Acta Geodaetica et Geophysica

, Volume 53, Issue 4, pp 639–659 | Cite as

Nonextensive and distance-based entropy analysis on the influence of sunspot variability in magnetospheric dynamics

  • Sumesh GopinathEmail author
  • P. R. Prince
Original Study


In the present study, information theoretic distance-based entropies have been employed for a better understanding of nonlinear features of the magnetosphere using proxies such as AE and Dst indices. Among the various distance-based entropies, approximate and sample entropies are considered as potential quantifiers which could track the nonlinear variations of the magnetospheric system. The generalized nonextensive Tsallis q-entropy and Fisher’s information measure are used to study the nonextensive entropy and complexity respectively of the magnetospheric dynamics. For the analysis, 1-min AE and Dst indices are considered during the period 1985–2007. The results indicate that nonlinearity and nonextensive entropy of Dst index are solar activity dependent. But, the nonlinearity and nonextensive measures of AE index are not having any solar activity dependence. This implies that, other than the modulating solar wind, certain other complex phenomena of internal origin are having influence on the dynamics of geomagnetic activity in the auroral zone.


Information theory Nonextensivity Magnetospheric dynamics Sunspot variability 



The authors are thankful to World Data Center for Geomagnetism, Kyoto, Japan for providing 1-min auroral index data ( and to USGS for providing 1-min Dst index data ( The authors acknowledge SIDC for providing sunspot numbers ( Authors would also like to thank the anonymous referee for providing valuable comments and suggestions which have led to significant improvement of the quality of the manuscript.

Supplementary material

40328_2018_235_MOESM1_ESM.docx (1.6 mb)
Supplementary material 1 (DOCX 1640 kb)


