3. EMD-HT Method

Empirical mode decomposition (EMD)-Hilbert transform (HT) is a signal processing technique simply called Hilbert-Huang Transform (HHT). This technique is local and adaptive in the analysis of time-frequency signal. The study of waves and optics in Physics has shown that a wavefield can be composed of superimposed monocomponent waves. Since Physics deals with signals such as nature, there is need to decompose the signal into series of mono-component called intrinsic mode functions

(imf) [5]. According to Huang et al.; 1998, a signal can be decomposed into imfs such that becomes

(1)

where, contains the shortest period component of the signal. is the residue which is constant.

The extracted intrinsic mode functions (imf) have the number of extrema and the number of zero crossings equal or differing at most by one, and the mean value between the upper and the lower envelope equal to zero at any point [5,8]. This means that the imfs are symmetric. One imf has a unique instantaneous frequency different from the others. Each imf is a mono-component signal. Hilbert Transform is easily applied to each imf for the determination of instantaneous frequency of each imf [3,9-11].

For, the Hilbert Transform [5,7,12-15] is given as

(2)

where H is the Hilbert Transform operator P. V means the Cauchy principal value of the integral.

Let be. and form a complex conjugate pair defining an analytic signal [16-18].

(3)

(4)

where is the instantaneous amplitude defined as

(5)

And is the instantaneous phase defined as

(6)

Therefore, instantaneous frequency is computed as

(7)

The original signal can be expressed as

(8)

Equation (8) enables the instantaneous envelopment and the instantaneous frequency to be represented in three dimensions as a function of time. R in Equation (8) is the real part operator.

4. Observations

In the analysis of space plasma data collected within the magnetosheath using Fluxgate Magnetometer (FGM) instrument of Cluster spacecraft, the existence of some remarkable difference between the upstream and the downstream shows the possibility of the generation of more waves. This could be due to instabilities arising from the upstream of bow shock, within the magnetosheath or responses of the magnetosphere. The magnetosphere responses to the variations in the magnetosheath through reconnection at the magnetopause which aid transport of mass, momentum, and energy into the magnetosphere which changes the dynamic pressure of the solar wind [19]. In this paper, ten (10) minutes data from FGM instrument taken between the period 02 00 00 and 02 00 10 UT on 01 01 2001 is selected to determine the instantaneous frequency. From magnetic field profile on 01 012001 showing magnitude and all components (x, y, z) of four Cluster spacecraft crossing the bow shock into the magnetosheath in Figure 1, the selected magnetic field profile of 10-mins data is shown below, featuring a waveform that is modulated in both amplitude and frequency (see Figure 2" target="_self"> Figure 2(a))

5. Results

EMD has been used for the extraction of the various constituent signals embedded in the data. This leads to the determination of instantaneous envelopment in Figure 2(b). This is achieved by forming envelopes on maxima and minima. The envelopes of the maxima and minima alongside their mean are shown in Figure 2(b) determined using cubic spline fitting. The subtraction of the mean from the original data gives the first imf (imf1) if the sifting conditions are met. This process continues until the sifting is completed. Using the y-components of the magnetic field selected with the magnetosheath region between the period 300 - 1000 s in Figure 1, the decomposition of the original data (signal) using EMD gives first, second, third, fourth imfs and the residue (constant) (see Figure 3). Using Hilbert Transform (HT) on the extracted imfs; the first, second, third and the

fourth imf, the instantaneous frequencies are determined as shown in Figure 4" target="_self"> Figure 4. The original data used in all these have been sampled with frequency having a nyquist frequency of about 0.13 Hz. This simply informs that the power spectra of the imf above this value will alias. The plot of instantaneous frequencies of imf2 and imf3 has been divided into portion A, B and C as shown in Table 2. Correlation and coherency table has been formed for the different portions A, B and C of the instantaneous frequencies of imf2 and imf3 as shown below.

6. Discussion

In Figure 4, the frequency axis allows us to see the time instant of the frequency of the imf. It is obvious from this figure that imf1 has frequency modulation that exceeds the Nyquist frequency (0.13 Hz). It is also very clear that

the instantaneous frequencies of imf2, imf3 and imf4 modulate with slow variation in the amplitude. Imf2 and imf3 show clear modulation in frequency. But imf4 shows a steady instantaneous very low frequency. The instantaneous frequencies (IF) of 2^nd and 3^rd imfs have been shown in Figure 4. The instantaneous frequency of the second imf varies between 0.00 and 0.04 Hz, while the instantaneous frequency of the third imf varies between 0.02 and 0.07 Hz. The mean value of frequency of imf2 is about 0.02 Hz (20 mHz) and the mean value of the frequency of imf3 is about 0.045 Hz (45 mHz) (see Figure 5" target="_self"> Figure 5). This plot has been divided into A, B, and C portions. Portion A covers 1 - 20 samples, B covers 20 - 80 samples, and C covers 100 - 125 samples. Portion A shows anti-phase superposition of two instantaneous frequencies, B shows in-phase superposition and C shows another anti-phase superposition. Figure 5" target="_self"> Figure 5: The instantaneous frequencies (IF) of 2nd and 3rd imfs of magnetic field data on 01 Feb, 2001 from 02 00 00 - 02 10 00 UT using simple Hilbert transform. From the Table 1, the correlation and coherency have been computed and for 2^nd and 3^rd imfs, and the result shows superposition between the two frequency modes. An obvious negative correlation and a strong coherency in portion A explains the attenuated waveform in that portion of the original 10-minute data which is due to out of phase superposition. Positive correlation and strong coherency in portion B shows an in-phase superposition which is also found in enhanced amplitude of the 10-minute magnetic field data. Portion C has a high negative correlation and a very strong coherency that explains the high attenuation in the waveform of the 10-minutes magnetic field data of Figure 2.

7. Conclusion

All of the above analysis confirms the statistical significance of the second and the third imf extracted using EMD and the subsequent application of the Hilbert transform to determine the instantaneous frequencies. The magnetic field data in Figure 1 show an active

magnetic field profile as the spacecraft move from the solar wind regime into a turbulent regime crossing the bow shock. The waveform informs that this region of magnetosheath requires special attention. The analysis above shows that it is possible that the complex behaviour (non-linear) is due to superposition of more waves. These waves could be due to the activity at both the up stream of the bow shock and the magnetosphere across magnetopause. We have shown that the complex waveform of the selected magnetic field has waves that have been superimposed. The determined instantaneous frequencies can be used to determine the wave vectors which will give detailed understandings of behavior of plasma waves in the region (magnetosheath).

REFERENCES

NOTES

^*Corresponding author.