Brazilian Journal of Physics

, Volume 47, Issue 2, pp 167–181 | Cite as

On the Effects of Gaps and Uses of Approximation Functions on the Time-Scale Signal Analysis: A Case Study Based on Space Geophysical Events

  • Luciano A. Magrini
  • Margarete O. Domingues
  • Odim Mendes
General and Applied Physics


The presence of gaps is quite common in signals related to space science phenomena. Usually, this presence prevents the direct use of standard time-scale analysis because this analysis needs equally spaced data; it is affected by the time series borders (boundaries), and gaps can cause an increase of internal borders. Numerical approximations can be used to estimate the records whose entries are gaps. However, their use has limitations. In many practical cases, these approximations cannot faithfully reproduce the original signal behaviour. Alternatively, in this work, we compare an adapted wavelet technique (gaped wavelet transform), based on the continuous wavelet transform with Morlet wavelet analysing function, with two other standard approximation methods, namely, spline and Hermite cubic polynomials. This wavelet method does not require an approximation of the data on the gap positions, but it adapts the analysing wavelet function to deal with the gaps. To perform our comparisons, we use 120 magnetic field time series from a well-known space geophysical phenomena and we select and classify their gaps. Then, we analyse the influence of these methods in two time-scale tools. As conclusions, we observe that when the gaps are small (very few points sequentially missing), all the methods work well. However, with large gaps, the adapted wavelet method presents a better performance in the time-scale representation. Nevertheless, the cubic Hermite polynomial approximation is also an option when a reconstruction of the data is also needed, with the price of having a worse time-scale representation than the adapted wavelet method.


Signals with gaps Adaptive wavelet Numeric approximations Data treatment Space Geophysics 



Authors acknowledge the support and grants received from CAPES, CNPq (grants: 306038/2015−3, and 312246/2013−7), FAPESP (grant 2015/25624−2), and FINEP. We also thank NASA for the data set — OmniWeb service ( —, and Prof. Peter Frick for the original version of the code used on this work and scientific discussions during his visits to INPE.


