The observations of in-situ spacecraft mission in the magnetosheath and a region of thermalized subsonic plasma behind the bow shock reveal a non-linear behaviour of plasma waves. The study of waves and optics in Physics has given the understanding of the effect of many waves coming together to form a wave field or wave packet. The common aspect of such study shows that two or more waves can superimpose constructively or destructively. The sudden high magnetic field data in the magnetosheath displays such possibility of superposition of waves. In this paper, we use the empirical mode decomposition (EMD) and Hilbert transform (HT) techniques to determine the instantaneous frequencies of low frequency plasma waves in the magnetosheath. Our analysis has shown that the turbulent behavior of magnetic field in the magnetosheath within the selected period is due to superposition of waves.
From the study of the waves arising from the events on the upstream of the bow shock, there is a process of events leading to another regime of waves with turbulent or complex behaviour in the downstream region otherwise called the magnetosheath. Magnetosheath is an interface between the bow shock and the magnetopause. It is a region of thermalized subsonic plasma behind the bow shock. The plasma in the magnetosheath is denser and hotter than that in the solar wind. Also the magnetic field strength in the magnetosheath is higher than the magnetic field strength in the solar wind [1,2].
Many analysts have chosen different techniques for the analysis of resultant waves of space plasma aimed at understanding these waves. Most analyses of plasma waves have been carried out with Fourier transform (FT) or Wavelet Transform (WT). Detailed investigations of the dynamic properties of space plasma have been limited by the use of these standard techniques. This limitation is due to the assumptions of linearity and stationary (using FT) or linearity and non-stationary (using WT) behaviours of these waves leading to wrong determination of the frequency and other properties of these waves.
The truth is that space plasma data are observational data that exhibit unsteady character (non-linearity) [3,4] in oscillations throughout the data. Therefore, the use of the standard spectral analysis techniques limits the possibility of investigating the details of the dynamics of such data. In order to investigate the details of the dynamics of space plasma especially the plasma waves in the magnetosheath, there is a need for an approach that will decompose the complex waves into simple or mono-component waves, an approach that is based upon the local characteristic time scale of the signal. There is also a need for an approach that will help construct the time evolutions of the signal.
In this paper, we use the combination of empirical mode decomposition (EMD) technique and Hilbert transform (HT) to determine the instantaneous frequencies of plasma waves in the magnetosheath which could be used in the detailed investigation of space plasma behaviour.
2. Brief Comparison of Fourier Transform (FT), Wavelet Transform (WT) and Empirical Mode Decomposition (EMD)-Hilbert Transform Combination Methods
Fourier Transform (FT) is a type of global transform that is most suitable for linear and stationary signals. It provides a general technique for the examination of the global energy-frequency distribution [5]. Huang has further revealed the dependence of FT on linearity. It is true that many natural events can be approximated by linear systems.
It is also true that they also have tendency to be nonlinear. The imperfection of our probes (or numerical schemes) can lead to non-linear behaviour when there is interaction between the imperfect probes and the linear system. Fourier Transform can deal with the linear case and not the non-linear case.
Wavelet Transform (WT) is an adjustable window Fourier Transform. It can supply localised information in time-frequency domain, as it possesses the multi-scale property and mathematical microscope ability that makes it to detect the sudden component of the signals [6]. WT is a better approach than Fourier Transform in the analysis of non-stationary signals.
EMD technique generates a collection of intrinsic mode functions (imf). The decomposition is based on the direct extraction of the energy associated with various intrinsic timescale. According to Huang et al., 1998, the decomposition can be viewed as an expansion of the data in terms of the imfs. After the extraction of the imfs using EMD, the Hilbert Transform (HT) Approach as used in Carozzi et al., 2004 can be applied on each imf. The local energy and the instantaneous frequency derived from each imf through Hilbert Transform can give a full energy-frequency-time distribution of data.
Table 1 [5,7] displays the comparison between empirical mode decomposition (EMD)-Hilbert Transform (HT) approach, Fourier and Wavelet Transform. Various qualities have been considered for this comparison between the second and the last rows. This table shows at a glance that the EMD-HT approach is robust for the nonlinear and non-stationary signal analysis.
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