Before applying r-DFA technique on SEP datasets, r-DFAn (n = 1–6) method is employed for calculating scaling exponents (SEs) of an artificial signal which are statistically significant, in the class of non-stationary stochastic processes, such as an uncorrelated fractional Brownian motion (fBm) series having a predefined Hurst exponent value of 0.5 and the result is presented in Fig. 1. The method of wavelet-based synthesizing of such a signal has been proposed earlier by Abry and Sellan, (1996) and is used in the present work. The fBm series with H = 0.5 is a process which cannot be distinguished from a memory-less random process. The values of the global scaling exponents (GSEs) from r-DFA analysis of artificial fractional Brownian noise (generated for Hurst exponents 0.3, 0.5 and 0.7) for different orders are shown in Table 1. The values of the GSEs for the first four orders are very close to the expected value, i.e. within a 95% significance level. The Hurst exponent less than 0.5 (here 0.3) signifies processes that has a memory exhibiting negative correlations while those greater than 0.5 (here 0.7) also possess memory but exhibiting positive correlations. The algorithm is able to find the exponent values both for correlated and anti-correlated processes as well as for uncorrelated ones. The GSE values for H = 0.5 is computed from the robust regression slopes of Fig. 1. Figure 2 shows robust regression of fBm series generated for H = 0.3 and 0.7. In the class of stationary stochastic processes, an artificial fractional Gaussian noise (white noise) which is also a memory-less sequence, whose self-similarity exhibits power-law scaling characterized by \(F\left(L\right) \approx {D}^{\alpha }\), where \(\alpha =H\), GSE values can be computed for different DFA orders as before. Figure 3 shows the DFA analysis of fGn series and the values of the GSEs (where \(H = \alpha\)) are given in Table 1. The fBm and fGn series show difference in scaling as observed in Figs. 2 and 3.
Table 1
r-DFA global scaling exponents of fBm and fGn series.
DFA order
|
DFA Scaling exponent (α)
|
|
H = 0.3
|
H = 0.5
|
H = 0.7
|
fGn
|
1
|
1.26444
|
1.513902
|
1.692922
|
0.486681
|
2
|
1.278788
|
1.480403
|
1.72749
|
0.492216
|
3
|
1.270822
|
1.480321
|
1.707267
|
0.493237
|
4
|
1.261962
|
1.481049
|
1.686055
|
0.49401
|
5
|
1.222058
|
1.418125
|
1.60591
|
0.498793
|
6
|
1.196915
|
1.365147
|
1.547726
|
0.49517
|
Figure 4 shows the different artificially synthesized series used for the verification of r-DFA scaling characteristics. The SEP data from the years 2000, 2001 (maximum activity years) and 2007, 2008 (deep minimum years) are considered for the comparison of global scaling exponents during different solar activity periods. The solar cycle 23 commenced in April 1996, peaking in early 2000 while the declining phase begins on 2002 [Hady, 2013].
For the solar cycle 23, the maximum is on April, 2000 and the minimum is on December 2008 [Hathaway, 2015; Miyahara et al., 2021]. The r-DFA analysis of SEP data for the year 2000 and year 2008 with energies in the range > 1 MeV to > 100 MeV are shown in Figs. 5 and 6. In 2000, the variation in the scaling regions are evident at all energy ranges where the algorithm has been able to calculate only up to fourth DFA order for energy range > 100 MeV. The global scaling exponents are calculated for each year and the results are compared for each energy range separately as shown in Fig. 7. It is seen that for all the computed DFA orders, the GSEs (\(\alpha\) exponents) that characterize the scaling of the SEPs are found to be higher during solar maxima when compared with those during solar minima. It should also be significant that for the energy band > 1 MeV, which falls on the low energy tail of the cosmic ray spectrum, the demarcation between both solar activity periods is not pronounced as seen in other energy bands. A large class of astrophysical time series can be represented by the behavior at the extremes of the family of random processes, known as fractional Gaussian noises or fractional Brownian motions, or a combination of both (i.e., noise superimposed on a random walk), which describe the fractal dynamics of solar physical time series.