  1. 1.
    J.P. Antoine, R. Murenzi, P. Vandergheynst, S.T. Ali. Two-dimensional wavelets and their relatives (Cambridge, Cambridge, 2008)MATHGoogle Scholar
  2. 2.
    W.S. Araujo, F.A.S. Souza, J.I.B. Brito, L.M. Lima. Revista Brasileira de Geografia Física. 4, 741 (2012)Google Scholar
  3. 3.
    K. Atkinson. An introduction to numerical analysis (Jonh Wiley & Sons, New York, 1978)MATHGoogle Scholar
  4. 4.
    E. Bernardes, M.O. Domingues, O Mendes, Transformadas integrais e suas aplicações em tecnologias espaciais. relatório final de projeto de iniciação científica, Maringá (2015)Google Scholar
  5. 5.
    J. Büchner, C.T. Dum, M. Scholer. (Eds.), Space Plasma Simulation (Springer, New York, 2003)CrossRefGoogle Scholar
  6. 6.
    R.L. Burden, J.D. Faires, Análise Numérica. CENGAGE learning, São Paulo (2015)Google Scholar
  7. 7.
    W.H. Campbell. Introduction to Geomagnetic Fields (Cambridge, Cambridge, 2003)CrossRefGoogle Scholar
  8. 8.
    M.O. Domingues, O. Mendes, J.E. Castilho, A. Pagamisse. Indrodução ao mundo das wavelets (SBMAC, São Carlos, 2012)Google Scholar
  9. 9.
    M.O. Domingues, O. Mendes, A.M. da Costa. Adv. Space Res. 35(5), 831 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    M.O. Domingues, O. Mendes, M. Kaibara, V.E. Menconi. Rev. Bras. Ensino Fís. 38(5), 1 (2016)Google Scholar
  11. 11.
    W.J. Emery, R.E. Thomson. Data analysis methods in physical oceanography, Chapter 11: Smoothing with cubic splines (Elsevier, Amsterdam, 2004)Google Scholar
  12. 12.
    M. Farge. Annu. Rev. Fluid Mech. 24, 395 (1992)ADSCrossRefGoogle Scholar
  13. 13.
    E. Foufoula-Georgiou, P. Kumar. (Eds.) Wavelets in Geophysics (Academic Press, New York, 1994)MATHGoogle Scholar
  14. 14.
    P. Frick, S.L. Baliunas, D. Galyagln, D. Sokoloff, W. Soon. Astrophys J. 483, 426 (1997)ADSCrossRefGoogle Scholar
  15. 15.
    P. Frick, A. Grossmann, P. Tchamitchian. J. Math. Phys. 39, 4091 (1998)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    W.D. Gonzalez, F.L. Guarnieri, A.L. Clua-Gonzalez, E. Echer, M.V. Alves, T. Ogino, B.T. Tsurutani. Geophysical Monograph Series. 167, 175 (2013)Google Scholar
  17. 17.
    P. Goupillaud, A. Grossmann, J. Morlet. Geoexploration. 23, 85 (1984)CrossRefGoogle Scholar
  18. 18.
    A. Grossmann, J. Morlet. SIAM J. Math. Anal. 15, 723 (1984)MathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Hajra, E. Echer, B.T. Tsurutani, W.D. Gonzalez. J. Geophys. Res. Space. 118, 5626 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    B.B. Hubbard. The World According to Wavelets: The Story of a Mathematical Technique in the Making (A K Peters/CRC Press, Massachusetts, 1998)MATHGoogle Scholar
  21. 21.
    M.G. Kivelson, C.T. Russell. (eds.), Introduction to Space Physics (Cambridge, New York, 1996)Google Scholar
  22. 22.
    L.A. Magrini, M.O. Domingues, O. Mendes, in Análise tempo-escala de séries temporais de geofísica espacial com lacunas: estudo de caso. Congresso Nacional de Matemática Aplicada e Computacional, (2016)Google Scholar
  23. 23.
    O. Mendes, A origem interplanetária e o desenvolvimento da fase principal das tempestades geomagnéticas moderadas (1978-1979). Tese de Doutoramento, Instituto Nacional de Pesquisas Espaciais, INPE-5445-TDI/491, São José dos Campos (1992)Google Scholar
  24. 24.
    O. Mendes, M.O. Domingues, A.M. da Costa. Adv. Space. Res. 35(5), 812 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    Y. Meyer. (Ed.), Wavelets and applications (Springer-Verlag, Berlin, 1992)Google Scholar
  26. 26.
    G. Micula, S. Micula. Handbook of splines, Chapter 11 Smoothing with cubic splines (Springer Sciences-Business Media, Dordrecht, 1999)MATHGoogle Scholar
  27. 27.
    J. Morlet, Sampling Theory and wave propagation. Issues in Acoustic Signal - Image Processing and Recognition NATO ASI Series. 1, 233 (1983)CrossRefGoogle Scholar
  28. 28.
    D.B. Percival, A.T. Walden. Wavelet methods for time series analysis (Cambridge, New York, 2000)CrossRefMATHGoogle Scholar
  29. 29.
    D.S.G. Pollock, Vol. I. A Handbook of Time-series Analysis Signal Processing and Dynamics (Academic, San Diego, 1999)Google Scholar
  30. 30.
    M.J.D. Powell. Approximation theory and methods (Cambridge, New York, 1981)MATHGoogle Scholar
  31. 31.
    M.B. Ruskai, G. Beylkin, R. Coifamn, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael. (Eds.), Wavelets and their applications (Jones and Bartlett, Boston, 1992)Google Scholar
  32. 32.
    L.R. Scott. Numerical Analysis (Princeton University, New Jersey, 2011)MATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Física 2017

Authors and Affiliations

  • Luciano A. Magrini
    • 1
  • Margarete O. Domingues
    • 2
  • Odim Mendes
    • 3
  1. 1.National Institute for Space Research (INPE), Associated Laboratory of Computation and Applied Mathematics (LAC), Graduation Program of Applied Computation (CAP)Instituto Federal de Educação, Ciência e Tecnologia de Sao Paulo.São José dos CamposBrazil
  2. 2.National Institute for Space Research (INPE)Associated Laboratory of Computation and Applied Mathematics (LAC)São José dos CamposBrazil
  3. 3.National Institute for Space Research (INPE)Space Geophysics Division (DGE)São José dos CamposBrazil

Personalised recommendations