  1. Ahn B-H, Kroehl HW, Kamide Y, Kihn EA (2000) Seasonal and solar cycle variations of the auroral electrojet indices. J Geophys Res 62:1301–1310Google Scholar
  2. Akasofu S-I (2013) The relationship between the magnetosphere and magnetospheric/auroral substorms. Ann Geophys 31:387–394CrossRefGoogle Scholar
  3. Angelopoulos V, Kennel CF, Coroniti FV, Pellat R, Kivelson MG, Walker RJ, Russell CT, Baumjohann W, Feldman WC, Gosling JT (1994) Statistical characteristics of bursty bulk flow events. J Geophys Res 99:21257CrossRefGoogle Scholar
  4. Aschwanden M (2011) Self-organized criticality in astrophysics. Springer, BerlinCrossRefGoogle Scholar
  5. Bak P, Tang C, Weisenfeld K (1987) Self-organized criticality: an explanation of 1/f noise. Phys Rev A 38:364–374CrossRefGoogle Scholar
  6. Bak P, Tang C, Wiesenfeld K (1988) Self–organized criticality. Phys Rev A 38:364–374CrossRefGoogle Scholar
  7. Balan N, Alleyne H, Walker S, Réme H, Décréau PME, Balogh A, André M, Fazakerley AN, Cornilleau-Wehrlin N, Gurnett D, Fraenz M (2006) Cluster observations of a structured magnetospheric cusp. Ann Geophys 24:1015–1027CrossRefGoogle Scholar
  8. Balasis G, Daglis IA, Kapiris P, Mandea M, Vassiliadis D, Eftaxias K (2006) From pre-storm activity to magnetic storms: a transition described in terms of fractal dynamics. Ann Geophys 24:3557–3567CrossRefGoogle Scholar
  9. Balasis G, Daglis IA, Papadimitriou C, Kalimeri M, Anastasiadis A, Eftaxias K (2008) Dynamical complexity in Dst time series using non-extensive Tsallis entropy. Geophys Res Lett 35:L14102CrossRefGoogle Scholar
  10. Balasis G, Daglis IA, Papadimitriou C, Kalimeri M, Anastasiadis A, Eftaxias K (2009) Investigating dynamical complexity in the magnetosphere using various entropy measures. J Geophys Res 114:A00D06CrossRefGoogle Scholar
  11. Balasis G, Papadimitriou C, Daglis IA, Anastasiadis A, Sandberg I, Eftaxias K (2011) Similarities between extreme events in the solar-terrestrial system by means of nonextensivity. Nonlinear Process Geophys 18:563–572CrossRefGoogle Scholar
  12. Balasis G, Potirakis SM, Mandea M (2016) Investigating dynamical complexity of geomagnetic jerks using various entropy measures. Front Earth Sci 4:71CrossRefGoogle Scholar
  13. Chen J, Sharma AS, Edwards JW, Shao X, Kamide Y (2008) Spatiotemporal dynamics of the magnetosphere during geospace storms: mutual information analysis. J Geophys Res 113:A05217Google Scholar
  14. Clilverd MA, Rodger CJ, Danskin D, Usanova ME, Raita T, Ulich T, Spanswick EL (2012) Energetic particle injection, acceleration, and loss during the geomagnetic disturbances which upset Galaxy. J Geophys Res 117:A12213Google Scholar
  15. Colominas MA, Schlotthauer G, Torres ME, Flandrin P (2012) Noise-assisted EMD methods in action. Adv Adapt Data Anal 4:1250025CrossRefGoogle Scholar
  16. Consolini G (2002) Self-organized criticality: a new paradigm for the magnetotail dynamics. Fractals 10:275–283CrossRefGoogle Scholar
  17. Consolini G, Chang TS (2001) Magnetic field topology and criticality in geotail dynamics: relevance to substorm phenomena. Space Sci Rev 95:309–321CrossRefGoogle Scholar
  18. Consolini G, Marcucci MF, Candidi M (1996) Multifractal structure of auroral electrojet index data. Phys Rev Lett 76(21):4082–4085CrossRefGoogle Scholar
  19. Consolini G, De Michelis P, Tozzi R (2008) On the Earth’s magnetospheric dynamics: nonequilibrium evolution and the fluctuation theorem. J Geophys Res 113:A08222. CrossRefGoogle Scholar
  20. Consolini G, De Marco R, De Michelis P (2013) Intermittency and multifractional Brownian character of geomagnetic time series. Nonlinear Process Geophys 20:455–466. CrossRefGoogle Scholar
  21. Consolini G, De Marco R, Carbone V (2015) On the emergence of a 1/k spectrum in the sub-inertial domain of turbulent media. Astrophys J 809:21CrossRefGoogle Scholar
  22. Davis TN, Sugiura M (1966) Auroral electrojet activity index AE and its universal time variations. J Geophys Res 71:785–801CrossRefGoogle Scholar
  23. Fisher RA (1925) Theory of statistical estimation. Proc Camb Philos Soc 22:700–725CrossRefGoogle Scholar
  24. Frieden BR (1998) Physics from Fisher information. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  25. Frieden BR (2004) Science from Fisher information. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  26. Fu HS, Cao JB, Zong Q-G, Lu HY, Huang SY, Wei XH, Ma YD (2012) The role of electrons during chorus intensification: energy source and energy loss. J Atm Sol Terr Phys 80:37CrossRefGoogle Scholar
  27. Ganon JL, Love JJ (2011) USGS 1-min Dst index. J Atm Sol Terr Phys 73:323–334CrossRefGoogle Scholar
  28. Gonzalez WD, Joselyn JA, Kamide Y, Kroehl HW, Rostoker G, Tsurutani BT, Vasyliunas VM (1994) What is a geomagnetic storm? J Geophys Res 99(A4):5771–5792CrossRefGoogle Scholar
