, September 1995. Available from ECMWF, Shinfield, Reading, UK 1997 'The maintenance of uncertainty'. Pp. 177-246 in Proceedings of the International School of Physics 'Enrico Fermi', Course CXXXIII. Italian Physical Society, Bologna, Italy
50. Smith, L. A. 1992 Identification and prediction of low-dimensional dynamics. Physica D, 58, 50-76 1994a Visualising predictability with chaotic ensembles. Pp. 293-304 in Advanced signal processing: Algorithms, architectures and implementations. Ed. F. T. Luk. SPIE, Bellingham, USA 1994b Local optimal prediction. Philos. Trans. R. Soc. London, A384, 371-381 1996 'Accountability and error in ensemble prediction in baroclinic flows'. Pp. 351-368 in Proceedings of seminar of predictability (vol. 1). European Centre for Medium-Range Weather Forecasts, September 1995. Available from ECMWF, Shinfield, Reading, UK 1997 'The maintenance of uncertainty'. Pp. 177-246 in Proceedings of the International School of Physics 'Enrico Fermi', Course CXXXIII. Italian Physical Society, Bologna, Italy
51. Smith, L. A. 1992 Identification and prediction of low-dimensional dynamics. Physica D, 58, 50-76 1994a Visualising predictability with chaotic ensembles. Pp. 293-304 in Advanced signal processing: Algorithms, architectures and implementations. Ed. F. T. Luk. SPIE, Bellingham, USA 1994b Local optimal prediction. Philos. Trans. R. Soc. London, A384, 371-381 1996 'Accountability and error in ensemble prediction in baroclinic flows'. Pp. 351-368 in Proceedings of seminar of predictability (vol. 1). European Centre for Medium-Range Weather Forecasts, September 1995. Available from ECMWF, Shinfield, Reading, UK 1997 'The maintenance of uncertainty'. Pp. 177-246 in Proceedings of the International School of Physics 'Enrico Fermi', Course CXXXIII. Italian Physical Society, Bologna, Italy
52. Smith, L. A. and Gilmour, I. 1998 'Accountability and internal consistency in ensemble formation'. In Proceedings of the European Centre for Medium-Range Weather Forecasts Workshop on predictability. Available from ECMWF, Shinfield, Reading, UK
53. Sparrow, C. 1982 The Lorenz equations. Springer-Verlag, New York, USA
54. Strang, G. 1988 Linear algebra and its application. Hartcourt Brace Jovanovich, San Diego, USA
55. Sugihara, G. and May, R. M. 1990 Nonlinear forecasting as a way of distinguishing chaos from measurement error in a time series. Nature, 344, 734-741
56. Talagrand, O., Vautard, R. and Strauss, B. 1997 'Evaluation of probabilistic prediction systems'. In Proceedings of the European Centre for Medium-Range Weather Forecasts Workshop on predictability. Available from ECMWF, Shinfield, Reading, UK
57. Tennekes, H. 1991 'Karl Popper and the accountability of numerical forecasting'. Proceedings of the European Centre for Medium-Range Forecasts Workshop on new developments in predictability. Available from ECMWF, Shinfield, Reading, UK
58. Thompson, P. D. 1957 Uncertainty of initial state as a factor in the predictability of large-scale atmospheric flow patterns. Tellus, 9, 275-295
59. Tong, H. and Moeanaddin, R. 1988 On multi-step non-linear least squares prediction. The Statistician, 37, 101-110
60. Toth, Z. 1991 Estimation of atmospheric predictability by circulation analogs. Mon. Weather Rev., 119, 65-72
61. Toth, Z. and Kalnay, E. 1993 Ensemble forecasting at NMC: The generation of perturbations. Bull. Am. Meteorol. Soc., 74(12), 2317-2330
62. Trefethen, L., Trefethen, A., Reddy, S. and Driscoll, T. 1993 Hydrodynamic stability without eigenvalues. Science, 261, 578-584
63. Trevisan, A. 1993 Impact of transient error growth on global average predictability measures. J. Atmos. Sci., 50, 1016-1028
64. Trevisan, A. and Legnani, R. 1995 Transient error growth and local predictability: a study in the Lorenz system. Tellus, 47A, 103-117
65. van den Dool, H. M. 1994 Searching for analogues, how long must we wait? Tellus, 46A, 314-324
66. van den Dool, H. M. and Rukhovets, L. 1994 On the weights for an ensemble-average 6-10-day forecast. Weather and Forecasting, 9, 457-465
67. Wolf, A., Swift, J., Swinney, H. L. and Vasrano, J. A. 1985 Determining Lyapunov exponents from a time series. Physica, 16D, 285
68. Yoden, S. and Nomura, M. 1993 Finite-time Lyapunov stability analysis and its application to atmospheric dynamics. J. Atmos. Sci., 50, 1531-1543
69. Ziehmann, C., Smith, L. A. and Kurths, J. 1999 The Bootstrap and Lyapunov exponents in deterministic chaos. Physica, D in press
70. Ziehmann-Schlumbohm, C. 1994 'Vorhersagestudien in chaotischen Systemen und in der Praxis'. PhD thesis, Freie Universität Berlin
71. Ziehmann-Schlumbohm, C., Fraedrich, K. and Smith, L. A. 1995 Ein internes Vorhersagbarkeitsexperiment im Lorenz-modell. Meteorologische Zeitschrift N.F., 4, 16-21