{'fraction_non_alphanumeric': 0.0323653962492438, 'fraction_numerical': 0.024924379915305505, 'mean_word_length': 5.129403040415276, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 19, 'https://': 2, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Wavelet and Cross-Correlation Analysis of Relativistic Electron Flux with Sunspot Number, Solar Flux, and Solar Wind Parameters

The Geostationary Operational Environmental Satellites (GOES) have been monitoring the Earth's radiation environment and is providing the electron flux data (of energy >0.8 MeV, >2 MeV, and >4 MeV) by means of a connected sensor subsystem. Relativistic electron flux is one of the components of the radiation belt which not only affects the electrical system in satellites but also has an impact on Earth’s upper atmospheric climatic variation. We have carried out a study to determine the relation of sunspot number (R), solar flux (F10.7), and solar wind parameters i.e., solar wind velocity (Vsw), plasma density Nsw), the southern component of the interplanetary magnetic field (IMF-Bz), Plasma temperature (Tsw) with relativistic electron flux of energy >0.8 MeV, >2 MeV, and >4 MeV in outer radiation belt using the data of 24 years (1996-2020) covering solar cycle 23 and 24. Time series analysis, Cross-correlation and wavelet analysis techniques have been used in this study. The time series plot displayed that the radiation is occupied mostly by electron flux of energy less than 4 Mev and solar cycle 23 (1996-2008) was strong to produce more intensity of relativistic electron flux of all energy in comparison to cycle 24 (2008-2019). Results from cross-correlation analysis illustrated that Bz has no significant impact on the enhancement of relativistic electron flux of any energy range in the radiation belt. Whereas other studied parameters have a positive correlation with relativistic electron flux, but with significantly different coefficient values for different energy. We found that electron flux >0.8 MeV and >2 MeV has a strong positive association with sunspot number, solar flux, solar wind velocity, plasma density and temperature whereas weak correlation with electron flux of energy >4 MeV. This result leads us to conclude that solar activity and solar parameters have greater influence in producing relativistic electron flux of energy ~ 0.8-4 MeV, than of flux > 4 MeV. The study made to observe the distribution of relativistic electrons in radiation belt with time through continuous wavelet analysis showed that electron flux of energy >0.8 has a higher periodicity in comparison to the flux of other energy ranger.