  29. Gonzalez WD, Guarnieri FL, Gonzalez ALC, Echer E, Alves MV, Ogino T, Tsurutani BT (2006) Magnetospheric energetics during HILDCAA. In: Tsurutani BT et al. (eds) Recurrent magnetic storms: corotating solar wind streams. Geophys Monogr Ser, vol 167. AGU, Washington, D. C., CrossRefGoogle Scholar
  30. Gopinath S (2016) Multifractal features of magnetospheric dynamics and their dependence on solar activity. Astrophys Space Sci 361:290CrossRefGoogle Scholar
  31. Gopinath S, Prince PR (2016) Multiscale and cross entropy analysis of auroral and polar cap indices during geomagnetic storms. Adv Space Res 57:289–301CrossRefGoogle Scholar
  32. Gopinath S, Prince PR (2017) Multifractal characteristics of magnetospheric dynamics and their relationship with sunspot cycle. Adv Space Res 59:2265–2278CrossRefGoogle Scholar
  33. Gopinath S, Suji KJ, Prince PR (2015) Information measures based analysis of complex solar wind-magnetosphere interaction dynamics during geomagnetic storms. Indian J Phys 89(8):759–772CrossRefGoogle Scholar
  34. Guarnieri FL (2005) A study of the interplanetary and solar origin of high intensity long duration and continuous auroral activity events. Ph.D. thesis, Instituto Nacional de Pesquisas Espaciais, Sao Jose dos Campos, BrazilGoogle Scholar
  35. Hajra R, Echer E, Tsurutani BT, Gonzalez WD (2013) Solar cycle dependence of high-intensity long-duration continuous AE activity (HILDCAA) events, relativistic electron predictors? J Geophys Res 118:5626–5638CrossRefGoogle Scholar
  36. Hiraki Y (2015) Auroral vortex street formed by the magnetosphere–ionosphere coupling instability. Ann Geophys 33:217–224CrossRefGoogle Scholar
  37. Huang NE, Wu Z (2008) A review on Hilbert–Huang transform: method and its applications to geophysical studies. Rev Geophys 46:1–23CrossRefGoogle Scholar
  38. Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, Yen N-C, Tung CC, Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc Lond Ser A 454:903–993CrossRefGoogle Scholar
  39. Huang C-S, Foster JC, Reeves GD, Le G, Frey HU, Pollock CJ, Jahn J-M (2003) Periodic magnetospheric substorms: multiple space-based and ground-based instrumental observations. J Geophys Res 108:1411–1428CrossRefGoogle Scholar
  40. Lakhina GS (1994) Solar wind-magnetosphere–ionosphere coupling and chaotic dynamics. Surv Geophys 15:703–754CrossRefGoogle Scholar
  41. Li W, Thorne RM, Angelopoulos V, Bonnell JW, McFadden JP, Carlson CW, LeContel O, Roux A, Glassmeier KH, Auster HU (2009) Evaluation of whistler-mode chorus intensification on the nightside during an injection event observed on the THEMIS spacecraft. J Geophys Res 114:A00C14Google Scholar
  42. Lui ATY (2002) Multiscale phenomena in the near-Earth magnetosphere. J Atmos Solar Terr Phys 64:125–143CrossRefGoogle Scholar
  43. Lui ATY, Chapman SC, Liou K, Newell PT, Meng CI, Brittnacher M, Parks GK (2000) Is the dynamic magnetosphere an avalanching system? Geophys Res Lett 27(7):911–914CrossRefGoogle Scholar
  44. Lundin R, Aparicio B, Yamauchi M (2001) On the solar wind flow control of the polar cusp. J Geophys Res 106:13023–13035CrossRefGoogle Scholar
  45. Mayer AL, Pawlowski CW, Cabezas H (2006) Fisher information and dynamic regime changes in ecological systems. Ecol Model 195:72–82CrossRefGoogle Scholar
  46. Milovanov AV, Zelenyi LM (2000) Functional background of the Tsallis entropy: “coarse-grained” systems and “kappa” distribution functions. Nonlinear Process Geophys 7:211CrossRefGoogle Scholar
  47. Nelson KP, Scannell BJ, Landau H (2011) A risk profile for information fusion algorithms. Entropy 13(8):1518–1532CrossRefGoogle Scholar
  48. Newell PT, Gjerloev JW, Mitchell EJ (2013) Space climate implications from substorm frequency. J Geophys Res 118:6254–6265CrossRefGoogle Scholar
  49. Osepian A, Kirkwood S, Smirnova N (1996) Energetic electron precipitation during auroral events observed by incoherent scatter radar. Adv Space Res 17:149CrossRefGoogle Scholar
  50. Pavlos GP, Karakatsanis LP, Xenakis MN, Sarafopoulos D, Pavlos EG (2012) Tsallis statistics and magnetospheric self-organization. Physica A 391(11):3069–3080CrossRefGoogle Scholar
  51. Pincus SM (1991) Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA 88:2297–2301CrossRefGoogle Scholar
  52. Pincus S (1995) Approximate entropy (ApEn) as a complexity measure. Chaos 5:110–117CrossRefGoogle Scholar
  53. Richman JS, Moorman JR (2000) Physiological time-series analysis using approximate and sample entropy. Am J Physiol 278:H2039–H2049Google Scholar
  54. Sharma AS (1995) Assessing the magnetosphere’s nonlinear behavior: its dimension is low, its predictability, high. Rev Geophys 33:645–650CrossRefGoogle Scholar
