Abstract
In nonlinear error growth dynamics, the initial error cannot be accurately determined, and the forecast error, which is also uncertain, can be considered to be a random variable. Entropy in information theory is a natural measure of the uncertainty of a random variable associated with a probability distribution. This paper effectively combines statistical information theory and nonlinear error growth dynamics, and introduces some fundamental concepts of entropy in information theory for nonlinear error growth dynamics. Entropy based on nonlinear error can be divided into time entropy and space entropy, which are used to estimate the predictabilities of the whole dynamical system and each of its variables. This is not only applicable for investigating the dependence between any two variables of a multivariable system, but also for measuring the influence of each variable on the predictability of the whole system. Taking the Lorenz system as an example, the entropy of nonlinear error is applied to estimate predictability. The time and space entropies are used to investigate the spatial distribution of predictability of the whole Lorenz system. The results show that when moving around two chaotic attractors or near the edge of system space, a Lorenz system with lower sensitivity to the initial field behaves with higher predictability and a longer predictability limit. The example analysis of predictability of the Lorenz system demonstrates that the predictability estimated by the entropy of nonlinear error is feasible and effective, especially for estimation of predictability of the whole system. This provides a theoretical foundation for further work in estimating real atmospheric multivariable joint predictability.
Similar content being viewed by others
References
Lorenz E N. Deterministic nonperiodic flow. J Atmos Sci, 1963, 20: 130–141
Chervin R M, Schneider S H. On determining the statistical significance of climate experiments with general circulation models. J Atmos Sci, 1976, 33: 405–412
Shukla J, Gutzler D S. Interannual variability and predictability of 500mb geopotential heights over the northern hemisphere. Mon Weather Rev, 1983, 111: 1273–1279
Leung L Y, North G R. Information theory and climate prediction. J Clim, 1990, 3: 5–14
DelSole T. Predictability and information theory. Part I: Measure of predictability. J Atmos Sci, 2004, 61: 2425–2440
Schneider T, Griffies S M. A conceptual framework for predictability studies. J Clim, 1999, 12: 3133–3155
Anderson J L, Stern W F. Evaluating the potential predictive utility of ensemble forecasts. J Clim, 1996, 9: 260–269
Yang X Q, Anderson J L, Stern W F. Reproducible forced modes in AGCM ensemble integration and potential predictability of atmospheric seasonal variations in the extratropics. J Clim, 1998, 11: 2942–2959
Sardeshmukh P D, Compo G P, Penland C. Changes of probability associated with El Niño. J Clim, 2000, 13: 4268–4286
Li J P, Ding R Q, Chen B H. Review and prospect on the predictability study of the atmosphere. Review and Prospects of the Developments of Atmosphere Science in Early 21st Century. Beijing: China Meteorological Press, 2006. 96–103
Li J P, Zeng Q C, Chou J F. Computational uncertainty principle in nonlinear ordinary differential equations I: Numerical results. Sci China Ser E-Technol Sci, 2000, 43: 449–460
Li J P, Zeng Q C, Chou J F. Computational uncertainty principle in nonlinear ordinary differential equations II: Theoretical analysis. Sci China Ser E-Technol Sci, 2000, 44: 55–74
Mu M, Duan W S, Wang J C. The predictability problems in numerical weather and climate prediction. Adv Atmos Sci, 2002, 19: 191–204
Mu M, Duan W S, Wang B. Conditional nonlinear optimal perturbation and its applications. Nonlinear Process Geophys, 2003, 10: 493–501
Mu M, Duan W S. A new approach to studying ENSO predictability: conditional nonlinear optimal perturbation. Chin Sci Bull, 2003, 48: 1045–1047
Mu M, Duan W, Wang Q, et al. An extension of conditional nonlinear optimal perturbation approach and its applications. Nonlinear Process Geophys, 2010, 12: 211–220
