Abstract
It has been demonstrated that ensemble mean forecasts, in the context of the sample mean, have higher forecasting skill than deterministic (or single) forecasts. However, few studies have focused on quantifying the relationship between their forecast errors, especially in individual prediction cases. Clarification of the characteristics of deterministic and ensemble mean forecasts from the perspective of attractors of dynamical systems has also rarely been involved. In this paper, two attractor statistics—namely, the global and local attractor radii (GAR and LAR, respectively)—are applied to reveal the relationship between deterministic and ensemble mean forecast errors. The practical forecast experiments are implemented in a perfect model scenario with the Lorenz96 model as the numerical results for verification. The sample mean errors of deterministic and ensemble mean forecasts can be expressed by GAR and LAR, respectively, and their ratio is found to approach \(\sqrt 2 \) with lead time. Meanwhile, the LAR can provide the expected ratio of the ensemble mean and deterministic forecast errors in individual cases.
摘要
前人研究表明集合平均预报在大样本平均的情况下比确定性(或单一)预报有更高的预报技巧.然而,很少研究关注它们预报误差之间的定量关系,尤其在一些个例预报中.同时,从动力系统吸引子的角度对确定性和集合平均预报的特征进行的研究也很少.本文利用吸引子的两个统计量即全局和局部吸引子半径来揭示确定性和集合平均预报误差的关系.基于Lorenz96模型的完美模式情景下的实际预报试验结果用来作为理论的检验.确定性预报和集合平均预报的样本平均误差可以分别用全局和局部吸引子半径来表达,它们的比值随着预报时间接近 \(\sqrt 2 \).同时,局部吸引子半径提供了确定性和集合平均预报误差在不同个例中的期望比值.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Basnarkov, L., and L. Kocarev, 2012: Forecast improvement in Lorenz 96 system. Nonlinear Processes in Geophysics, 19, 569–575, https://doi.org/10.5194/npg-19-569-2012.
Buizza, R., M. Milleer, and T. N. Palmer, 1999: Stochastic representation of model uncertainties in the ECMWF ensemble prediction system. Quart. J. Roy. Meteor. Soc., 125, 2887–2908, https://doi.org/10.1002/qj.49712556006.
Corazza, M., and Coauthors, 2003: Use of the breeding technique to estimate the structure of the analysis “errors of the day”. Nonlinear Processes in Geophysics, 10(3), 233–243, https://doi.org/10.5194/npg-10-233-2003.
Duan, W. S., and Z. H. Huo, 2016: An approach to generating mutually independent initial perturbations for ensemble forecasts: Orthogonal conditional nonlinear optimal perturbations. J. Atmos. Sci., 73, 997–1014, https://doi.org/10.1175/JAS-D-15-0138.1.
Eckmann, J.-P., and D. Ruelle, 1985: Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617–656, https://doi.org/10.1103/RevModPhys.57.617.
Farmer, J. D., E. Ott, and J. A. Yorke, 1983: The dimension of chaotic attractors. Physica D, 7, 153–180, https://doi.org/10.1016/0167-2789(83)90125-2.
Feng, J., R. Q. Ding, D. Q. Liu, and J. P. Li, 2014: The application of nonlinear local Lyapunov vectors to ensemble predictions in Lorenz systems. J. Atmos. Sci., 71, 3554–3567, https://doi.org/10.1175/JAS-D-13-0270.1.
Houtekamer, P. L., and J. Derome, 1995: Methods for ensemble prediction. Mon. Wea. Rev., 123, 2181–2196, https://doi.org/10.1175/1520-0493(1995)123<2181:MFEP>2.0.CO;2.
Kalnay, E., 2003: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 230 pp.
Leith, C. E., 1974: Theoretical skill of Monte Carlo forecasts. Mon. Wea. Rev., 102, 409–418, https://doi.org/10.1175/1520-0493(1974)102<0409:TSOMCF>2.0.CO;2.
Li, J. P., and J. F. Chou, 1997: Existence of the atmosphere attractor. Science in China Series D: Earth Sciences, 40(2), 215–220, https://doi.org/10.1007/BF02878381.