Furthermore, it is possible to proceed with the application of the r-DFA results to characterize self-similarity by taking that if this method is employed, a precise estimation of the Hurst parameter (H) in the set [0,1], only for SEP data that are indicative of a fractional Gaussian noise (fGn) process. Here, the value of H lies between 0 and 1, whereas for other SEP data, H may saturate at 1 or exceed 1, leading to an undefined value [Cadavid et al., 2019]. In turn, this limitation is exploited to determine datasets with fluctuations complying a \(1/f\) scaling law. In other words, during solar minimum periods, the proton flux series shows more of a Gaussian nature with scaling exponents in the range \(0.5 < \alpha =H < 1\), consistent with a stationary process. While during solar maximum periods, SEP portrays a time series with more of a non-Gaussian nature with scaling exponents in the range \(1.0<\alpha <1.5\), compatible with a non-stationary process. Thus, if we model the SEP scaling dynamics according to the two classes such as persistent (\(0.5 \le \alpha \le 1\)) and anti-persistent (\(1 < \alpha \le 1.5\)), then the variations can be treated as fingerprint related with dynamical phase change in the underlying system in relation with solar activity.
Singh and Bhargawa, (2017) have considered the monthly variation of proton flux > 1 MeV during the period 1976–2016 and reported the Hurst exponent value as 0.5516 using R/S analysis, which is more close to a Gaussian nature. In addition, they have computed the Hurst exponent for other solar parameters, such as the such as sunspot number, F10.7 radio flux, solar magnetic field and Alfvén Mach number, which were reported as 0.86, 0.82, 0.64 and 0.63 respectively. The self-affinity in the range of Hurst exponent values 0.7–1 for have been reported earlier where different investigators focused on different solar proxies such as sunspot numbers [Mandelbrot and Wallis, 1969; Kilcik et al., 2009; Xapsos and Burke, 2009; Zhou et al., 2014], sunspot areas [Zhou et al., 2014], solar radius [Kiliç and Golbasi, 2011], \(H\alpha\) flare activity [Lepreti et al., 2000; Kiliç and Golbasi, 2011] at different time intervals.
Using DFA method, Rypdal and Rypdal, (2012) demonstrated that sunspot number, total solar irradiance (TSI) show persistency for a long-range, with a Hurst exponent value H ≈ 0.7. Meanwhile, the process related with solar flare index whose stochastic component displays either a very low level of persistence (H < 0.6) or no correlation at all on timescales shorter than the sunspot period (on timescales of 10–1000 days). Earlier, for studying long-term persistence in solar activity, Ruzmaikin et al., (1994) have used cosmogenic radiocarbon data on which they got a Hurst exponent of 0.84 which indicates persistence of solar activity during the period from 100–3000 years. Their conclusion is that the stochasticity underlying the continuum is a correlated random process, which remains consistent across a broad range of time intervals [Ogurtsov, 2004]. Recently, Lepreti et al. (2021) utilized reconstructed sunspot number data obtained from the atmospheric activity of the cosmogenic isotope \({}_{ }{}^{14}C\), which was derived from tree rings, along with reconstructed total solar irradiance data obtained from various \({}_{ }{}^{10}Be\)ice core records from Greenland and Antarctica. Additionally, they employed a new multi-proxy sunspot number reconstructed data derived from \({}_{ }{}^{10}Be\)datasets and the global \({}_{ }{}^{14}C\), production series. These datasets were found to exhibit comparable scaling ranges, with similar DFA scaling exponents between 0.70 and 0.77.
The analysis of the Hurst exponent of Mount Wilson rotation measurements suggests that the temporal fluctuations of solar rotation on time-periods shorter than the 11-year cycle can be attributed to a stochastic process specified by persistence [Komm, 1995]. Later, while studying lunar and solar variability related persistence changes in North Atlantic Ocean (NAO) index, Alvarez-Ramirez et al., (2011) have reported changes from 1.0 (\(1/f\)-noise) for small time scales to 0.5 (uncorrelated noise) for large time scales. Recently, using Rescaled Range (R/S) analysis, Sumesh and Prince (2017) have reported increase of Hurst exponent values during solar maxima when compared with solar minima in geomagnetic indices such as AE, SYM-H and Dst index.