  55. Sharma AS, Vassiliadis DV, Papadopoulos K (1993) Reconstruction of low-dimensional magnetospheric dynamics by singular spectrum analysis. Geophys Res Lett 20:335–338CrossRefGoogle Scholar
  56. Sitnov MI, Sharma AS, Papadopoulos K, Valdivia JA, Klimas AJ, Baker DN (2000) Phase transition-like behavior of the magnetosphere during substorms. J Geophys Res 105:12955–12974CrossRefGoogle Scholar
  57. Sitnov MI, Sharma AS, Papadopoulos K, Vassiliadis D (2001) Modeling substorm dynamics of the magnetosphere: from self-organization and self-organized criticality to nonequilibrium phase transitions. Phys Rev E 65:016116CrossRefGoogle Scholar
  58. Torres ME, Colominas MA, Schlotthauer G, Flandrin P (2011) A complete ensemble empirical mode decomposition with adaptive noise. In: ICASSP IEEE, pp 4144–4147Google Scholar
  59. Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52:479–488CrossRefGoogle Scholar
  60. Tsurutani BT, Gonzalez WD (1987) The cause of high intensity long-duration continuous AE activity (HILDCAAs): interplanetary Alfvén wave trains. Planet Space Sci 35:405CrossRefGoogle Scholar
  61. Tsurutani BT, Gonzalez WD, Guarnieri F, Kamide Y, Zhou X, Arballo JK (2004) Are high-intensity long-duration continuous AE activity (HILDCAA) events substorm expansion events? J Atmos Sol Terr Phys 66:167CrossRefGoogle Scholar
  62. Ukhorskiy AY, Sitnov MI, Sharma AS, Papadopoulos K (2003) Combining global and multi-scale features in a description of the solar wind-magnetosphere coupling. Ann Geophys 21:1913–1929CrossRefGoogle Scholar
  63. Ukhorskiy AY, Sitnov MI, Sharma AS, Papadopoulos K (2004) Global and multiscale dynamics of the magnetosphere. Geophys Res Lett 31:L08802. CrossRefGoogle Scholar
  64. Unnikrishnan K (2008) Comparison of chaotic aspects of magnetosphere under various physical conditions using AE index time series. Ann Geophys 26:941–953CrossRefGoogle Scholar
  65. Unnikrishnan K, Richards P (2014) How does solar eclipse influence the complex behavior of midlatitude ionosphere? Two case studies. J Geophys Res 119(2):1157CrossRefGoogle Scholar
  66. Uritsky VM, Pudovkin MI (1998) Low frequency 1/f-like fluctuations of the AE-index as a possible manifestation of self-organized criticality in the magnetosphere. Ann Geophys 16:1580CrossRefGoogle Scholar
  67. Uritsky VM, Klimas AJ, Vassiliadis D (2001) Comparative study of dynamical critical scaling in the auroral electro jet index versus solar wind fluctuations. Geophys Res Lett 28(19):3809–3812CrossRefGoogle Scholar
  68. Uritsky VM, Klimas AJ, Vassiliadis D, Chua D, Parks G (2002) Scale-free statistics of spatiotemporal auroral emissions as depicted by POLAR UVI images: dynamic magnetosphere as an avalanching system. J Geophys Res 107:1426CrossRefGoogle Scholar
  69. Valdivia JA, Rogan J, Muñoz V, Gomberoff L, Klimas A, Vassiliadis D, Uritsky V, Sharma S, Toledo B, Wastavino L (2005) The magnetosphere as a complex system. Adv Space Res 35:961–971CrossRefGoogle Scholar
  70. Valenza G, Allegrini P, Lanatà A, Scilingo EP (2012) Dominant Lyapunov exponent and approximate entropy in heart rate variability during emotional visual elicitation. Front Neuroeng 5:3CrossRefGoogle Scholar
  71. Vassiliadis DV, Sharma AS, Eastman TE, Papadopoulos K (1990) Low-dimensional chaos in magnetospheric activity from AE time series. Geophys Res Lett 17:1841–1844CrossRefGoogle Scholar
  72. Vörös Z, Jankovičová D, Kovács P (2002) Scaling and singularity characteristics of solar wind and magnetospheric fluctuations. Nonlinear Process Geophys 9:149–162CrossRefGoogle Scholar
  73. Wanliss JA, Dobias P (2007) Space storm as a phase transition. J Atmos Sol Terr Phys 69:675–684CrossRefGoogle Scholar
  74. Watanabe T-H, Kurata H, Maeyama S (2016) Generation of auroral turbulence through the magnetosphere–ionosphere coupling. New J Phys 18:125010CrossRefGoogle Scholar
  75. Wu Z, Huang NE (2009) Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv Adapt Data Anal 01:1–41CrossRefGoogle Scholar
  76. Zelenyi LM, Milovanov AV, Zimbardo G (1998) Multiscale magnetic structure of the distant tail: self-consistent fractal approach. In: Nishida A, Baker DN, Cowley SWH (eds) New perspectives on the earth’s magnetotail, Geophys Mon Ser 105 AGU 321Google Scholar
  77. Živković T, Rypdal K (2012) Organization of the magnetosphere during substorms. J Geophys Res 117:A05212. CrossRefGoogle Scholar
  78. Zurek S, Guzik P, Pawlak S, Kosmider M, Piskorski J (2012) On the relation between correlation dimension, approximate entropy and sample entropy parameters, and a fast algorithm for their calculation. Phys A 391(24):6601–6610CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversity CollegeTrivandrumIndia

Personalised recommendations