Duan W S, Mu M. Conditional nonlinear optimal perturbation: Applications to stability, sensitivity, and predictability. Sci China Ser D-Earth Sci, 2009, 52: 883–906
Mu M, Duan W S. Conditional nonlinear optimal perturbation and its application to the studies of weather and climate predictability. Chin Sci Bull, 2005, 50: 2401–2407
Duan W S, Mu M. Applications of nonlinear optimization method to numerical studies of atmospheric and oceanic sciences. Appl Math Mech, 2005, 26: 636–646
Yu Y, Duan W, Xu H, et al. Dynamics of nonlinear error growth and season-dependent predictability of El Niño events in the Zebiak-Cane model. Q J R Meteorol Soc, 2009, 135: 2146–2160
Yu Y, Mu M, Duan W. Does model parameter error cause a significant “Spring Predictability Barrier” for El Niño events in the Zebiak-Cane Model? J Clim, 2012, 25: 1263–1277
Duan W, Chao W. The ’spring predictability barrier’ for ENSO predictions and its possible mechanism: Results from a fully coupled model. Int J Climatol, 2012, doi: 10.1002/joc.351
Chen B H, Li J P, Ding R Q. Nonlinear local Lyapunov exponent and atmospheric predictability research. Sci China Ser D-Earth Sci, 2006, 49: 11430–11436
Ding R Q, Li J P. Nonlinear finite-time Lyapunov exponent and predictability. Phys Lett A, 2007, 364: 396–400
Ding R Q, Li J P. Nonlinear error dynamics and predictability study (in Chinese). Chin J Atmos Sci, 2007, 31: 571–576
Ding R Q, Li J P. Comparison of the influences of initial error and model parameter error on the predictability of numerical forecast (in Chinese). Chin J Geophys, 2008, 51: 1007–1012
Li J P, Ding R Q. Studies of predictability of single variable from multi-dimensional chaotic dynamical system (in Chinese). Chin J Atmos Sci, 2009, 33: 551–556
Li J P, Ding R Q. Temporal-spatial distribution of predictability limit of short-term climate (in Chinese). Chin J Atmos Sci, 2008, 32: 975–986
Ding R Q, Li J P. Applicaition of nonlinear error growth dynamics in studies of atmospheric predictability (in Chinese). Acta Meteor Sin, 2009, 67: 241–249
Ding R Q, Li J P. The temporal-spatial distributions of weather predictability of different variables (in Chinese). Acta Meteorol Sin, 2009, 67: 343–354
Li J P, Ding R Q. Tempeoral-spatial distribution of atmospheric predictability limit by local dynamical analogs. Mon Weather Rev, 2011, 139: 3265–3283
Ding R Q, Li J P. Comparisons of two ensemble mean methods in measuring the average error growth and the predictability. Acta Meteorol Sin, 2011, 25: 395–404
Cover T M, Thomas J A. Elements of Information Theory. 2nd ed. New York: John Wiley, 2006. 1–12
Kleeman R. Measuring dynamical prediction utility using relative entropy. J Atmos Sci, 2002, 59: 2057–2072
Roulston M, Smith L. Evaluating probabilistic forecasts using information theory. Mon Weather Rev, 2002, 130: 1653–1660
Abramov R, Majda A, Kleeman R. Information theory and predictability for low-frequency variability. J Atmos Sci, 2005, 62: 65–87
DelSole T. Predictability and information theory. Part II: Imperfect Forecast. J Atmos Sci, 2005, 61: 3368–3381
Tang Y, Lin H, Derome J, et al. A predictability measure applied to seasonal predictions of the Arctic Oscillation. J Clim, 2007, 20: 4733–4750
DelSole T, Tippett M K. Predictability: Recent insights from information theory. Rev Geophys, 2007, 45: RG4002
Zhang J G, Liu X R. Information entropy analysis on nonuniformity of prediction distribution in time-space. I: Basic concept and data analysis (in Chinese). Adv Water Sci, 2000, 11: 133–137
Zhang J G, Liu X R. Information entropy analysis on nonuniformity of prediction distribution in time-space. II: Model evaluation and application (in Chinese). Adv Water Sci, 2000, 11: 138–143
Zhang J G. Information entropy study on precipitation distribution in time and space (in Chinese). Dissertation for the Doctoral Degree. Nanjing: Hohai University, 2004. 49–102
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, A., Zhang, L., Wang, Q. et al. Information theory in nonlinear error growth dynamics and its application to predictability: Taking the Lorenz system as an example. Sci. China Earth Sci. 56, 1413–1421 (2013). https://doi.org/10.1007/s11430-012-4506-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11430-012-4506-0