The local scaling exponents (\({\alpha }_{1})\) for the smaller scale region are calculated for each year and the results are compared for each energy range separately as shown in Fig. 8. Similar to the global scaling exponents, here also, for all the computed DFA orders, the \({\alpha }_{1}\) exponents that characterize the scaling of the SEPs in the smaller scale region are found to be higher during solar maxima when compared with those during solar minima. It is also interesting to see that for > 1 MeV range, the value of scaling exponents for the minimum year 2008 is greater than 1. This means that for the scales less than 18 hours the series became non-stationary and became a fractional Brownian series. Hence, it is evident that > 1 MeV proton flux is governed by different stochastic processes such as the energy changes attributed by trapped energetic protons where > 1 MeV (low energy protons) occupy those altitudes that extend beyond the geosynchronous Earth orbit (GEO). The low-energy protons enter the GEO region through the flanks of the magnetosphere, where they are temporarily trapped for a few drift periods [Blake et al., 1974]. Ivanova et al. (1985) studied polar orbiting satellite data and reported that the access of > 1 MeV solar protons to lower L-shells is correlated with Dst index, AE index, and dynamic pressure. Consequently, proton anisotropy with gyrocenters inside and outside the geosynchronous orbit is partially influenced by the Dst index and the IMF clock angle function. Supporting these findings, Rodriguez et al. (2010) discovered that the SEPs’ access is increased when there is high dynamic pressure (> 10 nPa) for protons inside the geosynchronous orbit, whereas low dynamic pressure leads to repeated undulations of proton and ion fluxes inside the orbit, which correlates with auroral electrojet indices. Due to the inconsistent variations of lower energy levels, i.e. those < 50 MeV, at higher scales, i.e. for scales having more than a day duration, which affect the significance threshold at higher DFA orders, a comparative analysis of those regions for different energy ranges are not performed.
As shown in Figs. 9 and 10, for detecting and comparing exact crossover regions, r-DFAs have been re-calculated with multiple yardsticks for the solar activity maximum (2000) and minimum (2008) years, taking only those fall on the limits of energy ranges, i.e. the least energetic protons, > 1 MeV and the most energetic protons, > 100 MeV. When compared with solar maximum, the solar minimum crossover points are highly apparent of which the prominent one is in the middle of the scaling region, with a significant protuberant shape, having a 19.5-hour crossover in > 1 MeV and 24.8-hour cross over in > 100 MeV for first order DFA. Around the central scaling region, albeit at subsequent times, similar distinct crossover points are also evident in other DFA orders for solar minimum period. Persistency in protuberant crossover across higher orders of r-DFAn denote the presence of harmonic components with periods less than those observed in first-order r-DFA where maximal bulge occurs [Kantelhardt et al., 2001; Movahed et al., 2006; Li and Zhang, 2007]. The amount of protuberance in the fractal domain is directly proportional to the amplitude of the cycle in the time domain [Habib et al., 2017]. Thus, it is clear that the absence of such harmonics could be due to the presence of underlying turbulence resulting in weak persistence in crossover across higher order r-DFAn of solar maximum periods.
When comparing > 1 MeV and > 100 MeV of solar minimum period, the time of crossovers also seen increasing as the crossover point advances over time where the first crossover point, for > 1 MeV, occurs at 4 hours while, for > 100 MeV, crossover point occurs at 5.17 hours. The second crossover point, for > 1 MeV occurs at 19.5 hours while, for > 100 MeV, crossover point occurs at a scale greater than a day. The third crossover point, for > 1 MeV occurs at 2.4 days while, for > 100 MeV, crossover point occurs at 3.8 days. The lack of distinct crossover points along different DFA orders (like the one seen for solar minimum period), for solar maximum period could be attributed to the weak persistency and underlying turbulent dynamics that dominate the SEP series during periods of maximum solar activity.
Now, as a confirmation for the existence of protuberant signatures during solar minimum and absence of such humpy regimes during solar maximum period, SEPs of years 2001 (a solar maximum year) and 2007 (a solar minimum year) have been compared and the results are shown in Figs. 11 and 12. It is evident from Fig. 11 that for both energy bands, > 1 MeV and > 100 MeV, characteristic protrusions exist during solar minimum while from Fig. 12, it is certain that such humpy features are rather absent (for r-DFA 1) or low (for higher order r-DFAs) during solar maximum. This confirms the argument that during periods of solar maximum, turbulent features emerge in SEP dynamics which disrupts underlying periodicities, if exist, to a large extent. Thus, the self-similarity features of SEP series during solar maxima and minima is attributed to the difference in solar modulation effects, to the degree of solar forcing rates and to the contribution of trapped particles within the geosynchronous earth orbit.
Figures 13 and 14 show the regressive analysis of > 1 MeV and > 100 MeV of 2008 for three identified crossovers from Fig. 9(b) and 10(b) respectively. The global crossover locations are identified from the y-intercept of the linear fit of crossover locations versus DFA order. The crossover locations are at 5.6 min, 10 hours and 1.1 days for > 1 MeV while 48.6 min, 16.6 hours and 1.47 days for > 100 MeV. It is clear that there is an increase in regime switching time for > 100 MeV than > 1 MeV proton flux for all crossovers.