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

  • General and Applied Physics
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Abstract

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.

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Notes

  1. http://omniweb.gsfc.nasa.gov/

  2. The Parseval theorem implies the conservation of energy in physical and Fourier space. In mathematical terms, it has been \(\displaystyle {\int }_{-\infty }^{\infty } |s(t)|^{2}\,dt={\int }_{-\infty }^{\infty } | \hat {s}(\xi )|^{2}\,d\xi \), where s(t) e \(\hat {s}(\xi )\) indicate any signal and its Fourier transform, respectively, where ξ is the frequency.

  3. http://omniweb.gsfc.nasa.gov/ow_min.html

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Acknowledgements

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 (http://omniweb.gsfc.nasa.gov) —, and Prof. Peter Frick for the original version of the code used on this work and scientific discussions during his visits to INPE.

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Authors and Affiliations

  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 Campos, Brazil

    Luciano A. Magrini

  2. National Institute for Space Research (INPE), Associated Laboratory of Computation and Applied Mathematics (LAC), São José dos Campos, Brazil

    Margarete O. Domingues

  3. National Institute for Space Research (INPE), Space Geophysics Division (DGE), São José dos Campos, Brazil

    Odim Mendes

Authors
  1. Luciano A. Magrini
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  2. Margarete O. Domingues
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  3. Odim Mendes
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Corresponding author

Correspondence to Margarete O. Domingues.

Appendix:

Appendix:

To allow work reproduction, a complete list of the geophysical events used in this study is shown in Table 4. Complementary, this information can be used for geophysicists in further studies. The events are related to the ones analysed in the work of [19]. Taking into account the geomagnetic disturbance on ground as an answer of the solar plasma-magnetosphere-ionosphere coupling, this work deals with the period of High-Intensity, Long-Duration, Continuous Auroral Electroject Activities (HILDCAAS), connected with the presence of interplanetary Alfvén wave trains in the interplanetary medium. The Auroral Electroject index, designated as AE, evaluates the geomagnetic disturbance level in the auroral regions established by magnetometer records collected on ground, as described in [7] . Relative to the Geocentric Solar Magnetospheric (GSM) coordinate frame, the north-south component, B z , of the interplanetary magnetic field is used for the signal analysis. For each event, they identify the initial time (Start), the final time (End), and we verify the number of gaps according to the gap pattern presented on the time series. Corresponding to each of the patterns discussed in this work, the count of the gap intervals was done. The not-shown events are those whose data are not available in OMNIWeb service siteFootnote 3 from NASA, and the events marked with an asterisk (∗) are those that do not have gaps in the edges.

Table 4 Time series intervals and number of gaps on the B z component of the interplanetary magnetic field
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Magrini, L.A., Domingues, M.O. & Mendes, O. On the Effects of Gaps and Uses of Approximation Functions on the Time-Scale Signal Analysis: A Case Study Based on Space Geophysical Events. Braz J Phys 47, 167–181 (2017). https://doi.org/10.1007/s13538-017-0486-z

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