INTRODUCTION
The instability in Sun's magnetic field causes a powerful eruption of plasma bubbles and clouds of magnetic fields from the sun's surface in interplanetary space known as coronal mass ejection (CME). When these erupted structures, mostly possessing charged particles hit our Planet, they get trapped in Earth's magnetic field forming a radiation belt in the magnetosphere called as Van Allen radiation belt [1,2]. Van Allen radiation belt covering the height around 3Re to 10Re consists of relativistic electrons and protons and most of the artificial satellites operates in this belt region [3]. The energetic electrons of this radiation belt are sometimes called killer electrons because they potentially harm satellites causing satellite anomalies through dielectric charging [4,5]. Due to the increasing importance of satellite and space technology for human well-being (communication, navigation, safety and emergency management, Earth observation, etc.), the research on radiation belt and relativistic electrons have received the considerable attention over last few decades. [6,7,8]. For the first time, the relativistic electrons of MeV energy were directly detected by Meyer and Vogt from solar proton flares by balloon observation [9]. Taking account of various processes, after being ejected into solar atmosphere with solar proton flares, both components of proton and electron diffuses into the outer space [10]. Having a negligible amount of energy loss while interacting with solar plasma, proton accelerates [11]. But in contrary to it, electron components decelerate as it loses a surplus amount of energy through ionization and synchrotron radiation [10]. Various researchers have studied the relation of relativistic electrons with other solar parameters and geomagnetic indices using different statistical tools and instruments. In 1979, George Paulikas and Bern Blake showed a correlation between solar wind velocity (Vsw) and geosynchronous relativistic electron fluxes which became the landmark on the study of radiation belt [12]. Paulikas & Blake (1979), Blake et al. (1997) provided results about the continuous correlation between solar wind properties and relativistic electron population in the Earth's magnetosphere [12,13]. They noted the important role the southward turning of the interplanetary magnetic field (IMF) plays in the process, with northward turning providing insignificant contribution in the modulation of the relativistic electron population by the high-speed solar wind and the leading pressure pulse. The role of the southward turning of the IMF has been studied by Baker et al. (1998) [14]. Baker et al. (1998) argued that the relativistic electron population is seeded by the electrons in the energy range of a few hundred KeV, which are introduced to the magnetosphere by the sub-storm injections associated with the southward turning of the IMF [14]. Reeves et al. (2011) repeated the analysis of Paulikas & Blake (1979) with a longer dataset from 1989 to 2010 and reported a more complex relationship between radiation belt electron fluxes and solar wind velocities than the linear relationship hinted by the previous statistical studies [15]. Particularly, they uncovered a triangular distribution of the electron fluxes with respect to the solar wind velocities. There were a velocity-dependent lower limit and a velocity independent upper limit for the electron fluxes in contrast with that predicted by an assumption of linear correlation between solar wind velocity (Vsw) and the logarithm of the electron fluxes [15]. Belian et al., 1996 examined electron flux data from 1979 to 1994, it's shown that high energy electron fluxes (E >300 keV) showed a cycle of 11 years. Also, they showed the relativistic electron cycle is out of phase with the sunspot cycle as electrons showed a minimum rate at the solar maximum instead of peaking [16].  analyzed the 30 most intense relativistic electron events from 1992 to 1995 in the transition period from solar maximum to solar minimum and discovered that every relativistic electron enhancement was associated with a magnetic storm and concluded that there is a low correlation between the strength of the magnetic storm and the strength of the relativistic electron flow [17]. The question of which magnetic storms produce relativistic electrons was further investigated by O'Brien et al. (2001) using cross-correlation analysis to determine which parameters of the solar wind influence the flux of relativistic electrons and found that sustained high solar wind velocity and long-duration ULF wave activity are more likely to be important mechanisms for electron flux enhancements than the sub-storms or VLF waves determined by the AE index [18]. The acceleration of relativistic electrons has been widely studied, but the mechanisms for electron loss during geomagnetic storms are also of particular interest. The decrease in the electron flux associated with the 'Dst effect' was studied by Kim & Chan (1997) and involves a temporary adiabatic dropout in electron fluxes [19]. Reeves et al. (2003) analyzed 276 geomagnetic storms between 1989 and 2000 and confirmed that geomagnetic storms can either increase or decrease electron fluxes in the radiation belts, also no correlations between the pre-storm and post-storm electron fluxes were reported [20].
Using correlation analysis, Lyatsky and Khazanov investigated the effect of solar wind density and solar wind velocity upon the relativistic electron population (>2 MeV) at geostationary orbit [21]. They found that the electron fluxes increase for higher solar wind speeds and decrease for increasing solar wind densities. Their results are consistent with results from Onsager et al. (2007), who had reported that an abrupt increase in solar wind densities can contribute to relativistic electron loss [22]. Lyatsky & Khazanov (2008) also reported that solar wind speeds display correlation with electron fluxes after two days like previously known, but solar wind densities show the best correlation with electron fluxes after about 15 hours and stressed the strong influence of solar wind density rather than just solar wind velocity upon the relativistic electron fluxes [21]. Although the various studies on radiation belt and relativistic electron flux has been made, still the adequate research and study is needed for a proper understanding of the dynamics of the radiation belt and factors that enhance the relativistic electrons. The observation of radiation belt and relativistic electron flux is always necessary because it has a direct impact on satellite technology, the health of astronauts, and space weather phenomena. Thus, we have realized to study the pattern of relativistic electron flux on geostationary orbit and other probable factors that affect it. In this paper, we will show the correlation of relativistic electrons with different solar parameters (sunspot number, solar flux and solar wind parameters) using 27 days averaged data of 24 years (1996-2020). The study covers two solar cycles-solar cycle 23 and solar cycle 24. The paper is organized as follows: In section 2, we have described the data set and methodology of our work. In section 3, we have presented our results and discussion. Finally, we have concluded our findings in section 4.

DATA SET AND METHODOLOGY
In this study, the internet-based supply of data supplied by OMNI (Operating Mission as Nodes on the Internet) and NOAA (National Oceanic and Atmospheric Administration) has been used. OMNI database (https://www.omniweb.gsfc.nasa.gov) provides near-Earth solar wind magnetic field and plasma parameter data from several spacecraft in geocentric orbits. NOAA is the American scientific agency that monitors Earth systems, manages the data collection network, and conducts scientific research on the oceans and atmosphere. Under NOAA's space weather monitoring mission, it has operated a series of GOES (Geostationary Operational Environmental Satellites) with Space Environment Monitor (SEM) instrument subsystem since 1974 [23]. The SEM is continuously providing Energetic particles and soft x-ray data, which is made available by NOAA's National Environmental Satellite Data and Information Service division through GOES data access link https://www.ngdc.noaa.gov/stp/satellite/goes/dataac cess.html. To date, 17 GOES satellites have been launched and currently four GOES satellites are in operational use, whereas the other 13 satellites in geostationary orbit are inactive or repurposed [24]. From the OMNI system, we have made use of 27days resolution data observation of Vsw, Nsw, IMF-Bz, R, F10.7, and Tsw. Whereas from NOAA's database, we have made the 1-minute interval data observation of relativistic electron fluxes with energies >0.8 MeV, >2MeV and >4 MeV. Later this observation was converted to a 27day resolution by taking the average of 1-minute resolution data of 27 days. The observation period is chosen for 24 years, from 1996 to February of 2020, which includes solar cycle 23 (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007) and solar cycle 24 (2008-2019). We have applied cross correlation and wavelet analysis techniques to observe the variation of electron flux in geostationary orbit with time and to evaluate the correlation between relativistic electron flux and other OMNI parameters respectively.