Li, J. P., and R. Q. Ding, 2015: Seasonal and interannual weather prediction. Encyclopedia of Atmospheric Sciences, 2nd ed., North et al., Eds., Academic Press and Elsevier, 303–312.
Li, J. P., J. Feng, and R. Q. Ding, 2018: Attractor radius and global attractor radius and their application to the quantification of predictability limits. Climate Dyn., 51(5–6), 2359–2374, https://doi.org/10.1007/s00382-017-4017-y.
Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141, https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
Lorenz, E. N., 1965: A study of the predictability of a 28-variable atmospheric model. Tellus, 17(3), 321–333, https://doi.org/10.1111/j.2153-3490.1965.tb01424.x.
Lorenz, E. N., 1996: Predictability: A problem partly solved. Proceedings of the Seminar on Predictability, Vol. I, Reading, UK, ECMWF Seminar, 1–18.
Lorenz, E. N., and K. A. Emanuel, 1998: Optimal sites for supplementary weather observations: Simulation with a small model. J. Atmos. Sci., 55, 399–414, https://doi.org/10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2.
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, 73–119, https://doi.org/10.1002/qj.49712252905.
Ott, E., and Coauthors, 2004: A local ensemble Kalman filter for atmospheric data assimilation. Tellus A, 56, 415–428, https://doi.org/10.3402/tellusa.v56i5.14462.
Szunyogh, I., and Z. Toth, 2002: The effect of increased horizontal resolution on the NCEP global ensemble mean forecasts. Mon. Wea. Rev., 130, 1125–1143, https://doi.org/10.1175/1520-0493(2002)130<1125:TEOIHR>2.0.CO;2.
Thompson, P. D., 1957: Uncertainty of initial state as a factor in the predictability of large scale atmospheric flow patterns. Tellus, 9(3), 275–295, https://doi.org/10.1111/j.2153-3490.1957.tb01885.x.
Toth, Z., and E. Kalnay, 1993: Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc., 74, 2317–2330, https://doi.org/10.1175/1520-0477(1993)074<2317:EFANTG>2.0.CO;2.
Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125, 3297–3319, https://doi.org/10.1175/1520-0493(1997)125<3297:EFANAT>2.0.CO;2.
Wang, X. G., and C. H. Bishop, 2003: A comparison of breeding and ensemble transform Kalman filter ensemble forecast schemes. J. Atmos. Sci., 60, 1140–1158, https://doi.org/10.1175/1520-0469(2003)060<1140:ACOBAE>2.0.CO;2.
Wei, M. Z., Z. Toth, R. Wobus, and Y. J. Zhu, 2008: Initial perturbations based on the ensemble transform (ET) technique in the NCEP global operational forecast system. Tellus A, 60, 62–79, https://doi.org/10.1111/j.1600-0870.2007.00273.x.
Zheng, F., J. Zhu, H. Wang, and R. H. Zhang, 2009: Ensemble hindcasts of ENSO events over the past 120 years using a large number of ensembles. Adv. Atmos. Sci., 26(2), 359–372, https://doi.org/10.1007/s00376-009-0359-7.
Ziehmann, C., L. A. Smith, and J. Kurths, 2000: Localized Lyapunov exponents and the prediction of predictability. Physics Letters A, 271(4), 237–251, https://doi.org/10.1016/S0375-9601(00)00336-4.
Zou, C. Z., X. J. Zhou, and P. C. Yang, 1985: The statistical structure of Lorenz strange attractors. Adv. Atmos. Sci., 2(2), 215–224, https://doi.org/10.1007/BF03179753.
Acknowledgements
The authors acknowledge funding from the National Natural Science Foundation of China (Grant Nos. 41375110 and 41522502).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feng, J., Li, J., Zhang, J. et al. The Relationship between Deterministic and Ensemble Mean Forecast Errors Revealed by Global and Local Attractor Radii. Adv. Atmos. Sci. 36, 271–278 (2019). https://doi.org/10.1007/s00376-018-8123-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00376-018-8123-5