Cross-Correlation
Cross correlation is the multi-time scale, statistical tool used to understand the level of relationship among different variables along with time-delay analysis and extracts the analogous nature relative to each other in time to scout out the new information [25,26]. The correlation coefficient ranges from -1 to +1 and the coefficient value around ±1 explains high correlation, whereas around zero reflects a moderate or poor correlation between the compared parameters [27,28]. The cross-correlation coefficient is plotted against time (years, in this paper) which helps to find out the lead or lag between the comparable indices after establishing their correlation [29].

Wavelet Analysis
Wavelet analysis is used to analyze changes in variance where we can determine dominant modes of variability and how these modes vary in time by decomposing time series into time-frequency space [30]. As, the signals are dynamic over time we have used Continuous Wavelet Analysis in our work.
Continuous Wavelet Transforms (CWT) can provide the time-frequency representation of the signal [31].
To extract the information about the behavior of the system, the local time-frequency energy density of a signal can be observed on a wavelet scalogram. Scalogram is a tool for wavelet analysis, which is obtained by taking the squared modulus of the wavelet coefficients [30]. Various researchers have used wavelet analysis tools in their work. Katsavrias et al. (2012) performed wavelet and Lomb-Scargle periodogram analysis of the time series data of solar wind parameters and magnetospheric indices to verify short-term, annual, semi-annual and intermittent periodicities observed in solar activity [32]. Adhikari et al. (2018) utilized wavelet analysis to study Polar Cap Voltage (PCV) and field-aligned currents [29]. Khanal et al. (2019) employed CWT analysis to study spectral characteristics of geomagnetically induced currents (GICs) associated with high-intensity long-duration auroral-electrojet activities [33]. The detailed explanation of the theory associated with CWT can be found in various papers [25,30]. According to the theory, we have programmed all the algorithms in MATLAB selecting the Morlet wavelet function to obtain meaningful information about the relativistic electron flux in geostationary orbit through scalogram and Global Wavelet Spectrum (GWS). Figure 1   The curve of plasma density ranged from a maximum value of 11 N/cc to a minimum value of 2.5 N/cc. During the period 1996-2020, the continuous variation between between -1 nT to 2 nT of IMF-Bz was observed.

RESULTS AND DISCUSSIONS 3.1 Time Series analysis
As per the result reflected by time series plots, the outer radiation belt in the magnetosphere is occupied mostly by relativistic electrons of energy less than 4 MeV. Also, it depicts that low relativistic electron flux is counted during the increasing phase of the solar cycle. But the plots are not sufficient to establish the relation between the relativistic electron flux and the solar parameters. Hence, we have used the statistical methods: cross-correlation analysis and wavelet analysis method in this study.

CWT analysis
To study the periodicities of relativistic electrons we apply wavelet analysis. Figures 3, 4, and 5 present the results of CWT. The peak contours regions seen in the power spectrum comprise a confidence level of more than 95% which concerns red noise processing levels [32]. To establish the null hypothesis for the significance of power regions in wavelet spectrum Red noise leveling is important [29]. We have set the lag 1 autocorrelation coefficient of 0.72 for construction of red noise. The red dotted line in GWS ( Figure  3(c), 4(c), 5(c)) corresponds to red noise spectrum. The vertical plane in the scalogram (Figure 3(b), 4(b), 5(b) is the time in years and the horizontal plane is the period in the year. Dealing with finitelength time series makes an error to occur at edges of wavelet power spectrum [29]. To overcome this, we padded the time series with appropriate zeros. This method of adding zeros, however, creates discontinuities at the edges of the time series, which is solved by inserting a cone of influence (represented in the figures with U-shaped black lines). The edge effects are negligible beyond the cone of influence [29]. The peaks observed in the GWS plot show the main periodicities associated with the relativistic electron flux in the period of 1996 to 2020. It is clear from the figures, flux >0.8MeV and >2MeV frequencies can be observed in both higher and lower frequency regions whereas electron flux >4MeV have values in higher frequency regions only. According to the figure 3(b)